%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue May 5 06:58:52 PM UTC 2026
% Result : Theorem 9.64s 2.82s
% Output : Refutation 9.64s
% Verified :
% SZS Type : Refutation
% Derivation depth : 48
% Number of leaves : 137
% Syntax : Number of formulae : 237 ( 27 unt; 0 typ; 0 def)
% Number of atoms : 11880 (5616 equ; 0 cnn)
% Maximal formula atoms : 530 ( 50 avg)
% Number of connectives : 29329 (3111 ~;2903 |;2460 &;19760 @)
% ( 507 <=>; 412 =>; 0 <=; 0 <~>)
% Maximal formula depth : 193 ( 14 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 2246 (2246 >; 0 *; 0 +; 0 <<)
% Number of symbols : 732 ( 728 usr; 293 con; 0-4 aty)
% ( 176 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 6078 ( 301 ^;4377 !;1400 ?;6078 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
term: $tType ).
thf(type_def_7,type,
subst: $tType ).
thf(type_def_8,type,
d_term: $tType ).
thf(type_def_9,type,
d_subst: $tType ).
thf(func_def_0,type,
term: $tType ).
thf(func_def_1,type,
subst: $tType ).
thf(func_def_2,type,
one: term ).
thf(func_def_3,type,
ap: term > term > term ).
thf(func_def_4,type,
lam: term > term ).
thf(func_def_5,type,
sub: term > subst > term ).
thf(func_def_6,type,
id: subst ).
thf(func_def_7,type,
sh: subst ).
thf(func_def_8,type,
push: term > subst > subst ).
thf(func_def_9,type,
comp: subst > subst > subst ).
thf(func_def_10,type,
var: term > $o ).
thf(func_def_18,type,
hoasap: subst > term > subst > term > term ).
thf(func_def_73,type,
hoaslam: subst > ( subst > term > term ) > term ).
thf(func_def_86,type,
hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
thf(func_def_107,type,
hoasvar: subst > term > subst > $o ).
thf(func_def_121,type,
pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
thf(func_def_124,type,
d_term: $tType ).
thf(func_def_125,type,
d_subst: $tType ).
thf(func_def_126,type,
d2term: d_term > term ).
thf(func_def_127,type,
d2subst: d_subst > subst ).
thf(func_def_128,type,
d_one: d_term ).
thf(func_def_129,type,
d_id: d_subst ).
thf(func_def_144,type,
sP2: ( subst > term > subst > $o ) > $o ).
thf(func_def_146,type,
sP4: ( subst > term > subst > $o ) > $o ).
thf(func_def_148,type,
sP6: ( subst > term > subst > $o ) > $o ).
thf(func_def_151,type,
sP9: ( subst > term > subst > $o ) > $o ).
thf(func_def_152,type,
sP10: ( subst > term > subst > $o ) > $o ).
thf(func_def_153,type,
sP11: ( subst > term > subst > $o ) > $o ).
thf(func_def_154,type,
sP12: ( subst > term > subst > $o ) > $o ).
thf(func_def_155,type,
sP13: ( term > $o ) > $o ).
thf(func_def_162,type,
sP20: ( term > $o ) > $o ).
thf(func_def_163,type,
sP21: ( term > $o ) > $o ).
thf(func_def_167,type,
sP25: ( term > $o ) > $o ).
thf(func_def_168,type,
sP26: ( term > $o ) > $o ).
thf(func_def_172,type,
sP30: ( term > $o ) > $o ).
thf(func_def_174,type,
sP32: ( term > $o ) > $o ).
thf(func_def_177,type,
sP35: ( subst > term > subst > $o ) > $o ).
thf(func_def_178,type,
sP36: ( subst > term > subst > $o ) > $o ).
thf(func_def_179,type,
sP37: ( term > $o ) > $o ).
thf(func_def_183,type,
sP41: ( term > $o ) > $o ).
thf(func_def_184,type,
sP42: ( subst > term > subst > $o ) > $o ).
thf(func_def_187,type,
sP45: ( subst > term > subst > $o ) > $o ).
thf(func_def_188,type,
sP46: ( subst > term > subst > $o ) > $o ).
thf(func_def_190,type,
sP48: ( term > $o ) > $o ).
thf(func_def_191,type,
sP49: ( subst > term > subst > $o ) > $o ).
thf(func_def_195,type,
sP53: ( subst > term > subst > $o ) > $o ).
thf(func_def_197,type,
sP55: ( subst > term > subst > $o ) > $o ).
thf(func_def_199,type,
sP57: ( subst > term > subst > $o ) > $o ).
thf(func_def_201,type,
sK59: term > $o ).
thf(func_def_202,type,
sK60: subst > term > subst > $o ).
thf(func_def_203,type,
sK61: term ).
thf(func_def_204,type,
sK62: term ).
thf(func_def_205,type,
sK63: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_206,type,
sK64: ( subst > term > subst > $o ) > subst ).
thf(func_def_207,type,
sK65: ( subst > term > subst > $o ) > term ).
thf(func_def_208,type,
sK66: ( subst > term > subst > $o ) > subst ).
thf(func_def_209,type,
sK67: ( subst > term > subst > $o ) > subst ).
thf(func_def_210,type,
sK68: ( subst > term > subst > $o ) > subst ).
thf(func_def_211,type,
sK69: ( subst > term > subst > $o ) > subst ).
thf(func_def_212,type,
sK70: ( subst > term > subst > $o ) > term ).
thf(func_def_213,type,
sK71: ( subst > term > subst > $o ) > subst ).
thf(func_def_214,type,
sK72: term > ( term > $o ) > term ).
thf(func_def_215,type,
sK73: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_216,type,
sK74: ( subst > term > term ) > subst ).
thf(func_def_217,type,
sK75: ( subst > term > term ) > subst ).
thf(func_def_218,type,
sK76: ( subst > term > term ) > term ).
thf(func_def_219,type,
sK77: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_220,type,
sK78: term ).
thf(func_def_221,type,
sK79: term > $o ).
thf(func_def_222,type,
sK80: subst > term > subst > $o ).
thf(func_def_223,type,
sK81: term ).
thf(func_def_224,type,
sK82: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_225,type,
sK83: ( subst > term > subst > $o ) > subst ).
thf(func_def_226,type,
sK84: ( subst > term > subst > $o ) > subst ).
thf(func_def_227,type,
sK85: ( subst > term > subst > $o ) > term ).
thf(func_def_228,type,
sK86: ( subst > term > subst > $o ) > subst ).
thf(func_def_229,type,
sK87: ( subst > term > subst > $o ) > term ).
thf(func_def_230,type,
sK88: ( subst > term > subst > $o ) > subst ).
thf(func_def_231,type,
sK89: ( subst > term > subst > $o ) > subst ).
thf(func_def_232,type,
sK90: ( subst > term > subst > $o ) > subst ).
thf(func_def_233,type,
sK91: ( subst > term > subst > $o ) > term ).
thf(func_def_234,type,
sK92: ( subst > term > subst > $o ) > term ).
thf(func_def_235,type,
sK93: subst > term > subst > $o ).
thf(func_def_236,type,
sK94: term ).
thf(func_def_237,type,
sK95: term > $o ).
thf(func_def_238,type,
sK96: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_239,type,
sK97: ( subst > term > subst > $o ) > subst ).
thf(func_def_240,type,
sK98: ( subst > term > subst > $o ) > subst ).
thf(func_def_241,type,
sK99: ( subst > term > subst > $o ) > term ).
thf(func_def_242,type,
sK100: ( subst > term > subst > $o ) > subst ).
thf(func_def_243,type,
sK101: ( subst > term > subst > $o ) > subst ).
thf(func_def_244,type,
sK102: ( subst > term > subst > $o ) > subst ).
thf(func_def_245,type,
sK103: ( subst > term > subst > $o ) > term ).
thf(func_def_246,type,
sK104: ( subst > term > subst > $o ) > subst ).
thf(func_def_247,type,
sK105: ( term > $o ) > term > term ).
thf(func_def_248,type,
sK106: ( subst > term > term ) > subst ).
thf(func_def_249,type,
sK107: ( subst > term > term ) > subst ).
thf(func_def_250,type,
sK108: ( subst > term > term ) > term ).
thf(func_def_251,type,
sK109: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_252,type,
sK110: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_253,type,
sK111: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_254,type,
sK112: subst > term > subst > $o ).
thf(func_def_255,type,
sK113: term ).
thf(func_def_256,type,
sK114: ( subst > term > term ) > term ).
thf(func_def_257,type,
sK115: ( subst > term > term ) > subst ).
thf(func_def_258,type,
sK116: ( subst > term > term ) > subst ).
thf(func_def_259,type,
sK117: ( subst > term > term ) > term ).
thf(func_def_260,type,
sK118: subst > term > subst > $o ).
thf(func_def_261,type,
sK119: ( term > $o ) > term ).
thf(func_def_262,type,
sK120: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_263,type,
sK121: term ).
thf(func_def_264,type,
sK122: ( term > $o ) > term ).
thf(func_def_265,type,
sK123: ( term > $o ) > term ).
thf(func_def_266,type,
sK124: term > $o ).
thf(func_def_267,type,
sK125: term ).
thf(func_def_268,type,
sK126: term > term ).
thf(func_def_269,type,
sK127: ( term > $o ) > term > term ).
thf(func_def_270,type,
sK128: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_271,type,
sK129: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_272,type,
sK130: subst > term > subst > $o ).
thf(func_def_273,type,
sK131: term ).
thf(func_def_274,type,
sK132: term > $o ).
thf(func_def_275,type,
sK133: ( subst > term > term ) > term ).
thf(func_def_276,type,
sK134: ( subst > term > term ) > subst ).
thf(func_def_277,type,
sK135: ( subst > term > term ) > term ).
thf(func_def_278,type,
sK136: ( subst > term > term ) > subst ).
thf(func_def_279,type,
sK137: ( subst > term > subst > $o ) > term ).
thf(func_def_280,type,
sK138: ( subst > term > subst > $o ) > term ).
thf(func_def_281,type,
sK139: ( term > $o ) > term ).
thf(func_def_282,type,
sK140: ( term > $o ) > term ).
thf(func_def_283,type,
sK141: term ).
thf(func_def_284,type,
sK142: subst > term > subst > $o ).
thf(func_def_285,type,
sK143: ( subst > term > subst > $o ) > term ).
thf(func_def_286,type,
sK144: ( subst > term > subst > $o ) > subst ).
thf(func_def_287,type,
sK145: ( subst > term > subst > $o ) > term ).
thf(func_def_288,type,
sK146: ( subst > term > subst > $o ) > subst ).
thf(func_def_289,type,
sK147: ( subst > term > subst > $o ) > subst ).
thf(func_def_290,type,
sK148: ( subst > term > subst > $o ) > term ).
thf(func_def_291,type,
sK149: ( subst > term > subst > $o ) > subst ).
thf(func_def_292,type,
sK150: ( subst > term > subst > $o ) > subst ).
thf(func_def_293,type,
sK151: ( subst > term > subst > $o ) > subst ).
thf(func_def_294,type,
sK152: ( subst > term > term ) > term ).
thf(func_def_295,type,
sK153: ( subst > term > term ) > subst ).
thf(func_def_296,type,
sK154: ( subst > term > term ) > subst ).
thf(func_def_297,type,
sK155: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_298,type,
sK156: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_299,type,
sK157: ( subst > term > subst > $o ) > term ).
thf(func_def_300,type,
sK158: ( subst > term > subst > $o ) > term ).
thf(func_def_301,type,
sK159: term ).
thf(func_def_302,type,
sK160: subst > term > subst > $o ).
thf(func_def_303,type,
sK161: term ).
thf(func_def_304,type,
sK162: ( subst > term > subst > $o ) > subst ).
thf(func_def_305,type,
sK163: ( subst > term > subst > $o ) > subst ).
thf(func_def_306,type,
sK164: ( subst > term > subst > $o ) > term ).
thf(func_def_307,type,
sK165: ( subst > term > subst > $o ) > subst ).
thf(func_def_308,type,
sK166: ( subst > term > subst > $o ) > term > term ).
thf(func_def_309,type,
sK167: ( subst > term > subst > $o ) > term ).
thf(func_def_310,type,
sK168: ( subst > term > subst > $o ) > subst ).
thf(func_def_311,type,
sK169: ( subst > term > subst > $o ) > subst ).
thf(func_def_312,type,
sK170: ( subst > term > subst > $o ) > subst ).
thf(func_def_313,type,
sK171: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_314,type,
sK172: term > ( subst > term > subst > $o ) > term ).
thf(func_def_315,type,
sK173: subst > term > subst > $o ).
thf(func_def_316,type,
sK174: term ).
thf(func_def_317,type,
sK175: ( subst > term > term ) > subst ).
thf(func_def_318,type,
sK176: ( subst > term > term ) > subst ).
thf(func_def_319,type,
sK177: ( subst > term > term ) > term ).
thf(func_def_320,type,
sK178: ( subst > term > term ) > term ).
thf(func_def_321,type,
sK179: ( subst > term > term ) > subst ).
thf(func_def_322,type,
sK180: ( subst > term > term ) > subst ).
thf(func_def_323,type,
sK181: ( subst > term > term ) > term ).
thf(func_def_324,type,
sK182: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_325,type,
sK183: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_326,type,
sK184: ( term > $o ) > term ).
thf(func_def_327,type,
sK185: ( term > $o ) > term ).
thf(func_def_328,type,
sK186: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_329,type,
sK187: ( term > $o ) > term ).
thf(func_def_330,type,
sK188: ( term > $o ) > term ).
thf(func_def_331,type,
sK189: subst > term > subst > $o ).
thf(func_def_332,type,
sK190: ( term > $o ) > term ).
thf(func_def_333,type,
sK191: term ).
thf(func_def_334,type,
sK192: term > $o ).
thf(func_def_335,type,
sK193: term > term ).
thf(func_def_336,type,
sK194: term ).
thf(func_def_337,type,
sK195: ( term > $o ) > term > term ).
thf(func_def_338,type,
sK196: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_339,type,
sK197: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_340,type,
sK198: term ).
thf(func_def_341,type,
sK199: term > $o ).
thf(func_def_342,type,
sK200: subst > term > subst > $o ).
thf(func_def_343,type,
sK201: ( subst > term > term ) > subst ).
thf(func_def_344,type,
sK202: ( subst > term > term ) > term ).
thf(func_def_345,type,
sK203: ( subst > term > term ) > subst ).
thf(func_def_346,type,
sK204: ( subst > term > term ) > term ).
thf(func_def_347,type,
sK205: subst > term > subst > $o ).
thf(func_def_348,type,
sK206: term ).
thf(func_def_349,type,
sK207: ( subst > term > subst > $o ) > term ).
thf(func_def_350,type,
sK208: ( subst > term > subst > $o ) > term ).
thf(func_def_351,type,
sK209: ( subst > term > subst > $o ) > subst ).
thf(func_def_352,type,
sK210: ( subst > term > subst > $o ) > subst ).
thf(func_def_353,type,
sK211: ( subst > term > subst > $o ) > subst ).
thf(func_def_354,type,
sK212: ( subst > term > subst > $o ) > subst ).
thf(func_def_355,type,
sK213: ( subst > term > subst > $o ) > subst ).
thf(func_def_356,type,
sK214: ( subst > term > subst > $o ) > term ).
thf(func_def_357,type,
sK215: ( subst > term > subst > $o ) > subst ).
thf(func_def_358,type,
sK216: ( term > $o ) > term ).
thf(func_def_359,type,
sK217: ( term > $o ) > term ).
thf(func_def_360,type,
sK218: ( subst > term > term ) > subst ).
thf(func_def_361,type,
sK219: ( subst > term > term ) > term ).
thf(func_def_362,type,
sK220: ( subst > term > term ) > subst ).
thf(func_def_363,type,
sK221: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_364,type,
sK222: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_365,type,
sK223: ( subst > term > subst > $o ) > term ).
thf(func_def_366,type,
sK224: ( subst > term > subst > $o ) > term ).
thf(func_def_367,type,
sK225: subst > term > term ).
thf(func_def_368,type,
sK226: subst ).
thf(func_def_369,type,
sK227: term ).
thf(func_def_370,type,
sK228: subst > term > term ).
thf(func_def_371,type,
sK229: term ).
thf(func_def_372,type,
sK230: subst ).
thf(func_def_373,type,
sK231: term ).
thf(func_def_374,type,
sK232: subst ).
thf(func_def_375,type,
sK233: term ).
thf(func_def_376,type,
sK234: term ).
thf(func_def_377,type,
sK235: ( subst > term > term ) > term ).
thf(func_def_378,type,
sK236: ( subst > term > term ) > subst ).
thf(func_def_379,type,
sK237: ( subst > term > term ) > subst ).
thf(func_def_380,type,
sK238: ( subst > term > term ) > subst ).
thf(func_def_381,type,
sK239: ( subst > term > term ) > term ).
thf(func_def_382,type,
sK240: ( subst > term > term ) > subst ).
thf(func_def_383,type,
sK241: term ).
thf(func_def_384,type,
sK242: term > $o ).
thf(func_def_385,type,
sK243: term > term ).
thf(func_def_386,type,
sK244: ( term > $o ) > term ).
thf(func_def_387,type,
sK245: ( term > $o ) > term ).
thf(func_def_388,type,
sK246: ( term > $o ) > term ).
thf(func_def_389,type,
sK247: ( term > $o ) > term ).
thf(func_def_390,type,
sK248: term > $o ).
thf(func_def_391,type,
sK249: subst ).
thf(func_def_392,type,
sK250: term ).
thf(func_def_393,type,
sK251: term > term ).
thf(func_def_394,type,
sK252: subst > ( term > $o ) > term ).
thf(func_def_395,type,
sK253: ( term > $o ) > term ).
thf(func_def_396,type,
sK254: ( term > $o ) > term ).
thf(func_def_397,type,
sK255: ( term > $o ) > term ).
thf(func_def_398,type,
sK256: subst > term > subst > $o ).
thf(func_def_399,type,
sK257: term ).
thf(func_def_400,type,
sK258: ( subst > term > term ) > subst ).
thf(func_def_401,type,
sK259: ( subst > term > term ) > subst ).
thf(func_def_402,type,
sK260: ( subst > term > term ) > term ).
thf(func_def_403,type,
sK261: ( subst > term > term ) > term ).
thf(func_def_404,type,
sK262: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_405,type,
sK263: term > ( subst > term > subst > $o ) > term ).
thf(func_def_406,type,
sK264: ( term > $o ) > term ).
thf(func_def_407,type,
sK265: ( term > $o ) > term ).
thf(func_def_408,type,
sK266: subst ).
thf(func_def_409,type,
sK267: term ).
thf(func_def_410,type,
sK268: term > $o ).
thf(func_def_411,type,
sK269: term ).
thf(func_def_412,type,
sK270: term ).
thf(func_def_413,type,
sK271: subst ).
thf(func_def_414,type,
sK272: subst ).
thf(func_def_415,type,
sK273: subst ).
thf(func_def_416,type,
sK274: term ).
thf(func_def_417,type,
sK275: term > $o ).
thf(func_def_418,type,
sK276: term > term ).
thf(func_def_419,type,
sK277: ( term > $o ) > subst > term ).
thf(func_def_420,type,
sK278: ( term > $o ) > term ).
thf(func_def_421,type,
sK279: ( term > $o ) > term ).
thf(func_def_422,type,
sK280: ( term > $o ) > term ).
thf(func_def_423,type,
sK281: ( term > $o ) > term ).
thf(func_def_424,type,
sK282: ( term > $o ) > term ).
thf(func_def_425,type,
sK283: subst > term > subst > $o ).
thf(func_def_426,type,
sK284: term ).
thf(func_def_427,type,
sK285: ( subst > term > term ) > term ).
thf(func_def_428,type,
sK286: ( subst > term > term ) > subst ).
thf(func_def_429,type,
sK287: ( subst > term > term ) > subst ).
thf(func_def_430,type,
sK288: ( subst > term > term ) > term ).
thf(func_def_431,type,
sK289: term > ( subst > term > subst > $o ) > term ).
thf(func_def_432,type,
sK290: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_433,type,
sK291: ( term > $o ) > term ).
thf(func_def_434,type,
sK292: ( term > $o ) > subst > term ).
thf(func_def_435,type,
sK293: subst ).
thf(func_def_436,type,
sK294: term > $o ).
thf(func_def_437,type,
sK295: term ).
thf(func_def_438,type,
sK296: term > term ).
thf(func_def_439,type,
sK297: term ).
thf(func_def_440,type,
sK298: term > $o ).
thf(func_def_441,type,
sK299: term ).
thf(func_def_442,type,
sK300: term > term ).
thf(func_def_443,type,
sK301: ( term > $o ) > term ).
thf(func_def_444,type,
sK302: ( term > $o ) > term ).
thf(func_def_445,type,
sK303: ( term > $o ) > term ).
thf(func_def_446,type,
sK304: ( term > $o ) > term ).
thf(func_def_447,type,
sK305: ( term > $o ) > term ).
thf(func_def_448,type,
sK306: ( term > $o ) > term ).
thf(func_def_449,type,
sK307: subst > term > subst > $o ).
thf(func_def_450,type,
sK308: term ).
thf(func_def_451,type,
sK309: ( subst > term > term ) > term > subst ).
thf(func_def_452,type,
sK310: ( subst > term > term ) > term > subst ).
thf(func_def_453,type,
sK311: ( subst > term > term ) > term > term ).
thf(func_def_454,type,
sK312: ( subst > term > term ) > term > subst ).
thf(func_def_455,type,
sK313: ( subst > term > term ) > term > subst ).
thf(func_def_456,type,
sK314: ( subst > term > term ) > term > term ).
thf(func_def_457,type,
sK315: ( subst > term > term ) > term ).
thf(func_def_458,type,
sK316: term > ( subst > term > subst > $o ) > term ).
thf(func_def_459,type,
sK317: ( subst > term > subst > $o ) > term ).
thf(func_def_460,type,
sK318: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_461,type,
sK319: term ).
thf(func_def_462,type,
sK320: subst > term > subst > $o ).
thf(func_def_463,type,
sK321: term > $o ).
thf(func_def_464,type,
sK322: ( subst > term > term ) > term ).
thf(func_def_465,type,
sK323: ( subst > term > term ) > term ).
thf(func_def_466,type,
sK324: ( subst > term > term ) > subst ).
thf(func_def_467,type,
sK325: ( subst > term > term ) > subst ).
thf(func_def_468,type,
sK326: term > subst > term > term ).
thf(func_def_469,type,
sK327: term ).
thf(func_def_470,type,
sK328: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_471,type,
sK329: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_472,type,
sK330: ( term > $o ) > term > term ).
thf(func_def_473,type,
sK331: term ).
thf(func_def_474,type,
sK332: ( subst > term > term ) > term ).
thf(func_def_475,type,
sK333: ( subst > term > term ) > subst ).
thf(func_def_476,type,
sK334: ( term > $o ) > term ).
thf(func_def_477,type,
sK335: ( term > $o ) > term ).
thf(func_def_478,type,
sK336: term ).
thf(func_def_479,type,
sK337: term > $o ).
thf(func_def_480,type,
sK338: term > term ).
thf(func_def_481,type,
sK339: ( subst > term > subst > $o ) > term ).
thf(func_def_482,type,
sK340: ( subst > term > subst > $o ) > subst ).
thf(func_def_483,type,
sK341: ( subst > term > subst > $o ) > subst ).
thf(func_def_484,type,
sK342: ( subst > term > subst > $o ) > subst ).
thf(func_def_485,type,
sK343: ( subst > term > subst > $o ) > subst ).
thf(func_def_486,type,
sK344: ( subst > term > subst > $o ) > subst ).
thf(func_def_487,type,
sK345: ( subst > term > subst > $o ) > term ).
thf(func_def_488,type,
sK346: ( subst > term > subst > $o ) > subst ).
thf(func_def_489,type,
sK347: term ).
thf(func_def_490,type,
sK348: term ).
thf(func_def_491,type,
sK349: subst > term > subst > $o ).
thf(func_def_492,type,
sK350: term ).
thf(func_def_493,type,
sK351: subst > term > subst > $o ).
thf(func_def_494,type,
sK352: ( subst > term > subst > $o ) > term ).
thf(func_def_495,type,
sK353: ( subst > term > subst > $o ) > subst ).
thf(func_def_496,type,
sK354: ( subst > term > subst > $o ) > subst ).
thf(func_def_497,type,
sK355: ( subst > term > subst > $o ) > subst ).
thf(func_def_498,type,
sK356: ( subst > term > subst > $o ) > term ).
thf(func_def_499,type,
sK357: ( subst > term > subst > $o ) > subst ).
thf(func_def_500,type,
sK358: ( subst > term > subst > $o ) > term ).
thf(func_def_501,type,
sK359: ( subst > term > subst > $o ) > subst ).
thf(func_def_502,type,
sK360: ( subst > term > subst > $o ) > subst ).
thf(func_def_503,type,
sK361: ( subst > term > subst > $o ) > term > term ).
thf(func_def_504,type,
sK362: ( subst > term > subst > $o ) > subst ).
thf(func_def_505,type,
sK363: ( subst > term > subst > $o ) > subst ).
thf(func_def_506,type,
sK364: ( subst > term > subst > $o ) > term ).
thf(func_def_507,type,
sK365: ( subst > term > subst > $o ) > subst ).
thf(func_def_508,type,
sK366: ( subst > term > subst > $o ) > subst ).
thf(func_def_509,type,
sK367: ( subst > term > subst > $o ) > subst ).
thf(func_def_510,type,
sK368: ( subst > term > subst > $o ) > term ).
thf(func_def_511,type,
sK369: ( subst > term > subst > $o ) > subst ).
thf(func_def_512,type,
sK370: term ).
thf(func_def_513,type,
sK371: subst > term > subst > $o ).
thf(func_def_514,type,
sK372: ( term > $o ) > term ).
thf(func_def_515,type,
sK373: ( term > $o ) > term ).
thf(func_def_516,type,
sK374: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_517,type,
sK375: ( subst > term > term ) > subst ).
thf(func_def_518,type,
sK376: ( subst > term > term ) > subst ).
thf(func_def_519,type,
sK377: ( subst > term > term ) > term ).
thf(func_def_520,type,
sK378: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_521,type,
sK379: ( subst > term > subst > $o ) > term ).
thf(func_def_522,type,
sK380: ( subst > term > subst > $o ) > term ).
thf(func_def_523,type,
sK381: ( subst > term > subst > $o ) > term ).
thf(func_def_524,type,
sK382: ( subst > term > subst > $o ) > term ).
thf(func_def_525,type,
sK383: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_526,type,
sK384: ( subst > term > term ) > subst ).
thf(func_def_527,type,
sK385: ( subst > term > term ) > subst ).
thf(func_def_528,type,
sK386: ( subst > term > term ) > term ).
thf(func_def_529,type,
sK387: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_530,type,
sK388: subst > term > subst > $o ).
thf(func_def_531,type,
sK389: term ).
thf(func_def_532,type,
sK390: term > $o ).
thf(func_def_533,type,
sK391: ( subst > term > term ) > term ).
thf(func_def_534,type,
sK392: ( subst > term > term ) > subst ).
thf(func_def_535,type,
sK393: ( subst > term > term ) > subst ).
thf(func_def_536,type,
sK394: ( subst > term > term ) > term ).
thf(func_def_537,type,
sK395: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_538,type,
sK396: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_539,type,
sK397: ( term > $o ) > term > term ).
thf(func_def_540,type,
sK398: term ).
thf(func_def_541,type,
sK399: subst > term > subst > $o ).
thf(func_def_542,type,
sK400: ( subst > term > subst > $o ) > term > term ).
thf(func_def_543,type,
sK401: ( subst > term > subst > $o ) > subst ).
thf(func_def_544,type,
sK402: ( subst > term > subst > $o ) > subst ).
thf(func_def_545,type,
sK403: ( subst > term > subst > $o ) > term ).
thf(func_def_546,type,
sK404: ( subst > term > subst > $o ) > subst ).
thf(func_def_547,type,
sK405: ( subst > term > subst > $o ) > subst ).
thf(func_def_548,type,
sK406: ( subst > term > subst > $o ) > subst ).
thf(func_def_549,type,
sK407: ( subst > term > subst > $o ) > subst ).
thf(func_def_550,type,
sK408: ( subst > term > subst > $o ) > term ).
thf(func_def_551,type,
sK409: ( subst > term > term ) > subst ).
thf(func_def_552,type,
sK410: ( subst > term > term ) > term ).
thf(func_def_553,type,
sK411: ( subst > term > term ) > subst ).
thf(func_def_554,type,
sK412: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_555,type,
sK413: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_556,type,
sK414: subst > term > subst > $o ).
thf(func_def_557,type,
sK415: term ).
thf(func_def_558,type,
sK416: ( subst > term > term ) > term ).
thf(func_def_559,type,
sK417: ( subst > term > term ) > subst ).
thf(func_def_560,type,
sK418: ( subst > term > term ) > term ).
thf(func_def_561,type,
sK419: ( subst > term > term ) > subst ).
thf(func_def_562,type,
sK420: term > ( subst > term > subst > $o ) > term ).
thf(func_def_563,type,
sK421: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_564,type,
sK422: ( subst > term > subst > $o ) > term ).
thf(func_def_565,type,
sK423: ( subst > term > subst > $o ) > term ).
thf(func_def_566,type,
sK424: term ).
thf(func_def_567,type,
sK425: subst > term > subst > $o ).
thf(func_def_568,type,
sK426: term ).
thf(func_def_569,type,
sK427: ( subst > term > subst > $o ) > subst ).
thf(func_def_570,type,
sK428: ( subst > term > subst > $o ) > subst ).
thf(func_def_571,type,
sK429: ( subst > term > subst > $o ) > term ).
thf(func_def_572,type,
sK430: ( subst > term > subst > $o ) > subst ).
thf(func_def_573,type,
sK431: ( subst > term > subst > $o ) > subst ).
thf(func_def_574,type,
sK432: ( subst > term > subst > $o ) > subst ).
thf(func_def_575,type,
sK433: ( subst > term > subst > $o ) > term ).
thf(func_def_576,type,
sK434: ( subst > term > subst > $o ) > subst ).
thf(func_def_577,type,
sK435: ( subst > term > subst > $o ) > term ).
thf(func_def_578,type,
sK436: ( subst > term > subst > $o ) > term ).
thf(func_def_579,type,
sK437: term ).
thf(func_def_580,type,
sK438: subst > term > subst > $o ).
thf(func_def_581,type,
sK439: ( subst > term > term ) > subst ).
thf(func_def_582,type,
sK440: ( subst > term > term ) > subst ).
thf(func_def_583,type,
sK441: ( subst > term > term ) > term ).
thf(func_def_584,type,
sK442: ( subst > term > term ) > term ).
thf(func_def_585,type,
sK443: ( subst > term > subst > $o ) > term > term ).
thf(func_def_586,type,
sK444: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_587,type,
sK445: term ).
thf(func_def_588,type,
sK446: ( subst > term > subst > $o ) > term > term ).
thf(func_def_589,type,
sK447: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_590,type,
sK448: term ).
thf(func_def_591,type,
sK449: subst > term > subst > $o ).
thf(func_def_592,type,
sK450: ( subst > term > term ) > term ).
thf(func_def_593,type,
sK451: ( subst > term > term ) > subst ).
thf(func_def_594,type,
sK452: ( subst > term > term ) > subst ).
thf(func_def_595,type,
sK453: ( subst > term > term ) > term ).
thf(func_def_596,type,
sK454: term > subst > term > term ).
thf(func_def_597,type,
sK455: term ).
thf(func_def_598,type,
sK456: ( subst > term > term ) > term ).
thf(func_def_599,type,
sK457: ( subst > term > term ) > subst ).
thf(func_def_600,type,
sK458: term ).
thf(func_def_601,type,
sK459: term ).
thf(func_def_602,type,
sK460: term ).
thf(func_def_603,type,
sK461: term ).
thf(func_def_604,type,
sK462: subst ).
thf(func_def_605,type,
sK463: subst > d_subst ).
thf(func_def_606,type,
sK464: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_607,type,
sK465: subst > term > subst > $o ).
thf(func_def_608,type,
sK466: ( term > $o ) > term ).
thf(func_def_609,type,
sK467: term ).
thf(func_def_610,type,
sK468: term ).
thf(func_def_611,type,
sK469: term ).
thf(func_def_612,type,
sK470: term ).
thf(func_def_613,type,
sK471: term ).
thf(func_def_614,type,
sK472: term ).
thf(func_def_615,type,
sK473: term ).
thf(func_def_616,type,
sK474: term ).
thf(func_def_617,type,
sK475: term ).
thf(func_def_618,type,
sK476: term > $o ).
thf(func_def_619,type,
sK477: term ).
thf(func_def_620,type,
sK478: subst ).
thf(func_def_621,type,
sK479: term > $o ).
thf(func_def_622,type,
sK480: term ).
thf(func_def_623,type,
sK481: subst > term > term ).
thf(func_def_624,type,
sK482: subst > term > term ).
thf(func_def_625,type,
sK483: subst ).
thf(func_def_626,type,
sK484: term ).
thf(func_def_627,type,
sK485: ( subst > term > term ) > subst ).
thf(func_def_628,type,
sK486: ( subst > term > term ) > subst ).
thf(func_def_629,type,
sK487: ( subst > term > term ) > term ).
thf(func_def_630,type,
sK488: ( subst > term > term ) > term ).
thf(func_def_631,type,
sK489: ( subst > term > term ) > subst ).
thf(func_def_632,type,
sK490: ( subst > term > term ) > subst ).
thf(func_def_633,type,
sK491: term ).
thf(func_def_634,type,
sK492: term > $o ).
thf(func_def_635,type,
sK493: subst ).
thf(func_def_636,type,
sK494: ( term > $o ) > term ).
thf(func_def_637,type,
sK495: subst > ( term > $o ) > term > term > $o ).
thf(func_def_638,type,
sK496: term ).
thf(func_def_639,type,
sK497: term ).
thf(func_def_640,type,
sK498: term ).
thf(func_def_641,type,
sK499: term ).
thf(func_def_642,type,
sK500: term > $o ).
thf(func_def_643,type,
sK501: subst ).
thf(func_def_644,type,
sK502: term ).
thf(func_def_645,type,
sK503: ( term > $o ) > term ).
thf(func_def_646,type,
sK504: term > subst > ( term > $o ) > term > $o ).
thf(func_def_647,type,
sK505: ( term > $o ) > term ).
thf(func_def_648,type,
sK506: ( term > $o ) > term ).
thf(func_def_649,type,
sK507: term > $o ).
thf(func_def_650,type,
sK508: term ).
thf(func_def_651,type,
sK509: term > subst > term > term ).
thf(func_def_652,type,
sK510: term ).
thf(func_def_653,type,
sK511: ( subst > term > term ) > subst ).
thf(func_def_654,type,
sK512: ( subst > term > term ) > term ).
thf(func_def_655,type,
sK513: subst ).
thf(func_def_656,type,
sK514: term ).
thf(func_def_657,type,
sK515: term ).
thf(func_def_658,type,
sK516: term ).
thf(func_def_659,type,
sK517: term ).
thf(func_def_660,type,
sK518: term > $o ).
thf(func_def_661,type,
sK519: term > $o ).
thf(func_def_662,type,
sK520: term ).
thf(func_def_663,type,
sK521: subst ).
thf(func_def_664,type,
sK522: subst > term > ( term > $o ) > ( term > $o ) > term ).
thf(func_def_665,type,
sK523: subst ).
thf(func_def_666,type,
sK524: term ).
thf(func_def_667,type,
sK525: term ).
thf(func_def_668,type,
sK526: term ).
thf(func_def_669,type,
sK527: term ).
thf(func_def_670,type,
sK528: term ).
thf(func_def_671,type,
sK529: term ).
thf(func_def_672,type,
sK530: term ).
thf(func_def_673,type,
sK531: term ).
thf(func_def_674,type,
sK532: term ).
thf(func_def_675,type,
sK533: term ).
thf(func_def_676,type,
sK534: term ).
thf(func_def_677,type,
sK535: term ).
thf(func_def_678,type,
sK536: term ).
thf(func_def_679,type,
sK537: term ).
thf(func_def_680,type,
sK538: term ).
thf(func_def_681,type,
sK539: term ).
thf(func_def_682,type,
sK540: term ).
thf(func_def_683,type,
sK541: term ).
thf(func_def_684,type,
sK542: term ).
thf(func_def_685,type,
sK543: term ).
thf(func_def_686,type,
sK544: term > d_term ).
thf(func_def_687,type,
sK545: term > $o ).
thf(func_def_688,type,
sK546: term > $o ).
thf(func_def_689,type,
sK547: term ).
thf(func_def_690,type,
sK548: subst ).
thf(func_def_691,type,
sK549: subst > term > ( term > $o ) > ( term > $o ) > term ).
thf(func_def_692,type,
sK550: subst ).
thf(func_def_693,type,
sK551: subst ).
thf(func_def_694,type,
sK552: subst > term > subst > $o ).
thf(func_def_695,type,
sK553: ( term > $o ) > term ).
thf(func_def_696,type,
sK554: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_697,type,
sK555: term ).
thf(func_def_698,type,
sK556: term > subst > term > term ).
thf(func_def_699,type,
sK557: subst > term > subst > $o ).
thf(func_def_700,type,
sK558: term ).
thf(func_def_701,type,
sK559: ( subst > term > term ) > term ).
thf(func_def_702,type,
sK560: ( subst > term > term ) > term ).
thf(func_def_703,type,
sK561: ( subst > term > term ) > subst ).
thf(func_def_704,type,
sK562: ( subst > term > term ) > subst ).
thf(func_def_705,type,
sK563: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_706,type,
sK564: term > subst > term > term ).
thf(func_def_707,type,
sK565: term ).
thf(func_def_708,type,
sK566: subst ).
thf(func_def_709,type,
sK567: term ).
thf(func_def_710,type,
sK568: term ).
thf(func_def_711,type,
sK569: subst ).
thf(func_def_712,type,
sK570: subst ).
thf(func_def_713,type,
sK571: subst ).
thf(func_def_714,type,
sK572: subst ).
thf(func_def_715,type,
sK573: subst ).
thf(func_def_716,type,
sK574: term ).
thf(func_def_717,type,
sK575: subst ).
thf(func_def_718,type,
sK576: term > $o ).
thf(func_def_719,type,
sK577: term ).
thf(func_def_720,type,
sK578: subst ).
thf(func_def_721,type,
sK579: term > $o ).
thf(func_def_722,type,
sK580: term ).
thf(func_def_723,type,
sK581: subst ).
thf(func_def_724,type,
sK582: term ).
thf(func_def_725,type,
sK583: term ).
thf(func_def_726,type,
sK584: subst ).
thf(func_def_727,type,
sK585: term ).
thf(func_def_728,type,
sK586: term ).
thf(func_def_729,type,
sK587: subst > term > term ).
thf(func_def_730,type,
sK588: ( subst > term > term ) > subst ).
thf(func_def_731,type,
sK589: ( subst > term > term ) > term ).
thf(func_def_732,type,
sK590: ( subst > term > term ) > subst ).
thf(func_def_733,type,
sK591: subst > term > subst > $o ).
thf(func_def_734,type,
sK592: ( subst > term > term ) > subst ).
thf(func_def_735,type,
sK593: ( subst > term > term ) > subst ).
thf(func_def_736,type,
sK594: ( subst > term > term ) > term ).
thf(func_def_737,type,
sK595: ( subst > term > term ) > term ).
thf(func_def_738,type,
sK596: term ).
thf(func_def_739,type,
sK597: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_740,type,
sK598: ( subst > term > subst > $o ) > term > term ).
thf(func_def_741,type,
sK599: ( subst > term > subst > $o ) > term > term ).
thf(func_def_742,type,
sK600: ( subst > term > subst > $o ) > term > term ).
thf(func_def_743,type,
ph601:
!>[X0: $tType] : X0 ).
thf(f5653,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f5652]) ).
thf(f5652,plain,
$true = $false,
inference(forward_demodulation,[status(thm)],[f5651,f2545]) ).
thf(f2545,plain,
! [X0: subst,X1: term] :
( $false
= ( sK283 @ X0 @ X1 @ X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2544]) ).
thf(f2544,plain,
! [X0: subst,X1: term] :
( ( $false
= ( sK283 @ X0 @ X1 @ X0 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2535,f2210]) ).
thf(f2210,plain,
! [X2: subst,X3: term,X0: term,X1: subst > term > subst > $o] :
( ( $true
= ( X1 @ X2 @ X3 @ X2 ) )
| ( ( X1 @ X2 @ X0 @ X2 )
= $false ) ),
inference(trivial_inequality_removal,[status(thm)],[f2207]) ).
thf(f2207,plain,
! [X2: subst,X3: term,X0: term,X1: subst > term > subst > $o] :
( ( $true = $false )
| ( $true
= ( X1 @ X2 @ X3 @ X2 ) )
| ( ( X1 @ X2 @ X0 @ X2 )
= $false ) ),
inference(backward_demodulation,[status(thm)],[f2107,f2194]) ).
thf(f2194,plain,
hoasinduction_lem3a = $false,
inference(trivial_inequality_removal,[status(thm)],[f2193]) ).
thf(f2193,plain,
( ( $true = $false )
| ( hoasinduction_lem3a = $false ) ),
inference(duplicate_literal_removal,[status(thm)],[f2190]) ).
thf(f2190,plain,
( ( hoasinduction_lem3a = $false )
| ( $true = $false )
| ( hoasinduction_lem3a = $false ) ),
inference(superposition,[status(thm)],[f1968,f2174]) ).
thf(f2174,plain,
( ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( hoasinduction_lem3a = $false ) ),
inference(duplicate_literal_removal,[status(thm)],[f2173]) ).
thf(f2173,plain,
( ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( hoasinduction_lem3a = $false ) ),
inference(forward_demodulation,[status(thm)],[f2172,f1886]) ).
thf(f1886,plain,
! [X183: term] :
( ( d2term @ d_one )
= X183 ),
inference(forward_demodulation,[status(thm)],[f1112,f1094]) ).
thf(f1094,plain,
! [X209: d_term] : ( d_one = X209 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f545,plain,
( ( ! [X0: term,X2: term,X3: subst] :
( ( sK454 @ X0 @ X3 @ X2 )
= ( sub @ X0 @ ( push @ X2 @ X3 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ! [X5: subst > term > term] :
( ( X5 @ ( sK457 @ X5 ) @ ( sK456 @ X5 ) )
!= ( sub @ sK455 @ ( push @ ( sK456 @ X5 ) @ ( sK457 @ X5 ) ) ) ) )
& ( ( apinj2 = $true )
| ( ( ( ap @ sK459 @ sK460 )
= ( ap @ sK458 @ sK461 ) )
& ( sK461 != sK460 ) ) )
& ( ! [X12: term,X13: term,X14: term,X15: term] :
( ( ( ap @ X13 @ X14 )
!= ( ap @ X12 @ X15 ) )
| ( X14 = X15 ) )
| ( apinj2 != $true ) )
& ( ! [X16: subst] :
( ( comp @ id @ X16 )
= X16 )
| ( axidl != $true ) )
& ( ( axidl = $true )
| ( sK462
!= ( comp @ id @ sK462 ) ) )
& ( ( sP58 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( sP58 != $true ) )
& ! [X18: subst] :
( ( d2subst @ ( sK463 @ X18 ) )
= X18 )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X20: subst > term > subst > $o,X22: term] :
( ( sK464 @ X20 @ X22 )
= ( X20 @ id @ X22 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ! [X24: term > $o] :
( ( sK465 @ id @ ( sK466 @ X24 ) @ id )
!= ( X24 @ ( sK466 @ X24 ) ) ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP56 ) )
& ( ( $true = sP56 )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( $true != sP54 ) )
& ( ( $true = sP54 )
| ( hoasinduction_lem3v2 != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X26: term,X27: term,X28: term,X29: term] :
( ( X26 = X27 )
| ( ( ap @ X26 @ X29 )
!= ( ap @ X27 @ X28 ) ) )
& ! [X30: term] :
( ( sub @ X30 @ id )
= X30 )
& ( ( ap @ ( sub @ sK470 @ id ) @ sK468 )
= ( ap @ ( sub @ sK469 @ id ) @ sK467 ) )
& ( sK469 != sK470 ) ) )
& ( ( ( sK471 != sK472 )
& ( ( ap @ sK471 @ sK474 )
= ( ap @ sK472 @ sK473 ) ) )
| ( ( sub @ sK475 @ id )
!= sK475 )
| ! [X40: term,X41: term,X42: term,X43: term] :
( ( ( ap @ ( sub @ X43 @ id ) @ X41 )
!= ( ap @ ( sub @ X42 @ id ) @ X40 ) )
| ( X42 = X43 ) )
| ( hoasapinj1_lthm != $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP52 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP52 != $true ) )
& ( induction2_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X44: term > $o,X45: term,X46: subst,X47: term > $o,X48: term] :
( ( $true
= ( X47 @ ( sub @ X45 @ ( push @ X48 @ X46 ) ) ) )
| ( $true
!= ( X44 @ X45 ) ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ( ( $true
!= ( sK479 @ ( sub @ sK477 @ ( push @ sK480 @ sK478 ) ) ) )
& ( ( sK476 @ sK477 )
= $true ) ) )
& ( ( hoaslaminj = $true )
| ( ( ( sK481 @ sK483 @ sK484 )
!= ( sK482 @ sK483 @ sK484 ) )
& ! [X58: subst,X59: subst,X60: term] :
( ( sub @ ( sK482 @ X59 @ X60 ) @ X58 )
= ( sK482 @ ( comp @ X59 @ X58 ) @ ( sub @ X60 @ X58 ) ) )
& ( ( lam @ ( sK481 @ sh @ one ) )
= ( lam @ ( sK482 @ sh @ one ) ) )
& ! [X61: term,X62: subst,X63: subst] :
( ( sub @ ( sK481 @ X62 @ X61 ) @ X63 )
= ( sK481 @ ( comp @ X62 @ X63 ) @ ( sub @ X61 @ X63 ) ) ) ) )
& ( ! [X64: subst > term > term,X65: subst > term > term,X66: subst,X67: term] :
( ( ( X64 @ X66 @ X67 )
= ( X65 @ X66 @ X67 ) )
| ( ( sub @ ( X65 @ ( sK486 @ X65 ) @ ( sK487 @ X65 ) ) @ ( sK485 @ X65 ) )
!= ( X65 @ ( comp @ ( sK486 @ X65 ) @ ( sK485 @ X65 ) ) @ ( sub @ ( sK487 @ X65 ) @ ( sK485 @ X65 ) ) ) )
| ( ( lam @ ( X64 @ sh @ one ) )
!= ( lam @ ( X65 @ sh @ one ) ) )
| ( ( X64 @ ( comp @ ( sK489 @ X64 ) @ ( sK490 @ X64 ) ) @ ( sub @ ( sK488 @ X64 ) @ ( sK490 @ X64 ) ) )
!= ( sub @ ( X64 @ ( sK489 @ X64 ) @ ( sK488 @ X64 ) ) @ ( sK490 @ X64 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( pushprop_lem0 = $true )
| ! [X77: term > $o] :
( ( sK492 @ ( sub @ ( sK494 @ X77 ) @ ( push @ sK491 @ sK493 ) ) )
!= ( X77 @ ( sK494 @ X77 ) ) ) )
& ( ! [X79: term,X80: term > $o,X81: subst,X83: term] :
( ( sK495 @ X81 @ X80 @ X79 @ X83 )
= ( X80 @ ( sub @ X83 @ ( push @ X79 @ X81 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ! [X84: term,X85: term,X86: term,X87: term] :
( ( ( ap @ ( sub @ X84 @ id ) @ X86 )
!= ( ap @ ( sub @ X85 @ id ) @ X87 ) )
| ( X84 = X85 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ( ( ( ap @ ( sub @ sK497 @ id ) @ sK499 )
= ( ap @ ( sub @ sK496 @ id ) @ sK498 ) )
& ( sK496 != sK497 ) ) )
& ! [X92: d_subst,X93: d_subst] :
( ( X92 = X93 )
| ( ( d2subst @ X93 )
!= ( d2subst @ X92 ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ! [X97: term > $o] :
( ( sK500 @ ( sub @ ( sK503 @ X97 ) @ ( push @ sK502 @ sK501 ) ) )
!= ( X97 @ ( sK503 @ X97 ) ) ) )
& ( ! [X99: term > $o,X100: subst,X101: term,X103: term] :
( ( X99 @ ( sub @ X103 @ ( push @ X101 @ X100 ) ) )
= ( sK504 @ X101 @ X100 @ X99 @ X103 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ( $true = sP47 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP47 ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP44 ) )
& ( ( $true = sP44 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ! [X104: term > $o,X105: term] :
( ( ( ( X104 @ ( sK505 @ X104 ) )
!= $true )
& ( ( var @ ( sK505 @ X104 ) )
= $true ) )
| ( $true
= ( X104 @ X105 ) )
| ( ( ( X104 @ ( lam @ ( sK506 @ X104 ) ) )
!= $true )
& ( $true
= ( X104 @ ( sK506 @ X104 ) ) ) )
| ( $true
= ( sP41 @ X104 ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ( ! [X110: term] :
( ( ( sK507 @ X110 )
= $true )
| ( ( var @ X110 )
!= $true ) )
& ( $true
!= ( sK507 @ sK508 ) )
& ! [X111: term] :
( ( ( sK507 @ ( lam @ X111 ) )
= $true )
| ( $true
!= ( sK507 @ X111 ) ) )
& ( ( sP41 @ sK507 )
!= $true ) ) )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP40 )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( $true != sP40 ) )
& ( ( hoaslaminj_lthm = $true )
| ( sP34 != $true ) )
& ( ( sP34 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP33 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP33 ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( $true = sP31 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP31 ) )
& ( axapp = $true )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3aa != $true ) )
& ( ! [X112: term,X114: subst,X115: term] :
( ( sub @ X112 @ ( push @ X115 @ X114 ) )
= ( sK509 @ X112 @ X114 @ X115 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ! [X117: subst > term > term] :
( ( X117 @ ( sK511 @ X117 ) @ ( sK512 @ X117 ) )
!= ( sub @ sK510 @ ( push @ ( sK512 @ X117 ) @ ( sK511 @ X117 ) ) ) ) )
& ( ( induction2lem_lthm = $true )
| ( $true != sP28 ) )
& ( ( $true = sP28 )
| ( induction2lem_lthm != $true ) )
& ( axvarshift = $true )
& ( ( axidr = $true )
| ( ( comp @ sK513 @ id )
!= sK513 ) )
& ( ! [X121: subst] :
( ( comp @ X121 @ id )
= X121 )
| ( axidr != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3a = $true )
| ( sP24 != $true ) )
& ( ( sP24 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X122: d_subst] : ( d_id = X122 )
& ( ( apinj1 = $true )
| ( ( sK517 != sK514 )
& ( ( ap @ sK514 @ sK516 )
= ( ap @ sK517 @ sK515 ) ) ) )
& ( ! [X127: term,X128: term,X129: term,X130: term] :
( ( X127 = X130 )
| ( ( ap @ X127 @ X129 )
!= ( ap @ X130 @ X128 ) ) )
| ( apinj1 != $true ) )
& ( ( induction2_lthm = $true )
| ( $true != sP23 ) )
& ( ( $true = sP23 )
| ( induction2_lthm != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP19 ) )
& ( ( $true = sP19 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ( ( sK518 @ sK520 )
= $true )
& ( $true
!= ( sK519 @ one ) )
& ! [X135: term] :
( ( sK519 @ X135 )
= ( sK518 @ ( sub @ X135 @ ( push @ sK520 @ sK521 ) ) ) )
& ! [X136: subst,X137: term] :
( ( sub @ one @ ( push @ X137 @ X136 ) )
= X137 ) ) )
& ( ! [X138: term > $o,X139: term > $o,X140: term,X141: subst] :
( ( $true
!= ( X138 @ X140 ) )
| ( ( X139 @ one )
= $true )
| ( ( X138 @ ( sub @ ( sK522 @ X141 @ X140 @ X139 @ X138 ) @ ( push @ X140 @ X141 ) ) )
!= ( X139 @ ( sK522 @ X141 @ X140 @ X139 @ X138 ) ) ) )
| ( sK524
!= ( sub @ one @ ( push @ sK524 @ sK523 ) ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( sP18 != $true ) )
& ( ( sP18 = $true )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( laminj = $true )
| ( ( sK525 != sK526 )
& ( ( lam @ sK526 )
= ( lam @ sK525 ) ) ) )
& ( ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) )
| ( laminj != $true ) )
& ( ! [X149: term,X150: term] :
( ( ( sub @ X149 @ sh )
!= ( sub @ X150 @ sh ) )
| ( X149 = X150 ) )
| ( shinj != $true ) )
& ( ( shinj = $true )
| ( ( ( sub @ sK528 @ sh )
= ( sub @ sK527 @ sh ) )
& ( sK527 != sK528 ) ) )
& ( hoasapnotvar_gthm = $true )
& ( ! [X153: term] :
( ( sub @ X153 @ id )
= X153 )
| ( sK529
!= ( sub @ sK529 @ id ) )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ( ( sub @ sK530 @ id )
!= sK530 )
& ! [X156: term] :
( ( sub @ X156 @ id )
= X156 ) ) )
& ( ! [X157: term] :
( ( sub @ X157 @ id )
= X157 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ( ( sub @ sK531 @ id )
!= sK531 ) )
& ( apnotvar = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lthm = $true )
| ( sP17 != $true ) )
& ( ( sP17 = $true )
| ( hoasinduction_lthm != $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ! [X159: term,X160: term,X161: term,X162: term] :
( ( X159 = X162 )
| ( ( ap @ X161 @ X162 )
!= ( ap @ X160 @ X159 ) ) )
& ( ( ap @ ( sub @ sK532 @ id ) @ sK533 )
= ( ap @ ( sub @ sK534 @ id ) @ sK535 ) )
& ( sK533 != sK535 ) ) )
& ( ( ( sK539 != sK536 )
& ( ( ap @ sK538 @ sK539 )
= ( ap @ sK537 @ sK536 ) ) )
| ! [X171: term,X172: term,X173: term,X174: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X171 @ id ) @ X172 ) )
| ( X172 = X174 ) )
| ( hoasapinj2_lthm != $true ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lthm = $true )
& ( ( sP8 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP8 != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( sP7 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( sP7 != $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( axassoc = $true )
& ( axabs = $true )
& ( lamnotap != $true )
& ( ! [X175: term,X176: term,X177: term,X178: term] :
( ( X176 = X177 )
| ( ( ap @ ( sub @ X178 @ id ) @ X177 )
!= ( ap @ ( sub @ X175 @ id ) @ X176 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ( ( sK541 != sK542 )
& ( ( ap @ ( sub @ sK540 @ id ) @ sK541 )
= ( ap @ ( sub @ sK543 @ id ) @ sK542 ) ) ) )
& ! [X183: term] :
( ( d2term @ ( sK544 @ X183 ) )
= X183 )
& ( pushprop = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
| ( ( $true
= ( sK545 @ sK547 ) )
& ( ( sK546 @ one )
!= $true )
& ! [X189: term] :
( ( sK545 @ ( sub @ X189 @ ( push @ sK547 @ sK548 ) ) )
= ( sK546 @ X189 ) ) ) )
& ( ! [X190: term > $o,X191: term > $o,X192: term,X193: subst] :
( ( ( X190 @ X192 )
!= $true )
| ( $true
= ( X191 @ one ) )
| ( ( X190 @ ( sub @ ( sK549 @ X193 @ X192 @ X191 @ X190 ) @ ( push @ X192 @ X193 ) ) )
!= ( X191 @ ( sK549 @ X193 @ X192 @ X191 @ X190 ) ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( ! [X195: subst] :
( ( comp @ X195 @ id )
= X195 )
& ! [X196: subst] :
( ( comp @ id @ X196 )
= X196 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ( ( comp @ sK550 @ id )
!= sK550 )
| ( ( comp @ id @ sK551 )
!= sK551 ) )
& ( ( hoasinduction_lem0 = $true )
| ! [X200: term > $o] :
( ( X200 @ ( sK553 @ X200 ) )
!= ( sK552 @ id @ ( sK553 @ X200 ) @ id ) ) )
& ( ! [X202: subst > term > subst > $o,X204: term] :
( ( sK554 @ X202 @ X204 )
= ( X202 @ id @ X204 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP5 ) )
& ( ( $true = sP5 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ! [X206: subst > term > term] :
( ( X206 @ sh @ one )
!= ( sub @ sK555 @ ( push @ one @ sh ) ) ) )
& ( ! [X207: term] :
( ( sub @ X207 @ ( push @ one @ sh ) )
= ( sK556 @ X207 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ulamvarsh = $true )
& ! [X209: d_term] : ( d_one = X209 )
& ( axclos = $true )
& ( substmonoid_gthm = $true )
& ( induction2lem_gthm = $true )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ( ! [X212: subst > term > term] :
( ( ( $true
= ( sK557 @ id @ ( sK559 @ X212 ) @ id ) )
& ( $true
!= ( sK557 @ id @ ( X212 @ id @ ( sK559 @ X212 ) ) @ id ) ) )
| ( ( sub @ ( X212 @ ( sK562 @ X212 ) @ ( sK560 @ X212 ) ) @ ( sK561 @ X212 ) )
!= ( X212 @ ( comp @ ( sK562 @ X212 ) @ ( sK561 @ X212 ) ) @ ( sub @ ( sK560 @ X212 ) @ ( sK561 @ X212 ) ) ) )
| ( $true
= ( sK557 @ id @ ( lam @ ( X212 @ sh @ one ) ) @ id ) ) )
& ( $true
!= ( sP4 @ sK557 ) )
& ( ( sK557 @ id @ sK558 @ id )
!= $true ) ) )
& ( ! [X217: subst > term > subst > $o,X218: term] :
( ( ! [X220: term] :
( ( ( X217 @ id @ X220 @ id )
!= $true )
| ( $true
= ( X217 @ id @ ( sK563 @ X217 @ id @ X220 ) @ id ) ) )
& ! [X221: term,X222: subst,X223: subst] :
( ( sub @ ( sK563 @ X217 @ X223 @ X221 ) @ X222 )
= ( sK563 @ X217 @ ( comp @ X223 @ X222 ) @ ( sub @ X221 @ X222 ) ) )
& ( ( X217 @ id @ ( lam @ ( sK563 @ X217 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( sP4 @ X217 )
= $true )
| ( $true
= ( X217 @ id @ X218 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( axmap = $true )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP3 ) )
& ( ( $true = sP3 )
| ( hoasinduction_lem2 != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasapinj2_gthm = $true )
& ( ! [X224: term] :
( ( sK564 @ X224 @ sh @ one )
= ( sub @ X224 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ! [X227: subst > term > term] :
( ( X227 @ sh @ one )
!= ( sub @ sK565 @ ( push @ one @ sh ) ) ) )
& ( one
= ( d2term @ d_one ) )
& ( termmset_gthm = $true )
& ( ! [X228: subst,X229: term] :
( ( sub @ one @ ( push @ X229 @ X228 ) )
= X229 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ( ( sub @ one @ ( push @ sK567 @ sK566 ) )
!= sK567 ) )
& ( ulamvarind = $true )
& ( hoasapinj1_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem3v2_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ! [X232: term] :
( ( sub @ X232 @ id )
= X232 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ( ( sub @ sK568 @ id )
!= sK568 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ! [X234: d_term,X235: d_term] :
( ( ( d2term @ X234 )
!= ( d2term @ X235 ) )
| ( X234 = X235 ) )
& ( ( ( comp @ sK569 @ id )
!= sK569 )
| ( ( comp @ id @ sK570 )
!= sK570 )
| ( ! [X238: subst] :
( ( comp @ id @ X238 )
= X238 )
& ! [X239: subst] :
( ( comp @ X239 @ id )
= X239 ) )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ! [X240: subst] :
( ( comp @ X240 @ id )
= X240 )
& ! [X241: subst] :
( ( comp @ id @ X241 )
= X241 )
& ( ( ( comp @ id @ sK571 )
!= sK571 )
| ( ( comp @ sK572 @ id )
!= sK572 ) ) ) )
& ( ulamvar1 != $true )
& ( pushprop_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( axscons = $true )
| ( sK573
!= ( push @ ( sub @ one @ sK573 ) @ ( comp @ sh @ sK573 ) ) ) )
& ( ! [X245: subst] :
( ( push @ ( sub @ one @ X245 ) @ ( comp @ sh @ X245 ) )
= X245 )
| ( axscons != $true ) )
& ( ( axshiftcons = $true )
| ( ( comp @ sh @ ( push @ sK574 @ sK575 ) )
!= sK575 ) )
& ( ! [X248: term,X249: subst] :
( ( comp @ sh @ ( push @ X248 @ X249 ) )
= X249 )
| ( axshiftcons != $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( $true = sP1 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( $true != sP1 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X250: subst,X251: term] :
( ( sub @ one @ ( push @ X251 @ X250 ) )
= X251 )
& ! [X252: term,X253: subst] :
( ( comp @ sh @ ( push @ X252 @ X253 ) )
= X253 )
& ( ( sK576 @ ( sub @ sK580 @ ( push @ sK577 @ sK578 ) ) )
!= $true )
& ( $true
= ( sK579 @ sK580 ) ) ) )
& ( ( ( sub @ one @ ( push @ sK582 @ sK581 ) )
!= sK582 )
| ( sK584
!= ( comp @ sh @ ( push @ sK583 @ sK584 ) ) )
| ! [X263: term > $o,X264: term,X265: subst,X266: term > $o,X267: term] :
( ( ( X263 @ ( sub @ X267 @ ( push @ X264 @ X265 ) ) )
= $true )
| ( $true
!= ( X266 @ X267 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoaslamnotap = $true )
| ( ( ( ap @ ( sub @ sK586 @ id ) @ sK585 )
= ( lam @ ( sK587 @ sh @ one ) ) )
& ! [X271: subst,X272: term,X273: subst] :
( ( sK587 @ ( comp @ X273 @ X271 ) @ ( sub @ X272 @ X271 ) )
= ( sub @ ( sK587 @ X273 @ X272 ) @ X271 ) ) ) )
& ( ! [X274: term,X275: term,X276: subst > term > term] :
( ( ( lam @ ( X276 @ sh @ one ) )
!= ( ap @ ( sub @ X275 @ id ) @ X274 ) )
| ( ( sub @ ( X276 @ ( sK590 @ X276 ) @ ( sK589 @ X276 ) ) @ ( sK588 @ X276 ) )
!= ( X276 @ ( comp @ ( sK590 @ X276 ) @ ( sK588 @ X276 ) ) @ ( sub @ ( sK589 @ X276 ) @ ( sK588 @ X276 ) ) ) ) )
| ( hoaslamnotap != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK454,sK455,sK456,sK457,sK458,sK459,sK460,sK461,sK462,sK463,sK464,sK465,sK466,sK467,sK468,sK469,sK470,sK471,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,sK487,sK488,sK489,sK490,sK491,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,sK509,sK510,sK511,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,sK521,sK522,sK523,sK524,sK525,sK526,sK527,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,sK536,sK537,sK538,sK539,sK540,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,sK562,sK563,sK564,sK565,sK566,sK567,sK568,sK569,sK570,sK571,sK572,sK573,sK574,sK575,sK576,sK577,sK578,sK579,sK580,sK581,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590])],[f473,f544,f543,f542,f541,f540,f539,f538,f537,f536,f535,f534,f533,f532,f531,f530,f529,f528,f527,f526,f525,f524,f523,f522,f521,f520,f519,f518,f517,f516,f515,f514,f513,f512,f511,f510,f509,f508,f507,f506,f505,f504,f503,f502,f501,f500,f499,f498,f497,f496,f495,f494,f493,f492,f491,f490,f489,f488,f487,f486,f485,f484,f483,f482,f481,f480,f479,f478,f477,f476,f475,f474]) ).
thf(f474,plain,
! [X0: term] :
( ? [X1: subst > term > term] :
! [X2: term,X3: subst] :
( ( X1 @ X3 @ X2 )
= ( sub @ X0 @ ( push @ X2 @ X3 ) ) )
=> ! [X3: subst,X2: term] :
( ( sK454 @ X0 @ X3 @ X2 )
= ( sub @ X0 @ ( push @ X2 @ X3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f475,plain,
( ? [X4: term] :
! [X5: subst > term > term] :
? [X6: term,X7: subst] :
( ( X5 @ X7 @ X6 )
!= ( sub @ X4 @ ( push @ X6 @ X7 ) ) )
=> ! [X5: subst > term > term] :
? [X7: subst,X6: term] :
( ( X5 @ X7 @ X6 )
!= ( sub @ sK455 @ ( push @ X6 @ X7 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f476,plain,
! [X5: subst > term > term] :
( ? [X7: subst,X6: term] :
( ( X5 @ X7 @ X6 )
!= ( sub @ sK455 @ ( push @ X6 @ X7 ) ) )
=> ( ( X5 @ ( sK457 @ X5 ) @ ( sK456 @ X5 ) )
!= ( sub @ sK455 @ ( push @ ( sK456 @ X5 ) @ ( sK457 @ X5 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f477,plain,
( ? [X8: term,X9: term,X10: term,X11: term] :
( ( ( ap @ X8 @ X11 )
= ( ap @ X9 @ X10 ) )
& ( X10 != X11 ) )
=> ( ( ( ap @ sK459 @ sK460 )
= ( ap @ sK458 @ sK461 ) )
& ( sK461 != sK460 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f478,plain,
( ? [X17: subst] :
( ( comp @ id @ X17 )
!= X17 )
=> ( sK462
!= ( comp @ id @ sK462 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f479,plain,
! [X18: subst] :
( ? [X19: d_subst] :
( ( d2subst @ X19 )
= X18 )
=> ( ( d2subst @ ( sK463 @ X18 ) )
= X18 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f480,plain,
! [X20: subst > term > subst > $o] :
( ? [X21: term > $o] :
! [X22: term] :
( ( X20 @ id @ X22 @ id )
= ( X21 @ X22 ) )
=> ! [X22: term] :
( ( sK464 @ X20 @ X22 )
= ( X20 @ id @ X22 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f481,plain,
( ? [X23: subst > term > subst > $o] :
! [X24: term > $o] :
? [X25: term] :
( ( X24 @ X25 )
!= ( X23 @ id @ X25 @ id ) )
=> ! [X24: term > $o] :
? [X25: term] :
( ( X24 @ X25 )
!= ( sK465 @ id @ X25 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f482,plain,
! [X24: term > $o] :
( ? [X25: term] :
( ( X24 @ X25 )
!= ( sK465 @ id @ X25 @ id ) )
=> ( ( sK465 @ id @ ( sK466 @ X24 ) @ id )
!= ( X24 @ ( sK466 @ X24 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f483,plain,
( ? [X31: term,X32: term,X33: term,X34: term] :
( ( ( ap @ ( sub @ X33 @ id ) @ X31 )
= ( ap @ ( sub @ X34 @ id ) @ X32 ) )
& ( X33 != X34 ) )
=> ( ( ( ap @ ( sub @ sK470 @ id ) @ sK468 )
= ( ap @ ( sub @ sK469 @ id ) @ sK467 ) )
& ( sK469 != sK470 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f484,plain,
( ? [X35: term,X36: term,X37: term,X38: term] :
( ( X35 != X36 )
& ( ( ap @ X35 @ X38 )
= ( ap @ X36 @ X37 ) ) )
=> ( ( sK471 != sK472 )
& ( ( ap @ sK471 @ sK474 )
= ( ap @ sK472 @ sK473 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f485,plain,
( ? [X39: term] :
( ( sub @ X39 @ id )
!= X39 )
=> ( ( sub @ sK475 @ id )
!= sK475 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f486,plain,
( ? [X49: term > $o,X50: term,X51: subst,X52: term > $o,X53: term] :
( ( $true
!= ( X52 @ ( sub @ X50 @ ( push @ X53 @ X51 ) ) ) )
& ( $true
= ( X49 @ X50 ) ) )
=> ( ( $true
!= ( sK479 @ ( sub @ sK477 @ ( push @ sK480 @ sK478 ) ) ) )
& ( ( sK476 @ sK477 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f487,plain,
( ? [X54: subst > term > term,X55: subst > term > term,X56: subst,X57: term] :
( ( ( X55 @ X56 @ X57 )
!= ( X54 @ X56 @ X57 ) )
& ! [X58: subst,X59: subst,X60: term] :
( ( sub @ ( X55 @ X59 @ X60 ) @ X58 )
= ( X55 @ ( comp @ X59 @ X58 ) @ ( sub @ X60 @ X58 ) ) )
& ( ( lam @ ( X55 @ sh @ one ) )
= ( lam @ ( X54 @ sh @ one ) ) )
& ! [X61: term,X62: subst,X63: subst] :
( ( sub @ ( X54 @ X62 @ X61 ) @ X63 )
= ( X54 @ ( comp @ X62 @ X63 ) @ ( sub @ X61 @ X63 ) ) ) )
=> ( ( ( sK481 @ sK483 @ sK484 )
!= ( sK482 @ sK483 @ sK484 ) )
& ! [X60: term,X59: subst,X58: subst] :
( ( sub @ ( sK482 @ X59 @ X60 ) @ X58 )
= ( sK482 @ ( comp @ X59 @ X58 ) @ ( sub @ X60 @ X58 ) ) )
& ( ( lam @ ( sK481 @ sh @ one ) )
= ( lam @ ( sK482 @ sh @ one ) ) )
& ! [X63: subst,X62: subst,X61: term] :
( ( sub @ ( sK481 @ X62 @ X61 ) @ X63 )
= ( sK481 @ ( comp @ X62 @ X63 ) @ ( sub @ X61 @ X63 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f488,plain,
! [X65: subst > term > term] :
( ? [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X65 @ X69 @ X70 ) @ X68 )
!= ( X65 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
=> ( ( sub @ ( X65 @ ( sK486 @ X65 ) @ ( sK487 @ X65 ) ) @ ( sK485 @ X65 ) )
!= ( X65 @ ( comp @ ( sK486 @ X65 ) @ ( sK485 @ X65 ) ) @ ( sub @ ( sK487 @ X65 ) @ ( sK485 @ X65 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f489,plain,
! [X64: subst > term > term] :
( ? [X71: term,X72: subst,X73: subst] :
( ( X64 @ ( comp @ X72 @ X73 ) @ ( sub @ X71 @ X73 ) )
!= ( sub @ ( X64 @ X72 @ X71 ) @ X73 ) )
=> ( ( X64 @ ( comp @ ( sK489 @ X64 ) @ ( sK490 @ X64 ) ) @ ( sub @ ( sK488 @ X64 ) @ ( sK490 @ X64 ) ) )
!= ( sub @ ( X64 @ ( sK489 @ X64 ) @ ( sK488 @ X64 ) ) @ ( sK490 @ X64 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f490,plain,
( ? [X74: term,X75: term > $o,X76: subst] :
! [X77: term > $o] :
? [X78: term] :
( ( X77 @ X78 )
!= ( X75 @ ( sub @ X78 @ ( push @ X74 @ X76 ) ) ) )
=> ! [X77: term > $o] :
? [X78: term] :
( ( X77 @ X78 )
!= ( sK492 @ ( sub @ X78 @ ( push @ sK491 @ sK493 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f491,plain,
! [X77: term > $o] :
( ? [X78: term] :
( ( X77 @ X78 )
!= ( sK492 @ ( sub @ X78 @ ( push @ sK491 @ sK493 ) ) ) )
=> ( ( sK492 @ ( sub @ ( sK494 @ X77 ) @ ( push @ sK491 @ sK493 ) ) )
!= ( X77 @ ( sK494 @ X77 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f492,plain,
! [X79: term,X80: term > $o,X81: subst] :
( ? [X82: term > $o] :
! [X83: term] :
( ( X82 @ X83 )
= ( X80 @ ( sub @ X83 @ ( push @ X79 @ X81 ) ) ) )
=> ! [X83: term] :
( ( sK495 @ X81 @ X80 @ X79 @ X83 )
= ( X80 @ ( sub @ X83 @ ( push @ X79 @ X81 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f493,plain,
( ? [X88: term,X89: term,X90: term,X91: term] :
( ( ( ap @ ( sub @ X88 @ id ) @ X90 )
= ( ap @ ( sub @ X89 @ id ) @ X91 ) )
& ( X88 != X89 ) )
=> ( ( ( ap @ ( sub @ sK497 @ id ) @ sK499 )
= ( ap @ ( sub @ sK496 @ id ) @ sK498 ) )
& ( sK496 != sK497 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f494,plain,
( ? [X94: term > $o,X95: subst,X96: term] :
! [X97: term > $o] :
? [X98: term] :
( ( X94 @ ( sub @ X98 @ ( push @ X96 @ X95 ) ) )
!= ( X97 @ X98 ) )
=> ! [X97: term > $o] :
? [X98: term] :
( ( sK500 @ ( sub @ X98 @ ( push @ sK502 @ sK501 ) ) )
!= ( X97 @ X98 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f495,plain,
! [X97: term > $o] :
( ? [X98: term] :
( ( sK500 @ ( sub @ X98 @ ( push @ sK502 @ sK501 ) ) )
!= ( X97 @ X98 ) )
=> ( ( sK500 @ ( sub @ ( sK503 @ X97 ) @ ( push @ sK502 @ sK501 ) ) )
!= ( X97 @ ( sK503 @ X97 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f496,plain,
! [X99: term > $o,X100: subst,X101: term] :
( ? [X102: term > $o] :
! [X103: term] :
( ( X99 @ ( sub @ X103 @ ( push @ X101 @ X100 ) ) )
= ( X102 @ X103 ) )
=> ! [X103: term] :
( ( X99 @ ( sub @ X103 @ ( push @ X101 @ X100 ) ) )
= ( sK504 @ X101 @ X100 @ X99 @ X103 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f497,plain,
! [X104: term > $o] :
( ? [X106: term] :
( ( $true
!= ( X104 @ X106 ) )
& ( $true
= ( var @ X106 ) ) )
=> ( ( ( X104 @ ( sK505 @ X104 ) )
!= $true )
& ( ( var @ ( sK505 @ X104 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f498,plain,
! [X104: term > $o] :
( ? [X107: term] :
( ( ( X104 @ ( lam @ X107 ) )
!= $true )
& ( $true
= ( X104 @ X107 ) ) )
=> ( ( ( X104 @ ( lam @ ( sK506 @ X104 ) ) )
!= $true )
& ( $true
= ( X104 @ ( sK506 @ X104 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f499,plain,
( ? [X108: term > $o,X109: term] :
( ! [X110: term] :
( ( $true
= ( X108 @ X110 ) )
| ( ( var @ X110 )
!= $true ) )
& ( $true
!= ( X108 @ X109 ) )
& ! [X111: term] :
( ( $true
= ( X108 @ ( lam @ X111 ) ) )
| ( ( X108 @ X111 )
!= $true ) )
& ( ( sP41 @ X108 )
!= $true ) )
=> ( ! [X110: term] :
( ( ( sK507 @ X110 )
= $true )
| ( ( var @ X110 )
!= $true ) )
& ( $true
!= ( sK507 @ sK508 ) )
& ! [X111: term] :
( ( ( sK507 @ ( lam @ X111 ) )
= $true )
| ( $true
!= ( sK507 @ X111 ) ) )
& ( ( sP41 @ sK507 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f500,plain,
! [X112: term] :
( ? [X113: subst > term > term] :
! [X114: subst,X115: term] :
( ( sub @ X112 @ ( push @ X115 @ X114 ) )
= ( X113 @ X114 @ X115 ) )
=> ! [X115: term,X114: subst] :
( ( sub @ X112 @ ( push @ X115 @ X114 ) )
= ( sK509 @ X112 @ X114 @ X115 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f501,plain,
( ? [X116: term] :
! [X117: subst > term > term] :
? [X118: subst,X119: term] :
( ( sub @ X116 @ ( push @ X119 @ X118 ) )
!= ( X117 @ X118 @ X119 ) )
=> ! [X117: subst > term > term] :
? [X119: term,X118: subst] :
( ( sub @ sK510 @ ( push @ X119 @ X118 ) )
!= ( X117 @ X118 @ X119 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f502,plain,
! [X117: subst > term > term] :
( ? [X119: term,X118: subst] :
( ( sub @ sK510 @ ( push @ X119 @ X118 ) )
!= ( X117 @ X118 @ X119 ) )
=> ( ( X117 @ ( sK511 @ X117 ) @ ( sK512 @ X117 ) )
!= ( sub @ sK510 @ ( push @ ( sK512 @ X117 ) @ ( sK511 @ X117 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f503,plain,
( ? [X120: subst] :
( ( comp @ X120 @ id )
!= X120 )
=> ( ( comp @ sK513 @ id )
!= sK513 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f504,plain,
( ? [X123: term,X124: term,X125: term,X126: term] :
( ( X123 != X126 )
& ( ( ap @ X126 @ X124 )
= ( ap @ X123 @ X125 ) ) )
=> ( ( sK517 != sK514 )
& ( ( ap @ sK514 @ sK516 )
= ( ap @ sK517 @ sK515 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f505,plain,
( ? [X131: term > $o,X132: term > $o,X133: term,X134: subst] :
( ( $true
= ( X131 @ X133 ) )
& ( $true
!= ( X132 @ one ) )
& ! [X135: term] :
( ( X132 @ X135 )
= ( X131 @ ( sub @ X135 @ ( push @ X133 @ X134 ) ) ) ) )
=> ( ( ( sK518 @ sK520 )
= $true )
& ( $true
!= ( sK519 @ one ) )
& ! [X135: term] :
( ( sK519 @ X135 )
= ( sK518 @ ( sub @ X135 @ ( push @ sK520 @ sK521 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f506,plain,
! [X138: term > $o,X139: term > $o,X140: term,X141: subst] :
( ? [X142: term] :
( ( X138 @ ( sub @ X142 @ ( push @ X140 @ X141 ) ) )
!= ( X139 @ X142 ) )
=> ( ( X138 @ ( sub @ ( sK522 @ X141 @ X140 @ X139 @ X138 ) @ ( push @ X140 @ X141 ) ) )
!= ( X139 @ ( sK522 @ X141 @ X140 @ X139 @ X138 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f507,plain,
( ? [X143: subst,X144: term] :
( ( sub @ one @ ( push @ X144 @ X143 ) )
!= X144 )
=> ( sK524
!= ( sub @ one @ ( push @ sK524 @ sK523 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f508,plain,
( ? [X145: term,X146: term] :
( ( X145 != X146 )
& ( ( lam @ X146 )
= ( lam @ X145 ) ) )
=> ( ( sK525 != sK526 )
& ( ( lam @ sK526 )
= ( lam @ sK525 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f509,plain,
( ? [X151: term,X152: term] :
( ( ( sub @ X152 @ sh )
= ( sub @ X151 @ sh ) )
& ( X151 != X152 ) )
=> ( ( ( sub @ sK528 @ sh )
= ( sub @ sK527 @ sh ) )
& ( sK527 != sK528 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f510,plain,
( ? [X154: term] :
( ( sub @ X154 @ id )
!= X154 )
=> ( sK529
!= ( sub @ sK529 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f511,plain,
( ? [X155: term] :
( ( sub @ X155 @ id )
!= X155 )
=> ( ( sub @ sK530 @ id )
!= sK530 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f512,plain,
( ? [X158: term] :
( ( sub @ X158 @ id )
!= X158 )
=> ( ( sub @ sK531 @ id )
!= sK531 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f513,plain,
( ? [X163: term,X164: term,X165: term,X166: term] :
( ( ( ap @ ( sub @ X163 @ id ) @ X164 )
= ( ap @ ( sub @ X165 @ id ) @ X166 ) )
& ( X164 != X166 ) )
=> ( ( ( ap @ ( sub @ sK532 @ id ) @ sK533 )
= ( ap @ ( sub @ sK534 @ id ) @ sK535 ) )
& ( sK533 != sK535 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f514,plain,
( ? [X167: term,X168: term,X169: term,X170: term] :
( ( X167 != X170 )
& ( ( ap @ X168 @ X167 )
= ( ap @ X169 @ X170 ) ) )
=> ( ( sK539 != sK536 )
& ( ( ap @ sK538 @ sK539 )
= ( ap @ sK537 @ sK536 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f515,plain,
( ? [X179: term,X180: term,X181: term,X182: term] :
( ( X180 != X181 )
& ( ( ap @ ( sub @ X182 @ id ) @ X181 )
= ( ap @ ( sub @ X179 @ id ) @ X180 ) ) )
=> ( ( sK541 != sK542 )
& ( ( ap @ ( sub @ sK540 @ id ) @ sK541 )
= ( ap @ ( sub @ sK543 @ id ) @ sK542 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f516,plain,
! [X183: term] :
( ? [X184: d_term] :
( ( d2term @ X184 )
= X183 )
=> ( ( d2term @ ( sK544 @ X183 ) )
= X183 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f517,plain,
( ? [X185: term > $o,X186: term > $o,X187: term,X188: subst] :
( ( $true
= ( X185 @ X187 ) )
& ( $true
!= ( X186 @ one ) )
& ! [X189: term] :
( ( X186 @ X189 )
= ( X185 @ ( sub @ X189 @ ( push @ X187 @ X188 ) ) ) ) )
=> ( ( $true
= ( sK545 @ sK547 ) )
& ( ( sK546 @ one )
!= $true )
& ! [X189: term] :
( ( sK545 @ ( sub @ X189 @ ( push @ sK547 @ sK548 ) ) )
= ( sK546 @ X189 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f518,plain,
! [X190: term > $o,X191: term > $o,X192: term,X193: subst] :
( ? [X194: term] :
( ( X190 @ ( sub @ X194 @ ( push @ X192 @ X193 ) ) )
!= ( X191 @ X194 ) )
=> ( ( X190 @ ( sub @ ( sK549 @ X193 @ X192 @ X191 @ X190 ) @ ( push @ X192 @ X193 ) ) )
!= ( X191 @ ( sK549 @ X193 @ X192 @ X191 @ X190 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f519,plain,
( ? [X197: subst] :
( ( comp @ X197 @ id )
!= X197 )
=> ( ( comp @ sK550 @ id )
!= sK550 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f520,plain,
( ? [X198: subst] :
( ( comp @ id @ X198 )
!= X198 )
=> ( ( comp @ id @ sK551 )
!= sK551 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f521,plain,
( ? [X199: subst > term > subst > $o] :
! [X200: term > $o] :
? [X201: term] :
( ( X200 @ X201 )
!= ( X199 @ id @ X201 @ id ) )
=> ! [X200: term > $o] :
? [X201: term] :
( ( sK552 @ id @ X201 @ id )
!= ( X200 @ X201 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f522,plain,
! [X200: term > $o] :
( ? [X201: term] :
( ( sK552 @ id @ X201 @ id )
!= ( X200 @ X201 ) )
=> ( ( X200 @ ( sK553 @ X200 ) )
!= ( sK552 @ id @ ( sK553 @ X200 ) @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f523,plain,
! [X202: subst > term > subst > $o] :
( ? [X203: term > $o] :
! [X204: term] :
( ( X202 @ id @ X204 @ id )
= ( X203 @ X204 ) )
=> ! [X204: term] :
( ( sK554 @ X202 @ X204 )
= ( X202 @ id @ X204 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f524,plain,
( ? [X205: term] :
! [X206: subst > term > term] :
( ( X206 @ sh @ one )
!= ( sub @ X205 @ ( push @ one @ sh ) ) )
=> ! [X206: subst > term > term] :
( ( X206 @ sh @ one )
!= ( sub @ sK555 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f525,plain,
! [X207: term] :
( ? [X208: subst > term > term] :
( ( sub @ X207 @ ( push @ one @ sh ) )
= ( X208 @ sh @ one ) )
=> ( ( sub @ X207 @ ( push @ one @ sh ) )
= ( sK556 @ X207 @ sh @ one ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f526,plain,
( ? [X210: subst > term > subst > $o,X211: term] :
( ! [X212: subst > term > term] :
( ? [X213: term] :
( ( $true
= ( X210 @ id @ X213 @ id ) )
& ( $true
!= ( X210 @ id @ ( X212 @ id @ X213 ) @ id ) ) )
| ? [X214: term,X215: subst,X216: subst] :
( ( sub @ ( X212 @ X216 @ X214 ) @ X215 )
!= ( X212 @ ( comp @ X216 @ X215 ) @ ( sub @ X214 @ X215 ) ) )
| ( $true
= ( X210 @ id @ ( lam @ ( X212 @ sh @ one ) ) @ id ) ) )
& ( $true
!= ( sP4 @ X210 ) )
& ( ( X210 @ id @ X211 @ id )
!= $true ) )
=> ( ! [X212: subst > term > term] :
( ? [X213: term] :
( ( ( sK557 @ id @ X213 @ id )
= $true )
& ( ( sK557 @ id @ ( X212 @ id @ X213 ) @ id )
!= $true ) )
| ? [X214: term,X215: subst,X216: subst] :
( ( sub @ ( X212 @ X216 @ X214 ) @ X215 )
!= ( X212 @ ( comp @ X216 @ X215 ) @ ( sub @ X214 @ X215 ) ) )
| ( $true
= ( sK557 @ id @ ( lam @ ( X212 @ sh @ one ) ) @ id ) ) )
& ( $true
!= ( sP4 @ sK557 ) )
& ( ( sK557 @ id @ sK558 @ id )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f527,plain,
! [X212: subst > term > term] :
( ? [X213: term] :
( ( ( sK557 @ id @ X213 @ id )
= $true )
& ( ( sK557 @ id @ ( X212 @ id @ X213 ) @ id )
!= $true ) )
=> ( ( $true
= ( sK557 @ id @ ( sK559 @ X212 ) @ id ) )
& ( $true
!= ( sK557 @ id @ ( X212 @ id @ ( sK559 @ X212 ) ) @ id ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f528,plain,
! [X212: subst > term > term] :
( ? [X214: term,X215: subst,X216: subst] :
( ( sub @ ( X212 @ X216 @ X214 ) @ X215 )
!= ( X212 @ ( comp @ X216 @ X215 ) @ ( sub @ X214 @ X215 ) ) )
=> ( ( sub @ ( X212 @ ( sK562 @ X212 ) @ ( sK560 @ X212 ) ) @ ( sK561 @ X212 ) )
!= ( X212 @ ( comp @ ( sK562 @ X212 ) @ ( sK561 @ X212 ) ) @ ( sub @ ( sK560 @ X212 ) @ ( sK561 @ X212 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f529,plain,
! [X217: subst > term > subst > $o] :
( ? [X219: subst > term > term] :
( ! [X220: term] :
( ( ( X217 @ id @ X220 @ id )
!= $true )
| ( $true
= ( X217 @ id @ ( X219 @ id @ X220 ) @ id ) ) )
& ! [X221: term,X222: subst,X223: subst] :
( ( sub @ ( X219 @ X223 @ X221 ) @ X222 )
= ( X219 @ ( comp @ X223 @ X222 ) @ ( sub @ X221 @ X222 ) ) )
& ( $true
!= ( X217 @ id @ ( lam @ ( X219 @ sh @ one ) ) @ id ) ) )
=> ( ! [X220: term] :
( ( ( X217 @ id @ X220 @ id )
!= $true )
| ( $true
= ( X217 @ id @ ( sK563 @ X217 @ id @ X220 ) @ id ) ) )
& ! [X223: subst,X222: subst,X221: term] :
( ( sub @ ( sK563 @ X217 @ X223 @ X221 ) @ X222 )
= ( sK563 @ X217 @ ( comp @ X223 @ X222 ) @ ( sub @ X221 @ X222 ) ) )
& ( ( X217 @ id @ ( lam @ ( sK563 @ X217 @ sh @ one ) ) @ id )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f530,plain,
! [X224: term] :
( ? [X225: subst > term > term] :
( ( X225 @ sh @ one )
= ( sub @ X224 @ ( push @ one @ sh ) ) )
=> ( ( sK564 @ X224 @ sh @ one )
= ( sub @ X224 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f531,plain,
( ? [X226: term] :
! [X227: subst > term > term] :
( ( X227 @ sh @ one )
!= ( sub @ X226 @ ( push @ one @ sh ) ) )
=> ! [X227: subst > term > term] :
( ( X227 @ sh @ one )
!= ( sub @ sK565 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f532,plain,
( ? [X230: subst,X231: term] :
( ( sub @ one @ ( push @ X231 @ X230 ) )
!= X231 )
=> ( ( sub @ one @ ( push @ sK567 @ sK566 ) )
!= sK567 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f533,plain,
( ? [X233: term] :
( ( sub @ X233 @ id )
!= X233 )
=> ( ( sub @ sK568 @ id )
!= sK568 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f534,plain,
( ? [X236: subst] :
( ( comp @ X236 @ id )
!= X236 )
=> ( ( comp @ sK569 @ id )
!= sK569 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f535,plain,
( ? [X237: subst] :
( ( comp @ id @ X237 )
!= X237 )
=> ( ( comp @ id @ sK570 )
!= sK570 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f536,plain,
( ? [X242: subst] :
( ( comp @ id @ X242 )
!= X242 )
=> ( ( comp @ id @ sK571 )
!= sK571 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f537,plain,
( ? [X243: subst] :
( ( comp @ X243 @ id )
!= X243 )
=> ( ( comp @ sK572 @ id )
!= sK572 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f538,plain,
( ? [X244: subst] :
( ( push @ ( sub @ one @ X244 ) @ ( comp @ sh @ X244 ) )
!= X244 )
=> ( sK573
!= ( push @ ( sub @ one @ sK573 ) @ ( comp @ sh @ sK573 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f539,plain,
( ? [X246: term,X247: subst] :
( ( comp @ sh @ ( push @ X246 @ X247 ) )
!= X247 )
=> ( ( comp @ sh @ ( push @ sK574 @ sK575 ) )
!= sK575 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f540,plain,
( ? [X254: term > $o,X255: term,X256: subst,X257: term > $o,X258: term] :
( ( $true
!= ( X254 @ ( sub @ X258 @ ( push @ X255 @ X256 ) ) ) )
& ( $true
= ( X257 @ X258 ) ) )
=> ( ( ( sK576 @ ( sub @ sK580 @ ( push @ sK577 @ sK578 ) ) )
!= $true )
& ( $true
= ( sK579 @ sK580 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f541,plain,
( ? [X259: subst,X260: term] :
( ( sub @ one @ ( push @ X260 @ X259 ) )
!= X260 )
=> ( ( sub @ one @ ( push @ sK582 @ sK581 ) )
!= sK582 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f542,plain,
( ? [X261: term,X262: subst] :
( ( comp @ sh @ ( push @ X261 @ X262 ) )
!= X262 )
=> ( sK584
!= ( comp @ sh @ ( push @ sK583 @ sK584 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f543,plain,
( ? [X268: term,X269: term,X270: subst > term > term] :
( ( ( lam @ ( X270 @ sh @ one ) )
= ( ap @ ( sub @ X269 @ id ) @ X268 ) )
& ! [X271: subst,X272: term,X273: subst] :
( ( sub @ ( X270 @ X273 @ X272 ) @ X271 )
= ( X270 @ ( comp @ X273 @ X271 ) @ ( sub @ X272 @ X271 ) ) ) )
=> ( ( ( ap @ ( sub @ sK586 @ id ) @ sK585 )
= ( lam @ ( sK587 @ sh @ one ) ) )
& ! [X273: subst,X272: term,X271: subst] :
( ( sK587 @ ( comp @ X273 @ X271 ) @ ( sub @ X272 @ X271 ) )
= ( sub @ ( sK587 @ X273 @ X272 ) @ X271 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f544,plain,
! [X276: subst > term > term] :
( ? [X277: subst,X278: term,X279: subst] :
( ( X276 @ ( comp @ X279 @ X277 ) @ ( sub @ X278 @ X277 ) )
!= ( sub @ ( X276 @ X279 @ X278 ) @ X277 ) )
=> ( ( sub @ ( X276 @ ( sK590 @ X276 ) @ ( sK589 @ X276 ) ) @ ( sK588 @ X276 ) )
!= ( X276 @ ( comp @ ( sK590 @ X276 ) @ ( sK588 @ X276 ) ) @ ( sub @ ( sK589 @ X276 ) @ ( sK588 @ X276 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f473,plain,
( ( ! [X0: term] :
? [X1: subst > term > term] :
! [X2: term,X3: subst] :
( ( X1 @ X3 @ X2 )
= ( sub @ X0 @ ( push @ X2 @ X3 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X4: term] :
! [X5: subst > term > term] :
? [X6: term,X7: subst] :
( ( X5 @ X7 @ X6 )
!= ( sub @ X4 @ ( push @ X6 @ X7 ) ) ) )
& ( ( apinj2 = $true )
| ? [X8: term,X9: term,X10: term,X11: term] :
( ( ( ap @ X8 @ X11 )
= ( ap @ X9 @ X10 ) )
& ( X10 != X11 ) ) )
& ( ! [X12: term,X13: term,X14: term,X15: term] :
( ( ( ap @ X13 @ X14 )
!= ( ap @ X12 @ X15 ) )
| ( X14 = X15 ) )
| ( apinj2 != $true ) )
& ( ! [X16: subst] :
( ( comp @ id @ X16 )
= X16 )
| ( axidl != $true ) )
& ( ( axidl = $true )
| ? [X17: subst] :
( ( comp @ id @ X17 )
!= X17 ) )
& ( ( sP58 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( sP58 != $true ) )
& ! [X18: subst] :
? [X19: d_subst] :
( ( d2subst @ X19 )
= X18 )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X20: subst > term > subst > $o] :
? [X21: term > $o] :
! [X22: term] :
( ( X20 @ id @ X22 @ id )
= ( X21 @ X22 ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X23: subst > term > subst > $o] :
! [X24: term > $o] :
? [X25: term] :
( ( X24 @ X25 )
!= ( X23 @ id @ X25 @ id ) ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP56 ) )
& ( ( $true = sP56 )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( $true != sP54 ) )
& ( ( $true = sP54 )
| ( hoasinduction_lem3v2 != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X26: term,X27: term,X28: term,X29: term] :
( ( X26 = X27 )
| ( ( ap @ X26 @ X29 )
!= ( ap @ X27 @ X28 ) ) )
& ! [X30: term] :
( ( sub @ X30 @ id )
= X30 )
& ? [X31: term,X32: term,X33: term,X34: term] :
( ( ( ap @ ( sub @ X33 @ id ) @ X31 )
= ( ap @ ( sub @ X34 @ id ) @ X32 ) )
& ( X33 != X34 ) ) ) )
& ( ? [X35: term,X36: term,X37: term,X38: term] :
( ( X35 != X36 )
& ( ( ap @ X35 @ X38 )
= ( ap @ X36 @ X37 ) ) )
| ? [X39: term] :
( ( sub @ X39 @ id )
!= X39 )
| ! [X40: term,X41: term,X42: term,X43: term] :
( ( ( ap @ ( sub @ X43 @ id ) @ X41 )
!= ( ap @ ( sub @ X42 @ id ) @ X40 ) )
| ( X42 = X43 ) )
| ( hoasapinj1_lthm != $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP52 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP52 != $true ) )
& ( induction2_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X44: term > $o,X45: term,X46: subst,X47: term > $o,X48: term] :
( ( $true
= ( X47 @ ( sub @ X45 @ ( push @ X48 @ X46 ) ) ) )
| ( $true
!= ( X44 @ X45 ) ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X49: term > $o,X50: term,X51: subst,X52: term > $o,X53: term] :
( ( $true
!= ( X52 @ ( sub @ X50 @ ( push @ X53 @ X51 ) ) ) )
& ( $true
= ( X49 @ X50 ) ) ) )
& ( ( hoaslaminj = $true )
| ? [X54: subst > term > term,X55: subst > term > term,X56: subst,X57: term] :
( ( ( X55 @ X56 @ X57 )
!= ( X54 @ X56 @ X57 ) )
& ! [X58: subst,X59: subst,X60: term] :
( ( sub @ ( X55 @ X59 @ X60 ) @ X58 )
= ( X55 @ ( comp @ X59 @ X58 ) @ ( sub @ X60 @ X58 ) ) )
& ( ( lam @ ( X55 @ sh @ one ) )
= ( lam @ ( X54 @ sh @ one ) ) )
& ! [X61: term,X62: subst,X63: subst] :
( ( sub @ ( X54 @ X62 @ X61 ) @ X63 )
= ( X54 @ ( comp @ X62 @ X63 ) @ ( sub @ X61 @ X63 ) ) ) ) )
& ( ! [X64: subst > term > term,X65: subst > term > term,X66: subst,X67: term] :
( ( ( X64 @ X66 @ X67 )
= ( X65 @ X66 @ X67 ) )
| ? [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X65 @ X69 @ X70 ) @ X68 )
!= ( X65 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
| ( ( lam @ ( X64 @ sh @ one ) )
!= ( lam @ ( X65 @ sh @ one ) ) )
| ? [X71: term,X72: subst,X73: subst] :
( ( X64 @ ( comp @ X72 @ X73 ) @ ( sub @ X71 @ X73 ) )
!= ( sub @ ( X64 @ X72 @ X71 ) @ X73 ) ) )
| ( hoaslaminj != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X74: term,X75: term > $o,X76: subst] :
! [X77: term > $o] :
? [X78: term] :
( ( X77 @ X78 )
!= ( X75 @ ( sub @ X78 @ ( push @ X74 @ X76 ) ) ) ) )
& ( ! [X79: term,X80: term > $o,X81: subst] :
? [X82: term > $o] :
! [X83: term] :
( ( X82 @ X83 )
= ( X80 @ ( sub @ X83 @ ( push @ X79 @ X81 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ! [X84: term,X85: term,X86: term,X87: term] :
( ( ( ap @ ( sub @ X84 @ id ) @ X86 )
!= ( ap @ ( sub @ X85 @ id ) @ X87 ) )
| ( X84 = X85 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X88: term,X89: term,X90: term,X91: term] :
( ( ( ap @ ( sub @ X88 @ id ) @ X90 )
= ( ap @ ( sub @ X89 @ id ) @ X91 ) )
& ( X88 != X89 ) ) )
& ! [X92: d_subst,X93: d_subst] :
( ( X92 = X93 )
| ( ( d2subst @ X93 )
!= ( d2subst @ X92 ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X94: term > $o,X95: subst,X96: term] :
! [X97: term > $o] :
? [X98: term] :
( ( X94 @ ( sub @ X98 @ ( push @ X96 @ X95 ) ) )
!= ( X97 @ X98 ) ) )
& ( ! [X99: term > $o,X100: subst,X101: term] :
? [X102: term > $o] :
! [X103: term] :
( ( X99 @ ( sub @ X103 @ ( push @ X101 @ X100 ) ) )
= ( X102 @ X103 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ( $true = sP47 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP47 ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP44 ) )
& ( ( $true = sP44 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ! [X104: term > $o,X105: term] :
( ? [X106: term] :
( ( $true
!= ( X104 @ X106 ) )
& ( $true
= ( var @ X106 ) ) )
| ( $true
= ( X104 @ X105 ) )
| ? [X107: term] :
( ( ( X104 @ ( lam @ X107 ) )
!= $true )
& ( $true
= ( X104 @ X107 ) ) )
| ( $true
= ( sP41 @ X104 ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X108: term > $o,X109: term] :
( ! [X110: term] :
( ( $true
= ( X108 @ X110 ) )
| ( ( var @ X110 )
!= $true ) )
& ( $true
!= ( X108 @ X109 ) )
& ! [X111: term] :
( ( $true
= ( X108 @ ( lam @ X111 ) ) )
| ( ( X108 @ X111 )
!= $true ) )
& ( ( sP41 @ X108 )
!= $true ) ) )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP40 )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( $true != sP40 ) )
& ( ( hoaslaminj_lthm = $true )
| ( sP34 != $true ) )
& ( ( sP34 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP33 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP33 ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( $true = sP31 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP31 ) )
& ( axapp = $true )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3aa != $true ) )
& ( ! [X112: term] :
? [X113: subst > term > term] :
! [X114: subst,X115: term] :
( ( sub @ X112 @ ( push @ X115 @ X114 ) )
= ( X113 @ X114 @ X115 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X116: term] :
! [X117: subst > term > term] :
? [X118: subst,X119: term] :
( ( sub @ X116 @ ( push @ X119 @ X118 ) )
!= ( X117 @ X118 @ X119 ) ) )
& ( ( induction2lem_lthm = $true )
| ( $true != sP28 ) )
& ( ( $true = sP28 )
| ( induction2lem_lthm != $true ) )
& ( axvarshift = $true )
& ( ( axidr = $true )
| ? [X120: subst] :
( ( comp @ X120 @ id )
!= X120 ) )
& ( ! [X121: subst] :
( ( comp @ X121 @ id )
= X121 )
| ( axidr != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3a = $true )
| ( sP24 != $true ) )
& ( ( sP24 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X122: d_subst] : ( d_id = X122 )
& ( ( apinj1 = $true )
| ? [X123: term,X124: term,X125: term,X126: term] :
( ( X123 != X126 )
& ( ( ap @ X126 @ X124 )
= ( ap @ X123 @ X125 ) ) ) )
& ( ! [X127: term,X128: term,X129: term,X130: term] :
( ( X127 = X130 )
| ( ( ap @ X127 @ X129 )
!= ( ap @ X130 @ X128 ) ) )
| ( apinj1 != $true ) )
& ( ( induction2_lthm = $true )
| ( $true != sP23 ) )
& ( ( $true = sP23 )
| ( induction2_lthm != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP19 ) )
& ( ( $true = sP19 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X131: term > $o,X132: term > $o,X133: term,X134: subst] :
( ( $true
= ( X131 @ X133 ) )
& ( $true
!= ( X132 @ one ) )
& ! [X135: term] :
( ( X132 @ X135 )
= ( X131 @ ( sub @ X135 @ ( push @ X133 @ X134 ) ) ) ) )
& ! [X136: subst,X137: term] :
( ( sub @ one @ ( push @ X137 @ X136 ) )
= X137 ) ) )
& ( ! [X138: term > $o,X139: term > $o,X140: term,X141: subst] :
( ( $true
!= ( X138 @ X140 ) )
| ( ( X139 @ one )
= $true )
| ? [X142: term] :
( ( X138 @ ( sub @ X142 @ ( push @ X140 @ X141 ) ) )
!= ( X139 @ X142 ) ) )
| ? [X143: subst,X144: term] :
( ( sub @ one @ ( push @ X144 @ X143 ) )
!= X144 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( sP18 != $true ) )
& ( ( sP18 = $true )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( laminj = $true )
| ? [X145: term,X146: term] :
( ( X145 != X146 )
& ( ( lam @ X146 )
= ( lam @ X145 ) ) ) )
& ( ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) )
| ( laminj != $true ) )
& ( ! [X149: term,X150: term] :
( ( ( sub @ X149 @ sh )
!= ( sub @ X150 @ sh ) )
| ( X149 = X150 ) )
| ( shinj != $true ) )
& ( ( shinj = $true )
| ? [X151: term,X152: term] :
( ( ( sub @ X152 @ sh )
= ( sub @ X151 @ sh ) )
& ( X151 != X152 ) ) )
& ( hoasapnotvar_gthm = $true )
& ( ! [X153: term] :
( ( sub @ X153 @ id )
= X153 )
| ? [X154: term] :
( ( sub @ X154 @ id )
!= X154 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ? [X155: term] :
( ( sub @ X155 @ id )
!= X155 )
& ! [X156: term] :
( ( sub @ X156 @ id )
= X156 ) ) )
& ( ! [X157: term] :
( ( sub @ X157 @ id )
= X157 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X158: term] :
( ( sub @ X158 @ id )
!= X158 ) )
& ( apnotvar = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lthm = $true )
| ( sP17 != $true ) )
& ( ( sP17 = $true )
| ( hoasinduction_lthm != $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ! [X159: term,X160: term,X161: term,X162: term] :
( ( X159 = X162 )
| ( ( ap @ X161 @ X162 )
!= ( ap @ X160 @ X159 ) ) )
& ? [X163: term,X164: term,X165: term,X166: term] :
( ( ( ap @ ( sub @ X163 @ id ) @ X164 )
= ( ap @ ( sub @ X165 @ id ) @ X166 ) )
& ( X164 != X166 ) ) ) )
& ( ? [X167: term,X168: term,X169: term,X170: term] :
( ( X167 != X170 )
& ( ( ap @ X168 @ X167 )
= ( ap @ X169 @ X170 ) ) )
| ! [X171: term,X172: term,X173: term,X174: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X171 @ id ) @ X172 ) )
| ( X172 = X174 ) )
| ( hoasapinj2_lthm != $true ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lthm = $true )
& ( ( sP8 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP8 != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( sP7 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( sP7 != $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( axassoc = $true )
& ( axabs = $true )
& ( lamnotap != $true )
& ( ! [X175: term,X176: term,X177: term,X178: term] :
( ( X176 = X177 )
| ( ( ap @ ( sub @ X178 @ id ) @ X177 )
!= ( ap @ ( sub @ X175 @ id ) @ X176 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X179: term,X180: term,X181: term,X182: term] :
( ( X180 != X181 )
& ( ( ap @ ( sub @ X182 @ id ) @ X181 )
= ( ap @ ( sub @ X179 @ id ) @ X180 ) ) ) )
& ! [X183: term] :
? [X184: d_term] :
( ( d2term @ X184 )
= X183 )
& ( pushprop = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
| ? [X185: term > $o,X186: term > $o,X187: term,X188: subst] :
( ( $true
= ( X185 @ X187 ) )
& ( $true
!= ( X186 @ one ) )
& ! [X189: term] :
( ( X186 @ X189 )
= ( X185 @ ( sub @ X189 @ ( push @ X187 @ X188 ) ) ) ) ) )
& ( ! [X190: term > $o,X191: term > $o,X192: term,X193: subst] :
( ( ( X190 @ X192 )
!= $true )
| ( $true
= ( X191 @ one ) )
| ? [X194: term] :
( ( X190 @ ( sub @ X194 @ ( push @ X192 @ X193 ) ) )
!= ( X191 @ X194 ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( ! [X195: subst] :
( ( comp @ X195 @ id )
= X195 )
& ! [X196: subst] :
( ( comp @ id @ X196 )
= X196 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X197: subst] :
( ( comp @ X197 @ id )
!= X197 )
| ? [X198: subst] :
( ( comp @ id @ X198 )
!= X198 ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X199: subst > term > subst > $o] :
! [X200: term > $o] :
? [X201: term] :
( ( X200 @ X201 )
!= ( X199 @ id @ X201 @ id ) ) )
& ( ! [X202: subst > term > subst > $o] :
? [X203: term > $o] :
! [X204: term] :
( ( X202 @ id @ X204 @ id )
= ( X203 @ X204 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP5 ) )
& ( ( $true = sP5 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X205: term] :
! [X206: subst > term > term] :
( ( X206 @ sh @ one )
!= ( sub @ X205 @ ( push @ one @ sh ) ) ) )
& ( ! [X207: term] :
? [X208: subst > term > term] :
( ( sub @ X207 @ ( push @ one @ sh ) )
= ( X208 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ulamvarsh = $true )
& ! [X209: d_term] : ( d_one = X209 )
& ( axclos = $true )
& ( substmonoid_gthm = $true )
& ( induction2lem_gthm = $true )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X210: subst > term > subst > $o,X211: term] :
( ! [X212: subst > term > term] :
( ? [X213: term] :
( ( $true
= ( X210 @ id @ X213 @ id ) )
& ( $true
!= ( X210 @ id @ ( X212 @ id @ X213 ) @ id ) ) )
| ? [X214: term,X215: subst,X216: subst] :
( ( sub @ ( X212 @ X216 @ X214 ) @ X215 )
!= ( X212 @ ( comp @ X216 @ X215 ) @ ( sub @ X214 @ X215 ) ) )
| ( $true
= ( X210 @ id @ ( lam @ ( X212 @ sh @ one ) ) @ id ) ) )
& ( $true
!= ( sP4 @ X210 ) )
& ( ( X210 @ id @ X211 @ id )
!= $true ) ) )
& ( ! [X217: subst > term > subst > $o,X218: term] :
( ? [X219: subst > term > term] :
( ! [X220: term] :
( ( ( X217 @ id @ X220 @ id )
!= $true )
| ( $true
= ( X217 @ id @ ( X219 @ id @ X220 ) @ id ) ) )
& ! [X221: term,X222: subst,X223: subst] :
( ( sub @ ( X219 @ X223 @ X221 ) @ X222 )
= ( X219 @ ( comp @ X223 @ X222 ) @ ( sub @ X221 @ X222 ) ) )
& ( $true
!= ( X217 @ id @ ( lam @ ( X219 @ sh @ one ) ) @ id ) ) )
| ( ( sP4 @ X217 )
= $true )
| ( $true
= ( X217 @ id @ X218 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( axmap = $true )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP3 ) )
& ( ( $true = sP3 )
| ( hoasinduction_lem2 != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasapinj2_gthm = $true )
& ( ! [X224: term] :
? [X225: subst > term > term] :
( ( X225 @ sh @ one )
= ( sub @ X224 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X226: term] :
! [X227: subst > term > term] :
( ( X227 @ sh @ one )
!= ( sub @ X226 @ ( push @ one @ sh ) ) ) )
& ( one
= ( d2term @ d_one ) )
& ( termmset_gthm = $true )
& ( ! [X228: subst,X229: term] :
( ( sub @ one @ ( push @ X229 @ X228 ) )
= X229 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X230: subst,X231: term] :
( ( sub @ one @ ( push @ X231 @ X230 ) )
!= X231 ) )
& ( ulamvarind = $true )
& ( hoasapinj1_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem3v2_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ! [X232: term] :
( ( sub @ X232 @ id )
= X232 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X233: term] :
( ( sub @ X233 @ id )
!= X233 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ! [X234: d_term,X235: d_term] :
( ( ( d2term @ X234 )
!= ( d2term @ X235 ) )
| ( X234 = X235 ) )
& ( ? [X236: subst] :
( ( comp @ X236 @ id )
!= X236 )
| ? [X237: subst] :
( ( comp @ id @ X237 )
!= X237 )
| ( ! [X238: subst] :
( ( comp @ id @ X238 )
= X238 )
& ! [X239: subst] :
( ( comp @ X239 @ id )
= X239 ) )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ! [X240: subst] :
( ( comp @ X240 @ id )
= X240 )
& ! [X241: subst] :
( ( comp @ id @ X241 )
= X241 )
& ( ? [X242: subst] :
( ( comp @ id @ X242 )
!= X242 )
| ? [X243: subst] :
( ( comp @ X243 @ id )
!= X243 ) ) ) )
& ( ulamvar1 != $true )
& ( pushprop_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( axscons = $true )
| ? [X244: subst] :
( ( push @ ( sub @ one @ X244 ) @ ( comp @ sh @ X244 ) )
!= X244 ) )
& ( ! [X245: subst] :
( ( push @ ( sub @ one @ X245 ) @ ( comp @ sh @ X245 ) )
= X245 )
| ( axscons != $true ) )
& ( ( axshiftcons = $true )
| ? [X246: term,X247: subst] :
( ( comp @ sh @ ( push @ X246 @ X247 ) )
!= X247 ) )
& ( ! [X248: term,X249: subst] :
( ( comp @ sh @ ( push @ X248 @ X249 ) )
= X249 )
| ( axshiftcons != $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( $true = sP1 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( $true != sP1 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X250: subst,X251: term] :
( ( sub @ one @ ( push @ X251 @ X250 ) )
= X251 )
& ! [X252: term,X253: subst] :
( ( comp @ sh @ ( push @ X252 @ X253 ) )
= X253 )
& ? [X254: term > $o,X255: term,X256: subst,X257: term > $o,X258: term] :
( ( $true
!= ( X254 @ ( sub @ X258 @ ( push @ X255 @ X256 ) ) ) )
& ( $true
= ( X257 @ X258 ) ) ) ) )
& ( ? [X259: subst,X260: term] :
( ( sub @ one @ ( push @ X260 @ X259 ) )
!= X260 )
| ? [X261: term,X262: subst] :
( ( comp @ sh @ ( push @ X261 @ X262 ) )
!= X262 )
| ! [X263: term > $o,X264: term,X265: subst,X266: term > $o,X267: term] :
( ( ( X263 @ ( sub @ X267 @ ( push @ X264 @ X265 ) ) )
= $true )
| ( $true
!= ( X266 @ X267 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoaslamnotap = $true )
| ? [X268: term,X269: term,X270: subst > term > term] :
( ( ( lam @ ( X270 @ sh @ one ) )
= ( ap @ ( sub @ X269 @ id ) @ X268 ) )
& ! [X271: subst,X272: term,X273: subst] :
( ( sub @ ( X270 @ X273 @ X272 ) @ X271 )
= ( X270 @ ( comp @ X273 @ X271 ) @ ( sub @ X272 @ X271 ) ) ) ) )
& ( ! [X274: term,X275: term,X276: subst > term > term] :
( ( ( lam @ ( X276 @ sh @ one ) )
!= ( ap @ ( sub @ X275 @ id ) @ X274 ) )
| ? [X277: subst,X278: term,X279: subst] :
( ( X276 @ ( comp @ X279 @ X277 ) @ ( sub @ X278 @ X277 ) )
!= ( sub @ ( X276 @ X279 @ X278 ) @ X277 ) ) )
| ( hoaslamnotap != $true ) ) ),
inference(rectify,[status(thm)],[f472]) ).
thf(f472,plain,
( ( ! [X138: term] :
? [X139: subst > term > term] :
! [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
= ( X139 @ X141 @ X140 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X138: term] :
! [X139: subst > term > term] :
? [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
!= ( X139 @ X141 @ X140 ) ) )
& ( ( apinj2 = $true )
| ? [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
= ( ap @ X79 @ X82 ) )
& ( X81 != X82 ) ) )
& ( ! [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
!= ( ap @ X79 @ X82 ) )
| ( X81 = X82 ) )
| ( apinj2 != $true ) )
& ( ! [X324: subst] :
( ( comp @ id @ X324 )
= X324 )
| ( axidl != $true ) )
& ( ( axidl = $true )
| ? [X324: subst] :
( ( comp @ id @ X324 )
!= X324 ) )
& ( ( sP58 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( sP58 != $true ) )
& ! [X356: subst] :
? [X357: d_subst] :
( ( d2subst @ X357 )
= X356 )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X26: subst > term > subst > $o] :
? [X27: term > $o] :
! [X28: term] :
( ( X27 @ X28 )
= ( X26 @ id @ X28 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X26: subst > term > subst > $o] :
! [X27: term > $o] :
? [X28: term] :
( ( X27 @ X28 )
!= ( X26 @ id @ X28 @ id ) ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP56 ) )
& ( ( $true = sP56 )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( $true != sP54 ) )
& ( ( $true = sP54 )
| ( hoasinduction_lem3v2 != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X527: term,X528: term,X529: term,X530: term] :
( ( X527 = X528 )
| ( ( ap @ X528 @ X529 )
!= ( ap @ X527 @ X530 ) ) )
& ! [X526: term] :
( ( sub @ X526 @ id )
= X526 )
& ? [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
& ( X533 != X534 ) ) ) )
& ( ? [X527: term,X528: term,X529: term,X530: term] :
( ( X527 != X528 )
& ( ( ap @ X528 @ X529 )
= ( ap @ X527 @ X530 ) ) )
| ? [X526: term] :
( ( sub @ X526 @ id )
!= X526 )
| ! [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
!= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
| ( X533 = X534 ) )
| ( hoasapinj1_lthm != $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP52 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP52 != $true ) )
& ( induction2_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) )
| ( $true
!= ( X472 @ X473 ) ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
!= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) )
& ( $true
= ( X472 @ X473 ) ) ) )
& ( ( hoaslaminj = $true )
| ? [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ( ( X144 @ X146 @ X147 )
!= ( X145 @ X146 @ X147 ) )
& ! [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
& ( ( lam @ ( X144 @ sh @ one ) )
= ( lam @ ( X145 @ sh @ one ) ) )
& ! [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) ) ) )
& ( ! [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ( ( X144 @ X146 @ X147 )
= ( X145 @ X146 @ X147 ) )
| ? [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
!= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
| ( ( lam @ ( X144 @ sh @ one ) )
!= ( lam @ ( X145 @ sh @ one ) ) )
| ? [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
!= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) ) )
| ( hoaslaminj != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X333: term,X334: term > $o,X335: subst] :
! [X336: term > $o] :
? [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
!= ( X336 @ X337 ) ) )
& ( ! [X333: term,X334: term > $o,X335: subst] :
? [X336: term > $o] :
! [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
= ( X336 @ X337 ) )
| ( pushprop_lem0 != $true ) )
& ( ! [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
!= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
| ( X507 = X508 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
& ( X507 != X508 ) ) )
& ! [X338: d_subst,X339: d_subst] :
( ( X338 = X339 )
| ( ( d2subst @ X338 )
!= ( d2subst @ X339 ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X535: term > $o,X536: subst,X537: term] :
! [X538: term > $o] :
? [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
!= ( X538 @ X539 ) ) )
& ( ! [X535: term > $o,X536: subst,X537: term] :
? [X538: term > $o] :
! [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
= ( X538 @ X539 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ( $true = sP47 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP47 ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP44 ) )
& ( ( $true = sP44 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ! [X230: term > $o,X231: term] :
( ? [X232: term] :
( ( $true
!= ( X230 @ X232 ) )
& ( ( var @ X232 )
= $true ) )
| ( ( X230 @ X231 )
= $true )
| ? [X235: term] :
( ( $true
!= ( X230 @ ( lam @ X235 ) ) )
& ( $true
= ( X230 @ X235 ) ) )
| ( $true
= ( sP41 @ X230 ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X230: term > $o,X231: term] :
( ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( ( var @ X232 )
!= $true ) )
& ( ( X230 @ X231 )
!= $true )
& ! [X235: term] :
( ( $true
= ( X230 @ ( lam @ X235 ) ) )
| ( $true
!= ( X230 @ X235 ) ) )
& ( $true
!= ( sP41 @ X230 ) ) ) )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP40 )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( $true != sP40 ) )
& ( ( hoaslaminj_lthm = $true )
| ( sP34 != $true ) )
& ( ( sP34 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP33 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP33 ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( $true = sP31 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP31 ) )
& ( axapp = $true )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3aa != $true ) )
& ( ! [X391: term] :
? [X392: subst > term > term] :
! [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
= ( X392 @ X393 @ X394 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X391: term] :
! [X392: subst > term > term] :
? [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
!= ( X392 @ X393 @ X394 ) ) )
& ( ( induction2lem_lthm = $true )
| ( $true != sP28 ) )
& ( ( $true = sP28 )
| ( induction2lem_lthm != $true ) )
& ( axvarshift = $true )
& ( ( axidr = $true )
| ? [X416: subst] :
( ( comp @ X416 @ id )
!= X416 ) )
& ( ! [X416: subst] :
( ( comp @ X416 @ id )
= X416 )
| ( axidr != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3a = $true )
| ( sP24 != $true ) )
& ( ( sP24 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X417: d_subst] : ( d_id = X417 )
& ( ( apinj1 = $true )
| ? [X236: term,X237: term,X238: term,X239: term] :
( ( X236 != X239 )
& ( ( ap @ X236 @ X238 )
= ( ap @ X239 @ X237 ) ) ) )
& ( ! [X236: term,X237: term,X238: term,X239: term] :
( ( X236 = X239 )
| ( ( ap @ X236 @ X238 )
!= ( ap @ X239 @ X237 ) ) )
| ( apinj1 != $true ) )
& ( ( induction2_lthm = $true )
| ( $true != sP23 ) )
& ( ( $true = sP23 )
| ( induction2_lthm != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP19 ) )
& ( ( $true = sP19 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ( ( X479 @ X481 )
= $true )
& ( $true
!= ( X480 @ one ) )
& ! [X483: term] :
( ( X480 @ X483 )
= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) ) )
& ! [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
= X478 ) ) )
& ( ! [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ( ( X479 @ X481 )
!= $true )
| ( $true
= ( X480 @ one ) )
| ? [X483: term] :
( ( X480 @ X483 )
!= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) ) )
| ? [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
!= X478 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( sP18 != $true ) )
& ( ( sP18 = $true )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( laminj = $true )
| ? [X366: term,X367: term] :
( ( X366 != X367 )
& ( ( lam @ X366 )
= ( lam @ X367 ) ) ) )
& ( ! [X366: term,X367: term] :
( ( X366 = X367 )
| ( ( lam @ X366 )
!= ( lam @ X367 ) ) )
| ( laminj != $true ) )
& ( ! [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
!= ( sub @ X77 @ sh ) )
| ( X77 = X78 ) )
| ( shinj != $true ) )
& ( ( shinj = $true )
| ? [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
= ( sub @ X77 @ sh ) )
& ( X77 != X78 ) ) )
& ( hoasapnotvar_gthm = $true )
& ( ! [X6: term] :
( ( sub @ X6 @ id )
= X6 )
| ? [X5: term] :
( ( sub @ X5 @ id )
!= X5 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ? [X6: term] :
( ( sub @ X6 @ id )
!= X6 )
& ! [X5: term] :
( ( sub @ X5 @ id )
= X5 ) ) )
& ( ! [X109: term] :
( ( sub @ X109 @ id )
= X109 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X109: term] :
( ( sub @ X109 @ id )
!= X109 ) )
& ( apnotvar = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lthm = $true )
| ( sP17 != $true ) )
& ( ( sP17 = $true )
| ( hoasinduction_lthm != $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ! [X358: term,X359: term,X360: term,X361: term] :
( ( X358 = X361 )
| ( ( ap @ X360 @ X361 )
!= ( ap @ X359 @ X358 ) ) )
& ? [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
& ( X363 != X365 ) ) ) )
& ( ? [X358: term,X359: term,X360: term,X361: term] :
( ( X358 != X361 )
& ( ( ap @ X360 @ X361 )
= ( ap @ X359 @ X358 ) ) )
| ! [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
| ( X363 = X365 ) )
| ( hoasapinj2_lthm != $true ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lthm = $true )
& ( ( sP8 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP8 != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( sP7 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( sP7 != $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( axassoc = $true )
& ( axabs = $true )
& ( lamnotap != $true )
& ( ! [X352: term,X353: term,X354: term,X355: term] :
( ( X353 = X354 )
| ( ( ap @ ( sub @ X352 @ id ) @ X353 )
!= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X352: term,X353: term,X354: term,X355: term] :
( ( X353 != X354 )
& ( ( ap @ ( sub @ X352 @ id ) @ X353 )
= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) ) )
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& ( pushprop = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
| ? [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
= ( X395 @ X397 ) )
& ( $true
!= ( X396 @ one ) )
& ! [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
= ( X396 @ X399 ) ) ) )
& ( ! [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
!= ( X395 @ X397 ) )
| ( $true
= ( X396 @ one ) )
| ? [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
!= ( X396 @ X399 ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( ! [X373: subst] :
( ( comp @ X373 @ id )
= X373 )
& ! [X372: subst] :
( ( comp @ id @ X372 )
= X372 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X373: subst] :
( ( comp @ X373 @ id )
!= X373 )
| ? [X372: subst] :
( ( comp @ id @ X372 )
!= X372 ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X110: subst > term > subst > $o] :
! [X111: term > $o] :
? [X112: term] :
( ( X110 @ id @ X112 @ id )
!= ( X111 @ X112 ) ) )
& ( ! [X110: subst > term > subst > $o] :
? [X111: term > $o] :
! [X112: term] :
( ( X110 @ id @ X112 @ id )
= ( X111 @ X112 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP5 ) )
& ( ( $true = sP5 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X1: term] :
! [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
!= ( X2 @ sh @ one ) ) )
& ( ! [X1: term] :
? [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
= ( X2 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ulamvarsh = $true )
& ! [X229: d_term] : ( d_one = X229 )
& ( axclos = $true )
& ( substmonoid_gthm = $true )
& ( induction2lem_gthm = $true )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X202: subst > term > subst > $o,X203: term] :
( ! [X206: subst > term > term] :
( ? [X207: term] :
( ( ( X202 @ id @ X207 @ id )
= $true )
& ( $true
!= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
| ? [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
!= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
| ( $true
= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) )
& ( $true
!= ( sP4 @ X202 ) )
& ( ( X202 @ id @ X203 @ id )
!= $true ) ) )
& ( ! [X202: subst > term > subst > $o,X203: term] :
( ? [X206: subst > term > term] :
( ! [X207: term] :
( ( ( X202 @ id @ X207 @ id )
!= $true )
| ( $true
= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
& ! [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
& ( $true
!= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( sP4 @ X202 ) )
| ( ( X202 @ id @ X203 @ id )
= $true ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( axmap = $true )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP3 ) )
& ( ( $true = sP3 )
| ( hoasinduction_lem2 != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasapinj2_gthm = $true )
& ( ! [X142: term] :
? [X143: subst > term > term] :
( ( X143 @ sh @ one )
= ( sub @ X142 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X142: term] :
! [X143: subst > term > term] :
( ( X143 @ sh @ one )
!= ( sub @ X142 @ ( push @ one @ sh ) ) ) )
& ( one
= ( d2term @ d_one ) )
& ( termmset_gthm = $true )
& ( ! [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
= X371 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
!= X371 ) )
& ( ulamvarind = $true )
& ( hoasapinj1_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem3v2_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X511: term] :
( ( sub @ X511 @ id )
!= X511 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ! [X368: d_term,X369: d_term] :
( ( ( d2term @ X368 )
!= ( d2term @ X369 ) )
| ( X368 = X369 ) )
& ( ? [X504: subst] :
( ( comp @ X504 @ id )
!= X504 )
| ? [X503: subst] :
( ( comp @ id @ X503 )
!= X503 )
| ( ! [X506: subst] :
( ( comp @ id @ X506 )
= X506 )
& ! [X505: subst] :
( ( comp @ X505 @ id )
= X505 ) )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ! [X504: subst] :
( ( comp @ X504 @ id )
= X504 )
& ! [X503: subst] :
( ( comp @ id @ X503 )
= X503 )
& ( ? [X506: subst] :
( ( comp @ id @ X506 )
!= X506 )
| ? [X505: subst] :
( ( comp @ X505 @ id )
!= X505 ) ) ) )
& ( ulamvar1 != $true )
& ( pushprop_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( axscons = $true )
| ? [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
!= X0 ) )
& ( ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 )
| ( axscons != $true ) )
& ( ( axshiftcons = $true )
| ? [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
!= X201 ) )
& ( ! [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
= X201 )
| ( axshiftcons != $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( $true = sP1 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( $true != sP1 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
= X114 )
& ! [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
= X116 )
& ? [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
!= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
& ( $true
= ( X120 @ X121 ) ) ) ) )
& ( ? [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
!= X114 )
| ? [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
!= X116 )
| ! [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
| ( $true
!= ( X120 @ X121 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoaslamnotap = $true )
| ? [X520: term,X521: term,X522: subst > term > term] :
( ( ( ap @ ( sub @ X521 @ id ) @ X520 )
= ( lam @ ( X522 @ sh @ one ) ) )
& ! [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) ) ) )
& ( ! [X520: term,X521: term,X522: subst > term > term] :
( ( ( ap @ ( sub @ X521 @ id ) @ X520 )
!= ( lam @ ( X522 @ sh @ one ) ) )
| ? [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
!= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) ) )
| ( hoaslamnotap != $true ) ) ),
inference(flattening,[status(thm)],[f471]) ).
thf(f471,plain,
( ( ! [X138: term] :
? [X139: subst > term > term] :
! [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
= ( X139 @ X141 @ X140 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X138: term] :
! [X139: subst > term > term] :
? [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
!= ( X139 @ X141 @ X140 ) ) )
& ( ( apinj2 = $true )
| ? [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
= ( ap @ X79 @ X82 ) )
& ( X81 != X82 ) ) )
& ( ! [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
!= ( ap @ X79 @ X82 ) )
| ( X81 = X82 ) )
| ( apinj2 != $true ) )
& ( ! [X324: subst] :
( ( comp @ id @ X324 )
= X324 )
| ( axidl != $true ) )
& ( ( axidl = $true )
| ? [X324: subst] :
( ( comp @ id @ X324 )
!= X324 ) )
& ( ( sP58 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( sP58 != $true ) )
& ! [X356: subst] :
? [X357: d_subst] :
( ( d2subst @ X357 )
= X356 )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X26: subst > term > subst > $o] :
? [X27: term > $o] :
! [X28: term] :
( ( X27 @ X28 )
= ( X26 @ id @ X28 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X26: subst > term > subst > $o] :
! [X27: term > $o] :
? [X28: term] :
( ( X27 @ X28 )
!= ( X26 @ id @ X28 @ id ) ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP56 ) )
& ( ( $true = sP56 )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( $true != sP54 ) )
& ( ( $true = sP54 )
| ( hoasinduction_lem3v2 != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X527: term,X528: term,X529: term,X530: term] :
( ( X527 = X528 )
| ( ( ap @ X528 @ X529 )
!= ( ap @ X527 @ X530 ) ) )
& ! [X526: term] :
( ( sub @ X526 @ id )
= X526 )
& ? [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
& ( X533 != X534 ) ) ) )
& ( ? [X527: term,X528: term,X529: term,X530: term] :
( ( X527 != X528 )
& ( ( ap @ X528 @ X529 )
= ( ap @ X527 @ X530 ) ) )
| ? [X526: term] :
( ( sub @ X526 @ id )
!= X526 )
| ! [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
!= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
| ( X533 = X534 ) )
| ( hoasapinj1_lthm != $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP52 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP52 != $true ) )
& ( induction2_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) )
| ( $true
!= ( X472 @ X473 ) ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
!= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) )
& ( $true
= ( X472 @ X473 ) ) ) )
& ( ( hoaslaminj = $true )
| ? [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ( ( X144 @ X146 @ X147 )
!= ( X145 @ X146 @ X147 ) )
& ! [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
& ( ( lam @ ( X144 @ sh @ one ) )
= ( lam @ ( X145 @ sh @ one ) ) )
& ! [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) ) ) )
& ( ! [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ( ( X144 @ X146 @ X147 )
= ( X145 @ X146 @ X147 ) )
| ? [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
!= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
| ( ( lam @ ( X144 @ sh @ one ) )
!= ( lam @ ( X145 @ sh @ one ) ) )
| ? [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
!= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) ) )
| ( hoaslaminj != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X333: term,X334: term > $o,X335: subst] :
! [X336: term > $o] :
? [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
!= ( X336 @ X337 ) ) )
& ( ! [X333: term,X334: term > $o,X335: subst] :
? [X336: term > $o] :
! [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
= ( X336 @ X337 ) )
| ( pushprop_lem0 != $true ) )
& ( ! [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
!= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
| ( X507 = X508 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
& ( X507 != X508 ) ) )
& ! [X338: d_subst,X339: d_subst] :
( ( X338 = X339 )
| ( ( d2subst @ X338 )
!= ( d2subst @ X339 ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X535: term > $o,X536: subst,X537: term] :
! [X538: term > $o] :
? [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
!= ( X538 @ X539 ) ) )
& ( ! [X535: term > $o,X536: subst,X537: term] :
? [X538: term > $o] :
! [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
= ( X538 @ X539 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ( $true = sP47 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP47 ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP44 ) )
& ( ( $true = sP44 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ! [X230: term > $o,X231: term] :
( ? [X232: term] :
( ( $true
!= ( X230 @ X232 ) )
& ( ( var @ X232 )
= $true ) )
| ( ( X230 @ X231 )
= $true )
| ? [X235: term] :
( ( $true
!= ( X230 @ ( lam @ X235 ) ) )
& ( $true
= ( X230 @ X235 ) ) )
| ( $true
= ( sP41 @ X230 ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X230: term > $o,X231: term] :
( ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( ( var @ X232 )
!= $true ) )
& ( ( X230 @ X231 )
!= $true )
& ! [X235: term] :
( ( $true
= ( X230 @ ( lam @ X235 ) ) )
| ( $true
!= ( X230 @ X235 ) ) )
& ( $true
!= ( sP41 @ X230 ) ) ) )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP40 )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( $true != sP40 ) )
& ( ( hoaslaminj_lthm = $true )
| ( sP34 != $true ) )
& ( ( sP34 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP33 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP33 ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( $true = sP31 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP31 ) )
& ( axapp = $true )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3aa != $true ) )
& ( ! [X391: term] :
? [X392: subst > term > term] :
! [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
= ( X392 @ X393 @ X394 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X391: term] :
! [X392: subst > term > term] :
? [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
!= ( X392 @ X393 @ X394 ) ) )
& ( ( induction2lem_lthm = $true )
| ( $true != sP28 ) )
& ( ( $true = sP28 )
| ( induction2lem_lthm != $true ) )
& ( axvarshift = $true )
& ( ( axidr = $true )
| ? [X416: subst] :
( ( comp @ X416 @ id )
!= X416 ) )
& ( ! [X416: subst] :
( ( comp @ X416 @ id )
= X416 )
| ( axidr != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3a = $true )
| ( sP24 != $true ) )
& ( ( sP24 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X417: d_subst] : ( d_id = X417 )
& ( ( apinj1 = $true )
| ? [X236: term,X237: term,X238: term,X239: term] :
( ( X236 != X239 )
& ( ( ap @ X236 @ X238 )
= ( ap @ X239 @ X237 ) ) ) )
& ( ! [X236: term,X237: term,X238: term,X239: term] :
( ( X236 = X239 )
| ( ( ap @ X236 @ X238 )
!= ( ap @ X239 @ X237 ) ) )
| ( apinj1 != $true ) )
& ( ( induction2_lthm = $true )
| ( $true != sP23 ) )
& ( ( $true = sP23 )
| ( induction2_lthm != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP19 ) )
& ( ( $true = sP19 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ( ( X479 @ X481 )
= $true )
& ( $true
!= ( X480 @ one ) )
& ! [X483: term] :
( ( X480 @ X483 )
= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) ) )
& ! [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
= X478 ) ) )
& ( ! [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ( ( X479 @ X481 )
!= $true )
| ( $true
= ( X480 @ one ) )
| ? [X483: term] :
( ( X480 @ X483 )
!= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) ) )
| ? [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
!= X478 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( sP18 != $true ) )
& ( ( sP18 = $true )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( laminj = $true )
| ? [X366: term,X367: term] :
( ( X366 != X367 )
& ( ( lam @ X366 )
= ( lam @ X367 ) ) ) )
& ( ! [X366: term,X367: term] :
( ( X366 = X367 )
| ( ( lam @ X366 )
!= ( lam @ X367 ) ) )
| ( laminj != $true ) )
& ( ! [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
!= ( sub @ X77 @ sh ) )
| ( X77 = X78 ) )
| ( shinj != $true ) )
& ( ( shinj = $true )
| ? [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
= ( sub @ X77 @ sh ) )
& ( X77 != X78 ) ) )
& ( hoasapnotvar_gthm = $true )
& ( ! [X6: term] :
( ( sub @ X6 @ id )
= X6 )
| ? [X5: term] :
( ( sub @ X5 @ id )
!= X5 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ? [X6: term] :
( ( sub @ X6 @ id )
!= X6 )
& ! [X5: term] :
( ( sub @ X5 @ id )
= X5 ) ) )
& ( ! [X109: term] :
( ( sub @ X109 @ id )
= X109 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X109: term] :
( ( sub @ X109 @ id )
!= X109 ) )
& ( apnotvar = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lthm = $true )
| ( sP17 != $true ) )
& ( ( sP17 = $true )
| ( hoasinduction_lthm != $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ! [X358: term,X359: term,X360: term,X361: term] :
( ( X358 = X361 )
| ( ( ap @ X360 @ X361 )
!= ( ap @ X359 @ X358 ) ) )
& ? [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
& ( X363 != X365 ) ) ) )
& ( ? [X358: term,X359: term,X360: term,X361: term] :
( ( X358 != X361 )
& ( ( ap @ X360 @ X361 )
= ( ap @ X359 @ X358 ) ) )
| ! [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
| ( X363 = X365 ) )
| ( hoasapinj2_lthm != $true ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lthm = $true )
& ( ( sP8 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP8 != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( sP7 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( sP7 != $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( axassoc = $true )
& ( axabs = $true )
& ( lamnotap != $true )
& ( ! [X352: term,X353: term,X354: term,X355: term] :
( ( X353 = X354 )
| ( ( ap @ ( sub @ X352 @ id ) @ X353 )
!= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X352: term,X353: term,X354: term,X355: term] :
( ( X353 != X354 )
& ( ( ap @ ( sub @ X352 @ id ) @ X353 )
= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) ) )
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& ( pushprop = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
| ? [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
= ( X395 @ X397 ) )
& ( $true
!= ( X396 @ one ) )
& ! [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
= ( X396 @ X399 ) ) ) )
& ( ! [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
!= ( X395 @ X397 ) )
| ( $true
= ( X396 @ one ) )
| ? [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
!= ( X396 @ X399 ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( ! [X373: subst] :
( ( comp @ X373 @ id )
= X373 )
& ! [X372: subst] :
( ( comp @ id @ X372 )
= X372 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X373: subst] :
( ( comp @ X373 @ id )
!= X373 )
| ? [X372: subst] :
( ( comp @ id @ X372 )
!= X372 ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X110: subst > term > subst > $o] :
! [X111: term > $o] :
? [X112: term] :
( ( X110 @ id @ X112 @ id )
!= ( X111 @ X112 ) ) )
& ( ! [X110: subst > term > subst > $o] :
? [X111: term > $o] :
! [X112: term] :
( ( X110 @ id @ X112 @ id )
= ( X111 @ X112 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP5 ) )
& ( ( $true = sP5 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X1: term] :
! [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
!= ( X2 @ sh @ one ) ) )
& ( ! [X1: term] :
? [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
= ( X2 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ulamvarsh = $true )
& ! [X229: d_term] : ( d_one = X229 )
& ( axclos = $true )
& ( substmonoid_gthm = $true )
& ( induction2lem_gthm = $true )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X202: subst > term > subst > $o,X203: term] :
( ! [X206: subst > term > term] :
( ? [X207: term] :
( ( ( X202 @ id @ X207 @ id )
= $true )
& ( $true
!= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
| ? [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
!= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
| ( $true
= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) )
& ( $true
!= ( sP4 @ X202 ) )
& ( ( X202 @ id @ X203 @ id )
!= $true ) ) )
& ( ! [X202: subst > term > subst > $o,X203: term] :
( ? [X206: subst > term > term] :
( ! [X207: term] :
( ( ( X202 @ id @ X207 @ id )
!= $true )
| ( $true
= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
& ! [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
& ( $true
!= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( sP4 @ X202 ) )
| ( ( X202 @ id @ X203 @ id )
= $true ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( axmap = $true )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP3 ) )
& ( ( $true = sP3 )
| ( hoasinduction_lem2 != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasapinj2_gthm = $true )
& ( ! [X142: term] :
? [X143: subst > term > term] :
( ( X143 @ sh @ one )
= ( sub @ X142 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X142: term] :
! [X143: subst > term > term] :
( ( X143 @ sh @ one )
!= ( sub @ X142 @ ( push @ one @ sh ) ) ) )
& ( one
= ( d2term @ d_one ) )
& ( termmset_gthm = $true )
& ( ! [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
= X371 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
!= X371 ) )
& ( ulamvarind = $true )
& ( hoasapinj1_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem3v2_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X511: term] :
( ( sub @ X511 @ id )
!= X511 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ! [X368: d_term,X369: d_term] :
( ( ( d2term @ X368 )
!= ( d2term @ X369 ) )
| ( X368 = X369 ) )
& ( ? [X504: subst] :
( ( comp @ X504 @ id )
!= X504 )
| ? [X503: subst] :
( ( comp @ id @ X503 )
!= X503 )
| ( ! [X506: subst] :
( ( comp @ id @ X506 )
= X506 )
& ! [X505: subst] :
( ( comp @ X505 @ id )
= X505 ) )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ! [X504: subst] :
( ( comp @ X504 @ id )
= X504 )
& ! [X503: subst] :
( ( comp @ id @ X503 )
= X503 )
& ( ? [X506: subst] :
( ( comp @ id @ X506 )
!= X506 )
| ? [X505: subst] :
( ( comp @ X505 @ id )
!= X505 ) ) ) )
& ( ulamvar1 != $true )
& ( pushprop_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( axscons = $true )
| ? [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
!= X0 ) )
& ( ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 )
| ( axscons != $true ) )
& ( ( axshiftcons = $true )
| ? [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
!= X201 ) )
& ( ! [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
= X201 )
| ( axshiftcons != $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( $true = sP1 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( $true != sP1 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
= X114 )
& ! [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
= X116 )
& ? [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
!= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
& ( $true
= ( X120 @ X121 ) ) ) ) )
& ( ? [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
!= X114 )
| ? [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
!= X116 )
| ! [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
| ( $true
!= ( X120 @ X121 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoaslamnotap = $true )
| ? [X520: term,X521: term,X522: subst > term > term] :
( ( ( ap @ ( sub @ X521 @ id ) @ X520 )
= ( lam @ ( X522 @ sh @ one ) ) )
& ! [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) ) ) )
& ( ! [X520: term,X521: term,X522: subst > term > term] :
( ( ( ap @ ( sub @ X521 @ id ) @ X520 )
!= ( lam @ ( X522 @ sh @ one ) ) )
| ? [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
!= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) ) )
| ( hoaslamnotap != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f73]) ).
thf(f73,plain,
( ( ! [X138: term] :
? [X139: subst > term > term] :
! [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
= ( X139 @ X141 @ X140 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( apinj2 = $true )
<=> ! [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
!= ( ap @ X79 @ X82 ) )
| ( X81 = X82 ) ) )
& ( ! [X324: subst] :
( ( comp @ id @ X324 )
= X324 )
<=> ( axidl = $true ) )
& ( ( sP58 = $true )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ! [X356: subst] :
? [X357: d_subst] :
( ( d2subst @ X357 )
= X356 )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X26: subst > term > subst > $o] :
? [X27: term > $o] :
! [X28: term] :
( ( X27 @ X28 )
= ( X26 @ id @ X28 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ( $true = sP56 ) )
& ( ( hoasinduction_lem3v2 = $true )
<=> ( $true = sP54 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
<=> ( ? [X527: term,X528: term,X529: term,X530: term] :
( ( X527 != X528 )
& ( ( ap @ X528 @ X529 )
= ( ap @ X527 @ X530 ) ) )
| ? [X526: term] :
( ( sub @ X526 @ id )
!= X526 )
| ! [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
!= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
| ( X533 = X534 ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP52 = $true )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( induction2_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) )
| ( $true
!= ( X472 @ X473 ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ( hoaslaminj = $true )
<=> ! [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ( ( X144 @ X146 @ X147 )
= ( X145 @ X146 @ X147 ) )
| ? [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
!= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
| ( ( lam @ ( X144 @ sh @ one ) )
!= ( lam @ ( X145 @ sh @ one ) ) )
| ? [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
!= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) ) ) )
& ( ( pushprop_lem0 = $true )
<=> ! [X333: term,X334: term > $o,X335: subst] :
? [X336: term > $o] :
! [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
= ( X336 @ X337 ) ) )
& ( ! [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
!= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
| ( X507 = X508 ) )
<=> ( hoasapinj1 = $true ) )
& ! [X338: d_subst,X339: d_subst] :
( ( X338 = X339 )
| ( ( d2subst @ X338 )
!= ( d2subst @ X339 ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X535: term > $o,X536: subst,X537: term] :
? [X538: term > $o] :
! [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
= ( X538 @ X539 ) ) )
& ( ( $true = sP47 )
<=> ( hoasinduction = $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( $true = sP44 ) )
& ( ! [X230: term > $o,X231: term] :
( ? [X232: term] :
( ( $true
!= ( X230 @ X232 ) )
& ( ( var @ X232 )
= $true ) )
| ( ( X230 @ X231 )
= $true )
| ? [X235: term] :
( ( $true
!= ( X230 @ ( lam @ X235 ) ) )
& ( $true
= ( X230 @ X235 ) ) )
| ( $true
= ( sP41 @ X230 ) ) )
<=> ( induction = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP40 )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( hoaslaminj_lthm = $true )
<=> ( sP34 = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP33 )
<=> ( induction2 = $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( $true = sP31 )
<=> ( induction2lem = $true ) )
& ( axapp = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ( $true = sP29 ) )
& ( ! [X391: term] :
? [X392: subst > term > term] :
! [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
= ( X392 @ X393 @ X394 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( induction2lem_lthm = $true )
<=> ( $true = sP28 ) )
& ( axvarshift = $true )
& ( ( axidr = $true )
<=> ! [X416: subst] :
( ( comp @ X416 @ id )
= X416 ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3a = $true )
<=> ( sP24 = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X417: d_subst] : ( d_id = X417 )
& ( ( apinj1 = $true )
<=> ! [X236: term,X237: term,X238: term,X239: term] :
( ( X236 = X239 )
| ( ( ap @ X236 @ X238 )
!= ( ap @ X239 @ X237 ) ) ) )
& ( ( induction2_lthm = $true )
<=> ( $true = sP23 ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ( $true = sP19 ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ( ( X479 @ X481 )
!= $true )
| ( $true
= ( X480 @ one ) )
| ? [X483: term] :
( ( X480 @ X483 )
!= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) ) )
| ? [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
!= X478 ) ) )
& ( pushprop_lem0_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( sP18 = $true ) )
& ( ( laminj = $true )
<=> ! [X366: term,X367: term] :
( ( X366 = X367 )
| ( ( lam @ X366 )
!= ( lam @ X367 ) ) ) )
& ( ! [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
!= ( sub @ X77 @ sh ) )
| ( X77 = X78 ) )
<=> ( shinj = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( ( ! [X6: term] :
( ( sub @ X6 @ id )
= X6 )
| ? [X5: term] :
( ( sub @ X5 @ id )
!= X5 ) )
<=> ( termmset_lthm = $true ) )
& ( ! [X109: term] :
( ( sub @ X109 @ id )
= X109 )
<=> ( termmset = $true ) )
& ( apnotvar = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lthm = $true )
<=> ( sP17 = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ? [X358: term,X359: term,X360: term,X361: term] :
( ( X358 != X361 )
& ( ( ap @ X360 @ X361 )
= ( ap @ X359 @ X358 ) ) )
| ! [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
| ( X363 = X365 ) ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lthm = $true )
& ( ( sP8 = $true )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( sP7 = $true )
<=> ( hoasinduction_lem3 = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( axassoc = $true )
& ( axabs = $true )
& ( lamnotap != $true )
& ( ! [X352: term,X353: term,X354: term,X355: term] :
( ( X353 = X354 )
| ( ( ap @ ( sub @ X352 @ id ) @ X353 )
!= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) )
<=> ( hoasapinj2 = $true ) )
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& ( pushprop = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
!= ( X395 @ X397 ) )
| ( $true
= ( X396 @ one ) )
| ? [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
!= ( X396 @ X399 ) ) ) )
& ( ( ! [X373: subst] :
( ( comp @ X373 @ id )
= X373 )
& ! [X372: subst] :
( ( comp @ id @ X372 )
= X372 ) )
<=> ( substmonoid = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X110: subst > term > subst > $o] :
? [X111: term > $o] :
! [X112: term] :
( ( X110 @ id @ X112 @ id )
= ( X111 @ X112 ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ( $true = sP5 ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X1: term] :
? [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
= ( X2 @ sh @ one ) ) )
& ( ulamvarsh = $true )
& ! [X229: d_term] : ( d_one = X229 )
& ( axclos = $true )
& ( substmonoid_gthm = $true )
& ( induction2lem_gthm = $true )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X202: subst > term > subst > $o,X203: term] :
( ? [X206: subst > term > term] :
( ! [X207: term] :
( ( ( X202 @ id @ X207 @ id )
!= $true )
| ( $true
= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
& ! [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
& ( $true
!= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( sP4 @ X202 ) )
| ( ( X202 @ id @ X203 @ id )
= $true ) ) )
& ( axmap = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ( $true = sP3 ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasapinj2_gthm = $true )
& ( ! [X142: term] :
? [X143: subst > term > term] :
( ( X143 @ sh @ one )
= ( sub @ X142 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( one
= ( d2term @ d_one ) )
& ( termmset_gthm = $true )
& ( ! [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
= X371 )
<=> ( axvarcons = $true ) )
& ( ulamvarind = $true )
& ( hoasapinj1_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem3v2_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
<=> ( axvarid = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ! [X368: d_term,X369: d_term] :
( ( ( d2term @ X368 )
!= ( d2term @ X369 ) )
| ( X368 = X369 ) )
& ( ( ? [X504: subst] :
( ( comp @ X504 @ id )
!= X504 )
| ? [X503: subst] :
( ( comp @ id @ X503 )
!= X503 )
| ( ! [X506: subst] :
( ( comp @ id @ X506 )
= X506 )
& ! [X505: subst] :
( ( comp @ X505 @ id )
= X505 ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( ulamvar1 != $true )
& ( pushprop_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( axscons = $true )
<=> ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 ) )
& ( ( axshiftcons = $true )
<=> ! [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
= X201 ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( $true = sP1 )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ? [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
!= X114 )
| ? [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
!= X116 )
| ! [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
| ( $true
!= ( X120 @ X121 ) ) ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoaslamnotap = $true )
<=> ! [X520: term,X521: term,X522: subst > term > term] :
( ( ( ap @ ( sub @ X521 @ id ) @ X520 )
!= ( lam @ ( X522 @ sh @ one ) ) )
| ? [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
!= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) ) ) ) ),
inference(definition_folding,[status(thm)],[f13,f72,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14]) ).
thf(f14,plain,
( ( $true = sP0 )
<=> ? [X375: term,X376: subst > term > subst > $o] :
( ! [X382: term] :
( ( $true
!= ( X376 @ id @ X382 @ id ) )
| ( $true
= ( X376 @ id @ ( sub @ X375 @ ( push @ X382 @ id ) ) @ id ) ) )
& ( ( X376 @ id @ ( lam @ ( sub @ X375 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X377: subst > term > term] :
( ( $true
= ( X376 @ id @ ( lam @ ( X377 @ sh @ one ) ) @ id ) )
| ? [X378: term] :
( ( $true
!= ( X376 @ id @ ( X377 @ id @ X378 ) @ id ) )
& ( $true
= ( X376 @ id @ X378 @ id ) ) )
| ? [X379: subst,X380: subst,X381: term] :
( ( X377 @ ( comp @ X379 @ X380 ) @ ( sub @ X381 @ X380 ) )
!= ( sub @ ( X377 @ X379 @ X381 ) @ X380 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f15,plain,
( ( $true = sP1 )
<=> ( ! [X383: term,X384: subst > term > subst > $o] :
( ( $true
= ( X384 @ id @ ( lam @ X383 ) @ id ) )
| ? [X385: term] :
( ( ( X384 @ id @ ( sub @ X383 @ ( push @ X385 @ id ) ) @ id )
!= $true )
& ( $true
= ( X384 @ id @ X385 @ id ) ) )
| ? [X386: subst > term > term] :
( ( $true
!= ( X384 @ id @ ( lam @ ( X386 @ sh @ one ) ) @ id ) )
& ! [X388: subst,X389: subst,X390: term] :
( ( sub @ ( X386 @ X389 @ X390 ) @ X388 )
= ( X386 @ ( comp @ X389 @ X388 ) @ ( sub @ X390 @ X388 ) ) )
& ! [X387: term] :
( ( $true
= ( X384 @ id @ ( X386 @ id @ X387 ) @ id ) )
| ( ( X384 @ id @ X387 @ id )
!= $true ) ) ) )
| ( $true = sP0 )
| ? [X374: term] :
( ( sub @ X374 @ id )
!= X374 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f16,plain,
! [X188: subst > term > subst > $o] :
( ( $true
= ( sP2 @ X188 ) )
<=> ? [X198: term,X199: term] :
( ( $true
= ( X188 @ id @ X199 @ id ) )
& ( $true
!= ( X188 @ id @ ( ap @ ( sub @ X199 @ id ) @ X198 ) @ id ) )
& ( ( X188 @ id @ X198 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f17,plain,
( ( $true = sP3 )
<=> ! [X187: term,X188: subst > term > subst > $o,X189: term] :
( ( $true
= ( X188 @ id @ ( ap @ X189 @ X187 ) @ id ) )
| ( $true
= ( sP2 @ X188 ) )
| ( $true
!= ( X188 @ id @ X187 @ id ) )
| ( $true
!= ( X188 @ id @ X189 @ id ) )
| ? [X190: subst,X191: subst,X192: term,X193: subst] :
( ( ( X188 @ X191 @ X192 @ ( comp @ X193 @ X190 ) )
!= $true )
& ( ( X188 @ ( comp @ X191 @ X193 ) @ ( sub @ X192 @ X193 ) @ X190 )
= $true ) )
| ? [X194: subst,X195: subst,X196: term,X197: subst] :
( ( $true
= ( X188 @ X194 @ X196 @ ( comp @ X197 @ X195 ) ) )
& ( ( X188 @ ( comp @ X194 @ X197 ) @ ( sub @ X196 @ X197 ) @ X195 )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f18,plain,
! [X202: subst > term > subst > $o] :
( ( $true
= ( sP4 @ X202 ) )
<=> ? [X204: term,X205: term] :
( ( ( X202 @ id @ ( ap @ ( sub @ X204 @ id ) @ X205 ) @ id )
!= $true )
& ( $true
= ( X202 @ id @ X205 @ id ) )
& ( $true
= ( X202 @ id @ X204 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f19,plain,
( ( $true = sP5 )
<=> ! [X512: subst > term > subst > $o,X513: term] :
( ? [X514: term] :
( ( ( X512 @ id @ X514 @ id )
= $true )
& ( ( X512 @ id @ ( sub @ X513 @ ( push @ X514 @ id ) ) @ id )
!= $true ) )
| ? [X515: subst > term > term] :
( ! [X519: term] :
( ( ( X512 @ id @ X519 @ id )
!= $true )
| ( ( X512 @ id @ ( X515 @ id @ X519 ) @ id )
= $true ) )
& ! [X516: subst,X517: term,X518: subst] :
( ( X515 @ ( comp @ X518 @ X516 ) @ ( sub @ X517 @ X516 ) )
= ( sub @ ( X515 @ X518 @ X517 ) @ X516 ) )
& ( ( X512 @ id @ ( lam @ ( X515 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X512 @ id @ ( lam @ ( sub @ X513 @ ( push @ one @ sh ) ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f20,plain,
! [X123: subst > term > subst > $o] :
( ( $true
= ( sP6 @ X123 ) )
<=> ? [X125: subst > term > term] :
( ! [X126: subst,X127: term,X128: subst] :
( ( X125 @ ( comp @ X128 @ X126 ) @ ( sub @ X127 @ X126 ) )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
& ! [X129: term] :
( ( ( X123 @ id @ X129 @ id )
!= $true )
| ( $true
= ( X123 @ id @ ( X125 @ id @ X129 ) @ id ) ) )
& ( $true
!= ( X123 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f21,plain,
( ( sP7 = $true )
<=> ! [X122: term,X123: subst > term > subst > $o] :
( ( $true
= ( sP6 @ X123 ) )
| ? [X124: term] :
( ( $true
!= ( X123 @ id @ ( sub @ X122 @ ( push @ X124 @ id ) ) @ id ) )
& ( $true
= ( X123 @ id @ X124 @ id ) ) )
| ( ( X123 @ id @ ( lam @ X122 ) @ id )
= $true )
| ? [X134: subst,X135: subst,X136: term,X137: subst] :
( ( ( X123 @ ( comp @ X135 @ X137 ) @ ( sub @ X136 @ X137 ) @ X134 )
= $true )
& ( ( X123 @ X135 @ X136 @ ( comp @ X137 @ X134 ) )
!= $true ) )
| ? [X130: subst,X131: subst,X132: subst,X133: term] :
( ( ( X123 @ ( comp @ X131 @ X130 ) @ ( sub @ X133 @ X130 ) @ X132 )
!= $true )
& ( ( X123 @ X131 @ X133 @ ( comp @ X130 @ X132 ) )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f22,plain,
( ( sP8 = $true )
<=> ! [X83: subst > term > subst > $o,X84: term,X85: term > $o] :
( ( $true
= ( X85 @ ( lam @ X84 ) ) )
| ? [X88: subst > term > term] :
( ! [X92: term] :
( ( ( X83 @ id @ X92 @ id )
!= $true )
| ( ( X83 @ id @ ( X88 @ id @ X92 ) @ id )
= $true ) )
& ! [X89: subst,X90: subst,X91: term] :
( ( X88 @ ( comp @ X90 @ X89 ) @ ( sub @ X91 @ X89 ) )
= ( sub @ ( X88 @ X90 @ X91 ) @ X89 ) )
& ( $true
!= ( X83 @ id @ ( lam @ ( X88 @ sh @ one ) ) @ id ) ) )
| ? [X87: term] :
( ( X83 @ id @ X87 @ id )
!= ( X85 @ X87 ) )
| ? [X86: term] :
( ( $true
= ( X85 @ X86 ) )
& ( $true
!= ( X85 @ ( sub @ X84 @ ( push @ X86 @ id ) ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f23,plain,
! [X439: subst > term > subst > $o] :
( ( $true
= ( sP9 @ X439 ) )
<=> ! [X440: subst > term > term] :
( ? [X441: subst,X442: subst,X443: term] :
( ( sub @ ( X440 @ X442 @ X443 ) @ X441 )
!= ( X440 @ ( comp @ X442 @ X441 ) @ ( sub @ X443 @ X441 ) ) )
| ? [X444: term] :
( ( ( X439 @ id @ X444 @ id )
= $true )
& ( $true
!= ( X439 @ id @ ( X440 @ id @ X444 ) @ id ) ) )
| ( ( X439 @ id @ ( lam @ ( X440 @ sh @ one ) ) @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f24,plain,
! [X427: subst > term > subst > $o] :
( ( $true
= ( sP10 @ X427 ) )
<=> ! [X436: term,X437: term] :
( ( $true
= ( X427 @ id @ ( ap @ ( sub @ X437 @ id ) @ X436 ) @ id ) )
| ( $true
!= ( X427 @ id @ X437 @ id ) )
| ( ( X427 @ id @ X436 @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f25,plain,
! [X455: subst > term > subst > $o] :
( ( ( sP11 @ X455 )
= $true )
<=> ? [X456: term,X457: term] :
( ( ( X455 @ id @ ( ap @ ( sub @ X456 @ id ) @ X457 ) @ id )
!= $true )
& ( $true
= ( X455 @ id @ X456 @ id ) )
& ( $true
= ( X455 @ id @ X457 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f26,plain,
! [X455: subst > term > subst > $o] :
( ( ( sP12 @ X455 )
= $true )
<=> ? [X458: subst > term > term] :
( ( ( X455 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id )
!= $true )
& ! [X459: term] :
( ( ( X455 @ id @ X459 @ id )
!= $true )
| ( ( X455 @ id @ ( X458 @ id @ X459 ) @ id )
= $true ) )
& ! [X460: subst,X461: subst,X462: term] :
( ( X458 @ ( comp @ X460 @ X461 ) @ ( sub @ X462 @ X461 ) )
= ( sub @ ( X458 @ X460 @ X462 ) @ X461 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f27,plain,
! [X419: term > $o] :
( ( ( sP13 @ X419 )
= $true )
<=> ! [X423: term,X424: term] :
( ( $true
= ( X419 @ ( ap @ X424 @ X423 ) ) )
| ( ( X419 @ X423 )
!= $true )
| ( ( X419 @ X424 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f28,plain,
( ( $true = sP14 )
<=> ? [X438: term,X439: subst > term > subst > $o] :
( ! [X445: term] :
( ( $true
!= ( X439 @ id @ X445 @ id ) )
| ( $true
= ( X439 @ id @ ( sub @ X438 @ ( push @ X445 @ id ) ) @ id ) ) )
& ( $true
= ( sP9 @ X439 ) )
& ( $true
!= ( X439 @ id @ ( lam @ X438 ) @ id ) )
& ! [X446: subst,X447: subst,X448: term,X449: subst] :
( ( ( X439 @ ( comp @ X449 @ X447 ) @ ( sub @ X448 @ X447 ) @ X446 )
= $true )
| ( ( X439 @ X449 @ X448 @ ( comp @ X447 @ X446 ) )
!= $true ) )
& ! [X450: subst,X451: subst,X452: term,X453: subst] :
( ( $true
= ( X439 @ X453 @ X452 @ ( comp @ X450 @ X451 ) ) )
| ( ( X439 @ ( comp @ X453 @ X450 ) @ ( sub @ X452 @ X450 ) @ X451 )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f29,plain,
( ( $true = sP15 )
<=> ! [X454: term,X455: subst > term > subst > $o] :
( ? [X467: term,X468: subst,X469: subst,X470: subst] :
( ( $true
= ( X455 @ ( comp @ X468 @ X469 ) @ ( sub @ X467 @ X469 ) @ X470 ) )
& ( ( X455 @ X468 @ X467 @ ( comp @ X469 @ X470 ) )
!= $true ) )
| ( $true
= ( X455 @ id @ X454 @ id ) )
| ( ( sP12 @ X455 )
= $true )
| ( ( sP11 @ X455 )
= $true )
| ? [X471: term] :
( ( ( X455 @ id @ X471 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X471 @ id ) ) ) )
| ? [X463: subst,X464: term,X465: subst,X466: subst] :
( ( ( X455 @ ( comp @ X466 @ X465 ) @ ( sub @ X464 @ X465 ) @ X463 )
!= $true )
& ( $true
= ( X455 @ X466 @ X464 @ ( comp @ X465 @ X463 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f30,plain,
( ( $true = sP16 )
<=> ? [X425: term,X426: term,X427: subst > term > subst > $o] :
( ( ( X427 @ id @ X425 @ id )
= $true )
& ! [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X427 @ ( comp @ X429 @ X430 ) @ ( sub @ X428 @ X430 ) @ X431 ) )
| ( ( X427 @ X429 @ X428 @ ( comp @ X430 @ X431 ) )
!= $true ) )
& ( ( X427 @ id @ ( ap @ X426 @ X425 ) @ id )
!= $true )
& ( $true
= ( X427 @ id @ X426 @ id ) )
& ( $true
= ( sP10 @ X427 ) )
& ! [X432: subst,X433: subst,X434: term,X435: subst] :
( ( ( X427 @ ( comp @ X432 @ X433 ) @ ( sub @ X434 @ X433 ) @ X435 )
!= $true )
| ( ( X427 @ X432 @ X434 @ ( comp @ X433 @ X435 ) )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f31,plain,
( ( sP17 = $true )
<=> ( ? [X418: term,X419: term > $o] :
( ! [X420: term] :
( ? [X421: term] :
( ( $true
!= ( X419 @ ( sub @ X420 @ ( push @ X421 @ id ) ) ) )
& ( $true
= ( X419 @ X421 ) ) )
| ( ( X419 @ ( lam @ X420 ) )
= $true ) )
& ! [X422: term] :
( ( $true
= ( X419 @ X422 ) )
| ( ( var @ X422 )
!= $true ) )
& ( ( sP13 @ X419 )
= $true )
& ( $true
!= ( X419 @ X418 ) ) )
| ( $true = sP15 )
| ( $true = sP16 )
| ( $true = sP14 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f32,plain,
( ( sP18 = $true )
<=> ( ? [X244: term] :
( ( sub @ X244 @ id )
!= X244 )
| ! [X245: term,X246: subst > term > subst > $o,X247: term > $o] :
( ? [X248: subst > term > term] :
( ! [X252: term] :
( ( $true
!= ( X246 @ id @ X252 @ id ) )
| ( $true
= ( X246 @ id @ ( X248 @ id @ X252 ) @ id ) ) )
& ! [X249: term,X250: subst,X251: subst] :
( ( sub @ ( X248 @ X251 @ X249 ) @ X250 )
= ( X248 @ ( comp @ X251 @ X250 ) @ ( sub @ X249 @ X250 ) ) )
& ( ( X246 @ id @ ( lam @ ( X248 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X253: term] :
( ( X247 @ X253 )
!= ( X246 @ id @ X253 @ id ) )
| ? [X254: term] :
( ( ( X247 @ X254 )
= $true )
& ( ( X247 @ ( sub @ X245 @ ( push @ X254 @ id ) ) )
!= $true ) )
| ( $true
= ( X247 @ ( lam @ X245 ) ) ) )
| ? [X240: term] :
! [X241: subst > term > term] :
? [X242: term,X243: subst] :
( ( X241 @ X243 @ X242 )
!= ( sub @ X240 @ ( push @ X242 @ X243 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f33,plain,
( ( $true = sP19 )
<=> ! [X340: subst > term > subst > $o,X341: term] :
( ? [X342: term] :
( ( $true
!= ( X340 @ id @ ( sub @ X341 @ ( push @ X342 @ id ) ) @ id ) )
& ( ( X340 @ id @ X342 @ id )
= $true ) )
| ? [X343: term,X344: subst > term > term] :
( ( ( X340 @ id @ ( lam @ ( X344 @ sh @ one ) ) @ id )
!= $true )
& ! [X345: subst,X346: subst,X347: term] :
( ( sub @ ( sub @ X343 @ ( push @ X347 @ X346 ) ) @ X345 )
= ( sub @ ( X344 @ X346 @ X347 ) @ X345 ) )
& ! [X349: subst,X350: subst,X351: term] :
( ( sub @ X343 @ ( push @ ( sub @ X351 @ X350 ) @ ( comp @ X349 @ X350 ) ) )
= ( X344 @ ( comp @ X349 @ X350 ) @ ( sub @ X351 @ X350 ) ) )
& ! [X348: term] :
( ( $true
= ( X340 @ id @ ( X344 @ id @ X348 ) @ id ) )
| ( $true
!= ( X340 @ id @ X348 @ id ) ) ) )
| ( $true
= ( X340 @ id @ ( lam @ ( sub @ X341 @ ( push @ one @ sh ) ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f34,plain,
! [X102: term > $o] :
( ( ( sP20 @ X102 )
= $true )
<=> ? [X104: term,X105: term] :
( ( ( X102 @ X105 )
= $true )
& ( ( X102 @ X104 )
= $true )
& ( ( X102 @ ( ap @ X104 @ X105 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f35,plain,
! [X95: term > $o] :
( ( ( sP21 @ X95 )
= $true )
<=> ! [X97: term,X98: term] :
( ( $true
= ( X95 @ ( ap @ X97 @ X98 ) ) )
| ( ( X95 @ X98 )
!= $true )
| ( $true
!= ( X95 @ X97 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f36,plain,
( ( sP22 = $true )
<=> ! [X102: term > $o,X103: term] :
( ( ( X102 @ X103 )
= $true )
| ( ( sP20 @ X102 )
= $true )
| ? [X106: term] :
( ( ( X102 @ ( lam @ X106 ) )
!= $true )
& ! [X107: term] :
( ( $true
= ( X102 @ ( sub @ X106 @ ( push @ X107 @ id ) ) ) )
| ( ( X102 @ X107 )
!= $true ) ) )
| ? [X108: term] :
( ( ( X102 @ X108 )
!= $true )
& ( $true
= ( var @ X108 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f37,plain,
( ( $true = sP23 )
<=> ( ? [X94: subst,X95: term > $o,X96: term] :
( ! [X100: term] :
( ( ( X95 @ ( lam @ X100 ) )
= $true )
| ? [X101: term] :
( ( ( X95 @ ( sub @ X100 @ ( push @ X101 @ id ) ) )
!= $true )
& ( ( X95 @ X101 )
= $true ) ) )
& ( ( X95 @ ( sub @ X96 @ X94 ) )
!= $true )
& ( ( sP21 @ X95 )
= $true )
& ! [X99: term] :
( ( ( var @ X99 )
!= $true )
| ( $true
= ( X95 @ ( sub @ X99 @ X94 ) ) ) ) )
| ? [X93: term] :
( ( sub @ X93 @ id )
!= X93 )
| ( sP22 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f38,plain,
( ( sP24 = $true )
<=> ! [X262: subst > term > subst > $o,X263: term] :
( ? [X264: term] :
( ( ( X262 @ id @ ( sub @ X263 @ ( push @ X264 @ id ) ) @ id )
!= $true )
& ( ( X262 @ id @ X264 @ id )
= $true ) )
| ( ( X262 @ id @ ( lam @ X263 ) @ id )
= $true )
| ? [X265: subst > term > term] :
( ! [X266: term] :
( ( $true
= ( X262 @ id @ ( X265 @ id @ X266 ) @ id ) )
| ( ( X262 @ id @ X266 @ id )
!= $true ) )
& ! [X267: subst,X268: subst,X269: term] :
( ( sub @ ( X265 @ X268 @ X269 ) @ X267 )
= ( X265 @ ( comp @ X268 @ X267 ) @ ( sub @ X269 @ X267 ) ) )
& ( $true
!= ( X262 @ id @ ( lam @ ( X265 @ sh @ one ) ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f39,plain,
! [X20: term > $o] :
( ( $true
= ( sP25 @ X20 ) )
<=> ? [X22: term,X23: term] :
( ( $true
!= ( X20 @ ( ap @ X23 @ X22 ) ) )
& ( $true
= ( X20 @ X23 ) )
& ( ( X20 @ X22 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f40,plain,
! [X12: term > $o] :
( ( $true
= ( sP26 @ X12 ) )
<=> ! [X16: term,X17: term] :
( ( ( X12 @ ( ap @ X17 @ X16 ) )
= $true )
| ( ( X12 @ X16 )
!= $true )
| ( $true
!= ( X12 @ X17 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f41,plain,
( ( $true = sP27 )
<=> ! [X18: subst,X19: term,X20: term > $o] :
( ( $true
= ( X20 @ ( sub @ X19 @ X18 ) ) )
| ? [X21: term] :
( ( $true
!= ( X20 @ ( sub @ X21 @ X18 ) ) )
& ( ( var @ X21 )
= $true ) )
| ? [X24: term] :
( ( ( X20 @ ( lam @ X24 ) )
!= $true )
& ! [X25: term] :
( ( ( X20 @ X25 )
!= $true )
| ( ( X20 @ ( sub @ X24 @ ( push @ X25 @ id ) ) )
= $true ) ) )
| ( $true
= ( sP25 @ X20 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f42,plain,
( ( $true = sP28 )
<=> ( ? [X9: subst,X10: term] :
( ( comp @ sh @ ( push @ X10 @ X9 ) )
!= X9 )
| ? [X12: term > $o,X13: term] :
( ! [X14: term] :
( ( $true
!= ( var @ X14 ) )
| ( $true
= ( X12 @ X14 ) ) )
& ( $true
= ( sP26 @ X12 ) )
& ( $true
!= ( X12 @ X13 ) )
& ! [X15: term] :
( ( $true
!= ( X12 @ X15 ) )
| ( ( X12 @ ( lam @ X15 ) )
= $true ) ) )
| ? [X7: term,X8: subst] :
( ( sub @ one @ ( push @ X7 @ X8 ) )
!= X7 )
| ? [X11: subst] :
( ( comp @ X11 @ id )
!= X11 )
| ( $true = sP27 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f43,plain,
( ( $true = sP29 )
<=> ! [X154: subst > term > subst > $o,X155: term] :
( ? [X157: subst > term > term] :
( ! [X159: subst,X160: subst,X161: term] :
( ( X157 @ ( comp @ X159 @ X160 ) @ ( sub @ X161 @ X160 ) )
= ( sub @ ( X157 @ X159 @ X161 ) @ X160 ) )
& ( $true
!= ( X154 @ id @ ( lam @ ( X157 @ sh @ one ) ) @ id ) )
& ! [X158: term] :
( ( $true
= ( X154 @ id @ ( X157 @ id @ X158 ) @ id ) )
| ( $true
!= ( X154 @ id @ X158 @ id ) ) ) )
| ( $true
= ( X154 @ id @ ( lam @ ( sub @ X155 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X156: term] :
( ( ( X154 @ id @ X156 @ id )
= $true )
& ( $true
!= ( X154 @ id @ ( sub @ X155 @ ( push @ X156 @ id ) ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f44,plain,
! [X325: term > $o] :
( ( $true
= ( sP30 @ X325 ) )
<=> ? [X328: term,X329: term] :
( ( $true
!= ( X325 @ ( ap @ X329 @ X328 ) ) )
& ( $true
= ( X325 @ X329 ) )
& ( $true
= ( X325 @ X328 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f45,plain,
( ( $true = sP31 )
<=> ! [X325: term > $o,X326: subst,X327: term] :
( ( $true
= ( sP30 @ X325 ) )
| ? [X330: term] :
( ( $true
= ( var @ X330 ) )
& ( $true
!= ( X325 @ ( sub @ X330 @ X326 ) ) ) )
| ? [X331: term] :
( ( ( X325 @ ( lam @ X331 ) )
!= $true )
& ! [X332: term] :
( ( $true
!= ( X325 @ X332 ) )
| ( $true
= ( X325 @ ( sub @ X331 @ ( push @ X332 @ id ) ) ) ) ) )
| ( $true
= ( X325 @ ( sub @ X327 @ X326 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f46,plain,
! [X256: term > $o] :
( ( $true
= ( sP32 @ X256 ) )
<=> ? [X257: term,X258: term] :
( ( ( X256 @ X258 )
= $true )
& ( $true
= ( X256 @ X257 ) )
& ( ( X256 @ ( ap @ X257 @ X258 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f47,plain,
( ( $true = sP33 )
<=> ! [X255: term,X256: term > $o] :
( ( ( X256 @ X255 )
= $true )
| ? [X260: term] :
( ! [X261: term] :
( ( $true
!= ( X256 @ X261 ) )
| ( $true
= ( X256 @ ( sub @ X260 @ ( push @ X261 @ id ) ) ) ) )
& ( $true
!= ( X256 @ ( lam @ X260 ) ) ) )
| ( $true
= ( sP32 @ X256 ) )
| ? [X259: term] :
( ( ( var @ X259 )
= $true )
& ( $true
!= ( X256 @ X259 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f48,plain,
( ( sP34 = $true )
<=> ( ? [X400: term,X401: subst] :
( ( sub @ one @ ( push @ X400 @ X401 ) )
!= X400 )
| ? [X402: term,X403: subst] :
( ( comp @ sh @ ( push @ X402 @ X403 ) )
!= X403 )
| ? [X404: term,X405: term] :
( ( ( lam @ X405 )
= ( lam @ X404 ) )
& ( X404 != X405 ) )
| ! [X406: subst > term > term,X407: subst,X408: term,X409: subst > term > term] :
( ? [X413: term,X414: subst,X415: subst] :
( ( sub @ ( X409 @ X415 @ X413 ) @ X414 )
!= ( X409 @ ( comp @ X415 @ X414 ) @ ( sub @ X413 @ X414 ) ) )
| ? [X410: subst,X411: term,X412: subst] :
( ( sub @ ( X406 @ X410 @ X411 ) @ X412 )
!= ( X406 @ ( comp @ X410 @ X412 ) @ ( sub @ X411 @ X412 ) ) )
| ( ( X409 @ X407 @ X408 )
= ( X406 @ X407 @ X408 ) )
| ( ( lam @ ( X409 @ sh @ one ) )
!= ( lam @ ( X406 @ sh @ one ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f49,plain,
! [X306: subst > term > subst > $o] :
( ( $true
= ( sP35 @ X306 ) )
<=> ? [X318: term,X319: term] :
( ( ( X306 @ id @ X318 @ id )
= $true )
& ( $true
= ( X306 @ id @ X319 @ id ) )
& ( ( X306 @ id @ ( ap @ ( sub @ X318 @ id ) @ X319 ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f50,plain,
! [X306: subst > term > subst > $o] :
( ( $true
= ( sP36 @ X306 ) )
<=> ? [X309: subst > term > term] :
( ( $true
!= ( X306 @ id @ ( lam @ ( X309 @ sh @ one ) ) @ id ) )
& ! [X311: subst,X312: term,X313: subst] :
( ( sub @ ( X309 @ X311 @ X312 ) @ X313 )
= ( X309 @ ( comp @ X311 @ X313 ) @ ( sub @ X312 @ X313 ) ) )
& ! [X310: term] :
( ( $true
= ( X306 @ id @ ( X309 @ id @ X310 ) @ id ) )
| ( $true
!= ( X306 @ id @ X310 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f51,plain,
! [X289: term > $o] :
( ( ( sP37 @ X289 )
= $true )
<=> ! [X293: term,X294: term] :
( ( ( X289 @ X293 )
!= $true )
| ( $true
= ( X289 @ ( ap @ X293 @ X294 ) ) )
| ( ( X289 @ X294 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f52,plain,
( ( $true = sP38 )
<=> ! [X306: subst > term > subst > $o,X307: term] :
( ( $true
= ( sP36 @ X306 ) )
| ? [X308: term] :
( ( ( X306 @ id @ X308 @ id )
!= $true )
& ( ( var @ ( sub @ X308 @ id ) )
= $true ) )
| ( ( X306 @ id @ X307 @ id )
= $true )
| ( $true
= ( sP35 @ X306 ) )
| ? [X320: term,X321: subst,X322: subst,X323: subst] :
( ( $true
!= ( X306 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) @ X323 ) )
& ( $true
= ( X306 @ X321 @ X320 @ ( comp @ X322 @ X323 ) ) ) )
| ? [X314: subst,X315: subst,X316: term,X317: subst] :
( ( ( X306 @ X314 @ X316 @ ( comp @ X315 @ X317 ) )
!= $true )
& ( ( X306 @ ( comp @ X314 @ X315 ) @ ( sub @ X316 @ X315 ) @ X317 )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f53,plain,
( ( $true = sP39 )
<=> ? [X296: term,X297: term > $o,X298: subst > term > subst > $o] :
( ! [X305: term] :
( ( ( X297 @ X305 )
!= $true )
| ( ( X297 @ ( sub @ X296 @ ( push @ X305 @ id ) ) )
= $true ) )
& ! [X300: subst > term > term] :
( ? [X301: subst,X302: term,X303: subst] :
( ( sub @ ( X300 @ X303 @ X302 ) @ X301 )
!= ( X300 @ ( comp @ X303 @ X301 ) @ ( sub @ X302 @ X301 ) ) )
| ? [X304: term] :
( ( ( X298 @ id @ ( X300 @ id @ X304 ) @ id )
!= $true )
& ( $true
= ( X298 @ id @ X304 @ id ) ) )
| ( $true
= ( X298 @ id @ ( lam @ ( X300 @ sh @ one ) ) @ id ) ) )
& ! [X299: term] :
( ( X297 @ X299 )
= ( X298 @ id @ X299 @ id ) )
& ( ( X297 @ ( lam @ X296 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f54,plain,
( ( $true = sP40 )
<=> ( ? [X285: subst > term > subst > $o] :
! [X286: term > $o] :
? [X287: term] :
( ( X285 @ id @ X287 @ id )
!= ( X286 @ X287 ) )
| ( $true = sP39 )
| ? [X288: term,X289: term > $o] :
( ( ( X289 @ X288 )
!= $true )
& ! [X290: term] :
( ( ( var @ X290 )
!= $true )
| ( $true
= ( X289 @ X290 ) ) )
& ! [X291: term] :
( ? [X292: term] :
( ( ( X289 @ X292 )
= $true )
& ( $true
!= ( X289 @ ( sub @ X291 @ ( push @ X292 @ id ) ) ) ) )
| ( ( X289 @ ( lam @ X291 ) )
= $true ) )
& ( ( sP37 @ X289 )
= $true ) )
| ? [X295: term] :
( ( sub @ X295 @ id )
!= X295 )
| ( $true = sP38 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f55,plain,
! [X230: term > $o] :
( ( $true
= ( sP41 @ X230 ) )
<=> ? [X233: term,X234: term] :
( ( $true
= ( X230 @ X234 ) )
& ( $true
= ( X230 @ X233 ) )
& ( $true
!= ( X230 @ ( ap @ X233 @ X234 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f56,plain,
! [X172: subst > term > subst > $o] :
( ( ( sP42 @ X172 )
= $true )
<=> ? [X174: subst > term > term] :
( ! [X176: subst,X177: subst,X178: term] :
( ( sub @ ( X174 @ X176 @ X178 ) @ X177 )
= ( X174 @ ( comp @ X176 @ X177 ) @ ( sub @ X178 @ X177 ) ) )
& ( ( X172 @ id @ ( lam @ ( X174 @ sh @ one ) ) @ id )
!= $true )
& ! [X175: term] :
( ( ( X172 @ id @ ( X174 @ id @ X175 ) @ id )
= $true )
| ( ( X172 @ id @ X175 @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f57,plain,
( ( $true = sP43 )
<=> ? [X163: subst > term > subst > $o,X164: term] :
( ! [X166: subst > term > term] :
( ? [X167: subst,X168: subst,X169: term] :
( ( X166 @ ( comp @ X168 @ X167 ) @ ( sub @ X169 @ X167 ) )
!= ( sub @ ( X166 @ X168 @ X169 ) @ X167 ) )
| ? [X170: term] :
( ( $true
= ( X163 @ id @ X170 @ id ) )
& ( ( X163 @ id @ ( X166 @ id @ X170 ) @ id )
!= $true ) )
| ( $true
= ( X163 @ id @ ( lam @ ( X166 @ sh @ one ) ) @ id ) ) )
& ! [X165: term] :
( ( $true
!= ( X163 @ id @ X165 @ id ) )
| ( $true
= ( X163 @ id @ ( sub @ X164 @ ( push @ X165 @ id ) ) @ id ) ) )
& ( ( X163 @ id @ ( lam @ ( sub @ X164 @ ( push @ one @ sh ) ) ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f58,plain,
( ( $true = sP44 )
<=> ( ( $true = sP43 )
| ? [X162: term] :
( ( sub @ X162 @ id )
!= X162 )
| ! [X171: term,X172: subst > term > subst > $o] :
( ? [X183: subst,X184: subst,X185: term,X186: subst] :
( ( $true
= ( X172 @ ( comp @ X184 @ X183 ) @ ( sub @ X185 @ X183 ) @ X186 ) )
& ( ( X172 @ X184 @ X185 @ ( comp @ X183 @ X186 ) )
!= $true ) )
| ? [X173: term] :
( ( $true
!= ( X172 @ id @ ( sub @ X171 @ ( push @ X173 @ id ) ) @ id ) )
& ( ( X172 @ id @ X173 @ id )
= $true ) )
| ? [X179: term,X180: subst,X181: subst,X182: subst] :
( ( ( X172 @ X182 @ X179 @ ( comp @ X180 @ X181 ) )
= $true )
& ( $true
!= ( X172 @ ( comp @ X182 @ X180 ) @ ( sub @ X179 @ X180 ) @ X181 ) ) )
| ( ( sP42 @ X172 )
= $true )
| ( $true
= ( X172 @ id @ ( lam @ X171 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f59,plain,
! [X212: subst > term > subst > $o] :
( ( ( sP45 @ X212 )
= $true )
<=> ? [X227: term,X228: term] :
( ( ( X212 @ id @ ( ap @ ( sub @ X227 @ id ) @ X228 ) @ id )
!= $true )
& ( ( X212 @ id @ X228 @ id )
= $true )
& ( ( X212 @ id @ X227 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f60,plain,
! [X212: subst > term > subst > $o] :
( ( $true
= ( sP46 @ X212 ) )
<=> ? [X222: subst > term > term] :
( ! [X224: term,X225: subst,X226: subst] :
( ( X222 @ ( comp @ X226 @ X225 ) @ ( sub @ X224 @ X225 ) )
= ( sub @ ( X222 @ X226 @ X224 ) @ X225 ) )
& ! [X223: term] :
( ( $true
= ( X212 @ id @ ( X222 @ id @ X223 ) @ id ) )
| ( ( X212 @ id @ X223 @ id )
!= $true ) )
& ( $true
!= ( X212 @ id @ ( lam @ ( X222 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f61,plain,
( ( $true = sP47 )
<=> ! [X211: term,X212: subst > term > subst > $o] :
( ? [X221: term] :
( ( ( var @ ( sub @ X221 @ id ) )
= $true )
& ( ( X212 @ id @ X221 @ id )
!= $true ) )
| ( $true
= ( sP46 @ X212 ) )
| ? [X217: subst,X218: term,X219: subst,X220: subst] :
( ( ( X212 @ X219 @ X218 @ ( comp @ X217 @ X220 ) )
= $true )
& ( ( X212 @ ( comp @ X219 @ X217 ) @ ( sub @ X218 @ X217 ) @ X220 )
!= $true ) )
| ( $true
= ( X212 @ id @ X211 @ id ) )
| ( ( sP45 @ X212 )
= $true )
| ? [X213: term,X214: subst,X215: subst,X216: subst] :
( ( ( X212 @ X214 @ X213 @ ( comp @ X216 @ X215 ) )
!= $true )
& ( $true
= ( X212 @ ( comp @ X214 @ X216 ) @ ( sub @ X213 @ X216 ) @ X215 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f62,plain,
! [X50: term > $o] :
( ( ( sP48 @ X50 )
= $true )
<=> ! [X52: term,X53: term] :
( ( $true
!= ( X50 @ X53 ) )
| ( ( X50 @ X52 )
!= $true )
| ( $true
= ( X50 @ ( ap @ X53 @ X52 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f63,plain,
! [X68: subst > term > subst > $o] :
( ( ( sP49 @ X68 )
= $true )
<=> ? [X70: term,X71: term] :
( ( ( X68 @ id @ X70 @ id )
= $true )
& ( $true
= ( X68 @ id @ X71 @ id ) )
& ( $true
!= ( X68 @ id @ ( ap @ ( sub @ X71 @ id ) @ X70 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f64,plain,
( ( sP50 = $true )
<=> ? [X58: subst > term > subst > $o,X59: term,X60: term > $o] :
( ! [X61: term] :
( ( ( X60 @ ( sub @ X59 @ ( push @ X61 @ id ) ) )
= $true )
| ( ( X60 @ X61 )
!= $true ) )
& ( $true
!= ( X60 @ ( lam @ X59 ) ) )
& ! [X62: term] :
( ( X58 @ id @ X62 @ id )
= ( X60 @ X62 ) )
& ! [X63: subst > term > term] :
( ? [X67: term] :
( ( ( X58 @ id @ ( X63 @ id @ X67 ) @ id )
!= $true )
& ( $true
= ( X58 @ id @ X67 @ id ) ) )
| ( ( X58 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true )
| ? [X64: subst,X65: term,X66: subst] :
( ( sub @ ( X63 @ X64 @ X65 ) @ X66 )
!= ( X63 @ ( comp @ X64 @ X66 ) @ ( sub @ X65 @ X66 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f65,plain,
( ( sP51 = $true )
<=> ? [X50: term > $o,X51: term] :
( ( ( sP48 @ X50 )
= $true )
& ( ( X50 @ X51 )
!= $true )
& ! [X54: term] :
( ( ( X50 @ ( lam @ X54 ) )
= $true )
| ? [X55: term] :
( ( ( X50 @ ( sub @ X54 @ ( push @ X55 @ id ) ) )
!= $true )
& ( ( X50 @ X55 )
= $true ) ) )
& ! [X56: term] :
( ( $true
!= ( var @ X56 ) )
| ( $true
= ( X50 @ X56 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f66,plain,
( ( sP52 = $true )
<=> ( ? [X47: subst > term > subst > $o] :
! [X48: term > $o] :
? [X49: term] :
( ( X47 @ id @ X49 @ id )
!= ( X48 @ X49 ) )
| ! [X68: subst > term > subst > $o,X69: term] :
( ? [X72: subst > term > term] :
( ! [X76: term] :
( ( $true
= ( X68 @ id @ ( X72 @ id @ X76 ) @ id ) )
| ( ( X68 @ id @ X76 @ id )
!= $true ) )
& ! [X73: subst,X74: subst,X75: term] :
( ( sub @ ( X72 @ X74 @ X75 ) @ X73 )
= ( X72 @ ( comp @ X74 @ X73 ) @ ( sub @ X75 @ X73 ) ) )
& ( ( X68 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X68 @ id @ X69 @ id ) )
| ( ( sP49 @ X68 )
= $true ) )
| ( sP51 = $true )
| ? [X57: term] :
( ( sub @ X57 @ id )
!= X57 )
| ( sP50 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f67,plain,
! [X29: subst > term > subst > $o] :
( ( $true
= ( sP53 @ X29 ) )
<=> ? [X37: subst > term > term] :
( ! [X38: subst,X39: subst,X40: term] :
( ( sub @ ( X37 @ X38 @ X40 ) @ X39 )
= ( X37 @ ( comp @ X38 @ X39 ) @ ( sub @ X40 @ X39 ) ) )
& ( ( X29 @ id @ ( lam @ ( X37 @ sh @ one ) ) @ id )
!= $true )
& ! [X41: term] :
( ( ( X29 @ id @ X41 @ id )
!= $true )
| ( $true
= ( X29 @ id @ ( X37 @ id @ X41 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f68,plain,
( ( $true = sP54 )
<=> ! [X29: subst > term > subst > $o,X30: term,X31: term > $o] :
( ? [X46: term] :
( ( X29 @ id @ X46 @ id )
!= ( X31 @ X46 ) )
| ? [X42: subst,X43: subst,X44: term,X45: subst] :
( ( $true
!= ( X29 @ X42 @ X44 @ ( comp @ X43 @ X45 ) ) )
& ( ( X29 @ ( comp @ X42 @ X43 ) @ ( sub @ X44 @ X43 ) @ X45 )
= $true ) )
| ? [X32: subst,X33: subst,X34: term,X35: subst] :
( ( $true
= ( X29 @ X35 @ X34 @ ( comp @ X32 @ X33 ) ) )
& ( $true
!= ( X29 @ ( comp @ X35 @ X32 ) @ ( sub @ X34 @ X32 ) @ X33 ) ) )
| ? [X36: term] :
( ( $true
= ( X31 @ X36 ) )
& ( $true
!= ( X31 @ ( sub @ X30 @ ( push @ X36 @ id ) ) ) ) )
| ( ( X31 @ ( lam @ X30 ) )
= $true )
| ( $true
= ( sP53 @ X29 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f69,plain,
! [X272: subst > term > subst > $o] :
( ( $true
= ( sP55 @ X272 ) )
<=> ? [X278: term,X279: term] :
( ( ( X272 @ id @ X278 @ id )
= $true )
& ( $true
!= ( X272 @ id @ ( ap @ ( sub @ X279 @ id ) @ X278 ) @ id ) )
& ( ( X272 @ id @ X279 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f70,plain,
( ( $true = sP56 )
<=> ! [X270: term,X271: term > $o,X272: subst > term > subst > $o,X273: term] :
( ( $true
!= ( X271 @ X273 ) )
| ? [X284: term] :
( ( X272 @ id @ X284 @ id )
!= ( X271 @ X284 ) )
| ( $true
= ( X271 @ ( ap @ X273 @ X270 ) ) )
| ( $true
!= ( X271 @ X270 ) )
| ? [X280: subst,X281: subst,X282: term,X283: subst] :
( ( $true
!= ( X272 @ X281 @ X282 @ ( comp @ X283 @ X280 ) ) )
& ( ( X272 @ ( comp @ X281 @ X283 ) @ ( sub @ X282 @ X283 ) @ X280 )
= $true ) )
| ? [X274: term,X275: subst,X276: subst,X277: subst] :
( ( $true
= ( X272 @ X277 @ X274 @ ( comp @ X276 @ X275 ) ) )
& ( $true
!= ( X272 @ ( comp @ X277 @ X276 ) @ ( sub @ X274 @ X276 ) @ X275 ) ) )
| ( $true
= ( sP55 @ X272 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f71,plain,
! [X486: subst > term > subst > $o] :
( ( $true
= ( sP57 @ X486 ) )
<=> ? [X498: subst > term > term] :
( ( ( X486 @ id @ ( lam @ ( X498 @ sh @ one ) ) @ id )
!= $true )
& ! [X499: term] :
( ( $true
!= ( X486 @ id @ X499 @ id ) )
| ( $true
= ( X486 @ id @ ( X498 @ id @ X499 ) @ id ) ) )
& ! [X500: subst,X501: subst,X502: term] :
( ( X498 @ ( comp @ X500 @ X501 ) @ ( sub @ X502 @ X501 ) )
= ( sub @ ( X498 @ X500 @ X502 ) @ X501 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f72,plain,
( ( sP58 = $true )
<=> ( ? [X484: term] :
( ( sub @ X484 @ id )
!= X484 )
| ! [X485: term > $o,X486: subst > term > subst > $o,X487: term] :
( ? [X497: term] :
( ( X485 @ X497 )
!= ( X486 @ id @ X497 @ id ) )
| ? [X492: subst,X493: term,X494: subst,X495: subst] :
( ( ( X486 @ X492 @ X493 @ ( comp @ X495 @ X494 ) )
!= $true )
& ( ( X486 @ ( comp @ X492 @ X495 ) @ ( sub @ X493 @ X495 ) @ X494 )
= $true ) )
| ? [X488: subst,X489: subst,X490: term,X491: subst] :
( ( ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X491 )
!= $true )
& ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X491 ) )
= $true ) )
| ? [X496: term] :
( ( ( X485 @ X496 )
= $true )
& ( ( X485 @ ( sub @ X487 @ ( push @ X496 @ id ) ) )
!= $true ) )
| ( ( X485 @ ( lam @ X487 ) )
= $true )
| ( $true
= ( sP57 @ X486 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f13,plain,
( ( ! [X138: term] :
? [X139: subst > term > term] :
! [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
= ( X139 @ X141 @ X140 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( apinj2 = $true )
<=> ! [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
!= ( ap @ X79 @ X82 ) )
| ( X81 = X82 ) ) )
& ( ! [X324: subst] :
( ( comp @ id @ X324 )
= X324 )
<=> ( axidl = $true ) )
& ( ( ? [X484: term] :
( ( sub @ X484 @ id )
!= X484 )
| ! [X485: term > $o,X486: subst > term > subst > $o,X487: term] :
( ? [X497: term] :
( ( X485 @ X497 )
!= ( X486 @ id @ X497 @ id ) )
| ? [X492: subst,X493: term,X494: subst,X495: subst] :
( ( ( X486 @ X492 @ X493 @ ( comp @ X495 @ X494 ) )
!= $true )
& ( ( X486 @ ( comp @ X492 @ X495 ) @ ( sub @ X493 @ X495 ) @ X494 )
= $true ) )
| ? [X488: subst,X489: subst,X490: term,X491: subst] :
( ( ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X491 )
!= $true )
& ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X491 ) )
= $true ) )
| ? [X496: term] :
( ( ( X485 @ X496 )
= $true )
& ( ( X485 @ ( sub @ X487 @ ( push @ X496 @ id ) ) )
!= $true ) )
| ( ( X485 @ ( lam @ X487 ) )
= $true )
| ? [X498: subst > term > term] :
( ( ( X486 @ id @ ( lam @ ( X498 @ sh @ one ) ) @ id )
!= $true )
& ! [X499: term] :
( ( $true
!= ( X486 @ id @ X499 @ id ) )
| ( $true
= ( X486 @ id @ ( X498 @ id @ X499 ) @ id ) ) )
& ! [X500: subst,X501: subst,X502: term] :
( ( X498 @ ( comp @ X500 @ X501 ) @ ( sub @ X502 @ X501 ) )
= ( sub @ ( X498 @ X500 @ X502 ) @ X501 ) ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ! [X356: subst] :
? [X357: d_subst] :
( ( d2subst @ X357 )
= X356 )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X26: subst > term > subst > $o] :
? [X27: term > $o] :
! [X28: term] :
( ( X27 @ X28 )
= ( X26 @ id @ X28 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X270: term,X271: term > $o,X272: subst > term > subst > $o,X273: term] :
( ( $true
!= ( X271 @ X273 ) )
| ? [X284: term] :
( ( X272 @ id @ X284 @ id )
!= ( X271 @ X284 ) )
| ( $true
= ( X271 @ ( ap @ X273 @ X270 ) ) )
| ( $true
!= ( X271 @ X270 ) )
| ? [X280: subst,X281: subst,X282: term,X283: subst] :
( ( $true
!= ( X272 @ X281 @ X282 @ ( comp @ X283 @ X280 ) ) )
& ( ( X272 @ ( comp @ X281 @ X283 ) @ ( sub @ X282 @ X283 ) @ X280 )
= $true ) )
| ? [X274: term,X275: subst,X276: subst,X277: subst] :
( ( $true
= ( X272 @ X277 @ X274 @ ( comp @ X276 @ X275 ) ) )
& ( $true
!= ( X272 @ ( comp @ X277 @ X276 ) @ ( sub @ X274 @ X276 ) @ X275 ) ) )
| ? [X278: term,X279: term] :
( ( ( X272 @ id @ X278 @ id )
= $true )
& ( $true
!= ( X272 @ id @ ( ap @ ( sub @ X279 @ id ) @ X278 ) @ id ) )
& ( ( X272 @ id @ X279 @ id )
= $true ) ) ) )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X29: subst > term > subst > $o,X30: term,X31: term > $o] :
( ? [X46: term] :
( ( X29 @ id @ X46 @ id )
!= ( X31 @ X46 ) )
| ? [X42: subst,X43: subst,X44: term,X45: subst] :
( ( $true
!= ( X29 @ X42 @ X44 @ ( comp @ X43 @ X45 ) ) )
& ( ( X29 @ ( comp @ X42 @ X43 ) @ ( sub @ X44 @ X43 ) @ X45 )
= $true ) )
| ? [X32: subst,X33: subst,X34: term,X35: subst] :
( ( $true
= ( X29 @ X35 @ X34 @ ( comp @ X32 @ X33 ) ) )
& ( $true
!= ( X29 @ ( comp @ X35 @ X32 ) @ ( sub @ X34 @ X32 ) @ X33 ) ) )
| ? [X36: term] :
( ( $true
= ( X31 @ X36 ) )
& ( $true
!= ( X31 @ ( sub @ X30 @ ( push @ X36 @ id ) ) ) ) )
| ( ( X31 @ ( lam @ X30 ) )
= $true )
| ? [X37: subst > term > term] :
( ! [X38: subst,X39: subst,X40: term] :
( ( sub @ ( X37 @ X38 @ X40 ) @ X39 )
= ( X37 @ ( comp @ X38 @ X39 ) @ ( sub @ X40 @ X39 ) ) )
& ( ( X29 @ id @ ( lam @ ( X37 @ sh @ one ) ) @ id )
!= $true )
& ! [X41: term] :
( ( ( X29 @ id @ X41 @ id )
!= $true )
| ( $true
= ( X29 @ id @ ( X37 @ id @ X41 ) @ id ) ) ) ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
<=> ( ? [X527: term,X528: term,X529: term,X530: term] :
( ( X527 != X528 )
& ( ( ap @ X528 @ X529 )
= ( ap @ X527 @ X530 ) ) )
| ? [X526: term] :
( ( sub @ X526 @ id )
!= X526 )
| ! [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
!= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
| ( X533 = X534 ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( ? [X47: subst > term > subst > $o] :
! [X48: term > $o] :
? [X49: term] :
( ( X47 @ id @ X49 @ id )
!= ( X48 @ X49 ) )
| ! [X68: subst > term > subst > $o,X69: term] :
( ? [X72: subst > term > term] :
( ! [X76: term] :
( ( $true
= ( X68 @ id @ ( X72 @ id @ X76 ) @ id ) )
| ( ( X68 @ id @ X76 @ id )
!= $true ) )
& ! [X73: subst,X74: subst,X75: term] :
( ( sub @ ( X72 @ X74 @ X75 ) @ X73 )
= ( X72 @ ( comp @ X74 @ X73 ) @ ( sub @ X75 @ X73 ) ) )
& ( ( X68 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X68 @ id @ X69 @ id ) )
| ? [X70: term,X71: term] :
( ( ( X68 @ id @ X70 @ id )
= $true )
& ( $true
= ( X68 @ id @ X71 @ id ) )
& ( $true
!= ( X68 @ id @ ( ap @ ( sub @ X71 @ id ) @ X70 ) @ id ) ) ) )
| ? [X50: term > $o,X51: term] :
( ! [X52: term,X53: term] :
( ( $true
!= ( X50 @ X53 ) )
| ( ( X50 @ X52 )
!= $true )
| ( $true
= ( X50 @ ( ap @ X53 @ X52 ) ) ) )
& ( ( X50 @ X51 )
!= $true )
& ! [X54: term] :
( ( ( X50 @ ( lam @ X54 ) )
= $true )
| ? [X55: term] :
( ( ( X50 @ ( sub @ X54 @ ( push @ X55 @ id ) ) )
!= $true )
& ( ( X50 @ X55 )
= $true ) ) )
& ! [X56: term] :
( ( $true
!= ( var @ X56 ) )
| ( $true
= ( X50 @ X56 ) ) ) )
| ? [X57: term] :
( ( sub @ X57 @ id )
!= X57 )
| ? [X58: subst > term > subst > $o,X59: term,X60: term > $o] :
( ! [X61: term] :
( ( ( X60 @ ( sub @ X59 @ ( push @ X61 @ id ) ) )
= $true )
| ( ( X60 @ X61 )
!= $true ) )
& ( $true
!= ( X60 @ ( lam @ X59 ) ) )
& ! [X62: term] :
( ( X58 @ id @ X62 @ id )
= ( X60 @ X62 ) )
& ! [X63: subst > term > term] :
( ? [X67: term] :
( ( ( X58 @ id @ ( X63 @ id @ X67 ) @ id )
!= $true )
& ( $true
= ( X58 @ id @ X67 @ id ) ) )
| ( ( X58 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true )
| ? [X64: subst,X65: term,X66: subst] :
( ( sub @ ( X63 @ X64 @ X65 ) @ X66 )
!= ( X63 @ ( comp @ X64 @ X66 ) @ ( sub @ X65 @ X66 ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( induction2_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) )
| ( $true
!= ( X472 @ X473 ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ( hoaslaminj = $true )
<=> ! [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ( ( X144 @ X146 @ X147 )
= ( X145 @ X146 @ X147 ) )
| ? [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
!= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
| ( ( lam @ ( X144 @ sh @ one ) )
!= ( lam @ ( X145 @ sh @ one ) ) )
| ? [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
!= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) ) ) )
& ( ( pushprop_lem0 = $true )
<=> ! [X333: term,X334: term > $o,X335: subst] :
? [X336: term > $o] :
! [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
= ( X336 @ X337 ) ) )
& ( ! [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
!= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
| ( X507 = X508 ) )
<=> ( hoasapinj1 = $true ) )
& ! [X338: d_subst,X339: d_subst] :
( ( X338 = X339 )
| ( ( d2subst @ X338 )
!= ( d2subst @ X339 ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X535: term > $o,X536: subst,X537: term] :
? [X538: term > $o] :
! [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
= ( X538 @ X539 ) ) )
& ( ! [X211: term,X212: subst > term > subst > $o] :
( ? [X221: term] :
( ( ( var @ ( sub @ X221 @ id ) )
= $true )
& ( ( X212 @ id @ X221 @ id )
!= $true ) )
| ? [X222: subst > term > term] :
( ! [X224: term,X225: subst,X226: subst] :
( ( X222 @ ( comp @ X226 @ X225 ) @ ( sub @ X224 @ X225 ) )
= ( sub @ ( X222 @ X226 @ X224 ) @ X225 ) )
& ! [X223: term] :
( ( $true
= ( X212 @ id @ ( X222 @ id @ X223 ) @ id ) )
| ( ( X212 @ id @ X223 @ id )
!= $true ) )
& ( $true
!= ( X212 @ id @ ( lam @ ( X222 @ sh @ one ) ) @ id ) ) )
| ? [X217: subst,X218: term,X219: subst,X220: subst] :
( ( ( X212 @ X219 @ X218 @ ( comp @ X217 @ X220 ) )
= $true )
& ( ( X212 @ ( comp @ X219 @ X217 ) @ ( sub @ X218 @ X217 ) @ X220 )
!= $true ) )
| ( $true
= ( X212 @ id @ X211 @ id ) )
| ? [X227: term,X228: term] :
( ( ( X212 @ id @ ( ap @ ( sub @ X227 @ id ) @ X228 ) @ id )
!= $true )
& ( ( X212 @ id @ X228 @ id )
= $true )
& ( ( X212 @ id @ X227 @ id )
= $true ) )
| ? [X213: term,X214: subst,X215: subst,X216: subst] :
( ( ( X212 @ X214 @ X213 @ ( comp @ X216 @ X215 ) )
!= $true )
& ( $true
= ( X212 @ ( comp @ X214 @ X216 ) @ ( sub @ X213 @ X216 ) @ X215 ) ) ) )
<=> ( hoasinduction = $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ? [X163: subst > term > subst > $o,X164: term] :
( ! [X166: subst > term > term] :
( ? [X167: subst,X168: subst,X169: term] :
( ( X166 @ ( comp @ X168 @ X167 ) @ ( sub @ X169 @ X167 ) )
!= ( sub @ ( X166 @ X168 @ X169 ) @ X167 ) )
| ? [X170: term] :
( ( $true
= ( X163 @ id @ X170 @ id ) )
& ( ( X163 @ id @ ( X166 @ id @ X170 ) @ id )
!= $true ) )
| ( $true
= ( X163 @ id @ ( lam @ ( X166 @ sh @ one ) ) @ id ) ) )
& ! [X165: term] :
( ( $true
!= ( X163 @ id @ X165 @ id ) )
| ( $true
= ( X163 @ id @ ( sub @ X164 @ ( push @ X165 @ id ) ) @ id ) ) )
& ( ( X163 @ id @ ( lam @ ( sub @ X164 @ ( push @ one @ sh ) ) ) @ id )
!= $true ) )
| ? [X162: term] :
( ( sub @ X162 @ id )
!= X162 )
| ! [X171: term,X172: subst > term > subst > $o] :
( ? [X183: subst,X184: subst,X185: term,X186: subst] :
( ( $true
= ( X172 @ ( comp @ X184 @ X183 ) @ ( sub @ X185 @ X183 ) @ X186 ) )
& ( ( X172 @ X184 @ X185 @ ( comp @ X183 @ X186 ) )
!= $true ) )
| ? [X173: term] :
( ( $true
!= ( X172 @ id @ ( sub @ X171 @ ( push @ X173 @ id ) ) @ id ) )
& ( ( X172 @ id @ X173 @ id )
= $true ) )
| ? [X179: term,X180: subst,X181: subst,X182: subst] :
( ( ( X172 @ X182 @ X179 @ ( comp @ X180 @ X181 ) )
= $true )
& ( $true
!= ( X172 @ ( comp @ X182 @ X180 ) @ ( sub @ X179 @ X180 ) @ X181 ) ) )
| ? [X174: subst > term > term] :
( ! [X176: subst,X177: subst,X178: term] :
( ( sub @ ( X174 @ X176 @ X178 ) @ X177 )
= ( X174 @ ( comp @ X176 @ X177 ) @ ( sub @ X178 @ X177 ) ) )
& ( ( X172 @ id @ ( lam @ ( X174 @ sh @ one ) ) @ id )
!= $true )
& ! [X175: term] :
( ( ( X172 @ id @ ( X174 @ id @ X175 ) @ id )
= $true )
| ( ( X172 @ id @ X175 @ id )
!= $true ) ) )
| ( $true
= ( X172 @ id @ ( lam @ X171 ) @ id ) ) ) ) )
& ( ! [X230: term > $o,X231: term] :
( ? [X232: term] :
( ( $true
!= ( X230 @ X232 ) )
& ( ( var @ X232 )
= $true ) )
| ( ( X230 @ X231 )
= $true )
| ? [X235: term] :
( ( $true
!= ( X230 @ ( lam @ X235 ) ) )
& ( $true
= ( X230 @ X235 ) ) )
| ? [X233: term,X234: term] :
( ( $true
= ( X230 @ X234 ) )
& ( $true
= ( X230 @ X233 ) )
& ( $true
!= ( X230 @ ( ap @ X233 @ X234 ) ) ) ) )
<=> ( induction = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ( ? [X285: subst > term > subst > $o] :
! [X286: term > $o] :
? [X287: term] :
( ( X285 @ id @ X287 @ id )
!= ( X286 @ X287 ) )
| ? [X296: term,X297: term > $o,X298: subst > term > subst > $o] :
( ! [X305: term] :
( ( ( X297 @ X305 )
!= $true )
| ( ( X297 @ ( sub @ X296 @ ( push @ X305 @ id ) ) )
= $true ) )
& ! [X300: subst > term > term] :
( ? [X301: subst,X302: term,X303: subst] :
( ( sub @ ( X300 @ X303 @ X302 ) @ X301 )
!= ( X300 @ ( comp @ X303 @ X301 ) @ ( sub @ X302 @ X301 ) ) )
| ? [X304: term] :
( ( ( X298 @ id @ ( X300 @ id @ X304 ) @ id )
!= $true )
& ( $true
= ( X298 @ id @ X304 @ id ) ) )
| ( $true
= ( X298 @ id @ ( lam @ ( X300 @ sh @ one ) ) @ id ) ) )
& ! [X299: term] :
( ( X297 @ X299 )
= ( X298 @ id @ X299 @ id ) )
& ( ( X297 @ ( lam @ X296 ) )
!= $true ) )
| ? [X288: term,X289: term > $o] :
( ( ( X289 @ X288 )
!= $true )
& ! [X290: term] :
( ( ( var @ X290 )
!= $true )
| ( $true
= ( X289 @ X290 ) ) )
& ! [X291: term] :
( ? [X292: term] :
( ( ( X289 @ X292 )
= $true )
& ( $true
!= ( X289 @ ( sub @ X291 @ ( push @ X292 @ id ) ) ) ) )
| ( ( X289 @ ( lam @ X291 ) )
= $true ) )
& ! [X293: term,X294: term] :
( ( ( X289 @ X293 )
!= $true )
| ( $true
= ( X289 @ ( ap @ X293 @ X294 ) ) )
| ( ( X289 @ X294 )
!= $true ) ) )
| ? [X295: term] :
( ( sub @ X295 @ id )
!= X295 )
| ! [X306: subst > term > subst > $o,X307: term] :
( ? [X309: subst > term > term] :
( ( $true
!= ( X306 @ id @ ( lam @ ( X309 @ sh @ one ) ) @ id ) )
& ! [X311: subst,X312: term,X313: subst] :
( ( sub @ ( X309 @ X311 @ X312 ) @ X313 )
= ( X309 @ ( comp @ X311 @ X313 ) @ ( sub @ X312 @ X313 ) ) )
& ! [X310: term] :
( ( $true
= ( X306 @ id @ ( X309 @ id @ X310 ) @ id ) )
| ( $true
!= ( X306 @ id @ X310 @ id ) ) ) )
| ? [X308: term] :
( ( ( X306 @ id @ X308 @ id )
!= $true )
& ( ( var @ ( sub @ X308 @ id ) )
= $true ) )
| ( ( X306 @ id @ X307 @ id )
= $true )
| ? [X318: term,X319: term] :
( ( ( X306 @ id @ X318 @ id )
= $true )
& ( $true
= ( X306 @ id @ X319 @ id ) )
& ( ( X306 @ id @ ( ap @ ( sub @ X318 @ id ) @ X319 ) @ id )
!= $true ) )
| ? [X320: term,X321: subst,X322: subst,X323: subst] :
( ( $true
!= ( X306 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) @ X323 ) )
& ( $true
= ( X306 @ X321 @ X320 @ ( comp @ X322 @ X323 ) ) ) )
| ? [X314: subst,X315: subst,X316: term,X317: subst] :
( ( ( X306 @ X314 @ X316 @ ( comp @ X315 @ X317 ) )
!= $true )
& ( ( X306 @ ( comp @ X314 @ X315 ) @ ( sub @ X316 @ X315 ) @ X317 )
= $true ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( hoaslaminj_lthm = $true )
<=> ( ? [X400: term,X401: subst] :
( ( sub @ one @ ( push @ X400 @ X401 ) )
!= X400 )
| ? [X402: term,X403: subst] :
( ( comp @ sh @ ( push @ X402 @ X403 ) )
!= X403 )
| ? [X404: term,X405: term] :
( ( ( lam @ X405 )
= ( lam @ X404 ) )
& ( X404 != X405 ) )
| ! [X406: subst > term > term,X407: subst,X408: term,X409: subst > term > term] :
( ? [X413: term,X414: subst,X415: subst] :
( ( sub @ ( X409 @ X415 @ X413 ) @ X414 )
!= ( X409 @ ( comp @ X415 @ X414 ) @ ( sub @ X413 @ X414 ) ) )
| ? [X410: subst,X411: term,X412: subst] :
( ( sub @ ( X406 @ X410 @ X411 ) @ X412 )
!= ( X406 @ ( comp @ X410 @ X412 ) @ ( sub @ X411 @ X412 ) ) )
| ( ( X409 @ X407 @ X408 )
= ( X406 @ X407 @ X408 ) )
| ( ( lam @ ( X409 @ sh @ one ) )
!= ( lam @ ( X406 @ sh @ one ) ) ) ) ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ! [X255: term,X256: term > $o] :
( ( ( X256 @ X255 )
= $true )
| ? [X260: term] :
( ! [X261: term] :
( ( $true
!= ( X256 @ X261 ) )
| ( $true
= ( X256 @ ( sub @ X260 @ ( push @ X261 @ id ) ) ) ) )
& ( $true
!= ( X256 @ ( lam @ X260 ) ) ) )
| ? [X257: term,X258: term] :
( ( ( X256 @ X258 )
= $true )
& ( $true
= ( X256 @ X257 ) )
& ( ( X256 @ ( ap @ X257 @ X258 ) )
!= $true ) )
| ? [X259: term] :
( ( ( var @ X259 )
= $true )
& ( $true
!= ( X256 @ X259 ) ) ) )
<=> ( induction2 = $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X325: term > $o,X326: subst,X327: term] :
( ? [X328: term,X329: term] :
( ( $true
!= ( X325 @ ( ap @ X329 @ X328 ) ) )
& ( $true
= ( X325 @ X329 ) )
& ( $true
= ( X325 @ X328 ) ) )
| ? [X330: term] :
( ( $true
= ( var @ X330 ) )
& ( $true
!= ( X325 @ ( sub @ X330 @ X326 ) ) ) )
| ? [X331: term] :
( ( ( X325 @ ( lam @ X331 ) )
!= $true )
& ! [X332: term] :
( ( $true
!= ( X325 @ X332 ) )
| ( $true
= ( X325 @ ( sub @ X331 @ ( push @ X332 @ id ) ) ) ) ) )
| ( $true
= ( X325 @ ( sub @ X327 @ X326 ) ) ) )
<=> ( induction2lem = $true ) )
& ( axapp = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X154: subst > term > subst > $o,X155: term] :
( ? [X157: subst > term > term] :
( ! [X159: subst,X160: subst,X161: term] :
( ( X157 @ ( comp @ X159 @ X160 ) @ ( sub @ X161 @ X160 ) )
= ( sub @ ( X157 @ X159 @ X161 ) @ X160 ) )
& ( $true
!= ( X154 @ id @ ( lam @ ( X157 @ sh @ one ) ) @ id ) )
& ! [X158: term] :
( ( $true
= ( X154 @ id @ ( X157 @ id @ X158 ) @ id ) )
| ( $true
!= ( X154 @ id @ X158 @ id ) ) ) )
| ( $true
= ( X154 @ id @ ( lam @ ( sub @ X155 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X156: term] :
( ( ( X154 @ id @ X156 @ id )
= $true )
& ( $true
!= ( X154 @ id @ ( sub @ X155 @ ( push @ X156 @ id ) ) @ id ) ) ) ) )
& ( ! [X391: term] :
? [X392: subst > term > term] :
! [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
= ( X392 @ X393 @ X394 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( induction2lem_lthm = $true )
<=> ( ? [X9: subst,X10: term] :
( ( comp @ sh @ ( push @ X10 @ X9 ) )
!= X9 )
| ? [X12: term > $o,X13: term] :
( ! [X14: term] :
( ( $true
!= ( var @ X14 ) )
| ( $true
= ( X12 @ X14 ) ) )
& ! [X16: term,X17: term] :
( ( ( X12 @ ( ap @ X17 @ X16 ) )
= $true )
| ( ( X12 @ X16 )
!= $true )
| ( $true
!= ( X12 @ X17 ) ) )
& ( $true
!= ( X12 @ X13 ) )
& ! [X15: term] :
( ( $true
!= ( X12 @ X15 ) )
| ( ( X12 @ ( lam @ X15 ) )
= $true ) ) )
| ? [X7: term,X8: subst] :
( ( sub @ one @ ( push @ X7 @ X8 ) )
!= X7 )
| ? [X11: subst] :
( ( comp @ X11 @ id )
!= X11 )
| ! [X18: subst,X19: term,X20: term > $o] :
( ( $true
= ( X20 @ ( sub @ X19 @ X18 ) ) )
| ? [X21: term] :
( ( $true
!= ( X20 @ ( sub @ X21 @ X18 ) ) )
& ( ( var @ X21 )
= $true ) )
| ? [X24: term] :
( ( ( X20 @ ( lam @ X24 ) )
!= $true )
& ! [X25: term] :
( ( ( X20 @ X25 )
!= $true )
| ( ( X20 @ ( sub @ X24 @ ( push @ X25 @ id ) ) )
= $true ) ) )
| ? [X22: term,X23: term] :
( ( $true
!= ( X20 @ ( ap @ X23 @ X22 ) ) )
& ( $true
= ( X20 @ X23 ) )
& ( ( X20 @ X22 )
= $true ) ) ) ) )
& ( axvarshift = $true )
& ( ( axidr = $true )
<=> ! [X416: subst] :
( ( comp @ X416 @ id )
= X416 ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X262: subst > term > subst > $o,X263: term] :
( ? [X264: term] :
( ( ( X262 @ id @ ( sub @ X263 @ ( push @ X264 @ id ) ) @ id )
!= $true )
& ( ( X262 @ id @ X264 @ id )
= $true ) )
| ( ( X262 @ id @ ( lam @ X263 ) @ id )
= $true )
| ? [X265: subst > term > term] :
( ! [X266: term] :
( ( $true
= ( X262 @ id @ ( X265 @ id @ X266 ) @ id ) )
| ( ( X262 @ id @ X266 @ id )
!= $true ) )
& ! [X267: subst,X268: subst,X269: term] :
( ( sub @ ( X265 @ X268 @ X269 ) @ X267 )
= ( X265 @ ( comp @ X268 @ X267 ) @ ( sub @ X269 @ X267 ) ) )
& ( $true
!= ( X262 @ id @ ( lam @ ( X265 @ sh @ one ) ) @ id ) ) ) ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X417: d_subst] : ( d_id = X417 )
& ( ( apinj1 = $true )
<=> ! [X236: term,X237: term,X238: term,X239: term] :
( ( X236 = X239 )
| ( ( ap @ X236 @ X238 )
!= ( ap @ X239 @ X237 ) ) ) )
& ( ( induction2_lthm = $true )
<=> ( ? [X94: subst,X95: term > $o,X96: term] :
( ! [X100: term] :
( ( ( X95 @ ( lam @ X100 ) )
= $true )
| ? [X101: term] :
( ( ( X95 @ ( sub @ X100 @ ( push @ X101 @ id ) ) )
!= $true )
& ( ( X95 @ X101 )
= $true ) ) )
& ( ( X95 @ ( sub @ X96 @ X94 ) )
!= $true )
& ! [X97: term,X98: term] :
( ( $true
= ( X95 @ ( ap @ X97 @ X98 ) ) )
| ( ( X95 @ X98 )
!= $true )
| ( $true
!= ( X95 @ X97 ) ) )
& ! [X99: term] :
( ( ( var @ X99 )
!= $true )
| ( $true
= ( X95 @ ( sub @ X99 @ X94 ) ) ) ) )
| ? [X93: term] :
( ( sub @ X93 @ id )
!= X93 )
| ! [X102: term > $o,X103: term] :
( ( ( X102 @ X103 )
= $true )
| ? [X104: term,X105: term] :
( ( ( X102 @ X105 )
= $true )
& ( ( X102 @ X104 )
= $true )
& ( ( X102 @ ( ap @ X104 @ X105 ) )
!= $true ) )
| ? [X106: term] :
( ( ( X102 @ ( lam @ X106 ) )
!= $true )
& ! [X107: term] :
( ( $true
= ( X102 @ ( sub @ X106 @ ( push @ X107 @ id ) ) ) )
| ( ( X102 @ X107 )
!= $true ) ) )
| ? [X108: term] :
( ( ( X102 @ X108 )
!= $true )
& ( $true
= ( var @ X108 ) ) ) ) ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X340: subst > term > subst > $o,X341: term] :
( ? [X342: term] :
( ( $true
!= ( X340 @ id @ ( sub @ X341 @ ( push @ X342 @ id ) ) @ id ) )
& ( ( X340 @ id @ X342 @ id )
= $true ) )
| ? [X343: term,X344: subst > term > term] :
( ( ( X340 @ id @ ( lam @ ( X344 @ sh @ one ) ) @ id )
!= $true )
& ! [X345: subst,X346: subst,X347: term] :
( ( sub @ ( sub @ X343 @ ( push @ X347 @ X346 ) ) @ X345 )
= ( sub @ ( X344 @ X346 @ X347 ) @ X345 ) )
& ! [X349: subst,X350: subst,X351: term] :
( ( sub @ X343 @ ( push @ ( sub @ X351 @ X350 ) @ ( comp @ X349 @ X350 ) ) )
= ( X344 @ ( comp @ X349 @ X350 ) @ ( sub @ X351 @ X350 ) ) )
& ! [X348: term] :
( ( $true
= ( X340 @ id @ ( X344 @ id @ X348 ) @ id ) )
| ( $true
!= ( X340 @ id @ X348 @ id ) ) ) )
| ( $true
= ( X340 @ id @ ( lam @ ( sub @ X341 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ( ( X479 @ X481 )
!= $true )
| ( $true
= ( X480 @ one ) )
| ? [X483: term] :
( ( X480 @ X483 )
!= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) ) )
| ? [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
!= X478 ) ) )
& ( pushprop_lem0_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ? [X244: term] :
( ( sub @ X244 @ id )
!= X244 )
| ! [X245: term,X246: subst > term > subst > $o,X247: term > $o] :
( ? [X248: subst > term > term] :
( ! [X252: term] :
( ( $true
!= ( X246 @ id @ X252 @ id ) )
| ( $true
= ( X246 @ id @ ( X248 @ id @ X252 ) @ id ) ) )
& ! [X249: term,X250: subst,X251: subst] :
( ( sub @ ( X248 @ X251 @ X249 ) @ X250 )
= ( X248 @ ( comp @ X251 @ X250 ) @ ( sub @ X249 @ X250 ) ) )
& ( ( X246 @ id @ ( lam @ ( X248 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X253: term] :
( ( X247 @ X253 )
!= ( X246 @ id @ X253 @ id ) )
| ? [X254: term] :
( ( ( X247 @ X254 )
= $true )
& ( ( X247 @ ( sub @ X245 @ ( push @ X254 @ id ) ) )
!= $true ) )
| ( $true
= ( X247 @ ( lam @ X245 ) ) ) )
| ? [X240: term] :
! [X241: subst > term > term] :
? [X242: term,X243: subst] :
( ( X241 @ X243 @ X242 )
!= ( sub @ X240 @ ( push @ X242 @ X243 ) ) ) ) )
& ( ( laminj = $true )
<=> ! [X366: term,X367: term] :
( ( X366 = X367 )
| ( ( lam @ X366 )
!= ( lam @ X367 ) ) ) )
& ( ! [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
!= ( sub @ X77 @ sh ) )
| ( X77 = X78 ) )
<=> ( shinj = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( ( ! [X6: term] :
( ( sub @ X6 @ id )
= X6 )
| ? [X5: term] :
( ( sub @ X5 @ id )
!= X5 ) )
<=> ( termmset_lthm = $true ) )
& ( ! [X109: term] :
( ( sub @ X109 @ id )
= X109 )
<=> ( termmset = $true ) )
& ( apnotvar = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lthm = $true )
<=> ( ? [X418: term,X419: term > $o] :
( ! [X420: term] :
( ? [X421: term] :
( ( $true
!= ( X419 @ ( sub @ X420 @ ( push @ X421 @ id ) ) ) )
& ( $true
= ( X419 @ X421 ) ) )
| ( ( X419 @ ( lam @ X420 ) )
= $true ) )
& ! [X422: term] :
( ( $true
= ( X419 @ X422 ) )
| ( ( var @ X422 )
!= $true ) )
& ! [X423: term,X424: term] :
( ( $true
= ( X419 @ ( ap @ X424 @ X423 ) ) )
| ( ( X419 @ X423 )
!= $true )
| ( ( X419 @ X424 )
!= $true ) )
& ( $true
!= ( X419 @ X418 ) ) )
| ! [X454: term,X455: subst > term > subst > $o] :
( ? [X467: term,X468: subst,X469: subst,X470: subst] :
( ( $true
= ( X455 @ ( comp @ X468 @ X469 ) @ ( sub @ X467 @ X469 ) @ X470 ) )
& ( ( X455 @ X468 @ X467 @ ( comp @ X469 @ X470 ) )
!= $true ) )
| ( $true
= ( X455 @ id @ X454 @ id ) )
| ? [X458: subst > term > term] :
( ( ( X455 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id )
!= $true )
& ! [X459: term] :
( ( ( X455 @ id @ X459 @ id )
!= $true )
| ( ( X455 @ id @ ( X458 @ id @ X459 ) @ id )
= $true ) )
& ! [X460: subst,X461: subst,X462: term] :
( ( X458 @ ( comp @ X460 @ X461 ) @ ( sub @ X462 @ X461 ) )
= ( sub @ ( X458 @ X460 @ X462 ) @ X461 ) ) )
| ? [X456: term,X457: term] :
( ( ( X455 @ id @ ( ap @ ( sub @ X456 @ id ) @ X457 ) @ id )
!= $true )
& ( $true
= ( X455 @ id @ X456 @ id ) )
& ( $true
= ( X455 @ id @ X457 @ id ) ) )
| ? [X471: term] :
( ( ( X455 @ id @ X471 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X471 @ id ) ) ) )
| ? [X463: subst,X464: term,X465: subst,X466: subst] :
( ( ( X455 @ ( comp @ X466 @ X465 ) @ ( sub @ X464 @ X465 ) @ X463 )
!= $true )
& ( $true
= ( X455 @ X466 @ X464 @ ( comp @ X465 @ X463 ) ) ) ) )
| ? [X425: term,X426: term,X427: subst > term > subst > $o] :
( ( ( X427 @ id @ X425 @ id )
= $true )
& ! [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X427 @ ( comp @ X429 @ X430 ) @ ( sub @ X428 @ X430 ) @ X431 ) )
| ( ( X427 @ X429 @ X428 @ ( comp @ X430 @ X431 ) )
!= $true ) )
& ( ( X427 @ id @ ( ap @ X426 @ X425 ) @ id )
!= $true )
& ( $true
= ( X427 @ id @ X426 @ id ) )
& ! [X436: term,X437: term] :
( ( $true
= ( X427 @ id @ ( ap @ ( sub @ X437 @ id ) @ X436 ) @ id ) )
| ( $true
!= ( X427 @ id @ X437 @ id ) )
| ( ( X427 @ id @ X436 @ id )
!= $true ) )
& ! [X432: subst,X433: subst,X434: term,X435: subst] :
( ( ( X427 @ ( comp @ X432 @ X433 ) @ ( sub @ X434 @ X433 ) @ X435 )
!= $true )
| ( ( X427 @ X432 @ X434 @ ( comp @ X433 @ X435 ) )
= $true ) ) )
| ? [X438: term,X439: subst > term > subst > $o] :
( ! [X445: term] :
( ( $true
!= ( X439 @ id @ X445 @ id ) )
| ( $true
= ( X439 @ id @ ( sub @ X438 @ ( push @ X445 @ id ) ) @ id ) ) )
& ! [X440: subst > term > term] :
( ? [X441: subst,X442: subst,X443: term] :
( ( sub @ ( X440 @ X442 @ X443 ) @ X441 )
!= ( X440 @ ( comp @ X442 @ X441 ) @ ( sub @ X443 @ X441 ) ) )
| ? [X444: term] :
( ( ( X439 @ id @ X444 @ id )
= $true )
& ( $true
!= ( X439 @ id @ ( X440 @ id @ X444 ) @ id ) ) )
| ( ( X439 @ id @ ( lam @ ( X440 @ sh @ one ) ) @ id )
= $true ) )
& ( $true
!= ( X439 @ id @ ( lam @ X438 ) @ id ) )
& ! [X446: subst,X447: subst,X448: term,X449: subst] :
( ( ( X439 @ ( comp @ X449 @ X447 ) @ ( sub @ X448 @ X447 ) @ X446 )
= $true )
| ( ( X439 @ X449 @ X448 @ ( comp @ X447 @ X446 ) )
!= $true ) )
& ! [X450: subst,X451: subst,X452: term,X453: subst] :
( ( $true
= ( X439 @ X453 @ X452 @ ( comp @ X450 @ X451 ) ) )
| ( ( X439 @ ( comp @ X453 @ X450 ) @ ( sub @ X452 @ X450 ) @ X451 )
!= $true ) ) ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ? [X358: term,X359: term,X360: term,X361: term] :
( ( X358 != X361 )
& ( ( ap @ X360 @ X361 )
= ( ap @ X359 @ X358 ) ) )
| ! [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
| ( X363 = X365 ) ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lthm = $true )
& ( ! [X83: subst > term > subst > $o,X84: term,X85: term > $o] :
( ( $true
= ( X85 @ ( lam @ X84 ) ) )
| ? [X88: subst > term > term] :
( ! [X92: term] :
( ( ( X83 @ id @ X92 @ id )
!= $true )
| ( ( X83 @ id @ ( X88 @ id @ X92 ) @ id )
= $true ) )
& ! [X89: subst,X90: subst,X91: term] :
( ( X88 @ ( comp @ X90 @ X89 ) @ ( sub @ X91 @ X89 ) )
= ( sub @ ( X88 @ X90 @ X91 ) @ X89 ) )
& ( $true
!= ( X83 @ id @ ( lam @ ( X88 @ sh @ one ) ) @ id ) ) )
| ? [X87: term] :
( ( X83 @ id @ X87 @ id )
!= ( X85 @ X87 ) )
| ? [X86: term] :
( ( $true
= ( X85 @ X86 ) )
& ( $true
!= ( X85 @ ( sub @ X84 @ ( push @ X86 @ id ) ) ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X122: term,X123: subst > term > subst > $o] :
( ? [X125: subst > term > term] :
( ! [X126: subst,X127: term,X128: subst] :
( ( X125 @ ( comp @ X128 @ X126 ) @ ( sub @ X127 @ X126 ) )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
& ! [X129: term] :
( ( ( X123 @ id @ X129 @ id )
!= $true )
| ( $true
= ( X123 @ id @ ( X125 @ id @ X129 ) @ id ) ) )
& ( $true
!= ( X123 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) ) )
| ? [X124: term] :
( ( $true
!= ( X123 @ id @ ( sub @ X122 @ ( push @ X124 @ id ) ) @ id ) )
& ( $true
= ( X123 @ id @ X124 @ id ) ) )
| ( ( X123 @ id @ ( lam @ X122 ) @ id )
= $true )
| ? [X134: subst,X135: subst,X136: term,X137: subst] :
( ( ( X123 @ ( comp @ X135 @ X137 ) @ ( sub @ X136 @ X137 ) @ X134 )
= $true )
& ( ( X123 @ X135 @ X136 @ ( comp @ X137 @ X134 ) )
!= $true ) )
| ? [X130: subst,X131: subst,X132: subst,X133: term] :
( ( ( X123 @ ( comp @ X131 @ X130 ) @ ( sub @ X133 @ X130 ) @ X132 )
!= $true )
& ( ( X123 @ X131 @ X133 @ ( comp @ X130 @ X132 ) )
= $true ) ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( axassoc = $true )
& ( axabs = $true )
& ( lamnotap != $true )
& ( ! [X352: term,X353: term,X354: term,X355: term] :
( ( X353 = X354 )
| ( ( ap @ ( sub @ X352 @ id ) @ X353 )
!= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) )
<=> ( hoasapinj2 = $true ) )
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& ( pushprop = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
!= ( X395 @ X397 ) )
| ( $true
= ( X396 @ one ) )
| ? [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
!= ( X396 @ X399 ) ) ) )
& ( ( ! [X373: subst] :
( ( comp @ X373 @ id )
= X373 )
& ! [X372: subst] :
( ( comp @ id @ X372 )
= X372 ) )
<=> ( substmonoid = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X110: subst > term > subst > $o] :
? [X111: term > $o] :
! [X112: term] :
( ( X110 @ id @ X112 @ id )
= ( X111 @ X112 ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X512: subst > term > subst > $o,X513: term] :
( ? [X514: term] :
( ( ( X512 @ id @ X514 @ id )
= $true )
& ( ( X512 @ id @ ( sub @ X513 @ ( push @ X514 @ id ) ) @ id )
!= $true ) )
| ? [X515: subst > term > term] :
( ! [X519: term] :
( ( ( X512 @ id @ X519 @ id )
!= $true )
| ( ( X512 @ id @ ( X515 @ id @ X519 ) @ id )
= $true ) )
& ! [X516: subst,X517: term,X518: subst] :
( ( X515 @ ( comp @ X518 @ X516 ) @ ( sub @ X517 @ X516 ) )
= ( sub @ ( X515 @ X518 @ X517 ) @ X516 ) )
& ( ( X512 @ id @ ( lam @ ( X515 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X512 @ id @ ( lam @ ( sub @ X513 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X1: term] :
? [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
= ( X2 @ sh @ one ) ) )
& ( ulamvarsh = $true )
& ! [X229: d_term] : ( d_one = X229 )
& ( axclos = $true )
& ( substmonoid_gthm = $true )
& ( induction2lem_gthm = $true )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X202: subst > term > subst > $o,X203: term] :
( ? [X206: subst > term > term] :
( ! [X207: term] :
( ( ( X202 @ id @ X207 @ id )
!= $true )
| ( $true
= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
& ! [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
& ( $true
!= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) )
| ? [X204: term,X205: term] :
( ( ( X202 @ id @ ( ap @ ( sub @ X204 @ id ) @ X205 ) @ id )
!= $true )
& ( $true
= ( X202 @ id @ X205 @ id ) )
& ( $true
= ( X202 @ id @ X204 @ id ) ) )
| ( ( X202 @ id @ X203 @ id )
= $true ) ) )
& ( axmap = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X187: term,X188: subst > term > subst > $o,X189: term] :
( ( $true
= ( X188 @ id @ ( ap @ X189 @ X187 ) @ id ) )
| ? [X198: term,X199: term] :
( ( $true
= ( X188 @ id @ X199 @ id ) )
& ( $true
!= ( X188 @ id @ ( ap @ ( sub @ X199 @ id ) @ X198 ) @ id ) )
& ( ( X188 @ id @ X198 @ id )
= $true ) )
| ( $true
!= ( X188 @ id @ X187 @ id ) )
| ( $true
!= ( X188 @ id @ X189 @ id ) )
| ? [X190: subst,X191: subst,X192: term,X193: subst] :
( ( ( X188 @ X191 @ X192 @ ( comp @ X193 @ X190 ) )
!= $true )
& ( ( X188 @ ( comp @ X191 @ X193 ) @ ( sub @ X192 @ X193 ) @ X190 )
= $true ) )
| ? [X194: subst,X195: subst,X196: term,X197: subst] :
( ( $true
= ( X188 @ X194 @ X196 @ ( comp @ X197 @ X195 ) ) )
& ( ( X188 @ ( comp @ X194 @ X197 ) @ ( sub @ X196 @ X197 ) @ X195 )
!= $true ) ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasapinj2_gthm = $true )
& ( ! [X142: term] :
? [X143: subst > term > term] :
( ( X143 @ sh @ one )
= ( sub @ X142 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( one
= ( d2term @ d_one ) )
& ( termmset_gthm = $true )
& ( ! [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
= X371 )
<=> ( axvarcons = $true ) )
& ( ulamvarind = $true )
& ( hoasapinj1_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem3v2_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
<=> ( axvarid = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ! [X368: d_term,X369: d_term] :
( ( ( d2term @ X368 )
!= ( d2term @ X369 ) )
| ( X368 = X369 ) )
& ( ( ? [X504: subst] :
( ( comp @ X504 @ id )
!= X504 )
| ? [X503: subst] :
( ( comp @ id @ X503 )
!= X503 )
| ( ! [X506: subst] :
( ( comp @ id @ X506 )
= X506 )
& ! [X505: subst] :
( ( comp @ X505 @ id )
= X505 ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( ulamvar1 != $true )
& ( pushprop_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( axscons = $true )
<=> ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 ) )
& ( ( axshiftcons = $true )
<=> ! [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
= X201 ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( ! [X383: term,X384: subst > term > subst > $o] :
( ( $true
= ( X384 @ id @ ( lam @ X383 ) @ id ) )
| ? [X385: term] :
( ( ( X384 @ id @ ( sub @ X383 @ ( push @ X385 @ id ) ) @ id )
!= $true )
& ( $true
= ( X384 @ id @ X385 @ id ) ) )
| ? [X386: subst > term > term] :
( ( $true
!= ( X384 @ id @ ( lam @ ( X386 @ sh @ one ) ) @ id ) )
& ! [X388: subst,X389: subst,X390: term] :
( ( sub @ ( X386 @ X389 @ X390 ) @ X388 )
= ( X386 @ ( comp @ X389 @ X388 ) @ ( sub @ X390 @ X388 ) ) )
& ! [X387: term] :
( ( $true
= ( X384 @ id @ ( X386 @ id @ X387 ) @ id ) )
| ( ( X384 @ id @ X387 @ id )
!= $true ) ) ) )
| ? [X375: term,X376: subst > term > subst > $o] :
( ! [X382: term] :
( ( $true
!= ( X376 @ id @ X382 @ id ) )
| ( $true
= ( X376 @ id @ ( sub @ X375 @ ( push @ X382 @ id ) ) @ id ) ) )
& ( ( X376 @ id @ ( lam @ ( sub @ X375 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X377: subst > term > term] :
( ( $true
= ( X376 @ id @ ( lam @ ( X377 @ sh @ one ) ) @ id ) )
| ? [X378: term] :
( ( $true
!= ( X376 @ id @ ( X377 @ id @ X378 ) @ id ) )
& ( $true
= ( X376 @ id @ X378 @ id ) ) )
| ? [X379: subst,X380: subst,X381: term] :
( ( X377 @ ( comp @ X379 @ X380 ) @ ( sub @ X381 @ X380 ) )
!= ( sub @ ( X377 @ X379 @ X381 ) @ X380 ) ) ) )
| ? [X374: term] :
( ( sub @ X374 @ id )
!= X374 ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ? [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
!= X114 )
| ? [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
!= X116 )
| ! [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
| ( $true
!= ( X120 @ X121 ) ) ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoaslamnotap = $true )
<=> ! [X520: term,X521: term,X522: subst > term > term] :
( ( ( ap @ ( sub @ X521 @ id ) @ X520 )
!= ( lam @ ( X522 @ sh @ one ) ) )
| ? [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
!= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) ) ) ) ),
inference(flattening,[status(thm)],[f12]) ).
thf(f12,plain,
( ( pushprop_lem1v2_gthm = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X383: term,X384: subst > term > subst > $o] :
( ( $true
= ( X384 @ id @ ( lam @ X383 ) @ id ) )
| ? [X385: term] :
( ( ( X384 @ id @ ( sub @ X383 @ ( push @ X385 @ id ) ) @ id )
!= $true )
& ( $true
= ( X384 @ id @ X385 @ id ) ) )
| ? [X386: subst > term > term] :
( ( $true
!= ( X384 @ id @ ( lam @ ( X386 @ sh @ one ) ) @ id ) )
& ! [X388: subst,X389: subst,X390: term] :
( ( sub @ ( X386 @ X389 @ X390 ) @ X388 )
= ( X386 @ ( comp @ X389 @ X388 ) @ ( sub @ X390 @ X388 ) ) )
& ! [X387: term] :
( ( $true
= ( X384 @ id @ ( X386 @ id @ X387 ) @ id ) )
| ( ( X384 @ id @ X387 @ id )
!= $true ) ) ) )
| ? [X375: term,X376: subst > term > subst > $o] :
( ! [X382: term] :
( ( $true
!= ( X376 @ id @ X382 @ id ) )
| ( $true
= ( X376 @ id @ ( sub @ X375 @ ( push @ X382 @ id ) ) @ id ) ) )
& ( ( X376 @ id @ ( lam @ ( sub @ X375 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X377: subst > term > term] :
( ( $true
= ( X376 @ id @ ( lam @ ( X377 @ sh @ one ) ) @ id ) )
| ? [X378: term] :
( ( $true
!= ( X376 @ id @ ( X377 @ id @ X378 ) @ id ) )
& ( $true
= ( X376 @ id @ X378 @ id ) ) )
| ? [X379: subst,X380: subst,X381: term] :
( ( X377 @ ( comp @ X379 @ X380 ) @ ( sub @ X381 @ X380 ) )
!= ( sub @ ( X377 @ X379 @ X381 ) @ X380 ) ) ) )
| ? [X374: term] :
( ( sub @ X374 @ id )
!= X374 ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ! [X211: term,X212: subst > term > subst > $o] :
( ? [X221: term] :
( ( ( var @ ( sub @ X221 @ id ) )
= $true )
& ( ( X212 @ id @ X221 @ id )
!= $true ) )
| ? [X222: subst > term > term] :
( ! [X224: term,X225: subst,X226: subst] :
( ( X222 @ ( comp @ X226 @ X225 ) @ ( sub @ X224 @ X225 ) )
= ( sub @ ( X222 @ X226 @ X224 ) @ X225 ) )
& ! [X223: term] :
( ( $true
= ( X212 @ id @ ( X222 @ id @ X223 ) @ id ) )
| ( ( X212 @ id @ X223 @ id )
!= $true ) )
& ( $true
!= ( X212 @ id @ ( lam @ ( X222 @ sh @ one ) ) @ id ) ) )
| ? [X217: subst,X218: term,X219: subst,X220: subst] :
( ( ( X212 @ X219 @ X218 @ ( comp @ X217 @ X220 ) )
= $true )
& ( ( X212 @ ( comp @ X219 @ X217 ) @ ( sub @ X218 @ X217 ) @ X220 )
!= $true ) )
| ( $true
= ( X212 @ id @ X211 @ id ) )
| ? [X227: term,X228: term] :
( ( ( X212 @ id @ ( ap @ ( sub @ X227 @ id ) @ X228 ) @ id )
!= $true )
& ( ( X212 @ id @ X228 @ id )
= $true )
& ( ( X212 @ id @ X227 @ id )
= $true ) )
| ? [X213: term,X214: subst,X215: subst,X216: subst] :
( ( ( X212 @ X214 @ X213 @ ( comp @ X216 @ X215 ) )
!= $true )
& ( $true
= ( X212 @ ( comp @ X214 @ X216 ) @ ( sub @ X213 @ X216 ) @ X215 ) ) ) )
<=> ( hoasinduction = $true ) )
& ( hoasapinj2_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( axvarshift = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X406: subst > term > term,X407: subst,X408: term,X409: subst > term > term] :
( ? [X413: term,X414: subst,X415: subst] :
( ( sub @ ( X409 @ X415 @ X413 ) @ X414 )
!= ( X409 @ ( comp @ X415 @ X414 ) @ ( sub @ X413 @ X414 ) ) )
| ? [X410: subst,X411: term,X412: subst] :
( ( sub @ ( X406 @ X410 @ X411 ) @ X412 )
!= ( X406 @ ( comp @ X410 @ X412 ) @ ( sub @ X411 @ X412 ) ) )
| ( ( X409 @ X407 @ X408 )
= ( X406 @ X407 @ X408 ) )
| ( ( lam @ ( X409 @ sh @ one ) )
!= ( lam @ ( X406 @ sh @ one ) ) ) )
| ? [X404: term,X405: term] :
( ( ( lam @ X405 )
= ( lam @ X404 ) )
& ( X404 != X405 ) )
| ? [X402: term,X403: subst] :
( ( comp @ sh @ ( push @ X402 @ X403 ) )
!= X403 )
| ? [X400: term,X401: subst] :
( ( sub @ one @ ( push @ X400 @ X401 ) )
!= X400 ) ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X454: term,X455: subst > term > subst > $o] :
( ? [X467: term,X468: subst,X469: subst,X470: subst] :
( ( $true
= ( X455 @ ( comp @ X468 @ X469 ) @ ( sub @ X467 @ X469 ) @ X470 ) )
& ( ( X455 @ X468 @ X467 @ ( comp @ X469 @ X470 ) )
!= $true ) )
| ( $true
= ( X455 @ id @ X454 @ id ) )
| ? [X458: subst > term > term] :
( ( ( X455 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id )
!= $true )
& ! [X459: term] :
( ( ( X455 @ id @ X459 @ id )
!= $true )
| ( ( X455 @ id @ ( X458 @ id @ X459 ) @ id )
= $true ) )
& ! [X460: subst,X461: subst,X462: term] :
( ( X458 @ ( comp @ X460 @ X461 ) @ ( sub @ X462 @ X461 ) )
= ( sub @ ( X458 @ X460 @ X462 ) @ X461 ) ) )
| ? [X456: term,X457: term] :
( ( ( X455 @ id @ ( ap @ ( sub @ X456 @ id ) @ X457 ) @ id )
!= $true )
& ( $true
= ( X455 @ id @ X456 @ id ) )
& ( $true
= ( X455 @ id @ X457 @ id ) ) )
| ? [X471: term] :
( ( ( X455 @ id @ X471 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X471 @ id ) ) ) )
| ? [X463: subst,X464: term,X465: subst,X466: subst] :
( ( ( X455 @ ( comp @ X466 @ X465 ) @ ( sub @ X464 @ X465 ) @ X463 )
!= $true )
& ( $true
= ( X455 @ X466 @ X464 @ ( comp @ X465 @ X463 ) ) ) ) )
| ? [X438: term,X439: subst > term > subst > $o] :
( ! [X445: term] :
( ( $true
!= ( X439 @ id @ X445 @ id ) )
| ( $true
= ( X439 @ id @ ( sub @ X438 @ ( push @ X445 @ id ) ) @ id ) ) )
& ! [X440: subst > term > term] :
( ? [X441: subst,X442: subst,X443: term] :
( ( sub @ ( X440 @ X442 @ X443 ) @ X441 )
!= ( X440 @ ( comp @ X442 @ X441 ) @ ( sub @ X443 @ X441 ) ) )
| ? [X444: term] :
( ( ( X439 @ id @ X444 @ id )
= $true )
& ( $true
!= ( X439 @ id @ ( X440 @ id @ X444 ) @ id ) ) )
| ( ( X439 @ id @ ( lam @ ( X440 @ sh @ one ) ) @ id )
= $true ) )
& ( $true
!= ( X439 @ id @ ( lam @ X438 ) @ id ) )
& ! [X446: subst,X447: subst,X448: term,X449: subst] :
( ( ( X439 @ ( comp @ X449 @ X447 ) @ ( sub @ X448 @ X447 ) @ X446 )
= $true )
| ( ( X439 @ X449 @ X448 @ ( comp @ X447 @ X446 ) )
!= $true ) )
& ! [X450: subst,X451: subst,X452: term,X453: subst] :
( ( $true
= ( X439 @ X453 @ X452 @ ( comp @ X450 @ X451 ) ) )
| ( ( X439 @ ( comp @ X453 @ X450 ) @ ( sub @ X452 @ X450 ) @ X451 )
!= $true ) ) )
| ? [X425: term,X426: term,X427: subst > term > subst > $o] :
( ( ( X427 @ id @ X425 @ id )
= $true )
& ! [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X427 @ ( comp @ X429 @ X430 ) @ ( sub @ X428 @ X430 ) @ X431 ) )
| ( ( X427 @ X429 @ X428 @ ( comp @ X430 @ X431 ) )
!= $true ) )
& ( ( X427 @ id @ ( ap @ X426 @ X425 ) @ id )
!= $true )
& ( $true
= ( X427 @ id @ X426 @ id ) )
& ! [X436: term,X437: term] :
( ( $true
= ( X427 @ id @ ( ap @ ( sub @ X437 @ id ) @ X436 ) @ id ) )
| ( $true
!= ( X427 @ id @ X437 @ id ) )
| ( ( X427 @ id @ X436 @ id )
!= $true ) )
& ! [X432: subst,X433: subst,X434: term,X435: subst] :
( ( ( X427 @ ( comp @ X432 @ X433 ) @ ( sub @ X434 @ X433 ) @ X435 )
!= $true )
| ( ( X427 @ X432 @ X434 @ ( comp @ X433 @ X435 ) )
= $true ) ) )
| ? [X418: term,X419: term > $o] :
( ! [X420: term] :
( ? [X421: term] :
( ( $true
!= ( X419 @ ( sub @ X420 @ ( push @ X421 @ id ) ) ) )
& ( $true
= ( X419 @ X421 ) ) )
| ( ( X419 @ ( lam @ X420 ) )
= $true ) )
& ! [X422: term] :
( ( $true
= ( X419 @ X422 ) )
| ( ( var @ X422 )
!= $true ) )
& ! [X423: term,X424: term] :
( ( $true
= ( X419 @ ( ap @ X424 @ X423 ) ) )
| ( ( X419 @ X423 )
!= $true )
| ( ( X419 @ X424 )
!= $true ) )
& ( $true
!= ( X419 @ X418 ) ) ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X535: term > $o,X536: subst,X537: term] :
? [X538: term > $o] :
! [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
= ( X538 @ X539 ) ) )
& ( axclos = $true )
& ! [X229: d_term] : ( d_one = X229 )
& ( pushprop_lthm = $true )
& ( ( ! [X373: subst] :
( ( comp @ X373 @ id )
= X373 )
& ! [X372: subst] :
( ( comp @ id @ X372 )
= X372 ) )
<=> ( substmonoid = $true ) )
& ( ! [X324: subst] :
( ( comp @ id @ X324 )
= X324 )
<=> ( axidl = $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X171: term,X172: subst > term > subst > $o] :
( ? [X183: subst,X184: subst,X185: term,X186: subst] :
( ( $true
= ( X172 @ ( comp @ X184 @ X183 ) @ ( sub @ X185 @ X183 ) @ X186 ) )
& ( ( X172 @ X184 @ X185 @ ( comp @ X183 @ X186 ) )
!= $true ) )
| ? [X173: term] :
( ( $true
!= ( X172 @ id @ ( sub @ X171 @ ( push @ X173 @ id ) ) @ id ) )
& ( ( X172 @ id @ X173 @ id )
= $true ) )
| ? [X179: term,X180: subst,X181: subst,X182: subst] :
( ( ( X172 @ X182 @ X179 @ ( comp @ X180 @ X181 ) )
= $true )
& ( $true
!= ( X172 @ ( comp @ X182 @ X180 ) @ ( sub @ X179 @ X180 ) @ X181 ) ) )
| ? [X174: subst > term > term] :
( ! [X176: subst,X177: subst,X178: term] :
( ( sub @ ( X174 @ X176 @ X178 ) @ X177 )
= ( X174 @ ( comp @ X176 @ X177 ) @ ( sub @ X178 @ X177 ) ) )
& ( ( X172 @ id @ ( lam @ ( X174 @ sh @ one ) ) @ id )
!= $true )
& ! [X175: term] :
( ( ( X172 @ id @ ( X174 @ id @ X175 ) @ id )
= $true )
| ( ( X172 @ id @ X175 @ id )
!= $true ) ) )
| ( $true
= ( X172 @ id @ ( lam @ X171 ) @ id ) ) )
| ? [X163: subst > term > subst > $o,X164: term] :
( ! [X166: subst > term > term] :
( ? [X167: subst,X168: subst,X169: term] :
( ( X166 @ ( comp @ X168 @ X167 ) @ ( sub @ X169 @ X167 ) )
!= ( sub @ ( X166 @ X168 @ X169 ) @ X167 ) )
| ? [X170: term] :
( ( $true
= ( X163 @ id @ X170 @ id ) )
& ( ( X163 @ id @ ( X166 @ id @ X170 ) @ id )
!= $true ) )
| ( $true
= ( X163 @ id @ ( lam @ ( X166 @ sh @ one ) ) @ id ) ) )
& ! [X165: term] :
( ( $true
!= ( X163 @ id @ X165 @ id ) )
| ( $true
= ( X163 @ id @ ( sub @ X164 @ ( push @ X165 @ id ) ) @ id ) ) )
& ( ( X163 @ id @ ( lam @ ( sub @ X164 @ ( push @ one @ sh ) ) ) @ id )
!= $true ) )
| ? [X162: term] :
( ( sub @ X162 @ id )
!= X162 ) ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X154: subst > term > subst > $o,X155: term] :
( ? [X157: subst > term > term] :
( ! [X159: subst,X160: subst,X161: term] :
( ( X157 @ ( comp @ X159 @ X160 ) @ ( sub @ X161 @ X160 ) )
= ( sub @ ( X157 @ X159 @ X161 ) @ X160 ) )
& ( $true
!= ( X154 @ id @ ( lam @ ( X157 @ sh @ one ) ) @ id ) )
& ! [X158: term] :
( ( $true
= ( X154 @ id @ ( X157 @ id @ X158 ) @ id ) )
| ( $true
!= ( X154 @ id @ X158 @ id ) ) ) )
| ( $true
= ( X154 @ id @ ( lam @ ( sub @ X155 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X156: term] :
( ( ( X154 @ id @ X156 @ id )
= $true )
& ( $true
!= ( X154 @ id @ ( sub @ X155 @ ( push @ X156 @ id ) ) @ id ) ) ) ) )
& ( ( apinj2 = $true )
<=> ! [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
!= ( ap @ X79 @ X82 ) )
| ( X81 = X82 ) ) )
& ( ! [X391: term] :
? [X392: subst > term > term] :
! [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
= ( X392 @ X393 @ X394 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ! [X255: term,X256: term > $o] :
( ( ( X256 @ X255 )
= $true )
| ? [X260: term] :
( ! [X261: term] :
( ( $true
!= ( X256 @ X261 ) )
| ( $true
= ( X256 @ ( sub @ X260 @ ( push @ X261 @ id ) ) ) ) )
& ( $true
!= ( X256 @ ( lam @ X260 ) ) ) )
| ? [X257: term,X258: term] :
( ( ( X256 @ X258 )
= $true )
& ( $true
= ( X256 @ X257 ) )
& ( ( X256 @ ( ap @ X257 @ X258 ) )
!= $true ) )
| ? [X259: term] :
( ( ( var @ X259 )
= $true )
& ( $true
!= ( X256 @ X259 ) ) ) )
<=> ( induction2 = $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X202: subst > term > subst > $o,X203: term] :
( ? [X206: subst > term > term] :
( ! [X207: term] :
( ( ( X202 @ id @ X207 @ id )
!= $true )
| ( $true
= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
& ! [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
& ( $true
!= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) )
| ? [X204: term,X205: term] :
( ( ( X202 @ id @ ( ap @ ( sub @ X204 @ id ) @ X205 ) @ id )
!= $true )
& ( $true
= ( X202 @ id @ X205 @ id ) )
& ( $true
= ( X202 @ id @ X204 @ id ) ) )
| ( ( X202 @ id @ X203 @ id )
= $true ) ) )
& ( ! [X138: term] :
? [X139: subst > term > term] :
! [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
= ( X139 @ X141 @ X140 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ( ( X479 @ X481 )
!= $true )
| ( $true
= ( X480 @ one ) )
| ? [X483: term] :
( ( X480 @ X483 )
!= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) ) )
| ? [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
!= X478 ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
!= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
| ( X533 = X534 ) )
| ? [X527: term,X528: term,X529: term,X530: term] :
( ( X527 != X528 )
& ( ( ap @ X528 @ X529 )
= ( ap @ X527 @ X530 ) ) )
| ? [X526: term] :
( ( sub @ X526 @ id )
!= X526 ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
| ( $true
!= ( X120 @ X121 ) ) )
| ? [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
!= X116 )
| ? [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
!= X114 ) ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ! [X325: term > $o,X326: subst,X327: term] :
( ? [X328: term,X329: term] :
( ( $true
!= ( X325 @ ( ap @ X329 @ X328 ) ) )
& ( $true
= ( X325 @ X329 ) )
& ( $true
= ( X325 @ X328 ) ) )
| ? [X330: term] :
( ( $true
= ( var @ X330 ) )
& ( $true
!= ( X325 @ ( sub @ X330 @ X326 ) ) ) )
| ? [X331: term] :
( ( ( X325 @ ( lam @ X331 ) )
!= $true )
& ! [X332: term] :
( ( $true
!= ( X325 @ X332 ) )
| ( $true
= ( X325 @ ( sub @ X331 @ ( push @ X332 @ id ) ) ) ) ) )
| ( $true
= ( X325 @ ( sub @ X327 @ X326 ) ) ) )
<=> ( induction2lem = $true ) )
& ( ( laminj = $true )
<=> ! [X366: term,X367: term] :
( ( X366 = X367 )
| ( ( lam @ X366 )
!= ( lam @ X367 ) ) ) )
& ( hoaslamnotap_lthm = $true )
& ( apnotvar = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X110: subst > term > subst > $o] :
? [X111: term > $o] :
! [X112: term] :
( ( X110 @ id @ X112 @ id )
= ( X111 @ X112 ) ) )
& ( ulamvarsh = $true )
& ! [X368: d_term,X369: d_term] :
( ( ( d2term @ X368 )
!= ( d2term @ X369 ) )
| ( X368 = X369 ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( induction2_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X352: term,X353: term,X354: term,X355: term] :
( ( X353 = X354 )
| ( ( ap @ ( sub @ X352 @ id ) @ X353 )
!= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( one
= ( d2term @ d_one ) )
& ( ! [X142: term] :
? [X143: subst > term > term] :
( ( X143 @ sh @ one )
= ( sub @ X142 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ? [X358: term,X359: term,X360: term,X361: term] :
( ( X358 != X361 )
& ( ( ap @ X360 @ X361 )
= ( ap @ X359 @ X358 ) ) )
| ! [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
| ( X363 = X365 ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasapinj1_gthm = $true )
& ( axassoc = $true )
& ( hoaslaminj_gthm = $true )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X262: subst > term > subst > $o,X263: term] :
( ? [X264: term] :
( ( ( X262 @ id @ ( sub @ X263 @ ( push @ X264 @ id ) ) @ id )
!= $true )
& ( ( X262 @ id @ X264 @ id )
= $true ) )
| ( ( X262 @ id @ ( lam @ X263 ) @ id )
= $true )
| ? [X265: subst > term > term] :
( ! [X266: term] :
( ( $true
= ( X262 @ id @ ( X265 @ id @ X266 ) @ id ) )
| ( ( X262 @ id @ X266 @ id )
!= $true ) )
& ! [X267: subst,X268: subst,X269: term] :
( ( sub @ ( X265 @ X268 @ X269 ) @ X267 )
= ( X265 @ ( comp @ X268 @ X267 ) @ ( sub @ X269 @ X267 ) ) )
& ( $true
!= ( X262 @ id @ ( lam @ ( X265 @ sh @ one ) ) @ id ) ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_lthm = $true )
& ( lamnotap != $true )
& ( axabs = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ulamvarind = $true )
& ( ! [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
= X371 )
<=> ( axvarcons = $true ) )
& ( ( axidr = $true )
<=> ! [X416: subst] :
( ( comp @ X416 @ id )
= X416 ) )
& ( ( d2subst @ d_id )
= id )
& ( ulamvar1 != $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
!= ( sub @ X77 @ sh ) )
| ( X77 = X78 ) )
<=> ( shinj = $true ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X270: term,X271: term > $o,X272: subst > term > subst > $o,X273: term] :
( ( $true
!= ( X271 @ X273 ) )
| ? [X284: term] :
( ( X272 @ id @ X284 @ id )
!= ( X271 @ X284 ) )
| ( $true
= ( X271 @ ( ap @ X273 @ X270 ) ) )
| ( $true
!= ( X271 @ X270 ) )
| ? [X280: subst,X281: subst,X282: term,X283: subst] :
( ( $true
!= ( X272 @ X281 @ X282 @ ( comp @ X283 @ X280 ) ) )
& ( ( X272 @ ( comp @ X281 @ X283 ) @ ( sub @ X282 @ X283 ) @ X280 )
= $true ) )
| ? [X274: term,X275: subst,X276: subst,X277: subst] :
( ( $true
= ( X272 @ X277 @ X274 @ ( comp @ X276 @ X275 ) ) )
& ( $true
!= ( X272 @ ( comp @ X277 @ X276 ) @ ( sub @ X274 @ X276 ) @ X275 ) ) )
| ? [X278: term,X279: term] :
( ( ( X272 @ id @ X278 @ id )
= $true )
& ( $true
!= ( X272 @ id @ ( ap @ ( sub @ X279 @ id ) @ X278 ) @ id ) )
& ( ( X272 @ id @ X279 @ id )
= $true ) ) ) )
& ( termmset_gthm = $true )
& ( ( ( ! [X506: subst] :
( ( comp @ id @ X506 )
= X506 )
& ! [X505: subst] :
( ( comp @ X505 @ id )
= X505 ) )
| ? [X504: subst] :
( ( comp @ X504 @ id )
!= X504 )
| ? [X503: subst] :
( ( comp @ id @ X503 )
!= X503 ) )
<=> ( substmonoid_lthm = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X417: d_subst] : ( d_id = X417 )
& ( pushprop_lem2v2 = $true )
& ( ( hoaslamnotap = $true )
<=> ! [X520: term,X521: term,X522: subst > term > term] :
( ( ( ap @ ( sub @ X521 @ id ) @ X520 )
!= ( lam @ ( X522 @ sh @ one ) ) )
| ? [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
!= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) ) ) )
& ( ! [X230: term > $o,X231: term] :
( ? [X232: term] :
( ( $true
!= ( X230 @ X232 ) )
& ( ( var @ X232 )
= $true ) )
| ( ( X230 @ X231 )
= $true )
| ? [X235: term] :
( ( $true
!= ( X230 @ ( lam @ X235 ) ) )
& ( $true
= ( X230 @ X235 ) ) )
| ? [X233: term,X234: term] :
( ( $true
= ( X230 @ X234 ) )
& ( $true
= ( X230 @ X233 ) )
& ( $true
!= ( X230 @ ( ap @ X233 @ X234 ) ) ) ) )
<=> ( induction = $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X1: term] :
? [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
= ( X2 @ sh @ one ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X512: subst > term > subst > $o,X513: term] :
( ? [X514: term] :
( ( ( X512 @ id @ X514 @ id )
= $true )
& ( ( X512 @ id @ ( sub @ X513 @ ( push @ X514 @ id ) ) @ id )
!= $true ) )
| ? [X515: subst > term > term] :
( ! [X519: term] :
( ( ( X512 @ id @ X519 @ id )
!= $true )
| ( ( X512 @ id @ ( X515 @ id @ X519 ) @ id )
= $true ) )
& ! [X516: subst,X517: term,X518: subst] :
( ( X515 @ ( comp @ X518 @ X516 ) @ ( sub @ X517 @ X516 ) )
= ( sub @ ( X515 @ X518 @ X517 ) @ X516 ) )
& ( ( X512 @ id @ ( lam @ ( X515 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X512 @ id @ ( lam @ ( sub @ X513 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( ! [X306: subst > term > subst > $o,X307: term] :
( ? [X309: subst > term > term] :
( ( $true
!= ( X306 @ id @ ( lam @ ( X309 @ sh @ one ) ) @ id ) )
& ! [X311: subst,X312: term,X313: subst] :
( ( sub @ ( X309 @ X311 @ X312 ) @ X313 )
= ( X309 @ ( comp @ X311 @ X313 ) @ ( sub @ X312 @ X313 ) ) )
& ! [X310: term] :
( ( $true
= ( X306 @ id @ ( X309 @ id @ X310 ) @ id ) )
| ( $true
!= ( X306 @ id @ X310 @ id ) ) ) )
| ? [X308: term] :
( ( ( X306 @ id @ X308 @ id )
!= $true )
& ( ( var @ ( sub @ X308 @ id ) )
= $true ) )
| ( ( X306 @ id @ X307 @ id )
= $true )
| ? [X318: term,X319: term] :
( ( ( X306 @ id @ X318 @ id )
= $true )
& ( $true
= ( X306 @ id @ X319 @ id ) )
& ( ( X306 @ id @ ( ap @ ( sub @ X318 @ id ) @ X319 ) @ id )
!= $true ) )
| ? [X320: term,X321: subst,X322: subst,X323: subst] :
( ( $true
!= ( X306 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) @ X323 ) )
& ( $true
= ( X306 @ X321 @ X320 @ ( comp @ X322 @ X323 ) ) ) )
| ? [X314: subst,X315: subst,X316: term,X317: subst] :
( ( ( X306 @ X314 @ X316 @ ( comp @ X315 @ X317 ) )
!= $true )
& ( ( X306 @ ( comp @ X314 @ X315 ) @ ( sub @ X316 @ X315 ) @ X317 )
= $true ) ) )
| ? [X296: term,X297: term > $o,X298: subst > term > subst > $o] :
( ! [X305: term] :
( ( ( X297 @ X305 )
!= $true )
| ( ( X297 @ ( sub @ X296 @ ( push @ X305 @ id ) ) )
= $true ) )
& ! [X300: subst > term > term] :
( ? [X301: subst,X302: term,X303: subst] :
( ( sub @ ( X300 @ X303 @ X302 ) @ X301 )
!= ( X300 @ ( comp @ X303 @ X301 ) @ ( sub @ X302 @ X301 ) ) )
| ? [X304: term] :
( ( ( X298 @ id @ ( X300 @ id @ X304 ) @ id )
!= $true )
& ( $true
= ( X298 @ id @ X304 @ id ) ) )
| ( $true
= ( X298 @ id @ ( lam @ ( X300 @ sh @ one ) ) @ id ) ) )
& ! [X299: term] :
( ( X297 @ X299 )
= ( X298 @ id @ X299 @ id ) )
& ( ( X297 @ ( lam @ X296 ) )
!= $true ) )
| ? [X295: term] :
( ( sub @ X295 @ id )
!= X295 )
| ? [X288: term,X289: term > $o] :
( ( ( X289 @ X288 )
!= $true )
& ! [X290: term] :
( ( ( var @ X290 )
!= $true )
| ( $true
= ( X289 @ X290 ) ) )
& ! [X291: term] :
( ? [X292: term] :
( ( ( X289 @ X292 )
= $true )
& ( $true
!= ( X289 @ ( sub @ X291 @ ( push @ X292 @ id ) ) ) ) )
| ( ( X289 @ ( lam @ X291 ) )
= $true ) )
& ! [X293: term,X294: term] :
( ( ( X289 @ X293 )
!= $true )
| ( $true
= ( X289 @ ( ap @ X293 @ X294 ) ) )
| ( ( X289 @ X294 )
!= $true ) ) )
| ? [X285: subst > term > subst > $o] :
! [X286: term > $o] :
? [X287: term] :
( ( X285 @ id @ X287 @ id )
!= ( X286 @ X287 ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ! [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
!= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
| ( X507 = X508 ) )
<=> ( hoasapinj1 = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& ( ( pushprop_lem0 = $true )
<=> ! [X333: term,X334: term > $o,X335: subst] :
? [X336: term > $o] :
! [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
= ( X336 @ X337 ) ) )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X187: term,X188: subst > term > subst > $o,X189: term] :
( ( $true
= ( X188 @ id @ ( ap @ X189 @ X187 ) @ id ) )
| ? [X198: term,X199: term] :
( ( $true
= ( X188 @ id @ X199 @ id ) )
& ( $true
!= ( X188 @ id @ ( ap @ ( sub @ X199 @ id ) @ X198 ) @ id ) )
& ( ( X188 @ id @ X198 @ id )
= $true ) )
| ( $true
!= ( X188 @ id @ X187 @ id ) )
| ( $true
!= ( X188 @ id @ X189 @ id ) )
| ? [X190: subst,X191: subst,X192: term,X193: subst] :
( ( ( X188 @ X191 @ X192 @ ( comp @ X193 @ X190 ) )
!= $true )
& ( ( X188 @ ( comp @ X191 @ X193 ) @ ( sub @ X192 @ X193 ) @ X190 )
= $true ) )
| ? [X194: subst,X195: subst,X196: term,X197: subst] :
( ( $true
= ( X188 @ X194 @ X196 @ ( comp @ X197 @ X195 ) ) )
& ( ( X188 @ ( comp @ X194 @ X197 ) @ ( sub @ X196 @ X197 ) @ X195 )
!= $true ) ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X245: term,X246: subst > term > subst > $o,X247: term > $o] :
( ? [X248: subst > term > term] :
( ! [X252: term] :
( ( $true
!= ( X246 @ id @ X252 @ id ) )
| ( $true
= ( X246 @ id @ ( X248 @ id @ X252 ) @ id ) ) )
& ! [X249: term,X250: subst,X251: subst] :
( ( sub @ ( X248 @ X251 @ X249 ) @ X250 )
= ( X248 @ ( comp @ X251 @ X250 ) @ ( sub @ X249 @ X250 ) ) )
& ( ( X246 @ id @ ( lam @ ( X248 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X253: term] :
( ( X247 @ X253 )
!= ( X246 @ id @ X253 @ id ) )
| ? [X254: term] :
( ( ( X247 @ X254 )
= $true )
& ( ( X247 @ ( sub @ X245 @ ( push @ X254 @ id ) ) )
!= $true ) )
| ( $true
= ( X247 @ ( lam @ X245 ) ) ) )
| ? [X244: term] :
( ( sub @ X244 @ id )
!= X244 )
| ? [X240: term] :
! [X241: subst > term > term] :
? [X242: term,X243: subst] :
( ( X241 @ X243 @ X242 )
!= ( sub @ X240 @ ( push @ X242 @ X243 ) ) ) ) )
& ( hoaslamnotvar = $true )
& ( ( ! [X6: term] :
( ( sub @ X6 @ id )
= X6 )
| ? [X5: term] :
( ( sub @ X5 @ id )
!= X5 ) )
<=> ( termmset_lthm = $true ) )
& ! [X356: subst] :
? [X357: d_subst] :
( ( d2subst @ X357 )
= X356 )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoaslaminj = $true )
<=> ! [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ( ( X144 @ X146 @ X147 )
= ( X145 @ X146 @ X147 ) )
| ? [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
!= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
| ( ( lam @ ( X144 @ sh @ one ) )
!= ( lam @ ( X145 @ sh @ one ) ) )
| ? [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
!= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) ) ) )
& ( pushprop = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslamnotvar_lthm = $true )
& ( ( induction2_lthm = $true )
<=> ( ! [X102: term > $o,X103: term] :
( ( ( X102 @ X103 )
= $true )
| ? [X104: term,X105: term] :
( ( ( X102 @ X105 )
= $true )
& ( ( X102 @ X104 )
= $true )
& ( ( X102 @ ( ap @ X104 @ X105 ) )
!= $true ) )
| ? [X106: term] :
( ( ( X102 @ ( lam @ X106 ) )
!= $true )
& ! [X107: term] :
( ( $true
= ( X102 @ ( sub @ X106 @ ( push @ X107 @ id ) ) ) )
| ( ( X102 @ X107 )
!= $true ) ) )
| ? [X108: term] :
( ( ( X102 @ X108 )
!= $true )
& ( $true
= ( var @ X108 ) ) ) )
| ? [X94: subst,X95: term > $o,X96: term] :
( ! [X100: term] :
( ( ( X95 @ ( lam @ X100 ) )
= $true )
| ? [X101: term] :
( ( ( X95 @ ( sub @ X100 @ ( push @ X101 @ id ) ) )
!= $true )
& ( ( X95 @ X101 )
= $true ) ) )
& ( ( X95 @ ( sub @ X96 @ X94 ) )
!= $true )
& ! [X97: term,X98: term] :
( ( $true
= ( X95 @ ( ap @ X97 @ X98 ) ) )
| ( ( X95 @ X98 )
!= $true )
| ( $true
!= ( X95 @ X97 ) ) )
& ! [X99: term] :
( ( ( var @ X99 )
!= $true )
| ( $true
= ( X95 @ ( sub @ X99 @ X94 ) ) ) ) )
| ? [X93: term] :
( ( sub @ X93 @ id )
!= X93 ) ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
!= ( X395 @ X397 ) )
| ( $true
= ( X396 @ one ) )
| ? [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
!= ( X396 @ X399 ) ) ) )
& ( lamnotvar = $true )
& ( ! [X109: term] :
( ( sub @ X109 @ id )
= X109 )
<=> ( termmset = $true ) )
& ( ! [X26: subst > term > subst > $o] :
? [X27: term > $o] :
! [X28: term] :
( ( X27 @ X28 )
= ( X26 @ id @ X28 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( pushprop_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( hoasinduction_lem1 = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( ! [X18: subst,X19: term,X20: term > $o] :
( ( $true
= ( X20 @ ( sub @ X19 @ X18 ) ) )
| ? [X21: term] :
( ( $true
!= ( X20 @ ( sub @ X21 @ X18 ) ) )
& ( ( var @ X21 )
= $true ) )
| ? [X24: term] :
( ( ( X20 @ ( lam @ X24 ) )
!= $true )
& ! [X25: term] :
( ( ( X20 @ X25 )
!= $true )
| ( ( X20 @ ( sub @ X24 @ ( push @ X25 @ id ) ) )
= $true ) ) )
| ? [X22: term,X23: term] :
( ( $true
!= ( X20 @ ( ap @ X23 @ X22 ) ) )
& ( $true
= ( X20 @ X23 ) )
& ( ( X20 @ X22 )
= $true ) ) )
| ? [X12: term > $o,X13: term] :
( ! [X14: term] :
( ( $true
!= ( var @ X14 ) )
| ( $true
= ( X12 @ X14 ) ) )
& ! [X16: term,X17: term] :
( ( ( X12 @ ( ap @ X17 @ X16 ) )
= $true )
| ( ( X12 @ X16 )
!= $true )
| ( $true
!= ( X12 @ X17 ) ) )
& ( $true
!= ( X12 @ X13 ) )
& ! [X15: term] :
( ( $true
!= ( X12 @ X15 ) )
| ( ( X12 @ ( lam @ X15 ) )
= $true ) ) )
| ? [X11: subst] :
( ( comp @ X11 @ id )
!= X11 )
| ? [X9: subst,X10: term] :
( ( comp @ sh @ ( push @ X10 @ X9 ) )
!= X9 )
| ? [X7: term,X8: subst] :
( ( sub @ one @ ( push @ X7 @ X8 ) )
!= X7 ) )
<=> ( induction2lem_lthm = $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( axshiftcons = $true )
<=> ! [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
= X201 ) )
& ( ( ? [X484: term] :
( ( sub @ X484 @ id )
!= X484 )
| ! [X485: term > $o,X486: subst > term > subst > $o,X487: term] :
( ? [X497: term] :
( ( X485 @ X497 )
!= ( X486 @ id @ X497 @ id ) )
| ? [X492: subst,X493: term,X494: subst,X495: subst] :
( ( ( X486 @ X492 @ X493 @ ( comp @ X495 @ X494 ) )
!= $true )
& ( ( X486 @ ( comp @ X492 @ X495 ) @ ( sub @ X493 @ X495 ) @ X494 )
= $true ) )
| ? [X488: subst,X489: subst,X490: term,X491: subst] :
( ( ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X491 )
!= $true )
& ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X491 ) )
= $true ) )
| ? [X496: term] :
( ( ( X485 @ X496 )
= $true )
& ( ( X485 @ ( sub @ X487 @ ( push @ X496 @ id ) ) )
!= $true ) )
| ( ( X485 @ ( lam @ X487 ) )
= $true )
| ? [X498: subst > term > term] :
( ( ( X486 @ id @ ( lam @ ( X498 @ sh @ one ) ) @ id )
!= $true )
& ! [X499: term] :
( ( $true
!= ( X486 @ id @ X499 @ id ) )
| ( $true
= ( X486 @ id @ ( X498 @ id @ X499 ) @ id ) ) )
& ! [X500: subst,X501: subst,X502: term] :
( ( X498 @ ( comp @ X500 @ X501 ) @ ( sub @ X502 @ X501 ) )
= ( sub @ ( X498 @ X500 @ X502 ) @ X501 ) ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ! [X338: d_subst,X339: d_subst] :
( ( X338 = X339 )
| ( ( d2subst @ X338 )
!= ( d2subst @ X339 ) ) )
& ( ! [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) )
| ( $true
!= ( X472 @ X473 ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
<=> ( axvarid = $true ) )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X29: subst > term > subst > $o,X30: term,X31: term > $o] :
( ? [X46: term] :
( ( X29 @ id @ X46 @ id )
!= ( X31 @ X46 ) )
| ? [X42: subst,X43: subst,X44: term,X45: subst] :
( ( $true
!= ( X29 @ X42 @ X44 @ ( comp @ X43 @ X45 ) ) )
& ( ( X29 @ ( comp @ X42 @ X43 ) @ ( sub @ X44 @ X43 ) @ X45 )
= $true ) )
| ? [X32: subst,X33: subst,X34: term,X35: subst] :
( ( $true
= ( X29 @ X35 @ X34 @ ( comp @ X32 @ X33 ) ) )
& ( $true
!= ( X29 @ ( comp @ X35 @ X32 ) @ ( sub @ X34 @ X32 ) @ X33 ) ) )
| ? [X36: term] :
( ( $true
= ( X31 @ X36 ) )
& ( $true
!= ( X31 @ ( sub @ X30 @ ( push @ X36 @ id ) ) ) ) )
| ( ( X31 @ ( lam @ X30 ) )
= $true )
| ? [X37: subst > term > term] :
( ! [X38: subst,X39: subst,X40: term] :
( ( sub @ ( X37 @ X38 @ X40 ) @ X39 )
= ( X37 @ ( comp @ X38 @ X39 ) @ ( sub @ X40 @ X39 ) ) )
& ( ( X29 @ id @ ( lam @ ( X37 @ sh @ one ) ) @ id )
!= $true )
& ! [X41: term] :
( ( ( X29 @ id @ X41 @ id )
!= $true )
| ( $true
= ( X29 @ id @ ( X37 @ id @ X41 ) @ id ) ) ) ) ) )
& ( axmap = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ! [X68: subst > term > subst > $o,X69: term] :
( ? [X72: subst > term > term] :
( ! [X76: term] :
( ( $true
= ( X68 @ id @ ( X72 @ id @ X76 ) @ id ) )
| ( ( X68 @ id @ X76 @ id )
!= $true ) )
& ! [X73: subst,X74: subst,X75: term] :
( ( sub @ ( X72 @ X74 @ X75 ) @ X73 )
= ( X72 @ ( comp @ X74 @ X73 ) @ ( sub @ X75 @ X73 ) ) )
& ( ( X68 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X68 @ id @ X69 @ id ) )
| ? [X70: term,X71: term] :
( ( ( X68 @ id @ X70 @ id )
= $true )
& ( $true
= ( X68 @ id @ X71 @ id ) )
& ( $true
!= ( X68 @ id @ ( ap @ ( sub @ X71 @ id ) @ X70 ) @ id ) ) ) )
| ? [X58: subst > term > subst > $o,X59: term,X60: term > $o] :
( ! [X61: term] :
( ( ( X60 @ ( sub @ X59 @ ( push @ X61 @ id ) ) )
= $true )
| ( ( X60 @ X61 )
!= $true ) )
& ( $true
!= ( X60 @ ( lam @ X59 ) ) )
& ! [X62: term] :
( ( X58 @ id @ X62 @ id )
= ( X60 @ X62 ) )
& ! [X63: subst > term > term] :
( ? [X67: term] :
( ( ( X58 @ id @ ( X63 @ id @ X67 ) @ id )
!= $true )
& ( $true
= ( X58 @ id @ X67 @ id ) ) )
| ( ( X58 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true )
| ? [X64: subst,X65: term,X66: subst] :
( ( sub @ ( X63 @ X64 @ X65 ) @ X66 )
!= ( X63 @ ( comp @ X64 @ X66 ) @ ( sub @ X65 @ X66 ) ) ) ) )
| ? [X57: term] :
( ( sub @ X57 @ id )
!= X57 )
| ? [X50: term > $o,X51: term] :
( ! [X52: term,X53: term] :
( ( $true
!= ( X50 @ X53 ) )
| ( ( X50 @ X52 )
!= $true )
| ( $true
= ( X50 @ ( ap @ X53 @ X52 ) ) ) )
& ( ( X50 @ X51 )
!= $true )
& ! [X54: term] :
( ( ( X50 @ ( lam @ X54 ) )
= $true )
| ? [X55: term] :
( ( ( X50 @ ( sub @ X54 @ ( push @ X55 @ id ) ) )
!= $true )
& ( ( X50 @ X55 )
= $true ) ) )
& ! [X56: term] :
( ( $true
!= ( var @ X56 ) )
| ( $true
= ( X50 @ X56 ) ) ) )
| ? [X47: subst > term > subst > $o] :
! [X48: term > $o] :
? [X49: term] :
( ( X47 @ id @ X49 @ id )
!= ( X48 @ X49 ) ) ) )
& ( induction2lem_gthm = $true )
& ( ( apinj1 = $true )
<=> ! [X236: term,X237: term,X238: term,X239: term] :
( ( X236 = X239 )
| ( ( ap @ X236 @ X238 )
!= ( ap @ X239 @ X237 ) ) ) )
& ( ! [X122: term,X123: subst > term > subst > $o] :
( ? [X125: subst > term > term] :
( ! [X126: subst,X127: term,X128: subst] :
( ( X125 @ ( comp @ X128 @ X126 ) @ ( sub @ X127 @ X126 ) )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
& ! [X129: term] :
( ( ( X123 @ id @ X129 @ id )
!= $true )
| ( $true
= ( X123 @ id @ ( X125 @ id @ X129 ) @ id ) ) )
& ( $true
!= ( X123 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) ) )
| ? [X124: term] :
( ( $true
!= ( X123 @ id @ ( sub @ X122 @ ( push @ X124 @ id ) ) @ id ) )
& ( $true
= ( X123 @ id @ X124 @ id ) ) )
| ( ( X123 @ id @ ( lam @ X122 ) @ id )
= $true )
| ? [X134: subst,X135: subst,X136: term,X137: subst] :
( ( ( X123 @ ( comp @ X135 @ X137 ) @ ( sub @ X136 @ X137 ) @ X134 )
= $true )
& ( ( X123 @ X135 @ X136 @ ( comp @ X137 @ X134 ) )
!= $true ) )
| ? [X130: subst,X131: subst,X132: subst,X133: term] :
( ( ( X123 @ ( comp @ X131 @ X130 ) @ ( sub @ X133 @ X130 ) @ X132 )
!= $true )
& ( ( X123 @ X131 @ X133 @ ( comp @ X130 @ X132 ) )
= $true ) ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( axapp = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X340: subst > term > subst > $o,X341: term] :
( ? [X342: term] :
( ( $true
!= ( X340 @ id @ ( sub @ X341 @ ( push @ X342 @ id ) ) @ id ) )
& ( ( X340 @ id @ X342 @ id )
= $true ) )
| ? [X343: term,X344: subst > term > term] :
( ( ( X340 @ id @ ( lam @ ( X344 @ sh @ one ) ) @ id )
!= $true )
& ! [X345: subst,X346: subst,X347: term] :
( ( sub @ ( sub @ X343 @ ( push @ X347 @ X346 ) ) @ X345 )
= ( sub @ ( X344 @ X346 @ X347 ) @ X345 ) )
& ! [X349: subst,X350: subst,X351: term] :
( ( sub @ X343 @ ( push @ ( sub @ X351 @ X350 ) @ ( comp @ X349 @ X350 ) ) )
= ( X344 @ ( comp @ X349 @ X350 ) @ ( sub @ X351 @ X350 ) ) )
& ! [X348: term] :
( ( $true
= ( X340 @ id @ ( X344 @ id @ X348 ) @ id ) )
| ( $true
!= ( X340 @ id @ X348 @ id ) ) ) )
| ( $true
= ( X340 @ id @ ( lam @ ( sub @ X341 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( hoasapnotvar = $true )
& ( ( axscons = $true )
<=> ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 ) )
& ( ! [X83: subst > term > subst > $o,X84: term,X85: term > $o] :
( ( $true
= ( X85 @ ( lam @ X84 ) ) )
| ? [X88: subst > term > term] :
( ! [X92: term] :
( ( ( X83 @ id @ X92 @ id )
!= $true )
| ( ( X83 @ id @ ( X88 @ id @ X92 ) @ id )
= $true ) )
& ! [X89: subst,X90: subst,X91: term] :
( ( X88 @ ( comp @ X90 @ X89 ) @ ( sub @ X91 @ X89 ) )
= ( sub @ ( X88 @ X90 @ X91 ) @ X89 ) )
& ( $true
!= ( X83 @ id @ ( lam @ ( X88 @ sh @ one ) ) @ id ) ) )
| ? [X87: term] :
( ( X83 @ id @ X87 @ id )
!= ( X85 @ X87 ) )
| ? [X86: term] :
( ( $true
= ( X85 @ X86 ) )
& ( $true
!= ( X85 @ ( sub @ X84 @ ( push @ X86 @ id ) ) ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( ( pushprop_lem1v2_gthm = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X374: term] :
( ( sub @ X374 @ id )
= X374 )
=> ( ! [X375: term,X376: subst > term > subst > $o] :
( ~ ! [X382: term] :
( ( $true
!= ( X376 @ id @ X382 @ id ) )
| ( $true
= ( X376 @ id @ ( sub @ X375 @ ( push @ X382 @ id ) ) @ id ) ) )
| ( ( X376 @ id @ ( lam @ ( sub @ X375 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X377: subst > term > term] :
( ~ ! [X379: subst,X380: subst,X381: term] :
( ( X377 @ ( comp @ X379 @ X380 ) @ ( sub @ X381 @ X380 ) )
= ( sub @ ( X377 @ X379 @ X381 ) @ X380 ) )
| ( $true
= ( X376 @ id @ ( lam @ ( X377 @ sh @ one ) ) @ id ) )
| ~ ! [X378: term] :
( ( $true
!= ( X376 @ id @ X378 @ id ) )
| ( $true
= ( X376 @ id @ ( X377 @ id @ X378 ) @ id ) ) ) ) )
=> ! [X383: term,X384: subst > term > subst > $o] :
( ~ ! [X386: subst > term > term] :
( ( $true
= ( X384 @ id @ ( lam @ ( X386 @ sh @ one ) ) @ id ) )
| ~ ! [X387: term] :
( ( $true
= ( X384 @ id @ ( X386 @ id @ X387 ) @ id ) )
| ( ( X384 @ id @ X387 @ id )
!= $true ) )
| ~ ! [X388: subst,X389: subst,X390: term] :
( ( sub @ ( X386 @ X389 @ X390 ) @ X388 )
= ( X386 @ ( comp @ X389 @ X388 ) @ ( sub @ X390 @ X388 ) ) ) )
| ( $true
= ( X384 @ id @ ( lam @ X383 ) @ id ) )
| ~ ! [X385: term] :
( ( ( X384 @ id @ ( sub @ X383 @ ( push @ X385 @ id ) ) @ id )
= $true )
| ( $true
!= ( X384 @ id @ X385 @ id ) ) ) ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( hoasinduction = $true )
<=> ! [X211: term,X212: subst > term > subst > $o] :
( ~ ! [X222: subst > term > term] :
( ( $true
= ( X212 @ id @ ( lam @ ( X222 @ sh @ one ) ) @ id ) )
| ~ ! [X224: term,X225: subst,X226: subst] :
( ( X222 @ ( comp @ X226 @ X225 ) @ ( sub @ X224 @ X225 ) )
= ( sub @ ( X222 @ X226 @ X224 ) @ X225 ) )
| ~ ! [X223: term] :
( ( $true
= ( X212 @ id @ ( X222 @ id @ X223 ) @ id ) )
| ( ( X212 @ id @ X223 @ id )
!= $true ) ) )
| ~ ! [X227: term,X228: term] :
( ( ( X212 @ id @ X228 @ id )
!= $true )
| ( ( X212 @ id @ ( ap @ ( sub @ X227 @ id ) @ X228 ) @ id )
= $true )
| ( ( X212 @ id @ X227 @ id )
!= $true ) )
| ~ ! [X213: term,X214: subst,X215: subst,X216: subst] :
( ( ( X212 @ X214 @ X213 @ ( comp @ X216 @ X215 ) )
= $true )
| ( $true
!= ( X212 @ ( comp @ X214 @ X216 ) @ ( sub @ X213 @ X216 ) @ X215 ) ) )
| ~ ! [X221: term] :
( ( ( var @ ( sub @ X221 @ id ) )
!= $true )
| ( ( X212 @ id @ X221 @ id )
= $true ) )
| ( $true
= ( X212 @ id @ X211 @ id ) )
| ~ ! [X217: subst,X218: term,X219: subst,X220: subst] :
( ( ( X212 @ X219 @ X218 @ ( comp @ X217 @ X220 ) )
!= $true )
| ( ( X212 @ ( comp @ X219 @ X217 ) @ ( sub @ X218 @ X217 ) @ X220 )
= $true ) ) ) )
& ( hoasapinj2_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( axvarshift = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X400: term,X401: subst] :
( ( sub @ one @ ( push @ X400 @ X401 ) )
= X400 )
=> ( ! [X402: term,X403: subst] :
( ( comp @ sh @ ( push @ X402 @ X403 ) )
= X403 )
=> ( ! [X404: term,X405: term] :
( ( X404 = X405 )
| ( ( lam @ X405 )
!= ( lam @ X404 ) ) )
=> ! [X406: subst > term > term,X407: subst,X408: term,X409: subst > term > term] :
( ( ( lam @ ( X409 @ sh @ one ) )
!= ( lam @ ( X406 @ sh @ one ) ) )
| ~ ! [X413: term,X414: subst,X415: subst] :
( ( sub @ ( X409 @ X415 @ X413 ) @ X414 )
= ( X409 @ ( comp @ X415 @ X414 ) @ ( sub @ X413 @ X414 ) ) )
| ( ( X409 @ X407 @ X408 )
= ( X406 @ X407 @ X408 ) )
| ~ ! [X410: subst,X411: term,X412: subst] :
( ( sub @ ( X406 @ X410 @ X411 ) @ X412 )
= ( X406 @ ( comp @ X410 @ X412 ) @ ( sub @ X411 @ X412 ) ) ) ) ) ) ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X418: term,X419: term > $o] :
( ( $true
= ( X419 @ X418 ) )
| ~ ! [X423: term,X424: term] :
( ( $true
= ( X419 @ ( ap @ X424 @ X423 ) ) )
| ( ( X419 @ X423 )
!= $true )
| ( ( X419 @ X424 )
!= $true ) )
| ~ ! [X420: term] :
( ( ( X419 @ ( lam @ X420 ) )
= $true )
| ~ ! [X421: term] :
( ( $true
!= ( X419 @ X421 ) )
| ( $true
= ( X419 @ ( sub @ X420 @ ( push @ X421 @ id ) ) ) ) ) )
| ~ ! [X422: term] :
( ( $true
= ( X419 @ X422 ) )
| ( ( var @ X422 )
!= $true ) ) )
=> ( ! [X425: term,X426: term,X427: subst > term > subst > $o] :
( ( ( X427 @ id @ ( ap @ X426 @ X425 ) @ id )
= $true )
| ~ ! [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X427 @ ( comp @ X429 @ X430 ) @ ( sub @ X428 @ X430 ) @ X431 ) )
| ( ( X427 @ X429 @ X428 @ ( comp @ X430 @ X431 ) )
!= $true ) )
| ~ ! [X436: term,X437: term] :
( ( $true
= ( X427 @ id @ ( ap @ ( sub @ X437 @ id ) @ X436 ) @ id ) )
| ( $true
!= ( X427 @ id @ X437 @ id ) )
| ( ( X427 @ id @ X436 @ id )
!= $true ) )
| ( ( X427 @ id @ X425 @ id )
!= $true )
| ( $true
!= ( X427 @ id @ X426 @ id ) )
| ~ ! [X432: subst,X433: subst,X434: term,X435: subst] :
( ( ( X427 @ ( comp @ X432 @ X433 ) @ ( sub @ X434 @ X433 ) @ X435 )
!= $true )
| ( ( X427 @ X432 @ X434 @ ( comp @ X433 @ X435 ) )
= $true ) ) )
=> ( ! [X438: term,X439: subst > term > subst > $o] :
( ~ ! [X446: subst,X447: subst,X448: term,X449: subst] :
( ( ( X439 @ ( comp @ X449 @ X447 ) @ ( sub @ X448 @ X447 ) @ X446 )
= $true )
| ( ( X439 @ X449 @ X448 @ ( comp @ X447 @ X446 ) )
!= $true ) )
| ~ ! [X450: subst,X451: subst,X452: term,X453: subst] :
( ( $true
= ( X439 @ X453 @ X452 @ ( comp @ X450 @ X451 ) ) )
| ( ( X439 @ ( comp @ X453 @ X450 ) @ ( sub @ X452 @ X450 ) @ X451 )
!= $true ) )
| ~ ! [X445: term] :
( ( $true
!= ( X439 @ id @ X445 @ id ) )
| ( $true
= ( X439 @ id @ ( sub @ X438 @ ( push @ X445 @ id ) ) @ id ) ) )
| ~ ! [X440: subst > term > term] :
( ~ ! [X441: subst,X442: subst,X443: term] :
( ( sub @ ( X440 @ X442 @ X443 ) @ X441 )
= ( X440 @ ( comp @ X442 @ X441 ) @ ( sub @ X443 @ X441 ) ) )
| ~ ! [X444: term] :
( ( $true
= ( X439 @ id @ ( X440 @ id @ X444 ) @ id ) )
| ( ( X439 @ id @ X444 @ id )
!= $true ) )
| ( ( X439 @ id @ ( lam @ ( X440 @ sh @ one ) ) @ id )
= $true ) )
| ( $true
= ( X439 @ id @ ( lam @ X438 ) @ id ) ) )
=> ! [X454: term,X455: subst > term > subst > $o] :
( ~ ! [X471: term] :
( ( $true
!= ( var @ ( sub @ X471 @ id ) ) )
| ( ( X455 @ id @ X471 @ id )
= $true ) )
| ~ ! [X458: subst > term > term] :
( ~ ! [X460: subst,X461: subst,X462: term] :
( ( X458 @ ( comp @ X460 @ X461 ) @ ( sub @ X462 @ X461 ) )
= ( sub @ ( X458 @ X460 @ X462 ) @ X461 ) )
| ( ( X455 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X459: term] :
( ( ( X455 @ id @ X459 @ id )
!= $true )
| ( ( X455 @ id @ ( X458 @ id @ X459 ) @ id )
= $true ) ) )
| ~ ! [X463: subst,X464: term,X465: subst,X466: subst] :
( ( $true
!= ( X455 @ X466 @ X464 @ ( comp @ X465 @ X463 ) ) )
| ( ( X455 @ ( comp @ X466 @ X465 ) @ ( sub @ X464 @ X465 ) @ X463 )
= $true ) )
| ( $true
= ( X455 @ id @ X454 @ id ) )
| ~ ! [X456: term,X457: term] :
( ( $true
!= ( X455 @ id @ X457 @ id ) )
| ( $true
!= ( X455 @ id @ X456 @ id ) )
| ( ( X455 @ id @ ( ap @ ( sub @ X456 @ id ) @ X457 ) @ id )
= $true ) )
| ~ ! [X467: term,X468: subst,X469: subst,X470: subst] :
( ( $true
!= ( X455 @ ( comp @ X468 @ X469 ) @ ( sub @ X467 @ X469 ) @ X470 ) )
| ( ( X455 @ X468 @ X467 @ ( comp @ X469 @ X470 ) )
= $true ) ) ) ) ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X535: term > $o,X536: subst,X537: term] :
~ ! [X538: term > $o] :
~ ! [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
= ( X538 @ X539 ) ) )
& ( axclos = $true )
& ! [X229: d_term] : ( d_one = X229 )
& ( pushprop_lthm = $true )
& ( ( ! [X373: subst] :
( ( comp @ X373 @ id )
= X373 )
& ! [X372: subst] :
( ( comp @ id @ X372 )
= X372 ) )
<=> ( substmonoid = $true ) )
& ( ! [X324: subst] :
( ( comp @ id @ X324 )
= X324 )
<=> ( axidl = $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X162: term] :
( ( sub @ X162 @ id )
= X162 )
=> ( ! [X163: subst > term > subst > $o,X164: term] :
( ~ ! [X165: term] :
( ( $true
!= ( X163 @ id @ X165 @ id ) )
| ( $true
= ( X163 @ id @ ( sub @ X164 @ ( push @ X165 @ id ) ) @ id ) ) )
| ( ( X163 @ id @ ( lam @ ( sub @ X164 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X166: subst > term > term] :
( ( $true
= ( X163 @ id @ ( lam @ ( X166 @ sh @ one ) ) @ id ) )
| ~ ! [X167: subst,X168: subst,X169: term] :
( ( X166 @ ( comp @ X168 @ X167 ) @ ( sub @ X169 @ X167 ) )
= ( sub @ ( X166 @ X168 @ X169 ) @ X167 ) )
| ~ ! [X170: term] :
( ( ( X163 @ id @ ( X166 @ id @ X170 ) @ id )
= $true )
| ( $true
!= ( X163 @ id @ X170 @ id ) ) ) ) )
=> ! [X171: term,X172: subst > term > subst > $o] :
( ~ ! [X174: subst > term > term] :
( ( ( X172 @ id @ ( lam @ ( X174 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X175: term] :
( ( ( X172 @ id @ ( X174 @ id @ X175 ) @ id )
= $true )
| ( ( X172 @ id @ X175 @ id )
!= $true ) )
| ~ ! [X176: subst,X177: subst,X178: term] :
( ( sub @ ( X174 @ X176 @ X178 ) @ X177 )
= ( X174 @ ( comp @ X176 @ X177 ) @ ( sub @ X178 @ X177 ) ) ) )
| ~ ! [X173: term] :
( ( $true
= ( X172 @ id @ ( sub @ X171 @ ( push @ X173 @ id ) ) @ id ) )
| ( ( X172 @ id @ X173 @ id )
!= $true ) )
| ( $true
= ( X172 @ id @ ( lam @ X171 ) @ id ) )
| ~ ! [X183: subst,X184: subst,X185: term,X186: subst] :
( ( ( X172 @ X184 @ X185 @ ( comp @ X183 @ X186 ) )
= $true )
| ( $true
!= ( X172 @ ( comp @ X184 @ X183 ) @ ( sub @ X185 @ X183 ) @ X186 ) ) )
| ~ ! [X179: term,X180: subst,X181: subst,X182: subst] :
( ( ( X172 @ X182 @ X179 @ ( comp @ X180 @ X181 ) )
!= $true )
| ( $true
= ( X172 @ ( comp @ X182 @ X180 ) @ ( sub @ X179 @ X180 ) @ X181 ) ) ) ) ) ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X154: subst > term > subst > $o,X155: term] :
( ( $true
= ( X154 @ id @ ( lam @ ( sub @ X155 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X157: subst > term > term] :
( ( $true
= ( X154 @ id @ ( lam @ ( X157 @ sh @ one ) ) @ id ) )
| ~ ! [X158: term] :
( ( $true
= ( X154 @ id @ ( X157 @ id @ X158 ) @ id ) )
| ( $true
!= ( X154 @ id @ X158 @ id ) ) )
| ~ ! [X159: subst,X160: subst,X161: term] :
( ( X157 @ ( comp @ X159 @ X160 ) @ ( sub @ X161 @ X160 ) )
= ( sub @ ( X157 @ X159 @ X161 ) @ X160 ) ) )
| ~ ! [X156: term] :
( ( $true
= ( X154 @ id @ ( sub @ X155 @ ( push @ X156 @ id ) ) @ id ) )
| ( ( X154 @ id @ X156 @ id )
!= $true ) ) ) )
& ( ( apinj2 = $true )
<=> ! [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
!= ( ap @ X79 @ X82 ) )
| ( X81 = X82 ) ) )
& ( ! [X391: term] :
~ ! [X392: subst > term > term] :
~ ! [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
= ( X392 @ X393 @ X394 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( induction2 = $true )
<=> ! [X255: term,X256: term > $o] :
( ~ ! [X260: term] :
( ~ ! [X261: term] :
( ( $true
!= ( X256 @ X261 ) )
| ( $true
= ( X256 @ ( sub @ X260 @ ( push @ X261 @ id ) ) ) ) )
| ( $true
= ( X256 @ ( lam @ X260 ) ) ) )
| ~ ! [X257: term,X258: term] :
( ( ( X256 @ X258 )
!= $true )
| ( $true
!= ( X256 @ X257 ) )
| ( ( X256 @ ( ap @ X257 @ X258 ) )
= $true ) )
| ~ ! [X259: term] :
( ( $true
= ( X256 @ X259 ) )
| ( ( var @ X259 )
!= $true ) )
| ( ( X256 @ X255 )
= $true ) ) )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X202: subst > term > subst > $o,X203: term] :
( ( ( X202 @ id @ X203 @ id )
= $true )
| ~ ! [X204: term,X205: term] :
( ( ( X202 @ id @ ( ap @ ( sub @ X204 @ id ) @ X205 ) @ id )
= $true )
| ( $true
!= ( X202 @ id @ X204 @ id ) )
| ( $true
!= ( X202 @ id @ X205 @ id ) ) )
| ~ ! [X206: subst > term > term] :
( ~ ! [X207: term] :
( ( ( X202 @ id @ X207 @ id )
!= $true )
| ( $true
= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
| ~ ! [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
| ( $true
= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) ) ) )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X138: term] :
~ ! [X139: subst > term > term] :
~ ! [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
= ( X139 @ X141 @ X140 ) ) )
& ( ( ! [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
= X478 )
=> ! [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ~ ! [X483: term] :
( ( X480 @ X483 )
= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) )
| ( $true
= ( X480 @ one ) )
| ( ( X479 @ X481 )
!= $true ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X526: term] :
( ( sub @ X526 @ id )
= X526 )
=> ( ! [X527: term,X528: term,X529: term,X530: term] :
( ( ( ap @ X528 @ X529 )
!= ( ap @ X527 @ X530 ) )
| ( X527 = X528 ) )
=> ! [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
!= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
| ( X533 = X534 ) ) ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
= X114 )
=> ( ! [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
= X116 )
=> ! [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
| ( $true
!= ( X120 @ X121 ) ) ) ) ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( induction2lem = $true )
<=> ! [X325: term > $o,X326: subst,X327: term] :
( ~ ! [X328: term,X329: term] :
( ( $true
!= ( X325 @ X328 ) )
| ( $true
= ( X325 @ ( ap @ X329 @ X328 ) ) )
| ( $true
!= ( X325 @ X329 ) ) )
| ( $true
= ( X325 @ ( sub @ X327 @ X326 ) ) )
| ~ ! [X330: term] :
( ( $true
!= ( var @ X330 ) )
| ( $true
= ( X325 @ ( sub @ X330 @ X326 ) ) ) )
| ~ ! [X331: term] :
( ~ ! [X332: term] :
( ( $true
!= ( X325 @ X332 ) )
| ( $true
= ( X325 @ ( sub @ X331 @ ( push @ X332 @ id ) ) ) ) )
| ( ( X325 @ ( lam @ X331 ) )
= $true ) ) ) )
& ( ( laminj = $true )
<=> ! [X366: term,X367: term] :
( ( X366 = X367 )
| ( ( lam @ X366 )
!= ( lam @ X367 ) ) ) )
& ( hoaslamnotap_lthm = $true )
& ( apnotvar = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X110: subst > term > subst > $o] :
~ ! [X111: term > $o] :
~ ! [X112: term] :
( ( X110 @ id @ X112 @ id )
= ( X111 @ X112 ) ) )
& ( ulamvarsh = $true )
& ! [X368: d_term,X369: d_term] :
( ( ( d2term @ X368 )
= ( d2term @ X369 ) )
=> ( X368 = X369 ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( induction2_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X352: term,X353: term,X354: term,X355: term] :
( ( X353 = X354 )
| ( ( ap @ ( sub @ X352 @ id ) @ X353 )
!= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( one
= ( d2term @ d_one ) )
& ( ! [X142: term] :
~ ! [X143: subst > term > term] :
( ( X143 @ sh @ one )
!= ( sub @ X142 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( ( ! [X358: term,X359: term,X360: term,X361: term] :
( ( ( ap @ X360 @ X361 )
!= ( ap @ X359 @ X358 ) )
| ( X358 = X361 ) )
=> ! [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
| ( X363 = X365 ) ) )
<=> ( hoasapinj2_lthm = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasapinj1_gthm = $true )
& ( axassoc = $true )
& ( hoaslaminj_gthm = $true )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X262: subst > term > subst > $o,X263: term] :
( ~ ! [X264: term] :
( ( ( X262 @ id @ X264 @ id )
!= $true )
| ( ( X262 @ id @ ( sub @ X263 @ ( push @ X264 @ id ) ) @ id )
= $true ) )
| ~ ! [X265: subst > term > term] :
( ~ ! [X267: subst,X268: subst,X269: term] :
( ( sub @ ( X265 @ X268 @ X269 ) @ X267 )
= ( X265 @ ( comp @ X268 @ X267 ) @ ( sub @ X269 @ X267 ) ) )
| ( $true
= ( X262 @ id @ ( lam @ ( X265 @ sh @ one ) ) @ id ) )
| ~ ! [X266: term] :
( ( $true
= ( X262 @ id @ ( X265 @ id @ X266 ) @ id ) )
| ( ( X262 @ id @ X266 @ id )
!= $true ) ) )
| ( ( X262 @ id @ ( lam @ X263 ) @ id )
= $true ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_lthm = $true )
& ( lamnotap != $true )
& ( axabs = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ulamvarind = $true )
& ( ! [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
= X371 )
<=> ( axvarcons = $true ) )
& ( ( axidr = $true )
<=> ! [X416: subst] :
( ( comp @ X416 @ id )
= X416 ) )
& ( ( d2subst @ d_id )
= id )
& ( ulamvar1 != $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
!= ( sub @ X77 @ sh ) )
| ( X77 = X78 ) )
<=> ( shinj = $true ) )
& ( ! [X270: term,X271: term > $o,X272: subst > term > subst > $o,X273: term] :
( ~ ! [X274: term,X275: subst,X276: subst,X277: subst] :
( ( $true
!= ( X272 @ X277 @ X274 @ ( comp @ X276 @ X275 ) ) )
| ( $true
= ( X272 @ ( comp @ X277 @ X276 ) @ ( sub @ X274 @ X276 ) @ X275 ) ) )
| ( $true
!= ( X271 @ X270 ) )
| ~ ! [X284: term] :
( ( X272 @ id @ X284 @ id )
= ( X271 @ X284 ) )
| ( $true
= ( X271 @ ( ap @ X273 @ X270 ) ) )
| ~ ! [X280: subst,X281: subst,X282: term,X283: subst] :
( ( $true
= ( X272 @ X281 @ X282 @ ( comp @ X283 @ X280 ) ) )
| ( ( X272 @ ( comp @ X281 @ X283 ) @ ( sub @ X282 @ X283 ) @ X280 )
!= $true ) )
| ~ ! [X278: term,X279: term] :
( ( ( X272 @ id @ X279 @ id )
!= $true )
| ( $true
= ( X272 @ id @ ( ap @ ( sub @ X279 @ id ) @ X278 ) @ id ) )
| ( ( X272 @ id @ X278 @ id )
!= $true ) )
| ( $true
!= ( X271 @ X273 ) ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( termmset_gthm = $true )
& ( ( ! [X503: subst] :
( ( comp @ id @ X503 )
= X503 )
=> ( ! [X504: subst] :
( ( comp @ X504 @ id )
= X504 )
=> ( ! [X506: subst] :
( ( comp @ id @ X506 )
= X506 )
& ! [X505: subst] :
( ( comp @ X505 @ id )
= X505 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X417: d_subst] : ( d_id = X417 )
& ( pushprop_lem2v2 = $true )
& ( ( hoaslamnotap = $true )
<=> ! [X520: term,X521: term,X522: subst > term > term] :
( ~ ! [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) )
| ( ( ap @ ( sub @ X521 @ id ) @ X520 )
!= ( lam @ ( X522 @ sh @ one ) ) ) ) )
& ( ! [X230: term > $o,X231: term] :
( ~ ! [X233: term,X234: term] :
( ( $true
!= ( X230 @ X233 ) )
| ( $true
= ( X230 @ ( ap @ X233 @ X234 ) ) )
| ( $true
!= ( X230 @ X234 ) ) )
| ( ( X230 @ X231 )
= $true )
| ~ ! [X235: term] :
( ( $true
!= ( X230 @ X235 ) )
| ( $true
= ( X230 @ ( lam @ X235 ) ) ) )
| ~ ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( ( var @ X232 )
!= $true ) ) )
<=> ( induction = $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X1: term] :
~ ! [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
!= ( X2 @ sh @ one ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X512: subst > term > subst > $o,X513: term] :
( ~ ! [X515: subst > term > term] :
( ~ ! [X516: subst,X517: term,X518: subst] :
( ( X515 @ ( comp @ X518 @ X516 ) @ ( sub @ X517 @ X516 ) )
= ( sub @ ( X515 @ X518 @ X517 ) @ X516 ) )
| ( ( X512 @ id @ ( lam @ ( X515 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X519: term] :
( ( ( X512 @ id @ X519 @ id )
!= $true )
| ( ( X512 @ id @ ( X515 @ id @ X519 ) @ id )
= $true ) ) )
| ( $true
= ( X512 @ id @ ( lam @ ( sub @ X513 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X514: term] :
( ( ( X512 @ id @ ( sub @ X513 @ ( push @ X514 @ id ) ) @ id )
= $true )
| ( ( X512 @ id @ X514 @ id )
!= $true ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( ! [X285: subst > term > subst > $o] :
~ ! [X286: term > $o] :
~ ! [X287: term] :
( ( X285 @ id @ X287 @ id )
= ( X286 @ X287 ) )
=> ( ! [X288: term,X289: term > $o] :
( ~ ! [X290: term] :
( ( ( var @ X290 )
!= $true )
| ( $true
= ( X289 @ X290 ) ) )
| ~ ! [X291: term] :
( ( ( X289 @ ( lam @ X291 ) )
= $true )
| ~ ! [X292: term] :
( ( ( X289 @ X292 )
!= $true )
| ( $true
= ( X289 @ ( sub @ X291 @ ( push @ X292 @ id ) ) ) ) ) )
| ( ( X289 @ X288 )
= $true )
| ~ ! [X293: term,X294: term] :
( ( ( X289 @ X293 )
!= $true )
| ( $true
= ( X289 @ ( ap @ X293 @ X294 ) ) )
| ( ( X289 @ X294 )
!= $true ) ) )
=> ( ! [X295: term] :
( ( sub @ X295 @ id )
= X295 )
=> ( ! [X296: term,X297: term > $o,X298: subst > term > subst > $o] :
( ~ ! [X299: term] :
( ( X297 @ X299 )
= ( X298 @ id @ X299 @ id ) )
| ~ ! [X305: term] :
( ( ( X297 @ X305 )
!= $true )
| ( ( X297 @ ( sub @ X296 @ ( push @ X305 @ id ) ) )
= $true ) )
| ( ( X297 @ ( lam @ X296 ) )
= $true )
| ~ ! [X300: subst > term > term] :
( ~ ! [X301: subst,X302: term,X303: subst] :
( ( sub @ ( X300 @ X303 @ X302 ) @ X301 )
= ( X300 @ ( comp @ X303 @ X301 ) @ ( sub @ X302 @ X301 ) ) )
| ( $true
= ( X298 @ id @ ( lam @ ( X300 @ sh @ one ) ) @ id ) )
| ~ ! [X304: term] :
( ( $true
!= ( X298 @ id @ X304 @ id ) )
| ( ( X298 @ id @ ( X300 @ id @ X304 ) @ id )
= $true ) ) ) )
=> ! [X306: subst > term > subst > $o,X307: term] :
( ~ ! [X308: term] :
( ( ( var @ ( sub @ X308 @ id ) )
!= $true )
| ( ( X306 @ id @ X308 @ id )
= $true ) )
| ~ ! [X314: subst,X315: subst,X316: term,X317: subst] :
( ( ( X306 @ ( comp @ X314 @ X315 ) @ ( sub @ X316 @ X315 ) @ X317 )
!= $true )
| ( ( X306 @ X314 @ X316 @ ( comp @ X315 @ X317 ) )
= $true ) )
| ( ( X306 @ id @ X307 @ id )
= $true )
| ~ ! [X309: subst > term > term] :
( ~ ! [X311: subst,X312: term,X313: subst] :
( ( sub @ ( X309 @ X311 @ X312 ) @ X313 )
= ( X309 @ ( comp @ X311 @ X313 ) @ ( sub @ X312 @ X313 ) ) )
| ( $true
= ( X306 @ id @ ( lam @ ( X309 @ sh @ one ) ) @ id ) )
| ~ ! [X310: term] :
( ( $true
= ( X306 @ id @ ( X309 @ id @ X310 ) @ id ) )
| ( $true
!= ( X306 @ id @ X310 @ id ) ) ) )
| ~ ! [X320: term,X321: subst,X322: subst,X323: subst] :
( ( $true
= ( X306 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) @ X323 ) )
| ( $true
!= ( X306 @ X321 @ X320 @ ( comp @ X322 @ X323 ) ) ) )
| ~ ! [X318: term,X319: term] :
( ( $true
!= ( X306 @ id @ X319 @ id ) )
| ( ( X306 @ id @ X318 @ id )
!= $true )
| ( ( X306 @ id @ ( ap @ ( sub @ X318 @ id ) @ X319 ) @ id )
= $true ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ! [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
!= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
| ( X507 = X508 ) )
<=> ( hoasapinj1 = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& ( ( pushprop_lem0 = $true )
<=> ! [X333: term,X334: term > $o,X335: subst] :
~ ! [X336: term > $o] :
~ ! [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
= ( X336 @ X337 ) ) )
& ( ! [X187: term,X188: subst > term > subst > $o,X189: term] :
( ( $true
!= ( X188 @ id @ X189 @ id ) )
| ( $true
= ( X188 @ id @ ( ap @ X189 @ X187 ) @ id ) )
| ( $true
!= ( X188 @ id @ X187 @ id ) )
| ~ ! [X198: term,X199: term] :
( ( $true
!= ( X188 @ id @ X199 @ id ) )
| ( $true
= ( X188 @ id @ ( ap @ ( sub @ X199 @ id ) @ X198 ) @ id ) )
| ( ( X188 @ id @ X198 @ id )
!= $true ) )
| ~ ! [X194: subst,X195: subst,X196: term,X197: subst] :
( ( $true
!= ( X188 @ X194 @ X196 @ ( comp @ X197 @ X195 ) ) )
| ( ( X188 @ ( comp @ X194 @ X197 ) @ ( sub @ X196 @ X197 ) @ X195 )
= $true ) )
| ~ ! [X190: subst,X191: subst,X192: term,X193: subst] :
( ( ( X188 @ X191 @ X192 @ ( comp @ X193 @ X190 ) )
= $true )
| ( ( X188 @ ( comp @ X191 @ X193 ) @ ( sub @ X192 @ X193 ) @ X190 )
!= $true ) ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X240: term] :
~ ! [X241: subst > term > term] :
~ ! [X242: term,X243: subst] :
( ( X241 @ X243 @ X242 )
= ( sub @ X240 @ ( push @ X242 @ X243 ) ) )
=> ( ! [X244: term] :
( ( sub @ X244 @ id )
= X244 )
=> ! [X245: term,X246: subst > term > subst > $o,X247: term > $o] :
( ~ ! [X248: subst > term > term] :
( ~ ! [X252: term] :
( ( $true
!= ( X246 @ id @ X252 @ id ) )
| ( $true
= ( X246 @ id @ ( X248 @ id @ X252 ) @ id ) ) )
| ~ ! [X249: term,X250: subst,X251: subst] :
( ( sub @ ( X248 @ X251 @ X249 ) @ X250 )
= ( X248 @ ( comp @ X251 @ X250 ) @ ( sub @ X249 @ X250 ) ) )
| ( ( X246 @ id @ ( lam @ ( X248 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X254: term] :
( ( ( X247 @ X254 )
!= $true )
| ( ( X247 @ ( sub @ X245 @ ( push @ X254 @ id ) ) )
= $true ) )
| ~ ! [X253: term] :
( ( X247 @ X253 )
= ( X246 @ id @ X253 @ id ) )
| ( $true
= ( X247 @ ( lam @ X245 ) ) ) ) ) ) )
& ( hoaslamnotvar = $true )
& ( ( termmset_lthm = $true )
<=> ( ! [X5: term] :
( ( sub @ X5 @ id )
= X5 )
=> ! [X6: term] :
( ( sub @ X6 @ id )
= X6 ) ) )
& ! [X356: subst] :
? [X357: d_subst] :
( ( d2subst @ X357 )
= X356 )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ~ ! [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) )
| ~ ! [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
| ( ( X144 @ X146 @ X147 )
= ( X145 @ X146 @ X147 ) )
| ( ( lam @ ( X144 @ sh @ one ) )
!= ( lam @ ( X145 @ sh @ one ) ) ) )
<=> ( hoaslaminj = $true ) )
& ( pushprop = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslamnotvar_lthm = $true )
& ( ( induction2_lthm = $true )
<=> ( ! [X93: term] :
( ( sub @ X93 @ id )
= X93 )
=> ( ! [X94: subst,X95: term > $o,X96: term] :
( ( ( X95 @ ( sub @ X96 @ X94 ) )
= $true )
| ~ ! [X97: term,X98: term] :
( ( $true
= ( X95 @ ( ap @ X97 @ X98 ) ) )
| ( ( X95 @ X98 )
!= $true )
| ( $true
!= ( X95 @ X97 ) ) )
| ~ ! [X100: term] :
( ~ ! [X101: term] :
( ( ( X95 @ ( sub @ X100 @ ( push @ X101 @ id ) ) )
= $true )
| ( ( X95 @ X101 )
!= $true ) )
| ( ( X95 @ ( lam @ X100 ) )
= $true ) )
| ~ ! [X99: term] :
( ( ( var @ X99 )
!= $true )
| ( $true
= ( X95 @ ( sub @ X99 @ X94 ) ) ) ) )
=> ! [X102: term > $o,X103: term] :
( ~ ! [X108: term] :
( ( ( X102 @ X108 )
= $true )
| ( $true
!= ( var @ X108 ) ) )
| ( ( X102 @ X103 )
= $true )
| ~ ! [X106: term] :
( ( ( X102 @ ( lam @ X106 ) )
= $true )
| ~ ! [X107: term] :
( ( $true
= ( X102 @ ( sub @ X106 @ ( push @ X107 @ id ) ) ) )
| ( ( X102 @ X107 )
!= $true ) ) )
| ~ ! [X104: term,X105: term] :
( ( ( X102 @ ( ap @ X104 @ X105 ) )
= $true )
| ( ( X102 @ X105 )
!= $true )
| ( ( X102 @ X104 )
!= $true ) ) ) ) ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ! [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
!= ( X395 @ X397 ) )
| ( $true
= ( X396 @ one ) )
| ~ ! [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
= ( X396 @ X399 ) ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( lamnotvar = $true )
& ( ! [X109: term] :
( ( sub @ X109 @ id )
= X109 )
<=> ( termmset = $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X26: subst > term > subst > $o] :
~ ! [X27: term > $o] :
~ ! [X28: term] :
( ( X27 @ X28 )
= ( X26 @ id @ X28 @ id ) ) )
& ( pushprop_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( hoasinduction_lem1 = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( ! [X7: term,X8: subst] :
( ( sub @ one @ ( push @ X7 @ X8 ) )
= X7 )
=> ( ! [X9: subst,X10: term] :
( ( comp @ sh @ ( push @ X10 @ X9 ) )
= X9 )
=> ( ! [X11: subst] :
( ( comp @ X11 @ id )
= X11 )
=> ( ! [X12: term > $o,X13: term] :
( ~ ! [X16: term,X17: term] :
( ( ( X12 @ ( ap @ X17 @ X16 ) )
= $true )
| ( ( X12 @ X16 )
!= $true )
| ( $true
!= ( X12 @ X17 ) ) )
| ~ ! [X15: term] :
( ( $true
!= ( X12 @ X15 ) )
| ( ( X12 @ ( lam @ X15 ) )
= $true ) )
| ( $true
= ( X12 @ X13 ) )
| ~ ! [X14: term] :
( ( $true
!= ( var @ X14 ) )
| ( $true
= ( X12 @ X14 ) ) ) )
=> ! [X18: subst,X19: term,X20: term > $o] :
( ( $true
= ( X20 @ ( sub @ X19 @ X18 ) ) )
| ~ ! [X21: term] :
( ( $true
= ( X20 @ ( sub @ X21 @ X18 ) ) )
| ( ( var @ X21 )
!= $true ) )
| ~ ! [X24: term] :
( ~ ! [X25: term] :
( ( ( X20 @ X25 )
!= $true )
| ( ( X20 @ ( sub @ X24 @ ( push @ X25 @ id ) ) )
= $true ) )
| ( ( X20 @ ( lam @ X24 ) )
= $true ) )
| ~ ! [X22: term,X23: term] :
( ( $true
= ( X20 @ ( ap @ X23 @ X22 ) ) )
| ( $true
!= ( X20 @ X23 ) )
| ( ( X20 @ X22 )
!= $true ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( axshiftcons = $true )
<=> ! [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
= X201 ) )
& ( ( ! [X484: term] :
( ( sub @ X484 @ id )
= X484 )
=> ! [X485: term > $o,X486: subst > term > subst > $o,X487: term] :
( ~ ! [X498: subst > term > term] :
( ~ ! [X500: subst,X501: subst,X502: term] :
( ( X498 @ ( comp @ X500 @ X501 ) @ ( sub @ X502 @ X501 ) )
= ( sub @ ( X498 @ X500 @ X502 ) @ X501 ) )
| ~ ! [X499: term] :
( ( $true
!= ( X486 @ id @ X499 @ id ) )
| ( $true
= ( X486 @ id @ ( X498 @ id @ X499 ) @ id ) ) )
| ( ( X486 @ id @ ( lam @ ( X498 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X488: subst,X489: subst,X490: term,X491: subst] :
( ( ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X491 )
= $true )
| ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X491 ) )
!= $true ) )
| ~ ! [X492: subst,X493: term,X494: subst,X495: subst] :
( ( ( X486 @ ( comp @ X492 @ X495 ) @ ( sub @ X493 @ X495 ) @ X494 )
!= $true )
| ( ( X486 @ X492 @ X493 @ ( comp @ X495 @ X494 ) )
= $true ) )
| ( ( X485 @ ( lam @ X487 ) )
= $true )
| ~ ! [X496: term] :
( ( ( X485 @ ( sub @ X487 @ ( push @ X496 @ id ) ) )
= $true )
| ( ( X485 @ X496 )
!= $true ) )
| ~ ! [X497: term] :
( ( X485 @ X497 )
= ( X486 @ id @ X497 @ id ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ! [X338: d_subst,X339: d_subst] :
( ( ( d2subst @ X338 )
= ( d2subst @ X339 ) )
=> ( X338 = X339 ) )
& ( ! [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) )
| ( $true
!= ( X472 @ X473 ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
<=> ( axvarid = $true ) )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X29: subst > term > subst > $o,X30: term,X31: term > $o] :
( ( ( X31 @ ( lam @ X30 ) )
= $true )
| ~ ! [X46: term] :
( ( X29 @ id @ X46 @ id )
= ( X31 @ X46 ) )
| ~ ! [X36: term] :
( ( $true
!= ( X31 @ X36 ) )
| ( $true
= ( X31 @ ( sub @ X30 @ ( push @ X36 @ id ) ) ) ) )
| ~ ! [X42: subst,X43: subst,X44: term,X45: subst] :
( ( ( X29 @ ( comp @ X42 @ X43 ) @ ( sub @ X44 @ X43 ) @ X45 )
!= $true )
| ( $true
= ( X29 @ X42 @ X44 @ ( comp @ X43 @ X45 ) ) ) )
| ~ ! [X32: subst,X33: subst,X34: term,X35: subst] :
( ( $true
= ( X29 @ ( comp @ X35 @ X32 ) @ ( sub @ X34 @ X32 ) @ X33 ) )
| ( $true
!= ( X29 @ X35 @ X34 @ ( comp @ X32 @ X33 ) ) ) )
| ~ ! [X37: subst > term > term] :
( ~ ! [X41: term] :
( ( ( X29 @ id @ X41 @ id )
!= $true )
| ( $true
= ( X29 @ id @ ( X37 @ id @ X41 ) @ id ) ) )
| ( ( X29 @ id @ ( lam @ ( X37 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X38: subst,X39: subst,X40: term] :
( ( sub @ ( X37 @ X38 @ X40 ) @ X39 )
= ( X37 @ ( comp @ X38 @ X39 ) @ ( sub @ X40 @ X39 ) ) ) ) ) )
& ( axmap = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ! [X47: subst > term > subst > $o] :
~ ! [X48: term > $o] :
~ ! [X49: term] :
( ( X47 @ id @ X49 @ id )
= ( X48 @ X49 ) )
=> ( ! [X50: term > $o,X51: term] :
( ~ ! [X52: term,X53: term] :
( ( $true
!= ( X50 @ X53 ) )
| ( ( X50 @ X52 )
!= $true )
| ( $true
= ( X50 @ ( ap @ X53 @ X52 ) ) ) )
| ( ( X50 @ X51 )
= $true )
| ~ ! [X56: term] :
( ( $true
!= ( var @ X56 ) )
| ( $true
= ( X50 @ X56 ) ) )
| ~ ! [X54: term] :
( ( ( X50 @ ( lam @ X54 ) )
= $true )
| ~ ! [X55: term] :
( ( ( X50 @ X55 )
!= $true )
| ( ( X50 @ ( sub @ X54 @ ( push @ X55 @ id ) ) )
= $true ) ) ) )
=> ( ! [X57: term] :
( ( sub @ X57 @ id )
= X57 )
=> ( ! [X58: subst > term > subst > $o,X59: term,X60: term > $o] :
( ( $true
= ( X60 @ ( lam @ X59 ) ) )
| ~ ! [X63: subst > term > term] :
( ( ( X58 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X67: term] :
( ( $true
!= ( X58 @ id @ X67 @ id ) )
| ( ( X58 @ id @ ( X63 @ id @ X67 ) @ id )
= $true ) )
| ~ ! [X64: subst,X65: term,X66: subst] :
( ( sub @ ( X63 @ X64 @ X65 ) @ X66 )
= ( X63 @ ( comp @ X64 @ X66 ) @ ( sub @ X65 @ X66 ) ) ) )
| ~ ! [X61: term] :
( ( ( X60 @ ( sub @ X59 @ ( push @ X61 @ id ) ) )
= $true )
| ( ( X60 @ X61 )
!= $true ) )
| ~ ! [X62: term] :
( ( X58 @ id @ X62 @ id )
= ( X60 @ X62 ) ) )
=> ! [X68: subst > term > subst > $o,X69: term] :
( ~ ! [X70: term,X71: term] :
( ( $true
!= ( X68 @ id @ X71 @ id ) )
| ( $true
= ( X68 @ id @ ( ap @ ( sub @ X71 @ id ) @ X70 ) @ id ) )
| ( ( X68 @ id @ X70 @ id )
!= $true ) )
| ~ ! [X72: subst > term > term] :
( ( ( X68 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X73: subst,X74: subst,X75: term] :
( ( sub @ ( X72 @ X74 @ X75 ) @ X73 )
= ( X72 @ ( comp @ X74 @ X73 ) @ ( sub @ X75 @ X73 ) ) )
| ~ ! [X76: term] :
( ( $true
= ( X68 @ id @ ( X72 @ id @ X76 ) @ id ) )
| ( ( X68 @ id @ X76 @ id )
!= $true ) ) )
| ( $true
= ( X68 @ id @ X69 @ id ) ) ) ) ) ) ) )
& ( induction2lem_gthm = $true )
& ( ( apinj1 = $true )
<=> ! [X236: term,X237: term,X238: term,X239: term] :
( ( X236 = X239 )
| ( ( ap @ X236 @ X238 )
!= ( ap @ X239 @ X237 ) ) ) )
& ( ! [X122: term,X123: subst > term > subst > $o] :
( ~ ! [X125: subst > term > term] :
( ( $true
= ( X123 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) )
| ~ ! [X126: subst,X127: term,X128: subst] :
( ( X125 @ ( comp @ X128 @ X126 ) @ ( sub @ X127 @ X126 ) )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
| ~ ! [X129: term] :
( ( ( X123 @ id @ X129 @ id )
!= $true )
| ( $true
= ( X123 @ id @ ( X125 @ id @ X129 ) @ id ) ) ) )
| ~ ! [X130: subst,X131: subst,X132: subst,X133: term] :
( ( ( X123 @ ( comp @ X131 @ X130 ) @ ( sub @ X133 @ X130 ) @ X132 )
= $true )
| ( ( X123 @ X131 @ X133 @ ( comp @ X130 @ X132 ) )
!= $true ) )
| ~ ! [X134: subst,X135: subst,X136: term,X137: subst] :
( ( ( X123 @ X135 @ X136 @ ( comp @ X137 @ X134 ) )
= $true )
| ( ( X123 @ ( comp @ X135 @ X137 ) @ ( sub @ X136 @ X137 ) @ X134 )
!= $true ) )
| ~ ! [X124: term] :
( ( $true
!= ( X123 @ id @ X124 @ id ) )
| ( $true
= ( X123 @ id @ ( sub @ X122 @ ( push @ X124 @ id ) ) @ id ) ) )
| ( ( X123 @ id @ ( lam @ X122 ) @ id )
= $true ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( axapp = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( ! [X340: subst > term > subst > $o,X341: term] :
( ( $true
= ( X340 @ id @ ( lam @ ( sub @ X341 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X342: term] :
( ( ( X340 @ id @ X342 @ id )
!= $true )
| ( $true
= ( X340 @ id @ ( sub @ X341 @ ( push @ X342 @ id ) ) @ id ) ) )
| ~ ! [X343: term,X344: subst > term > term] :
( ~ ! [X349: subst,X350: subst,X351: term] :
( ( sub @ X343 @ ( push @ ( sub @ X351 @ X350 ) @ ( comp @ X349 @ X350 ) ) )
= ( X344 @ ( comp @ X349 @ X350 ) @ ( sub @ X351 @ X350 ) ) )
| ( ( X340 @ id @ ( lam @ ( X344 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X345: subst,X346: subst,X347: term] :
( ( sub @ ( sub @ X343 @ ( push @ X347 @ X346 ) ) @ X345 )
= ( sub @ ( X344 @ X346 @ X347 ) @ X345 ) )
| ~ ! [X348: term] :
( ( $true
= ( X340 @ id @ ( X344 @ id @ X348 ) @ id ) )
| ( $true
!= ( X340 @ id @ X348 @ id ) ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( hoasapnotvar = $true )
& ( ( axscons = $true )
<=> ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 ) )
& ( ! [X83: subst > term > subst > $o,X84: term,X85: term > $o] :
( ~ ! [X88: subst > term > term] :
( ~ ! [X92: term] :
( ( ( X83 @ id @ X92 @ id )
!= $true )
| ( ( X83 @ id @ ( X88 @ id @ X92 ) @ id )
= $true ) )
| ( $true
= ( X83 @ id @ ( lam @ ( X88 @ sh @ one ) ) @ id ) )
| ~ ! [X89: subst,X90: subst,X91: term] :
( ( X88 @ ( comp @ X90 @ X89 ) @ ( sub @ X91 @ X89 ) )
= ( sub @ ( X88 @ X90 @ X91 ) @ X89 ) ) )
| ~ ! [X87: term] :
( ( X83 @ id @ X87 @ id )
= ( X85 @ X87 ) )
| ~ ! [X86: term] :
( ( $true
!= ( X85 @ X86 ) )
| ( $true
= ( X85 @ ( sub @ X84 @ ( push @ X86 @ id ) ) ) ) )
| ( $true
= ( X85 @ ( lam @ X84 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) ) ),
inference(flattening,[status(thm)],[f10]) ).
thf(f10,plain,
( ( ( axscons = $true )
<=> ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 ) )
& ( axabs = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X1: term] :
~ ! [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
!= ( X2 @ sh @ one ) ) )
& ( axclos = $true )
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( termmset_lthm = $true )
<=> ( ! [X5: term] :
( ( sub @ X5 @ id )
= X5 )
=> ! [X6: term] :
( ( sub @ X6 @ id )
= X6 ) ) )
& ( axvarshift = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( ( ! [X7: term,X8: subst] :
( ( sub @ one @ ( push @ X7 @ X8 ) )
= X7 )
=> ( ! [X9: subst,X10: term] :
( ( comp @ sh @ ( push @ X10 @ X9 ) )
= X9 )
=> ( ! [X11: subst] :
( ( comp @ X11 @ id )
= X11 )
=> ( ! [X12: term > $o,X13: term] :
( ~ ! [X14: term] :
( ( $true
= ( X12 @ X14 ) )
| ( $true
!= ( var @ X14 ) ) )
| ( $true
= ( X12 @ X13 ) )
| ~ ! [X15: term] :
( ( $true
!= ( X12 @ X15 ) )
| ( ( X12 @ ( lam @ X15 ) )
= $true ) )
| ~ ! [X16: term,X17: term] :
( ( ( X12 @ ( ap @ X17 @ X16 ) )
= $true )
| ( $true
!= ( X12 @ X17 ) )
| ( ( X12 @ X16 )
!= $true ) ) )
=> ! [X18: subst,X19: term,X20: term > $o] :
( ~ ! [X21: term] :
( ( $true
= ( X20 @ ( sub @ X21 @ X18 ) ) )
| ( ( var @ X21 )
!= $true ) )
| ( $true
= ( X20 @ ( sub @ X19 @ X18 ) ) )
| ~ ! [X22: term,X23: term] :
( ( $true
!= ( X20 @ X23 ) )
| ( ( X20 @ X22 )
!= $true )
| ( $true
= ( X20 @ ( ap @ X23 @ X22 ) ) ) )
| ~ ! [X24: term] :
( ( ( X20 @ ( lam @ X24 ) )
= $true )
| ~ ! [X25: term] :
( ( ( X20 @ ( sub @ X24 @ ( push @ X25 @ id ) ) )
= $true )
| ( ( X20 @ X25 )
!= $true ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( ulamvarsh = $true )
& ( hoasinduction_gthm = $true )
& ( pushprop_lthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X26: subst > term > subst > $o] :
~ ! [X27: term > $o] :
~ ! [X28: term] :
( ( X27 @ X28 )
= ( X26 @ id @ X28 @ id ) ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X29: subst > term > subst > $o,X30: term,X31: term > $o] :
( ~ ! [X32: subst,X33: subst,X34: term,X35: subst] :
( ( $true
= ( X29 @ ( comp @ X35 @ X32 ) @ ( sub @ X34 @ X32 ) @ X33 ) )
| ( $true
!= ( X29 @ X35 @ X34 @ ( comp @ X32 @ X33 ) ) ) )
| ~ ! [X36: term] :
( ( $true
= ( X31 @ ( sub @ X30 @ ( push @ X36 @ id ) ) ) )
| ( $true
!= ( X31 @ X36 ) ) )
| ~ ! [X37: subst > term > term] :
( ( ( X29 @ id @ ( lam @ ( X37 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X38: subst,X39: subst,X40: term] :
( ( sub @ ( X37 @ X38 @ X40 ) @ X39 )
= ( X37 @ ( comp @ X38 @ X39 ) @ ( sub @ X40 @ X39 ) ) )
| ~ ! [X41: term] :
( ( $true
= ( X29 @ id @ ( X37 @ id @ X41 ) @ id ) )
| ( ( X29 @ id @ X41 @ id )
!= $true ) ) )
| ~ ! [X42: subst,X43: subst,X44: term,X45: subst] :
( ( ( X29 @ ( comp @ X42 @ X43 ) @ ( sub @ X44 @ X43 ) @ X45 )
!= $true )
| ( $true
= ( X29 @ X42 @ X44 @ ( comp @ X43 @ X45 ) ) ) )
| ~ ! [X46: term] :
( ( X29 @ id @ X46 @ id )
= ( X31 @ X46 ) )
| ( ( X31 @ ( lam @ X30 ) )
= $true ) ) )
& ( ( ! [X47: subst > term > subst > $o] :
~ ! [X48: term > $o] :
~ ! [X49: term] :
( ( X47 @ id @ X49 @ id )
= ( X48 @ X49 ) )
=> ( ! [X50: term > $o,X51: term] :
( ~ ! [X52: term,X53: term] :
( ( $true
= ( X50 @ ( ap @ X53 @ X52 ) ) )
| ( $true
!= ( X50 @ X53 ) )
| ( ( X50 @ X52 )
!= $true ) )
| ~ ! [X54: term] :
( ~ ! [X55: term] :
( ( ( X50 @ ( sub @ X54 @ ( push @ X55 @ id ) ) )
= $true )
| ( ( X50 @ X55 )
!= $true ) )
| ( ( X50 @ ( lam @ X54 ) )
= $true ) )
| ( ( X50 @ X51 )
= $true )
| ~ ! [X56: term] :
( ( $true
= ( X50 @ X56 ) )
| ( $true
!= ( var @ X56 ) ) ) )
=> ( ! [X57: term] :
( ( sub @ X57 @ id )
= X57 )
=> ( ! [X58: subst > term > subst > $o,X59: term,X60: term > $o] :
( ~ ! [X61: term] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X60 @ ( sub @ X59 @ ( push @ X61 @ id ) ) )
= $true ) )
| ~ ! [X62: term] :
( ( X58 @ id @ X62 @ id )
= ( X60 @ X62 ) )
| ( $true
= ( X60 @ ( lam @ X59 ) ) )
| ~ ! [X63: subst > term > term] :
( ~ ! [X64: subst,X65: term,X66: subst] :
( ( sub @ ( X63 @ X64 @ X65 ) @ X66 )
= ( X63 @ ( comp @ X64 @ X66 ) @ ( sub @ X65 @ X66 ) ) )
| ~ ! [X67: term] :
( ( $true
!= ( X58 @ id @ X67 @ id ) )
| ( ( X58 @ id @ ( X63 @ id @ X67 ) @ id )
= $true ) )
| ( ( X58 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true ) ) )
=> ! [X68: subst > term > subst > $o,X69: term] :
( ~ ! [X70: term,X71: term] :
( ( $true
!= ( X68 @ id @ X71 @ id ) )
| ( $true
= ( X68 @ id @ ( ap @ ( sub @ X71 @ id ) @ X70 ) @ id ) )
| ( ( X68 @ id @ X70 @ id )
!= $true ) )
| ~ ! [X72: subst > term > term] :
( ( ( X68 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X73: subst,X74: subst,X75: term] :
( ( sub @ ( X72 @ X74 @ X75 ) @ X73 )
= ( X72 @ ( comp @ X74 @ X73 ) @ ( sub @ X75 @ X73 ) ) )
| ~ ! [X76: term] :
( ( ( X68 @ id @ X76 @ id )
!= $true )
| ( $true
= ( X68 @ id @ ( X72 @ id @ X76 ) @ id ) ) ) )
| ( $true
= ( X68 @ id @ X69 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( ! [X77: term,X78: term] :
( ( ( sub @ X78 @ sh )
!= ( sub @ X77 @ sh ) )
| ( X77 = X78 ) )
<=> ( shinj = $true ) )
& ( ( apinj2 = $true )
<=> ! [X79: term,X80: term,X81: term,X82: term] :
( ( ( ap @ X80 @ X81 )
!= ( ap @ X79 @ X82 ) )
| ( X81 = X82 ) ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X83: subst > term > subst > $o,X84: term,X85: term > $o] :
( ~ ! [X86: term] :
( ( $true
= ( X85 @ ( sub @ X84 @ ( push @ X86 @ id ) ) ) )
| ( $true
!= ( X85 @ X86 ) ) )
| ~ ! [X87: term] :
( ( X83 @ id @ X87 @ id )
= ( X85 @ X87 ) )
| ~ ! [X88: subst > term > term] :
( ~ ! [X89: subst,X90: subst,X91: term] :
( ( X88 @ ( comp @ X90 @ X89 ) @ ( sub @ X91 @ X89 ) )
= ( sub @ ( X88 @ X90 @ X91 ) @ X89 ) )
| ~ ! [X92: term] :
( ( ( X83 @ id @ X92 @ id )
!= $true )
| ( ( X83 @ id @ ( X88 @ id @ X92 ) @ id )
= $true ) )
| ( $true
= ( X83 @ id @ ( lam @ ( X88 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X85 @ ( lam @ X84 ) ) ) ) )
& ( pushprop = $true )
& ( ( induction2_lthm = $true )
<=> ( ! [X93: term] :
( ( sub @ X93 @ id )
= X93 )
=> ( ! [X94: subst,X95: term > $o,X96: term] :
( ~ ! [X97: term,X98: term] :
( ( $true
!= ( X95 @ X97 ) )
| ( ( X95 @ X98 )
!= $true )
| ( $true
= ( X95 @ ( ap @ X97 @ X98 ) ) ) )
| ~ ! [X99: term] :
( ( ( var @ X99 )
!= $true )
| ( $true
= ( X95 @ ( sub @ X99 @ X94 ) ) ) )
| ( ( X95 @ ( sub @ X96 @ X94 ) )
= $true )
| ~ ! [X100: term] :
( ( ( X95 @ ( lam @ X100 ) )
= $true )
| ~ ! [X101: term] :
( ( ( X95 @ ( sub @ X100 @ ( push @ X101 @ id ) ) )
= $true )
| ( ( X95 @ X101 )
!= $true ) ) ) )
=> ! [X102: term > $o,X103: term] :
( ~ ! [X104: term,X105: term] :
( ( ( X102 @ X105 )
!= $true )
| ( ( X102 @ ( ap @ X104 @ X105 ) )
= $true )
| ( ( X102 @ X104 )
!= $true ) )
| ( ( X102 @ X103 )
= $true )
| ~ ! [X106: term] :
( ~ ! [X107: term] :
( ( $true
= ( X102 @ ( sub @ X106 @ ( push @ X107 @ id ) ) ) )
| ( ( X102 @ X107 )
!= $true ) )
| ( ( X102 @ ( lam @ X106 ) )
= $true ) )
| ~ ! [X108: term] :
( ( ( X102 @ X108 )
= $true )
| ( $true
!= ( var @ X108 ) ) ) ) ) ) )
& ( ! [X109: term] :
( ( sub @ X109 @ id )
= X109 )
<=> ( termmset = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X110: subst > term > subst > $o] :
~ ! [X111: term > $o] :
~ ! [X112: term] :
( ( X110 @ id @ X112 @ id )
= ( X111 @ X112 ) ) )
& ( substmonoid_gthm = $true )
& ( ( ! [X113: subst,X114: term] :
( ( sub @ one @ ( push @ X114 @ X113 ) )
= X114 )
=> ( ! [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
= X116 )
=> ! [X117: term > $o,X118: term,X119: subst,X120: term > $o,X121: term] :
( ( $true
= ( X117 @ ( sub @ X121 @ ( push @ X118 @ X119 ) ) ) )
| ( $true
!= ( X120 @ X121 ) ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ! [X122: term,X123: subst > term > subst > $o] :
( ~ ! [X124: term] :
( ( $true
= ( X123 @ id @ ( sub @ X122 @ ( push @ X124 @ id ) ) @ id ) )
| ( $true
!= ( X123 @ id @ X124 @ id ) ) )
| ~ ! [X125: subst > term > term] :
( ~ ! [X126: subst,X127: term,X128: subst] :
( ( X125 @ ( comp @ X128 @ X126 ) @ ( sub @ X127 @ X126 ) )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
| ~ ! [X129: term] :
( ( $true
= ( X123 @ id @ ( X125 @ id @ X129 ) @ id ) )
| ( ( X123 @ id @ X129 @ id )
!= $true ) )
| ( $true
= ( X123 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) ) )
| ~ ! [X130: subst,X131: subst,X132: subst,X133: term] :
( ( ( X123 @ X131 @ X133 @ ( comp @ X130 @ X132 ) )
!= $true )
| ( ( X123 @ ( comp @ X131 @ X130 ) @ ( sub @ X133 @ X130 ) @ X132 )
= $true ) )
| ~ ! [X134: subst,X135: subst,X136: term,X137: subst] :
( ( ( X123 @ ( comp @ X135 @ X137 ) @ ( sub @ X136 @ X137 ) @ X134 )
!= $true )
| ( ( X123 @ X135 @ X136 @ ( comp @ X137 @ X134 ) )
= $true ) )
| ( ( X123 @ id @ ( lam @ X122 ) @ id )
= $true ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X138: term] :
~ ! [X139: subst > term > term] :
~ ! [X140: term,X141: subst] :
( ( sub @ X138 @ ( push @ X140 @ X141 ) )
= ( X139 @ X141 @ X140 ) ) )
& ( ! [X142: term] :
~ ! [X143: subst > term > term] :
( ( X143 @ sh @ one )
!= ( sub @ X142 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( hoaslaminj_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( ! [X144: subst > term > term,X145: subst > term > term,X146: subst,X147: term] :
( ~ ! [X148: term,X149: subst,X150: subst] :
( ( X144 @ ( comp @ X149 @ X150 ) @ ( sub @ X148 @ X150 ) )
= ( sub @ ( X144 @ X149 @ X148 ) @ X150 ) )
| ~ ! [X151: subst,X152: subst,X153: term] :
( ( sub @ ( X145 @ X152 @ X153 ) @ X151 )
= ( X145 @ ( comp @ X152 @ X151 ) @ ( sub @ X153 @ X151 ) ) )
| ( ( X144 @ X146 @ X147 )
= ( X145 @ X146 @ X147 ) )
| ( ( lam @ ( X144 @ sh @ one ) )
!= ( lam @ ( X145 @ sh @ one ) ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ! [X154: subst > term > subst > $o,X155: term] :
( ~ ! [X156: term] :
( ( $true
= ( X154 @ id @ ( sub @ X155 @ ( push @ X156 @ id ) ) @ id ) )
| ( ( X154 @ id @ X156 @ id )
!= $true ) )
| ~ ! [X157: subst > term > term] :
( ( $true
= ( X154 @ id @ ( lam @ ( X157 @ sh @ one ) ) @ id ) )
| ~ ! [X158: term] :
( ( $true
!= ( X154 @ id @ X158 @ id ) )
| ( $true
= ( X154 @ id @ ( X157 @ id @ X158 ) @ id ) ) )
| ~ ! [X159: subst,X160: subst,X161: term] :
( ( X157 @ ( comp @ X159 @ X160 ) @ ( sub @ X161 @ X160 ) )
= ( sub @ ( X157 @ X159 @ X161 ) @ X160 ) ) )
| ( $true
= ( X154 @ id @ ( lam @ ( sub @ X155 @ ( push @ one @ sh ) ) ) @ id ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( axmap = $true )
& ( apnotvar = $true )
& ( ( ! [X162: term] :
( ( sub @ X162 @ id )
= X162 )
=> ( ! [X163: subst > term > subst > $o,X164: term] :
( ~ ! [X165: term] :
( ( $true
= ( X163 @ id @ ( sub @ X164 @ ( push @ X165 @ id ) ) @ id ) )
| ( $true
!= ( X163 @ id @ X165 @ id ) ) )
| ( ( X163 @ id @ ( lam @ ( sub @ X164 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X166: subst > term > term] :
( ~ ! [X167: subst,X168: subst,X169: term] :
( ( X166 @ ( comp @ X168 @ X167 ) @ ( sub @ X169 @ X167 ) )
= ( sub @ ( X166 @ X168 @ X169 ) @ X167 ) )
| ~ ! [X170: term] :
( ( ( X163 @ id @ ( X166 @ id @ X170 ) @ id )
= $true )
| ( $true
!= ( X163 @ id @ X170 @ id ) ) )
| ( $true
= ( X163 @ id @ ( lam @ ( X166 @ sh @ one ) ) @ id ) ) ) )
=> ! [X171: term,X172: subst > term > subst > $o] :
( ~ ! [X173: term] :
( ( ( X172 @ id @ X173 @ id )
!= $true )
| ( $true
= ( X172 @ id @ ( sub @ X171 @ ( push @ X173 @ id ) ) @ id ) ) )
| ~ ! [X174: subst > term > term] :
( ~ ! [X175: term] :
( ( ( X172 @ id @ ( X174 @ id @ X175 ) @ id )
= $true )
| ( ( X172 @ id @ X175 @ id )
!= $true ) )
| ~ ! [X176: subst,X177: subst,X178: term] :
( ( sub @ ( X174 @ X176 @ X178 ) @ X177 )
= ( X174 @ ( comp @ X176 @ X177 ) @ ( sub @ X178 @ X177 ) ) )
| ( ( X172 @ id @ ( lam @ ( X174 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X179: term,X180: subst,X181: subst,X182: subst] :
( ( $true
= ( X172 @ ( comp @ X182 @ X180 ) @ ( sub @ X179 @ X180 ) @ X181 ) )
| ( ( X172 @ X182 @ X179 @ ( comp @ X180 @ X181 ) )
!= $true ) )
| ( $true
= ( X172 @ id @ ( lam @ X171 ) @ id ) )
| ~ ! [X183: subst,X184: subst,X185: term,X186: subst] :
( ( $true
!= ( X172 @ ( comp @ X184 @ X183 ) @ ( sub @ X185 @ X183 ) @ X186 ) )
| ( ( X172 @ X184 @ X185 @ ( comp @ X183 @ X186 ) )
= $true ) ) ) ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( lamnotvar = $true )
& ( hoasinduction_lem1_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X187: term,X188: subst > term > subst > $o,X189: term] :
( ~ ! [X190: subst,X191: subst,X192: term,X193: subst] :
( ( ( X188 @ ( comp @ X191 @ X193 ) @ ( sub @ X192 @ X193 ) @ X190 )
!= $true )
| ( ( X188 @ X191 @ X192 @ ( comp @ X193 @ X190 ) )
= $true ) )
| ( $true
= ( X188 @ id @ ( ap @ X189 @ X187 ) @ id ) )
| ~ ! [X194: subst,X195: subst,X196: term,X197: subst] :
( ( $true
!= ( X188 @ X194 @ X196 @ ( comp @ X197 @ X195 ) ) )
| ( ( X188 @ ( comp @ X194 @ X197 ) @ ( sub @ X196 @ X197 ) @ X195 )
= $true ) )
| ( $true
!= ( X188 @ id @ X189 @ id ) )
| ( $true
!= ( X188 @ id @ X187 @ id ) )
| ~ ! [X198: term,X199: term] :
( ( $true
= ( X188 @ id @ ( ap @ ( sub @ X199 @ id ) @ X198 ) @ id ) )
| ( ( X188 @ id @ X198 @ id )
!= $true )
| ( $true
!= ( X188 @ id @ X199 @ id ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ( axshiftcons = $true )
<=> ! [X200: term,X201: subst] :
( ( comp @ sh @ ( push @ X200 @ X201 ) )
= X201 ) )
& ( ! [X202: subst > term > subst > $o,X203: term] :
( ( ( X202 @ id @ X203 @ id )
= $true )
| ~ ! [X204: term,X205: term] :
( ( $true
!= ( X202 @ id @ X205 @ id ) )
| ( $true
!= ( X202 @ id @ X204 @ id ) )
| ( ( X202 @ id @ ( ap @ ( sub @ X204 @ id ) @ X205 ) @ id )
= $true ) )
| ~ ! [X206: subst > term > term] :
( ~ ! [X207: term] :
( ( ( X202 @ id @ X207 @ id )
!= $true )
| ( $true
= ( X202 @ id @ ( X206 @ id @ X207 ) @ id ) ) )
| ~ ! [X208: term,X209: subst,X210: subst] :
( ( X206 @ ( comp @ X210 @ X209 ) @ ( sub @ X208 @ X209 ) )
= ( sub @ ( X206 @ X210 @ X208 ) @ X209 ) )
| ( $true
= ( X202 @ id @ ( lam @ ( X206 @ sh @ one ) ) @ id ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction = $true )
<=> ! [X211: term,X212: subst > term > subst > $o] :
( ( $true
= ( X212 @ id @ X211 @ id ) )
| ~ ! [X213: term,X214: subst,X215: subst,X216: subst] :
( ( ( X212 @ X214 @ X213 @ ( comp @ X216 @ X215 ) )
= $true )
| ( $true
!= ( X212 @ ( comp @ X214 @ X216 ) @ ( sub @ X213 @ X216 ) @ X215 ) ) )
| ~ ! [X217: subst,X218: term,X219: subst,X220: subst] :
( ( ( X212 @ ( comp @ X219 @ X217 ) @ ( sub @ X218 @ X217 ) @ X220 )
= $true )
| ( ( X212 @ X219 @ X218 @ ( comp @ X217 @ X220 ) )
!= $true ) )
| ~ ! [X221: term] :
( ( ( var @ ( sub @ X221 @ id ) )
!= $true )
| ( ( X212 @ id @ X221 @ id )
= $true ) )
| ~ ! [X222: subst > term > term] :
( ~ ! [X223: term] :
( ( ( X212 @ id @ X223 @ id )
!= $true )
| ( $true
= ( X212 @ id @ ( X222 @ id @ X223 ) @ id ) ) )
| ~ ! [X224: term,X225: subst,X226: subst] :
( ( X222 @ ( comp @ X226 @ X225 ) @ ( sub @ X224 @ X225 ) )
= ( sub @ ( X222 @ X226 @ X224 ) @ X225 ) )
| ( $true
= ( X212 @ id @ ( lam @ ( X222 @ sh @ one ) ) @ id ) ) )
| ~ ! [X227: term,X228: term] :
( ( ( X212 @ id @ X228 @ id )
!= $true )
| ( ( X212 @ id @ X227 @ id )
!= $true )
| ( ( X212 @ id @ ( ap @ ( sub @ X227 @ id ) @ X228 ) @ id )
= $true ) ) ) )
& ( hoasinduction_lem1 = $true )
& ! [X229: d_term] : ( d_one = X229 )
& ( hoaslamnotap_lthm = $true )
& ( ( induction = $true )
<=> ! [X230: term > $o,X231: term] :
( ( ( X230 @ X231 )
= $true )
| ~ ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( ( var @ X232 )
!= $true ) )
| ~ ! [X233: term,X234: term] :
( ( $true
!= ( X230 @ X234 ) )
| ( $true
= ( X230 @ ( ap @ X233 @ X234 ) ) )
| ( $true
!= ( X230 @ X233 ) ) )
| ~ ! [X235: term] :
( ( $true
!= ( X230 @ X235 ) )
| ( $true
= ( X230 @ ( lam @ X235 ) ) ) ) ) )
& ( ( apinj1 = $true )
<=> ! [X236: term,X237: term,X238: term,X239: term] :
( ( X236 = X239 )
| ( ( ap @ X236 @ X238 )
!= ( ap @ X239 @ X237 ) ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X240: term] :
~ ! [X241: subst > term > term] :
~ ! [X242: term,X243: subst] :
( ( X241 @ X243 @ X242 )
= ( sub @ X240 @ ( push @ X242 @ X243 ) ) )
=> ( ! [X244: term] :
( ( sub @ X244 @ id )
= X244 )
=> ! [X245: term,X246: subst > term > subst > $o,X247: term > $o] :
( ~ ! [X248: subst > term > term] :
( ~ ! [X249: term,X250: subst,X251: subst] :
( ( sub @ ( X248 @ X251 @ X249 ) @ X250 )
= ( X248 @ ( comp @ X251 @ X250 ) @ ( sub @ X249 @ X250 ) ) )
| ( ( X246 @ id @ ( lam @ ( X248 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X252: term] :
( ( $true
!= ( X246 @ id @ X252 @ id ) )
| ( $true
= ( X246 @ id @ ( X248 @ id @ X252 ) @ id ) ) ) )
| ~ ! [X253: term] :
( ( X247 @ X253 )
= ( X246 @ id @ X253 @ id ) )
| ( $true
= ( X247 @ ( lam @ X245 ) ) )
| ~ ! [X254: term] :
( ( ( X247 @ X254 )
!= $true )
| ( ( X247 @ ( sub @ X245 @ ( push @ X254 @ id ) ) )
= $true ) ) ) ) ) )
& ( ! [X255: term,X256: term > $o] :
( ~ ! [X257: term,X258: term] :
( ( ( X256 @ ( ap @ X257 @ X258 ) )
= $true )
| ( $true
!= ( X256 @ X257 ) )
| ( ( X256 @ X258 )
!= $true ) )
| ~ ! [X259: term] :
( ( $true
= ( X256 @ X259 ) )
| ( ( var @ X259 )
!= $true ) )
| ( ( X256 @ X255 )
= $true )
| ~ ! [X260: term] :
( ( $true
= ( X256 @ ( lam @ X260 ) ) )
| ~ ! [X261: term] :
( ( $true
!= ( X256 @ X261 ) )
| ( $true
= ( X256 @ ( sub @ X260 @ ( push @ X261 @ id ) ) ) ) ) ) )
<=> ( induction2 = $true ) )
& ( ! [X262: subst > term > subst > $o,X263: term] :
( ~ ! [X264: term] :
( ( ( X262 @ id @ ( sub @ X263 @ ( push @ X264 @ id ) ) @ id )
= $true )
| ( ( X262 @ id @ X264 @ id )
!= $true ) )
| ~ ! [X265: subst > term > term] :
( ~ ! [X266: term] :
( ( $true
= ( X262 @ id @ ( X265 @ id @ X266 ) @ id ) )
| ( ( X262 @ id @ X266 @ id )
!= $true ) )
| ( $true
= ( X262 @ id @ ( lam @ ( X265 @ sh @ one ) ) @ id ) )
| ~ ! [X267: subst,X268: subst,X269: term] :
( ( sub @ ( X265 @ X268 @ X269 ) @ X267 )
= ( X265 @ ( comp @ X268 @ X267 ) @ ( sub @ X269 @ X267 ) ) ) )
| ( ( X262 @ id @ ( lam @ X263 ) @ id )
= $true ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ! [X270: term,X271: term > $o,X272: subst > term > subst > $o,X273: term] :
( ~ ! [X274: term,X275: subst,X276: subst,X277: subst] :
( ( $true
= ( X272 @ ( comp @ X277 @ X276 ) @ ( sub @ X274 @ X276 ) @ X275 ) )
| ( $true
!= ( X272 @ X277 @ X274 @ ( comp @ X276 @ X275 ) ) ) )
| ( $true
!= ( X271 @ X273 ) )
| ( $true
= ( X271 @ ( ap @ X273 @ X270 ) ) )
| ~ ! [X278: term,X279: term] :
( ( ( X272 @ id @ X279 @ id )
!= $true )
| ( ( X272 @ id @ X278 @ id )
!= $true )
| ( $true
= ( X272 @ id @ ( ap @ ( sub @ X279 @ id ) @ X278 ) @ id ) ) )
| ~ ! [X280: subst,X281: subst,X282: term,X283: subst] :
( ( $true
= ( X272 @ X281 @ X282 @ ( comp @ X283 @ X280 ) ) )
| ( ( X272 @ ( comp @ X281 @ X283 ) @ ( sub @ X282 @ X283 ) @ X280 )
!= $true ) )
| ~ ! [X284: term] :
( ( X272 @ id @ X284 @ id )
= ( X271 @ X284 ) )
| ( $true
!= ( X271 @ X270 ) ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ! [X285: subst > term > subst > $o] :
~ ! [X286: term > $o] :
~ ! [X287: term] :
( ( X285 @ id @ X287 @ id )
= ( X286 @ X287 ) )
=> ( ! [X288: term,X289: term > $o] :
( ~ ! [X290: term] :
( ( $true
= ( X289 @ X290 ) )
| ( ( var @ X290 )
!= $true ) )
| ( ( X289 @ X288 )
= $true )
| ~ ! [X291: term] :
( ~ ! [X292: term] :
( ( ( X289 @ X292 )
!= $true )
| ( $true
= ( X289 @ ( sub @ X291 @ ( push @ X292 @ id ) ) ) ) )
| ( ( X289 @ ( lam @ X291 ) )
= $true ) )
| ~ ! [X293: term,X294: term] :
( ( ( X289 @ X294 )
!= $true )
| ( $true
= ( X289 @ ( ap @ X293 @ X294 ) ) )
| ( ( X289 @ X293 )
!= $true ) ) )
=> ( ! [X295: term] :
( ( sub @ X295 @ id )
= X295 )
=> ( ! [X296: term,X297: term > $o,X298: subst > term > subst > $o] :
( ~ ! [X299: term] :
( ( X297 @ X299 )
= ( X298 @ id @ X299 @ id ) )
| ( ( X297 @ ( lam @ X296 ) )
= $true )
| ~ ! [X300: subst > term > term] :
( ( $true
= ( X298 @ id @ ( lam @ ( X300 @ sh @ one ) ) @ id ) )
| ~ ! [X301: subst,X302: term,X303: subst] :
( ( sub @ ( X300 @ X303 @ X302 ) @ X301 )
= ( X300 @ ( comp @ X303 @ X301 ) @ ( sub @ X302 @ X301 ) ) )
| ~ ! [X304: term] :
( ( $true
!= ( X298 @ id @ X304 @ id ) )
| ( ( X298 @ id @ ( X300 @ id @ X304 ) @ id )
= $true ) ) )
| ~ ! [X305: term] :
( ( ( X297 @ X305 )
!= $true )
| ( ( X297 @ ( sub @ X296 @ ( push @ X305 @ id ) ) )
= $true ) ) )
=> ! [X306: subst > term > subst > $o,X307: term] :
( ~ ! [X308: term] :
( ( ( var @ ( sub @ X308 @ id ) )
!= $true )
| ( ( X306 @ id @ X308 @ id )
= $true ) )
| ~ ! [X309: subst > term > term] :
( ~ ! [X310: term] :
( ( $true
= ( X306 @ id @ ( X309 @ id @ X310 ) @ id ) )
| ( $true
!= ( X306 @ id @ X310 @ id ) ) )
| ( $true
= ( X306 @ id @ ( lam @ ( X309 @ sh @ one ) ) @ id ) )
| ~ ! [X311: subst,X312: term,X313: subst] :
( ( sub @ ( X309 @ X311 @ X312 ) @ X313 )
= ( X309 @ ( comp @ X311 @ X313 ) @ ( sub @ X312 @ X313 ) ) ) )
| ~ ! [X314: subst,X315: subst,X316: term,X317: subst] :
( ( ( X306 @ X314 @ X316 @ ( comp @ X315 @ X317 ) )
= $true )
| ( ( X306 @ ( comp @ X314 @ X315 ) @ ( sub @ X316 @ X315 ) @ X317 )
!= $true ) )
| ~ ! [X318: term,X319: term] :
( ( ( X306 @ id @ ( ap @ ( sub @ X318 @ id ) @ X319 ) @ id )
= $true )
| ( $true
!= ( X306 @ id @ X319 @ id ) )
| ( ( X306 @ id @ X318 @ id )
!= $true ) )
| ~ ! [X320: term,X321: subst,X322: subst,X323: subst] :
( ( $true
!= ( X306 @ X321 @ X320 @ ( comp @ X322 @ X323 ) ) )
| ( $true
= ( X306 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) @ X323 ) ) )
| ( ( X306 @ id @ X307 @ id )
= $true ) ) ) ) ) ) )
& ( lamnotap != $true )
& ( ! [X324: subst] :
( ( comp @ id @ X324 )
= X324 )
<=> ( axidl = $true ) )
& ( ! [X325: term > $o,X326: subst,X327: term] :
( ~ ! [X328: term,X329: term] :
( ( $true
!= ( X325 @ X329 ) )
| ( $true
!= ( X325 @ X328 ) )
| ( $true
= ( X325 @ ( ap @ X329 @ X328 ) ) ) )
| ~ ! [X330: term] :
( ( $true
!= ( var @ X330 ) )
| ( $true
= ( X325 @ ( sub @ X330 @ X326 ) ) ) )
| ~ ! [X331: term] :
( ~ ! [X332: term] :
( ( $true
!= ( X325 @ X332 ) )
| ( $true
= ( X325 @ ( sub @ X331 @ ( push @ X332 @ id ) ) ) ) )
| ( ( X325 @ ( lam @ X331 ) )
= $true ) )
| ( $true
= ( X325 @ ( sub @ X327 @ X326 ) ) ) )
<=> ( induction2lem = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( pushprop_lem0 = $true )
<=> ! [X333: term,X334: term > $o,X335: subst] :
~ ! [X336: term > $o] :
~ ! [X337: term] :
( ( X334 @ ( sub @ X337 @ ( push @ X333 @ X335 ) ) )
= ( X336 @ X337 ) ) )
& ( hoasapnotvar = $true )
& ! [X338: d_subst,X339: d_subst] :
( ( ( d2subst @ X338 )
= ( d2subst @ X339 ) )
=> ( X338 = X339 ) )
& ( pushprop_gthm = $true )
& ( ulamvarind = $true )
& ( ! [X340: subst > term > subst > $o,X341: term] :
( ~ ! [X342: term] :
( ( ( X340 @ id @ X342 @ id )
!= $true )
| ( $true
= ( X340 @ id @ ( sub @ X341 @ ( push @ X342 @ id ) ) @ id ) ) )
| ~ ! [X343: term,X344: subst > term > term] :
( ~ ! [X345: subst,X346: subst,X347: term] :
( ( sub @ ( sub @ X343 @ ( push @ X347 @ X346 ) ) @ X345 )
= ( sub @ ( X344 @ X346 @ X347 ) @ X345 ) )
| ~ ! [X348: term] :
( ( $true
= ( X340 @ id @ ( X344 @ id @ X348 ) @ id ) )
| ( $true
!= ( X340 @ id @ X348 @ id ) ) )
| ( ( X340 @ id @ ( lam @ ( X344 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X349: subst,X350: subst,X351: term] :
( ( sub @ X343 @ ( push @ ( sub @ X351 @ X350 ) @ ( comp @ X349 @ X350 ) ) )
= ( X344 @ ( comp @ X349 @ X350 ) @ ( sub @ X351 @ X350 ) ) ) )
| ( $true
= ( X340 @ id @ ( lam @ ( sub @ X341 @ ( push @ one @ sh ) ) ) @ id ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X352: term,X353: term,X354: term,X355: term] :
( ( X353 = X354 )
| ( ( ap @ ( sub @ X352 @ id ) @ X353 )
!= ( ap @ ( sub @ X355 @ id ) @ X354 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( axapp = $true )
& ! [X356: subst] :
? [X357: d_subst] :
( ( d2subst @ X357 )
= X356 )
& ( ( ! [X358: term,X359: term,X360: term,X361: term] :
( ( ( ap @ X360 @ X361 )
!= ( ap @ X359 @ X358 ) )
| ( X358 = X361 ) )
=> ! [X362: term,X363: term,X364: term,X365: term] :
( ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X364 @ id ) @ X365 ) )
| ( X363 = X365 ) ) )
<=> ( hoasapinj2_lthm = $true ) )
& ( induction2_gthm = $true )
& ( ( laminj = $true )
<=> ! [X366: term,X367: term] :
( ( X366 = X367 )
| ( ( lam @ X366 )
!= ( lam @ X367 ) ) ) )
& ! [X368: d_term,X369: d_term] :
( ( ( d2term @ X368 )
= ( d2term @ X369 ) )
=> ( X368 = X369 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ! [X370: subst,X371: term] :
( ( sub @ one @ ( push @ X371 @ X370 ) )
= X371 )
<=> ( axvarcons = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( ! [X373: subst] :
( ( comp @ X373 @ id )
= X373 )
& ! [X372: subst] :
( ( comp @ id @ X372 )
= X372 ) )
<=> ( substmonoid = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X374: term] :
( ( sub @ X374 @ id )
= X374 )
=> ( ! [X375: term,X376: subst > term > subst > $o] :
( ~ ! [X377: subst > term > term] :
( ~ ! [X378: term] :
( ( $true
= ( X376 @ id @ ( X377 @ id @ X378 ) @ id ) )
| ( $true
!= ( X376 @ id @ X378 @ id ) ) )
| ( $true
= ( X376 @ id @ ( lam @ ( X377 @ sh @ one ) ) @ id ) )
| ~ ! [X379: subst,X380: subst,X381: term] :
( ( X377 @ ( comp @ X379 @ X380 ) @ ( sub @ X381 @ X380 ) )
= ( sub @ ( X377 @ X379 @ X381 ) @ X380 ) ) )
| ~ ! [X382: term] :
( ( $true
= ( X376 @ id @ ( sub @ X375 @ ( push @ X382 @ id ) ) @ id ) )
| ( $true
!= ( X376 @ id @ X382 @ id ) ) )
| ( ( X376 @ id @ ( lam @ ( sub @ X375 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X383: term,X384: subst > term > subst > $o] :
( ~ ! [X385: term] :
( ( $true
!= ( X384 @ id @ X385 @ id ) )
| ( ( X384 @ id @ ( sub @ X383 @ ( push @ X385 @ id ) ) @ id )
= $true ) )
| ~ ! [X386: subst > term > term] :
( ~ ! [X387: term] :
( ( $true
= ( X384 @ id @ ( X386 @ id @ X387 ) @ id ) )
| ( ( X384 @ id @ X387 @ id )
!= $true ) )
| ( $true
= ( X384 @ id @ ( lam @ ( X386 @ sh @ one ) ) @ id ) )
| ~ ! [X388: subst,X389: subst,X390: term] :
( ( sub @ ( X386 @ X389 @ X390 ) @ X388 )
= ( X386 @ ( comp @ X389 @ X388 ) @ ( sub @ X390 @ X388 ) ) ) )
| ( $true
= ( X384 @ id @ ( lam @ X383 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X391: term] :
~ ! [X392: subst > term > term] :
~ ! [X393: subst,X394: term] :
( ( sub @ X391 @ ( push @ X394 @ X393 ) )
= ( X392 @ X393 @ X394 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X395: term > $o,X396: term > $o,X397: term,X398: subst] :
( ( $true
= ( X396 @ one ) )
| ( $true
!= ( X395 @ X397 ) )
| ~ ! [X399: term] :
( ( X395 @ ( sub @ X399 @ ( push @ X397 @ X398 ) ) )
= ( X396 @ X399 ) ) ) )
& ( one
= ( d2term @ d_one ) )
& ( ulamvar1 != $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X400: term,X401: subst] :
( ( sub @ one @ ( push @ X400 @ X401 ) )
= X400 )
=> ( ! [X402: term,X403: subst] :
( ( comp @ sh @ ( push @ X402 @ X403 ) )
= X403 )
=> ( ! [X404: term,X405: term] :
( ( X404 = X405 )
| ( ( lam @ X405 )
!= ( lam @ X404 ) ) )
=> ! [X406: subst > term > term,X407: subst,X408: term,X409: subst > term > term] :
( ( ( lam @ ( X409 @ sh @ one ) )
!= ( lam @ ( X406 @ sh @ one ) ) )
| ~ ! [X413: term,X414: subst,X415: subst] :
( ( sub @ ( X409 @ X415 @ X413 ) @ X414 )
= ( X409 @ ( comp @ X415 @ X414 ) @ ( sub @ X413 @ X414 ) ) )
| ( ( X409 @ X407 @ X408 )
= ( X406 @ X407 @ X408 ) )
| ~ ! [X410: subst,X411: term,X412: subst] :
( ( sub @ ( X406 @ X410 @ X411 ) @ X412 )
= ( X406 @ ( comp @ X410 @ X412 ) @ ( sub @ X411 @ X412 ) ) ) ) ) ) ) )
& ( axassoc = $true )
& ( ( axidr = $true )
<=> ! [X416: subst] :
( ( comp @ X416 @ id )
= X416 ) )
& ! [X417: d_subst] : ( d_id = X417 )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X418: term,X419: term > $o] :
( ( $true
= ( X419 @ X418 ) )
| ~ ! [X420: term] :
( ( ( X419 @ ( lam @ X420 ) )
= $true )
| ~ ! [X421: term] :
( ( $true
!= ( X419 @ X421 ) )
| ( $true
= ( X419 @ ( sub @ X420 @ ( push @ X421 @ id ) ) ) ) ) )
| ~ ! [X422: term] :
( ( $true
= ( X419 @ X422 ) )
| ( ( var @ X422 )
!= $true ) )
| ~ ! [X423: term,X424: term] :
( ( $true
= ( X419 @ ( ap @ X424 @ X423 ) ) )
| ( ( X419 @ X424 )
!= $true )
| ( ( X419 @ X423 )
!= $true ) ) )
=> ( ! [X425: term,X426: term,X427: subst > term > subst > $o] :
( ~ ! [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X427 @ ( comp @ X429 @ X430 ) @ ( sub @ X428 @ X430 ) @ X431 ) )
| ( ( X427 @ X429 @ X428 @ ( comp @ X430 @ X431 ) )
!= $true ) )
| ( ( X427 @ id @ ( ap @ X426 @ X425 ) @ id )
= $true )
| ( $true
!= ( X427 @ id @ X426 @ id ) )
| ~ ! [X432: subst,X433: subst,X434: term,X435: subst] :
( ( ( X427 @ X432 @ X434 @ ( comp @ X433 @ X435 ) )
= $true )
| ( ( X427 @ ( comp @ X432 @ X433 ) @ ( sub @ X434 @ X433 ) @ X435 )
!= $true ) )
| ~ ! [X436: term,X437: term] :
( ( ( X427 @ id @ X436 @ id )
!= $true )
| ( $true
!= ( X427 @ id @ X437 @ id ) )
| ( $true
= ( X427 @ id @ ( ap @ ( sub @ X437 @ id ) @ X436 ) @ id ) ) )
| ( ( X427 @ id @ X425 @ id )
!= $true ) )
=> ( ! [X438: term,X439: subst > term > subst > $o] :
( ~ ! [X440: subst > term > term] :
( ( ( X439 @ id @ ( lam @ ( X440 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X441: subst,X442: subst,X443: term] :
( ( sub @ ( X440 @ X442 @ X443 ) @ X441 )
= ( X440 @ ( comp @ X442 @ X441 ) @ ( sub @ X443 @ X441 ) ) )
| ~ ! [X444: term] :
( ( ( X439 @ id @ X444 @ id )
!= $true )
| ( $true
= ( X439 @ id @ ( X440 @ id @ X444 ) @ id ) ) ) )
| ( $true
= ( X439 @ id @ ( lam @ X438 ) @ id ) )
| ~ ! [X445: term] :
( ( $true
= ( X439 @ id @ ( sub @ X438 @ ( push @ X445 @ id ) ) @ id ) )
| ( $true
!= ( X439 @ id @ X445 @ id ) ) )
| ~ ! [X446: subst,X447: subst,X448: term,X449: subst] :
( ( ( X439 @ ( comp @ X449 @ X447 ) @ ( sub @ X448 @ X447 ) @ X446 )
= $true )
| ( ( X439 @ X449 @ X448 @ ( comp @ X447 @ X446 ) )
!= $true ) )
| ~ ! [X450: subst,X451: subst,X452: term,X453: subst] :
( ( ( X439 @ ( comp @ X453 @ X450 ) @ ( sub @ X452 @ X450 ) @ X451 )
!= $true )
| ( $true
= ( X439 @ X453 @ X452 @ ( comp @ X450 @ X451 ) ) ) ) )
=> ! [X454: term,X455: subst > term > subst > $o] :
( ~ ! [X456: term,X457: term] :
( ( $true
!= ( X455 @ id @ X456 @ id ) )
| ( $true
!= ( X455 @ id @ X457 @ id ) )
| ( ( X455 @ id @ ( ap @ ( sub @ X456 @ id ) @ X457 ) @ id )
= $true ) )
| ~ ! [X458: subst > term > term] :
( ( ( X455 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X459: term] :
( ( ( X455 @ id @ X459 @ id )
!= $true )
| ( ( X455 @ id @ ( X458 @ id @ X459 ) @ id )
= $true ) )
| ~ ! [X460: subst,X461: subst,X462: term] :
( ( X458 @ ( comp @ X460 @ X461 ) @ ( sub @ X462 @ X461 ) )
= ( sub @ ( X458 @ X460 @ X462 ) @ X461 ) ) )
| ( $true
= ( X455 @ id @ X454 @ id ) )
| ~ ! [X463: subst,X464: term,X465: subst,X466: subst] :
( ( $true
!= ( X455 @ X466 @ X464 @ ( comp @ X465 @ X463 ) ) )
| ( ( X455 @ ( comp @ X466 @ X465 ) @ ( sub @ X464 @ X465 ) @ X463 )
= $true ) )
| ~ ! [X467: term,X468: subst,X469: subst,X470: subst] :
( ( $true
!= ( X455 @ ( comp @ X468 @ X469 ) @ ( sub @ X467 @ X469 ) @ X470 ) )
| ( ( X455 @ X468 @ X467 @ ( comp @ X469 @ X470 ) )
= $true ) )
| ~ ! [X471: term] :
( ( ( X455 @ id @ X471 @ id )
= $true )
| ( $true
!= ( var @ ( sub @ X471 @ id ) ) ) ) ) ) ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( termmset_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X472: term > $o,X473: term,X474: subst,X475: term > $o,X476: term] :
( ( $true
!= ( X472 @ X473 ) )
| ( $true
= ( X475 @ ( sub @ X473 @ ( push @ X476 @ X474 ) ) ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ( ! [X477: subst,X478: term] :
( ( sub @ one @ ( push @ X478 @ X477 ) )
= X478 )
=> ! [X479: term > $o,X480: term > $o,X481: term,X482: subst] :
( ( $true
= ( X480 @ one ) )
| ( ( X479 @ X481 )
!= $true )
| ~ ! [X483: term] :
( ( X480 @ X483 )
= ( X479 @ ( sub @ X483 @ ( push @ X481 @ X482 ) ) ) ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( pushprop_lem0_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ( ( ! [X484: term] :
( ( sub @ X484 @ id )
= X484 )
=> ! [X485: term > $o,X486: subst > term > subst > $o,X487: term] :
( ~ ! [X488: subst,X489: subst,X490: term,X491: subst] :
( ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X491 ) )
!= $true )
| ( ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X491 )
= $true ) )
| ( ( X485 @ ( lam @ X487 ) )
= $true )
| ~ ! [X492: subst,X493: term,X494: subst,X495: subst] :
( ( ( X486 @ ( comp @ X492 @ X495 ) @ ( sub @ X493 @ X495 ) @ X494 )
!= $true )
| ( ( X486 @ X492 @ X493 @ ( comp @ X495 @ X494 ) )
= $true ) )
| ~ ! [X496: term] :
( ( ( X485 @ ( sub @ X487 @ ( push @ X496 @ id ) ) )
= $true )
| ( ( X485 @ X496 )
!= $true ) )
| ~ ! [X497: term] :
( ( X485 @ X497 )
= ( X486 @ id @ X497 @ id ) )
| ~ ! [X498: subst > term > term] :
( ~ ! [X499: term] :
( ( $true
= ( X486 @ id @ ( X498 @ id @ X499 ) @ id ) )
| ( $true
!= ( X486 @ id @ X499 @ id ) ) )
| ( ( X486 @ id @ ( lam @ ( X498 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X500: subst,X501: subst,X502: term] :
( ( X498 @ ( comp @ X500 @ X501 ) @ ( sub @ X502 @ X501 ) )
= ( sub @ ( X498 @ X500 @ X502 ) @ X501 ) ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( ( ! [X503: subst] :
( ( comp @ id @ X503 )
= X503 )
=> ( ! [X504: subst] :
( ( comp @ X504 @ id )
= X504 )
=> ( ! [X506: subst] :
( ( comp @ id @ X506 )
= X506 )
& ! [X505: subst] :
( ( comp @ X505 @ id )
= X505 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( hoasapinj1_gthm = $true )
& ( ! [X507: term,X508: term,X509: term,X510: term] :
( ( ( ap @ ( sub @ X507 @ id ) @ X509 )
!= ( ap @ ( sub @ X508 @ id ) @ X510 ) )
| ( X507 = X508 ) )
<=> ( hoasapinj1 = $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
<=> ( axvarid = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X512: subst > term > subst > $o,X513: term] :
( ( $true
= ( X512 @ id @ ( lam @ ( sub @ X513 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X514: term] :
( ( ( X512 @ id @ ( sub @ X513 @ ( push @ X514 @ id ) ) @ id )
= $true )
| ( ( X512 @ id @ X514 @ id )
!= $true ) )
| ~ ! [X515: subst > term > term] :
( ( ( X512 @ id @ ( lam @ ( X515 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X516: subst,X517: term,X518: subst] :
( ( X515 @ ( comp @ X518 @ X516 ) @ ( sub @ X517 @ X516 ) )
= ( sub @ ( X515 @ X518 @ X517 ) @ X516 ) )
| ~ ! [X519: term] :
( ( ( X512 @ id @ X519 @ id )
!= $true )
| ( ( X512 @ id @ ( X515 @ id @ X519 ) @ id )
= $true ) ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X520: term,X521: term,X522: subst > term > term] :
( ~ ! [X523: subst,X524: term,X525: subst] :
( ( X522 @ ( comp @ X525 @ X523 ) @ ( sub @ X524 @ X523 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X523 ) )
| ( ( ap @ ( sub @ X521 @ id ) @ X520 )
!= ( lam @ ( X522 @ sh @ one ) ) ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X526: term] :
( ( sub @ X526 @ id )
= X526 )
=> ( ! [X527: term,X528: term,X529: term,X530: term] :
( ( ( ap @ X528 @ X529 )
!= ( ap @ X527 @ X530 ) )
| ( X527 = X528 ) )
=> ! [X531: term,X532: term,X533: term,X534: term] :
( ( ( ap @ ( sub @ X533 @ id ) @ X531 )
!= ( ap @ ( sub @ X534 @ id ) @ X532 ) )
| ( X533 = X534 ) ) ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X535: term > $o,X536: subst,X537: term] :
~ ! [X538: term > $o] :
~ ! [X539: term] :
( ( X535 @ ( sub @ X539 @ ( push @ X537 @ X536 ) ) )
= ( X538 @ X539 ) ) )
& ( hoaslamnotvar = $true ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,plain,
( ( ( axscons = $true )
<=> ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 ) )
& ( axabs = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X1: term] :
~ ! [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
!= ( X2 @ sh @ one ) ) )
& ( axclos = $true )
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( termmset_lthm = $true )
<=> ( ! [X5: term] :
( ( sub @ X5 @ id )
= X5 )
=> ! [X6: term] :
( ( sub @ X6 @ id )
= X6 ) ) )
& ( axvarshift = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( ( ! [X10: term,X11: subst] :
( ( sub @ one @ ( push @ X10 @ X11 ) )
= X10 )
=> ( ! [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
= X12 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X15: term > $o,X16: term] :
( ~ ! [X17: term] :
( ( $true
= ( X15 @ X17 ) )
| ( $true
!= ( var @ X17 ) ) )
| ( $true
= ( X15 @ X16 ) )
| ~ ! [X18: term] :
( ( $true
!= ( X15 @ X18 ) )
| ( $true
= ( X15 @ ( lam @ X18 ) ) ) )
| ~ ! [X19: term,X20: term] :
( ( ( X15 @ ( ap @ X20 @ X19 ) )
= $true )
| ( $true
!= ( X15 @ X20 ) )
| ( $true
!= ( X15 @ X19 ) ) ) )
=> ! [X21: subst,X22: term,X23: term > $o] :
( ~ ! [X24: term] :
( ( $true
= ( X23 @ ( sub @ X24 @ X21 ) ) )
| ( ( var @ X24 )
!= $true ) )
| ( ( X23 @ ( sub @ X22 @ X21 ) )
= $true )
| ~ ! [X25: term,X26: term] :
( ( ( X23 @ X26 )
!= $true )
| ( ( X23 @ X25 )
!= $true )
| ( $true
= ( X23 @ ( ap @ X26 @ X25 ) ) ) )
| ~ ! [X27: term] :
( ( ( X23 @ ( lam @ X27 ) )
= $true )
| ~ ! [X28: term] :
( ( ( X23 @ ( sub @ X27 @ ( push @ X28 @ id ) ) )
= $true )
| ( ( X23 @ X28 )
!= $true ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( ulamvarsh = $true )
& ( hoasinduction_gthm = $true )
& ( pushprop_lthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( ! [X29: subst > term > subst > $o] :
~ ! [X30: term > $o] :
~ ! [X31: term] :
( ( X30 @ X31 )
= ( X29 @ id @ X31 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X32: subst > term > subst > $o,X33: term,X34: term > $o] :
( ~ ! [X35: subst,X36: subst,X37: term,X38: subst] :
( ( $true
= ( X32 @ ( comp @ X38 @ X35 ) @ ( sub @ X37 @ X35 ) @ X36 ) )
| ( ( X32 @ X38 @ X37 @ ( comp @ X35 @ X36 ) )
!= $true ) )
| ~ ! [X39: term] :
( ( ( X34 @ ( sub @ X33 @ ( push @ X39 @ id ) ) )
= $true )
| ( $true
!= ( X34 @ X39 ) ) )
| ~ ! [X40: subst > term > term] :
( ( ( X32 @ id @ ( lam @ ( X40 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X41: subst,X42: subst,X43: term] :
( ( sub @ ( X40 @ X41 @ X43 ) @ X42 )
= ( X40 @ ( comp @ X41 @ X42 ) @ ( sub @ X43 @ X42 ) ) )
| ~ ! [X44: term] :
( ( $true
= ( X32 @ id @ ( X40 @ id @ X44 ) @ id ) )
| ( ( X32 @ id @ X44 @ id )
!= $true ) ) )
| ~ ! [X45: subst,X46: subst,X47: term,X48: subst] :
( ( ( X32 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) @ X48 )
!= $true )
| ( ( X32 @ X45 @ X47 @ ( comp @ X46 @ X48 ) )
= $true ) )
| ~ ! [X49: term] :
( ( X32 @ id @ X49 @ id )
= ( X34 @ X49 ) )
| ( ( X34 @ ( lam @ X33 ) )
= $true ) ) )
& ( ( ! [X50: subst > term > subst > $o] :
~ ! [X51: term > $o] :
~ ! [X52: term] :
( ( X51 @ X52 )
= ( X50 @ id @ X52 @ id ) )
=> ( ! [X53: term > $o,X54: term] :
( ~ ! [X55: term,X56: term] :
( ( $true
= ( X53 @ ( ap @ X56 @ X55 ) ) )
| ( $true
!= ( X53 @ X56 ) )
| ( $true
!= ( X53 @ X55 ) ) )
| ~ ! [X57: term] :
( ~ ! [X58: term] :
( ( ( X53 @ ( sub @ X57 @ ( push @ X58 @ id ) ) )
= $true )
| ( $true
!= ( X53 @ X58 ) ) )
| ( ( X53 @ ( lam @ X57 ) )
= $true ) )
| ( ( X53 @ X54 )
= $true )
| ~ ! [X59: term] :
( ( ( X53 @ X59 )
= $true )
| ( $true
!= ( var @ X59 ) ) ) )
=> ( ! [X60: term] :
( ( sub @ X60 @ id )
= X60 )
=> ( ! [X61: subst > term > subst > $o,X62: term,X63: term > $o] :
( ~ ! [X64: term] :
( ( ( X63 @ X64 )
!= $true )
| ( ( X63 @ ( sub @ X62 @ ( push @ X64 @ id ) ) )
= $true ) )
| ~ ! [X65: term] :
( ( X63 @ X65 )
= ( X61 @ id @ X65 @ id ) )
| ( $true
= ( X63 @ ( lam @ X62 ) ) )
| ~ ! [X66: subst > term > term] :
( ~ ! [X67: subst,X68: term,X69: subst] :
( ( X66 @ ( comp @ X67 @ X69 ) @ ( sub @ X68 @ X69 ) )
= ( sub @ ( X66 @ X67 @ X68 ) @ X69 ) )
| ~ ! [X70: term] :
( ( $true
!= ( X61 @ id @ X70 @ id ) )
| ( ( X61 @ id @ ( X66 @ id @ X70 ) @ id )
= $true ) )
| ( $true
= ( X61 @ id @ ( lam @ ( X66 @ sh @ one ) ) @ id ) ) ) )
=> ! [X71: subst > term > subst > $o,X72: term] :
( ~ ! [X73: term,X74: term] :
( ( ( X71 @ id @ X74 @ id )
!= $true )
| ( ( X71 @ id @ ( ap @ ( sub @ X74 @ id ) @ X73 ) @ id )
= $true )
| ( ( X71 @ id @ X73 @ id )
!= $true ) )
| ~ ! [X75: subst > term > term] :
( ( $true
= ( X71 @ id @ ( lam @ ( X75 @ sh @ one ) ) @ id ) )
| ~ ! [X76: subst,X77: subst,X78: term] :
( ( sub @ ( X75 @ X77 @ X78 ) @ X76 )
= ( X75 @ ( comp @ X77 @ X76 ) @ ( sub @ X78 @ X76 ) ) )
| ~ ! [X79: term] :
( ( ( X71 @ id @ X79 @ id )
!= $true )
| ( $true
= ( X71 @ id @ ( X75 @ id @ X79 ) @ id ) ) ) )
| ( $true
= ( X71 @ id @ X72 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( ( shinj = $true )
<=> ! [X80: term,X81: term] :
( ( ( sub @ X80 @ sh )
!= ( sub @ X81 @ sh ) )
| ( X80 = X81 ) ) )
& ( ! [X82: term,X83: term,X84: term,X85: term] :
( ( X84 = X85 )
| ( ( ap @ X83 @ X84 )
!= ( ap @ X82 @ X85 ) ) )
<=> ( apinj2 = $true ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X86: subst > term > subst > $o,X87: term,X88: term > $o] :
( ~ ! [X89: term] :
( ( $true
= ( X88 @ ( sub @ X87 @ ( push @ X89 @ id ) ) ) )
| ( ( X88 @ X89 )
!= $true ) )
| ~ ! [X90: term] :
( ( X86 @ id @ X90 @ id )
= ( X88 @ X90 ) )
| ~ ! [X91: subst > term > term] :
( ~ ! [X92: subst,X93: subst,X94: term] :
( ( X91 @ ( comp @ X93 @ X92 ) @ ( sub @ X94 @ X92 ) )
= ( sub @ ( X91 @ X93 @ X94 ) @ X92 ) )
| ~ ! [X95: term] :
( ( $true
!= ( X86 @ id @ X95 @ id ) )
| ( $true
= ( X86 @ id @ ( X91 @ id @ X95 ) @ id ) ) )
| ( $true
= ( X86 @ id @ ( lam @ ( X91 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X88 @ ( lam @ X87 ) ) ) ) )
& ( pushprop = $true )
& ( ( induction2_lthm = $true )
<=> ( ! [X96: term] :
( ( sub @ X96 @ id )
= X96 )
=> ( ! [X97: subst,X98: term > $o,X99: term] :
( ~ ! [X100: term,X101: term] :
( ( ( X98 @ X100 )
!= $true )
| ( $true
!= ( X98 @ X101 ) )
| ( ( X98 @ ( ap @ X100 @ X101 ) )
= $true ) )
| ~ ! [X102: term] :
( ( $true
!= ( var @ X102 ) )
| ( $true
= ( X98 @ ( sub @ X102 @ X97 ) ) ) )
| ( $true
= ( X98 @ ( sub @ X99 @ X97 ) ) )
| ~ ! [X103: term] :
( ( ( X98 @ ( lam @ X103 ) )
= $true )
| ~ ! [X104: term] :
( ( ( X98 @ ( sub @ X103 @ ( push @ X104 @ id ) ) )
= $true )
| ( ( X98 @ X104 )
!= $true ) ) ) )
=> ! [X105: term > $o,X106: term] :
( ~ ! [X107: term,X108: term] :
( ( $true
!= ( X105 @ X108 ) )
| ( ( X105 @ ( ap @ X107 @ X108 ) )
= $true )
| ( $true
!= ( X105 @ X107 ) ) )
| ( $true
= ( X105 @ X106 ) )
| ~ ! [X109: term] :
( ~ ! [X110: term] :
( ( ( X105 @ ( sub @ X109 @ ( push @ X110 @ id ) ) )
= $true )
| ( ( X105 @ X110 )
!= $true ) )
| ( ( X105 @ ( lam @ X109 ) )
= $true ) )
| ~ ! [X111: term] :
( ( $true
= ( X105 @ X111 ) )
| ( ( var @ X111 )
!= $true ) ) ) ) ) )
& ( ! [X112: term] :
( ( sub @ X112 @ id )
= X112 )
<=> ( termmset = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X113: subst > term > subst > $o] :
~ ! [X114: term > $o] :
~ ! [X115: term] :
( ( X113 @ id @ X115 @ id )
= ( X114 @ X115 ) ) )
& ( substmonoid_gthm = $true )
& ( ( ! [X116: subst,X117: term] :
( ( sub @ one @ ( push @ X117 @ X116 ) )
= X117 )
=> ( ! [X118: term,X119: subst] :
( ( comp @ sh @ ( push @ X118 @ X119 ) )
= X119 )
=> ! [X120: term > $o,X121: term,X122: subst,X123: term > $o,X124: term] :
( ( $true
= ( X120 @ ( sub @ X124 @ ( push @ X121 @ X122 ) ) ) )
| ( ( X123 @ X124 )
!= $true ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ! [X125: term,X126: subst > term > subst > $o] :
( ~ ! [X127: term] :
( ( ( X126 @ id @ ( sub @ X125 @ ( push @ X127 @ id ) ) @ id )
= $true )
| ( $true
!= ( X126 @ id @ X127 @ id ) ) )
| ~ ! [X128: subst > term > term] :
( ~ ! [X129: subst,X130: term,X131: subst] :
( ( sub @ ( X128 @ X131 @ X130 ) @ X129 )
= ( X128 @ ( comp @ X131 @ X129 ) @ ( sub @ X130 @ X129 ) ) )
| ~ ! [X132: term] :
( ( $true
= ( X126 @ id @ ( X128 @ id @ X132 ) @ id ) )
| ( ( X126 @ id @ X132 @ id )
!= $true ) )
| ( ( X126 @ id @ ( lam @ ( X128 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X133: subst,X134: subst,X135: subst,X136: term] :
( ( ( X126 @ X134 @ X136 @ ( comp @ X133 @ X135 ) )
!= $true )
| ( ( X126 @ ( comp @ X134 @ X133 ) @ ( sub @ X136 @ X133 ) @ X135 )
= $true ) )
| ~ ! [X137: subst,X138: subst,X139: term,X140: subst] :
( ( ( X126 @ ( comp @ X138 @ X140 ) @ ( sub @ X139 @ X140 ) @ X137 )
!= $true )
| ( $true
= ( X126 @ X138 @ X139 @ ( comp @ X140 @ X137 ) ) ) )
| ( ( X126 @ id @ ( lam @ X125 ) @ id )
= $true ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X143: term] :
~ ! [X144: subst > term > term] :
~ ! [X145: term,X146: subst] :
( ( sub @ X143 @ ( push @ X145 @ X146 ) )
= ( X144 @ X146 @ X145 ) ) )
& ( ! [X147: term] :
~ ! [X148: subst > term > term] :
( ( X148 @ sh @ one )
!= ( sub @ X147 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( hoaslaminj_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( ( hoaslaminj = $true )
<=> ! [X154: subst > term > term,X155: subst > term > term,X156: subst,X157: term] :
( ~ ! [X158: term,X159: subst,X160: subst] :
( ( X154 @ ( comp @ X159 @ X160 ) @ ( sub @ X158 @ X160 ) )
= ( sub @ ( X154 @ X159 @ X158 ) @ X160 ) )
| ( ( X154 @ X156 @ X157 )
= ( X155 @ X156 @ X157 ) )
| ~ ! [X161: subst,X162: subst,X163: term] :
( ( X155 @ ( comp @ X162 @ X161 ) @ ( sub @ X163 @ X161 ) )
= ( sub @ ( X155 @ X162 @ X163 ) @ X161 ) )
| ( ( lam @ ( X154 @ sh @ one ) )
!= ( lam @ ( X155 @ sh @ one ) ) ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ! [X164: subst > term > subst > $o,X165: term] :
( ~ ! [X166: term] :
( ( ( X164 @ id @ ( sub @ X165 @ ( push @ X166 @ id ) ) @ id )
= $true )
| ( $true
!= ( X164 @ id @ X166 @ id ) ) )
| ~ ! [X167: subst > term > term] :
( ( $true
= ( X164 @ id @ ( lam @ ( X167 @ sh @ one ) ) @ id ) )
| ~ ! [X168: term] :
( ( $true
!= ( X164 @ id @ X168 @ id ) )
| ( $true
= ( X164 @ id @ ( X167 @ id @ X168 ) @ id ) ) )
| ~ ! [X169: subst,X170: subst,X171: term] :
( ( X167 @ ( comp @ X169 @ X170 ) @ ( sub @ X171 @ X170 ) )
= ( sub @ ( X167 @ X169 @ X171 ) @ X170 ) ) )
| ( $true
= ( X164 @ id @ ( lam @ ( sub @ X165 @ ( push @ one @ sh ) ) ) @ id ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( axmap = $true )
& ( apnotvar = $true )
& ( ( ! [X172: term] :
( ( sub @ X172 @ id )
= X172 )
=> ( ! [X173: subst > term > subst > $o,X174: term] :
( ~ ! [X175: term] :
( ( $true
= ( X173 @ id @ ( sub @ X174 @ ( push @ X175 @ id ) ) @ id ) )
| ( ( X173 @ id @ X175 @ id )
!= $true ) )
| ( $true
= ( X173 @ id @ ( lam @ ( sub @ X174 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X176: subst > term > term] :
( ~ ! [X177: subst,X178: subst,X179: term] :
( ( sub @ ( X176 @ X178 @ X179 ) @ X177 )
= ( X176 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) ) )
| ~ ! [X180: term] :
( ( $true
= ( X173 @ id @ ( X176 @ id @ X180 ) @ id ) )
| ( ( X173 @ id @ X180 @ id )
!= $true ) )
| ( $true
= ( X173 @ id @ ( lam @ ( X176 @ sh @ one ) ) @ id ) ) ) )
=> ! [X181: term,X182: subst > term > subst > $o] :
( ~ ! [X183: term] :
( ( $true
!= ( X182 @ id @ X183 @ id ) )
| ( $true
= ( X182 @ id @ ( sub @ X181 @ ( push @ X183 @ id ) ) @ id ) ) )
| ~ ! [X184: subst > term > term] :
( ~ ! [X185: term] :
( ( $true
= ( X182 @ id @ ( X184 @ id @ X185 ) @ id ) )
| ( $true
!= ( X182 @ id @ X185 @ id ) ) )
| ~ ! [X186: subst,X187: subst,X188: term] :
( ( X184 @ ( comp @ X186 @ X187 ) @ ( sub @ X188 @ X187 ) )
= ( sub @ ( X184 @ X186 @ X188 ) @ X187 ) )
| ( $true
= ( X182 @ id @ ( lam @ ( X184 @ sh @ one ) ) @ id ) ) )
| ~ ! [X189: term,X190: subst,X191: subst,X192: subst] :
( ( $true
= ( X182 @ ( comp @ X192 @ X190 ) @ ( sub @ X189 @ X190 ) @ X191 ) )
| ( $true
!= ( X182 @ X192 @ X189 @ ( comp @ X190 @ X191 ) ) ) )
| ( $true
= ( X182 @ id @ ( lam @ X181 ) @ id ) )
| ~ ! [X193: subst,X194: subst,X195: term,X196: subst] :
( ( ( X182 @ ( comp @ X194 @ X193 ) @ ( sub @ X195 @ X193 ) @ X196 )
!= $true )
| ( $true
= ( X182 @ X194 @ X195 @ ( comp @ X193 @ X196 ) ) ) ) ) ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( lamnotvar = $true )
& ( hoasinduction_lem1_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X197: term,X198: subst > term > subst > $o,X199: term] :
( ~ ! [X200: subst,X201: subst,X202: term,X203: subst] :
( ( ( X198 @ ( comp @ X201 @ X203 ) @ ( sub @ X202 @ X203 ) @ X200 )
!= $true )
| ( ( X198 @ X201 @ X202 @ ( comp @ X203 @ X200 ) )
= $true ) )
| ( $true
= ( X198 @ id @ ( ap @ X199 @ X197 ) @ id ) )
| ~ ! [X204: subst,X205: subst,X206: term,X207: subst] :
( ( $true
!= ( X198 @ X204 @ X206 @ ( comp @ X207 @ X205 ) ) )
| ( ( X198 @ ( comp @ X204 @ X207 ) @ ( sub @ X206 @ X207 ) @ X205 )
= $true ) )
| ( $true
!= ( X198 @ id @ X199 @ id ) )
| ( $true
!= ( X198 @ id @ X197 @ id ) )
| ~ ! [X208: term,X209: term] :
( ( ( X198 @ id @ ( ap @ ( sub @ X209 @ id ) @ X208 ) @ id )
= $true )
| ( $true
!= ( X198 @ id @ X208 @ id ) )
| ( $true
!= ( X198 @ id @ X209 @ id ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ! [X210: term,X211: subst] :
( ( comp @ sh @ ( push @ X210 @ X211 ) )
= X211 )
<=> ( axshiftcons = $true ) )
& ( ! [X212: subst > term > subst > $o,X213: term] :
( ( ( X212 @ id @ X213 @ id )
= $true )
| ~ ! [X214: term,X215: term] :
( ( $true
!= ( X212 @ id @ X215 @ id ) )
| ( $true
!= ( X212 @ id @ X214 @ id ) )
| ( ( X212 @ id @ ( ap @ ( sub @ X214 @ id ) @ X215 ) @ id )
= $true ) )
| ~ ! [X216: subst > term > term] :
( ~ ! [X217: term] :
( ( ( X212 @ id @ X217 @ id )
!= $true )
| ( ( X212 @ id @ ( X216 @ id @ X217 ) @ id )
= $true ) )
| ~ ! [X218: term,X219: subst,X220: subst] :
( ( sub @ ( X216 @ X220 @ X218 ) @ X219 )
= ( X216 @ ( comp @ X220 @ X219 ) @ ( sub @ X218 @ X219 ) ) )
| ( ( X212 @ id @ ( lam @ ( X216 @ sh @ one ) ) @ id )
= $true ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction = $true )
<=> ! [X221: term,X222: subst > term > subst > $o] :
( ( $true
= ( X222 @ id @ X221 @ id ) )
| ~ ! [X223: term,X224: subst,X225: subst,X226: subst] :
( ( ( X222 @ X224 @ X223 @ ( comp @ X226 @ X225 ) )
= $true )
| ( ( X222 @ ( comp @ X224 @ X226 ) @ ( sub @ X223 @ X226 ) @ X225 )
!= $true ) )
| ~ ! [X227: subst,X228: term,X229: subst,X230: subst] :
( ( $true
= ( X222 @ ( comp @ X229 @ X227 ) @ ( sub @ X228 @ X227 ) @ X230 ) )
| ( ( X222 @ X229 @ X228 @ ( comp @ X227 @ X230 ) )
!= $true ) )
| ~ ! [X231: term] :
( ( ( var @ ( sub @ X231 @ id ) )
!= $true )
| ( ( X222 @ id @ X231 @ id )
= $true ) )
| ~ ! [X232: subst > term > term] :
( ~ ! [X233: term] :
( ( ( X222 @ id @ X233 @ id )
!= $true )
| ( $true
= ( X222 @ id @ ( X232 @ id @ X233 ) @ id ) ) )
| ~ ! [X234: term,X235: subst,X236: subst] :
( ( X232 @ ( comp @ X236 @ X235 ) @ ( sub @ X234 @ X235 ) )
= ( sub @ ( X232 @ X236 @ X234 ) @ X235 ) )
| ( $true
= ( X222 @ id @ ( lam @ ( X232 @ sh @ one ) ) @ id ) ) )
| ~ ! [X237: term,X238: term] :
( ( $true
!= ( X222 @ id @ X238 @ id ) )
| ( $true
!= ( X222 @ id @ X237 @ id ) )
| ( $true
= ( X222 @ id @ ( ap @ ( sub @ X237 @ id ) @ X238 ) @ id ) ) ) ) )
& ( hoasinduction_lem1 = $true )
& ! [X239: d_term] : ( d_one = X239 )
& ( hoaslamnotap_lthm = $true )
& ( ( induction = $true )
<=> ! [X240: term > $o,X241: term] :
( ( $true
= ( X240 @ X241 ) )
| ~ ! [X242: term] :
( ( $true
= ( X240 @ X242 ) )
| ( $true
!= ( var @ X242 ) ) )
| ~ ! [X243: term,X244: term] :
( ( ( X240 @ X244 )
!= $true )
| ( ( X240 @ ( ap @ X243 @ X244 ) )
= $true )
| ( $true
!= ( X240 @ X243 ) ) )
| ~ ! [X245: term] :
( ( ( X240 @ X245 )
!= $true )
| ( ( X240 @ ( lam @ X245 ) )
= $true ) ) ) )
& ( ( apinj1 = $true )
<=> ! [X246: term,X247: term,X248: term,X249: term] :
( ( X246 = X249 )
| ( ( ap @ X249 @ X247 )
!= ( ap @ X246 @ X248 ) ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X250: term] :
~ ! [X251: subst > term > term] :
~ ! [X252: term,X253: subst] :
( ( sub @ X250 @ ( push @ X252 @ X253 ) )
= ( X251 @ X253 @ X252 ) )
=> ( ! [X254: term] :
( ( sub @ X254 @ id )
= X254 )
=> ! [X255: term,X256: subst > term > subst > $o,X257: term > $o] :
( ~ ! [X258: subst > term > term] :
( ~ ! [X259: term,X260: subst,X261: subst] :
( ( sub @ ( X258 @ X261 @ X259 ) @ X260 )
= ( X258 @ ( comp @ X261 @ X260 ) @ ( sub @ X259 @ X260 ) ) )
| ( ( X256 @ id @ ( lam @ ( X258 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X262: term] :
( ( $true
!= ( X256 @ id @ X262 @ id ) )
| ( $true
= ( X256 @ id @ ( X258 @ id @ X262 ) @ id ) ) ) )
| ~ ! [X263: term] :
( ( X257 @ X263 )
= ( X256 @ id @ X263 @ id ) )
| ( $true
= ( X257 @ ( lam @ X255 ) ) )
| ~ ! [X264: term] :
( ( ( X257 @ X264 )
!= $true )
| ( $true
= ( X257 @ ( sub @ X255 @ ( push @ X264 @ id ) ) ) ) ) ) ) ) )
& ( ! [X265: term,X266: term > $o] :
( ~ ! [X267: term,X268: term] :
( ( $true
= ( X266 @ ( ap @ X267 @ X268 ) ) )
| ( $true
!= ( X266 @ X267 ) )
| ( ( X266 @ X268 )
!= $true ) )
| ~ ! [X269: term] :
( ( $true
= ( X266 @ X269 ) )
| ( ( var @ X269 )
!= $true ) )
| ( $true
= ( X266 @ X265 ) )
| ~ ! [X270: term] :
( ( $true
= ( X266 @ ( lam @ X270 ) ) )
| ~ ! [X271: term] :
( ( ( X266 @ X271 )
!= $true )
| ( ( X266 @ ( sub @ X270 @ ( push @ X271 @ id ) ) )
= $true ) ) ) )
<=> ( induction2 = $true ) )
& ( ! [X272: subst > term > subst > $o,X273: term] :
( ~ ! [X274: term] :
( ( ( X272 @ id @ ( sub @ X273 @ ( push @ X274 @ id ) ) @ id )
= $true )
| ( $true
!= ( X272 @ id @ X274 @ id ) ) )
| ~ ! [X275: subst > term > term] :
( ~ ! [X276: term] :
( ( $true
= ( X272 @ id @ ( X275 @ id @ X276 ) @ id ) )
| ( $true
!= ( X272 @ id @ X276 @ id ) ) )
| ( $true
= ( X272 @ id @ ( lam @ ( X275 @ sh @ one ) ) @ id ) )
| ~ ! [X277: subst,X278: subst,X279: term] :
( ( sub @ ( X275 @ X278 @ X279 ) @ X277 )
= ( X275 @ ( comp @ X278 @ X277 ) @ ( sub @ X279 @ X277 ) ) ) )
| ( ( X272 @ id @ ( lam @ X273 ) @ id )
= $true ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ! [X280: term,X281: term > $o,X282: subst > term > subst > $o,X283: term] :
( ~ ! [X284: term,X285: subst,X286: subst,X287: subst] :
( ( $true
= ( X282 @ ( comp @ X287 @ X286 ) @ ( sub @ X284 @ X286 ) @ X285 ) )
| ( ( X282 @ X287 @ X284 @ ( comp @ X286 @ X285 ) )
!= $true ) )
| ( $true
!= ( X281 @ X283 ) )
| ( $true
= ( X281 @ ( ap @ X283 @ X280 ) ) )
| ~ ! [X288: term,X289: term] :
( ( ( X282 @ id @ X289 @ id )
!= $true )
| ( $true
!= ( X282 @ id @ X288 @ id ) )
| ( ( X282 @ id @ ( ap @ ( sub @ X289 @ id ) @ X288 ) @ id )
= $true ) )
| ~ ! [X290: subst,X291: subst,X292: term,X293: subst] :
( ( ( X282 @ X291 @ X292 @ ( comp @ X293 @ X290 ) )
= $true )
| ( ( X282 @ ( comp @ X291 @ X293 ) @ ( sub @ X292 @ X293 ) @ X290 )
!= $true ) )
| ~ ! [X294: term] :
( ( X282 @ id @ X294 @ id )
= ( X281 @ X294 ) )
| ( ( X281 @ X280 )
!= $true ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ! [X295: subst > term > subst > $o] :
~ ! [X296: term > $o] :
~ ! [X297: term] :
( ( X295 @ id @ X297 @ id )
= ( X296 @ X297 ) )
=> ( ! [X298: term,X299: term > $o] :
( ~ ! [X300: term] :
( ( $true
= ( X299 @ X300 ) )
| ( $true
!= ( var @ X300 ) ) )
| ( $true
= ( X299 @ X298 ) )
| ~ ! [X301: term] :
( ~ ! [X302: term] :
( ( $true
!= ( X299 @ X302 ) )
| ( ( X299 @ ( sub @ X301 @ ( push @ X302 @ id ) ) )
= $true ) )
| ( ( X299 @ ( lam @ X301 ) )
= $true ) )
| ~ ! [X303: term,X304: term] :
( ( $true
!= ( X299 @ X304 ) )
| ( $true
= ( X299 @ ( ap @ X303 @ X304 ) ) )
| ( ( X299 @ X303 )
!= $true ) ) )
=> ( ! [X305: term] :
( ( sub @ X305 @ id )
= X305 )
=> ( ! [X306: term,X307: term > $o,X308: subst > term > subst > $o] :
( ~ ! [X309: term] :
( ( X308 @ id @ X309 @ id )
= ( X307 @ X309 ) )
| ( $true
= ( X307 @ ( lam @ X306 ) ) )
| ~ ! [X310: subst > term > term] :
( ( $true
= ( X308 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id ) )
| ~ ! [X311: subst,X312: term,X313: subst] :
( ( sub @ ( X310 @ X313 @ X312 ) @ X311 )
= ( X310 @ ( comp @ X313 @ X311 ) @ ( sub @ X312 @ X311 ) ) )
| ~ ! [X314: term] :
( ( ( X308 @ id @ X314 @ id )
!= $true )
| ( ( X308 @ id @ ( X310 @ id @ X314 ) @ id )
= $true ) ) )
| ~ ! [X315: term] :
( ( $true
!= ( X307 @ X315 ) )
| ( ( X307 @ ( sub @ X306 @ ( push @ X315 @ id ) ) )
= $true ) ) )
=> ! [X316: subst > term > subst > $o,X317: term] :
( ~ ! [X318: term] :
( ( ( var @ ( sub @ X318 @ id ) )
!= $true )
| ( ( X316 @ id @ X318 @ id )
= $true ) )
| ~ ! [X319: subst > term > term] :
( ~ ! [X320: term] :
( ( $true
= ( X316 @ id @ ( X319 @ id @ X320 ) @ id ) )
| ( $true
!= ( X316 @ id @ X320 @ id ) ) )
| ( ( X316 @ id @ ( lam @ ( X319 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X321: subst,X322: term,X323: subst] :
( ( X319 @ ( comp @ X321 @ X323 ) @ ( sub @ X322 @ X323 ) )
= ( sub @ ( X319 @ X321 @ X322 ) @ X323 ) ) )
| ~ ! [X324: subst,X325: subst,X326: term,X327: subst] :
( ( ( X316 @ X324 @ X326 @ ( comp @ X325 @ X327 ) )
= $true )
| ( $true
!= ( X316 @ ( comp @ X324 @ X325 ) @ ( sub @ X326 @ X325 ) @ X327 ) ) )
| ~ ! [X328: term,X329: term] :
( ( $true
= ( X316 @ id @ ( ap @ ( sub @ X328 @ id ) @ X329 ) @ id ) )
| ( ( X316 @ id @ X329 @ id )
!= $true )
| ( ( X316 @ id @ X328 @ id )
!= $true ) )
| ~ ! [X330: term,X331: subst,X332: subst,X333: subst] :
( ( ( X316 @ X331 @ X330 @ ( comp @ X332 @ X333 ) )
!= $true )
| ( $true
= ( X316 @ ( comp @ X331 @ X332 ) @ ( sub @ X330 @ X332 ) @ X333 ) ) )
| ( ( X316 @ id @ X317 @ id )
= $true ) ) ) ) ) ) )
& ( lamnotap != $true )
& ( ( axidl = $true )
<=> ! [X334: subst] :
( ( comp @ id @ X334 )
= X334 ) )
& ( ! [X335: term > $o,X336: subst,X337: term] :
( ~ ! [X338: term,X339: term] :
( ( ( X335 @ X339 )
!= $true )
| ( ( X335 @ X338 )
!= $true )
| ( $true
= ( X335 @ ( ap @ X339 @ X338 ) ) ) )
| ~ ! [X340: term] :
( ( $true
!= ( var @ X340 ) )
| ( ( X335 @ ( sub @ X340 @ X336 ) )
= $true ) )
| ~ ! [X341: term] :
( ~ ! [X342: term] :
( ( ( X335 @ X342 )
!= $true )
| ( ( X335 @ ( sub @ X341 @ ( push @ X342 @ id ) ) )
= $true ) )
| ( $true
= ( X335 @ ( lam @ X341 ) ) ) )
| ( $true
= ( X335 @ ( sub @ X337 @ X336 ) ) ) )
<=> ( induction2lem = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( d2subst @ d_id )
= id )
& ( ! [X343: term,X344: term > $o,X345: subst] :
~ ! [X346: term > $o] :
~ ! [X347: term] :
( ( X344 @ ( sub @ X347 @ ( push @ X343 @ X345 ) ) )
= ( X346 @ X347 ) )
<=> ( pushprop_lem0 = $true ) )
& ( hoasapnotvar = $true )
& ! [X348: d_subst,X349: d_subst] :
( ( ( d2subst @ X348 )
= ( d2subst @ X349 ) )
=> ( X348 = X349 ) )
& ( pushprop_gthm = $true )
& ( ulamvarind = $true )
& ( ! [X350: subst > term > subst > $o,X351: term] :
( ~ ! [X352: term] :
( ( $true
!= ( X350 @ id @ X352 @ id ) )
| ( $true
= ( X350 @ id @ ( sub @ X351 @ ( push @ X352 @ id ) ) @ id ) ) )
| ~ ! [X353: term,X354: subst > term > term] :
( ~ ! [X355: subst,X356: subst,X357: term] :
( ( sub @ ( sub @ X353 @ ( push @ X357 @ X356 ) ) @ X355 )
= ( sub @ ( X354 @ X356 @ X357 ) @ X355 ) )
| ~ ! [X358: term] :
( ( $true
= ( X350 @ id @ ( X354 @ id @ X358 ) @ id ) )
| ( $true
!= ( X350 @ id @ X358 @ id ) ) )
| ( ( X350 @ id @ ( lam @ ( X354 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X359: subst,X360: subst,X361: term] :
( ( sub @ X353 @ ( push @ ( sub @ X361 @ X360 ) @ ( comp @ X359 @ X360 ) ) )
= ( X354 @ ( comp @ X359 @ X360 ) @ ( sub @ X361 @ X360 ) ) ) )
| ( $true
= ( X350 @ id @ ( lam @ ( sub @ X351 @ ( push @ one @ sh ) ) ) @ id ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X362: term,X363: term,X364: term,X365: term] :
( ( X363 = X364 )
| ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X365 @ id ) @ X364 ) ) ) )
& ( axapp = $true )
& ! [X366: subst] :
? [X367: d_subst] :
( ( d2subst @ X367 )
= X366 )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X368: term,X369: term,X370: term,X371: term] :
( ( ( ap @ X369 @ X368 )
!= ( ap @ X370 @ X371 ) )
| ( X368 = X371 ) )
=> ! [X372: term,X373: term,X374: term,X375: term] :
( ( X373 = X375 )
| ( ( ap @ ( sub @ X374 @ id ) @ X375 )
!= ( ap @ ( sub @ X372 @ id ) @ X373 ) ) ) ) )
& ( induction2_gthm = $true )
& ( ! [X376: term,X377: term] :
( ( ( lam @ X376 )
!= ( lam @ X377 ) )
| ( X376 = X377 ) )
<=> ( laminj = $true ) )
& ! [X378: d_term,X379: d_term] :
( ( ( d2term @ X378 )
= ( d2term @ X379 ) )
=> ( X378 = X379 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( axvarcons = $true )
<=> ! [X380: subst,X381: term] :
( ( sub @ one @ ( push @ X381 @ X380 ) )
= X381 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( ! [X382: subst] :
( ( comp @ id @ X382 )
= X382 )
& ! [X383: subst] :
( ( comp @ X383 @ id )
= X383 ) )
<=> ( substmonoid = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X384: term] :
( ( sub @ X384 @ id )
= X384 )
=> ( ! [X385: term,X386: subst > term > subst > $o] :
( ~ ! [X387: subst > term > term] :
( ~ ! [X388: term] :
( ( $true
= ( X386 @ id @ ( X387 @ id @ X388 ) @ id ) )
| ( ( X386 @ id @ X388 @ id )
!= $true ) )
| ( $true
= ( X386 @ id @ ( lam @ ( X387 @ sh @ one ) ) @ id ) )
| ~ ! [X389: subst,X390: subst,X391: term] :
( ( X387 @ ( comp @ X389 @ X390 ) @ ( sub @ X391 @ X390 ) )
= ( sub @ ( X387 @ X389 @ X391 ) @ X390 ) ) )
| ~ ! [X392: term] :
( ( ( X386 @ id @ ( sub @ X385 @ ( push @ X392 @ id ) ) @ id )
= $true )
| ( ( X386 @ id @ X392 @ id )
!= $true ) )
| ( ( X386 @ id @ ( lam @ ( sub @ X385 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X393: term,X394: subst > term > subst > $o] :
( ~ ! [X395: term] :
( ( ( X394 @ id @ X395 @ id )
!= $true )
| ( $true
= ( X394 @ id @ ( sub @ X393 @ ( push @ X395 @ id ) ) @ id ) ) )
| ~ ! [X396: subst > term > term] :
( ~ ! [X397: term] :
( ( $true
= ( X394 @ id @ ( X396 @ id @ X397 ) @ id ) )
| ( ( X394 @ id @ X397 @ id )
!= $true ) )
| ( $true
= ( X394 @ id @ ( lam @ ( X396 @ sh @ one ) ) @ id ) )
| ~ ! [X398: subst,X399: subst,X400: term] :
( ( X396 @ ( comp @ X399 @ X398 ) @ ( sub @ X400 @ X398 ) )
= ( sub @ ( X396 @ X399 @ X400 ) @ X398 ) ) )
| ( ( X394 @ id @ ( lam @ X393 ) @ id )
= $true ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X401: term] :
~ ! [X402: subst > term > term] :
~ ! [X403: subst,X404: term] :
( ( X402 @ X403 @ X404 )
= ( sub @ X401 @ ( push @ X404 @ X403 ) ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X405: term > $o,X406: term > $o,X407: term,X408: subst] :
( ( ( X406 @ one )
= $true )
| ( ( X405 @ X407 )
!= $true )
| ~ ! [X409: term] :
( ( X406 @ X409 )
= ( X405 @ ( sub @ X409 @ ( push @ X407 @ X408 ) ) ) ) ) )
& ( one
= ( d2term @ d_one ) )
& ( ulamvar1 != $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X410: term,X411: subst] :
( ( sub @ one @ ( push @ X410 @ X411 ) )
= X410 )
=> ( ! [X412: term,X413: subst] :
( ( comp @ sh @ ( push @ X412 @ X413 ) )
= X413 )
=> ( ! [X414: term,X415: term] :
( ( X414 = X415 )
| ( ( lam @ X414 )
!= ( lam @ X415 ) ) )
=> ! [X416: subst > term > term,X417: subst,X418: term,X419: subst > term > term] :
( ( ( lam @ ( X416 @ sh @ one ) )
!= ( lam @ ( X419 @ sh @ one ) ) )
| ( ( X416 @ X417 @ X418 )
= ( X419 @ X417 @ X418 ) )
| ~ ! [X420: subst,X421: term,X422: subst] :
( ( X416 @ ( comp @ X420 @ X422 ) @ ( sub @ X421 @ X422 ) )
= ( sub @ ( X416 @ X420 @ X421 ) @ X422 ) )
| ~ ! [X423: term,X424: subst,X425: subst] :
( ( X419 @ ( comp @ X425 @ X424 ) @ ( sub @ X423 @ X424 ) )
= ( sub @ ( X419 @ X425 @ X423 ) @ X424 ) ) ) ) ) ) )
& ( axassoc = $true )
& ( ! [X426: subst] :
( ( comp @ X426 @ id )
= X426 )
<=> ( axidr = $true ) )
& ! [X427: d_subst] : ( d_id = X427 )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X428: term,X429: term > $o] :
( ( ( X429 @ X428 )
= $true )
| ~ ! [X430: term] :
( ( $true
= ( X429 @ ( lam @ X430 ) ) )
| ~ ! [X431: term] :
( ( ( X429 @ X431 )
!= $true )
| ( ( X429 @ ( sub @ X430 @ ( push @ X431 @ id ) ) )
= $true ) ) )
| ~ ! [X432: term] :
( ( $true
= ( X429 @ X432 ) )
| ( ( var @ X432 )
!= $true ) )
| ~ ! [X433: term,X434: term] :
( ( ( X429 @ ( ap @ X434 @ X433 ) )
= $true )
| ( ( X429 @ X434 )
!= $true )
| ( $true
!= ( X429 @ X433 ) ) ) )
=> ( ! [X435: term,X436: term,X437: subst > term > subst > $o] :
( ~ ! [X438: term,X439: subst,X440: subst,X441: subst] :
( ( $true
= ( X437 @ ( comp @ X439 @ X440 ) @ ( sub @ X438 @ X440 ) @ X441 ) )
| ( $true
!= ( X437 @ X439 @ X438 @ ( comp @ X440 @ X441 ) ) ) )
| ( ( X437 @ id @ ( ap @ X436 @ X435 ) @ id )
= $true )
| ( ( X437 @ id @ X436 @ id )
!= $true )
| ~ ! [X442: subst,X443: subst,X444: term,X445: subst] :
( ( ( X437 @ X442 @ X444 @ ( comp @ X443 @ X445 ) )
= $true )
| ( $true
!= ( X437 @ ( comp @ X442 @ X443 ) @ ( sub @ X444 @ X443 ) @ X445 ) ) )
| ~ ! [X446: term,X447: term] :
( ( ( X437 @ id @ X446 @ id )
!= $true )
| ( $true
!= ( X437 @ id @ X447 @ id ) )
| ( $true
= ( X437 @ id @ ( ap @ ( sub @ X447 @ id ) @ X446 ) @ id ) ) )
| ( $true
!= ( X437 @ id @ X435 @ id ) ) )
=> ( ! [X448: term,X449: subst > term > subst > $o] :
( ~ ! [X450: subst > term > term] :
( ( ( X449 @ id @ ( lam @ ( X450 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X451: subst,X452: subst,X453: term] :
( ( X450 @ ( comp @ X452 @ X451 ) @ ( sub @ X453 @ X451 ) )
= ( sub @ ( X450 @ X452 @ X453 ) @ X451 ) )
| ~ ! [X454: term] :
( ( $true
!= ( X449 @ id @ X454 @ id ) )
| ( ( X449 @ id @ ( X450 @ id @ X454 ) @ id )
= $true ) ) )
| ( ( X449 @ id @ ( lam @ X448 ) @ id )
= $true )
| ~ ! [X455: term] :
( ( ( X449 @ id @ ( sub @ X448 @ ( push @ X455 @ id ) ) @ id )
= $true )
| ( $true
!= ( X449 @ id @ X455 @ id ) ) )
| ~ ! [X456: subst,X457: subst,X458: term,X459: subst] :
( ( ( X449 @ ( comp @ X459 @ X457 ) @ ( sub @ X458 @ X457 ) @ X456 )
= $true )
| ( ( X449 @ X459 @ X458 @ ( comp @ X457 @ X456 ) )
!= $true ) )
| ~ ! [X460: subst,X461: subst,X462: term,X463: subst] :
( ( $true
!= ( X449 @ ( comp @ X463 @ X460 ) @ ( sub @ X462 @ X460 ) @ X461 ) )
| ( $true
= ( X449 @ X463 @ X462 @ ( comp @ X460 @ X461 ) ) ) ) )
=> ! [X464: term,X465: subst > term > subst > $o] :
( ~ ! [X466: term,X467: term] :
( ( $true
!= ( X465 @ id @ X466 @ id ) )
| ( $true
!= ( X465 @ id @ X467 @ id ) )
| ( ( X465 @ id @ ( ap @ ( sub @ X466 @ id ) @ X467 ) @ id )
= $true ) )
| ~ ! [X468: subst > term > term] :
( ( ( X465 @ id @ ( lam @ ( X468 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X469: term] :
( ( $true
!= ( X465 @ id @ X469 @ id ) )
| ( $true
= ( X465 @ id @ ( X468 @ id @ X469 ) @ id ) ) )
| ~ ! [X470: subst,X471: subst,X472: term] :
( ( sub @ ( X468 @ X470 @ X472 ) @ X471 )
= ( X468 @ ( comp @ X470 @ X471 ) @ ( sub @ X472 @ X471 ) ) ) )
| ( ( X465 @ id @ X464 @ id )
= $true )
| ~ ! [X473: subst,X474: term,X475: subst,X476: subst] :
( ( $true
!= ( X465 @ X476 @ X474 @ ( comp @ X475 @ X473 ) ) )
| ( $true
= ( X465 @ ( comp @ X476 @ X475 ) @ ( sub @ X474 @ X475 ) @ X473 ) ) )
| ~ ! [X477: term,X478: subst,X479: subst,X480: subst] :
( ( ( X465 @ ( comp @ X478 @ X479 ) @ ( sub @ X477 @ X479 ) @ X480 )
!= $true )
| ( $true
= ( X465 @ X478 @ X477 @ ( comp @ X479 @ X480 ) ) ) )
| ~ ! [X481: term] :
( ( ( X465 @ id @ X481 @ id )
= $true )
| ( $true
!= ( var @ ( sub @ X481 @ id ) ) ) ) ) ) ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( termmset_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X482: term > $o,X483: term,X484: subst,X485: term > $o,X486: term] :
( ( ( X482 @ X483 )
!= $true )
| ( ( X485 @ ( sub @ X483 @ ( push @ X486 @ X484 ) ) )
= $true ) )
<=> ( pushprop_lem1 = $true ) )
& ( ( ! [X487: subst,X488: term] :
( ( sub @ one @ ( push @ X488 @ X487 ) )
= X488 )
=> ! [X489: term > $o,X490: term > $o,X491: term,X492: subst] :
( ( $true
= ( X490 @ one ) )
| ( ( X489 @ X491 )
!= $true )
| ~ ! [X493: term] :
( ( X490 @ X493 )
= ( X489 @ ( sub @ X493 @ ( push @ X491 @ X492 ) ) ) ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( pushprop_lem0_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ( ( ! [X494: term] :
( ( sub @ X494 @ id )
= X494 )
=> ! [X495: term > $o,X496: subst > term > subst > $o,X497: term] :
( ~ ! [X498: subst,X499: subst,X500: term,X501: subst] :
( ( ( X496 @ X498 @ X500 @ ( comp @ X499 @ X501 ) )
!= $true )
| ( ( X496 @ ( comp @ X498 @ X499 ) @ ( sub @ X500 @ X499 ) @ X501 )
= $true ) )
| ( $true
= ( X495 @ ( lam @ X497 ) ) )
| ~ ! [X502: subst,X503: term,X504: subst,X505: subst] :
( ( ( X496 @ ( comp @ X502 @ X505 ) @ ( sub @ X503 @ X505 ) @ X504 )
!= $true )
| ( $true
= ( X496 @ X502 @ X503 @ ( comp @ X505 @ X504 ) ) ) )
| ~ ! [X506: term] :
( ( ( X495 @ ( sub @ X497 @ ( push @ X506 @ id ) ) )
= $true )
| ( ( X495 @ X506 )
!= $true ) )
| ~ ! [X507: term] :
( ( X496 @ id @ X507 @ id )
= ( X495 @ X507 ) )
| ~ ! [X508: subst > term > term] :
( ~ ! [X509: term] :
( ( $true
= ( X496 @ id @ ( X508 @ id @ X509 ) @ id ) )
| ( ( X496 @ id @ X509 @ id )
!= $true ) )
| ( $true
= ( X496 @ id @ ( lam @ ( X508 @ sh @ one ) ) @ id ) )
| ~ ! [X510: subst,X511: subst,X512: term] :
( ( sub @ ( X508 @ X510 @ X512 ) @ X511 )
= ( X508 @ ( comp @ X510 @ X511 ) @ ( sub @ X512 @ X511 ) ) ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X513: subst] :
( ( comp @ id @ X513 )
= X513 )
=> ( ! [X514: subst] :
( ( comp @ X514 @ id )
= X514 )
=> ( ! [X515: subst] :
( ( comp @ X515 @ id )
= X515 )
& ! [X516: subst] :
( ( comp @ id @ X516 )
= X516 ) ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( hoasapinj1 = $true )
<=> ! [X517: term,X518: term,X519: term,X520: term] :
( ( X517 = X518 )
| ( ( ap @ ( sub @ X518 @ id ) @ X520 )
!= ( ap @ ( sub @ X517 @ id ) @ X519 ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( axvarid = $true )
<=> ! [X521: term] :
( ( sub @ X521 @ id )
= X521 ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X522: subst > term > subst > $o,X523: term] :
( ( ( X522 @ id @ ( lam @ ( sub @ X523 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X524: term] :
( ( $true
= ( X522 @ id @ ( sub @ X523 @ ( push @ X524 @ id ) ) @ id ) )
| ( ( X522 @ id @ X524 @ id )
!= $true ) )
| ~ ! [X525: subst > term > term] :
( ( $true
= ( X522 @ id @ ( lam @ ( X525 @ sh @ one ) ) @ id ) )
| ~ ! [X526: subst,X527: term,X528: subst] :
( ( X525 @ ( comp @ X528 @ X526 ) @ ( sub @ X527 @ X526 ) )
= ( sub @ ( X525 @ X528 @ X527 ) @ X526 ) )
| ~ ! [X529: term] :
( ( $true
!= ( X522 @ id @ X529 @ id ) )
| ( ( X522 @ id @ ( X525 @ id @ X529 ) @ id )
= $true ) ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X530: term,X531: term,X532: subst > term > term] :
( ~ ! [X533: subst,X534: term,X535: subst] :
( ( X532 @ ( comp @ X535 @ X533 ) @ ( sub @ X534 @ X533 ) )
= ( sub @ ( X532 @ X535 @ X534 ) @ X533 ) )
| ( ( ap @ ( sub @ X531 @ id ) @ X530 )
!= ( lam @ ( X532 @ sh @ one ) ) ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X536: term] :
( ( sub @ X536 @ id )
= X536 )
=> ( ! [X537: term,X538: term,X539: term,X540: term] :
( ( ( ap @ X537 @ X540 )
!= ( ap @ X538 @ X539 ) )
| ( X537 = X538 ) )
=> ! [X541: term,X542: term,X543: term,X544: term] :
( ( ( ap @ ( sub @ X544 @ id ) @ X542 )
!= ( ap @ ( sub @ X543 @ id ) @ X541 ) )
| ( X543 = X544 ) ) ) ) )
& ( ! [X545: term > $o,X546: subst,X547: term] :
~ ! [X548: term > $o] :
~ ! [X549: term] :
( ( X548 @ X549 )
= ( X545 @ ( sub @ X549 @ ( push @ X547 @ X546 ) ) ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( hoaslamnotvar = $true ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
( ( ! [X0: subst] :
( ( push @ ( sub @ one @ X0 ) @ ( comp @ sh @ X0 ) )
= X0 )
<=> axscons )
& axabs
& hoasinduction_lem1_lthm
& ( hoasinduction_lem3b_lthm
<=> ! [X1: term] :
~ ! [X2: subst > term > term] :
( ( sub @ X1 @ ( push @ one @ sh ) )
!= ( X2 @ sh @ one ) ) )
& axclos
& ! [X3: term] :
? [X4: d_term] :
( ( d2term @ X4 )
= X3 )
& hoasinduction_lem1v2_gthm
& ( termmset_lthm
<=> ( ! [X5: term] :
( ( sub @ X5 @ id )
= X5 )
=> ! [X6: term] :
( ( sub @ X6 @ id )
= X6 ) ) )
& axvarshift
& ( hoasinduction_p_and_p_prime
= ( ^ [X7: subst > term > subst > $o,X8: term > $o] :
! [X9: term] :
( ( X7 @ id @ X9 @ id )
= ( X8 @ X9 ) ) ) )
& ( ( ! [X10: term,X11: subst] :
( ( sub @ one @ ( push @ X10 @ X11 ) )
= X10 )
=> ( ! [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
= X12 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X15: term > $o,X16: term] :
( ~ ! [X17: term] :
( ( X15 @ X17 )
| ~ ( var @ X17 ) )
| ( X15 @ X16 )
| ~ ! [X18: term] :
( ~ ( X15 @ X18 )
| ( X15 @ ( lam @ X18 ) ) )
| ~ ! [X19: term,X20: term] :
( ( X15 @ ( ap @ X20 @ X19 ) )
| ~ ( X15 @ X20 )
| ~ ( X15 @ X19 ) ) )
=> ! [X21: subst,X22: term,X23: term > $o] :
( ~ ! [X24: term] :
( ( X23 @ ( sub @ X24 @ X21 ) )
| ~ ( var @ X24 ) )
| ( X23 @ ( sub @ X22 @ X21 ) )
| ~ ! [X25: term,X26: term] :
( ~ ( X23 @ X26 )
| ~ ( X23 @ X25 )
| ( X23 @ ( ap @ X26 @ X25 ) ) )
| ~ ! [X27: term] :
( ( X23 @ ( lam @ X27 ) )
| ~ ! [X28: term] :
( ( X23 @ ( sub @ X27 @ ( push @ X28 @ id ) ) )
| ~ ( X23 @ X28 ) ) ) ) ) ) ) )
<=> induction2lem_lthm )
& ulamvarsh
& hoasinduction_gthm
& pushprop_lthm
& hoaslamnotvar_lthm
& ( ! [X29: subst > term > subst > $o] :
~ ! [X30: term > $o] :
~ ! [X31: term] :
( ( X30 @ X31 )
= ( X29 @ id @ X31 @ id ) )
<=> hoasinduction_lem0_lthm )
& hoasinduction_lem3v2_gthm
& hoaslamnotvar_gthm
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lem3v2
<=> ! [X32: subst > term > subst > $o,X33: term,X34: term > $o] :
( ~ ! [X35: subst,X36: subst,X37: term,X38: subst] :
( ( X32 @ ( comp @ X38 @ X35 ) @ ( sub @ X37 @ X35 ) @ X36 )
| ~ ( X32 @ X38 @ X37 @ ( comp @ X35 @ X36 ) ) )
| ~ ! [X39: term] :
( ( X34 @ ( sub @ X33 @ ( push @ X39 @ id ) ) )
| ~ ( X34 @ X39 ) )
| ~ ! [X40: subst > term > term] :
( ( X32 @ id @ ( lam @ ( X40 @ sh @ one ) ) @ id )
| ~ ! [X41: subst,X42: subst,X43: term] :
( ( sub @ ( X40 @ X41 @ X43 ) @ X42 )
= ( X40 @ ( comp @ X41 @ X42 ) @ ( sub @ X43 @ X42 ) ) )
| ~ ! [X44: term] :
( ( X32 @ id @ ( X40 @ id @ X44 ) @ id )
| ~ ( X32 @ id @ X44 @ id ) ) )
| ~ ! [X45: subst,X46: subst,X47: term,X48: subst] :
( ~ ( X32 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) @ X48 )
| ( X32 @ X45 @ X47 @ ( comp @ X46 @ X48 ) ) )
| ~ ! [X49: term] :
( ( X32 @ id @ X49 @ id )
= ( X34 @ X49 ) )
| ( X34 @ ( lam @ X33 ) ) ) )
& ( ( ! [X50: subst > term > subst > $o] :
~ ! [X51: term > $o] :
~ ! [X52: term] :
( ( X51 @ X52 )
= ( X50 @ id @ X52 @ id ) )
=> ( ! [X53: term > $o,X54: term] :
( ~ ! [X55: term,X56: term] :
( ( X53 @ ( ap @ X56 @ X55 ) )
| ~ ( X53 @ X56 )
| ~ ( X53 @ X55 ) )
| ~ ! [X57: term] :
( ~ ! [X58: term] :
( ( X53 @ ( sub @ X57 @ ( push @ X58 @ id ) ) )
| ~ ( X53 @ X58 ) )
| ( X53 @ ( lam @ X57 ) ) )
| ( X53 @ X54 )
| ~ ! [X59: term] :
( ( X53 @ X59 )
| ~ ( var @ X59 ) ) )
=> ( ! [X60: term] :
( ( sub @ X60 @ id )
= X60 )
=> ( ! [X61: subst > term > subst > $o,X62: term,X63: term > $o] :
( ~ ! [X64: term] :
( ~ ( X63 @ X64 )
| ( X63 @ ( sub @ X62 @ ( push @ X64 @ id ) ) ) )
| ~ ! [X65: term] :
( ( X63 @ X65 )
= ( X61 @ id @ X65 @ id ) )
| ( X63 @ ( lam @ X62 ) )
| ~ ! [X66: subst > term > term] :
( ~ ! [X67: subst,X68: term,X69: subst] :
( ( X66 @ ( comp @ X67 @ X69 ) @ ( sub @ X68 @ X69 ) )
= ( sub @ ( X66 @ X67 @ X68 ) @ X69 ) )
| ~ ! [X70: term] :
( ~ ( X61 @ id @ X70 @ id )
| ( X61 @ id @ ( X66 @ id @ X70 ) @ id ) )
| ( X61 @ id @ ( lam @ ( X66 @ sh @ one ) ) @ id ) ) )
=> ! [X71: subst > term > subst > $o,X72: term] :
( ~ ! [X73: term,X74: term] :
( ~ ( X71 @ id @ X74 @ id )
| ( X71 @ id @ ( ap @ ( sub @ X74 @ id ) @ X73 ) @ id )
| ~ ( X71 @ id @ X73 @ id ) )
| ~ ! [X75: subst > term > term] :
( ( X71 @ id @ ( lam @ ( X75 @ sh @ one ) ) @ id )
| ~ ! [X76: subst,X77: subst,X78: term] :
( ( sub @ ( X75 @ X77 @ X78 ) @ X76 )
= ( X75 @ ( comp @ X77 @ X76 ) @ ( sub @ X78 @ X76 ) ) )
| ~ ! [X79: term] :
( ~ ( X71 @ id @ X79 @ id )
| ( X71 @ id @ ( X75 @ id @ X79 ) @ id ) ) )
| ( X71 @ id @ X72 @ id ) ) ) ) ) )
<=> hoasinduction_no_psi_cond_lthm )
& pushprop_lem2v2_lthm
& ( shinj
<=> ! [X80: term,X81: term] :
( ( ( sub @ X80 @ sh )
!= ( sub @ X81 @ sh ) )
| ( X80 = X81 ) ) )
& ( ! [X82: term,X83: term,X84: term,X85: term] :
( ( X84 = X85 )
| ( ( ap @ X83 @ X84 )
!= ( ap @ X82 @ X85 ) ) )
<=> apinj2 )
& ( hoasinduction_lem3v2a
<=> ! [X86: subst > term > subst > $o,X87: term,X88: term > $o] :
( ~ ! [X89: term] :
( ( X88 @ ( sub @ X87 @ ( push @ X89 @ id ) ) )
| ~ ( X88 @ X89 ) )
| ~ ! [X90: term] :
( ( X86 @ id @ X90 @ id )
= ( X88 @ X90 ) )
| ~ ! [X91: subst > term > term] :
( ~ ! [X92: subst,X93: subst,X94: term] :
( ( X91 @ ( comp @ X93 @ X92 ) @ ( sub @ X94 @ X92 ) )
= ( sub @ ( X91 @ X93 @ X94 ) @ X92 ) )
| ~ ! [X95: term] :
( ~ ( X86 @ id @ X95 @ id )
| ( X86 @ id @ ( X91 @ id @ X95 ) @ id ) )
| ( X86 @ id @ ( lam @ ( X91 @ sh @ one ) ) @ id ) )
| ( X88 @ ( lam @ X87 ) ) ) )
& pushprop
& ( induction2_lthm
<=> ( ! [X96: term] :
( ( sub @ X96 @ id )
= X96 )
=> ( ! [X97: subst,X98: term > $o,X99: term] :
( ~ ! [X100: term,X101: term] :
( ~ ( X98 @ X100 )
| ~ ( X98 @ X101 )
| ( X98 @ ( ap @ X100 @ X101 ) ) )
| ~ ! [X102: term] :
( ~ ( var @ X102 )
| ( X98 @ ( sub @ X102 @ X97 ) ) )
| ( X98 @ ( sub @ X99 @ X97 ) )
| ~ ! [X103: term] :
( ( X98 @ ( lam @ X103 ) )
| ~ ! [X104: term] :
( ( X98 @ ( sub @ X103 @ ( push @ X104 @ id ) ) )
| ~ ( X98 @ X104 ) ) ) )
=> ! [X105: term > $o,X106: term] :
( ~ ! [X107: term,X108: term] :
( ~ ( X105 @ X108 )
| ( X105 @ ( ap @ X107 @ X108 ) )
| ~ ( X105 @ X107 ) )
| ( X105 @ X106 )
| ~ ! [X109: term] :
( ~ ! [X110: term] :
( ( X105 @ ( sub @ X109 @ ( push @ X110 @ id ) ) )
| ~ ( X105 @ X110 ) )
| ( X105 @ ( lam @ X109 ) ) )
| ~ ! [X111: term] :
( ( X105 @ X111 )
| ~ ( var @ X111 ) ) ) ) ) )
& ( ! [X112: term] :
( ( sub @ X112 @ id )
= X112 )
<=> termmset )
& ( hoasinduction_lem0
<=> ! [X113: subst > term > subst > $o] :
~ ! [X114: term > $o] :
~ ! [X115: term] :
( ( X113 @ id @ X115 @ id )
= ( X114 @ X115 ) ) )
& substmonoid_gthm
& ( ( ! [X116: subst,X117: term] :
( ( sub @ one @ ( push @ X117 @ X116 ) )
= X117 )
=> ( ! [X118: term,X119: subst] :
( ( comp @ sh @ ( push @ X118 @ X119 ) )
= X119 )
=> ! [X120: term > $o,X121: term,X122: subst,X123: term > $o,X124: term] :
( ( X120 @ ( sub @ X124 @ ( push @ X121 @ X122 ) ) )
| ~ ( X123 @ X124 ) ) ) )
<=> pushprop_lem1_lthm )
& ( ! [X125: term,X126: subst > term > subst > $o] :
( ~ ! [X127: term] :
( ( X126 @ id @ ( sub @ X125 @ ( push @ X127 @ id ) ) @ id )
| ~ ( X126 @ id @ X127 @ id ) )
| ~ ! [X128: subst > term > term] :
( ~ ! [X129: subst,X130: term,X131: subst] :
( ( sub @ ( X128 @ X131 @ X130 ) @ X129 )
= ( X128 @ ( comp @ X131 @ X129 ) @ ( sub @ X130 @ X129 ) ) )
| ~ ! [X132: term] :
( ( X126 @ id @ ( X128 @ id @ X132 ) @ id )
| ~ ( X126 @ id @ X132 @ id ) )
| ( X126 @ id @ ( lam @ ( X128 @ sh @ one ) ) @ id ) )
| ~ ! [X133: subst,X134: subst,X135: subst,X136: term] :
( ~ ( X126 @ X134 @ X136 @ ( comp @ X133 @ X135 ) )
| ( X126 @ ( comp @ X134 @ X133 ) @ ( sub @ X136 @ X133 ) @ X135 ) )
| ~ ! [X137: subst,X138: subst,X139: term,X140: subst] :
( ~ ( X126 @ ( comp @ X138 @ X140 ) @ ( sub @ X139 @ X140 ) @ X137 )
| ( X126 @ X138 @ X139 @ ( comp @ X140 @ X137 ) ) )
| ( X126 @ id @ ( lam @ X125 ) @ id ) )
<=> hoasinduction_lem3 )
& ( ( ^ [X141: subst,X142: subst > term > term] : ( d2term @ d_one ) )
= hoaslam )
& ( hoasinduction_lem3v2_f
<=> ! [X143: term] :
~ ! [X144: subst > term > term] :
~ ! [X145: term,X146: subst] :
( ( sub @ X143 @ ( push @ X145 @ X146 ) )
= ( X144 @ X146 @ X145 ) ) )
& ( ! [X147: term] :
~ ! [X148: subst > term > term] :
( ( X148 @ sh @ one )
!= ( sub @ X147 @ ( push @ one @ sh ) ) )
<=> hoasinduction_lem3b )
& hoaslaminj_gthm
& pushprop_lem1_gthm
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_p_and_p_prime
= ( ^ [X149: term,X150: subst,X151: term > $o,X152: term > $o] :
! [X153: term] :
( ( X152 @ X153 )
= ( X151 @ ( sub @ X153 @ ( push @ X149 @ X150 ) ) ) ) ) )
& pushprop_lem3v2
& ( hoaslaminj
<=> ! [X154: subst > term > term,X155: subst > term > term,X156: subst,X157: term] :
( ~ ! [X158: term,X159: subst,X160: subst] :
( ( X154 @ ( comp @ X159 @ X160 ) @ ( sub @ X158 @ X160 ) )
= ( sub @ ( X154 @ X159 @ X158 ) @ X160 ) )
| ( ( X154 @ X156 @ X157 )
= ( X155 @ X156 @ X157 ) )
| ~ ! [X161: subst,X162: subst,X163: term] :
( ( X155 @ ( comp @ X162 @ X161 ) @ ( sub @ X163 @ X161 ) )
= ( sub @ ( X155 @ X162 @ X163 ) @ X161 ) )
| ( ( lam @ ( X154 @ sh @ one ) )
!= ( lam @ ( X155 @ sh @ one ) ) ) ) )
& ~ ( var @ ( d2term @ d_one ) )
& ( ! [X164: subst > term > subst > $o,X165: term] :
( ~ ! [X166: term] :
( ( X164 @ id @ ( sub @ X165 @ ( push @ X166 @ id ) ) @ id )
| ~ ( X164 @ id @ X166 @ id ) )
| ~ ! [X167: subst > term > term] :
( ( X164 @ id @ ( lam @ ( X167 @ sh @ one ) ) @ id )
| ~ ! [X168: term] :
( ~ ( X164 @ id @ X168 @ id )
| ( X164 @ id @ ( X167 @ id @ X168 ) @ id ) )
| ~ ! [X169: subst,X170: subst,X171: term] :
( ( X167 @ ( comp @ X169 @ X170 ) @ ( sub @ X171 @ X170 ) )
= ( sub @ ( X167 @ X169 @ X171 ) @ X170 ) ) )
| ( X164 @ id @ ( lam @ ( sub @ X165 @ ( push @ one @ sh ) ) ) @ id ) )
<=> hoasinduction_lem3aa )
& axmap
& apnotvar
& ( ( ! [X172: term] :
( ( sub @ X172 @ id )
= X172 )
=> ( ! [X173: subst > term > subst > $o,X174: term] :
( ~ ! [X175: term] :
( ( X173 @ id @ ( sub @ X174 @ ( push @ X175 @ id ) ) @ id )
| ~ ( X173 @ id @ X175 @ id ) )
| ( X173 @ id @ ( lam @ ( sub @ X174 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X176: subst > term > term] :
( ~ ! [X177: subst,X178: subst,X179: term] :
( ( sub @ ( X176 @ X178 @ X179 ) @ X177 )
= ( X176 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) ) )
| ~ ! [X180: term] :
( ( X173 @ id @ ( X176 @ id @ X180 ) @ id )
| ~ ( X173 @ id @ X180 @ id ) )
| ( X173 @ id @ ( lam @ ( X176 @ sh @ one ) ) @ id ) ) )
=> ! [X181: term,X182: subst > term > subst > $o] :
( ~ ! [X183: term] :
( ~ ( X182 @ id @ X183 @ id )
| ( X182 @ id @ ( sub @ X181 @ ( push @ X183 @ id ) ) @ id ) )
| ~ ! [X184: subst > term > term] :
( ~ ! [X185: term] :
( ( X182 @ id @ ( X184 @ id @ X185 ) @ id )
| ~ ( X182 @ id @ X185 @ id ) )
| ~ ! [X186: subst,X187: subst,X188: term] :
( ( X184 @ ( comp @ X186 @ X187 ) @ ( sub @ X188 @ X187 ) )
= ( sub @ ( X184 @ X186 @ X188 ) @ X187 ) )
| ( X182 @ id @ ( lam @ ( X184 @ sh @ one ) ) @ id ) )
| ~ ! [X189: term,X190: subst,X191: subst,X192: subst] :
( ( X182 @ ( comp @ X192 @ X190 ) @ ( sub @ X189 @ X190 ) @ X191 )
| ~ ( X182 @ X192 @ X189 @ ( comp @ X190 @ X191 ) ) )
| ( X182 @ id @ ( lam @ X181 ) @ id )
| ~ ! [X193: subst,X194: subst,X195: term,X196: subst] :
( ~ ( X182 @ ( comp @ X194 @ X193 ) @ ( sub @ X195 @ X193 ) @ X196 )
| ( X182 @ X194 @ X195 @ ( comp @ X193 @ X196 ) ) ) ) ) )
<=> hoasinduction_lem3_lthm )
& lamnotvar
& hoasinduction_lem1_gthm
& hoasinduction_lem2v2_gthm
& ( hoasinduction_lem2
<=> ! [X197: term,X198: subst > term > subst > $o,X199: term] :
( ~ ! [X200: subst,X201: subst,X202: term,X203: subst] :
( ~ ( X198 @ ( comp @ X201 @ X203 ) @ ( sub @ X202 @ X203 ) @ X200 )
| ( X198 @ X201 @ X202 @ ( comp @ X203 @ X200 ) ) )
| ( X198 @ id @ ( ap @ X199 @ X197 ) @ id )
| ~ ! [X204: subst,X205: subst,X206: term,X207: subst] :
( ~ ( X198 @ X204 @ X206 @ ( comp @ X207 @ X205 ) )
| ( X198 @ ( comp @ X204 @ X207 ) @ ( sub @ X206 @ X207 ) @ X205 ) )
| ~ ( X198 @ id @ X199 @ id )
| ~ ( X198 @ id @ X197 @ id )
| ~ ! [X208: term,X209: term] :
( ( X198 @ id @ ( ap @ ( sub @ X209 @ id ) @ X208 ) @ id )
| ~ ( X198 @ id @ X208 @ id )
| ~ ( X198 @ id @ X209 @ id ) ) ) )
& pushprop_lem2v2
& ( ! [X210: term,X211: subst] :
( ( comp @ sh @ ( push @ X210 @ X211 ) )
= X211 )
<=> axshiftcons )
& ( ! [X212: subst > term > subst > $o,X213: term] :
( ( X212 @ id @ X213 @ id )
| ~ ! [X214: term,X215: term] :
( ~ ( X212 @ id @ X215 @ id )
| ~ ( X212 @ id @ X214 @ id )
| ( X212 @ id @ ( ap @ ( sub @ X214 @ id ) @ X215 ) @ id ) )
| ~ ! [X216: subst > term > term] :
( ~ ! [X217: term] :
( ~ ( X212 @ id @ X217 @ id )
| ( X212 @ id @ ( X216 @ id @ X217 ) @ id ) )
| ~ ! [X218: term,X219: subst,X220: subst] :
( ( sub @ ( X216 @ X220 @ X218 ) @ X219 )
= ( X216 @ ( comp @ X220 @ X219 ) @ ( sub @ X218 @ X219 ) ) )
| ( X212 @ id @ ( lam @ ( X216 @ sh @ one ) ) @ id ) ) )
<=> hoasinduction_no_psi_cond )
& hoasinduction_lem3_gthm
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( hoasinduction
<=> ! [X221: term,X222: subst > term > subst > $o] :
( ( X222 @ id @ X221 @ id )
| ~ ! [X223: term,X224: subst,X225: subst,X226: subst] :
( ( X222 @ X224 @ X223 @ ( comp @ X226 @ X225 ) )
| ~ ( X222 @ ( comp @ X224 @ X226 ) @ ( sub @ X223 @ X226 ) @ X225 ) )
| ~ ! [X227: subst,X228: term,X229: subst,X230: subst] :
( ( X222 @ ( comp @ X229 @ X227 ) @ ( sub @ X228 @ X227 ) @ X230 )
| ~ ( X222 @ X229 @ X228 @ ( comp @ X227 @ X230 ) ) )
| ~ ! [X231: term] :
( ~ ( var @ ( sub @ X231 @ id ) )
| ( X222 @ id @ X231 @ id ) )
| ~ ! [X232: subst > term > term] :
( ~ ! [X233: term] :
( ~ ( X222 @ id @ X233 @ id )
| ( X222 @ id @ ( X232 @ id @ X233 ) @ id ) )
| ~ ! [X234: term,X235: subst,X236: subst] :
( ( X232 @ ( comp @ X236 @ X235 ) @ ( sub @ X234 @ X235 ) )
= ( sub @ ( X232 @ X236 @ X234 ) @ X235 ) )
| ( X222 @ id @ ( lam @ ( X232 @ sh @ one ) ) @ id ) )
| ~ ! [X237: term,X238: term] :
( ~ ( X222 @ id @ X238 @ id )
| ~ ( X222 @ id @ X237 @ id )
| ( X222 @ id @ ( ap @ ( sub @ X237 @ id ) @ X238 ) @ id ) ) ) )
& hoasinduction_lem1
& ! [X239: d_term] : ( d_one = X239 )
& hoaslamnotap_lthm
& ( induction
<=> ! [X240: term > $o,X241: term] :
( ( X240 @ X241 )
| ~ ! [X242: term] :
( ( X240 @ X242 )
| ~ ( var @ X242 ) )
| ~ ! [X243: term,X244: term] :
( ~ ( X240 @ X244 )
| ( X240 @ ( ap @ X243 @ X244 ) )
| ~ ( X240 @ X243 ) )
| ~ ! [X245: term] :
( ~ ( X240 @ X245 )
| ( X240 @ ( lam @ X245 ) ) ) ) )
& ( apinj1
<=> ! [X246: term,X247: term,X248: term,X249: term] :
( ( X246 = X249 )
| ( ( ap @ X249 @ X247 )
!= ( ap @ X246 @ X248 ) ) ) )
& ( hoasinduction_lem3v2a_lthm
<=> ( ! [X250: term] :
~ ! [X251: subst > term > term] :
~ ! [X252: term,X253: subst] :
( ( sub @ X250 @ ( push @ X252 @ X253 ) )
= ( X251 @ X253 @ X252 ) )
=> ( ! [X254: term] :
( ( sub @ X254 @ id )
= X254 )
=> ! [X255: term,X256: subst > term > subst > $o,X257: term > $o] :
( ~ ! [X258: subst > term > term] :
( ~ ! [X259: term,X260: subst,X261: subst] :
( ( sub @ ( X258 @ X261 @ X259 ) @ X260 )
= ( X258 @ ( comp @ X261 @ X260 ) @ ( sub @ X259 @ X260 ) ) )
| ( X256 @ id @ ( lam @ ( X258 @ sh @ one ) ) @ id )
| ~ ! [X262: term] :
( ~ ( X256 @ id @ X262 @ id )
| ( X256 @ id @ ( X258 @ id @ X262 ) @ id ) ) )
| ~ ! [X263: term] :
( ( X257 @ X263 )
= ( X256 @ id @ X263 @ id ) )
| ( X257 @ ( lam @ X255 ) )
| ~ ! [X264: term] :
( ~ ( X257 @ X264 )
| ( X257 @ ( sub @ X255 @ ( push @ X264 @ id ) ) ) ) ) ) ) )
& ( ! [X265: term,X266: term > $o] :
( ~ ! [X267: term,X268: term] :
( ( X266 @ ( ap @ X267 @ X268 ) )
| ~ ( X266 @ X267 )
| ~ ( X266 @ X268 ) )
| ~ ! [X269: term] :
( ( X266 @ X269 )
| ~ ( var @ X269 ) )
| ( X266 @ X265 )
| ~ ! [X270: term] :
( ( X266 @ ( lam @ X270 ) )
| ~ ! [X271: term] :
( ~ ( X266 @ X271 )
| ( X266 @ ( sub @ X270 @ ( push @ X271 @ id ) ) ) ) ) )
<=> induction2 )
& ( ! [X272: subst > term > subst > $o,X273: term] :
( ~ ! [X274: term] :
( ( X272 @ id @ ( sub @ X273 @ ( push @ X274 @ id ) ) @ id )
| ~ ( X272 @ id @ X274 @ id ) )
| ~ ! [X275: subst > term > term] :
( ~ ! [X276: term] :
( ( X272 @ id @ ( X275 @ id @ X276 ) @ id )
| ~ ( X272 @ id @ X276 @ id ) )
| ( X272 @ id @ ( lam @ ( X275 @ sh @ one ) ) @ id )
| ~ ! [X277: subst,X278: subst,X279: term] :
( ( sub @ ( X275 @ X278 @ X279 ) @ X277 )
= ( X275 @ ( comp @ X278 @ X277 ) @ ( sub @ X279 @ X277 ) ) ) )
| ( X272 @ id @ ( lam @ X273 ) @ id ) )
<=> hoasinduction_lem3a )
& pushprop_lem1v2_gthm
& pushprop_lthm_orig
& ( ! [X280: term,X281: term > $o,X282: subst > term > subst > $o,X283: term] :
( ~ ! [X284: term,X285: subst,X286: subst,X287: subst] :
( ( X282 @ ( comp @ X287 @ X286 ) @ ( sub @ X284 @ X286 ) @ X285 )
| ~ ( X282 @ X287 @ X284 @ ( comp @ X286 @ X285 ) ) )
| ~ ( X281 @ X283 )
| ( X281 @ ( ap @ X283 @ X280 ) )
| ~ ! [X288: term,X289: term] :
( ~ ( X282 @ id @ X289 @ id )
| ~ ( X282 @ id @ X288 @ id )
| ( X282 @ id @ ( ap @ ( sub @ X289 @ id ) @ X288 ) @ id ) )
| ~ ! [X290: subst,X291: subst,X292: term,X293: subst] :
( ( X282 @ X291 @ X292 @ ( comp @ X293 @ X290 ) )
| ~ ( X282 @ ( comp @ X291 @ X293 ) @ ( sub @ X292 @ X293 ) @ X290 ) )
| ~ ! [X294: term] :
( ( X282 @ id @ X294 @ id )
= ( X281 @ X294 ) )
| ~ ( X281 @ X280 ) )
<=> hoasinduction_lem2v2 )
& ( hoasinduction_lthm_3
<=> ( ! [X295: subst > term > subst > $o] :
~ ! [X296: term > $o] :
~ ! [X297: term] :
( ( X295 @ id @ X297 @ id )
= ( X296 @ X297 ) )
=> ( ! [X298: term,X299: term > $o] :
( ~ ! [X300: term] :
( ( X299 @ X300 )
| ~ ( var @ X300 ) )
| ( X299 @ X298 )
| ~ ! [X301: term] :
( ~ ! [X302: term] :
( ~ ( X299 @ X302 )
| ( X299 @ ( sub @ X301 @ ( push @ X302 @ id ) ) ) )
| ( X299 @ ( lam @ X301 ) ) )
| ~ ! [X303: term,X304: term] :
( ~ ( X299 @ X304 )
| ( X299 @ ( ap @ X303 @ X304 ) )
| ~ ( X299 @ X303 ) ) )
=> ( ! [X305: term] :
( ( sub @ X305 @ id )
= X305 )
=> ( ! [X306: term,X307: term > $o,X308: subst > term > subst > $o] :
( ~ ! [X309: term] :
( ( X308 @ id @ X309 @ id )
= ( X307 @ X309 ) )
| ( X307 @ ( lam @ X306 ) )
| ~ ! [X310: subst > term > term] :
( ( X308 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id )
| ~ ! [X311: subst,X312: term,X313: subst] :
( ( sub @ ( X310 @ X313 @ X312 ) @ X311 )
= ( X310 @ ( comp @ X313 @ X311 ) @ ( sub @ X312 @ X311 ) ) )
| ~ ! [X314: term] :
( ~ ( X308 @ id @ X314 @ id )
| ( X308 @ id @ ( X310 @ id @ X314 ) @ id ) ) )
| ~ ! [X315: term] :
( ~ ( X307 @ X315 )
| ( X307 @ ( sub @ X306 @ ( push @ X315 @ id ) ) ) ) )
=> ! [X316: subst > term > subst > $o,X317: term] :
( ~ ! [X318: term] :
( ~ ( var @ ( sub @ X318 @ id ) )
| ( X316 @ id @ X318 @ id ) )
| ~ ! [X319: subst > term > term] :
( ~ ! [X320: term] :
( ( X316 @ id @ ( X319 @ id @ X320 ) @ id )
| ~ ( X316 @ id @ X320 @ id ) )
| ( X316 @ id @ ( lam @ ( X319 @ sh @ one ) ) @ id )
| ~ ! [X321: subst,X322: term,X323: subst] :
( ( X319 @ ( comp @ X321 @ X323 ) @ ( sub @ X322 @ X323 ) )
= ( sub @ ( X319 @ X321 @ X322 ) @ X323 ) ) )
| ~ ! [X324: subst,X325: subst,X326: term,X327: subst] :
( ( X316 @ X324 @ X326 @ ( comp @ X325 @ X327 ) )
| ~ ( X316 @ ( comp @ X324 @ X325 ) @ ( sub @ X326 @ X325 ) @ X327 ) )
| ~ ! [X328: term,X329: term] :
( ( X316 @ id @ ( ap @ ( sub @ X328 @ id ) @ X329 ) @ id )
| ~ ( X316 @ id @ X329 @ id )
| ~ ( X316 @ id @ X328 @ id ) )
| ~ ! [X330: term,X331: subst,X332: subst,X333: subst] :
( ~ ( X316 @ X331 @ X330 @ ( comp @ X332 @ X333 ) )
| ( X316 @ ( comp @ X331 @ X332 ) @ ( sub @ X330 @ X332 ) @ X333 ) )
| ( X316 @ id @ X317 @ id ) ) ) ) ) ) )
& ~ lamnotap
& ( axidl
<=> ! [X334: subst] :
( ( comp @ id @ X334 )
= X334 ) )
& ( ! [X335: term > $o,X336: subst,X337: term] :
( ~ ! [X338: term,X339: term] :
( ~ ( X335 @ X339 )
| ~ ( X335 @ X338 )
| ( X335 @ ( ap @ X339 @ X338 ) ) )
| ~ ! [X340: term] :
( ~ ( var @ X340 )
| ( X335 @ ( sub @ X340 @ X336 ) ) )
| ~ ! [X341: term] :
( ~ ! [X342: term] :
( ~ ( X335 @ X342 )
| ( X335 @ ( sub @ X341 @ ( push @ X342 @ id ) ) ) )
| ( X335 @ ( lam @ X341 ) ) )
| ( X335 @ ( sub @ X337 @ X336 ) ) )
<=> induction2lem )
& hoasinduction_lem3b_gthm
& hoasinduction_lem3a_gthm
& hoasapnotvar_gthm
& ( ( d2subst @ d_id )
= id )
& ( ! [X343: term,X344: term > $o,X345: subst] :
~ ! [X346: term > $o] :
~ ! [X347: term] :
( ( X344 @ ( sub @ X347 @ ( push @ X343 @ X345 ) ) )
= ( X346 @ X347 ) )
<=> pushprop_lem0 )
& hoasapnotvar
& ! [X348: d_subst,X349: d_subst] :
( ( ( d2subst @ X348 )
= ( d2subst @ X349 ) )
=> ( X348 = X349 ) )
& pushprop_gthm
& ulamvarind
& ( ! [X350: subst > term > subst > $o,X351: term] :
( ~ ! [X352: term] :
( ~ ( X350 @ id @ X352 @ id )
| ( X350 @ id @ ( sub @ X351 @ ( push @ X352 @ id ) ) @ id ) )
| ~ ! [X353: term,X354: subst > term > term] :
( ~ ! [X355: subst,X356: subst,X357: term] :
( ( sub @ ( sub @ X353 @ ( push @ X357 @ X356 ) ) @ X355 )
= ( sub @ ( X354 @ X356 @ X357 ) @ X355 ) )
| ~ ! [X358: term] :
( ( X350 @ id @ ( X354 @ id @ X358 ) @ id )
| ~ ( X350 @ id @ X358 @ id ) )
| ( X350 @ id @ ( lam @ ( X354 @ sh @ one ) ) @ id )
| ~ ! [X359: subst,X360: subst,X361: term] :
( ( sub @ X353 @ ( push @ ( sub @ X361 @ X360 ) @ ( comp @ X359 @ X360 ) ) )
= ( X354 @ ( comp @ X359 @ X360 ) @ ( sub @ X361 @ X360 ) ) ) )
| ( X350 @ id @ ( lam @ ( sub @ X351 @ ( push @ one @ sh ) ) ) @ id ) )
<=> hoasinduction_lem3aaa )
& hoasinduction_lem1v2
& ( hoasapinj2
<=> ! [X362: term,X363: term,X364: term,X365: term] :
( ( X363 = X364 )
| ( ( ap @ ( sub @ X362 @ id ) @ X363 )
!= ( ap @ ( sub @ X365 @ id ) @ X364 ) ) ) )
& axapp
& ! [X366: subst] :
? [X367: d_subst] :
( ( d2subst @ X367 )
= X366 )
& ( hoasapinj2_lthm
<=> ( ! [X368: term,X369: term,X370: term,X371: term] :
( ( ( ap @ X369 @ X368 )
!= ( ap @ X370 @ X371 ) )
| ( X368 = X371 ) )
=> ! [X372: term,X373: term,X374: term,X375: term] :
( ( X373 = X375 )
| ( ( ap @ ( sub @ X374 @ id ) @ X375 )
!= ( ap @ ( sub @ X372 @ id ) @ X373 ) ) ) ) )
& induction2_gthm
& ( ! [X376: term,X377: term] :
( ( ( lam @ X376 )
!= ( lam @ X377 ) )
| ( X376 = X377 ) )
<=> laminj )
& ! [X378: d_term,X379: d_term] :
( ( ( d2term @ X378 )
= ( d2term @ X379 ) )
=> ( X378 = X379 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ( axvarcons
<=> ! [X380: subst,X381: term] :
( ( sub @ one @ ( push @ X381 @ X380 ) )
= X381 ) )
& hoasinduction_lem2_gthm
& hoaslamnotap_gthm
& ( ( ! [X382: subst] :
( ( comp @ id @ X382 )
= X382 )
& ! [X383: subst] :
( ( comp @ X383 @ id )
= X383 ) )
<=> substmonoid )
& hoasapnotvar_lthm
& ( ( ! [X384: term] :
( ( sub @ X384 @ id )
= X384 )
=> ( ! [X385: term,X386: subst > term > subst > $o] :
( ~ ! [X387: subst > term > term] :
( ~ ! [X388: term] :
( ( X386 @ id @ ( X387 @ id @ X388 ) @ id )
| ~ ( X386 @ id @ X388 @ id ) )
| ( X386 @ id @ ( lam @ ( X387 @ sh @ one ) ) @ id )
| ~ ! [X389: subst,X390: subst,X391: term] :
( ( X387 @ ( comp @ X389 @ X390 ) @ ( sub @ X391 @ X390 ) )
= ( sub @ ( X387 @ X389 @ X391 ) @ X390 ) ) )
| ~ ! [X392: term] :
( ( X386 @ id @ ( sub @ X385 @ ( push @ X392 @ id ) ) @ id )
| ~ ( X386 @ id @ X392 @ id ) )
| ( X386 @ id @ ( lam @ ( sub @ X385 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X393: term,X394: subst > term > subst > $o] :
( ~ ! [X395: term] :
( ~ ( X394 @ id @ X395 @ id )
| ( X394 @ id @ ( sub @ X393 @ ( push @ X395 @ id ) ) @ id ) )
| ~ ! [X396: subst > term > term] :
( ~ ! [X397: term] :
( ( X394 @ id @ ( X396 @ id @ X397 ) @ id )
| ~ ( X394 @ id @ X397 @ id ) )
| ( X394 @ id @ ( lam @ ( X396 @ sh @ one ) ) @ id )
| ~ ! [X398: subst,X399: subst,X400: term] :
( ( X396 @ ( comp @ X399 @ X398 ) @ ( sub @ X400 @ X398 ) )
= ( sub @ ( X396 @ X399 @ X400 ) @ X398 ) ) )
| ( X394 @ id @ ( lam @ X393 ) @ id ) ) ) )
<=> hoasinduction_lem3a_lthm )
& hoasinduction_lem2_lthm
& ( ! [X401: term] :
~ ! [X402: subst > term > term] :
~ ! [X403: subst,X404: term] :
( ( X402 @ X403 @ X404 )
= ( sub @ X401 @ ( push @ X404 @ X403 ) ) )
<=> hoasinduction_lem3v2_f_lthm )
& ( pushprop_lem1v2
<=> ! [X405: term > $o,X406: term > $o,X407: term,X408: subst] :
( ( X406 @ one )
| ~ ( X405 @ X407 )
| ~ ! [X409: term] :
( ( X406 @ X409 )
= ( X405 @ ( sub @ X409 @ ( push @ X407 @ X408 ) ) ) ) ) )
& ( one
= ( d2term @ d_one ) )
& ~ ulamvar1
& ( hoaslaminj_lthm
<=> ( ! [X410: term,X411: subst] :
( ( sub @ one @ ( push @ X410 @ X411 ) )
= X410 )
=> ( ! [X412: term,X413: subst] :
( ( comp @ sh @ ( push @ X412 @ X413 ) )
= X413 )
=> ( ! [X414: term,X415: term] :
( ( X414 = X415 )
| ( ( lam @ X414 )
!= ( lam @ X415 ) ) )
=> ! [X416: subst > term > term,X417: subst,X418: term,X419: subst > term > term] :
( ( ( lam @ ( X416 @ sh @ one ) )
!= ( lam @ ( X419 @ sh @ one ) ) )
| ( ( X416 @ X417 @ X418 )
= ( X419 @ X417 @ X418 ) )
| ~ ! [X420: subst,X421: term,X422: subst] :
( ( X416 @ ( comp @ X420 @ X422 ) @ ( sub @ X421 @ X422 ) )
= ( sub @ ( X416 @ X420 @ X421 ) @ X422 ) )
| ~ ! [X423: term,X424: subst,X425: subst] :
( ( X419 @ ( comp @ X425 @ X424 ) @ ( sub @ X423 @ X424 ) )
= ( sub @ ( X419 @ X425 @ X423 ) @ X424 ) ) ) ) ) ) )
& axassoc
& ( ! [X426: subst] :
( ( comp @ X426 @ id )
= X426 )
<=> axidr )
& ! [X427: d_subst] : ( d_id = X427 )
& ( hoasinduction_lthm
<=> ( ! [X428: term,X429: term > $o] :
( ( X429 @ X428 )
| ~ ! [X430: term] :
( ( X429 @ ( lam @ X430 ) )
| ~ ! [X431: term] :
( ~ ( X429 @ X431 )
| ( X429 @ ( sub @ X430 @ ( push @ X431 @ id ) ) ) ) )
| ~ ! [X432: term] :
( ( X429 @ X432 )
| ~ ( var @ X432 ) )
| ~ ! [X433: term,X434: term] :
( ( X429 @ ( ap @ X434 @ X433 ) )
| ~ ( X429 @ X434 )
| ~ ( X429 @ X433 ) ) )
=> ( ! [X435: term,X436: term,X437: subst > term > subst > $o] :
( ~ ! [X438: term,X439: subst,X440: subst,X441: subst] :
( ( X437 @ ( comp @ X439 @ X440 ) @ ( sub @ X438 @ X440 ) @ X441 )
| ~ ( X437 @ X439 @ X438 @ ( comp @ X440 @ X441 ) ) )
| ( X437 @ id @ ( ap @ X436 @ X435 ) @ id )
| ~ ( X437 @ id @ X436 @ id )
| ~ ! [X442: subst,X443: subst,X444: term,X445: subst] :
( ( X437 @ X442 @ X444 @ ( comp @ X443 @ X445 ) )
| ~ ( X437 @ ( comp @ X442 @ X443 ) @ ( sub @ X444 @ X443 ) @ X445 ) )
| ~ ! [X446: term,X447: term] :
( ~ ( X437 @ id @ X446 @ id )
| ~ ( X437 @ id @ X447 @ id )
| ( X437 @ id @ ( ap @ ( sub @ X447 @ id ) @ X446 ) @ id ) )
| ~ ( X437 @ id @ X435 @ id ) )
=> ( ! [X448: term,X449: subst > term > subst > $o] :
( ~ ! [X450: subst > term > term] :
( ( X449 @ id @ ( lam @ ( X450 @ sh @ one ) ) @ id )
| ~ ! [X451: subst,X452: subst,X453: term] :
( ( X450 @ ( comp @ X452 @ X451 ) @ ( sub @ X453 @ X451 ) )
= ( sub @ ( X450 @ X452 @ X453 ) @ X451 ) )
| ~ ! [X454: term] :
( ~ ( X449 @ id @ X454 @ id )
| ( X449 @ id @ ( X450 @ id @ X454 ) @ id ) ) )
| ( X449 @ id @ ( lam @ X448 ) @ id )
| ~ ! [X455: term] :
( ( X449 @ id @ ( sub @ X448 @ ( push @ X455 @ id ) ) @ id )
| ~ ( X449 @ id @ X455 @ id ) )
| ~ ! [X456: subst,X457: subst,X458: term,X459: subst] :
( ( X449 @ ( comp @ X459 @ X457 ) @ ( sub @ X458 @ X457 ) @ X456 )
| ~ ( X449 @ X459 @ X458 @ ( comp @ X457 @ X456 ) ) )
| ~ ! [X460: subst,X461: subst,X462: term,X463: subst] :
( ~ ( X449 @ ( comp @ X463 @ X460 ) @ ( sub @ X462 @ X460 ) @ X461 )
| ( X449 @ X463 @ X462 @ ( comp @ X460 @ X461 ) ) ) )
=> ! [X464: term,X465: subst > term > subst > $o] :
( ~ ! [X466: term,X467: term] :
( ~ ( X465 @ id @ X466 @ id )
| ~ ( X465 @ id @ X467 @ id )
| ( X465 @ id @ ( ap @ ( sub @ X466 @ id ) @ X467 ) @ id ) )
| ~ ! [X468: subst > term > term] :
( ( X465 @ id @ ( lam @ ( X468 @ sh @ one ) ) @ id )
| ~ ! [X469: term] :
( ~ ( X465 @ id @ X469 @ id )
| ( X465 @ id @ ( X468 @ id @ X469 ) @ id ) )
| ~ ! [X470: subst,X471: subst,X472: term] :
( ( sub @ ( X468 @ X470 @ X472 ) @ X471 )
= ( X468 @ ( comp @ X470 @ X471 ) @ ( sub @ X472 @ X471 ) ) ) )
| ( X465 @ id @ X464 @ id )
| ~ ! [X473: subst,X474: term,X475: subst,X476: subst] :
( ~ ( X465 @ X476 @ X474 @ ( comp @ X475 @ X473 ) )
| ( X465 @ ( comp @ X476 @ X475 ) @ ( sub @ X474 @ X475 ) @ X473 ) )
| ~ ! [X477: term,X478: subst,X479: subst,X480: subst] :
( ~ ( X465 @ ( comp @ X478 @ X479 ) @ ( sub @ X477 @ X479 ) @ X480 )
| ( X465 @ X478 @ X477 @ ( comp @ X479 @ X480 ) ) )
| ~ ! [X481: term] :
( ( X465 @ id @ X481 @ id )
| ~ ( var @ ( sub @ X481 @ id ) ) ) ) ) ) ) )
& pushprop_lem2v2_gthm
& termmset_gthm
& induction2lem_gthm
& ( ! [X482: term > $o,X483: term,X484: subst,X485: term > $o,X486: term] :
( ~ ( X482 @ X483 )
| ( X485 @ ( sub @ X483 @ ( push @ X486 @ X484 ) ) ) )
<=> pushprop_lem1 )
& ( ( ! [X487: subst,X488: term] :
( ( sub @ one @ ( push @ X488 @ X487 ) )
= X488 )
=> ! [X489: term > $o,X490: term > $o,X491: term,X492: subst] :
( ( X490 @ one )
| ~ ( X489 @ X491 )
| ~ ! [X493: term] :
( ( X490 @ X493 )
= ( X489 @ ( sub @ X493 @ ( push @ X491 @ X492 ) ) ) ) ) )
<=> pushprop_lem1v2_lthm )
& pushprop_lem0_gthm
& hoasapinj2_gthm
& ( ( ! [X494: term] :
( ( sub @ X494 @ id )
= X494 )
=> ! [X495: term > $o,X496: subst > term > subst > $o,X497: term] :
( ~ ! [X498: subst,X499: subst,X500: term,X501: subst] :
( ~ ( X496 @ X498 @ X500 @ ( comp @ X499 @ X501 ) )
| ( X496 @ ( comp @ X498 @ X499 ) @ ( sub @ X500 @ X499 ) @ X501 ) )
| ( X495 @ ( lam @ X497 ) )
| ~ ! [X502: subst,X503: term,X504: subst,X505: subst] :
( ~ ( X496 @ ( comp @ X502 @ X505 ) @ ( sub @ X503 @ X505 ) @ X504 )
| ( X496 @ X502 @ X503 @ ( comp @ X505 @ X504 ) ) )
| ~ ! [X506: term] :
( ( X495 @ ( sub @ X497 @ ( push @ X506 @ id ) ) )
| ~ ( X495 @ X506 ) )
| ~ ! [X507: term] :
( ( X496 @ id @ X507 @ id )
= ( X495 @ X507 ) )
| ~ ! [X508: subst > term > term] :
( ~ ! [X509: term] :
( ( X496 @ id @ ( X508 @ id @ X509 ) @ id )
| ~ ( X496 @ id @ X509 @ id ) )
| ( X496 @ id @ ( lam @ ( X508 @ sh @ one ) ) @ id )
| ~ ! [X510: subst,X511: subst,X512: term] :
( ( sub @ ( X508 @ X510 @ X512 ) @ X511 )
= ( X508 @ ( comp @ X510 @ X511 ) @ ( sub @ X512 @ X511 ) ) ) ) ) )
<=> hoasinduction_lem3v2_lthm )
& ( substmonoid_lthm
<=> ( ! [X513: subst] :
( ( comp @ id @ X513 )
= X513 )
=> ( ! [X514: subst] :
( ( comp @ X514 @ id )
= X514 )
=> ( ! [X515: subst] :
( ( comp @ X515 @ id )
= X515 )
& ! [X516: subst] :
( ( comp @ id @ X516 )
= X516 ) ) ) ) )
& hoasapinj1_gthm
& ( hoasapinj1
<=> ! [X517: term,X518: term,X519: term,X520: term] :
( ( X517 = X518 )
| ( ( ap @ ( sub @ X518 @ id ) @ X520 )
!= ( ap @ ( sub @ X517 @ id ) @ X519 ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( axvarid
<=> ! [X521: term] :
( ( sub @ X521 @ id )
= X521 ) )
& pushprop_lem3v2_lthm
& ( hoasinduction_lem3aa_lthm
<=> ! [X522: subst > term > subst > $o,X523: term] :
( ( X522 @ id @ ( lam @ ( sub @ X523 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X524: term] :
( ( X522 @ id @ ( sub @ X523 @ ( push @ X524 @ id ) ) @ id )
| ~ ( X522 @ id @ X524 @ id ) )
| ~ ! [X525: subst > term > term] :
( ( X522 @ id @ ( lam @ ( X525 @ sh @ one ) ) @ id )
| ~ ! [X526: subst,X527: term,X528: subst] :
( ( X525 @ ( comp @ X528 @ X526 ) @ ( sub @ X527 @ X526 ) )
= ( sub @ ( X525 @ X528 @ X527 ) @ X526 ) )
| ~ ! [X529: term] :
( ~ ( X522 @ id @ X529 @ id )
| ( X522 @ id @ ( X525 @ id @ X529 ) @ id ) ) ) ) )
& ( hoaslamnotap
<=> ! [X530: term,X531: term,X532: subst > term > term] :
( ~ ! [X533: subst,X534: term,X535: subst] :
( ( X532 @ ( comp @ X535 @ X533 ) @ ( sub @ X534 @ X533 ) )
= ( sub @ ( X532 @ X535 @ X534 ) @ X533 ) )
| ( ( ap @ ( sub @ X531 @ id ) @ X530 )
!= ( lam @ ( X532 @ sh @ one ) ) ) ) )
& ( hoasapinj1_lthm
<=> ( ! [X536: term] :
( ( sub @ X536 @ id )
= X536 )
=> ( ! [X537: term,X538: term,X539: term,X540: term] :
( ( ( ap @ X537 @ X540 )
!= ( ap @ X538 @ X539 ) )
| ( X537 = X538 ) )
=> ! [X541: term,X542: term,X543: term,X544: term] :
( ( ( ap @ ( sub @ X544 @ id ) @ X542 )
!= ( ap @ ( sub @ X543 @ id ) @ X541 ) )
| ( X543 = X544 ) ) ) ) )
& ( ! [X545: term > $o,X546: subst,X547: term] :
~ ! [X548: term > $o] :
~ ! [X549: term] :
( ( X548 @ X549 )
= ( X545 @ ( sub @ X549 @ ( push @ X547 @ X546 ) ) ) )
<=> pushprop_lem0_lthm )
& hoaslamnotvar ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ! [X14: subst] :
( ( push @ ( sub @ one @ X14 ) @ ( comp @ sh @ X14 ) )
= X14 )
<=> axscons )
& axabs
& hoasinduction_lem1_lthm
& ( hoasinduction_lem3b_lthm
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) ) )
& axclos
& ! [X0: term] :
? [X1: d_term] :
( ( d2term @ X1 )
= X0 )
& hoasinduction_lem1v2_gthm
& ( termmset_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) ) )
& axvarshift
& ( hoasinduction_p_and_p_prime
= ( ^ [X10: subst > term > subst > $o,X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) ) )
& ( ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X10: term > $o,X21: term] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ( X10 @ X21 )
| ~ ! [X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) ) )
=> ! [X17: subst,X16: term,X10: term > $o] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X15 @ X17 ) )
| ~ ( var @ X15 ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) ) ) ) ) ) ) )
<=> induction2lem_lthm )
& ulamvarsh
& hoasinduction_gthm
& pushprop_lthm
& hoaslamnotvar_lthm
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0_lthm )
& hoasinduction_lem3v2_gthm
& hoaslamnotvar_gthm
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lem3v2
<=> ! [X10: subst > term > subst > $o,X48: term,X11: term > $o] :
( ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X48 ) ) ) )
& ( ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X10: term > $o,X22: term] :
( ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: subst > term > subst > $o,X29: term,X11: term > $o] :
( ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) ) )
=> ! [X10: subst > term > subst > $o,X30: term] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ( X10 @ id @ X30 @ id ) ) ) ) ) )
<=> hoasinduction_no_psi_cond_lthm )
& pushprop_lem2v2_lthm
& ( shinj
<=> ! [X15: term,X13: term] :
( ( ( sub @ X13 @ sh )
!= ( sub @ X15 @ sh ) )
| ( X13 = X15 ) ) )
& ( ! [X13: term,X15: term,X20: term,X19: term] :
( ( X19 = X20 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
<=> apinj2 )
& ( hoasinduction_lem3v2a
<=> ! [X10: subst > term > subst > $o,X29: term,X11: term > $o] :
( ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X11 @ ( lam @ X29 ) ) ) )
& pushprop
& ( induction2_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X17: subst,X10: term > $o,X16: term] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) ) ) )
=> ! [X10: term > $o,X22: term] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) ) ) ) ) )
& ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
<=> termmset )
& ( hoasinduction_lem0
<=> ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
& substmonoid_gthm
& ( ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ! [X26: term > $o,X15: term,X14: subst,X10: term > $o,X13: term] :
( ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) )
| ~ ( X10 @ X13 ) ) ) )
<=> pushprop_lem1_lthm )
& ( ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X26: subst,X14: subst,X25: subst,X13: term] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X25: subst,X14: subst,X13: term,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ ( lam @ X27 ) @ id ) )
<=> hoasinduction_lem3 )
& ( ( ^ [X8: subst,X9: subst > term > term] : ( d2term @ d_one ) )
= hoaslam )
& ( hoasinduction_lem3v2_f
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) ) )
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) )
<=> hoasinduction_lem3b )
& hoaslaminj_gthm
& pushprop_lem1_gthm
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_p_and_p_prime
= ( ^ [X13: term,X14: subst,X10: term > $o,X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) ) )
& pushprop_lem3v2
& ( hoaslaminj
<=> ! [X31: subst > term > term,X18: subst > term > term,X32: subst,X33: term] :
( ~ ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) ) ) )
& ~ ( var @ ( d2term @ d_one ) )
& ( ! [X10: subst > term > subst > $o,X42: term] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
<=> hoasinduction_lem3aa )
& axmap
& apnotvar
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: subst > term > subst > $o,X42: term] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X18: subst > term > term] :
( ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) ) )
=> ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X13: term,X26: subst,X25: subst,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) ) ) )
<=> hoasinduction_lem3_lthm )
& lamnotvar
& hoasinduction_lem1_gthm
& hoasinduction_lem2v2_gthm
& ( hoasinduction_lem2
<=> ! [X24: term,X10: subst > term > subst > $o,X23: term] :
( ~ ! [X25: subst,X14: subst,X13: term,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ! [X14: subst,X25: subst,X13: term,X26: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ( X10 @ id @ X23 @ id )
| ~ ( X10 @ id @ X24 @ id )
| ~ ! [X15: term,X13: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id ) ) ) )
& pushprop_lem2v2
& ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
<=> axshiftcons )
& ( ! [X10: subst > term > subst > $o,X30: term] :
( ( X10 @ id @ X30 @ id )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) ) )
<=> hoasinduction_no_psi_cond )
& hoasinduction_lem3_gthm
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( hoasinduction
<=> ! [X28: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ X28 @ id )
| ~ ! [X13: term,X14: subst,X25: subst,X26: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X26: subst,X13: term,X14: subst,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) ) ) )
& hoasinduction_lem1
& ! [X1: d_term] : ( d_one = X1 )
& hoaslamnotap_lthm
& ( induction
<=> ! [X10: term > $o,X21: term] :
( ( X10 @ X21 )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( lam @ X13 ) ) ) ) )
& ( apinj1
<=> ! [X15: term,X19: term,X20: term,X13: term] :
( ( X13 = X15 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) ) )
& ( hoasinduction_lem3v2a_lthm
<=> ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X29: term,X10: subst > term > subst > $o,X11: term > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) ) ) ) ) )
& ( ! [X22: term,X10: term > $o] :
( ~ ! [X13: term,X15: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) ) )
<=> induction2 )
& ( ! [X10: subst > term > subst > $o,X43: term] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X10 @ id @ ( lam @ X43 ) @ id ) )
<=> hoasinduction_lem3a )
& pushprop_lem1v2_gthm
& pushprop_lthm_orig
& ( ! [X45: term,X11: term > $o,X10: subst > term > subst > $o,X44: term] :
( ~ ! [X13: term,X25: subst,X26: subst,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ( X11 @ X44 )
| ( X11 @ ( ap @ X44 @ X45 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ! [X25: subst,X14: subst,X13: term,X26: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ( X11 @ X45 ) )
<=> hoasinduction_lem2v2 )
& ( hoasinduction_lthm_3
<=> ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X22: term,X10: term > $o] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X29: term,X11: term > $o,X10: subst > term > subst > $o] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) ) )
=> ! [X10: subst > term > subst > $o,X28: term] :
( ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term,X14: subst,X26: subst,X25: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X10 @ id @ X28 @ id ) ) ) ) ) ) )
& ~ lamnotap
& ( axidl
<=> ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) )
& ( ! [X10: term > $o,X17: subst,X16: term] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) ) )
<=> induction2lem )
& hoasinduction_lem3b_gthm
& hoasinduction_lem3a_gthm
& hoasapnotvar_gthm
& ( ( d2subst @ d_id )
= id )
& ( ! [X13: term,X10: term > $o,X14: subst] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
<=> pushprop_lem0 )
& hoasapnotvar
& ! [X6: d_subst,X7: d_subst] :
( ( ( d2subst @ X6 )
= ( d2subst @ X7 ) )
=> ( X6 = X7 ) )
& pushprop_gthm
& ulamvarind
& ( ! [X10: subst > term > subst > $o,X34: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X34 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X35: term,X18: subst > term > term] :
( ~ ! [X38: subst,X36: subst,X37: term] :
( ( sub @ ( X18 @ X36 @ X37 ) @ X38 )
= ( sub @ ( sub @ X35 @ ( push @ X37 @ X36 ) ) @ X38 ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X39: subst,X41: subst,X40: term] :
( ( X18 @ ( comp @ X39 @ X41 ) @ ( sub @ X40 @ X41 ) )
= ( sub @ X35 @ ( push @ ( sub @ X40 @ X41 ) @ ( comp @ X39 @ X41 ) ) ) ) )
| ( X10 @ id @ ( lam @ ( sub @ X34 @ ( push @ one @ sh ) ) ) @ id ) )
<=> hoasinduction_lem3aaa )
& hoasinduction_lem1v2
& ( hoasapinj2
<=> ! [X15: term,X20: term,X19: term,X13: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) )
& axapp
& ! [X4: subst] :
? [X5: d_subst] :
( ( d2subst @ X5 )
= X4 )
& ( hoasapinj2_lthm
<=> ( ! [X20: term,X15: term,X13: term,X19: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X19 = X20 ) )
=> ! [X15: term,X20: term,X13: term,X19: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) ) )
& induction2_gthm
& ( ! [X15: term,X13: term] :
( ( ( lam @ X13 )
!= ( lam @ X15 ) )
| ( X13 = X15 ) )
<=> laminj )
& ! [X3: d_term,X2: d_term] :
( ( ( d2term @ X2 )
= ( d2term @ X3 ) )
=> ( X2 = X3 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ( axvarcons
<=> ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 ) )
& hoasinduction_lem2_gthm
& hoaslamnotap_gthm
& ( ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
& ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 ) )
<=> substmonoid )
& hoasapnotvar_lthm
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X43: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X10 @ id @ ( lam @ X43 ) @ id ) ) ) )
<=> hoasinduction_lem3a_lthm )
& hoasinduction_lem2_lthm
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
<=> hoasinduction_lem3v2_f_lthm )
& ( pushprop_lem1v2
<=> ! [X10: term > $o,X11: term > $o,X13: term,X14: subst] :
( ( X11 @ one )
| ~ ( X10 @ X13 )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) ) )
& ( one
= ( d2term @ d_one ) )
& ~ ulamvar1
& ( hoaslaminj_lthm
<=> ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X13: term,X15: term] :
( ( X13 = X15 )
| ( ( lam @ X13 )
!= ( lam @ X15 ) ) )
=> ! [X31: subst > term > term,X32: subst,X33: term,X18: subst > term > term] :
( ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) )
| ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) ) ) )
& axassoc
& ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
<=> axidr )
& ! [X5: d_subst] : ( d_id = X5 )
& ( hoasinduction_lthm
<=> ( ! [X22: term,X10: term > $o] :
( ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) ) )
=> ( ! [X24: term,X23: term,X10: subst > term > subst > $o] :
( ~ ! [X13: term,X14: subst,X26: subst,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ( X10 @ id @ X23 @ id )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ( X10 @ id @ X24 @ id ) )
=> ( ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) )
=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X10 @ id @ X28 @ id )
| ~ ! [X25: subst,X13: term,X26: subst,X14: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term,X14: subst,X26: subst,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ X13 @ id )
| ~ ( var @ ( sub @ X13 @ id ) ) ) ) ) ) ) )
& pushprop_lem2v2_gthm
& termmset_gthm
& induction2lem_gthm
& ( ! [X10: term > $o,X13: term,X14: subst,X26: term > $o,X15: term] :
( ~ ( X10 @ X13 )
| ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) ) )
<=> pushprop_lem1 )
& ( ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ! [X10: term > $o,X11: term > $o,X13: term,X14: subst] :
( ( X11 @ one )
| ~ ( X10 @ X13 )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) ) )
<=> pushprop_lem1v2_lthm )
& pushprop_lem0_gthm
& hoasapinj2_gthm
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X11: term > $o,X10: subst > term > subst > $o,X48: term] :
( ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X14: subst,X13: term,X25: subst,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) )
<=> hoasinduction_lem3v2_lthm )
& ( substmonoid_lthm
<=> ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) ) ) ) )
& hoasapinj1_gthm
& ( hoasapinj1
<=> ! [X13: term,X15: term,X19: term,X20: term] :
( ( X13 = X15 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( axvarid
<=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) )
& pushprop_lem3v2_lthm
& ( hoasinduction_lem3aa_lthm
<=> ! [X10: subst > term > subst > $o,X42: term] :
( ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) ) ) )
& ( hoaslamnotap
<=> ! [X47: term,X46: term,X18: subst > term > term] :
( ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( ap @ ( sub @ X46 @ id ) @ X47 ) ) ) )
& ( hoasapinj1_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X13: term,X15: term,X20: term,X19: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X13 = X15 ) )
=> ! [X19: term,X20: term,X13: term,X15: term] :
( ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) )
| ( X13 = X15 ) ) ) ) )
& ( ! [X10: term > $o,X14: subst,X13: term] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
<=> pushprop_lem0_lthm )
& hoaslamnotvar ),
file('/export/starexec/sandbox2/tmp/tmp.u2UgYFbzEX/DTF2THF_17811.p',alg444_1) ).
thf(f1112,plain,
! [X183: term] :
( ( d2term @ ( sK544 @ X183 ) )
= X183 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f2172,plain,
! [X1: subst > term > term] :
( ( ( sK591 @ ( d2subst @ d_id ) @ ( sK595 @ X1 ) @ ( d2subst @ d_id ) )
= $true )
| ( hoasinduction_lem3a = $false )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f1696,f1892]) ).
thf(f1892,plain,
! [X0: term,X1: term] : ( X0 = X1 ),
inference(superposition,[status(thm)],[f1886,f1886]) ).
thf(f1696,plain,
! [X1: subst > term > term] :
( ( ( sK591 @ ( d2subst @ d_id ) @ ( sK595 @ X1 ) @ ( d2subst @ d_id ) )
= $true )
| ( hoasinduction_lem3a = $false )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( ( X1 @ ( comp @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ ( sK594 @ X1 ) @ ( sK592 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK593 @ X1 ) @ ( sK594 @ X1 ) ) @ ( sK592 @ X1 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1694]) ).
thf(f1694,plain,
! [X1: subst > term > term] :
( ( hoasinduction_lem3a = $false )
| ( ( X1 @ ( comp @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ ( sK594 @ X1 ) @ ( sK592 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK593 @ X1 ) @ ( sK594 @ X1 ) ) @ ( sK592 @ X1 ) ) )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( $false
= ( ( sK591 @ ( d2subst @ d_id ) @ ( sK595 @ X1 ) @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ ( sK595 @ X1 ) ) @ ( d2subst @ d_id ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1693]) ).
thf(f1693,plain,
! [X1: subst > term > term] :
( ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( hoasinduction_lem3a = $false )
| ( ( ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) )
@ ( sK595 @ X1 ) )
= $false )
| ( ( X1 @ ( comp @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ ( sK594 @ X1 ) @ ( sK592 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK593 @ X1 ) @ ( sK594 @ X1 ) ) @ ( sK592 @ X1 ) ) ) ),
inference(sigma_clausification,[status(thm)],[f1692]) ).
thf(f1692,plain,
! [X1: subst > term > term] :
( ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( hoasinduction_lem3a = $false )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( ( X1 @ ( comp @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ ( sK594 @ X1 ) @ ( sK592 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK593 @ X1 ) @ ( sK594 @ X1 ) ) @ ( sK592 @ X1 ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1691]) ).
thf(f1691,plain,
! [X1: subst > term > term] :
( ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( $false
= ( ( sub @ ( X1 @ ( sK593 @ X1 ) @ ( sK594 @ X1 ) ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ ( sK594 @ X1 ) @ ( sK592 @ X1 ) ) ) ) )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1690]) ).
thf(f1690,plain,
! [X1: subst > term > term] :
( ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( ( ^ [Y0: term] :
( ( sub @ ( X1 @ ( sK593 @ X1 ) @ Y0 ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ Y0 @ ( sK592 @ X1 ) ) ) )
@ ( sK594 @ X1 ) )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(sigma_clausification,[status(thm)],[f1689]) ).
thf(f1689,plain,
! [X1: subst > term > term] :
( ( ( !! @ term
@ ^ [Y0: term] :
( ( sub @ ( X1 @ ( sK593 @ X1 ) @ Y0 ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ Y0 @ ( sK592 @ X1 ) ) ) ) )
= $false )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1688]) ).
thf(f1688,plain,
! [X1: subst > term > term] :
( ( $false
= ( ^ [Y0: subst] :
( !! @ term
@ ^ [Y1: term] :
( ( sub @ ( X1 @ Y0 @ Y1 ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y0 @ ( sK592 @ X1 ) ) @ ( sub @ Y1 @ ( sK592 @ X1 ) ) ) ) )
@ ( sK593 @ X1 ) ) )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(sigma_clausification,[status(thm)],[f1687]) ).
thf(f1687,plain,
! [X1: subst > term > term] :
( ( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ term
@ ^ [Y1: term] :
( ( sub @ ( X1 @ Y0 @ Y1 ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y0 @ ( sK592 @ X1 ) ) @ ( sub @ Y1 @ ( sK592 @ X1 ) ) ) ) ) )
= $false )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1686]) ).
thf(f1686,plain,
! [X1: subst > term > term] :
( ( $false
= ( ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ ( X1 @ Y1 @ Y2 ) @ Y0 )
= ( X1 @ ( comp @ Y1 @ Y0 ) @ ( sub @ Y2 @ Y0 ) ) ) ) )
@ ( sK592 @ X1 ) ) )
| ( hoasinduction_lem3a = $false )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ),
inference(sigma_clausification,[status(thm)],[f1685]) ).
thf(f1685,plain,
! [X1: subst > term > term] :
( ( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ ( X1 @ Y1 @ Y2 ) @ Y0 )
= ( X1 @ ( comp @ Y1 @ Y0 ) @ ( sub @ Y2 @ Y0 ) ) ) ) ) )
= $false )
| ( hoasinduction_lem3a = $false )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( $true
= ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1684]) ).
thf(f1684,plain,
! [X1: subst > term > term] :
( ( $true
= ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) )
| ( hoasinduction_lem3a = $false )
| ( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ ( X1 @ Y1 @ Y2 ) @ Y0 )
= ( X1 @ ( comp @ Y1 @ Y0 ) @ ( sub @ Y2 @ Y0 ) ) ) ) ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1683]) ).
thf(f1683,plain,
! [X1: subst > term > term] :
( ( $true
= ( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ ( X1 @ Y1 @ Y2 ) @ Y0 )
= ( X1 @ ( comp @ Y1 @ Y0 ) @ ( sub @ Y2 @ Y0 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) )
| ( hoasinduction_lem3a = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1682]) ).
thf(f1682,plain,
! [X1: subst > term > term] :
( ( hoasinduction_lem3a = $false )
| ( $true
= ( ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( ( sub @ ( Y0 @ Y2 @ Y3 ) @ Y1 )
= ( Y0 @ ( comp @ Y2 @ Y1 ) @ ( sub @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( Y0 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) )
@ X1 ) ) ),
inference(pi_clausification,[status(thm)],[f1681]) ).
thf(f1681,plain,
( ( $true
= ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( ( sub @ ( Y0 @ Y2 @ Y3 ) @ Y1 )
= ( Y0 @ ( comp @ Y2 @ Y1 ) @ ( sub @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( Y0 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) ) )
| ( hoasinduction_lem3a = $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1679]) ).
thf(f1679,plain,
( ( hoasinduction_lem3a = $false )
| ( ( ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( ( sub @ ( Y0 @ Y2 @ Y3 ) @ Y1 )
= ( Y0 @ ( comp @ Y2 @ Y1 ) @ ( sub @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( Y0 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) )
=> ( !! @ term
@ ^ [Y0: term] :
( ( !! @ term
@ ^ [Y1: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( sub @ Y0 @ ( push @ Y1 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( lam @ Y0 ) @ ( d2subst @ d_id ) ) ) ) )
= $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1678]) ).
thf(f1678,plain,
( ( hoasinduction_lem3a = $false )
| ( ( ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( Y1 @ ( d2subst @ d_id ) @ Y2 ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( sub @ Y1 @ ( push @ Y2 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( lam @ Y1 ) @ ( d2subst @ d_id ) ) ) ) )
@ sK591 )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f1677]) ).
thf(f1677,plain,
( ( hoasinduction_lem3a = $false )
| ( $false
= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( Y1 @ ( d2subst @ d_id ) @ Y2 ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( sub @ Y1 @ ( push @ Y2 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( lam @ Y1 ) @ ( d2subst @ d_id ) ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1675]) ).
thf(f1675,plain,
( hoasinduction_lem3a
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( Y1 @ ( d2subst @ d_id ) @ Y2 ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( sub @ Y1 @ ( push @ Y2 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( lam @ Y1 ) @ ( d2subst @ d_id ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1660]) ).
thf(f1660,plain,
( hoasinduction_lem3a
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( Y1 @ ( d2subst @ d_id ) @ Y2 ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id )
@ ( ^ [Y2: subst,Y3: subst > term > term] : ( d2term @ d_one )
@ ( d2subst @ d_id )
@ ^ [Y2: subst,Y3: term] : ( Y1 @ Y2 @ Y3 ) )
@ ( d2subst @ d_id ) ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( sub @ Y1 @ ( push @ Y2 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( lam @ Y1 ) @ ( d2subst @ d_id ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1260,f1137,f1137,f1137,f1137,f1137,f1137,f1078,f1137,f1137,f1137,f1137,f1137,f1137,f1137,f1137,f1137]) ).
thf(f1078,plain,
( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1137,plain,
( ( d2subst @ d_id )
= id ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1260,plain,
( hoasinduction_lem3a
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( Y0 @ id @ ( Y1 @ id @ Y2 ) @ id ) ) )
=> ( Y0 @ id
@ ( hoaslam @ id
@ ^ [Y2: subst,Y3: term] : ( Y1 @ Y2 @ Y3 ) )
@ id ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( Y0 @ id @ ( sub @ Y1 @ ( push @ Y2 @ id ) ) @ id ) ) )
=> ( Y0 @ id @ ( lam @ Y1 ) @ id ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( hoasinduction_lem3a
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( Y0 @ id @ ( Y1 @ id @ Y2 ) @ id ) ) )
=> ( Y0 @ id
@ ( hoaslam @ id
@ ^ [Y2: subst,Y3: term] : ( Y1 @ Y2 @ Y3 ) )
@ id ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( Y0 @ id @ ( sub @ Y1 @ ( push @ Y2 @ id ) ) @ id ) ) )
=> ( Y0 @ id @ ( lam @ Y1 ) @ id ) ) ) ) ) ),
inference(flattening,[status(thm)],[f6]) ).
thf(f6,plain,
( hoasinduction_lem3a
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( Y0 @ id @ ( Y1 @ id @ Y2 ) @ id ) ) )
=> ( Y0 @ id
@ ( hoaslam @ id
@ ^ [Y2: subst,Y3: term] : ( Y1 @ Y2 @ Y3 ) )
@ id ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( Y0 @ id @ ( sub @ Y1 @ ( push @ Y2 @ id ) ) @ id ) ) )
=> ( Y0 @ id @ ( lam @ Y1 ) @ id ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
( hoasinduction_lem3a
!= ( ! [X0: subst > term > subst > $o] :
( ! [X1: subst > term > term] :
( ! [X2: term,X3: subst,X4: subst] :
( ( sub @ ( X1 @ X3 @ X2 ) @ X4 )
= ( X1 @ ( comp @ X3 @ X4 ) @ ( sub @ X2 @ X4 ) ) )
=> ( ! [X5: term] :
( ( X0 @ id @ X5 @ id )
=> ( X0 @ id @ ( X1 @ id @ X5 ) @ id ) )
=> ( X0 @ id
@ ( hoaslam @ id
@ ^ [X6: subst,X7: term] : ( X1 @ X6 @ X7 ) )
@ id ) ) )
=> ! [X8: term] :
( ! [X9: term] :
( ( X0 @ id @ X9 @ id )
=> ( X0 @ id @ ( sub @ X8 @ ( push @ X9 @ id ) ) @ id ) )
=> ( X0 @ id @ ( lam @ X8 ) @ id ) ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
( hoasinduction_lem3a
!= ( ! [X10: subst > term > subst > $o] :
( ! [X18: subst > term > term] :
( ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
=> ( ! [X13: term] :
( ( X10 @ id @ X13 @ id )
=> ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
=> ( X10 @ id
@ ( hoaslam @ id
@ ^ [X14: subst,X13: term] : ( X18 @ X14 @ X13 ) )
@ id ) ) )
=> ! [X13: term] :
( ! [X15: term] :
( ( X10 @ id @ X15 @ id )
=> ( X10 @ id @ ( sub @ X13 @ ( push @ X15 @ id ) ) @ id ) )
=> ( X10 @ id @ ( lam @ X13 ) @ id ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
( hoasinduction_lem3a
= ( ! [X10: subst > term > subst > $o] :
( ! [X18: subst > term > term] :
( ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
=> ( ! [X13: term] :
( ( X10 @ id @ X13 @ id )
=> ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
=> ( X10 @ id
@ ( hoaslam @ id
@ ^ [X14: subst,X13: term] : ( X18 @ X14 @ X13 ) )
@ id ) ) )
=> ! [X13: term] :
( ! [X15: term] :
( ( X10 @ id @ X15 @ id )
=> ( X10 @ id @ ( sub @ X13 @ ( push @ X15 @ id ) ) @ id ) )
=> ( X10 @ id @ ( lam @ X13 ) @ id ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.u2UgYFbzEX/DTF2THF_17811.p',hoasinduction_lem3a) ).
thf(f1968,plain,
! [X0: term,X1: subst] :
( ( ( sK591 @ X1 @ X0 @ X1 )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(superposition,[status(thm)],[f1963,f1892]) ).
thf(f1963,plain,
! [X0: subst] :
( ( $false
= ( sK591 @ X0 @ ( lam @ sK596 ) @ X0 ) )
| ( hoasinduction_lem3a = $false ) ),
inference(superposition,[status(thm)],[f1699,f1894]) ).
thf(f1894,plain,
! [X18: subst] :
( ( d2subst @ d_id )
= X18 ),
inference(forward_demodulation,[status(thm)],[f1250,f1170]) ).
thf(f1170,plain,
! [X122: d_subst] : ( d_id = X122 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1250,plain,
! [X18: subst] :
( ( d2subst @ ( sK463 @ X18 ) )
= X18 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1699,plain,
( ( ( sK591 @ ( d2subst @ d_id ) @ ( lam @ sK596 ) @ ( d2subst @ d_id ) )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1698]) ).
thf(f1698,plain,
( ( ( ( !! @ term
@ ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( sub @ sK596 @ ( push @ Y0 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( lam @ sK596 ) @ ( d2subst @ d_id ) ) )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1697]) ).
thf(f1697,plain,
( ( ( ^ [Y0: term] :
( ( !! @ term
@ ^ [Y1: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( sub @ Y0 @ ( push @ Y1 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( lam @ Y0 ) @ ( d2subst @ d_id ) ) )
@ sK596 )
= $false )
| ( hoasinduction_lem3a = $false ) ),
inference(sigma_clausification,[status(thm)],[f1680]) ).
thf(f1680,plain,
( ( hoasinduction_lem3a = $false )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( !! @ term
@ ^ [Y1: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( sub @ Y0 @ ( push @ Y1 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( lam @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1679]) ).
thf(f2107,plain,
! [X2: subst,X3: term,X0: term,X1: subst > term > subst > $o] :
( ( $true
= ( X1 @ X2 @ X3 @ X2 ) )
| ( ( X1 @ X2 @ X0 @ X2 )
= $false )
| ( hoasinduction_lem3a = $true ) ),
inference(superposition,[status(thm)],[f2084,f1886]) ).
thf(f2084,plain,
! [X2: subst > term > subst > $o,X3: subst,X1: term] :
( ( ( X2 @ X3 @ ( d2term @ d_one ) @ X3 )
= $false )
| ( $true
= ( X2 @ X3 @ X1 @ X3 ) )
| ( hoasinduction_lem3a = $true ) ),
inference(superposition,[status(thm)],[f2065,f1892]) ).
thf(f2065,plain,
! [X2: term,X0: subst,X1: subst > term > subst > $o] :
( ( $true
= ( X1 @ X0 @ ( lam @ X2 ) @ X0 ) )
| ( hoasinduction_lem3a = $true )
| ( ( X1 @ X0 @ ( d2term @ d_one ) @ X0 )
= $false ) ),
inference(superposition,[status(thm)],[f2058,f1894]) ).
thf(f2058,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( $true
= ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) )
| ( ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $false )
| ( hoasinduction_lem3a = $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f2057]) ).
thf(f2057,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( $true
= ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) )
| ( hoasinduction_lem3a = $true )
| ( ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $false )
| ( ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $false ) ),
inference(forward_demodulation,[status(thm)],[f1737,f1886]) ).
thf(f1737,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ ( sK600 @ X1 @ X2 ) @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) )
= $false )
| ( hoasinduction_lem3a = $true )
| ( $true
= ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) )
| ( ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1736]) ).
thf(f1736,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $false )
| ( hoasinduction_lem3a = $true )
| ( $true
= ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) )
| ( $false
= ( ( X1 @ ( d2subst @ d_id ) @ ( sK600 @ X1 @ X2 ) @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ ( sK600 @ X1 @ X2 ) @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1735]) ).
thf(f1735,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( $false
= ( ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ Y0 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) )
@ ( sK600 @ X1 @ X2 ) ) )
| ( ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $false )
| ( hoasinduction_lem3a = $true )
| ( $true
= ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) ) ),
inference(sigma_clausification,[status(thm)],[f1726]) ).
thf(f1726,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $false )
| ( $true
= ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) )
| ( hoasinduction_lem3a = $true )
| ( $false
= ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ Y0 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1725]) ).
thf(f1725,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( $true
= ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) )
| ( hoasinduction_lem3a = $true )
| ( $false
= ( ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sK597 @ X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) )
| ( $false
= ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ Y0 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1711]) ).
thf(f1711,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( hoasinduction_lem3a = $true )
| ( $true
= ( ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ Y0 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) ) )
| ( $false
= ( ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sK597 @ X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1710]) ).
thf(f1710,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( ( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ ( sK597 @ X1 @ Y1 @ Y2 ) @ Y0 )
= ( sK597 @ X1 @ ( comp @ Y1 @ Y0 ) @ ( sub @ Y2 @ Y0 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sK597 @ X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) )
= $false )
| ( hoasinduction_lem3a = $true )
| ( $true
= ( ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ Y0 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1709]) ).
thf(f1709,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( $true
= ( ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ Y0 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) ) )
| ( $false
= ( ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( ( sub @ ( Y0 @ Y2 @ Y3 ) @ Y1 )
= ( Y0 @ ( comp @ Y2 @ Y1 ) @ ( sub @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( Y0 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) )
@ ( sK597 @ X1 ) ) )
| ( hoasinduction_lem3a = $true ) ),
inference(sigma_clausification,[status(thm)],[f1708]) ).
thf(f1708,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( $false
= ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( ( sub @ ( Y0 @ Y2 @ Y3 ) @ Y1 )
= ( Y0 @ ( comp @ Y2 @ Y1 ) @ ( sub @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( Y0 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) ) )
| ( $true
= ( ( !! @ term
@ ^ [Y0: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ X2 @ ( push @ Y0 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( lam @ X2 ) @ ( d2subst @ d_id ) ) ) )
| ( hoasinduction_lem3a = $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1707]) ).
thf(f1707,plain,
! [X2: term,X1: subst > term > subst > $o] :
( ( hoasinduction_lem3a = $true )
| ( $false
= ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( ( sub @ ( Y0 @ Y2 @ Y3 ) @ Y1 )
= ( Y0 @ ( comp @ Y2 @ Y1 ) @ ( sub @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( Y0 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) ) )
| ( $true
= ( ^ [Y0: term] :
( ( !! @ term
@ ^ [Y1: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ Y0 @ ( push @ Y1 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( lam @ Y0 ) @ ( d2subst @ d_id ) ) )
@ X2 ) ) ),
inference(pi_clausification,[status(thm)],[f1706]) ).
thf(f1706,plain,
! [X1: subst > term > subst > $o] :
( ( ( !! @ term
@ ^ [Y0: term] :
( ( !! @ term
@ ^ [Y1: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ Y0 @ ( push @ Y1 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( lam @ Y0 ) @ ( d2subst @ d_id ) ) ) )
= $true )
| ( hoasinduction_lem3a = $true )
| ( $false
= ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( ( sub @ ( Y0 @ Y2 @ Y3 ) @ Y1 )
= ( Y0 @ ( comp @ Y2 @ Y1 ) @ ( sub @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( Y0 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1705]) ).
thf(f1705,plain,
! [X1: subst > term > subst > $o] :
( ( hoasinduction_lem3a = $true )
| ( $true
= ( ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( ( sub @ ( Y0 @ Y2 @ Y3 ) @ Y1 )
= ( Y0 @ ( comp @ Y2 @ Y1 ) @ ( sub @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( Y0 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) )
=> ( !! @ term
@ ^ [Y0: term] :
( ( !! @ term
@ ^ [Y1: term] :
( ( X1 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( sub @ Y0 @ ( push @ Y1 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( X1 @ ( d2subst @ d_id ) @ ( lam @ Y0 ) @ ( d2subst @ d_id ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1704]) ).
thf(f1704,plain,
! [X1: subst > term > subst > $o] :
( ( hoasinduction_lem3a = $true )
| ( $true
= ( ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( Y1 @ ( d2subst @ d_id ) @ Y2 ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( sub @ Y1 @ ( push @ Y2 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( lam @ Y1 ) @ ( d2subst @ d_id ) ) ) ) )
@ X1 ) ) ),
inference(pi_clausification,[status(thm)],[f1676]) ).
thf(f1676,plain,
( ( $true
= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ subst
@ ^ [Y3: subst] :
( !! @ term
@ ^ [Y4: term] :
( ( sub @ ( Y1 @ Y3 @ Y4 ) @ Y2 )
= ( Y1 @ ( comp @ Y3 @ Y2 ) @ ( sub @ Y4 @ Y2 ) ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( Y1 @ ( d2subst @ d_id ) @ Y2 ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( sub @ Y1 @ ( push @ Y2 @ ( d2subst @ d_id ) ) ) @ ( d2subst @ d_id ) ) ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( lam @ Y1 ) @ ( d2subst @ d_id ) ) ) ) ) ) )
| ( hoasinduction_lem3a = $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f1675]) ).
thf(f2535,plain,
! [X0: subst] :
( ( sK283 @ X0 @ ( lam @ sK284 ) @ X0 )
!= $true ),
inference(superposition,[status(thm)],[f2527,f1894]) ).
thf(f2527,plain,
( $true
!= ( sK283 @ ( d2subst @ d_id ) @ ( lam @ sK284 ) @ ( d2subst @ d_id ) ) ),
inference(subsumption_resolution,[status(thm)],[f1437,f2209]) ).
thf(f2209,plain,
sP24 != $true,
inference(trivial_inequality_removal,[status(thm)],[f2199]) ).
thf(f2199,plain,
( ( sP24 != $true )
| ( $true = $false ) ),
inference(backward_demodulation,[status(thm)],[f1173,f2194]) ).
thf(f1173,plain,
( ( sP24 != $true )
| ( hoasinduction_lem3a = $true ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1437,plain,
( ( sP24 = $true )
| ( $true
!= ( sK283 @ ( d2subst @ d_id ) @ ( lam @ sK284 ) @ ( d2subst @ d_id ) ) ) ),
inference(definition_unfolding,[status(thm)],[f829,f1137,f1137]) ).
thf(f829,plain,
( ( sP24 = $true )
| ( ( sK283 @ id @ ( lam @ sK284 ) @ id )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f309]) ).
thf(f309,plain,
( ( ( sP24 = $true )
| ( ! [X2: term] :
( ( ( sK283 @ id @ ( sub @ sK284 @ ( push @ X2 @ id ) ) @ id )
= $true )
| ( $true
!= ( sK283 @ id @ X2 @ id ) ) )
& ( ( sK283 @ id @ ( lam @ sK284 ) @ id )
!= $true )
& ! [X3: subst > term > term] :
( ( ( ( sK283 @ id @ ( X3 @ id @ ( sK285 @ X3 ) ) @ id )
!= $true )
& ( ( sK283 @ id @ ( sK285 @ X3 ) @ id )
= $true ) )
| ( ( X3 @ ( comp @ ( sK287 @ X3 ) @ ( sK286 @ X3 ) ) @ ( sub @ ( sK288 @ X3 ) @ ( sK286 @ X3 ) ) )
!= ( sub @ ( X3 @ ( sK287 @ X3 ) @ ( sK288 @ X3 ) ) @ ( sK286 @ X3 ) ) )
| ( ( sK283 @ id @ ( lam @ ( X3 @ sh @ one ) ) @ id )
= $true ) ) ) )
& ( ! [X8: subst > term > subst > $o,X9: term] :
( ( ( $true
!= ( X8 @ id @ ( sub @ X9 @ ( push @ ( sK289 @ X9 @ X8 ) @ id ) ) @ id ) )
& ( ( X8 @ id @ ( sK289 @ X9 @ X8 ) @ id )
= $true ) )
| ( ( X8 @ id @ ( lam @ X9 ) @ id )
= $true )
| ( ! [X12: term] :
( ( $true
= ( X8 @ id @ ( sK290 @ X8 @ id @ X12 ) @ id ) )
| ( $true
!= ( X8 @ id @ X12 @ id ) ) )
& ! [X13: subst,X14: subst,X15: term] :
( ( sK290 @ X8 @ ( comp @ X14 @ X13 ) @ ( sub @ X15 @ X13 ) )
= ( sub @ ( sK290 @ X8 @ X14 @ X15 ) @ X13 ) )
& ( $true
!= ( X8 @ id @ ( lam @ ( sK290 @ X8 @ sh @ one ) ) @ id ) ) ) )
| ( sP24 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK283,sK284,sK285,sK286,sK287,sK288,sK289,sK290])],[f303,f308,f307,f306,f305,f304]) ).
thf(f304,plain,
( ? [X0: subst > term > subst > $o,X1: term] :
( ! [X2: term] :
( ( $true
= ( X0 @ id @ ( sub @ X1 @ ( push @ X2 @ id ) ) @ id ) )
| ( $true
!= ( X0 @ id @ X2 @ id ) ) )
& ( $true
!= ( X0 @ id @ ( lam @ X1 ) @ id ) )
& ! [X3: subst > term > term] :
( ? [X4: term] :
( ( $true
!= ( X0 @ id @ ( X3 @ id @ X4 ) @ id ) )
& ( ( X0 @ id @ X4 @ id )
= $true ) )
| ? [X5: subst,X6: subst,X7: term] :
( ( sub @ ( X3 @ X6 @ X7 ) @ X5 )
!= ( X3 @ ( comp @ X6 @ X5 ) @ ( sub @ X7 @ X5 ) ) )
| ( $true
= ( X0 @ id @ ( lam @ ( X3 @ sh @ one ) ) @ id ) ) ) )
=> ( ! [X2: term] :
( ( ( sK283 @ id @ ( sub @ sK284 @ ( push @ X2 @ id ) ) @ id )
= $true )
| ( $true
!= ( sK283 @ id @ X2 @ id ) ) )
& ( ( sK283 @ id @ ( lam @ sK284 ) @ id )
!= $true )
& ! [X3: subst > term > term] :
( ? [X4: term] :
( ( $true
!= ( sK283 @ id @ ( X3 @ id @ X4 ) @ id ) )
& ( ( sK283 @ id @ X4 @ id )
= $true ) )
| ? [X5: subst,X6: subst,X7: term] :
( ( sub @ ( X3 @ X6 @ X7 ) @ X5 )
!= ( X3 @ ( comp @ X6 @ X5 ) @ ( sub @ X7 @ X5 ) ) )
| ( ( sK283 @ id @ ( lam @ ( X3 @ sh @ one ) ) @ id )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f305,plain,
! [X3: subst > term > term] :
( ? [X4: term] :
( ( $true
!= ( sK283 @ id @ ( X3 @ id @ X4 ) @ id ) )
& ( ( sK283 @ id @ X4 @ id )
= $true ) )
=> ( ( ( sK283 @ id @ ( X3 @ id @ ( sK285 @ X3 ) ) @ id )
!= $true )
& ( ( sK283 @ id @ ( sK285 @ X3 ) @ id )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f306,plain,
! [X3: subst > term > term] :
( ? [X5: subst,X6: subst,X7: term] :
( ( sub @ ( X3 @ X6 @ X7 ) @ X5 )
!= ( X3 @ ( comp @ X6 @ X5 ) @ ( sub @ X7 @ X5 ) ) )
=> ( ( X3 @ ( comp @ ( sK287 @ X3 ) @ ( sK286 @ X3 ) ) @ ( sub @ ( sK288 @ X3 ) @ ( sK286 @ X3 ) ) )
!= ( sub @ ( X3 @ ( sK287 @ X3 ) @ ( sK288 @ X3 ) ) @ ( sK286 @ X3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f307,plain,
! [X8: subst > term > subst > $o,X9: term] :
( ? [X10: term] :
( ( ( X8 @ id @ ( sub @ X9 @ ( push @ X10 @ id ) ) @ id )
!= $true )
& ( $true
= ( X8 @ id @ X10 @ id ) ) )
=> ( ( $true
!= ( X8 @ id @ ( sub @ X9 @ ( push @ ( sK289 @ X9 @ X8 ) @ id ) ) @ id ) )
& ( ( X8 @ id @ ( sK289 @ X9 @ X8 ) @ id )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f308,plain,
! [X8: subst > term > subst > $o] :
( ? [X11: subst > term > term] :
( ! [X12: term] :
( ( $true
= ( X8 @ id @ ( X11 @ id @ X12 ) @ id ) )
| ( $true
!= ( X8 @ id @ X12 @ id ) ) )
& ! [X13: subst,X14: subst,X15: term] :
( ( sub @ ( X11 @ X14 @ X15 ) @ X13 )
= ( X11 @ ( comp @ X14 @ X13 ) @ ( sub @ X15 @ X13 ) ) )
& ( $true
!= ( X8 @ id @ ( lam @ ( X11 @ sh @ one ) ) @ id ) ) )
=> ( ! [X12: term] :
( ( $true
= ( X8 @ id @ ( sK290 @ X8 @ id @ X12 ) @ id ) )
| ( $true
!= ( X8 @ id @ X12 @ id ) ) )
& ! [X15: term,X14: subst,X13: subst] :
( ( sK290 @ X8 @ ( comp @ X14 @ X13 ) @ ( sub @ X15 @ X13 ) )
= ( sub @ ( sK290 @ X8 @ X14 @ X15 ) @ X13 ) )
& ( $true
!= ( X8 @ id @ ( lam @ ( sK290 @ X8 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f303,plain,
( ( ( sP24 = $true )
| ? [X0: subst > term > subst > $o,X1: term] :
( ! [X2: term] :
( ( $true
= ( X0 @ id @ ( sub @ X1 @ ( push @ X2 @ id ) ) @ id ) )
| ( $true
!= ( X0 @ id @ X2 @ id ) ) )
& ( $true
!= ( X0 @ id @ ( lam @ X1 ) @ id ) )
& ! [X3: subst > term > term] :
( ? [X4: term] :
( ( $true
!= ( X0 @ id @ ( X3 @ id @ X4 ) @ id ) )
& ( ( X0 @ id @ X4 @ id )
= $true ) )
| ? [X5: subst,X6: subst,X7: term] :
( ( sub @ ( X3 @ X6 @ X7 ) @ X5 )
!= ( X3 @ ( comp @ X6 @ X5 ) @ ( sub @ X7 @ X5 ) ) )
| ( $true
= ( X0 @ id @ ( lam @ ( X3 @ sh @ one ) ) @ id ) ) ) ) )
& ( ! [X8: subst > term > subst > $o,X9: term] :
( ? [X10: term] :
( ( ( X8 @ id @ ( sub @ X9 @ ( push @ X10 @ id ) ) @ id )
!= $true )
& ( $true
= ( X8 @ id @ X10 @ id ) ) )
| ( ( X8 @ id @ ( lam @ X9 ) @ id )
= $true )
| ? [X11: subst > term > term] :
( ! [X12: term] :
( ( $true
= ( X8 @ id @ ( X11 @ id @ X12 ) @ id ) )
| ( $true
!= ( X8 @ id @ X12 @ id ) ) )
& ! [X13: subst,X14: subst,X15: term] :
( ( sub @ ( X11 @ X14 @ X15 ) @ X13 )
= ( X11 @ ( comp @ X14 @ X13 ) @ ( sub @ X15 @ X13 ) ) )
& ( $true
!= ( X8 @ id @ ( lam @ ( X11 @ sh @ one ) ) @ id ) ) ) )
| ( sP24 != $true ) ) ),
inference(rectify,[status(thm)],[f302]) ).
thf(f302,plain,
( ( ( sP24 = $true )
| ? [X262: subst > term > subst > $o,X263: term] :
( ! [X264: term] :
( ( ( X262 @ id @ ( sub @ X263 @ ( push @ X264 @ id ) ) @ id )
= $true )
| ( ( X262 @ id @ X264 @ id )
!= $true ) )
& ( ( X262 @ id @ ( lam @ X263 ) @ id )
!= $true )
& ! [X265: subst > term > term] :
( ? [X266: term] :
( ( $true
!= ( X262 @ id @ ( X265 @ id @ X266 ) @ id ) )
& ( ( X262 @ id @ X266 @ id )
= $true ) )
| ? [X267: subst,X268: subst,X269: term] :
( ( sub @ ( X265 @ X268 @ X269 ) @ X267 )
!= ( X265 @ ( comp @ X268 @ X267 ) @ ( sub @ X269 @ X267 ) ) )
| ( $true
= ( X262 @ id @ ( lam @ ( X265 @ sh @ one ) ) @ id ) ) ) ) )
& ( ! [X262: subst > term > subst > $o,X263: term] :
( ? [X264: term] :
( ( ( X262 @ id @ ( sub @ X263 @ ( push @ X264 @ id ) ) @ id )
!= $true )
& ( ( X262 @ id @ X264 @ id )
= $true ) )
| ( ( X262 @ id @ ( lam @ X263 ) @ id )
= $true )
| ? [X265: subst > term > term] :
( ! [X266: term] :
( ( $true
= ( X262 @ id @ ( X265 @ id @ X266 ) @ id ) )
| ( ( X262 @ id @ X266 @ id )
!= $true ) )
& ! [X267: subst,X268: subst,X269: term] :
( ( sub @ ( X265 @ X268 @ X269 ) @ X267 )
= ( X265 @ ( comp @ X268 @ X267 ) @ ( sub @ X269 @ X267 ) ) )
& ( $true
!= ( X262 @ id @ ( lam @ ( X265 @ sh @ one ) ) @ id ) ) ) )
| ( sP24 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f38]) ).
thf(f5651,plain,
! [X3: subst > term > term] :
( $true
= ( sK283 @ ( d2subst @ d_id ) @ ( lam @ ( X3 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5650]) ).
thf(f5650,plain,
! [X3: subst > term > term] :
( ( $true
= ( sK283 @ ( d2subst @ d_id ) @ ( lam @ ( X3 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) ) )
| ( $true = $false ) ),
inference(forward_demodulation,[status(thm)],[f5649,f2545]) ).
thf(f5649,plain,
! [X3: subst > term > term] :
( ( $true
= ( sK283 @ ( d2subst @ d_id ) @ ( sK285 @ X3 ) @ ( d2subst @ d_id ) ) )
| ( $true
= ( sK283 @ ( d2subst @ d_id ) @ ( lam @ ( X3 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f5648,f1892]) ).
thf(f5648,plain,
! [X3: subst > term > term] :
( ( $true
= ( sK283 @ ( d2subst @ d_id ) @ ( sK285 @ X3 ) @ ( d2subst @ d_id ) ) )
| ( $true
= ( sK283 @ ( d2subst @ d_id ) @ ( lam @ ( X3 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) ) )
| ( ( X3 @ ( comp @ ( sK287 @ X3 ) @ ( sK286 @ X3 ) ) @ ( sub @ ( sK288 @ X3 ) @ ( sK286 @ X3 ) ) )
!= ( sub @ ( X3 @ ( sK287 @ X3 ) @ ( sK288 @ X3 ) ) @ ( sK286 @ X3 ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f1439,f2209]) ).
thf(f1439,plain,
! [X3: subst > term > term] :
( ( sP24 = $true )
| ( ( X3 @ ( comp @ ( sK287 @ X3 ) @ ( sK286 @ X3 ) ) @ ( sub @ ( sK288 @ X3 ) @ ( sK286 @ X3 ) ) )
!= ( sub @ ( X3 @ ( sK287 @ X3 ) @ ( sK288 @ X3 ) ) @ ( sK286 @ X3 ) ) )
| ( $true
= ( sK283 @ ( d2subst @ d_id ) @ ( lam @ ( X3 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) ) )
| ( $true
= ( sK283 @ ( d2subst @ d_id ) @ ( sK285 @ X3 ) @ ( d2subst @ d_id ) ) ) ),
inference(definition_unfolding,[status(thm)],[f827,f1137,f1137,f1137,f1053,f1074,f1137]) ).
thf(f1074,plain,
( one
= ( d2term @ d_one ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1053,plain,
( ( d2subst @ d_id )
= sh ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f827,plain,
! [X3: subst > term > term] :
( ( sP24 = $true )
| ( ( sK283 @ id @ ( sK285 @ X3 ) @ id )
= $true )
| ( ( X3 @ ( comp @ ( sK287 @ X3 ) @ ( sK286 @ X3 ) ) @ ( sub @ ( sK288 @ X3 ) @ ( sK286 @ X3 ) ) )
!= ( sub @ ( X3 @ ( sK287 @ X3 ) @ ( sK288 @ X3 ) ) @ ( sK286 @ X3 ) ) )
| ( ( sK283 @ id @ ( lam @ ( X3 @ sh @ one ) ) @ id )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f309]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n010.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue May 5 09:11:36 EDT 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.26/0.47 ---- Original DTF file ---
% 0.26/0.47 thf(spec,logic,$$dhol).
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 % File : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.26/0.47 % Domain : Software Verification
% 0.26/0.47 % Problem : Benchmark: ALG444^1, Active target: hoasinduction_lem3a
% 0.26/0.47 % Version : Especial.
% 0.26/0.47 % English :
% 0.26/0.47
% 0.26/0.47 % Refs : [Kon26] Kondylidou (2026), Email to Geoff Sutcliffe
% 0.26/0.47 % Source : [Kon26]
% 0.26/0.47 % Names : ALG444^1__005__axclos.verify [Kon26]
% 0.26/0.47
% 0.26/0.47 % Status : Theorem
% 0.26/0.47 % Rating : ? v9.3.0
% 0.26/0.47 % Syntax : Number of formulae : 132 ( 1 unt; 130 typ; 0 def)
% 0.26/0.47 % Number of atoms : 255 ( 129 equ; 0 cnn)
% 0.26/0.47 % Maximal formula atoms : 248 ( 127 avg)
% 0.26/0.47 % Number of connectives : 2614 ( 333 ~; 306 |; 129 &;1746 @)
% 0.26/0.47 % ( 56 <=>; 44 =>; 0 <=; 0 <~>)
% 0.26/0.47 % Maximal formula depth : 151 ( 76 avg)
% 0.26/0.47 % Number of types : 5 ( 4 usr)
% 0.26/0.47 % Number of type conns : 227 ( 227 >; 0 *; 0 +; 0 <<)
% 0.26/0.47 % Number of symbols : 127 ( 126 usr; 113 con; 0-4 aty)
% 0.26/0.47 % Number of variables : 560 ( 10 ^; 548 !; 2 ?; 560 :)
% 0.26/0.47 % SPC : TH0_THM_EQU_NAR
% 0.26/0.47
% 0.26/0.47 % Comments :
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 thf(term_type,type,
% 0.26/0.47 term: $tType ).
% 0.26/0.47
% 0.26/0.47 thf(subst_type,type,
% 0.26/0.47 subst: $tType ).
% 0.26/0.47
% 0.26/0.47 thf(one_decl,type,
% 0.26/0.47 one: term ).
% 0.26/0.47
% 0.26/0.47 thf(ap_decl,type,
% 0.26/0.47 ap: term > term > term ).
% 0.26/0.47
% 0.26/0.47 thf(lam_decl,type,
% 0.26/0.47 lam: term > term ).
% 0.26/0.47
% 0.26/0.47 thf(sub_decl,type,
% 0.26/0.47 sub: term > subst > term ).
% 0.26/0.47
% 0.26/0.47 thf(id_decl,type,
% 0.26/0.47 id: subst ).
% 0.26/0.47
% 0.26/0.47 thf(sh_decl,type,
% 0.26/0.47 sh: subst ).
% 0.26/0.47
% 0.26/0.47 thf(push_decl,type,
% 0.26/0.47 push: term > subst > subst ).
% 0.26/0.47
% 0.26/0.47 thf(comp_decl,type,
% 0.26/0.47 comp: subst > subst > subst ).
% 0.26/0.47
% 0.26/0.47 thf(var_decl,type,
% 0.26/0.47 var: term > $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem1v2_decl,type,
% 0.26/0.47 pushprop_lem1v2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem1_gthm_decl,type,
% 0.26/0.47 pushprop_lem1_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axmap_decl,type,
% 0.26/0.47 axmap: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem0_gthm_decl,type,
% 0.26/0.47 pushprop_lem0_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(shinj_decl,type,
% 0.26/0.47 shinj: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem1v2_decl,type,
% 0.26/0.47 hoasinduction_lem1v2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem1v2_gthm_decl,type,
% 0.26/0.47 hoasinduction_lem1v2_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasap_decl,type,
% 0.26/0.47 hoasap: subst > term > subst > term > term ).
% 0.26/0.47
% 0.26/0.47 thf(induction2lem_decl,type,
% 0.26/0.47 induction2lem: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3v2_f_decl,type,
% 0.26/0.47 hoasinduction_lem3v2_f: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axvarshift_decl,type,
% 0.26/0.47 axvarshift: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapinj2_decl,type,
% 0.26/0.47 hoasapinj2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapnotvar_gthm_decl,type,
% 0.26/0.47 hoasapnotvar_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapinj1_decl,type,
% 0.26/0.47 hoasapinj1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(ulamvar1_decl,type,
% 0.26/0.47 ulamvar1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(induction2lem_lthm_decl,type,
% 0.26/0.47 induction2lem_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3v2_gthm_decl,type,
% 0.26/0.47 hoasinduction_lem3v2_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(apnotvar_decl,type,
% 0.26/0.47 apnotvar: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lthm_orig_decl,type,
% 0.26/0.47 pushprop_lthm_orig: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem3v2_f_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lthm_decl,type,
% 0.26/0.47 hoasinduction_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 0.26/0.47 hoasinduction_no_psi_cond_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslaminj_decl,type,
% 0.26/0.47 hoaslaminj: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3aaa_decl,type,
% 0.26/0.47 hoasinduction_lem3aaa: $o ).
% 0.26/0.47
% 0.26/0.47 thf(induction2lem_gthm_decl,type,
% 0.26/0.47 induction2lem_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3aa_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem3aa_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3_decl,type,
% 0.26/0.47 hoasinduction_lem3: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem2_decl,type,
% 0.26/0.47 hoasinduction_lem2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(termmset_lthm_decl,type,
% 0.26/0.47 termmset_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem1_decl,type,
% 0.26/0.47 hoasinduction_lem1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslamnotap_lthm_decl,type,
% 0.26/0.47 hoaslamnotap_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem1v2_lthm_decl,type,
% 0.26/0.47 pushprop_lem1v2_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapnotvar_decl,type,
% 0.26/0.47 hoasapnotvar: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem0_decl,type,
% 0.26/0.47 hoasinduction_lem0: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_decl,type,
% 0.26/0.47 hoasinduction: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_gthm_decl,type,
% 0.26/0.47 hoasinduction_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axapp_decl,type,
% 0.26/0.47 axapp: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslamnotvar_lthm_decl,type,
% 0.26/0.47 hoaslamnotvar_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem3v2_lthm_decl,type,
% 0.26/0.47 pushprop_lem3v2_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3b_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem3b_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(ulamvarind_decl,type,
% 0.26/0.47 ulamvarind: $o ).
% 0.26/0.47
% 0.26/0.47 thf(induction_decl,type,
% 0.26/0.47 induction: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3a_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem3a_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(termmset_gthm_decl,type,
% 0.26/0.47 termmset_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3aa_decl,type,
% 0.26/0.47 hoasinduction_lem3aa: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem1v2_gthm_decl,type,
% 0.26/0.47 pushprop_lem1v2_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslamnotap_gthm_decl,type,
% 0.26/0.47 hoaslamnotap_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslamnotvar_gthm_decl,type,
% 0.26/0.47 hoaslamnotvar_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3b_gthm_decl,type,
% 0.26/0.47 hoasinduction_lem3b_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem2v2_decl,type,
% 0.26/0.47 pushprop_lem2v2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3a_gthm_decl,type,
% 0.26/0.47 hoasinduction_lem3a_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axclos_decl,type,
% 0.26/0.47 axclos: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axassoc_decl,type,
% 0.26/0.47 axassoc: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem2v2_decl,type,
% 0.26/0.47 hoasinduction_lem2v2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lthm_decl,type,
% 0.26/0.47 pushprop_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(apinj2_decl,type,
% 0.26/0.47 apinj2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(apinj1_decl,type,
% 0.26/0.47 apinj1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapinj2_lthm_decl,type,
% 0.26/0.47 hoasapinj2_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3v2a_decl,type,
% 0.26/0.47 hoasinduction_lem3v2a: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapinj1_lthm_decl,type,
% 0.26/0.47 hoasapinj1_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslaminj_lthm_decl,type,
% 0.26/0.47 hoaslaminj_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axvarcons_decl,type,
% 0.26/0.47 axvarcons: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslam_decl,type,
% 0.26/0.47 hoaslam: subst > ( subst > term > term ) > term ).
% 0.26/0.47
% 0.26/0.47 thf(axscons_decl,type,
% 0.26/0.47 axscons: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem2v2_gthm_decl,type,
% 0.26/0.47 hoasinduction_lem2v2_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axidr_decl,type,
% 0.26/0.47 axidr: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem1_decl,type,
% 0.26/0.47 pushprop_lem1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(laminj_decl,type,
% 0.26/0.47 laminj: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem3_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem0_decl,type,
% 0.26/0.47 pushprop_lem0: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_gthm_decl,type,
% 0.26/0.47 pushprop_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axabs_decl,type,
% 0.26/0.47 axabs: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem3v2a_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem2_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem2_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapinj2_gthm_decl,type,
% 0.26/0.47 hoasapinj2_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_p_and_p_prime_decl,type,
% 0.26/0.47 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem1_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem1_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(lamnotap_decl,type,
% 0.26/0.47 lamnotap: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapinj1_gthm_decl,type,
% 0.26/0.47 hoasapinj1_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslamnotvar_decl,type,
% 0.26/0.47 hoaslamnotvar: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axidl_decl,type,
% 0.26/0.47 axidl: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslaminj_gthm_decl,type,
% 0.26/0.47 hoaslaminj_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(induction2_lthm_decl,type,
% 0.26/0.47 induction2_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem0_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem0_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(substmonoid_lthm_decl,type,
% 0.26/0.47 substmonoid_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_decl,type,
% 0.26/0.47 pushprop: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3_gthm_decl,type,
% 0.26/0.47 hoasinduction_lem3_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem2_gthm_decl,type,
% 0.26/0.47 hoasinduction_lem2_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3b_decl,type,
% 0.26/0.47 hoasinduction_lem3b: $o ).
% 0.26/0.47
% 0.26/0.47 thf(substmonoid_decl,type,
% 0.26/0.47 substmonoid: $o ).
% 0.26/0.47
% 0.26/0.47 thf(lamnotvar_decl,type,
% 0.26/0.47 lamnotvar: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3a_decl,type,
% 0.26/0.47 hoasinduction_lem3a: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem1_gthm_decl,type,
% 0.26/0.47 hoasinduction_lem1_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_no_psi_cond_decl,type,
% 0.26/0.47 hoasinduction_no_psi_cond: $o ).
% 0.26/0.47
% 0.26/0.47 thf(induction2_gthm_decl,type,
% 0.26/0.47 induction2_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem2v2_lthm_decl,type,
% 0.26/0.47 pushprop_lem2v2_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasvar_decl,type,
% 0.26/0.47 hoasvar: subst > term > subst > $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoaslamnotap_decl,type,
% 0.26/0.47 hoaslamnotap: $o ).
% 0.26/0.47
% 0.26/0.47 thf(substmonoid_gthm_decl,type,
% 0.26/0.47 substmonoid_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(ulamvarsh_decl,type,
% 0.26/0.47 ulamvarsh: $o ).
% 0.26/0.47
% 0.26/0.47 thf(induction2_decl,type,
% 0.26/0.47 induction2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem3v2_decl,type,
% 0.26/0.47 pushprop_lem3v2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem2v2_gthm_decl,type,
% 0.26/0.47 pushprop_lem2v2_gthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem1_lthm_decl,type,
% 0.26/0.47 pushprop_lem1_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3v2_decl,type,
% 0.26/0.47 hoasinduction_lem3v2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(axshiftcons_decl,type,
% 0.26/0.47 axshiftcons: $o ).
% 0.26/0.47
% 0.26/0.47 thf(termmset_decl,type,
% 0.26/0.47 termmset: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_lem0_lthm_decl,type,
% 0.26/0.47 pushprop_lem0_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasapnotvar_lthm_decl,type,
% 0.26/0.47 hoasapnotvar_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3v2_lthm_decl,type,
% 0.26/0.47 hoasinduction_lem3v2_lthm: $o ).
% 0.26/0.47
% 0.26/0.47 thf(pushprop_p_and_p_prime_decl,type,
% 0.26/0.47 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(axvarid_decl,type,
% 0.26/0.47 axvarid: $o ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lthm_3_decl,type,
% 0.26/0.47 hoasinduction_lthm_3: $o ).
% 0.26/0.47
% 0.26/0.47 %----Types of the domains
% 0.26/0.47 thf(d_term_type,type,
% 0.26/0.47 d_term: $tType ).
% 0.26/0.47
% 0.26/0.47 thf(d_subst_type,type,
% 0.26/0.47 d_subst: $tType ).
% 0.26/0.47
% 0.26/0.47 %----Types of the promotion functions
% 0.26/0.47 thf(d2term_decl,type,
% 0.26/0.47 d2term: d_term > term ).
% 0.26/0.47
% 0.26/0.47 thf(d2subst_decl,type,
% 0.26/0.47 d2subst: d_subst > subst ).
% 0.26/0.47
% 0.26/0.47 %----Types of the domain elements
% 0.26/0.47 thf(d_one_decl,type,
% 0.26/0.47 d_one: d_term ).
% 0.26/0.47
% 0.26/0.47 thf(d_id_decl,type,
% 0.26/0.47 d_id: d_subst ).
% 0.26/0.47
% 0.26/0.47 thf(alg444_1,axiom,
% 0.26/0.47 ( ! [T: term] :
% 0.26/0.47 ? [DT: d_term] :
% 0.26/0.47 ( T
% 0.26/0.47 = ( d2term @ DT ) )
% 0.26/0.47 & ! [DT: d_term] : ( DT = d_one )
% 0.26/0.47 & ! [DT1: d_term,DT2: d_term] :
% 0.26/0.47 ( ( ( d2term @ DT1 )
% 0.26/0.47 = ( d2term @ DT2 ) )
% 0.26/0.47 => ( DT1 = DT2 ) )
% 0.26/0.47 & ! [S: subst] :
% 0.26/0.47 ? [DS: d_subst] :
% 0.26/0.47 ( S
% 0.26/0.47 = ( d2subst @ DS ) )
% 0.26/0.47 & ! [DS: d_subst] : ( DS = d_id )
% 0.26/0.47 & ! [DS1: d_subst,DS2: d_subst] :
% 0.26/0.47 ( ( ( d2subst @ DS1 )
% 0.26/0.47 = ( d2subst @ DS2 ) )
% 0.26/0.47 => ( DS1 = DS2 ) )
% 0.26/0.47 & ( one
% 0.26/0.47 = ( d2term @ d_one ) )
% 0.26/0.47 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 0.26/0.47 = ( d2term @ d_one ) )
% 0.26/0.47 & ( ( lam @ ( d2term @ d_one ) )
% 0.26/0.47 = ( d2term @ d_one ) )
% 0.26/0.47 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.26/0.47 = ( d2term @ d_one ) )
% 0.26/0.47 & ( id
% 0.26/0.47 = ( d2subst @ d_id ) )
% 0.26/0.47 & ( sh
% 0.26/0.47 = ( d2subst @ d_id ) )
% 0.26/0.47 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.26/0.47 = ( d2subst @ d_id ) )
% 0.26/0.47 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 0.26/0.47 = ( d2subst @ d_id ) )
% 0.26/0.47 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 0.26/0.47 = ( d2term @ d_one ) )
% 0.26/0.47 & ( hoaslam
% 0.26/0.47 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 0.26/0.47 & ( hoasinduction_p_and_p_prime
% 0.26/0.47 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 0.26/0.47 ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) ) ) )
% 0.26/0.47 & ( pushprop_p_and_p_prime
% 0.26/0.47 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 0.26/0.47 ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 0.26/0.47 & ~ ( var @ ( d2term @ d_one ) )
% 0.26/0.47 & ( pushprop_lem1v2
% 0.26/0.47 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.26/0.47 | ( Q @ one ) ) )
% 0.26/0.47 & pushprop_lem1_gthm
% 0.26/0.47 & axmap
% 0.26/0.47 & pushprop_lem0_gthm
% 0.26/0.47 & ( shinj
% 0.26/0.47 <=> ! [A: term,B: term] :
% 0.26/0.47 ( ( ( sub @ A @ sh )
% 0.26/0.47 != ( sub @ B @ sh ) )
% 0.26/0.47 | ( A = B ) ) )
% 0.26/0.47 & hoasinduction_lem1v2
% 0.26/0.47 & hoasinduction_lem1v2_gthm
% 0.26/0.47 & ( induction2lem
% 0.26/0.47 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.26/0.47 ( ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ B )
% 0.26/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( var @ B )
% 0.26/0.47 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.26/0.47 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 0.26/0.47 & ( hoasinduction_lem3v2_f
% 0.26/0.47 <=> ! [B: term] :
% 0.26/0.47 ~ ! [F: subst > term > term] :
% 0.26/0.47 ~ ! [A: term,M: subst] :
% 0.26/0.47 ( ( sub @ B @ ( push @ A @ M ) )
% 0.26/0.47 = ( F @ M @ A ) ) )
% 0.26/0.47 & axvarshift
% 0.26/0.47 & ( hoasapinj2
% 0.26/0.47 <=> ! [A: term,B: term,C: term,D: term] :
% 0.26/0.47 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.26/0.47 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.26/0.47 | ( C = D ) ) )
% 0.26/0.47 & hoasapnotvar_gthm
% 0.26/0.47 & ( hoasapinj1
% 0.26/0.47 <=> ! [A: term,B: term,C: term,D: term] :
% 0.26/0.47 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.26/0.47 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.26/0.47 | ( A = B ) ) )
% 0.26/0.47 & ~ ulamvar1
% 0.26/0.47 & ( induction2lem_lthm
% 0.26/0.47 <=> ( ! [A: term,M: subst] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.26/0.47 => ( ! [A: term,M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.26/0.47 => ( ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ M @ id ) )
% 0.26/0.47 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 0.26/0.47 ( ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ A )
% 0.26/0.47 | ( P @ A ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ( P @ Bound_variable_2140 ) )
% 0.26/0.47 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.26/0.47 ( ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ B )
% 0.26/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( var @ B )
% 0.26/0.47 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.26/0.47 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 0.26/0.47 & hoasinduction_lem3v2_gthm
% 0.26/0.47 & apnotvar
% 0.26/0.47 & pushprop_lthm_orig
% 0.26/0.47 & ( hoasinduction_lem3v2_f_lthm
% 0.26/0.47 <=> ! [B: term] :
% 0.26/0.47 ~ ! [F: subst > term > term] :
% 0.26/0.47 ~ ! [A: term,M: subst] :
% 0.26/0.47 ( ( sub @ B @ ( push @ A @ M ) )
% 0.26/0.47 = ( F @ M @ A ) ) )
% 0.26/0.47 & ( hoasinduction_lthm
% 0.26/0.47 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.26/0.47 ( ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ A )
% 0.26/0.47 | ( P @ A ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ B )
% 0.26/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ( P @ Bound_variable_2367 ) )
% 0.26/0.47 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.26/0.47 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.26/0.47 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 0.26/0.47 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 0.26/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ ( sub @ A @ id ) )
% 0.26/0.47 | ( P @ id @ A @ id ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 0.26/0.47 & ( hoasinduction_no_psi_cond_lthm
% 0.26/0.47 <=> ( ! [P: subst > term > subst > $o] :
% 0.26/0.47 ~ ! [Q: term > $o] :
% 0.26/0.47 ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.26/0.47 ( ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ A )
% 0.26/0.47 | ( P @ A ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ B )
% 0.26/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ( P @ Bound_variable_2367 ) )
% 0.26/0.47 => ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( Q @ B )
% 0.26/0.47 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.26/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.26/0.47 ( ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 0.26/0.47 & ( hoaslaminj
% 0.26/0.47 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.26/0.47 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ( ( lam @ ( F @ sh @ one ) )
% 0.26/0.47 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.26/0.47 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.26/0.47 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 0.26/0.47 & ( hoasinduction_lem3aaa
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 0.26/0.47 ( ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 0.26/0.47 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 0.26/0.47 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 0.26/0.47 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 0.26/0.47 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 0.26/0.47 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 0.26/0.47 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.26/0.47 & induction2lem_gthm
% 0.26/0.47 & ( hoasinduction_lem3aa_lthm
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.26/0.47 & ( hoasinduction_lem3
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 0.26/0.47 & ( hoasinduction_lem2
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.26/0.47 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.26/0.47 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 0.26/0.47 & ( termmset_lthm
% 0.26/0.47 <=> ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) ) ) )
% 0.26/0.47 & hoasinduction_lem1
% 0.26/0.47 & hoaslamnotap_lthm
% 0.26/0.47 & ( pushprop_lem1v2_lthm
% 0.26/0.47 <=> ( ! [A: term,M: subst] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.26/0.47 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.26/0.47 | ( Q @ one ) ) ) )
% 0.26/0.47 & hoasapnotvar
% 0.26/0.47 & ( hoasinduction_lem0
% 0.26/0.47 <=> ! [P: subst > term > subst > $o] :
% 0.26/0.47 ~ ! [Q: term > $o] :
% 0.26/0.47 ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) ) )
% 0.26/0.47 & ( hoasinduction
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ ( sub @ A @ id ) )
% 0.26/0.47 | ( P @ id @ A @ id ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 0.26/0.47 & hoasinduction_gthm
% 0.26/0.47 & axapp
% 0.26/0.47 & hoaslamnotvar_lthm
% 0.26/0.47 & pushprop_lem3v2_lthm
% 0.26/0.47 & ( hoasinduction_lem3b_lthm
% 0.26/0.47 <=> ! [B: term] :
% 0.26/0.47 ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ( F @ sh @ one )
% 0.26/0.47 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.26/0.47 & ulamvarind
% 0.26/0.47 & ( induction
% 0.26/0.47 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 0.26/0.47 ( ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ A )
% 0.26/0.47 | ( P @ A ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ( P @ Bound_variable_2140 ) ) )
% 0.26/0.47 & ( hoasinduction_lem3a_lthm
% 0.26/0.47 <=> ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.26/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 0.26/0.47 & termmset_gthm
% 0.26/0.47 & ( hoasinduction_lem3aa
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.26/0.47 & pushprop_lem1v2_gthm
% 0.26/0.47 & hoaslamnotap_gthm
% 0.26/0.47 & hoaslamnotvar_gthm
% 0.26/0.47 & hoasinduction_lem3b_gthm
% 0.26/0.47 & pushprop_lem2v2
% 0.26/0.47 & hoasinduction_lem3a_gthm
% 0.26/0.47 & axclos
% 0.26/0.47 & axassoc
% 0.26/0.47 & ( hoasinduction_lem2v2
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 | ~ ( Q @ Bound_variable_2820 )
% 0.26/0.47 | ~ ( Q @ Bound_variable_2822 )
% 0.26/0.47 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 0.26/0.47 & pushprop_lthm
% 0.26/0.47 & ( apinj2
% 0.26/0.47 <=> ! [A: term,B: term,C: term,D: term] :
% 0.26/0.47 ( ( ( ap @ A @ C )
% 0.26/0.47 != ( ap @ B @ D ) )
% 0.26/0.47 | ( C = D ) ) )
% 0.26/0.47 & ( apinj1
% 0.26/0.47 <=> ! [A: term,B: term,C: term,D: term] :
% 0.26/0.47 ( ( ( ap @ A @ C )
% 0.26/0.47 != ( ap @ B @ D ) )
% 0.26/0.47 | ( A = B ) ) )
% 0.26/0.47 & ( hoasapinj2_lthm
% 0.26/0.47 <=> ( ! [A: term,B: term,C: term,D: term] :
% 0.26/0.47 ( ( ( ap @ A @ C )
% 0.26/0.47 != ( ap @ B @ D ) )
% 0.26/0.47 | ( C = D ) )
% 0.26/0.47 => ! [A: term,B: term,C: term,D: term] :
% 0.26/0.47 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.26/0.47 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.26/0.47 | ( C = D ) ) ) )
% 0.26/0.47 & ( hoasinduction_lem3v2a
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( Q @ B )
% 0.26/0.47 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 0.26/0.47 & ( hoasapinj1_lthm
% 0.26/0.47 <=> ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ( ! [A: term,B: term,C: term,D: term] :
% 0.26/0.47 ( ( ( ap @ A @ C )
% 0.26/0.47 != ( ap @ B @ D ) )
% 0.26/0.47 | ( A = B ) )
% 0.26/0.47 => ! [A: term,B: term,C: term,D: term] :
% 0.26/0.47 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.26/0.47 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.26/0.47 | ( A = B ) ) ) ) )
% 0.26/0.47 & ( hoaslaminj_lthm
% 0.26/0.47 <=> ( ! [A: term,M: subst] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.26/0.47 => ( ! [A: term,M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.26/0.47 => ( ! [A: term,B: term] :
% 0.26/0.47 ( ( ( lam @ A )
% 0.26/0.47 != ( lam @ B ) )
% 0.26/0.47 | ( A = B ) )
% 0.26/0.47 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.26/0.47 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ( ( lam @ ( F @ sh @ one ) )
% 0.26/0.47 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.26/0.47 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.26/0.47 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 0.26/0.47 & ( axvarcons
% 0.26/0.47 <=> ! [A: term,M: subst] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 0.26/0.47 & ( axscons
% 0.26/0.47 <=> ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 0.26/0.47 & hoasinduction_lem2v2_gthm
% 0.26/0.47 & ( axidr
% 0.26/0.47 <=> ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ M @ id ) ) )
% 0.26/0.47 & ( pushprop_lem1
% 0.26/0.47 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 0.26/0.47 & ( laminj
% 0.26/0.47 <=> ! [A: term,B: term] :
% 0.26/0.47 ( ( ( lam @ A )
% 0.26/0.47 != ( lam @ B ) )
% 0.26/0.47 | ( A = B ) ) )
% 0.26/0.47 & ( hoasinduction_lem3_lthm
% 0.26/0.47 <=> ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.26/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 0.26/0.47 & ( pushprop_lem0
% 0.26/0.47 <=> ! [P: term > $o,A: term,M: subst] :
% 0.26/0.47 ~ ! [Q: term > $o] :
% 0.26/0.47 ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.26/0.47 & pushprop_gthm
% 0.26/0.47 & axabs
% 0.26/0.47 & ( hoasinduction_lem3v2a_lthm
% 0.26/0.47 <=> ( ! [B: term] :
% 0.26/0.47 ~ ! [F: subst > term > term] :
% 0.26/0.47 ~ ! [A: term,M: subst] :
% 0.26/0.47 ( ( sub @ B @ ( push @ A @ M ) )
% 0.26/0.47 = ( F @ M @ A ) )
% 0.26/0.47 => ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( Q @ B )
% 0.26/0.47 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 0.26/0.47 & hoasinduction_lem2_lthm
% 0.26/0.47 & hoasapinj2_gthm
% 0.26/0.47 & hoasinduction_lem1_lthm
% 0.26/0.47 & ~ lamnotap
% 0.26/0.47 & hoasapinj1_gthm
% 0.26/0.47 & hoaslamnotvar
% 0.26/0.47 & ( axidl
% 0.26/0.47 <=> ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ id @ M ) ) )
% 0.26/0.47 & hoaslaminj_gthm
% 0.26/0.47 & ( induction2_lthm
% 0.26/0.47 <=> ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.26/0.47 ( ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ B )
% 0.26/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( var @ B )
% 0.26/0.47 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.26/0.47 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 0.26/0.47 => ! [P: term > $o,Bound_variable_2367: term] :
% 0.26/0.47 ( ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ A )
% 0.26/0.47 | ( P @ A ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ B )
% 0.26/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ( P @ Bound_variable_2367 ) ) ) ) )
% 0.26/0.47 & ( hoasinduction_lem0_lthm
% 0.26/0.47 <=> ! [P: subst > term > subst > $o] :
% 0.26/0.47 ~ ! [Q: term > $o] :
% 0.26/0.47 ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) ) )
% 0.26/0.47 & ( substmonoid_lthm
% 0.26/0.47 <=> ( ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ id @ M ) )
% 0.26/0.47 => ( ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ M @ id ) )
% 0.26/0.47 => ( ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ id @ M ) )
% 0.26/0.47 & ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ M @ id ) ) ) ) ) )
% 0.26/0.47 & pushprop
% 0.26/0.47 & hoasinduction_lem3_gthm
% 0.26/0.47 & hoasinduction_lem2_gthm
% 0.26/0.47 & ( hoasinduction_lem3b
% 0.26/0.47 <=> ! [B: term] :
% 0.26/0.47 ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ( F @ sh @ one )
% 0.26/0.47 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.26/0.47 & ( substmonoid
% 0.26/0.47 <=> ( ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ id @ M ) )
% 0.26/0.47 & ! [M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ M @ id ) ) ) )
% 0.26/0.47 & lamnotvar
% 0.26/0.47 & ( hoasinduction_lem3a
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 0.26/0.47 & hoasinduction_lem1_gthm
% 0.26/0.47 & ( hoasinduction_no_psi_cond
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.26/0.47 ( ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 0.26/0.47 & induction2_gthm
% 0.26/0.47 & pushprop_lem2v2_lthm
% 0.26/0.47 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.26/0.47 & ( hoaslamnotap
% 0.26/0.47 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ( ( lam @ ( F @ sh @ one ) )
% 0.26/0.47 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 0.26/0.47 & substmonoid_gthm
% 0.26/0.47 & ulamvarsh
% 0.26/0.47 & ( induction2
% 0.26/0.47 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 0.26/0.47 ( ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ A )
% 0.26/0.47 | ( P @ A ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ B )
% 0.26/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ( P @ Bound_variable_2367 ) ) )
% 0.26/0.47 & pushprop_lem3v2
% 0.26/0.47 & pushprop_lem2v2_gthm
% 0.26/0.47 & ( pushprop_lem1_lthm
% 0.26/0.47 <=> ( ! [A: term,M: subst] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.26/0.47 => ( ! [A: term,M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.26/0.47 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 0.26/0.47 & ( hoasinduction_lem3v2
% 0.26/0.47 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( Q @ B )
% 0.26/0.47 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 0.26/0.47 & ( axshiftcons
% 0.26/0.47 <=> ! [A: term,M: subst] :
% 0.26/0.47 ( M
% 0.26/0.47 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 0.26/0.47 & ( termmset
% 0.26/0.47 <=> ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) ) )
% 0.26/0.47 & ( pushprop_lem0_lthm
% 0.26/0.47 <=> ! [P: term > $o,A: term,M: subst] :
% 0.26/0.47 ~ ! [Q: term > $o] :
% 0.26/0.47 ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.26/0.47 & hoasapnotvar_lthm
% 0.26/0.47 & ( hoasinduction_lem3v2_lthm
% 0.26/0.47 <=> ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( Q @ B )
% 0.26/0.47 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 0.26/0.47 & ( axvarid
% 0.26/0.47 <=> ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) ) )
% 0.26/0.47 & ( hoasinduction_lthm_3
% 0.26/0.47 <=> ( ! [P: subst > term > subst > $o] :
% 0.26/0.47 ~ ! [Q: term > $o] :
% 0.26/0.47 ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.26/0.47 ( ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ A )
% 0.26/0.47 | ( P @ A ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ A )
% 0.26/0.47 | ~ ( P @ B )
% 0.26/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ! [B: term] :
% 0.26/0.47 ( ~ ( P @ B )
% 0.26/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( P @ ( lam @ A ) ) )
% 0.26/0.47 | ( P @ Bound_variable_2367 ) )
% 0.26/0.47 => ( ! [A: term] :
% 0.26/0.47 ( A
% 0.26/0.47 = ( sub @ A @ id ) )
% 0.26/0.47 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.26/0.47 ( ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ~ ! [X: term] :
% 0.26/0.47 ( ( Q @ X )
% 0.26/0.47 = ( P @ id @ X @ id ) )
% 0.26/0.47 | ~ ! [B: term] :
% 0.26/0.47 ( ~ ( Q @ B )
% 0.26/0.47 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.26/0.47 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.26/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.26/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.26/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.26/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.26/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( var @ ( sub @ A @ id ) )
% 0.26/0.47 | ( P @ id @ A @ id ) )
% 0.26/0.47 | ~ ! [A: term,B: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ~ ( P @ id @ B @ id )
% 0.26/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.26/0.47 | ~ ! [F: subst > term > term] :
% 0.26/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 | ~ ! [A: term] :
% 0.26/0.47 ( ~ ( P @ id @ A @ id )
% 0.26/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.26/0.47 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(hoasinduction_lem3a,conjecture,
% 0.26/0.47 ( hoasinduction_lem3a
% 0.26/0.47 = ( ! [P: subst > term > subst > $o] :
% 0.26/0.47 ( ! [F: subst > term > term] :
% 0.26/0.47 ( ! [M: subst,A: term,N: subst] :
% 0.26/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.26/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.26/0.47 => ( ! [A: term] :
% 0.26/0.47 ( ( P @ id @ A @ id )
% 0.26/0.47 => ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.26/0.47 => ( P @ id
% 0.26/0.47 @ ( hoaslam @ id
% 0.26/0.47 @ ^ [M: subst,A: term] : ( F @ M @ A ) )
% 0.26/0.47 @ id ) ) )
% 0.26/0.47 => ! [A: term] :
% 0.26/0.47 ( ! [B: term] :
% 0.26/0.47 ( ( P @ id @ B @ id )
% 0.26/0.47 => ( P @ id @ ( sub @ A @ ( push @ B @ id ) ) @ id ) )
% 0.26/0.47 => ( P @ id @ ( lam @ A ) @ id ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 ------------------------
% 2.37/1.46 ---- Embedded in THF ---
% 2.37/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.37/1.46 %%% Generated on Tue May 05 09:11:37 EDT 2026
% 2.37/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.37/1.46 %%% Logic specification used:
% 2.37/1.46 %%% thf(spec, logic, $$dhol).
% 2.37/1.46
% 2.37/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.u2UgYFbzEX/DTF2DTF17811.p
% See solution above
% 2.37/1.46 ------------------------
% 2.37/1.48 ---- Cleaned THF ---
% 2.37/1.48 thf(term_type,type,
% 2.37/1.48 term: $tType ).
% 2.37/1.48
% 2.37/1.48 thf(subst_type,type,
% 2.37/1.48 subst: $tType ).
% 2.37/1.48
% 2.37/1.48 thf(one_decl,type,
% 2.37/1.48 one: term ).
% 2.37/1.48
% 2.37/1.48 thf(ap_decl,type,
% 2.37/1.48 ap: term > term > term ).
% 2.37/1.48
% 2.37/1.48 thf(lam_decl,type,
% 2.37/1.48 lam: term > term ).
% 2.37/1.48
% 2.37/1.48 thf(sub_decl,type,
% 2.37/1.48 sub: term > subst > term ).
% 2.37/1.48
% 2.37/1.48 thf(id_decl,type,
% 2.37/1.48 id: subst ).
% 2.37/1.48
% 2.37/1.48 thf(sh_decl,type,
% 2.37/1.48 sh: subst ).
% 2.37/1.48
% 2.37/1.48 thf(push_decl,type,
% 2.37/1.48 push: term > subst > subst ).
% 2.37/1.48
% 2.37/1.48 thf(comp_decl,type,
% 2.37/1.48 comp: subst > subst > subst ).
% 2.37/1.48
% 2.37/1.48 thf(var_decl,type,
% 2.37/1.48 var: term > $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem1v2_decl,type,
% 2.37/1.48 pushprop_lem1v2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem1_gthm_decl,type,
% 2.37/1.48 pushprop_lem1_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axmap_decl,type,
% 2.37/1.48 axmap: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem0_gthm_decl,type,
% 2.37/1.48 pushprop_lem0_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(shinj_decl,type,
% 2.37/1.48 shinj: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem1v2_decl,type,
% 2.37/1.48 hoasinduction_lem1v2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem1v2_gthm_decl,type,
% 2.37/1.48 hoasinduction_lem1v2_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasap_decl,type,
% 2.37/1.48 hoasap: subst > term > subst > term > term ).
% 2.37/1.48
% 2.37/1.48 thf(induction2lem_decl,type,
% 2.37/1.48 induction2lem: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3v2_f_decl,type,
% 2.37/1.48 hoasinduction_lem3v2_f: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axvarshift_decl,type,
% 2.37/1.48 axvarshift: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapinj2_decl,type,
% 2.37/1.48 hoasapinj2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapnotvar_gthm_decl,type,
% 2.37/1.48 hoasapnotvar_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapinj1_decl,type,
% 2.37/1.48 hoasapinj1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ulamvar1_decl,type,
% 2.37/1.48 ulamvar1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(induction2lem_lthm_decl,type,
% 2.37/1.48 induction2lem_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3v2_gthm_decl,type,
% 2.37/1.48 hoasinduction_lem3v2_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(apnotvar_decl,type,
% 2.37/1.48 apnotvar: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lthm_orig_decl,type,
% 2.37/1.48 pushprop_lthm_orig: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem3v2_f_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lthm_decl,type,
% 2.37/1.48 hoasinduction_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 2.37/1.48 hoasinduction_no_psi_cond_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslaminj_decl,type,
% 2.37/1.48 hoaslaminj: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3aaa_decl,type,
% 2.37/1.48 hoasinduction_lem3aaa: $o ).
% 2.37/1.48
% 2.37/1.48 thf(induction2lem_gthm_decl,type,
% 2.37/1.48 induction2lem_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3aa_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem3aa_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3_decl,type,
% 2.37/1.48 hoasinduction_lem3: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem2_decl,type,
% 2.37/1.48 hoasinduction_lem2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(termmset_lthm_decl,type,
% 2.37/1.48 termmset_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem1_decl,type,
% 2.37/1.48 hoasinduction_lem1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslamnotap_lthm_decl,type,
% 2.37/1.48 hoaslamnotap_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem1v2_lthm_decl,type,
% 2.37/1.48 pushprop_lem1v2_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapnotvar_decl,type,
% 2.37/1.48 hoasapnotvar: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem0_decl,type,
% 2.37/1.48 hoasinduction_lem0: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_decl,type,
% 2.37/1.48 hoasinduction: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_gthm_decl,type,
% 2.37/1.48 hoasinduction_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axapp_decl,type,
% 2.37/1.48 axapp: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslamnotvar_lthm_decl,type,
% 2.37/1.48 hoaslamnotvar_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem3v2_lthm_decl,type,
% 2.37/1.48 pushprop_lem3v2_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3b_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem3b_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ulamvarind_decl,type,
% 2.37/1.48 ulamvarind: $o ).
% 2.37/1.48
% 2.37/1.48 thf(induction_decl,type,
% 2.37/1.48 induction: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3a_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem3a_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(termmset_gthm_decl,type,
% 2.37/1.48 termmset_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3aa_decl,type,
% 2.37/1.48 hoasinduction_lem3aa: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem1v2_gthm_decl,type,
% 2.37/1.48 pushprop_lem1v2_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslamnotap_gthm_decl,type,
% 2.37/1.48 hoaslamnotap_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslamnotvar_gthm_decl,type,
% 2.37/1.48 hoaslamnotvar_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3b_gthm_decl,type,
% 2.37/1.48 hoasinduction_lem3b_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem2v2_decl,type,
% 2.37/1.48 pushprop_lem2v2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3a_gthm_decl,type,
% 2.37/1.48 hoasinduction_lem3a_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axclos_decl,type,
% 2.37/1.48 axclos: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axassoc_decl,type,
% 2.37/1.48 axassoc: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem2v2_decl,type,
% 2.37/1.48 hoasinduction_lem2v2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lthm_decl,type,
% 2.37/1.48 pushprop_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(apinj2_decl,type,
% 2.37/1.48 apinj2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(apinj1_decl,type,
% 2.37/1.48 apinj1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapinj2_lthm_decl,type,
% 2.37/1.48 hoasapinj2_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3v2a_decl,type,
% 2.37/1.48 hoasinduction_lem3v2a: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapinj1_lthm_decl,type,
% 2.37/1.48 hoasapinj1_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslaminj_lthm_decl,type,
% 2.37/1.48 hoaslaminj_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axvarcons_decl,type,
% 2.37/1.48 axvarcons: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslam_decl,type,
% 2.37/1.48 hoaslam: subst > ( subst > term > term ) > term ).
% 2.37/1.48
% 2.37/1.48 thf(axscons_decl,type,
% 2.37/1.48 axscons: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem2v2_gthm_decl,type,
% 2.37/1.48 hoasinduction_lem2v2_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axidr_decl,type,
% 2.37/1.48 axidr: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem1_decl,type,
% 2.37/1.48 pushprop_lem1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(laminj_decl,type,
% 2.37/1.48 laminj: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem3_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem0_decl,type,
% 2.37/1.48 pushprop_lem0: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_gthm_decl,type,
% 2.37/1.48 pushprop_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axabs_decl,type,
% 2.37/1.48 axabs: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem3v2a_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem2_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem2_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapinj2_gthm_decl,type,
% 2.37/1.48 hoasapinj2_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_p_and_p_prime_decl,type,
% 2.37/1.48 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem1_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem1_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(lamnotap_decl,type,
% 2.37/1.48 lamnotap: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapinj1_gthm_decl,type,
% 2.37/1.48 hoasapinj1_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslamnotvar_decl,type,
% 2.37/1.48 hoaslamnotvar: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axidl_decl,type,
% 2.37/1.48 axidl: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslaminj_gthm_decl,type,
% 2.37/1.48 hoaslaminj_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(induction2_lthm_decl,type,
% 2.37/1.48 induction2_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem0_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem0_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(substmonoid_lthm_decl,type,
% 2.37/1.48 substmonoid_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_decl,type,
% 2.37/1.48 pushprop: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3_gthm_decl,type,
% 2.37/1.48 hoasinduction_lem3_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem2_gthm_decl,type,
% 2.37/1.48 hoasinduction_lem2_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3b_decl,type,
% 2.37/1.48 hoasinduction_lem3b: $o ).
% 2.37/1.48
% 2.37/1.48 thf(substmonoid_decl,type,
% 2.37/1.48 substmonoid: $o ).
% 2.37/1.48
% 2.37/1.48 thf(lamnotvar_decl,type,
% 2.37/1.48 lamnotvar: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3a_decl,type,
% 2.37/1.48 hoasinduction_lem3a: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem1_gthm_decl,type,
% 2.37/1.48 hoasinduction_lem1_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_no_psi_cond_decl,type,
% 2.37/1.48 hoasinduction_no_psi_cond: $o ).
% 2.37/1.48
% 2.37/1.48 thf(induction2_gthm_decl,type,
% 2.37/1.48 induction2_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem2v2_lthm_decl,type,
% 2.37/1.48 pushprop_lem2v2_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasvar_decl,type,
% 2.37/1.48 hoasvar: subst > term > subst > $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoaslamnotap_decl,type,
% 2.37/1.48 hoaslamnotap: $o ).
% 2.37/1.48
% 2.37/1.48 thf(substmonoid_gthm_decl,type,
% 2.37/1.48 substmonoid_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ulamvarsh_decl,type,
% 2.37/1.48 ulamvarsh: $o ).
% 2.37/1.48
% 2.37/1.48 thf(induction2_decl,type,
% 2.37/1.48 induction2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem3v2_decl,type,
% 2.37/1.48 pushprop_lem3v2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem2v2_gthm_decl,type,
% 2.37/1.48 pushprop_lem2v2_gthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem1_lthm_decl,type,
% 2.37/1.48 pushprop_lem1_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3v2_decl,type,
% 2.37/1.48 hoasinduction_lem3v2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(axshiftcons_decl,type,
% 2.37/1.48 axshiftcons: $o ).
% 2.37/1.48
% 2.37/1.48 thf(termmset_decl,type,
% 2.37/1.48 termmset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_lem0_lthm_decl,type,
% 2.37/1.48 pushprop_lem0_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasapnotvar_lthm_decl,type,
% 2.37/1.48 hoasapnotvar_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lem3v2_lthm_decl,type,
% 2.37/1.48 hoasinduction_lem3v2_lthm: $o ).
% 2.37/1.48
% 2.37/1.48 thf(pushprop_p_and_p_prime_decl,type,
% 2.37/1.48 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 2.37/1.48
% 2.37/1.48 thf(axvarid_decl,type,
% 2.37/1.48 axvarid: $o ).
% 2.37/1.48
% 2.37/1.48 thf(hoasinduction_lthm_3_decl,type,
% 2.37/1.48 hoasinduction_lthm_3: $o ).
% 2.37/1.48
% 2.37/1.48 thf(d_term_type,type,
% 2.37/1.48 d_term: $tType ).
% 2.37/1.48
% 2.37/1.48 thf(d_subst_type,type,
% 2.37/1.48 d_subst: $tType ).
% 2.37/1.48
% 2.37/1.48 thf(d2term_decl,type,
% 2.37/1.48 d2term: d_term > term ).
% 2.37/1.48
% 2.37/1.48 thf(d2subst_decl,type,
% 2.37/1.48 d2subst: d_subst > subst ).
% 2.37/1.48
% 2.37/1.48 thf(d_one_decl,type,
% 2.37/1.48 d_one: d_term ).
% 2.37/1.48
% 2.37/1.48 thf(d_id_decl,type,
% 2.37/1.48 d_id: d_subst ).
% 2.37/1.48
% 2.37/1.48 thf(alg444_1,axiom,
% 2.37/1.48 ( ! [T: term] :
% 2.37/1.48 ? [DT: d_term] :
% 2.37/1.48 ( T
% 2.37/1.48 = ( d2term @ DT ) )
% 2.37/1.48 & ! [DT: d_term] : ( DT = d_one )
% 2.37/1.48 & ! [DT1: d_term,DT2: d_term] :
% 2.37/1.48 ( ( ( d2term @ DT1 )
% 2.37/1.48 = ( d2term @ DT2 ) )
% 2.37/1.48 => ( DT1 = DT2 ) )
% 2.37/1.48 & ! [S: subst] :
% 2.37/1.48 ? [DS: d_subst] :
% 2.37/1.48 ( S
% 2.37/1.48 = ( d2subst @ DS ) )
% 2.37/1.48 & ! [DS: d_subst] : ( DS = d_id )
% 2.37/1.48 & ! [DS1: d_subst,DS2: d_subst] :
% 2.37/1.48 ( ( ( d2subst @ DS1 )
% 2.37/1.48 = ( d2subst @ DS2 ) )
% 2.37/1.48 => ( DS1 = DS2 ) )
% 2.37/1.48 & ( one
% 2.37/1.48 = ( d2term @ d_one ) )
% 2.37/1.48 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 2.37/1.48 = ( d2term @ d_one ) )
% 2.37/1.48 & ( ( lam @ ( d2term @ d_one ) )
% 2.37/1.48 = ( d2term @ d_one ) )
% 2.37/1.48 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.37/1.48 = ( d2term @ d_one ) )
% 2.37/1.48 & ( id
% 2.37/1.48 = ( d2subst @ d_id ) )
% 2.37/1.48 & ( sh
% 2.37/1.48 = ( d2subst @ d_id ) )
% 2.37/1.48 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.37/1.48 = ( d2subst @ d_id ) )
% 2.37/1.48 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 2.37/1.48 = ( d2subst @ d_id ) )
% 2.37/1.48 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 2.37/1.48 = ( d2term @ d_one ) )
% 2.37/1.48 & ( hoaslam
% 2.37/1.48 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 2.37/1.48 & ( hoasinduction_p_and_p_prime
% 2.37/1.48 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 2.37/1.48 ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ id @ X @ id ) ) ) )
% 2.37/1.48 & ( pushprop_p_and_p_prime
% 2.37/1.48 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 2.37/1.48 ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 2.37/1.48 & ~ ( var @ ( d2term @ d_one ) )
% 2.37/1.48 & ( pushprop_lem1v2
% 2.37/1.48 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.37/1.48 | ( Q @ one ) ) )
% 2.37/1.48 & pushprop_lem1_gthm
% 2.37/1.48 & axmap
% 2.37/1.48 & pushprop_lem0_gthm
% 2.37/1.48 & ( shinj
% 2.37/1.48 <=> ! [A: term,B: term] :
% 2.37/1.48 ( ( ( sub @ A @ sh )
% 2.37/1.48 != ( sub @ B @ sh ) )
% 2.37/1.48 | ( A = B ) ) )
% 2.37/1.48 & hoasinduction_lem1v2
% 2.37/1.48 & hoasinduction_lem1v2_gthm
% 2.37/1.48 & ( induction2lem
% 2.37/1.48 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.37/1.48 ( ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ( P @ B )
% 2.37/1.48 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ B )
% 2.37/1.48 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( P @ ( lam @ A ) ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( var @ B )
% 2.37/1.48 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.37/1.48 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 2.37/1.48 & ( hoasinduction_lem3v2_f
% 2.37/1.48 <=> ! [B: term] :
% 2.37/1.48 ~ ! [F: subst > term > term] :
% 2.37/1.48 ~ ! [A: term,M: subst] :
% 2.37/1.48 ( ( sub @ B @ ( push @ A @ M ) )
% 2.37/1.48 = ( F @ M @ A ) ) )
% 2.37/1.48 & axvarshift
% 2.37/1.48 & ( hoasapinj2
% 2.37/1.48 <=> ! [A: term,B: term,C: term,D: term] :
% 2.37/1.48 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.37/1.48 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.37/1.48 | ( C = D ) ) )
% 2.37/1.48 & hoasapnotvar_gthm
% 2.37/1.48 & ( hoasapinj1
% 2.37/1.48 <=> ! [A: term,B: term,C: term,D: term] :
% 2.37/1.48 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.37/1.48 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.37/1.48 | ( A = B ) ) )
% 2.37/1.48 & ~ ulamvar1
% 2.37/1.48 & ( induction2lem_lthm
% 2.37/1.48 <=> ( ! [A: term,M: subst] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.37/1.48 => ( ! [A: term,M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.37/1.48 => ( ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ M @ id ) )
% 2.37/1.48 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 2.37/1.48 ( ~ ! [A: term] :
% 2.37/1.48 ( ~ ( var @ A )
% 2.37/1.48 | ( P @ A ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ( P @ B )
% 2.37/1.48 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ( P @ ( lam @ A ) ) )
% 2.37/1.48 | ( P @ Bound_variable_2140 ) )
% 2.37/1.48 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.37/1.48 ( ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ( P @ B )
% 2.37/1.48 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ B )
% 2.37/1.48 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( P @ ( lam @ A ) ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( var @ B )
% 2.37/1.48 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.37/1.48 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 2.37/1.48 & hoasinduction_lem3v2_gthm
% 2.37/1.48 & apnotvar
% 2.37/1.48 & pushprop_lthm_orig
% 2.37/1.48 & ( hoasinduction_lem3v2_f_lthm
% 2.37/1.48 <=> ! [B: term] :
% 2.37/1.48 ~ ! [F: subst > term > term] :
% 2.37/1.48 ~ ! [A: term,M: subst] :
% 2.37/1.48 ( ( sub @ B @ ( push @ A @ M ) )
% 2.37/1.48 = ( F @ M @ A ) ) )
% 2.37/1.48 & ( hoasinduction_lthm
% 2.37/1.48 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.37/1.48 ( ~ ! [A: term] :
% 2.37/1.48 ( ~ ( var @ A )
% 2.37/1.48 | ( P @ A ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ( P @ B )
% 2.37/1.48 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ B )
% 2.37/1.48 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( P @ ( lam @ A ) ) )
% 2.37/1.48 | ( P @ Bound_variable_2367 ) )
% 2.37/1.48 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.48 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.48 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.37/1.48 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.37/1.48 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.37/1.48 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 2.37/1.48 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.48 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.48 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.48 | ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 2.37/1.48 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.48 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.48 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( var @ ( sub @ A @ id ) )
% 2.37/1.48 | ( P @ id @ A @ id ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.37/1.48 | ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 2.37/1.48 & ( hoasinduction_no_psi_cond_lthm
% 2.37/1.48 <=> ( ! [P: subst > term > subst > $o] :
% 2.37/1.48 ~ ! [Q: term > $o] :
% 2.37/1.48 ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ id @ X @ id ) )
% 2.37/1.48 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.37/1.48 ( ~ ! [A: term] :
% 2.37/1.48 ( ~ ( var @ A )
% 2.37/1.48 | ( P @ A ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ( P @ B )
% 2.37/1.48 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ B )
% 2.37/1.48 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( P @ ( lam @ A ) ) )
% 2.37/1.48 | ( P @ Bound_variable_2367 ) )
% 2.37/1.48 => ( ! [A: term] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ A @ id ) )
% 2.37/1.48 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ id @ X @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( Q @ B )
% 2.37/1.48 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.37/1.48 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.37/1.48 ( ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.37/1.48 | ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 2.37/1.48 & ( hoaslaminj
% 2.37/1.48 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.37/1.48 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ( ( lam @ ( F @ sh @ one ) )
% 2.37/1.48 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.37/1.48 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.37/1.48 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 2.37/1.48 & ( hoasinduction_lem3aaa
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 2.37/1.48 ( ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 2.37/1.48 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 2.37/1.48 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 2.37/1.48 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 2.37/1.48 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 2.37/1.48 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 2.37/1.48 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.37/1.48 & induction2lem_gthm
% 2.37/1.48 & ( hoasinduction_lem3aa_lthm
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.37/1.48 & ( hoasinduction_lem3
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.48 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.48 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.48 | ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 2.37/1.48 & ( hoasinduction_lem2
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.48 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.48 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.37/1.48 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.37/1.48 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.37/1.48 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 2.37/1.48 & ( termmset_lthm
% 2.37/1.48 <=> ( ! [A: term] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ A @ id ) )
% 2.37/1.48 => ! [A: term] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ A @ id ) ) ) )
% 2.37/1.48 & hoasinduction_lem1
% 2.37/1.48 & hoaslamnotap_lthm
% 2.37/1.48 & ( pushprop_lem1v2_lthm
% 2.37/1.48 <=> ( ! [A: term,M: subst] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.37/1.48 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.37/1.48 | ( Q @ one ) ) ) )
% 2.37/1.48 & hoasapnotvar
% 2.37/1.48 & ( hoasinduction_lem0
% 2.37/1.48 <=> ! [P: subst > term > subst > $o] :
% 2.37/1.48 ~ ! [Q: term > $o] :
% 2.37/1.48 ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ id @ X @ id ) ) )
% 2.37/1.48 & ( hoasinduction
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.48 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.48 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( var @ ( sub @ A @ id ) )
% 2.37/1.48 | ( P @ id @ A @ id ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.37/1.48 | ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 2.37/1.48 & hoasinduction_gthm
% 2.37/1.48 & axapp
% 2.37/1.48 & hoaslamnotvar_lthm
% 2.37/1.48 & pushprop_lem3v2_lthm
% 2.37/1.48 & ( hoasinduction_lem3b_lthm
% 2.37/1.48 <=> ! [B: term] :
% 2.37/1.48 ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ( F @ sh @ one )
% 2.37/1.48 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.37/1.48 & ulamvarind
% 2.37/1.48 & ( induction
% 2.37/1.48 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 2.37/1.48 ( ~ ! [A: term] :
% 2.37/1.48 ( ~ ( var @ A )
% 2.37/1.48 | ( P @ A ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ( P @ B )
% 2.37/1.48 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ( P @ ( lam @ A ) ) )
% 2.37/1.48 | ( P @ Bound_variable_2140 ) ) )
% 2.37/1.48 & ( hoasinduction_lem3a_lthm
% 2.37/1.48 <=> ( ! [A: term] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ A @ id ) )
% 2.37/1.48 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.37/1.48 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 2.37/1.48 & termmset_gthm
% 2.37/1.48 & ( hoasinduction_lem3aa
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.37/1.48 & pushprop_lem1v2_gthm
% 2.37/1.48 & hoaslamnotap_gthm
% 2.37/1.48 & hoaslamnotvar_gthm
% 2.37/1.48 & hoasinduction_lem3b_gthm
% 2.37/1.48 & pushprop_lem2v2
% 2.37/1.48 & hoasinduction_lem3a_gthm
% 2.37/1.48 & axclos
% 2.37/1.48 & axassoc
% 2.37/1.48 & ( hoasinduction_lem2v2
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.48 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.48 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.37/1.48 | ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ id @ X @ id ) )
% 2.37/1.48 | ~ ( Q @ Bound_variable_2820 )
% 2.37/1.48 | ~ ( Q @ Bound_variable_2822 )
% 2.37/1.48 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 2.37/1.48 & pushprop_lthm
% 2.37/1.48 & ( apinj2
% 2.37/1.48 <=> ! [A: term,B: term,C: term,D: term] :
% 2.37/1.48 ( ( ( ap @ A @ C )
% 2.37/1.48 != ( ap @ B @ D ) )
% 2.37/1.48 | ( C = D ) ) )
% 2.37/1.48 & ( apinj1
% 2.37/1.48 <=> ! [A: term,B: term,C: term,D: term] :
% 2.37/1.48 ( ( ( ap @ A @ C )
% 2.37/1.48 != ( ap @ B @ D ) )
% 2.37/1.48 | ( A = B ) ) )
% 2.37/1.48 & ( hoasapinj2_lthm
% 2.37/1.48 <=> ( ! [A: term,B: term,C: term,D: term] :
% 2.37/1.48 ( ( ( ap @ A @ C )
% 2.37/1.48 != ( ap @ B @ D ) )
% 2.37/1.48 | ( C = D ) )
% 2.37/1.48 => ! [A: term,B: term,C: term,D: term] :
% 2.37/1.48 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.37/1.48 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.37/1.48 | ( C = D ) ) ) )
% 2.37/1.48 & ( hoasinduction_lem3v2a
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ id @ X @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( Q @ B )
% 2.37/1.48 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 2.37/1.48 & ( hoasapinj1_lthm
% 2.37/1.48 <=> ( ! [A: term] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ A @ id ) )
% 2.37/1.48 => ( ! [A: term,B: term,C: term,D: term] :
% 2.37/1.48 ( ( ( ap @ A @ C )
% 2.37/1.48 != ( ap @ B @ D ) )
% 2.37/1.48 | ( A = B ) )
% 2.37/1.48 => ! [A: term,B: term,C: term,D: term] :
% 2.37/1.48 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.37/1.48 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.37/1.48 | ( A = B ) ) ) ) )
% 2.37/1.48 & ( hoaslaminj_lthm
% 2.37/1.48 <=> ( ! [A: term,M: subst] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.37/1.48 => ( ! [A: term,M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.37/1.48 => ( ! [A: term,B: term] :
% 2.37/1.48 ( ( ( lam @ A )
% 2.37/1.48 != ( lam @ B ) )
% 2.37/1.48 | ( A = B ) )
% 2.37/1.48 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.37/1.48 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ( ( lam @ ( F @ sh @ one ) )
% 2.37/1.48 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.37/1.48 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.37/1.48 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 2.37/1.48 & ( axvarcons
% 2.37/1.48 <=> ! [A: term,M: subst] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 2.37/1.48 & ( axscons
% 2.37/1.48 <=> ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 2.37/1.48 & hoasinduction_lem2v2_gthm
% 2.37/1.48 & ( axidr
% 2.37/1.48 <=> ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ M @ id ) ) )
% 2.37/1.48 & ( pushprop_lem1
% 2.37/1.48 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 2.37/1.48 & ( laminj
% 2.37/1.48 <=> ! [A: term,B: term] :
% 2.37/1.48 ( ( ( lam @ A )
% 2.37/1.48 != ( lam @ B ) )
% 2.37/1.48 | ( A = B ) ) )
% 2.37/1.48 & ( hoasinduction_lem3_lthm
% 2.37/1.48 <=> ( ! [A: term] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ A @ id ) )
% 2.37/1.48 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.37/1.48 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.48 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.48 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.48 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.48 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.48 | ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ id @ B @ id )
% 2.37/1.48 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 2.37/1.48 & ( pushprop_lem0
% 2.37/1.48 <=> ! [P: term > $o,A: term,M: subst] :
% 2.37/1.48 ~ ! [Q: term > $o] :
% 2.37/1.48 ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.37/1.48 & pushprop_gthm
% 2.37/1.48 & axabs
% 2.37/1.48 & ( hoasinduction_lem3v2a_lthm
% 2.37/1.48 <=> ( ! [B: term] :
% 2.37/1.48 ~ ! [F: subst > term > term] :
% 2.37/1.48 ~ ! [A: term,M: subst] :
% 2.37/1.48 ( ( sub @ B @ ( push @ A @ M ) )
% 2.37/1.48 = ( F @ M @ A ) )
% 2.37/1.48 => ( ! [A: term] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ A @ id ) )
% 2.37/1.48 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.48 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ( P @ id @ A @ id )
% 2.37/1.48 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.48 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.48 | ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ id @ X @ id ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( Q @ B )
% 2.37/1.48 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 2.37/1.48 & hoasinduction_lem2_lthm
% 2.37/1.48 & hoasapinj2_gthm
% 2.37/1.48 & hoasinduction_lem1_lthm
% 2.37/1.48 & ~ lamnotap
% 2.37/1.48 & hoasapinj1_gthm
% 2.37/1.48 & hoaslamnotvar
% 2.37/1.48 & ( axidl
% 2.37/1.48 <=> ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ id @ M ) ) )
% 2.37/1.48 & hoaslaminj_gthm
% 2.37/1.48 & ( induction2_lthm
% 2.37/1.48 <=> ( ! [A: term] :
% 2.37/1.48 ( A
% 2.37/1.48 = ( sub @ A @ id ) )
% 2.37/1.48 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.37/1.48 ( ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ( P @ B )
% 2.37/1.48 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ B )
% 2.37/1.48 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( P @ ( lam @ A ) ) )
% 2.37/1.48 | ~ ! [B: term] :
% 2.37/1.48 ( ~ ( var @ B )
% 2.37/1.48 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.37/1.48 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 2.37/1.48 => ! [P: term > $o,Bound_variable_2367: term] :
% 2.37/1.48 ( ~ ! [A: term] :
% 2.37/1.48 ( ~ ( var @ A )
% 2.37/1.48 | ( P @ A ) )
% 2.37/1.48 | ~ ! [A: term,B: term] :
% 2.37/1.48 ( ~ ( P @ A )
% 2.37/1.48 | ~ ( P @ B )
% 2.37/1.48 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.48 | ~ ! [A: term] :
% 2.37/1.48 ( ~ ! [B: term] :
% 2.37/1.48 ( ~ ( P @ B )
% 2.37/1.48 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.37/1.48 | ( P @ ( lam @ A ) ) )
% 2.37/1.48 | ( P @ Bound_variable_2367 ) ) ) ) )
% 2.37/1.48 & ( hoasinduction_lem0_lthm
% 2.37/1.48 <=> ! [P: subst > term > subst > $o] :
% 2.37/1.48 ~ ! [Q: term > $o] :
% 2.37/1.48 ~ ! [X: term] :
% 2.37/1.48 ( ( Q @ X )
% 2.37/1.48 = ( P @ id @ X @ id ) ) )
% 2.37/1.48 & ( substmonoid_lthm
% 2.37/1.48 <=> ( ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ id @ M ) )
% 2.37/1.48 => ( ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ M @ id ) )
% 2.37/1.48 => ( ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ id @ M ) )
% 2.37/1.48 & ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ M @ id ) ) ) ) ) )
% 2.37/1.48 & pushprop
% 2.37/1.48 & hoasinduction_lem3_gthm
% 2.37/1.48 & hoasinduction_lem2_gthm
% 2.37/1.48 & ( hoasinduction_lem3b
% 2.37/1.48 <=> ! [B: term] :
% 2.37/1.48 ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ( F @ sh @ one )
% 2.37/1.48 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.37/1.48 & ( substmonoid
% 2.37/1.48 <=> ( ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ id @ M ) )
% 2.37/1.48 & ! [M: subst] :
% 2.37/1.48 ( M
% 2.37/1.48 = ( comp @ M @ id ) ) ) )
% 2.37/1.48 & lamnotvar
% 2.37/1.48 & ( hoasinduction_lem3a
% 2.37/1.48 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.37/1.48 ( ~ ! [F: subst > term > term] :
% 2.37/1.48 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.48 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.49 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ( P @ id @ A @ id )
% 2.37/1.49 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.49 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.49 | ~ ! [B: term] :
% 2.37/1.49 ( ~ ( P @ id @ B @ id )
% 2.37/1.49 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.49 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 2.37/1.49 & hoasinduction_lem1_gthm
% 2.37/1.49 & ( hoasinduction_no_psi_cond
% 2.37/1.49 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.37/1.49 ( ~ ! [A: term,B: term] :
% 2.37/1.49 ( ~ ( P @ id @ A @ id )
% 2.37/1.49 | ~ ( P @ id @ B @ id )
% 2.37/1.49 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.37/1.49 | ~ ! [F: subst > term > term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.49 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.49 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ( P @ id @ A @ id )
% 2.37/1.49 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.49 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.49 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 2.37/1.49 & induction2_gthm
% 2.37/1.49 & pushprop_lem2v2_lthm
% 2.37/1.49 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.37/1.49 & ( hoaslamnotap
% 2.37/1.49 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.49 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.49 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.49 | ( ( lam @ ( F @ sh @ one ) )
% 2.37/1.49 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 2.37/1.49 & substmonoid_gthm
% 2.37/1.49 & ulamvarsh
% 2.37/1.49 & ( induction2
% 2.37/1.49 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 2.37/1.49 ( ~ ! [A: term] :
% 2.37/1.49 ( ~ ( var @ A )
% 2.37/1.49 | ( P @ A ) )
% 2.37/1.49 | ~ ! [A: term,B: term] :
% 2.37/1.49 ( ~ ( P @ A )
% 2.37/1.49 | ~ ( P @ B )
% 2.37/1.49 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ! [B: term] :
% 2.37/1.49 ( ~ ( P @ B )
% 2.37/1.49 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.37/1.49 | ( P @ ( lam @ A ) ) )
% 2.37/1.49 | ( P @ Bound_variable_2367 ) ) )
% 2.37/1.49 & pushprop_lem3v2
% 2.37/1.49 & pushprop_lem2v2_gthm
% 2.37/1.49 & ( pushprop_lem1_lthm
% 2.37/1.49 <=> ( ! [A: term,M: subst] :
% 2.37/1.49 ( A
% 2.37/1.49 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.37/1.49 => ( ! [A: term,M: subst] :
% 2.37/1.49 ( M
% 2.37/1.49 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.37/1.49 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.37/1.49 ( ~ ( P @ A )
% 2.37/1.49 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 2.37/1.49 & ( hoasinduction_lem3v2
% 2.37/1.49 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.49 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.49 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.49 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.49 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.49 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.49 | ~ ! [F: subst > term > term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.49 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.49 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ( P @ id @ A @ id )
% 2.37/1.49 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.49 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.49 | ~ ! [X: term] :
% 2.37/1.49 ( ( Q @ X )
% 2.37/1.49 = ( P @ id @ X @ id ) )
% 2.37/1.49 | ~ ! [B: term] :
% 2.37/1.49 ( ~ ( Q @ B )
% 2.37/1.49 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.37/1.49 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 2.37/1.49 & ( axshiftcons
% 2.37/1.49 <=> ! [A: term,M: subst] :
% 2.37/1.49 ( M
% 2.37/1.49 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 2.37/1.49 & ( termmset
% 2.37/1.49 <=> ! [A: term] :
% 2.37/1.49 ( A
% 2.37/1.49 = ( sub @ A @ id ) ) )
% 2.37/1.49 & ( pushprop_lem0_lthm
% 2.37/1.49 <=> ! [P: term > $o,A: term,M: subst] :
% 2.37/1.49 ~ ! [Q: term > $o] :
% 2.37/1.49 ~ ! [X: term] :
% 2.37/1.49 ( ( Q @ X )
% 2.37/1.49 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.37/1.49 & hoasapnotvar_lthm
% 2.37/1.49 & ( hoasinduction_lem3v2_lthm
% 2.37/1.49 <=> ( ! [A: term] :
% 2.37/1.49 ( A
% 2.37/1.49 = ( sub @ A @ id ) )
% 2.37/1.49 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.49 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.49 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.49 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.49 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.49 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.49 | ~ ! [F: subst > term > term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.49 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.49 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ( P @ id @ A @ id )
% 2.37/1.49 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.49 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.49 | ~ ! [X: term] :
% 2.37/1.49 ( ( Q @ X )
% 2.37/1.49 = ( P @ id @ X @ id ) )
% 2.37/1.49 | ~ ! [B: term] :
% 2.37/1.49 ( ~ ( Q @ B )
% 2.37/1.49 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.37/1.49 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 2.37/1.49 & ( axvarid
% 2.37/1.49 <=> ! [A: term] :
% 2.37/1.49 ( A
% 2.37/1.49 = ( sub @ A @ id ) ) )
% 2.37/1.49 & ( hoasinduction_lthm_3
% 2.37/1.49 <=> ( ! [P: subst > term > subst > $o] :
% 2.37/1.49 ~ ! [Q: term > $o] :
% 2.37/1.49 ~ ! [X: term] :
% 2.37/1.49 ( ( Q @ X )
% 2.37/1.49 = ( P @ id @ X @ id ) )
% 2.37/1.49 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.37/1.49 ( ~ ! [A: term] :
% 2.37/1.49 ( ~ ( var @ A )
% 2.37/1.49 | ( P @ A ) )
% 2.37/1.49 | ~ ! [A: term,B: term] :
% 2.37/1.49 ( ~ ( P @ A )
% 2.37/1.49 | ~ ( P @ B )
% 2.37/1.49 | ( P @ ( ap @ A @ B ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ! [B: term] :
% 2.37/1.49 ( ~ ( P @ B )
% 2.37/1.49 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.37/1.49 | ( P @ ( lam @ A ) ) )
% 2.37/1.49 | ( P @ Bound_variable_2367 ) )
% 2.37/1.49 => ( ! [A: term] :
% 2.37/1.49 ( A
% 2.37/1.49 = ( sub @ A @ id ) )
% 2.37/1.49 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.37/1.49 ( ~ ! [F: subst > term > term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.49 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.49 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ( P @ id @ A @ id )
% 2.37/1.49 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.49 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.49 | ~ ! [X: term] :
% 2.37/1.49 ( ( Q @ X )
% 2.37/1.49 = ( P @ id @ X @ id ) )
% 2.37/1.49 | ~ ! [B: term] :
% 2.37/1.49 ( ~ ( Q @ B )
% 2.37/1.49 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.37/1.49 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.37/1.49 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.49 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.37/1.49 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.37/1.49 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.37/1.49 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.37/1.49 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ( var @ ( sub @ A @ id ) )
% 2.37/1.49 | ( P @ id @ A @ id ) )
% 2.37/1.49 | ~ ! [A: term,B: term] :
% 2.37/1.49 ( ~ ( P @ id @ A @ id )
% 2.37/1.49 | ~ ( P @ id @ B @ id )
% 2.37/1.49 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.37/1.49 | ~ ! [F: subst > term > term] :
% 2.37/1.49 ( ~ ! [M: subst,A: term,N: subst] :
% 2.37/1.49 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.49 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.49 | ~ ! [A: term] :
% 2.37/1.49 ( ~ ( P @ id @ A @ id )
% 2.37/1.49 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.49 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.37/1.49 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 2.37/1.49
% 2.37/1.49 thf(hoasinduction_lem3a,conjecture,
% 2.37/1.49 ( hoasinduction_lem3a
% 2.37/1.49 = ( ! [P: subst > term > subst > $o] :
% 2.37/1.49 ( ! [F: subst > term > term] :
% 2.37/1.49 ( ! [M: subst,A: term,N: subst] :
% 2.37/1.49 ( ( sub @ ( F @ M @ A ) @ N )
% 2.37/1.49 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.37/1.49 => ( ! [A: term] :
% 2.37/1.49 ( ( P @ id @ A @ id )
% 2.37/1.49 => ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.37/1.49 => ( P @ id
% 2.37/1.49 @ ( hoaslam @ id
% 2.37/1.49 @ ^ [M: subst,A: term] : ( F @ M @ A ) )
% 2.37/1.49 @ id ) ) )
% 2.37/1.49 => ! [A: term] :
% 2.37/1.49 ( ! [B: term] :
% 2.37/1.49 ( ( P @ id @ B @ id )
% 2.37/1.49 => ( P @ id @ ( sub @ A @ ( push @ B @ id ) ) @ id ) )
% 2.37/1.49 => ( P @ id @ ( lam @ A ) @ id ) ) ) ) ) ).
% 2.37/1.49 ------------------------
% 2.55/1.50 % (17913)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/2Mi)
% 2.55/1.50 % (17913)Instruction limit reached!
% 2.55/1.50 % (17913)------------------------------
% 2.55/1.50 % (17913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.50 % (17913)Termination reason: Unknown
% 2.55/1.50 % (17913)Termination phase: shuffling
% 2.55/1.50
% 2.55/1.50 % (17913)Memory used [KB]: 1151
% 2.55/1.50 % (17913)Time elapsed: 0.002 s
% 2.55/1.50 % (17911)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/4Mi)
% 2.55/1.50 % (17913)Instructions burned: 2 (million)
% 2.55/1.50 % (17913)------------------------------
% 2.55/1.50 % (17913)------------------------------
% 2.55/1.50 % (17916)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/18Mi)
% 2.55/1.50 % (17911)Instruction limit reached!
% 2.55/1.50 % (17911)------------------------------
% 2.55/1.50 % (17911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.50 % (17911)Termination reason: Unknown
% 2.55/1.50 % (17911)Termination phase: shuffling
% 2.55/1.50
% 2.55/1.50 % (17911)Memory used [KB]: 1407
% 2.55/1.50 % (17911)Time elapsed: 0.003 s
% 2.55/1.50 % (17911)Instructions burned: 6 (million)
% 2.55/1.50 % (17911)------------------------------
% 2.55/1.50 % (17911)------------------------------
% 2.55/1.51 % (17916)Instruction limit reached!
% 2.55/1.51 % (17916)------------------------------
% 2.55/1.51 % (17916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (17916)Termination reason: Unknown
% 2.55/1.51 % (17916)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (17916)Memory used [KB]: 2046
% 2.55/1.51 % (17916)Time elapsed: 0.007 s
% 2.55/1.51 % (17916)Instructions burned: 18 (million)
% 2.55/1.51 % (17916)------------------------------
% 2.55/1.51 % (17916)------------------------------
% 2.55/1.51 % (17910)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/183Mi)
% 2.55/1.51 % (17912)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/27Mi)
% 2.55/1.51 % (17915)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/275Mi)
% 2.55/1.51 % (17914)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/2Mi)
% 2.55/1.51 % (17914)Instruction limit reached!
% 2.55/1.51 % (17914)------------------------------
% 2.55/1.51 % (17914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (17914)Termination reason: Unknown
% 2.55/1.51 % (17914)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (17914)Memory used [KB]: 1151
% 2.55/1.51 % (17914)Time elapsed: 0.003 s
% 2.55/1.51 % (17914)Instructions burned: 3 (million)
% 2.55/1.51 % (17914)------------------------------
% 2.55/1.51 % (17914)------------------------------
% 2.55/1.51 % (17917)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/3Mi)
% 2.55/1.51 % (17917)Instruction limit reached!
% 2.55/1.51 % (17917)------------------------------
% 2.55/1.51 % (17917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (17917)Termination reason: Unknown
% 2.55/1.51 % (17917)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (17917)Memory used [KB]: 1279
% 2.55/1.51 % (17917)Time elapsed: 0.002 s
% 2.55/1.51 % (17917)Instructions burned: 4 (million)
% 2.55/1.51 % (17917)------------------------------
% 2.55/1.51 % (17917)------------------------------
% 2.55/1.51 % (17918)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/37Mi)
% 2.55/1.52 % (17919)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/15Mi)
% 2.55/1.52 % (17921)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/1041Mi)
% 2.55/1.52 % (17919)Instruction limit reached!
% 2.55/1.52 % (17919)------------------------------
% 2.55/1.52 % (17919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.52 % (17919)Termination reason: Unknown
% 2.55/1.52 % (17919)Termination phase: shuffling
% 2.55/1.52
% 2.55/1.52 % (17919)Memory used [KB]: 1918
% 2.55/1.52 % (17919)Time elapsed: 0.007 s
% 2.55/1.52 % (17919)Instructions burned: 16 (million)
% 2.55/1.52 % (17919)------------------------------
% 2.55/1.52 % (17919)------------------------------
% 2.55/1.53 % (17912)Instruction limit reached!
% 2.55/1.53 % (17912)------------------------------
% 2.55/1.53 % (17912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (17912)Termination reason: Unknown
% 2.55/1.53 % (17912)Termination phase: Property scanning
% 2.55/1.53
% 2.55/1.53 % (17912)Memory used [KB]: 2174
% 2.55/1.53 % (17912)Time elapsed: 0.018 s
% 2.55/1.53 % (17912)Instructions burned: 27 (million)
% 2.55/1.53 % (17912)------------------------------
% 2.55/1.53 % (17912)------------------------------
% 2.55/1.53 % (17918)Instruction limit reached!
% 2.55/1.53 % (17918)------------------------------
% 2.55/1.53 % (17918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (17918)Termination reason: Unknown
% 2.55/1.53 % (17918)Termination phase: Property scanning
% 2.55/1.53
% 2.55/1.53 % (17918)Memory used [KB]: 1918
% 2.55/1.53 % (17918)Time elapsed: 0.013 s
% 2.55/1.53 % (17918)Instructions burned: 40 (million)
% 2.55/1.53 % (17918)------------------------------
% 2.55/1.53 % (17918)------------------------------
% 2.55/1.53 % (17920)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/3Mi)
% 2.55/1.53 % (17920)Instruction limit reached!
% 2.55/1.53 % (17920)------------------------------
% 2.55/1.53 % (17920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (17920)Termination reason: Unknown
% 2.55/1.53 % (17920)Termination phase: shuffling
% 2.55/1.53
% 2.55/1.53 % (17920)Memory used [KB]: 1279
% 2.55/1.53 % (17920)Time elapsed: 0.004 s
% 2.55/1.53 % (17920)Instructions burned: 3 (million)
% 2.55/1.53 % (17920)------------------------------
% 2.55/1.53 % (17920)------------------------------
% 2.55/1.53 % (17915)Refutation not found, incomplete strategy
% 2.55/1.53 % (17915)------------------------------
% 2.55/1.53 % (17915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (17915)Termination reason: Refutation not found, incomplete strategy
% 2.55/1.53
% 2.55/1.53
% 2.55/1.53 % (17915)Memory used [KB]: 6780
% 2.55/1.53 % (17915)Time elapsed: 0.026 s
% 2.55/1.53 % (17915)Instructions burned: 41 (million)
% 2.55/1.53 % (17915)------------------------------
% 2.55/1.53 % (17915)------------------------------
% 2.55/1.53 % (17922)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/7Mi)
% 2.55/1.54 % (17923)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/16Mi)
% 2.55/1.54 % (17922)Instruction limit reached!
% 2.55/1.54 % (17922)------------------------------
% 2.55/1.54 % (17922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.54 % (17922)Termination reason: Unknown
% 2.55/1.54 % (17922)Termination phase: shuffling
% 2.55/1.54
% 2.55/1.54 % (17922)Memory used [KB]: 1407
% 2.55/1.54 % (17922)Time elapsed: 0.004 s
% 2.55/1.54 % (17922)Instructions burned: 7 (million)
% 2.55/1.54 % (17922)------------------------------
% 2.55/1.54 % (17922)------------------------------
% 2.55/1.54 % (17923)Instruction limit reached!
% 2.55/1.54 % (17923)------------------------------
% 2.55/1.54 % (17923)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.54 % (17923)Termination reason: Unknown
% 2.55/1.54 % (17923)Termination phase: shuffling
% 2.55/1.54
% 2.55/1.54 % (17923)Memory used [KB]: 1918
% 2.55/1.54 % (17923)Time elapsed: 0.007 s
% 2.55/1.54 % (17923)Instructions burned: 16 (million)
% 2.55/1.54 % (17923)------------------------------
% 2.55/1.54 % (17923)------------------------------
% 2.55/1.54 % (17924)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/3Mi)
% 2.55/1.55 % (17924)Instruction limit reached!
% 2.55/1.55 % (17924)------------------------------
% 2.55/1.55 % (17924)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.55 % (17924)Termination reason: Unknown
% 2.55/1.55 % (17924)Termination phase: shuffling
% 2.55/1.55
% 2.55/1.55 % (17924)Memory used [KB]: 1279
% 2.55/1.55 % (17924)Time elapsed: 0.004 s
% 2.55/1.55 % (17924)Instructions burned: 4 (million)
% 2.55/1.55 % (17924)------------------------------
% 2.55/1.55 % (17924)------------------------------
% 2.55/1.55 % (17927)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/3Mi)
% 2.55/1.55 % (17927)Instruction limit reached!
% 2.55/1.55 % (17927)------------------------------
% 2.55/1.55 % (17927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.55 % (17927)Termination reason: Unknown
% 2.55/1.55 % (17927)Termination phase: shuffling
% 2.55/1.55
% 2.55/1.55 % (17927)Memory used [KB]: 1279
% 2.55/1.55 % (17927)Time elapsed: 0.002 s
% 2.55/1.55 % (17927)Instructions burned: 3 (million)
% 2.55/1.55 % (17927)------------------------------
% 2.55/1.55 % (17927)------------------------------
% 2.55/1.55 % (17925)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/3Mi)
% 2.55/1.55 % (17925)Instruction limit reached!
% 2.55/1.55 % (17925)------------------------------
% 2.55/1.55 % (17925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.55 % (17925)Termination reason: Unknown
% 2.55/1.55 % (17925)Termination phase: shuffling
% 2.55/1.55
% 2.55/1.55 % (17925)Memory used [KB]: 1279
% 2.55/1.55 % (17925)Time elapsed: 0.003 s
% 2.55/1.55 % (17925)Instructions burned: 3 (million)
% 2.55/1.55 % (17925)------------------------------
% 2.55/1.55 % (17925)------------------------------
% 2.55/1.55 % (17928)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/4Mi)
% 2.55/1.55 % (17926)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/7Mi)
% 2.55/1.55 % (17928)Instruction limit reached!
% 2.55/1.55 % (17928)------------------------------
% 2.55/1.55 % (17928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.55 % (17928)Termination reason: Unknown
% 2.55/1.55 % (17928)Termination phase: shuffling
% 2.55/1.55
% 2.55/1.55 % (17928)Memory used [KB]: 1279
% 2.55/1.55 % (17928)Time elapsed: 0.003 s
% 2.55/1.55 % (17928)Instructions burned: 4 (million)
% 2.55/1.55 % (17928)------------------------------
% 2.55/1.55 % (17928)------------------------------
% 2.55/1.56 % (17930)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/710Mi)
% 2.55/1.56 % (17926)Instruction limit reached!
% 2.55/1.56 % (17926)------------------------------
% 2.55/1.56 % (17926)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.56 % (17926)Termination reason: Unknown
% 2.55/1.56 % (17926)Termination phase: shuffling
% 2.55/1.56
% 2.55/1.56 % (17926)Memory used [KB]: 1407
% 2.55/1.56 % (17926)Time elapsed: 0.006 s
% 2.55/1.56 % (17926)Instructions burned: 7 (million)
% 2.55/1.56 % (17926)------------------------------
% 2.55/1.56 % (17926)------------------------------
% 2.55/1.56 % (17931)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/6Mi)
% 2.55/1.56 % (17931)Instruction limit reached!
% 2.55/1.56 % (17931)------------------------------
% 2.55/1.56 % (17931)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.56 % (17931)Termination reason: Unknown
% 2.55/1.56 % (17931)Termination phase: shuffling
% 2.55/1.56
% 2.55/1.56 % (17931)Memory used [KB]: 1407
% 2.55/1.56 % (17931)Time elapsed: 0.004 s
% 2.55/1.56 % (17931)Instructions burned: 8 (million)
% 2.55/1.56 % (17931)------------------------------
% 2.55/1.56 % (17931)------------------------------
% 2.55/1.56 % (17932)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/902Mi)
% 2.55/1.56 % (17929)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/18Mi)
% 2.55/1.57 % (17934)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/5Mi)
% 2.55/1.57 % (17929)Instruction limit reached!
% 2.55/1.57 % (17929)------------------------------
% 2.55/1.57 % (17929)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.57 % (17929)Termination reason: Unknown
% 2.55/1.57 % (17929)Termination phase: shuffling
% 2.55/1.57
% 2.55/1.57 % (17929)Memory used [KB]: 1918
% 2.55/1.57 % (17929)Time elapsed: 0.012 s
% 2.55/1.57 % (17929)Instructions burned: 18 (million)
% 2.55/1.57 % (17929)------------------------------
% 2.55/1.57 % (17929)------------------------------
% 2.55/1.57 % (17934)Instruction limit reached!
% 2.55/1.57 % (17934)------------------------------
% 2.55/1.57 % (17934)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.57 % (17934)Termination reason: Unknown
% 2.55/1.57 % (17934)Termination phase: shuffling
% 2.55/1.57
% 2.55/1.57 % (17934)Memory used [KB]: 1407
% 2.55/1.57 % (17934)Time elapsed: 0.004 s
% 2.55/1.57 % (17934)Instructions burned: 7 (million)
% 2.55/1.57 % (17934)------------------------------
% 2.55/1.57 % (17934)------------------------------
% 2.55/1.57 % (17933)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/21Mi)
% 2.55/1.58 % (17933)Instruction limit reached!
% 2.55/1.58 % (17933)------------------------------
% 2.55/1.58 % (17933)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.58 % (17933)Termination reason: Unknown
% 2.55/1.58 % (17933)Termination phase: shuffling
% 2.55/1.58
% 2.55/1.58 % (17933)Memory used [KB]: 2046
% 2.55/1.58 % (17933)Time elapsed: 0.011 s
% 2.55/1.58 % (17933)Instructions burned: 21 (million)
% 2.55/1.58 % (17933)------------------------------
% 2.55/1.58 % (17933)------------------------------
% 2.55/1.58 % (17936)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/377Mi)
% 2.55/1.59 % (17935)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/6Mi)
% 2.55/1.59 % (17910)Instruction limit reached!
% 2.55/1.59 % (17910)------------------------------
% 2.55/1.59 % (17910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.59 % (17910)Termination reason: Unknown
% 2.55/1.59 % (17910)Termination phase: Property scanning
% 2.55/1.59
% 2.55/1.59 % (17910)Memory used [KB]: 4349
% 2.55/1.59 % (17910)Time elapsed: 0.084 s
% 2.55/1.59 % (17910)Instructions burned: 184 (million)
% 2.55/1.59 % (17910)------------------------------
% 2.55/1.59 % (17910)------------------------------
% 2.55/1.59 % (17937)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/779Mi)
% 2.55/1.59 % (17935)Instruction limit reached!
% 2.55/1.59 % (17935)------------------------------
% 2.55/1.59 % (17935)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.59 % (17935)Termination reason: Unknown
% 2.55/1.59 % (17935)Termination phase: shuffling
% 2.55/1.59
% 2.55/1.59 % (17935)Memory used [KB]: 1407
% 2.55/1.59 % (17935)Time elapsed: 0.005 s
% 2.55/1.59 % (17935)Instructions burned: 6 (million)
% 2.55/1.59 % (17935)------------------------------
% 2.55/1.59 % (17935)------------------------------
% 2.55/1.60 % (17938)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/19Mi)
% 2.55/1.61 % (17939)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_17811 for (2999ds/879Mi)
% 2.55/1.61 % (17938)Instruction limit reached!
% 2.55/1.61 % (17938)------------------------------
% 2.55/1.61 % (17938)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.61 % (17938)Termination reason: Unknown
% 2.55/1.61 % (17938)Termination phase: shuffling
% 2.55/1.61
% 2.55/1.61 % (17938)Memory used [KB]: 1663
% 2.55/1.61 % (17938)Time elapsed: 0.007 s
% 2.55/1.61 % (17938)Instructions burned: 21 (million)
% 2.55/1.61 % (17938)------------------------------
% 2.55/1.61 % (17938)------------------------------
% 2.55/1.62 % (17940)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_17811 for (2998ds/17Mi)
% 2.55/1.62 % (17940)Instruction limit reached!
% 2.55/1.62 % (17940)------------------------------
% 2.55/1.62 % (17940)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.62 % (17940)Termination reason: Unknown
% 2.55/1.62 % (17940)Termination phase: shuffling
% 2.55/1.62
% 2.55/1.62 % (17940)Memory used [KB]: 2046
% 2.55/1.62 % (17940)Time elapsed: 0.008 s
% 2.55/1.62 % (17940)Instructions burned: 18 (million)
% 2.55/1.62 % (17940)------------------------------
% 2.55/1.62 % (17940)------------------------------
% 2.55/1.63 % (17932)Refutation not found, incomplete strategy
% 2.55/1.63 % (17932)------------------------------
% 2.55/1.63 % (17932)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.63 % (17932)Termination reason: Refutation not found, incomplete strategy
% 2.55/1.63
% 2.55/1.63
% 2.55/1.63 % (17932)Memory used [KB]: 9338
% 2.55/1.63 % (17932)Time elapsed: 0.064 s
% 2.55/1.63 % (17932)Instructions burned: 205 (million)
% 2.55/1.63 % (17932)------------------------------
% 2.55/1.63 % (17932)------------------------------
% 2.55/1.64 % (17941)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17811 for (2998ds/3Mi)
% 2.55/1.64 % (17941)Instruction limit reached!
% 2.55/1.64 % (17941)------------------------------
% 2.55/1.64 % (17941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.64 % (17941)Termination reason: Unknown
% 2.55/1.64 % (17941)Termination phase: shuffling
% 2.55/1.64
% 2.55/1.64 % (17941)Memory used [KB]: 1279
% 2.55/1.64 % (17941)Time elapsed: 0.003 s
% 2.55/1.64 % (17941)Instructions burned: 4 (million)
% 2.55/1.64 % (17941)------------------------------
% 2.55/1.64 % (17941)------------------------------
% 2.55/1.64 % (17942)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_17811 for (2998ds/30Mi)
% 2.55/1.65 % (17942)Instruction limit reached!
% 2.55/1.65 % (17942)------------------------------
% 2.55/1.65 % (17942)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.65 % (17942)Termination reason: Unknown
% 2.55/1.65 % (17942)Termination phase: Property scanning
% 2.55/1.65
% 2.55/1.65 % (17942)Memory used [KB]: 2174
% 2.55/1.65 % (17942)Time elapsed: 0.010 s
% 2.55/1.65 % (17942)Instructions burned: 30 (million)
% 2.55/1.65 % (17942)------------------------------
% 2.55/1.65 % (17942)------------------------------
% 2.55/1.65 % (17943)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_17811 for (2998ds/127Mi)
% 2.55/1.66 % (17944)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_17811 for (2998ds/100Mi)
% 2.55/1.68 % (17944)Instruction limit reached!
% 2.55/1.68 % (17944)------------------------------
% 2.55/1.68 % (17944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.68 % (17944)Termination reason: Unknown
% 2.55/1.68 % (17944)Termination phase: Saturation
% 2.55/1.68
% 2.55/1.68 % (17944)Memory used [KB]: 6524
% 2.55/1.68 % (17944)Time elapsed: 0.026 s
% 2.55/1.68 % (17944)Instructions burned: 100 (million)
% 2.55/1.68 % (17944)------------------------------
% 2.55/1.68 % (17944)------------------------------
% 2.55/1.68 % (17943)Instruction limit reached!
% 2.55/1.68 % (17943)------------------------------
% 2.55/1.68 % (17943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.68 % (17943)Termination reason: Unknown
% 2.55/1.68 % (17943)Termination phase: Property scanning
% 2.55/1.68
% 2.55/1.68 % (17943)Memory used [KB]: 4221
% 2.55/1.68 % (17943)Time elapsed: 0.039 s
% 2.55/1.68 % (17943)Instructions burned: 128 (million)
% 2.55/1.68 % (17943)------------------------------
% 2.55/1.68 % (17943)------------------------------
% 2.55/1.69 % (17945)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_17811 for (2998ds/3Mi)
% 2.55/1.69 % (17945)Instruction limit reached!
% 2.55/1.69 % (17945)------------------------------
% 2.55/1.69 % (17945)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.69 % (17945)Termination reason: Unknown
% 2.55/1.69 % (17945)Termination phase: shuffling
% 2.55/1.69
% 2.55/1.69 % (17945)Memory used [KB]: 1279
% 2.55/1.69 % (17945)Time elapsed: 0.002 s
% 2.55/1.69 % (17945)Instructions burned: 3 (million)
% 2.55/1.69 % (17945)------------------------------
% 2.55/1.69 % (17945)------------------------------
% 2.55/1.70 % (17946)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_17811 for (2998ds/20Mi)
% 2.55/1.70 % (17947)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_17811 for (2998ds/86Mi)
% 2.55/1.70 % (17936)Instruction limit reached!
% 2.55/1.70 % (17936)------------------------------
% 2.55/1.70 % (17936)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.70 % (17936)Termination reason: Unknown
% 2.55/1.70 % (17936)Termination phase: Saturation
% 2.55/1.70
% 2.55/1.70 % (17936)Memory used [KB]: 11001
% 2.55/1.70 % (17936)Time elapsed: 0.118 s
% 2.55/1.70 % (17936)Instructions burned: 379 (million)
% 2.55/1.70 % (17936)------------------------------
% 2.55/1.70 % (17936)------------------------------
% 2.55/1.70 % (17946)Instruction limit reached!
% 2.55/1.70 % (17946)------------------------------
% 2.55/1.70 % (17946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.70 % (17946)Termination reason: Unknown
% 2.55/1.70 % (17946)Termination phase: shuffling
% 2.55/1.70
% 2.55/1.70 % (17946)Memory used [KB]: 2046
% 2.55/1.70 % (17946)Time elapsed: 0.008 s
% 2.55/1.70 % (17946)Instructions burned: 21 (million)
% 2.55/1.70 % (17946)------------------------------
% 2.55/1.70 % (17946)------------------------------
% 2.55/1.71 % (17948)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/107Mi)
% 2.55/1.71 % (17949)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/39Mi)
% 2.55/1.72 % (17949)Instruction limit reached!
% 2.55/1.72 % (17949)------------------------------
% 2.55/1.72 % (17949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.72 % (17949)Termination reason: Unknown
% 2.55/1.72 % (17949)Termination phase: Property scanning
% 2.55/1.72
% 2.55/1.72 % (17949)Memory used [KB]: 1918
% 2.55/1.72 % (17949)Time elapsed: 0.013 s
% 2.55/1.72 % (17949)Instructions burned: 39 (million)
% 2.55/1.72 % (17949)------------------------------
% 2.55/1.72 % (17949)------------------------------
% 2.55/1.73 % (17947)Instruction limit reached!
% 2.55/1.73 % (17947)------------------------------
% 2.55/1.73 % (17947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.73 % (17947)Termination reason: Unknown
% 2.55/1.73 % (17947)Termination phase: Preprocessing 3
% 2.55/1.73
% 2.55/1.73 % (17947)Memory used [KB]: 4093
% 2.55/1.73 % (17947)Time elapsed: 0.026 s
% 2.55/1.73 % (17947)Instructions burned: 88 (million)
% 2.55/1.73 % (17947)------------------------------
% 2.55/1.73 % (17947)------------------------------
% 2.55/1.73 % (17950)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/448Mi)
% 2.55/1.74 % (17951)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/46Mi)
% 2.55/1.75 % (17948)Instruction limit reached!
% 2.55/1.75 % (17948)------------------------------
% 2.55/1.75 % (17948)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.75 % (17948)Termination reason: Unknown
% 2.55/1.75 % (17948)Termination phase: Property scanning
% 2.55/1.75
% 2.55/1.75 % (17948)Memory used [KB]: 4477
% 2.55/1.75 % (17948)Time elapsed: 0.035 s
% 2.55/1.75 % (17948)Instructions burned: 107 (million)
% 2.55/1.75 % (17948)------------------------------
% 2.55/1.75 % (17948)------------------------------
% 2.55/1.75 % (17951)Instruction limit reached!
% 2.55/1.75 % (17951)------------------------------
% 2.55/1.75 % (17951)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.75 % (17951)Termination reason: Unknown
% 2.55/1.75 % (17951)Termination phase: Preprocessing 3
% 2.55/1.75
% 2.55/1.75 % (17951)Memory used [KB]: 2430
% 2.55/1.75 % (17951)Time elapsed: 0.015 s
% 2.55/1.75 % (17951)Instructions burned: 48 (million)
% 2.55/1.75 % (17951)------------------------------
% 2.55/1.75 % (17951)------------------------------
% 2.55/1.76 % (17952)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/98Mi)
% 2.55/1.76 % (17953)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/507Mi)
% 2.55/1.76 % (17930)Instruction limit reached!
% 2.55/1.76 % (17930)------------------------------
% 2.55/1.76 % (17930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.76 % (17930)Termination reason: Unknown
% 2.55/1.76 % (17930)Termination phase: Saturation
% 2.55/1.76
% 2.55/1.76 % (17930)Memory used [KB]: 11129
% 2.55/1.76 % (17930)Time elapsed: 0.208 s
% 2.55/1.76 % (17930)Instructions burned: 711 (million)
% 2.55/1.76 % (17930)------------------------------
% 2.55/1.76 % (17930)------------------------------
% 2.55/1.77 % (17954)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/149Mi)
% 2.55/1.78 % (17921)Instruction limit reached!
% 2.55/1.78 % (17921)------------------------------
% 2.55/1.78 % (17921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.78 % (17921)Termination reason: Unknown
% 2.55/1.78 % (17921)Termination phase: Saturation
% 2.55/1.78
% 2.55/1.78 % (17921)Memory used [KB]: 7675
% 2.55/1.78 % (17921)Time elapsed: 0.257 s
% 2.55/1.78 % (17921)Instructions burned: 1042 (million)
% 2.55/1.78 % (17921)------------------------------
% 2.55/1.78 % (17921)------------------------------
% 2.55/1.78 % (17952)Instruction limit reached!
% 2.55/1.78 % (17952)------------------------------
% 2.55/1.78 % (17952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.78 % (17952)Termination reason: Unknown
% 2.55/1.78 % (17952)Termination phase: Saturation
% 2.55/1.78
% 2.55/1.78 % (17952)Memory used [KB]: 2046
% 2.55/1.78 % (17952)Time elapsed: 0.029 s
% 2.55/1.78 % (17952)Instructions burned: 101 (million)
% 2.55/1.78 % (17952)------------------------------
% 2.55/1.78 % (17952)------------------------------
% 2.55/1.79 % (17955)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/39Mi)
% 2.55/1.79 % (17956)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/6Mi)
% 2.55/1.80 % (17956)Instruction limit reached!
% 2.55/1.80 % (17956)------------------------------
% 2.55/1.80 % (17956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.80 % (17956)Termination reason: Unknown
% 2.55/1.80 % (17956)Termination phase: shuffling
% 2.55/1.80
% 2.55/1.80 % (17956)Memory used [KB]: 1407
% 2.55/1.80 % (17956)Time elapsed: 0.004 s
% 2.55/1.80 % (17956)Instructions burned: 7 (million)
% 2.55/1.80 % (17956)------------------------------
% 2.55/1.80 % (17956)------------------------------
% 2.55/1.80 % (17937)Instruction limit reached!
% 2.55/1.80 % (17937)------------------------------
% 2.55/1.80 % (17937)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.80 % (17937)Termination reason: Unknown
% 2.55/1.80 % (17937)Termination phase: Saturation
% 2.55/1.80
% 2.55/1.80 % (17937)Memory used [KB]: 7036
% 2.55/1.80 % (17937)Time elapsed: 0.206 s
% 2.55/1.80 % (17937)Instructions burned: 782 (million)
% 2.55/1.80 % (17937)------------------------------
% 2.55/1.80 % (17937)------------------------------
% 2.55/1.80 % (17955)Instruction limit reached!
% 2.55/1.80 % (17955)------------------------------
% 2.55/1.80 % (17955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.80 % (17955)Termination reason: Unknown
% 2.55/1.80 % (17955)Termination phase: shuffling
% 2.55/1.80
% 2.55/1.80 % (17955)Memory used [KB]: 2302
% 2.55/1.80 % (17955)Time elapsed: 0.013 s
% 2.55/1.80 % (17955)Instructions burned: 41 (million)
% 2.55/1.80 % (17955)------------------------------
% 2.55/1.80 % (17955)------------------------------
% 2.55/1.81 % (17957)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/1087Mi)
% 2.55/1.81 % (17958)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/47Mi)
% 2.55/1.81 % (17959)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_17811 for (2997ds/44Mi)
% 2.55/1.82 % (17954)Instruction limit reached!
% 2.55/1.82 % (17954)------------------------------
% 2.55/1.82 % (17954)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.82 % (17954)Termination reason: Unknown
% 2.55/1.82 % (17954)Termination phase: Property scanning
% 2.55/1.82
% 2.55/1.82 % (17954)Memory used [KB]: 7931
% 2.55/1.82 % (17954)Time elapsed: 0.043 s
% 2.55/1.82 % (17954)Instructions burned: 152 (million)
% 2.55/1.82 % (17954)------------------------------
% 2.55/1.82 % (17954)------------------------------
% 2.55/1.82 % (17959)Instruction limit reached!
% 2.55/1.82 % (17959)------------------------------
% 2.55/1.82 % (17959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.82 % (17959)Termination reason: Unknown
% 2.55/1.82 % (17959)Termination phase: Property scanning
% 2.55/1.82
% 2.55/1.82 % (17959)Memory used [KB]: 1918
% 2.55/1.82 % (17959)Time elapsed: 0.013 s
% 2.55/1.82 % (17959)Instructions burned: 46 (million)
% 2.55/1.82 % (17959)------------------------------
% 2.55/1.82 % (17959)------------------------------
% 2.55/1.82 % (17958)Instruction limit reached!
% 2.55/1.82 % (17958)------------------------------
% 2.55/1.82 % (17958)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.82 % (17958)Termination reason: Unknown
% 2.55/1.82 % (17958)Termination phase: Naming
% 2.55/1.82
% 2.55/1.82 % (17958)Memory used [KB]: 2686
% 2.55/1.82 % (17958)Time elapsed: 0.016 s
% 2.55/1.82 % (17958)Instructions burned: 47 (million)
% 2.55/1.82 % (17958)------------------------------
% 2.55/1.82 % (17958)------------------------------
% 2.55/1.83 % (17960)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/432Mi)
% 2.55/1.83 % (17961)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/62Mi)
% 2.55/1.83 % (17962)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/31Mi)
% 2.55/1.84 % (17953)Refutation not found, incomplete strategy
% 2.55/1.84 % (17953)------------------------------
% 2.55/1.84 % (17953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.84 % (17953)Termination reason: Refutation not found, incomplete strategy
% 2.55/1.84
% 2.55/1.84
% 2.55/1.84 % (17953)Memory used [KB]: 9850
% 2.55/1.84 % (17953)Time elapsed: 0.077 s
% 2.55/1.84 % (17953)Instructions burned: 277 (million)
% 2.55/1.84 % (17953)------------------------------
% 2.55/1.84 % (17953)------------------------------
% 2.55/1.85 % (17963)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/96Mi)
% 2.55/1.85 % (17962)Instruction limit reached!
% 2.55/1.85 % (17962)------------------------------
% 2.55/1.85 % (17962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.85 % (17962)Termination reason: Unknown
% 2.55/1.85 % (17962)Termination phase: Property scanning
% 2.55/1.85
% 2.55/1.85 % (17962)Memory used [KB]: 2174
% 2.55/1.85 % (17962)Time elapsed: 0.012 s
% 2.55/1.85 % (17962)Instructions burned: 34 (million)
% 2.55/1.85 % (17962)------------------------------
% 2.55/1.85 % (17962)------------------------------
% 2.55/1.85 % (17961)Instruction limit reached!
% 2.55/1.85 % (17961)------------------------------
% 2.55/1.85 % (17961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.85 % (17961)Termination reason: Unknown
% 2.55/1.85 % (17961)Termination phase: Property scanning
% 2.55/1.85
% 2.55/1.85 % (17961)Memory used [KB]: 1918
% 2.55/1.85 % (17961)Time elapsed: 0.017 s
% 2.55/1.85 % (17961)Instructions burned: 64 (million)
% 2.55/1.85 % (17961)------------------------------
% 2.55/1.85 % (17961)------------------------------
% 2.55/1.86 % (17964)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/453Mi)
% 2.55/1.86 % (17965)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/71Mi)
% 2.55/1.87 % (17950)Instruction limit reached!
% 2.55/1.87 % (17950)------------------------------
% 2.55/1.87 % (17950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.87 % (17950)Termination reason: Unknown
% 2.55/1.87 % (17950)Termination phase: Saturation
% 2.55/1.87
% 2.55/1.87 % (17950)Memory used [KB]: 7291
% 2.55/1.87 % (17950)Time elapsed: 0.139 s
% 2.55/1.87 % (17950)Instructions burned: 449 (million)
% 2.55/1.87 % (17950)------------------------------
% 2.55/1.87 % (17950)------------------------------
% 2.55/1.87 % (17963)Instruction limit reached!
% 2.55/1.87 % (17963)------------------------------
% 2.55/1.87 % (17963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.87 % (17963)Termination reason: Unknown
% 2.55/1.87 % (17963)Termination phase: Property scanning
% 2.55/1.87
% 2.55/1.87 % (17963)Memory used [KB]: 4221
% 2.55/1.87 % (17963)Time elapsed: 0.029 s
% 2.55/1.87 % (17963)Instructions burned: 98 (million)
% 2.55/1.87 % (17963)------------------------------
% 2.55/1.87 % (17963)------------------------------
% 2.55/1.88 % (17965)Instruction limit reached!
% 2.55/1.88 % (17965)------------------------------
% 2.55/1.88 % (17965)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.88 % (17965)Termination reason: Unknown
% 2.55/1.88 % (17965)Termination phase: Preprocessing 3
% 2.55/1.88
% 2.55/1.88 % (17965)Memory used [KB]: 3454
% 2.55/1.88 % (17965)Time elapsed: 0.022 s
% 2.55/1.88 % (17965)Instructions burned: 71 (million)
% 2.55/1.88 % (17965)------------------------------
% 2.55/1.88 % (17965)------------------------------
% 2.55/1.88 % (17966)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/83Mi)
% 2.55/1.88 % (17967)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/11Mi)
% 2.55/1.89 % (17967)Instruction limit reached!
% 2.55/1.89 % (17967)------------------------------
% 2.55/1.89 % (17967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.89 % (17967)Termination reason: Unknown
% 2.55/1.89 % (17967)Termination phase: shuffling
% 2.55/1.89
% 2.55/1.89 % (17967)Memory used [KB]: 1535
% 2.55/1.89 % (17967)Time elapsed: 0.004 s
% 2.55/1.89 % (17967)Instructions burned: 11 (million)
% 2.55/1.89 % (17967)------------------------------
% 2.55/1.89 % (17967)------------------------------
% 2.55/1.89 % (17968)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/41Mi)
% 2.55/1.89 % (17939)Instruction limit reached!
% 2.55/1.89 % (17939)------------------------------
% 2.55/1.89 % (17939)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.89 % (17939)Termination reason: Unknown
% 2.55/1.89 % (17939)Termination phase: Saturation
% 2.55/1.89
% 2.55/1.89 % (17939)Memory used [KB]: 11641
% 2.55/1.89 % (17939)Time elapsed: 0.287 s
% 2.55/1.89 % (17939)Instructions burned: 880 (million)
% 2.55/1.89 % (17939)------------------------------
% 2.55/1.89 % (17939)------------------------------
% 2.55/1.90 % (17969)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17811 for (2996ds/6Mi)
% 2.55/1.90 % (17969)Instruction limit reached!
% 2.55/1.90 % (17969)------------------------------
% 2.55/1.90 % (17969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.90 % (17969)Termination reason: Unknown
% 2.55/1.90 % (17969)Termination phase: shuffling
% 2.55/1.90
% 2.55/1.90 % (17969)Memory used [KB]: 1407
% 2.55/1.90 % (17969)Time elapsed: 0.003 s
% 2.55/1.90 % (17969)Instructions burned: 7 (million)
% 2.55/1.90 % (17969)------------------------------
% 2.55/1.90 % (17969)------------------------------
% 2.55/1.90 % (17968)Instruction limit reached!
% 2.55/1.90 % (17968)------------------------------
% 2.55/1.90 % (17968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.90 % (17968)Termination reason: Unknown
% 2.55/1.90 % (17968)Termination phase: Property scanning
% 2.55/1.90
% 2.55/1.90 % (17968)Memory used [KB]: 1918
% 2.55/1.90 % (17968)Time elapsed: 0.012 s
% 2.55/1.90 % (17968)Instructions burned: 42 (million)
% 2.55/1.90 % (17968)------------------------------
% 2.55/1.90 % (17968)------------------------------
% 2.55/1.90 % (17970)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/150Mi)
% 2.55/1.91 % (17971)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/343Mi)
% 2.55/1.91 % (17972)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/109Mi)
% 2.55/1.91 % (17966)Instruction limit reached!
% 2.55/1.91 % (17966)------------------------------
% 2.55/1.91 % (17966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.91 % (17966)Termination reason: Unknown
% 2.55/1.91 % (17966)Termination phase: Saturation
% 2.55/1.91
% 2.55/1.91 % (17966)Memory used [KB]: 7036
% 2.55/1.91 % (17966)Time elapsed: 0.027 s
% 2.55/1.91 % (17966)Instructions burned: 83 (million)
% 2.55/1.91 % (17966)------------------------------
% 2.55/1.91 % (17966)------------------------------
% 2.55/1.92 % (17973)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/253Mi)
% 2.55/1.93 % (17971)Refutation not found, incomplete strategy
% 2.55/1.93 % (17971)------------------------------
% 2.55/1.93 % (17971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.93 % (17971)Termination reason: Refutation not found, incomplete strategy
% 2.55/1.93
% 2.55/1.93
% 2.55/1.93 % (17971)Memory used [KB]: 7036
% 2.55/1.93 % (17971)Time elapsed: 0.025 s
% 2.55/1.93 % (17971)Instructions burned: 81 (million)
% 2.55/1.93 % (17971)------------------------------
% 2.55/1.93 % (17971)------------------------------
% 2.55/1.93 % (17960)Instruction limit reached!
% 2.55/1.93 % (17960)------------------------------
% 2.55/1.93 % (17960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.93 % (17960)Termination reason: Unknown
% 2.55/1.93 % (17960)Termination phase: Saturation
% 2.55/1.93
% 2.55/1.93 % (17960)Memory used [KB]: 2430
% 2.55/1.93 % (17960)Time elapsed: 0.130 s
% 2.55/1.93 % (17960)Instructions burned: 432 (million)
% 2.55/1.93 % (17960)------------------------------
% 2.55/1.93 % (17960)------------------------------
% 2.55/1.94 % (17972)Instruction limit reached!
% 2.55/1.94 % (17972)------------------------------
% 2.55/1.94 % (17972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.94 % (17972)Termination reason: Unknown
% 2.55/1.94 % (17972)Termination phase: Property scanning
% 2.55/1.94
% 2.55/1.94 % (17972)Memory used [KB]: 4221
% 2.55/1.94 % (17972)Time elapsed: 0.032 s
% 2.55/1.94 % (17972)Instructions burned: 109 (million)
% 2.55/1.94 % (17972)------------------------------
% 2.55/1.94 % (17972)------------------------------
% 2.55/1.94 % (17974)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/140Mi)
% 2.55/1.94 % (17975)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/614Mi)
% 2.55/1.95 % (17970)Instruction limit reached!
% 2.55/1.95 % (17970)------------------------------
% 2.55/1.95 % (17970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.95 % (17970)Termination reason: Unknown
% 2.55/1.95 % (17970)Termination phase: Property scanning
% 2.55/1.95
% 2.55/1.95 % (17970)Memory used [KB]: 4221
% 2.55/1.95 % (17970)Time elapsed: 0.045 s
% 2.55/1.95 % (17970)Instructions burned: 153 (million)
% 2.55/1.95 % (17970)------------------------------
% 2.55/1.95 % (17970)------------------------------
% 2.55/1.95 % (17976)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/202Mi)
% 2.55/1.96 % (17977)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/339Mi)
% 2.55/1.97 % (17964)Instruction limit reached!
% 2.55/1.97 % (17964)------------------------------
% 2.55/1.97 % (17964)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.97 % (17964)Termination reason: Unknown
% 2.55/1.97 % (17964)Termination phase: Saturation
% 2.55/1.97
% 2.55/1.97 % (17964)Memory used [KB]: 6652
% 2.55/1.97 % (17964)Time elapsed: 0.114 s
% 2.55/1.97 % (17964)Instructions burned: 453 (million)
% 2.55/1.97 % (17964)------------------------------
% 2.55/1.97 % (17964)------------------------------
% 2.55/1.98 % (17974)Instruction limit reached!
% 2.55/1.98 % (17974)------------------------------
% 2.55/1.98 % (17974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.98 % (17974)Termination reason: Unknown
% 2.55/1.98 % (17974)Termination phase: Property scanning
% 2.55/1.98
% 2.55/1.98 % (17974)Memory used [KB]: 4221
% 2.55/1.98 % (17974)Time elapsed: 0.039 s
% 2.55/1.98 % (17974)Instructions burned: 140 (million)
% 2.55/1.98 % (17974)------------------------------
% 2.55/1.98 % (17974)------------------------------
% 2.55/1.98 % (17978)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/283Mi)
% 2.55/1.99 % (17973)Instruction limit reached!
% 2.55/1.99 % (17973)------------------------------
% 2.55/1.99 % (17973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.99 % (17973)Termination reason: Unknown
% 2.55/1.99 % (17973)Termination phase: Saturation
% 2.55/1.99
% 2.55/1.99 % (17973)Memory used [KB]: 6908
% 2.55/1.99 % (17973)Time elapsed: 0.070 s
% 2.55/1.99 % (17973)Instructions burned: 256 (million)
% 2.55/1.99 % (17973)------------------------------
% 2.55/1.99 % (17973)------------------------------
% 2.55/1.99 % (17979)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/134Mi)
% 2.55/2.00 % (17980)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_17811 for (2995ds/298Mi)
% 2.55/2.00 % (17976)Instruction limit reached!
% 2.55/2.00 % (17976)------------------------------
% 2.55/2.00 % (17976)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/2.00 % (17976)Termination reason: Unknown
% 2.55/2.00 % (17976)Termination phase: Saturation
% 2.55/2.00
% 2.55/2.00 % (17976)Memory used [KB]: 4477
% 2.55/2.00 % (17976)Time elapsed: 0.055 s
% 2.55/2.00 % (17976)Instructions burned: 203 (million)
% 2.55/2.00 % (17976)------------------------------
% 2.55/2.00 % (17976)------------------------------
% 6.10/2.01 % (17981)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_17811 for (2994ds/192Mi)
% 6.10/2.05 % (17979)Instruction limit reached!
% 6.10/2.05 % (17979)------------------------------
% 6.10/2.05 % (17979)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.10/2.05 % (17979)Termination reason: Unknown
% 6.10/2.05 % (17979)Termination phase: Property scanning
% 6.10/2.05
% 6.10/2.05 % (17979)Memory used [KB]: 4349
% 6.10/2.05 % (17979)Time elapsed: 0.057 s
% 6.10/2.05 % (17979)Instructions burned: 134 (million)
% 6.10/2.05 % (17979)------------------------------
% 6.10/2.05 % (17979)------------------------------
% 6.10/2.05 % (17977)Instruction limit reached!
% 6.10/2.05 % (17977)------------------------------
% 6.10/2.05 % (17977)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.10/2.05 % (17977)Termination reason: Unknown
% 6.10/2.05 % (17977)Termination phase: Saturation
% 6.10/2.05
% 6.10/2.05 % (17977)Memory used [KB]: 9594
% 6.10/2.05 % (17977)Time elapsed: 0.092 s
% 6.10/2.05 % (17977)Instructions burned: 341 (million)
% 6.10/2.05 % (17977)------------------------------
% 6.10/2.05 % (17977)------------------------------
% 6.10/2.05 % (17978)Instruction limit reached!
% 6.10/2.05 % (17978)------------------------------
% 6.10/2.05 % (17978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.10/2.05 % (17978)Termination reason: Unknown
% 6.10/2.05 % (17978)Termination phase: Saturation
% 6.10/2.05
% 6.10/2.05 % (17978)Memory used [KB]: 6780
% 6.10/2.05 % (17978)Time elapsed: 0.074 s
% 6.10/2.05 % (17978)Instructions burned: 283 (million)
% 6.10/2.05 % (17978)------------------------------
% 6.10/2.05 % (17978)------------------------------
% 6.10/2.06 % (17982)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_17811 for (2994ds/258Mi)
% 6.10/2.06 % (17984)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_17811 for (2994ds/231Mi)
% 6.10/2.06 % (17983)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_17811 for (2994ds/90Mi)
% 6.10/2.07 % (17981)Instruction limit reached!
% 6.10/2.07 % (17981)------------------------------
% 6.10/2.07 % (17981)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.10/2.07 % (17981)Termination reason: Unknown
% 6.10/2.07 % (17981)Termination phase: Property scanning
% 6.10/2.07
% 6.10/2.07 % (17981)Memory used [KB]: 4605
% 6.10/2.07 % (17981)Time elapsed: 0.054 s
% 6.10/2.07 % (17981)Instructions burned: 196 (million)
% 6.10/2.07 % (17981)------------------------------
% 6.10/2.07 % (17981)------------------------------
% 6.10/2.07 % (17980)Instruction limit reached!
% 6.10/2.07 % (17980)------------------------------
% 6.10/2.07 % (17980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.10/2.07 % (17980)Termination reason: Unknown
% 6.10/2.07 % (17980)Termination phase: Saturation
% 6.10/2.07
% 6.10/2.07 % (17980)Memory used [KB]: 6652
% 6.10/2.07 % (17980)Time elapsed: 0.077 s
% 6.10/2.07 % (17980)Instructions burned: 299 (million)
% 6.10/2.07 % (17980)------------------------------
% 6.10/2.07 % (17980)------------------------------
% 6.10/2.07 % (17985)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_17811 for (2994ds/458Mi)
% 6.10/2.08 % (17986)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_17811 for (2994ds/31Mi)
% 6.10/2.09 % (17986)Instruction limit reached!
% 6.10/2.09 % (17986)------------------------------
% 6.10/2.09 % (17986)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.10/2.09 % (17986)Termination reason: Unknown
% 6.10/2.09 % (17986)Termination phase: Property scanning
% 6.10/2.09
% 6.10/2.09 % (17986)Memory used [KB]: 2174
% 6.10/2.09 % (17986)Time elapsed: 0.011 s
% 6.10/2.09 % (17986)Instructions burned: 33 (million)
% 6.10/2.09 % (17986)------------------------------
% 6.10/2.09 % (17986)------------------------------
% 6.10/2.10 % (17987)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_17811 for (2994ds/297Mi)
% 6.10/2.11 % (17983)Instruction limit reached!
% 6.10/2.11 % (17983)------------------------------
% 6.10/2.11 % (17983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.10/2.11 % (17983)Termination reason: Unknown
% 6.10/2.11 % (17983)Termination phase: Preprocessing 3
% 6.10/2.11
% 6.10/2.11 % (17983)Memory used [KB]: 3965
% 6.10/2.11 % (17983)Time elapsed: 0.045 s
% 6.10/2.11 % (17983)Instructions burned: 90 (million)
% 6.10/2.11 % (17983)------------------------------
% 6.10/2.11 % (17983)------------------------------
% 6.65/2.12 % (17988)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/987Mi)
% 6.65/2.12 % (17984)Instruction limit reached!
% 6.65/2.12 % (17984)------------------------------
% 6.65/2.12 % (17984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.65/2.12 % (17984)Termination reason: Unknown
% 6.65/2.12 % (17984)Termination phase: Saturation
% 6.65/2.12
% 6.65/2.12 % (17984)Memory used [KB]: 7036
% 6.65/2.12 % (17984)Time elapsed: 0.059 s
% 6.65/2.12 % (17984)Instructions burned: 233 (million)
% 6.65/2.12 % (17984)------------------------------
% 6.65/2.12 % (17984)------------------------------
% 6.65/2.12 % (17975)Instruction limit reached!
% 6.65/2.12 % (17975)------------------------------
% 6.65/2.12 % (17975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.65/2.12 % (17975)Termination reason: Unknown
% 6.65/2.12 % (17975)Termination phase: Saturation
% 6.65/2.12
% 6.65/2.12 % (17975)Memory used [KB]: 12281
% 6.65/2.12 % (17975)Time elapsed: 0.182 s
% 6.65/2.12 % (17975)Instructions burned: 615 (million)
% 6.65/2.12 % (17975)------------------------------
% 6.65/2.12 % (17975)------------------------------
% 6.65/2.13 % (17982)Instruction limit reached!
% 6.65/2.13 % (17982)------------------------------
% 6.65/2.13 % (17982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.65/2.13 % (17982)Termination reason: Unknown
% 6.65/2.13 % (17982)Termination phase: Property scanning
% 6.65/2.13
% 6.65/2.13 % (17982)Memory used [KB]: 7803
% 6.65/2.13 % (17982)Time elapsed: 0.069 s
% 6.65/2.13 % (17982)Instructions burned: 260 (million)
% 6.65/2.13 % (17982)------------------------------
% 6.65/2.13 % (17982)------------------------------
% 6.65/2.13 % (17989)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/182Mi)
% 6.65/2.13 % (17990)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/182Mi)
% 6.65/2.14 % (17991)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/154Mi)
% 6.65/2.17 % (17957)Instruction limit reached!
% 6.65/2.17 % (17957)------------------------------
% 6.65/2.17 % (17957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.65/2.17 % (17957)Termination reason: Unknown
% 6.65/2.17 % (17957)Termination phase: Saturation
% 6.65/2.17
% 6.65/2.17 % (17957)Memory used [KB]: 12281
% 6.65/2.17 % (17957)Time elapsed: 0.381 s
% 6.65/2.17 % (17957)Instructions burned: 1089 (million)
% 6.65/2.17 % (17957)------------------------------
% 6.65/2.17 % (17957)------------------------------
% 6.65/2.17 % (17991)Instruction limit reached!
% 6.65/2.17 % (17991)------------------------------
% 6.65/2.17 % (17991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.65/2.17 % (17991)Termination reason: Unknown
% 6.65/2.17 % (17991)Termination phase: Saturation
% 6.65/2.17
% 6.65/2.17 % (17991)Memory used [KB]: 2558
% 6.65/2.17 % (17991)Time elapsed: 0.040 s
% 6.65/2.17 % (17991)Instructions burned: 154 (million)
% 6.65/2.17 % (17991)------------------------------
% 6.65/2.17 % (17991)------------------------------
% 6.65/2.18 % (17992)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/79Mi)
% 6.65/2.18 % (17989)Instruction limit reached!
% 6.65/2.18 % (17989)------------------------------
% 6.65/2.18 % (17989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.65/2.18 % (17989)Termination reason: Unknown
% 6.65/2.18 % (17989)Termination phase: Property scanning
% 6.65/2.18
% 6.65/2.18 % (17989)Memory used [KB]: 7803
% 6.65/2.18 % (17989)Time elapsed: 0.050 s
% 6.65/2.18 % (17989)Instructions burned: 183 (million)
% 6.65/2.18 % (17989)------------------------------
% 6.65/2.18 % (17989)------------------------------
% 6.65/2.18 % (17990)Instruction limit reached!
% 6.65/2.18 % (17990)------------------------------
% 6.65/2.18 % (17990)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.65/2.18 % (17990)Termination reason: Unknown
% 6.65/2.18 % (17990)Termination phase: Saturation
% 6.65/2.18
% 6.65/2.18 % (17990)Memory used [KB]: 9083
% 6.65/2.18 % (17990)Time elapsed: 0.051 s
% 6.65/2.18 % (17990)Instructions burned: 182 (million)
% 6.65/2.18 % (17990)------------------------------
% 6.65/2.18 % (17990)------------------------------
% 6.65/2.18 % (17993)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/216Mi)
% 6.65/2.19 % (17994)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/114Mi)
% 6.65/2.19 % (17987)Instruction limit reached!
% 6.65/2.19 % (17987)------------------------------
% 6.65/2.19 % (17987)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.65/2.19 % (17987)Termination reason: Unknown
% 6.65/2.19 % (17987)Termination phase: Saturation
% 6.65/2.19
% 6.65/2.19 % (17987)Memory used [KB]: 7291
% 6.65/2.19 % (17987)Time elapsed: 0.088 s
% 6.65/2.19 % (17987)Instructions burned: 298 (million)
% 6.65/2.19 % (17987)------------------------------
% 6.65/2.19 % (17987)------------------------------
% 6.65/2.19 % (17995)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/922Mi)
% 7.14/2.20 % (17996)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/182Mi)
% 7.14/2.20 % (17992)Instruction limit reached!
% 7.14/2.20 % (17992)------------------------------
% 7.14/2.20 % (17992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.20 % (17992)Termination reason: Unknown
% 7.14/2.20 % (17992)Termination phase: Preprocessing 3
% 7.14/2.20
% 7.14/2.20 % (17992)Memory used [KB]: 3837
% 7.14/2.20 % (17992)Time elapsed: 0.025 s
% 7.14/2.20 % (17992)Instructions burned: 81 (million)
% 7.14/2.20 % (17992)------------------------------
% 7.14/2.20 % (17992)------------------------------
% 7.14/2.21 % (17985)Instruction limit reached!
% 7.14/2.21 % (17985)------------------------------
% 7.14/2.21 % (17985)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.21 % (17985)Termination reason: Unknown
% 7.14/2.21 % (17985)Termination phase: Saturation
% 7.14/2.21
% 7.14/2.21 % (17985)Memory used [KB]: 10362
% 7.14/2.21 % (17985)Time elapsed: 0.132 s
% 7.14/2.21 % (17985)Instructions burned: 458 (million)
% 7.14/2.21 % (17985)------------------------------
% 7.14/2.21 % (17985)------------------------------
% 7.14/2.21 % (17997)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_17811 for (2993ds/164Mi)
% 7.14/2.22 % (17998)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_17811 for (2992ds/50Mi)
% 7.14/2.22 % (17994)Instruction limit reached!
% 7.14/2.22 % (17994)------------------------------
% 7.14/2.22 % (17994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.22 % (17994)Termination reason: Unknown
% 7.14/2.22 % (17994)Termination phase: Property scanning
% 7.14/2.22
% 7.14/2.22 % (17994)Memory used [KB]: 4221
% 7.14/2.22 % (17994)Time elapsed: 0.033 s
% 7.14/2.22 % (17994)Instructions burned: 115 (million)
% 7.14/2.22 % (17994)------------------------------
% 7.14/2.22 % (17994)------------------------------
% 7.14/2.23 % (17998)Instruction limit reached!
% 7.14/2.23 % (17998)------------------------------
% 7.14/2.23 % (17998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.23 % (17998)Termination reason: Unknown
% 7.14/2.23 % (17998)Termination phase: Property scanning
% 7.14/2.23
% 7.14/2.23 % (17998)Memory used [KB]: 1918
% 7.14/2.23 % (17998)Time elapsed: 0.014 s
% 7.14/2.23 % (17998)Instructions burned: 50 (million)
% 7.14/2.23 % (17998)------------------------------
% 7.14/2.23 % (17998)------------------------------
% 7.14/2.23 % (17999)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_17811 for (2992ds/368Mi)
% 7.14/2.24 % (18000)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_17811 for (2992ds/1021Mi)
% 7.14/2.24 % (17993)Instruction limit reached!
% 7.14/2.24 % (17993)------------------------------
% 7.14/2.24 % (17993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.24 % (17993)Termination reason: Unknown
% 7.14/2.24 % (17993)Termination phase: Saturation
% 7.14/2.24
% 7.14/2.24 % (17993)Memory used [KB]: 7036
% 7.14/2.24 % (17993)Time elapsed: 0.061 s
% 7.14/2.24 % (17993)Instructions burned: 216 (million)
% 7.14/2.24 % (17993)------------------------------
% 7.14/2.24 % (17993)------------------------------
% 7.14/2.24 % (17996)Instruction limit reached!
% 7.14/2.24 % (17996)------------------------------
% 7.14/2.24 % (17996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.24 % (17996)Termination reason: Unknown
% 7.14/2.24 % (17996)Termination phase: Saturation
% 7.14/2.24
% 7.14/2.24 % (17996)Memory used [KB]: 6652
% 7.14/2.24 % (17996)Time elapsed: 0.046 s
% 7.14/2.24 % (17996)Instructions burned: 183 (million)
% 7.14/2.24 % (17996)------------------------------
% 7.14/2.24 % (17996)------------------------------
% 7.14/2.25 % (18002)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_17811 for (2992ds/204Mi)
% 7.14/2.25 % (18001)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_17811 for (2992ds/267Mi)
% 7.14/2.25 % (17997)Instruction limit reached!
% 7.14/2.25 % (17997)------------------------------
% 7.14/2.25 % (17997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.25 % (17997)Termination reason: Unknown
% 7.14/2.25 % (17997)Termination phase: Saturation
% 7.14/2.25
% 7.14/2.25 % (17997)Memory used [KB]: 2302
% 7.14/2.25 % (17997)Time elapsed: 0.046 s
% 7.14/2.25 % (17997)Instructions burned: 164 (million)
% 7.14/2.26 % (17997)------------------------------
% 7.14/2.26 % (17997)------------------------------
% 7.14/2.26 % (18003)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_17811 for (2992ds/26Mi)
% 7.14/2.27 % (18003)Instruction limit reached!
% 7.14/2.27 % (18003)------------------------------
% 7.14/2.27 % (18003)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.27 % (18003)Termination reason: Unknown
% 7.14/2.27 % (18003)Termination phase: shuffling
% 7.14/2.27
% 7.14/2.27 % (18003)Memory used [KB]: 1791
% 7.14/2.27 % (18003)Time elapsed: 0.009 s
% 7.14/2.27 % (18003)Instructions burned: 29 (million)
% 7.14/2.27 % (18003)------------------------------
% 7.14/2.27 % (18003)------------------------------
% 7.14/2.28 % (18004)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_17811 for (2992ds/53Mi)
% 7.14/2.30 % (18004)Instruction limit reached!
% 7.14/2.30 % (18004)------------------------------
% 7.14/2.30 % (18004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.14/2.30 % (18004)Termination reason: Unknown
% 7.14/2.30 % (18004)Termination phase: Property scanning
% 7.14/2.30
% 7.14/2.30 % (18004)Memory used [KB]: 1918
% 7.14/2.30 % (18004)Time elapsed: 0.015 s
% 7.57/2.30 % (18004)Instructions burned: 53 (million)
% 7.57/2.30 % (18004)------------------------------
% 7.57/2.30 % (18004)------------------------------
% 7.57/2.31 % (18005)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_17811 for (2992ds/4436Mi)
% 7.57/2.31 % (18002)Instruction limit reached!
% 7.57/2.31 % (18002)------------------------------
% 7.57/2.31 % (18002)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.31 % (18002)Termination reason: Unknown
% 7.57/2.31 % (18002)Termination phase: Property scanning
% 7.57/2.31
% 7.57/2.31 % (18002)Memory used [KB]: 4477
% 7.57/2.31 % (18002)Time elapsed: 0.055 s
% 7.57/2.31 % (18002)Instructions burned: 206 (million)
% 7.57/2.31 % (18002)------------------------------
% 7.57/2.31 % (18002)------------------------------
% 7.57/2.32 % (18006)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_17811 for (2991ds/1539Mi)
% 7.57/2.32 % (18001)Instruction limit reached!
% 7.57/2.32 % (18001)------------------------------
% 7.57/2.32 % (18001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.32 % (18001)Termination reason: Unknown
% 7.57/2.32 % (18001)Termination phase: Saturation
% 7.57/2.32
% 7.57/2.32 % (18001)Memory used [KB]: 6652
% 7.57/2.32 % (18001)Time elapsed: 0.068 s
% 7.57/2.32 % (18001)Instructions burned: 271 (million)
% 7.57/2.32 % (18001)------------------------------
% 7.57/2.32 % (18001)------------------------------
% 7.57/2.33 % (18007)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_17811 for (2991ds/408Mi)
% 7.57/2.33 % (17999)Instruction limit reached!
% 7.57/2.33 % (17999)------------------------------
% 7.57/2.33 % (17999)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.33 % (17999)Termination reason: Unknown
% 7.57/2.33 % (17999)Termination phase: Saturation
% 7.57/2.33
% 7.57/2.33 % (17999)Memory used [KB]: 11513
% 7.57/2.33 % (17999)Time elapsed: 0.105 s
% 7.57/2.33 % (17999)Instructions burned: 369 (million)
% 7.57/2.33 % (17999)------------------------------
% 7.57/2.33 % (17999)------------------------------
% 7.57/2.34 % (18008)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_17811 for (2991ds/5205Mi)
% 7.57/2.36 % (17988)Instruction limit reached!
% 7.57/2.36 % (17988)------------------------------
% 7.57/2.36 % (17988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.36 % (17988)Termination reason: Unknown
% 7.57/2.36 % (17988)Termination phase: Saturation
% 7.57/2.36
% 7.57/2.36 % (17988)Memory used [KB]: 7803
% 7.57/2.36 % (17988)Time elapsed: 0.242 s
% 7.57/2.36 % (17988)Instructions burned: 989 (million)
% 7.57/2.36 % (17988)------------------------------
% 7.57/2.36 % (17988)------------------------------
% 7.57/2.37 % (18009)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_17811 for (2991ds/1913Mi)
% 8.10/2.45 % (18007)Instruction limit reached!
% 8.10/2.45 % (18007)------------------------------
% 8.10/2.45 % (18007)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.45 % (18007)Termination reason: Unknown
% 8.10/2.45 % (18007)Termination phase: Saturation
% 8.10/2.45
% 8.10/2.45 % (18007)Memory used [KB]: 11001
% 8.10/2.45 % (18007)Time elapsed: 0.120 s
% 8.10/2.45 % (18007)Instructions burned: 410 (million)
% 8.10/2.45 % (18007)------------------------------
% 8.10/2.45 % (18007)------------------------------
% 8.10/2.46 % (18010)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_17811 for (2990ds/4350Mi)
% 8.10/2.47 % (17995)Instruction limit reached!
% 8.10/2.47 % (17995)------------------------------
% 8.10/2.47 % (17995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.47 % (17995)Termination reason: Unknown
% 8.10/2.47 % (17995)Termination phase: Saturation
% 8.10/2.47
% 8.10/2.47 % (17995)Memory used [KB]: 12537
% 8.10/2.47 % (17995)Time elapsed: 0.277 s
% 8.10/2.47 % (17995)Instructions burned: 922 (million)
% 8.10/2.47 % (17995)------------------------------
% 8.10/2.47 % (17995)------------------------------
% 8.10/2.48 % (18011)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_17811 for (2990ds/68Mi)
% 8.52/2.50 % (18011)Instruction limit reached!
% 8.52/2.50 % (18011)------------------------------
% 8.52/2.50 % (18011)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.52/2.50 % (18011)Termination reason: Unknown
% 8.52/2.50 % (18011)Termination phase: Preprocessing 3
% 8.52/2.50
% 8.52/2.50 % (18011)Memory used [KB]: 3198
% 8.52/2.50 % (18011)Time elapsed: 0.021 s
% 8.52/2.50 % (18011)Instructions burned: 68 (million)
% 8.52/2.50 % (18011)------------------------------
% 8.52/2.50 % (18011)------------------------------
% 8.52/2.51 % (18012)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_17811 for (2989ds/358Mi)
% 8.52/2.54 % (18000)Instruction limit reached!
% 8.52/2.54 % (18000)------------------------------
% 8.52/2.54 % (18000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.52/2.54 % (18000)Termination reason: Unknown
% 8.52/2.54 % (18000)Termination phase: Saturation
% 8.52/2.54
% 8.52/2.54 % (18000)Memory used [KB]: 12409
% 8.52/2.54 % (18000)Time elapsed: 0.304 s
% 8.52/2.54 % (18000)Instructions burned: 1023 (million)
% 8.52/2.54 % (18000)------------------------------
% 8.52/2.54 % (18000)------------------------------
% 8.52/2.55 % (18013)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_17811 for (2989ds/31Mi)
% 8.52/2.56 % (18013)Instruction limit reached!
% 8.52/2.56 % (18013)------------------------------
% 8.52/2.56 % (18013)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.52/2.56 % (18013)Termination reason: Unknown
% 8.52/2.56 % (18013)Termination phase: Property scanning
% 8.52/2.56
% 8.52/2.56 % (18013)Memory used [KB]: 2174
% 8.52/2.56 % (18013)Time elapsed: 0.011 s
% 8.52/2.56 % (18013)Instructions burned: 31 (million)
% 8.52/2.56 % (18013)------------------------------
% 8.52/2.56 % (18013)------------------------------
% 8.52/2.57 % (18014)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_17811 for (2989ds/279Mi)
% 8.88/2.61 % (18012)Instruction limit reached!
% 8.88/2.61 % (18012)------------------------------
% 8.88/2.61 % (18012)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.61 % (18012)Termination reason: Unknown
% 8.88/2.61 % (18012)Termination phase: Saturation
% 8.88/2.61
% 8.88/2.61 % (18012)Memory used [KB]: 9594
% 8.88/2.61 % (18012)Time elapsed: 0.103 s
% 8.88/2.61 % (18012)Instructions burned: 359 (million)
% 8.88/2.61 % (18012)------------------------------
% 8.88/2.61 % (18012)------------------------------
% 8.88/2.62 % (18015)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_17811 for (2988ds/258Mi)
% 8.88/2.64 % (18014)Instruction limit reached!
% 8.88/2.64 % (18014)------------------------------
% 8.88/2.64 % (18014)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.64 % (18014)Termination reason: Unknown
% 8.88/2.64 % (18014)Termination phase: Saturation
% 8.88/2.64
% 8.88/2.64 % (18014)Memory used [KB]: 7675
% 8.88/2.64 % (18014)Time elapsed: 0.070 s
% 8.88/2.64 % (18014)Instructions burned: 280 (million)
% 8.88/2.64 % (18014)------------------------------
% 8.88/2.64 % (18014)------------------------------
% 8.88/2.65 % (18016)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_17811 for (2988ds/26Mi)
% 8.88/2.66 % (18016)Instruction limit reached!
% 8.88/2.66 % (18016)------------------------------
% 8.88/2.66 % (18016)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.66 % (18016)Termination reason: Unknown
% 8.88/2.66 % (18016)Termination phase: shuffling
% 8.88/2.66
% 8.88/2.66 % (18016)Memory used [KB]: 1791
% 8.88/2.66 % (18016)Time elapsed: 0.009 s
% 8.88/2.66 % (18016)Instructions burned: 29 (million)
% 8.88/2.66 % (18016)------------------------------
% 8.88/2.66 % (18016)------------------------------
% 8.88/2.67 % (18017)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_17811 for (2988ds/995Mi)
% 8.88/2.69 % (18015)Instruction limit reached!
% 8.88/2.69 % (18015)------------------------------
% 8.88/2.69 % (18015)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.69 % (18015)Termination reason: Unknown
% 8.88/2.69 % (18015)Termination phase: Saturation
% 8.88/2.69
% 8.88/2.69 % (18015)Memory used [KB]: 9722
% 8.88/2.69 % (18015)Time elapsed: 0.075 s
% 8.88/2.69 % (18015)Instructions burned: 259 (million)
% 8.88/2.69 % (18015)------------------------------
% 8.88/2.69 % (18015)------------------------------
% 9.17/2.70 % (18018)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_17811 for (2987ds/373Mi)
% 9.17/2.76 % (18006)Instruction limit reached!
% 9.17/2.76 % (18006)------------------------------
% 9.17/2.76 % (18006)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.17/2.76 % (18006)Termination reason: Unknown
% 9.17/2.76 % (18006)Termination phase: Saturation
% 9.17/2.76
% 9.17/2.76 % (18006)Memory used [KB]: 12792
% 9.17/2.76 % (18006)Time elapsed: 0.469 s
% 9.17/2.76 % (18006)Instructions burned: 1541 (million)
% 9.17/2.76 % (18006)------------------------------
% 9.17/2.76 % (18006)------------------------------
% 9.17/2.77 % (18019)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_17811 for (2987ds/253Mi)
% 9.17/2.78 % (18009)First to succeed.
% 9.64/2.81 % (18018)Instruction limit reached!
% 9.64/2.81 % (18018)------------------------------
% 9.64/2.81 % (18018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.64/2.81 % (18018)Termination reason: Unknown
% 9.64/2.81 % (18018)Termination phase: Saturation
% 9.64/2.81
% 9.64/2.81 % (18018)Memory used [KB]: 9978
% 9.64/2.81 % (18018)Time elapsed: 0.110 s
% 9.64/2.81 % (18018)Instructions burned: 375 (million)
% 9.64/2.81 % (18018)------------------------------
% 9.64/2.81 % (18018)------------------------------
% 9.64/2.82 % (18009)Refutation found. Thanks to Tanya!
% 9.64/2.82 % SZS status Theorem for DTF2THF_17811
% 9.64/2.82 % SZS output start Proof for DTF2THF_17811
% 9.64/2.82 thf(type_def_5, type, term: $tType).
% 9.64/2.82 thf(type_def_7, type, subst: $tType).
% 9.64/2.82 thf(type_def_8, type, d_term: $tType).
% 9.64/2.82 thf(type_def_9, type, d_subst: $tType).
% 9.64/2.82 thf(func_def_0, type, term: $tType).
% 9.64/2.82 thf(func_def_1, type, subst: $tType).
% 9.64/2.82 thf(func_def_2, type, one: term).
% 9.64/2.82 thf(func_def_3, type, ap: term > term > term).
% 9.64/2.82 thf(func_def_4, type, lam: term > term).
% 9.64/2.82 thf(func_def_5, type, sub: term > subst > term).
% 9.64/2.82 thf(func_def_6, type, id: subst).
% 9.64/2.82 thf(func_def_7, type, sh: subst).
% 9.64/2.82 thf(func_def_8, type, push: term > subst > subst).
% 9.64/2.82 thf(func_def_9, type, comp: subst > subst > subst).
% 9.64/2.82 thf(func_def_10, type, var: term > $o).
% 9.64/2.82 thf(func_def_18, type, hoasap: subst > term > subst > term > term).
% 9.64/2.82 thf(func_def_73, type, hoaslam: subst > (subst > term > term) > term).
% 9.64/2.82 thf(func_def_86, type, hoasinduction_p_and_p_prime: (subst > term > subst > $o) > (term > $o) > $o).
% 9.64/2.82 thf(func_def_107, type, hoasvar: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_121, type, pushprop_p_and_p_prime: term > subst > (term > $o) > (term > $o) > $o).
% 9.64/2.82 thf(func_def_124, type, d_term: $tType).
% 9.64/2.82 thf(func_def_125, type, d_subst: $tType).
% 9.64/2.82 thf(func_def_126, type, d2term: d_term > term).
% 9.64/2.82 thf(func_def_127, type, d2subst: d_subst > subst).
% 9.64/2.82 thf(func_def_128, type, d_one: d_term).
% 9.64/2.82 thf(func_def_129, type, d_id: d_subst).
% 9.64/2.82 thf(func_def_144, type, sP2: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_146, type, sP4: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_148, type, sP6: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_151, type, sP9: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_152, type, sP10: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_153, type, sP11: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_154, type, sP12: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_155, type, sP13: (term > $o) > $o).
% 9.64/2.82 thf(func_def_162, type, sP20: (term > $o) > $o).
% 9.64/2.82 thf(func_def_163, type, sP21: (term > $o) > $o).
% 9.64/2.82 thf(func_def_167, type, sP25: (term > $o) > $o).
% 9.64/2.82 thf(func_def_168, type, sP26: (term > $o) > $o).
% 9.64/2.82 thf(func_def_172, type, sP30: (term > $o) > $o).
% 9.64/2.82 thf(func_def_174, type, sP32: (term > $o) > $o).
% 9.64/2.82 thf(func_def_177, type, sP35: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_178, type, sP36: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_179, type, sP37: (term > $o) > $o).
% 9.64/2.82 thf(func_def_183, type, sP41: (term > $o) > $o).
% 9.64/2.82 thf(func_def_184, type, sP42: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_187, type, sP45: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_188, type, sP46: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_190, type, sP48: (term > $o) > $o).
% 9.64/2.82 thf(func_def_191, type, sP49: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_195, type, sP53: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_197, type, sP55: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_199, type, sP57: (subst > term > subst > $o) > $o).
% 9.64/2.82 thf(func_def_201, type, sK59: term > $o).
% 9.64/2.82 thf(func_def_202, type, sK60: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_203, type, sK61: term).
% 9.64/2.82 thf(func_def_204, type, sK62: term).
% 9.64/2.82 thf(func_def_205, type, sK63: (subst > term > subst > $o) > (term > $o) > term).
% 9.64/2.82 thf(func_def_206, type, sK64: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_207, type, sK65: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_208, type, sK66: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_209, type, sK67: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_210, type, sK68: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_211, type, sK69: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_212, type, sK70: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_213, type, sK71: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_214, type, sK72: term > (term > $o) > term).
% 9.64/2.82 thf(func_def_215, type, sK73: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_216, type, sK74: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_217, type, sK75: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_218, type, sK76: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_219, type, sK77: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_220, type, sK78: term).
% 9.64/2.82 thf(func_def_221, type, sK79: term > $o).
% 9.64/2.82 thf(func_def_222, type, sK80: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_223, type, sK81: term).
% 9.64/2.82 thf(func_def_224, type, sK82: (subst > term > subst > $o) > (term > $o) > term).
% 9.64/2.82 thf(func_def_225, type, sK83: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_226, type, sK84: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_227, type, sK85: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_228, type, sK86: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_229, type, sK87: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_230, type, sK88: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_231, type, sK89: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_232, type, sK90: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_233, type, sK91: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_234, type, sK92: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_235, type, sK93: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_236, type, sK94: term).
% 9.64/2.82 thf(func_def_237, type, sK95: term > $o).
% 9.64/2.82 thf(func_def_238, type, sK96: (term > $o) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_239, type, sK97: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_240, type, sK98: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_241, type, sK99: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_242, type, sK100: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_243, type, sK101: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_244, type, sK102: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_245, type, sK103: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_246, type, sK104: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_247, type, sK105: (term > $o) > term > term).
% 9.64/2.82 thf(func_def_248, type, sK106: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_249, type, sK107: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_250, type, sK108: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_251, type, sK109: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_252, type, sK110: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_253, type, sK111: (subst > term > subst > $o) > term > $o).
% 9.64/2.82 thf(func_def_254, type, sK112: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_255, type, sK113: term).
% 9.64/2.82 thf(func_def_256, type, sK114: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_257, type, sK115: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_258, type, sK116: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_259, type, sK117: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_260, type, sK118: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_261, type, sK119: (term > $o) > term).
% 9.64/2.82 thf(func_def_262, type, sK120: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_263, type, sK121: term).
% 9.64/2.82 thf(func_def_264, type, sK122: (term > $o) > term).
% 9.64/2.82 thf(func_def_265, type, sK123: (term > $o) > term).
% 9.64/2.82 thf(func_def_266, type, sK124: term > $o).
% 9.64/2.82 thf(func_def_267, type, sK125: term).
% 9.64/2.82 thf(func_def_268, type, sK126: term > term).
% 9.64/2.82 thf(func_def_269, type, sK127: (term > $o) > term > term).
% 9.64/2.82 thf(func_def_270, type, sK128: (term > $o) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_271, type, sK129: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_272, type, sK130: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_273, type, sK131: term).
% 9.64/2.82 thf(func_def_274, type, sK132: term > $o).
% 9.64/2.82 thf(func_def_275, type, sK133: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_276, type, sK134: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_277, type, sK135: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_278, type, sK136: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_279, type, sK137: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_280, type, sK138: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_281, type, sK139: (term > $o) > term).
% 9.64/2.82 thf(func_def_282, type, sK140: (term > $o) > term).
% 9.64/2.82 thf(func_def_283, type, sK141: term).
% 9.64/2.82 thf(func_def_284, type, sK142: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_285, type, sK143: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_286, type, sK144: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_287, type, sK145: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_288, type, sK146: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_289, type, sK147: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_290, type, sK148: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_291, type, sK149: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_292, type, sK150: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_293, type, sK151: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_294, type, sK152: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_295, type, sK153: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_296, type, sK154: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_297, type, sK155: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_298, type, sK156: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_299, type, sK157: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_300, type, sK158: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_301, type, sK159: term).
% 9.64/2.82 thf(func_def_302, type, sK160: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_303, type, sK161: term).
% 9.64/2.82 thf(func_def_304, type, sK162: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_305, type, sK163: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_306, type, sK164: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_307, type, sK165: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_308, type, sK166: (subst > term > subst > $o) > term > term).
% 9.64/2.82 thf(func_def_309, type, sK167: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_310, type, sK168: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_311, type, sK169: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_312, type, sK170: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_313, type, sK171: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_314, type, sK172: term > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_315, type, sK173: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_316, type, sK174: term).
% 9.64/2.82 thf(func_def_317, type, sK175: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_318, type, sK176: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_319, type, sK177: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_320, type, sK178: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_321, type, sK179: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_322, type, sK180: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_323, type, sK181: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_324, type, sK182: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_325, type, sK183: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_326, type, sK184: (term > $o) > term).
% 9.64/2.82 thf(func_def_327, type, sK185: (term > $o) > term).
% 9.64/2.82 thf(func_def_328, type, sK186: (subst > term > subst > $o) > term > $o).
% 9.64/2.82 thf(func_def_329, type, sK187: (term > $o) > term).
% 9.64/2.82 thf(func_def_330, type, sK188: (term > $o) > term).
% 9.64/2.82 thf(func_def_331, type, sK189: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_332, type, sK190: (term > $o) > term).
% 9.64/2.82 thf(func_def_333, type, sK191: term).
% 9.64/2.82 thf(func_def_334, type, sK192: term > $o).
% 9.64/2.82 thf(func_def_335, type, sK193: term > term).
% 9.64/2.82 thf(func_def_336, type, sK194: term).
% 9.64/2.82 thf(func_def_337, type, sK195: (term > $o) > term > term).
% 9.64/2.82 thf(func_def_338, type, sK196: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_339, type, sK197: (subst > term > subst > $o) > (term > $o) > term).
% 9.64/2.82 thf(func_def_340, type, sK198: term).
% 9.64/2.82 thf(func_def_341, type, sK199: term > $o).
% 9.64/2.82 thf(func_def_342, type, sK200: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_343, type, sK201: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_344, type, sK202: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_345, type, sK203: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_346, type, sK204: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_347, type, sK205: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_348, type, sK206: term).
% 9.64/2.82 thf(func_def_349, type, sK207: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_350, type, sK208: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_351, type, sK209: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_352, type, sK210: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_353, type, sK211: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_354, type, sK212: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_355, type, sK213: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_356, type, sK214: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_357, type, sK215: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_358, type, sK216: (term > $o) > term).
% 9.64/2.82 thf(func_def_359, type, sK217: (term > $o) > term).
% 9.64/2.82 thf(func_def_360, type, sK218: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_361, type, sK219: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_362, type, sK220: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_363, type, sK221: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_364, type, sK222: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_365, type, sK223: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_366, type, sK224: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_367, type, sK225: subst > term > term).
% 9.64/2.82 thf(func_def_368, type, sK226: subst).
% 9.64/2.82 thf(func_def_369, type, sK227: term).
% 9.64/2.82 thf(func_def_370, type, sK228: subst > term > term).
% 9.64/2.82 thf(func_def_371, type, sK229: term).
% 9.64/2.82 thf(func_def_372, type, sK230: subst).
% 9.64/2.82 thf(func_def_373, type, sK231: term).
% 9.64/2.82 thf(func_def_374, type, sK232: subst).
% 9.64/2.82 thf(func_def_375, type, sK233: term).
% 9.64/2.82 thf(func_def_376, type, sK234: term).
% 9.64/2.82 thf(func_def_377, type, sK235: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_378, type, sK236: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_379, type, sK237: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_380, type, sK238: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_381, type, sK239: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_382, type, sK240: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_383, type, sK241: term).
% 9.64/2.82 thf(func_def_384, type, sK242: term > $o).
% 9.64/2.82 thf(func_def_385, type, sK243: term > term).
% 9.64/2.82 thf(func_def_386, type, sK244: (term > $o) > term).
% 9.64/2.82 thf(func_def_387, type, sK245: (term > $o) > term).
% 9.64/2.82 thf(func_def_388, type, sK246: (term > $o) > term).
% 9.64/2.82 thf(func_def_389, type, sK247: (term > $o) > term).
% 9.64/2.82 thf(func_def_390, type, sK248: term > $o).
% 9.64/2.82 thf(func_def_391, type, sK249: subst).
% 9.64/2.82 thf(func_def_392, type, sK250: term).
% 9.64/2.82 thf(func_def_393, type, sK251: term > term).
% 9.64/2.82 thf(func_def_394, type, sK252: subst > (term > $o) > term).
% 9.64/2.82 thf(func_def_395, type, sK253: (term > $o) > term).
% 9.64/2.82 thf(func_def_396, type, sK254: (term > $o) > term).
% 9.64/2.82 thf(func_def_397, type, sK255: (term > $o) > term).
% 9.64/2.82 thf(func_def_398, type, sK256: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_399, type, sK257: term).
% 9.64/2.82 thf(func_def_400, type, sK258: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_401, type, sK259: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_402, type, sK260: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_403, type, sK261: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_404, type, sK262: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_405, type, sK263: term > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_406, type, sK264: (term > $o) > term).
% 9.64/2.82 thf(func_def_407, type, sK265: (term > $o) > term).
% 9.64/2.82 thf(func_def_408, type, sK266: subst).
% 9.64/2.82 thf(func_def_409, type, sK267: term).
% 9.64/2.82 thf(func_def_410, type, sK268: term > $o).
% 9.64/2.82 thf(func_def_411, type, sK269: term).
% 9.64/2.82 thf(func_def_412, type, sK270: term).
% 9.64/2.82 thf(func_def_413, type, sK271: subst).
% 9.64/2.82 thf(func_def_414, type, sK272: subst).
% 9.64/2.82 thf(func_def_415, type, sK273: subst).
% 9.64/2.82 thf(func_def_416, type, sK274: term).
% 9.64/2.82 thf(func_def_417, type, sK275: term > $o).
% 9.64/2.82 thf(func_def_418, type, sK276: term > term).
% 9.64/2.82 thf(func_def_419, type, sK277: (term > $o) > subst > term).
% 9.64/2.82 thf(func_def_420, type, sK278: (term > $o) > term).
% 9.64/2.82 thf(func_def_421, type, sK279: (term > $o) > term).
% 9.64/2.82 thf(func_def_422, type, sK280: (term > $o) > term).
% 9.64/2.82 thf(func_def_423, type, sK281: (term > $o) > term).
% 9.64/2.82 thf(func_def_424, type, sK282: (term > $o) > term).
% 9.64/2.82 thf(func_def_425, type, sK283: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_426, type, sK284: term).
% 9.64/2.82 thf(func_def_427, type, sK285: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_428, type, sK286: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_429, type, sK287: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_430, type, sK288: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_431, type, sK289: term > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_432, type, sK290: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_433, type, sK291: (term > $o) > term).
% 9.64/2.82 thf(func_def_434, type, sK292: (term > $o) > subst > term).
% 9.64/2.82 thf(func_def_435, type, sK293: subst).
% 9.64/2.82 thf(func_def_436, type, sK294: term > $o).
% 9.64/2.82 thf(func_def_437, type, sK295: term).
% 9.64/2.82 thf(func_def_438, type, sK296: term > term).
% 9.64/2.82 thf(func_def_439, type, sK297: term).
% 9.64/2.82 thf(func_def_440, type, sK298: term > $o).
% 9.64/2.82 thf(func_def_441, type, sK299: term).
% 9.64/2.82 thf(func_def_442, type, sK300: term > term).
% 9.64/2.82 thf(func_def_443, type, sK301: (term > $o) > term).
% 9.64/2.82 thf(func_def_444, type, sK302: (term > $o) > term).
% 9.64/2.82 thf(func_def_445, type, sK303: (term > $o) > term).
% 9.64/2.82 thf(func_def_446, type, sK304: (term > $o) > term).
% 9.64/2.82 thf(func_def_447, type, sK305: (term > $o) > term).
% 9.64/2.82 thf(func_def_448, type, sK306: (term > $o) > term).
% 9.64/2.82 thf(func_def_449, type, sK307: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_450, type, sK308: term).
% 9.64/2.82 thf(func_def_451, type, sK309: (subst > term > term) > term > subst).
% 9.64/2.82 thf(func_def_452, type, sK310: (subst > term > term) > term > subst).
% 9.64/2.82 thf(func_def_453, type, sK311: (subst > term > term) > term > term).
% 9.64/2.82 thf(func_def_454, type, sK312: (subst > term > term) > term > subst).
% 9.64/2.82 thf(func_def_455, type, sK313: (subst > term > term) > term > subst).
% 9.64/2.82 thf(func_def_456, type, sK314: (subst > term > term) > term > term).
% 9.64/2.82 thf(func_def_457, type, sK315: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_458, type, sK316: term > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_459, type, sK317: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_460, type, sK318: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_461, type, sK319: term).
% 9.64/2.82 thf(func_def_462, type, sK320: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_463, type, sK321: term > $o).
% 9.64/2.82 thf(func_def_464, type, sK322: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_465, type, sK323: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_466, type, sK324: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_467, type, sK325: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_468, type, sK326: term > subst > term > term).
% 9.64/2.82 thf(func_def_469, type, sK327: term).
% 9.64/2.82 thf(func_def_470, type, sK328: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_471, type, sK329: (term > $o) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_472, type, sK330: (term > $o) > term > term).
% 9.64/2.82 thf(func_def_473, type, sK331: term).
% 9.64/2.82 thf(func_def_474, type, sK332: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_475, type, sK333: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_476, type, sK334: (term > $o) > term).
% 9.64/2.82 thf(func_def_477, type, sK335: (term > $o) > term).
% 9.64/2.82 thf(func_def_478, type, sK336: term).
% 9.64/2.82 thf(func_def_479, type, sK337: term > $o).
% 9.64/2.82 thf(func_def_480, type, sK338: term > term).
% 9.64/2.82 thf(func_def_481, type, sK339: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_482, type, sK340: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_483, type, sK341: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_484, type, sK342: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_485, type, sK343: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_486, type, sK344: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_487, type, sK345: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_488, type, sK346: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_489, type, sK347: term).
% 9.64/2.82 thf(func_def_490, type, sK348: term).
% 9.64/2.82 thf(func_def_491, type, sK349: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_492, type, sK350: term).
% 9.64/2.82 thf(func_def_493, type, sK351: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_494, type, sK352: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_495, type, sK353: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_496, type, sK354: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_497, type, sK355: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_498, type, sK356: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_499, type, sK357: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_500, type, sK358: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_501, type, sK359: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_502, type, sK360: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_503, type, sK361: (subst > term > subst > $o) > term > term).
% 9.64/2.82 thf(func_def_504, type, sK362: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_505, type, sK363: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_506, type, sK364: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_507, type, sK365: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_508, type, sK366: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_509, type, sK367: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_510, type, sK368: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_511, type, sK369: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_512, type, sK370: term).
% 9.64/2.82 thf(func_def_513, type, sK371: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_514, type, sK372: (term > $o) > term).
% 9.64/2.82 thf(func_def_515, type, sK373: (term > $o) > term).
% 9.64/2.82 thf(func_def_516, type, sK374: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_517, type, sK375: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_518, type, sK376: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_519, type, sK377: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_520, type, sK378: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_521, type, sK379: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_522, type, sK380: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_523, type, sK381: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_524, type, sK382: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_525, type, sK383: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_526, type, sK384: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_527, type, sK385: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_528, type, sK386: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_529, type, sK387: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_530, type, sK388: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_531, type, sK389: term).
% 9.64/2.82 thf(func_def_532, type, sK390: term > $o).
% 9.64/2.82 thf(func_def_533, type, sK391: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_534, type, sK392: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_535, type, sK393: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_536, type, sK394: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_537, type, sK395: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_538, type, sK396: (term > $o) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_539, type, sK397: (term > $o) > term > term).
% 9.64/2.82 thf(func_def_540, type, sK398: term).
% 9.64/2.82 thf(func_def_541, type, sK399: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_542, type, sK400: (subst > term > subst > $o) > term > term).
% 9.64/2.82 thf(func_def_543, type, sK401: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_544, type, sK402: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_545, type, sK403: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_546, type, sK404: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_547, type, sK405: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_548, type, sK406: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_549, type, sK407: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_550, type, sK408: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_551, type, sK409: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_552, type, sK410: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_553, type, sK411: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_554, type, sK412: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_555, type, sK413: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_556, type, sK414: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_557, type, sK415: term).
% 9.64/2.82 thf(func_def_558, type, sK416: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_559, type, sK417: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_560, type, sK418: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_561, type, sK419: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_562, type, sK420: term > (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_563, type, sK421: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_564, type, sK422: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_565, type, sK423: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_566, type, sK424: term).
% 9.64/2.82 thf(func_def_567, type, sK425: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_568, type, sK426: term).
% 9.64/2.82 thf(func_def_569, type, sK427: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_570, type, sK428: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_571, type, sK429: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_572, type, sK430: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_573, type, sK431: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_574, type, sK432: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_575, type, sK433: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_576, type, sK434: (subst > term > subst > $o) > subst).
% 9.64/2.82 thf(func_def_577, type, sK435: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_578, type, sK436: (subst > term > subst > $o) > term).
% 9.64/2.82 thf(func_def_579, type, sK437: term).
% 9.64/2.82 thf(func_def_580, type, sK438: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_581, type, sK439: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_582, type, sK440: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_583, type, sK441: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_584, type, sK442: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_585, type, sK443: (subst > term > subst > $o) > term > term).
% 9.64/2.82 thf(func_def_586, type, sK444: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_587, type, sK445: term).
% 9.64/2.82 thf(func_def_588, type, sK446: (subst > term > subst > $o) > term > term).
% 9.64/2.82 thf(func_def_589, type, sK447: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_590, type, sK448: term).
% 9.64/2.82 thf(func_def_591, type, sK449: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_592, type, sK450: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_593, type, sK451: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_594, type, sK452: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_595, type, sK453: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_596, type, sK454: term > subst > term > term).
% 9.64/2.82 thf(func_def_597, type, sK455: term).
% 9.64/2.82 thf(func_def_598, type, sK456: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_599, type, sK457: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_600, type, sK458: term).
% 9.64/2.82 thf(func_def_601, type, sK459: term).
% 9.64/2.82 thf(func_def_602, type, sK460: term).
% 9.64/2.82 thf(func_def_603, type, sK461: term).
% 9.64/2.82 thf(func_def_604, type, sK462: subst).
% 9.64/2.82 thf(func_def_605, type, sK463: subst > d_subst).
% 9.64/2.82 thf(func_def_606, type, sK464: (subst > term > subst > $o) > term > $o).
% 9.64/2.82 thf(func_def_607, type, sK465: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_608, type, sK466: (term > $o) > term).
% 9.64/2.82 thf(func_def_609, type, sK467: term).
% 9.64/2.82 thf(func_def_610, type, sK468: term).
% 9.64/2.82 thf(func_def_611, type, sK469: term).
% 9.64/2.82 thf(func_def_612, type, sK470: term).
% 9.64/2.82 thf(func_def_613, type, sK471: term).
% 9.64/2.82 thf(func_def_614, type, sK472: term).
% 9.64/2.82 thf(func_def_615, type, sK473: term).
% 9.64/2.82 thf(func_def_616, type, sK474: term).
% 9.64/2.82 thf(func_def_617, type, sK475: term).
% 9.64/2.82 thf(func_def_618, type, sK476: term > $o).
% 9.64/2.82 thf(func_def_619, type, sK477: term).
% 9.64/2.82 thf(func_def_620, type, sK478: subst).
% 9.64/2.82 thf(func_def_621, type, sK479: term > $o).
% 9.64/2.82 thf(func_def_622, type, sK480: term).
% 9.64/2.82 thf(func_def_623, type, sK481: subst > term > term).
% 9.64/2.82 thf(func_def_624, type, sK482: subst > term > term).
% 9.64/2.82 thf(func_def_625, type, sK483: subst).
% 9.64/2.82 thf(func_def_626, type, sK484: term).
% 9.64/2.82 thf(func_def_627, type, sK485: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_628, type, sK486: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_629, type, sK487: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_630, type, sK488: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_631, type, sK489: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_632, type, sK490: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_633, type, sK491: term).
% 9.64/2.82 thf(func_def_634, type, sK492: term > $o).
% 9.64/2.82 thf(func_def_635, type, sK493: subst).
% 9.64/2.82 thf(func_def_636, type, sK494: (term > $o) > term).
% 9.64/2.82 thf(func_def_637, type, sK495: subst > (term > $o) > term > term > $o).
% 9.64/2.82 thf(func_def_638, type, sK496: term).
% 9.64/2.82 thf(func_def_639, type, sK497: term).
% 9.64/2.82 thf(func_def_640, type, sK498: term).
% 9.64/2.82 thf(func_def_641, type, sK499: term).
% 9.64/2.82 thf(func_def_642, type, sK500: term > $o).
% 9.64/2.82 thf(func_def_643, type, sK501: subst).
% 9.64/2.82 thf(func_def_644, type, sK502: term).
% 9.64/2.82 thf(func_def_645, type, sK503: (term > $o) > term).
% 9.64/2.82 thf(func_def_646, type, sK504: term > subst > (term > $o) > term > $o).
% 9.64/2.82 thf(func_def_647, type, sK505: (term > $o) > term).
% 9.64/2.82 thf(func_def_648, type, sK506: (term > $o) > term).
% 9.64/2.82 thf(func_def_649, type, sK507: term > $o).
% 9.64/2.82 thf(func_def_650, type, sK508: term).
% 9.64/2.82 thf(func_def_651, type, sK509: term > subst > term > term).
% 9.64/2.82 thf(func_def_652, type, sK510: term).
% 9.64/2.82 thf(func_def_653, type, sK511: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_654, type, sK512: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_655, type, sK513: subst).
% 9.64/2.82 thf(func_def_656, type, sK514: term).
% 9.64/2.82 thf(func_def_657, type, sK515: term).
% 9.64/2.82 thf(func_def_658, type, sK516: term).
% 9.64/2.82 thf(func_def_659, type, sK517: term).
% 9.64/2.82 thf(func_def_660, type, sK518: term > $o).
% 9.64/2.82 thf(func_def_661, type, sK519: term > $o).
% 9.64/2.82 thf(func_def_662, type, sK520: term).
% 9.64/2.82 thf(func_def_663, type, sK521: subst).
% 9.64/2.82 thf(func_def_664, type, sK522: subst > term > (term > $o) > (term > $o) > term).
% 9.64/2.82 thf(func_def_665, type, sK523: subst).
% 9.64/2.82 thf(func_def_666, type, sK524: term).
% 9.64/2.82 thf(func_def_667, type, sK525: term).
% 9.64/2.82 thf(func_def_668, type, sK526: term).
% 9.64/2.82 thf(func_def_669, type, sK527: term).
% 9.64/2.82 thf(func_def_670, type, sK528: term).
% 9.64/2.82 thf(func_def_671, type, sK529: term).
% 9.64/2.82 thf(func_def_672, type, sK530: term).
% 9.64/2.82 thf(func_def_673, type, sK531: term).
% 9.64/2.82 thf(func_def_674, type, sK532: term).
% 9.64/2.82 thf(func_def_675, type, sK533: term).
% 9.64/2.82 thf(func_def_676, type, sK534: term).
% 9.64/2.82 thf(func_def_677, type, sK535: term).
% 9.64/2.82 thf(func_def_678, type, sK536: term).
% 9.64/2.82 thf(func_def_679, type, sK537: term).
% 9.64/2.82 thf(func_def_680, type, sK538: term).
% 9.64/2.82 thf(func_def_681, type, sK539: term).
% 9.64/2.82 thf(func_def_682, type, sK540: term).
% 9.64/2.82 thf(func_def_683, type, sK541: term).
% 9.64/2.82 thf(func_def_684, type, sK542: term).
% 9.64/2.82 thf(func_def_685, type, sK543: term).
% 9.64/2.82 thf(func_def_686, type, sK544: term > d_term).
% 9.64/2.82 thf(func_def_687, type, sK545: term > $o).
% 9.64/2.82 thf(func_def_688, type, sK546: term > $o).
% 9.64/2.82 thf(func_def_689, type, sK547: term).
% 9.64/2.82 thf(func_def_690, type, sK548: subst).
% 9.64/2.82 thf(func_def_691, type, sK549: subst > term > (term > $o) > (term > $o) > term).
% 9.64/2.82 thf(func_def_692, type, sK550: subst).
% 9.64/2.82 thf(func_def_693, type, sK551: subst).
% 9.64/2.82 thf(func_def_694, type, sK552: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_695, type, sK553: (term > $o) > term).
% 9.64/2.82 thf(func_def_696, type, sK554: (subst > term > subst > $o) > term > $o).
% 9.64/2.82 thf(func_def_697, type, sK555: term).
% 9.64/2.82 thf(func_def_698, type, sK556: term > subst > term > term).
% 9.64/2.82 thf(func_def_699, type, sK557: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_700, type, sK558: term).
% 9.64/2.82 thf(func_def_701, type, sK559: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_702, type, sK560: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_703, type, sK561: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_704, type, sK562: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_705, type, sK563: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_706, type, sK564: term > subst > term > term).
% 9.64/2.82 thf(func_def_707, type, sK565: term).
% 9.64/2.82 thf(func_def_708, type, sK566: subst).
% 9.64/2.82 thf(func_def_709, type, sK567: term).
% 9.64/2.82 thf(func_def_710, type, sK568: term).
% 9.64/2.82 thf(func_def_711, type, sK569: subst).
% 9.64/2.82 thf(func_def_712, type, sK570: subst).
% 9.64/2.82 thf(func_def_713, type, sK571: subst).
% 9.64/2.82 thf(func_def_714, type, sK572: subst).
% 9.64/2.82 thf(func_def_715, type, sK573: subst).
% 9.64/2.82 thf(func_def_716, type, sK574: term).
% 9.64/2.82 thf(func_def_717, type, sK575: subst).
% 9.64/2.82 thf(func_def_718, type, sK576: term > $o).
% 9.64/2.82 thf(func_def_719, type, sK577: term).
% 9.64/2.82 thf(func_def_720, type, sK578: subst).
% 9.64/2.82 thf(func_def_721, type, sK579: term > $o).
% 9.64/2.82 thf(func_def_722, type, sK580: term).
% 9.64/2.82 thf(func_def_723, type, sK581: subst).
% 9.64/2.82 thf(func_def_724, type, sK582: term).
% 9.64/2.82 thf(func_def_725, type, sK583: term).
% 9.64/2.82 thf(func_def_726, type, sK584: subst).
% 9.64/2.82 thf(func_def_727, type, sK585: term).
% 9.64/2.82 thf(func_def_728, type, sK586: term).
% 9.64/2.82 thf(func_def_729, type, sK587: subst > term > term).
% 9.64/2.82 thf(func_def_730, type, sK588: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_731, type, sK589: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_732, type, sK590: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_733, type, sK591: subst > term > subst > $o).
% 9.64/2.82 thf(func_def_734, type, sK592: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_735, type, sK593: (subst > term > term) > subst).
% 9.64/2.82 thf(func_def_736, type, sK594: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_737, type, sK595: (subst > term > term) > term).
% 9.64/2.82 thf(func_def_738, type, sK596: term).
% 9.64/2.82 thf(func_def_739, type, sK597: (subst > term > subst > $o) > subst > term > term).
% 9.64/2.82 thf(func_def_740, type, sK598: (subst > term > subst > $o) > term > term).
% 9.64/2.82 thf(func_def_741, type, sK599: (subst > term > subst > $o) > term > term).
% 9.64/2.82 thf(func_def_742, type, sK600: (subst > term > subst > $o) > term > term).
% 9.64/2.82 thf(func_def_743, type, ph601: !>[X0: $tType]:(X0)).
% 9.64/2.82 thf(f5653,plain,(
% 9.64/2.82 $false),
% 9.64/2.82 inference(trivial_inequality_removal,[status(thm)],[f5652])).
% 9.64/2.82 thf(f5652,plain,(
% 9.64/2.82 ($true = $false)),
% 9.64/2.82 inference(forward_demodulation,[status(thm)],[f5651,f2545])).
% 9.64/2.82 thf(f2545,plain,(
% 9.64/2.82 ( ! [X0 : subst,X1 : term] : (($false = (sK283 @ X0 @ X1 @ X0))) )),
% 9.64/2.82 inference(trivial_inequality_removal,[status(thm)],[f2544])).
% 9.64/2.82 thf(f2544,plain,(
% 9.64/2.82 ( ! [X0 : subst,X1 : term] : (($false = (sK283 @ X0 @ X1 @ X0)) | ($true != $true)) )),
% 9.64/2.82 inference(superposition,[status(thm)],[f2535,f2210])).
% 9.64/2.82 thf(f2210,plain,(
% 9.64/2.82 ( ! [X2 : subst,X3 : term,X0 : term,X1 : subst > term > subst > $o] : (($true = (X1 @ X2 @ X3 @ X2)) | ((X1 @ X2 @ X0 @ X2) = $false)) )),
% 9.64/2.82 inference(trivial_inequality_removal,[status(thm)],[f2207])).
% 9.64/2.82 thf(f2207,plain,(
% 9.64/2.82 ( ! [X2 : subst,X3 : term,X0 : term,X1 : subst > term > subst > $o] : (($true = $false) | ($true = (X1 @ X2 @ X3 @ X2)) | ((X1 @ X2 @ X0 @ X2) = $false)) )),
% 9.64/2.82 inference(backward_demodulation,[status(thm)],[f2107,f2194])).
% 9.64/2.82 thf(f2194,plain,(
% 9.64/2.82 (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(trivial_inequality_removal,[status(thm)],[f2193])).
% 9.64/2.82 thf(f2193,plain,(
% 9.64/2.82 ($true = $false) | (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(duplicate_literal_removal,[status(thm)],[f2190])).
% 9.64/2.82 thf(f2190,plain,(
% 9.64/2.82 (hoasinduction_lem3a = $false) | ($true = $false) | (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(superposition,[status(thm)],[f1968,f2174])).
% 9.64/2.82 thf(f2174,plain,(
% 9.64/2.82 ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(duplicate_literal_removal,[status(thm)],[f2173])).
% 9.64/2.82 thf(f2173,plain,(
% 9.64/2.82 ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(forward_demodulation,[status(thm)],[f2172,f1886])).
% 9.64/2.82 thf(f1886,plain,(
% 9.64/2.82 ( ! [X183 : term] : (((d2term @ d_one) = X183)) )),
% 9.64/2.82 inference(forward_demodulation,[status(thm)],[f1112,f1094])).
% 9.64/2.82 thf(f1094,plain,(
% 9.64/2.82 ( ! [X209 : d_term] : ((d_one = X209)) )),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f545,plain,(
% 9.64/2.82 (! [X0 : term] : ! [X2 : term,X3 : subst] : ((sK454 @ X0 @ X3 @ X2) = (sub @ X0 @ (push @ X2 @ X3))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X5 : subst > term > term] : ((X5 @ (sK457 @ X5) @ (sK456 @ X5)) != (sub @ sK455 @ (push @ (sK456 @ X5) @ (sK457 @ X5))))) & ((apinj2 = $true) | (((ap @ sK459 @ sK460) = (ap @ sK458 @ sK461)) & (sK461 != sK460))) & (! [X12 : term,X13 : term,X14 : term,X15 : term] : (((ap @ X13 @ X14) != (ap @ X12 @ X15)) | (X14 = X15)) | (apinj2 != $true)) & (! [X16 : subst] : ((comp @ id @ X16) = X16) | (axidl != $true)) & ((axidl = $true) | (sK462 != (comp @ id @ sK462))) & ((sP58 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP58 != $true)) & ! [X18 : subst] : ((d2subst @ (sK463 @ X18)) = X18) & (pushprop_lem2v2_lthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X20 : subst > term > subst > $o] : ! [X22 : term] : ((sK464 @ X20 @ X22) = (X20 @ id @ X22 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X24 : term > $o] : ((sK465 @ id @ (sK466 @ X24) @ id) != (X24 @ (sK466 @ X24)))) & ((hoasinduction_lem2v2 = $true) | ($true != sP56)) & (($true = sP56) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP54)) & (($true = sP54) | (hoasinduction_lem3v2 != $true)) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) | (! [X26 : term,X27 : term,X28 : term,X29 : term] : ((X26 = X27) | ((ap @ X26 @ X29) != (ap @ X27 @ X28))) & ! [X30 : term] : ((sub @ X30 @ id) = X30) & (((ap @ (sub @ sK470 @ id) @ sK468) = (ap @ (sub @ sK469 @ id) @ sK467)) & (sK469 != sK470)))) & (((sK471 != sK472) & ((ap @ sK471 @ sK474) = (ap @ sK472 @ sK473))) | ((sub @ sK475 @ id) != sK475) | ! [X40 : term,X41 : term,X42 : term,X43 : term] : (((ap @ (sub @ X43 @ id) @ X41) != (ap @ (sub @ X42 @ id) @ X40)) | (X42 = X43)) | (hoasapinj1_lthm != $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & (hoaslamnotvar_gthm = $true) & ((sP52 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP52 != $true)) & (induction2_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoaslamnotap_gthm = $true) & (! [X44 : term > $o,X45 : term,X46 : subst,X47 : term > $o,X48 : term] : (($true = (X47 @ (sub @ X45 @ (push @ X48 @ X46)))) | ($true != (X44 @ X45))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | (($true != (sK479 @ (sub @ sK477 @ (push @ sK480 @ sK478)))) & ((sK476 @ sK477) = $true))) & ((hoaslaminj = $true) | (((sK481 @ sK483 @ sK484) != (sK482 @ sK483 @ sK484)) & ! [X58 : subst,X59 : subst,X60 : term] : ((sub @ (sK482 @ X59 @ X60) @ X58) = (sK482 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) & ((lam @ (sK481 @ sh @ one)) = (lam @ (sK482 @ sh @ one))) & ! [X61 : term,X62 : subst,X63 : subst] : ((sub @ (sK481 @ X62 @ X61) @ X63) = (sK481 @ (comp @ X62 @ X63) @ (sub @ X61 @ X63))))) & (! [X64 : subst > term > term,X65 : subst > term > term,X66 : subst,X67 : term] : (((X64 @ X66 @ X67) = (X65 @ X66 @ X67)) | ((sub @ (X65 @ (sK486 @ X65) @ (sK487 @ X65)) @ (sK485 @ X65)) != (X65 @ (comp @ (sK486 @ X65) @ (sK485 @ X65)) @ (sub @ (sK487 @ X65) @ (sK485 @ X65)))) | ((lam @ (X64 @ sh @ one)) != (lam @ (X65 @ sh @ one))) | ((X64 @ (comp @ (sK489 @ X64) @ (sK490 @ X64)) @ (sub @ (sK488 @ X64) @ (sK490 @ X64))) != (sub @ (X64 @ (sK489 @ X64) @ (sK488 @ X64)) @ (sK490 @ X64)))) | (hoaslaminj != $true)) & ((pushprop_lem0 = $true) | ! [X77 : term > $o] : ((sK492 @ (sub @ (sK494 @ X77) @ (push @ sK491 @ sK493))) != (X77 @ (sK494 @ X77)))) & (! [X79 : term,X80 : term > $o,X81 : subst] : ! [X83 : term] : ((sK495 @ X81 @ X80 @ X79 @ X83) = (X80 @ (sub @ X83 @ (push @ X79 @ X81)))) | (pushprop_lem0 != $true)) & (! [X84 : term,X85 : term,X86 : term,X87 : term] : (((ap @ (sub @ X84 @ id) @ X86) != (ap @ (sub @ X85 @ id) @ X87)) | (X84 = X85)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | (((ap @ (sub @ sK497 @ id) @ sK499) = (ap @ (sub @ sK496 @ id) @ sK498)) & (sK496 != sK497))) & ! [X92 : d_subst,X93 : d_subst] : ((X92 = X93) | ((d2subst @ X93) != (d2subst @ X92))) & ((pushprop_lem0_lthm = $true) | ! [X97 : term > $o] : ((sK500 @ (sub @ (sK503 @ X97) @ (push @ sK502 @ sK501))) != (X97 @ (sK503 @ X97)))) & (! [X99 : term > $o,X100 : subst,X101 : term] : ! [X103 : term] : ((X99 @ (sub @ X103 @ (push @ X101 @ X100))) = (sK504 @ X101 @ X100 @ X99 @ X103)) | (pushprop_lem0_lthm != $true)) & (($true = sP47) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP47)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP44)) & (($true = sP44) | (hoasinduction_lem3_lthm != $true)) & (! [X104 : term > $o,X105 : term] : ((((X104 @ (sK505 @ X104)) != $true) & ((var @ (sK505 @ X104)) = $true)) | ($true = (X104 @ X105)) | (((X104 @ (lam @ (sK506 @ X104))) != $true) & ($true = (X104 @ (sK506 @ X104)))) | ($true = (sP41 @ X104))) | (induction != $true)) & ((induction = $true) | (! [X110 : term] : (((sK507 @ X110) = $true) | ((var @ X110) != $true)) & ($true != (sK507 @ sK508)) & ! [X111 : term] : (((sK507 @ (lam @ X111)) = $true) | ($true != (sK507 @ X111))) & ((sP41 @ sK507) != $true))) & (hoasinduction_lem1 = $true) & (($true = sP40) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP40)) & ((hoaslaminj_lthm = $true) | (sP34 != $true)) & ((sP34 = $true) | (hoaslaminj_lthm != $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (($true = sP33) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP33)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (hoasapnotvar_lthm = $true) & (($true = sP31) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP31)) & (axapp = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3aa != $true)) & (! [X112 : term] : ! [X114 : subst,X115 : term] : ((sub @ X112 @ (push @ X115 @ X114)) = (sK509 @ X112 @ X114 @ X115)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X117 : subst > term > term] : ((X117 @ (sK511 @ X117) @ (sK512 @ X117)) != (sub @ sK510 @ (push @ (sK512 @ X117) @ (sK511 @ X117))))) & ((induction2lem_lthm = $true) | ($true != sP28)) & (($true = sP28) | (induction2lem_lthm != $true)) & (axvarshift = $true) & ((axidr = $true) | ((comp @ sK513 @ id) != sK513)) & (! [X121 : subst] : ((comp @ X121 @ id) = X121) | (axidr != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (lamnotvar = $true) & ((hoasinduction_lem3a = $true) | (sP24 != $true)) & ((sP24 = $true) | (hoasinduction_lem3a != $true)) & (hoasinduction_lem3v2_gthm = $true) & ! [X122 : d_subst] : (d_id = X122) & ((apinj1 = $true) | ((sK517 != sK514) & ((ap @ sK514 @ sK516) = (ap @ sK517 @ sK515)))) & (! [X127 : term,X128 : term,X129 : term,X130 : term] : ((X127 = X130) | ((ap @ X127 @ X129) != (ap @ X130 @ X128))) | (apinj1 != $true)) & ((induction2_lthm = $true) | ($true != sP23)) & (($true = sP23) | (induction2_lthm != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP19)) & (($true = sP19) | (hoasinduction_lem3aaa != $true)) & ((pushprop_lem1v2_lthm = $true) | ((((sK518 @ sK520) = $true) & ($true != (sK519 @ one)) & ! [X135 : term] : ((sK519 @ X135) = (sK518 @ (sub @ X135 @ (push @ sK520 @ sK521))))) & ! [X136 : subst,X137 : term] : ((sub @ one @ (push @ X137 @ X136)) = X137))) & (! [X138 : term > $o,X139 : term > $o,X140 : term,X141 : subst] : (($true != (X138 @ X140)) | ((X139 @ one) = $true) | ((X138 @ (sub @ (sK522 @ X141 @ X140 @ X139 @ X138) @ (push @ X140 @ X141))) != (X139 @ (sK522 @ X141 @ X140 @ X139 @ X138)))) | (sK524 != (sub @ one @ (push @ sK524 @ sK523))) | (pushprop_lem1v2_lthm != $true)) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction_lem3v2a_lthm = $true) | (sP18 != $true)) & ((sP18 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((laminj = $true) | ((sK525 != sK526) & ((lam @ sK526) = (lam @ sK525)))) & (! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148))) | (laminj != $true)) & (! [X149 : term,X150 : term] : (((sub @ X149 @ sh) != (sub @ X150 @ sh)) | (X149 = X150)) | (shinj != $true)) & ((shinj = $true) | (((sub @ sK528 @ sh) = (sub @ sK527 @ sh)) & (sK527 != sK528))) & (hoasapnotvar_gthm = $true) & (! [X153 : term] : ((sub @ X153 @ id) = X153) | (sK529 != (sub @ sK529 @ id)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (((sub @ sK530 @ id) != sK530) & ! [X156 : term] : ((sub @ X156 @ id) = X156))) & (! [X157 : term] : ((sub @ X157 @ id) = X157) | (termmset != $true)) & ((termmset = $true) | ((sub @ sK531 @ id) != sK531)) & (apnotvar = $true) & ((d2subst @ d_id) = id) & ((hoasinduction_lthm = $true) | (sP17 != $true)) & ((sP17 = $true) | (hoasinduction_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lthm_orig = $true) & ((hoasapinj2_lthm = $true) | (! [X159 : term,X160 : term,X161 : term,X162 : term] : ((X159 = X162) | ((ap @ X161 @ X162) != (ap @ X160 @ X159))) & (((ap @ (sub @ sK532 @ id) @ sK533) = (ap @ (sub @ sK534 @ id) @ sK535)) & (sK533 != sK535)))) & (((sK539 != sK536) & ((ap @ sK538 @ sK539) = (ap @ sK537 @ sK536))) | ! [X171 : term,X172 : term,X173 : term,X174 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X171 @ id) @ X172)) | (X172 = X174)) | (hoasapinj2_lthm != $true)) & (hoaslamnotvar = $true) & (pushprop_lthm = $true) & ((sP8 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP8 != $true)) & (pushprop_lem2v2_gthm = $true) & ((sP7 = $true) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (sP7 != $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (axassoc = $true) & (axabs = $true) & (lamnotap != $true) & (! [X175 : term,X176 : term,X177 : term,X178 : term] : ((X176 = X177) | ((ap @ (sub @ X178 @ id) @ X177) != (ap @ (sub @ X175 @ id) @ X176))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ((sK541 != sK542) & ((ap @ (sub @ sK540 @ id) @ sK541) = (ap @ (sub @ sK543 @ id) @ sK542)))) & ! [X183 : term] : ((d2term @ (sK544 @ X183)) = X183) & (pushprop = $true) & (hoasinduction_lem3_gthm = $true) & ((pushprop_lem1v2 = $true) | (($true = (sK545 @ sK547)) & ((sK546 @ one) != $true) & ! [X189 : term] : ((sK545 @ (sub @ X189 @ (push @ sK547 @ sK548))) = (sK546 @ X189)))) & (! [X190 : term > $o,X191 : term > $o,X192 : term,X193 : subst] : (((X190 @ X192) != $true) | ($true = (X191 @ one)) | ((X190 @ (sub @ (sK549 @ X193 @ X192 @ X191 @ X190) @ (push @ X192 @ X193))) != (X191 @ (sK549 @ X193 @ X192 @ X191 @ X190)))) | (pushprop_lem1v2 != $true)) & ((! [X195 : subst] : ((comp @ X195 @ id) = X195) & ! [X196 : subst] : ((comp @ id @ X196) = X196)) | (substmonoid != $true)) & ((substmonoid = $true) | ((comp @ sK550 @ id) != sK550) | ((comp @ id @ sK551) != sK551)) & ((hoasinduction_lem0 = $true) | ! [X200 : term > $o] : ((X200 @ (sK553 @ X200)) != (sK552 @ id @ (sK553 @ X200) @ id))) & (! [X202 : subst > term > subst > $o] : ! [X204 : term] : ((sK554 @ X202 @ X204) = (X202 @ id @ X204 @ id)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP5)) & (($true = sP5) | (hoasinduction_lem3aa_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3b_lthm = $true) | ! [X206 : subst > term > term] : ((X206 @ sh @ one) != (sub @ sK555 @ (push @ one @ sh)))) & (! [X207 : term] : ((sub @ X207 @ (push @ one @ sh)) = (sK556 @ X207 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & (ulamvarsh = $true) & ! [X209 : d_term] : (d_one = X209) & (axclos = $true) & (substmonoid_gthm = $true) & (induction2lem_gthm = $true) & (pushprop_lem2v2 = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_no_psi_cond = $true) | (! [X212 : subst > term > term] : ((($true = (sK557 @ id @ (sK559 @ X212) @ id)) & ($true != (sK557 @ id @ (X212 @ id @ (sK559 @ X212)) @ id))) | ((sub @ (X212 @ (sK562 @ X212) @ (sK560 @ X212)) @ (sK561 @ X212)) != (X212 @ (comp @ (sK562 @ X212) @ (sK561 @ X212)) @ (sub @ (sK560 @ X212) @ (sK561 @ X212)))) | ($true = (sK557 @ id @ (lam @ (X212 @ sh @ one)) @ id))) & ($true != (sP4 @ sK557)) & ((sK557 @ id @ sK558 @ id) != $true))) & (! [X217 : subst > term > subst > $o,X218 : term] : ((! [X220 : term] : (((X217 @ id @ X220 @ id) != $true) | ($true = (X217 @ id @ (sK563 @ X217 @ id @ X220) @ id))) & ! [X221 : term,X222 : subst,X223 : subst] : ((sub @ (sK563 @ X217 @ X223 @ X221) @ X222) = (sK563 @ X217 @ (comp @ X223 @ X222) @ (sub @ X221 @ X222))) & ((X217 @ id @ (lam @ (sK563 @ X217 @ sh @ one)) @ id) != $true)) | ((sP4 @ X217) = $true) | ($true = (X217 @ id @ X218 @ id))) | (hoasinduction_no_psi_cond != $true)) & (axmap = $true) & ((hoasinduction_lem2 = $true) | ($true != sP3)) & (($true = sP3) | (hoasinduction_lem2 != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (! [X224 : term] : ((sK564 @ X224 @ sh @ one) = (sub @ X224 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X227 : subst > term > term] : ((X227 @ sh @ one) != (sub @ sK565 @ (push @ one @ sh)))) & (one = (d2term @ d_one)) & (termmset_gthm = $true) & (! [X228 : subst,X229 : term] : ((sub @ one @ (push @ X229 @ X228)) = X229) | (axvarcons != $true)) & ((axvarcons = $true) | ((sub @ one @ (push @ sK567 @ sK566)) != sK567)) & (ulamvarind = $true) & (hoasapinj1_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X232 : term] : ((sub @ X232 @ id) = X232) | (axvarid != $true)) & ((axvarid = $true) | ((sub @ sK568 @ id) != sK568)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ! [X234 : d_term,X235 : d_term] : (((d2term @ X234) != (d2term @ X235)) | (X234 = X235)) & (((comp @ sK569 @ id) != sK569) | ((comp @ id @ sK570) != sK570) | (! [X238 : subst] : ((comp @ id @ X238) = X238) & ! [X239 : subst] : ((comp @ X239 @ id) = X239)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X240 : subst] : ((comp @ X240 @ id) = X240) & ! [X241 : subst] : ((comp @ id @ X241) = X241) & (((comp @ id @ sK571) != sK571) | ((comp @ sK572 @ id) != sK572)))) & (ulamvar1 != $true) & (pushprop_gthm = $true) & ((d2subst @ d_id) = sh) & ((axscons = $true) | (sK573 != (push @ (sub @ one @ sK573) @ (comp @ sh @ sK573)))) & (! [X245 : subst] : ((push @ (sub @ one @ X245) @ (comp @ sh @ X245)) = X245) | (axscons != $true)) & ((axshiftcons = $true) | ((comp @ sh @ (push @ sK574 @ sK575)) != sK575)) & (! [X248 : term,X249 : subst] : ((comp @ sh @ (push @ X248 @ X249)) = X249) | (axshiftcons != $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP1) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP1)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X250 : subst,X251 : term] : ((sub @ one @ (push @ X251 @ X250)) = X251) & ! [X252 : term,X253 : subst] : ((comp @ sh @ (push @ X252 @ X253)) = X253) & (((sK576 @ (sub @ sK580 @ (push @ sK577 @ sK578))) != $true) & ($true = (sK579 @ sK580))))) & (((sub @ one @ (push @ sK582 @ sK581)) != sK582) | (sK584 != (comp @ sh @ (push @ sK583 @ sK584))) | ! [X263 : term > $o,X264 : term,X265 : subst,X266 : term > $o,X267 : term] : (((X263 @ (sub @ X267 @ (push @ X264 @ X265))) = $true) | ($true != (X266 @ X267))) | (pushprop_lem1_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ((hoaslamnotap = $true) | (((ap @ (sub @ sK586 @ id) @ sK585) = (lam @ (sK587 @ sh @ one))) & ! [X271 : subst,X272 : term,X273 : subst] : ((sK587 @ (comp @ X273 @ X271) @ (sub @ X272 @ X271)) = (sub @ (sK587 @ X273 @ X272) @ X271)))) & (! [X274 : term,X275 : term,X276 : subst > term > term] : (((lam @ (X276 @ sh @ one)) != (ap @ (sub @ X275 @ id) @ X274)) | ((sub @ (X276 @ (sK590 @ X276) @ (sK589 @ X276)) @ (sK588 @ X276)) != (X276 @ (comp @ (sK590 @ X276) @ (sK588 @ X276)) @ (sub @ (sK589 @ X276) @ (sK588 @ X276))))) | (hoaslamnotap != $true))),
% 9.64/2.82 inference(skolemisation,[status(esa),new_symbols(skolem,[sK454,sK455,sK456,sK457,sK458,sK459,sK460,sK461,sK462,sK463,sK464,sK465,sK466,sK467,sK468,sK469,sK470,sK471,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,sK487,sK488,sK489,sK490,sK491,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,sK509,sK510,sK511,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,sK521,sK522,sK523,sK524,sK525,sK526,sK527,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,sK536,sK537,sK538,sK539,sK540,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,sK562,sK563,sK564,sK565,sK566,sK567,sK568,sK569,sK570,sK571,sK572,sK573,sK574,sK575,sK576,sK577,sK578,sK579,sK580,sK581,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590])],[f473,f544,f543,f542,f541,f540,f539,f538,f537,f536,f535,f534,f533,f532,f531,f530,f529,f528,f527,f526,f525,f524,f523,f522,f521,f520,f519,f518,f517,f516,f515,f514,f513,f512,f511,f510,f509,f508,f507,f506,f505,f504,f503,f502,f501,f500,f499,f498,f497,f496,f495,f494,f493,f492,f491,f490,f489,f488,f487,f486,f485,f484,f483,f482,f481,f480,f479,f478,f477,f476,f475,f474])).
% 9.64/2.82 thf(f474,plain,(
% 9.64/2.82 ! [X0 : term] : (? [X1 : subst > term > term] : ! [X2 : term,X3 : subst] : ((X1 @ X3 @ X2) = (sub @ X0 @ (push @ X2 @ X3))) => ! [X3 : subst,X2 : term] : ((sK454 @ X0 @ X3 @ X2) = (sub @ X0 @ (push @ X2 @ X3))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f475,plain,(
% 9.64/2.82 ? [X4 : term] : ! [X5 : subst > term > term] : ? [X6 : term,X7 : subst] : ((X5 @ X7 @ X6) != (sub @ X4 @ (push @ X6 @ X7))) => ! [X5 : subst > term > term] : ? [X7 : subst,X6 : term] : ((X5 @ X7 @ X6) != (sub @ sK455 @ (push @ X6 @ X7)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f476,plain,(
% 9.64/2.82 ! [X5 : subst > term > term] : (? [X7 : subst,X6 : term] : ((X5 @ X7 @ X6) != (sub @ sK455 @ (push @ X6 @ X7))) => ((X5 @ (sK457 @ X5) @ (sK456 @ X5)) != (sub @ sK455 @ (push @ (sK456 @ X5) @ (sK457 @ X5)))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f477,plain,(
% 9.64/2.82 ? [X8 : term,X9 : term,X10 : term,X11 : term] : (((ap @ X8 @ X11) = (ap @ X9 @ X10)) & (X10 != X11)) => (((ap @ sK459 @ sK460) = (ap @ sK458 @ sK461)) & (sK461 != sK460))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f478,plain,(
% 9.64/2.82 ? [X17 : subst] : ((comp @ id @ X17) != X17) => (sK462 != (comp @ id @ sK462))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f479,plain,(
% 9.64/2.82 ! [X18 : subst] : (? [X19 : d_subst] : ((d2subst @ X19) = X18) => ((d2subst @ (sK463 @ X18)) = X18))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f480,plain,(
% 9.64/2.82 ! [X20 : subst > term > subst > $o] : (? [X21 : term > $o] : ! [X22 : term] : ((X20 @ id @ X22 @ id) = (X21 @ X22)) => ! [X22 : term] : ((sK464 @ X20 @ X22) = (X20 @ id @ X22 @ id)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f481,plain,(
% 9.64/2.82 ? [X23 : subst > term > subst > $o] : ! [X24 : term > $o] : ? [X25 : term] : ((X24 @ X25) != (X23 @ id @ X25 @ id)) => ! [X24 : term > $o] : ? [X25 : term] : ((X24 @ X25) != (sK465 @ id @ X25 @ id))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f482,plain,(
% 9.64/2.82 ! [X24 : term > $o] : (? [X25 : term] : ((X24 @ X25) != (sK465 @ id @ X25 @ id)) => ((sK465 @ id @ (sK466 @ X24) @ id) != (X24 @ (sK466 @ X24))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f483,plain,(
% 9.64/2.82 ? [X31 : term,X32 : term,X33 : term,X34 : term] : (((ap @ (sub @ X33 @ id) @ X31) = (ap @ (sub @ X34 @ id) @ X32)) & (X33 != X34)) => (((ap @ (sub @ sK470 @ id) @ sK468) = (ap @ (sub @ sK469 @ id) @ sK467)) & (sK469 != sK470))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f484,plain,(
% 9.64/2.82 ? [X35 : term,X36 : term,X37 : term,X38 : term] : ((X35 != X36) & ((ap @ X35 @ X38) = (ap @ X36 @ X37))) => ((sK471 != sK472) & ((ap @ sK471 @ sK474) = (ap @ sK472 @ sK473)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f485,plain,(
% 9.64/2.82 ? [X39 : term] : ((sub @ X39 @ id) != X39) => ((sub @ sK475 @ id) != sK475)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f486,plain,(
% 9.64/2.82 ? [X49 : term > $o,X50 : term,X51 : subst,X52 : term > $o,X53 : term] : (($true != (X52 @ (sub @ X50 @ (push @ X53 @ X51)))) & ($true = (X49 @ X50))) => (($true != (sK479 @ (sub @ sK477 @ (push @ sK480 @ sK478)))) & ((sK476 @ sK477) = $true))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f487,plain,(
% 9.64/2.82 ? [X54 : subst > term > term,X55 : subst > term > term,X56 : subst,X57 : term] : (((X55 @ X56 @ X57) != (X54 @ X56 @ X57)) & ! [X58 : subst,X59 : subst,X60 : term] : ((sub @ (X55 @ X59 @ X60) @ X58) = (X55 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) & ((lam @ (X55 @ sh @ one)) = (lam @ (X54 @ sh @ one))) & ! [X61 : term,X62 : subst,X63 : subst] : ((sub @ (X54 @ X62 @ X61) @ X63) = (X54 @ (comp @ X62 @ X63) @ (sub @ X61 @ X63)))) => (((sK481 @ sK483 @ sK484) != (sK482 @ sK483 @ sK484)) & ! [X60 : term,X59 : subst,X58 : subst] : ((sub @ (sK482 @ X59 @ X60) @ X58) = (sK482 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) & ((lam @ (sK481 @ sh @ one)) = (lam @ (sK482 @ sh @ one))) & ! [X63 : subst,X62 : subst,X61 : term] : ((sub @ (sK481 @ X62 @ X61) @ X63) = (sK481 @ (comp @ X62 @ X63) @ (sub @ X61 @ X63))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f488,plain,(
% 9.64/2.82 ! [X65 : subst > term > term] : (? [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X65 @ X69 @ X70) @ X68) != (X65 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) => ((sub @ (X65 @ (sK486 @ X65) @ (sK487 @ X65)) @ (sK485 @ X65)) != (X65 @ (comp @ (sK486 @ X65) @ (sK485 @ X65)) @ (sub @ (sK487 @ X65) @ (sK485 @ X65)))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f489,plain,(
% 9.64/2.82 ! [X64 : subst > term > term] : (? [X71 : term,X72 : subst,X73 : subst] : ((X64 @ (comp @ X72 @ X73) @ (sub @ X71 @ X73)) != (sub @ (X64 @ X72 @ X71) @ X73)) => ((X64 @ (comp @ (sK489 @ X64) @ (sK490 @ X64)) @ (sub @ (sK488 @ X64) @ (sK490 @ X64))) != (sub @ (X64 @ (sK489 @ X64) @ (sK488 @ X64)) @ (sK490 @ X64))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f490,plain,(
% 9.64/2.82 ? [X74 : term,X75 : term > $o,X76 : subst] : ! [X77 : term > $o] : ? [X78 : term] : ((X77 @ X78) != (X75 @ (sub @ X78 @ (push @ X74 @ X76)))) => ! [X77 : term > $o] : ? [X78 : term] : ((X77 @ X78) != (sK492 @ (sub @ X78 @ (push @ sK491 @ sK493))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f491,plain,(
% 9.64/2.82 ! [X77 : term > $o] : (? [X78 : term] : ((X77 @ X78) != (sK492 @ (sub @ X78 @ (push @ sK491 @ sK493)))) => ((sK492 @ (sub @ (sK494 @ X77) @ (push @ sK491 @ sK493))) != (X77 @ (sK494 @ X77))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f492,plain,(
% 9.64/2.82 ! [X79 : term,X80 : term > $o,X81 : subst] : (? [X82 : term > $o] : ! [X83 : term] : ((X82 @ X83) = (X80 @ (sub @ X83 @ (push @ X79 @ X81)))) => ! [X83 : term] : ((sK495 @ X81 @ X80 @ X79 @ X83) = (X80 @ (sub @ X83 @ (push @ X79 @ X81)))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f493,plain,(
% 9.64/2.82 ? [X88 : term,X89 : term,X90 : term,X91 : term] : (((ap @ (sub @ X88 @ id) @ X90) = (ap @ (sub @ X89 @ id) @ X91)) & (X88 != X89)) => (((ap @ (sub @ sK497 @ id) @ sK499) = (ap @ (sub @ sK496 @ id) @ sK498)) & (sK496 != sK497))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f494,plain,(
% 9.64/2.82 ? [X94 : term > $o,X95 : subst,X96 : term] : ! [X97 : term > $o] : ? [X98 : term] : ((X94 @ (sub @ X98 @ (push @ X96 @ X95))) != (X97 @ X98)) => ! [X97 : term > $o] : ? [X98 : term] : ((sK500 @ (sub @ X98 @ (push @ sK502 @ sK501))) != (X97 @ X98))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f495,plain,(
% 9.64/2.82 ! [X97 : term > $o] : (? [X98 : term] : ((sK500 @ (sub @ X98 @ (push @ sK502 @ sK501))) != (X97 @ X98)) => ((sK500 @ (sub @ (sK503 @ X97) @ (push @ sK502 @ sK501))) != (X97 @ (sK503 @ X97))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f496,plain,(
% 9.64/2.82 ! [X99 : term > $o,X100 : subst,X101 : term] : (? [X102 : term > $o] : ! [X103 : term] : ((X99 @ (sub @ X103 @ (push @ X101 @ X100))) = (X102 @ X103)) => ! [X103 : term] : ((X99 @ (sub @ X103 @ (push @ X101 @ X100))) = (sK504 @ X101 @ X100 @ X99 @ X103)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f497,plain,(
% 9.64/2.82 ! [X104 : term > $o] : (? [X106 : term] : (($true != (X104 @ X106)) & ($true = (var @ X106))) => (((X104 @ (sK505 @ X104)) != $true) & ((var @ (sK505 @ X104)) = $true)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f498,plain,(
% 9.64/2.82 ! [X104 : term > $o] : (? [X107 : term] : (((X104 @ (lam @ X107)) != $true) & ($true = (X104 @ X107))) => (((X104 @ (lam @ (sK506 @ X104))) != $true) & ($true = (X104 @ (sK506 @ X104)))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f499,plain,(
% 9.64/2.82 ? [X108 : term > $o,X109 : term] : (! [X110 : term] : (($true = (X108 @ X110)) | ((var @ X110) != $true)) & ($true != (X108 @ X109)) & ! [X111 : term] : (($true = (X108 @ (lam @ X111))) | ((X108 @ X111) != $true)) & ((sP41 @ X108) != $true)) => (! [X110 : term] : (((sK507 @ X110) = $true) | ((var @ X110) != $true)) & ($true != (sK507 @ sK508)) & ! [X111 : term] : (((sK507 @ (lam @ X111)) = $true) | ($true != (sK507 @ X111))) & ((sP41 @ sK507) != $true))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f500,plain,(
% 9.64/2.82 ! [X112 : term] : (? [X113 : subst > term > term] : ! [X114 : subst,X115 : term] : ((sub @ X112 @ (push @ X115 @ X114)) = (X113 @ X114 @ X115)) => ! [X115 : term,X114 : subst] : ((sub @ X112 @ (push @ X115 @ X114)) = (sK509 @ X112 @ X114 @ X115)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f501,plain,(
% 9.64/2.82 ? [X116 : term] : ! [X117 : subst > term > term] : ? [X118 : subst,X119 : term] : ((sub @ X116 @ (push @ X119 @ X118)) != (X117 @ X118 @ X119)) => ! [X117 : subst > term > term] : ? [X119 : term,X118 : subst] : ((sub @ sK510 @ (push @ X119 @ X118)) != (X117 @ X118 @ X119))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f502,plain,(
% 9.64/2.82 ! [X117 : subst > term > term] : (? [X119 : term,X118 : subst] : ((sub @ sK510 @ (push @ X119 @ X118)) != (X117 @ X118 @ X119)) => ((X117 @ (sK511 @ X117) @ (sK512 @ X117)) != (sub @ sK510 @ (push @ (sK512 @ X117) @ (sK511 @ X117)))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f503,plain,(
% 9.64/2.82 ? [X120 : subst] : ((comp @ X120 @ id) != X120) => ((comp @ sK513 @ id) != sK513)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f504,plain,(
% 9.64/2.82 ? [X123 : term,X124 : term,X125 : term,X126 : term] : ((X123 != X126) & ((ap @ X126 @ X124) = (ap @ X123 @ X125))) => ((sK517 != sK514) & ((ap @ sK514 @ sK516) = (ap @ sK517 @ sK515)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f505,plain,(
% 9.64/2.82 ? [X131 : term > $o,X132 : term > $o,X133 : term,X134 : subst] : (($true = (X131 @ X133)) & ($true != (X132 @ one)) & ! [X135 : term] : ((X132 @ X135) = (X131 @ (sub @ X135 @ (push @ X133 @ X134))))) => (((sK518 @ sK520) = $true) & ($true != (sK519 @ one)) & ! [X135 : term] : ((sK519 @ X135) = (sK518 @ (sub @ X135 @ (push @ sK520 @ sK521)))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f506,plain,(
% 9.64/2.82 ! [X138 : term > $o,X139 : term > $o,X140 : term,X141 : subst] : (? [X142 : term] : ((X138 @ (sub @ X142 @ (push @ X140 @ X141))) != (X139 @ X142)) => ((X138 @ (sub @ (sK522 @ X141 @ X140 @ X139 @ X138) @ (push @ X140 @ X141))) != (X139 @ (sK522 @ X141 @ X140 @ X139 @ X138))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f507,plain,(
% 9.64/2.82 ? [X143 : subst,X144 : term] : ((sub @ one @ (push @ X144 @ X143)) != X144) => (sK524 != (sub @ one @ (push @ sK524 @ sK523)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f508,plain,(
% 9.64/2.82 ? [X145 : term,X146 : term] : ((X145 != X146) & ((lam @ X146) = (lam @ X145))) => ((sK525 != sK526) & ((lam @ sK526) = (lam @ sK525)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f509,plain,(
% 9.64/2.82 ? [X151 : term,X152 : term] : (((sub @ X152 @ sh) = (sub @ X151 @ sh)) & (X151 != X152)) => (((sub @ sK528 @ sh) = (sub @ sK527 @ sh)) & (sK527 != sK528))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f510,plain,(
% 9.64/2.82 ? [X154 : term] : ((sub @ X154 @ id) != X154) => (sK529 != (sub @ sK529 @ id))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f511,plain,(
% 9.64/2.82 ? [X155 : term] : ((sub @ X155 @ id) != X155) => ((sub @ sK530 @ id) != sK530)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f512,plain,(
% 9.64/2.82 ? [X158 : term] : ((sub @ X158 @ id) != X158) => ((sub @ sK531 @ id) != sK531)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f513,plain,(
% 9.64/2.82 ? [X163 : term,X164 : term,X165 : term,X166 : term] : (((ap @ (sub @ X163 @ id) @ X164) = (ap @ (sub @ X165 @ id) @ X166)) & (X164 != X166)) => (((ap @ (sub @ sK532 @ id) @ sK533) = (ap @ (sub @ sK534 @ id) @ sK535)) & (sK533 != sK535))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f514,plain,(
% 9.64/2.82 ? [X167 : term,X168 : term,X169 : term,X170 : term] : ((X167 != X170) & ((ap @ X168 @ X167) = (ap @ X169 @ X170))) => ((sK539 != sK536) & ((ap @ sK538 @ sK539) = (ap @ sK537 @ sK536)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f515,plain,(
% 9.64/2.82 ? [X179 : term,X180 : term,X181 : term,X182 : term] : ((X180 != X181) & ((ap @ (sub @ X182 @ id) @ X181) = (ap @ (sub @ X179 @ id) @ X180))) => ((sK541 != sK542) & ((ap @ (sub @ sK540 @ id) @ sK541) = (ap @ (sub @ sK543 @ id) @ sK542)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f516,plain,(
% 9.64/2.82 ! [X183 : term] : (? [X184 : d_term] : ((d2term @ X184) = X183) => ((d2term @ (sK544 @ X183)) = X183))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f517,plain,(
% 9.64/2.82 ? [X185 : term > $o,X186 : term > $o,X187 : term,X188 : subst] : (($true = (X185 @ X187)) & ($true != (X186 @ one)) & ! [X189 : term] : ((X186 @ X189) = (X185 @ (sub @ X189 @ (push @ X187 @ X188))))) => (($true = (sK545 @ sK547)) & ((sK546 @ one) != $true) & ! [X189 : term] : ((sK545 @ (sub @ X189 @ (push @ sK547 @ sK548))) = (sK546 @ X189)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f518,plain,(
% 9.64/2.82 ! [X190 : term > $o,X191 : term > $o,X192 : term,X193 : subst] : (? [X194 : term] : ((X190 @ (sub @ X194 @ (push @ X192 @ X193))) != (X191 @ X194)) => ((X190 @ (sub @ (sK549 @ X193 @ X192 @ X191 @ X190) @ (push @ X192 @ X193))) != (X191 @ (sK549 @ X193 @ X192 @ X191 @ X190))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f519,plain,(
% 9.64/2.82 ? [X197 : subst] : ((comp @ X197 @ id) != X197) => ((comp @ sK550 @ id) != sK550)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f520,plain,(
% 9.64/2.82 ? [X198 : subst] : ((comp @ id @ X198) != X198) => ((comp @ id @ sK551) != sK551)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f521,plain,(
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% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f522,plain,(
% 9.64/2.82 ! [X200 : term > $o] : (? [X201 : term] : ((sK552 @ id @ X201 @ id) != (X200 @ X201)) => ((X200 @ (sK553 @ X200)) != (sK552 @ id @ (sK553 @ X200) @ id)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f523,plain,(
% 9.64/2.82 ! [X202 : subst > term > subst > $o] : (? [X203 : term > $o] : ! [X204 : term] : ((X202 @ id @ X204 @ id) = (X203 @ X204)) => ! [X204 : term] : ((sK554 @ X202 @ X204) = (X202 @ id @ X204 @ id)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f524,plain,(
% 9.64/2.82 ? [X205 : term] : ! [X206 : subst > term > term] : ((X206 @ sh @ one) != (sub @ X205 @ (push @ one @ sh))) => ! [X206 : subst > term > term] : ((X206 @ sh @ one) != (sub @ sK555 @ (push @ one @ sh)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f525,plain,(
% 9.64/2.82 ! [X207 : term] : (? [X208 : subst > term > term] : ((sub @ X207 @ (push @ one @ sh)) = (X208 @ sh @ one)) => ((sub @ X207 @ (push @ one @ sh)) = (sK556 @ X207 @ sh @ one)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f526,plain,(
% 9.64/2.82 ? [X210 : subst > term > subst > $o,X211 : term] : (! [X212 : subst > term > term] : (? [X213 : term] : (($true = (X210 @ id @ X213 @ id)) & ($true != (X210 @ id @ (X212 @ id @ X213) @ id))) | ? [X214 : term,X215 : subst,X216 : subst] : ((sub @ (X212 @ X216 @ X214) @ X215) != (X212 @ (comp @ X216 @ X215) @ (sub @ X214 @ X215))) | ($true = (X210 @ id @ (lam @ (X212 @ sh @ one)) @ id))) & ($true != (sP4 @ X210)) & ((X210 @ id @ X211 @ id) != $true)) => (! [X212 : subst > term > term] : (? [X213 : term] : (((sK557 @ id @ X213 @ id) = $true) & ((sK557 @ id @ (X212 @ id @ X213) @ id) != $true)) | ? [X214 : term,X215 : subst,X216 : subst] : ((sub @ (X212 @ X216 @ X214) @ X215) != (X212 @ (comp @ X216 @ X215) @ (sub @ X214 @ X215))) | ($true = (sK557 @ id @ (lam @ (X212 @ sh @ one)) @ id))) & ($true != (sP4 @ sK557)) & ((sK557 @ id @ sK558 @ id) != $true))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f527,plain,(
% 9.64/2.82 ! [X212 : subst > term > term] : (? [X213 : term] : (((sK557 @ id @ X213 @ id) = $true) & ((sK557 @ id @ (X212 @ id @ X213) @ id) != $true)) => (($true = (sK557 @ id @ (sK559 @ X212) @ id)) & ($true != (sK557 @ id @ (X212 @ id @ (sK559 @ X212)) @ id))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f528,plain,(
% 9.64/2.82 ! [X212 : subst > term > term] : (? [X214 : term,X215 : subst,X216 : subst] : ((sub @ (X212 @ X216 @ X214) @ X215) != (X212 @ (comp @ X216 @ X215) @ (sub @ X214 @ X215))) => ((sub @ (X212 @ (sK562 @ X212) @ (sK560 @ X212)) @ (sK561 @ X212)) != (X212 @ (comp @ (sK562 @ X212) @ (sK561 @ X212)) @ (sub @ (sK560 @ X212) @ (sK561 @ X212)))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f529,plain,(
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% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f530,plain,(
% 9.64/2.82 ! [X224 : term] : (? [X225 : subst > term > term] : ((X225 @ sh @ one) = (sub @ X224 @ (push @ one @ sh))) => ((sK564 @ X224 @ sh @ one) = (sub @ X224 @ (push @ one @ sh))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f531,plain,(
% 9.64/2.82 ? [X226 : term] : ! [X227 : subst > term > term] : ((X227 @ sh @ one) != (sub @ X226 @ (push @ one @ sh))) => ! [X227 : subst > term > term] : ((X227 @ sh @ one) != (sub @ sK565 @ (push @ one @ sh)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
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% 9.64/2.82 ? [X230 : subst,X231 : term] : ((sub @ one @ (push @ X231 @ X230)) != X231) => ((sub @ one @ (push @ sK567 @ sK566)) != sK567)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
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% 9.64/2.82 ? [X233 : term] : ((sub @ X233 @ id) != X233) => ((sub @ sK568 @ id) != sK568)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f534,plain,(
% 9.64/2.82 ? [X236 : subst] : ((comp @ X236 @ id) != X236) => ((comp @ sK569 @ id) != sK569)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f535,plain,(
% 9.64/2.82 ? [X237 : subst] : ((comp @ id @ X237) != X237) => ((comp @ id @ sK570) != sK570)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f536,plain,(
% 9.64/2.82 ? [X242 : subst] : ((comp @ id @ X242) != X242) => ((comp @ id @ sK571) != sK571)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f537,plain,(
% 9.64/2.82 ? [X243 : subst] : ((comp @ X243 @ id) != X243) => ((comp @ sK572 @ id) != sK572)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f538,plain,(
% 9.64/2.82 ? [X244 : subst] : ((push @ (sub @ one @ X244) @ (comp @ sh @ X244)) != X244) => (sK573 != (push @ (sub @ one @ sK573) @ (comp @ sh @ sK573)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f539,plain,(
% 9.64/2.82 ? [X246 : term,X247 : subst] : ((comp @ sh @ (push @ X246 @ X247)) != X247) => ((comp @ sh @ (push @ sK574 @ sK575)) != sK575)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f540,plain,(
% 9.64/2.82 ? [X254 : term > $o,X255 : term,X256 : subst,X257 : term > $o,X258 : term] : (($true != (X254 @ (sub @ X258 @ (push @ X255 @ X256)))) & ($true = (X257 @ X258))) => (((sK576 @ (sub @ sK580 @ (push @ sK577 @ sK578))) != $true) & ($true = (sK579 @ sK580)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f541,plain,(
% 9.64/2.82 ? [X259 : subst,X260 : term] : ((sub @ one @ (push @ X260 @ X259)) != X260) => ((sub @ one @ (push @ sK582 @ sK581)) != sK582)),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f542,plain,(
% 9.64/2.82 ? [X261 : term,X262 : subst] : ((comp @ sh @ (push @ X261 @ X262)) != X262) => (sK584 != (comp @ sh @ (push @ sK583 @ sK584)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f543,plain,(
% 9.64/2.82 ? [X268 : term,X269 : term,X270 : subst > term > term] : (((lam @ (X270 @ sh @ one)) = (ap @ (sub @ X269 @ id) @ X268)) & ! [X271 : subst,X272 : term,X273 : subst] : ((sub @ (X270 @ X273 @ X272) @ X271) = (X270 @ (comp @ X273 @ X271) @ (sub @ X272 @ X271)))) => (((ap @ (sub @ sK586 @ id) @ sK585) = (lam @ (sK587 @ sh @ one))) & ! [X273 : subst,X272 : term,X271 : subst] : ((sK587 @ (comp @ X273 @ X271) @ (sub @ X272 @ X271)) = (sub @ (sK587 @ X273 @ X272) @ X271)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f544,plain,(
% 9.64/2.82 ! [X276 : subst > term > term] : (? [X277 : subst,X278 : term,X279 : subst] : ((X276 @ (comp @ X279 @ X277) @ (sub @ X278 @ X277)) != (sub @ (X276 @ X279 @ X278) @ X277)) => ((sub @ (X276 @ (sK590 @ X276) @ (sK589 @ X276)) @ (sK588 @ X276)) != (X276 @ (comp @ (sK590 @ X276) @ (sK588 @ X276)) @ (sub @ (sK589 @ X276) @ (sK588 @ X276)))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f473,plain,(
% 9.64/2.82 (! [X0 : term] : ? [X1 : subst > term > term] : ! [X2 : term,X3 : subst] : ((X1 @ X3 @ X2) = (sub @ X0 @ (push @ X2 @ X3))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X4 : term] : ! [X5 : subst > term > term] : ? [X6 : term,X7 : subst] : ((X5 @ X7 @ X6) != (sub @ X4 @ (push @ X6 @ X7)))) & ((apinj2 = $true) | ? [X8 : term,X9 : term,X10 : term,X11 : term] : (((ap @ X8 @ X11) = (ap @ X9 @ X10)) & (X10 != X11))) & (! [X12 : term,X13 : term,X14 : term,X15 : term] : (((ap @ X13 @ X14) != (ap @ X12 @ X15)) | (X14 = X15)) | (apinj2 != $true)) & (! [X16 : subst] : ((comp @ id @ X16) = X16) | (axidl != $true)) & ((axidl = $true) | ? [X17 : subst] : ((comp @ id @ X17) != X17)) & ((sP58 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP58 != $true)) & ! [X18 : subst] : ? [X19 : d_subst] : ((d2subst @ X19) = X18) & (pushprop_lem2v2_lthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X20 : subst > term > subst > $o] : ? [X21 : term > $o] : ! [X22 : term] : ((X20 @ id @ X22 @ id) = (X21 @ X22)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X23 : subst > term > subst > $o] : ! [X24 : term > $o] : ? [X25 : term] : ((X24 @ X25) != (X23 @ id @ X25 @ id))) & ((hoasinduction_lem2v2 = $true) | ($true != sP56)) & (($true = sP56) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP54)) & (($true = sP54) | (hoasinduction_lem3v2 != $true)) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) | (! [X26 : term,X27 : term,X28 : term,X29 : term] : ((X26 = X27) | ((ap @ X26 @ X29) != (ap @ X27 @ X28))) & ! [X30 : term] : ((sub @ X30 @ id) = X30) & ? [X31 : term,X32 : term,X33 : term,X34 : term] : (((ap @ (sub @ X33 @ id) @ X31) = (ap @ (sub @ X34 @ id) @ X32)) & (X33 != X34)))) & (? [X35 : term,X36 : term,X37 : term,X38 : term] : ((X35 != X36) & ((ap @ X35 @ X38) = (ap @ X36 @ X37))) | ? [X39 : term] : ((sub @ X39 @ id) != X39) | ! [X40 : term,X41 : term,X42 : term,X43 : term] : (((ap @ (sub @ X43 @ id) @ X41) != (ap @ (sub @ X42 @ id) @ X40)) | (X42 = X43)) | (hoasapinj1_lthm != $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & (hoaslamnotvar_gthm = $true) & ((sP52 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP52 != $true)) & (induction2_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoaslamnotap_gthm = $true) & (! [X44 : term > $o,X45 : term,X46 : subst,X47 : term > $o,X48 : term] : (($true = (X47 @ (sub @ X45 @ (push @ X48 @ X46)))) | ($true != (X44 @ X45))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X49 : term > $o,X50 : term,X51 : subst,X52 : term > $o,X53 : term] : (($true != (X52 @ (sub @ X50 @ (push @ X53 @ X51)))) & ($true = (X49 @ X50)))) & ((hoaslaminj = $true) | ? [X54 : subst > term > term,X55 : subst > term > term,X56 : subst,X57 : term] : (((X55 @ X56 @ X57) != (X54 @ X56 @ X57)) & ! [X58 : subst,X59 : subst,X60 : term] : ((sub @ (X55 @ X59 @ X60) @ X58) = (X55 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) & ((lam @ (X55 @ sh @ one)) = (lam @ (X54 @ sh @ one))) & ! [X61 : term,X62 : subst,X63 : subst] : ((sub @ (X54 @ X62 @ X61) @ X63) = (X54 @ (comp @ X62 @ X63) @ (sub @ X61 @ X63))))) & (! [X64 : subst > term > term,X65 : subst > term > term,X66 : subst,X67 : term] : (((X64 @ X66 @ X67) = (X65 @ X66 @ X67)) | ? [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X65 @ X69 @ X70) @ X68) != (X65 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) | ((lam @ (X64 @ sh @ one)) != (lam @ (X65 @ sh @ one))) | ? [X71 : term,X72 : subst,X73 : subst] : ((X64 @ (comp @ X72 @ X73) @ (sub @ X71 @ X73)) != (sub @ (X64 @ X72 @ X71) @ X73))) | (hoaslaminj != $true)) & ((pushprop_lem0 = $true) | ? [X74 : term,X75 : term > $o,X76 : subst] : ! [X77 : term > $o] : ? [X78 : term] : ((X77 @ X78) != (X75 @ (sub @ X78 @ (push @ X74 @ X76))))) & (! [X79 : term,X80 : term > $o,X81 : subst] : ? [X82 : term > $o] : ! [X83 : term] : ((X82 @ X83) = (X80 @ (sub @ X83 @ (push @ X79 @ X81)))) | (pushprop_lem0 != $true)) & (! [X84 : term,X85 : term,X86 : term,X87 : term] : (((ap @ (sub @ X84 @ id) @ X86) != (ap @ (sub @ X85 @ id) @ X87)) | (X84 = X85)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X88 : term,X89 : term,X90 : term,X91 : term] : (((ap @ (sub @ X88 @ id) @ X90) = (ap @ (sub @ X89 @ id) @ X91)) & (X88 != X89))) & ! [X92 : d_subst,X93 : d_subst] : ((X92 = X93) | ((d2subst @ X93) != (d2subst @ X92))) & ((pushprop_lem0_lthm = $true) | ? [X94 : term > $o,X95 : subst,X96 : term] : ! [X97 : term > $o] : ? [X98 : term] : ((X94 @ (sub @ X98 @ (push @ X96 @ X95))) != (X97 @ X98))) & (! [X99 : term > $o,X100 : subst,X101 : term] : ? [X102 : term > $o] : ! [X103 : term] : ((X99 @ (sub @ X103 @ (push @ X101 @ X100))) = (X102 @ X103)) | (pushprop_lem0_lthm != $true)) & (($true = sP47) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP47)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP44)) & (($true = sP44) | (hoasinduction_lem3_lthm != $true)) & (! [X104 : term > $o,X105 : term] : (? [X106 : term] : (($true != (X104 @ X106)) & ($true = (var @ X106))) | ($true = (X104 @ X105)) | ? [X107 : term] : (((X104 @ (lam @ X107)) != $true) & ($true = (X104 @ X107))) | ($true = (sP41 @ X104))) | (induction != $true)) & ((induction = $true) | ? [X108 : term > $o,X109 : term] : (! [X110 : term] : (($true = (X108 @ X110)) | ((var @ X110) != $true)) & ($true != (X108 @ X109)) & ! [X111 : term] : (($true = (X108 @ (lam @ X111))) | ((X108 @ X111) != $true)) & ((sP41 @ X108) != $true))) & (hoasinduction_lem1 = $true) & (($true = sP40) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP40)) & ((hoaslaminj_lthm = $true) | (sP34 != $true)) & ((sP34 = $true) | (hoaslaminj_lthm != $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (($true = sP33) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP33)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (hoasapnotvar_lthm = $true) & (($true = sP31) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP31)) & (axapp = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3aa != $true)) & (! [X112 : term] : ? [X113 : subst > term > term] : ! [X114 : subst,X115 : term] : ((sub @ X112 @ (push @ X115 @ X114)) = (X113 @ X114 @ X115)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X116 : term] : ! [X117 : subst > term > term] : ? [X118 : subst,X119 : term] : ((sub @ X116 @ (push @ X119 @ X118)) != (X117 @ X118 @ X119))) & ((induction2lem_lthm = $true) | ($true != sP28)) & (($true = sP28) | (induction2lem_lthm != $true)) & (axvarshift = $true) & ((axidr = $true) | ? [X120 : subst] : ((comp @ X120 @ id) != X120)) & (! [X121 : subst] : ((comp @ X121 @ id) = X121) | (axidr != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (lamnotvar = $true) & ((hoasinduction_lem3a = $true) | (sP24 != $true)) & ((sP24 = $true) | (hoasinduction_lem3a != $true)) & (hoasinduction_lem3v2_gthm = $true) & ! [X122 : d_subst] : (d_id = X122) & ((apinj1 = $true) | ? [X123 : term,X124 : term,X125 : term,X126 : term] : ((X123 != X126) & ((ap @ X126 @ X124) = (ap @ X123 @ X125)))) & (! [X127 : term,X128 : term,X129 : term,X130 : term] : ((X127 = X130) | ((ap @ X127 @ X129) != (ap @ X130 @ X128))) | (apinj1 != $true)) & ((induction2_lthm = $true) | ($true != sP23)) & (($true = sP23) | (induction2_lthm != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP19)) & (($true = sP19) | (hoasinduction_lem3aaa != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X131 : term > $o,X132 : term > $o,X133 : term,X134 : subst] : (($true = (X131 @ X133)) & ($true != (X132 @ one)) & ! [X135 : term] : ((X132 @ X135) = (X131 @ (sub @ X135 @ (push @ X133 @ X134))))) & ! [X136 : subst,X137 : term] : ((sub @ one @ (push @ X137 @ X136)) = X137))) & (! [X138 : term > $o,X139 : term > $o,X140 : term,X141 : subst] : (($true != (X138 @ X140)) | ((X139 @ one) = $true) | ? [X142 : term] : ((X138 @ (sub @ X142 @ (push @ X140 @ X141))) != (X139 @ X142))) | ? [X143 : subst,X144 : term] : ((sub @ one @ (push @ X144 @ X143)) != X144) | (pushprop_lem1v2_lthm != $true)) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction_lem3v2a_lthm = $true) | (sP18 != $true)) & ((sP18 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((laminj = $true) | ? [X145 : term,X146 : term] : ((X145 != X146) & ((lam @ X146) = (lam @ X145)))) & (! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148))) | (laminj != $true)) & (! [X149 : term,X150 : term] : (((sub @ X149 @ sh) != (sub @ X150 @ sh)) | (X149 = X150)) | (shinj != $true)) & ((shinj = $true) | ? [X151 : term,X152 : term] : (((sub @ X152 @ sh) = (sub @ X151 @ sh)) & (X151 != X152))) & (hoasapnotvar_gthm = $true) & (! [X153 : term] : ((sub @ X153 @ id) = X153) | ? [X154 : term] : ((sub @ X154 @ id) != X154) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X155 : term] : ((sub @ X155 @ id) != X155) & ! [X156 : term] : ((sub @ X156 @ id) = X156))) & (! [X157 : term] : ((sub @ X157 @ id) = X157) | (termmset != $true)) & ((termmset = $true) | ? [X158 : term] : ((sub @ X158 @ id) != X158)) & (apnotvar = $true) & ((d2subst @ d_id) = id) & ((hoasinduction_lthm = $true) | (sP17 != $true)) & ((sP17 = $true) | (hoasinduction_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lthm_orig = $true) & ((hoasapinj2_lthm = $true) | (! [X159 : term,X160 : term,X161 : term,X162 : term] : ((X159 = X162) | ((ap @ X161 @ X162) != (ap @ X160 @ X159))) & ? [X163 : term,X164 : term,X165 : term,X166 : term] : (((ap @ (sub @ X163 @ id) @ X164) = (ap @ (sub @ X165 @ id) @ X166)) & (X164 != X166)))) & (? [X167 : term,X168 : term,X169 : term,X170 : term] : ((X167 != X170) & ((ap @ X168 @ X167) = (ap @ X169 @ X170))) | ! [X171 : term,X172 : term,X173 : term,X174 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X171 @ id) @ X172)) | (X172 = X174)) | (hoasapinj2_lthm != $true)) & (hoaslamnotvar = $true) & (pushprop_lthm = $true) & ((sP8 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP8 != $true)) & (pushprop_lem2v2_gthm = $true) & ((sP7 = $true) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (sP7 != $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (axassoc = $true) & (axabs = $true) & (lamnotap != $true) & (! [X175 : term,X176 : term,X177 : term,X178 : term] : ((X176 = X177) | ((ap @ (sub @ X178 @ id) @ X177) != (ap @ (sub @ X175 @ id) @ X176))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X179 : term,X180 : term,X181 : term,X182 : term] : ((X180 != X181) & ((ap @ (sub @ X182 @ id) @ X181) = (ap @ (sub @ X179 @ id) @ X180)))) & ! [X183 : term] : ? [X184 : d_term] : ((d2term @ X184) = X183) & (pushprop = $true) & (hoasinduction_lem3_gthm = $true) & ((pushprop_lem1v2 = $true) | ? [X185 : term > $o,X186 : term > $o,X187 : term,X188 : subst] : (($true = (X185 @ X187)) & ($true != (X186 @ one)) & ! [X189 : term] : ((X186 @ X189) = (X185 @ (sub @ X189 @ (push @ X187 @ X188)))))) & (! [X190 : term > $o,X191 : term > $o,X192 : term,X193 : subst] : (((X190 @ X192) != $true) | ($true = (X191 @ one)) | ? [X194 : term] : ((X190 @ (sub @ X194 @ (push @ X192 @ X193))) != (X191 @ X194))) | (pushprop_lem1v2 != $true)) & ((! [X195 : subst] : ((comp @ X195 @ id) = X195) & ! [X196 : subst] : ((comp @ id @ X196) = X196)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X197 : subst] : ((comp @ X197 @ id) != X197) | ? [X198 : subst] : ((comp @ id @ X198) != X198)) & ((hoasinduction_lem0 = $true) | ? [X199 : subst > term > subst > $o] : ! [X200 : term > $o] : ? [X201 : term] : ((X200 @ X201) != (X199 @ id @ X201 @ id))) & (! [X202 : subst > term > subst > $o] : ? [X203 : term > $o] : ! [X204 : term] : ((X202 @ id @ X204 @ id) = (X203 @ X204)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP5)) & (($true = sP5) | (hoasinduction_lem3aa_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3b_lthm = $true) | ? [X205 : term] : ! [X206 : subst > term > term] : ((X206 @ sh @ one) != (sub @ X205 @ (push @ one @ sh)))) & (! [X207 : term] : ? [X208 : subst > term > term] : ((sub @ X207 @ (push @ one @ sh)) = (X208 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & (ulamvarsh = $true) & ! [X209 : d_term] : (d_one = X209) & (axclos = $true) & (substmonoid_gthm = $true) & (induction2lem_gthm = $true) & (pushprop_lem2v2 = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X210 : subst > term > subst > $o,X211 : term] : (! [X212 : subst > term > term] : (? [X213 : term] : (($true = (X210 @ id @ X213 @ id)) & ($true != (X210 @ id @ (X212 @ id @ X213) @ id))) | ? [X214 : term,X215 : subst,X216 : subst] : ((sub @ (X212 @ X216 @ X214) @ X215) != (X212 @ (comp @ X216 @ X215) @ (sub @ X214 @ X215))) | ($true = (X210 @ id @ (lam @ (X212 @ sh @ one)) @ id))) & ($true != (sP4 @ X210)) & ((X210 @ id @ X211 @ id) != $true))) & (! [X217 : subst > term > subst > $o,X218 : term] : (? [X219 : subst > term > term] : (! [X220 : term] : (((X217 @ id @ X220 @ id) != $true) | ($true = (X217 @ id @ (X219 @ id @ X220) @ id))) & ! [X221 : term,X222 : subst,X223 : subst] : ((sub @ (X219 @ X223 @ X221) @ X222) = (X219 @ (comp @ X223 @ X222) @ (sub @ X221 @ X222))) & ($true != (X217 @ id @ (lam @ (X219 @ sh @ one)) @ id))) | ((sP4 @ X217) = $true) | ($true = (X217 @ id @ X218 @ id))) | (hoasinduction_no_psi_cond != $true)) & (axmap = $true) & ((hoasinduction_lem2 = $true) | ($true != sP3)) & (($true = sP3) | (hoasinduction_lem2 != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (! [X224 : term] : ? [X225 : subst > term > term] : ((X225 @ sh @ one) = (sub @ X224 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X226 : term] : ! [X227 : subst > term > term] : ((X227 @ sh @ one) != (sub @ X226 @ (push @ one @ sh)))) & (one = (d2term @ d_one)) & (termmset_gthm = $true) & (! [X228 : subst,X229 : term] : ((sub @ one @ (push @ X229 @ X228)) = X229) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X230 : subst,X231 : term] : ((sub @ one @ (push @ X231 @ X230)) != X231)) & (ulamvarind = $true) & (hoasapinj1_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X232 : term] : ((sub @ X232 @ id) = X232) | (axvarid != $true)) & ((axvarid = $true) | ? [X233 : term] : ((sub @ X233 @ id) != X233)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ! [X234 : d_term,X235 : d_term] : (((d2term @ X234) != (d2term @ X235)) | (X234 = X235)) & (? [X236 : subst] : ((comp @ X236 @ id) != X236) | ? [X237 : subst] : ((comp @ id @ X237) != X237) | (! [X238 : subst] : ((comp @ id @ X238) = X238) & ! [X239 : subst] : ((comp @ X239 @ id) = X239)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X240 : subst] : ((comp @ X240 @ id) = X240) & ! [X241 : subst] : ((comp @ id @ X241) = X241) & (? [X242 : subst] : ((comp @ id @ X242) != X242) | ? [X243 : subst] : ((comp @ X243 @ id) != X243)))) & (ulamvar1 != $true) & (pushprop_gthm = $true) & ((d2subst @ d_id) = sh) & ((axscons = $true) | ? [X244 : subst] : ((push @ (sub @ one @ X244) @ (comp @ sh @ X244)) != X244)) & (! [X245 : subst] : ((push @ (sub @ one @ X245) @ (comp @ sh @ X245)) = X245) | (axscons != $true)) & ((axshiftcons = $true) | ? [X246 : term,X247 : subst] : ((comp @ sh @ (push @ X246 @ X247)) != X247)) & (! [X248 : term,X249 : subst] : ((comp @ sh @ (push @ X248 @ X249)) = X249) | (axshiftcons != $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP1) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP1)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X250 : subst,X251 : term] : ((sub @ one @ (push @ X251 @ X250)) = X251) & ! [X252 : term,X253 : subst] : ((comp @ sh @ (push @ X252 @ X253)) = X253) & ? [X254 : term > $o,X255 : term,X256 : subst,X257 : term > $o,X258 : term] : (($true != (X254 @ (sub @ X258 @ (push @ X255 @ X256)))) & ($true = (X257 @ X258))))) & (? [X259 : subst,X260 : term] : ((sub @ one @ (push @ X260 @ X259)) != X260) | ? [X261 : term,X262 : subst] : ((comp @ sh @ (push @ X261 @ X262)) != X262) | ! [X263 : term > $o,X264 : term,X265 : subst,X266 : term > $o,X267 : term] : (((X263 @ (sub @ X267 @ (push @ X264 @ X265))) = $true) | ($true != (X266 @ X267))) | (pushprop_lem1_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ((hoaslamnotap = $true) | ? [X268 : term,X269 : term,X270 : subst > term > term] : (((lam @ (X270 @ sh @ one)) = (ap @ (sub @ X269 @ id) @ X268)) & ! [X271 : subst,X272 : term,X273 : subst] : ((sub @ (X270 @ X273 @ X272) @ X271) = (X270 @ (comp @ X273 @ X271) @ (sub @ X272 @ X271))))) & (! [X274 : term,X275 : term,X276 : subst > term > term] : (((lam @ (X276 @ sh @ one)) != (ap @ (sub @ X275 @ id) @ X274)) | ? [X277 : subst,X278 : term,X279 : subst] : ((X276 @ (comp @ X279 @ X277) @ (sub @ X278 @ X277)) != (sub @ (X276 @ X279 @ X278) @ X277))) | (hoaslamnotap != $true))),
% 9.64/2.82 inference(rectify,[status(thm)],[f472])).
% 9.64/2.82 thf(f472,plain,(
% 9.64/2.82 (! [X138 : term] : ? [X139 : subst > term > term] : ! [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) = (X139 @ X141 @ X140)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X138 : term] : ! [X139 : subst > term > term] : ? [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) != (X139 @ X141 @ X140))) & ((apinj2 = $true) | ? [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) = (ap @ X79 @ X82)) & (X81 != X82))) & (! [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) != (ap @ X79 @ X82)) | (X81 = X82)) | (apinj2 != $true)) & (! [X324 : subst] : ((comp @ id @ X324) = X324) | (axidl != $true)) & ((axidl = $true) | ? [X324 : subst] : ((comp @ id @ X324) != X324)) & ((sP58 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP58 != $true)) & ! [X356 : subst] : ? [X357 : d_subst] : ((d2subst @ X357) = X356) & (pushprop_lem2v2_lthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X26 : subst > term > subst > $o] : ? [X27 : term > $o] : ! [X28 : term] : ((X27 @ X28) = (X26 @ id @ X28 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X26 : subst > term > subst > $o] : ! [X27 : term > $o] : ? [X28 : term] : ((X27 @ X28) != (X26 @ id @ X28 @ id))) & ((hoasinduction_lem2v2 = $true) | ($true != sP56)) & (($true = sP56) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP54)) & (($true = sP54) | (hoasinduction_lem3v2 != $true)) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) | (! [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 = X528) | ((ap @ X528 @ X529) != (ap @ X527 @ X530))) & ! [X526 : term] : ((sub @ X526 @ id) = X526) & ? [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) = (ap @ (sub @ X534 @ id) @ X532)) & (X533 != X534)))) & (? [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 != X528) & ((ap @ X528 @ X529) = (ap @ X527 @ X530))) | ? [X526 : term] : ((sub @ X526 @ id) != X526) | ! [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) != (ap @ (sub @ X534 @ id) @ X532)) | (X533 = X534)) | (hoasapinj1_lthm != $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & (hoaslamnotvar_gthm = $true) & ((sP52 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP52 != $true)) & (induction2_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoaslamnotap_gthm = $true) & (! [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (($true = (X475 @ (sub @ X473 @ (push @ X476 @ X474)))) | ($true != (X472 @ X473))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (($true != (X475 @ (sub @ X473 @ (push @ X476 @ X474)))) & ($true = (X472 @ X473)))) & ((hoaslaminj = $true) | ? [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (((X144 @ X146 @ X147) != (X145 @ X146 @ X147)) & ! [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) = (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) & ((lam @ (X144 @ sh @ one)) = (lam @ (X145 @ sh @ one))) & ! [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) = (sub @ (X144 @ X149 @ X148) @ X150)))) & (! [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (((X144 @ X146 @ X147) = (X145 @ X146 @ X147)) | ? [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) != (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) | ((lam @ (X144 @ sh @ one)) != (lam @ (X145 @ sh @ one))) | ? [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) != (sub @ (X144 @ X149 @ X148) @ X150))) | (hoaslaminj != $true)) & ((pushprop_lem0 = $true) | ? [X333 : term,X334 : term > $o,X335 : subst] : ! [X336 : term > $o] : ? [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) != (X336 @ X337))) & (! [X333 : term,X334 : term > $o,X335 : subst] : ? [X336 : term > $o] : ! [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) = (X336 @ X337)) | (pushprop_lem0 != $true)) & (! [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) != (ap @ (sub @ X508 @ id) @ X510)) | (X507 = X508)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) = (ap @ (sub @ X508 @ id) @ X510)) & (X507 != X508))) & ! [X338 : d_subst,X339 : d_subst] : ((X338 = X339) | ((d2subst @ X338) != (d2subst @ X339))) & ((pushprop_lem0_lthm = $true) | ? [X535 : term > $o,X536 : subst,X537 : term] : ! [X538 : term > $o] : ? [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) != (X538 @ X539))) & (! [X535 : term > $o,X536 : subst,X537 : term] : ? [X538 : term > $o] : ! [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) = (X538 @ X539)) | (pushprop_lem0_lthm != $true)) & (($true = sP47) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP47)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP44)) & (($true = sP44) | (hoasinduction_lem3_lthm != $true)) & (! [X230 : term > $o,X231 : term] : (? [X232 : term] : (($true != (X230 @ X232)) & ((var @ X232) = $true)) | ((X230 @ X231) = $true) | ? [X235 : term] : (($true != (X230 @ (lam @ X235))) & ($true = (X230 @ X235))) | ($true = (sP41 @ X230))) | (induction != $true)) & ((induction = $true) | ? [X230 : term > $o,X231 : term] : (! [X232 : term] : (($true = (X230 @ X232)) | ((var @ X232) != $true)) & ((X230 @ X231) != $true) & ! [X235 : term] : (($true = (X230 @ (lam @ X235))) | ($true != (X230 @ X235))) & ($true != (sP41 @ X230)))) & (hoasinduction_lem1 = $true) & (($true = sP40) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP40)) & ((hoaslaminj_lthm = $true) | (sP34 != $true)) & ((sP34 = $true) | (hoaslaminj_lthm != $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (($true = sP33) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP33)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (hoasapnotvar_lthm = $true) & (($true = sP31) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP31)) & (axapp = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3aa != $true)) & (! [X391 : term] : ? [X392 : subst > term > term] : ! [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) = (X392 @ X393 @ X394)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X391 : term] : ! [X392 : subst > term > term] : ? [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) != (X392 @ X393 @ X394))) & ((induction2lem_lthm = $true) | ($true != sP28)) & (($true = sP28) | (induction2lem_lthm != $true)) & (axvarshift = $true) & ((axidr = $true) | ? [X416 : subst] : ((comp @ X416 @ id) != X416)) & (! [X416 : subst] : ((comp @ X416 @ id) = X416) | (axidr != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (lamnotvar = $true) & ((hoasinduction_lem3a = $true) | (sP24 != $true)) & ((sP24 = $true) | (hoasinduction_lem3a != $true)) & (hoasinduction_lem3v2_gthm = $true) & ! [X417 : d_subst] : (d_id = X417) & ((apinj1 = $true) | ? [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 != X239) & ((ap @ X236 @ X238) = (ap @ X239 @ X237)))) & (! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 = X239) | ((ap @ X236 @ X238) != (ap @ X239 @ X237))) | (apinj1 != $true)) & ((induction2_lthm = $true) | ($true != sP23)) & (($true = sP23) | (induction2_lthm != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP19)) & (($true = sP19) | (hoasinduction_lem3aaa != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (((X479 @ X481) = $true) & ($true != (X480 @ one)) & ! [X483 : term] : ((X480 @ X483) = (X479 @ (sub @ X483 @ (push @ X481 @ X482))))) & ! [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) = X478))) & (! [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (((X479 @ X481) != $true) | ($true = (X480 @ one)) | ? [X483 : term] : ((X480 @ X483) != (X479 @ (sub @ X483 @ (push @ X481 @ X482))))) | ? [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) != X478) | (pushprop_lem1v2_lthm != $true)) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction_lem3v2a_lthm = $true) | (sP18 != $true)) & ((sP18 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((laminj = $true) | ? [X366 : term,X367 : term] : ((X366 != X367) & ((lam @ X366) = (lam @ X367)))) & (! [X366 : term,X367 : term] : ((X366 = X367) | ((lam @ X366) != (lam @ X367))) | (laminj != $true)) & (! [X77 : term,X78 : term] : (((sub @ X78 @ sh) != (sub @ X77 @ sh)) | (X77 = X78)) | (shinj != $true)) & ((shinj = $true) | ? [X77 : term,X78 : term] : (((sub @ X78 @ sh) = (sub @ X77 @ sh)) & (X77 != X78))) & (hoasapnotvar_gthm = $true) & (! [X6 : term] : ((sub @ X6 @ id) = X6) | ? [X5 : term] : ((sub @ X5 @ id) != X5) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X6 : term] : ((sub @ X6 @ id) != X6) & ! [X5 : term] : ((sub @ X5 @ id) = X5))) & (! [X109 : term] : ((sub @ X109 @ id) = X109) | (termmset != $true)) & ((termmset = $true) | ? [X109 : term] : ((sub @ X109 @ id) != X109)) & (apnotvar = $true) & ((d2subst @ d_id) = id) & ((hoasinduction_lthm = $true) | (sP17 != $true)) & ((sP17 = $true) | (hoasinduction_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lthm_orig = $true) & ((hoasapinj2_lthm = $true) | (! [X358 : term,X359 : term,X360 : term,X361 : term] : ((X358 = X361) | ((ap @ X360 @ X361) != (ap @ X359 @ X358))) & ? [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) = (ap @ (sub @ X364 @ id) @ X365)) & (X363 != X365)))) & (? [X358 : term,X359 : term,X360 : term,X361 : term] : ((X358 != X361) & ((ap @ X360 @ X361) = (ap @ X359 @ X358))) | ! [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X364 @ id) @ X365)) | (X363 = X365)) | (hoasapinj2_lthm != $true)) & (hoaslamnotvar = $true) & (pushprop_lthm = $true) & ((sP8 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP8 != $true)) & (pushprop_lem2v2_gthm = $true) & ((sP7 = $true) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (sP7 != $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (axassoc = $true) & (axabs = $true) & (lamnotap != $true) & (! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 = X354) | ((ap @ (sub @ X352 @ id) @ X353) != (ap @ (sub @ X355 @ id) @ X354))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 != X354) & ((ap @ (sub @ X352 @ id) @ X353) = (ap @ (sub @ X355 @ id) @ X354)))) & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & (pushprop = $true) & (hoasinduction_lem3_gthm = $true) & ((pushprop_lem1v2 = $true) | ? [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true = (X395 @ X397)) & ($true != (X396 @ one)) & ! [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) = (X396 @ X399)))) & (! [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true != (X395 @ X397)) | ($true = (X396 @ one)) | ? [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) != (X396 @ X399))) | (pushprop_lem1v2 != $true)) & ((! [X373 : subst] : ((comp @ X373 @ id) = X373) & ! [X372 : subst] : ((comp @ id @ X372) = X372)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X373 : subst] : ((comp @ X373 @ id) != X373) | ? [X372 : subst] : ((comp @ id @ X372) != X372)) & ((hoasinduction_lem0 = $true) | ? [X110 : subst > term > subst > $o] : ! [X111 : term > $o] : ? [X112 : term] : ((X110 @ id @ X112 @ id) != (X111 @ X112))) & (! [X110 : subst > term > subst > $o] : ? [X111 : term > $o] : ! [X112 : term] : ((X110 @ id @ X112 @ id) = (X111 @ X112)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP5)) & (($true = sP5) | (hoasinduction_lem3aa_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3b_lthm = $true) | ? [X1 : term] : ! [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) != (X2 @ sh @ one))) & (! [X1 : term] : ? [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) = (X2 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & (ulamvarsh = $true) & ! [X229 : d_term] : (d_one = X229) & (axclos = $true) & (substmonoid_gthm = $true) & (induction2lem_gthm = $true) & (pushprop_lem2v2 = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X202 : subst > term > subst > $o,X203 : term] : (! [X206 : subst > term > term] : (? [X207 : term] : (((X202 @ id @ X207 @ id) = $true) & ($true != (X202 @ id @ (X206 @ id @ X207) @ id))) | ? [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) != (sub @ (X206 @ X210 @ X208) @ X209)) | ($true = (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id))) & ($true != (sP4 @ X202)) & ((X202 @ id @ X203 @ id) != $true))) & (! [X202 : subst > term > subst > $o,X203 : term] : (? [X206 : subst > term > term] : (! [X207 : term] : (((X202 @ id @ X207 @ id) != $true) | ($true = (X202 @ id @ (X206 @ id @ X207) @ id))) & ! [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) = (sub @ (X206 @ X210 @ X208) @ X209)) & ($true != (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id))) | ($true = (sP4 @ X202)) | ((X202 @ id @ X203 @ id) = $true)) | (hoasinduction_no_psi_cond != $true)) & (axmap = $true) & ((hoasinduction_lem2 = $true) | ($true != sP3)) & (($true = sP3) | (hoasinduction_lem2 != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (! [X142 : term] : ? [X143 : subst > term > term] : ((X143 @ sh @ one) = (sub @ X142 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X142 : term] : ! [X143 : subst > term > term] : ((X143 @ sh @ one) != (sub @ X142 @ (push @ one @ sh)))) & (one = (d2term @ d_one)) & (termmset_gthm = $true) & (! [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) = X371) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) != X371)) & (ulamvarind = $true) & (hoasapinj1_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X511 : term] : ((sub @ X511 @ id) = X511) | (axvarid != $true)) & ((axvarid = $true) | ? [X511 : term] : ((sub @ X511 @ id) != X511)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ! [X368 : d_term,X369 : d_term] : (((d2term @ X368) != (d2term @ X369)) | (X368 = X369)) & (? [X504 : subst] : ((comp @ X504 @ id) != X504) | ? [X503 : subst] : ((comp @ id @ X503) != X503) | (! [X506 : subst] : ((comp @ id @ X506) = X506) & ! [X505 : subst] : ((comp @ X505 @ id) = X505)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X504 : subst] : ((comp @ X504 @ id) = X504) & ! [X503 : subst] : ((comp @ id @ X503) = X503) & (? [X506 : subst] : ((comp @ id @ X506) != X506) | ? [X505 : subst] : ((comp @ X505 @ id) != X505)))) & (ulamvar1 != $true) & (pushprop_gthm = $true) & ((d2subst @ d_id) = sh) & ((axscons = $true) | ? [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) != X0)) & (! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0) | (axscons != $true)) & ((axshiftcons = $true) | ? [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) != X201)) & (! [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) = X201) | (axshiftcons != $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP1) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP1)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) = X114) & ! [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) = X116) & ? [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true != (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) & ($true = (X120 @ X121))))) & (? [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) != X114) | ? [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) != X116) | ! [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true = (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) | ($true != (X120 @ X121))) | (pushprop_lem1_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ((hoaslamnotap = $true) | ? [X520 : term,X521 : term,X522 : subst > term > term] : (((ap @ (sub @ X521 @ id) @ X520) = (lam @ (X522 @ sh @ one))) & ! [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) = (sub @ (X522 @ X525 @ X524) @ X523)))) & (! [X520 : term,X521 : term,X522 : subst > term > term] : (((ap @ (sub @ X521 @ id) @ X520) != (lam @ (X522 @ sh @ one))) | ? [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) != (sub @ (X522 @ X525 @ X524) @ X523))) | (hoaslamnotap != $true))),
% 9.64/2.82 inference(flattening,[status(thm)],[f471])).
% 9.64/2.82 thf(f471,plain,(
% 9.64/2.82 ((! [X138 : term] : ? [X139 : subst > term > term] : ! [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) = (X139 @ X141 @ X140)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X138 : term] : ! [X139 : subst > term > term] : ? [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) != (X139 @ X141 @ X140)))) & (((apinj2 = $true) | ? [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) = (ap @ X79 @ X82)) & (X81 != X82))) & (! [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) != (ap @ X79 @ X82)) | (X81 = X82)) | (apinj2 != $true))) & ((! [X324 : subst] : ((comp @ id @ X324) = X324) | (axidl != $true)) & ((axidl = $true) | ? [X324 : subst] : ((comp @ id @ X324) != X324))) & (((sP58 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP58 != $true))) & ! [X356 : subst] : ? [X357 : d_subst] : ((d2subst @ X357) = X356) & (pushprop_lem2v2_lthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((! [X26 : subst > term > subst > $o] : ? [X27 : term > $o] : ! [X28 : term] : ((X27 @ X28) = (X26 @ id @ X28 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X26 : subst > term > subst > $o] : ! [X27 : term > $o] : ? [X28 : term] : ((X27 @ X28) != (X26 @ id @ X28 @ id)))) & (((hoasinduction_lem2v2 = $true) | ($true != sP56)) & (($true = sP56) | (hoasinduction_lem2v2 != $true))) & (((hoasinduction_lem3v2 = $true) | ($true != sP54)) & (($true = sP54) | (hoasinduction_lem3v2 != $true))) & (hoasinduction_lem1_gthm = $true) & (((hoasapinj1_lthm = $true) | (! [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 = X528) | ((ap @ X528 @ X529) != (ap @ X527 @ X530))) & ! [X526 : term] : ((sub @ X526 @ id) = X526) & ? [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) = (ap @ (sub @ X534 @ id) @ X532)) & (X533 != X534)))) & ((? [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 != X528) & ((ap @ X528 @ X529) = (ap @ X527 @ X530))) | ? [X526 : term] : ((sub @ X526 @ id) != X526) | ! [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) != (ap @ (sub @ X534 @ id) @ X532)) | (X533 = X534))) | (hoasapinj1_lthm != $true))) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & (hoaslamnotvar_gthm = $true) & (((sP52 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP52 != $true))) & (induction2_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoaslamnotap_gthm = $true) & ((! [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (($true = (X475 @ (sub @ X473 @ (push @ X476 @ X474)))) | ($true != (X472 @ X473))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (($true != (X475 @ (sub @ X473 @ (push @ X476 @ X474)))) & ($true = (X472 @ X473))))) & (((hoaslaminj = $true) | ? [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (((X144 @ X146 @ X147) != (X145 @ X146 @ X147)) & ! [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) = (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) & ((lam @ (X144 @ sh @ one)) = (lam @ (X145 @ sh @ one))) & ! [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) = (sub @ (X144 @ X149 @ X148) @ X150)))) & (! [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (((X144 @ X146 @ X147) = (X145 @ X146 @ X147)) | ? [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) != (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) | ((lam @ (X144 @ sh @ one)) != (lam @ (X145 @ sh @ one))) | ? [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) != (sub @ (X144 @ X149 @ X148) @ X150))) | (hoaslaminj != $true))) & (((pushprop_lem0 = $true) | ? [X333 : term,X334 : term > $o,X335 : subst] : ! [X336 : term > $o] : ? [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) != (X336 @ X337))) & (! [X333 : term,X334 : term > $o,X335 : subst] : ? [X336 : term > $o] : ! [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) = (X336 @ X337)) | (pushprop_lem0 != $true))) & ((! [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) != (ap @ (sub @ X508 @ id) @ X510)) | (X507 = X508)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) = (ap @ (sub @ X508 @ id) @ X510)) & (X507 != X508)))) & ! [X338 : d_subst,X339 : d_subst] : ((X338 = X339) | ((d2subst @ X338) != (d2subst @ X339))) & (((pushprop_lem0_lthm = $true) | ? [X535 : term > $o,X536 : subst,X537 : term] : ! [X538 : term > $o] : ? [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) != (X538 @ X539))) & (! [X535 : term > $o,X536 : subst,X537 : term] : ? [X538 : term > $o] : ! [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) = (X538 @ X539)) | (pushprop_lem0_lthm != $true))) & ((($true = sP47) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP47))) & (((hoasinduction_lem3_lthm = $true) | ($true != sP44)) & (($true = sP44) | (hoasinduction_lem3_lthm != $true))) & ((! [X230 : term > $o,X231 : term] : (? [X232 : term] : (($true != (X230 @ X232)) & ((var @ X232) = $true)) | ((X230 @ X231) = $true) | ? [X235 : term] : (($true != (X230 @ (lam @ X235))) & ($true = (X230 @ X235))) | ($true = (sP41 @ X230))) | (induction != $true)) & ((induction = $true) | ? [X230 : term > $o,X231 : term] : (! [X232 : term] : (($true = (X230 @ X232)) | ((var @ X232) != $true)) & ((X230 @ X231) != $true) & ! [X235 : term] : (($true = (X230 @ (lam @ X235))) | ($true != (X230 @ X235))) & ($true != (sP41 @ X230))))) & (hoasinduction_lem1 = $true) & ((($true = sP40) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP40))) & (((hoaslaminj_lthm = $true) | (sP34 != $true)) & ((sP34 = $true) | (hoaslaminj_lthm != $true))) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & ((($true = sP33) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP33))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (hoasapnotvar_lthm = $true) & ((($true = sP31) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP31))) & (axapp = $true) & (((hoasinduction_lem3aa = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3aa != $true))) & ((! [X391 : term] : ? [X392 : subst > term > term] : ! [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) = (X392 @ X393 @ X394)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X391 : term] : ! [X392 : subst > term > term] : ? [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) != (X392 @ X393 @ X394)))) & (((induction2lem_lthm = $true) | ($true != sP28)) & (($true = sP28) | (induction2lem_lthm != $true))) & (axvarshift = $true) & (((axidr = $true) | ? [X416 : subst] : ((comp @ X416 @ id) != X416)) & (! [X416 : subst] : ((comp @ X416 @ id) = X416) | (axidr != $true))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (lamnotvar = $true) & (((hoasinduction_lem3a = $true) | (sP24 != $true)) & ((sP24 = $true) | (hoasinduction_lem3a != $true))) & (hoasinduction_lem3v2_gthm = $true) & ! [X417 : d_subst] : (d_id = X417) & (((apinj1 = $true) | ? [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 != X239) & ((ap @ X236 @ X238) = (ap @ X239 @ X237)))) & (! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 = X239) | ((ap @ X236 @ X238) != (ap @ X239 @ X237))) | (apinj1 != $true))) & (((induction2_lthm = $true) | ($true != sP23)) & (($true = sP23) | (induction2_lthm != $true))) & (((hoasinduction_lem3aaa = $true) | ($true != sP19)) & (($true = sP19) | (hoasinduction_lem3aaa != $true))) & (((pushprop_lem1v2_lthm = $true) | (? [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (((X479 @ X481) = $true) & ($true != (X480 @ one)) & ! [X483 : term] : ((X480 @ X483) = (X479 @ (sub @ X483 @ (push @ X481 @ X482))))) & ! [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) = X478))) & ((! [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (((X479 @ X481) != $true) | ($true = (X480 @ one)) | ? [X483 : term] : ((X480 @ X483) != (X479 @ (sub @ X483 @ (push @ X481 @ X482))))) | ? [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) != X478)) | (pushprop_lem1v2_lthm != $true))) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (((hoasinduction_lem3v2a_lthm = $true) | (sP18 != $true)) & ((sP18 = $true) | (hoasinduction_lem3v2a_lthm != $true))) & (((laminj = $true) | ? [X366 : term,X367 : term] : ((X366 != X367) & ((lam @ X366) = (lam @ X367)))) & (! [X366 : term,X367 : term] : ((X366 = X367) | ((lam @ X366) != (lam @ X367))) | (laminj != $true))) & ((! [X77 : term,X78 : term] : (((sub @ X78 @ sh) != (sub @ X77 @ sh)) | (X77 = X78)) | (shinj != $true)) & ((shinj = $true) | ? [X77 : term,X78 : term] : (((sub @ X78 @ sh) = (sub @ X77 @ sh)) & (X77 != X78)))) & (hoasapnotvar_gthm = $true) & (((! [X6 : term] : ((sub @ X6 @ id) = X6) | ? [X5 : term] : ((sub @ X5 @ id) != X5)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X6 : term] : ((sub @ X6 @ id) != X6) & ! [X5 : term] : ((sub @ X5 @ id) = X5)))) & ((! [X109 : term] : ((sub @ X109 @ id) = X109) | (termmset != $true)) & ((termmset = $true) | ? [X109 : term] : ((sub @ X109 @ id) != X109))) & (apnotvar = $true) & ((d2subst @ d_id) = id) & (((hoasinduction_lthm = $true) | (sP17 != $true)) & ((sP17 = $true) | (hoasinduction_lthm != $true))) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lthm_orig = $true) & (((hoasapinj2_lthm = $true) | (! [X358 : term,X359 : term,X360 : term,X361 : term] : ((X358 = X361) | ((ap @ X360 @ X361) != (ap @ X359 @ X358))) & ? [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) = (ap @ (sub @ X364 @ id) @ X365)) & (X363 != X365)))) & ((? [X358 : term,X359 : term,X360 : term,X361 : term] : ((X358 != X361) & ((ap @ X360 @ X361) = (ap @ X359 @ X358))) | ! [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X364 @ id) @ X365)) | (X363 = X365))) | (hoasapinj2_lthm != $true))) & (hoaslamnotvar = $true) & (pushprop_lthm = $true) & (((sP8 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP8 != $true))) & (pushprop_lem2v2_gthm = $true) & (((sP7 = $true) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (sP7 != $true))) & (hoasinduction_lem3a_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (axassoc = $true) & (axabs = $true) & (lamnotap != $true) & ((! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 = X354) | ((ap @ (sub @ X352 @ id) @ X353) != (ap @ (sub @ X355 @ id) @ X354))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 != X354) & ((ap @ (sub @ X352 @ id) @ X353) = (ap @ (sub @ X355 @ id) @ X354))))) & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & (pushprop = $true) & (hoasinduction_lem3_gthm = $true) & (((pushprop_lem1v2 = $true) | ? [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true = (X395 @ X397)) & ($true != (X396 @ one)) & ! [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) = (X396 @ X399)))) & (! [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true != (X395 @ X397)) | ($true = (X396 @ one)) | ? [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) != (X396 @ X399))) | (pushprop_lem1v2 != $true))) & (((! [X373 : subst] : ((comp @ X373 @ id) = X373) & ! [X372 : subst] : ((comp @ id @ X372) = X372)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X373 : subst] : ((comp @ X373 @ id) != X373) | ? [X372 : subst] : ((comp @ id @ X372) != X372)))) & (((hoasinduction_lem0 = $true) | ? [X110 : subst > term > subst > $o] : ! [X111 : term > $o] : ? [X112 : term] : ((X110 @ id @ X112 @ id) != (X111 @ X112))) & (! [X110 : subst > term > subst > $o] : ? [X111 : term > $o] : ! [X112 : term] : ((X110 @ id @ X112 @ id) = (X111 @ X112)) | (hoasinduction_lem0 != $true))) & (((hoasinduction_lem3aa_lthm = $true) | ($true != sP5)) & (($true = sP5) | (hoasinduction_lem3aa_lthm != $true))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (((hoasinduction_lem3b_lthm = $true) | ? [X1 : term] : ! [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) != (X2 @ sh @ one))) & (! [X1 : term] : ? [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) = (X2 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true))) & (ulamvarsh = $true) & ! [X229 : d_term] : (d_one = X229) & (axclos = $true) & (substmonoid_gthm = $true) & (induction2lem_gthm = $true) & (pushprop_lem2v2 = $true) & (hoasinduction_gthm = $true) & (((hoasinduction_no_psi_cond = $true) | ? [X202 : subst > term > subst > $o,X203 : term] : (! [X206 : subst > term > term] : (? [X207 : term] : (((X202 @ id @ X207 @ id) = $true) & ($true != (X202 @ id @ (X206 @ id @ X207) @ id))) | ? [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) != (sub @ (X206 @ X210 @ X208) @ X209)) | ($true = (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id))) & ($true != (sP4 @ X202)) & ((X202 @ id @ X203 @ id) != $true))) & (! [X202 : subst > term > subst > $o,X203 : term] : (? [X206 : subst > term > term] : (! [X207 : term] : (((X202 @ id @ X207 @ id) != $true) | ($true = (X202 @ id @ (X206 @ id @ X207) @ id))) & ! [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) = (sub @ (X206 @ X210 @ X208) @ X209)) & ($true != (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id))) | ($true = (sP4 @ X202)) | ((X202 @ id @ X203 @ id) = $true)) | (hoasinduction_no_psi_cond != $true))) & (axmap = $true) & (((hoasinduction_lem2 = $true) | ($true != sP3)) & (($true = sP3) | (hoasinduction_lem2 != $true))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & ((! [X142 : term] : ? [X143 : subst > term > term] : ((X143 @ sh @ one) = (sub @ X142 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X142 : term] : ! [X143 : subst > term > term] : ((X143 @ sh @ one) != (sub @ X142 @ (push @ one @ sh))))) & (one = (d2term @ d_one)) & (termmset_gthm = $true) & ((! [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) = X371) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) != X371))) & (ulamvarind = $true) & (hoasapinj1_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((! [X511 : term] : ((sub @ X511 @ id) = X511) | (axvarid != $true)) & ((axvarid = $true) | ? [X511 : term] : ((sub @ X511 @ id) != X511))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ! [X368 : d_term,X369 : d_term] : (((d2term @ X368) != (d2term @ X369)) | (X368 = X369)) & (((? [X504 : subst] : ((comp @ X504 @ id) != X504) | ? [X503 : subst] : ((comp @ id @ X503) != X503) | (! [X506 : subst] : ((comp @ id @ X506) = X506) & ! [X505 : subst] : ((comp @ X505 @ id) = X505))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X504 : subst] : ((comp @ X504 @ id) = X504) & ! [X503 : subst] : ((comp @ id @ X503) = X503) & (? [X506 : subst] : ((comp @ id @ X506) != X506) | ? [X505 : subst] : ((comp @ X505 @ id) != X505))))) & (ulamvar1 != $true) & (pushprop_gthm = $true) & ((d2subst @ d_id) = sh) & (((axscons = $true) | ? [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) != X0)) & (! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0) | (axscons != $true))) & (((axshiftcons = $true) | ? [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) != X201)) & (! [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) = X201) | (axshiftcons != $true))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem2v2_gthm = $true) & ((($true = sP1) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP1))) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (((pushprop_lem1_lthm = $true) | (! [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) = X114) & ! [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) = X116) & ? [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true != (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) & ($true = (X120 @ X121))))) & ((? [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) != X114) | ? [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) != X116) | ! [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true = (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) | ($true != (X120 @ X121)))) | (pushprop_lem1_lthm != $true))) & (hoaslamnotvar_lthm = $true) & (((hoaslamnotap = $true) | ? [X520 : term,X521 : term,X522 : subst > term > term] : (((ap @ (sub @ X521 @ id) @ X520) = (lam @ (X522 @ sh @ one))) & ! [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) = (sub @ (X522 @ X525 @ X524) @ X523)))) & (! [X520 : term,X521 : term,X522 : subst > term > term] : (((ap @ (sub @ X521 @ id) @ X520) != (lam @ (X522 @ sh @ one))) | ? [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) != (sub @ (X522 @ X525 @ X524) @ X523))) | (hoaslamnotap != $true)))),
% 9.64/2.82 inference(nnf_transformation,[status(thm)],[f73])).
% 9.64/2.82 thf(f73,plain,(
% 9.64/2.82 (! [X138 : term] : ? [X139 : subst > term > term] : ! [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) = (X139 @ X141 @ X140)) <=> (hoasinduction_lem3v2_f = $true)) & ((apinj2 = $true) <=> ! [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) != (ap @ X79 @ X82)) | (X81 = X82))) & (! [X324 : subst] : ((comp @ id @ X324) = X324) <=> (axidl = $true)) & ((sP58 = $true) <=> (hoasinduction_lem3v2_lthm = $true)) & ! [X356 : subst] : ? [X357 : d_subst] : ((d2subst @ X357) = X356) & (pushprop_lem2v2_lthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X26 : subst > term > subst > $o] : ? [X27 : term > $o] : ! [X28 : term] : ((X27 @ X28) = (X26 @ id @ X28 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & ((hoasinduction_lem2v2 = $true) <=> ($true = sP56)) & ((hoasinduction_lem3v2 = $true) <=> ($true = sP54)) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) <=> (? [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 != X528) & ((ap @ X528 @ X529) = (ap @ X527 @ X530))) | ? [X526 : term] : ((sub @ X526 @ id) != X526) | ! [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) != (ap @ (sub @ X534 @ id) @ X532)) | (X533 = X534)))) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & (hoaslamnotvar_gthm = $true) & ((sP52 = $true) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (induction2_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoaslamnotap_gthm = $true) & (! [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (($true = (X475 @ (sub @ X473 @ (push @ X476 @ X474)))) | ($true != (X472 @ X473))) <=> (pushprop_lem1 = $true)) & ((hoaslaminj = $true) <=> ! [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (((X144 @ X146 @ X147) = (X145 @ X146 @ X147)) | ? [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) != (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) | ((lam @ (X144 @ sh @ one)) != (lam @ (X145 @ sh @ one))) | ? [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) != (sub @ (X144 @ X149 @ X148) @ X150)))) & ((pushprop_lem0 = $true) <=> ! [X333 : term,X334 : term > $o,X335 : subst] : ? [X336 : term > $o] : ! [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) = (X336 @ X337))) & (! [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) != (ap @ (sub @ X508 @ id) @ X510)) | (X507 = X508)) <=> (hoasapinj1 = $true)) & ! [X338 : d_subst,X339 : d_subst] : ((X338 = X339) | ((d2subst @ X338) != (d2subst @ X339))) & ((pushprop_lem0_lthm = $true) <=> ! [X535 : term > $o,X536 : subst,X537 : term] : ? [X538 : term > $o] : ! [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) = (X538 @ X539))) & (($true = sP47) <=> (hoasinduction = $true)) & ((hoasinduction_lem3_lthm = $true) <=> ($true = sP44)) & (! [X230 : term > $o,X231 : term] : (? [X232 : term] : (($true != (X230 @ X232)) & ((var @ X232) = $true)) | ((X230 @ X231) = $true) | ? [X235 : term] : (($true != (X230 @ (lam @ X235))) & ($true = (X230 @ X235))) | ($true = (sP41 @ X230))) <=> (induction = $true)) & (hoasinduction_lem1 = $true) & (($true = sP40) <=> (hoasinduction_lthm_3 = $true)) & ((hoaslaminj_lthm = $true) <=> (sP34 = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (($true = sP33) <=> (induction2 = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (hoasapnotvar_lthm = $true) & (($true = sP31) <=> (induction2lem = $true)) & (axapp = $true) & ((hoasinduction_lem3aa = $true) <=> ($true = sP29)) & (! [X391 : term] : ? [X392 : subst > term > term] : ! [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) = (X392 @ X393 @ X394)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((induction2lem_lthm = $true) <=> ($true = sP28)) & (axvarshift = $true) & ((axidr = $true) <=> ! [X416 : subst] : ((comp @ X416 @ id) = X416)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (lamnotvar = $true) & ((hoasinduction_lem3a = $true) <=> (sP24 = $true)) & (hoasinduction_lem3v2_gthm = $true) & ! [X417 : d_subst] : (d_id = X417) & ((apinj1 = $true) <=> ! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 = X239) | ((ap @ X236 @ X238) != (ap @ X239 @ X237)))) & ((induction2_lthm = $true) <=> ($true = sP23)) & ((hoasinduction_lem3aaa = $true) <=> ($true = sP19)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (((X479 @ X481) != $true) | ($true = (X480 @ one)) | ? [X483 : term] : ((X480 @ X483) != (X479 @ (sub @ X483 @ (push @ X481 @ X482))))) | ? [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) != X478))) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (sP18 = $true)) & ((laminj = $true) <=> ! [X366 : term,X367 : term] : ((X366 = X367) | ((lam @ X366) != (lam @ X367)))) & (! [X77 : term,X78 : term] : (((sub @ X78 @ sh) != (sub @ X77 @ sh)) | (X77 = X78)) <=> (shinj = $true)) & (hoasapnotvar_gthm = $true) & ((! [X6 : term] : ((sub @ X6 @ id) = X6) | ? [X5 : term] : ((sub @ X5 @ id) != X5)) <=> (termmset_lthm = $true)) & (! [X109 : term] : ((sub @ X109 @ id) = X109) <=> (termmset = $true)) & (apnotvar = $true) & ((d2subst @ d_id) = id) & ((hoasinduction_lthm = $true) <=> (sP17 = $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lthm_orig = $true) & ((hoasapinj2_lthm = $true) <=> (? [X358 : term,X359 : term,X360 : term,X361 : term] : ((X358 != X361) & ((ap @ X360 @ X361) = (ap @ X359 @ X358))) | ! [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X364 @ id) @ X365)) | (X363 = X365)))) & (hoaslamnotvar = $true) & (pushprop_lthm = $true) & ((sP8 = $true) <=> (hoasinduction_lem3v2a = $true)) & (pushprop_lem2v2_gthm = $true) & ((sP7 = $true) <=> (hoasinduction_lem3 = $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (axassoc = $true) & (axabs = $true) & (lamnotap != $true) & (! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 = X354) | ((ap @ (sub @ X352 @ id) @ X353) != (ap @ (sub @ X355 @ id) @ X354))) <=> (hoasapinj2 = $true)) & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & (pushprop = $true) & (hoasinduction_lem3_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true != (X395 @ X397)) | ($true = (X396 @ one)) | ? [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) != (X396 @ X399)))) & ((! [X373 : subst] : ((comp @ X373 @ id) = X373) & ! [X372 : subst] : ((comp @ id @ X372) = X372)) <=> (substmonoid = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X110 : subst > term > subst > $o] : ? [X111 : term > $o] : ! [X112 : term] : ((X110 @ id @ X112 @ id) = (X111 @ X112))) & ((hoasinduction_lem3aa_lthm = $true) <=> ($true = sP5)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X1 : term] : ? [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) = (X2 @ sh @ one))) & (ulamvarsh = $true) & ! [X229 : d_term] : (d_one = X229) & (axclos = $true) & (substmonoid_gthm = $true) & (induction2lem_gthm = $true) & (pushprop_lem2v2 = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X202 : subst > term > subst > $o,X203 : term] : (? [X206 : subst > term > term] : (! [X207 : term] : (((X202 @ id @ X207 @ id) != $true) | ($true = (X202 @ id @ (X206 @ id @ X207) @ id))) & ! [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) = (sub @ (X206 @ X210 @ X208) @ X209)) & ($true != (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id))) | ($true = (sP4 @ X202)) | ((X202 @ id @ X203 @ id) = $true))) & (axmap = $true) & ((hoasinduction_lem2 = $true) <=> ($true = sP3)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (! [X142 : term] : ? [X143 : subst > term > term] : ((X143 @ sh @ one) = (sub @ X142 @ (push @ one @ sh))) <=> (hoasinduction_lem3b = $true)) & (one = (d2term @ d_one)) & (termmset_gthm = $true) & (! [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) = X371) <=> (axvarcons = $true)) & (ulamvarind = $true) & (hoasapinj1_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X511 : term] : ((sub @ X511 @ id) = X511) <=> (axvarid = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ! [X368 : d_term,X369 : d_term] : (((d2term @ X368) != (d2term @ X369)) | (X368 = X369)) & ((? [X504 : subst] : ((comp @ X504 @ id) != X504) | ? [X503 : subst] : ((comp @ id @ X503) != X503) | (! [X506 : subst] : ((comp @ id @ X506) = X506) & ! [X505 : subst] : ((comp @ X505 @ id) = X505))) <=> (substmonoid_lthm = $true)) & (ulamvar1 != $true) & (pushprop_gthm = $true) & ((d2subst @ d_id) = sh) & ((axscons = $true) <=> ! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0)) & ((axshiftcons = $true) <=> ! [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) = X201)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP1) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) != X114) | ? [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) != X116) | ! [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true = (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) | ($true != (X120 @ X121))))) & (hoaslamnotvar_lthm = $true) & ((hoaslamnotap = $true) <=> ! [X520 : term,X521 : term,X522 : subst > term > term] : (((ap @ (sub @ X521 @ id) @ X520) != (lam @ (X522 @ sh @ one))) | ? [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) != (sub @ (X522 @ X525 @ X524) @ X523))))),
% 9.64/2.82 inference(definition_folding,[status(thm)],[f13,f72,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14])).
% 9.64/2.82 thf(f14,plain,(
% 9.64/2.82 ($true = sP0) <=> ? [X375 : term,X376 : subst > term > subst > $o] : (! [X382 : term] : (($true != (X376 @ id @ X382 @ id)) | ($true = (X376 @ id @ (sub @ X375 @ (push @ X382 @ id)) @ id))) & ((X376 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id) != $true) & ! [X377 : subst > term > term] : (($true = (X376 @ id @ (lam @ (X377 @ sh @ one)) @ id)) | ? [X378 : term] : (($true != (X376 @ id @ (X377 @ id @ X378) @ id)) & ($true = (X376 @ id @ X378 @ id))) | ? [X379 : subst,X380 : subst,X381 : term] : ((X377 @ (comp @ X379 @ X380) @ (sub @ X381 @ X380)) != (sub @ (X377 @ X379 @ X381) @ X380))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f15,plain,(
% 9.64/2.82 ($true = sP1) <=> (! [X383 : term,X384 : subst > term > subst > $o] : (($true = (X384 @ id @ (lam @ X383) @ id)) | ? [X385 : term] : (((X384 @ id @ (sub @ X383 @ (push @ X385 @ id)) @ id) != $true) & ($true = (X384 @ id @ X385 @ id))) | ? [X386 : subst > term > term] : (($true != (X384 @ id @ (lam @ (X386 @ sh @ one)) @ id)) & ! [X388 : subst,X389 : subst,X390 : term] : ((sub @ (X386 @ X389 @ X390) @ X388) = (X386 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388))) & ! [X387 : term] : (($true = (X384 @ id @ (X386 @ id @ X387) @ id)) | ((X384 @ id @ X387 @ id) != $true)))) | ($true = sP0) | ? [X374 : term] : ((sub @ X374 @ id) != X374))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f16,plain,(
% 9.64/2.82 ! [X188 : subst > term > subst > $o] : (($true = (sP2 @ X188)) <=> ? [X198 : term,X199 : term] : (($true = (X188 @ id @ X199 @ id)) & ($true != (X188 @ id @ (ap @ (sub @ X199 @ id) @ X198) @ id)) & ((X188 @ id @ X198 @ id) = $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f17,plain,(
% 9.64/2.82 ($true = sP3) <=> ! [X187 : term,X188 : subst > term > subst > $o,X189 : term] : (($true = (X188 @ id @ (ap @ X189 @ X187) @ id)) | ($true = (sP2 @ X188)) | ($true != (X188 @ id @ X187 @ id)) | ($true != (X188 @ id @ X189 @ id)) | ? [X190 : subst,X191 : subst,X192 : term,X193 : subst] : (((X188 @ X191 @ X192 @ (comp @ X193 @ X190)) != $true) & ((X188 @ (comp @ X191 @ X193) @ (sub @ X192 @ X193) @ X190) = $true)) | ? [X194 : subst,X195 : subst,X196 : term,X197 : subst] : (($true = (X188 @ X194 @ X196 @ (comp @ X197 @ X195))) & ((X188 @ (comp @ X194 @ X197) @ (sub @ X196 @ X197) @ X195) != $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f18,plain,(
% 9.64/2.82 ! [X202 : subst > term > subst > $o] : (($true = (sP4 @ X202)) <=> ? [X204 : term,X205 : term] : (((X202 @ id @ (ap @ (sub @ X204 @ id) @ X205) @ id) != $true) & ($true = (X202 @ id @ X205 @ id)) & ($true = (X202 @ id @ X204 @ id))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f19,plain,(
% 9.64/2.82 ($true = sP5) <=> ! [X512 : subst > term > subst > $o,X513 : term] : (? [X514 : term] : (((X512 @ id @ X514 @ id) = $true) & ((X512 @ id @ (sub @ X513 @ (push @ X514 @ id)) @ id) != $true)) | ? [X515 : subst > term > term] : (! [X519 : term] : (((X512 @ id @ X519 @ id) != $true) | ((X512 @ id @ (X515 @ id @ X519) @ id) = $true)) & ! [X516 : subst,X517 : term,X518 : subst] : ((X515 @ (comp @ X518 @ X516) @ (sub @ X517 @ X516)) = (sub @ (X515 @ X518 @ X517) @ X516)) & ((X512 @ id @ (lam @ (X515 @ sh @ one)) @ id) != $true)) | ($true = (X512 @ id @ (lam @ (sub @ X513 @ (push @ one @ sh))) @ id)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f20,plain,(
% 9.64/2.82 ! [X123 : subst > term > subst > $o] : (($true = (sP6 @ X123)) <=> ? [X125 : subst > term > term] : (! [X126 : subst,X127 : term,X128 : subst] : ((X125 @ (comp @ X128 @ X126) @ (sub @ X127 @ X126)) = (sub @ (X125 @ X128 @ X127) @ X126)) & ! [X129 : term] : (((X123 @ id @ X129 @ id) != $true) | ($true = (X123 @ id @ (X125 @ id @ X129) @ id))) & ($true != (X123 @ id @ (lam @ (X125 @ sh @ one)) @ id))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f21,plain,(
% 9.64/2.82 (sP7 = $true) <=> ! [X122 : term,X123 : subst > term > subst > $o] : (($true = (sP6 @ X123)) | ? [X124 : term] : (($true != (X123 @ id @ (sub @ X122 @ (push @ X124 @ id)) @ id)) & ($true = (X123 @ id @ X124 @ id))) | ((X123 @ id @ (lam @ X122) @ id) = $true) | ? [X134 : subst,X135 : subst,X136 : term,X137 : subst] : (((X123 @ (comp @ X135 @ X137) @ (sub @ X136 @ X137) @ X134) = $true) & ((X123 @ X135 @ X136 @ (comp @ X137 @ X134)) != $true)) | ? [X130 : subst,X131 : subst,X132 : subst,X133 : term] : (((X123 @ (comp @ X131 @ X130) @ (sub @ X133 @ X130) @ X132) != $true) & ((X123 @ X131 @ X133 @ (comp @ X130 @ X132)) = $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f22,plain,(
% 9.64/2.82 (sP8 = $true) <=> ! [X83 : subst > term > subst > $o,X84 : term,X85 : term > $o] : (($true = (X85 @ (lam @ X84))) | ? [X88 : subst > term > term] : (! [X92 : term] : (((X83 @ id @ X92 @ id) != $true) | ((X83 @ id @ (X88 @ id @ X92) @ id) = $true)) & ! [X89 : subst,X90 : subst,X91 : term] : ((X88 @ (comp @ X90 @ X89) @ (sub @ X91 @ X89)) = (sub @ (X88 @ X90 @ X91) @ X89)) & ($true != (X83 @ id @ (lam @ (X88 @ sh @ one)) @ id))) | ? [X87 : term] : ((X83 @ id @ X87 @ id) != (X85 @ X87)) | ? [X86 : term] : (($true = (X85 @ X86)) & ($true != (X85 @ (sub @ X84 @ (push @ X86 @ id))))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f23,plain,(
% 9.64/2.82 ! [X439 : subst > term > subst > $o] : (($true = (sP9 @ X439)) <=> ! [X440 : subst > term > term] : (? [X441 : subst,X442 : subst,X443 : term] : ((sub @ (X440 @ X442 @ X443) @ X441) != (X440 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441))) | ? [X444 : term] : (((X439 @ id @ X444 @ id) = $true) & ($true != (X439 @ id @ (X440 @ id @ X444) @ id))) | ((X439 @ id @ (lam @ (X440 @ sh @ one)) @ id) = $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f24,plain,(
% 9.64/2.82 ! [X427 : subst > term > subst > $o] : (($true = (sP10 @ X427)) <=> ! [X436 : term,X437 : term] : (($true = (X427 @ id @ (ap @ (sub @ X437 @ id) @ X436) @ id)) | ($true != (X427 @ id @ X437 @ id)) | ((X427 @ id @ X436 @ id) != $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f25,plain,(
% 9.64/2.82 ! [X455 : subst > term > subst > $o] : (((sP11 @ X455) = $true) <=> ? [X456 : term,X457 : term] : (((X455 @ id @ (ap @ (sub @ X456 @ id) @ X457) @ id) != $true) & ($true = (X455 @ id @ X456 @ id)) & ($true = (X455 @ id @ X457 @ id))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f26,plain,(
% 9.64/2.82 ! [X455 : subst > term > subst > $o] : (((sP12 @ X455) = $true) <=> ? [X458 : subst > term > term] : (((X455 @ id @ (lam @ (X458 @ sh @ one)) @ id) != $true) & ! [X459 : term] : (((X455 @ id @ X459 @ id) != $true) | ((X455 @ id @ (X458 @ id @ X459) @ id) = $true)) & ! [X460 : subst,X461 : subst,X462 : term] : ((X458 @ (comp @ X460 @ X461) @ (sub @ X462 @ X461)) = (sub @ (X458 @ X460 @ X462) @ X461))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f27,plain,(
% 9.64/2.82 ! [X419 : term > $o] : (((sP13 @ X419) = $true) <=> ! [X423 : term,X424 : term] : (($true = (X419 @ (ap @ X424 @ X423))) | ((X419 @ X423) != $true) | ((X419 @ X424) != $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f28,plain,(
% 9.64/2.82 ($true = sP14) <=> ? [X438 : term,X439 : subst > term > subst > $o] : (! [X445 : term] : (($true != (X439 @ id @ X445 @ id)) | ($true = (X439 @ id @ (sub @ X438 @ (push @ X445 @ id)) @ id))) & ($true = (sP9 @ X439)) & ($true != (X439 @ id @ (lam @ X438) @ id)) & ! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (((X439 @ (comp @ X449 @ X447) @ (sub @ X448 @ X447) @ X446) = $true) | ((X439 @ X449 @ X448 @ (comp @ X447 @ X446)) != $true)) & ! [X450 : subst,X451 : subst,X452 : term,X453 : subst] : (($true = (X439 @ X453 @ X452 @ (comp @ X450 @ X451))) | ((X439 @ (comp @ X453 @ X450) @ (sub @ X452 @ X450) @ X451) != $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f29,plain,(
% 9.64/2.82 ($true = sP15) <=> ! [X454 : term,X455 : subst > term > subst > $o] : (? [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (($true = (X455 @ (comp @ X468 @ X469) @ (sub @ X467 @ X469) @ X470)) & ((X455 @ X468 @ X467 @ (comp @ X469 @ X470)) != $true)) | ($true = (X455 @ id @ X454 @ id)) | ((sP12 @ X455) = $true) | ((sP11 @ X455) = $true) | ? [X471 : term] : (((X455 @ id @ X471 @ id) != $true) & ($true = (var @ (sub @ X471 @ id)))) | ? [X463 : subst,X464 : term,X465 : subst,X466 : subst] : (((X455 @ (comp @ X466 @ X465) @ (sub @ X464 @ X465) @ X463) != $true) & ($true = (X455 @ X466 @ X464 @ (comp @ X465 @ X463)))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f30,plain,(
% 9.64/2.82 ($true = sP16) <=> ? [X425 : term,X426 : term,X427 : subst > term > subst > $o] : (((X427 @ id @ X425 @ id) = $true) & ! [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X427 @ (comp @ X429 @ X430) @ (sub @ X428 @ X430) @ X431)) | ((X427 @ X429 @ X428 @ (comp @ X430 @ X431)) != $true)) & ((X427 @ id @ (ap @ X426 @ X425) @ id) != $true) & ($true = (X427 @ id @ X426 @ id)) & ($true = (sP10 @ X427)) & ! [X432 : subst,X433 : subst,X434 : term,X435 : subst] : (((X427 @ (comp @ X432 @ X433) @ (sub @ X434 @ X433) @ X435) != $true) | ((X427 @ X432 @ X434 @ (comp @ X433 @ X435)) = $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f31,plain,(
% 9.64/2.82 (sP17 = $true) <=> (? [X418 : term,X419 : term > $o] : (! [X420 : term] : (? [X421 : term] : (($true != (X419 @ (sub @ X420 @ (push @ X421 @ id)))) & ($true = (X419 @ X421))) | ((X419 @ (lam @ X420)) = $true)) & ! [X422 : term] : (($true = (X419 @ X422)) | ((var @ X422) != $true)) & ((sP13 @ X419) = $true) & ($true != (X419 @ X418))) | ($true = sP15) | ($true = sP16) | ($true = sP14))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f32,plain,(
% 9.64/2.82 (sP18 = $true) <=> (? [X244 : term] : ((sub @ X244 @ id) != X244) | ! [X245 : term,X246 : subst > term > subst > $o,X247 : term > $o] : (? [X248 : subst > term > term] : (! [X252 : term] : (($true != (X246 @ id @ X252 @ id)) | ($true = (X246 @ id @ (X248 @ id @ X252) @ id))) & ! [X249 : term,X250 : subst,X251 : subst] : ((sub @ (X248 @ X251 @ X249) @ X250) = (X248 @ (comp @ X251 @ X250) @ (sub @ X249 @ X250))) & ((X246 @ id @ (lam @ (X248 @ sh @ one)) @ id) != $true)) | ? [X253 : term] : ((X247 @ X253) != (X246 @ id @ X253 @ id)) | ? [X254 : term] : (((X247 @ X254) = $true) & ((X247 @ (sub @ X245 @ (push @ X254 @ id))) != $true)) | ($true = (X247 @ (lam @ X245)))) | ? [X240 : term] : ! [X241 : subst > term > term] : ? [X242 : term,X243 : subst] : ((X241 @ X243 @ X242) != (sub @ X240 @ (push @ X242 @ X243))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f33,plain,(
% 9.64/2.82 ($true = sP19) <=> ! [X340 : subst > term > subst > $o,X341 : term] : (? [X342 : term] : (($true != (X340 @ id @ (sub @ X341 @ (push @ X342 @ id)) @ id)) & ((X340 @ id @ X342 @ id) = $true)) | ? [X343 : term,X344 : subst > term > term] : (((X340 @ id @ (lam @ (X344 @ sh @ one)) @ id) != $true) & ! [X345 : subst,X346 : subst,X347 : term] : ((sub @ (sub @ X343 @ (push @ X347 @ X346)) @ X345) = (sub @ (X344 @ X346 @ X347) @ X345)) & ! [X349 : subst,X350 : subst,X351 : term] : ((sub @ X343 @ (push @ (sub @ X351 @ X350) @ (comp @ X349 @ X350))) = (X344 @ (comp @ X349 @ X350) @ (sub @ X351 @ X350))) & ! [X348 : term] : (($true = (X340 @ id @ (X344 @ id @ X348) @ id)) | ($true != (X340 @ id @ X348 @ id)))) | ($true = (X340 @ id @ (lam @ (sub @ X341 @ (push @ one @ sh))) @ id)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f34,plain,(
% 9.64/2.82 ! [X102 : term > $o] : (((sP20 @ X102) = $true) <=> ? [X104 : term,X105 : term] : (((X102 @ X105) = $true) & ((X102 @ X104) = $true) & ((X102 @ (ap @ X104 @ X105)) != $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f35,plain,(
% 9.64/2.82 ! [X95 : term > $o] : (((sP21 @ X95) = $true) <=> ! [X97 : term,X98 : term] : (($true = (X95 @ (ap @ X97 @ X98))) | ((X95 @ X98) != $true) | ($true != (X95 @ X97))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f36,plain,(
% 9.64/2.82 (sP22 = $true) <=> ! [X102 : term > $o,X103 : term] : (((X102 @ X103) = $true) | ((sP20 @ X102) = $true) | ? [X106 : term] : (((X102 @ (lam @ X106)) != $true) & ! [X107 : term] : (($true = (X102 @ (sub @ X106 @ (push @ X107 @ id)))) | ((X102 @ X107) != $true))) | ? [X108 : term] : (((X102 @ X108) != $true) & ($true = (var @ X108))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f37,plain,(
% 9.64/2.82 ($true = sP23) <=> (? [X94 : subst,X95 : term > $o,X96 : term] : (! [X100 : term] : (((X95 @ (lam @ X100)) = $true) | ? [X101 : term] : (((X95 @ (sub @ X100 @ (push @ X101 @ id))) != $true) & ((X95 @ X101) = $true))) & ((X95 @ (sub @ X96 @ X94)) != $true) & ((sP21 @ X95) = $true) & ! [X99 : term] : (((var @ X99) != $true) | ($true = (X95 @ (sub @ X99 @ X94))))) | ? [X93 : term] : ((sub @ X93 @ id) != X93) | (sP22 = $true))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f38,plain,(
% 9.64/2.82 (sP24 = $true) <=> ! [X262 : subst > term > subst > $o,X263 : term] : (? [X264 : term] : (((X262 @ id @ (sub @ X263 @ (push @ X264 @ id)) @ id) != $true) & ((X262 @ id @ X264 @ id) = $true)) | ((X262 @ id @ (lam @ X263) @ id) = $true) | ? [X265 : subst > term > term] : (! [X266 : term] : (($true = (X262 @ id @ (X265 @ id @ X266) @ id)) | ((X262 @ id @ X266 @ id) != $true)) & ! [X267 : subst,X268 : subst,X269 : term] : ((sub @ (X265 @ X268 @ X269) @ X267) = (X265 @ (comp @ X268 @ X267) @ (sub @ X269 @ X267))) & ($true != (X262 @ id @ (lam @ (X265 @ sh @ one)) @ id))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f39,plain,(
% 9.64/2.82 ! [X20 : term > $o] : (($true = (sP25 @ X20)) <=> ? [X22 : term,X23 : term] : (($true != (X20 @ (ap @ X23 @ X22))) & ($true = (X20 @ X23)) & ((X20 @ X22) = $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f40,plain,(
% 9.64/2.82 ! [X12 : term > $o] : (($true = (sP26 @ X12)) <=> ! [X16 : term,X17 : term] : (((X12 @ (ap @ X17 @ X16)) = $true) | ((X12 @ X16) != $true) | ($true != (X12 @ X17))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f41,plain,(
% 9.64/2.82 ($true = sP27) <=> ! [X18 : subst,X19 : term,X20 : term > $o] : (($true = (X20 @ (sub @ X19 @ X18))) | ? [X21 : term] : (($true != (X20 @ (sub @ X21 @ X18))) & ((var @ X21) = $true)) | ? [X24 : term] : (((X20 @ (lam @ X24)) != $true) & ! [X25 : term] : (((X20 @ X25) != $true) | ((X20 @ (sub @ X24 @ (push @ X25 @ id))) = $true))) | ($true = (sP25 @ X20)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f42,plain,(
% 9.64/2.82 ($true = sP28) <=> (? [X9 : subst,X10 : term] : ((comp @ sh @ (push @ X10 @ X9)) != X9) | ? [X12 : term > $o,X13 : term] : (! [X14 : term] : (($true != (var @ X14)) | ($true = (X12 @ X14))) & ($true = (sP26 @ X12)) & ($true != (X12 @ X13)) & ! [X15 : term] : (($true != (X12 @ X15)) | ((X12 @ (lam @ X15)) = $true))) | ? [X7 : term,X8 : subst] : ((sub @ one @ (push @ X7 @ X8)) != X7) | ? [X11 : subst] : ((comp @ X11 @ id) != X11) | ($true = sP27))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f43,plain,(
% 9.64/2.82 ($true = sP29) <=> ! [X154 : subst > term > subst > $o,X155 : term] : (? [X157 : subst > term > term] : (! [X159 : subst,X160 : subst,X161 : term] : ((X157 @ (comp @ X159 @ X160) @ (sub @ X161 @ X160)) = (sub @ (X157 @ X159 @ X161) @ X160)) & ($true != (X154 @ id @ (lam @ (X157 @ sh @ one)) @ id)) & ! [X158 : term] : (($true = (X154 @ id @ (X157 @ id @ X158) @ id)) | ($true != (X154 @ id @ X158 @ id)))) | ($true = (X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) | ? [X156 : term] : (((X154 @ id @ X156 @ id) = $true) & ($true != (X154 @ id @ (sub @ X155 @ (push @ X156 @ id)) @ id))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f44,plain,(
% 9.64/2.82 ! [X325 : term > $o] : (($true = (sP30 @ X325)) <=> ? [X328 : term,X329 : term] : (($true != (X325 @ (ap @ X329 @ X328))) & ($true = (X325 @ X329)) & ($true = (X325 @ X328))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f45,plain,(
% 9.64/2.82 ($true = sP31) <=> ! [X325 : term > $o,X326 : subst,X327 : term] : (($true = (sP30 @ X325)) | ? [X330 : term] : (($true = (var @ X330)) & ($true != (X325 @ (sub @ X330 @ X326)))) | ? [X331 : term] : (((X325 @ (lam @ X331)) != $true) & ! [X332 : term] : (($true != (X325 @ X332)) | ($true = (X325 @ (sub @ X331 @ (push @ X332 @ id)))))) | ($true = (X325 @ (sub @ X327 @ X326))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f46,plain,(
% 9.64/2.82 ! [X256 : term > $o] : (($true = (sP32 @ X256)) <=> ? [X257 : term,X258 : term] : (((X256 @ X258) = $true) & ($true = (X256 @ X257)) & ((X256 @ (ap @ X257 @ X258)) != $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f47,plain,(
% 9.64/2.82 ($true = sP33) <=> ! [X255 : term,X256 : term > $o] : (((X256 @ X255) = $true) | ? [X260 : term] : (! [X261 : term] : (($true != (X256 @ X261)) | ($true = (X256 @ (sub @ X260 @ (push @ X261 @ id))))) & ($true != (X256 @ (lam @ X260)))) | ($true = (sP32 @ X256)) | ? [X259 : term] : (((var @ X259) = $true) & ($true != (X256 @ X259))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f48,plain,(
% 9.64/2.82 (sP34 = $true) <=> (? [X400 : term,X401 : subst] : ((sub @ one @ (push @ X400 @ X401)) != X400) | ? [X402 : term,X403 : subst] : ((comp @ sh @ (push @ X402 @ X403)) != X403) | ? [X404 : term,X405 : term] : (((lam @ X405) = (lam @ X404)) & (X404 != X405)) | ! [X406 : subst > term > term,X407 : subst,X408 : term,X409 : subst > term > term] : (? [X413 : term,X414 : subst,X415 : subst] : ((sub @ (X409 @ X415 @ X413) @ X414) != (X409 @ (comp @ X415 @ X414) @ (sub @ X413 @ X414))) | ? [X410 : subst,X411 : term,X412 : subst] : ((sub @ (X406 @ X410 @ X411) @ X412) != (X406 @ (comp @ X410 @ X412) @ (sub @ X411 @ X412))) | ((X409 @ X407 @ X408) = (X406 @ X407 @ X408)) | ((lam @ (X409 @ sh @ one)) != (lam @ (X406 @ sh @ one)))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f49,plain,(
% 9.64/2.82 ! [X306 : subst > term > subst > $o] : (($true = (sP35 @ X306)) <=> ? [X318 : term,X319 : term] : (((X306 @ id @ X318 @ id) = $true) & ($true = (X306 @ id @ X319 @ id)) & ((X306 @ id @ (ap @ (sub @ X318 @ id) @ X319) @ id) != $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f50,plain,(
% 9.64/2.82 ! [X306 : subst > term > subst > $o] : (($true = (sP36 @ X306)) <=> ? [X309 : subst > term > term] : (($true != (X306 @ id @ (lam @ (X309 @ sh @ one)) @ id)) & ! [X311 : subst,X312 : term,X313 : subst] : ((sub @ (X309 @ X311 @ X312) @ X313) = (X309 @ (comp @ X311 @ X313) @ (sub @ X312 @ X313))) & ! [X310 : term] : (($true = (X306 @ id @ (X309 @ id @ X310) @ id)) | ($true != (X306 @ id @ X310 @ id)))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f51,plain,(
% 9.64/2.82 ! [X289 : term > $o] : (((sP37 @ X289) = $true) <=> ! [X293 : term,X294 : term] : (((X289 @ X293) != $true) | ($true = (X289 @ (ap @ X293 @ X294))) | ((X289 @ X294) != $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f52,plain,(
% 9.64/2.82 ($true = sP38) <=> ! [X306 : subst > term > subst > $o,X307 : term] : (($true = (sP36 @ X306)) | ? [X308 : term] : (((X306 @ id @ X308 @ id) != $true) & ((var @ (sub @ X308 @ id)) = $true)) | ((X306 @ id @ X307 @ id) = $true) | ($true = (sP35 @ X306)) | ? [X320 : term,X321 : subst,X322 : subst,X323 : subst] : (($true != (X306 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322) @ X323)) & ($true = (X306 @ X321 @ X320 @ (comp @ X322 @ X323)))) | ? [X314 : subst,X315 : subst,X316 : term,X317 : subst] : (((X306 @ X314 @ X316 @ (comp @ X315 @ X317)) != $true) & ((X306 @ (comp @ X314 @ X315) @ (sub @ X316 @ X315) @ X317) = $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f53,plain,(
% 9.64/2.82 ($true = sP39) <=> ? [X296 : term,X297 : term > $o,X298 : subst > term > subst > $o] : (! [X305 : term] : (((X297 @ X305) != $true) | ((X297 @ (sub @ X296 @ (push @ X305 @ id))) = $true)) & ! [X300 : subst > term > term] : (? [X301 : subst,X302 : term,X303 : subst] : ((sub @ (X300 @ X303 @ X302) @ X301) != (X300 @ (comp @ X303 @ X301) @ (sub @ X302 @ X301))) | ? [X304 : term] : (((X298 @ id @ (X300 @ id @ X304) @ id) != $true) & ($true = (X298 @ id @ X304 @ id))) | ($true = (X298 @ id @ (lam @ (X300 @ sh @ one)) @ id))) & ! [X299 : term] : ((X297 @ X299) = (X298 @ id @ X299 @ id)) & ((X297 @ (lam @ X296)) != $true))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f54,plain,(
% 9.64/2.82 ($true = sP40) <=> (? [X285 : subst > term > subst > $o] : ! [X286 : term > $o] : ? [X287 : term] : ((X285 @ id @ X287 @ id) != (X286 @ X287)) | ($true = sP39) | ? [X288 : term,X289 : term > $o] : (((X289 @ X288) != $true) & ! [X290 : term] : (((var @ X290) != $true) | ($true = (X289 @ X290))) & ! [X291 : term] : (? [X292 : term] : (((X289 @ X292) = $true) & ($true != (X289 @ (sub @ X291 @ (push @ X292 @ id))))) | ((X289 @ (lam @ X291)) = $true)) & ((sP37 @ X289) = $true)) | ? [X295 : term] : ((sub @ X295 @ id) != X295) | ($true = sP38))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f55,plain,(
% 9.64/2.82 ! [X230 : term > $o] : (($true = (sP41 @ X230)) <=> ? [X233 : term,X234 : term] : (($true = (X230 @ X234)) & ($true = (X230 @ X233)) & ($true != (X230 @ (ap @ X233 @ X234)))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f56,plain,(
% 9.64/2.82 ! [X172 : subst > term > subst > $o] : (((sP42 @ X172) = $true) <=> ? [X174 : subst > term > term] : (! [X176 : subst,X177 : subst,X178 : term] : ((sub @ (X174 @ X176 @ X178) @ X177) = (X174 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177))) & ((X172 @ id @ (lam @ (X174 @ sh @ one)) @ id) != $true) & ! [X175 : term] : (((X172 @ id @ (X174 @ id @ X175) @ id) = $true) | ((X172 @ id @ X175 @ id) != $true))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f57,plain,(
% 9.64/2.82 ($true = sP43) <=> ? [X163 : subst > term > subst > $o,X164 : term] : (! [X166 : subst > term > term] : (? [X167 : subst,X168 : subst,X169 : term] : ((X166 @ (comp @ X168 @ X167) @ (sub @ X169 @ X167)) != (sub @ (X166 @ X168 @ X169) @ X167)) | ? [X170 : term] : (($true = (X163 @ id @ X170 @ id)) & ((X163 @ id @ (X166 @ id @ X170) @ id) != $true)) | ($true = (X163 @ id @ (lam @ (X166 @ sh @ one)) @ id))) & ! [X165 : term] : (($true != (X163 @ id @ X165 @ id)) | ($true = (X163 @ id @ (sub @ X164 @ (push @ X165 @ id)) @ id))) & ((X163 @ id @ (lam @ (sub @ X164 @ (push @ one @ sh))) @ id) != $true))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f58,plain,(
% 9.64/2.82 ($true = sP44) <=> (($true = sP43) | ? [X162 : term] : ((sub @ X162 @ id) != X162) | ! [X171 : term,X172 : subst > term > subst > $o] : (? [X183 : subst,X184 : subst,X185 : term,X186 : subst] : (($true = (X172 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183) @ X186)) & ((X172 @ X184 @ X185 @ (comp @ X183 @ X186)) != $true)) | ? [X173 : term] : (($true != (X172 @ id @ (sub @ X171 @ (push @ X173 @ id)) @ id)) & ((X172 @ id @ X173 @ id) = $true)) | ? [X179 : term,X180 : subst,X181 : subst,X182 : subst] : (((X172 @ X182 @ X179 @ (comp @ X180 @ X181)) = $true) & ($true != (X172 @ (comp @ X182 @ X180) @ (sub @ X179 @ X180) @ X181))) | ((sP42 @ X172) = $true) | ($true = (X172 @ id @ (lam @ X171) @ id))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f59,plain,(
% 9.64/2.82 ! [X212 : subst > term > subst > $o] : (((sP45 @ X212) = $true) <=> ? [X227 : term,X228 : term] : (((X212 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id) != $true) & ((X212 @ id @ X228 @ id) = $true) & ((X212 @ id @ X227 @ id) = $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f60,plain,(
% 9.64/2.82 ! [X212 : subst > term > subst > $o] : (($true = (sP46 @ X212)) <=> ? [X222 : subst > term > term] : (! [X224 : term,X225 : subst,X226 : subst] : ((X222 @ (comp @ X226 @ X225) @ (sub @ X224 @ X225)) = (sub @ (X222 @ X226 @ X224) @ X225)) & ! [X223 : term] : (($true = (X212 @ id @ (X222 @ id @ X223) @ id)) | ((X212 @ id @ X223 @ id) != $true)) & ($true != (X212 @ id @ (lam @ (X222 @ sh @ one)) @ id))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f61,plain,(
% 9.64/2.82 ($true = sP47) <=> ! [X211 : term,X212 : subst > term > subst > $o] : (? [X221 : term] : (((var @ (sub @ X221 @ id)) = $true) & ((X212 @ id @ X221 @ id) != $true)) | ($true = (sP46 @ X212)) | ? [X217 : subst,X218 : term,X219 : subst,X220 : subst] : (((X212 @ X219 @ X218 @ (comp @ X217 @ X220)) = $true) & ((X212 @ (comp @ X219 @ X217) @ (sub @ X218 @ X217) @ X220) != $true)) | ($true = (X212 @ id @ X211 @ id)) | ((sP45 @ X212) = $true) | ? [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (((X212 @ X214 @ X213 @ (comp @ X216 @ X215)) != $true) & ($true = (X212 @ (comp @ X214 @ X216) @ (sub @ X213 @ X216) @ X215))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f62,plain,(
% 9.64/2.82 ! [X50 : term > $o] : (((sP48 @ X50) = $true) <=> ! [X52 : term,X53 : term] : (($true != (X50 @ X53)) | ((X50 @ X52) != $true) | ($true = (X50 @ (ap @ X53 @ X52)))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f63,plain,(
% 9.64/2.82 ! [X68 : subst > term > subst > $o] : (((sP49 @ X68) = $true) <=> ? [X70 : term,X71 : term] : (((X68 @ id @ X70 @ id) = $true) & ($true = (X68 @ id @ X71 @ id)) & ($true != (X68 @ id @ (ap @ (sub @ X71 @ id) @ X70) @ id))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f64,plain,(
% 9.64/2.82 (sP50 = $true) <=> ? [X58 : subst > term > subst > $o,X59 : term,X60 : term > $o] : (! [X61 : term] : (((X60 @ (sub @ X59 @ (push @ X61 @ id))) = $true) | ((X60 @ X61) != $true)) & ($true != (X60 @ (lam @ X59))) & ! [X62 : term] : ((X58 @ id @ X62 @ id) = (X60 @ X62)) & ! [X63 : subst > term > term] : (? [X67 : term] : (((X58 @ id @ (X63 @ id @ X67) @ id) != $true) & ($true = (X58 @ id @ X67 @ id))) | ((X58 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true) | ? [X64 : subst,X65 : term,X66 : subst] : ((sub @ (X63 @ X64 @ X65) @ X66) != (X63 @ (comp @ X64 @ X66) @ (sub @ X65 @ X66)))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f65,plain,(
% 9.64/2.82 (sP51 = $true) <=> ? [X50 : term > $o,X51 : term] : (((sP48 @ X50) = $true) & ((X50 @ X51) != $true) & ! [X54 : term] : (((X50 @ (lam @ X54)) = $true) | ? [X55 : term] : (((X50 @ (sub @ X54 @ (push @ X55 @ id))) != $true) & ((X50 @ X55) = $true))) & ! [X56 : term] : (($true != (var @ X56)) | ($true = (X50 @ X56))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f66,plain,(
% 9.64/2.82 (sP52 = $true) <=> (? [X47 : subst > term > subst > $o] : ! [X48 : term > $o] : ? [X49 : term] : ((X47 @ id @ X49 @ id) != (X48 @ X49)) | ! [X68 : subst > term > subst > $o,X69 : term] : (? [X72 : subst > term > term] : (! [X76 : term] : (($true = (X68 @ id @ (X72 @ id @ X76) @ id)) | ((X68 @ id @ X76 @ id) != $true)) & ! [X73 : subst,X74 : subst,X75 : term] : ((sub @ (X72 @ X74 @ X75) @ X73) = (X72 @ (comp @ X74 @ X73) @ (sub @ X75 @ X73))) & ((X68 @ id @ (lam @ (X72 @ sh @ one)) @ id) != $true)) | ($true = (X68 @ id @ X69 @ id)) | ((sP49 @ X68) = $true)) | (sP51 = $true) | ? [X57 : term] : ((sub @ X57 @ id) != X57) | (sP50 = $true))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f67,plain,(
% 9.64/2.82 ! [X29 : subst > term > subst > $o] : (($true = (sP53 @ X29)) <=> ? [X37 : subst > term > term] : (! [X38 : subst,X39 : subst,X40 : term] : ((sub @ (X37 @ X38 @ X40) @ X39) = (X37 @ (comp @ X38 @ X39) @ (sub @ X40 @ X39))) & ((X29 @ id @ (lam @ (X37 @ sh @ one)) @ id) != $true) & ! [X41 : term] : (((X29 @ id @ X41 @ id) != $true) | ($true = (X29 @ id @ (X37 @ id @ X41) @ id)))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f68,plain,(
% 9.64/2.82 ($true = sP54) <=> ! [X29 : subst > term > subst > $o,X30 : term,X31 : term > $o] : (? [X46 : term] : ((X29 @ id @ X46 @ id) != (X31 @ X46)) | ? [X42 : subst,X43 : subst,X44 : term,X45 : subst] : (($true != (X29 @ X42 @ X44 @ (comp @ X43 @ X45))) & ((X29 @ (comp @ X42 @ X43) @ (sub @ X44 @ X43) @ X45) = $true)) | ? [X32 : subst,X33 : subst,X34 : term,X35 : subst] : (($true = (X29 @ X35 @ X34 @ (comp @ X32 @ X33))) & ($true != (X29 @ (comp @ X35 @ X32) @ (sub @ X34 @ X32) @ X33))) | ? [X36 : term] : (($true = (X31 @ X36)) & ($true != (X31 @ (sub @ X30 @ (push @ X36 @ id))))) | ((X31 @ (lam @ X30)) = $true) | ($true = (sP53 @ X29)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f69,plain,(
% 9.64/2.82 ! [X272 : subst > term > subst > $o] : (($true = (sP55 @ X272)) <=> ? [X278 : term,X279 : term] : (((X272 @ id @ X278 @ id) = $true) & ($true != (X272 @ id @ (ap @ (sub @ X279 @ id) @ X278) @ id)) & ((X272 @ id @ X279 @ id) = $true)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f70,plain,(
% 9.64/2.82 ($true = sP56) <=> ! [X270 : term,X271 : term > $o,X272 : subst > term > subst > $o,X273 : term] : (($true != (X271 @ X273)) | ? [X284 : term] : ((X272 @ id @ X284 @ id) != (X271 @ X284)) | ($true = (X271 @ (ap @ X273 @ X270))) | ($true != (X271 @ X270)) | ? [X280 : subst,X281 : subst,X282 : term,X283 : subst] : (($true != (X272 @ X281 @ X282 @ (comp @ X283 @ X280))) & ((X272 @ (comp @ X281 @ X283) @ (sub @ X282 @ X283) @ X280) = $true)) | ? [X274 : term,X275 : subst,X276 : subst,X277 : subst] : (($true = (X272 @ X277 @ X274 @ (comp @ X276 @ X275))) & ($true != (X272 @ (comp @ X277 @ X276) @ (sub @ X274 @ X276) @ X275))) | ($true = (sP55 @ X272)))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f71,plain,(
% 9.64/2.82 ! [X486 : subst > term > subst > $o] : (($true = (sP57 @ X486)) <=> ? [X498 : subst > term > term] : (((X486 @ id @ (lam @ (X498 @ sh @ one)) @ id) != $true) & ! [X499 : term] : (($true != (X486 @ id @ X499 @ id)) | ($true = (X486 @ id @ (X498 @ id @ X499) @ id))) & ! [X500 : subst,X501 : subst,X502 : term] : ((X498 @ (comp @ X500 @ X501) @ (sub @ X502 @ X501)) = (sub @ (X498 @ X500 @ X502) @ X501))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f72,plain,(
% 9.64/2.82 (sP58 = $true) <=> (? [X484 : term] : ((sub @ X484 @ id) != X484) | ! [X485 : term > $o,X486 : subst > term > subst > $o,X487 : term] : (? [X497 : term] : ((X485 @ X497) != (X486 @ id @ X497 @ id)) | ? [X492 : subst,X493 : term,X494 : subst,X495 : subst] : (((X486 @ X492 @ X493 @ (comp @ X495 @ X494)) != $true) & ((X486 @ (comp @ X492 @ X495) @ (sub @ X493 @ X495) @ X494) = $true)) | ? [X488 : subst,X489 : subst,X490 : term,X491 : subst] : (((X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X491) != $true) & ((X486 @ X488 @ X490 @ (comp @ X489 @ X491)) = $true)) | ? [X496 : term] : (((X485 @ X496) = $true) & ((X485 @ (sub @ X487 @ (push @ X496 @ id))) != $true)) | ((X485 @ (lam @ X487)) = $true) | ($true = (sP57 @ X486))))),
% 9.64/2.82 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.64/2.82 thf(f13,plain,(
% 9.64/2.82 (! [X138 : term] : ? [X139 : subst > term > term] : ! [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) = (X139 @ X141 @ X140)) <=> (hoasinduction_lem3v2_f = $true)) & ((apinj2 = $true) <=> ! [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) != (ap @ X79 @ X82)) | (X81 = X82))) & (! [X324 : subst] : ((comp @ id @ X324) = X324) <=> (axidl = $true)) & ((? [X484 : term] : ((sub @ X484 @ id) != X484) | ! [X485 : term > $o,X486 : subst > term > subst > $o,X487 : term] : (? [X497 : term] : ((X485 @ X497) != (X486 @ id @ X497 @ id)) | ? [X492 : subst,X493 : term,X494 : subst,X495 : subst] : (((X486 @ X492 @ X493 @ (comp @ X495 @ X494)) != $true) & ((X486 @ (comp @ X492 @ X495) @ (sub @ X493 @ X495) @ X494) = $true)) | ? [X488 : subst,X489 : subst,X490 : term,X491 : subst] : (((X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X491) != $true) & ((X486 @ X488 @ X490 @ (comp @ X489 @ X491)) = $true)) | ? [X496 : term] : (((X485 @ X496) = $true) & ((X485 @ (sub @ X487 @ (push @ X496 @ id))) != $true)) | ((X485 @ (lam @ X487)) = $true) | ? [X498 : subst > term > term] : (((X486 @ id @ (lam @ (X498 @ sh @ one)) @ id) != $true) & ! [X499 : term] : (($true != (X486 @ id @ X499 @ id)) | ($true = (X486 @ id @ (X498 @ id @ X499) @ id))) & ! [X500 : subst,X501 : subst,X502 : term] : ((X498 @ (comp @ X500 @ X501) @ (sub @ X502 @ X501)) = (sub @ (X498 @ X500 @ X502) @ X501))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ! [X356 : subst] : ? [X357 : d_subst] : ((d2subst @ X357) = X356) & (pushprop_lem2v2_lthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X26 : subst > term > subst > $o] : ? [X27 : term > $o] : ! [X28 : term] : ((X27 @ X28) = (X26 @ id @ X28 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & ((hoasinduction_lem2v2 = $true) <=> ! [X270 : term,X271 : term > $o,X272 : subst > term > subst > $o,X273 : term] : (($true != (X271 @ X273)) | ? [X284 : term] : ((X272 @ id @ X284 @ id) != (X271 @ X284)) | ($true = (X271 @ (ap @ X273 @ X270))) | ($true != (X271 @ X270)) | ? [X280 : subst,X281 : subst,X282 : term,X283 : subst] : (($true != (X272 @ X281 @ X282 @ (comp @ X283 @ X280))) & ((X272 @ (comp @ X281 @ X283) @ (sub @ X282 @ X283) @ X280) = $true)) | ? [X274 : term,X275 : subst,X276 : subst,X277 : subst] : (($true = (X272 @ X277 @ X274 @ (comp @ X276 @ X275))) & ($true != (X272 @ (comp @ X277 @ X276) @ (sub @ X274 @ X276) @ X275))) | ? [X278 : term,X279 : term] : (((X272 @ id @ X278 @ id) = $true) & ($true != (X272 @ id @ (ap @ (sub @ X279 @ id) @ X278) @ id)) & ((X272 @ id @ X279 @ id) = $true)))) & ((hoasinduction_lem3v2 = $true) <=> ! [X29 : subst > term > subst > $o,X30 : term,X31 : term > $o] : (? [X46 : term] : ((X29 @ id @ X46 @ id) != (X31 @ X46)) | ? [X42 : subst,X43 : subst,X44 : term,X45 : subst] : (($true != (X29 @ X42 @ X44 @ (comp @ X43 @ X45))) & ((X29 @ (comp @ X42 @ X43) @ (sub @ X44 @ X43) @ X45) = $true)) | ? [X32 : subst,X33 : subst,X34 : term,X35 : subst] : (($true = (X29 @ X35 @ X34 @ (comp @ X32 @ X33))) & ($true != (X29 @ (comp @ X35 @ X32) @ (sub @ X34 @ X32) @ X33))) | ? [X36 : term] : (($true = (X31 @ X36)) & ($true != (X31 @ (sub @ X30 @ (push @ X36 @ id))))) | ((X31 @ (lam @ X30)) = $true) | ? [X37 : subst > term > term] : (! [X38 : subst,X39 : subst,X40 : term] : ((sub @ (X37 @ X38 @ X40) @ X39) = (X37 @ (comp @ X38 @ X39) @ (sub @ X40 @ X39))) & ((X29 @ id @ (lam @ (X37 @ sh @ one)) @ id) != $true) & ! [X41 : term] : (((X29 @ id @ X41 @ id) != $true) | ($true = (X29 @ id @ (X37 @ id @ X41) @ id)))))) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) <=> (? [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 != X528) & ((ap @ X528 @ X529) = (ap @ X527 @ X530))) | ? [X526 : term] : ((sub @ X526 @ id) != X526) | ! [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) != (ap @ (sub @ X534 @ id) @ X532)) | (X533 = X534)))) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & (hoaslamnotvar_gthm = $true) & ((? [X47 : subst > term > subst > $o] : ! [X48 : term > $o] : ? [X49 : term] : ((X47 @ id @ X49 @ id) != (X48 @ X49)) | ! [X68 : subst > term > subst > $o,X69 : term] : (? [X72 : subst > term > term] : (! [X76 : term] : (($true = (X68 @ id @ (X72 @ id @ X76) @ id)) | ((X68 @ id @ X76 @ id) != $true)) & ! [X73 : subst,X74 : subst,X75 : term] : ((sub @ (X72 @ X74 @ X75) @ X73) = (X72 @ (comp @ X74 @ X73) @ (sub @ X75 @ X73))) & ((X68 @ id @ (lam @ (X72 @ sh @ one)) @ id) != $true)) | ($true = (X68 @ id @ X69 @ id)) | ? [X70 : term,X71 : term] : (((X68 @ id @ X70 @ id) = $true) & ($true = (X68 @ id @ X71 @ id)) & ($true != (X68 @ id @ (ap @ (sub @ X71 @ id) @ X70) @ id)))) | ? [X50 : term > $o,X51 : term] : (! [X52 : term,X53 : term] : (($true != (X50 @ X53)) | ((X50 @ X52) != $true) | ($true = (X50 @ (ap @ X53 @ X52)))) & ((X50 @ X51) != $true) & ! [X54 : term] : (((X50 @ (lam @ X54)) = $true) | ? [X55 : term] : (((X50 @ (sub @ X54 @ (push @ X55 @ id))) != $true) & ((X50 @ X55) = $true))) & ! [X56 : term] : (($true != (var @ X56)) | ($true = (X50 @ X56)))) | ? [X57 : term] : ((sub @ X57 @ id) != X57) | ? [X58 : subst > term > subst > $o,X59 : term,X60 : term > $o] : (! [X61 : term] : (((X60 @ (sub @ X59 @ (push @ X61 @ id))) = $true) | ((X60 @ X61) != $true)) & ($true != (X60 @ (lam @ X59))) & ! [X62 : term] : ((X58 @ id @ X62 @ id) = (X60 @ X62)) & ! [X63 : subst > term > term] : (? [X67 : term] : (((X58 @ id @ (X63 @ id @ X67) @ id) != $true) & ($true = (X58 @ id @ X67 @ id))) | ((X58 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true) | ? [X64 : subst,X65 : term,X66 : subst] : ((sub @ (X63 @ X64 @ X65) @ X66) != (X63 @ (comp @ X64 @ X66) @ (sub @ X65 @ X66)))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (induction2_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoaslamnotap_gthm = $true) & (! [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (($true = (X475 @ (sub @ X473 @ (push @ X476 @ X474)))) | ($true != (X472 @ X473))) <=> (pushprop_lem1 = $true)) & ((hoaslaminj = $true) <=> ! [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (((X144 @ X146 @ X147) = (X145 @ X146 @ X147)) | ? [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) != (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) | ((lam @ (X144 @ sh @ one)) != (lam @ (X145 @ sh @ one))) | ? [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) != (sub @ (X144 @ X149 @ X148) @ X150)))) & ((pushprop_lem0 = $true) <=> ! [X333 : term,X334 : term > $o,X335 : subst] : ? [X336 : term > $o] : ! [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) = (X336 @ X337))) & (! [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) != (ap @ (sub @ X508 @ id) @ X510)) | (X507 = X508)) <=> (hoasapinj1 = $true)) & ! [X338 : d_subst,X339 : d_subst] : ((X338 = X339) | ((d2subst @ X338) != (d2subst @ X339))) & ((pushprop_lem0_lthm = $true) <=> ! [X535 : term > $o,X536 : subst,X537 : term] : ? [X538 : term > $o] : ! [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) = (X538 @ X539))) & (! [X211 : term,X212 : subst > term > subst > $o] : (? [X221 : term] : (((var @ (sub @ X221 @ id)) = $true) & ((X212 @ id @ X221 @ id) != $true)) | ? [X222 : subst > term > term] : (! [X224 : term,X225 : subst,X226 : subst] : ((X222 @ (comp @ X226 @ X225) @ (sub @ X224 @ X225)) = (sub @ (X222 @ X226 @ X224) @ X225)) & ! [X223 : term] : (($true = (X212 @ id @ (X222 @ id @ X223) @ id)) | ((X212 @ id @ X223 @ id) != $true)) & ($true != (X212 @ id @ (lam @ (X222 @ sh @ one)) @ id))) | ? [X217 : subst,X218 : term,X219 : subst,X220 : subst] : (((X212 @ X219 @ X218 @ (comp @ X217 @ X220)) = $true) & ((X212 @ (comp @ X219 @ X217) @ (sub @ X218 @ X217) @ X220) != $true)) | ($true = (X212 @ id @ X211 @ id)) | ? [X227 : term,X228 : term] : (((X212 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id) != $true) & ((X212 @ id @ X228 @ id) = $true) & ((X212 @ id @ X227 @ id) = $true)) | ? [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (((X212 @ X214 @ X213 @ (comp @ X216 @ X215)) != $true) & ($true = (X212 @ (comp @ X214 @ X216) @ (sub @ X213 @ X216) @ X215)))) <=> (hoasinduction = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (? [X163 : subst > term > subst > $o,X164 : term] : (! [X166 : subst > term > term] : (? [X167 : subst,X168 : subst,X169 : term] : ((X166 @ (comp @ X168 @ X167) @ (sub @ X169 @ X167)) != (sub @ (X166 @ X168 @ X169) @ X167)) | ? [X170 : term] : (($true = (X163 @ id @ X170 @ id)) & ((X163 @ id @ (X166 @ id @ X170) @ id) != $true)) | ($true = (X163 @ id @ (lam @ (X166 @ sh @ one)) @ id))) & ! [X165 : term] : (($true != (X163 @ id @ X165 @ id)) | ($true = (X163 @ id @ (sub @ X164 @ (push @ X165 @ id)) @ id))) & ((X163 @ id @ (lam @ (sub @ X164 @ (push @ one @ sh))) @ id) != $true)) | ? [X162 : term] : ((sub @ X162 @ id) != X162) | ! [X171 : term,X172 : subst > term > subst > $o] : (? [X183 : subst,X184 : subst,X185 : term,X186 : subst] : (($true = (X172 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183) @ X186)) & ((X172 @ X184 @ X185 @ (comp @ X183 @ X186)) != $true)) | ? [X173 : term] : (($true != (X172 @ id @ (sub @ X171 @ (push @ X173 @ id)) @ id)) & ((X172 @ id @ X173 @ id) = $true)) | ? [X179 : term,X180 : subst,X181 : subst,X182 : subst] : (((X172 @ X182 @ X179 @ (comp @ X180 @ X181)) = $true) & ($true != (X172 @ (comp @ X182 @ X180) @ (sub @ X179 @ X180) @ X181))) | ? [X174 : subst > term > term] : (! [X176 : subst,X177 : subst,X178 : term] : ((sub @ (X174 @ X176 @ X178) @ X177) = (X174 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177))) & ((X172 @ id @ (lam @ (X174 @ sh @ one)) @ id) != $true) & ! [X175 : term] : (((X172 @ id @ (X174 @ id @ X175) @ id) = $true) | ((X172 @ id @ X175 @ id) != $true))) | ($true = (X172 @ id @ (lam @ X171) @ id))))) & (! [X230 : term > $o,X231 : term] : (? [X232 : term] : (($true != (X230 @ X232)) & ((var @ X232) = $true)) | ((X230 @ X231) = $true) | ? [X235 : term] : (($true != (X230 @ (lam @ X235))) & ($true = (X230 @ X235))) | ? [X233 : term,X234 : term] : (($true = (X230 @ X234)) & ($true = (X230 @ X233)) & ($true != (X230 @ (ap @ X233 @ X234))))) <=> (induction = $true)) & (hoasinduction_lem1 = $true) & ((? [X285 : subst > term > subst > $o] : ! [X286 : term > $o] : ? [X287 : term] : ((X285 @ id @ X287 @ id) != (X286 @ X287)) | ? [X296 : term,X297 : term > $o,X298 : subst > term > subst > $o] : (! [X305 : term] : (((X297 @ X305) != $true) | ((X297 @ (sub @ X296 @ (push @ X305 @ id))) = $true)) & ! [X300 : subst > term > term] : (? [X301 : subst,X302 : term,X303 : subst] : ((sub @ (X300 @ X303 @ X302) @ X301) != (X300 @ (comp @ X303 @ X301) @ (sub @ X302 @ X301))) | ? [X304 : term] : (((X298 @ id @ (X300 @ id @ X304) @ id) != $true) & ($true = (X298 @ id @ X304 @ id))) | ($true = (X298 @ id @ (lam @ (X300 @ sh @ one)) @ id))) & ! [X299 : term] : ((X297 @ X299) = (X298 @ id @ X299 @ id)) & ((X297 @ (lam @ X296)) != $true)) | ? [X288 : term,X289 : term > $o] : (((X289 @ X288) != $true) & ! [X290 : term] : (((var @ X290) != $true) | ($true = (X289 @ X290))) & ! [X291 : term] : (? [X292 : term] : (((X289 @ X292) = $true) & ($true != (X289 @ (sub @ X291 @ (push @ X292 @ id))))) | ((X289 @ (lam @ X291)) = $true)) & ! [X293 : term,X294 : term] : (((X289 @ X293) != $true) | ($true = (X289 @ (ap @ X293 @ X294))) | ((X289 @ X294) != $true))) | ? [X295 : term] : ((sub @ X295 @ id) != X295) | ! [X306 : subst > term > subst > $o,X307 : term] : (? [X309 : subst > term > term] : (($true != (X306 @ id @ (lam @ (X309 @ sh @ one)) @ id)) & ! [X311 : subst,X312 : term,X313 : subst] : ((sub @ (X309 @ X311 @ X312) @ X313) = (X309 @ (comp @ X311 @ X313) @ (sub @ X312 @ X313))) & ! [X310 : term] : (($true = (X306 @ id @ (X309 @ id @ X310) @ id)) | ($true != (X306 @ id @ X310 @ id)))) | ? [X308 : term] : (((X306 @ id @ X308 @ id) != $true) & ((var @ (sub @ X308 @ id)) = $true)) | ((X306 @ id @ X307 @ id) = $true) | ? [X318 : term,X319 : term] : (((X306 @ id @ X318 @ id) = $true) & ($true = (X306 @ id @ X319 @ id)) & ((X306 @ id @ (ap @ (sub @ X318 @ id) @ X319) @ id) != $true)) | ? [X320 : term,X321 : subst,X322 : subst,X323 : subst] : (($true != (X306 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322) @ X323)) & ($true = (X306 @ X321 @ X320 @ (comp @ X322 @ X323)))) | ? [X314 : subst,X315 : subst,X316 : term,X317 : subst] : (((X306 @ X314 @ X316 @ (comp @ X315 @ X317)) != $true) & ((X306 @ (comp @ X314 @ X315) @ (sub @ X316 @ X315) @ X317) = $true)))) <=> (hoasinduction_lthm_3 = $true)) & ((hoaslaminj_lthm = $true) <=> (? [X400 : term,X401 : subst] : ((sub @ one @ (push @ X400 @ X401)) != X400) | ? [X402 : term,X403 : subst] : ((comp @ sh @ (push @ X402 @ X403)) != X403) | ? [X404 : term,X405 : term] : (((lam @ X405) = (lam @ X404)) & (X404 != X405)) | ! [X406 : subst > term > term,X407 : subst,X408 : term,X409 : subst > term > term] : (? [X413 : term,X414 : subst,X415 : subst] : ((sub @ (X409 @ X415 @ X413) @ X414) != (X409 @ (comp @ X415 @ X414) @ (sub @ X413 @ X414))) | ? [X410 : subst,X411 : term,X412 : subst] : ((sub @ (X406 @ X410 @ X411) @ X412) != (X406 @ (comp @ X410 @ X412) @ (sub @ X411 @ X412))) | ((X409 @ X407 @ X408) = (X406 @ X407 @ X408)) | ((lam @ (X409 @ sh @ one)) != (lam @ (X406 @ sh @ one)))))) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (! [X255 : term,X256 : term > $o] : (((X256 @ X255) = $true) | ? [X260 : term] : (! [X261 : term] : (($true != (X256 @ X261)) | ($true = (X256 @ (sub @ X260 @ (push @ X261 @ id))))) & ($true != (X256 @ (lam @ X260)))) | ? [X257 : term,X258 : term] : (((X256 @ X258) = $true) & ($true = (X256 @ X257)) & ((X256 @ (ap @ X257 @ X258)) != $true)) | ? [X259 : term] : (((var @ X259) = $true) & ($true != (X256 @ X259)))) <=> (induction2 = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (hoasapnotvar_lthm = $true) & (! [X325 : term > $o,X326 : subst,X327 : term] : (? [X328 : term,X329 : term] : (($true != (X325 @ (ap @ X329 @ X328))) & ($true = (X325 @ X329)) & ($true = (X325 @ X328))) | ? [X330 : term] : (($true = (var @ X330)) & ($true != (X325 @ (sub @ X330 @ X326)))) | ? [X331 : term] : (((X325 @ (lam @ X331)) != $true) & ! [X332 : term] : (($true != (X325 @ X332)) | ($true = (X325 @ (sub @ X331 @ (push @ X332 @ id)))))) | ($true = (X325 @ (sub @ X327 @ X326)))) <=> (induction2lem = $true)) & (axapp = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X154 : subst > term > subst > $o,X155 : term] : (? [X157 : subst > term > term] : (! [X159 : subst,X160 : subst,X161 : term] : ((X157 @ (comp @ X159 @ X160) @ (sub @ X161 @ X160)) = (sub @ (X157 @ X159 @ X161) @ X160)) & ($true != (X154 @ id @ (lam @ (X157 @ sh @ one)) @ id)) & ! [X158 : term] : (($true = (X154 @ id @ (X157 @ id @ X158) @ id)) | ($true != (X154 @ id @ X158 @ id)))) | ($true = (X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) | ? [X156 : term] : (((X154 @ id @ X156 @ id) = $true) & ($true != (X154 @ id @ (sub @ X155 @ (push @ X156 @ id)) @ id))))) & (! [X391 : term] : ? [X392 : subst > term > term] : ! [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) = (X392 @ X393 @ X394)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((induction2lem_lthm = $true) <=> (? [X9 : subst,X10 : term] : ((comp @ sh @ (push @ X10 @ X9)) != X9) | ? [X12 : term > $o,X13 : term] : (! [X14 : term] : (($true != (var @ X14)) | ($true = (X12 @ X14))) & ! [X16 : term,X17 : term] : (((X12 @ (ap @ X17 @ X16)) = $true) | ((X12 @ X16) != $true) | ($true != (X12 @ X17))) & ($true != (X12 @ X13)) & ! [X15 : term] : (($true != (X12 @ X15)) | ((X12 @ (lam @ X15)) = $true))) | ? [X7 : term,X8 : subst] : ((sub @ one @ (push @ X7 @ X8)) != X7) | ? [X11 : subst] : ((comp @ X11 @ id) != X11) | ! [X18 : subst,X19 : term,X20 : term > $o] : (($true = (X20 @ (sub @ X19 @ X18))) | ? [X21 : term] : (($true != (X20 @ (sub @ X21 @ X18))) & ((var @ X21) = $true)) | ? [X24 : term] : (((X20 @ (lam @ X24)) != $true) & ! [X25 : term] : (((X20 @ X25) != $true) | ((X20 @ (sub @ X24 @ (push @ X25 @ id))) = $true))) | ? [X22 : term,X23 : term] : (($true != (X20 @ (ap @ X23 @ X22))) & ($true = (X20 @ X23)) & ((X20 @ X22) = $true))))) & (axvarshift = $true) & ((axidr = $true) <=> ! [X416 : subst] : ((comp @ X416 @ id) = X416)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (lamnotvar = $true) & ((hoasinduction_lem3a = $true) <=> ! [X262 : subst > term > subst > $o,X263 : term] : (? [X264 : term] : (((X262 @ id @ (sub @ X263 @ (push @ X264 @ id)) @ id) != $true) & ((X262 @ id @ X264 @ id) = $true)) | ((X262 @ id @ (lam @ X263) @ id) = $true) | ? [X265 : subst > term > term] : (! [X266 : term] : (($true = (X262 @ id @ (X265 @ id @ X266) @ id)) | ((X262 @ id @ X266 @ id) != $true)) & ! [X267 : subst,X268 : subst,X269 : term] : ((sub @ (X265 @ X268 @ X269) @ X267) = (X265 @ (comp @ X268 @ X267) @ (sub @ X269 @ X267))) & ($true != (X262 @ id @ (lam @ (X265 @ sh @ one)) @ id))))) & (hoasinduction_lem3v2_gthm = $true) & ! [X417 : d_subst] : (d_id = X417) & ((apinj1 = $true) <=> ! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 = X239) | ((ap @ X236 @ X238) != (ap @ X239 @ X237)))) & ((induction2_lthm = $true) <=> (? [X94 : subst,X95 : term > $o,X96 : term] : (! [X100 : term] : (((X95 @ (lam @ X100)) = $true) | ? [X101 : term] : (((X95 @ (sub @ X100 @ (push @ X101 @ id))) != $true) & ((X95 @ X101) = $true))) & ((X95 @ (sub @ X96 @ X94)) != $true) & ! [X97 : term,X98 : term] : (($true = (X95 @ (ap @ X97 @ X98))) | ((X95 @ X98) != $true) | ($true != (X95 @ X97))) & ! [X99 : term] : (((var @ X99) != $true) | ($true = (X95 @ (sub @ X99 @ X94))))) | ? [X93 : term] : ((sub @ X93 @ id) != X93) | ! [X102 : term > $o,X103 : term] : (((X102 @ X103) = $true) | ? [X104 : term,X105 : term] : (((X102 @ X105) = $true) & ((X102 @ X104) = $true) & ((X102 @ (ap @ X104 @ X105)) != $true)) | ? [X106 : term] : (((X102 @ (lam @ X106)) != $true) & ! [X107 : term] : (($true = (X102 @ (sub @ X106 @ (push @ X107 @ id)))) | ((X102 @ X107) != $true))) | ? [X108 : term] : (((X102 @ X108) != $true) & ($true = (var @ X108)))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X340 : subst > term > subst > $o,X341 : term] : (? [X342 : term] : (($true != (X340 @ id @ (sub @ X341 @ (push @ X342 @ id)) @ id)) & ((X340 @ id @ X342 @ id) = $true)) | ? [X343 : term,X344 : subst > term > term] : (((X340 @ id @ (lam @ (X344 @ sh @ one)) @ id) != $true) & ! [X345 : subst,X346 : subst,X347 : term] : ((sub @ (sub @ X343 @ (push @ X347 @ X346)) @ X345) = (sub @ (X344 @ X346 @ X347) @ X345)) & ! [X349 : subst,X350 : subst,X351 : term] : ((sub @ X343 @ (push @ (sub @ X351 @ X350) @ (comp @ X349 @ X350))) = (X344 @ (comp @ X349 @ X350) @ (sub @ X351 @ X350))) & ! [X348 : term] : (($true = (X340 @ id @ (X344 @ id @ X348) @ id)) | ($true != (X340 @ id @ X348 @ id)))) | ($true = (X340 @ id @ (lam @ (sub @ X341 @ (push @ one @ sh))) @ id)))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (((X479 @ X481) != $true) | ($true = (X480 @ one)) | ? [X483 : term] : ((X480 @ X483) != (X479 @ (sub @ X483 @ (push @ X481 @ X482))))) | ? [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) != X478))) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (? [X244 : term] : ((sub @ X244 @ id) != X244) | ! [X245 : term,X246 : subst > term > subst > $o,X247 : term > $o] : (? [X248 : subst > term > term] : (! [X252 : term] : (($true != (X246 @ id @ X252 @ id)) | ($true = (X246 @ id @ (X248 @ id @ X252) @ id))) & ! [X249 : term,X250 : subst,X251 : subst] : ((sub @ (X248 @ X251 @ X249) @ X250) = (X248 @ (comp @ X251 @ X250) @ (sub @ X249 @ X250))) & ((X246 @ id @ (lam @ (X248 @ sh @ one)) @ id) != $true)) | ? [X253 : term] : ((X247 @ X253) != (X246 @ id @ X253 @ id)) | ? [X254 : term] : (((X247 @ X254) = $true) & ((X247 @ (sub @ X245 @ (push @ X254 @ id))) != $true)) | ($true = (X247 @ (lam @ X245)))) | ? [X240 : term] : ! [X241 : subst > term > term] : ? [X242 : term,X243 : subst] : ((X241 @ X243 @ X242) != (sub @ X240 @ (push @ X242 @ X243))))) & ((laminj = $true) <=> ! [X366 : term,X367 : term] : ((X366 = X367) | ((lam @ X366) != (lam @ X367)))) & (! [X77 : term,X78 : term] : (((sub @ X78 @ sh) != (sub @ X77 @ sh)) | (X77 = X78)) <=> (shinj = $true)) & (hoasapnotvar_gthm = $true) & ((! [X6 : term] : ((sub @ X6 @ id) = X6) | ? [X5 : term] : ((sub @ X5 @ id) != X5)) <=> (termmset_lthm = $true)) & (! [X109 : term] : ((sub @ X109 @ id) = X109) <=> (termmset = $true)) & (apnotvar = $true) & ((d2subst @ d_id) = id) & ((hoasinduction_lthm = $true) <=> (? [X418 : term,X419 : term > $o] : (! [X420 : term] : (? [X421 : term] : (($true != (X419 @ (sub @ X420 @ (push @ X421 @ id)))) & ($true = (X419 @ X421))) | ((X419 @ (lam @ X420)) = $true)) & ! [X422 : term] : (($true = (X419 @ X422)) | ((var @ X422) != $true)) & ! [X423 : term,X424 : term] : (($true = (X419 @ (ap @ X424 @ X423))) | ((X419 @ X423) != $true) | ((X419 @ X424) != $true)) & ($true != (X419 @ X418))) | ! [X454 : term,X455 : subst > term > subst > $o] : (? [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (($true = (X455 @ (comp @ X468 @ X469) @ (sub @ X467 @ X469) @ X470)) & ((X455 @ X468 @ X467 @ (comp @ X469 @ X470)) != $true)) | ($true = (X455 @ id @ X454 @ id)) | ? [X458 : subst > term > term] : (((X455 @ id @ (lam @ (X458 @ sh @ one)) @ id) != $true) & ! [X459 : term] : (((X455 @ id @ X459 @ id) != $true) | ((X455 @ id @ (X458 @ id @ X459) @ id) = $true)) & ! [X460 : subst,X461 : subst,X462 : term] : ((X458 @ (comp @ X460 @ X461) @ (sub @ X462 @ X461)) = (sub @ (X458 @ X460 @ X462) @ X461))) | ? [X456 : term,X457 : term] : (((X455 @ id @ (ap @ (sub @ X456 @ id) @ X457) @ id) != $true) & ($true = (X455 @ id @ X456 @ id)) & ($true = (X455 @ id @ X457 @ id))) | ? [X471 : term] : (((X455 @ id @ X471 @ id) != $true) & ($true = (var @ (sub @ X471 @ id)))) | ? [X463 : subst,X464 : term,X465 : subst,X466 : subst] : (((X455 @ (comp @ X466 @ X465) @ (sub @ X464 @ X465) @ X463) != $true) & ($true = (X455 @ X466 @ X464 @ (comp @ X465 @ X463))))) | ? [X425 : term,X426 : term,X427 : subst > term > subst > $o] : (((X427 @ id @ X425 @ id) = $true) & ! [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X427 @ (comp @ X429 @ X430) @ (sub @ X428 @ X430) @ X431)) | ((X427 @ X429 @ X428 @ (comp @ X430 @ X431)) != $true)) & ((X427 @ id @ (ap @ X426 @ X425) @ id) != $true) & ($true = (X427 @ id @ X426 @ id)) & ! [X436 : term,X437 : term] : (($true = (X427 @ id @ (ap @ (sub @ X437 @ id) @ X436) @ id)) | ($true != (X427 @ id @ X437 @ id)) | ((X427 @ id @ X436 @ id) != $true)) & ! [X432 : subst,X433 : subst,X434 : term,X435 : subst] : (((X427 @ (comp @ X432 @ X433) @ (sub @ X434 @ X433) @ X435) != $true) | ((X427 @ X432 @ X434 @ (comp @ X433 @ X435)) = $true))) | ? [X438 : term,X439 : subst > term > subst > $o] : (! [X445 : term] : (($true != (X439 @ id @ X445 @ id)) | ($true = (X439 @ id @ (sub @ X438 @ (push @ X445 @ id)) @ id))) & ! [X440 : subst > term > term] : (? [X441 : subst,X442 : subst,X443 : term] : ((sub @ (X440 @ X442 @ X443) @ X441) != (X440 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441))) | ? [X444 : term] : (((X439 @ id @ X444 @ id) = $true) & ($true != (X439 @ id @ (X440 @ id @ X444) @ id))) | ((X439 @ id @ (lam @ (X440 @ sh @ one)) @ id) = $true)) & ($true != (X439 @ id @ (lam @ X438) @ id)) & ! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (((X439 @ (comp @ X449 @ X447) @ (sub @ X448 @ X447) @ X446) = $true) | ((X439 @ X449 @ X448 @ (comp @ X447 @ X446)) != $true)) & ! [X450 : subst,X451 : subst,X452 : term,X453 : subst] : (($true = (X439 @ X453 @ X452 @ (comp @ X450 @ X451))) | ((X439 @ (comp @ X453 @ X450) @ (sub @ X452 @ X450) @ X451) != $true))))) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lthm_orig = $true) & ((hoasapinj2_lthm = $true) <=> (? [X358 : term,X359 : term,X360 : term,X361 : term] : ((X358 != X361) & ((ap @ X360 @ X361) = (ap @ X359 @ X358))) | ! [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X364 @ id) @ X365)) | (X363 = X365)))) & (hoaslamnotvar = $true) & (pushprop_lthm = $true) & (! [X83 : subst > term > subst > $o,X84 : term,X85 : term > $o] : (($true = (X85 @ (lam @ X84))) | ? [X88 : subst > term > term] : (! [X92 : term] : (((X83 @ id @ X92 @ id) != $true) | ((X83 @ id @ (X88 @ id @ X92) @ id) = $true)) & ! [X89 : subst,X90 : subst,X91 : term] : ((X88 @ (comp @ X90 @ X89) @ (sub @ X91 @ X89)) = (sub @ (X88 @ X90 @ X91) @ X89)) & ($true != (X83 @ id @ (lam @ (X88 @ sh @ one)) @ id))) | ? [X87 : term] : ((X83 @ id @ X87 @ id) != (X85 @ X87)) | ? [X86 : term] : (($true = (X85 @ X86)) & ($true != (X85 @ (sub @ X84 @ (push @ X86 @ id)))))) <=> (hoasinduction_lem3v2a = $true)) & (pushprop_lem2v2_gthm = $true) & (! [X122 : term,X123 : subst > term > subst > $o] : (? [X125 : subst > term > term] : (! [X126 : subst,X127 : term,X128 : subst] : ((X125 @ (comp @ X128 @ X126) @ (sub @ X127 @ X126)) = (sub @ (X125 @ X128 @ X127) @ X126)) & ! [X129 : term] : (((X123 @ id @ X129 @ id) != $true) | ($true = (X123 @ id @ (X125 @ id @ X129) @ id))) & ($true != (X123 @ id @ (lam @ (X125 @ sh @ one)) @ id))) | ? [X124 : term] : (($true != (X123 @ id @ (sub @ X122 @ (push @ X124 @ id)) @ id)) & ($true = (X123 @ id @ X124 @ id))) | ((X123 @ id @ (lam @ X122) @ id) = $true) | ? [X134 : subst,X135 : subst,X136 : term,X137 : subst] : (((X123 @ (comp @ X135 @ X137) @ (sub @ X136 @ X137) @ X134) = $true) & ((X123 @ X135 @ X136 @ (comp @ X137 @ X134)) != $true)) | ? [X130 : subst,X131 : subst,X132 : subst,X133 : term] : (((X123 @ (comp @ X131 @ X130) @ (sub @ X133 @ X130) @ X132) != $true) & ((X123 @ X131 @ X133 @ (comp @ X130 @ X132)) = $true))) <=> (hoasinduction_lem3 = $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (axassoc = $true) & (axabs = $true) & (lamnotap != $true) & (! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 = X354) | ((ap @ (sub @ X352 @ id) @ X353) != (ap @ (sub @ X355 @ id) @ X354))) <=> (hoasapinj2 = $true)) & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & (pushprop = $true) & (hoasinduction_lem3_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true != (X395 @ X397)) | ($true = (X396 @ one)) | ? [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) != (X396 @ X399)))) & ((! [X373 : subst] : ((comp @ X373 @ id) = X373) & ! [X372 : subst] : ((comp @ id @ X372) = X372)) <=> (substmonoid = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X110 : subst > term > subst > $o] : ? [X111 : term > $o] : ! [X112 : term] : ((X110 @ id @ X112 @ id) = (X111 @ X112))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X512 : subst > term > subst > $o,X513 : term] : (? [X514 : term] : (((X512 @ id @ X514 @ id) = $true) & ((X512 @ id @ (sub @ X513 @ (push @ X514 @ id)) @ id) != $true)) | ? [X515 : subst > term > term] : (! [X519 : term] : (((X512 @ id @ X519 @ id) != $true) | ((X512 @ id @ (X515 @ id @ X519) @ id) = $true)) & ! [X516 : subst,X517 : term,X518 : subst] : ((X515 @ (comp @ X518 @ X516) @ (sub @ X517 @ X516)) = (sub @ (X515 @ X518 @ X517) @ X516)) & ((X512 @ id @ (lam @ (X515 @ sh @ one)) @ id) != $true)) | ($true = (X512 @ id @ (lam @ (sub @ X513 @ (push @ one @ sh))) @ id)))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X1 : term] : ? [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) = (X2 @ sh @ one))) & (ulamvarsh = $true) & ! [X229 : d_term] : (d_one = X229) & (axclos = $true) & (substmonoid_gthm = $true) & (induction2lem_gthm = $true) & (pushprop_lem2v2 = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X202 : subst > term > subst > $o,X203 : term] : (? [X206 : subst > term > term] : (! [X207 : term] : (((X202 @ id @ X207 @ id) != $true) | ($true = (X202 @ id @ (X206 @ id @ X207) @ id))) & ! [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) = (sub @ (X206 @ X210 @ X208) @ X209)) & ($true != (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id))) | ? [X204 : term,X205 : term] : (((X202 @ id @ (ap @ (sub @ X204 @ id) @ X205) @ id) != $true) & ($true = (X202 @ id @ X205 @ id)) & ($true = (X202 @ id @ X204 @ id))) | ((X202 @ id @ X203 @ id) = $true))) & (axmap = $true) & ((hoasinduction_lem2 = $true) <=> ! [X187 : term,X188 : subst > term > subst > $o,X189 : term] : (($true = (X188 @ id @ (ap @ X189 @ X187) @ id)) | ? [X198 : term,X199 : term] : (($true = (X188 @ id @ X199 @ id)) & ($true != (X188 @ id @ (ap @ (sub @ X199 @ id) @ X198) @ id)) & ((X188 @ id @ X198 @ id) = $true)) | ($true != (X188 @ id @ X187 @ id)) | ($true != (X188 @ id @ X189 @ id)) | ? [X190 : subst,X191 : subst,X192 : term,X193 : subst] : (((X188 @ X191 @ X192 @ (comp @ X193 @ X190)) != $true) & ((X188 @ (comp @ X191 @ X193) @ (sub @ X192 @ X193) @ X190) = $true)) | ? [X194 : subst,X195 : subst,X196 : term,X197 : subst] : (($true = (X188 @ X194 @ X196 @ (comp @ X197 @ X195))) & ((X188 @ (comp @ X194 @ X197) @ (sub @ X196 @ X197) @ X195) != $true)))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (! [X142 : term] : ? [X143 : subst > term > term] : ((X143 @ sh @ one) = (sub @ X142 @ (push @ one @ sh))) <=> (hoasinduction_lem3b = $true)) & (one = (d2term @ d_one)) & (termmset_gthm = $true) & (! [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) = X371) <=> (axvarcons = $true)) & (ulamvarind = $true) & (hoasapinj1_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X511 : term] : ((sub @ X511 @ id) = X511) <=> (axvarid = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ! [X368 : d_term,X369 : d_term] : (((d2term @ X368) != (d2term @ X369)) | (X368 = X369)) & ((? [X504 : subst] : ((comp @ X504 @ id) != X504) | ? [X503 : subst] : ((comp @ id @ X503) != X503) | (! [X506 : subst] : ((comp @ id @ X506) = X506) & ! [X505 : subst] : ((comp @ X505 @ id) = X505))) <=> (substmonoid_lthm = $true)) & (ulamvar1 != $true) & (pushprop_gthm = $true) & ((d2subst @ d_id) = sh) & ((axscons = $true) <=> ! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0)) & ((axshiftcons = $true) <=> ! [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) = X201)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem2v2_gthm = $true) & ((! [X383 : term,X384 : subst > term > subst > $o] : (($true = (X384 @ id @ (lam @ X383) @ id)) | ? [X385 : term] : (((X384 @ id @ (sub @ X383 @ (push @ X385 @ id)) @ id) != $true) & ($true = (X384 @ id @ X385 @ id))) | ? [X386 : subst > term > term] : (($true != (X384 @ id @ (lam @ (X386 @ sh @ one)) @ id)) & ! [X388 : subst,X389 : subst,X390 : term] : ((sub @ (X386 @ X389 @ X390) @ X388) = (X386 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388))) & ! [X387 : term] : (($true = (X384 @ id @ (X386 @ id @ X387) @ id)) | ((X384 @ id @ X387 @ id) != $true)))) | ? [X375 : term,X376 : subst > term > subst > $o] : (! [X382 : term] : (($true != (X376 @ id @ X382 @ id)) | ($true = (X376 @ id @ (sub @ X375 @ (push @ X382 @ id)) @ id))) & ((X376 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id) != $true) & ! [X377 : subst > term > term] : (($true = (X376 @ id @ (lam @ (X377 @ sh @ one)) @ id)) | ? [X378 : term] : (($true != (X376 @ id @ (X377 @ id @ X378) @ id)) & ($true = (X376 @ id @ X378 @ id))) | ? [X379 : subst,X380 : subst,X381 : term] : ((X377 @ (comp @ X379 @ X380) @ (sub @ X381 @ X380)) != (sub @ (X377 @ X379 @ X381) @ X380)))) | ? [X374 : term] : ((sub @ X374 @ id) != X374)) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) != X114) | ? [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) != X116) | ! [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true = (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) | ($true != (X120 @ X121))))) & (hoaslamnotvar_lthm = $true) & ((hoaslamnotap = $true) <=> ! [X520 : term,X521 : term,X522 : subst > term > term] : (((ap @ (sub @ X521 @ id) @ X520) != (lam @ (X522 @ sh @ one))) | ? [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) != (sub @ (X522 @ X525 @ X524) @ X523))))),
% 9.64/2.82 inference(flattening,[status(thm)],[f12])).
% 9.64/2.82 thf(f12,plain,(
% 9.64/2.82 (pushprop_lem1v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X383 : term,X384 : subst > term > subst > $o] : (($true = (X384 @ id @ (lam @ X383) @ id)) | ? [X385 : term] : (((X384 @ id @ (sub @ X383 @ (push @ X385 @ id)) @ id) != $true) & ($true = (X384 @ id @ X385 @ id))) | ? [X386 : subst > term > term] : (($true != (X384 @ id @ (lam @ (X386 @ sh @ one)) @ id)) & ! [X388 : subst,X389 : subst,X390 : term] : ((sub @ (X386 @ X389 @ X390) @ X388) = (X386 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388))) & ! [X387 : term] : (($true = (X384 @ id @ (X386 @ id @ X387) @ id)) | ((X384 @ id @ X387 @ id) != $true)))) | ? [X375 : term,X376 : subst > term > subst > $o] : (! [X382 : term] : (($true != (X376 @ id @ X382 @ id)) | ($true = (X376 @ id @ (sub @ X375 @ (push @ X382 @ id)) @ id))) & ((X376 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id) != $true) & ! [X377 : subst > term > term] : (($true = (X376 @ id @ (lam @ (X377 @ sh @ one)) @ id)) | ? [X378 : term] : (($true != (X376 @ id @ (X377 @ id @ X378) @ id)) & ($true = (X376 @ id @ X378 @ id))) | ? [X379 : subst,X380 : subst,X381 : term] : ((X377 @ (comp @ X379 @ X380) @ (sub @ X381 @ X380)) != (sub @ (X377 @ X379 @ X381) @ X380))))) | ? [X374 : term] : ((sub @ X374 @ id) != X374))) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotap_gthm = $true) & (! [X211 : term,X212 : subst > term > subst > $o] : (? [X221 : term] : (((var @ (sub @ X221 @ id)) = $true) & ((X212 @ id @ X221 @ id) != $true)) | ? [X222 : subst > term > term] : (! [X224 : term,X225 : subst,X226 : subst] : ((X222 @ (comp @ X226 @ X225) @ (sub @ X224 @ X225)) = (sub @ (X222 @ X226 @ X224) @ X225)) & ! [X223 : term] : (($true = (X212 @ id @ (X222 @ id @ X223) @ id)) | ((X212 @ id @ X223 @ id) != $true)) & ($true != (X212 @ id @ (lam @ (X222 @ sh @ one)) @ id))) | ? [X217 : subst,X218 : term,X219 : subst,X220 : subst] : (((X212 @ X219 @ X218 @ (comp @ X217 @ X220)) = $true) & ((X212 @ (comp @ X219 @ X217) @ (sub @ X218 @ X217) @ X220) != $true)) | ($true = (X212 @ id @ X211 @ id)) | ? [X227 : term,X228 : term] : (((X212 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id) != $true) & ((X212 @ id @ X228 @ id) = $true) & ((X212 @ id @ X227 @ id) = $true)) | ? [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (((X212 @ X214 @ X213 @ (comp @ X216 @ X215)) != $true) & ($true = (X212 @ (comp @ X214 @ X216) @ (sub @ X213 @ X216) @ X215)))) <=> (hoasinduction = $true)) & (hoasapinj2_gthm = $true) & ((d2subst @ d_id) = sh) & (axvarshift = $true) & ((hoaslaminj_lthm = $true) <=> (((! [X406 : subst > term > term,X407 : subst,X408 : term,X409 : subst > term > term] : (? [X413 : term,X414 : subst,X415 : subst] : ((sub @ (X409 @ X415 @ X413) @ X414) != (X409 @ (comp @ X415 @ X414) @ (sub @ X413 @ X414))) | ? [X410 : subst,X411 : term,X412 : subst] : ((sub @ (X406 @ X410 @ X411) @ X412) != (X406 @ (comp @ X410 @ X412) @ (sub @ X411 @ X412))) | ((X409 @ X407 @ X408) = (X406 @ X407 @ X408)) | ((lam @ (X409 @ sh @ one)) != (lam @ (X406 @ sh @ one)))) | ? [X404 : term,X405 : term] : (((lam @ X405) = (lam @ X404)) & (X404 != X405))) | ? [X402 : term,X403 : subst] : ((comp @ sh @ (push @ X402 @ X403)) != X403)) | ? [X400 : term,X401 : subst] : ((sub @ one @ (push @ X400 @ X401)) != X400))) & ((hoasinduction_lthm = $true) <=> (((! [X454 : term,X455 : subst > term > subst > $o] : (? [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (($true = (X455 @ (comp @ X468 @ X469) @ (sub @ X467 @ X469) @ X470)) & ((X455 @ X468 @ X467 @ (comp @ X469 @ X470)) != $true)) | ($true = (X455 @ id @ X454 @ id)) | ? [X458 : subst > term > term] : (((X455 @ id @ (lam @ (X458 @ sh @ one)) @ id) != $true) & ! [X459 : term] : (((X455 @ id @ X459 @ id) != $true) | ((X455 @ id @ (X458 @ id @ X459) @ id) = $true)) & ! [X460 : subst,X461 : subst,X462 : term] : ((X458 @ (comp @ X460 @ X461) @ (sub @ X462 @ X461)) = (sub @ (X458 @ X460 @ X462) @ X461))) | ? [X456 : term,X457 : term] : (((X455 @ id @ (ap @ (sub @ X456 @ id) @ X457) @ id) != $true) & ($true = (X455 @ id @ X456 @ id)) & ($true = (X455 @ id @ X457 @ id))) | ? [X471 : term] : (((X455 @ id @ X471 @ id) != $true) & ($true = (var @ (sub @ X471 @ id)))) | ? [X463 : subst,X464 : term,X465 : subst,X466 : subst] : (((X455 @ (comp @ X466 @ X465) @ (sub @ X464 @ X465) @ X463) != $true) & ($true = (X455 @ X466 @ X464 @ (comp @ X465 @ X463))))) | ? [X438 : term,X439 : subst > term > subst > $o] : (! [X445 : term] : (($true != (X439 @ id @ X445 @ id)) | ($true = (X439 @ id @ (sub @ X438 @ (push @ X445 @ id)) @ id))) & ! [X440 : subst > term > term] : (? [X441 : subst,X442 : subst,X443 : term] : ((sub @ (X440 @ X442 @ X443) @ X441) != (X440 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441))) | ? [X444 : term] : (((X439 @ id @ X444 @ id) = $true) & ($true != (X439 @ id @ (X440 @ id @ X444) @ id))) | ((X439 @ id @ (lam @ (X440 @ sh @ one)) @ id) = $true)) & ($true != (X439 @ id @ (lam @ X438) @ id)) & ! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (((X439 @ (comp @ X449 @ X447) @ (sub @ X448 @ X447) @ X446) = $true) | ((X439 @ X449 @ X448 @ (comp @ X447 @ X446)) != $true)) & ! [X450 : subst,X451 : subst,X452 : term,X453 : subst] : (($true = (X439 @ X453 @ X452 @ (comp @ X450 @ X451))) | ((X439 @ (comp @ X453 @ X450) @ (sub @ X452 @ X450) @ X451) != $true)))) | ? [X425 : term,X426 : term,X427 : subst > term > subst > $o] : (((X427 @ id @ X425 @ id) = $true) & ! [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X427 @ (comp @ X429 @ X430) @ (sub @ X428 @ X430) @ X431)) | ((X427 @ X429 @ X428 @ (comp @ X430 @ X431)) != $true)) & ((X427 @ id @ (ap @ X426 @ X425) @ id) != $true) & ($true = (X427 @ id @ X426 @ id)) & ! [X436 : term,X437 : term] : (($true = (X427 @ id @ (ap @ (sub @ X437 @ id) @ X436) @ id)) | ($true != (X427 @ id @ X437 @ id)) | ((X427 @ id @ X436 @ id) != $true)) & ! [X432 : subst,X433 : subst,X434 : term,X435 : subst] : (((X427 @ (comp @ X432 @ X433) @ (sub @ X434 @ X433) @ X435) != $true) | ((X427 @ X432 @ X434 @ (comp @ X433 @ X435)) = $true)))) | ? [X418 : term,X419 : term > $o] : (! [X420 : term] : (? [X421 : term] : (($true != (X419 @ (sub @ X420 @ (push @ X421 @ id)))) & ($true = (X419 @ X421))) | ((X419 @ (lam @ X420)) = $true)) & ! [X422 : term] : (($true = (X419 @ X422)) | ((var @ X422) != $true)) & ! [X423 : term,X424 : term] : (($true = (X419 @ (ap @ X424 @ X423))) | ((X419 @ X423) != $true) | ((X419 @ X424) != $true)) & ($true != (X419 @ X418))))) & ((pushprop_lem0_lthm = $true) <=> ! [X535 : term > $o,X536 : subst,X537 : term] : ? [X538 : term > $o] : ! [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) = (X538 @ X539))) & (axclos = $true) & ! [X229 : d_term] : (d_one = X229) & (pushprop_lthm = $true) & ((! [X373 : subst] : ((comp @ X373 @ id) = X373) & ! [X372 : subst] : ((comp @ id @ X372) = X372)) <=> (substmonoid = $true)) & (! [X324 : subst] : ((comp @ id @ X324) = X324) <=> (axidl = $true)) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X171 : term,X172 : subst > term > subst > $o] : (? [X183 : subst,X184 : subst,X185 : term,X186 : subst] : (($true = (X172 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183) @ X186)) & ((X172 @ X184 @ X185 @ (comp @ X183 @ X186)) != $true)) | ? [X173 : term] : (($true != (X172 @ id @ (sub @ X171 @ (push @ X173 @ id)) @ id)) & ((X172 @ id @ X173 @ id) = $true)) | ? [X179 : term,X180 : subst,X181 : subst,X182 : subst] : (((X172 @ X182 @ X179 @ (comp @ X180 @ X181)) = $true) & ($true != (X172 @ (comp @ X182 @ X180) @ (sub @ X179 @ X180) @ X181))) | ? [X174 : subst > term > term] : (! [X176 : subst,X177 : subst,X178 : term] : ((sub @ (X174 @ X176 @ X178) @ X177) = (X174 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177))) & ((X172 @ id @ (lam @ (X174 @ sh @ one)) @ id) != $true) & ! [X175 : term] : (((X172 @ id @ (X174 @ id @ X175) @ id) = $true) | ((X172 @ id @ X175 @ id) != $true))) | ($true = (X172 @ id @ (lam @ X171) @ id))) | ? [X163 : subst > term > subst > $o,X164 : term] : (! [X166 : subst > term > term] : (? [X167 : subst,X168 : subst,X169 : term] : ((X166 @ (comp @ X168 @ X167) @ (sub @ X169 @ X167)) != (sub @ (X166 @ X168 @ X169) @ X167)) | ? [X170 : term] : (($true = (X163 @ id @ X170 @ id)) & ((X163 @ id @ (X166 @ id @ X170) @ id) != $true)) | ($true = (X163 @ id @ (lam @ (X166 @ sh @ one)) @ id))) & ! [X165 : term] : (($true != (X163 @ id @ X165 @ id)) | ($true = (X163 @ id @ (sub @ X164 @ (push @ X165 @ id)) @ id))) & ((X163 @ id @ (lam @ (sub @ X164 @ (push @ one @ sh))) @ id) != $true))) | ? [X162 : term] : ((sub @ X162 @ id) != X162))) & ((hoasinduction_lem3aa = $true) <=> ! [X154 : subst > term > subst > $o,X155 : term] : (? [X157 : subst > term > term] : (! [X159 : subst,X160 : subst,X161 : term] : ((X157 @ (comp @ X159 @ X160) @ (sub @ X161 @ X160)) = (sub @ (X157 @ X159 @ X161) @ X160)) & ($true != (X154 @ id @ (lam @ (X157 @ sh @ one)) @ id)) & ! [X158 : term] : (($true = (X154 @ id @ (X157 @ id @ X158) @ id)) | ($true != (X154 @ id @ X158 @ id)))) | ($true = (X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) | ? [X156 : term] : (((X154 @ id @ X156 @ id) = $true) & ($true != (X154 @ id @ (sub @ X155 @ (push @ X156 @ id)) @ id))))) & ((apinj2 = $true) <=> ! [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) != (ap @ X79 @ X82)) | (X81 = X82))) & (! [X391 : term] : ? [X392 : subst > term > term] : ! [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) = (X392 @ X393 @ X394)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X255 : term,X256 : term > $o] : (((X256 @ X255) = $true) | ? [X260 : term] : (! [X261 : term] : (($true != (X256 @ X261)) | ($true = (X256 @ (sub @ X260 @ (push @ X261 @ id))))) & ($true != (X256 @ (lam @ X260)))) | ? [X257 : term,X258 : term] : (((X256 @ X258) = $true) & ($true = (X256 @ X257)) & ((X256 @ (ap @ X257 @ X258)) != $true)) | ? [X259 : term] : (((var @ X259) = $true) & ($true != (X256 @ X259)))) <=> (induction2 = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X202 : subst > term > subst > $o,X203 : term] : (? [X206 : subst > term > term] : (! [X207 : term] : (((X202 @ id @ X207 @ id) != $true) | ($true = (X202 @ id @ (X206 @ id @ X207) @ id))) & ! [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) = (sub @ (X206 @ X210 @ X208) @ X209)) & ($true != (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id))) | ? [X204 : term,X205 : term] : (((X202 @ id @ (ap @ (sub @ X204 @ id) @ X205) @ id) != $true) & ($true = (X202 @ id @ X205 @ id)) & ($true = (X202 @ id @ X204 @ id))) | ((X202 @ id @ X203 @ id) = $true))) & (! [X138 : term] : ? [X139 : subst > term > term] : ! [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) = (X139 @ X141 @ X140)) <=> (hoasinduction_lem3v2_f = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (((X479 @ X481) != $true) | ($true = (X480 @ one)) | ? [X483 : term] : ((X480 @ X483) != (X479 @ (sub @ X483 @ (push @ X481 @ X482))))) | ? [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) != X478))) & ((hoasapinj1_lthm = $true) <=> ((! [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) != (ap @ (sub @ X534 @ id) @ X532)) | (X533 = X534)) | ? [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 != X528) & ((ap @ X528 @ X529) = (ap @ X527 @ X530)))) | ? [X526 : term] : ((sub @ X526 @ id) != X526))) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> ((! [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true = (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) | ($true != (X120 @ X121))) | ? [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) != X116)) | ? [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) != X114))) & (hoasinduction_gthm = $true) & (pushprop_lem3v2 = $true) & (hoasinduction_lem2v2_gthm = $true) & (! [X325 : term > $o,X326 : subst,X327 : term] : (? [X328 : term,X329 : term] : (($true != (X325 @ (ap @ X329 @ X328))) & ($true = (X325 @ X329)) & ($true = (X325 @ X328))) | ? [X330 : term] : (($true = (var @ X330)) & ($true != (X325 @ (sub @ X330 @ X326)))) | ? [X331 : term] : (((X325 @ (lam @ X331)) != $true) & ! [X332 : term] : (($true != (X325 @ X332)) | ($true = (X325 @ (sub @ X331 @ (push @ X332 @ id)))))) | ($true = (X325 @ (sub @ X327 @ X326)))) <=> (induction2lem = $true)) & ((laminj = $true) <=> ! [X366 : term,X367 : term] : ((X366 = X367) | ((lam @ X366) != (lam @ X367)))) & (hoaslamnotap_lthm = $true) & (apnotvar = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem0 = $true) <=> ! [X110 : subst > term > subst > $o] : ? [X111 : term > $o] : ! [X112 : term] : ((X110 @ id @ X112 @ id) = (X111 @ X112))) & (ulamvarsh = $true) & ! [X368 : d_term,X369 : d_term] : (((d2term @ X368) != (d2term @ X369)) | (X368 = X369)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (induction2_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 = X354) | ((ap @ (sub @ X352 @ id) @ X353) != (ap @ (sub @ X355 @ id) @ X354))) <=> (hoasapinj2 = $true)) & (one = (d2term @ d_one)) & (! [X142 : term] : ? [X143 : subst > term > term] : ((X143 @ sh @ one) = (sub @ X142 @ (push @ one @ sh))) <=> (hoasinduction_lem3b = $true)) & ((hoasapinj2_lthm = $true) <=> (? [X358 : term,X359 : term,X360 : term,X361 : term] : ((X358 != X361) & ((ap @ X360 @ X361) = (ap @ X359 @ X358))) | ! [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X364 @ id) @ X365)) | (X363 = X365)))) & (hoaslamnotvar_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj1_gthm = $true) & (axassoc = $true) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X262 : subst > term > subst > $o,X263 : term] : (? [X264 : term] : (((X262 @ id @ (sub @ X263 @ (push @ X264 @ id)) @ id) != $true) & ((X262 @ id @ X264 @ id) = $true)) | ((X262 @ id @ (lam @ X263) @ id) = $true) | ? [X265 : subst > term > term] : (! [X266 : term] : (($true = (X262 @ id @ (X265 @ id @ X266) @ id)) | ((X262 @ id @ X266 @ id) != $true)) & ! [X267 : subst,X268 : subst,X269 : term] : ((sub @ (X265 @ X268 @ X269) @ X267) = (X265 @ (comp @ X268 @ X267) @ (sub @ X269 @ X267))) & ($true != (X262 @ id @ (lam @ (X265 @ sh @ one)) @ id))))) & (hoasinduction_lem3a_gthm = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem2v2_lthm = $true) & (lamnotap != $true) & (axabs = $true) & (hoasinduction_lem2_lthm = $true) & (pushprop_lem3v2_lthm = $true) & (ulamvarind = $true) & (! [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) = X371) <=> (axvarcons = $true)) & ((axidr = $true) <=> ! [X416 : subst] : ((comp @ X416 @ id) = X416)) & ((d2subst @ d_id) = id) & (ulamvar1 != $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X77 : term,X78 : term] : (((sub @ X78 @ sh) != (sub @ X77 @ sh)) | (X77 = X78)) <=> (shinj = $true)) & ((hoasinduction_lem2v2 = $true) <=> ! [X270 : term,X271 : term > $o,X272 : subst > term > subst > $o,X273 : term] : (($true != (X271 @ X273)) | ? [X284 : term] : ((X272 @ id @ X284 @ id) != (X271 @ X284)) | ($true = (X271 @ (ap @ X273 @ X270))) | ($true != (X271 @ X270)) | ? [X280 : subst,X281 : subst,X282 : term,X283 : subst] : (($true != (X272 @ X281 @ X282 @ (comp @ X283 @ X280))) & ((X272 @ (comp @ X281 @ X283) @ (sub @ X282 @ X283) @ X280) = $true)) | ? [X274 : term,X275 : subst,X276 : subst,X277 : subst] : (($true = (X272 @ X277 @ X274 @ (comp @ X276 @ X275))) & ($true != (X272 @ (comp @ X277 @ X276) @ (sub @ X274 @ X276) @ X275))) | ? [X278 : term,X279 : term] : (((X272 @ id @ X278 @ id) = $true) & ($true != (X272 @ id @ (ap @ (sub @ X279 @ id) @ X278) @ id)) & ((X272 @ id @ X279 @ id) = $true)))) & (termmset_gthm = $true) & ((((! [X506 : subst] : ((comp @ id @ X506) = X506) & ! [X505 : subst] : ((comp @ X505 @ id) = X505)) | ? [X504 : subst] : ((comp @ X504 @ id) != X504)) | ? [X503 : subst] : ((comp @ id @ X503) != X503)) <=> (substmonoid_lthm = $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lthm_orig = $true) & (hoasinduction_lem3v2_gthm = $true) & ! [X417 : d_subst] : (d_id = X417) & (pushprop_lem2v2 = $true) & ((hoaslamnotap = $true) <=> ! [X520 : term,X521 : term,X522 : subst > term > term] : (((ap @ (sub @ X521 @ id) @ X520) != (lam @ (X522 @ sh @ one))) | ? [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) != (sub @ (X522 @ X525 @ X524) @ X523)))) & (! [X230 : term > $o,X231 : term] : (? [X232 : term] : (($true != (X230 @ X232)) & ((var @ X232) = $true)) | ((X230 @ X231) = $true) | ? [X235 : term] : (($true != (X230 @ (lam @ X235))) & ($true = (X230 @ X235))) | ? [X233 : term,X234 : term] : (($true = (X230 @ X234)) & ($true = (X230 @ X233)) & ($true != (X230 @ (ap @ X233 @ X234))))) <=> (induction = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X1 : term] : ? [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) = (X2 @ sh @ one))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X512 : subst > term > subst > $o,X513 : term] : (? [X514 : term] : (((X512 @ id @ X514 @ id) = $true) & ((X512 @ id @ (sub @ X513 @ (push @ X514 @ id)) @ id) != $true)) | ? [X515 : subst > term > term] : (! [X519 : term] : (((X512 @ id @ X519 @ id) != $true) | ((X512 @ id @ (X515 @ id @ X519) @ id) = $true)) & ! [X516 : subst,X517 : term,X518 : subst] : ((X515 @ (comp @ X518 @ X516) @ (sub @ X517 @ X516)) = (sub @ (X515 @ X518 @ X517) @ X516)) & ((X512 @ id @ (lam @ (X515 @ sh @ one)) @ id) != $true)) | ($true = (X512 @ id @ (lam @ (sub @ X513 @ (push @ one @ sh))) @ id)))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem1_lthm = $true) & (((((! [X306 : subst > term > subst > $o,X307 : term] : (? [X309 : subst > term > term] : (($true != (X306 @ id @ (lam @ (X309 @ sh @ one)) @ id)) & ! [X311 : subst,X312 : term,X313 : subst] : ((sub @ (X309 @ X311 @ X312) @ X313) = (X309 @ (comp @ X311 @ X313) @ (sub @ X312 @ X313))) & ! [X310 : term] : (($true = (X306 @ id @ (X309 @ id @ X310) @ id)) | ($true != (X306 @ id @ X310 @ id)))) | ? [X308 : term] : (((X306 @ id @ X308 @ id) != $true) & ((var @ (sub @ X308 @ id)) = $true)) | ((X306 @ id @ X307 @ id) = $true) | ? [X318 : term,X319 : term] : (((X306 @ id @ X318 @ id) = $true) & ($true = (X306 @ id @ X319 @ id)) & ((X306 @ id @ (ap @ (sub @ X318 @ id) @ X319) @ id) != $true)) | ? [X320 : term,X321 : subst,X322 : subst,X323 : subst] : (($true != (X306 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322) @ X323)) & ($true = (X306 @ X321 @ X320 @ (comp @ X322 @ X323)))) | ? [X314 : subst,X315 : subst,X316 : term,X317 : subst] : (((X306 @ X314 @ X316 @ (comp @ X315 @ X317)) != $true) & ((X306 @ (comp @ X314 @ X315) @ (sub @ X316 @ X315) @ X317) = $true))) | ? [X296 : term,X297 : term > $o,X298 : subst > term > subst > $o] : (! [X305 : term] : (((X297 @ X305) != $true) | ((X297 @ (sub @ X296 @ (push @ X305 @ id))) = $true)) & ! [X300 : subst > term > term] : (? [X301 : subst,X302 : term,X303 : subst] : ((sub @ (X300 @ X303 @ X302) @ X301) != (X300 @ (comp @ X303 @ X301) @ (sub @ X302 @ X301))) | ? [X304 : term] : (((X298 @ id @ (X300 @ id @ X304) @ id) != $true) & ($true = (X298 @ id @ X304 @ id))) | ($true = (X298 @ id @ (lam @ (X300 @ sh @ one)) @ id))) & ! [X299 : term] : ((X297 @ X299) = (X298 @ id @ X299 @ id)) & ((X297 @ (lam @ X296)) != $true))) | ? [X295 : term] : ((sub @ X295 @ id) != X295)) | ? [X288 : term,X289 : term > $o] : (((X289 @ X288) != $true) & ! [X290 : term] : (((var @ X290) != $true) | ($true = (X289 @ X290))) & ! [X291 : term] : (? [X292 : term] : (((X289 @ X292) = $true) & ($true != (X289 @ (sub @ X291 @ (push @ X292 @ id))))) | ((X289 @ (lam @ X291)) = $true)) & ! [X293 : term,X294 : term] : (((X289 @ X293) != $true) | ($true = (X289 @ (ap @ X293 @ X294))) | ((X289 @ X294) != $true)))) | ? [X285 : subst > term > subst > $o] : ! [X286 : term > $o] : ? [X287 : term] : ((X285 @ id @ X287 @ id) != (X286 @ X287))) <=> (hoasinduction_lthm_3 = $true)) & (! [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) != (ap @ (sub @ X508 @ id) @ X510)) | (X507 = X508)) <=> (hoasapinj1 = $true)) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & ((pushprop_lem0 = $true) <=> ! [X333 : term,X334 : term > $o,X335 : subst] : ? [X336 : term > $o] : ! [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) = (X336 @ X337))) & ((hoasinduction_lem2 = $true) <=> ! [X187 : term,X188 : subst > term > subst > $o,X189 : term] : (($true = (X188 @ id @ (ap @ X189 @ X187) @ id)) | ? [X198 : term,X199 : term] : (($true = (X188 @ id @ X199 @ id)) & ($true != (X188 @ id @ (ap @ (sub @ X199 @ id) @ X198) @ id)) & ((X188 @ id @ X198 @ id) = $true)) | ($true != (X188 @ id @ X187 @ id)) | ($true != (X188 @ id @ X189 @ id)) | ? [X190 : subst,X191 : subst,X192 : term,X193 : subst] : (((X188 @ X191 @ X192 @ (comp @ X193 @ X190)) != $true) & ((X188 @ (comp @ X191 @ X193) @ (sub @ X192 @ X193) @ X190) = $true)) | ? [X194 : subst,X195 : subst,X196 : term,X197 : subst] : (($true = (X188 @ X194 @ X196 @ (comp @ X197 @ X195))) & ((X188 @ (comp @ X194 @ X197) @ (sub @ X196 @ X197) @ X195) != $true)))) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X245 : term,X246 : subst > term > subst > $o,X247 : term > $o] : (? [X248 : subst > term > term] : (! [X252 : term] : (($true != (X246 @ id @ X252 @ id)) | ($true = (X246 @ id @ (X248 @ id @ X252) @ id))) & ! [X249 : term,X250 : subst,X251 : subst] : ((sub @ (X248 @ X251 @ X249) @ X250) = (X248 @ (comp @ X251 @ X250) @ (sub @ X249 @ X250))) & ((X246 @ id @ (lam @ (X248 @ sh @ one)) @ id) != $true)) | ? [X253 : term] : ((X247 @ X253) != (X246 @ id @ X253 @ id)) | ? [X254 : term] : (((X247 @ X254) = $true) & ((X247 @ (sub @ X245 @ (push @ X254 @ id))) != $true)) | ($true = (X247 @ (lam @ X245)))) | ? [X244 : term] : ((sub @ X244 @ id) != X244)) | ? [X240 : term] : ! [X241 : subst > term > term] : ? [X242 : term,X243 : subst] : ((X241 @ X243 @ X242) != (sub @ X240 @ (push @ X242 @ X243))))) & (hoaslamnotvar = $true) & ((! [X6 : term] : ((sub @ X6 @ id) = X6) | ? [X5 : term] : ((sub @ X5 @ id) != X5)) <=> (termmset_lthm = $true)) & ! [X356 : subst] : ? [X357 : d_subst] : ((d2subst @ X357) = X356) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoaslaminj = $true) <=> ! [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (((X144 @ X146 @ X147) = (X145 @ X146 @ X147)) | ? [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) != (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) | ((lam @ (X144 @ sh @ one)) != (lam @ (X145 @ sh @ one))) | ? [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) != (sub @ (X144 @ X149 @ X148) @ X150)))) & (pushprop = $true) & ((var @ (d2term @ d_one)) != $true) & (hoaslamnotvar_lthm = $true) & ((induction2_lthm = $true) <=> ((! [X102 : term > $o,X103 : term] : (((X102 @ X103) = $true) | ? [X104 : term,X105 : term] : (((X102 @ X105) = $true) & ((X102 @ X104) = $true) & ((X102 @ (ap @ X104 @ X105)) != $true)) | ? [X106 : term] : (((X102 @ (lam @ X106)) != $true) & ! [X107 : term] : (($true = (X102 @ (sub @ X106 @ (push @ X107 @ id)))) | ((X102 @ X107) != $true))) | ? [X108 : term] : (((X102 @ X108) != $true) & ($true = (var @ X108)))) | ? [X94 : subst,X95 : term > $o,X96 : term] : (! [X100 : term] : (((X95 @ (lam @ X100)) = $true) | ? [X101 : term] : (((X95 @ (sub @ X100 @ (push @ X101 @ id))) != $true) & ((X95 @ X101) = $true))) & ((X95 @ (sub @ X96 @ X94)) != $true) & ! [X97 : term,X98 : term] : (($true = (X95 @ (ap @ X97 @ X98))) | ((X95 @ X98) != $true) | ($true != (X95 @ X97))) & ! [X99 : term] : (((var @ X99) != $true) | ($true = (X95 @ (sub @ X99 @ X94)))))) | ? [X93 : term] : ((sub @ X93 @ id) != X93))) & (substmonoid_gthm = $true) & (pushprop_lem1_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true != (X395 @ X397)) | ($true = (X396 @ one)) | ? [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) != (X396 @ X399)))) & (lamnotvar = $true) & (! [X109 : term] : ((sub @ X109 @ id) = X109) <=> (termmset = $true)) & (! [X26 : subst > term > subst > $o] : ? [X27 : term > $o] : ! [X28 : term] : ((X27 @ X28) = (X26 @ id @ X28 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (pushprop_gthm = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem1 = $true) & (hoasapnotvar_gthm = $true) & (((((! [X18 : subst,X19 : term,X20 : term > $o] : (($true = (X20 @ (sub @ X19 @ X18))) | ? [X21 : term] : (($true != (X20 @ (sub @ X21 @ X18))) & ((var @ X21) = $true)) | ? [X24 : term] : (((X20 @ (lam @ X24)) != $true) & ! [X25 : term] : (((X20 @ X25) != $true) | ((X20 @ (sub @ X24 @ (push @ X25 @ id))) = $true))) | ? [X22 : term,X23 : term] : (($true != (X20 @ (ap @ X23 @ X22))) & ($true = (X20 @ X23)) & ((X20 @ X22) = $true))) | ? [X12 : term > $o,X13 : term] : (! [X14 : term] : (($true != (var @ X14)) | ($true = (X12 @ X14))) & ! [X16 : term,X17 : term] : (((X12 @ (ap @ X17 @ X16)) = $true) | ((X12 @ X16) != $true) | ($true != (X12 @ X17))) & ($true != (X12 @ X13)) & ! [X15 : term] : (($true != (X12 @ X15)) | ((X12 @ (lam @ X15)) = $true)))) | ? [X11 : subst] : ((comp @ X11 @ id) != X11)) | ? [X9 : subst,X10 : term] : ((comp @ sh @ (push @ X10 @ X9)) != X9)) | ? [X7 : term,X8 : subst] : ((sub @ one @ (push @ X7 @ X8)) != X7)) <=> (induction2lem_lthm = $true)) & (pushprop_lem2v2_gthm = $true) & ((axshiftcons = $true) <=> ! [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) = X201)) & ((? [X484 : term] : ((sub @ X484 @ id) != X484) | ! [X485 : term > $o,X486 : subst > term > subst > $o,X487 : term] : (? [X497 : term] : ((X485 @ X497) != (X486 @ id @ X497 @ id)) | ? [X492 : subst,X493 : term,X494 : subst,X495 : subst] : (((X486 @ X492 @ X493 @ (comp @ X495 @ X494)) != $true) & ((X486 @ (comp @ X492 @ X495) @ (sub @ X493 @ X495) @ X494) = $true)) | ? [X488 : subst,X489 : subst,X490 : term,X491 : subst] : (((X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X491) != $true) & ((X486 @ X488 @ X490 @ (comp @ X489 @ X491)) = $true)) | ? [X496 : term] : (((X485 @ X496) = $true) & ((X485 @ (sub @ X487 @ (push @ X496 @ id))) != $true)) | ((X485 @ (lam @ X487)) = $true) | ? [X498 : subst > term > term] : (((X486 @ id @ (lam @ (X498 @ sh @ one)) @ id) != $true) & ! [X499 : term] : (($true != (X486 @ id @ X499 @ id)) | ($true = (X486 @ id @ (X498 @ id @ X499) @ id))) & ! [X500 : subst,X501 : subst,X502 : term] : ((X498 @ (comp @ X500 @ X501) @ (sub @ X502 @ X501)) = (sub @ (X498 @ X500 @ X502) @ X501))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ! [X338 : d_subst,X339 : d_subst] : ((X338 = X339) | ((d2subst @ X338) != (d2subst @ X339))) & (! [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (($true = (X475 @ (sub @ X473 @ (push @ X476 @ X474)))) | ($true != (X472 @ X473))) <=> (pushprop_lem1 = $true)) & (! [X511 : term] : ((sub @ X511 @ id) = X511) <=> (axvarid = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X29 : subst > term > subst > $o,X30 : term,X31 : term > $o] : (? [X46 : term] : ((X29 @ id @ X46 @ id) != (X31 @ X46)) | ? [X42 : subst,X43 : subst,X44 : term,X45 : subst] : (($true != (X29 @ X42 @ X44 @ (comp @ X43 @ X45))) & ((X29 @ (comp @ X42 @ X43) @ (sub @ X44 @ X43) @ X45) = $true)) | ? [X32 : subst,X33 : subst,X34 : term,X35 : subst] : (($true = (X29 @ X35 @ X34 @ (comp @ X32 @ X33))) & ($true != (X29 @ (comp @ X35 @ X32) @ (sub @ X34 @ X32) @ X33))) | ? [X36 : term] : (($true = (X31 @ X36)) & ($true != (X31 @ (sub @ X30 @ (push @ X36 @ id))))) | ((X31 @ (lam @ X30)) = $true) | ? [X37 : subst > term > term] : (! [X38 : subst,X39 : subst,X40 : term] : ((sub @ (X37 @ X38 @ X40) @ X39) = (X37 @ (comp @ X38 @ X39) @ (sub @ X40 @ X39))) & ((X29 @ id @ (lam @ (X37 @ sh @ one)) @ id) != $true) & ! [X41 : term] : (((X29 @ id @ X41 @ id) != $true) | ($true = (X29 @ id @ (X37 @ id @ X41) @ id)))))) & (axmap = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ((((! [X68 : subst > term > subst > $o,X69 : term] : (? [X72 : subst > term > term] : (! [X76 : term] : (($true = (X68 @ id @ (X72 @ id @ X76) @ id)) | ((X68 @ id @ X76 @ id) != $true)) & ! [X73 : subst,X74 : subst,X75 : term] : ((sub @ (X72 @ X74 @ X75) @ X73) = (X72 @ (comp @ X74 @ X73) @ (sub @ X75 @ X73))) & ((X68 @ id @ (lam @ (X72 @ sh @ one)) @ id) != $true)) | ($true = (X68 @ id @ X69 @ id)) | ? [X70 : term,X71 : term] : (((X68 @ id @ X70 @ id) = $true) & ($true = (X68 @ id @ X71 @ id)) & ($true != (X68 @ id @ (ap @ (sub @ X71 @ id) @ X70) @ id)))) | ? [X58 : subst > term > subst > $o,X59 : term,X60 : term > $o] : (! [X61 : term] : (((X60 @ (sub @ X59 @ (push @ X61 @ id))) = $true) | ((X60 @ X61) != $true)) & ($true != (X60 @ (lam @ X59))) & ! [X62 : term] : ((X58 @ id @ X62 @ id) = (X60 @ X62)) & ! [X63 : subst > term > term] : (? [X67 : term] : (((X58 @ id @ (X63 @ id @ X67) @ id) != $true) & ($true = (X58 @ id @ X67 @ id))) | ((X58 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true) | ? [X64 : subst,X65 : term,X66 : subst] : ((sub @ (X63 @ X64 @ X65) @ X66) != (X63 @ (comp @ X64 @ X66) @ (sub @ X65 @ X66)))))) | ? [X57 : term] : ((sub @ X57 @ id) != X57)) | ? [X50 : term > $o,X51 : term] : (! [X52 : term,X53 : term] : (($true != (X50 @ X53)) | ((X50 @ X52) != $true) | ($true = (X50 @ (ap @ X53 @ X52)))) & ((X50 @ X51) != $true) & ! [X54 : term] : (((X50 @ (lam @ X54)) = $true) | ? [X55 : term] : (((X50 @ (sub @ X54 @ (push @ X55 @ id))) != $true) & ((X50 @ X55) = $true))) & ! [X56 : term] : (($true != (var @ X56)) | ($true = (X50 @ X56))))) | ? [X47 : subst > term > subst > $o] : ! [X48 : term > $o] : ? [X49 : term] : ((X47 @ id @ X49 @ id) != (X48 @ X49)))) & (induction2lem_gthm = $true) & ((apinj1 = $true) <=> ! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 = X239) | ((ap @ X236 @ X238) != (ap @ X239 @ X237)))) & (! [X122 : term,X123 : subst > term > subst > $o] : (? [X125 : subst > term > term] : (! [X126 : subst,X127 : term,X128 : subst] : ((X125 @ (comp @ X128 @ X126) @ (sub @ X127 @ X126)) = (sub @ (X125 @ X128 @ X127) @ X126)) & ! [X129 : term] : (((X123 @ id @ X129 @ id) != $true) | ($true = (X123 @ id @ (X125 @ id @ X129) @ id))) & ($true != (X123 @ id @ (lam @ (X125 @ sh @ one)) @ id))) | ? [X124 : term] : (($true != (X123 @ id @ (sub @ X122 @ (push @ X124 @ id)) @ id)) & ($true = (X123 @ id @ X124 @ id))) | ((X123 @ id @ (lam @ X122) @ id) = $true) | ? [X134 : subst,X135 : subst,X136 : term,X137 : subst] : (((X123 @ (comp @ X135 @ X137) @ (sub @ X136 @ X137) @ X134) = $true) & ((X123 @ X135 @ X136 @ (comp @ X137 @ X134)) != $true)) | ? [X130 : subst,X131 : subst,X132 : subst,X133 : term] : (((X123 @ (comp @ X131 @ X130) @ (sub @ X133 @ X130) @ X132) != $true) & ((X123 @ X131 @ X133 @ (comp @ X130 @ X132)) = $true))) <=> (hoasinduction_lem3 = $true)) & (axapp = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X340 : subst > term > subst > $o,X341 : term] : (? [X342 : term] : (($true != (X340 @ id @ (sub @ X341 @ (push @ X342 @ id)) @ id)) & ((X340 @ id @ X342 @ id) = $true)) | ? [X343 : term,X344 : subst > term > term] : (((X340 @ id @ (lam @ (X344 @ sh @ one)) @ id) != $true) & ! [X345 : subst,X346 : subst,X347 : term] : ((sub @ (sub @ X343 @ (push @ X347 @ X346)) @ X345) = (sub @ (X344 @ X346 @ X347) @ X345)) & ! [X349 : subst,X350 : subst,X351 : term] : ((sub @ X343 @ (push @ (sub @ X351 @ X350) @ (comp @ X349 @ X350))) = (X344 @ (comp @ X349 @ X350) @ (sub @ X351 @ X350))) & ! [X348 : term] : (($true = (X340 @ id @ (X344 @ id @ X348) @ id)) | ($true != (X340 @ id @ X348 @ id)))) | ($true = (X340 @ id @ (lam @ (sub @ X341 @ (push @ one @ sh))) @ id)))) & (hoasapnotvar = $true) & ((axscons = $true) <=> ! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0)) & (! [X83 : subst > term > subst > $o,X84 : term,X85 : term > $o] : (($true = (X85 @ (lam @ X84))) | ? [X88 : subst > term > term] : (! [X92 : term] : (((X83 @ id @ X92 @ id) != $true) | ((X83 @ id @ (X88 @ id @ X92) @ id) = $true)) & ! [X89 : subst,X90 : subst,X91 : term] : ((X88 @ (comp @ X90 @ X89) @ (sub @ X91 @ X89)) = (sub @ (X88 @ X90 @ X91) @ X89)) & ($true != (X83 @ id @ (lam @ (X88 @ sh @ one)) @ id))) | ? [X87 : term] : ((X83 @ id @ X87 @ id) != (X85 @ X87)) | ? [X86 : term] : (($true = (X85 @ X86)) & ($true != (X85 @ (sub @ X84 @ (push @ X86 @ id)))))) <=> (hoasinduction_lem3v2a = $true))),
% 9.64/2.82 inference(ennf_transformation,[status(thm)],[f11])).
% 9.64/2.82 thf(f11,plain,(
% 9.64/2.82 (pushprop_lem1v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X374 : term] : ((sub @ X374 @ id) = X374) => (! [X375 : term,X376 : subst > term > subst > $o] : (~! [X382 : term] : (($true != (X376 @ id @ X382 @ id)) | ($true = (X376 @ id @ (sub @ X375 @ (push @ X382 @ id)) @ id))) | ((X376 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id) = $true) | ~! [X377 : subst > term > term] : (~! [X379 : subst,X380 : subst,X381 : term] : ((X377 @ (comp @ X379 @ X380) @ (sub @ X381 @ X380)) = (sub @ (X377 @ X379 @ X381) @ X380)) | ($true = (X376 @ id @ (lam @ (X377 @ sh @ one)) @ id)) | ~! [X378 : term] : (($true != (X376 @ id @ X378 @ id)) | ($true = (X376 @ id @ (X377 @ id @ X378) @ id))))) => ! [X383 : term,X384 : subst > term > subst > $o] : (~! [X386 : subst > term > term] : (($true = (X384 @ id @ (lam @ (X386 @ sh @ one)) @ id)) | ~! [X387 : term] : (($true = (X384 @ id @ (X386 @ id @ X387) @ id)) | ((X384 @ id @ X387 @ id) != $true)) | ~! [X388 : subst,X389 : subst,X390 : term] : ((sub @ (X386 @ X389 @ X390) @ X388) = (X386 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388)))) | ($true = (X384 @ id @ (lam @ X383) @ id)) | ~! [X385 : term] : (((X384 @ id @ (sub @ X383 @ (push @ X385 @ id)) @ id) = $true) | ($true != (X384 @ id @ X385 @ id))))))) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction = $true) <=> ! [X211 : term,X212 : subst > term > subst > $o] : (~! [X222 : subst > term > term] : (($true = (X212 @ id @ (lam @ (X222 @ sh @ one)) @ id)) | ~! [X224 : term,X225 : subst,X226 : subst] : ((X222 @ (comp @ X226 @ X225) @ (sub @ X224 @ X225)) = (sub @ (X222 @ X226 @ X224) @ X225)) | ~! [X223 : term] : (($true = (X212 @ id @ (X222 @ id @ X223) @ id)) | ((X212 @ id @ X223 @ id) != $true))) | ~! [X227 : term,X228 : term] : (((X212 @ id @ X228 @ id) != $true) | ((X212 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id) = $true) | ((X212 @ id @ X227 @ id) != $true)) | ~! [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (((X212 @ X214 @ X213 @ (comp @ X216 @ X215)) = $true) | ($true != (X212 @ (comp @ X214 @ X216) @ (sub @ X213 @ X216) @ X215))) | ~! [X221 : term] : (((var @ (sub @ X221 @ id)) != $true) | ((X212 @ id @ X221 @ id) = $true)) | ($true = (X212 @ id @ X211 @ id)) | ~! [X217 : subst,X218 : term,X219 : subst,X220 : subst] : (((X212 @ X219 @ X218 @ (comp @ X217 @ X220)) != $true) | ((X212 @ (comp @ X219 @ X217) @ (sub @ X218 @ X217) @ X220) = $true)))) & (hoasapinj2_gthm = $true) & ((d2subst @ d_id) = sh) & (axvarshift = $true) & ((hoaslaminj_lthm = $true) <=> (! [X400 : term,X401 : subst] : ((sub @ one @ (push @ X400 @ X401)) = X400) => (! [X402 : term,X403 : subst] : ((comp @ sh @ (push @ X402 @ X403)) = X403) => (! [X404 : term,X405 : term] : ((X404 = X405) | ((lam @ X405) != (lam @ X404))) => ! [X406 : subst > term > term,X407 : subst,X408 : term,X409 : subst > term > term] : (((lam @ (X409 @ sh @ one)) != (lam @ (X406 @ sh @ one))) | ~! [X413 : term,X414 : subst,X415 : subst] : ((sub @ (X409 @ X415 @ X413) @ X414) = (X409 @ (comp @ X415 @ X414) @ (sub @ X413 @ X414))) | ((X409 @ X407 @ X408) = (X406 @ X407 @ X408)) | ~! [X410 : subst,X411 : term,X412 : subst] : ((sub @ (X406 @ X410 @ X411) @ X412) = (X406 @ (comp @ X410 @ X412) @ (sub @ X411 @ X412)))))))) & ((hoasinduction_lthm = $true) <=> (! [X418 : term,X419 : term > $o] : (($true = (X419 @ X418)) | ~! [X423 : term,X424 : term] : (($true = (X419 @ (ap @ X424 @ X423))) | ((X419 @ X423) != $true) | ((X419 @ X424) != $true)) | ~! [X420 : term] : (((X419 @ (lam @ X420)) = $true) | ~! [X421 : term] : (($true != (X419 @ X421)) | ($true = (X419 @ (sub @ X420 @ (push @ X421 @ id)))))) | ~! [X422 : term] : (($true = (X419 @ X422)) | ((var @ X422) != $true))) => (! [X425 : term,X426 : term,X427 : subst > term > subst > $o] : (((X427 @ id @ (ap @ X426 @ X425) @ id) = $true) | ~! [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X427 @ (comp @ X429 @ X430) @ (sub @ X428 @ X430) @ X431)) | ((X427 @ X429 @ X428 @ (comp @ X430 @ X431)) != $true)) | ~! [X436 : term,X437 : term] : (($true = (X427 @ id @ (ap @ (sub @ X437 @ id) @ X436) @ id)) | ($true != (X427 @ id @ X437 @ id)) | ((X427 @ id @ X436 @ id) != $true)) | ((X427 @ id @ X425 @ id) != $true) | ($true != (X427 @ id @ X426 @ id)) | ~! [X432 : subst,X433 : subst,X434 : term,X435 : subst] : (((X427 @ (comp @ X432 @ X433) @ (sub @ X434 @ X433) @ X435) != $true) | ((X427 @ X432 @ X434 @ (comp @ X433 @ X435)) = $true))) => (! [X438 : term,X439 : subst > term > subst > $o] : (~! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (((X439 @ (comp @ X449 @ X447) @ (sub @ X448 @ X447) @ X446) = $true) | ((X439 @ X449 @ X448 @ (comp @ X447 @ X446)) != $true)) | ~! [X450 : subst,X451 : subst,X452 : term,X453 : subst] : (($true = (X439 @ X453 @ X452 @ (comp @ X450 @ X451))) | ((X439 @ (comp @ X453 @ X450) @ (sub @ X452 @ X450) @ X451) != $true)) | ~! [X445 : term] : (($true != (X439 @ id @ X445 @ id)) | ($true = (X439 @ id @ (sub @ X438 @ (push @ X445 @ id)) @ id))) | ~! [X440 : subst > term > term] : (~! [X441 : subst,X442 : subst,X443 : term] : ((sub @ (X440 @ X442 @ X443) @ X441) = (X440 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441))) | ~! [X444 : term] : (($true = (X439 @ id @ (X440 @ id @ X444) @ id)) | ((X439 @ id @ X444 @ id) != $true)) | ((X439 @ id @ (lam @ (X440 @ sh @ one)) @ id) = $true)) | ($true = (X439 @ id @ (lam @ X438) @ id))) => ! [X454 : term,X455 : subst > term > subst > $o] : (~! [X471 : term] : (($true != (var @ (sub @ X471 @ id))) | ((X455 @ id @ X471 @ id) = $true)) | ~! [X458 : subst > term > term] : (~! [X460 : subst,X461 : subst,X462 : term] : ((X458 @ (comp @ X460 @ X461) @ (sub @ X462 @ X461)) = (sub @ (X458 @ X460 @ X462) @ X461)) | ((X455 @ id @ (lam @ (X458 @ sh @ one)) @ id) = $true) | ~! [X459 : term] : (((X455 @ id @ X459 @ id) != $true) | ((X455 @ id @ (X458 @ id @ X459) @ id) = $true))) | ~! [X463 : subst,X464 : term,X465 : subst,X466 : subst] : (($true != (X455 @ X466 @ X464 @ (comp @ X465 @ X463))) | ((X455 @ (comp @ X466 @ X465) @ (sub @ X464 @ X465) @ X463) = $true)) | ($true = (X455 @ id @ X454 @ id)) | ~! [X456 : term,X457 : term] : (($true != (X455 @ id @ X457 @ id)) | ($true != (X455 @ id @ X456 @ id)) | ((X455 @ id @ (ap @ (sub @ X456 @ id) @ X457) @ id) = $true)) | ~! [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (($true != (X455 @ (comp @ X468 @ X469) @ (sub @ X467 @ X469) @ X470)) | ((X455 @ X468 @ X467 @ (comp @ X469 @ X470)) = $true))))))) & ((pushprop_lem0_lthm = $true) <=> ! [X535 : term > $o,X536 : subst,X537 : term] : ~! [X538 : term > $o] : ~! [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) = (X538 @ X539))) & (axclos = $true) & ! [X229 : d_term] : (d_one = X229) & (pushprop_lthm = $true) & ((! [X373 : subst] : ((comp @ X373 @ id) = X373) & ! [X372 : subst] : ((comp @ id @ X372) = X372)) <=> (substmonoid = $true)) & (! [X324 : subst] : ((comp @ id @ X324) = X324) <=> (axidl = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X162 : term] : ((sub @ X162 @ id) = X162) => (! [X163 : subst > term > subst > $o,X164 : term] : (~! [X165 : term] : (($true != (X163 @ id @ X165 @ id)) | ($true = (X163 @ id @ (sub @ X164 @ (push @ X165 @ id)) @ id))) | ((X163 @ id @ (lam @ (sub @ X164 @ (push @ one @ sh))) @ id) = $true) | ~! [X166 : subst > term > term] : (($true = (X163 @ id @ (lam @ (X166 @ sh @ one)) @ id)) | ~! [X167 : subst,X168 : subst,X169 : term] : ((X166 @ (comp @ X168 @ X167) @ (sub @ X169 @ X167)) = (sub @ (X166 @ X168 @ X169) @ X167)) | ~! [X170 : term] : (((X163 @ id @ (X166 @ id @ X170) @ id) = $true) | ($true != (X163 @ id @ X170 @ id))))) => ! [X171 : term,X172 : subst > term > subst > $o] : (~! [X174 : subst > term > term] : (((X172 @ id @ (lam @ (X174 @ sh @ one)) @ id) = $true) | ~! [X175 : term] : (((X172 @ id @ (X174 @ id @ X175) @ id) = $true) | ((X172 @ id @ X175 @ id) != $true)) | ~! [X176 : subst,X177 : subst,X178 : term] : ((sub @ (X174 @ X176 @ X178) @ X177) = (X174 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177)))) | ~! [X173 : term] : (($true = (X172 @ id @ (sub @ X171 @ (push @ X173 @ id)) @ id)) | ((X172 @ id @ X173 @ id) != $true)) | ($true = (X172 @ id @ (lam @ X171) @ id)) | ~! [X183 : subst,X184 : subst,X185 : term,X186 : subst] : (((X172 @ X184 @ X185 @ (comp @ X183 @ X186)) = $true) | ($true != (X172 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183) @ X186))) | ~! [X179 : term,X180 : subst,X181 : subst,X182 : subst] : (((X172 @ X182 @ X179 @ (comp @ X180 @ X181)) != $true) | ($true = (X172 @ (comp @ X182 @ X180) @ (sub @ X179 @ X180) @ X181))))))) & ((hoasinduction_lem3aa = $true) <=> ! [X154 : subst > term > subst > $o,X155 : term] : (($true = (X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) | ~! [X157 : subst > term > term] : (($true = (X154 @ id @ (lam @ (X157 @ sh @ one)) @ id)) | ~! [X158 : term] : (($true = (X154 @ id @ (X157 @ id @ X158) @ id)) | ($true != (X154 @ id @ X158 @ id))) | ~! [X159 : subst,X160 : subst,X161 : term] : ((X157 @ (comp @ X159 @ X160) @ (sub @ X161 @ X160)) = (sub @ (X157 @ X159 @ X161) @ X160))) | ~! [X156 : term] : (($true = (X154 @ id @ (sub @ X155 @ (push @ X156 @ id)) @ id)) | ((X154 @ id @ X156 @ id) != $true)))) & ((apinj2 = $true) <=> ! [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) != (ap @ X79 @ X82)) | (X81 = X82))) & (! [X391 : term] : ~! [X392 : subst > term > term] : ~! [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) = (X392 @ X393 @ X394)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((induction2 = $true) <=> ! [X255 : term,X256 : term > $o] : (~! [X260 : term] : (~! [X261 : term] : (($true != (X256 @ X261)) | ($true = (X256 @ (sub @ X260 @ (push @ X261 @ id))))) | ($true = (X256 @ (lam @ X260)))) | ~! [X257 : term,X258 : term] : (((X256 @ X258) != $true) | ($true != (X256 @ X257)) | ((X256 @ (ap @ X257 @ X258)) = $true)) | ~! [X259 : term] : (($true = (X256 @ X259)) | ((var @ X259) != $true)) | ((X256 @ X255) = $true))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X202 : subst > term > subst > $o,X203 : term] : (((X202 @ id @ X203 @ id) = $true) | ~! [X204 : term,X205 : term] : (((X202 @ id @ (ap @ (sub @ X204 @ id) @ X205) @ id) = $true) | ($true != (X202 @ id @ X204 @ id)) | ($true != (X202 @ id @ X205 @ id))) | ~! [X206 : subst > term > term] : (~! [X207 : term] : (((X202 @ id @ X207 @ id) != $true) | ($true = (X202 @ id @ (X206 @ id @ X207) @ id))) | ~! [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) = (sub @ (X206 @ X210 @ X208) @ X209)) | ($true = (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id))))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X138 : term] : ~! [X139 : subst > term > term] : ~! [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) = (X139 @ X141 @ X140))) & ((! [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) = X478) => ! [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (~! [X483 : term] : ((X480 @ X483) = (X479 @ (sub @ X483 @ (push @ X481 @ X482)))) | ($true = (X480 @ one)) | ((X479 @ X481) != $true))) <=> (pushprop_lem1v2_lthm = $true)) & ((hoasapinj1_lthm = $true) <=> (! [X526 : term] : ((sub @ X526 @ id) = X526) => (! [X527 : term,X528 : term,X529 : term,X530 : term] : (((ap @ X528 @ X529) != (ap @ X527 @ X530)) | (X527 = X528)) => ! [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) != (ap @ (sub @ X534 @ id) @ X532)) | (X533 = X534))))) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) = X114) => (! [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) = X116) => ! [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true = (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) | ($true != (X120 @ X121)))))) & (hoasinduction_gthm = $true) & (pushprop_lem3v2 = $true) & (hoasinduction_lem2v2_gthm = $true) & ((induction2lem = $true) <=> ! [X325 : term > $o,X326 : subst,X327 : term] : (~! [X328 : term,X329 : term] : (($true != (X325 @ X328)) | ($true = (X325 @ (ap @ X329 @ X328))) | ($true != (X325 @ X329))) | ($true = (X325 @ (sub @ X327 @ X326))) | ~! [X330 : term] : (($true != (var @ X330)) | ($true = (X325 @ (sub @ X330 @ X326)))) | ~! [X331 : term] : (~! [X332 : term] : (($true != (X325 @ X332)) | ($true = (X325 @ (sub @ X331 @ (push @ X332 @ id))))) | ((X325 @ (lam @ X331)) = $true)))) & ((laminj = $true) <=> ! [X366 : term,X367 : term] : ((X366 = X367) | ((lam @ X366) != (lam @ X367)))) & (hoaslamnotap_lthm = $true) & (apnotvar = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem0 = $true) <=> ! [X110 : subst > term > subst > $o] : ~! [X111 : term > $o] : ~! [X112 : term] : ((X110 @ id @ X112 @ id) = (X111 @ X112))) & (ulamvarsh = $true) & ! [X368 : d_term,X369 : d_term] : (((d2term @ X368) = (d2term @ X369)) => (X368 = X369)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (induction2_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 = X354) | ((ap @ (sub @ X352 @ id) @ X353) != (ap @ (sub @ X355 @ id) @ X354))) <=> (hoasapinj2 = $true)) & (one = (d2term @ d_one)) & (! [X142 : term] : ~! [X143 : subst > term > term] : ((X143 @ sh @ one) != (sub @ X142 @ (push @ one @ sh))) <=> (hoasinduction_lem3b = $true)) & ((! [X358 : term,X359 : term,X360 : term,X361 : term] : (((ap @ X360 @ X361) != (ap @ X359 @ X358)) | (X358 = X361)) => ! [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X364 @ id) @ X365)) | (X363 = X365))) <=> (hoasapinj2_lthm = $true)) & (hoaslamnotvar_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj1_gthm = $true) & (axassoc = $true) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X262 : subst > term > subst > $o,X263 : term] : (~! [X264 : term] : (((X262 @ id @ X264 @ id) != $true) | ((X262 @ id @ (sub @ X263 @ (push @ X264 @ id)) @ id) = $true)) | ~! [X265 : subst > term > term] : (~! [X267 : subst,X268 : subst,X269 : term] : ((sub @ (X265 @ X268 @ X269) @ X267) = (X265 @ (comp @ X268 @ X267) @ (sub @ X269 @ X267))) | ($true = (X262 @ id @ (lam @ (X265 @ sh @ one)) @ id)) | ~! [X266 : term] : (($true = (X262 @ id @ (X265 @ id @ X266) @ id)) | ((X262 @ id @ X266 @ id) != $true))) | ((X262 @ id @ (lam @ X263) @ id) = $true))) & (hoasinduction_lem3a_gthm = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem2v2_lthm = $true) & (lamnotap != $true) & (axabs = $true) & (hoasinduction_lem2_lthm = $true) & (pushprop_lem3v2_lthm = $true) & (ulamvarind = $true) & (! [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) = X371) <=> (axvarcons = $true)) & ((axidr = $true) <=> ! [X416 : subst] : ((comp @ X416 @ id) = X416)) & ((d2subst @ d_id) = id) & (ulamvar1 != $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X77 : term,X78 : term] : (((sub @ X78 @ sh) != (sub @ X77 @ sh)) | (X77 = X78)) <=> (shinj = $true)) & (! [X270 : term,X271 : term > $o,X272 : subst > term > subst > $o,X273 : term] : (~! [X274 : term,X275 : subst,X276 : subst,X277 : subst] : (($true != (X272 @ X277 @ X274 @ (comp @ X276 @ X275))) | ($true = (X272 @ (comp @ X277 @ X276) @ (sub @ X274 @ X276) @ X275))) | ($true != (X271 @ X270)) | ~! [X284 : term] : ((X272 @ id @ X284 @ id) = (X271 @ X284)) | ($true = (X271 @ (ap @ X273 @ X270))) | ~! [X280 : subst,X281 : subst,X282 : term,X283 : subst] : (($true = (X272 @ X281 @ X282 @ (comp @ X283 @ X280))) | ((X272 @ (comp @ X281 @ X283) @ (sub @ X282 @ X283) @ X280) != $true)) | ~! [X278 : term,X279 : term] : (((X272 @ id @ X279 @ id) != $true) | ($true = (X272 @ id @ (ap @ (sub @ X279 @ id) @ X278) @ id)) | ((X272 @ id @ X278 @ id) != $true)) | ($true != (X271 @ X273))) <=> (hoasinduction_lem2v2 = $true)) & (termmset_gthm = $true) & ((! [X503 : subst] : ((comp @ id @ X503) = X503) => (! [X504 : subst] : ((comp @ X504 @ id) = X504) => (! [X506 : subst] : ((comp @ id @ X506) = X506) & ! [X505 : subst] : ((comp @ X505 @ id) = X505)))) <=> (substmonoid_lthm = $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lthm_orig = $true) & (hoasinduction_lem3v2_gthm = $true) & ! [X417 : d_subst] : (d_id = X417) & (pushprop_lem2v2 = $true) & ((hoaslamnotap = $true) <=> ! [X520 : term,X521 : term,X522 : subst > term > term] : (~! [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) = (sub @ (X522 @ X525 @ X524) @ X523)) | ((ap @ (sub @ X521 @ id) @ X520) != (lam @ (X522 @ sh @ one))))) & (! [X230 : term > $o,X231 : term] : (~! [X233 : term,X234 : term] : (($true != (X230 @ X233)) | ($true = (X230 @ (ap @ X233 @ X234))) | ($true != (X230 @ X234))) | ((X230 @ X231) = $true) | ~! [X235 : term] : (($true != (X230 @ X235)) | ($true = (X230 @ (lam @ X235)))) | ~! [X232 : term] : (($true = (X230 @ X232)) | ((var @ X232) != $true))) <=> (induction = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X1 : term] : ~! [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) != (X2 @ sh @ one))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X512 : subst > term > subst > $o,X513 : term] : (~! [X515 : subst > term > term] : (~! [X516 : subst,X517 : term,X518 : subst] : ((X515 @ (comp @ X518 @ X516) @ (sub @ X517 @ X516)) = (sub @ (X515 @ X518 @ X517) @ X516)) | ((X512 @ id @ (lam @ (X515 @ sh @ one)) @ id) = $true) | ~! [X519 : term] : (((X512 @ id @ X519 @ id) != $true) | ((X512 @ id @ (X515 @ id @ X519) @ id) = $true))) | ($true = (X512 @ id @ (lam @ (sub @ X513 @ (push @ one @ sh))) @ id)) | ~! [X514 : term] : (((X512 @ id @ (sub @ X513 @ (push @ X514 @ id)) @ id) = $true) | ((X512 @ id @ X514 @ id) != $true)))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem1_lthm = $true) & ((! [X285 : subst > term > subst > $o] : ~! [X286 : term > $o] : ~! [X287 : term] : ((X285 @ id @ X287 @ id) = (X286 @ X287)) => (! [X288 : term,X289 : term > $o] : (~! [X290 : term] : (((var @ X290) != $true) | ($true = (X289 @ X290))) | ~! [X291 : term] : (((X289 @ (lam @ X291)) = $true) | ~! [X292 : term] : (((X289 @ X292) != $true) | ($true = (X289 @ (sub @ X291 @ (push @ X292 @ id)))))) | ((X289 @ X288) = $true) | ~! [X293 : term,X294 : term] : (((X289 @ X293) != $true) | ($true = (X289 @ (ap @ X293 @ X294))) | ((X289 @ X294) != $true))) => (! [X295 : term] : ((sub @ X295 @ id) = X295) => (! [X296 : term,X297 : term > $o,X298 : subst > term > subst > $o] : (~! [X299 : term] : ((X297 @ X299) = (X298 @ id @ X299 @ id)) | ~! [X305 : term] : (((X297 @ X305) != $true) | ((X297 @ (sub @ X296 @ (push @ X305 @ id))) = $true)) | ((X297 @ (lam @ X296)) = $true) | ~! [X300 : subst > term > term] : (~! [X301 : subst,X302 : term,X303 : subst] : ((sub @ (X300 @ X303 @ X302) @ X301) = (X300 @ (comp @ X303 @ X301) @ (sub @ X302 @ X301))) | ($true = (X298 @ id @ (lam @ (X300 @ sh @ one)) @ id)) | ~! [X304 : term] : (($true != (X298 @ id @ X304 @ id)) | ((X298 @ id @ (X300 @ id @ X304) @ id) = $true)))) => ! [X306 : subst > term > subst > $o,X307 : term] : (~! [X308 : term] : (((var @ (sub @ X308 @ id)) != $true) | ((X306 @ id @ X308 @ id) = $true)) | ~! [X314 : subst,X315 : subst,X316 : term,X317 : subst] : (((X306 @ (comp @ X314 @ X315) @ (sub @ X316 @ X315) @ X317) != $true) | ((X306 @ X314 @ X316 @ (comp @ X315 @ X317)) = $true)) | ((X306 @ id @ X307 @ id) = $true) | ~! [X309 : subst > term > term] : (~! [X311 : subst,X312 : term,X313 : subst] : ((sub @ (X309 @ X311 @ X312) @ X313) = (X309 @ (comp @ X311 @ X313) @ (sub @ X312 @ X313))) | ($true = (X306 @ id @ (lam @ (X309 @ sh @ one)) @ id)) | ~! [X310 : term] : (($true = (X306 @ id @ (X309 @ id @ X310) @ id)) | ($true != (X306 @ id @ X310 @ id)))) | ~! [X320 : term,X321 : subst,X322 : subst,X323 : subst] : (($true = (X306 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322) @ X323)) | ($true != (X306 @ X321 @ X320 @ (comp @ X322 @ X323)))) | ~! [X318 : term,X319 : term] : (($true != (X306 @ id @ X319 @ id)) | ((X306 @ id @ X318 @ id) != $true) | ((X306 @ id @ (ap @ (sub @ X318 @ id) @ X319) @ id) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (! [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) != (ap @ (sub @ X508 @ id) @ X510)) | (X507 = X508)) <=> (hoasapinj1 = $true)) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & ((pushprop_lem0 = $true) <=> ! [X333 : term,X334 : term > $o,X335 : subst] : ~! [X336 : term > $o] : ~! [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) = (X336 @ X337))) & (! [X187 : term,X188 : subst > term > subst > $o,X189 : term] : (($true != (X188 @ id @ X189 @ id)) | ($true = (X188 @ id @ (ap @ X189 @ X187) @ id)) | ($true != (X188 @ id @ X187 @ id)) | ~! [X198 : term,X199 : term] : (($true != (X188 @ id @ X199 @ id)) | ($true = (X188 @ id @ (ap @ (sub @ X199 @ id) @ X198) @ id)) | ((X188 @ id @ X198 @ id) != $true)) | ~! [X194 : subst,X195 : subst,X196 : term,X197 : subst] : (($true != (X188 @ X194 @ X196 @ (comp @ X197 @ X195))) | ((X188 @ (comp @ X194 @ X197) @ (sub @ X196 @ X197) @ X195) = $true)) | ~! [X190 : subst,X191 : subst,X192 : term,X193 : subst] : (((X188 @ X191 @ X192 @ (comp @ X193 @ X190)) = $true) | ((X188 @ (comp @ X191 @ X193) @ (sub @ X192 @ X193) @ X190) != $true))) <=> (hoasinduction_lem2 = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X240 : term] : ~! [X241 : subst > term > term] : ~! [X242 : term,X243 : subst] : ((X241 @ X243 @ X242) = (sub @ X240 @ (push @ X242 @ X243))) => (! [X244 : term] : ((sub @ X244 @ id) = X244) => ! [X245 : term,X246 : subst > term > subst > $o,X247 : term > $o] : (~! [X248 : subst > term > term] : (~! [X252 : term] : (($true != (X246 @ id @ X252 @ id)) | ($true = (X246 @ id @ (X248 @ id @ X252) @ id))) | ~! [X249 : term,X250 : subst,X251 : subst] : ((sub @ (X248 @ X251 @ X249) @ X250) = (X248 @ (comp @ X251 @ X250) @ (sub @ X249 @ X250))) | ((X246 @ id @ (lam @ (X248 @ sh @ one)) @ id) = $true)) | ~! [X254 : term] : (((X247 @ X254) != $true) | ((X247 @ (sub @ X245 @ (push @ X254 @ id))) = $true)) | ~! [X253 : term] : ((X247 @ X253) = (X246 @ id @ X253 @ id)) | ($true = (X247 @ (lam @ X245))))))) & (hoaslamnotvar = $true) & ((termmset_lthm = $true) <=> (! [X5 : term] : ((sub @ X5 @ id) = X5) => ! [X6 : term] : ((sub @ X6 @ id) = X6))) & ! [X356 : subst] : ? [X357 : d_subst] : ((d2subst @ X357) = X356) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (~! [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) = (sub @ (X144 @ X149 @ X148) @ X150)) | ~! [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) = (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) | ((X144 @ X146 @ X147) = (X145 @ X146 @ X147)) | ((lam @ (X144 @ sh @ one)) != (lam @ (X145 @ sh @ one)))) <=> (hoaslaminj = $true)) & (pushprop = $true) & ((var @ (d2term @ d_one)) != $true) & (hoaslamnotvar_lthm = $true) & ((induction2_lthm = $true) <=> (! [X93 : term] : ((sub @ X93 @ id) = X93) => (! [X94 : subst,X95 : term > $o,X96 : term] : (((X95 @ (sub @ X96 @ X94)) = $true) | ~! [X97 : term,X98 : term] : (($true = (X95 @ (ap @ X97 @ X98))) | ((X95 @ X98) != $true) | ($true != (X95 @ X97))) | ~! [X100 : term] : (~! [X101 : term] : (((X95 @ (sub @ X100 @ (push @ X101 @ id))) = $true) | ((X95 @ X101) != $true)) | ((X95 @ (lam @ X100)) = $true)) | ~! [X99 : term] : (((var @ X99) != $true) | ($true = (X95 @ (sub @ X99 @ X94))))) => ! [X102 : term > $o,X103 : term] : (~! [X108 : term] : (((X102 @ X108) = $true) | ($true != (var @ X108))) | ((X102 @ X103) = $true) | ~! [X106 : term] : (((X102 @ (lam @ X106)) = $true) | ~! [X107 : term] : (($true = (X102 @ (sub @ X106 @ (push @ X107 @ id)))) | ((X102 @ X107) != $true))) | ~! [X104 : term,X105 : term] : (((X102 @ (ap @ X104 @ X105)) = $true) | ((X102 @ X105) != $true) | ((X102 @ X104) != $true)))))) & (substmonoid_gthm = $true) & (pushprop_lem1_gthm = $true) & (! [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true != (X395 @ X397)) | ($true = (X396 @ one)) | ~! [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) = (X396 @ X399))) <=> (pushprop_lem1v2 = $true)) & (lamnotvar = $true) & (! [X109 : term] : ((sub @ X109 @ id) = X109) <=> (termmset = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X26 : subst > term > subst > $o] : ~! [X27 : term > $o] : ~! [X28 : term] : ((X27 @ X28) = (X26 @ id @ X28 @ id))) & (pushprop_gthm = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem1 = $true) & (hoasapnotvar_gthm = $true) & ((! [X7 : term,X8 : subst] : ((sub @ one @ (push @ X7 @ X8)) = X7) => (! [X9 : subst,X10 : term] : ((comp @ sh @ (push @ X10 @ X9)) = X9) => (! [X11 : subst] : ((comp @ X11 @ id) = X11) => (! [X12 : term > $o,X13 : term] : (~! [X16 : term,X17 : term] : (((X12 @ (ap @ X17 @ X16)) = $true) | ((X12 @ X16) != $true) | ($true != (X12 @ X17))) | ~! [X15 : term] : (($true != (X12 @ X15)) | ((X12 @ (lam @ X15)) = $true)) | ($true = (X12 @ X13)) | ~! [X14 : term] : (($true != (var @ X14)) | ($true = (X12 @ X14)))) => ! [X18 : subst,X19 : term,X20 : term > $o] : (($true = (X20 @ (sub @ X19 @ X18))) | ~! [X21 : term] : (($true = (X20 @ (sub @ X21 @ X18))) | ((var @ X21) != $true)) | ~! [X24 : term] : (~! [X25 : term] : (((X20 @ X25) != $true) | ((X20 @ (sub @ X24 @ (push @ X25 @ id))) = $true)) | ((X20 @ (lam @ X24)) = $true)) | ~! [X22 : term,X23 : term] : (($true = (X20 @ (ap @ X23 @ X22))) | ($true != (X20 @ X23)) | ((X20 @ X22) != $true))))))) <=> (induction2lem_lthm = $true)) & (pushprop_lem2v2_gthm = $true) & ((axshiftcons = $true) <=> ! [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) = X201)) & ((! [X484 : term] : ((sub @ X484 @ id) = X484) => ! [X485 : term > $o,X486 : subst > term > subst > $o,X487 : term] : (~! [X498 : subst > term > term] : (~! [X500 : subst,X501 : subst,X502 : term] : ((X498 @ (comp @ X500 @ X501) @ (sub @ X502 @ X501)) = (sub @ (X498 @ X500 @ X502) @ X501)) | ~! [X499 : term] : (($true != (X486 @ id @ X499 @ id)) | ($true = (X486 @ id @ (X498 @ id @ X499) @ id))) | ((X486 @ id @ (lam @ (X498 @ sh @ one)) @ id) = $true)) | ~! [X488 : subst,X489 : subst,X490 : term,X491 : subst] : (((X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X491) = $true) | ((X486 @ X488 @ X490 @ (comp @ X489 @ X491)) != $true)) | ~! [X492 : subst,X493 : term,X494 : subst,X495 : subst] : (((X486 @ (comp @ X492 @ X495) @ (sub @ X493 @ X495) @ X494) != $true) | ((X486 @ X492 @ X493 @ (comp @ X495 @ X494)) = $true)) | ((X485 @ (lam @ X487)) = $true) | ~! [X496 : term] : (((X485 @ (sub @ X487 @ (push @ X496 @ id))) = $true) | ((X485 @ X496) != $true)) | ~! [X497 : term] : ((X485 @ X497) = (X486 @ id @ X497 @ id)))) <=> (hoasinduction_lem3v2_lthm = $true)) & ! [X338 : d_subst,X339 : d_subst] : (((d2subst @ X338) = (d2subst @ X339)) => (X338 = X339)) & (! [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (($true = (X475 @ (sub @ X473 @ (push @ X476 @ X474)))) | ($true != (X472 @ X473))) <=> (pushprop_lem1 = $true)) & (! [X511 : term] : ((sub @ X511 @ id) = X511) <=> (axvarid = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X29 : subst > term > subst > $o,X30 : term,X31 : term > $o] : (((X31 @ (lam @ X30)) = $true) | ~! [X46 : term] : ((X29 @ id @ X46 @ id) = (X31 @ X46)) | ~! [X36 : term] : (($true != (X31 @ X36)) | ($true = (X31 @ (sub @ X30 @ (push @ X36 @ id))))) | ~! [X42 : subst,X43 : subst,X44 : term,X45 : subst] : (((X29 @ (comp @ X42 @ X43) @ (sub @ X44 @ X43) @ X45) != $true) | ($true = (X29 @ X42 @ X44 @ (comp @ X43 @ X45)))) | ~! [X32 : subst,X33 : subst,X34 : term,X35 : subst] : (($true = (X29 @ (comp @ X35 @ X32) @ (sub @ X34 @ X32) @ X33)) | ($true != (X29 @ X35 @ X34 @ (comp @ X32 @ X33)))) | ~! [X37 : subst > term > term] : (~! [X41 : term] : (((X29 @ id @ X41 @ id) != $true) | ($true = (X29 @ id @ (X37 @ id @ X41) @ id))) | ((X29 @ id @ (lam @ (X37 @ sh @ one)) @ id) = $true) | ~! [X38 : subst,X39 : subst,X40 : term] : ((sub @ (X37 @ X38 @ X40) @ X39) = (X37 @ (comp @ X38 @ X39) @ (sub @ X40 @ X39)))))) & (axmap = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X47 : subst > term > subst > $o] : ~! [X48 : term > $o] : ~! [X49 : term] : ((X47 @ id @ X49 @ id) = (X48 @ X49)) => (! [X50 : term > $o,X51 : term] : (~! [X52 : term,X53 : term] : (($true != (X50 @ X53)) | ((X50 @ X52) != $true) | ($true = (X50 @ (ap @ X53 @ X52)))) | ((X50 @ X51) = $true) | ~! [X56 : term] : (($true != (var @ X56)) | ($true = (X50 @ X56))) | ~! [X54 : term] : (((X50 @ (lam @ X54)) = $true) | ~! [X55 : term] : (((X50 @ X55) != $true) | ((X50 @ (sub @ X54 @ (push @ X55 @ id))) = $true)))) => (! [X57 : term] : ((sub @ X57 @ id) = X57) => (! [X58 : subst > term > subst > $o,X59 : term,X60 : term > $o] : (($true = (X60 @ (lam @ X59))) | ~! [X63 : subst > term > term] : (((X58 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true) | ~! [X67 : term] : (($true != (X58 @ id @ X67 @ id)) | ((X58 @ id @ (X63 @ id @ X67) @ id) = $true)) | ~! [X64 : subst,X65 : term,X66 : subst] : ((sub @ (X63 @ X64 @ X65) @ X66) = (X63 @ (comp @ X64 @ X66) @ (sub @ X65 @ X66)))) | ~! [X61 : term] : (((X60 @ (sub @ X59 @ (push @ X61 @ id))) = $true) | ((X60 @ X61) != $true)) | ~! [X62 : term] : ((X58 @ id @ X62 @ id) = (X60 @ X62))) => ! [X68 : subst > term > subst > $o,X69 : term] : (~! [X70 : term,X71 : term] : (($true != (X68 @ id @ X71 @ id)) | ($true = (X68 @ id @ (ap @ (sub @ X71 @ id) @ X70) @ id)) | ((X68 @ id @ X70 @ id) != $true)) | ~! [X72 : subst > term > term] : (((X68 @ id @ (lam @ (X72 @ sh @ one)) @ id) = $true) | ~! [X73 : subst,X74 : subst,X75 : term] : ((sub @ (X72 @ X74 @ X75) @ X73) = (X72 @ (comp @ X74 @ X73) @ (sub @ X75 @ X73))) | ~! [X76 : term] : (($true = (X68 @ id @ (X72 @ id @ X76) @ id)) | ((X68 @ id @ X76 @ id) != $true))) | ($true = (X68 @ id @ X69 @ id)))))))) & (induction2lem_gthm = $true) & ((apinj1 = $true) <=> ! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 = X239) | ((ap @ X236 @ X238) != (ap @ X239 @ X237)))) & (! [X122 : term,X123 : subst > term > subst > $o] : (~! [X125 : subst > term > term] : (($true = (X123 @ id @ (lam @ (X125 @ sh @ one)) @ id)) | ~! [X126 : subst,X127 : term,X128 : subst] : ((X125 @ (comp @ X128 @ X126) @ (sub @ X127 @ X126)) = (sub @ (X125 @ X128 @ X127) @ X126)) | ~! [X129 : term] : (((X123 @ id @ X129 @ id) != $true) | ($true = (X123 @ id @ (X125 @ id @ X129) @ id)))) | ~! [X130 : subst,X131 : subst,X132 : subst,X133 : term] : (((X123 @ (comp @ X131 @ X130) @ (sub @ X133 @ X130) @ X132) = $true) | ((X123 @ X131 @ X133 @ (comp @ X130 @ X132)) != $true)) | ~! [X134 : subst,X135 : subst,X136 : term,X137 : subst] : (((X123 @ X135 @ X136 @ (comp @ X137 @ X134)) = $true) | ((X123 @ (comp @ X135 @ X137) @ (sub @ X136 @ X137) @ X134) != $true)) | ~! [X124 : term] : (($true != (X123 @ id @ X124 @ id)) | ($true = (X123 @ id @ (sub @ X122 @ (push @ X124 @ id)) @ id))) | ((X123 @ id @ (lam @ X122) @ id) = $true)) <=> (hoasinduction_lem3 = $true)) & (axapp = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (! [X340 : subst > term > subst > $o,X341 : term] : (($true = (X340 @ id @ (lam @ (sub @ X341 @ (push @ one @ sh))) @ id)) | ~! [X342 : term] : (((X340 @ id @ X342 @ id) != $true) | ($true = (X340 @ id @ (sub @ X341 @ (push @ X342 @ id)) @ id))) | ~! [X343 : term,X344 : subst > term > term] : (~! [X349 : subst,X350 : subst,X351 : term] : ((sub @ X343 @ (push @ (sub @ X351 @ X350) @ (comp @ X349 @ X350))) = (X344 @ (comp @ X349 @ X350) @ (sub @ X351 @ X350))) | ((X340 @ id @ (lam @ (X344 @ sh @ one)) @ id) = $true) | ~! [X345 : subst,X346 : subst,X347 : term] : ((sub @ (sub @ X343 @ (push @ X347 @ X346)) @ X345) = (sub @ (X344 @ X346 @ X347) @ X345)) | ~! [X348 : term] : (($true = (X340 @ id @ (X344 @ id @ X348) @ id)) | ($true != (X340 @ id @ X348 @ id))))) <=> (hoasinduction_lem3aaa = $true)) & (hoasapnotvar = $true) & ((axscons = $true) <=> ! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0)) & (! [X83 : subst > term > subst > $o,X84 : term,X85 : term > $o] : (~! [X88 : subst > term > term] : (~! [X92 : term] : (((X83 @ id @ X92 @ id) != $true) | ((X83 @ id @ (X88 @ id @ X92) @ id) = $true)) | ($true = (X83 @ id @ (lam @ (X88 @ sh @ one)) @ id)) | ~! [X89 : subst,X90 : subst,X91 : term] : ((X88 @ (comp @ X90 @ X89) @ (sub @ X91 @ X89)) = (sub @ (X88 @ X90 @ X91) @ X89))) | ~! [X87 : term] : ((X83 @ id @ X87 @ id) = (X85 @ X87)) | ~! [X86 : term] : (($true != (X85 @ X86)) | ($true = (X85 @ (sub @ X84 @ (push @ X86 @ id))))) | ($true = (X85 @ (lam @ X84)))) <=> (hoasinduction_lem3v2a = $true))),
% 9.64/2.82 inference(flattening,[status(thm)],[f10])).
% 9.64/2.82 thf(f10,plain,(
% 9.64/2.82 ((axscons = $true) <=> ! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0)) & (axabs = $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X1 : term] : ~! [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) != (X2 @ sh @ one))) & (axclos = $true) & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & (hoasinduction_lem1v2_gthm = $true) & ((termmset_lthm = $true) <=> (! [X5 : term] : ((sub @ X5 @ id) = X5) => ! [X6 : term] : ((sub @ X6 @ id) = X6))) & (axvarshift = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((! [X7 : term,X8 : subst] : ((sub @ one @ (push @ X7 @ X8)) = X7) => (! [X9 : subst,X10 : term] : ((comp @ sh @ (push @ X10 @ X9)) = X9) => (! [X11 : subst] : ((comp @ X11 @ id) = X11) => (! [X12 : term > $o,X13 : term] : (~! [X14 : term] : (($true = (X12 @ X14)) | ~($true = (var @ X14))) | ($true = (X12 @ X13)) | ~! [X15 : term] : (~($true = (X12 @ X15)) | ((X12 @ (lam @ X15)) = $true)) | ~! [X16 : term,X17 : term] : (((X12 @ (ap @ X17 @ X16)) = $true) | ~($true = (X12 @ X17)) | ~((X12 @ X16) = $true))) => ! [X18 : subst,X19 : term,X20 : term > $o] : (~! [X21 : term] : (($true = (X20 @ (sub @ X21 @ X18))) | ~((var @ X21) = $true)) | ($true = (X20 @ (sub @ X19 @ X18))) | ~! [X22 : term,X23 : term] : (~($true = (X20 @ X23)) | ~((X20 @ X22) = $true) | ($true = (X20 @ (ap @ X23 @ X22)))) | ~! [X24 : term] : (((X20 @ (lam @ X24)) = $true) | ~! [X25 : term] : (((X20 @ (sub @ X24 @ (push @ X25 @ id))) = $true) | ~((X20 @ X25) = $true)))))))) <=> (induction2lem_lthm = $true)) & (ulamvarsh = $true) & (hoasinduction_gthm = $true) & (pushprop_lthm = $true) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X26 : subst > term > subst > $o] : ~! [X27 : term > $o] : ~! [X28 : term] : ((X27 @ X28) = (X26 @ id @ X28 @ id))) & (hoasinduction_lem3v2_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((d2subst @ d_id) = sh) & ((hoasinduction_lem3v2 = $true) <=> ! [X29 : subst > term > subst > $o,X30 : term,X31 : term > $o] : (~! [X32 : subst,X33 : subst,X34 : term,X35 : subst] : (($true = (X29 @ (comp @ X35 @ X32) @ (sub @ X34 @ X32) @ X33)) | ~($true = (X29 @ X35 @ X34 @ (comp @ X32 @ X33)))) | ~! [X36 : term] : (($true = (X31 @ (sub @ X30 @ (push @ X36 @ id)))) | ~($true = (X31 @ X36))) | ~! [X37 : subst > term > term] : (((X29 @ id @ (lam @ (X37 @ sh @ one)) @ id) = $true) | ~! [X38 : subst,X39 : subst,X40 : term] : ((sub @ (X37 @ X38 @ X40) @ X39) = (X37 @ (comp @ X38 @ X39) @ (sub @ X40 @ X39))) | ~! [X41 : term] : (($true = (X29 @ id @ (X37 @ id @ X41) @ id)) | ~((X29 @ id @ X41 @ id) = $true))) | ~! [X42 : subst,X43 : subst,X44 : term,X45 : subst] : (~((X29 @ (comp @ X42 @ X43) @ (sub @ X44 @ X43) @ X45) = $true) | ($true = (X29 @ X42 @ X44 @ (comp @ X43 @ X45)))) | ~! [X46 : term] : ((X29 @ id @ X46 @ id) = (X31 @ X46)) | ((X31 @ (lam @ X30)) = $true))) & ((! [X47 : subst > term > subst > $o] : ~! [X48 : term > $o] : ~! [X49 : term] : ((X47 @ id @ X49 @ id) = (X48 @ X49)) => (! [X50 : term > $o,X51 : term] : (~! [X52 : term,X53 : term] : (($true = (X50 @ (ap @ X53 @ X52))) | ~($true = (X50 @ X53)) | ~((X50 @ X52) = $true)) | ~! [X54 : term] : (~! [X55 : term] : (((X50 @ (sub @ X54 @ (push @ X55 @ id))) = $true) | ~((X50 @ X55) = $true)) | ((X50 @ (lam @ X54)) = $true)) | ((X50 @ X51) = $true) | ~! [X56 : term] : (($true = (X50 @ X56)) | ~($true = (var @ X56)))) => (! [X57 : term] : ((sub @ X57 @ id) = X57) => (! [X58 : subst > term > subst > $o,X59 : term,X60 : term > $o] : (~! [X61 : term] : (~((X60 @ X61) = $true) | ((X60 @ (sub @ X59 @ (push @ X61 @ id))) = $true)) | ~! [X62 : term] : ((X58 @ id @ X62 @ id) = (X60 @ X62)) | ($true = (X60 @ (lam @ X59))) | ~! [X63 : subst > term > term] : (~! [X64 : subst,X65 : term,X66 : subst] : ((sub @ (X63 @ X64 @ X65) @ X66) = (X63 @ (comp @ X64 @ X66) @ (sub @ X65 @ X66))) | ~! [X67 : term] : (~($true = (X58 @ id @ X67 @ id)) | ((X58 @ id @ (X63 @ id @ X67) @ id) = $true)) | ((X58 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true))) => ! [X68 : subst > term > subst > $o,X69 : term] : (~! [X70 : term,X71 : term] : (~($true = (X68 @ id @ X71 @ id)) | ($true = (X68 @ id @ (ap @ (sub @ X71 @ id) @ X70) @ id)) | ~((X68 @ id @ X70 @ id) = $true)) | ~! [X72 : subst > term > term] : (((X68 @ id @ (lam @ (X72 @ sh @ one)) @ id) = $true) | ~! [X73 : subst,X74 : subst,X75 : term] : ((sub @ (X72 @ X74 @ X75) @ X73) = (X72 @ (comp @ X74 @ X73) @ (sub @ X75 @ X73))) | ~! [X76 : term] : (~((X68 @ id @ X76 @ id) = $true) | ($true = (X68 @ id @ (X72 @ id @ X76) @ id)))) | ($true = (X68 @ id @ X69 @ id))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & (! [X77 : term,X78 : term] : (((sub @ X78 @ sh) != (sub @ X77 @ sh)) | (X77 = X78)) <=> (shinj = $true)) & ((apinj2 = $true) <=> ! [X79 : term,X80 : term,X81 : term,X82 : term] : (((ap @ X80 @ X81) != (ap @ X79 @ X82)) | (X81 = X82))) & ((hoasinduction_lem3v2a = $true) <=> ! [X83 : subst > term > subst > $o,X84 : term,X85 : term > $o] : (~! [X86 : term] : (($true = (X85 @ (sub @ X84 @ (push @ X86 @ id)))) | ~($true = (X85 @ X86))) | ~! [X87 : term] : ((X83 @ id @ X87 @ id) = (X85 @ X87)) | ~! [X88 : subst > term > term] : (~! [X89 : subst,X90 : subst,X91 : term] : ((X88 @ (comp @ X90 @ X89) @ (sub @ X91 @ X89)) = (sub @ (X88 @ X90 @ X91) @ X89)) | ~! [X92 : term] : (~((X83 @ id @ X92 @ id) = $true) | ((X83 @ id @ (X88 @ id @ X92) @ id) = $true)) | ($true = (X83 @ id @ (lam @ (X88 @ sh @ one)) @ id))) | ($true = (X85 @ (lam @ X84))))) & (pushprop = $true) & ((induction2_lthm = $true) <=> (! [X93 : term] : ((sub @ X93 @ id) = X93) => (! [X94 : subst,X95 : term > $o,X96 : term] : (~! [X97 : term,X98 : term] : (~($true = (X95 @ X97)) | ~((X95 @ X98) = $true) | ($true = (X95 @ (ap @ X97 @ X98)))) | ~! [X99 : term] : (~((var @ X99) = $true) | ($true = (X95 @ (sub @ X99 @ X94)))) | ((X95 @ (sub @ X96 @ X94)) = $true) | ~! [X100 : term] : (((X95 @ (lam @ X100)) = $true) | ~! [X101 : term] : (((X95 @ (sub @ X100 @ (push @ X101 @ id))) = $true) | ~((X95 @ X101) = $true)))) => ! [X102 : term > $o,X103 : term] : (~! [X104 : term,X105 : term] : (~((X102 @ X105) = $true) | ((X102 @ (ap @ X104 @ X105)) = $true) | ~((X102 @ X104) = $true)) | ((X102 @ X103) = $true) | ~! [X106 : term] : (~! [X107 : term] : (($true = (X102 @ (sub @ X106 @ (push @ X107 @ id)))) | ~((X102 @ X107) = $true)) | ((X102 @ (lam @ X106)) = $true)) | ~! [X108 : term] : (((X102 @ X108) = $true) | ~($true = (var @ X108))))))) & (! [X109 : term] : ((sub @ X109 @ id) = X109) <=> (termmset = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X110 : subst > term > subst > $o] : ~! [X111 : term > $o] : ~! [X112 : term] : ((X110 @ id @ X112 @ id) = (X111 @ X112))) & (substmonoid_gthm = $true) & ((! [X113 : subst,X114 : term] : ((sub @ one @ (push @ X114 @ X113)) = X114) => (! [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) = X116) => ! [X117 : term > $o,X118 : term,X119 : subst,X120 : term > $o,X121 : term] : (($true = (X117 @ (sub @ X121 @ (push @ X118 @ X119)))) | ~($true = (X120 @ X121))))) <=> (pushprop_lem1_lthm = $true)) & (! [X122 : term,X123 : subst > term > subst > $o] : (~! [X124 : term] : (($true = (X123 @ id @ (sub @ X122 @ (push @ X124 @ id)) @ id)) | ~($true = (X123 @ id @ X124 @ id))) | ~! [X125 : subst > term > term] : (~! [X126 : subst,X127 : term,X128 : subst] : ((X125 @ (comp @ X128 @ X126) @ (sub @ X127 @ X126)) = (sub @ (X125 @ X128 @ X127) @ X126)) | ~! [X129 : term] : (($true = (X123 @ id @ (X125 @ id @ X129) @ id)) | ~((X123 @ id @ X129 @ id) = $true)) | ($true = (X123 @ id @ (lam @ (X125 @ sh @ one)) @ id))) | ~! [X130 : subst,X131 : subst,X132 : subst,X133 : term] : (~((X123 @ X131 @ X133 @ (comp @ X130 @ X132)) = $true) | ((X123 @ (comp @ X131 @ X130) @ (sub @ X133 @ X130) @ X132) = $true)) | ~! [X134 : subst,X135 : subst,X136 : term,X137 : subst] : (~((X123 @ (comp @ X135 @ X137) @ (sub @ X136 @ X137) @ X134) = $true) | ((X123 @ X135 @ X136 @ (comp @ X137 @ X134)) = $true)) | ((X123 @ id @ (lam @ X122) @ id) = $true)) <=> (hoasinduction_lem3 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X138 : term] : ~! [X139 : subst > term > term] : ~! [X140 : term,X141 : subst] : ((sub @ X138 @ (push @ X140 @ X141)) = (X139 @ X141 @ X140))) & (! [X142 : term] : ~! [X143 : subst > term > term] : ((X143 @ sh @ one) != (sub @ X142 @ (push @ one @ sh))) <=> (hoasinduction_lem3b = $true)) & (hoaslaminj_gthm = $true) & (pushprop_lem1_gthm = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lem3v2 = $true) & (! [X144 : subst > term > term,X145 : subst > term > term,X146 : subst,X147 : term] : (~! [X148 : term,X149 : subst,X150 : subst] : ((X144 @ (comp @ X149 @ X150) @ (sub @ X148 @ X150)) = (sub @ (X144 @ X149 @ X148) @ X150)) | ~! [X151 : subst,X152 : subst,X153 : term] : ((sub @ (X145 @ X152 @ X153) @ X151) = (X145 @ (comp @ X152 @ X151) @ (sub @ X153 @ X151))) | ((X144 @ X146 @ X147) = (X145 @ X146 @ X147)) | ((lam @ (X144 @ sh @ one)) != (lam @ (X145 @ sh @ one)))) <=> (hoaslaminj = $true)) & ~((var @ (d2term @ d_one)) = $true) & (! [X154 : subst > term > subst > $o,X155 : term] : (~! [X156 : term] : (($true = (X154 @ id @ (sub @ X155 @ (push @ X156 @ id)) @ id)) | ~((X154 @ id @ X156 @ id) = $true)) | ~! [X157 : subst > term > term] : (($true = (X154 @ id @ (lam @ (X157 @ sh @ one)) @ id)) | ~! [X158 : term] : (~($true = (X154 @ id @ X158 @ id)) | ($true = (X154 @ id @ (X157 @ id @ X158) @ id))) | ~! [X159 : subst,X160 : subst,X161 : term] : ((X157 @ (comp @ X159 @ X160) @ (sub @ X161 @ X160)) = (sub @ (X157 @ X159 @ X161) @ X160))) | ($true = (X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id))) <=> (hoasinduction_lem3aa = $true)) & (axmap = $true) & (apnotvar = $true) & ((! [X162 : term] : ((sub @ X162 @ id) = X162) => (! [X163 : subst > term > subst > $o,X164 : term] : (~! [X165 : term] : (($true = (X163 @ id @ (sub @ X164 @ (push @ X165 @ id)) @ id)) | ~($true = (X163 @ id @ X165 @ id))) | ((X163 @ id @ (lam @ (sub @ X164 @ (push @ one @ sh))) @ id) = $true) | ~! [X166 : subst > term > term] : (~! [X167 : subst,X168 : subst,X169 : term] : ((X166 @ (comp @ X168 @ X167) @ (sub @ X169 @ X167)) = (sub @ (X166 @ X168 @ X169) @ X167)) | ~! [X170 : term] : (((X163 @ id @ (X166 @ id @ X170) @ id) = $true) | ~($true = (X163 @ id @ X170 @ id))) | ($true = (X163 @ id @ (lam @ (X166 @ sh @ one)) @ id)))) => ! [X171 : term,X172 : subst > term > subst > $o] : (~! [X173 : term] : (~((X172 @ id @ X173 @ id) = $true) | ($true = (X172 @ id @ (sub @ X171 @ (push @ X173 @ id)) @ id))) | ~! [X174 : subst > term > term] : (~! [X175 : term] : (((X172 @ id @ (X174 @ id @ X175) @ id) = $true) | ~((X172 @ id @ X175 @ id) = $true)) | ~! [X176 : subst,X177 : subst,X178 : term] : ((sub @ (X174 @ X176 @ X178) @ X177) = (X174 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177))) | ((X172 @ id @ (lam @ (X174 @ sh @ one)) @ id) = $true)) | ~! [X179 : term,X180 : subst,X181 : subst,X182 : subst] : (($true = (X172 @ (comp @ X182 @ X180) @ (sub @ X179 @ X180) @ X181)) | ~((X172 @ X182 @ X179 @ (comp @ X180 @ X181)) = $true)) | ($true = (X172 @ id @ (lam @ X171) @ id)) | ~! [X183 : subst,X184 : subst,X185 : term,X186 : subst] : (~($true = (X172 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183) @ X186)) | ((X172 @ X184 @ X185 @ (comp @ X183 @ X186)) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & (lamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X187 : term,X188 : subst > term > subst > $o,X189 : term] : (~! [X190 : subst,X191 : subst,X192 : term,X193 : subst] : (~((X188 @ (comp @ X191 @ X193) @ (sub @ X192 @ X193) @ X190) = $true) | ((X188 @ X191 @ X192 @ (comp @ X193 @ X190)) = $true)) | ($true = (X188 @ id @ (ap @ X189 @ X187) @ id)) | ~! [X194 : subst,X195 : subst,X196 : term,X197 : subst] : (~($true = (X188 @ X194 @ X196 @ (comp @ X197 @ X195))) | ((X188 @ (comp @ X194 @ X197) @ (sub @ X196 @ X197) @ X195) = $true)) | ~($true = (X188 @ id @ X189 @ id)) | ~($true = (X188 @ id @ X187 @ id)) | ~! [X198 : term,X199 : term] : (($true = (X188 @ id @ (ap @ (sub @ X199 @ id) @ X198) @ id)) | ~((X188 @ id @ X198 @ id) = $true) | ~($true = (X188 @ id @ X199 @ id))))) & (pushprop_lem2v2 = $true) & ((axshiftcons = $true) <=> ! [X200 : term,X201 : subst] : ((comp @ sh @ (push @ X200 @ X201)) = X201)) & (! [X202 : subst > term > subst > $o,X203 : term] : (((X202 @ id @ X203 @ id) = $true) | ~! [X204 : term,X205 : term] : (~($true = (X202 @ id @ X205 @ id)) | ~($true = (X202 @ id @ X204 @ id)) | ((X202 @ id @ (ap @ (sub @ X204 @ id) @ X205) @ id) = $true)) | ~! [X206 : subst > term > term] : (~! [X207 : term] : (~((X202 @ id @ X207 @ id) = $true) | ($true = (X202 @ id @ (X206 @ id @ X207) @ id))) | ~! [X208 : term,X209 : subst,X210 : subst] : ((X206 @ (comp @ X210 @ X209) @ (sub @ X208 @ X209)) = (sub @ (X206 @ X210 @ X208) @ X209)) | ($true = (X202 @ id @ (lam @ (X206 @ sh @ one)) @ id)))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasinduction_lem3_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction = $true) <=> ! [X211 : term,X212 : subst > term > subst > $o] : (($true = (X212 @ id @ X211 @ id)) | ~! [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (((X212 @ X214 @ X213 @ (comp @ X216 @ X215)) = $true) | ~($true = (X212 @ (comp @ X214 @ X216) @ (sub @ X213 @ X216) @ X215))) | ~! [X217 : subst,X218 : term,X219 : subst,X220 : subst] : (((X212 @ (comp @ X219 @ X217) @ (sub @ X218 @ X217) @ X220) = $true) | ~((X212 @ X219 @ X218 @ (comp @ X217 @ X220)) = $true)) | ~! [X221 : term] : (~((var @ (sub @ X221 @ id)) = $true) | ((X212 @ id @ X221 @ id) = $true)) | ~! [X222 : subst > term > term] : (~! [X223 : term] : (~((X212 @ id @ X223 @ id) = $true) | ($true = (X212 @ id @ (X222 @ id @ X223) @ id))) | ~! [X224 : term,X225 : subst,X226 : subst] : ((X222 @ (comp @ X226 @ X225) @ (sub @ X224 @ X225)) = (sub @ (X222 @ X226 @ X224) @ X225)) | ($true = (X212 @ id @ (lam @ (X222 @ sh @ one)) @ id))) | ~! [X227 : term,X228 : term] : (~((X212 @ id @ X228 @ id) = $true) | ~((X212 @ id @ X227 @ id) = $true) | ((X212 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id) = $true)))) & (hoasinduction_lem1 = $true) & ! [X229 : d_term] : (d_one = X229) & (hoaslamnotap_lthm = $true) & ((induction = $true) <=> ! [X230 : term > $o,X231 : term] : (((X230 @ X231) = $true) | ~! [X232 : term] : (($true = (X230 @ X232)) | ~((var @ X232) = $true)) | ~! [X233 : term,X234 : term] : (~($true = (X230 @ X234)) | ($true = (X230 @ (ap @ X233 @ X234))) | ~($true = (X230 @ X233))) | ~! [X235 : term] : (~($true = (X230 @ X235)) | ($true = (X230 @ (lam @ X235)))))) & ((apinj1 = $true) <=> ! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X236 = X239) | ((ap @ X236 @ X238) != (ap @ X239 @ X237)))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X240 : term] : ~! [X241 : subst > term > term] : ~! [X242 : term,X243 : subst] : ((X241 @ X243 @ X242) = (sub @ X240 @ (push @ X242 @ X243))) => (! [X244 : term] : ((sub @ X244 @ id) = X244) => ! [X245 : term,X246 : subst > term > subst > $o,X247 : term > $o] : (~! [X248 : subst > term > term] : (~! [X249 : term,X250 : subst,X251 : subst] : ((sub @ (X248 @ X251 @ X249) @ X250) = (X248 @ (comp @ X251 @ X250) @ (sub @ X249 @ X250))) | ((X246 @ id @ (lam @ (X248 @ sh @ one)) @ id) = $true) | ~! [X252 : term] : (~($true = (X246 @ id @ X252 @ id)) | ($true = (X246 @ id @ (X248 @ id @ X252) @ id)))) | ~! [X253 : term] : ((X247 @ X253) = (X246 @ id @ X253 @ id)) | ($true = (X247 @ (lam @ X245))) | ~! [X254 : term] : (~((X247 @ X254) = $true) | ((X247 @ (sub @ X245 @ (push @ X254 @ id))) = $true)))))) & (! [X255 : term,X256 : term > $o] : (~! [X257 : term,X258 : term] : (((X256 @ (ap @ X257 @ X258)) = $true) | ~($true = (X256 @ X257)) | ~((X256 @ X258) = $true)) | ~! [X259 : term] : (($true = (X256 @ X259)) | ~((var @ X259) = $true)) | ((X256 @ X255) = $true) | ~! [X260 : term] : (($true = (X256 @ (lam @ X260))) | ~! [X261 : term] : (~($true = (X256 @ X261)) | ($true = (X256 @ (sub @ X260 @ (push @ X261 @ id))))))) <=> (induction2 = $true)) & (! [X262 : subst > term > subst > $o,X263 : term] : (~! [X264 : term] : (((X262 @ id @ (sub @ X263 @ (push @ X264 @ id)) @ id) = $true) | ~((X262 @ id @ X264 @ id) = $true)) | ~! [X265 : subst > term > term] : (~! [X266 : term] : (($true = (X262 @ id @ (X265 @ id @ X266) @ id)) | ~((X262 @ id @ X266 @ id) = $true)) | ($true = (X262 @ id @ (lam @ (X265 @ sh @ one)) @ id)) | ~! [X267 : subst,X268 : subst,X269 : term] : ((sub @ (X265 @ X268 @ X269) @ X267) = (X265 @ (comp @ X268 @ X267) @ (sub @ X269 @ X267)))) | ((X262 @ id @ (lam @ X263) @ id) = $true)) <=> (hoasinduction_lem3a = $true)) & (pushprop_lem1v2_gthm = $true) & (pushprop_lthm_orig = $true) & (! [X270 : term,X271 : term > $o,X272 : subst > term > subst > $o,X273 : term] : (~! [X274 : term,X275 : subst,X276 : subst,X277 : subst] : (($true = (X272 @ (comp @ X277 @ X276) @ (sub @ X274 @ X276) @ X275)) | ~($true = (X272 @ X277 @ X274 @ (comp @ X276 @ X275)))) | ~($true = (X271 @ X273)) | ($true = (X271 @ (ap @ X273 @ X270))) | ~! [X278 : term,X279 : term] : (~((X272 @ id @ X279 @ id) = $true) | ~((X272 @ id @ X278 @ id) = $true) | ($true = (X272 @ id @ (ap @ (sub @ X279 @ id) @ X278) @ id))) | ~! [X280 : subst,X281 : subst,X282 : term,X283 : subst] : (($true = (X272 @ X281 @ X282 @ (comp @ X283 @ X280))) | ~((X272 @ (comp @ X281 @ X283) @ (sub @ X282 @ X283) @ X280) = $true)) | ~! [X284 : term] : ((X272 @ id @ X284 @ id) = (X271 @ X284)) | ~($true = (X271 @ X270))) <=> (hoasinduction_lem2v2 = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X285 : subst > term > subst > $o] : ~! [X286 : term > $o] : ~! [X287 : term] : ((X285 @ id @ X287 @ id) = (X286 @ X287)) => (! [X288 : term,X289 : term > $o] : (~! [X290 : term] : (($true = (X289 @ X290)) | ~((var @ X290) = $true)) | ((X289 @ X288) = $true) | ~! [X291 : term] : (~! [X292 : term] : (~((X289 @ X292) = $true) | ($true = (X289 @ (sub @ X291 @ (push @ X292 @ id))))) | ((X289 @ (lam @ X291)) = $true)) | ~! [X293 : term,X294 : term] : (~((X289 @ X294) = $true) | ($true = (X289 @ (ap @ X293 @ X294))) | ~((X289 @ X293) = $true))) => (! [X295 : term] : ((sub @ X295 @ id) = X295) => (! [X296 : term,X297 : term > $o,X298 : subst > term > subst > $o] : (~! [X299 : term] : ((X297 @ X299) = (X298 @ id @ X299 @ id)) | ((X297 @ (lam @ X296)) = $true) | ~! [X300 : subst > term > term] : (($true = (X298 @ id @ (lam @ (X300 @ sh @ one)) @ id)) | ~! [X301 : subst,X302 : term,X303 : subst] : ((sub @ (X300 @ X303 @ X302) @ X301) = (X300 @ (comp @ X303 @ X301) @ (sub @ X302 @ X301))) | ~! [X304 : term] : (~($true = (X298 @ id @ X304 @ id)) | ((X298 @ id @ (X300 @ id @ X304) @ id) = $true))) | ~! [X305 : term] : (~((X297 @ X305) = $true) | ((X297 @ (sub @ X296 @ (push @ X305 @ id))) = $true))) => ! [X306 : subst > term > subst > $o,X307 : term] : (~! [X308 : term] : (~((var @ (sub @ X308 @ id)) = $true) | ((X306 @ id @ X308 @ id) = $true)) | ~! [X309 : subst > term > term] : (~! [X310 : term] : (($true = (X306 @ id @ (X309 @ id @ X310) @ id)) | ~($true = (X306 @ id @ X310 @ id))) | ($true = (X306 @ id @ (lam @ (X309 @ sh @ one)) @ id)) | ~! [X311 : subst,X312 : term,X313 : subst] : ((sub @ (X309 @ X311 @ X312) @ X313) = (X309 @ (comp @ X311 @ X313) @ (sub @ X312 @ X313)))) | ~! [X314 : subst,X315 : subst,X316 : term,X317 : subst] : (((X306 @ X314 @ X316 @ (comp @ X315 @ X317)) = $true) | ~((X306 @ (comp @ X314 @ X315) @ (sub @ X316 @ X315) @ X317) = $true)) | ~! [X318 : term,X319 : term] : (((X306 @ id @ (ap @ (sub @ X318 @ id) @ X319) @ id) = $true) | ~($true = (X306 @ id @ X319 @ id)) | ~((X306 @ id @ X318 @ id) = $true)) | ~! [X320 : term,X321 : subst,X322 : subst,X323 : subst] : (~($true = (X306 @ X321 @ X320 @ (comp @ X322 @ X323))) | ($true = (X306 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322) @ X323))) | ((X306 @ id @ X307 @ id) = $true))))))) & ~(lamnotap = $true) & (! [X324 : subst] : ((comp @ id @ X324) = X324) <=> (axidl = $true)) & (! [X325 : term > $o,X326 : subst,X327 : term] : (~! [X328 : term,X329 : term] : (~($true = (X325 @ X329)) | ~($true = (X325 @ X328)) | ($true = (X325 @ (ap @ X329 @ X328)))) | ~! [X330 : term] : (~($true = (var @ X330)) | ($true = (X325 @ (sub @ X330 @ X326)))) | ~! [X331 : term] : (~! [X332 : term] : (~($true = (X325 @ X332)) | ($true = (X325 @ (sub @ X331 @ (push @ X332 @ id))))) | ((X325 @ (lam @ X331)) = $true)) | ($true = (X325 @ (sub @ X327 @ X326)))) <=> (induction2lem = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (hoasapnotvar_gthm = $true) & ((d2subst @ d_id) = id) & ((pushprop_lem0 = $true) <=> ! [X333 : term,X334 : term > $o,X335 : subst] : ~! [X336 : term > $o] : ~! [X337 : term] : ((X334 @ (sub @ X337 @ (push @ X333 @ X335))) = (X336 @ X337))) & (hoasapnotvar = $true) & ! [X338 : d_subst,X339 : d_subst] : (((d2subst @ X338) = (d2subst @ X339)) => (X338 = X339)) & (pushprop_gthm = $true) & (ulamvarind = $true) & (! [X340 : subst > term > subst > $o,X341 : term] : (~! [X342 : term] : (~((X340 @ id @ X342 @ id) = $true) | ($true = (X340 @ id @ (sub @ X341 @ (push @ X342 @ id)) @ id))) | ~! [X343 : term,X344 : subst > term > term] : (~! [X345 : subst,X346 : subst,X347 : term] : ((sub @ (sub @ X343 @ (push @ X347 @ X346)) @ X345) = (sub @ (X344 @ X346 @ X347) @ X345)) | ~! [X348 : term] : (($true = (X340 @ id @ (X344 @ id @ X348) @ id)) | ~($true = (X340 @ id @ X348 @ id))) | ((X340 @ id @ (lam @ (X344 @ sh @ one)) @ id) = $true) | ~! [X349 : subst,X350 : subst,X351 : term] : ((sub @ X343 @ (push @ (sub @ X351 @ X350) @ (comp @ X349 @ X350))) = (X344 @ (comp @ X349 @ X350) @ (sub @ X351 @ X350)))) | ($true = (X340 @ id @ (lam @ (sub @ X341 @ (push @ one @ sh))) @ id))) <=> (hoasinduction_lem3aaa = $true)) & (hoasinduction_lem1v2 = $true) & (! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X353 = X354) | ((ap @ (sub @ X352 @ id) @ X353) != (ap @ (sub @ X355 @ id) @ X354))) <=> (hoasapinj2 = $true)) & (axapp = $true) & ! [X356 : subst] : ? [X357 : d_subst] : ((d2subst @ X357) = X356) & ((! [X358 : term,X359 : term,X360 : term,X361 : term] : (((ap @ X360 @ X361) != (ap @ X359 @ X358)) | (X358 = X361)) => ! [X362 : term,X363 : term,X364 : term,X365 : term] : (((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X364 @ id) @ X365)) | (X363 = X365))) <=> (hoasapinj2_lthm = $true)) & (induction2_gthm = $true) & ((laminj = $true) <=> ! [X366 : term,X367 : term] : ((X366 = X367) | ((lam @ X366) != (lam @ X367)))) & ! [X368 : d_term,X369 : d_term] : (((d2term @ X368) = (d2term @ X369)) => (X368 = X369)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & (! [X370 : subst,X371 : term] : ((sub @ one @ (push @ X371 @ X370)) = X371) <=> (axvarcons = $true)) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotap_gthm = $true) & ((! [X373 : subst] : ((comp @ X373 @ id) = X373) & ! [X372 : subst] : ((comp @ id @ X372) = X372)) <=> (substmonoid = $true)) & (hoasapnotvar_lthm = $true) & ((! [X374 : term] : ((sub @ X374 @ id) = X374) => (! [X375 : term,X376 : subst > term > subst > $o] : (~! [X377 : subst > term > term] : (~! [X378 : term] : (($true = (X376 @ id @ (X377 @ id @ X378) @ id)) | ~($true = (X376 @ id @ X378 @ id))) | ($true = (X376 @ id @ (lam @ (X377 @ sh @ one)) @ id)) | ~! [X379 : subst,X380 : subst,X381 : term] : ((X377 @ (comp @ X379 @ X380) @ (sub @ X381 @ X380)) = (sub @ (X377 @ X379 @ X381) @ X380))) | ~! [X382 : term] : (($true = (X376 @ id @ (sub @ X375 @ (push @ X382 @ id)) @ id)) | ~($true = (X376 @ id @ X382 @ id))) | ((X376 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id) = $true)) => ! [X383 : term,X384 : subst > term > subst > $o] : (~! [X385 : term] : (~($true = (X384 @ id @ X385 @ id)) | ((X384 @ id @ (sub @ X383 @ (push @ X385 @ id)) @ id) = $true)) | ~! [X386 : subst > term > term] : (~! [X387 : term] : (($true = (X384 @ id @ (X386 @ id @ X387) @ id)) | ~((X384 @ id @ X387 @ id) = $true)) | ($true = (X384 @ id @ (lam @ (X386 @ sh @ one)) @ id)) | ~! [X388 : subst,X389 : subst,X390 : term] : ((sub @ (X386 @ X389 @ X390) @ X388) = (X386 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388)))) | ($true = (X384 @ id @ (lam @ X383) @ id))))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & (! [X391 : term] : ~! [X392 : subst > term > term] : ~! [X393 : subst,X394 : term] : ((sub @ X391 @ (push @ X394 @ X393)) = (X392 @ X393 @ X394)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X395 : term > $o,X396 : term > $o,X397 : term,X398 : subst] : (($true = (X396 @ one)) | ~($true = (X395 @ X397)) | ~! [X399 : term] : ((X395 @ (sub @ X399 @ (push @ X397 @ X398))) = (X396 @ X399)))) & (one = (d2term @ d_one)) & ~(ulamvar1 = $true) & ((hoaslaminj_lthm = $true) <=> (! [X400 : term,X401 : subst] : ((sub @ one @ (push @ X400 @ X401)) = X400) => (! [X402 : term,X403 : subst] : ((comp @ sh @ (push @ X402 @ X403)) = X403) => (! [X404 : term,X405 : term] : ((X404 = X405) | ((lam @ X405) != (lam @ X404))) => ! [X406 : subst > term > term,X407 : subst,X408 : term,X409 : subst > term > term] : (((lam @ (X409 @ sh @ one)) != (lam @ (X406 @ sh @ one))) | ~! [X413 : term,X414 : subst,X415 : subst] : ((sub @ (X409 @ X415 @ X413) @ X414) = (X409 @ (comp @ X415 @ X414) @ (sub @ X413 @ X414))) | ((X409 @ X407 @ X408) = (X406 @ X407 @ X408)) | ~! [X410 : subst,X411 : term,X412 : subst] : ((sub @ (X406 @ X410 @ X411) @ X412) = (X406 @ (comp @ X410 @ X412) @ (sub @ X411 @ X412)))))))) & (axassoc = $true) & ((axidr = $true) <=> ! [X416 : subst] : ((comp @ X416 @ id) = X416)) & ! [X417 : d_subst] : (d_id = X417) & ((hoasinduction_lthm = $true) <=> (! [X418 : term,X419 : term > $o] : (($true = (X419 @ X418)) | ~! [X420 : term] : (((X419 @ (lam @ X420)) = $true) | ~! [X421 : term] : (~($true = (X419 @ X421)) | ($true = (X419 @ (sub @ X420 @ (push @ X421 @ id)))))) | ~! [X422 : term] : (($true = (X419 @ X422)) | ~((var @ X422) = $true)) | ~! [X423 : term,X424 : term] : (($true = (X419 @ (ap @ X424 @ X423))) | ~((X419 @ X424) = $true) | ~((X419 @ X423) = $true))) => (! [X425 : term,X426 : term,X427 : subst > term > subst > $o] : (~! [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X427 @ (comp @ X429 @ X430) @ (sub @ X428 @ X430) @ X431)) | ~((X427 @ X429 @ X428 @ (comp @ X430 @ X431)) = $true)) | ((X427 @ id @ (ap @ X426 @ X425) @ id) = $true) | ~($true = (X427 @ id @ X426 @ id)) | ~! [X432 : subst,X433 : subst,X434 : term,X435 : subst] : (((X427 @ X432 @ X434 @ (comp @ X433 @ X435)) = $true) | ~((X427 @ (comp @ X432 @ X433) @ (sub @ X434 @ X433) @ X435) = $true)) | ~! [X436 : term,X437 : term] : (~((X427 @ id @ X436 @ id) = $true) | ~($true = (X427 @ id @ X437 @ id)) | ($true = (X427 @ id @ (ap @ (sub @ X437 @ id) @ X436) @ id))) | ~((X427 @ id @ X425 @ id) = $true)) => (! [X438 : term,X439 : subst > term > subst > $o] : (~! [X440 : subst > term > term] : (((X439 @ id @ (lam @ (X440 @ sh @ one)) @ id) = $true) | ~! [X441 : subst,X442 : subst,X443 : term] : ((sub @ (X440 @ X442 @ X443) @ X441) = (X440 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441))) | ~! [X444 : term] : (~((X439 @ id @ X444 @ id) = $true) | ($true = (X439 @ id @ (X440 @ id @ X444) @ id)))) | ($true = (X439 @ id @ (lam @ X438) @ id)) | ~! [X445 : term] : (($true = (X439 @ id @ (sub @ X438 @ (push @ X445 @ id)) @ id)) | ~($true = (X439 @ id @ X445 @ id))) | ~! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (((X439 @ (comp @ X449 @ X447) @ (sub @ X448 @ X447) @ X446) = $true) | ~((X439 @ X449 @ X448 @ (comp @ X447 @ X446)) = $true)) | ~! [X450 : subst,X451 : subst,X452 : term,X453 : subst] : (~((X439 @ (comp @ X453 @ X450) @ (sub @ X452 @ X450) @ X451) = $true) | ($true = (X439 @ X453 @ X452 @ (comp @ X450 @ X451))))) => ! [X454 : term,X455 : subst > term > subst > $o] : (~! [X456 : term,X457 : term] : (~($true = (X455 @ id @ X456 @ id)) | ~($true = (X455 @ id @ X457 @ id)) | ((X455 @ id @ (ap @ (sub @ X456 @ id) @ X457) @ id) = $true)) | ~! [X458 : subst > term > term] : (((X455 @ id @ (lam @ (X458 @ sh @ one)) @ id) = $true) | ~! [X459 : term] : (~((X455 @ id @ X459 @ id) = $true) | ((X455 @ id @ (X458 @ id @ X459) @ id) = $true)) | ~! [X460 : subst,X461 : subst,X462 : term] : ((X458 @ (comp @ X460 @ X461) @ (sub @ X462 @ X461)) = (sub @ (X458 @ X460 @ X462) @ X461))) | ($true = (X455 @ id @ X454 @ id)) | ~! [X463 : subst,X464 : term,X465 : subst,X466 : subst] : (~($true = (X455 @ X466 @ X464 @ (comp @ X465 @ X463))) | ((X455 @ (comp @ X466 @ X465) @ (sub @ X464 @ X465) @ X463) = $true)) | ~! [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (~($true = (X455 @ (comp @ X468 @ X469) @ (sub @ X467 @ X469) @ X470)) | ((X455 @ X468 @ X467 @ (comp @ X469 @ X470)) = $true)) | ~! [X471 : term] : (((X455 @ id @ X471 @ id) = $true) | ~($true = (var @ (sub @ X471 @ id))))))))) & (pushprop_lem2v2_gthm = $true) & (termmset_gthm = $true) & (induction2lem_gthm = $true) & (! [X472 : term > $o,X473 : term,X474 : subst,X475 : term > $o,X476 : term] : (~($true = (X472 @ X473)) | ($true = (X475 @ (sub @ X473 @ (push @ X476 @ X474))))) <=> (pushprop_lem1 = $true)) & ((! [X477 : subst,X478 : term] : ((sub @ one @ (push @ X478 @ X477)) = X478) => ! [X479 : term > $o,X480 : term > $o,X481 : term,X482 : subst] : (($true = (X480 @ one)) | ~((X479 @ X481) = $true) | ~! [X483 : term] : ((X480 @ X483) = (X479 @ (sub @ X483 @ (push @ X481 @ X482)))))) <=> (pushprop_lem1v2_lthm = $true)) & (pushprop_lem0_gthm = $true) & (hoasapinj2_gthm = $true) & ((! [X484 : term] : ((sub @ X484 @ id) = X484) => ! [X485 : term > $o,X486 : subst > term > subst > $o,X487 : term] : (~! [X488 : subst,X489 : subst,X490 : term,X491 : subst] : (~((X486 @ X488 @ X490 @ (comp @ X489 @ X491)) = $true) | ((X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X491) = $true)) | ((X485 @ (lam @ X487)) = $true) | ~! [X492 : subst,X493 : term,X494 : subst,X495 : subst] : (~((X486 @ (comp @ X492 @ X495) @ (sub @ X493 @ X495) @ X494) = $true) | ((X486 @ X492 @ X493 @ (comp @ X495 @ X494)) = $true)) | ~! [X496 : term] : (((X485 @ (sub @ X487 @ (push @ X496 @ id))) = $true) | ~((X485 @ X496) = $true)) | ~! [X497 : term] : ((X485 @ X497) = (X486 @ id @ X497 @ id)) | ~! [X498 : subst > term > term] : (~! [X499 : term] : (($true = (X486 @ id @ (X498 @ id @ X499) @ id)) | ~($true = (X486 @ id @ X499 @ id))) | ((X486 @ id @ (lam @ (X498 @ sh @ one)) @ id) = $true) | ~! [X500 : subst,X501 : subst,X502 : term] : ((X498 @ (comp @ X500 @ X501) @ (sub @ X502 @ X501)) = (sub @ (X498 @ X500 @ X502) @ X501))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((! [X503 : subst] : ((comp @ id @ X503) = X503) => (! [X504 : subst] : ((comp @ X504 @ id) = X504) => (! [X506 : subst] : ((comp @ id @ X506) = X506) & ! [X505 : subst] : ((comp @ X505 @ id) = X505)))) <=> (substmonoid_lthm = $true)) & (hoasapinj1_gthm = $true) & (! [X507 : term,X508 : term,X509 : term,X510 : term] : (((ap @ (sub @ X507 @ id) @ X509) != (ap @ (sub @ X508 @ id) @ X510)) | (X507 = X508)) <=> (hoasapinj1 = $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X511 : term] : ((sub @ X511 @ id) = X511) <=> (axvarid = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X512 : subst > term > subst > $o,X513 : term] : (($true = (X512 @ id @ (lam @ (sub @ X513 @ (push @ one @ sh))) @ id)) | ~! [X514 : term] : (((X512 @ id @ (sub @ X513 @ (push @ X514 @ id)) @ id) = $true) | ~((X512 @ id @ X514 @ id) = $true)) | ~! [X515 : subst > term > term] : (((X512 @ id @ (lam @ (X515 @ sh @ one)) @ id) = $true) | ~! [X516 : subst,X517 : term,X518 : subst] : ((X515 @ (comp @ X518 @ X516) @ (sub @ X517 @ X516)) = (sub @ (X515 @ X518 @ X517) @ X516)) | ~! [X519 : term] : (~((X512 @ id @ X519 @ id) = $true) | ((X512 @ id @ (X515 @ id @ X519) @ id) = $true))))) & ((hoaslamnotap = $true) <=> ! [X520 : term,X521 : term,X522 : subst > term > term] : (~! [X523 : subst,X524 : term,X525 : subst] : ((X522 @ (comp @ X525 @ X523) @ (sub @ X524 @ X523)) = (sub @ (X522 @ X525 @ X524) @ X523)) | ((ap @ (sub @ X521 @ id) @ X520) != (lam @ (X522 @ sh @ one))))) & ((hoasapinj1_lthm = $true) <=> (! [X526 : term] : ((sub @ X526 @ id) = X526) => (! [X527 : term,X528 : term,X529 : term,X530 : term] : (((ap @ X528 @ X529) != (ap @ X527 @ X530)) | (X527 = X528)) => ! [X531 : term,X532 : term,X533 : term,X534 : term] : (((ap @ (sub @ X533 @ id) @ X531) != (ap @ (sub @ X534 @ id) @ X532)) | (X533 = X534))))) & ((pushprop_lem0_lthm = $true) <=> ! [X535 : term > $o,X536 : subst,X537 : term] : ~! [X538 : term > $o] : ~! [X539 : term] : ((X535 @ (sub @ X539 @ (push @ X537 @ X536))) = (X538 @ X539))) & (hoaslamnotvar = $true)),
% 9.64/2.82 inference(rectify,[status(thm)],[f8])).
% 9.64/2.82 thf(f8,plain,(
% 9.64/2.82 ((axscons = $true) <=> ! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0)) & (axabs = $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X1 : term] : ~! [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) != (X2 @ sh @ one))) & (axclos = $true) & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & (hoasinduction_lem1v2_gthm = $true) & ((termmset_lthm = $true) <=> (! [X5 : term] : ((sub @ X5 @ id) = X5) => ! [X6 : term] : ((sub @ X6 @ id) = X6))) & (axvarshift = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((! [X10 : term,X11 : subst] : ((sub @ one @ (push @ X10 @ X11)) = X10) => (! [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) = X12) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X15 : term > $o,X16 : term] : (~! [X17 : term] : (($true = (X15 @ X17)) | ~($true = (var @ X17))) | ($true = (X15 @ X16)) | ~! [X18 : term] : (~($true = (X15 @ X18)) | ($true = (X15 @ (lam @ X18)))) | ~! [X19 : term,X20 : term] : (((X15 @ (ap @ X20 @ X19)) = $true) | ~($true = (X15 @ X20)) | ~($true = (X15 @ X19)))) => ! [X21 : subst,X22 : term,X23 : term > $o] : (~! [X24 : term] : (($true = (X23 @ (sub @ X24 @ X21))) | ~((var @ X24) = $true)) | ((X23 @ (sub @ X22 @ X21)) = $true) | ~! [X25 : term,X26 : term] : (~((X23 @ X26) = $true) | ~((X23 @ X25) = $true) | ($true = (X23 @ (ap @ X26 @ X25)))) | ~! [X27 : term] : (((X23 @ (lam @ X27)) = $true) | ~! [X28 : term] : (((X23 @ (sub @ X27 @ (push @ X28 @ id))) = $true) | ~((X23 @ X28) = $true)))))))) <=> (induction2lem_lthm = $true)) & (ulamvarsh = $true) & (hoasinduction_gthm = $true) & (pushprop_lthm = $true) & (hoaslamnotvar_lthm = $true) & (! [X29 : subst > term > subst > $o] : ~! [X30 : term > $o] : ~! [X31 : term] : ((X30 @ X31) = (X29 @ id @ X31 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((d2subst @ d_id) = sh) & ((hoasinduction_lem3v2 = $true) <=> ! [X32 : subst > term > subst > $o,X33 : term,X34 : term > $o] : (~! [X35 : subst,X36 : subst,X37 : term,X38 : subst] : (($true = (X32 @ (comp @ X38 @ X35) @ (sub @ X37 @ X35) @ X36)) | ~((X32 @ X38 @ X37 @ (comp @ X35 @ X36)) = $true)) | ~! [X39 : term] : (((X34 @ (sub @ X33 @ (push @ X39 @ id))) = $true) | ~($true = (X34 @ X39))) | ~! [X40 : subst > term > term] : (((X32 @ id @ (lam @ (X40 @ sh @ one)) @ id) = $true) | ~! [X41 : subst,X42 : subst,X43 : term] : ((sub @ (X40 @ X41 @ X43) @ X42) = (X40 @ (comp @ X41 @ X42) @ (sub @ X43 @ X42))) | ~! [X44 : term] : (($true = (X32 @ id @ (X40 @ id @ X44) @ id)) | ~((X32 @ id @ X44 @ id) = $true))) | ~! [X45 : subst,X46 : subst,X47 : term,X48 : subst] : (~((X32 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46) @ X48) = $true) | ((X32 @ X45 @ X47 @ (comp @ X46 @ X48)) = $true)) | ~! [X49 : term] : ((X32 @ id @ X49 @ id) = (X34 @ X49)) | ((X34 @ (lam @ X33)) = $true))) & ((! [X50 : subst > term > subst > $o] : ~! [X51 : term > $o] : ~! [X52 : term] : ((X51 @ X52) = (X50 @ id @ X52 @ id)) => (! [X53 : term > $o,X54 : term] : (~! [X55 : term,X56 : term] : (($true = (X53 @ (ap @ X56 @ X55))) | ~($true = (X53 @ X56)) | ~($true = (X53 @ X55))) | ~! [X57 : term] : (~! [X58 : term] : (((X53 @ (sub @ X57 @ (push @ X58 @ id))) = $true) | ~($true = (X53 @ X58))) | ((X53 @ (lam @ X57)) = $true)) | ((X53 @ X54) = $true) | ~! [X59 : term] : (((X53 @ X59) = $true) | ~($true = (var @ X59)))) => (! [X60 : term] : ((sub @ X60 @ id) = X60) => (! [X61 : subst > term > subst > $o,X62 : term,X63 : term > $o] : (~! [X64 : term] : (~((X63 @ X64) = $true) | ((X63 @ (sub @ X62 @ (push @ X64 @ id))) = $true)) | ~! [X65 : term] : ((X63 @ X65) = (X61 @ id @ X65 @ id)) | ($true = (X63 @ (lam @ X62))) | ~! [X66 : subst > term > term] : (~! [X67 : subst,X68 : term,X69 : subst] : ((X66 @ (comp @ X67 @ X69) @ (sub @ X68 @ X69)) = (sub @ (X66 @ X67 @ X68) @ X69)) | ~! [X70 : term] : (~($true = (X61 @ id @ X70 @ id)) | ((X61 @ id @ (X66 @ id @ X70) @ id) = $true)) | ($true = (X61 @ id @ (lam @ (X66 @ sh @ one)) @ id)))) => ! [X71 : subst > term > subst > $o,X72 : term] : (~! [X73 : term,X74 : term] : (~((X71 @ id @ X74 @ id) = $true) | ((X71 @ id @ (ap @ (sub @ X74 @ id) @ X73) @ id) = $true) | ~((X71 @ id @ X73 @ id) = $true)) | ~! [X75 : subst > term > term] : (($true = (X71 @ id @ (lam @ (X75 @ sh @ one)) @ id)) | ~! [X76 : subst,X77 : subst,X78 : term] : ((sub @ (X75 @ X77 @ X78) @ X76) = (X75 @ (comp @ X77 @ X76) @ (sub @ X78 @ X76))) | ~! [X79 : term] : (~((X71 @ id @ X79 @ id) = $true) | ($true = (X71 @ id @ (X75 @ id @ X79) @ id)))) | ($true = (X71 @ id @ X72 @ id))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & ((shinj = $true) <=> ! [X80 : term,X81 : term] : (((sub @ X80 @ sh) != (sub @ X81 @ sh)) | (X80 = X81))) & (! [X82 : term,X83 : term,X84 : term,X85 : term] : ((X84 = X85) | ((ap @ X83 @ X84) != (ap @ X82 @ X85))) <=> (apinj2 = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X86 : subst > term > subst > $o,X87 : term,X88 : term > $o] : (~! [X89 : term] : (($true = (X88 @ (sub @ X87 @ (push @ X89 @ id)))) | ~((X88 @ X89) = $true)) | ~! [X90 : term] : ((X86 @ id @ X90 @ id) = (X88 @ X90)) | ~! [X91 : subst > term > term] : (~! [X92 : subst,X93 : subst,X94 : term] : ((X91 @ (comp @ X93 @ X92) @ (sub @ X94 @ X92)) = (sub @ (X91 @ X93 @ X94) @ X92)) | ~! [X95 : term] : (~($true = (X86 @ id @ X95 @ id)) | ($true = (X86 @ id @ (X91 @ id @ X95) @ id))) | ($true = (X86 @ id @ (lam @ (X91 @ sh @ one)) @ id))) | ($true = (X88 @ (lam @ X87))))) & (pushprop = $true) & ((induction2_lthm = $true) <=> (! [X96 : term] : ((sub @ X96 @ id) = X96) => (! [X97 : subst,X98 : term > $o,X99 : term] : (~! [X100 : term,X101 : term] : (~((X98 @ X100) = $true) | ~($true = (X98 @ X101)) | ((X98 @ (ap @ X100 @ X101)) = $true)) | ~! [X102 : term] : (~($true = (var @ X102)) | ($true = (X98 @ (sub @ X102 @ X97)))) | ($true = (X98 @ (sub @ X99 @ X97))) | ~! [X103 : term] : (((X98 @ (lam @ X103)) = $true) | ~! [X104 : term] : (((X98 @ (sub @ X103 @ (push @ X104 @ id))) = $true) | ~((X98 @ X104) = $true)))) => ! [X105 : term > $o,X106 : term] : (~! [X107 : term,X108 : term] : (~($true = (X105 @ X108)) | ((X105 @ (ap @ X107 @ X108)) = $true) | ~($true = (X105 @ X107))) | ($true = (X105 @ X106)) | ~! [X109 : term] : (~! [X110 : term] : (((X105 @ (sub @ X109 @ (push @ X110 @ id))) = $true) | ~((X105 @ X110) = $true)) | ((X105 @ (lam @ X109)) = $true)) | ~! [X111 : term] : (($true = (X105 @ X111)) | ~((var @ X111) = $true)))))) & (! [X112 : term] : ((sub @ X112 @ id) = X112) <=> (termmset = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X113 : subst > term > subst > $o] : ~! [X114 : term > $o] : ~! [X115 : term] : ((X113 @ id @ X115 @ id) = (X114 @ X115))) & (substmonoid_gthm = $true) & ((! [X116 : subst,X117 : term] : ((sub @ one @ (push @ X117 @ X116)) = X117) => (! [X118 : term,X119 : subst] : ((comp @ sh @ (push @ X118 @ X119)) = X119) => ! [X120 : term > $o,X121 : term,X122 : subst,X123 : term > $o,X124 : term] : (($true = (X120 @ (sub @ X124 @ (push @ X121 @ X122)))) | ~((X123 @ X124) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (! [X125 : term,X126 : subst > term > subst > $o] : (~! [X127 : term] : (((X126 @ id @ (sub @ X125 @ (push @ X127 @ id)) @ id) = $true) | ~($true = (X126 @ id @ X127 @ id))) | ~! [X128 : subst > term > term] : (~! [X129 : subst,X130 : term,X131 : subst] : ((sub @ (X128 @ X131 @ X130) @ X129) = (X128 @ (comp @ X131 @ X129) @ (sub @ X130 @ X129))) | ~! [X132 : term] : (($true = (X126 @ id @ (X128 @ id @ X132) @ id)) | ~((X126 @ id @ X132 @ id) = $true)) | ((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id) = $true)) | ~! [X133 : subst,X134 : subst,X135 : subst,X136 : term] : (~((X126 @ X134 @ X136 @ (comp @ X133 @ X135)) = $true) | ((X126 @ (comp @ X134 @ X133) @ (sub @ X136 @ X133) @ X135) = $true)) | ~! [X137 : subst,X138 : subst,X139 : term,X140 : subst] : (~((X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137) = $true) | ($true = (X126 @ X138 @ X139 @ (comp @ X140 @ X137)))) | ((X126 @ id @ (lam @ X125) @ id) = $true)) <=> (hoasinduction_lem3 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X143 : term] : ~! [X144 : subst > term > term] : ~! [X145 : term,X146 : subst] : ((sub @ X143 @ (push @ X145 @ X146)) = (X144 @ X146 @ X145))) & (! [X147 : term] : ~! [X148 : subst > term > term] : ((X148 @ sh @ one) != (sub @ X147 @ (push @ one @ sh))) <=> (hoasinduction_lem3b = $true)) & (hoaslaminj_gthm = $true) & (pushprop_lem1_gthm = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lem3v2 = $true) & ((hoaslaminj = $true) <=> ! [X154 : subst > term > term,X155 : subst > term > term,X156 : subst,X157 : term] : (~! [X158 : term,X159 : subst,X160 : subst] : ((X154 @ (comp @ X159 @ X160) @ (sub @ X158 @ X160)) = (sub @ (X154 @ X159 @ X158) @ X160)) | ((X154 @ X156 @ X157) = (X155 @ X156 @ X157)) | ~! [X161 : subst,X162 : subst,X163 : term] : ((X155 @ (comp @ X162 @ X161) @ (sub @ X163 @ X161)) = (sub @ (X155 @ X162 @ X163) @ X161)) | ((lam @ (X154 @ sh @ one)) != (lam @ (X155 @ sh @ one))))) & ~((var @ (d2term @ d_one)) = $true) & (! [X164 : subst > term > subst > $o,X165 : term] : (~! [X166 : term] : (((X164 @ id @ (sub @ X165 @ (push @ X166 @ id)) @ id) = $true) | ~($true = (X164 @ id @ X166 @ id))) | ~! [X167 : subst > term > term] : (($true = (X164 @ id @ (lam @ (X167 @ sh @ one)) @ id)) | ~! [X168 : term] : (~($true = (X164 @ id @ X168 @ id)) | ($true = (X164 @ id @ (X167 @ id @ X168) @ id))) | ~! [X169 : subst,X170 : subst,X171 : term] : ((X167 @ (comp @ X169 @ X170) @ (sub @ X171 @ X170)) = (sub @ (X167 @ X169 @ X171) @ X170))) | ($true = (X164 @ id @ (lam @ (sub @ X165 @ (push @ one @ sh))) @ id))) <=> (hoasinduction_lem3aa = $true)) & (axmap = $true) & (apnotvar = $true) & ((! [X172 : term] : ((sub @ X172 @ id) = X172) => (! [X173 : subst > term > subst > $o,X174 : term] : (~! [X175 : term] : (($true = (X173 @ id @ (sub @ X174 @ (push @ X175 @ id)) @ id)) | ~((X173 @ id @ X175 @ id) = $true)) | ($true = (X173 @ id @ (lam @ (sub @ X174 @ (push @ one @ sh))) @ id)) | ~! [X176 : subst > term > term] : (~! [X177 : subst,X178 : subst,X179 : term] : ((sub @ (X176 @ X178 @ X179) @ X177) = (X176 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177))) | ~! [X180 : term] : (($true = (X173 @ id @ (X176 @ id @ X180) @ id)) | ~((X173 @ id @ X180 @ id) = $true)) | ($true = (X173 @ id @ (lam @ (X176 @ sh @ one)) @ id)))) => ! [X181 : term,X182 : subst > term > subst > $o] : (~! [X183 : term] : (~($true = (X182 @ id @ X183 @ id)) | ($true = (X182 @ id @ (sub @ X181 @ (push @ X183 @ id)) @ id))) | ~! [X184 : subst > term > term] : (~! [X185 : term] : (($true = (X182 @ id @ (X184 @ id @ X185) @ id)) | ~($true = (X182 @ id @ X185 @ id))) | ~! [X186 : subst,X187 : subst,X188 : term] : ((X184 @ (comp @ X186 @ X187) @ (sub @ X188 @ X187)) = (sub @ (X184 @ X186 @ X188) @ X187)) | ($true = (X182 @ id @ (lam @ (X184 @ sh @ one)) @ id))) | ~! [X189 : term,X190 : subst,X191 : subst,X192 : subst] : (($true = (X182 @ (comp @ X192 @ X190) @ (sub @ X189 @ X190) @ X191)) | ~($true = (X182 @ X192 @ X189 @ (comp @ X190 @ X191)))) | ($true = (X182 @ id @ (lam @ X181) @ id)) | ~! [X193 : subst,X194 : subst,X195 : term,X196 : subst] : (~((X182 @ (comp @ X194 @ X193) @ (sub @ X195 @ X193) @ X196) = $true) | ($true = (X182 @ X194 @ X195 @ (comp @ X193 @ X196))))))) <=> (hoasinduction_lem3_lthm = $true)) & (lamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X197 : term,X198 : subst > term > subst > $o,X199 : term] : (~! [X200 : subst,X201 : subst,X202 : term,X203 : subst] : (~((X198 @ (comp @ X201 @ X203) @ (sub @ X202 @ X203) @ X200) = $true) | ((X198 @ X201 @ X202 @ (comp @ X203 @ X200)) = $true)) | ($true = (X198 @ id @ (ap @ X199 @ X197) @ id)) | ~! [X204 : subst,X205 : subst,X206 : term,X207 : subst] : (~($true = (X198 @ X204 @ X206 @ (comp @ X207 @ X205))) | ((X198 @ (comp @ X204 @ X207) @ (sub @ X206 @ X207) @ X205) = $true)) | ~($true = (X198 @ id @ X199 @ id)) | ~($true = (X198 @ id @ X197 @ id)) | ~! [X208 : term,X209 : term] : (((X198 @ id @ (ap @ (sub @ X209 @ id) @ X208) @ id) = $true) | ~($true = (X198 @ id @ X208 @ id)) | ~($true = (X198 @ id @ X209 @ id))))) & (pushprop_lem2v2 = $true) & (! [X210 : term,X211 : subst] : ((comp @ sh @ (push @ X210 @ X211)) = X211) <=> (axshiftcons = $true)) & (! [X212 : subst > term > subst > $o,X213 : term] : (((X212 @ id @ X213 @ id) = $true) | ~! [X214 : term,X215 : term] : (~($true = (X212 @ id @ X215 @ id)) | ~($true = (X212 @ id @ X214 @ id)) | ((X212 @ id @ (ap @ (sub @ X214 @ id) @ X215) @ id) = $true)) | ~! [X216 : subst > term > term] : (~! [X217 : term] : (~((X212 @ id @ X217 @ id) = $true) | ((X212 @ id @ (X216 @ id @ X217) @ id) = $true)) | ~! [X218 : term,X219 : subst,X220 : subst] : ((sub @ (X216 @ X220 @ X218) @ X219) = (X216 @ (comp @ X220 @ X219) @ (sub @ X218 @ X219))) | ((X212 @ id @ (lam @ (X216 @ sh @ one)) @ id) = $true))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasinduction_lem3_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction = $true) <=> ! [X221 : term,X222 : subst > term > subst > $o] : (($true = (X222 @ id @ X221 @ id)) | ~! [X223 : term,X224 : subst,X225 : subst,X226 : subst] : (((X222 @ X224 @ X223 @ (comp @ X226 @ X225)) = $true) | ~((X222 @ (comp @ X224 @ X226) @ (sub @ X223 @ X226) @ X225) = $true)) | ~! [X227 : subst,X228 : term,X229 : subst,X230 : subst] : (($true = (X222 @ (comp @ X229 @ X227) @ (sub @ X228 @ X227) @ X230)) | ~((X222 @ X229 @ X228 @ (comp @ X227 @ X230)) = $true)) | ~! [X231 : term] : (~((var @ (sub @ X231 @ id)) = $true) | ((X222 @ id @ X231 @ id) = $true)) | ~! [X232 : subst > term > term] : (~! [X233 : term] : (~((X222 @ id @ X233 @ id) = $true) | ($true = (X222 @ id @ (X232 @ id @ X233) @ id))) | ~! [X234 : term,X235 : subst,X236 : subst] : ((X232 @ (comp @ X236 @ X235) @ (sub @ X234 @ X235)) = (sub @ (X232 @ X236 @ X234) @ X235)) | ($true = (X222 @ id @ (lam @ (X232 @ sh @ one)) @ id))) | ~! [X237 : term,X238 : term] : (~($true = (X222 @ id @ X238 @ id)) | ~($true = (X222 @ id @ X237 @ id)) | ($true = (X222 @ id @ (ap @ (sub @ X237 @ id) @ X238) @ id))))) & (hoasinduction_lem1 = $true) & ! [X239 : d_term] : (d_one = X239) & (hoaslamnotap_lthm = $true) & ((induction = $true) <=> ! [X240 : term > $o,X241 : term] : (($true = (X240 @ X241)) | ~! [X242 : term] : (($true = (X240 @ X242)) | ~($true = (var @ X242))) | ~! [X243 : term,X244 : term] : (~((X240 @ X244) = $true) | ((X240 @ (ap @ X243 @ X244)) = $true) | ~($true = (X240 @ X243))) | ~! [X245 : term] : (~((X240 @ X245) = $true) | ((X240 @ (lam @ X245)) = $true)))) & ((apinj1 = $true) <=> ! [X246 : term,X247 : term,X248 : term,X249 : term] : ((X246 = X249) | ((ap @ X249 @ X247) != (ap @ X246 @ X248)))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X250 : term] : ~! [X251 : subst > term > term] : ~! [X252 : term,X253 : subst] : ((sub @ X250 @ (push @ X252 @ X253)) = (X251 @ X253 @ X252)) => (! [X254 : term] : ((sub @ X254 @ id) = X254) => ! [X255 : term,X256 : subst > term > subst > $o,X257 : term > $o] : (~! [X258 : subst > term > term] : (~! [X259 : term,X260 : subst,X261 : subst] : ((sub @ (X258 @ X261 @ X259) @ X260) = (X258 @ (comp @ X261 @ X260) @ (sub @ X259 @ X260))) | ((X256 @ id @ (lam @ (X258 @ sh @ one)) @ id) = $true) | ~! [X262 : term] : (~($true = (X256 @ id @ X262 @ id)) | ($true = (X256 @ id @ (X258 @ id @ X262) @ id)))) | ~! [X263 : term] : ((X257 @ X263) = (X256 @ id @ X263 @ id)) | ($true = (X257 @ (lam @ X255))) | ~! [X264 : term] : (~((X257 @ X264) = $true) | ($true = (X257 @ (sub @ X255 @ (push @ X264 @ id))))))))) & (! [X265 : term,X266 : term > $o] : (~! [X267 : term,X268 : term] : (($true = (X266 @ (ap @ X267 @ X268))) | ~($true = (X266 @ X267)) | ~((X266 @ X268) = $true)) | ~! [X269 : term] : (($true = (X266 @ X269)) | ~((var @ X269) = $true)) | ($true = (X266 @ X265)) | ~! [X270 : term] : (($true = (X266 @ (lam @ X270))) | ~! [X271 : term] : (~((X266 @ X271) = $true) | ((X266 @ (sub @ X270 @ (push @ X271 @ id))) = $true)))) <=> (induction2 = $true)) & (! [X272 : subst > term > subst > $o,X273 : term] : (~! [X274 : term] : (((X272 @ id @ (sub @ X273 @ (push @ X274 @ id)) @ id) = $true) | ~($true = (X272 @ id @ X274 @ id))) | ~! [X275 : subst > term > term] : (~! [X276 : term] : (($true = (X272 @ id @ (X275 @ id @ X276) @ id)) | ~($true = (X272 @ id @ X276 @ id))) | ($true = (X272 @ id @ (lam @ (X275 @ sh @ one)) @ id)) | ~! [X277 : subst,X278 : subst,X279 : term] : ((sub @ (X275 @ X278 @ X279) @ X277) = (X275 @ (comp @ X278 @ X277) @ (sub @ X279 @ X277)))) | ((X272 @ id @ (lam @ X273) @ id) = $true)) <=> (hoasinduction_lem3a = $true)) & (pushprop_lem1v2_gthm = $true) & (pushprop_lthm_orig = $true) & (! [X280 : term,X281 : term > $o,X282 : subst > term > subst > $o,X283 : term] : (~! [X284 : term,X285 : subst,X286 : subst,X287 : subst] : (($true = (X282 @ (comp @ X287 @ X286) @ (sub @ X284 @ X286) @ X285)) | ~((X282 @ X287 @ X284 @ (comp @ X286 @ X285)) = $true)) | ~($true = (X281 @ X283)) | ($true = (X281 @ (ap @ X283 @ X280))) | ~! [X288 : term,X289 : term] : (~((X282 @ id @ X289 @ id) = $true) | ~($true = (X282 @ id @ X288 @ id)) | ((X282 @ id @ (ap @ (sub @ X289 @ id) @ X288) @ id) = $true)) | ~! [X290 : subst,X291 : subst,X292 : term,X293 : subst] : (((X282 @ X291 @ X292 @ (comp @ X293 @ X290)) = $true) | ~((X282 @ (comp @ X291 @ X293) @ (sub @ X292 @ X293) @ X290) = $true)) | ~! [X294 : term] : ((X282 @ id @ X294 @ id) = (X281 @ X294)) | ~((X281 @ X280) = $true)) <=> (hoasinduction_lem2v2 = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X295 : subst > term > subst > $o] : ~! [X296 : term > $o] : ~! [X297 : term] : ((X295 @ id @ X297 @ id) = (X296 @ X297)) => (! [X298 : term,X299 : term > $o] : (~! [X300 : term] : (($true = (X299 @ X300)) | ~($true = (var @ X300))) | ($true = (X299 @ X298)) | ~! [X301 : term] : (~! [X302 : term] : (~($true = (X299 @ X302)) | ((X299 @ (sub @ X301 @ (push @ X302 @ id))) = $true)) | ((X299 @ (lam @ X301)) = $true)) | ~! [X303 : term,X304 : term] : (~($true = (X299 @ X304)) | ($true = (X299 @ (ap @ X303 @ X304))) | ~((X299 @ X303) = $true))) => (! [X305 : term] : ((sub @ X305 @ id) = X305) => (! [X306 : term,X307 : term > $o,X308 : subst > term > subst > $o] : (~! [X309 : term] : ((X308 @ id @ X309 @ id) = (X307 @ X309)) | ($true = (X307 @ (lam @ X306))) | ~! [X310 : subst > term > term] : (($true = (X308 @ id @ (lam @ (X310 @ sh @ one)) @ id)) | ~! [X311 : subst,X312 : term,X313 : subst] : ((sub @ (X310 @ X313 @ X312) @ X311) = (X310 @ (comp @ X313 @ X311) @ (sub @ X312 @ X311))) | ~! [X314 : term] : (~((X308 @ id @ X314 @ id) = $true) | ((X308 @ id @ (X310 @ id @ X314) @ id) = $true))) | ~! [X315 : term] : (~($true = (X307 @ X315)) | ((X307 @ (sub @ X306 @ (push @ X315 @ id))) = $true))) => ! [X316 : subst > term > subst > $o,X317 : term] : (~! [X318 : term] : (~((var @ (sub @ X318 @ id)) = $true) | ((X316 @ id @ X318 @ id) = $true)) | ~! [X319 : subst > term > term] : (~! [X320 : term] : (($true = (X316 @ id @ (X319 @ id @ X320) @ id)) | ~($true = (X316 @ id @ X320 @ id))) | ((X316 @ id @ (lam @ (X319 @ sh @ one)) @ id) = $true) | ~! [X321 : subst,X322 : term,X323 : subst] : ((X319 @ (comp @ X321 @ X323) @ (sub @ X322 @ X323)) = (sub @ (X319 @ X321 @ X322) @ X323))) | ~! [X324 : subst,X325 : subst,X326 : term,X327 : subst] : (((X316 @ X324 @ X326 @ (comp @ X325 @ X327)) = $true) | ~($true = (X316 @ (comp @ X324 @ X325) @ (sub @ X326 @ X325) @ X327))) | ~! [X328 : term,X329 : term] : (($true = (X316 @ id @ (ap @ (sub @ X328 @ id) @ X329) @ id)) | ~((X316 @ id @ X329 @ id) = $true) | ~((X316 @ id @ X328 @ id) = $true)) | ~! [X330 : term,X331 : subst,X332 : subst,X333 : subst] : (~((X316 @ X331 @ X330 @ (comp @ X332 @ X333)) = $true) | ($true = (X316 @ (comp @ X331 @ X332) @ (sub @ X330 @ X332) @ X333))) | ((X316 @ id @ X317 @ id) = $true))))))) & ~(lamnotap = $true) & ((axidl = $true) <=> ! [X334 : subst] : ((comp @ id @ X334) = X334)) & (! [X335 : term > $o,X336 : subst,X337 : term] : (~! [X338 : term,X339 : term] : (~((X335 @ X339) = $true) | ~((X335 @ X338) = $true) | ($true = (X335 @ (ap @ X339 @ X338)))) | ~! [X340 : term] : (~($true = (var @ X340)) | ((X335 @ (sub @ X340 @ X336)) = $true)) | ~! [X341 : term] : (~! [X342 : term] : (~((X335 @ X342) = $true) | ((X335 @ (sub @ X341 @ (push @ X342 @ id))) = $true)) | ($true = (X335 @ (lam @ X341)))) | ($true = (X335 @ (sub @ X337 @ X336)))) <=> (induction2lem = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (hoasapnotvar_gthm = $true) & ((d2subst @ d_id) = id) & (! [X343 : term,X344 : term > $o,X345 : subst] : ~! [X346 : term > $o] : ~! [X347 : term] : ((X344 @ (sub @ X347 @ (push @ X343 @ X345))) = (X346 @ X347)) <=> (pushprop_lem0 = $true)) & (hoasapnotvar = $true) & ! [X348 : d_subst,X349 : d_subst] : (((d2subst @ X348) = (d2subst @ X349)) => (X348 = X349)) & (pushprop_gthm = $true) & (ulamvarind = $true) & (! [X350 : subst > term > subst > $o,X351 : term] : (~! [X352 : term] : (~($true = (X350 @ id @ X352 @ id)) | ($true = (X350 @ id @ (sub @ X351 @ (push @ X352 @ id)) @ id))) | ~! [X353 : term,X354 : subst > term > term] : (~! [X355 : subst,X356 : subst,X357 : term] : ((sub @ (sub @ X353 @ (push @ X357 @ X356)) @ X355) = (sub @ (X354 @ X356 @ X357) @ X355)) | ~! [X358 : term] : (($true = (X350 @ id @ (X354 @ id @ X358) @ id)) | ~($true = (X350 @ id @ X358 @ id))) | ((X350 @ id @ (lam @ (X354 @ sh @ one)) @ id) = $true) | ~! [X359 : subst,X360 : subst,X361 : term] : ((sub @ X353 @ (push @ (sub @ X361 @ X360) @ (comp @ X359 @ X360))) = (X354 @ (comp @ X359 @ X360) @ (sub @ X361 @ X360)))) | ($true = (X350 @ id @ (lam @ (sub @ X351 @ (push @ one @ sh))) @ id))) <=> (hoasinduction_lem3aaa = $true)) & (hoasinduction_lem1v2 = $true) & ((hoasapinj2 = $true) <=> ! [X362 : term,X363 : term,X364 : term,X365 : term] : ((X363 = X364) | ((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X365 @ id) @ X364)))) & (axapp = $true) & ! [X366 : subst] : ? [X367 : d_subst] : ((d2subst @ X367) = X366) & ((hoasapinj2_lthm = $true) <=> (! [X368 : term,X369 : term,X370 : term,X371 : term] : (((ap @ X369 @ X368) != (ap @ X370 @ X371)) | (X368 = X371)) => ! [X372 : term,X373 : term,X374 : term,X375 : term] : ((X373 = X375) | ((ap @ (sub @ X374 @ id) @ X375) != (ap @ (sub @ X372 @ id) @ X373))))) & (induction2_gthm = $true) & (! [X376 : term,X377 : term] : (((lam @ X376) != (lam @ X377)) | (X376 = X377)) <=> (laminj = $true)) & ! [X378 : d_term,X379 : d_term] : (((d2term @ X378) = (d2term @ X379)) => (X378 = X379)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ((axvarcons = $true) <=> ! [X380 : subst,X381 : term] : ((sub @ one @ (push @ X381 @ X380)) = X381)) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotap_gthm = $true) & ((! [X382 : subst] : ((comp @ id @ X382) = X382) & ! [X383 : subst] : ((comp @ X383 @ id) = X383)) <=> (substmonoid = $true)) & (hoasapnotvar_lthm = $true) & ((! [X384 : term] : ((sub @ X384 @ id) = X384) => (! [X385 : term,X386 : subst > term > subst > $o] : (~! [X387 : subst > term > term] : (~! [X388 : term] : (($true = (X386 @ id @ (X387 @ id @ X388) @ id)) | ~((X386 @ id @ X388 @ id) = $true)) | ($true = (X386 @ id @ (lam @ (X387 @ sh @ one)) @ id)) | ~! [X389 : subst,X390 : subst,X391 : term] : ((X387 @ (comp @ X389 @ X390) @ (sub @ X391 @ X390)) = (sub @ (X387 @ X389 @ X391) @ X390))) | ~! [X392 : term] : (((X386 @ id @ (sub @ X385 @ (push @ X392 @ id)) @ id) = $true) | ~((X386 @ id @ X392 @ id) = $true)) | ((X386 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id) = $true)) => ! [X393 : term,X394 : subst > term > subst > $o] : (~! [X395 : term] : (~((X394 @ id @ X395 @ id) = $true) | ($true = (X394 @ id @ (sub @ X393 @ (push @ X395 @ id)) @ id))) | ~! [X396 : subst > term > term] : (~! [X397 : term] : (($true = (X394 @ id @ (X396 @ id @ X397) @ id)) | ~((X394 @ id @ X397 @ id) = $true)) | ($true = (X394 @ id @ (lam @ (X396 @ sh @ one)) @ id)) | ~! [X398 : subst,X399 : subst,X400 : term] : ((X396 @ (comp @ X399 @ X398) @ (sub @ X400 @ X398)) = (sub @ (X396 @ X399 @ X400) @ X398))) | ((X394 @ id @ (lam @ X393) @ id) = $true)))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & (! [X401 : term] : ~! [X402 : subst > term > term] : ~! [X403 : subst,X404 : term] : ((X402 @ X403 @ X404) = (sub @ X401 @ (push @ X404 @ X403))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X405 : term > $o,X406 : term > $o,X407 : term,X408 : subst] : (((X406 @ one) = $true) | ~((X405 @ X407) = $true) | ~! [X409 : term] : ((X406 @ X409) = (X405 @ (sub @ X409 @ (push @ X407 @ X408)))))) & (one = (d2term @ d_one)) & ~(ulamvar1 = $true) & ((hoaslaminj_lthm = $true) <=> (! [X410 : term,X411 : subst] : ((sub @ one @ (push @ X410 @ X411)) = X410) => (! [X412 : term,X413 : subst] : ((comp @ sh @ (push @ X412 @ X413)) = X413) => (! [X414 : term,X415 : term] : ((X414 = X415) | ((lam @ X414) != (lam @ X415))) => ! [X416 : subst > term > term,X417 : subst,X418 : term,X419 : subst > term > term] : (((lam @ (X416 @ sh @ one)) != (lam @ (X419 @ sh @ one))) | ((X416 @ X417 @ X418) = (X419 @ X417 @ X418)) | ~! [X420 : subst,X421 : term,X422 : subst] : ((X416 @ (comp @ X420 @ X422) @ (sub @ X421 @ X422)) = (sub @ (X416 @ X420 @ X421) @ X422)) | ~! [X423 : term,X424 : subst,X425 : subst] : ((X419 @ (comp @ X425 @ X424) @ (sub @ X423 @ X424)) = (sub @ (X419 @ X425 @ X423) @ X424))))))) & (axassoc = $true) & (! [X426 : subst] : ((comp @ X426 @ id) = X426) <=> (axidr = $true)) & ! [X427 : d_subst] : (d_id = X427) & ((hoasinduction_lthm = $true) <=> (! [X428 : term,X429 : term > $o] : (((X429 @ X428) = $true) | ~! [X430 : term] : (($true = (X429 @ (lam @ X430))) | ~! [X431 : term] : (~((X429 @ X431) = $true) | ((X429 @ (sub @ X430 @ (push @ X431 @ id))) = $true))) | ~! [X432 : term] : (($true = (X429 @ X432)) | ~((var @ X432) = $true)) | ~! [X433 : term,X434 : term] : (((X429 @ (ap @ X434 @ X433)) = $true) | ~((X429 @ X434) = $true) | ~($true = (X429 @ X433)))) => (! [X435 : term,X436 : term,X437 : subst > term > subst > $o] : (~! [X438 : term,X439 : subst,X440 : subst,X441 : subst] : (($true = (X437 @ (comp @ X439 @ X440) @ (sub @ X438 @ X440) @ X441)) | ~($true = (X437 @ X439 @ X438 @ (comp @ X440 @ X441)))) | ((X437 @ id @ (ap @ X436 @ X435) @ id) = $true) | ~((X437 @ id @ X436 @ id) = $true) | ~! [X442 : subst,X443 : subst,X444 : term,X445 : subst] : (((X437 @ X442 @ X444 @ (comp @ X443 @ X445)) = $true) | ~($true = (X437 @ (comp @ X442 @ X443) @ (sub @ X444 @ X443) @ X445))) | ~! [X446 : term,X447 : term] : (~((X437 @ id @ X446 @ id) = $true) | ~($true = (X437 @ id @ X447 @ id)) | ($true = (X437 @ id @ (ap @ (sub @ X447 @ id) @ X446) @ id))) | ~($true = (X437 @ id @ X435 @ id))) => (! [X448 : term,X449 : subst > term > subst > $o] : (~! [X450 : subst > term > term] : (((X449 @ id @ (lam @ (X450 @ sh @ one)) @ id) = $true) | ~! [X451 : subst,X452 : subst,X453 : term] : ((X450 @ (comp @ X452 @ X451) @ (sub @ X453 @ X451)) = (sub @ (X450 @ X452 @ X453) @ X451)) | ~! [X454 : term] : (~($true = (X449 @ id @ X454 @ id)) | ((X449 @ id @ (X450 @ id @ X454) @ id) = $true))) | ((X449 @ id @ (lam @ X448) @ id) = $true) | ~! [X455 : term] : (((X449 @ id @ (sub @ X448 @ (push @ X455 @ id)) @ id) = $true) | ~($true = (X449 @ id @ X455 @ id))) | ~! [X456 : subst,X457 : subst,X458 : term,X459 : subst] : (((X449 @ (comp @ X459 @ X457) @ (sub @ X458 @ X457) @ X456) = $true) | ~((X449 @ X459 @ X458 @ (comp @ X457 @ X456)) = $true)) | ~! [X460 : subst,X461 : subst,X462 : term,X463 : subst] : (~($true = (X449 @ (comp @ X463 @ X460) @ (sub @ X462 @ X460) @ X461)) | ($true = (X449 @ X463 @ X462 @ (comp @ X460 @ X461))))) => ! [X464 : term,X465 : subst > term > subst > $o] : (~! [X466 : term,X467 : term] : (~($true = (X465 @ id @ X466 @ id)) | ~($true = (X465 @ id @ X467 @ id)) | ((X465 @ id @ (ap @ (sub @ X466 @ id) @ X467) @ id) = $true)) | ~! [X468 : subst > term > term] : (((X465 @ id @ (lam @ (X468 @ sh @ one)) @ id) = $true) | ~! [X469 : term] : (~($true = (X465 @ id @ X469 @ id)) | ($true = (X465 @ id @ (X468 @ id @ X469) @ id))) | ~! [X470 : subst,X471 : subst,X472 : term] : ((sub @ (X468 @ X470 @ X472) @ X471) = (X468 @ (comp @ X470 @ X471) @ (sub @ X472 @ X471)))) | ((X465 @ id @ X464 @ id) = $true) | ~! [X473 : subst,X474 : term,X475 : subst,X476 : subst] : (~($true = (X465 @ X476 @ X474 @ (comp @ X475 @ X473))) | ($true = (X465 @ (comp @ X476 @ X475) @ (sub @ X474 @ X475) @ X473))) | ~! [X477 : term,X478 : subst,X479 : subst,X480 : subst] : (~((X465 @ (comp @ X478 @ X479) @ (sub @ X477 @ X479) @ X480) = $true) | ($true = (X465 @ X478 @ X477 @ (comp @ X479 @ X480)))) | ~! [X481 : term] : (((X465 @ id @ X481 @ id) = $true) | ~($true = (var @ (sub @ X481 @ id))))))))) & (pushprop_lem2v2_gthm = $true) & (termmset_gthm = $true) & (induction2lem_gthm = $true) & (! [X482 : term > $o,X483 : term,X484 : subst,X485 : term > $o,X486 : term] : (~((X482 @ X483) = $true) | ((X485 @ (sub @ X483 @ (push @ X486 @ X484))) = $true)) <=> (pushprop_lem1 = $true)) & ((! [X487 : subst,X488 : term] : ((sub @ one @ (push @ X488 @ X487)) = X488) => ! [X489 : term > $o,X490 : term > $o,X491 : term,X492 : subst] : (($true = (X490 @ one)) | ~((X489 @ X491) = $true) | ~! [X493 : term] : ((X490 @ X493) = (X489 @ (sub @ X493 @ (push @ X491 @ X492)))))) <=> (pushprop_lem1v2_lthm = $true)) & (pushprop_lem0_gthm = $true) & (hoasapinj2_gthm = $true) & ((! [X494 : term] : ((sub @ X494 @ id) = X494) => ! [X495 : term > $o,X496 : subst > term > subst > $o,X497 : term] : (~! [X498 : subst,X499 : subst,X500 : term,X501 : subst] : (~((X496 @ X498 @ X500 @ (comp @ X499 @ X501)) = $true) | ((X496 @ (comp @ X498 @ X499) @ (sub @ X500 @ X499) @ X501) = $true)) | ($true = (X495 @ (lam @ X497))) | ~! [X502 : subst,X503 : term,X504 : subst,X505 : subst] : (~((X496 @ (comp @ X502 @ X505) @ (sub @ X503 @ X505) @ X504) = $true) | ($true = (X496 @ X502 @ X503 @ (comp @ X505 @ X504)))) | ~! [X506 : term] : (((X495 @ (sub @ X497 @ (push @ X506 @ id))) = $true) | ~((X495 @ X506) = $true)) | ~! [X507 : term] : ((X496 @ id @ X507 @ id) = (X495 @ X507)) | ~! [X508 : subst > term > term] : (~! [X509 : term] : (($true = (X496 @ id @ (X508 @ id @ X509) @ id)) | ~((X496 @ id @ X509 @ id) = $true)) | ($true = (X496 @ id @ (lam @ (X508 @ sh @ one)) @ id)) | ~! [X510 : subst,X511 : subst,X512 : term] : ((sub @ (X508 @ X510 @ X512) @ X511) = (X508 @ (comp @ X510 @ X511) @ (sub @ X512 @ X511)))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((substmonoid_lthm = $true) <=> (! [X513 : subst] : ((comp @ id @ X513) = X513) => (! [X514 : subst] : ((comp @ X514 @ id) = X514) => (! [X515 : subst] : ((comp @ X515 @ id) = X515) & ! [X516 : subst] : ((comp @ id @ X516) = X516))))) & (hoasapinj1_gthm = $true) & ((hoasapinj1 = $true) <=> ! [X517 : term,X518 : term,X519 : term,X520 : term] : ((X517 = X518) | ((ap @ (sub @ X518 @ id) @ X520) != (ap @ (sub @ X517 @ id) @ X519)))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((axvarid = $true) <=> ! [X521 : term] : ((sub @ X521 @ id) = X521)) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X522 : subst > term > subst > $o,X523 : term] : (((X522 @ id @ (lam @ (sub @ X523 @ (push @ one @ sh))) @ id) = $true) | ~! [X524 : term] : (($true = (X522 @ id @ (sub @ X523 @ (push @ X524 @ id)) @ id)) | ~((X522 @ id @ X524 @ id) = $true)) | ~! [X525 : subst > term > term] : (($true = (X522 @ id @ (lam @ (X525 @ sh @ one)) @ id)) | ~! [X526 : subst,X527 : term,X528 : subst] : ((X525 @ (comp @ X528 @ X526) @ (sub @ X527 @ X526)) = (sub @ (X525 @ X528 @ X527) @ X526)) | ~! [X529 : term] : (~($true = (X522 @ id @ X529 @ id)) | ((X522 @ id @ (X525 @ id @ X529) @ id) = $true))))) & ((hoaslamnotap = $true) <=> ! [X530 : term,X531 : term,X532 : subst > term > term] : (~! [X533 : subst,X534 : term,X535 : subst] : ((X532 @ (comp @ X535 @ X533) @ (sub @ X534 @ X533)) = (sub @ (X532 @ X535 @ X534) @ X533)) | ((ap @ (sub @ X531 @ id) @ X530) != (lam @ (X532 @ sh @ one))))) & ((hoasapinj1_lthm = $true) <=> (! [X536 : term] : ((sub @ X536 @ id) = X536) => (! [X537 : term,X538 : term,X539 : term,X540 : term] : (((ap @ X537 @ X540) != (ap @ X538 @ X539)) | (X537 = X538)) => ! [X541 : term,X542 : term,X543 : term,X544 : term] : (((ap @ (sub @ X544 @ id) @ X542) != (ap @ (sub @ X543 @ id) @ X541)) | (X543 = X544))))) & (! [X545 : term > $o,X546 : subst,X547 : term] : ~! [X548 : term > $o] : ~! [X549 : term] : ((X548 @ X549) = (X545 @ (sub @ X549 @ (push @ X547 @ X546)))) <=> (pushprop_lem0_lthm = $true)) & (hoaslamnotvar = $true)),
% 9.64/2.82 inference(fool_elimination,[status(thm)],[f7])).
% 9.64/2.82 thf(f7,plain,(
% 9.64/2.82 (! [X0 : subst] : ((push @ (sub @ one @ X0) @ (comp @ sh @ X0)) = X0) <=> axscons) & axabs & hoasinduction_lem1_lthm & (hoasinduction_lem3b_lthm <=> ! [X1 : term] : ~! [X2 : subst > term > term] : ((sub @ X1 @ (push @ one @ sh)) != (X2 @ sh @ one))) & axclos & ! [X3 : term] : ? [X4 : d_term] : ((d2term @ X4) = X3) & hoasinduction_lem1v2_gthm & (termmset_lthm <=> (! [X5 : term] : ((sub @ X5 @ id) = X5) => ! [X6 : term] : ((sub @ X6 @ id) = X6))) & axvarshift & (hoasinduction_p_and_p_prime = (^[X7 : subst > term > subst > $o, X8 : term > $o] : (! [X9 : term] : ((X7 @ id @ X9 @ id) = (X8 @ X9))))) & ((! [X10 : term,X11 : subst] : ((sub @ one @ (push @ X10 @ X11)) = X10) => (! [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) = X12) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X15 : term > $o,X16 : term] : (~! [X17 : term] : ((X15 @ X17) | ~(var @ X17)) | (X15 @ X16) | ~! [X18 : term] : (~(X15 @ X18) | (X15 @ (lam @ X18))) | ~! [X19 : term,X20 : term] : ((X15 @ (ap @ X20 @ X19)) | ~(X15 @ X20) | ~(X15 @ X19))) => ! [X21 : subst,X22 : term,X23 : term > $o] : (~! [X24 : term] : ((X23 @ (sub @ X24 @ X21)) | ~(var @ X24)) | (X23 @ (sub @ X22 @ X21)) | ~! [X25 : term,X26 : term] : (~(X23 @ X26) | ~(X23 @ X25) | (X23 @ (ap @ X26 @ X25))) | ~! [X27 : term] : ((X23 @ (lam @ X27)) | ~! [X28 : term] : ((X23 @ (sub @ X27 @ (push @ X28 @ id))) | ~(X23 @ X28)))))))) <=> induction2lem_lthm) & ulamvarsh & hoasinduction_gthm & pushprop_lthm & hoaslamnotvar_lthm & (! [X29 : subst > term > subst > $o] : ~! [X30 : term > $o] : ~! [X31 : term] : ((X30 @ X31) = (X29 @ id @ X31 @ id)) <=> hoasinduction_lem0_lthm) & hoasinduction_lem3v2_gthm & hoaslamnotvar_gthm & ((d2subst @ d_id) = sh) & (hoasinduction_lem3v2 <=> ! [X32 : subst > term > subst > $o,X33 : term,X34 : term > $o] : (~! [X35 : subst,X36 : subst,X37 : term,X38 : subst] : ((X32 @ (comp @ X38 @ X35) @ (sub @ X37 @ X35) @ X36) | ~(X32 @ X38 @ X37 @ (comp @ X35 @ X36))) | ~! [X39 : term] : ((X34 @ (sub @ X33 @ (push @ X39 @ id))) | ~(X34 @ X39)) | ~! [X40 : subst > term > term] : ((X32 @ id @ (lam @ (X40 @ sh @ one)) @ id) | ~! [X41 : subst,X42 : subst,X43 : term] : ((sub @ (X40 @ X41 @ X43) @ X42) = (X40 @ (comp @ X41 @ X42) @ (sub @ X43 @ X42))) | ~! [X44 : term] : ((X32 @ id @ (X40 @ id @ X44) @ id) | ~(X32 @ id @ X44 @ id))) | ~! [X45 : subst,X46 : subst,X47 : term,X48 : subst] : (~(X32 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46) @ X48) | (X32 @ X45 @ X47 @ (comp @ X46 @ X48))) | ~! [X49 : term] : ((X32 @ id @ X49 @ id) = (X34 @ X49)) | (X34 @ (lam @ X33)))) & ((! [X50 : subst > term > subst > $o] : ~! [X51 : term > $o] : ~! [X52 : term] : ((X51 @ X52) = (X50 @ id @ X52 @ id)) => (! [X53 : term > $o,X54 : term] : (~! [X55 : term,X56 : term] : ((X53 @ (ap @ X56 @ X55)) | ~(X53 @ X56) | ~(X53 @ X55)) | ~! [X57 : term] : (~! [X58 : term] : ((X53 @ (sub @ X57 @ (push @ X58 @ id))) | ~(X53 @ X58)) | (X53 @ (lam @ X57))) | (X53 @ X54) | ~! [X59 : term] : ((X53 @ X59) | ~(var @ X59))) => (! [X60 : term] : ((sub @ X60 @ id) = X60) => (! [X61 : subst > term > subst > $o,X62 : term,X63 : term > $o] : (~! [X64 : term] : (~(X63 @ X64) | (X63 @ (sub @ X62 @ (push @ X64 @ id)))) | ~! [X65 : term] : ((X63 @ X65) = (X61 @ id @ X65 @ id)) | (X63 @ (lam @ X62)) | ~! [X66 : subst > term > term] : (~! [X67 : subst,X68 : term,X69 : subst] : ((X66 @ (comp @ X67 @ X69) @ (sub @ X68 @ X69)) = (sub @ (X66 @ X67 @ X68) @ X69)) | ~! [X70 : term] : (~(X61 @ id @ X70 @ id) | (X61 @ id @ (X66 @ id @ X70) @ id)) | (X61 @ id @ (lam @ (X66 @ sh @ one)) @ id))) => ! [X71 : subst > term > subst > $o,X72 : term] : (~! [X73 : term,X74 : term] : (~(X71 @ id @ X74 @ id) | (X71 @ id @ (ap @ (sub @ X74 @ id) @ X73) @ id) | ~(X71 @ id @ X73 @ id)) | ~! [X75 : subst > term > term] : ((X71 @ id @ (lam @ (X75 @ sh @ one)) @ id) | ~! [X76 : subst,X77 : subst,X78 : term] : ((sub @ (X75 @ X77 @ X78) @ X76) = (X75 @ (comp @ X77 @ X76) @ (sub @ X78 @ X76))) | ~! [X79 : term] : (~(X71 @ id @ X79 @ id) | (X71 @ id @ (X75 @ id @ X79) @ id))) | (X71 @ id @ X72 @ id)))))) <=> hoasinduction_no_psi_cond_lthm) & pushprop_lem2v2_lthm & (shinj <=> ! [X80 : term,X81 : term] : (((sub @ X80 @ sh) != (sub @ X81 @ sh)) | (X80 = X81))) & (! [X82 : term,X83 : term,X84 : term,X85 : term] : ((X84 = X85) | ((ap @ X83 @ X84) != (ap @ X82 @ X85))) <=> apinj2) & (hoasinduction_lem3v2a <=> ! [X86 : subst > term > subst > $o,X87 : term,X88 : term > $o] : (~! [X89 : term] : ((X88 @ (sub @ X87 @ (push @ X89 @ id))) | ~(X88 @ X89)) | ~! [X90 : term] : ((X86 @ id @ X90 @ id) = (X88 @ X90)) | ~! [X91 : subst > term > term] : (~! [X92 : subst,X93 : subst,X94 : term] : ((X91 @ (comp @ X93 @ X92) @ (sub @ X94 @ X92)) = (sub @ (X91 @ X93 @ X94) @ X92)) | ~! [X95 : term] : (~(X86 @ id @ X95 @ id) | (X86 @ id @ (X91 @ id @ X95) @ id)) | (X86 @ id @ (lam @ (X91 @ sh @ one)) @ id)) | (X88 @ (lam @ X87)))) & pushprop & (induction2_lthm <=> (! [X96 : term] : ((sub @ X96 @ id) = X96) => (! [X97 : subst,X98 : term > $o,X99 : term] : (~! [X100 : term,X101 : term] : (~(X98 @ X100) | ~(X98 @ X101) | (X98 @ (ap @ X100 @ X101))) | ~! [X102 : term] : (~(var @ X102) | (X98 @ (sub @ X102 @ X97))) | (X98 @ (sub @ X99 @ X97)) | ~! [X103 : term] : ((X98 @ (lam @ X103)) | ~! [X104 : term] : ((X98 @ (sub @ X103 @ (push @ X104 @ id))) | ~(X98 @ X104)))) => ! [X105 : term > $o,X106 : term] : (~! [X107 : term,X108 : term] : (~(X105 @ X108) | (X105 @ (ap @ X107 @ X108)) | ~(X105 @ X107)) | (X105 @ X106) | ~! [X109 : term] : (~! [X110 : term] : ((X105 @ (sub @ X109 @ (push @ X110 @ id))) | ~(X105 @ X110)) | (X105 @ (lam @ X109))) | ~! [X111 : term] : ((X105 @ X111) | ~(var @ X111)))))) & (! [X112 : term] : ((sub @ X112 @ id) = X112) <=> termmset) & (hoasinduction_lem0 <=> ! [X113 : subst > term > subst > $o] : ~! [X114 : term > $o] : ~! [X115 : term] : ((X113 @ id @ X115 @ id) = (X114 @ X115))) & substmonoid_gthm & ((! [X116 : subst,X117 : term] : ((sub @ one @ (push @ X117 @ X116)) = X117) => (! [X118 : term,X119 : subst] : ((comp @ sh @ (push @ X118 @ X119)) = X119) => ! [X120 : term > $o,X121 : term,X122 : subst,X123 : term > $o,X124 : term] : ((X120 @ (sub @ X124 @ (push @ X121 @ X122))) | ~(X123 @ X124)))) <=> pushprop_lem1_lthm) & (! [X125 : term,X126 : subst > term > subst > $o] : (~! [X127 : term] : ((X126 @ id @ (sub @ X125 @ (push @ X127 @ id)) @ id) | ~(X126 @ id @ X127 @ id)) | ~! [X128 : subst > term > term] : (~! [X129 : subst,X130 : term,X131 : subst] : ((sub @ (X128 @ X131 @ X130) @ X129) = (X128 @ (comp @ X131 @ X129) @ (sub @ X130 @ X129))) | ~! [X132 : term] : ((X126 @ id @ (X128 @ id @ X132) @ id) | ~(X126 @ id @ X132 @ id)) | (X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) | ~! [X133 : subst,X134 : subst,X135 : subst,X136 : term] : (~(X126 @ X134 @ X136 @ (comp @ X133 @ X135)) | (X126 @ (comp @ X134 @ X133) @ (sub @ X136 @ X133) @ X135)) | ~! [X137 : subst,X138 : subst,X139 : term,X140 : subst] : (~(X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137) | (X126 @ X138 @ X139 @ (comp @ X140 @ X137))) | (X126 @ id @ (lam @ X125) @ id)) <=> hoasinduction_lem3) & ((^[X141 : subst, X142 : subst > term > term] : (d2term @ d_one)) = hoaslam) & (hoasinduction_lem3v2_f <=> ! [X143 : term] : ~! [X144 : subst > term > term] : ~! [X145 : term,X146 : subst] : ((sub @ X143 @ (push @ X145 @ X146)) = (X144 @ X146 @ X145))) & (! [X147 : term] : ~! [X148 : subst > term > term] : ((X148 @ sh @ one) != (sub @ X147 @ (push @ one @ sh))) <=> hoasinduction_lem3b) & hoaslaminj_gthm & pushprop_lem1_gthm & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_p_and_p_prime = (^[X149 : term, X150 : subst, X151 : term > $o, X152 : term > $o] : (! [X153 : term] : ((X152 @ X153) = (X151 @ (sub @ X153 @ (push @ X149 @ X150))))))) & pushprop_lem3v2 & (hoaslaminj <=> ! [X154 : subst > term > term,X155 : subst > term > term,X156 : subst,X157 : term] : (~! [X158 : term,X159 : subst,X160 : subst] : ((X154 @ (comp @ X159 @ X160) @ (sub @ X158 @ X160)) = (sub @ (X154 @ X159 @ X158) @ X160)) | ((X154 @ X156 @ X157) = (X155 @ X156 @ X157)) | ~! [X161 : subst,X162 : subst,X163 : term] : ((X155 @ (comp @ X162 @ X161) @ (sub @ X163 @ X161)) = (sub @ (X155 @ X162 @ X163) @ X161)) | ((lam @ (X154 @ sh @ one)) != (lam @ (X155 @ sh @ one))))) & ~(var @ (d2term @ d_one)) & (! [X164 : subst > term > subst > $o,X165 : term] : (~! [X166 : term] : ((X164 @ id @ (sub @ X165 @ (push @ X166 @ id)) @ id) | ~(X164 @ id @ X166 @ id)) | ~! [X167 : subst > term > term] : ((X164 @ id @ (lam @ (X167 @ sh @ one)) @ id) | ~! [X168 : term] : (~(X164 @ id @ X168 @ id) | (X164 @ id @ (X167 @ id @ X168) @ id)) | ~! [X169 : subst,X170 : subst,X171 : term] : ((X167 @ (comp @ X169 @ X170) @ (sub @ X171 @ X170)) = (sub @ (X167 @ X169 @ X171) @ X170))) | (X164 @ id @ (lam @ (sub @ X165 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aa) & axmap & apnotvar & ((! [X172 : term] : ((sub @ X172 @ id) = X172) => (! [X173 : subst > term > subst > $o,X174 : term] : (~! [X175 : term] : ((X173 @ id @ (sub @ X174 @ (push @ X175 @ id)) @ id) | ~(X173 @ id @ X175 @ id)) | (X173 @ id @ (lam @ (sub @ X174 @ (push @ one @ sh))) @ id) | ~! [X176 : subst > term > term] : (~! [X177 : subst,X178 : subst,X179 : term] : ((sub @ (X176 @ X178 @ X179) @ X177) = (X176 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177))) | ~! [X180 : term] : ((X173 @ id @ (X176 @ id @ X180) @ id) | ~(X173 @ id @ X180 @ id)) | (X173 @ id @ (lam @ (X176 @ sh @ one)) @ id))) => ! [X181 : term,X182 : subst > term > subst > $o] : (~! [X183 : term] : (~(X182 @ id @ X183 @ id) | (X182 @ id @ (sub @ X181 @ (push @ X183 @ id)) @ id)) | ~! [X184 : subst > term > term] : (~! [X185 : term] : ((X182 @ id @ (X184 @ id @ X185) @ id) | ~(X182 @ id @ X185 @ id)) | ~! [X186 : subst,X187 : subst,X188 : term] : ((X184 @ (comp @ X186 @ X187) @ (sub @ X188 @ X187)) = (sub @ (X184 @ X186 @ X188) @ X187)) | (X182 @ id @ (lam @ (X184 @ sh @ one)) @ id)) | ~! [X189 : term,X190 : subst,X191 : subst,X192 : subst] : ((X182 @ (comp @ X192 @ X190) @ (sub @ X189 @ X190) @ X191) | ~(X182 @ X192 @ X189 @ (comp @ X190 @ X191))) | (X182 @ id @ (lam @ X181) @ id) | ~! [X193 : subst,X194 : subst,X195 : term,X196 : subst] : (~(X182 @ (comp @ X194 @ X193) @ (sub @ X195 @ X193) @ X196) | (X182 @ X194 @ X195 @ (comp @ X193 @ X196)))))) <=> hoasinduction_lem3_lthm) & lamnotvar & hoasinduction_lem1_gthm & hoasinduction_lem2v2_gthm & (hoasinduction_lem2 <=> ! [X197 : term,X198 : subst > term > subst > $o,X199 : term] : (~! [X200 : subst,X201 : subst,X202 : term,X203 : subst] : (~(X198 @ (comp @ X201 @ X203) @ (sub @ X202 @ X203) @ X200) | (X198 @ X201 @ X202 @ (comp @ X203 @ X200))) | (X198 @ id @ (ap @ X199 @ X197) @ id) | ~! [X204 : subst,X205 : subst,X206 : term,X207 : subst] : (~(X198 @ X204 @ X206 @ (comp @ X207 @ X205)) | (X198 @ (comp @ X204 @ X207) @ (sub @ X206 @ X207) @ X205)) | ~(X198 @ id @ X199 @ id) | ~(X198 @ id @ X197 @ id) | ~! [X208 : term,X209 : term] : ((X198 @ id @ (ap @ (sub @ X209 @ id) @ X208) @ id) | ~(X198 @ id @ X208 @ id) | ~(X198 @ id @ X209 @ id)))) & pushprop_lem2v2 & (! [X210 : term,X211 : subst] : ((comp @ sh @ (push @ X210 @ X211)) = X211) <=> axshiftcons) & (! [X212 : subst > term > subst > $o,X213 : term] : ((X212 @ id @ X213 @ id) | ~! [X214 : term,X215 : term] : (~(X212 @ id @ X215 @ id) | ~(X212 @ id @ X214 @ id) | (X212 @ id @ (ap @ (sub @ X214 @ id) @ X215) @ id)) | ~! [X216 : subst > term > term] : (~! [X217 : term] : (~(X212 @ id @ X217 @ id) | (X212 @ id @ (X216 @ id @ X217) @ id)) | ~! [X218 : term,X219 : subst,X220 : subst] : ((sub @ (X216 @ X220 @ X218) @ X219) = (X216 @ (comp @ X220 @ X219) @ (sub @ X218 @ X219))) | (X212 @ id @ (lam @ (X216 @ sh @ one)) @ id))) <=> hoasinduction_no_psi_cond) & hoasinduction_lem3_gthm & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (hoasinduction <=> ! [X221 : term,X222 : subst > term > subst > $o] : ((X222 @ id @ X221 @ id) | ~! [X223 : term,X224 : subst,X225 : subst,X226 : subst] : ((X222 @ X224 @ X223 @ (comp @ X226 @ X225)) | ~(X222 @ (comp @ X224 @ X226) @ (sub @ X223 @ X226) @ X225)) | ~! [X227 : subst,X228 : term,X229 : subst,X230 : subst] : ((X222 @ (comp @ X229 @ X227) @ (sub @ X228 @ X227) @ X230) | ~(X222 @ X229 @ X228 @ (comp @ X227 @ X230))) | ~! [X231 : term] : (~(var @ (sub @ X231 @ id)) | (X222 @ id @ X231 @ id)) | ~! [X232 : subst > term > term] : (~! [X233 : term] : (~(X222 @ id @ X233 @ id) | (X222 @ id @ (X232 @ id @ X233) @ id)) | ~! [X234 : term,X235 : subst,X236 : subst] : ((X232 @ (comp @ X236 @ X235) @ (sub @ X234 @ X235)) = (sub @ (X232 @ X236 @ X234) @ X235)) | (X222 @ id @ (lam @ (X232 @ sh @ one)) @ id)) | ~! [X237 : term,X238 : term] : (~(X222 @ id @ X238 @ id) | ~(X222 @ id @ X237 @ id) | (X222 @ id @ (ap @ (sub @ X237 @ id) @ X238) @ id)))) & hoasinduction_lem1 & ! [X239 : d_term] : (d_one = X239) & hoaslamnotap_lthm & (induction <=> ! [X240 : term > $o,X241 : term] : ((X240 @ X241) | ~! [X242 : term] : ((X240 @ X242) | ~(var @ X242)) | ~! [X243 : term,X244 : term] : (~(X240 @ X244) | (X240 @ (ap @ X243 @ X244)) | ~(X240 @ X243)) | ~! [X245 : term] : (~(X240 @ X245) | (X240 @ (lam @ X245))))) & (apinj1 <=> ! [X246 : term,X247 : term,X248 : term,X249 : term] : ((X246 = X249) | ((ap @ X249 @ X247) != (ap @ X246 @ X248)))) & (hoasinduction_lem3v2a_lthm <=> (! [X250 : term] : ~! [X251 : subst > term > term] : ~! [X252 : term,X253 : subst] : ((sub @ X250 @ (push @ X252 @ X253)) = (X251 @ X253 @ X252)) => (! [X254 : term] : ((sub @ X254 @ id) = X254) => ! [X255 : term,X256 : subst > term > subst > $o,X257 : term > $o] : (~! [X258 : subst > term > term] : (~! [X259 : term,X260 : subst,X261 : subst] : ((sub @ (X258 @ X261 @ X259) @ X260) = (X258 @ (comp @ X261 @ X260) @ (sub @ X259 @ X260))) | (X256 @ id @ (lam @ (X258 @ sh @ one)) @ id) | ~! [X262 : term] : (~(X256 @ id @ X262 @ id) | (X256 @ id @ (X258 @ id @ X262) @ id))) | ~! [X263 : term] : ((X257 @ X263) = (X256 @ id @ X263 @ id)) | (X257 @ (lam @ X255)) | ~! [X264 : term] : (~(X257 @ X264) | (X257 @ (sub @ X255 @ (push @ X264 @ id)))))))) & (! [X265 : term,X266 : term > $o] : (~! [X267 : term,X268 : term] : ((X266 @ (ap @ X267 @ X268)) | ~(X266 @ X267) | ~(X266 @ X268)) | ~! [X269 : term] : ((X266 @ X269) | ~(var @ X269)) | (X266 @ X265) | ~! [X270 : term] : ((X266 @ (lam @ X270)) | ~! [X271 : term] : (~(X266 @ X271) | (X266 @ (sub @ X270 @ (push @ X271 @ id)))))) <=> induction2) & (! [X272 : subst > term > subst > $o,X273 : term] : (~! [X274 : term] : ((X272 @ id @ (sub @ X273 @ (push @ X274 @ id)) @ id) | ~(X272 @ id @ X274 @ id)) | ~! [X275 : subst > term > term] : (~! [X276 : term] : ((X272 @ id @ (X275 @ id @ X276) @ id) | ~(X272 @ id @ X276 @ id)) | (X272 @ id @ (lam @ (X275 @ sh @ one)) @ id) | ~! [X277 : subst,X278 : subst,X279 : term] : ((sub @ (X275 @ X278 @ X279) @ X277) = (X275 @ (comp @ X278 @ X277) @ (sub @ X279 @ X277)))) | (X272 @ id @ (lam @ X273) @ id)) <=> hoasinduction_lem3a) & pushprop_lem1v2_gthm & pushprop_lthm_orig & (! [X280 : term,X281 : term > $o,X282 : subst > term > subst > $o,X283 : term] : (~! [X284 : term,X285 : subst,X286 : subst,X287 : subst] : ((X282 @ (comp @ X287 @ X286) @ (sub @ X284 @ X286) @ X285) | ~(X282 @ X287 @ X284 @ (comp @ X286 @ X285))) | ~(X281 @ X283) | (X281 @ (ap @ X283 @ X280)) | ~! [X288 : term,X289 : term] : (~(X282 @ id @ X289 @ id) | ~(X282 @ id @ X288 @ id) | (X282 @ id @ (ap @ (sub @ X289 @ id) @ X288) @ id)) | ~! [X290 : subst,X291 : subst,X292 : term,X293 : subst] : ((X282 @ X291 @ X292 @ (comp @ X293 @ X290)) | ~(X282 @ (comp @ X291 @ X293) @ (sub @ X292 @ X293) @ X290)) | ~! [X294 : term] : ((X282 @ id @ X294 @ id) = (X281 @ X294)) | ~(X281 @ X280)) <=> hoasinduction_lem2v2) & (hoasinduction_lthm_3 <=> (! [X295 : subst > term > subst > $o] : ~! [X296 : term > $o] : ~! [X297 : term] : ((X295 @ id @ X297 @ id) = (X296 @ X297)) => (! [X298 : term,X299 : term > $o] : (~! [X300 : term] : ((X299 @ X300) | ~(var @ X300)) | (X299 @ X298) | ~! [X301 : term] : (~! [X302 : term] : (~(X299 @ X302) | (X299 @ (sub @ X301 @ (push @ X302 @ id)))) | (X299 @ (lam @ X301))) | ~! [X303 : term,X304 : term] : (~(X299 @ X304) | (X299 @ (ap @ X303 @ X304)) | ~(X299 @ X303))) => (! [X305 : term] : ((sub @ X305 @ id) = X305) => (! [X306 : term,X307 : term > $o,X308 : subst > term > subst > $o] : (~! [X309 : term] : ((X308 @ id @ X309 @ id) = (X307 @ X309)) | (X307 @ (lam @ X306)) | ~! [X310 : subst > term > term] : ((X308 @ id @ (lam @ (X310 @ sh @ one)) @ id) | ~! [X311 : subst,X312 : term,X313 : subst] : ((sub @ (X310 @ X313 @ X312) @ X311) = (X310 @ (comp @ X313 @ X311) @ (sub @ X312 @ X311))) | ~! [X314 : term] : (~(X308 @ id @ X314 @ id) | (X308 @ id @ (X310 @ id @ X314) @ id))) | ~! [X315 : term] : (~(X307 @ X315) | (X307 @ (sub @ X306 @ (push @ X315 @ id))))) => ! [X316 : subst > term > subst > $o,X317 : term] : (~! [X318 : term] : (~(var @ (sub @ X318 @ id)) | (X316 @ id @ X318 @ id)) | ~! [X319 : subst > term > term] : (~! [X320 : term] : ((X316 @ id @ (X319 @ id @ X320) @ id) | ~(X316 @ id @ X320 @ id)) | (X316 @ id @ (lam @ (X319 @ sh @ one)) @ id) | ~! [X321 : subst,X322 : term,X323 : subst] : ((X319 @ (comp @ X321 @ X323) @ (sub @ X322 @ X323)) = (sub @ (X319 @ X321 @ X322) @ X323))) | ~! [X324 : subst,X325 : subst,X326 : term,X327 : subst] : ((X316 @ X324 @ X326 @ (comp @ X325 @ X327)) | ~(X316 @ (comp @ X324 @ X325) @ (sub @ X326 @ X325) @ X327)) | ~! [X328 : term,X329 : term] : ((X316 @ id @ (ap @ (sub @ X328 @ id) @ X329) @ id) | ~(X316 @ id @ X329 @ id) | ~(X316 @ id @ X328 @ id)) | ~! [X330 : term,X331 : subst,X332 : subst,X333 : subst] : (~(X316 @ X331 @ X330 @ (comp @ X332 @ X333)) | (X316 @ (comp @ X331 @ X332) @ (sub @ X330 @ X332) @ X333)) | (X316 @ id @ X317 @ id))))))) & ~lamnotap & (axidl <=> ! [X334 : subst] : ((comp @ id @ X334) = X334)) & (! [X335 : term > $o,X336 : subst,X337 : term] : (~! [X338 : term,X339 : term] : (~(X335 @ X339) | ~(X335 @ X338) | (X335 @ (ap @ X339 @ X338))) | ~! [X340 : term] : (~(var @ X340) | (X335 @ (sub @ X340 @ X336))) | ~! [X341 : term] : (~! [X342 : term] : (~(X335 @ X342) | (X335 @ (sub @ X341 @ (push @ X342 @ id)))) | (X335 @ (lam @ X341))) | (X335 @ (sub @ X337 @ X336))) <=> induction2lem) & hoasinduction_lem3b_gthm & hoasinduction_lem3a_gthm & hoasapnotvar_gthm & ((d2subst @ d_id) = id) & (! [X343 : term,X344 : term > $o,X345 : subst] : ~! [X346 : term > $o] : ~! [X347 : term] : ((X344 @ (sub @ X347 @ (push @ X343 @ X345))) = (X346 @ X347)) <=> pushprop_lem0) & hoasapnotvar & ! [X348 : d_subst,X349 : d_subst] : (((d2subst @ X348) = (d2subst @ X349)) => (X348 = X349)) & pushprop_gthm & ulamvarind & (! [X350 : subst > term > subst > $o,X351 : term] : (~! [X352 : term] : (~(X350 @ id @ X352 @ id) | (X350 @ id @ (sub @ X351 @ (push @ X352 @ id)) @ id)) | ~! [X353 : term,X354 : subst > term > term] : (~! [X355 : subst,X356 : subst,X357 : term] : ((sub @ (sub @ X353 @ (push @ X357 @ X356)) @ X355) = (sub @ (X354 @ X356 @ X357) @ X355)) | ~! [X358 : term] : ((X350 @ id @ (X354 @ id @ X358) @ id) | ~(X350 @ id @ X358 @ id)) | (X350 @ id @ (lam @ (X354 @ sh @ one)) @ id) | ~! [X359 : subst,X360 : subst,X361 : term] : ((sub @ X353 @ (push @ (sub @ X361 @ X360) @ (comp @ X359 @ X360))) = (X354 @ (comp @ X359 @ X360) @ (sub @ X361 @ X360)))) | (X350 @ id @ (lam @ (sub @ X351 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aaa) & hoasinduction_lem1v2 & (hoasapinj2 <=> ! [X362 : term,X363 : term,X364 : term,X365 : term] : ((X363 = X364) | ((ap @ (sub @ X362 @ id) @ X363) != (ap @ (sub @ X365 @ id) @ X364)))) & axapp & ! [X366 : subst] : ? [X367 : d_subst] : ((d2subst @ X367) = X366) & (hoasapinj2_lthm <=> (! [X368 : term,X369 : term,X370 : term,X371 : term] : (((ap @ X369 @ X368) != (ap @ X370 @ X371)) | (X368 = X371)) => ! [X372 : term,X373 : term,X374 : term,X375 : term] : ((X373 = X375) | ((ap @ (sub @ X374 @ id) @ X375) != (ap @ (sub @ X372 @ id) @ X373))))) & induction2_gthm & (! [X376 : term,X377 : term] : (((lam @ X376) != (lam @ X377)) | (X376 = X377)) <=> laminj) & ! [X378 : d_term,X379 : d_term] : (((d2term @ X378) = (d2term @ X379)) => (X378 = X379)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (axvarcons <=> ! [X380 : subst,X381 : term] : ((sub @ one @ (push @ X381 @ X380)) = X381)) & hoasinduction_lem2_gthm & hoaslamnotap_gthm & ((! [X382 : subst] : ((comp @ id @ X382) = X382) & ! [X383 : subst] : ((comp @ X383 @ id) = X383)) <=> substmonoid) & hoasapnotvar_lthm & ((! [X384 : term] : ((sub @ X384 @ id) = X384) => (! [X385 : term,X386 : subst > term > subst > $o] : (~! [X387 : subst > term > term] : (~! [X388 : term] : ((X386 @ id @ (X387 @ id @ X388) @ id) | ~(X386 @ id @ X388 @ id)) | (X386 @ id @ (lam @ (X387 @ sh @ one)) @ id) | ~! [X389 : subst,X390 : subst,X391 : term] : ((X387 @ (comp @ X389 @ X390) @ (sub @ X391 @ X390)) = (sub @ (X387 @ X389 @ X391) @ X390))) | ~! [X392 : term] : ((X386 @ id @ (sub @ X385 @ (push @ X392 @ id)) @ id) | ~(X386 @ id @ X392 @ id)) | (X386 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id)) => ! [X393 : term,X394 : subst > term > subst > $o] : (~! [X395 : term] : (~(X394 @ id @ X395 @ id) | (X394 @ id @ (sub @ X393 @ (push @ X395 @ id)) @ id)) | ~! [X396 : subst > term > term] : (~! [X397 : term] : ((X394 @ id @ (X396 @ id @ X397) @ id) | ~(X394 @ id @ X397 @ id)) | (X394 @ id @ (lam @ (X396 @ sh @ one)) @ id) | ~! [X398 : subst,X399 : subst,X400 : term] : ((X396 @ (comp @ X399 @ X398) @ (sub @ X400 @ X398)) = (sub @ (X396 @ X399 @ X400) @ X398))) | (X394 @ id @ (lam @ X393) @ id)))) <=> hoasinduction_lem3a_lthm) & hoasinduction_lem2_lthm & (! [X401 : term] : ~! [X402 : subst > term > term] : ~! [X403 : subst,X404 : term] : ((X402 @ X403 @ X404) = (sub @ X401 @ (push @ X404 @ X403))) <=> hoasinduction_lem3v2_f_lthm) & (pushprop_lem1v2 <=> ! [X405 : term > $o,X406 : term > $o,X407 : term,X408 : subst] : ((X406 @ one) | ~(X405 @ X407) | ~! [X409 : term] : ((X406 @ X409) = (X405 @ (sub @ X409 @ (push @ X407 @ X408)))))) & (one = (d2term @ d_one)) & ~ulamvar1 & (hoaslaminj_lthm <=> (! [X410 : term,X411 : subst] : ((sub @ one @ (push @ X410 @ X411)) = X410) => (! [X412 : term,X413 : subst] : ((comp @ sh @ (push @ X412 @ X413)) = X413) => (! [X414 : term,X415 : term] : ((X414 = X415) | ((lam @ X414) != (lam @ X415))) => ! [X416 : subst > term > term,X417 : subst,X418 : term,X419 : subst > term > term] : (((lam @ (X416 @ sh @ one)) != (lam @ (X419 @ sh @ one))) | ((X416 @ X417 @ X418) = (X419 @ X417 @ X418)) | ~! [X420 : subst,X421 : term,X422 : subst] : ((X416 @ (comp @ X420 @ X422) @ (sub @ X421 @ X422)) = (sub @ (X416 @ X420 @ X421) @ X422)) | ~! [X423 : term,X424 : subst,X425 : subst] : ((X419 @ (comp @ X425 @ X424) @ (sub @ X423 @ X424)) = (sub @ (X419 @ X425 @ X423) @ X424))))))) & axassoc & (! [X426 : subst] : ((comp @ X426 @ id) = X426) <=> axidr) & ! [X427 : d_subst] : (d_id = X427) & (hoasinduction_lthm <=> (! [X428 : term,X429 : term > $o] : ((X429 @ X428) | ~! [X430 : term] : ((X429 @ (lam @ X430)) | ~! [X431 : term] : (~(X429 @ X431) | (X429 @ (sub @ X430 @ (push @ X431 @ id))))) | ~! [X432 : term] : ((X429 @ X432) | ~(var @ X432)) | ~! [X433 : term,X434 : term] : ((X429 @ (ap @ X434 @ X433)) | ~(X429 @ X434) | ~(X429 @ X433))) => (! [X435 : term,X436 : term,X437 : subst > term > subst > $o] : (~! [X438 : term,X439 : subst,X440 : subst,X441 : subst] : ((X437 @ (comp @ X439 @ X440) @ (sub @ X438 @ X440) @ X441) | ~(X437 @ X439 @ X438 @ (comp @ X440 @ X441))) | (X437 @ id @ (ap @ X436 @ X435) @ id) | ~(X437 @ id @ X436 @ id) | ~! [X442 : subst,X443 : subst,X444 : term,X445 : subst] : ((X437 @ X442 @ X444 @ (comp @ X443 @ X445)) | ~(X437 @ (comp @ X442 @ X443) @ (sub @ X444 @ X443) @ X445)) | ~! [X446 : term,X447 : term] : (~(X437 @ id @ X446 @ id) | ~(X437 @ id @ X447 @ id) | (X437 @ id @ (ap @ (sub @ X447 @ id) @ X446) @ id)) | ~(X437 @ id @ X435 @ id)) => (! [X448 : term,X449 : subst > term > subst > $o] : (~! [X450 : subst > term > term] : ((X449 @ id @ (lam @ (X450 @ sh @ one)) @ id) | ~! [X451 : subst,X452 : subst,X453 : term] : ((X450 @ (comp @ X452 @ X451) @ (sub @ X453 @ X451)) = (sub @ (X450 @ X452 @ X453) @ X451)) | ~! [X454 : term] : (~(X449 @ id @ X454 @ id) | (X449 @ id @ (X450 @ id @ X454) @ id))) | (X449 @ id @ (lam @ X448) @ id) | ~! [X455 : term] : ((X449 @ id @ (sub @ X448 @ (push @ X455 @ id)) @ id) | ~(X449 @ id @ X455 @ id)) | ~! [X456 : subst,X457 : subst,X458 : term,X459 : subst] : ((X449 @ (comp @ X459 @ X457) @ (sub @ X458 @ X457) @ X456) | ~(X449 @ X459 @ X458 @ (comp @ X457 @ X456))) | ~! [X460 : subst,X461 : subst,X462 : term,X463 : subst] : (~(X449 @ (comp @ X463 @ X460) @ (sub @ X462 @ X460) @ X461) | (X449 @ X463 @ X462 @ (comp @ X460 @ X461)))) => ! [X464 : term,X465 : subst > term > subst > $o] : (~! [X466 : term,X467 : term] : (~(X465 @ id @ X466 @ id) | ~(X465 @ id @ X467 @ id) | (X465 @ id @ (ap @ (sub @ X466 @ id) @ X467) @ id)) | ~! [X468 : subst > term > term] : ((X465 @ id @ (lam @ (X468 @ sh @ one)) @ id) | ~! [X469 : term] : (~(X465 @ id @ X469 @ id) | (X465 @ id @ (X468 @ id @ X469) @ id)) | ~! [X470 : subst,X471 : subst,X472 : term] : ((sub @ (X468 @ X470 @ X472) @ X471) = (X468 @ (comp @ X470 @ X471) @ (sub @ X472 @ X471)))) | (X465 @ id @ X464 @ id) | ~! [X473 : subst,X474 : term,X475 : subst,X476 : subst] : (~(X465 @ X476 @ X474 @ (comp @ X475 @ X473)) | (X465 @ (comp @ X476 @ X475) @ (sub @ X474 @ X475) @ X473)) | ~! [X477 : term,X478 : subst,X479 : subst,X480 : subst] : (~(X465 @ (comp @ X478 @ X479) @ (sub @ X477 @ X479) @ X480) | (X465 @ X478 @ X477 @ (comp @ X479 @ X480))) | ~! [X481 : term] : ((X465 @ id @ X481 @ id) | ~(var @ (sub @ X481 @ id)))))))) & pushprop_lem2v2_gthm & termmset_gthm & induction2lem_gthm & (! [X482 : term > $o,X483 : term,X484 : subst,X485 : term > $o,X486 : term] : (~(X482 @ X483) | (X485 @ (sub @ X483 @ (push @ X486 @ X484)))) <=> pushprop_lem1) & ((! [X487 : subst,X488 : term] : ((sub @ one @ (push @ X488 @ X487)) = X488) => ! [X489 : term > $o,X490 : term > $o,X491 : term,X492 : subst] : ((X490 @ one) | ~(X489 @ X491) | ~! [X493 : term] : ((X490 @ X493) = (X489 @ (sub @ X493 @ (push @ X491 @ X492)))))) <=> pushprop_lem1v2_lthm) & pushprop_lem0_gthm & hoasapinj2_gthm & ((! [X494 : term] : ((sub @ X494 @ id) = X494) => ! [X495 : term > $o,X496 : subst > term > subst > $o,X497 : term] : (~! [X498 : subst,X499 : subst,X500 : term,X501 : subst] : (~(X496 @ X498 @ X500 @ (comp @ X499 @ X501)) | (X496 @ (comp @ X498 @ X499) @ (sub @ X500 @ X499) @ X501)) | (X495 @ (lam @ X497)) | ~! [X502 : subst,X503 : term,X504 : subst,X505 : subst] : (~(X496 @ (comp @ X502 @ X505) @ (sub @ X503 @ X505) @ X504) | (X496 @ X502 @ X503 @ (comp @ X505 @ X504))) | ~! [X506 : term] : ((X495 @ (sub @ X497 @ (push @ X506 @ id))) | ~(X495 @ X506)) | ~! [X507 : term] : ((X496 @ id @ X507 @ id) = (X495 @ X507)) | ~! [X508 : subst > term > term] : (~! [X509 : term] : ((X496 @ id @ (X508 @ id @ X509) @ id) | ~(X496 @ id @ X509 @ id)) | (X496 @ id @ (lam @ (X508 @ sh @ one)) @ id) | ~! [X510 : subst,X511 : subst,X512 : term] : ((sub @ (X508 @ X510 @ X512) @ X511) = (X508 @ (comp @ X510 @ X511) @ (sub @ X512 @ X511)))))) <=> hoasinduction_lem3v2_lthm) & (substmonoid_lthm <=> (! [X513 : subst] : ((comp @ id @ X513) = X513) => (! [X514 : subst] : ((comp @ X514 @ id) = X514) => (! [X515 : subst] : ((comp @ X515 @ id) = X515) & ! [X516 : subst] : ((comp @ id @ X516) = X516))))) & hoasapinj1_gthm & (hoasapinj1 <=> ! [X517 : term,X518 : term,X519 : term,X520 : term] : ((X517 = X518) | ((ap @ (sub @ X518 @ id) @ X520) != (ap @ (sub @ X517 @ id) @ X519)))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (axvarid <=> ! [X521 : term] : ((sub @ X521 @ id) = X521)) & pushprop_lem3v2_lthm & (hoasinduction_lem3aa_lthm <=> ! [X522 : subst > term > subst > $o,X523 : term] : ((X522 @ id @ (lam @ (sub @ X523 @ (push @ one @ sh))) @ id) | ~! [X524 : term] : ((X522 @ id @ (sub @ X523 @ (push @ X524 @ id)) @ id) | ~(X522 @ id @ X524 @ id)) | ~! [X525 : subst > term > term] : ((X522 @ id @ (lam @ (X525 @ sh @ one)) @ id) | ~! [X526 : subst,X527 : term,X528 : subst] : ((X525 @ (comp @ X528 @ X526) @ (sub @ X527 @ X526)) = (sub @ (X525 @ X528 @ X527) @ X526)) | ~! [X529 : term] : (~(X522 @ id @ X529 @ id) | (X522 @ id @ (X525 @ id @ X529) @ id))))) & (hoaslamnotap <=> ! [X530 : term,X531 : term,X532 : subst > term > term] : (~! [X533 : subst,X534 : term,X535 : subst] : ((X532 @ (comp @ X535 @ X533) @ (sub @ X534 @ X533)) = (sub @ (X532 @ X535 @ X534) @ X533)) | ((ap @ (sub @ X531 @ id) @ X530) != (lam @ (X532 @ sh @ one))))) & (hoasapinj1_lthm <=> (! [X536 : term] : ((sub @ X536 @ id) = X536) => (! [X537 : term,X538 : term,X539 : term,X540 : term] : (((ap @ X537 @ X540) != (ap @ X538 @ X539)) | (X537 = X538)) => ! [X541 : term,X542 : term,X543 : term,X544 : term] : (((ap @ (sub @ X544 @ id) @ X542) != (ap @ (sub @ X543 @ id) @ X541)) | (X543 = X544))))) & (! [X545 : term > $o,X546 : subst,X547 : term] : ~! [X548 : term > $o] : ~! [X549 : term] : ((X548 @ X549) = (X545 @ (sub @ X549 @ (push @ X547 @ X546)))) <=> pushprop_lem0_lthm) & hoaslamnotvar),
% 9.64/2.82 inference(rectify,[status(thm)],[f1])).
% 9.64/2.82 thf(f1,axiom,(
% 9.64/2.82 (! [X14 : subst] : ((push @ (sub @ one @ X14) @ (comp @ sh @ X14)) = X14) <=> axscons) & axabs & hoasinduction_lem1_lthm & (hoasinduction_lem3b_lthm <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh)))) & axclos & ! [X0 : term] : ? [X1 : d_term] : ((d2term @ X1) = X0) & hoasinduction_lem1v2_gthm & (termmset_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X13 : term] : ((sub @ X13 @ id) = X13))) & axvarshift & (hoasinduction_p_and_p_prime = (^[X10 : subst > term > subst > $o, X11 : term > $o] : (! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))))) & ((! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X10 : term > $o,X21 : term] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | (X10 @ X21) | ~! [X13 : term] : (~(X10 @ X13) | (X10 @ (lam @ X13))) | ~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15))) => ! [X17 : subst,X16 : term,X10 : term > $o] : (~! [X15 : term] : ((X10 @ (sub @ X15 @ X17)) | ~(var @ X15)) | (X10 @ (sub @ X16 @ X17)) | ~! [X15 : term,X13 : term] : (~(X10 @ X13) | ~(X10 @ X15) | (X10 @ (ap @ X13 @ X15))) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)))))))) <=> induction2lem_lthm) & ulamvarsh & hoasinduction_gthm & pushprop_lthm & hoaslamnotvar_lthm & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0_lthm) & hoasinduction_lem3v2_gthm & hoaslamnotvar_gthm & ((d2subst @ d_id) = sh) & (hoasinduction_lem3v2 <=> ! [X10 : subst > term > subst > $o,X48 : term,X11 : term > $o] : (~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X15 : term] : ((X11 @ (sub @ X48 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X48)))) & ((! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X10 : term > $o,X22 : term] : (~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | (X10 @ X22) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : subst > term > subst > $o,X29 : term,X11 : term > $o] : (~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X29)) | ~! [X18 : subst > term > term] : (~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id))) => ! [X10 : subst > term > subst > $o,X30 : term] : (~! [X15 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | (X10 @ id @ X30 @ id)))))) <=> hoasinduction_no_psi_cond_lthm) & pushprop_lem2v2_lthm & (shinj <=> ! [X15 : term,X13 : term] : (((sub @ X13 @ sh) != (sub @ X15 @ sh)) | (X13 = X15))) & (! [X13 : term,X15 : term,X20 : term,X19 : term] : ((X19 = X20) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) <=> apinj2) & (hoasinduction_lem3v2a <=> ! [X10 : subst > term > subst > $o,X29 : term,X11 : term > $o] : (~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X18 : subst > term > term] : (~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X11 @ (lam @ X29)))) & pushprop & (induction2_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X17 : subst,X10 : term > $o,X16 : term] : (~! [X13 : term,X15 : term] : (~(X10 @ X13) | ~(X10 @ X15) | (X10 @ (ap @ X13 @ X15))) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | (X10 @ (sub @ X16 @ X17)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)))) => ! [X10 : term > $o,X22 : term] : (~! [X13 : term,X15 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13)) | (X10 @ X22) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)))))) & (! [X13 : term] : ((sub @ X13 @ id) = X13) <=> termmset) & (hoasinduction_lem0 <=> ! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))) & substmonoid_gthm & ((! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => ! [X26 : term > $o,X15 : term,X14 : subst,X10 : term > $o,X13 : term] : ((X26 @ (sub @ X13 @ (push @ X15 @ X14))) | ~(X10 @ X13)))) <=> pushprop_lem1_lthm) & (! [X27 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : (~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X26 : subst,X14 : subst,X25 : subst,X13 : term] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X25 : subst,X14 : subst,X13 : term,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ (lam @ X27) @ id)) <=> hoasinduction_lem3) & ((^[X8 : subst, X9 : subst > term > term] : (d2term @ d_one)) = hoaslam) & (hoasinduction_lem3v2_f <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13))) & (! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh))) <=> hoasinduction_lem3b) & hoaslaminj_gthm & pushprop_lem1_gthm & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_p_and_p_prime = (^[X13 : term, X14 : subst, X10 : term > $o, X11 : term > $o] : (! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14))))))) & pushprop_lem3v2 & (hoaslaminj <=> ! [X31 : subst > term > term,X18 : subst > term > term,X32 : subst,X33 : term] : (~! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33)) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))))) & ~(var @ (d2term @ d_one)) & (! [X10 : subst > term > subst > $o,X42 : term] : (~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aa) & axmap & apnotvar & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : subst > term > subst > $o,X42 : term] : (~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X18 : subst > term > term] : (~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id))) => ! [X27 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X13 : term,X26 : subst,X25 : subst,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ (lam @ X27) @ id) | ~! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25)))))) <=> hoasinduction_lem3_lthm) & lamnotvar & hoasinduction_lem1_gthm & hoasinduction_lem2v2_gthm & (hoasinduction_lem2 <=> ! [X24 : term,X10 : subst > term > subst > $o,X23 : term] : (~! [X25 : subst,X14 : subst,X13 : term,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ (ap @ X23 @ X24) @ id) | ~! [X14 : subst,X25 : subst,X13 : term,X26 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~(X10 @ id @ X23 @ id) | ~(X10 @ id @ X24 @ id) | ~! [X15 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id)))) & pushprop_lem2v2 & (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) <=> axshiftcons) & (! [X10 : subst > term > subst > $o,X30 : term] : ((X10 @ id @ X30 @ id) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id))) <=> hoasinduction_no_psi_cond) & hoasinduction_lem3_gthm & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (hoasinduction <=> ! [X28 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ X28 @ id) | ~! [X13 : term,X14 : subst,X25 : subst,X26 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X26 : subst,X13 : term,X14 : subst,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)))) & hoasinduction_lem1 & ! [X1 : d_term] : (d_one = X1) & hoaslamnotap_lthm & (induction <=> ! [X10 : term > $o,X21 : term] : ((X10 @ X21) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X13 : term,X15 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13)) | ~! [X13 : term] : (~(X10 @ X13) | (X10 @ (lam @ X13))))) & (apinj1 <=> ! [X15 : term,X19 : term,X20 : term,X13 : term] : ((X13 = X15) | ((ap @ X13 @ X19) != (ap @ X15 @ X20)))) & (hoasinduction_lem3v2a_lthm <=> (! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X29 : term,X10 : subst > term > subst > $o,X11 : term > $o] : (~! [X18 : subst > term > term] : (~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X29)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))))))) & (! [X22 : term,X10 : term > $o] : (~! [X13 : term,X15 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | (X10 @ X22) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))))) <=> induction2) & (! [X10 : subst > term > subst > $o,X43 : term] : (~! [X15 : term] : ((X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X10 @ id @ (lam @ X43) @ id)) <=> hoasinduction_lem3a) & pushprop_lem1v2_gthm & pushprop_lthm_orig & (! [X45 : term,X11 : term > $o,X10 : subst > term > subst > $o,X44 : term] : (~! [X13 : term,X25 : subst,X26 : subst,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~(X11 @ X44) | (X11 @ (ap @ X44 @ X45)) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~! [X25 : subst,X14 : subst,X13 : term,X26 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~(X11 @ X45)) <=> hoasinduction_lem2v2) & (hoasinduction_lthm_3 <=> (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X22 : term,X10 : term > $o] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | (X10 @ X22) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | ~! [X13 : term,X15 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X29 : term,X11 : term > $o,X10 : subst > term > subst > $o] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X29)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id))))) => ! [X10 : subst > term > subst > $o,X28 : term] : (~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X13 : term,X14 : subst,X26 : subst,X25 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X10 @ id @ X28 @ id))))))) & ~lamnotap & (axidl <=> ! [X14 : subst] : ((comp @ id @ X14) = X14)) & (! [X10 : term > $o,X17 : subst,X16 : term] : (~! [X15 : term,X13 : term] : (~(X10 @ X13) | ~(X10 @ X15) | (X10 @ (ap @ X13 @ X15))) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | (X10 @ (sub @ X16 @ X17))) <=> induction2lem) & hoasinduction_lem3b_gthm & hoasinduction_lem3a_gthm & hoasapnotvar_gthm & ((d2subst @ d_id) = id) & (! [X13 : term,X10 : term > $o,X14 : subst] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) <=> pushprop_lem0) & hoasapnotvar & ! [X6 : d_subst,X7 : d_subst] : (((d2subst @ X6) = (d2subst @ X7)) => (X6 = X7)) & pushprop_gthm & ulamvarind & (! [X10 : subst > term > subst > $o,X34 : term] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X34 @ (push @ X15 @ id)) @ id)) | ~! [X35 : term,X18 : subst > term > term] : (~! [X38 : subst,X36 : subst,X37 : term] : ((sub @ (X18 @ X36 @ X37) @ X38) = (sub @ (sub @ X35 @ (push @ X37 @ X36)) @ X38)) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X39 : subst,X41 : subst,X40 : term] : ((X18 @ (comp @ X39 @ X41) @ (sub @ X40 @ X41)) = (sub @ X35 @ (push @ (sub @ X40 @ X41) @ (comp @ X39 @ X41))))) | (X10 @ id @ (lam @ (sub @ X34 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aaa) & hoasinduction_lem1v2 & (hoasapinj2 <=> ! [X15 : term,X20 : term,X19 : term,X13 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)))) & axapp & ! [X4 : subst] : ? [X5 : d_subst] : ((d2subst @ X5) = X4) & (hoasapinj2_lthm <=> (! [X20 : term,X15 : term,X13 : term,X19 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X19 = X20)) => ! [X15 : term,X20 : term,X13 : term,X19 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))))) & induction2_gthm & (! [X15 : term,X13 : term] : (((lam @ X13) != (lam @ X15)) | (X13 = X15)) <=> laminj) & ! [X3 : d_term,X2 : d_term] : (((d2term @ X2) = (d2term @ X3)) => (X2 = X3)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (axvarcons <=> ! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13)) & hoasinduction_lem2_gthm & hoaslamnotap_gthm & ((! [X14 : subst] : ((comp @ id @ X14) = X14) & ! [X14 : subst] : ((comp @ X14 @ id) = X14)) <=> substmonoid) & hoasapnotvar_lthm & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) => ! [X43 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X10 @ id @ (lam @ X43) @ id)))) <=> hoasinduction_lem3a_lthm) & hoasinduction_lem2_lthm & (! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) <=> hoasinduction_lem3v2_f_lthm) & (pushprop_lem1v2 <=> ! [X10 : term > $o,X11 : term > $o,X13 : term,X14 : subst] : ((X11 @ one) | ~(X10 @ X13) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))))) & (one = (d2term @ d_one)) & ~ulamvar1 & (hoaslaminj_lthm <=> (! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X13 : term,X15 : term] : ((X13 = X15) | ((lam @ X13) != (lam @ X15))) => ! [X31 : subst > term > term,X32 : subst,X33 : term,X18 : subst > term > term] : (((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33)) | ~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))))))) & axassoc & (! [X14 : subst] : ((comp @ X14 @ id) = X14) <=> axidr) & ! [X5 : d_subst] : (d_id = X5) & (hoasinduction_lthm <=> (! [X22 : term,X10 : term > $o] : ((X10 @ X22) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id))))) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15))) => (! [X24 : term,X23 : term,X10 : subst > term > subst > $o] : (~! [X13 : term,X14 : subst,X26 : subst,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ (ap @ X23 @ X24) @ id) | ~(X10 @ id @ X23 @ id) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~(X10 @ id @ X24 @ id)) => (! [X27 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | (X10 @ id @ (lam @ X27) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25)))) => ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X13 : term,X15 : term] : (~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X10 @ id @ X28 @ id) | ~! [X25 : subst,X13 : term,X26 : subst,X14 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term,X14 : subst,X26 : subst,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term] : ((X10 @ id @ X13 @ id) | ~(var @ (sub @ X13 @ id)))))))) & pushprop_lem2v2_gthm & termmset_gthm & induction2lem_gthm & (! [X10 : term > $o,X13 : term,X14 : subst,X26 : term > $o,X15 : term] : (~(X10 @ X13) | (X26 @ (sub @ X13 @ (push @ X15 @ X14)))) <=> pushprop_lem1) & ((! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => ! [X10 : term > $o,X11 : term > $o,X13 : term,X14 : subst] : ((X11 @ one) | ~(X10 @ X13) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))))) <=> pushprop_lem1v2_lthm) & pushprop_lem0_gthm & hoasapinj2_gthm & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X11 : term > $o,X10 : subst > term > subst > $o,X48 : term] : (~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X11 @ (lam @ X48)) | ~! [X14 : subst,X13 : term,X25 : subst,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X15 : term] : ((X11 @ (sub @ X48 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))))) <=> hoasinduction_lem3v2_lthm) & (substmonoid_lthm <=> (! [X14 : subst] : ((comp @ id @ X14) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X14 : subst] : ((comp @ id @ X14) = X14))))) & hoasapinj1_gthm & (hoasapinj1 <=> ! [X13 : term,X15 : term,X19 : term,X20 : term] : ((X13 = X15) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (axvarid <=> ! [X13 : term] : ((sub @ X13 @ id) = X13)) & pushprop_lem3v2_lthm & (hoasinduction_lem3aa_lthm <=> ! [X10 : subst > term > subst > $o,X42 : term] : ((X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))))) & (hoaslamnotap <=> ! [X47 : term,X46 : term,X18 : subst > term > term] : (~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (ap @ (sub @ X46 @ id) @ X47)))) & (hoasapinj1_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X13 : term,X15 : term,X20 : term,X19 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X13 = X15)) => ! [X19 : term,X20 : term,X13 : term,X15 : term] : (((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)) | (X13 = X15))))) & (! [X10 : term > $o,X14 : subst,X13 : term] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) <=> pushprop_lem0_lthm) & hoaslamnotvar),
% 9.64/2.82 file('/export/starexec/sandbox2/tmp/tmp.u2UgYFbzEX/DTF2THF_17811.p',alg444_1)).
% 9.64/2.82 thf(f1112,plain,(
% 9.64/2.82 ( ! [X183 : term] : (((d2term @ (sK544 @ X183)) = X183)) )),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f2172,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (((sK591 @ (d2subst @ d_id) @ (sK595 @ X1) @ (d2subst @ d_id)) = $true) | (hoasinduction_lem3a = $false) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))) )),
% 9.64/2.82 inference(subsumption_resolution,[status(thm)],[f1696,f1892])).
% 9.64/2.82 thf(f1892,plain,(
% 9.64/2.82 ( ! [X0 : term,X1 : term] : ((X0 = X1)) )),
% 9.64/2.82 inference(superposition,[status(thm)],[f1886,f1886])).
% 9.64/2.82 thf(f1696,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (((sK591 @ (d2subst @ d_id) @ (sK595 @ X1) @ (d2subst @ d_id)) = $true) | (hoasinduction_lem3a = $false) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | ((X1 @ (comp @ (sK593 @ X1) @ (sK592 @ X1)) @ (sub @ (sK594 @ X1) @ (sK592 @ X1))) != (sub @ (X1 @ (sK593 @ X1) @ (sK594 @ X1)) @ (sK592 @ X1)))) )),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1694])).
% 9.64/2.82 thf(f1694,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : ((hoasinduction_lem3a = $false) | ((X1 @ (comp @ (sK593 @ X1) @ (sK592 @ X1)) @ (sub @ (sK594 @ X1) @ (sK592 @ X1))) != (sub @ (X1 @ (sK593 @ X1) @ (sK594 @ X1)) @ (sK592 @ X1))) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | ($false = ((sK591 @ (d2subst @ d_id) @ (sK595 @ X1) @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ (sK595 @ X1)) @ (d2subst @ d_id))))) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1693])).
% 9.64/2.82 thf(f1693,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | (hoasinduction_lem3a = $false) | (((^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id)))) @ (sK595 @ X1)) = $false) | ((X1 @ (comp @ (sK593 @ X1) @ (sK592 @ X1)) @ (sub @ (sK594 @ X1) @ (sK592 @ X1))) != (sub @ (X1 @ (sK593 @ X1) @ (sK594 @ X1)) @ (sK592 @ X1)))) )),
% 9.64/2.82 inference(sigma_clausification,[status(thm)],[f1692])).
% 9.64/2.82 thf(f1692,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | (hoasinduction_lem3a = $false) | ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) = $false) | ((X1 @ (comp @ (sK593 @ X1) @ (sK592 @ X1)) @ (sub @ (sK594 @ X1) @ (sK592 @ X1))) != (sub @ (X1 @ (sK593 @ X1) @ (sK594 @ X1)) @ (sK592 @ X1)))) )),
% 9.64/2.82 inference(equality_proxy_clausification,[status(thm)],[f1691])).
% 9.64/2.82 thf(f1691,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | ($false = ((sub @ (X1 @ (sK593 @ X1) @ (sK594 @ X1)) @ (sK592 @ X1)) = (X1 @ (comp @ (sK593 @ X1) @ (sK592 @ X1)) @ (sub @ (sK594 @ X1) @ (sK592 @ X1))))) | ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) = $false) | (hoasinduction_lem3a = $false)) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1690])).
% 9.64/2.82 thf(f1690,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) = $false) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | (((^[Y0 : term]: ((sub @ (X1 @ (sK593 @ X1) @ Y0) @ (sK592 @ X1)) = (X1 @ (comp @ (sK593 @ X1) @ (sK592 @ X1)) @ (sub @ Y0 @ (sK592 @ X1))))) @ (sK594 @ X1)) = $false) | (hoasinduction_lem3a = $false)) )),
% 9.64/2.82 inference(sigma_clausification,[status(thm)],[f1689])).
% 9.64/2.82 thf(f1689,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (((!! @ term @ (^[Y0 : term]: ((sub @ (X1 @ (sK593 @ X1) @ Y0) @ (sK592 @ X1)) = (X1 @ (comp @ (sK593 @ X1) @ (sK592 @ X1)) @ (sub @ Y0 @ (sK592 @ X1)))))) = $false) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) = $false) | (hoasinduction_lem3a = $false)) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1688])).
% 9.64/2.82 thf(f1688,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (($false = ((^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sub @ (X1 @ Y0 @ Y1) @ (sK592 @ X1)) = (X1 @ (comp @ Y0 @ (sK592 @ X1)) @ (sub @ Y1 @ (sK592 @ X1))))))) @ (sK593 @ X1))) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) = $false) | (hoasinduction_lem3a = $false)) )),
% 9.64/2.82 inference(sigma_clausification,[status(thm)],[f1687])).
% 9.64/2.82 thf(f1687,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sub @ (X1 @ Y0 @ Y1) @ (sK592 @ X1)) = (X1 @ (comp @ Y0 @ (sK592 @ X1)) @ (sub @ Y1 @ (sK592 @ X1)))))))) = $false) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) | ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) = $false) | (hoasinduction_lem3a = $false)) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1686])).
% 9.64/2.82 thf(f1686,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (($false = ((^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (X1 @ Y1 @ Y2) @ Y0) = (X1 @ (comp @ Y1 @ Y0) @ (sub @ Y2 @ Y0)))))))) @ (sK592 @ X1))) | (hoasinduction_lem3a = $false) | ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) = $false) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))) )),
% 9.64/2.82 inference(sigma_clausification,[status(thm)],[f1685])).
% 9.64/2.82 thf(f1685,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (X1 @ Y1 @ Y2) @ Y0) = (X1 @ (comp @ Y1 @ Y0) @ (sub @ Y2 @ Y0))))))))) = $false) | (hoasinduction_lem3a = $false) | ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) = $false) | ($true = (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))) )),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1684])).
% 9.64/2.82 thf(f1684,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (($true = ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))) | (hoasinduction_lem3a = $false) | ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (X1 @ Y1 @ Y2) @ Y0) = (X1 @ (comp @ Y1 @ Y0) @ (sub @ Y2 @ Y0))))))))) = $false)) )),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1683])).
% 9.64/2.82 thf(f1683,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : (($true = ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (X1 @ Y1 @ Y2) @ Y0) = (X1 @ (comp @ Y1 @ Y0) @ (sub @ Y2 @ Y0))))))))) => ((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))))) | (hoasinduction_lem3a = $false)) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1682])).
% 9.64/2.82 thf(f1682,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > term] : ((hoasinduction_lem3a = $false) | ($true = ((^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => ((!! @ term @ (^[Y1 : term]: ((sK591 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (Y0 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))))) @ X1))) )),
% 9.64/2.82 inference(pi_clausification,[status(thm)],[f1681])).
% 9.64/2.82 thf(f1681,plain,(
% 9.64/2.82 ($true = (!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => ((!! @ term @ (^[Y1 : term]: ((sK591 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (Y0 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))))))) | (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1679])).
% 9.64/2.82 thf(f1679,plain,(
% 9.64/2.82 (hoasinduction_lem3a = $false) | (((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => ((!! @ term @ (^[Y1 : term]: ((sK591 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (Y0 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))))) => (!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK591 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (sub @ Y0 @ (push @ Y1 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (lam @ Y0) @ (d2subst @ d_id)))))) = $false)),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1678])).
% 9.64/2.82 thf(f1678,plain,(
% 9.64/2.82 (hoasinduction_lem3a = $false) | (((^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (Y1 @ (d2subst @ d_id) @ Y2) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (sub @ Y1 @ (push @ Y2 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (lam @ Y1) @ (d2subst @ d_id))))))) @ sK591) = $false)),
% 9.64/2.82 inference(sigma_clausification,[status(thm)],[f1677])).
% 9.64/2.82 thf(f1677,plain,(
% 9.64/2.82 (hoasinduction_lem3a = $false) | ($false = (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (Y1 @ (d2subst @ d_id) @ Y2) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (sub @ Y1 @ (push @ Y2 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (lam @ Y1) @ (d2subst @ d_id)))))))))),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1675])).
% 9.64/2.82 thf(f1675,plain,(
% 9.64/2.82 (hoasinduction_lem3a != (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (Y1 @ (d2subst @ d_id) @ Y2) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (sub @ Y1 @ (push @ Y2 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (lam @ Y1) @ (d2subst @ d_id)))))))))),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1660])).
% 9.64/2.82 thf(f1660,plain,(
% 9.64/2.82 (hoasinduction_lem3a != (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (Y1 @ (d2subst @ d_id) @ Y2) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ ((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (d2term @ d_one)))) @ (d2subst @ d_id) @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ (d2subst @ d_id)))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (sub @ Y1 @ (push @ Y2 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (lam @ Y1) @ (d2subst @ d_id)))))))))),
% 9.64/2.82 inference(definition_unfolding,[status(thm)],[f1260,f1137,f1137,f1137,f1137,f1137,f1137,f1078,f1137,f1137,f1137,f1137,f1137,f1137,f1137,f1137,f1137])).
% 9.64/2.82 thf(f1078,plain,(
% 9.64/2.82 (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one)))))),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f1137,plain,(
% 9.64/2.82 ((d2subst @ d_id) = id)),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f1260,plain,(
% 9.64/2.82 (hoasinduction_lem3a != (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ Y1) @ id))))))))),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f9])).
% 9.64/2.82 thf(f9,plain,(
% 9.64/2.82 (hoasinduction_lem3a != (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ Y1) @ id))))))))),
% 9.64/2.82 inference(flattening,[status(thm)],[f6])).
% 9.64/2.82 thf(f6,plain,(
% 9.64/2.82 ~(hoasinduction_lem3a = (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ Y1) @ id))))))))),
% 9.64/2.82 inference(fool_elimination,[status(thm)],[f5])).
% 9.64/2.82 thf(f5,plain,(
% 9.64/2.82 ~(hoasinduction_lem3a = ! [X0 : subst > term > subst > $o] : (! [X1 : subst > term > term] : (! [X2 : term,X3 : subst,X4 : subst] : ((sub @ (X1 @ X3 @ X2) @ X4) = (X1 @ (comp @ X3 @ X4) @ (sub @ X2 @ X4))) => (! [X5 : term] : ((X0 @ id @ X5 @ id) => (X0 @ id @ (X1 @ id @ X5) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X6 : subst, X7 : term] : (X1 @ X6 @ X7))) @ id))) => ! [X8 : term] : (! [X9 : term] : ((X0 @ id @ X9 @ id) => (X0 @ id @ (sub @ X8 @ (push @ X9 @ id)) @ id)) => (X0 @ id @ (lam @ X8) @ id))))),
% 9.64/2.82 inference(rectify,[status(thm)],[f3])).
% 9.64/2.82 thf(f3,negated_conjecture,(
% 9.64/2.82 ~(hoasinduction_lem3a = ! [X10 : subst > term > subst > $o] : (! [X18 : subst > term > term] : (! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) => (! [X13 : term] : ((X10 @ id @ X13 @ id) => (X10 @ id @ (X18 @ id @ X13) @ id)) => (X10 @ id @ (hoaslam @ id @ (^[X14 : subst, X13 : term] : (X18 @ X14 @ X13))) @ id))) => ! [X13 : term] : (! [X15 : term] : ((X10 @ id @ X15 @ id) => (X10 @ id @ (sub @ X13 @ (push @ X15 @ id)) @ id)) => (X10 @ id @ (lam @ X13) @ id))))),
% 9.64/2.82 inference(negated_conjecture,[status(cth)],[f2])).
% 9.64/2.82 thf(f2,conjecture,(
% 9.64/2.82 (hoasinduction_lem3a = ! [X10 : subst > term > subst > $o] : (! [X18 : subst > term > term] : (! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) => (! [X13 : term] : ((X10 @ id @ X13 @ id) => (X10 @ id @ (X18 @ id @ X13) @ id)) => (X10 @ id @ (hoaslam @ id @ (^[X14 : subst, X13 : term] : (X18 @ X14 @ X13))) @ id))) => ! [X13 : term] : (! [X15 : term] : ((X10 @ id @ X15 @ id) => (X10 @ id @ (sub @ X13 @ (push @ X15 @ id)) @ id)) => (X10 @ id @ (lam @ X13) @ id))))),
% 9.64/2.82 file('/export/starexec/sandbox2/tmp/tmp.u2UgYFbzEX/DTF2THF_17811.p',hoasinduction_lem3a)).
% 9.64/2.82 thf(f1968,plain,(
% 9.64/2.82 ( ! [X0 : term,X1 : subst] : (((sK591 @ X1 @ X0 @ X1) = $false) | (hoasinduction_lem3a = $false)) )),
% 9.64/2.82 inference(superposition,[status(thm)],[f1963,f1892])).
% 9.64/2.82 thf(f1963,plain,(
% 9.64/2.82 ( ! [X0 : subst] : (($false = (sK591 @ X0 @ (lam @ sK596) @ X0)) | (hoasinduction_lem3a = $false)) )),
% 9.64/2.82 inference(superposition,[status(thm)],[f1699,f1894])).
% 9.64/2.82 thf(f1894,plain,(
% 9.64/2.82 ( ! [X18 : subst] : (((d2subst @ d_id) = X18)) )),
% 9.64/2.82 inference(forward_demodulation,[status(thm)],[f1250,f1170])).
% 9.64/2.82 thf(f1170,plain,(
% 9.64/2.82 ( ! [X122 : d_subst] : ((d_id = X122)) )),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f1250,plain,(
% 9.64/2.82 ( ! [X18 : subst] : (((d2subst @ (sK463 @ X18)) = X18)) )),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f1699,plain,(
% 9.64/2.82 ((sK591 @ (d2subst @ d_id) @ (lam @ sK596) @ (d2subst @ d_id)) = $false) | (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1698])).
% 9.64/2.82 thf(f1698,plain,(
% 9.64/2.82 (((!! @ term @ (^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (sub @ sK596 @ (push @ Y0 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (lam @ sK596) @ (d2subst @ d_id))) = $false) | (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1697])).
% 9.64/2.82 thf(f1697,plain,(
% 9.64/2.82 (((^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK591 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (sub @ Y0 @ (push @ Y1 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (lam @ Y0) @ (d2subst @ d_id)))) @ sK596) = $false) | (hoasinduction_lem3a = $false)),
% 9.64/2.82 inference(sigma_clausification,[status(thm)],[f1680])).
% 9.64/2.82 thf(f1680,plain,(
% 9.64/2.82 (hoasinduction_lem3a = $false) | ((!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK591 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (sub @ Y0 @ (push @ Y1 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (sK591 @ (d2subst @ d_id) @ (lam @ Y0) @ (d2subst @ d_id))))) = $false)),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1679])).
% 9.64/2.82 thf(f2107,plain,(
% 9.64/2.82 ( ! [X2 : subst,X3 : term,X0 : term,X1 : subst > term > subst > $o] : (($true = (X1 @ X2 @ X3 @ X2)) | ((X1 @ X2 @ X0 @ X2) = $false) | (hoasinduction_lem3a = $true)) )),
% 9.64/2.82 inference(superposition,[status(thm)],[f2084,f1886])).
% 9.64/2.82 thf(f2084,plain,(
% 9.64/2.82 ( ! [X2 : subst > term > subst > $o,X3 : subst,X1 : term] : (((X2 @ X3 @ (d2term @ d_one) @ X3) = $false) | ($true = (X2 @ X3 @ X1 @ X3)) | (hoasinduction_lem3a = $true)) )),
% 9.64/2.82 inference(superposition,[status(thm)],[f2065,f1892])).
% 9.64/2.82 thf(f2065,plain,(
% 9.64/2.82 ( ! [X2 : term,X0 : subst,X1 : subst > term > subst > $o] : (($true = (X1 @ X0 @ (lam @ X2) @ X0)) | (hoasinduction_lem3a = $true) | ((X1 @ X0 @ (d2term @ d_one) @ X0) = $false)) )),
% 9.64/2.82 inference(superposition,[status(thm)],[f2058,f1894])).
% 9.64/2.82 thf(f2058,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (($true = (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id))) | ((X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $false) | (hoasinduction_lem3a = $true)) )),
% 9.64/2.82 inference(duplicate_literal_removal,[status(thm)],[f2057])).
% 9.64/2.82 thf(f2057,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (($true = (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id))) | (hoasinduction_lem3a = $true) | ((X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $false) | ((X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $false)) )),
% 9.64/2.82 inference(forward_demodulation,[status(thm)],[f1737,f1886])).
% 9.64/2.82 thf(f1737,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (((X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ (sK600 @ X1 @ X2) @ (d2subst @ d_id))) @ (d2subst @ d_id)) = $false) | (hoasinduction_lem3a = $true) | ($true = (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id))) | ((X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $false)) )),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1736])).
% 9.64/2.82 thf(f1736,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (((X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $false) | (hoasinduction_lem3a = $true) | ($true = (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id))) | ($false = ((X1 @ (d2subst @ d_id) @ (sK600 @ X1 @ X2) @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ (sK600 @ X1 @ X2) @ (d2subst @ d_id))) @ (d2subst @ d_id))))) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1735])).
% 9.64/2.82 thf(f1735,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (($false = ((^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ Y0 @ (d2subst @ d_id))) @ (d2subst @ d_id)))) @ (sK600 @ X1 @ X2))) | ((X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $false) | (hoasinduction_lem3a = $true) | ($true = (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id)))) )),
% 9.64/2.82 inference(sigma_clausification,[status(thm)],[f1726])).
% 9.64/2.82 thf(f1726,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (((X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $false) | ($true = (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id))) | (hoasinduction_lem3a = $true) | ($false = (!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ Y0 @ (d2subst @ d_id))) @ (d2subst @ d_id))))))) )),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1725])).
% 9.64/2.82 thf(f1725,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (($true = (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id))) | (hoasinduction_lem3a = $true) | ($false = ((!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sK597 @ X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))) | ($false = (!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ Y0 @ (d2subst @ d_id))) @ (d2subst @ d_id))))))) )),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1711])).
% 9.64/2.82 thf(f1711,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : ((hoasinduction_lem3a = $true) | ($true = ((!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ Y0 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id)))) | ($false = ((!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sK597 @ X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))))) )),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1710])).
% 9.64/2.82 thf(f1710,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : ((((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (sK597 @ X1 @ Y1 @ Y2) @ Y0) = (sK597 @ X1 @ (comp @ Y1 @ Y0) @ (sub @ Y2 @ Y0))))))))) => ((!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sK597 @ X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))) = $false) | (hoasinduction_lem3a = $true) | ($true = ((!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ Y0 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id))))) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1709])).
% 9.64/2.82 thf(f1709,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (($true = ((!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ Y0 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id)))) | ($false = ((^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => ((!! @ term @ (^[Y1 : term]: ((X1 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (Y0 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))))) @ (sK597 @ X1))) | (hoasinduction_lem3a = $true)) )),
% 9.64/2.82 inference(sigma_clausification,[status(thm)],[f1708])).
% 9.64/2.82 thf(f1708,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : (($false = (!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => ((!! @ term @ (^[Y1 : term]: ((X1 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (Y0 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))))))) | ($true = ((!! @ term @ (^[Y0 : term]: ((X1 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ X2 @ (push @ Y0 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (lam @ X2) @ (d2subst @ d_id)))) | (hoasinduction_lem3a = $true)) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1707])).
% 9.64/2.82 thf(f1707,plain,(
% 9.64/2.82 ( ! [X2 : term,X1 : subst > term > subst > $o] : ((hoasinduction_lem3a = $true) | ($false = (!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => ((!! @ term @ (^[Y1 : term]: ((X1 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (Y0 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))))))) | ($true = ((^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((X1 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ Y0 @ (push @ Y1 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (lam @ Y0) @ (d2subst @ d_id)))) @ X2))) )),
% 9.64/2.82 inference(pi_clausification,[status(thm)],[f1706])).
% 9.64/2.82 thf(f1706,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > subst > $o] : (((!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((X1 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ Y0 @ (push @ Y1 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (lam @ Y0) @ (d2subst @ d_id))))) = $true) | (hoasinduction_lem3a = $true) | ($false = (!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => ((!! @ term @ (^[Y1 : term]: ((X1 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (Y0 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))))))) )),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1705])).
% 9.64/2.82 thf(f1705,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > subst > $o] : ((hoasinduction_lem3a = $true) | ($true = ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => ((!! @ term @ (^[Y1 : term]: ((X1 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (Y0 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))))) => (!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((X1 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (X1 @ (d2subst @ d_id) @ (sub @ Y0 @ (push @ Y1 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (X1 @ (d2subst @ d_id) @ (lam @ Y0) @ (d2subst @ d_id)))))))) )),
% 9.64/2.82 inference(beta_eta_normalization,[status(thm)],[f1704])).
% 9.64/2.82 thf(f1704,plain,(
% 9.64/2.82 ( ! [X1 : subst > term > subst > $o] : ((hoasinduction_lem3a = $true) | ($true = ((^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (Y1 @ (d2subst @ d_id) @ Y2) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (sub @ Y1 @ (push @ Y2 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (lam @ Y1) @ (d2subst @ d_id))))))) @ X1))) )),
% 9.64/2.82 inference(pi_clausification,[status(thm)],[f1676])).
% 9.64/2.82 thf(f1676,plain,(
% 9.64/2.82 ($true = (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((sub @ (Y1 @ Y3 @ Y4) @ Y2) = (Y1 @ (comp @ Y3 @ Y2) @ (sub @ Y4 @ Y2))))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (Y1 @ (d2subst @ d_id) @ Y2) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (sub @ Y1 @ (push @ Y2 @ (d2subst @ d_id))) @ (d2subst @ d_id))))) => (Y0 @ (d2subst @ d_id) @ (lam @ Y1) @ (d2subst @ d_id))))))))) | (hoasinduction_lem3a = $true)),
% 9.64/2.82 inference(binary_proxy_clausification,[status(thm)],[f1675])).
% 9.64/2.82 thf(f2535,plain,(
% 9.64/2.82 ( ! [X0 : subst] : (((sK283 @ X0 @ (lam @ sK284) @ X0) != $true)) )),
% 9.64/2.82 inference(superposition,[status(thm)],[f2527,f1894])).
% 9.64/2.82 thf(f2527,plain,(
% 9.64/2.82 ($true != (sK283 @ (d2subst @ d_id) @ (lam @ sK284) @ (d2subst @ d_id)))),
% 9.64/2.82 inference(subsumption_resolution,[status(thm)],[f1437,f2209])).
% 9.64/2.82 thf(f2209,plain,(
% 9.64/2.82 (sP24 != $true)),
% 9.64/2.82 inference(trivial_inequality_removal,[status(thm)],[f2199])).
% 9.64/2.82 thf(f2199,plain,(
% 9.64/2.82 (sP24 != $true) | ($true = $false)),
% 9.64/2.82 inference(backward_demodulation,[status(thm)],[f1173,f2194])).
% 9.64/2.82 thf(f1173,plain,(
% 9.64/2.82 (sP24 != $true) | (hoasinduction_lem3a = $true)),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f1437,plain,(
% 9.64/2.82 (sP24 = $true) | ($true != (sK283 @ (d2subst @ d_id) @ (lam @ sK284) @ (d2subst @ d_id)))),
% 9.64/2.82 inference(definition_unfolding,[status(thm)],[f829,f1137,f1137])).
% 9.64/2.82 thf(f829,plain,(
% 9.64/2.82 (sP24 = $true) | ((sK283 @ id @ (lam @ sK284) @ id) != $true)),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f309])).
% 9.64/2.82 thf(f309,plain,(
% 9.64/2.82 ((sP24 = $true) | (! [X2 : term] : (((sK283 @ id @ (sub @ sK284 @ (push @ X2 @ id)) @ id) = $true) | ($true != (sK283 @ id @ X2 @ id))) & ((sK283 @ id @ (lam @ sK284) @ id) != $true) & ! [X3 : subst > term > term] : ((((sK283 @ id @ (X3 @ id @ (sK285 @ X3)) @ id) != $true) & ((sK283 @ id @ (sK285 @ X3) @ id) = $true)) | ((X3 @ (comp @ (sK287 @ X3) @ (sK286 @ X3)) @ (sub @ (sK288 @ X3) @ (sK286 @ X3))) != (sub @ (X3 @ (sK287 @ X3) @ (sK288 @ X3)) @ (sK286 @ X3))) | ((sK283 @ id @ (lam @ (X3 @ sh @ one)) @ id) = $true)))) & (! [X8 : subst > term > subst > $o,X9 : term] : ((($true != (X8 @ id @ (sub @ X9 @ (push @ (sK289 @ X9 @ X8) @ id)) @ id)) & ((X8 @ id @ (sK289 @ X9 @ X8) @ id) = $true)) | ((X8 @ id @ (lam @ X9) @ id) = $true) | (! [X12 : term] : (($true = (X8 @ id @ (sK290 @ X8 @ id @ X12) @ id)) | ($true != (X8 @ id @ X12 @ id))) & ! [X13 : subst,X14 : subst,X15 : term] : ((sK290 @ X8 @ (comp @ X14 @ X13) @ (sub @ X15 @ X13)) = (sub @ (sK290 @ X8 @ X14 @ X15) @ X13)) & ($true != (X8 @ id @ (lam @ (sK290 @ X8 @ sh @ one)) @ id)))) | (sP24 != $true))),
% 9.64/2.82 inference(skolemisation,[status(esa),new_symbols(skolem,[sK283,sK284,sK285,sK286,sK287,sK288,sK289,sK290])],[f303,f308,f307,f306,f305,f304])).
% 9.64/2.82 thf(f304,plain,(
% 9.64/2.82 ? [X0 : subst > term > subst > $o,X1 : term] : (! [X2 : term] : (($true = (X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)) | ($true != (X0 @ id @ X2 @ id))) & ($true != (X0 @ id @ (lam @ X1) @ id)) & ! [X3 : subst > term > term] : (? [X4 : term] : (($true != (X0 @ id @ (X3 @ id @ X4) @ id)) & ((X0 @ id @ X4 @ id) = $true)) | ? [X5 : subst,X6 : subst,X7 : term] : ((sub @ (X3 @ X6 @ X7) @ X5) != (X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) | ($true = (X0 @ id @ (lam @ (X3 @ sh @ one)) @ id)))) => (! [X2 : term] : (((sK283 @ id @ (sub @ sK284 @ (push @ X2 @ id)) @ id) = $true) | ($true != (sK283 @ id @ X2 @ id))) & ((sK283 @ id @ (lam @ sK284) @ id) != $true) & ! [X3 : subst > term > term] : (? [X4 : term] : (($true != (sK283 @ id @ (X3 @ id @ X4) @ id)) & ((sK283 @ id @ X4 @ id) = $true)) | ? [X5 : subst,X6 : subst,X7 : term] : ((sub @ (X3 @ X6 @ X7) @ X5) != (X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) | ((sK283 @ id @ (lam @ (X3 @ sh @ one)) @ id) = $true)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f305,plain,(
% 9.64/2.82 ! [X3 : subst > term > term] : (? [X4 : term] : (($true != (sK283 @ id @ (X3 @ id @ X4) @ id)) & ((sK283 @ id @ X4 @ id) = $true)) => (((sK283 @ id @ (X3 @ id @ (sK285 @ X3)) @ id) != $true) & ((sK283 @ id @ (sK285 @ X3) @ id) = $true)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f306,plain,(
% 9.64/2.82 ! [X3 : subst > term > term] : (? [X5 : subst,X6 : subst,X7 : term] : ((sub @ (X3 @ X6 @ X7) @ X5) != (X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) => ((X3 @ (comp @ (sK287 @ X3) @ (sK286 @ X3)) @ (sub @ (sK288 @ X3) @ (sK286 @ X3))) != (sub @ (X3 @ (sK287 @ X3) @ (sK288 @ X3)) @ (sK286 @ X3))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f307,plain,(
% 9.64/2.82 ! [X8 : subst > term > subst > $o,X9 : term] : (? [X10 : term] : (((X8 @ id @ (sub @ X9 @ (push @ X10 @ id)) @ id) != $true) & ($true = (X8 @ id @ X10 @ id))) => (($true != (X8 @ id @ (sub @ X9 @ (push @ (sK289 @ X9 @ X8) @ id)) @ id)) & ((X8 @ id @ (sK289 @ X9 @ X8) @ id) = $true)))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f308,plain,(
% 9.64/2.82 ! [X8 : subst > term > subst > $o] : (? [X11 : subst > term > term] : (! [X12 : term] : (($true = (X8 @ id @ (X11 @ id @ X12) @ id)) | ($true != (X8 @ id @ X12 @ id))) & ! [X13 : subst,X14 : subst,X15 : term] : ((sub @ (X11 @ X14 @ X15) @ X13) = (X11 @ (comp @ X14 @ X13) @ (sub @ X15 @ X13))) & ($true != (X8 @ id @ (lam @ (X11 @ sh @ one)) @ id))) => (! [X12 : term] : (($true = (X8 @ id @ (sK290 @ X8 @ id @ X12) @ id)) | ($true != (X8 @ id @ X12 @ id))) & ! [X15 : term,X14 : subst,X13 : subst] : ((sK290 @ X8 @ (comp @ X14 @ X13) @ (sub @ X15 @ X13)) = (sub @ (sK290 @ X8 @ X14 @ X15) @ X13)) & ($true != (X8 @ id @ (lam @ (sK290 @ X8 @ sh @ one)) @ id))))),
% 9.64/2.82 introduced(definition,[],[choice_axiom])).
% 9.64/2.82 thf(f303,plain,(
% 9.64/2.82 ((sP24 = $true) | ? [X0 : subst > term > subst > $o,X1 : term] : (! [X2 : term] : (($true = (X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)) | ($true != (X0 @ id @ X2 @ id))) & ($true != (X0 @ id @ (lam @ X1) @ id)) & ! [X3 : subst > term > term] : (? [X4 : term] : (($true != (X0 @ id @ (X3 @ id @ X4) @ id)) & ((X0 @ id @ X4 @ id) = $true)) | ? [X5 : subst,X6 : subst,X7 : term] : ((sub @ (X3 @ X6 @ X7) @ X5) != (X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) | ($true = (X0 @ id @ (lam @ (X3 @ sh @ one)) @ id))))) & (! [X8 : subst > term > subst > $o,X9 : term] : (? [X10 : term] : (((X8 @ id @ (sub @ X9 @ (push @ X10 @ id)) @ id) != $true) & ($true = (X8 @ id @ X10 @ id))) | ((X8 @ id @ (lam @ X9) @ id) = $true) | ? [X11 : subst > term > term] : (! [X12 : term] : (($true = (X8 @ id @ (X11 @ id @ X12) @ id)) | ($true != (X8 @ id @ X12 @ id))) & ! [X13 : subst,X14 : subst,X15 : term] : ((sub @ (X11 @ X14 @ X15) @ X13) = (X11 @ (comp @ X14 @ X13) @ (sub @ X15 @ X13))) & ($true != (X8 @ id @ (lam @ (X11 @ sh @ one)) @ id)))) | (sP24 != $true))),
% 9.64/2.82 inference(rectify,[status(thm)],[f302])).
% 9.64/2.82 thf(f302,plain,(
% 9.64/2.82 ((sP24 = $true) | ? [X262 : subst > term > subst > $o,X263 : term] : (! [X264 : term] : (((X262 @ id @ (sub @ X263 @ (push @ X264 @ id)) @ id) = $true) | ((X262 @ id @ X264 @ id) != $true)) & ((X262 @ id @ (lam @ X263) @ id) != $true) & ! [X265 : subst > term > term] : (? [X266 : term] : (($true != (X262 @ id @ (X265 @ id @ X266) @ id)) & ((X262 @ id @ X266 @ id) = $true)) | ? [X267 : subst,X268 : subst,X269 : term] : ((sub @ (X265 @ X268 @ X269) @ X267) != (X265 @ (comp @ X268 @ X267) @ (sub @ X269 @ X267))) | ($true = (X262 @ id @ (lam @ (X265 @ sh @ one)) @ id))))) & (! [X262 : subst > term > subst > $o,X263 : term] : (? [X264 : term] : (((X262 @ id @ (sub @ X263 @ (push @ X264 @ id)) @ id) != $true) & ((X262 @ id @ X264 @ id) = $true)) | ((X262 @ id @ (lam @ X263) @ id) = $true) | ? [X265 : subst > term > term] : (! [X266 : term] : (($true = (X262 @ id @ (X265 @ id @ X266) @ id)) | ((X262 @ id @ X266 @ id) != $true)) & ! [X267 : subst,X268 : subst,X269 : term] : ((sub @ (X265 @ X268 @ X269) @ X267) = (X265 @ (comp @ X268 @ X267) @ (sub @ X269 @ X267))) & ($true != (X262 @ id @ (lam @ (X265 @ sh @ one)) @ id)))) | (sP24 != $true))),
% 9.64/2.82 inference(nnf_transformation,[status(thm)],[f38])).
% 9.64/2.82 thf(f5651,plain,(
% 9.64/2.82 ( ! [X3 : subst > term > term] : (($true = (sK283 @ (d2subst @ d_id) @ (lam @ (X3 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id)))) )),
% 9.64/2.82 inference(trivial_inequality_removal,[status(thm)],[f5650])).
% 9.64/2.82 thf(f5650,plain,(
% 9.64/2.82 ( ! [X3 : subst > term > term] : (($true = (sK283 @ (d2subst @ d_id) @ (lam @ (X3 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id))) | ($true = $false)) )),
% 9.64/2.82 inference(forward_demodulation,[status(thm)],[f5649,f2545])).
% 9.64/2.82 thf(f5649,plain,(
% 9.64/2.82 ( ! [X3 : subst > term > term] : (($true = (sK283 @ (d2subst @ d_id) @ (sK285 @ X3) @ (d2subst @ d_id))) | ($true = (sK283 @ (d2subst @ d_id) @ (lam @ (X3 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id)))) )),
% 9.64/2.82 inference(subsumption_resolution,[status(thm)],[f5648,f1892])).
% 9.64/2.82 thf(f5648,plain,(
% 9.64/2.82 ( ! [X3 : subst > term > term] : (($true = (sK283 @ (d2subst @ d_id) @ (sK285 @ X3) @ (d2subst @ d_id))) | ($true = (sK283 @ (d2subst @ d_id) @ (lam @ (X3 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id))) | ((X3 @ (comp @ (sK287 @ X3) @ (sK286 @ X3)) @ (sub @ (sK288 @ X3) @ (sK286 @ X3))) != (sub @ (X3 @ (sK287 @ X3) @ (sK288 @ X3)) @ (sK286 @ X3)))) )),
% 9.64/2.82 inference(subsumption_resolution,[status(thm)],[f1439,f2209])).
% 9.64/2.82 thf(f1439,plain,(
% 9.64/2.82 ( ! [X3 : subst > term > term] : ((sP24 = $true) | ((X3 @ (comp @ (sK287 @ X3) @ (sK286 @ X3)) @ (sub @ (sK288 @ X3) @ (sK286 @ X3))) != (sub @ (X3 @ (sK287 @ X3) @ (sK288 @ X3)) @ (sK286 @ X3))) | ($true = (sK283 @ (d2subst @ d_id) @ (lam @ (X3 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id))) | ($true = (sK283 @ (d2subst @ d_id) @ (sK285 @ X3) @ (d2subst @ d_id)))) )),
% 9.64/2.82 inference(definition_unfolding,[status(thm)],[f827,f1137,f1137,f1137,f1053,f1074,f1137])).
% 9.64/2.82 thf(f1074,plain,(
% 9.64/2.82 (one = (d2term @ d_one))),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f1053,plain,(
% 9.64/2.82 ((d2subst @ d_id) = sh)),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f545])).
% 9.64/2.82 thf(f827,plain,(
% 9.64/2.82 ( ! [X3 : subst > term > term] : ((sP24 = $true) | ((sK283 @ id @ (sK285 @ X3) @ id) = $true) | ((X3 @ (comp @ (sK287 @ X3) @ (sK286 @ X3)) @ (sub @ (sK288 @ X3) @ (sK286 @ X3))) != (sub @ (X3 @ (sK287 @ X3) @ (sK288 @ X3)) @ (sK286 @ X3))) | ((sK283 @ id @ (lam @ (X3 @ sh @ one)) @ id) = $true)) )),
% 9.64/2.82 inference(cnf_transformation,[status(thm)],[f309])).
% 9.64/2.82 % SZS output end Proof for DTF2THF_17811
% 9.64/2.82 % (18009)------------------------------
% 9.64/2.82 % (18009)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.64/2.82 % (18009)Termination reason: Refutation
% 9.64/2.82
% 9.64/2.82 % (18009)Memory used [KB]: 7675
% 9.64/2.82 % (18009)Time elapsed: 0.448 s
% 9.64/2.82 % (18009)Instructions burned: 1359 (million)
% 9.64/2.82 % (18009)------------------------------
% 9.64/2.82 % (18009)------------------------------
% 9.64/2.85 % (17909)Success in time 1.329 s
%------------------------------------------------------------------------------