%------------------------------------------------------------------------------
% 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 : n026.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 06:58:53 PM UTC 2026
% Result : Theorem 9.05s 2.74s
% Output : Refutation 9.05s
% Verified :
% SZS Type : Refutation
% Derivation depth : 46
% Number of leaves : 139
% Syntax : Number of formulae : 251 ( 28 unt; 0 typ; 0 def)
% Number of atoms : 11987 (5679 equ; 0 cnn)
% Maximal formula atoms : 530 ( 47 avg)
% Number of connectives : 29110 (3173 ~;2931 |;2483 &;19517 @)
% ( 507 <=>; 369 =>; 0 <=; 0 <~>)
% Maximal formula depth : 188 ( 14 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 2247 (2247 >; 0 *; 0 +; 0 <<)
% Number of symbols : 732 ( 728 usr; 293 con; 0-4 aty)
% ( 130 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 6052 ( 251 ^;4400 !;1401 ?;6052 :)
% 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_142,type,
sP0: ( subst > term > subst > $o ) > $o ).
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: ( term > $o ) > $o ).
thf(func_def_149,type,
sP7: ( term > $o ) > $o ).
thf(func_def_152,type,
sP10: ( term > $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_160,type,
sP18: ( subst > term > subst > $o ) > $o ).
thf(func_def_164,type,
sP22: ( subst > term > subst > $o ) > $o ).
thf(func_def_165,type,
sP23: ( term > $o ) > $o ).
thf(func_def_166,type,
sP24: ( subst > term > subst > $o ) > $o ).
thf(func_def_167,type,
sP25: ( subst > term > subst > $o ) > $o ).
thf(func_def_168,type,
sP26: ( subst > term > subst > $o ) > $o ).
thf(func_def_174,type,
sP32: ( term > $o ) > $o ).
thf(func_def_180,type,
sP38: ( term > $o ) > $o ).
thf(func_def_182,type,
sP40: ( subst > term > subst > $o ) > $o ).
thf(func_def_183,type,
sP41: ( subst > term > subst > $o ) > $o ).
thf(func_def_185,type,
sP43: ( subst > term > subst > $o ) > $o ).
thf(func_def_187,type,
sP45: ( subst > term > subst > $o ) > $o ).
thf(func_def_191,type,
sP49: ( term > $o ) > $o ).
thf(func_def_192,type,
sP50: ( subst > term > subst > $o ) > $o ).
thf(func_def_196,type,
sP54: ( term > $o ) > $o ).
thf(func_def_197,type,
sP55: ( term > $o ) > $o ).
thf(func_def_200,type,
sP58: ( subst > term > subst > $o ) > $o ).
thf(func_def_201,type,
sK59: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_202,type,
sK60: ( subst > term > term ) > subst ).
thf(func_def_203,type,
sK61: ( subst > term > term ) > subst ).
thf(func_def_204,type,
sK62: ( subst > term > term ) > term ).
thf(func_def_205,type,
sK63: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_206,type,
sK64: ( term > $o ) > subst > term ).
thf(func_def_207,type,
sK65: ( term > $o ) > term ).
thf(func_def_208,type,
sK66: subst ).
thf(func_def_209,type,
sK67: term > $o ).
thf(func_def_210,type,
sK68: term ).
thf(func_def_211,type,
sK69: term > term ).
thf(func_def_212,type,
sK70: term ).
thf(func_def_213,type,
sK71: term ).
thf(func_def_214,type,
sK72: term > $o ).
thf(func_def_215,type,
sK73: term > term ).
thf(func_def_216,type,
sK74: ( term > $o ) > term ).
thf(func_def_217,type,
sK75: ( term > $o ) > term ).
thf(func_def_218,type,
sK76: ( term > $o ) > term ).
thf(func_def_219,type,
sK77: ( term > $o ) > term ).
thf(func_def_220,type,
sK78: ( term > $o ) > term ).
thf(func_def_221,type,
sK79: ( term > $o ) > term ).
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 > term ) > subst ).
thf(func_def_225,type,
sK83: ( subst > term > term ) > term ).
thf(func_def_226,type,
sK84: ( subst > term > term ) > subst ).
thf(func_def_227,type,
sK85: ( subst > term > term ) > term ).
thf(func_def_228,type,
sK86: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_229,type,
sK87: term ).
thf(func_def_230,type,
sK88: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_231,type,
sK89: subst > term > subst > $o ).
thf(func_def_232,type,
sK90: ( term > $o ) > term ).
thf(func_def_233,type,
sK91: ( term > $o ) > term ).
thf(func_def_234,type,
sK92: ( term > $o ) > term ).
thf(func_def_235,type,
sK93: term > $o ).
thf(func_def_236,type,
sK94: term ).
thf(func_def_237,type,
sK95: term > term ).
thf(func_def_238,type,
sK96: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_239,type,
sK97: term > ( term > $o ) > term ).
thf(func_def_240,type,
sK98: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_241,type,
sK99: subst > term > subst > $o ).
thf(func_def_242,type,
sK100: term > $o ).
thf(func_def_243,type,
sK101: term ).
thf(func_def_244,type,
sK102: ( subst > term > term ) > subst ).
thf(func_def_245,type,
sK103: ( subst > term > term ) > subst ).
thf(func_def_246,type,
sK104: ( subst > term > term ) > term ).
thf(func_def_247,type,
sK105: ( subst > term > term ) > term ).
thf(func_def_248,type,
sK106: ( subst > term > subst > $o ) > term ).
thf(func_def_249,type,
sK107: ( subst > term > subst > $o ) > term ).
thf(func_def_250,type,
sK108: ( term > $o ) > term ).
thf(func_def_251,type,
sK109: ( term > $o ) > term ).
thf(func_def_252,type,
sK110: subst > term > subst > $o ).
thf(func_def_253,type,
sK111: term ).
thf(func_def_254,type,
sK112: term > ( subst > term > term ) > subst ).
thf(func_def_255,type,
sK113: term > ( subst > term > term ) > term ).
thf(func_def_256,type,
sK114: term > ( subst > term > term ) > subst ).
thf(func_def_257,type,
sK115: term > ( subst > term > term ) > subst ).
thf(func_def_258,type,
sK116: term > ( subst > term > term ) > term ).
thf(func_def_259,type,
sK117: term > ( subst > term > term ) > subst ).
thf(func_def_260,type,
sK118: ( subst > term > term ) > term ).
thf(func_def_261,type,
sK119: term > ( subst > term > subst > $o ) > term ).
thf(func_def_262,type,
sK120: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_263,type,
sK121: ( subst > term > subst > $o ) > term ).
thf(func_def_264,type,
sK122: subst > term > subst > $o ).
thf(func_def_265,type,
sK123: term ).
thf(func_def_266,type,
sK124: ( subst > term > term ) > term ).
thf(func_def_267,type,
sK125: ( subst > term > term ) > subst ).
thf(func_def_268,type,
sK126: ( subst > term > term ) > subst ).
thf(func_def_269,type,
sK127: ( subst > term > term ) > term ).
thf(func_def_270,type,
sK128: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_271,type,
sK129: term > ( subst > term > subst > $o ) > term ).
thf(func_def_272,type,
sK130: subst > term > subst > $o ).
thf(func_def_273,type,
sK131: term > $o ).
thf(func_def_274,type,
sK132: term ).
thf(func_def_275,type,
sK133: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_276,type,
sK134: ( subst > term > subst > $o ) > subst ).
thf(func_def_277,type,
sK135: ( subst > term > subst > $o ) > subst ).
thf(func_def_278,type,
sK136: ( subst > term > subst > $o ) > subst ).
thf(func_def_279,type,
sK137: ( subst > term > subst > $o ) > term ).
thf(func_def_280,type,
sK138: term > ( term > $o ) > term ).
thf(func_def_281,type,
sK139: ( subst > term > subst > $o ) > subst ).
thf(func_def_282,type,
sK140: ( subst > term > subst > $o ) > subst ).
thf(func_def_283,type,
sK141: ( subst > term > subst > $o ) > subst ).
thf(func_def_284,type,
sK142: ( subst > term > subst > $o ) > term ).
thf(func_def_285,type,
sK143: ( subst > term > term ) > term ).
thf(func_def_286,type,
sK144: ( subst > term > term ) > subst ).
thf(func_def_287,type,
sK145: ( subst > term > term ) > subst ).
thf(func_def_288,type,
sK146: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_289,type,
sK147: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_290,type,
sK148: term > $o ).
thf(func_def_291,type,
sK149: term ).
thf(func_def_292,type,
sK150: term ).
thf(func_def_293,type,
sK151: subst > term > subst > $o ).
thf(func_def_294,type,
sK152: ( subst > term > subst > $o ) > subst ).
thf(func_def_295,type,
sK153: ( subst > term > subst > $o ) > subst ).
thf(func_def_296,type,
sK154: ( subst > term > subst > $o ) > term ).
thf(func_def_297,type,
sK155: ( subst > term > subst > $o ) > subst ).
thf(func_def_298,type,
sK156: ( subst > term > subst > $o ) > subst ).
thf(func_def_299,type,
sK157: ( subst > term > subst > $o ) > subst ).
thf(func_def_300,type,
sK158: ( subst > term > subst > $o ) > term ).
thf(func_def_301,type,
sK159: ( subst > term > subst > $o ) > subst ).
thf(func_def_302,type,
sK160: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_303,type,
sK161: ( subst > term > subst > $o ) > term ).
thf(func_def_304,type,
sK162: ( subst > term > subst > $o ) > term ).
thf(func_def_305,type,
sK163: subst > term > subst > $o ).
thf(func_def_306,type,
sK164: term ).
thf(func_def_307,type,
sK165: ( subst > term > subst > $o ) > term ).
thf(func_def_308,type,
sK166: ( subst > term > subst > $o ) > subst ).
thf(func_def_309,type,
sK167: ( subst > term > subst > $o ) > subst ).
thf(func_def_310,type,
sK168: ( subst > term > subst > $o ) > subst ).
thf(func_def_311,type,
sK169: ( subst > term > subst > $o ) > term ).
thf(func_def_312,type,
sK170: ( subst > term > subst > $o ) > subst ).
thf(func_def_313,type,
sK171: ( subst > term > subst > $o ) > term ).
thf(func_def_314,type,
sK172: ( subst > term > subst > $o ) > subst ).
thf(func_def_315,type,
sK173: ( subst > term > subst > $o ) > subst ).
thf(func_def_316,type,
sK174: ( subst > term > subst > $o ) > term ).
thf(func_def_317,type,
sK175: ( subst > term > subst > $o ) > term ).
thf(func_def_318,type,
sK176: ( subst > term > term ) > term ).
thf(func_def_319,type,
sK177: ( subst > term > term ) > subst ).
thf(func_def_320,type,
sK178: ( subst > term > term ) > subst ).
thf(func_def_321,type,
sK179: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_322,type,
sK180: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_323,type,
sK181: term ).
thf(func_def_324,type,
sK182: term > $o ).
thf(func_def_325,type,
sK183: 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: ( term > $o ) > term ).
thf(func_def_329,type,
sK187: ( term > $o ) > term ).
thf(func_def_330,type,
sK188: subst > term > subst > $o ).
thf(func_def_331,type,
sK189: term ).
thf(func_def_332,type,
sK190: ( subst > term > term ) > subst ).
thf(func_def_333,type,
sK191: ( subst > term > term ) > subst ).
thf(func_def_334,type,
sK192: ( subst > term > term ) > term ).
thf(func_def_335,type,
sK193: ( subst > term > term ) > term ).
thf(func_def_336,type,
sK194: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_337,type,
sK195: term > ( subst > term > subst > $o ) > term ).
thf(func_def_338,type,
sK196: subst > term > subst > $o ).
thf(func_def_339,type,
sK197: term ).
thf(func_def_340,type,
sK198: ( subst > term > term ) > term ).
thf(func_def_341,type,
sK199: ( subst > term > term ) > term ).
thf(func_def_342,type,
sK200: ( subst > term > term ) > subst ).
thf(func_def_343,type,
sK201: ( subst > term > term ) > subst ).
thf(func_def_344,type,
sK202: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_345,type,
sK203: term > ( subst > term > subst > $o ) > term ).
thf(func_def_346,type,
sK204: term ).
thf(func_def_347,type,
sK205: subst > term > subst > $o ).
thf(func_def_348,type,
sK206: ( subst > term > term ) > term ).
thf(func_def_349,type,
sK207: ( subst > term > term ) > subst ).
thf(func_def_350,type,
sK208: ( subst > term > term ) > term ).
thf(func_def_351,type,
sK209: ( subst > term > term ) > subst ).
thf(func_def_352,type,
sK210: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_353,type,
sK211: ( subst > term > subst > $o ) > term > term ).
thf(func_def_354,type,
sK212: term ).
thf(func_def_355,type,
sK213: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_356,type,
sK214: ( subst > term > subst > $o ) > term > term ).
thf(func_def_357,type,
sK215: term ).
thf(func_def_358,type,
sK216: subst > term > subst > $o ).
thf(func_def_359,type,
sK217: ( subst > term > term ) > subst ).
thf(func_def_360,type,
sK218: ( subst > term > term ) > term ).
thf(func_def_361,type,
sK219: ( subst > term > term ) > subst ).
thf(func_def_362,type,
sK220: ( subst > term > term ) > term ).
thf(func_def_363,type,
sK221: term ).
thf(func_def_364,type,
sK222: term > $o ).
thf(func_def_365,type,
sK223: subst ).
thf(func_def_366,type,
sK224: term > term ).
thf(func_def_367,type,
sK225: subst > ( term > $o ) > term ).
thf(func_def_368,type,
sK226: ( term > $o ) > term ).
thf(func_def_369,type,
sK227: ( term > $o ) > term ).
thf(func_def_370,type,
sK228: ( term > $o ) > term ).
thf(func_def_371,type,
sK229: subst > term > term ).
thf(func_def_372,type,
sK230: subst > term > term ).
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: subst ).
thf(func_def_385,type,
sK243: term ).
thf(func_def_386,type,
sK244: subst ).
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 ).
thf(func_def_390,type,
sK248: term ).
thf(func_def_391,type,
sK249: term > term ).
thf(func_def_392,type,
sK250: ( subst > term > subst > $o ) > subst ).
thf(func_def_393,type,
sK251: ( subst > term > subst > $o ) > term ).
thf(func_def_394,type,
sK252: ( subst > term > subst > $o ) > subst ).
thf(func_def_395,type,
sK253: ( subst > term > subst > $o ) > subst ).
thf(func_def_396,type,
sK254: ( subst > term > subst > $o ) > subst ).
thf(func_def_397,type,
sK255: ( subst > term > subst > $o ) > subst ).
thf(func_def_398,type,
sK256: ( subst > term > subst > $o ) > term ).
thf(func_def_399,type,
sK257: ( subst > term > subst > $o ) > subst ).
thf(func_def_400,type,
sK258: term ).
thf(func_def_401,type,
sK259: subst > term > subst > $o ).
thf(func_def_402,type,
sK260: term ).
thf(func_def_403,type,
sK261: term ).
thf(func_def_404,type,
sK262: subst > term > subst > $o ).
thf(func_def_405,type,
sK263: ( subst > term > subst > $o ) > subst ).
thf(func_def_406,type,
sK264: ( subst > term > subst > $o ) > subst ).
thf(func_def_407,type,
sK265: ( subst > term > subst > $o ) > subst ).
thf(func_def_408,type,
sK266: ( subst > term > subst > $o ) > term ).
thf(func_def_409,type,
sK267: ( subst > term > subst > $o ) > term ).
thf(func_def_410,type,
sK268: ( subst > term > subst > $o ) > subst ).
thf(func_def_411,type,
sK269: ( subst > term > subst > $o ) > subst ).
thf(func_def_412,type,
sK270: ( subst > term > subst > $o ) > term ).
thf(func_def_413,type,
sK271: ( subst > term > subst > $o ) > subst ).
thf(func_def_414,type,
sK272: ( subst > term > subst > $o ) > subst ).
thf(func_def_415,type,
sK273: ( subst > term > subst > $o ) > subst ).
thf(func_def_416,type,
sK274: ( subst > term > subst > $o ) > term ).
thf(func_def_417,type,
sK275: ( subst > term > subst > $o ) > subst ).
thf(func_def_418,type,
sK276: ( subst > term > subst > $o ) > subst ).
thf(func_def_419,type,
sK277: ( subst > term > subst > $o ) > subst ).
thf(func_def_420,type,
sK278: ( subst > term > subst > $o ) > term ).
thf(func_def_421,type,
sK279: ( subst > term > subst > $o ) > subst ).
thf(func_def_422,type,
sK280: term > ( subst > term > subst > $o ) > term ).
thf(func_def_423,type,
sK281: subst > term > subst > $o ).
thf(func_def_424,type,
sK282: term ).
thf(func_def_425,type,
sK283: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_426,type,
sK284: ( subst > term > term ) > subst ).
thf(func_def_427,type,
sK285: ( subst > term > term ) > subst ).
thf(func_def_428,type,
sK286: ( subst > term > term ) > term ).
thf(func_def_429,type,
sK287: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_430,type,
sK288: ( subst > term > term ) > subst ).
thf(func_def_431,type,
sK289: ( subst > term > term ) > term ).
thf(func_def_432,type,
sK290: ( subst > term > term ) > subst ).
thf(func_def_433,type,
sK291: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_434,type,
sK292: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_435,type,
sK293: ( subst > term > subst > $o ) > term ).
thf(func_def_436,type,
sK294: ( subst > term > subst > $o ) > term ).
thf(func_def_437,type,
sK295: ( term > $o ) > term ).
thf(func_def_438,type,
sK296: ( term > $o ) > term ).
thf(func_def_439,type,
sK297: ( subst > term > subst > $o ) > term ).
thf(func_def_440,type,
sK298: ( subst > term > subst > $o ) > term ).
thf(func_def_441,type,
sK299: subst > term > subst > $o ).
thf(func_def_442,type,
sK300: term ).
thf(func_def_443,type,
sK301: term > $o ).
thf(func_def_444,type,
sK302: ( subst > term > term ) > subst ).
thf(func_def_445,type,
sK303: ( subst > term > term ) > term ).
thf(func_def_446,type,
sK304: ( subst > term > term ) > subst ).
thf(func_def_447,type,
sK305: ( subst > term > term ) > term ).
thf(func_def_448,type,
sK306: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_449,type,
sK307: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_450,type,
sK308: ( term > $o ) > term > term ).
thf(func_def_451,type,
sK309: term ).
thf(func_def_452,type,
sK310: subst > term > subst > $o ).
thf(func_def_453,type,
sK311: ( subst > term > subst > $o ) > term ).
thf(func_def_454,type,
sK312: ( subst > term > subst > $o ) > subst ).
thf(func_def_455,type,
sK313: ( subst > term > subst > $o ) > subst ).
thf(func_def_456,type,
sK314: ( subst > term > subst > $o ) > subst ).
thf(func_def_457,type,
sK315: ( subst > term > subst > $o ) > subst ).
thf(func_def_458,type,
sK316: ( subst > term > subst > $o ) > subst ).
thf(func_def_459,type,
sK317: ( subst > term > subst > $o ) > subst ).
thf(func_def_460,type,
sK318: ( subst > term > subst > $o ) > term ).
thf(func_def_461,type,
sK319: ( subst > term > subst > $o ) > term > term ).
thf(func_def_462,type,
sK320: term ).
thf(func_def_463,type,
sK321: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_464,type,
sK322: ( subst > term > subst > $o ) > term > term ).
thf(func_def_465,type,
sK323: term ).
thf(func_def_466,type,
sK324: subst > term > subst > $o ).
thf(func_def_467,type,
sK325: ( subst > term > term ) > term ).
thf(func_def_468,type,
sK326: ( subst > term > term ) > term ).
thf(func_def_469,type,
sK327: ( subst > term > term ) > subst ).
thf(func_def_470,type,
sK328: ( subst > term > term ) > subst ).
thf(func_def_471,type,
sK329: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_472,type,
sK330: ( subst > term > term ) > term ).
thf(func_def_473,type,
sK331: ( subst > term > term ) > subst ).
thf(func_def_474,type,
sK332: ( subst > term > term ) > subst ).
thf(func_def_475,type,
sK333: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_476,type,
sK334: term ).
thf(func_def_477,type,
sK335: subst > term > subst > $o ).
thf(func_def_478,type,
sK336: term > $o ).
thf(func_def_479,type,
sK337: ( subst > term > term ) > subst ).
thf(func_def_480,type,
sK338: ( subst > term > term ) > term ).
thf(func_def_481,type,
sK339: ( subst > term > term ) > subst ).
thf(func_def_482,type,
sK340: ( subst > term > term ) > term ).
thf(func_def_483,type,
sK341: term > subst > term > term ).
thf(func_def_484,type,
sK342: ( term > $o ) > term > term ).
thf(func_def_485,type,
sK343: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_486,type,
sK344: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_487,type,
sK345: term ).
thf(func_def_488,type,
sK346: ( subst > term > term ) > subst ).
thf(func_def_489,type,
sK347: ( subst > term > term ) > term ).
thf(func_def_490,type,
sK348: term ).
thf(func_def_491,type,
sK349: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_492,type,
sK350: ( term > $o ) > term ).
thf(func_def_493,type,
sK351: ( term > $o ) > term ).
thf(func_def_494,type,
sK352: subst > term > subst > $o ).
thf(func_def_495,type,
sK353: ( term > $o ) > term ).
thf(func_def_496,type,
sK354: term ).
thf(func_def_497,type,
sK355: term > $o ).
thf(func_def_498,type,
sK356: term > term ).
thf(func_def_499,type,
sK357: term ).
thf(func_def_500,type,
sK358: term > ( term > $o ) > term ).
thf(func_def_501,type,
sK359: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_502,type,
sK360: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_503,type,
sK361: term > $o ).
thf(func_def_504,type,
sK362: subst > term > subst > $o ).
thf(func_def_505,type,
sK363: term ).
thf(func_def_506,type,
sK364: ( subst > term > term ) > subst ).
thf(func_def_507,type,
sK365: ( subst > term > term ) > term ).
thf(func_def_508,type,
sK366: ( subst > term > term ) > subst ).
thf(func_def_509,type,
sK367: ( subst > term > term ) > term ).
thf(func_def_510,type,
sK368: subst > term > subst > $o ).
thf(func_def_511,type,
sK369: term ).
thf(func_def_512,type,
sK370: ( subst > term > subst > $o ) > subst ).
thf(func_def_513,type,
sK371: ( subst > term > subst > $o ) > term ).
thf(func_def_514,type,
sK372: ( subst > term > subst > $o ) > subst ).
thf(func_def_515,type,
sK373: ( subst > term > subst > $o ) > subst ).
thf(func_def_516,type,
sK374: ( subst > term > subst > $o ) > term ).
thf(func_def_517,type,
sK375: ( subst > term > subst > $o ) > subst ).
thf(func_def_518,type,
sK376: ( subst > term > subst > $o ) > subst ).
thf(func_def_519,type,
sK377: ( subst > term > subst > $o ) > subst ).
thf(func_def_520,type,
sK378: ( subst > term > subst > $o ) > term ).
thf(func_def_521,type,
sK379: ( term > $o ) > term ).
thf(func_def_522,type,
sK380: ( term > $o ) > term ).
thf(func_def_523,type,
sK381: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_524,type,
sK382: ( subst > term > term ) > subst ).
thf(func_def_525,type,
sK383: ( subst > term > term ) > subst ).
thf(func_def_526,type,
sK384: ( subst > term > term ) > term ).
thf(func_def_527,type,
sK385: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_528,type,
sK386: ( subst > term > subst > $o ) > term ).
thf(func_def_529,type,
sK387: ( subst > term > subst > $o ) > term ).
thf(func_def_530,type,
sK388: ( term > $o ) > term ).
thf(func_def_531,type,
sK389: ( term > $o ) > term ).
thf(func_def_532,type,
sK390: ( term > $o ) > term ).
thf(func_def_533,type,
sK391: ( term > $o ) > term ).
thf(func_def_534,type,
sK392: subst ).
thf(func_def_535,type,
sK393: term ).
thf(func_def_536,type,
sK394: subst ).
thf(func_def_537,type,
sK395: term ).
thf(func_def_538,type,
sK396: subst ).
thf(func_def_539,type,
sK397: term > $o ).
thf(func_def_540,type,
sK398: term ).
thf(func_def_541,type,
sK399: term ).
thf(func_def_542,type,
sK400: subst ).
thf(func_def_543,type,
sK401: term > $o ).
thf(func_def_544,type,
sK402: term > term ).
thf(func_def_545,type,
sK403: ( term > $o ) > subst > term ).
thf(func_def_546,type,
sK404: ( term > $o ) > term ).
thf(func_def_547,type,
sK405: ( term > $o ) > term ).
thf(func_def_548,type,
sK406: ( term > $o ) > term ).
thf(func_def_549,type,
sK407: ( term > $o ) > term ).
thf(func_def_550,type,
sK408: ( term > $o ) > term ).
thf(func_def_551,type,
sK409: term ).
thf(func_def_552,type,
sK410: term ).
thf(func_def_553,type,
sK411: subst > term > subst > $o ).
thf(func_def_554,type,
sK412: ( subst > term > subst > $o ) > subst ).
thf(func_def_555,type,
sK413: ( subst > term > subst > $o ) > subst ).
thf(func_def_556,type,
sK414: ( subst > term > subst > $o ) > term ).
thf(func_def_557,type,
sK415: ( subst > term > subst > $o ) > subst ).
thf(func_def_558,type,
sK416: ( subst > term > subst > $o ) > subst ).
thf(func_def_559,type,
sK417: ( subst > term > subst > $o ) > subst ).
thf(func_def_560,type,
sK418: ( subst > term > subst > $o ) > term ).
thf(func_def_561,type,
sK419: ( subst > term > subst > $o ) > subst ).
thf(func_def_562,type,
sK420: ( subst > term > subst > $o ) > term ).
thf(func_def_563,type,
sK421: ( subst > term > subst > $o ) > term ).
thf(func_def_564,type,
sK422: term ).
thf(func_def_565,type,
sK423: subst > term > subst > $o ).
thf(func_def_566,type,
sK424: ( subst > term > subst > $o ) > subst ).
thf(func_def_567,type,
sK425: ( subst > term > subst > $o ) > subst ).
thf(func_def_568,type,
sK426: ( subst > term > subst > $o ) > subst ).
thf(func_def_569,type,
sK427: ( subst > term > subst > $o ) > term ).
thf(func_def_570,type,
sK428: ( subst > term > subst > $o ) > term > term ).
thf(func_def_571,type,
sK429: ( subst > term > subst > $o ) > subst ).
thf(func_def_572,type,
sK430: ( subst > term > subst > $o ) > subst ).
thf(func_def_573,type,
sK431: ( subst > term > subst > $o ) > term ).
thf(func_def_574,type,
sK432: ( subst > term > subst > $o ) > subst ).
thf(func_def_575,type,
sK433: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_576,type,
sK434: ( subst > term > term ) > subst ).
thf(func_def_577,type,
sK435: ( subst > term > term ) > subst ).
thf(func_def_578,type,
sK436: ( subst > term > term ) > term ).
thf(func_def_579,type,
sK437: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_580,type,
sK438: term > $o ).
thf(func_def_581,type,
sK439: subst > term > subst > $o ).
thf(func_def_582,type,
sK440: term ).
thf(func_def_583,type,
sK441: ( subst > term > subst > $o ) > subst ).
thf(func_def_584,type,
sK442: ( subst > term > subst > $o ) > subst ).
thf(func_def_585,type,
sK443: ( subst > term > subst > $o ) > subst ).
thf(func_def_586,type,
sK444: ( subst > term > subst > $o ) > term ).
thf(func_def_587,type,
sK445: term > ( term > $o ) > term ).
thf(func_def_588,type,
sK446: ( subst > term > subst > $o ) > subst ).
thf(func_def_589,type,
sK447: ( subst > term > subst > $o ) > subst ).
thf(func_def_590,type,
sK448: ( subst > term > subst > $o ) > subst ).
thf(func_def_591,type,
sK449: ( subst > term > subst > $o ) > term ).
thf(func_def_592,type,
sK450: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_593,type,
sK451: term ).
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: ( subst > term > term ) > subst ).
thf(func_def_597,type,
sK455: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_598,type,
sK456: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_599,type,
sK457: term ).
thf(func_def_600,type,
sK458: subst > term > subst > $o ).
thf(func_def_601,type,
sK459: ( subst > term > subst > $o ) > term ).
thf(func_def_602,type,
sK460: ( subst > term > subst > $o ) > term ).
thf(func_def_603,type,
sK461: term ).
thf(func_def_604,type,
sK462: term ).
thf(func_def_605,type,
sK463: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_606,type,
sK464: subst > term > subst > $o ).
thf(func_def_607,type,
sK465: ( term > $o ) > term ).
thf(func_def_608,type,
sK466: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_609,type,
sK467: subst > term > subst > $o ).
thf(func_def_610,type,
sK468: ( term > $o ) > term ).
thf(func_def_611,type,
sK469: subst ).
thf(func_def_612,type,
sK470: subst ).
thf(func_def_613,type,
sK471: subst ).
thf(func_def_614,type,
sK472: subst ).
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 ).
thf(func_def_619,type,
sK477: term ).
thf(func_def_620,type,
sK478: term ).
thf(func_def_621,type,
sK479: term ).
thf(func_def_622,type,
sK480: term ).
thf(func_def_623,type,
sK481: term ).
thf(func_def_624,type,
sK482: term ).
thf(func_def_625,type,
sK483: term ).
thf(func_def_626,type,
sK484: term ).
thf(func_def_627,type,
sK485: term ).
thf(func_def_628,type,
sK486: subst > term > term ).
thf(func_def_629,type,
sK487: subst ).
thf(func_def_630,type,
sK488: subst > term > term ).
thf(func_def_631,type,
sK489: ( subst > term > term ) > term ).
thf(func_def_632,type,
sK490: ( subst > term > term ) > subst ).
thf(func_def_633,type,
sK491: ( subst > term > term ) > subst ).
thf(func_def_634,type,
sK492: ( subst > term > term ) > subst ).
thf(func_def_635,type,
sK493: ( subst > term > term ) > term ).
thf(func_def_636,type,
sK494: ( subst > term > term ) > subst ).
thf(func_def_637,type,
sK495: term ).
thf(func_def_638,type,
sK496: subst > term > term ).
thf(func_def_639,type,
sK497: term ).
thf(func_def_640,type,
sK498: ( subst > term > term ) > subst ).
thf(func_def_641,type,
sK499: ( subst > term > term ) > subst ).
thf(func_def_642,type,
sK500: ( subst > term > term ) > term ).
thf(func_def_643,type,
sK501: term ).
thf(func_def_644,type,
sK502: term ).
thf(func_def_645,type,
sK503: term ).
thf(func_def_646,type,
sK504: term ).
thf(func_def_647,type,
sK505: term ).
thf(func_def_648,type,
sK506: term ).
thf(func_def_649,type,
sK507: term ).
thf(func_def_650,type,
sK508: term ).
thf(func_def_651,type,
sK509: term ).
thf(func_def_652,type,
sK510: term ).
thf(func_def_653,type,
sK511: term > $o ).
thf(func_def_654,type,
sK512: term ).
thf(func_def_655,type,
sK513: term > $o ).
thf(func_def_656,type,
sK514: term ).
thf(func_def_657,type,
sK515: subst ).
thf(func_def_658,type,
sK516: subst ).
thf(func_def_659,type,
sK517: subst ).
thf(func_def_660,type,
sK518: term ).
thf(func_def_661,type,
sK519: subst ).
thf(func_def_662,type,
sK520: term ).
thf(func_def_663,type,
sK521: subst ).
thf(func_def_664,type,
sK522: term > $o ).
thf(func_def_665,type,
sK523: subst ).
thf(func_def_666,type,
sK524: term ).
thf(func_def_667,type,
sK525: term > $o ).
thf(func_def_668,type,
sK526: term ).
thf(func_def_669,type,
sK527: term ).
thf(func_def_670,type,
sK528: term > $o ).
thf(func_def_671,type,
sK529: subst ).
thf(func_def_672,type,
sK530: ( term > $o ) > term ).
thf(func_def_673,type,
sK531: subst > ( term > $o ) > term > term > $o ).
thf(func_def_674,type,
sK532: term ).
thf(func_def_675,type,
sK533: term ).
thf(func_def_676,type,
sK534: subst > d_subst ).
thf(func_def_677,type,
sK535: subst ).
thf(func_def_678,type,
sK536: term ).
thf(func_def_679,type,
sK537: ( subst > term > term ) > subst ).
thf(func_def_680,type,
sK538: ( subst > term > term ) > term ).
thf(func_def_681,type,
sK539: term > subst > term > term ).
thf(func_def_682,type,
sK540: term ).
thf(func_def_683,type,
sK541: term ).
thf(func_def_684,type,
sK542: term > subst > term > term ).
thf(func_def_685,type,
sK543: term ).
thf(func_def_686,type,
sK544: term ).
thf(func_def_687,type,
sK545: term ).
thf(func_def_688,type,
sK546: term ).
thf(func_def_689,type,
sK547: term ).
thf(func_def_690,type,
sK548: term ).
thf(func_def_691,type,
sK549: term ).
thf(func_def_692,type,
sK550: term ).
thf(func_def_693,type,
sK551: term ).
thf(func_def_694,type,
sK552: subst ).
thf(func_def_695,type,
sK553: term ).
thf(func_def_696,type,
sK554: term > $o ).
thf(func_def_697,type,
sK555: term > $o ).
thf(func_def_698,type,
sK556: ( term > $o ) > ( term > $o ) > term > subst > term ).
thf(func_def_699,type,
sK557: term ).
thf(func_def_700,type,
sK558: subst ).
thf(func_def_701,type,
sK559: term > ( term > $o ) > subst > ( term > $o ) > term ).
thf(func_def_702,type,
sK560: term > $o ).
thf(func_def_703,type,
sK561: subst ).
thf(func_def_704,type,
sK562: term > $o ).
thf(func_def_705,type,
sK563: term ).
thf(func_def_706,type,
sK564: term ).
thf(func_def_707,type,
sK565: term > subst > term > term ).
thf(func_def_708,type,
sK566: term ).
thf(func_def_709,type,
sK567: ( subst > term > term ) > term ).
thf(func_def_710,type,
sK568: ( subst > term > term ) > subst ).
thf(func_def_711,type,
sK569: term > subst > term > term ).
thf(func_def_712,type,
sK570: subst ).
thf(func_def_713,type,
sK571: ( term > $o ) > term ).
thf(func_def_714,type,
sK572: ( term > $o ) > term ).
thf(func_def_715,type,
sK573: term > $o ).
thf(func_def_716,type,
sK574: term ).
thf(func_def_717,type,
sK575: term > ( term > $o ) > subst > term > $o ).
thf(func_def_718,type,
sK576: subst ).
thf(func_def_719,type,
sK577: term > $o ).
thf(func_def_720,type,
sK578: term ).
thf(func_def_721,type,
sK579: ( term > $o ) > term ).
thf(func_def_722,type,
sK580: term ).
thf(func_def_723,type,
sK581: term ).
thf(func_def_724,type,
sK582: subst ).
thf(func_def_725,type,
sK583: term > d_term ).
thf(func_def_726,type,
sK584: subst ).
thf(func_def_727,type,
sK585: subst ).
thf(func_def_728,type,
sK586: term ).
thf(func_def_729,type,
sK587: term ).
thf(func_def_730,type,
sK588: term ).
thf(func_def_731,type,
sK589: term ).
thf(func_def_732,type,
sK590: term ).
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 ) > term ).
thf(func_def_736,type,
sK594: ( subst > term > term ) > term ).
thf(func_def_737,type,
sK595: ( subst > term > term ) > subst ).
thf(func_def_738,type,
sK596: ( subst > term > term ) > subst ).
thf(func_def_739,type,
sK597: term ).
thf(func_def_740,type,
sK598: ( subst > term > subst > $o ) > term ).
thf(func_def_741,type,
sK599: ( subst > term > subst > $o ) > term ).
thf(func_def_742,type,
sK600: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_743,type,
ph601:
!>[X0: $tType] : X0 ).
thf(f5150,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f5107,f2180]) ).
thf(f2180,plain,
( ( sP58 @ sK458 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2169]) ).
thf(f2169,plain,
( ( $false = $true )
| ( ( sP58 @ sK458 )
!= $true ) ),
inference(backward_demodulation,[status(thm)],[f1259,f2159]) ).
thf(f2159,plain,
hoasinduction_no_psi_cond = $false,
inference(trivial_inequality_removal,[status(thm)],[f2158]) ).
thf(f2158,plain,
( ( hoasinduction_no_psi_cond = $false )
| ( $false = $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f2153]) ).
thf(f2153,plain,
( ( hoasinduction_no_psi_cond = $false )
| ( hoasinduction_no_psi_cond = $false )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f1959,f2135]) ).
thf(f2135,plain,
( ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(duplicate_literal_removal,[status(thm)],[f2134]) ).
thf(f2134,plain,
( ( hoasinduction_no_psi_cond = $false )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f2133,f1894]) ).
thf(f1894,plain,
! [X258: term] :
( ( d2term @ d_one )
= X258 ),
inference(forward_demodulation,[status(thm)],[f1053,f1212]) ).
thf(f1212,plain,
! [X44: d_term] : ( d_one = X44 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f545,plain,
( ( ( hoasinduction_no_psi_cond = $true )
| ( ! [X2: term,X3: term] :
( ( ( sK458 @ id @ X3 @ id )
!= $true )
| ( ( sK458 @ id @ X2 @ id )
!= $true )
| ( ( sK458 @ id @ ( ap @ ( sub @ X2 @ id ) @ X3 ) @ id )
= $true ) )
& ( ( sP58 @ sK458 )
!= $true )
& ( ( sK458 @ id @ sK457 @ id )
!= $true ) ) )
& ( ! [X4: term,X5: subst > term > subst > $o] :
( ( ( ( X5 @ id @ ( sK460 @ X5 ) @ id )
= $true )
& ( ( X5 @ id @ ( sK459 @ X5 ) @ id )
= $true )
& ( ( X5 @ id @ ( ap @ ( sub @ ( sK459 @ X5 ) @ id ) @ ( sK460 @ X5 ) ) @ id )
!= $true ) )
| ( ( sP58 @ X5 )
= $true )
| ( ( X5 @ id @ X4 @ id )
= $true ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( sP57 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP57 != $true ) )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP53 != $true ) )
& ( ( sP53 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( termmset_lthm = $true )
| ( ! [X8: term] :
( ( sub @ X8 @ id )
= X8 )
& ( ( sub @ sK461 @ id )
!= sK461 ) ) )
& ( ( sK462
!= ( sub @ sK462 @ id ) )
| ! [X11: term] :
( ( sub @ X11 @ id )
= X11 )
| ( termmset_lthm != $true ) )
& ( ! [X12: subst > term > subst > $o,X14: term] :
( ( sK463 @ X12 @ X14 )
= ( X12 @ id @ X14 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ! [X16: term > $o] :
( ( X16 @ ( sK465 @ X16 ) )
!= ( sK464 @ id @ ( sK465 @ X16 ) @ id ) ) )
& ( axmap = $true )
& ( ulamvarsh = $true )
& ( ! [X18: subst > term > subst > $o,X20: term] :
( ( sK466 @ X18 @ X20 )
= ( X18 @ id @ X20 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ! [X22: term > $o] :
( ( sK467 @ id @ ( sK468 @ X22 ) @ id )
!= ( X22 @ ( sK468 @ X22 ) ) ) )
& ( ( ( comp @ sK469 @ id )
!= sK469 )
| ( ( comp @ id @ sK470 )
!= sK470 )
| ( ! [X26: subst] :
( ( comp @ id @ X26 )
= X26 )
& ! [X27: subst] :
( ( comp @ X27 @ id )
= X27 ) )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ! [X28: subst] :
( ( comp @ X28 @ id )
= X28 )
& ! [X29: subst] :
( ( comp @ id @ X29 )
= X29 )
& ( ( ( comp @ id @ sK471 )
!= sK471 )
| ( sK472
!= ( comp @ sK472 @ id ) ) ) ) )
& ( ( sP48 = $true )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( sP48 != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( sP47 != $true ) )
& ( ( sP47 = $true )
| ( hoasinduction_lem3aa != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( substmonoid_gthm = $true )
& ! [X32: d_term,X33: d_term] :
( ( ( d2term @ X32 )
!= ( d2term @ X33 ) )
| ( X32 = X33 ) )
& ( ulamvarind = $true )
& ( pushprop_lthm_orig = $true )
& ( ! [X34: term,X35: term,X36: term,X37: term] :
( ( X34 = X35 )
| ( ( ap @ X34 @ X36 )
!= ( ap @ X35 @ X37 ) ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ( ( sK473 != sK474 )
& ( ( ap @ sK474 @ sK476 )
= ( ap @ sK473 @ sK475 ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ! [X42: d_subst,X43: d_subst] :
( ( X42 = X43 )
| ( ( d2subst @ X43 )
!= ( d2subst @ X42 ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP46 != $true ) )
& ( ( sP46 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ! [X44: d_term] : ( d_one = X44 )
& ( ( sP44 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP44 != $true ) )
& ( ! [X45: term,X46: term,X47: term,X48: term] :
( ( ( ap @ X46 @ X48 )
!= ( ap @ X47 @ X45 ) )
| ( X45 = X48 ) )
| ( apinj2 != $true ) )
& ( ( apinj2 = $true )
| ( ( ( ap @ sK478 @ sK480 )
= ( ap @ sK479 @ sK477 ) )
& ( sK477 != sK480 ) ) )
& ( lamnotvar = $true )
& ( axabs = $true )
& ( ( hoasinduction = $true )
| ( sP42 != $true ) )
& ( ( sP42 = $true )
| ( hoasinduction != $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( pushprop_lem0_gthm = $true )
& ( ! [X53: term,X54: term,X55: term,X56: term] :
( ( ( ap @ ( sub @ X55 @ id ) @ X53 )
!= ( ap @ ( sub @ X54 @ id ) @ X56 ) )
| ( X54 = X55 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ( ( ( ap @ ( sub @ sK482 @ id ) @ sK484 )
= ( ap @ ( sub @ sK483 @ id ) @ sK481 ) )
& ( sK483 != sK482 ) ) )
& ( hoasapnotvar_gthm = $true )
& ( ( hoaslaminj = $true )
| ( ( ( lam @ ( sK486 @ sh @ one ) )
= ( lam @ ( sK488 @ sh @ one ) ) )
& ( ( sK488 @ sK487 @ sK485 )
!= ( sK486 @ sK487 @ sK485 ) )
& ! [X65: term,X66: subst,X67: subst] :
( ( sK486 @ ( comp @ X67 @ X66 ) @ ( sub @ X65 @ X66 ) )
= ( sub @ ( sK486 @ X67 @ X65 ) @ X66 ) )
& ! [X68: subst,X69: term,X70: subst] :
( ( sub @ ( sK488 @ X70 @ X69 ) @ X68 )
= ( sK488 @ ( comp @ X70 @ X68 ) @ ( sub @ X69 @ X68 ) ) ) ) )
& ( ! [X71: term,X72: subst > term > term,X73: subst,X74: subst > term > term] :
( ( ( lam @ ( X72 @ sh @ one ) )
!= ( lam @ ( X74 @ sh @ one ) ) )
| ( ( X72 @ X73 @ X71 )
= ( X74 @ X73 @ X71 ) )
| ( ( X72 @ ( comp @ ( sK491 @ X72 ) @ ( sK490 @ X72 ) ) @ ( sub @ ( sK489 @ X72 ) @ ( sK490 @ X72 ) ) )
!= ( sub @ ( X72 @ ( sK491 @ X72 ) @ ( sK489 @ X72 ) ) @ ( sK490 @ X72 ) ) )
| ( ( X74 @ ( comp @ ( sK494 @ X74 ) @ ( sK492 @ X74 ) ) @ ( sub @ ( sK493 @ X74 ) @ ( sK492 @ X74 ) ) )
!= ( sub @ ( X74 @ ( sK494 @ X74 ) @ ( sK493 @ X74 ) ) @ ( sK492 @ X74 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslamnotap = $true )
| ( ! [X84: subst,X85: subst,X86: term] :
( ( sub @ ( sK496 @ X84 @ X86 ) @ X85 )
= ( sK496 @ ( comp @ X84 @ X85 ) @ ( sub @ X86 @ X85 ) ) )
& ( ( ap @ ( sub @ sK495 @ id ) @ sK497 )
= ( lam @ ( sK496 @ sh @ one ) ) ) ) )
& ( ! [X87: term,X88: subst > term > term,X89: term] :
( ( ( sub @ ( X88 @ ( sK498 @ X88 ) @ ( sK500 @ X88 ) ) @ ( sK499 @ X88 ) )
!= ( X88 @ ( comp @ ( sK498 @ X88 ) @ ( sK499 @ X88 ) ) @ ( sub @ ( sK500 @ X88 ) @ ( sK499 @ X88 ) ) ) )
| ( ( ap @ ( sub @ X87 @ id ) @ X89 )
!= ( lam @ ( X88 @ sh @ one ) ) ) )
| ( hoaslamnotap != $true ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
| ( ( sK504 != sK501 )
& ( ( ap @ ( sub @ sK504 @ id ) @ sK502 )
= ( ap @ ( sub @ sK501 @ id ) @ sK503 ) )
& ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X97 @ X99 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X100 ) )
& ! [X101: term] :
( ( sub @ X101 @ id )
= X101 ) ) )
& ( ! [X102: term,X103: term,X104: term,X105: term] :
( ( X102 = X105 )
| ( ( ap @ ( sub @ X105 @ id ) @ X103 )
!= ( ap @ ( sub @ X102 @ id ) @ X104 ) ) )
| ( ( ( ap @ sK505 @ sK507 )
= ( ap @ sK508 @ sK506 ) )
& ( sK508 != sK505 ) )
| ( ( sub @ sK509 @ id )
!= sK509 )
| ( hoasapinj1_lthm != $true ) )
& ( ! [X111: term] :
( ( sub @ X111 @ id )
= X111 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ( ( sub @ sK510 @ id )
!= sK510 ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( pushprop_lem1 = $true )
| ( ( ( sK511 @ sK512 )
= $true )
& ( $true
!= ( sK513 @ ( sub @ sK512 @ ( push @ sK514 @ sK515 ) ) ) ) ) )
& ( ! [X118: term > $o,X119: term,X120: term > $o,X121: term,X122: subst] :
( ( ( X118 @ X119 )
!= $true )
| ( ( X120 @ ( sub @ X119 @ ( push @ X121 @ X122 ) ) )
= $true ) )
| ( pushprop_lem1 != $true ) )
& ( ( sP39 = $true )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( sP39 != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP37 != $true ) )
& ( ( sP37 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( hoasinduction_lem1 = $true )
& ( hoaslaminj_gthm = $true )
& ( pushprop = $true )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( substmonoid = $true )
| ( ( comp @ id @ sK516 )
!= sK516 )
| ( ( comp @ sK517 @ id )
!= sK517 ) )
& ( ( ! [X125: subst] :
( ( comp @ id @ X125 )
= X125 )
& ! [X126: subst] :
( ( comp @ X126 @ id )
= X126 ) )
| ( substmonoid != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP36 ) )
& ( ( $true = sP36 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ulamvar1 != $true )
& ( ( ( comp @ sh @ ( push @ sK518 @ sK519 ) )
!= sK519 )
| ! [X129: term > $o,X130: subst,X131: term,X132: term > $o,X133: term] :
( ( ( X129 @ X131 )
!= $true )
| ( ( X132 @ ( sub @ X131 @ ( push @ X133 @ X130 ) ) )
= $true ) )
| ( sK520
!= ( sub @ one @ ( push @ sK520 @ sK521 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X136: term,X137: subst] :
( ( comp @ sh @ ( push @ X136 @ X137 ) )
= X137 )
& ( ( sK522 @ sK524 )
= $true )
& ( ( sK525 @ ( sub @ sK524 @ ( push @ sK526 @ sK523 ) ) )
!= $true )
& ! [X143: term,X144: subst] :
( ( sub @ one @ ( push @ X143 @ X144 ) )
= X143 ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ! [X148: term > $o] :
( ( sK528 @ ( sub @ ( sK530 @ X148 ) @ ( push @ sK527 @ sK529 ) ) )
!= ( X148 @ ( sK530 @ X148 ) ) ) )
& ( ! [X150: term,X151: term > $o,X152: subst,X154: term] :
( ( sK531 @ X152 @ X151 @ X150 @ X154 )
= ( X151 @ ( sub @ X154 @ ( push @ X150 @ X152 ) ) ) )
| ( pushprop_lem0_lthm != $true ) )
& ( one
= ( d2term @ d_one ) )
& ( ( shinj = $true )
| ( ( sK532 != sK533 )
& ( ( sub @ sK533 @ sh )
= ( sub @ sK532 @ sh ) ) ) )
& ( ! [X157: term,X158: term] :
( ( X157 = X158 )
| ( ( sub @ X158 @ sh )
!= ( sub @ X157 @ sh ) ) )
| ( shinj != $true ) )
& ( axvarshift = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2subst @ d_id )
= 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 ) ) ) ) ) ) )
& ! [X159: subst] :
( ( d2subst @ ( sK534 @ X159 ) )
= X159 )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP35 != $true ) )
& ( ( sP35 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ! [X161: subst] :
( ( push @ ( sub @ one @ X161 ) @ ( comp @ sh @ X161 ) )
= X161 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ( ( push @ ( sub @ one @ sK535 ) @ ( comp @ sh @ sK535 ) )
!= sK535 ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ! [X164: subst > term > term] :
( ( X164 @ ( sK537 @ X164 ) @ ( sK538 @ X164 ) )
!= ( sub @ sK536 @ ( push @ ( sK538 @ X164 ) @ ( sK537 @ X164 ) ) ) ) )
& ( ! [X167: term,X169: subst,X170: term] :
( ( sub @ X167 @ ( push @ X170 @ X169 ) )
= ( sK539 @ X167 @ X169 @ X170 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ! [X171: term,X172: term] :
( ( X171 = X172 )
| ( ( lam @ X172 )
!= ( lam @ X171 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ( ( sK540 != sK541 )
& ( ( lam @ sK540 )
= ( lam @ sK541 ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X175: term] :
( ( sub @ X175 @ ( push @ one @ sh ) )
= ( sK542 @ X175 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ! [X178: subst > term > term] :
( ( X178 @ sh @ one )
!= ( sub @ sK543 @ ( push @ one @ sh ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ( sK545 != sK544 )
& ( ( ap @ ( sub @ sK546 @ id ) @ sK544 )
= ( ap @ ( sub @ sK547 @ id ) @ sK545 ) )
& ! [X183: term,X184: term,X185: term,X186: term] :
( ( ( ap @ X186 @ X183 )
!= ( ap @ X185 @ X184 ) )
| ( X183 = X184 ) ) ) )
& ( ! [X187: term,X188: term,X189: term,X190: term] :
( ( X187 = X188 )
| ( ( ap @ ( sub @ X190 @ id ) @ X188 )
!= ( ap @ ( sub @ X189 @ id ) @ X187 ) ) )
| ( ( ( ap @ sK550 @ sK549 )
= ( ap @ sK551 @ sK548 ) )
& ( sK548 != sK549 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( sP33 = $true )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( sP33 != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP31 ) )
& ( ( $true = sP31 )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( sP30 = $true )
| ( hoasinduction_lthm != $true ) )
& ( ( hoasinduction_lthm = $true )
| ( sP30 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ( ( ( sK555 @ one )
!= $true )
& ( ( sK554 @ sK553 )
= $true )
& ! [X199: term] :
( ( sK554 @ ( sub @ X199 @ ( push @ sK553 @ sK552 ) ) )
= ( sK555 @ X199 ) ) ) )
& ( ! [X200: subst,X201: term,X202: term > $o,X203: term > $o] :
( ( ( X203 @ one )
= $true )
| ( ( X202 @ X201 )
!= $true )
| ( ( X203 @ ( sK556 @ X203 @ X202 @ X201 @ X200 ) )
!= ( X202 @ ( sub @ ( sK556 @ X203 @ X202 @ X201 @ X200 ) @ ( push @ X201 @ X200 ) ) ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP21 != $true ) )
& ( ( sP21 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( sP20 != $true ) )
& ( ( sP20 = $true )
| ( hoasinduction_lem3_lthm != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( sK557
!= ( sub @ one @ ( push @ sK557 @ sK558 ) ) )
| ! [X207: term > $o,X208: subst,X209: term > $o,X210: term] :
( ( ( X207 @ X210 )
!= $true )
| ( ( X207 @ ( sub @ ( sK559 @ X210 @ X209 @ X208 @ X207 ) @ ( push @ X210 @ X208 ) ) )
!= ( X209 @ ( sK559 @ X210 @ X209 @ X208 @ X207 ) ) )
| ( ( X209 @ one )
= $true ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ! [X212: term,X213: subst] :
( ( sub @ one @ ( push @ X212 @ X213 ) )
= X212 )
& ( ( sK560 @ sK563 )
= $true )
& ! [X218: term] :
( ( sK562 @ X218 )
= ( sK560 @ ( sub @ X218 @ ( push @ sK563 @ sK561 ) ) ) )
& ( ( sK562 @ one )
!= $true ) ) )
& ! [X219: d_subst] : ( d_id = X219 )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( apnotvar = $true )
& ( ( sP17 = $true )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( sP17 != $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP16 != $true ) )
& ( ( sP16 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( axclos = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( hoasinduction_lem3b = $true )
| ! [X221: subst > term > term] :
( ( sub @ sK564 @ ( push @ one @ sh ) )
!= ( X221 @ sh @ one ) ) )
& ( ! [X222: term] :
( ( sub @ X222 @ ( push @ one @ sh ) )
= ( sK565 @ X222 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ! [X225: subst > term > term] :
( ( X225 @ ( sK568 @ X225 ) @ ( sK567 @ X225 ) )
!= ( sub @ sK566 @ ( push @ ( sK567 @ X225 ) @ ( sK568 @ X225 ) ) ) ) )
& ( ! [X228: term,X230: term,X231: subst] :
( ( sK569 @ X228 @ X231 @ X230 )
= ( sub @ X228 @ ( push @ X230 @ X231 ) ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X232: subst] :
( ( comp @ X232 @ id )
= X232 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ( sK570
!= ( comp @ sK570 @ id ) ) )
& ( axassoc = $true )
& ( ! [X234: term > $o,X235: term] :
( ( ( sP10 @ X234 )
= $true )
| ( ( ( var @ ( sK571 @ X234 ) )
= $true )
& ( ( X234 @ ( sK571 @ X234 ) )
!= $true ) )
| ( ( X234 @ X235 )
= $true )
| ( ( ( X234 @ ( lam @ ( sK572 @ X234 ) ) )
!= $true )
& ( ( X234 @ ( sK572 @ X234 ) )
= $true ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ( ( ( sP10 @ sK573 )
!= $true )
& ! [X240: term] :
( ( ( var @ X240 )
!= $true )
| ( ( sK573 @ X240 )
= $true ) )
& ( ( sK573 @ sK574 )
!= $true )
& ! [X241: term] :
( ( $true
= ( sK573 @ ( lam @ X241 ) ) )
| ( ( sK573 @ X241 )
!= $true ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( sP9 = $true )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( sP9 != $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X242: subst,X243: term > $o,X244: term,X246: term] :
( ( sK575 @ X244 @ X243 @ X242 @ X246 )
= ( X243 @ ( sub @ X246 @ ( push @ X244 @ X242 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ! [X250: term > $o] :
( ( X250 @ ( sK579 @ X250 ) )
!= ( sK577 @ ( sub @ ( sK579 @ X250 ) @ ( push @ sK578 @ sK576 ) ) ) ) )
& ( ! [X252: term] :
( ( sub @ X252 @ id )
= X252 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ( ( sub @ sK580 @ id )
!= sK580 ) )
& ( ! [X254: term,X255: subst] :
( ( sub @ one @ ( push @ X254 @ X255 ) )
= X254 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ( sK581
!= ( sub @ one @ ( push @ sK581 @ sK582 ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP5 = $true )
| ( hoasinduction_lem2 != $true ) )
& ( ( hoasinduction_lem2 = $true )
| ( sP5 != $true ) )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ! [X258: term] :
( ( d2term @ ( sK583 @ X258 ) )
= X258 )
& ( ( axidl = $true )
| ( ( comp @ id @ sK584 )
!= sK584 ) )
& ( ! [X261: subst] :
( ( comp @ id @ X261 )
= X261 )
| ( axidl != $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( axshiftcons = $true )
| ( ( comp @ sh @ ( push @ sK586 @ sK585 ) )
!= sK585 ) )
& ( ! [X264: subst,X265: term] :
( ( comp @ sh @ ( push @ X265 @ X264 ) )
= X264 )
| ( axshiftcons != $true ) )
& ( pushprop_gthm = $true )
& ( induction2_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
| ( sP3 != $true ) )
& ( ( sP3 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ! [X266: term,X267: term,X268: term,X269: term] :
( ( X267 = X269 )
| ( ( ap @ ( sub @ X266 @ id ) @ X267 )
!= ( ap @ ( sub @ X268 @ id ) @ X269 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ( ( sK590 != sK588 )
& ( ( ap @ ( sub @ sK587 @ id ) @ sK588 )
= ( ap @ ( sub @ sK589 @ id ) @ sK590 ) ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( sP1 != $true ) )
& ( ( sP1 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( axapp = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[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])],[f475,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]) ).
thf(f476,plain,
( ? [X0: term,X1: subst > term > subst > $o] :
( ! [X2: term,X3: term] :
( ( ( X1 @ id @ X3 @ id )
!= $true )
| ( ( X1 @ id @ X2 @ id )
!= $true )
| ( ( X1 @ id @ ( ap @ ( sub @ X2 @ id ) @ X3 ) @ id )
= $true ) )
& ( $true
!= ( sP58 @ X1 ) )
& ( ( X1 @ id @ X0 @ id )
!= $true ) )
=> ( ! [X3: term,X2: term] :
( ( ( sK458 @ id @ X3 @ id )
!= $true )
| ( ( sK458 @ id @ X2 @ id )
!= $true )
| ( ( sK458 @ id @ ( ap @ ( sub @ X2 @ id ) @ X3 ) @ id )
= $true ) )
& ( ( sP58 @ sK458 )
!= $true )
& ( ( sK458 @ id @ sK457 @ id )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f477,plain,
! [X5: subst > term > subst > $o] :
( ? [X6: term,X7: term] :
( ( ( X5 @ id @ X7 @ id )
= $true )
& ( ( X5 @ id @ X6 @ id )
= $true )
& ( ( X5 @ id @ ( ap @ ( sub @ X6 @ id ) @ X7 ) @ id )
!= $true ) )
=> ( ( ( X5 @ id @ ( sK460 @ X5 ) @ id )
= $true )
& ( ( X5 @ id @ ( sK459 @ X5 ) @ id )
= $true )
& ( ( X5 @ id @ ( ap @ ( sub @ ( sK459 @ X5 ) @ id ) @ ( sK460 @ X5 ) ) @ id )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f478,plain,
( ? [X9: term] :
( ( sub @ X9 @ id )
!= X9 )
=> ( ( sub @ sK461 @ id )
!= sK461 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f479,plain,
( ? [X10: term] :
( ( sub @ X10 @ id )
!= X10 )
=> ( sK462
!= ( sub @ sK462 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f480,plain,
! [X12: subst > term > subst > $o] :
( ? [X13: term > $o] :
! [X14: term] :
( ( X13 @ X14 )
= ( X12 @ id @ X14 @ id ) )
=> ! [X14: term] :
( ( sK463 @ X12 @ X14 )
= ( X12 @ id @ X14 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f481,plain,
( ? [X15: subst > term > subst > $o] :
! [X16: term > $o] :
? [X17: term] :
( ( X16 @ X17 )
!= ( X15 @ id @ X17 @ id ) )
=> ! [X16: term > $o] :
? [X17: term] :
( ( X16 @ X17 )
!= ( sK464 @ id @ X17 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f482,plain,
! [X16: term > $o] :
( ? [X17: term] :
( ( X16 @ X17 )
!= ( sK464 @ id @ X17 @ id ) )
=> ( ( X16 @ ( sK465 @ X16 ) )
!= ( sK464 @ id @ ( sK465 @ X16 ) @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f483,plain,
! [X18: subst > term > subst > $o] :
( ? [X19: term > $o] :
! [X20: term] :
( ( X18 @ id @ X20 @ id )
= ( X19 @ X20 ) )
=> ! [X20: term] :
( ( sK466 @ X18 @ X20 )
= ( X18 @ id @ X20 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f484,plain,
( ? [X21: subst > term > subst > $o] :
! [X22: term > $o] :
? [X23: term] :
( ( X21 @ id @ X23 @ id )
!= ( X22 @ X23 ) )
=> ! [X22: term > $o] :
? [X23: term] :
( ( sK467 @ id @ X23 @ id )
!= ( X22 @ X23 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f485,plain,
! [X22: term > $o] :
( ? [X23: term] :
( ( sK467 @ id @ X23 @ id )
!= ( X22 @ X23 ) )
=> ( ( sK467 @ id @ ( sK468 @ X22 ) @ id )
!= ( X22 @ ( sK468 @ X22 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f486,plain,
( ? [X24: subst] :
( ( comp @ X24 @ id )
!= X24 )
=> ( ( comp @ sK469 @ id )
!= sK469 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f487,plain,
( ? [X25: subst] :
( ( comp @ id @ X25 )
!= X25 )
=> ( ( comp @ id @ sK470 )
!= sK470 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f488,plain,
( ? [X30: subst] :
( ( comp @ id @ X30 )
!= X30 )
=> ( ( comp @ id @ sK471 )
!= sK471 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f489,plain,
( ? [X31: subst] :
( ( comp @ X31 @ id )
!= X31 )
=> ( sK472
!= ( comp @ sK472 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f490,plain,
( ? [X38: term,X39: term,X40: term,X41: term] :
( ( X38 != X39 )
& ( ( ap @ X39 @ X41 )
= ( ap @ X38 @ X40 ) ) )
=> ( ( sK473 != sK474 )
& ( ( ap @ sK474 @ sK476 )
= ( ap @ sK473 @ sK475 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f491,plain,
( ? [X49: term,X50: term,X51: term,X52: term] :
( ( ( ap @ X50 @ X52 )
= ( ap @ X51 @ X49 ) )
& ( X49 != X52 ) )
=> ( ( ( ap @ sK478 @ sK480 )
= ( ap @ sK479 @ sK477 ) )
& ( sK477 != sK480 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f492,plain,
( ? [X57: term,X58: term,X59: term,X60: term] :
( ( ( ap @ ( sub @ X58 @ id ) @ X60 )
= ( ap @ ( sub @ X59 @ id ) @ X57 ) )
& ( X58 != X59 ) )
=> ( ( ( ap @ ( sub @ sK482 @ id ) @ sK484 )
= ( ap @ ( sub @ sK483 @ id ) @ sK481 ) )
& ( sK483 != sK482 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f493,plain,
( ? [X61: term,X62: subst > term > term,X63: subst,X64: subst > term > term] :
( ( ( lam @ ( X64 @ sh @ one ) )
= ( lam @ ( X62 @ sh @ one ) ) )
& ( ( X62 @ X63 @ X61 )
!= ( X64 @ X63 @ X61 ) )
& ! [X65: term,X66: subst,X67: subst] :
( ( X62 @ ( comp @ X67 @ X66 ) @ ( sub @ X65 @ X66 ) )
= ( sub @ ( X62 @ X67 @ X65 ) @ X66 ) )
& ! [X68: subst,X69: term,X70: subst] :
( ( sub @ ( X64 @ X70 @ X69 ) @ X68 )
= ( X64 @ ( comp @ X70 @ X68 ) @ ( sub @ X69 @ X68 ) ) ) )
=> ( ( ( lam @ ( sK486 @ sh @ one ) )
= ( lam @ ( sK488 @ sh @ one ) ) )
& ( ( sK488 @ sK487 @ sK485 )
!= ( sK486 @ sK487 @ sK485 ) )
& ! [X67: subst,X66: subst,X65: term] :
( ( sK486 @ ( comp @ X67 @ X66 ) @ ( sub @ X65 @ X66 ) )
= ( sub @ ( sK486 @ X67 @ X65 ) @ X66 ) )
& ! [X70: subst,X69: term,X68: subst] :
( ( sub @ ( sK488 @ X70 @ X69 ) @ X68 )
= ( sK488 @ ( comp @ X70 @ X68 ) @ ( sub @ X69 @ X68 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f494,plain,
! [X72: subst > term > term] :
( ? [X75: term,X76: subst,X77: subst] :
( ( sub @ ( X72 @ X77 @ X75 ) @ X76 )
!= ( X72 @ ( comp @ X77 @ X76 ) @ ( sub @ X75 @ X76 ) ) )
=> ( ( X72 @ ( comp @ ( sK491 @ X72 ) @ ( sK490 @ X72 ) ) @ ( sub @ ( sK489 @ X72 ) @ ( sK490 @ X72 ) ) )
!= ( sub @ ( X72 @ ( sK491 @ X72 ) @ ( sK489 @ X72 ) ) @ ( sK490 @ X72 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f495,plain,
! [X74: subst > term > term] :
( ? [X78: subst,X79: term,X80: subst] :
( ( X74 @ ( comp @ X80 @ X78 ) @ ( sub @ X79 @ X78 ) )
!= ( sub @ ( X74 @ X80 @ X79 ) @ X78 ) )
=> ( ( X74 @ ( comp @ ( sK494 @ X74 ) @ ( sK492 @ X74 ) ) @ ( sub @ ( sK493 @ X74 ) @ ( sK492 @ X74 ) ) )
!= ( sub @ ( X74 @ ( sK494 @ X74 ) @ ( sK493 @ X74 ) ) @ ( sK492 @ X74 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f496,plain,
( ? [X81: term,X82: subst > term > term,X83: term] :
( ! [X84: subst,X85: subst,X86: term] :
( ( sub @ ( X82 @ X84 @ X86 ) @ X85 )
= ( X82 @ ( comp @ X84 @ X85 ) @ ( sub @ X86 @ X85 ) ) )
& ( ( lam @ ( X82 @ sh @ one ) )
= ( ap @ ( sub @ X81 @ id ) @ X83 ) ) )
=> ( ! [X86: term,X85: subst,X84: subst] :
( ( sub @ ( sK496 @ X84 @ X86 ) @ X85 )
= ( sK496 @ ( comp @ X84 @ X85 ) @ ( sub @ X86 @ X85 ) ) )
& ( ( ap @ ( sub @ sK495 @ id ) @ sK497 )
= ( lam @ ( sK496 @ sh @ one ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f497,plain,
! [X88: subst > term > term] :
( ? [X90: subst,X91: subst,X92: term] :
( ( sub @ ( X88 @ X90 @ X92 ) @ X91 )
!= ( X88 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) ) )
=> ( ( sub @ ( X88 @ ( sK498 @ X88 ) @ ( sK500 @ X88 ) ) @ ( sK499 @ X88 ) )
!= ( X88 @ ( comp @ ( sK498 @ X88 ) @ ( sK499 @ X88 ) ) @ ( sub @ ( sK500 @ X88 ) @ ( sK499 @ X88 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f498,plain,
( ? [X93: term,X94: term,X95: term,X96: term] :
( ( X93 != X96 )
& ( ( ap @ ( sub @ X96 @ id ) @ X94 )
= ( ap @ ( sub @ X93 @ id ) @ X95 ) ) )
=> ( ( sK504 != sK501 )
& ( ( ap @ ( sub @ sK504 @ id ) @ sK502 )
= ( ap @ ( sub @ sK501 @ id ) @ sK503 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f499,plain,
( ? [X106: term,X107: term,X108: term,X109: term] :
( ( ( ap @ X109 @ X107 )
= ( ap @ X106 @ X108 ) )
& ( X106 != X109 ) )
=> ( ( ( ap @ sK505 @ sK507 )
= ( ap @ sK508 @ sK506 ) )
& ( sK508 != sK505 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f500,plain,
( ? [X110: term] :
( ( sub @ X110 @ id )
!= X110 )
=> ( ( sub @ sK509 @ id )
!= sK509 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f501,plain,
( ? [X112: term] :
( ( sub @ X112 @ id )
!= X112 )
=> ( ( sub @ sK510 @ id )
!= sK510 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f502,plain,
( ? [X113: term > $o,X114: term,X115: term > $o,X116: term,X117: subst] :
( ( $true
= ( X113 @ X114 ) )
& ( ( X115 @ ( sub @ X114 @ ( push @ X116 @ X117 ) ) )
!= $true ) )
=> ( ( ( sK511 @ sK512 )
= $true )
& ( $true
!= ( sK513 @ ( sub @ sK512 @ ( push @ sK514 @ sK515 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f503,plain,
( ? [X123: subst] :
( ( comp @ id @ X123 )
!= X123 )
=> ( ( comp @ id @ sK516 )
!= sK516 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f504,plain,
( ? [X124: subst] :
( ( comp @ X124 @ id )
!= X124 )
=> ( ( comp @ sK517 @ id )
!= sK517 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f505,plain,
( ? [X127: term,X128: subst] :
( ( comp @ sh @ ( push @ X127 @ X128 ) )
!= X128 )
=> ( ( comp @ sh @ ( push @ sK518 @ sK519 ) )
!= sK519 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f506,plain,
( ? [X134: term,X135: subst] :
( ( sub @ one @ ( push @ X134 @ X135 ) )
!= X134 )
=> ( sK520
!= ( sub @ one @ ( push @ sK520 @ sK521 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f507,plain,
( ? [X138: term > $o,X139: subst,X140: term,X141: term > $o,X142: term] :
( ( ( X138 @ X140 )
= $true )
& ( ( X141 @ ( sub @ X140 @ ( push @ X142 @ X139 ) ) )
!= $true ) )
=> ( ( ( sK522 @ sK524 )
= $true )
& ( ( sK525 @ ( sub @ sK524 @ ( push @ sK526 @ sK523 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f508,plain,
( ? [X145: term,X146: term > $o,X147: subst] :
! [X148: term > $o] :
? [X149: term] :
( ( X148 @ X149 )
!= ( X146 @ ( sub @ X149 @ ( push @ X145 @ X147 ) ) ) )
=> ! [X148: term > $o] :
? [X149: term] :
( ( X148 @ X149 )
!= ( sK528 @ ( sub @ X149 @ ( push @ sK527 @ sK529 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f509,plain,
! [X148: term > $o] :
( ? [X149: term] :
( ( X148 @ X149 )
!= ( sK528 @ ( sub @ X149 @ ( push @ sK527 @ sK529 ) ) ) )
=> ( ( sK528 @ ( sub @ ( sK530 @ X148 ) @ ( push @ sK527 @ sK529 ) ) )
!= ( X148 @ ( sK530 @ X148 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f510,plain,
! [X150: term,X151: term > $o,X152: subst] :
( ? [X153: term > $o] :
! [X154: term] :
( ( X153 @ X154 )
= ( X151 @ ( sub @ X154 @ ( push @ X150 @ X152 ) ) ) )
=> ! [X154: term] :
( ( sK531 @ X152 @ X151 @ X150 @ X154 )
= ( X151 @ ( sub @ X154 @ ( push @ X150 @ X152 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f511,plain,
( ? [X155: term,X156: term] :
( ( X155 != X156 )
& ( ( sub @ X156 @ sh )
= ( sub @ X155 @ sh ) ) )
=> ( ( sK532 != sK533 )
& ( ( sub @ sK533 @ sh )
= ( sub @ sK532 @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f512,plain,
! [X159: subst] :
( ? [X160: d_subst] :
( ( d2subst @ X160 )
= X159 )
=> ( ( d2subst @ ( sK534 @ X159 ) )
= X159 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f513,plain,
( ? [X162: subst] :
( ( push @ ( sub @ one @ X162 ) @ ( comp @ sh @ X162 ) )
!= X162 )
=> ( ( push @ ( sub @ one @ sK535 ) @ ( comp @ sh @ sK535 ) )
!= sK535 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f514,plain,
( ? [X163: term] :
! [X164: subst > term > term] :
? [X165: subst,X166: term] :
( ( sub @ X163 @ ( push @ X166 @ X165 ) )
!= ( X164 @ X165 @ X166 ) )
=> ! [X164: subst > term > term] :
? [X166: term,X165: subst] :
( ( X164 @ X165 @ X166 )
!= ( sub @ sK536 @ ( push @ X166 @ X165 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f515,plain,
! [X164: subst > term > term] :
( ? [X166: term,X165: subst] :
( ( X164 @ X165 @ X166 )
!= ( sub @ sK536 @ ( push @ X166 @ X165 ) ) )
=> ( ( X164 @ ( sK537 @ X164 ) @ ( sK538 @ X164 ) )
!= ( sub @ sK536 @ ( push @ ( sK538 @ X164 ) @ ( sK537 @ X164 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f516,plain,
! [X167: term] :
( ? [X168: subst > term > term] :
! [X169: subst,X170: term] :
( ( sub @ X167 @ ( push @ X170 @ X169 ) )
= ( X168 @ X169 @ X170 ) )
=> ! [X170: term,X169: subst] :
( ( sub @ X167 @ ( push @ X170 @ X169 ) )
= ( sK539 @ X167 @ X169 @ X170 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f517,plain,
( ? [X173: term,X174: term] :
( ( X173 != X174 )
& ( ( lam @ X174 )
= ( lam @ X173 ) ) )
=> ( ( sK540 != sK541 )
& ( ( lam @ sK540 )
= ( lam @ sK541 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f518,plain,
! [X175: term] :
( ? [X176: subst > term > term] :
( ( sub @ X175 @ ( push @ one @ sh ) )
= ( X176 @ sh @ one ) )
=> ( ( sub @ X175 @ ( push @ one @ sh ) )
= ( sK542 @ X175 @ sh @ one ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f519,plain,
( ? [X177: term] :
! [X178: subst > term > term] :
( ( X178 @ sh @ one )
!= ( sub @ X177 @ ( push @ one @ sh ) ) )
=> ! [X178: subst > term > term] :
( ( X178 @ sh @ one )
!= ( sub @ sK543 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f520,plain,
( ? [X179: term,X180: term,X181: term,X182: term] :
( ( X179 != X180 )
& ( ( ap @ ( sub @ X181 @ id ) @ X179 )
= ( ap @ ( sub @ X182 @ id ) @ X180 ) ) )
=> ( ( sK545 != sK544 )
& ( ( ap @ ( sub @ sK546 @ id ) @ sK544 )
= ( ap @ ( sub @ sK547 @ id ) @ sK545 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f521,plain,
( ? [X191: term,X192: term,X193: term,X194: term] :
( ( ( ap @ X194 @ X191 )
= ( ap @ X193 @ X192 ) )
& ( X191 != X192 ) )
=> ( ( ( ap @ sK550 @ sK549 )
= ( ap @ sK551 @ sK548 ) )
& ( sK548 != sK549 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f522,plain,
( ? [X195: subst,X196: term,X197: term > $o,X198: term > $o] :
( ( ( X198 @ one )
!= $true )
& ( ( X197 @ X196 )
= $true )
& ! [X199: term] :
( ( X198 @ X199 )
= ( X197 @ ( sub @ X199 @ ( push @ X196 @ X195 ) ) ) ) )
=> ( ( ( sK555 @ one )
!= $true )
& ( ( sK554 @ sK553 )
= $true )
& ! [X199: term] :
( ( sK554 @ ( sub @ X199 @ ( push @ sK553 @ sK552 ) ) )
= ( sK555 @ X199 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f523,plain,
! [X200: subst,X201: term,X202: term > $o,X203: term > $o] :
( ? [X204: term] :
( ( X202 @ ( sub @ X204 @ ( push @ X201 @ X200 ) ) )
!= ( X203 @ X204 ) )
=> ( ( X203 @ ( sK556 @ X203 @ X202 @ X201 @ X200 ) )
!= ( X202 @ ( sub @ ( sK556 @ X203 @ X202 @ X201 @ X200 ) @ ( push @ X201 @ X200 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f524,plain,
( ? [X205: term,X206: subst] :
( ( sub @ one @ ( push @ X205 @ X206 ) )
!= X205 )
=> ( sK557
!= ( sub @ one @ ( push @ sK557 @ sK558 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f525,plain,
! [X207: term > $o,X208: subst,X209: term > $o,X210: term] :
( ? [X211: term] :
( ( X209 @ X211 )
!= ( X207 @ ( sub @ X211 @ ( push @ X210 @ X208 ) ) ) )
=> ( ( X207 @ ( sub @ ( sK559 @ X210 @ X209 @ X208 @ X207 ) @ ( push @ X210 @ X208 ) ) )
!= ( X209 @ ( sK559 @ X210 @ X209 @ X208 @ X207 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f526,plain,
( ? [X214: term > $o,X215: subst,X216: term > $o,X217: term] :
( ( ( X214 @ X217 )
= $true )
& ! [X218: term] :
( ( X214 @ ( sub @ X218 @ ( push @ X217 @ X215 ) ) )
= ( X216 @ X218 ) )
& ( ( X216 @ one )
!= $true ) )
=> ( ( ( sK560 @ sK563 )
= $true )
& ! [X218: term] :
( ( sK562 @ X218 )
= ( sK560 @ ( sub @ X218 @ ( push @ sK563 @ sK561 ) ) ) )
& ( ( sK562 @ one )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f527,plain,
( ? [X220: term] :
! [X221: subst > term > term] :
( ( sub @ X220 @ ( push @ one @ sh ) )
!= ( X221 @ sh @ one ) )
=> ! [X221: subst > term > term] :
( ( sub @ sK564 @ ( push @ one @ sh ) )
!= ( X221 @ sh @ one ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f528,plain,
! [X222: term] :
( ? [X223: subst > term > term] :
( ( sub @ X222 @ ( push @ one @ sh ) )
= ( X223 @ sh @ one ) )
=> ( ( sub @ X222 @ ( push @ one @ sh ) )
= ( sK565 @ X222 @ sh @ one ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f529,plain,
( ? [X224: term] :
! [X225: subst > term > term] :
? [X226: term,X227: subst] :
( ( X225 @ X227 @ X226 )
!= ( sub @ X224 @ ( push @ X226 @ X227 ) ) )
=> ! [X225: subst > term > term] :
? [X227: subst,X226: term] :
( ( sub @ sK566 @ ( push @ X226 @ X227 ) )
!= ( X225 @ X227 @ X226 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f530,plain,
! [X225: subst > term > term] :
( ? [X227: subst,X226: term] :
( ( sub @ sK566 @ ( push @ X226 @ X227 ) )
!= ( X225 @ X227 @ X226 ) )
=> ( ( X225 @ ( sK568 @ X225 ) @ ( sK567 @ X225 ) )
!= ( sub @ sK566 @ ( push @ ( sK567 @ X225 ) @ ( sK568 @ X225 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f531,plain,
! [X228: term] :
( ? [X229: subst > term > term] :
! [X230: term,X231: subst] :
( ( X229 @ X231 @ X230 )
= ( sub @ X228 @ ( push @ X230 @ X231 ) ) )
=> ! [X231: subst,X230: term] :
( ( sK569 @ X228 @ X231 @ X230 )
= ( sub @ X228 @ ( push @ X230 @ X231 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f532,plain,
( ? [X233: subst] :
( ( comp @ X233 @ id )
!= X233 )
=> ( sK570
!= ( comp @ sK570 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f533,plain,
! [X234: term > $o] :
( ? [X236: term] :
( ( $true
= ( var @ X236 ) )
& ( ( X234 @ X236 )
!= $true ) )
=> ( ( ( var @ ( sK571 @ X234 ) )
= $true )
& ( ( X234 @ ( sK571 @ X234 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f534,plain,
! [X234: term > $o] :
( ? [X237: term] :
( ( ( X234 @ ( lam @ X237 ) )
!= $true )
& ( ( X234 @ X237 )
= $true ) )
=> ( ( ( X234 @ ( lam @ ( sK572 @ X234 ) ) )
!= $true )
& ( ( X234 @ ( sK572 @ X234 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f535,plain,
( ? [X238: term > $o,X239: term] :
( ( ( sP10 @ X238 )
!= $true )
& ! [X240: term] :
( ( ( var @ X240 )
!= $true )
| ( ( X238 @ X240 )
= $true ) )
& ( ( X238 @ X239 )
!= $true )
& ! [X241: term] :
( ( ( X238 @ ( lam @ X241 ) )
= $true )
| ( ( X238 @ X241 )
!= $true ) ) )
=> ( ( ( sP10 @ sK573 )
!= $true )
& ! [X240: term] :
( ( ( var @ X240 )
!= $true )
| ( ( sK573 @ X240 )
= $true ) )
& ( ( sK573 @ sK574 )
!= $true )
& ! [X241: term] :
( ( $true
= ( sK573 @ ( lam @ X241 ) ) )
| ( ( sK573 @ X241 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f536,plain,
! [X242: subst,X243: term > $o,X244: term] :
( ? [X245: term > $o] :
! [X246: term] :
( ( X245 @ X246 )
= ( X243 @ ( sub @ X246 @ ( push @ X244 @ X242 ) ) ) )
=> ! [X246: term] :
( ( sK575 @ X244 @ X243 @ X242 @ X246 )
= ( X243 @ ( sub @ X246 @ ( push @ X244 @ X242 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f537,plain,
( ? [X247: subst,X248: term > $o,X249: term] :
! [X250: term > $o] :
? [X251: term] :
( ( X250 @ X251 )
!= ( X248 @ ( sub @ X251 @ ( push @ X249 @ X247 ) ) ) )
=> ! [X250: term > $o] :
? [X251: term] :
( ( X250 @ X251 )
!= ( sK577 @ ( sub @ X251 @ ( push @ sK578 @ sK576 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f538,plain,
! [X250: term > $o] :
( ? [X251: term] :
( ( X250 @ X251 )
!= ( sK577 @ ( sub @ X251 @ ( push @ sK578 @ sK576 ) ) ) )
=> ( ( X250 @ ( sK579 @ X250 ) )
!= ( sK577 @ ( sub @ ( sK579 @ X250 ) @ ( push @ sK578 @ sK576 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f539,plain,
( ? [X253: term] :
( ( sub @ X253 @ id )
!= X253 )
=> ( ( sub @ sK580 @ id )
!= sK580 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f540,plain,
( ? [X256: term,X257: subst] :
( ( sub @ one @ ( push @ X256 @ X257 ) )
!= X256 )
=> ( sK581
!= ( sub @ one @ ( push @ sK581 @ sK582 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f541,plain,
! [X258: term] :
( ? [X259: d_term] :
( ( d2term @ X259 )
= X258 )
=> ( ( d2term @ ( sK583 @ X258 ) )
= X258 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f542,plain,
( ? [X260: subst] :
( ( comp @ id @ X260 )
!= X260 )
=> ( ( comp @ id @ sK584 )
!= sK584 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f543,plain,
( ? [X262: subst,X263: term] :
( ( comp @ sh @ ( push @ X263 @ X262 ) )
!= X262 )
=> ( ( comp @ sh @ ( push @ sK586 @ sK585 ) )
!= sK585 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f544,plain,
( ? [X270: term,X271: term,X272: term,X273: term] :
( ( X271 != X273 )
& ( ( ap @ ( sub @ X270 @ id ) @ X271 )
= ( ap @ ( sub @ X272 @ id ) @ X273 ) ) )
=> ( ( sK590 != sK588 )
& ( ( ap @ ( sub @ sK587 @ id ) @ sK588 )
= ( ap @ ( sub @ sK589 @ id ) @ sK590 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f475,plain,
( ( ( hoasinduction_no_psi_cond = $true )
| ? [X0: term,X1: subst > term > subst > $o] :
( ! [X2: term,X3: term] :
( ( ( X1 @ id @ X3 @ id )
!= $true )
| ( ( X1 @ id @ X2 @ id )
!= $true )
| ( ( X1 @ id @ ( ap @ ( sub @ X2 @ id ) @ X3 ) @ id )
= $true ) )
& ( $true
!= ( sP58 @ X1 ) )
& ( ( X1 @ id @ X0 @ id )
!= $true ) ) )
& ( ! [X4: term,X5: subst > term > subst > $o] :
( ? [X6: term,X7: term] :
( ( ( X5 @ id @ X7 @ id )
= $true )
& ( ( X5 @ id @ X6 @ id )
= $true )
& ( ( X5 @ id @ ( ap @ ( sub @ X6 @ id ) @ X7 ) @ id )
!= $true ) )
| ( ( sP58 @ X5 )
= $true )
| ( ( X5 @ id @ X4 @ id )
= $true ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( sP57 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP57 != $true ) )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP53 != $true ) )
& ( ( sP53 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( termmset_lthm = $true )
| ( ! [X8: term] :
( ( sub @ X8 @ id )
= X8 )
& ? [X9: term] :
( ( sub @ X9 @ id )
!= X9 ) ) )
& ( ? [X10: term] :
( ( sub @ X10 @ id )
!= X10 )
| ! [X11: term] :
( ( sub @ X11 @ id )
= X11 )
| ( termmset_lthm != $true ) )
& ( ! [X12: subst > term > subst > $o] :
? [X13: term > $o] :
! [X14: term] :
( ( X13 @ X14 )
= ( X12 @ id @ X14 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X15: subst > term > subst > $o] :
! [X16: term > $o] :
? [X17: term] :
( ( X16 @ X17 )
!= ( X15 @ id @ X17 @ id ) ) )
& ( axmap = $true )
& ( ulamvarsh = $true )
& ( ! [X18: subst > term > subst > $o] :
? [X19: term > $o] :
! [X20: term] :
( ( X18 @ id @ X20 @ id )
= ( X19 @ X20 ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X21: subst > term > subst > $o] :
! [X22: term > $o] :
? [X23: term] :
( ( X21 @ id @ X23 @ id )
!= ( X22 @ X23 ) ) )
& ( ? [X24: subst] :
( ( comp @ X24 @ id )
!= X24 )
| ? [X25: subst] :
( ( comp @ id @ X25 )
!= X25 )
| ( ! [X26: subst] :
( ( comp @ id @ X26 )
= X26 )
& ! [X27: subst] :
( ( comp @ X27 @ id )
= X27 ) )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ! [X28: subst] :
( ( comp @ X28 @ id )
= X28 )
& ! [X29: subst] :
( ( comp @ id @ X29 )
= X29 )
& ( ? [X30: subst] :
( ( comp @ id @ X30 )
!= X30 )
| ? [X31: subst] :
( ( comp @ X31 @ id )
!= X31 ) ) ) )
& ( ( sP48 = $true )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( sP48 != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( sP47 != $true ) )
& ( ( sP47 = $true )
| ( hoasinduction_lem3aa != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( substmonoid_gthm = $true )
& ! [X32: d_term,X33: d_term] :
( ( ( d2term @ X32 )
!= ( d2term @ X33 ) )
| ( X32 = X33 ) )
& ( ulamvarind = $true )
& ( pushprop_lthm_orig = $true )
& ( ! [X34: term,X35: term,X36: term,X37: term] :
( ( X34 = X35 )
| ( ( ap @ X34 @ X36 )
!= ( ap @ X35 @ X37 ) ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X38: term,X39: term,X40: term,X41: term] :
( ( X38 != X39 )
& ( ( ap @ X39 @ X41 )
= ( ap @ X38 @ X40 ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ! [X42: d_subst,X43: d_subst] :
( ( X42 = X43 )
| ( ( d2subst @ X43 )
!= ( d2subst @ X42 ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP46 != $true ) )
& ( ( sP46 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ! [X44: d_term] : ( d_one = X44 )
& ( ( sP44 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP44 != $true ) )
& ( ! [X45: term,X46: term,X47: term,X48: term] :
( ( ( ap @ X46 @ X48 )
!= ( ap @ X47 @ X45 ) )
| ( X45 = X48 ) )
| ( apinj2 != $true ) )
& ( ( apinj2 = $true )
| ? [X49: term,X50: term,X51: term,X52: term] :
( ( ( ap @ X50 @ X52 )
= ( ap @ X51 @ X49 ) )
& ( X49 != X52 ) ) )
& ( lamnotvar = $true )
& ( axabs = $true )
& ( ( hoasinduction = $true )
| ( sP42 != $true ) )
& ( ( sP42 = $true )
| ( hoasinduction != $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( pushprop_lem0_gthm = $true )
& ( ! [X53: term,X54: term,X55: term,X56: term] :
( ( ( ap @ ( sub @ X55 @ id ) @ X53 )
!= ( ap @ ( sub @ X54 @ id ) @ X56 ) )
| ( X54 = X55 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X57: term,X58: term,X59: term,X60: term] :
( ( ( ap @ ( sub @ X58 @ id ) @ X60 )
= ( ap @ ( sub @ X59 @ id ) @ X57 ) )
& ( X58 != X59 ) ) )
& ( hoasapnotvar_gthm = $true )
& ( ( hoaslaminj = $true )
| ? [X61: term,X62: subst > term > term,X63: subst,X64: subst > term > term] :
( ( ( lam @ ( X64 @ sh @ one ) )
= ( lam @ ( X62 @ sh @ one ) ) )
& ( ( X62 @ X63 @ X61 )
!= ( X64 @ X63 @ X61 ) )
& ! [X65: term,X66: subst,X67: subst] :
( ( X62 @ ( comp @ X67 @ X66 ) @ ( sub @ X65 @ X66 ) )
= ( sub @ ( X62 @ X67 @ X65 ) @ X66 ) )
& ! [X68: subst,X69: term,X70: subst] :
( ( sub @ ( X64 @ X70 @ X69 ) @ X68 )
= ( X64 @ ( comp @ X70 @ X68 ) @ ( sub @ X69 @ X68 ) ) ) ) )
& ( ! [X71: term,X72: subst > term > term,X73: subst,X74: subst > term > term] :
( ( ( lam @ ( X72 @ sh @ one ) )
!= ( lam @ ( X74 @ sh @ one ) ) )
| ( ( X72 @ X73 @ X71 )
= ( X74 @ X73 @ X71 ) )
| ? [X75: term,X76: subst,X77: subst] :
( ( sub @ ( X72 @ X77 @ X75 ) @ X76 )
!= ( X72 @ ( comp @ X77 @ X76 ) @ ( sub @ X75 @ X76 ) ) )
| ? [X78: subst,X79: term,X80: subst] :
( ( X74 @ ( comp @ X80 @ X78 ) @ ( sub @ X79 @ X78 ) )
!= ( sub @ ( X74 @ X80 @ X79 ) @ X78 ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X81: term,X82: subst > term > term,X83: term] :
( ! [X84: subst,X85: subst,X86: term] :
( ( sub @ ( X82 @ X84 @ X86 ) @ X85 )
= ( X82 @ ( comp @ X84 @ X85 ) @ ( sub @ X86 @ X85 ) ) )
& ( ( lam @ ( X82 @ sh @ one ) )
= ( ap @ ( sub @ X81 @ id ) @ X83 ) ) ) )
& ( ! [X87: term,X88: subst > term > term,X89: term] :
( ? [X90: subst,X91: subst,X92: term] :
( ( sub @ ( X88 @ X90 @ X92 ) @ X91 )
!= ( X88 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) ) )
| ( ( ap @ ( sub @ X87 @ id ) @ X89 )
!= ( lam @ ( X88 @ sh @ one ) ) ) )
| ( hoaslamnotap != $true ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
| ( ? [X93: term,X94: term,X95: term,X96: term] :
( ( X93 != X96 )
& ( ( ap @ ( sub @ X96 @ id ) @ X94 )
= ( ap @ ( sub @ X93 @ id ) @ X95 ) ) )
& ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X97 @ X99 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X100 ) )
& ! [X101: term] :
( ( sub @ X101 @ id )
= X101 ) ) )
& ( ! [X102: term,X103: term,X104: term,X105: term] :
( ( X102 = X105 )
| ( ( ap @ ( sub @ X105 @ id ) @ X103 )
!= ( ap @ ( sub @ X102 @ id ) @ X104 ) ) )
| ? [X106: term,X107: term,X108: term,X109: term] :
( ( ( ap @ X109 @ X107 )
= ( ap @ X106 @ X108 ) )
& ( X106 != X109 ) )
| ? [X110: term] :
( ( sub @ X110 @ id )
!= X110 )
| ( hoasapinj1_lthm != $true ) )
& ( ! [X111: term] :
( ( sub @ X111 @ id )
= X111 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X112: term] :
( ( sub @ X112 @ id )
!= X112 ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( pushprop_lem1 = $true )
| ? [X113: term > $o,X114: term,X115: term > $o,X116: term,X117: subst] :
( ( $true
= ( X113 @ X114 ) )
& ( ( X115 @ ( sub @ X114 @ ( push @ X116 @ X117 ) ) )
!= $true ) ) )
& ( ! [X118: term > $o,X119: term,X120: term > $o,X121: term,X122: subst] :
( ( ( X118 @ X119 )
!= $true )
| ( ( X120 @ ( sub @ X119 @ ( push @ X121 @ X122 ) ) )
= $true ) )
| ( pushprop_lem1 != $true ) )
& ( ( sP39 = $true )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( sP39 != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP37 != $true ) )
& ( ( sP37 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( hoasinduction_lem1 = $true )
& ( hoaslaminj_gthm = $true )
& ( pushprop = $true )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( substmonoid = $true )
| ? [X123: subst] :
( ( comp @ id @ X123 )
!= X123 )
| ? [X124: subst] :
( ( comp @ X124 @ id )
!= X124 ) )
& ( ( ! [X125: subst] :
( ( comp @ id @ X125 )
= X125 )
& ! [X126: subst] :
( ( comp @ X126 @ id )
= X126 ) )
| ( substmonoid != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP36 ) )
& ( ( $true = sP36 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ulamvar1 != $true )
& ( ? [X127: term,X128: subst] :
( ( comp @ sh @ ( push @ X127 @ X128 ) )
!= X128 )
| ! [X129: term > $o,X130: subst,X131: term,X132: term > $o,X133: term] :
( ( ( X129 @ X131 )
!= $true )
| ( ( X132 @ ( sub @ X131 @ ( push @ X133 @ X130 ) ) )
= $true ) )
| ? [X134: term,X135: subst] :
( ( sub @ one @ ( push @ X134 @ X135 ) )
!= X134 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X136: term,X137: subst] :
( ( comp @ sh @ ( push @ X136 @ X137 ) )
= X137 )
& ? [X138: term > $o,X139: subst,X140: term,X141: term > $o,X142: term] :
( ( ( X138 @ X140 )
= $true )
& ( ( X141 @ ( sub @ X140 @ ( push @ X142 @ X139 ) ) )
!= $true ) )
& ! [X143: term,X144: subst] :
( ( sub @ one @ ( push @ X143 @ X144 ) )
= X143 ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X145: term,X146: term > $o,X147: subst] :
! [X148: term > $o] :
? [X149: term] :
( ( X148 @ X149 )
!= ( X146 @ ( sub @ X149 @ ( push @ X145 @ X147 ) ) ) ) )
& ( ! [X150: term,X151: term > $o,X152: subst] :
? [X153: term > $o] :
! [X154: term] :
( ( X153 @ X154 )
= ( X151 @ ( sub @ X154 @ ( push @ X150 @ X152 ) ) ) )
| ( pushprop_lem0_lthm != $true ) )
& ( one
= ( d2term @ d_one ) )
& ( ( shinj = $true )
| ? [X155: term,X156: term] :
( ( X155 != X156 )
& ( ( sub @ X156 @ sh )
= ( sub @ X155 @ sh ) ) ) )
& ( ! [X157: term,X158: term] :
( ( X157 = X158 )
| ( ( sub @ X158 @ sh )
!= ( sub @ X157 @ sh ) ) )
| ( shinj != $true ) )
& ( axvarshift = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2subst @ d_id )
= 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 ) ) ) ) ) ) )
& ! [X159: subst] :
? [X160: d_subst] :
( ( d2subst @ X160 )
= X159 )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP35 != $true ) )
& ( ( sP35 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ! [X161: subst] :
( ( push @ ( sub @ one @ X161 ) @ ( comp @ sh @ X161 ) )
= X161 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X162: subst] :
( ( push @ ( sub @ one @ X162 ) @ ( comp @ sh @ X162 ) )
!= X162 ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X163: term] :
! [X164: subst > term > term] :
? [X165: subst,X166: term] :
( ( sub @ X163 @ ( push @ X166 @ X165 ) )
!= ( X164 @ X165 @ X166 ) ) )
& ( ! [X167: term] :
? [X168: subst > term > term] :
! [X169: subst,X170: term] :
( ( sub @ X167 @ ( push @ X170 @ X169 ) )
= ( X168 @ X169 @ X170 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ! [X171: term,X172: term] :
( ( X171 = X172 )
| ( ( lam @ X172 )
!= ( lam @ X171 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X173: term,X174: term] :
( ( X173 != X174 )
& ( ( lam @ X174 )
= ( lam @ X173 ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X175: term] :
? [X176: subst > term > term] :
( ( sub @ X175 @ ( push @ one @ sh ) )
= ( X176 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X177: term] :
! [X178: subst > term > term] :
( ( X178 @ sh @ one )
!= ( sub @ X177 @ ( push @ one @ sh ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X179: term,X180: term,X181: term,X182: term] :
( ( X179 != X180 )
& ( ( ap @ ( sub @ X181 @ id ) @ X179 )
= ( ap @ ( sub @ X182 @ id ) @ X180 ) ) )
& ! [X183: term,X184: term,X185: term,X186: term] :
( ( ( ap @ X186 @ X183 )
!= ( ap @ X185 @ X184 ) )
| ( X183 = X184 ) ) ) )
& ( ! [X187: term,X188: term,X189: term,X190: term] :
( ( X187 = X188 )
| ( ( ap @ ( sub @ X190 @ id ) @ X188 )
!= ( ap @ ( sub @ X189 @ id ) @ X187 ) ) )
| ? [X191: term,X192: term,X193: term,X194: term] :
( ( ( ap @ X194 @ X191 )
= ( ap @ X193 @ X192 ) )
& ( X191 != X192 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( sP33 = $true )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( sP33 != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP31 ) )
& ( ( $true = sP31 )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( sP30 = $true )
| ( hoasinduction_lthm != $true ) )
& ( ( hoasinduction_lthm = $true )
| ( sP30 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X195: subst,X196: term,X197: term > $o,X198: term > $o] :
( ( ( X198 @ one )
!= $true )
& ( ( X197 @ X196 )
= $true )
& ! [X199: term] :
( ( X198 @ X199 )
= ( X197 @ ( sub @ X199 @ ( push @ X196 @ X195 ) ) ) ) ) )
& ( ! [X200: subst,X201: term,X202: term > $o,X203: term > $o] :
( ( ( X203 @ one )
= $true )
| ( ( X202 @ X201 )
!= $true )
| ? [X204: term] :
( ( X202 @ ( sub @ X204 @ ( push @ X201 @ X200 ) ) )
!= ( X203 @ X204 ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP21 != $true ) )
& ( ( sP21 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( sP20 != $true ) )
& ( ( sP20 = $true )
| ( hoasinduction_lem3_lthm != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ? [X205: term,X206: subst] :
( ( sub @ one @ ( push @ X205 @ X206 ) )
!= X205 )
| ! [X207: term > $o,X208: subst,X209: term > $o,X210: term] :
( ( ( X207 @ X210 )
!= $true )
| ? [X211: term] :
( ( X209 @ X211 )
!= ( X207 @ ( sub @ X211 @ ( push @ X210 @ X208 ) ) ) )
| ( ( X209 @ one )
= $true ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ! [X212: term,X213: subst] :
( ( sub @ one @ ( push @ X212 @ X213 ) )
= X212 )
& ? [X214: term > $o,X215: subst,X216: term > $o,X217: term] :
( ( ( X214 @ X217 )
= $true )
& ! [X218: term] :
( ( X214 @ ( sub @ X218 @ ( push @ X217 @ X215 ) ) )
= ( X216 @ X218 ) )
& ( ( X216 @ one )
!= $true ) ) ) )
& ! [X219: d_subst] : ( d_id = X219 )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( apnotvar = $true )
& ( ( sP17 = $true )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( sP17 != $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP16 != $true ) )
& ( ( sP16 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( axclos = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( hoasinduction_lem3b = $true )
| ? [X220: term] :
! [X221: subst > term > term] :
( ( sub @ X220 @ ( push @ one @ sh ) )
!= ( X221 @ sh @ one ) ) )
& ( ! [X222: term] :
? [X223: subst > term > term] :
( ( sub @ X222 @ ( push @ one @ sh ) )
= ( X223 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X224: term] :
! [X225: subst > term > term] :
? [X226: term,X227: subst] :
( ( X225 @ X227 @ X226 )
!= ( sub @ X224 @ ( push @ X226 @ X227 ) ) ) )
& ( ! [X228: term] :
? [X229: subst > term > term] :
! [X230: term,X231: subst] :
( ( X229 @ X231 @ X230 )
= ( sub @ X228 @ ( push @ X230 @ X231 ) ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X232: subst] :
( ( comp @ X232 @ id )
= X232 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X233: subst] :
( ( comp @ X233 @ id )
!= X233 ) )
& ( axassoc = $true )
& ( ! [X234: term > $o,X235: term] :
( ( ( sP10 @ X234 )
= $true )
| ? [X236: term] :
( ( $true
= ( var @ X236 ) )
& ( ( X234 @ X236 )
!= $true ) )
| ( ( X234 @ X235 )
= $true )
| ? [X237: term] :
( ( ( X234 @ ( lam @ X237 ) )
!= $true )
& ( ( X234 @ X237 )
= $true ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X238: term > $o,X239: term] :
( ( ( sP10 @ X238 )
!= $true )
& ! [X240: term] :
( ( ( var @ X240 )
!= $true )
| ( ( X238 @ X240 )
= $true ) )
& ( ( X238 @ X239 )
!= $true )
& ! [X241: term] :
( ( ( X238 @ ( lam @ X241 ) )
= $true )
| ( ( X238 @ X241 )
!= $true ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( sP9 = $true )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( sP9 != $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X242: subst,X243: term > $o,X244: term] :
? [X245: term > $o] :
! [X246: term] :
( ( X245 @ X246 )
= ( X243 @ ( sub @ X246 @ ( push @ X244 @ X242 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X247: subst,X248: term > $o,X249: term] :
! [X250: term > $o] :
? [X251: term] :
( ( X250 @ X251 )
!= ( X248 @ ( sub @ X251 @ ( push @ X249 @ X247 ) ) ) ) )
& ( ! [X252: term] :
( ( sub @ X252 @ id )
= X252 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X253: term] :
( ( sub @ X253 @ id )
!= X253 ) )
& ( ! [X254: term,X255: subst] :
( ( sub @ one @ ( push @ X254 @ X255 ) )
= X254 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X256: term,X257: subst] :
( ( sub @ one @ ( push @ X256 @ X257 ) )
!= X256 ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP5 = $true )
| ( hoasinduction_lem2 != $true ) )
& ( ( hoasinduction_lem2 = $true )
| ( sP5 != $true ) )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ! [X258: term] :
? [X259: d_term] :
( ( d2term @ X259 )
= X258 )
& ( ( axidl = $true )
| ? [X260: subst] :
( ( comp @ id @ X260 )
!= X260 ) )
& ( ! [X261: subst] :
( ( comp @ id @ X261 )
= X261 )
| ( axidl != $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( axshiftcons = $true )
| ? [X262: subst,X263: term] :
( ( comp @ sh @ ( push @ X263 @ X262 ) )
!= X262 ) )
& ( ! [X264: subst,X265: term] :
( ( comp @ sh @ ( push @ X265 @ X264 ) )
= X264 )
| ( axshiftcons != $true ) )
& ( pushprop_gthm = $true )
& ( induction2_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
| ( sP3 != $true ) )
& ( ( sP3 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ! [X266: term,X267: term,X268: term,X269: term] :
( ( X267 = X269 )
| ( ( ap @ ( sub @ X266 @ id ) @ X267 )
!= ( ap @ ( sub @ X268 @ id ) @ X269 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X270: term,X271: term,X272: term,X273: term] :
( ( X271 != X273 )
& ( ( ap @ ( sub @ X270 @ id ) @ X271 )
= ( ap @ ( sub @ X272 @ id ) @ X273 ) ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( sP1 != $true ) )
& ( ( sP1 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( axapp = $true ) ),
inference(rectify,[status(thm)],[f474]) ).
thf(f474,plain,
( ( ( hoasinduction_no_psi_cond = $true )
| ? [X183: term,X184: subst > term > subst > $o] :
( ! [X185: term,X186: term] :
( ( ( X184 @ id @ X186 @ id )
!= $true )
| ( ( X184 @ id @ X185 @ id )
!= $true )
| ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
= $true ) )
& ( ( sP58 @ X184 )
!= $true )
& ( ( X184 @ id @ X183 @ id )
!= $true ) ) )
& ( ! [X183: term,X184: subst > term > subst > $o] :
( ? [X185: term,X186: term] :
( ( ( X184 @ id @ X186 @ id )
= $true )
& ( ( X184 @ id @ X185 @ id )
= $true )
& ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
!= $true ) )
| ( ( sP58 @ X184 )
= $true )
| ( ( X184 @ id @ X183 @ id )
= $true ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( sP57 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP57 != $true ) )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP53 != $true ) )
& ( ( sP53 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( termmset_lthm = $true )
| ( ! [X272: term] :
( ( sub @ X272 @ id )
= X272 )
& ? [X273: term] :
( ( sub @ X273 @ id )
!= X273 ) ) )
& ( ? [X272: term] :
( ( sub @ X272 @ id )
!= X272 )
| ! [X273: term] :
( ( sub @ X273 @ id )
= X273 )
| ( termmset_lthm != $true ) )
& ( ! [X165: subst > term > subst > $o] :
? [X166: term > $o] :
! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X165: subst > term > subst > $o] :
! [X166: term > $o] :
? [X167: term] :
( ( X165 @ id @ X167 @ id )
!= ( X166 @ X167 ) ) )
& ( axmap = $true )
& ( ulamvarsh = $true )
& ( ! [X2: subst > term > subst > $o] :
? [X3: term > $o] :
! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X2: subst > term > subst > $o] :
! [X3: term > $o] :
? [X4: term] :
( ( X2 @ id @ X4 @ id )
!= ( X3 @ X4 ) ) )
& ( ? [X410: subst] :
( ( comp @ X410 @ id )
!= X410 )
| ? [X409: subst] :
( ( comp @ id @ X409 )
!= X409 )
| ( ! [X411: subst] :
( ( comp @ id @ X411 )
= X411 )
& ! [X412: subst] :
( ( comp @ X412 @ id )
= X412 ) )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ! [X410: subst] :
( ( comp @ X410 @ id )
= X410 )
& ! [X409: subst] :
( ( comp @ id @ X409 )
= X409 )
& ( ? [X411: subst] :
( ( comp @ id @ X411 )
!= X411 )
| ? [X412: subst] :
( ( comp @ X412 @ id )
!= X412 ) ) ) )
& ( ( sP48 = $true )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( sP48 != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( sP47 != $true ) )
& ( ( sP47 = $true )
| ( hoasinduction_lem3aa != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( substmonoid_gthm = $true )
& ! [X245: d_term,X246: d_term] :
( ( ( d2term @ X246 )
!= ( d2term @ X245 ) )
| ( X245 = X246 ) )
& ( ulamvarind = $true )
& ( pushprop_lthm_orig = $true )
& ( ! [X261: term,X262: term,X263: term,X264: term] :
( ( X261 = X262 )
| ( ( ap @ X261 @ X263 )
!= ( ap @ X262 @ X264 ) ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X261: term,X262: term,X263: term,X264: term] :
( ( X261 != X262 )
& ( ( ap @ X261 @ X263 )
= ( ap @ X262 @ X264 ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ! [X181: d_subst,X182: d_subst] :
( ( X181 = X182 )
| ( ( d2subst @ X182 )
!= ( d2subst @ X181 ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP46 != $true ) )
& ( ( sP46 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ! [X265: d_term] : ( d_one = X265 )
& ( ( sP44 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP44 != $true ) )
& ( ! [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
!= ( ap @ X326 @ X328 ) )
| ( X325 = X328 ) )
| ( apinj2 != $true ) )
& ( ( apinj2 = $true )
| ? [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
= ( ap @ X326 @ X328 ) )
& ( X325 != X328 ) ) )
& ( lamnotvar = $true )
& ( axabs = $true )
& ( ( hoasinduction = $true )
| ( sP42 != $true ) )
& ( ( sP42 = $true )
| ( hoasinduction != $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( pushprop_lem0_gthm = $true )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
| ( X193 = X194 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
& ( X193 != X194 ) ) )
& ( hoasapnotvar_gthm = $true )
& ( ( hoaslaminj = $true )
| ? [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
= ( lam @ ( X465 @ sh @ one ) ) )
& ( ( X465 @ X466 @ X464 )
!= ( X467 @ X466 @ X464 ) )
& ! [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
& ! [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) ) )
& ( ! [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
!= ( lam @ ( X465 @ sh @ one ) ) )
| ( ( X465 @ X466 @ X464 )
= ( X467 @ X466 @ X464 ) )
| ? [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
!= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
| ? [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
!= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X247: term,X248: subst > term > term,X249: term] :
( ! [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) )
& ( ( ap @ ( sub @ X247 @ id ) @ X249 )
= ( lam @ ( X248 @ sh @ one ) ) ) ) )
& ( ! [X247: term,X248: subst > term > term,X249: term] :
( ? [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
!= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) )
| ( ( ap @ ( sub @ X247 @ id ) @ X249 )
!= ( lam @ ( X248 @ sh @ one ) ) ) )
| ( hoaslamnotap != $true ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
| ( ? [X405: term,X406: term,X407: term,X408: term] :
( ( X405 != X408 )
& ( ( ap @ ( sub @ X408 @ id ) @ X406 )
= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) )
& ! [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
!= ( ap @ X404 @ X402 ) )
| ( X401 = X404 ) )
& ! [X400: term] :
( ( sub @ X400 @ id )
= X400 ) ) )
& ( ! [X405: term,X406: term,X407: term,X408: term] :
( ( X405 = X408 )
| ( ( ap @ ( sub @ X408 @ id ) @ X406 )
!= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) )
| ? [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
= ( ap @ X404 @ X402 ) )
& ( X401 != X404 ) )
| ? [X400: term] :
( ( sub @ X400 @ id )
!= X400 )
| ( hoasapinj1_lthm != $true ) )
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X164: term] :
( ( sub @ X164 @ id )
!= X164 ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( pushprop_lem1 = $true )
| ? [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
= $true )
& ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
!= $true ) ) )
& ( ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
= $true ) )
| ( pushprop_lem1 != $true ) )
& ( ( sP39 = $true )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( sP39 != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP37 != $true ) )
& ( ( sP37 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( hoasinduction_lem1 = $true )
& ( hoaslaminj_gthm = $true )
& ( pushprop = $true )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( substmonoid = $true )
| ? [X1: subst] :
( ( comp @ id @ X1 )
!= X1 )
| ? [X0: subst] :
( ( comp @ X0 @ id )
!= X0 ) )
& ( ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
& ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 ) )
| ( substmonoid != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP36 ) )
& ( ( $true = sP36 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ulamvar1 != $true )
& ( ? [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
!= X121 )
| ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
!= $true )
| ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
= $true ) )
| ? [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
!= X118 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
= X121 )
& ? [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
= $true )
& ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
!= $true ) )
& ! [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
= X118 ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X139: term,X140: term > $o,X141: subst] :
! [X142: term > $o] :
? [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
!= ( X142 @ X143 ) ) )
& ( ! [X139: term,X140: term > $o,X141: subst] :
? [X142: term > $o] :
! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( one
= ( d2term @ d_one ) )
& ( ( shinj = $true )
| ? [X69: term,X70: term] :
( ( X69 != X70 )
& ( ( sub @ X70 @ sh )
= ( sub @ X69 @ sh ) ) ) )
& ( ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) )
| ( shinj != $true ) )
& ( axvarshift = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2subst @ d_id )
= 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 ) ) ) ) ) ) )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP35 != $true ) )
& ( ( sP35 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
!= X102 ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X135: term] :
! [X136: subst > term > term] :
? [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
!= ( X136 @ X137 @ X138 ) ) )
& ( ! [X135: term] :
? [X136: subst > term > term] :
! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ! [X292: term,X293: term] :
( ( X292 = X293 )
| ( ( lam @ X293 )
!= ( lam @ X292 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X292: term,X293: term] :
( ( X292 != X293 )
& ( ( lam @ X293 )
= ( lam @ X292 ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X144: term] :
? [X145: subst > term > term] :
( ( X145 @ sh @ one )
= ( sub @ X144 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X144: term] :
! [X145: subst > term > term] :
( ( X145 @ sh @ one )
!= ( sub @ X144 @ ( push @ one @ sh ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X257: term,X258: term,X259: term,X260: term] :
( ( X257 != X258 )
& ( ( ap @ ( sub @ X260 @ id ) @ X258 )
= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) )
& ! [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
!= ( ap @ X255 @ X254 ) )
| ( X253 = X254 ) ) ) )
& ( ! [X257: term,X258: term,X259: term,X260: term] :
( ( X257 = X258 )
| ( ( ap @ ( sub @ X260 @ id ) @ X258 )
!= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) )
| ? [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
= ( ap @ X255 @ X254 ) )
& ( X253 != X254 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( sP33 = $true )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( sP33 != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP31 ) )
& ( ( $true = sP31 )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( sP30 = $true )
| ( hoasinduction_lthm != $true ) )
& ( ( hoasinduction_lthm = $true )
| ( sP30 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
!= $true )
& ( ( X522 @ X521 )
= $true )
& ! [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
= ( X523 @ X524 ) ) ) )
& ( ! [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
= $true )
| ( ( X522 @ X521 )
!= $true )
| ? [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
!= ( X523 @ X524 ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP21 != $true ) )
& ( ( sP21 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( sP20 != $true ) )
& ( ( sP20 = $true )
| ( hoasinduction_lem3_lthm != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ? [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
!= X228 )
| ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ( ( X230 @ X233 )
!= $true )
| ? [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
!= ( X232 @ X234 ) )
| ( ( X232 @ one )
= $true ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ! [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
= X228 )
& ? [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ( ( X230 @ X233 )
= $true )
& ! [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
= ( X232 @ X234 ) )
& ( ( X232 @ one )
!= $true ) ) ) )
& ! [X271: d_subst] : ( d_id = X271 )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( apnotvar = $true )
& ( ( sP17 = $true )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( sP17 != $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP16 != $true ) )
& ( ( sP16 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( axclos = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( hoasinduction_lem3b = $true )
| ? [X73: term] :
! [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
!= ( X74 @ sh @ one ) ) )
& ( ! [X73: term] :
? [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
= ( X74 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X65: term] :
! [X66: subst > term > term] :
? [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
!= ( sub @ X65 @ ( push @ X67 @ X68 ) ) ) )
& ( ! [X65: term] :
? [X66: subst > term > term] :
! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X71: subst] :
( ( comp @ X71 @ id )
!= X71 ) )
& ( axassoc = $true )
& ( ! [X495: term > $o,X496: term] :
( ( ( sP10 @ X495 )
= $true )
| ? [X499: term] :
( ( ( var @ X499 )
= $true )
& ( ( X495 @ X499 )
!= $true ) )
| ( ( X495 @ X496 )
= $true )
| ? [X500: term] :
( ( ( X495 @ ( lam @ X500 ) )
!= $true )
& ( ( X495 @ X500 )
= $true ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X495: term > $o,X496: term] :
( ( ( sP10 @ X495 )
!= $true )
& ! [X499: term] :
( ( ( var @ X499 )
!= $true )
| ( ( X495 @ X499 )
= $true ) )
& ( ( X495 @ X496 )
!= $true )
& ! [X500: term] :
( ( ( X495 @ ( lam @ X500 ) )
= $true )
| ( ( X495 @ X500 )
!= $true ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( sP9 = $true )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( sP9 != $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X266: subst,X267: term > $o,X268: term] :
? [X269: term > $o] :
! [X270: term] :
( ( X269 @ X270 )
= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X266: subst,X267: term > $o,X268: term] :
! [X269: term > $o] :
? [X270: term] :
( ( X269 @ X270 )
!= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) ) )
& ( ! [X383: term] :
( ( sub @ X383 @ id )
= X383 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X383: term] :
( ( sub @ X383 @ id )
!= X383 ) )
& ( ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
!= X100 ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP5 = $true )
| ( hoasinduction_lem2 != $true ) )
& ( ( hoasinduction_lem2 = $true )
| ( sP5 != $true ) )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ! [X413: term] :
? [X414: d_term] :
( ( d2term @ X414 )
= X413 )
& ( ( axidl = $true )
| ? [X72: subst] :
( ( comp @ id @ X72 )
!= X72 ) )
& ( ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 )
| ( axidl != $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( axshiftcons = $true )
| ? [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
!= X493 ) )
& ( ! [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
= X493 )
| ( axshiftcons != $true ) )
& ( pushprop_gthm = $true )
& ( induction2_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
| ( sP3 != $true ) )
& ( ( sP3 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ! [X415: term,X416: term,X417: term,X418: term] :
( ( X416 = X418 )
| ( ( ap @ ( sub @ X417 @ id ) @ X418 )
!= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X415: term,X416: term,X417: term,X418: term] :
( ( X416 != X418 )
& ( ( ap @ ( sub @ X417 @ id ) @ X418 )
= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( sP1 != $true ) )
& ( ( sP1 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( axapp = $true ) ),
inference(flattening,[status(thm)],[f473]) ).
thf(f473,plain,
( ( ( hoasinduction_no_psi_cond = $true )
| ? [X183: term,X184: subst > term > subst > $o] :
( ! [X185: term,X186: term] :
( ( ( X184 @ id @ X186 @ id )
!= $true )
| ( ( X184 @ id @ X185 @ id )
!= $true )
| ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
= $true ) )
& ( ( sP58 @ X184 )
!= $true )
& ( ( X184 @ id @ X183 @ id )
!= $true ) ) )
& ( ! [X183: term,X184: subst > term > subst > $o] :
( ? [X185: term,X186: term] :
( ( ( X184 @ id @ X186 @ id )
= $true )
& ( ( X184 @ id @ X185 @ id )
= $true )
& ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
!= $true ) )
| ( ( sP58 @ X184 )
= $true )
| ( ( X184 @ id @ X183 @ id )
= $true ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( sP57 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP57 != $true ) )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP53 != $true ) )
& ( ( sP53 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( termmset_lthm = $true )
| ( ! [X272: term] :
( ( sub @ X272 @ id )
= X272 )
& ? [X273: term] :
( ( sub @ X273 @ id )
!= X273 ) ) )
& ( ? [X272: term] :
( ( sub @ X272 @ id )
!= X272 )
| ! [X273: term] :
( ( sub @ X273 @ id )
= X273 )
| ( termmset_lthm != $true ) )
& ( ! [X165: subst > term > subst > $o] :
? [X166: term > $o] :
! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X165: subst > term > subst > $o] :
! [X166: term > $o] :
? [X167: term] :
( ( X165 @ id @ X167 @ id )
!= ( X166 @ X167 ) ) )
& ( axmap = $true )
& ( ulamvarsh = $true )
& ( ! [X2: subst > term > subst > $o] :
? [X3: term > $o] :
! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X2: subst > term > subst > $o] :
! [X3: term > $o] :
? [X4: term] :
( ( X2 @ id @ X4 @ id )
!= ( X3 @ X4 ) ) )
& ( ? [X410: subst] :
( ( comp @ X410 @ id )
!= X410 )
| ? [X409: subst] :
( ( comp @ id @ X409 )
!= X409 )
| ( ! [X411: subst] :
( ( comp @ id @ X411 )
= X411 )
& ! [X412: subst] :
( ( comp @ X412 @ id )
= X412 ) )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ! [X410: subst] :
( ( comp @ X410 @ id )
= X410 )
& ! [X409: subst] :
( ( comp @ id @ X409 )
= X409 )
& ( ? [X411: subst] :
( ( comp @ id @ X411 )
!= X411 )
| ? [X412: subst] :
( ( comp @ X412 @ id )
!= X412 ) ) ) )
& ( ( sP48 = $true )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( sP48 != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( sP47 != $true ) )
& ( ( sP47 = $true )
| ( hoasinduction_lem3aa != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( substmonoid_gthm = $true )
& ! [X245: d_term,X246: d_term] :
( ( ( d2term @ X246 )
!= ( d2term @ X245 ) )
| ( X245 = X246 ) )
& ( ulamvarind = $true )
& ( pushprop_lthm_orig = $true )
& ( ! [X261: term,X262: term,X263: term,X264: term] :
( ( X261 = X262 )
| ( ( ap @ X261 @ X263 )
!= ( ap @ X262 @ X264 ) ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X261: term,X262: term,X263: term,X264: term] :
( ( X261 != X262 )
& ( ( ap @ X261 @ X263 )
= ( ap @ X262 @ X264 ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ! [X181: d_subst,X182: d_subst] :
( ( X181 = X182 )
| ( ( d2subst @ X182 )
!= ( d2subst @ X181 ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP46 != $true ) )
& ( ( sP46 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ! [X265: d_term] : ( d_one = X265 )
& ( ( sP44 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP44 != $true ) )
& ( ! [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
!= ( ap @ X326 @ X328 ) )
| ( X325 = X328 ) )
| ( apinj2 != $true ) )
& ( ( apinj2 = $true )
| ? [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
= ( ap @ X326 @ X328 ) )
& ( X325 != X328 ) ) )
& ( lamnotvar = $true )
& ( axabs = $true )
& ( ( hoasinduction = $true )
| ( sP42 != $true ) )
& ( ( sP42 = $true )
| ( hoasinduction != $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( pushprop_lem0_gthm = $true )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
| ( X193 = X194 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
& ( X193 != X194 ) ) )
& ( hoasapnotvar_gthm = $true )
& ( ( hoaslaminj = $true )
| ? [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
= ( lam @ ( X465 @ sh @ one ) ) )
& ( ( X465 @ X466 @ X464 )
!= ( X467 @ X466 @ X464 ) )
& ! [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
& ! [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) ) )
& ( ! [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
!= ( lam @ ( X465 @ sh @ one ) ) )
| ( ( X465 @ X466 @ X464 )
= ( X467 @ X466 @ X464 ) )
| ? [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
!= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
| ? [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
!= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X247: term,X248: subst > term > term,X249: term] :
( ! [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) )
& ( ( ap @ ( sub @ X247 @ id ) @ X249 )
= ( lam @ ( X248 @ sh @ one ) ) ) ) )
& ( ! [X247: term,X248: subst > term > term,X249: term] :
( ? [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
!= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) )
| ( ( ap @ ( sub @ X247 @ id ) @ X249 )
!= ( lam @ ( X248 @ sh @ one ) ) ) )
| ( hoaslamnotap != $true ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
| ( ? [X405: term,X406: term,X407: term,X408: term] :
( ( X405 != X408 )
& ( ( ap @ ( sub @ X408 @ id ) @ X406 )
= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) )
& ! [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
!= ( ap @ X404 @ X402 ) )
| ( X401 = X404 ) )
& ! [X400: term] :
( ( sub @ X400 @ id )
= X400 ) ) )
& ( ! [X405: term,X406: term,X407: term,X408: term] :
( ( X405 = X408 )
| ( ( ap @ ( sub @ X408 @ id ) @ X406 )
!= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) )
| ? [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
= ( ap @ X404 @ X402 ) )
& ( X401 != X404 ) )
| ? [X400: term] :
( ( sub @ X400 @ id )
!= X400 )
| ( hoasapinj1_lthm != $true ) )
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X164: term] :
( ( sub @ X164 @ id )
!= X164 ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( pushprop_lem1 = $true )
| ? [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
= $true )
& ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
!= $true ) ) )
& ( ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
= $true ) )
| ( pushprop_lem1 != $true ) )
& ( ( sP39 = $true )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( sP39 != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP37 != $true ) )
& ( ( sP37 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( hoasinduction_lem1 = $true )
& ( hoaslaminj_gthm = $true )
& ( pushprop = $true )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( substmonoid = $true )
| ? [X1: subst] :
( ( comp @ id @ X1 )
!= X1 )
| ? [X0: subst] :
( ( comp @ X0 @ id )
!= X0 ) )
& ( ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
& ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 ) )
| ( substmonoid != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP36 ) )
& ( ( $true = sP36 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ulamvar1 != $true )
& ( ? [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
!= X121 )
| ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
!= $true )
| ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
= $true ) )
| ? [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
!= X118 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
= X121 )
& ? [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
= $true )
& ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
!= $true ) )
& ! [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
= X118 ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X139: term,X140: term > $o,X141: subst] :
! [X142: term > $o] :
? [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
!= ( X142 @ X143 ) ) )
& ( ! [X139: term,X140: term > $o,X141: subst] :
? [X142: term > $o] :
! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( one
= ( d2term @ d_one ) )
& ( ( shinj = $true )
| ? [X69: term,X70: term] :
( ( X69 != X70 )
& ( ( sub @ X70 @ sh )
= ( sub @ X69 @ sh ) ) ) )
& ( ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) )
| ( shinj != $true ) )
& ( axvarshift = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2subst @ d_id )
= 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 ) ) ) ) ) ) )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP35 != $true ) )
& ( ( sP35 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
!= X102 ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X135: term] :
! [X136: subst > term > term] :
? [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
!= ( X136 @ X137 @ X138 ) ) )
& ( ! [X135: term] :
? [X136: subst > term > term] :
! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ! [X292: term,X293: term] :
( ( X292 = X293 )
| ( ( lam @ X293 )
!= ( lam @ X292 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X292: term,X293: term] :
( ( X292 != X293 )
& ( ( lam @ X293 )
= ( lam @ X292 ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X144: term] :
? [X145: subst > term > term] :
( ( X145 @ sh @ one )
= ( sub @ X144 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X144: term] :
! [X145: subst > term > term] :
( ( X145 @ sh @ one )
!= ( sub @ X144 @ ( push @ one @ sh ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X257: term,X258: term,X259: term,X260: term] :
( ( X257 != X258 )
& ( ( ap @ ( sub @ X260 @ id ) @ X258 )
= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) )
& ! [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
!= ( ap @ X255 @ X254 ) )
| ( X253 = X254 ) ) ) )
& ( ! [X257: term,X258: term,X259: term,X260: term] :
( ( X257 = X258 )
| ( ( ap @ ( sub @ X260 @ id ) @ X258 )
!= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) )
| ? [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
= ( ap @ X255 @ X254 ) )
& ( X253 != X254 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( sP33 = $true )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( sP33 != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP31 ) )
& ( ( $true = sP31 )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( sP30 = $true )
| ( hoasinduction_lthm != $true ) )
& ( ( hoasinduction_lthm = $true )
| ( sP30 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
!= $true )
& ( ( X522 @ X521 )
= $true )
& ! [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
= ( X523 @ X524 ) ) ) )
& ( ! [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
= $true )
| ( ( X522 @ X521 )
!= $true )
| ? [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
!= ( X523 @ X524 ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP21 != $true ) )
& ( ( sP21 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( sP20 != $true ) )
& ( ( sP20 = $true )
| ( hoasinduction_lem3_lthm != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ? [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
!= X228 )
| ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ( ( X230 @ X233 )
!= $true )
| ? [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
!= ( X232 @ X234 ) )
| ( ( X232 @ one )
= $true ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ! [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
= X228 )
& ? [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ( ( X230 @ X233 )
= $true )
& ! [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
= ( X232 @ X234 ) )
& ( ( X232 @ one )
!= $true ) ) ) )
& ! [X271: d_subst] : ( d_id = X271 )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( apnotvar = $true )
& ( ( sP17 = $true )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( sP17 != $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP16 != $true ) )
& ( ( sP16 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( axclos = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( hoasinduction_lem3b = $true )
| ? [X73: term] :
! [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
!= ( X74 @ sh @ one ) ) )
& ( ! [X73: term] :
? [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
= ( X74 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X65: term] :
! [X66: subst > term > term] :
? [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
!= ( sub @ X65 @ ( push @ X67 @ X68 ) ) ) )
& ( ! [X65: term] :
? [X66: subst > term > term] :
! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X71: subst] :
( ( comp @ X71 @ id )
!= X71 ) )
& ( axassoc = $true )
& ( ! [X495: term > $o,X496: term] :
( ( ( sP10 @ X495 )
= $true )
| ? [X499: term] :
( ( ( var @ X499 )
= $true )
& ( ( X495 @ X499 )
!= $true ) )
| ( ( X495 @ X496 )
= $true )
| ? [X500: term] :
( ( ( X495 @ ( lam @ X500 ) )
!= $true )
& ( ( X495 @ X500 )
= $true ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X495: term > $o,X496: term] :
( ( ( sP10 @ X495 )
!= $true )
& ! [X499: term] :
( ( ( var @ X499 )
!= $true )
| ( ( X495 @ X499 )
= $true ) )
& ( ( X495 @ X496 )
!= $true )
& ! [X500: term] :
( ( ( X495 @ ( lam @ X500 ) )
= $true )
| ( ( X495 @ X500 )
!= $true ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( sP9 = $true )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( sP9 != $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X266: subst,X267: term > $o,X268: term] :
? [X269: term > $o] :
! [X270: term] :
( ( X269 @ X270 )
= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X266: subst,X267: term > $o,X268: term] :
! [X269: term > $o] :
? [X270: term] :
( ( X269 @ X270 )
!= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) ) )
& ( ! [X383: term] :
( ( sub @ X383 @ id )
= X383 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X383: term] :
( ( sub @ X383 @ id )
!= X383 ) )
& ( ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
!= X100 ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP5 = $true )
| ( hoasinduction_lem2 != $true ) )
& ( ( hoasinduction_lem2 = $true )
| ( sP5 != $true ) )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ! [X413: term] :
? [X414: d_term] :
( ( d2term @ X414 )
= X413 )
& ( ( axidl = $true )
| ? [X72: subst] :
( ( comp @ id @ X72 )
!= X72 ) )
& ( ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 )
| ( axidl != $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( axshiftcons = $true )
| ? [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
!= X493 ) )
& ( ! [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
= X493 )
| ( axshiftcons != $true ) )
& ( pushprop_gthm = $true )
& ( induction2_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
| ( sP3 != $true ) )
& ( ( sP3 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ! [X415: term,X416: term,X417: term,X418: term] :
( ( X416 = X418 )
| ( ( ap @ ( sub @ X417 @ id ) @ X418 )
!= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X415: term,X416: term,X417: term,X418: term] :
( ( X416 != X418 )
& ( ( ap @ ( sub @ X417 @ id ) @ X418 )
= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( sP1 != $true ) )
& ( ( sP1 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( axapp = $true ) ),
inference(nnf_transformation,[status(thm)],[f73]) ).
thf(f73,plain,
( ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X183: term,X184: subst > term > subst > $o] :
( ? [X185: term,X186: term] :
( ( ( X184 @ id @ X186 @ id )
= $true )
& ( ( X184 @ id @ X185 @ id )
= $true )
& ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
!= $true ) )
| ( ( sP58 @ X184 )
= $true )
| ( ( X184 @ id @ X183 @ id )
= $true ) ) )
& ( ( sP57 = $true )
<=> ( induction2_lthm = $true ) )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( sP53 = $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( termmset_lthm = $true )
<=> ( ? [X272: term] :
( ( sub @ X272 @ id )
!= X272 )
| ! [X273: term] :
( ( sub @ X273 @ id )
= X273 ) ) )
& ( ! [X165: subst > term > subst > $o] :
? [X166: term > $o] :
! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( axmap = $true )
& ( ulamvarsh = $true )
& ( ! [X2: subst > term > subst > $o] :
? [X3: term > $o] :
! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( ? [X410: subst] :
( ( comp @ X410 @ id )
!= X410 )
| ? [X409: subst] :
( ( comp @ id @ X409 )
!= X409 )
| ( ! [X411: subst] :
( ( comp @ id @ X411 )
= X411 )
& ! [X412: subst] :
( ( comp @ X412 @ id )
= X412 ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( ( sP48 = $true )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ( sP47 = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( substmonoid_gthm = $true )
& ! [X245: d_term,X246: d_term] :
( ( ( d2term @ X246 )
!= ( d2term @ X245 ) )
| ( X245 = X246 ) )
& ( ulamvarind = $true )
& ( pushprop_lthm_orig = $true )
& ( ! [X261: term,X262: term,X263: term,X264: term] :
( ( X261 = X262 )
| ( ( ap @ X261 @ X263 )
!= ( ap @ X262 @ X264 ) ) )
<=> ( apinj1 = $true ) )
& ( hoaslamnotap_gthm = $true )
& ! [X181: d_subst,X182: d_subst] :
( ( X181 = X182 )
| ( ( d2subst @ X182 )
!= ( d2subst @ X181 ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ( sP46 = $true ) )
& ! [X265: d_term] : ( d_one = X265 )
& ( ( sP44 = $true )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( ! [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
!= ( ap @ X326 @ X328 ) )
| ( X325 = X328 ) )
<=> ( apinj2 = $true ) )
& ( lamnotvar = $true )
& ( axabs = $true )
& ( ( hoasinduction = $true )
<=> ( sP42 = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( pushprop_lem0_gthm = $true )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
| ( X193 = X194 ) )
<=> ( hoasapinj1 = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( ( hoaslaminj = $true )
<=> ! [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
!= ( lam @ ( X465 @ sh @ one ) ) )
| ( ( X465 @ X466 @ X464 )
= ( X467 @ X466 @ X464 ) )
| ? [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
!= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
| ? [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
!= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X247: term,X248: subst > term > term,X249: term] :
( ? [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
!= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) )
| ( ( ap @ ( sub @ X247 @ id ) @ X249 )
!= ( lam @ ( X248 @ sh @ one ) ) ) ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X405: term,X406: term,X407: term,X408: term] :
( ( X405 = X408 )
| ( ( ap @ ( sub @ X408 @ id ) @ X406 )
!= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) )
| ? [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
= ( ap @ X404 @ X402 ) )
& ( X401 != X404 ) )
| ? [X400: term] :
( ( sub @ X400 @ id )
!= X400 ) ) )
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
<=> ( termmset = $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
= $true ) ) )
& ( ( sP39 = $true )
<=> ( induction2 = $true ) )
& ( ( hoasinduction_lem3a = $true )
<=> ( sP37 = $true ) )
& ( hoasinduction_lem1 = $true )
& ( hoaslaminj_gthm = $true )
& ( pushprop = $true )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( substmonoid = $true )
<=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
& ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ( $true = sP36 ) )
& ( ulamvar1 != $true )
& ( ( ? [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
!= X121 )
| ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
!= $true )
| ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
= $true ) )
| ? [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
!= X118 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X139: term,X140: term > $o,X141: subst] :
? [X142: term > $o] :
! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) ) )
& ( one
= ( d2term @ d_one ) )
& ( ( shinj = $true )
<=> ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) ) )
& ( axvarshift = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2subst @ d_id )
= 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 ) ) ) ) ) ) )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( sP35 = $true ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X135: term] :
? [X136: subst > term > term] :
! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) ) )
& ( ! [X292: term,X293: term] :
( ( X292 = X293 )
| ( ( lam @ X293 )
!= ( lam @ X292 ) ) )
<=> ( laminj = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X144: term] :
? [X145: subst > term > term] :
( ( X145 @ sh @ one )
= ( sub @ X144 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X257: term,X258: term,X259: term,X260: term] :
( ( X257 = X258 )
| ( ( ap @ ( sub @ X260 @ id ) @ X258 )
!= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) )
| ? [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
= ( ap @ X255 @ X254 ) )
& ( X253 != X254 ) ) ) )
& ( ( sP33 = $true )
<=> ( induction2lem = $true ) )
& ( ( hoaslaminj_lthm = $true )
<=> ( $true = sP31 ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( sP30 = $true )
<=> ( hoasinduction_lthm = $true ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
= $true )
| ( ( X522 @ X521 )
!= $true )
| ? [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
!= ( X523 @ X524 ) ) ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ( sP21 = $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( sP20 = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( ? [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
!= X228 )
| ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ( ( X230 @ X233 )
!= $true )
| ? [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
!= ( X232 @ X234 ) )
| ( ( X232 @ one )
= $true ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ! [X271: d_subst] : ( d_id = X271 )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( apnotvar = $true )
& ( ( sP17 = $true )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( sP16 = $true ) )
& ( axclos = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X73: term] :
? [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
= ( X74 @ sh @ one ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X65: term] :
? [X66: subst > term > term] :
! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
<=> ( axidr = $true ) )
& ( axassoc = $true )
& ( ! [X495: term > $o,X496: term] :
( ( ( sP10 @ X495 )
= $true )
| ? [X499: term] :
( ( ( var @ X499 )
= $true )
& ( ( X495 @ X499 )
!= $true ) )
| ( ( X495 @ X496 )
= $true )
| ? [X500: term] :
( ( ( X495 @ ( lam @ X500 ) )
!= $true )
& ( ( X495 @ X500 )
= $true ) ) )
<=> ( induction = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( sP9 = $true )
<=> ( induction2lem_lthm = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X266: subst,X267: term > $o,X268: term] :
? [X269: term > $o] :
! [X270: term] :
( ( X269 @ X270 )
= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ( ! [X383: term] :
( ( sub @ X383 @ id )
= X383 )
<=> ( axvarid = $true ) )
& ( ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 )
<=> ( axvarcons = $true ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP5 = $true )
<=> ( hoasinduction_lem2 = $true ) )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ! [X413: term] :
? [X414: d_term] :
( ( d2term @ X414 )
= X413 )
& ( ( axidl = $true )
<=> ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 ) )
& ( hoasinduction_lem1v2 = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( axshiftcons = $true )
<=> ! [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
= X493 ) )
& ( pushprop_gthm = $true )
& ( induction2_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ( sP3 = $true ) )
& ( ! [X415: term,X416: term,X417: term,X418: term] :
( ( X416 = X418 )
| ( ( ap @ ( sub @ X417 @ id ) @ X418 )
!= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( sP1 = $true ) )
& ( axapp = $true ) ),
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,
! [X503: subst > term > subst > $o] :
( ( ( sP0 @ X503 )
= $true )
<=> ? [X510: subst > term > term] :
( ! [X512: subst,X513: term,X514: subst] :
( ( sub @ ( X510 @ X514 @ X513 ) @ X512 )
= ( X510 @ ( comp @ X514 @ X512 ) @ ( sub @ X513 @ X512 ) ) )
& ! [X511: term] :
( ( ( X503 @ id @ ( X510 @ id @ X511 ) @ id )
= $true )
| ( ( X503 @ id @ X511 @ id )
!= $true ) )
& ( ( X503 @ id @ ( lam @ ( X510 @ sh @ one ) ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f15,plain,
( ( sP1 = $true )
<=> ( ! [X502: term > $o,X503: subst > term > subst > $o,X504: term] :
( ? [X515: subst,X516: subst,X517: subst,X518: term] :
( ( ( X503 @ X516 @ X518 @ ( comp @ X515 @ X517 ) )
!= $true )
& ( ( X503 @ ( comp @ X516 @ X515 ) @ ( sub @ X518 @ X515 ) @ X517 )
= $true ) )
| ? [X519: term] :
( ( ( X502 @ ( sub @ X504 @ ( push @ X519 @ id ) ) )
!= $true )
& ( ( X502 @ X519 )
= $true ) )
| ( ( sP0 @ X503 )
= $true )
| ? [X506: subst,X507: subst,X508: subst,X509: term] :
( ( ( X503 @ ( comp @ X506 @ X507 ) @ ( sub @ X509 @ X507 ) @ X508 )
!= $true )
& ( ( X503 @ X506 @ X509 @ ( comp @ X507 @ X508 ) )
= $true ) )
| ( ( X502 @ ( lam @ X504 ) )
= $true )
| ? [X505: term] :
( ( X503 @ id @ X505 @ id )
!= ( X502 @ X505 ) ) )
| ? [X501: term] :
( ( sub @ X501 @ id )
!= X501 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f16,plain,
! [X385: subst > term > subst > $o] :
( ( ( sP2 @ X385 )
= $true )
<=> ? [X391: subst > term > term] :
( ! [X395: term] :
( ( ( X385 @ id @ X395 @ id )
!= $true )
| ( ( X385 @ id @ ( X391 @ id @ X395 ) @ id )
= $true ) )
& ! [X392: subst,X393: subst,X394: term] :
( ( sub @ ( X391 @ X393 @ X394 ) @ X392 )
= ( X391 @ ( comp @ X393 @ X392 ) @ ( sub @ X394 @ X392 ) ) )
& ( ( X385 @ id @ ( lam @ ( X391 @ sh @ one ) ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f17,plain,
( ( sP3 = $true )
<=> ! [X384: term,X385: subst > term > subst > $o] :
( ? [X386: subst,X387: subst,X388: subst,X389: term] :
( ( ( X385 @ ( comp @ X386 @ X388 ) @ ( sub @ X389 @ X388 ) @ X387 )
!= $true )
& ( ( X385 @ X386 @ X389 @ ( comp @ X388 @ X387 ) )
= $true ) )
| ( ( sP2 @ X385 )
= $true )
| ? [X390: term] :
( ( ( X385 @ id @ X390 @ id )
= $true )
& ( ( X385 @ id @ ( sub @ X384 @ ( push @ X390 @ id ) ) @ id )
!= $true ) )
| ? [X396: subst,X397: subst,X398: term,X399: subst] :
( ( ( X385 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X399 )
= $true )
& ( ( X385 @ X396 @ X398 @ ( comp @ X397 @ X399 ) )
!= $true ) )
| ( ( X385 @ id @ ( lam @ X384 ) @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f18,plain,
! [X170: subst > term > subst > $o] :
( ( ( sP4 @ X170 )
= $true )
<=> ? [X171: term,X172: term] :
( ( ( X170 @ id @ ( ap @ ( sub @ X171 @ id ) @ X172 ) @ id )
!= $true )
& ( ( X170 @ id @ X172 @ id )
= $true )
& ( ( X170 @ id @ X171 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f19,plain,
( ( sP5 = $true )
<=> ! [X168: term,X169: term,X170: subst > term > subst > $o] :
( ( ( X170 @ id @ ( ap @ X168 @ X169 ) @ id )
= $true )
| ( ( X170 @ id @ X169 @ id )
!= $true )
| ( ( X170 @ id @ X168 @ id )
!= $true )
| ? [X177: subst,X178: subst,X179: term,X180: subst] :
( ( ( X170 @ X178 @ X179 @ ( comp @ X177 @ X180 ) )
= $true )
& ( ( X170 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) @ X180 )
!= $true ) )
| ? [X173: subst,X174: subst,X175: term,X176: subst] :
( ( ( X170 @ ( comp @ X176 @ X173 ) @ ( sub @ X175 @ X173 ) @ X174 )
= $true )
& ( ( X170 @ X176 @ X175 @ ( comp @ X173 @ X174 ) )
!= $true ) )
| ( ( sP4 @ X170 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f20,plain,
! [X479: term > $o] :
( ( ( sP6 @ X479 )
= $true )
<=> ! [X483: term,X484: term] :
( ( ( X479 @ ( ap @ X484 @ X483 ) )
= $true )
| ( ( X479 @ X483 )
!= $true )
| ( ( X479 @ X484 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f21,plain,
! [X487: term > $o] :
( ( ( sP7 @ X487 )
= $true )
<=> ? [X491: term,X492: term] :
( ( ( X487 @ X491 )
= $true )
& ( ( X487 @ X492 )
= $true )
& ( ( X487 @ ( ap @ X491 @ X492 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f22,plain,
( ( sP8 = $true )
<=> ! [X485: term,X486: subst,X487: term > $o] :
( ? [X490: term] :
( ( ( X487 @ ( sub @ X490 @ X486 ) )
!= $true )
& ( ( var @ X490 )
= $true ) )
| ( ( sP7 @ X487 )
= $true )
| ? [X488: term] :
( ! [X489: term] :
( ( ( X487 @ X489 )
!= $true )
| ( ( X487 @ ( sub @ X488 @ ( push @ X489 @ id ) ) )
= $true ) )
& ( ( X487 @ ( lam @ X488 ) )
!= $true ) )
| ( ( X487 @ ( sub @ X485 @ X486 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f23,plain,
( ( sP9 = $true )
<=> ( ( sP8 = $true )
| ? [X476: subst,X477: term] :
( ( comp @ sh @ ( push @ X477 @ X476 ) )
!= X476 )
| ? [X474: subst,X475: term] :
( ( sub @ one @ ( push @ X475 @ X474 ) )
!= X475 )
| ? [X478: subst] :
( ( comp @ X478 @ id )
!= X478 )
| ? [X479: term > $o,X480: term] :
( ! [X482: term] :
( ( ( X479 @ X482 )
= $true )
| ( ( var @ X482 )
!= $true ) )
& ( ( X479 @ X480 )
!= $true )
& ! [X481: term] :
( ( ( X479 @ X481 )
!= $true )
| ( ( X479 @ ( lam @ X481 ) )
= $true ) )
& ( ( sP6 @ X479 )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f24,plain,
! [X495: term > $o] :
( ( ( sP10 @ X495 )
= $true )
<=> ? [X497: term,X498: term] :
( ( ( X495 @ ( ap @ X497 @ X498 ) )
!= $true )
& ( ( X495 @ X497 )
= $true )
& ( ( X495 @ X498 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f25,plain,
! [X42: subst > term > subst > $o] :
( ( ( sP11 @ X42 )
= $true )
<=> ? [X54: term,X55: term] :
( ( ( X42 @ id @ X55 @ id )
= $true )
& ( ( X42 @ id @ X54 @ id )
= $true )
& ( ( X42 @ id @ ( ap @ ( sub @ X54 @ id ) @ X55 ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f26,plain,
! [X42: subst > term > subst > $o] :
( ( ( sP12 @ X42 )
= $true )
<=> ? [X44: subst > term > term] :
( ( ( X42 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id )
!= $true )
& ! [X48: term] :
( ( ( X42 @ id @ ( X44 @ id @ X48 ) @ id )
= $true )
| ( ( X42 @ id @ X48 @ id )
!= $true ) )
& ! [X45: subst,X46: subst,X47: term] :
( ( sub @ ( X44 @ X45 @ X47 ) @ X46 )
= ( X44 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f27,plain,
! [X25: term > $o] :
( ( ( sP13 @ X25 )
= $true )
<=> ! [X26: term,X27: term] :
( ( ( X25 @ ( ap @ X27 @ X26 ) )
= $true )
| ( ( X25 @ X27 )
!= $true )
| ( ( X25 @ X26 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f28,plain,
( ( sP14 = $true )
<=> ! [X42: subst > term > subst > $o,X43: term] :
( ( ( sP12 @ X42 )
= $true )
| ? [X50: subst,X51: term,X52: subst,X53: subst] :
( ( ( X42 @ ( comp @ X53 @ X52 ) @ ( sub @ X51 @ X52 ) @ X50 )
!= $true )
& ( ( X42 @ X53 @ X51 @ ( comp @ X52 @ X50 ) )
= $true ) )
| ? [X49: term] :
( ( ( X42 @ id @ X49 @ id )
!= $true )
& ( ( var @ ( sub @ X49 @ id ) )
= $true ) )
| ( ( X42 @ id @ X43 @ id )
= $true )
| ( ( sP11 @ X42 )
= $true )
| ? [X56: subst,X57: subst,X58: subst,X59: term] :
( ( ( X42 @ X58 @ X59 @ ( comp @ X57 @ X56 ) )
!= $true )
& ( ( X42 @ ( comp @ X58 @ X57 ) @ ( sub @ X59 @ X57 ) @ X56 )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f29,plain,
( ( sP15 = $true )
<=> ? [X32: term > $o,X33: subst > term > subst > $o,X34: term] :
( ! [X35: term] :
( ( ( X32 @ ( sub @ X34 @ ( push @ X35 @ id ) ) )
= $true )
| ( ( X32 @ X35 )
!= $true ) )
& ( ( X32 @ ( lam @ X34 ) )
!= $true )
& ! [X41: term] :
( ( X33 @ id @ X41 @ id )
= ( X32 @ X41 ) )
& ! [X36: subst > term > term] :
( ( ( X33 @ id @ ( lam @ ( X36 @ sh @ one ) ) @ id )
= $true )
| ? [X38: subst,X39: term,X40: subst] :
( ( X36 @ ( comp @ X38 @ X40 ) @ ( sub @ X39 @ X40 ) )
!= ( sub @ ( X36 @ X38 @ X39 ) @ X40 ) )
| ? [X37: term] :
( ( ( X33 @ id @ X37 @ id )
= $true )
& ( ( X33 @ id @ ( X36 @ id @ X37 ) @ id )
!= $true ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f30,plain,
( ( sP16 = $true )
<=> ( ( sP15 = $true )
| ? [X21: subst > term > subst > $o] :
! [X22: term > $o] :
? [X23: term] :
( ( X21 @ id @ X23 @ id )
!= ( X22 @ X23 ) )
| ? [X24: term,X25: term > $o] :
( ! [X30: term] :
( ( ( var @ X30 )
!= $true )
| ( ( X25 @ X30 )
= $true ) )
& ( ( X25 @ X24 )
!= $true )
& ( ( sP13 @ X25 )
= $true )
& ! [X28: term] :
( ( ( X25 @ ( lam @ X28 ) )
= $true )
| ? [X29: term] :
( ( ( X25 @ ( sub @ X28 @ ( push @ X29 @ id ) ) )
!= $true )
& ( ( X25 @ X29 )
= $true ) ) ) )
| ( sP14 = $true )
| ? [X31: term] :
( ( sub @ X31 @ id )
!= X31 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f31,plain,
( ( sP17 = $true )
<=> ( ! [X108: term,X109: subst > term > subst > $o,X110: term > $o] :
( ? [X111: term] :
( ( ( X110 @ X111 )
= $true )
& ( $true
!= ( X110 @ ( sub @ X108 @ ( push @ X111 @ id ) ) ) ) )
| ( ( X110 @ ( lam @ X108 ) )
= $true )
| ? [X113: subst > term > term] :
( ( ( X109 @ id @ ( lam @ ( X113 @ sh @ one ) ) @ id )
!= $true )
& ! [X115: subst,X116: term,X117: subst] :
( ( X113 @ ( comp @ X115 @ X117 ) @ ( sub @ X116 @ X117 ) )
= ( sub @ ( X113 @ X115 @ X116 ) @ X117 ) )
& ! [X114: term] :
( ( ( X109 @ id @ X114 @ id )
!= $true )
| ( ( X109 @ id @ ( X113 @ id @ X114 ) @ id )
= $true ) ) )
| ? [X112: term] :
( ( X109 @ id @ X112 @ id )
!= ( X110 @ X112 ) ) )
| ? [X103: term] :
! [X104: subst > term > term] :
? [X105: subst,X106: term] :
( ( sub @ X103 @ ( push @ X106 @ X105 ) )
!= ( X104 @ X105 @ X106 ) )
| ? [X107: term] :
( ( sub @ X107 @ id )
!= X107 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f32,plain,
! [X437: subst > term > subst > $o] :
( ( ( sP18 @ X437 )
= $true )
<=> ? [X438: subst > term > term] :
( ( ( X437 @ id @ ( lam @ ( X438 @ sh @ one ) ) @ id )
!= $true )
& ! [X439: term] :
( ( ( X437 @ id @ ( X438 @ id @ X439 ) @ id )
= $true )
| ( ( X437 @ id @ X439 @ id )
!= $true ) )
& ! [X440: term,X441: subst,X442: subst] :
( ( sub @ ( X438 @ X441 @ X440 ) @ X442 )
= ( X438 @ ( comp @ X441 @ X442 ) @ ( sub @ X440 @ X442 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f33,plain,
( ( sP19 = $true )
<=> ? [X428: term,X429: subst > term > subst > $o] :
( ! [X431: subst > term > term] :
( ? [X432: term] :
( ( ( X429 @ id @ X432 @ id )
= $true )
& ( ( X429 @ id @ ( X431 @ id @ X432 ) @ id )
!= $true ) )
| ( ( X429 @ id @ ( lam @ ( X431 @ sh @ one ) ) @ id )
= $true )
| ? [X433: term,X434: subst,X435: subst] :
( ( X431 @ ( comp @ X435 @ X434 ) @ ( sub @ X433 @ X434 ) )
!= ( sub @ ( X431 @ X435 @ X433 ) @ X434 ) ) )
& ! [X430: term] :
( ( ( X429 @ id @ X430 @ id )
!= $true )
| ( ( X429 @ id @ ( sub @ X428 @ ( push @ X430 @ id ) ) @ id )
= $true ) )
& ( ( X429 @ id @ ( lam @ ( sub @ X428 @ ( push @ one @ sh ) ) ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f34,plain,
( ( sP20 = $true )
<=> ( ! [X436: term,X437: subst > term > subst > $o] :
( ? [X443: term,X444: subst,X445: subst,X446: subst] :
( ( ( X437 @ ( comp @ X446 @ X445 ) @ ( sub @ X443 @ X445 ) @ X444 )
= $true )
& ( $true
!= ( X437 @ X446 @ X443 @ ( comp @ X445 @ X444 ) ) ) )
| ( ( X437 @ id @ ( lam @ X436 ) @ id )
= $true )
| ( ( sP18 @ X437 )
= $true )
| ? [X448: subst,X449: subst,X450: subst,X451: term] :
( ( ( X437 @ ( comp @ X450 @ X449 ) @ ( sub @ X451 @ X449 ) @ X448 )
!= $true )
& ( ( X437 @ X450 @ X451 @ ( comp @ X449 @ X448 ) )
= $true ) )
| ? [X447: term] :
( ( ( X437 @ id @ ( sub @ X436 @ ( push @ X447 @ id ) ) @ id )
!= $true )
& ( ( X437 @ id @ X447 @ id )
= $true ) ) )
| ? [X427: term] :
( ( sub @ X427 @ id )
!= X427 )
| ( sP19 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f35,plain,
( ( sP21 = $true )
<=> ! [X235: subst > term > subst > $o,X236: term,X237: term > $o] :
( ? [X244: term] :
( ( X237 @ X244 )
!= ( X235 @ id @ X244 @ id ) )
| ( $true
= ( X237 @ ( lam @ X236 ) ) )
| ? [X239: subst > term > term] :
( ! [X240: subst,X241: term,X242: subst] :
( ( X239 @ ( comp @ X242 @ X240 ) @ ( sub @ X241 @ X240 ) )
= ( sub @ ( X239 @ X242 @ X241 ) @ X240 ) )
& ! [X243: term] :
( ( ( X235 @ id @ X243 @ id )
!= $true )
| ( ( X235 @ id @ ( X239 @ id @ X243 ) @ id )
= $true ) )
& ( ( X235 @ id @ ( lam @ ( X239 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X238: term] :
( ( ( X237 @ X238 )
= $true )
& ( ( X237 @ ( sub @ X236 @ ( push @ X238 @ id ) ) )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f36,plain,
! [X337: subst > term > subst > $o] :
( ( ( sP22 @ X337 )
= $true )
<=> ! [X343: term,X344: term] :
( ( ( X337 @ id @ X344 @ id )
!= $true )
| ( ( X337 @ id @ ( ap @ ( sub @ X343 @ id ) @ X344 ) @ id )
= $true )
| ( ( X337 @ id @ X343 @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f37,plain,
! [X329: term > $o] :
( ( ( sP23 @ X329 )
= $true )
<=> ! [X332: term,X333: term] :
( ( ( X329 @ ( ap @ X332 @ X333 ) )
= $true )
| ( ( X329 @ X332 )
!= $true )
| ( ( X329 @ X333 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f38,plain,
! [X366: subst > term > subst > $o] :
( ( ( sP24 @ X366 )
= $true )
<=> ? [X372: term,X373: term] :
( ( ( X366 @ id @ ( ap @ ( sub @ X373 @ id ) @ X372 ) @ id )
!= $true )
& ( ( X366 @ id @ X372 @ id )
= $true )
& ( ( X366 @ id @ X373 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f39,plain,
! [X366: subst > term > subst > $o] :
( ( ( sP25 @ X366 )
= $true )
<=> ? [X374: subst > term > term] :
( ! [X375: subst,X376: term,X377: subst] :
( ( X374 @ ( comp @ X377 @ X375 ) @ ( sub @ X376 @ X375 ) )
= ( sub @ ( X374 @ X377 @ X376 ) @ X375 ) )
& ! [X378: term] :
( ( ( X366 @ id @ X378 @ id )
!= $true )
| ( ( X366 @ id @ ( X374 @ id @ X378 ) @ id )
= $true ) )
& ( ( X366 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f40,plain,
! [X349: subst > term > subst > $o] :
( ( ( sP26 @ X349 )
= $true )
<=> ! [X355: subst > term > term] :
( ( ( X349 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id )
= $true )
| ? [X357: subst,X358: subst,X359: term] :
( ( sub @ ( X355 @ X358 @ X359 ) @ X357 )
!= ( X355 @ ( comp @ X358 @ X357 ) @ ( sub @ X359 @ X357 ) ) )
| ? [X356: term] :
( ( ( X349 @ id @ X356 @ id )
= $true )
& ( ( X349 @ id @ ( X355 @ id @ X356 ) @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f41,plain,
( ( sP27 = $true )
<=> ? [X349: subst > term > subst > $o,X350: term] :
( ! [X361: subst,X362: subst,X363: term,X364: subst] :
( ( ( X349 @ X361 @ X363 @ ( comp @ X362 @ X364 ) )
!= $true )
| ( ( X349 @ ( comp @ X361 @ X362 ) @ ( sub @ X363 @ X362 ) @ X364 )
= $true ) )
& ! [X351: subst,X352: subst,X353: term,X354: subst] :
( ( ( X349 @ ( comp @ X351 @ X352 ) @ ( sub @ X353 @ X352 ) @ X354 )
!= $true )
| ( ( X349 @ X351 @ X353 @ ( comp @ X352 @ X354 ) )
= $true ) )
& ( ( X349 @ id @ ( lam @ X350 ) @ id )
!= $true )
& ( ( sP26 @ X349 )
= $true )
& ! [X360: term] :
( ( ( X349 @ id @ ( sub @ X350 @ ( push @ X360 @ id ) ) @ id )
= $true )
| ( ( X349 @ id @ X360 @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f42,plain,
( ( sP28 = $true )
<=> ! [X365: term,X366: subst > term > subst > $o] :
( ? [X368: subst,X369: subst,X370: subst,X371: term] :
( ( ( X366 @ ( comp @ X369 @ X370 ) @ ( sub @ X371 @ X370 ) @ X368 )
!= $true )
& ( ( X366 @ X369 @ X371 @ ( comp @ X370 @ X368 ) )
= $true ) )
| ( ( sP25 @ X366 )
= $true )
| ? [X367: term] :
( ( ( X366 @ id @ X367 @ id )
!= $true )
& ( ( var @ ( sub @ X367 @ id ) )
= $true ) )
| ( ( sP24 @ X366 )
= $true )
| ? [X379: subst,X380: subst,X381: term,X382: subst] :
( ( ( X366 @ X380 @ X381 @ ( comp @ X379 @ X382 ) )
!= $true )
& ( ( X366 @ ( comp @ X380 @ X379 ) @ ( sub @ X381 @ X379 ) @ X382 )
= $true ) )
| ( ( X366 @ id @ X365 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f43,plain,
( ( sP29 = $true )
<=> ? [X336: term,X337: subst > term > subst > $o,X338: term] :
( ! [X345: subst,X346: term,X347: subst,X348: subst] :
( ( ( X337 @ ( comp @ X348 @ X347 ) @ ( sub @ X346 @ X347 ) @ X345 )
!= $true )
| ( ( X337 @ X348 @ X346 @ ( comp @ X347 @ X345 ) )
= $true ) )
& ! [X339: subst,X340: subst,X341: term,X342: subst] :
( ( ( X337 @ ( comp @ X342 @ X339 ) @ ( sub @ X341 @ X339 ) @ X340 )
= $true )
| ( ( X337 @ X342 @ X341 @ ( comp @ X339 @ X340 ) )
!= $true ) )
& ( ( X337 @ id @ X338 @ id )
= $true )
& ( ( X337 @ id @ ( ap @ X338 @ X336 ) @ id )
!= $true )
& ( ( sP22 @ X337 )
= $true )
& ( ( X337 @ id @ X336 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f44,plain,
( ( sP30 = $true )
<=> ( ( sP27 = $true )
| ( sP28 = $true )
| ? [X329: term > $o,X330: term] :
( ( ( X329 @ X330 )
!= $true )
& ( ( sP23 @ X329 )
= $true )
& ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( ( var @ X331 )
!= $true ) )
& ! [X334: term] :
( ? [X335: term] :
( ( ( X329 @ X335 )
= $true )
& ( ( X329 @ ( sub @ X334 @ ( push @ X335 @ id ) ) )
!= $true ) )
| ( ( X329 @ ( lam @ X334 ) )
= $true ) ) )
| ( sP29 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f45,plain,
( ( $true = sP31 )
<=> ( ? [X9: term,X10: term] :
( ( ( lam @ X10 )
= ( lam @ X9 ) )
& ( X9 != X10 ) )
| ! [X11: subst > term > term,X12: subst > term > term,X13: term,X14: subst] :
( ( ( lam @ ( X11 @ sh @ one ) )
!= ( lam @ ( X12 @ sh @ one ) ) )
| ? [X15: term,X16: subst,X17: subst] :
( ( sub @ ( X11 @ X17 @ X15 ) @ X16 )
!= ( X11 @ ( comp @ X17 @ X16 ) @ ( sub @ X15 @ X16 ) ) )
| ? [X18: subst,X19: term,X20: subst] :
( ( sub @ ( X12 @ X20 @ X19 ) @ X18 )
!= ( X12 @ ( comp @ X20 @ X18 ) @ ( sub @ X19 @ X18 ) ) )
| ( ( X11 @ X14 @ X13 )
= ( X12 @ X14 @ X13 ) ) )
| ? [X5: term,X6: subst] :
( ( sub @ one @ ( push @ X5 @ X6 ) )
!= X5 )
| ? [X7: term,X8: subst] :
( ( comp @ sh @ ( push @ X7 @ X8 ) )
!= X8 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f46,plain,
! [X318: term > $o] :
( ( ( sP32 @ X318 )
= $true )
<=> ? [X322: term,X323: term] :
( ( ( X318 @ X322 )
= $true )
& ( ( X318 @ ( ap @ X323 @ X322 ) )
!= $true )
& ( ( X318 @ X323 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f47,plain,
( ( sP33 = $true )
<=> ! [X317: term,X318: term > $o,X319: subst] :
( ? [X324: term] :
( ( ( var @ X324 )
= $true )
& ( ( X318 @ ( sub @ X324 @ X319 ) )
!= $true ) )
| ? [X320: term] :
( ! [X321: term] :
( ( ( X318 @ ( sub @ X320 @ ( push @ X321 @ id ) ) )
= $true )
| ( ( X318 @ X321 )
!= $true ) )
& ( ( X318 @ ( lam @ X320 ) )
!= $true ) )
| ( ( sP32 @ X318 )
= $true )
| ( ( X318 @ ( sub @ X317 @ X319 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f48,plain,
( ( sP34 = $true )
<=> ? [X84: term,X85: subst > term > subst > $o] :
( ( ( X85 @ id @ ( lam @ ( sub @ X84 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X87: subst > term > term] :
( ? [X89: subst,X90: term,X91: subst] :
( ( X87 @ ( comp @ X91 @ X89 ) @ ( sub @ X90 @ X89 ) )
!= ( sub @ ( X87 @ X91 @ X90 ) @ X89 ) )
| ( ( X85 @ id @ ( lam @ ( X87 @ sh @ one ) ) @ id )
= $true )
| ? [X88: term] :
( ( ( X85 @ id @ X88 @ id )
= $true )
& ( $true
!= ( X85 @ id @ ( X87 @ id @ X88 ) @ id ) ) ) )
& ! [X86: term] :
( ( ( X85 @ id @ ( sub @ X84 @ ( push @ X86 @ id ) ) @ id )
= $true )
| ( ( X85 @ id @ X86 @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f49,plain,
( ( sP35 = $true )
<=> ( ( sP34 = $true )
| ! [X92: term,X93: subst > term > subst > $o] :
( ? [X94: subst > term > term] :
( ! [X98: term] :
( ( ( X93 @ id @ X98 @ id )
!= $true )
| ( ( X93 @ id @ ( X94 @ id @ X98 ) @ id )
= $true ) )
& ! [X95: subst,X96: term,X97: subst] :
( ( X94 @ ( comp @ X95 @ X97 ) @ ( sub @ X96 @ X97 ) )
= ( sub @ ( X94 @ X95 @ X96 ) @ X97 ) )
& ( ( X93 @ id @ ( lam @ ( X94 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X93 @ id @ ( lam @ X92 ) @ id ) )
| ? [X99: term] :
( ( ( X93 @ id @ ( sub @ X92 @ ( push @ X99 @ id ) ) @ id )
!= $true )
& ( ( X93 @ id @ X99 @ id )
= $true ) ) )
| ? [X83: term] :
( ( sub @ X83 @ id )
!= X83 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f50,plain,
( ( $true = sP36 )
<=> ! [X127: subst > term > subst > $o,X128: term] :
( ( ( X127 @ id @ ( lam @ ( sub @ X128 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X130: subst > term > term] :
( ! [X134: term] :
( ( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
= $true )
| ( ( X127 @ id @ X134 @ id )
!= $true ) )
& ( ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id )
!= $true )
& ! [X131: term,X132: subst,X133: subst] :
( ( X130 @ ( comp @ X132 @ X133 ) @ ( sub @ X131 @ X133 ) )
= ( sub @ ( X130 @ X132 @ X131 ) @ X133 ) ) )
| ? [X129: term] :
( ( ( X127 @ id @ X129 @ id )
= $true )
& ( ( X127 @ id @ ( sub @ X128 @ ( push @ X129 @ id ) ) @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f51,plain,
( ( sP37 = $true )
<=> ! [X75: subst > term > subst > $o,X76: term] :
( ? [X77: subst > term > term] :
( ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
!= $true )
& ! [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) )
& ! [X81: term] :
( ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
= $true )
| ( ( X75 @ id @ X81 @ id )
!= $true ) ) )
| ? [X82: term] :
( ( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
!= $true )
& ( ( X75 @ id @ X82 @ id )
= $true ) )
| ( ( X75 @ id @ ( lam @ X76 ) @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f52,plain,
! [X295: term > $o] :
( ( ( sP38 @ X295 )
= $true )
<=> ? [X297: term,X298: term] :
( ( $true
!= ( X295 @ ( ap @ X297 @ X298 ) ) )
& ( ( X295 @ X297 )
= $true )
& ( ( X295 @ X298 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f53,plain,
( ( sP39 = $true )
<=> ! [X294: term,X295: term > $o] :
( ? [X299: term] :
( ( ( X295 @ ( lam @ X299 ) )
!= $true )
& ! [X300: term] :
( ( ( X295 @ X300 )
!= $true )
| ( ( X295 @ ( sub @ X299 @ ( push @ X300 @ id ) ) )
= $true ) ) )
| ( ( X295 @ X294 )
= $true )
| ? [X296: term] :
( ( ( var @ X296 )
= $true )
& ( ( X295 @ X296 )
!= $true ) )
| ( ( sP38 @ X295 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f54,plain,
! [X274: subst > term > subst > $o] :
( ( ( sP40 @ X274 )
= $true )
<=> ? [X285: subst > term > term] :
( ! [X286: term,X287: subst,X288: subst] :
( ( X285 @ ( comp @ X287 @ X288 ) @ ( sub @ X286 @ X288 ) )
= ( sub @ ( X285 @ X287 @ X286 ) @ X288 ) )
& ! [X289: term] :
( ( ( X274 @ id @ ( X285 @ id @ X289 ) @ id )
= $true )
| ( ( X274 @ id @ X289 @ id )
!= $true ) )
& ( ( X274 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f55,plain,
! [X274: subst > term > subst > $o] :
( ( ( sP41 @ X274 )
= $true )
<=> ? [X290: term,X291: term] :
( ( ( X274 @ id @ X290 @ id )
= $true )
& ( ( X274 @ id @ ( ap @ ( sub @ X291 @ id ) @ X290 ) @ id )
!= $true )
& ( ( X274 @ id @ X291 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f56,plain,
( ( sP42 = $true )
<=> ! [X274: subst > term > subst > $o,X275: term] :
( ( ( sP41 @ X274 )
= $true )
| ( ( X274 @ id @ X275 @ id )
= $true )
| ( ( sP40 @ X274 )
= $true )
| ? [X277: term,X278: subst,X279: subst,X280: subst] :
( ( ( X274 @ X279 @ X277 @ ( comp @ X278 @ X280 ) )
!= $true )
& ( ( X274 @ ( comp @ X279 @ X278 ) @ ( sub @ X277 @ X278 ) @ X280 )
= $true ) )
| ? [X276: term] :
( ( ( X274 @ id @ X276 @ id )
!= $true )
& ( ( var @ ( sub @ X276 @ id ) )
= $true ) )
| ? [X281: subst,X282: term,X283: subst,X284: subst] :
( ( ( X274 @ ( comp @ X283 @ X284 ) @ ( sub @ X282 @ X284 ) @ X281 )
!= $true )
& ( ( X274 @ X283 @ X282 @ ( comp @ X284 @ X281 ) )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f57,plain,
! [X528: subst > term > subst > $o] :
( ( ( sP43 @ X528 )
= $true )
<=> ? [X530: term,X531: term] :
( ( ( X528 @ id @ X531 @ id )
= $true )
& ( ( X528 @ id @ X530 @ id )
= $true )
& ( ( X528 @ id @ ( ap @ ( sub @ X531 @ id ) @ X530 ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f58,plain,
( ( sP44 = $true )
<=> ! [X525: term > $o,X526: term,X527: term,X528: subst > term > subst > $o] :
( ? [X532: subst,X533: subst,X534: term,X535: subst] :
( ( ( X528 @ X533 @ X534 @ ( comp @ X532 @ X535 ) )
!= $true )
& ( ( X528 @ ( comp @ X533 @ X532 ) @ ( sub @ X534 @ X532 ) @ X535 )
= $true ) )
| ( ( X525 @ X527 )
!= $true )
| ? [X536: subst,X537: subst,X538: term,X539: subst] :
( ( ( X528 @ X536 @ X538 @ ( comp @ X537 @ X539 ) )
= $true )
& ( ( X528 @ ( comp @ X536 @ X537 ) @ ( sub @ X538 @ X537 ) @ X539 )
!= $true ) )
| ? [X529: term] :
( ( X528 @ id @ X529 @ id )
!= ( X525 @ X529 ) )
| ( ( X525 @ X526 )
!= $true )
| ( ( X525 @ ( ap @ X527 @ X526 ) )
= $true )
| ( ( sP43 @ X528 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f59,plain,
! [X146: subst > term > subst > $o] :
( ( ( sP45 @ X146 )
= $true )
<=> ? [X159: subst > term > term] :
( ( ( X146 @ id @ ( lam @ ( X159 @ sh @ one ) ) @ id )
!= $true )
& ! [X161: term,X162: subst,X163: subst] :
( ( X159 @ ( comp @ X163 @ X162 ) @ ( sub @ X161 @ X162 ) )
= ( sub @ ( X159 @ X163 @ X161 ) @ X162 ) )
& ! [X160: term] :
( ( ( X146 @ id @ X160 @ id )
!= $true )
| ( ( X146 @ id @ ( X159 @ id @ X160 ) @ id )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f60,plain,
( ( sP46 = $true )
<=> ! [X146: subst > term > subst > $o,X147: term > $o,X148: term] :
( ( ( sP45 @ X146 )
= $true )
| ( ( X147 @ ( lam @ X148 ) )
= $true )
| ? [X149: term] :
( ( X147 @ X149 )
!= ( X146 @ id @ X149 @ id ) )
| ? [X154: subst,X155: subst,X156: subst,X157: term] :
( ( ( X146 @ X155 @ X157 @ ( comp @ X154 @ X156 ) )
= $true )
& ( ( X146 @ ( comp @ X155 @ X154 ) @ ( sub @ X157 @ X154 ) @ X156 )
!= $true ) )
| ? [X158: term] :
( ( ( X147 @ X158 )
= $true )
& ( ( X147 @ ( sub @ X148 @ ( push @ X158 @ id ) ) )
!= $true ) )
| ? [X150: subst,X151: subst,X152: subst,X153: term] :
( ( ( X146 @ X151 @ X153 @ ( comp @ X150 @ X152 ) )
!= $true )
& ( ( X146 @ ( comp @ X151 @ X150 ) @ ( sub @ X153 @ X150 ) @ X152 )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f61,plain,
( ( sP47 = $true )
<=> ! [X419: subst > term > subst > $o,X420: term] :
( ( ( X419 @ id @ ( lam @ ( sub @ X420 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X422: subst > term > term] :
( ! [X423: term,X424: subst,X425: subst] :
( ( X422 @ ( comp @ X425 @ X424 ) @ ( sub @ X423 @ X424 ) )
= ( sub @ ( X422 @ X425 @ X423 ) @ X424 ) )
& ( ( X419 @ id @ ( lam @ ( X422 @ sh @ one ) ) @ id )
!= $true )
& ! [X426: term] :
( ( ( X419 @ id @ ( X422 @ id @ X426 ) @ id )
= $true )
| ( ( X419 @ id @ X426 @ id )
!= $true ) ) )
| ? [X421: term] :
( ( ( X419 @ id @ X421 @ id )
= $true )
& ( ( X419 @ id @ ( sub @ X420 @ ( push @ X421 @ id ) ) @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f62,plain,
( ( sP48 = $true )
<=> ! [X452: subst > term > subst > $o,X453: term] :
( ( ( X452 @ id @ ( lam @ ( sub @ X453 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X454: term] :
( ( ( X452 @ id @ ( sub @ X453 @ ( push @ X454 @ id ) ) @ id )
!= $true )
& ( ( X452 @ id @ X454 @ id )
= $true ) )
| ? [X455: subst > term > term,X456: term] :
( ! [X461: subst,X462: term,X463: subst] :
( ( sub @ ( sub @ X456 @ ( push @ X462 @ X461 ) ) @ X463 )
= ( sub @ ( X455 @ X461 @ X462 ) @ X463 ) )
& ! [X457: subst,X458: term,X459: subst] :
( ( sub @ X456 @ ( push @ ( sub @ X458 @ X457 ) @ ( comp @ X459 @ X457 ) ) )
= ( X455 @ ( comp @ X459 @ X457 ) @ ( sub @ X458 @ X457 ) ) )
& ! [X460: term] :
( ( ( X452 @ id @ X460 @ id )
!= $true )
| ( ( X452 @ id @ ( X455 @ id @ X460 ) @ id )
= $true ) )
& ( ( X452 @ id @ ( lam @ ( X455 @ sh @ one ) ) @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f63,plain,
! [X201: term > $o] :
( ( ( sP49 @ X201 )
= $true )
<=> ! [X204: term,X205: term] :
( ( ( X201 @ X205 )
!= $true )
| ( ( X201 @ ( ap @ X205 @ X204 ) )
= $true )
| ( ( X201 @ X204 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f64,plain,
! [X219: subst > term > subst > $o] :
( ( ( sP50 @ X219 )
= $true )
<=> ? [X221: term,X222: term] :
( ( ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id )
!= $true )
& ( ( X219 @ id @ X221 @ id )
= $true )
& ( ( X219 @ id @ X222 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f65,plain,
( ( sP51 = $true )
<=> ? [X209: subst > term > subst > $o,X210: term > $o,X211: term] :
( ! [X218: term] :
( ( X209 @ id @ X218 @ id )
= ( X210 @ X218 ) )
& ( ( X210 @ ( lam @ X211 ) )
!= $true )
& ! [X212: term] :
( ( ( X210 @ ( sub @ X211 @ ( push @ X212 @ id ) ) )
= $true )
| ( ( X210 @ X212 )
!= $true ) )
& ! [X213: subst > term > term] :
( ? [X215: subst,X216: subst,X217: term] :
( ( sub @ ( X213 @ X216 @ X217 ) @ X215 )
!= ( X213 @ ( comp @ X216 @ X215 ) @ ( sub @ X217 @ X215 ) ) )
| ( ( X209 @ id @ ( lam @ ( X213 @ sh @ one ) ) @ id )
= $true )
| ? [X214: term] :
( ( ( X209 @ id @ ( X213 @ id @ X214 ) @ id )
!= $true )
& ( ( X209 @ id @ X214 @ id )
= $true ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f66,plain,
( ( sP52 = $true )
<=> ? [X201: term > $o,X202: term] :
( ( ( X201 @ X202 )
!= $true )
& ( ( sP49 @ X201 )
= $true )
& ! [X206: term] :
( ? [X207: term] :
( ( ( X201 @ ( sub @ X206 @ ( push @ X207 @ id ) ) )
!= $true )
& ( ( X201 @ X207 )
= $true ) )
| ( ( X201 @ ( lam @ X206 ) )
= $true ) )
& ! [X203: term] :
( ( ( X201 @ X203 )
= $true )
| ( ( var @ X203 )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f67,plain,
( ( sP53 = $true )
<=> ( ? [X208: term] :
( ( sub @ X208 @ id )
!= X208 )
| ! [X219: subst > term > subst > $o,X220: term] :
( ? [X223: subst > term > term] :
( ( $true
!= ( X219 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id ) )
& ! [X225: subst,X226: term,X227: subst] :
( ( X223 @ ( comp @ X227 @ X225 ) @ ( sub @ X226 @ X225 ) )
= ( sub @ ( X223 @ X227 @ X226 ) @ X225 ) )
& ! [X224: term] :
( ( ( X219 @ id @ ( X223 @ id @ X224 ) @ id )
= $true )
| ( ( X219 @ id @ X224 @ id )
!= $true ) ) )
| ( ( sP50 @ X219 )
= $true )
| ( ( X219 @ id @ X220 @ id )
= $true ) )
| ( sP52 = $true )
| ? [X198: subst > term > subst > $o] :
! [X199: term > $o] :
? [X200: term] :
( ( X198 @ id @ X200 @ id )
!= ( X199 @ X200 ) )
| ( sP51 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f68,plain,
! [X311: term > $o] :
( ( ( sP54 @ X311 )
= $true )
<=> ? [X315: term,X316: term] :
( ( ( X311 @ X315 )
= $true )
& ( ( X311 @ ( ap @ X315 @ X316 ) )
!= $true )
& ( ( X311 @ X316 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f69,plain,
! [X303: term > $o] :
( ( ( sP55 @ X303 )
= $true )
<=> ! [X305: term,X306: term] :
( ( ( X303 @ ( ap @ X305 @ X306 ) )
= $true )
| ( ( X303 @ X305 )
!= $true )
| ( ( X303 @ X306 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f70,plain,
( ( sP56 = $true )
<=> ! [X310: term,X311: term > $o] :
( ? [X313: term] :
( ! [X314: term] :
( ( ( X311 @ X314 )
!= $true )
| ( ( X311 @ ( sub @ X313 @ ( push @ X314 @ id ) ) )
= $true ) )
& ( ( X311 @ ( lam @ X313 ) )
!= $true ) )
| ( ( sP54 @ X311 )
= $true )
| ? [X312: term] :
( ( ( var @ X312 )
= $true )
& ( ( X311 @ X312 )
!= $true ) )
| ( ( X311 @ X310 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f71,plain,
( ( sP57 = $true )
<=> ( ? [X302: subst,X303: term > $o,X304: term] :
( ! [X309: term] :
( ( ( var @ X309 )
!= $true )
| ( ( X303 @ ( sub @ X309 @ X302 ) )
= $true ) )
& ( ( X303 @ ( sub @ X304 @ X302 ) )
!= $true )
& ! [X307: term] :
( ? [X308: term] :
( ( ( X303 @ ( sub @ X307 @ ( push @ X308 @ id ) ) )
!= $true )
& ( ( X303 @ X308 )
= $true ) )
| ( ( X303 @ ( lam @ X307 ) )
= $true ) )
& ( ( sP55 @ X303 )
= $true ) )
| ? [X301: term] :
( ( sub @ X301 @ id )
!= X301 )
| ( sP56 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f72,plain,
! [X184: subst > term > subst > $o] :
( ( ( sP58 @ X184 )
= $true )
<=> ? [X187: subst > term > term] :
( ! [X188: term] :
( ( ( X184 @ id @ ( X187 @ id @ X188 ) @ id )
= $true )
| ( ( X184 @ id @ X188 @ id )
!= $true ) )
& ( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
!= $true )
& ! [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f13,plain,
( ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X183: term,X184: subst > term > subst > $o] :
( ? [X185: term,X186: term] :
( ( ( X184 @ id @ X186 @ id )
= $true )
& ( ( X184 @ id @ X185 @ id )
= $true )
& ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
!= $true ) )
| ? [X187: subst > term > term] :
( ! [X188: term] :
( ( ( X184 @ id @ ( X187 @ id @ X188 ) @ id )
= $true )
| ( ( X184 @ id @ X188 @ id )
!= $true ) )
& ( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
!= $true )
& ! [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) ) )
| ( ( X184 @ id @ X183 @ id )
= $true ) ) )
& ( ( ? [X302: subst,X303: term > $o,X304: term] :
( ! [X309: term] :
( ( ( var @ X309 )
!= $true )
| ( ( X303 @ ( sub @ X309 @ X302 ) )
= $true ) )
& ( ( X303 @ ( sub @ X304 @ X302 ) )
!= $true )
& ! [X307: term] :
( ? [X308: term] :
( ( ( X303 @ ( sub @ X307 @ ( push @ X308 @ id ) ) )
!= $true )
& ( ( X303 @ X308 )
= $true ) )
| ( ( X303 @ ( lam @ X307 ) )
= $true ) )
& ! [X305: term,X306: term] :
( ( ( X303 @ ( ap @ X305 @ X306 ) )
= $true )
| ( ( X303 @ X305 )
!= $true )
| ( ( X303 @ X306 )
!= $true ) ) )
| ? [X301: term] :
( ( sub @ X301 @ id )
!= X301 )
| ! [X310: term,X311: term > $o] :
( ? [X313: term] :
( ! [X314: term] :
( ( ( X311 @ X314 )
!= $true )
| ( ( X311 @ ( sub @ X313 @ ( push @ X314 @ id ) ) )
= $true ) )
& ( ( X311 @ ( lam @ X313 ) )
!= $true ) )
| ? [X315: term,X316: term] :
( ( ( X311 @ X315 )
= $true )
& ( ( X311 @ ( ap @ X315 @ X316 ) )
!= $true )
& ( ( X311 @ X316 )
= $true ) )
| ? [X312: term] :
( ( ( var @ X312 )
= $true )
& ( ( X311 @ X312 )
!= $true ) )
| ( ( X311 @ X310 )
= $true ) ) )
<=> ( induction2_lthm = $true ) )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ? [X208: term] :
( ( sub @ X208 @ id )
!= X208 )
| ! [X219: subst > term > subst > $o,X220: term] :
( ? [X223: subst > term > term] :
( ( $true
!= ( X219 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id ) )
& ! [X225: subst,X226: term,X227: subst] :
( ( X223 @ ( comp @ X227 @ X225 ) @ ( sub @ X226 @ X225 ) )
= ( sub @ ( X223 @ X227 @ X226 ) @ X225 ) )
& ! [X224: term] :
( ( ( X219 @ id @ ( X223 @ id @ X224 ) @ id )
= $true )
| ( ( X219 @ id @ X224 @ id )
!= $true ) ) )
| ? [X221: term,X222: term] :
( ( ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id )
!= $true )
& ( ( X219 @ id @ X221 @ id )
= $true )
& ( ( X219 @ id @ X222 @ id )
= $true ) )
| ( ( X219 @ id @ X220 @ id )
= $true ) )
| ? [X201: term > $o,X202: term] :
( ( ( X201 @ X202 )
!= $true )
& ! [X204: term,X205: term] :
( ( ( X201 @ X205 )
!= $true )
| ( ( X201 @ ( ap @ X205 @ X204 ) )
= $true )
| ( ( X201 @ X204 )
!= $true ) )
& ! [X206: term] :
( ? [X207: term] :
( ( ( X201 @ ( sub @ X206 @ ( push @ X207 @ id ) ) )
!= $true )
& ( ( X201 @ X207 )
= $true ) )
| ( ( X201 @ ( lam @ X206 ) )
= $true ) )
& ! [X203: term] :
( ( ( X201 @ X203 )
= $true )
| ( ( var @ X203 )
!= $true ) ) )
| ? [X198: subst > term > subst > $o] :
! [X199: term > $o] :
? [X200: term] :
( ( X198 @ id @ X200 @ id )
!= ( X199 @ X200 ) )
| ? [X209: subst > term > subst > $o,X210: term > $o,X211: term] :
( ! [X218: term] :
( ( X209 @ id @ X218 @ id )
= ( X210 @ X218 ) )
& ( ( X210 @ ( lam @ X211 ) )
!= $true )
& ! [X212: term] :
( ( ( X210 @ ( sub @ X211 @ ( push @ X212 @ id ) ) )
= $true )
| ( ( X210 @ X212 )
!= $true ) )
& ! [X213: subst > term > term] :
( ? [X215: subst,X216: subst,X217: term] :
( ( sub @ ( X213 @ X216 @ X217 ) @ X215 )
!= ( X213 @ ( comp @ X216 @ X215 ) @ ( sub @ X217 @ X215 ) ) )
| ( ( X209 @ id @ ( lam @ ( X213 @ sh @ one ) ) @ id )
= $true )
| ? [X214: term] :
( ( ( X209 @ id @ ( X213 @ id @ X214 ) @ id )
!= $true )
& ( ( X209 @ id @ X214 @ id )
= $true ) ) ) ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( termmset_lthm = $true )
<=> ( ? [X272: term] :
( ( sub @ X272 @ id )
!= X272 )
| ! [X273: term] :
( ( sub @ X273 @ id )
= X273 ) ) )
& ( ! [X165: subst > term > subst > $o] :
? [X166: term > $o] :
! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( axmap = $true )
& ( ulamvarsh = $true )
& ( ! [X2: subst > term > subst > $o] :
? [X3: term > $o] :
! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( ? [X410: subst] :
( ( comp @ X410 @ id )
!= X410 )
| ? [X409: subst] :
( ( comp @ id @ X409 )
!= X409 )
| ( ! [X411: subst] :
( ( comp @ id @ X411 )
= X411 )
& ! [X412: subst] :
( ( comp @ X412 @ id )
= X412 ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( ! [X452: subst > term > subst > $o,X453: term] :
( ( ( X452 @ id @ ( lam @ ( sub @ X453 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X454: term] :
( ( ( X452 @ id @ ( sub @ X453 @ ( push @ X454 @ id ) ) @ id )
!= $true )
& ( ( X452 @ id @ X454 @ id )
= $true ) )
| ? [X455: subst > term > term,X456: term] :
( ! [X461: subst,X462: term,X463: subst] :
( ( sub @ ( sub @ X456 @ ( push @ X462 @ X461 ) ) @ X463 )
= ( sub @ ( X455 @ X461 @ X462 ) @ X463 ) )
& ! [X457: subst,X458: term,X459: subst] :
( ( sub @ X456 @ ( push @ ( sub @ X458 @ X457 ) @ ( comp @ X459 @ X457 ) ) )
= ( X455 @ ( comp @ X459 @ X457 ) @ ( sub @ X458 @ X457 ) ) )
& ! [X460: term] :
( ( ( X452 @ id @ X460 @ id )
!= $true )
| ( ( X452 @ id @ ( X455 @ id @ X460 ) @ id )
= $true ) )
& ( ( X452 @ id @ ( lam @ ( X455 @ sh @ one ) ) @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X419: subst > term > subst > $o,X420: term] :
( ( ( X419 @ id @ ( lam @ ( sub @ X420 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X422: subst > term > term] :
( ! [X423: term,X424: subst,X425: subst] :
( ( X422 @ ( comp @ X425 @ X424 ) @ ( sub @ X423 @ X424 ) )
= ( sub @ ( X422 @ X425 @ X423 ) @ X424 ) )
& ( ( X419 @ id @ ( lam @ ( X422 @ sh @ one ) ) @ id )
!= $true )
& ! [X426: term] :
( ( ( X419 @ id @ ( X422 @ id @ X426 ) @ id )
= $true )
| ( ( X419 @ id @ X426 @ id )
!= $true ) ) )
| ? [X421: term] :
( ( ( X419 @ id @ X421 @ id )
= $true )
& ( ( X419 @ id @ ( sub @ X420 @ ( push @ X421 @ id ) ) @ id )
!= $true ) ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( substmonoid_gthm = $true )
& ! [X245: d_term,X246: d_term] :
( ( ( d2term @ X246 )
!= ( d2term @ X245 ) )
| ( X245 = X246 ) )
& ( ulamvarind = $true )
& ( pushprop_lthm_orig = $true )
& ( ! [X261: term,X262: term,X263: term,X264: term] :
( ( X261 = X262 )
| ( ( ap @ X261 @ X263 )
!= ( ap @ X262 @ X264 ) ) )
<=> ( apinj1 = $true ) )
& ( hoaslamnotap_gthm = $true )
& ! [X181: d_subst,X182: d_subst] :
( ( X181 = X182 )
| ( ( d2subst @ X182 )
!= ( d2subst @ X181 ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X146: subst > term > subst > $o,X147: term > $o,X148: term] :
( ? [X159: subst > term > term] :
( ( ( X146 @ id @ ( lam @ ( X159 @ sh @ one ) ) @ id )
!= $true )
& ! [X161: term,X162: subst,X163: subst] :
( ( X159 @ ( comp @ X163 @ X162 ) @ ( sub @ X161 @ X162 ) )
= ( sub @ ( X159 @ X163 @ X161 ) @ X162 ) )
& ! [X160: term] :
( ( ( X146 @ id @ X160 @ id )
!= $true )
| ( ( X146 @ id @ ( X159 @ id @ X160 ) @ id )
= $true ) ) )
| ( ( X147 @ ( lam @ X148 ) )
= $true )
| ? [X149: term] :
( ( X147 @ X149 )
!= ( X146 @ id @ X149 @ id ) )
| ? [X154: subst,X155: subst,X156: subst,X157: term] :
( ( ( X146 @ X155 @ X157 @ ( comp @ X154 @ X156 ) )
= $true )
& ( ( X146 @ ( comp @ X155 @ X154 ) @ ( sub @ X157 @ X154 ) @ X156 )
!= $true ) )
| ? [X158: term] :
( ( ( X147 @ X158 )
= $true )
& ( ( X147 @ ( sub @ X148 @ ( push @ X158 @ id ) ) )
!= $true ) )
| ? [X150: subst,X151: subst,X152: subst,X153: term] :
( ( ( X146 @ X151 @ X153 @ ( comp @ X150 @ X152 ) )
!= $true )
& ( ( X146 @ ( comp @ X151 @ X150 ) @ ( sub @ X153 @ X150 ) @ X152 )
= $true ) ) ) )
& ! [X265: d_term] : ( d_one = X265 )
& ( ! [X525: term > $o,X526: term,X527: term,X528: subst > term > subst > $o] :
( ? [X532: subst,X533: subst,X534: term,X535: subst] :
( ( ( X528 @ X533 @ X534 @ ( comp @ X532 @ X535 ) )
!= $true )
& ( ( X528 @ ( comp @ X533 @ X532 ) @ ( sub @ X534 @ X532 ) @ X535 )
= $true ) )
| ( ( X525 @ X527 )
!= $true )
| ? [X536: subst,X537: subst,X538: term,X539: subst] :
( ( ( X528 @ X536 @ X538 @ ( comp @ X537 @ X539 ) )
= $true )
& ( ( X528 @ ( comp @ X536 @ X537 ) @ ( sub @ X538 @ X537 ) @ X539 )
!= $true ) )
| ? [X529: term] :
( ( X528 @ id @ X529 @ id )
!= ( X525 @ X529 ) )
| ( ( X525 @ X526 )
!= $true )
| ( ( X525 @ ( ap @ X527 @ X526 ) )
= $true )
| ? [X530: term,X531: term] :
( ( ( X528 @ id @ X531 @ id )
= $true )
& ( ( X528 @ id @ X530 @ id )
= $true )
& ( ( X528 @ id @ ( ap @ ( sub @ X531 @ id ) @ X530 ) @ id )
!= $true ) ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( ! [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
!= ( ap @ X326 @ X328 ) )
| ( X325 = X328 ) )
<=> ( apinj2 = $true ) )
& ( lamnotvar = $true )
& ( axabs = $true )
& ( ( hoasinduction = $true )
<=> ! [X274: subst > term > subst > $o,X275: term] :
( ? [X290: term,X291: term] :
( ( ( X274 @ id @ X290 @ id )
= $true )
& ( ( X274 @ id @ ( ap @ ( sub @ X291 @ id ) @ X290 ) @ id )
!= $true )
& ( ( X274 @ id @ X291 @ id )
= $true ) )
| ( ( X274 @ id @ X275 @ id )
= $true )
| ? [X285: subst > term > term] :
( ! [X286: term,X287: subst,X288: subst] :
( ( X285 @ ( comp @ X287 @ X288 ) @ ( sub @ X286 @ X288 ) )
= ( sub @ ( X285 @ X287 @ X286 ) @ X288 ) )
& ! [X289: term] :
( ( ( X274 @ id @ ( X285 @ id @ X289 ) @ id )
= $true )
| ( ( X274 @ id @ X289 @ id )
!= $true ) )
& ( ( X274 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X277: term,X278: subst,X279: subst,X280: subst] :
( ( ( X274 @ X279 @ X277 @ ( comp @ X278 @ X280 ) )
!= $true )
& ( ( X274 @ ( comp @ X279 @ X278 ) @ ( sub @ X277 @ X278 ) @ X280 )
= $true ) )
| ? [X276: term] :
( ( ( X274 @ id @ X276 @ id )
!= $true )
& ( ( var @ ( sub @ X276 @ id ) )
= $true ) )
| ? [X281: subst,X282: term,X283: subst,X284: subst] :
( ( ( X274 @ ( comp @ X283 @ X284 ) @ ( sub @ X282 @ X284 ) @ X281 )
!= $true )
& ( ( X274 @ X283 @ X282 @ ( comp @ X284 @ X281 ) )
= $true ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( pushprop_lem0_gthm = $true )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
| ( X193 = X194 ) )
<=> ( hoasapinj1 = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( ( hoaslaminj = $true )
<=> ! [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
!= ( lam @ ( X465 @ sh @ one ) ) )
| ( ( X465 @ X466 @ X464 )
= ( X467 @ X466 @ X464 ) )
| ? [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
!= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
| ? [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
!= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X247: term,X248: subst > term > term,X249: term] :
( ? [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
!= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) )
| ( ( ap @ ( sub @ X247 @ id ) @ X249 )
!= ( lam @ ( X248 @ sh @ one ) ) ) ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_lthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X405: term,X406: term,X407: term,X408: term] :
( ( X405 = X408 )
| ( ( ap @ ( sub @ X408 @ id ) @ X406 )
!= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) )
| ? [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
= ( ap @ X404 @ X402 ) )
& ( X401 != X404 ) )
| ? [X400: term] :
( ( sub @ X400 @ id )
!= X400 ) ) )
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
<=> ( termmset = $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
= $true ) ) )
& ( ! [X294: term,X295: term > $o] :
( ? [X299: term] :
( ( ( X295 @ ( lam @ X299 ) )
!= $true )
& ! [X300: term] :
( ( ( X295 @ X300 )
!= $true )
| ( ( X295 @ ( sub @ X299 @ ( push @ X300 @ id ) ) )
= $true ) ) )
| ( ( X295 @ X294 )
= $true )
| ? [X296: term] :
( ( ( var @ X296 )
= $true )
& ( ( X295 @ X296 )
!= $true ) )
| ? [X297: term,X298: term] :
( ( $true
!= ( X295 @ ( ap @ X297 @ X298 ) ) )
& ( ( X295 @ X297 )
= $true )
& ( ( X295 @ X298 )
= $true ) ) )
<=> ( induction2 = $true ) )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X75: subst > term > subst > $o,X76: term] :
( ? [X77: subst > term > term] :
( ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
!= $true )
& ! [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) )
& ! [X81: term] :
( ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
= $true )
| ( ( X75 @ id @ X81 @ id )
!= $true ) ) )
| ? [X82: term] :
( ( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
!= $true )
& ( ( X75 @ id @ X82 @ id )
= $true ) )
| ( ( X75 @ id @ ( lam @ X76 ) @ id )
= $true ) ) )
& ( hoasinduction_lem1 = $true )
& ( hoaslaminj_gthm = $true )
& ( pushprop = $true )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( substmonoid = $true )
<=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
& ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X127: subst > term > subst > $o,X128: term] :
( ( ( X127 @ id @ ( lam @ ( sub @ X128 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X130: subst > term > term] :
( ! [X134: term] :
( ( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
= $true )
| ( ( X127 @ id @ X134 @ id )
!= $true ) )
& ( ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id )
!= $true )
& ! [X131: term,X132: subst,X133: subst] :
( ( X130 @ ( comp @ X132 @ X133 ) @ ( sub @ X131 @ X133 ) )
= ( sub @ ( X130 @ X132 @ X131 ) @ X133 ) ) )
| ? [X129: term] :
( ( ( X127 @ id @ X129 @ id )
= $true )
& ( ( X127 @ id @ ( sub @ X128 @ ( push @ X129 @ id ) ) @ id )
!= $true ) ) ) )
& ( ulamvar1 != $true )
& ( ( ? [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
!= X121 )
| ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
!= $true )
| ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
= $true ) )
| ? [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
!= X118 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X139: term,X140: term > $o,X141: subst] :
? [X142: term > $o] :
! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) ) )
& ( one
= ( d2term @ d_one ) )
& ( ( shinj = $true )
<=> ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) ) )
& ( axvarshift = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2subst @ d_id )
= 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 ) ) ) ) ) ) )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ? [X84: term,X85: subst > term > subst > $o] :
( ( ( X85 @ id @ ( lam @ ( sub @ X84 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X87: subst > term > term] :
( ? [X89: subst,X90: term,X91: subst] :
( ( X87 @ ( comp @ X91 @ X89 ) @ ( sub @ X90 @ X89 ) )
!= ( sub @ ( X87 @ X91 @ X90 ) @ X89 ) )
| ( ( X85 @ id @ ( lam @ ( X87 @ sh @ one ) ) @ id )
= $true )
| ? [X88: term] :
( ( ( X85 @ id @ X88 @ id )
= $true )
& ( $true
!= ( X85 @ id @ ( X87 @ id @ X88 ) @ id ) ) ) )
& ! [X86: term] :
( ( ( X85 @ id @ ( sub @ X84 @ ( push @ X86 @ id ) ) @ id )
= $true )
| ( ( X85 @ id @ X86 @ id )
!= $true ) ) )
| ! [X92: term,X93: subst > term > subst > $o] :
( ? [X94: subst > term > term] :
( ! [X98: term] :
( ( ( X93 @ id @ X98 @ id )
!= $true )
| ( ( X93 @ id @ ( X94 @ id @ X98 ) @ id )
= $true ) )
& ! [X95: subst,X96: term,X97: subst] :
( ( X94 @ ( comp @ X95 @ X97 ) @ ( sub @ X96 @ X97 ) )
= ( sub @ ( X94 @ X95 @ X96 ) @ X97 ) )
& ( ( X93 @ id @ ( lam @ ( X94 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X93 @ id @ ( lam @ X92 ) @ id ) )
| ? [X99: term] :
( ( ( X93 @ id @ ( sub @ X92 @ ( push @ X99 @ id ) ) @ id )
!= $true )
& ( ( X93 @ id @ X99 @ id )
= $true ) ) )
| ? [X83: term] :
( ( sub @ X83 @ id )
!= X83 ) ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X135: term] :
? [X136: subst > term > term] :
! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) ) )
& ( ! [X292: term,X293: term] :
( ( X292 = X293 )
| ( ( lam @ X293 )
!= ( lam @ X292 ) ) )
<=> ( laminj = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X144: term] :
? [X145: subst > term > term] :
( ( X145 @ sh @ one )
= ( sub @ X144 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X257: term,X258: term,X259: term,X260: term] :
( ( X257 = X258 )
| ( ( ap @ ( sub @ X260 @ id ) @ X258 )
!= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) )
| ? [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
= ( ap @ X255 @ X254 ) )
& ( X253 != X254 ) ) ) )
& ( ! [X317: term,X318: term > $o,X319: subst] :
( ? [X324: term] :
( ( ( var @ X324 )
= $true )
& ( ( X318 @ ( sub @ X324 @ X319 ) )
!= $true ) )
| ? [X320: term] :
( ! [X321: term] :
( ( ( X318 @ ( sub @ X320 @ ( push @ X321 @ id ) ) )
= $true )
| ( ( X318 @ X321 )
!= $true ) )
& ( ( X318 @ ( lam @ X320 ) )
!= $true ) )
| ? [X322: term,X323: term] :
( ( ( X318 @ X322 )
= $true )
& ( ( X318 @ ( ap @ X323 @ X322 ) )
!= $true )
& ( ( X318 @ X323 )
= $true ) )
| ( ( X318 @ ( sub @ X317 @ X319 ) )
= $true ) )
<=> ( induction2lem = $true ) )
& ( ( hoaslaminj_lthm = $true )
<=> ( ? [X9: term,X10: term] :
( ( ( lam @ X10 )
= ( lam @ X9 ) )
& ( X9 != X10 ) )
| ! [X11: subst > term > term,X12: subst > term > term,X13: term,X14: subst] :
( ( ( lam @ ( X11 @ sh @ one ) )
!= ( lam @ ( X12 @ sh @ one ) ) )
| ? [X15: term,X16: subst,X17: subst] :
( ( sub @ ( X11 @ X17 @ X15 ) @ X16 )
!= ( X11 @ ( comp @ X17 @ X16 ) @ ( sub @ X15 @ X16 ) ) )
| ? [X18: subst,X19: term,X20: subst] :
( ( sub @ ( X12 @ X20 @ X19 ) @ X18 )
!= ( X12 @ ( comp @ X20 @ X18 ) @ ( sub @ X19 @ X18 ) ) )
| ( ( X11 @ X14 @ X13 )
= ( X12 @ X14 @ X13 ) ) )
| ? [X5: term,X6: subst] :
( ( sub @ one @ ( push @ X5 @ X6 ) )
!= X5 )
| ? [X7: term,X8: subst] :
( ( comp @ sh @ ( push @ X7 @ X8 ) )
!= X8 ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( ? [X349: subst > term > subst > $o,X350: term] :
( ! [X361: subst,X362: subst,X363: term,X364: subst] :
( ( ( X349 @ X361 @ X363 @ ( comp @ X362 @ X364 ) )
!= $true )
| ( ( X349 @ ( comp @ X361 @ X362 ) @ ( sub @ X363 @ X362 ) @ X364 )
= $true ) )
& ! [X351: subst,X352: subst,X353: term,X354: subst] :
( ( ( X349 @ ( comp @ X351 @ X352 ) @ ( sub @ X353 @ X352 ) @ X354 )
!= $true )
| ( ( X349 @ X351 @ X353 @ ( comp @ X352 @ X354 ) )
= $true ) )
& ( ( X349 @ id @ ( lam @ X350 ) @ id )
!= $true )
& ! [X355: subst > term > term] :
( ( ( X349 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id )
= $true )
| ? [X357: subst,X358: subst,X359: term] :
( ( sub @ ( X355 @ X358 @ X359 ) @ X357 )
!= ( X355 @ ( comp @ X358 @ X357 ) @ ( sub @ X359 @ X357 ) ) )
| ? [X356: term] :
( ( ( X349 @ id @ X356 @ id )
= $true )
& ( ( X349 @ id @ ( X355 @ id @ X356 ) @ id )
!= $true ) ) )
& ! [X360: term] :
( ( ( X349 @ id @ ( sub @ X350 @ ( push @ X360 @ id ) ) @ id )
= $true )
| ( ( X349 @ id @ X360 @ id )
!= $true ) ) )
| ! [X365: term,X366: subst > term > subst > $o] :
( ? [X368: subst,X369: subst,X370: subst,X371: term] :
( ( ( X366 @ ( comp @ X369 @ X370 ) @ ( sub @ X371 @ X370 ) @ X368 )
!= $true )
& ( ( X366 @ X369 @ X371 @ ( comp @ X370 @ X368 ) )
= $true ) )
| ? [X374: subst > term > term] :
( ! [X375: subst,X376: term,X377: subst] :
( ( X374 @ ( comp @ X377 @ X375 ) @ ( sub @ X376 @ X375 ) )
= ( sub @ ( X374 @ X377 @ X376 ) @ X375 ) )
& ! [X378: term] :
( ( ( X366 @ id @ X378 @ id )
!= $true )
| ( ( X366 @ id @ ( X374 @ id @ X378 ) @ id )
= $true ) )
& ( ( X366 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X367: term] :
( ( ( X366 @ id @ X367 @ id )
!= $true )
& ( ( var @ ( sub @ X367 @ id ) )
= $true ) )
| ? [X372: term,X373: term] :
( ( ( X366 @ id @ ( ap @ ( sub @ X373 @ id ) @ X372 ) @ id )
!= $true )
& ( ( X366 @ id @ X372 @ id )
= $true )
& ( ( X366 @ id @ X373 @ id )
= $true ) )
| ? [X379: subst,X380: subst,X381: term,X382: subst] :
( ( ( X366 @ X380 @ X381 @ ( comp @ X379 @ X382 ) )
!= $true )
& ( ( X366 @ ( comp @ X380 @ X379 ) @ ( sub @ X381 @ X379 ) @ X382 )
= $true ) )
| ( ( X366 @ id @ X365 @ id )
= $true ) )
| ? [X329: term > $o,X330: term] :
( ( ( X329 @ X330 )
!= $true )
& ! [X332: term,X333: term] :
( ( ( X329 @ ( ap @ X332 @ X333 ) )
= $true )
| ( ( X329 @ X332 )
!= $true )
| ( ( X329 @ X333 )
!= $true ) )
& ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( ( var @ X331 )
!= $true ) )
& ! [X334: term] :
( ? [X335: term] :
( ( ( X329 @ X335 )
= $true )
& ( ( X329 @ ( sub @ X334 @ ( push @ X335 @ id ) ) )
!= $true ) )
| ( ( X329 @ ( lam @ X334 ) )
= $true ) ) )
| ? [X336: term,X337: subst > term > subst > $o,X338: term] :
( ! [X345: subst,X346: term,X347: subst,X348: subst] :
( ( ( X337 @ ( comp @ X348 @ X347 ) @ ( sub @ X346 @ X347 ) @ X345 )
!= $true )
| ( ( X337 @ X348 @ X346 @ ( comp @ X347 @ X345 ) )
= $true ) )
& ! [X339: subst,X340: subst,X341: term,X342: subst] :
( ( ( X337 @ ( comp @ X342 @ X339 ) @ ( sub @ X341 @ X339 ) @ X340 )
= $true )
| ( ( X337 @ X342 @ X341 @ ( comp @ X339 @ X340 ) )
!= $true ) )
& ( ( X337 @ id @ X338 @ id )
= $true )
& ( ( X337 @ id @ ( ap @ X338 @ X336 ) @ id )
!= $true )
& ! [X343: term,X344: term] :
( ( ( X337 @ id @ X344 @ id )
!= $true )
| ( ( X337 @ id @ ( ap @ ( sub @ X343 @ id ) @ X344 ) @ id )
= $true )
| ( ( X337 @ id @ X343 @ id )
!= $true ) )
& ( ( X337 @ id @ X336 @ id )
= $true ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
= $true )
| ( ( X522 @ X521 )
!= $true )
| ? [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
!= ( X523 @ X524 ) ) ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X235: subst > term > subst > $o,X236: term,X237: term > $o] :
( ? [X244: term] :
( ( X237 @ X244 )
!= ( X235 @ id @ X244 @ id ) )
| ( $true
= ( X237 @ ( lam @ X236 ) ) )
| ? [X239: subst > term > term] :
( ! [X240: subst,X241: term,X242: subst] :
( ( X239 @ ( comp @ X242 @ X240 ) @ ( sub @ X241 @ X240 ) )
= ( sub @ ( X239 @ X242 @ X241 ) @ X240 ) )
& ! [X243: term] :
( ( ( X235 @ id @ X243 @ id )
!= $true )
| ( ( X235 @ id @ ( X239 @ id @ X243 ) @ id )
= $true ) )
& ( ( X235 @ id @ ( lam @ ( X239 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X238: term] :
( ( ( X237 @ X238 )
= $true )
& ( ( X237 @ ( sub @ X236 @ ( push @ X238 @ id ) ) )
!= $true ) ) ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X436: term,X437: subst > term > subst > $o] :
( ? [X443: term,X444: subst,X445: subst,X446: subst] :
( ( ( X437 @ ( comp @ X446 @ X445 ) @ ( sub @ X443 @ X445 ) @ X444 )
= $true )
& ( $true
!= ( X437 @ X446 @ X443 @ ( comp @ X445 @ X444 ) ) ) )
| ( ( X437 @ id @ ( lam @ X436 ) @ id )
= $true )
| ? [X438: subst > term > term] :
( ( ( X437 @ id @ ( lam @ ( X438 @ sh @ one ) ) @ id )
!= $true )
& ! [X439: term] :
( ( ( X437 @ id @ ( X438 @ id @ X439 ) @ id )
= $true )
| ( ( X437 @ id @ X439 @ id )
!= $true ) )
& ! [X440: term,X441: subst,X442: subst] :
( ( sub @ ( X438 @ X441 @ X440 ) @ X442 )
= ( X438 @ ( comp @ X441 @ X442 ) @ ( sub @ X440 @ X442 ) ) ) )
| ? [X448: subst,X449: subst,X450: subst,X451: term] :
( ( ( X437 @ ( comp @ X450 @ X449 ) @ ( sub @ X451 @ X449 ) @ X448 )
!= $true )
& ( ( X437 @ X450 @ X451 @ ( comp @ X449 @ X448 ) )
= $true ) )
| ? [X447: term] :
( ( ( X437 @ id @ ( sub @ X436 @ ( push @ X447 @ id ) ) @ id )
!= $true )
& ( ( X437 @ id @ X447 @ id )
= $true ) ) )
| ? [X427: term] :
( ( sub @ X427 @ id )
!= X427 )
| ? [X428: term,X429: subst > term > subst > $o] :
( ! [X431: subst > term > term] :
( ? [X432: term] :
( ( ( X429 @ id @ X432 @ id )
= $true )
& ( ( X429 @ id @ ( X431 @ id @ X432 ) @ id )
!= $true ) )
| ( ( X429 @ id @ ( lam @ ( X431 @ sh @ one ) ) @ id )
= $true )
| ? [X433: term,X434: subst,X435: subst] :
( ( X431 @ ( comp @ X435 @ X434 ) @ ( sub @ X433 @ X434 ) )
!= ( sub @ ( X431 @ X435 @ X433 ) @ X434 ) ) )
& ! [X430: term] :
( ( ( X429 @ id @ X430 @ id )
!= $true )
| ( ( X429 @ id @ ( sub @ X428 @ ( push @ X430 @ id ) ) @ id )
= $true ) )
& ( ( X429 @ id @ ( lam @ ( sub @ X428 @ ( push @ one @ sh ) ) ) @ id )
!= $true ) ) ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( ? [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
!= X228 )
| ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ( ( X230 @ X233 )
!= $true )
| ? [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
!= ( X232 @ X234 ) )
| ( ( X232 @ one )
= $true ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ! [X271: d_subst] : ( d_id = X271 )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( apnotvar = $true )
& ( ( ! [X108: term,X109: subst > term > subst > $o,X110: term > $o] :
( ? [X111: term] :
( ( ( X110 @ X111 )
= $true )
& ( $true
!= ( X110 @ ( sub @ X108 @ ( push @ X111 @ id ) ) ) ) )
| ( ( X110 @ ( lam @ X108 ) )
= $true )
| ? [X113: subst > term > term] :
( ( ( X109 @ id @ ( lam @ ( X113 @ sh @ one ) ) @ id )
!= $true )
& ! [X115: subst,X116: term,X117: subst] :
( ( X113 @ ( comp @ X115 @ X117 ) @ ( sub @ X116 @ X117 ) )
= ( sub @ ( X113 @ X115 @ X116 ) @ X117 ) )
& ! [X114: term] :
( ( ( X109 @ id @ X114 @ id )
!= $true )
| ( ( X109 @ id @ ( X113 @ id @ X114 ) @ id )
= $true ) ) )
| ? [X112: term] :
( ( X109 @ id @ X112 @ id )
!= ( X110 @ X112 ) ) )
| ? [X103: term] :
! [X104: subst > term > term] :
? [X105: subst,X106: term] :
( ( sub @ X103 @ ( push @ X106 @ X105 ) )
!= ( X104 @ X105 @ X106 ) )
| ? [X107: term] :
( ( sub @ X107 @ id )
!= X107 ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ? [X32: term > $o,X33: subst > term > subst > $o,X34: term] :
( ! [X35: term] :
( ( ( X32 @ ( sub @ X34 @ ( push @ X35 @ id ) ) )
= $true )
| ( ( X32 @ X35 )
!= $true ) )
& ( ( X32 @ ( lam @ X34 ) )
!= $true )
& ! [X41: term] :
( ( X33 @ id @ X41 @ id )
= ( X32 @ X41 ) )
& ! [X36: subst > term > term] :
( ( ( X33 @ id @ ( lam @ ( X36 @ sh @ one ) ) @ id )
= $true )
| ? [X38: subst,X39: term,X40: subst] :
( ( X36 @ ( comp @ X38 @ X40 ) @ ( sub @ X39 @ X40 ) )
!= ( sub @ ( X36 @ X38 @ X39 ) @ X40 ) )
| ? [X37: term] :
( ( ( X33 @ id @ X37 @ id )
= $true )
& ( ( X33 @ id @ ( X36 @ id @ X37 ) @ id )
!= $true ) ) ) )
| ? [X21: subst > term > subst > $o] :
! [X22: term > $o] :
? [X23: term] :
( ( X21 @ id @ X23 @ id )
!= ( X22 @ X23 ) )
| ? [X24: term,X25: term > $o] :
( ! [X30: term] :
( ( ( var @ X30 )
!= $true )
| ( ( X25 @ X30 )
= $true ) )
& ( ( X25 @ X24 )
!= $true )
& ! [X26: term,X27: term] :
( ( ( X25 @ ( ap @ X27 @ X26 ) )
= $true )
| ( ( X25 @ X27 )
!= $true )
| ( ( X25 @ X26 )
!= $true ) )
& ! [X28: term] :
( ( ( X25 @ ( lam @ X28 ) )
= $true )
| ? [X29: term] :
( ( ( X25 @ ( sub @ X28 @ ( push @ X29 @ id ) ) )
!= $true )
& ( ( X25 @ X29 )
= $true ) ) ) )
| ! [X42: subst > term > subst > $o,X43: term] :
( ? [X44: subst > term > term] :
( ( ( X42 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id )
!= $true )
& ! [X48: term] :
( ( ( X42 @ id @ ( X44 @ id @ X48 ) @ id )
= $true )
| ( ( X42 @ id @ X48 @ id )
!= $true ) )
& ! [X45: subst,X46: subst,X47: term] :
( ( sub @ ( X44 @ X45 @ X47 ) @ X46 )
= ( X44 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) ) ) )
| ? [X50: subst,X51: term,X52: subst,X53: subst] :
( ( ( X42 @ ( comp @ X53 @ X52 ) @ ( sub @ X51 @ X52 ) @ X50 )
!= $true )
& ( ( X42 @ X53 @ X51 @ ( comp @ X52 @ X50 ) )
= $true ) )
| ? [X49: term] :
( ( ( X42 @ id @ X49 @ id )
!= $true )
& ( ( var @ ( sub @ X49 @ id ) )
= $true ) )
| ( ( X42 @ id @ X43 @ id )
= $true )
| ? [X54: term,X55: term] :
( ( ( X42 @ id @ X55 @ id )
= $true )
& ( ( X42 @ id @ X54 @ id )
= $true )
& ( ( X42 @ id @ ( ap @ ( sub @ X54 @ id ) @ X55 ) @ id )
!= $true ) )
| ? [X56: subst,X57: subst,X58: subst,X59: term] :
( ( ( X42 @ X58 @ X59 @ ( comp @ X57 @ X56 ) )
!= $true )
& ( ( X42 @ ( comp @ X58 @ X57 ) @ ( sub @ X59 @ X57 ) @ X56 )
= $true ) ) )
| ? [X31: term] :
( ( sub @ X31 @ id )
!= X31 ) ) )
& ( axclos = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X73: term] :
? [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
= ( X74 @ sh @ one ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X65: term] :
? [X66: subst > term > term] :
! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
<=> ( axidr = $true ) )
& ( axassoc = $true )
& ( ! [X495: term > $o,X496: term] :
( ? [X497: term,X498: term] :
( ( ( X495 @ ( ap @ X497 @ X498 ) )
!= $true )
& ( ( X495 @ X497 )
= $true )
& ( ( X495 @ X498 )
= $true ) )
| ? [X499: term] :
( ( ( var @ X499 )
= $true )
& ( ( X495 @ X499 )
!= $true ) )
| ( ( X495 @ X496 )
= $true )
| ? [X500: term] :
( ( ( X495 @ ( lam @ X500 ) )
!= $true )
& ( ( X495 @ X500 )
= $true ) ) )
<=> ( induction = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( ! [X485: term,X486: subst,X487: term > $o] :
( ? [X490: term] :
( ( ( X487 @ ( sub @ X490 @ X486 ) )
!= $true )
& ( ( var @ X490 )
= $true ) )
| ? [X491: term,X492: term] :
( ( ( X487 @ X491 )
= $true )
& ( ( X487 @ X492 )
= $true )
& ( ( X487 @ ( ap @ X491 @ X492 ) )
!= $true ) )
| ? [X488: term] :
( ! [X489: term] :
( ( ( X487 @ X489 )
!= $true )
| ( ( X487 @ ( sub @ X488 @ ( push @ X489 @ id ) ) )
= $true ) )
& ( ( X487 @ ( lam @ X488 ) )
!= $true ) )
| ( ( X487 @ ( sub @ X485 @ X486 ) )
= $true ) )
| ? [X476: subst,X477: term] :
( ( comp @ sh @ ( push @ X477 @ X476 ) )
!= X476 )
| ? [X474: subst,X475: term] :
( ( sub @ one @ ( push @ X475 @ X474 ) )
!= X475 )
| ? [X478: subst] :
( ( comp @ X478 @ id )
!= X478 )
| ? [X479: term > $o,X480: term] :
( ! [X482: term] :
( ( ( X479 @ X482 )
= $true )
| ( ( var @ X482 )
!= $true ) )
& ( ( X479 @ X480 )
!= $true )
& ! [X481: term] :
( ( ( X479 @ X481 )
!= $true )
| ( ( X479 @ ( lam @ X481 ) )
= $true ) )
& ! [X483: term,X484: term] :
( ( ( X479 @ ( ap @ X484 @ X483 ) )
= $true )
| ( ( X479 @ X483 )
!= $true )
| ( ( X479 @ X484 )
!= $true ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X266: subst,X267: term > $o,X268: term] :
? [X269: term > $o] :
! [X270: term] :
( ( X269 @ X270 )
= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ( ! [X383: term] :
( ( sub @ X383 @ id )
= X383 )
<=> ( axvarid = $true ) )
& ( ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 )
<=> ( axvarcons = $true ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X168: term,X169: term,X170: subst > term > subst > $o] :
( ( ( X170 @ id @ ( ap @ X168 @ X169 ) @ id )
= $true )
| ( ( X170 @ id @ X169 @ id )
!= $true )
| ( ( X170 @ id @ X168 @ id )
!= $true )
| ? [X177: subst,X178: subst,X179: term,X180: subst] :
( ( ( X170 @ X178 @ X179 @ ( comp @ X177 @ X180 ) )
= $true )
& ( ( X170 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) @ X180 )
!= $true ) )
| ? [X173: subst,X174: subst,X175: term,X176: subst] :
( ( ( X170 @ ( comp @ X176 @ X173 ) @ ( sub @ X175 @ X173 ) @ X174 )
= $true )
& ( ( X170 @ X176 @ X175 @ ( comp @ X173 @ X174 ) )
!= $true ) )
| ? [X171: term,X172: term] :
( ( ( X170 @ id @ ( ap @ ( sub @ X171 @ id ) @ X172 ) @ id )
!= $true )
& ( ( X170 @ id @ X172 @ id )
= $true )
& ( ( X170 @ id @ X171 @ id )
= $true ) ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ! [X413: term] :
? [X414: d_term] :
( ( d2term @ X414 )
= X413 )
& ( ( axidl = $true )
<=> ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 ) )
& ( hoasinduction_lem1v2 = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( axshiftcons = $true )
<=> ! [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
= X493 ) )
& ( pushprop_gthm = $true )
& ( induction2_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X384: term,X385: subst > term > subst > $o] :
( ? [X386: subst,X387: subst,X388: subst,X389: term] :
( ( ( X385 @ ( comp @ X386 @ X388 ) @ ( sub @ X389 @ X388 ) @ X387 )
!= $true )
& ( ( X385 @ X386 @ X389 @ ( comp @ X388 @ X387 ) )
= $true ) )
| ? [X391: subst > term > term] :
( ! [X395: term] :
( ( ( X385 @ id @ X395 @ id )
!= $true )
| ( ( X385 @ id @ ( X391 @ id @ X395 ) @ id )
= $true ) )
& ! [X392: subst,X393: subst,X394: term] :
( ( sub @ ( X391 @ X393 @ X394 ) @ X392 )
= ( X391 @ ( comp @ X393 @ X392 ) @ ( sub @ X394 @ X392 ) ) )
& ( ( X385 @ id @ ( lam @ ( X391 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X390: term] :
( ( ( X385 @ id @ X390 @ id )
= $true )
& ( ( X385 @ id @ ( sub @ X384 @ ( push @ X390 @ id ) ) @ id )
!= $true ) )
| ? [X396: subst,X397: subst,X398: term,X399: subst] :
( ( ( X385 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X399 )
= $true )
& ( ( X385 @ X396 @ X398 @ ( comp @ X397 @ X399 ) )
!= $true ) )
| ( ( X385 @ id @ ( lam @ X384 ) @ id )
= $true ) ) )
& ( ! [X415: term,X416: term,X417: term,X418: term] :
( ( X416 = X418 )
| ( ( ap @ ( sub @ X417 @ id ) @ X418 )
!= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X502: term > $o,X503: subst > term > subst > $o,X504: term] :
( ? [X515: subst,X516: subst,X517: subst,X518: term] :
( ( ( X503 @ X516 @ X518 @ ( comp @ X515 @ X517 ) )
!= $true )
& ( ( X503 @ ( comp @ X516 @ X515 ) @ ( sub @ X518 @ X515 ) @ X517 )
= $true ) )
| ? [X519: term] :
( ( ( X502 @ ( sub @ X504 @ ( push @ X519 @ id ) ) )
!= $true )
& ( ( X502 @ X519 )
= $true ) )
| ? [X510: subst > term > term] :
( ! [X512: subst,X513: term,X514: subst] :
( ( sub @ ( X510 @ X514 @ X513 ) @ X512 )
= ( X510 @ ( comp @ X514 @ X512 ) @ ( sub @ X513 @ X512 ) ) )
& ! [X511: term] :
( ( ( X503 @ id @ ( X510 @ id @ X511 ) @ id )
= $true )
| ( ( X503 @ id @ X511 @ id )
!= $true ) )
& ( ( X503 @ id @ ( lam @ ( X510 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X506: subst,X507: subst,X508: subst,X509: term] :
( ( ( X503 @ ( comp @ X506 @ X507 ) @ ( sub @ X509 @ X507 ) @ X508 )
!= $true )
& ( ( X503 @ X506 @ X509 @ ( comp @ X507 @ X508 ) )
= $true ) )
| ( ( X502 @ ( lam @ X504 ) )
= $true )
| ? [X505: term] :
( ( X503 @ id @ X505 @ id )
!= ( X502 @ X505 ) ) )
| ? [X501: term] :
( ( sub @ X501 @ id )
!= X501 ) ) )
& ( axapp = $true ) ),
inference(flattening,[status(thm)],[f12]) ).
thf(f12,plain,
( ( 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 ) ) ) ) ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X127: subst > term > subst > $o,X128: term] :
( ( ( X127 @ id @ ( lam @ ( sub @ X128 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X130: subst > term > term] :
( ! [X134: term] :
( ( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
= $true )
| ( ( X127 @ id @ X134 @ id )
!= $true ) )
& ( ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id )
!= $true )
& ! [X131: term,X132: subst,X133: subst] :
( ( X130 @ ( comp @ X132 @ X133 ) @ ( sub @ X131 @ X133 ) )
= ( sub @ ( X130 @ X132 @ X131 ) @ X133 ) ) )
| ? [X129: term] :
( ( ( X127 @ id @ X129 @ id )
= $true )
& ( ( X127 @ id @ ( sub @ X128 @ ( push @ X129 @ id ) ) @ id )
!= $true ) ) ) )
& ( axabs = $true )
& ( ( shinj = $true )
<=> ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) ) )
& ( ( ! [X485: term,X486: subst,X487: term > $o] :
( ? [X490: term] :
( ( ( X487 @ ( sub @ X490 @ X486 ) )
!= $true )
& ( ( var @ X490 )
= $true ) )
| ? [X491: term,X492: term] :
( ( ( X487 @ X491 )
= $true )
& ( ( X487 @ X492 )
= $true )
& ( ( X487 @ ( ap @ X491 @ X492 ) )
!= $true ) )
| ? [X488: term] :
( ! [X489: term] :
( ( ( X487 @ X489 )
!= $true )
| ( ( X487 @ ( sub @ X488 @ ( push @ X489 @ id ) ) )
= $true ) )
& ( ( X487 @ ( lam @ X488 ) )
!= $true ) )
| ( ( X487 @ ( sub @ X485 @ X486 ) )
= $true ) )
| ? [X479: term > $o,X480: term] :
( ! [X482: term] :
( ( ( X479 @ X482 )
= $true )
| ( ( var @ X482 )
!= $true ) )
& ( ( X479 @ X480 )
!= $true )
& ! [X481: term] :
( ( ( X479 @ X481 )
!= $true )
| ( ( X479 @ ( lam @ X481 ) )
= $true ) )
& ! [X483: term,X484: term] :
( ( ( X479 @ ( ap @ X484 @ X483 ) )
= $true )
| ( ( X479 @ X483 )
!= $true )
| ( ( X479 @ X484 )
!= $true ) ) )
| ? [X478: subst] :
( ( comp @ X478 @ id )
!= X478 )
| ? [X476: subst,X477: term] :
( ( comp @ sh @ ( push @ X477 @ X476 ) )
!= X476 )
| ? [X474: subst,X475: term] :
( ( sub @ one @ ( push @ X475 @ X474 ) )
!= X475 ) )
<=> ( induction2lem_lthm = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ulamvarind = $true )
& ( ( termmset_lthm = $true )
<=> ( ? [X272: term] :
( ( sub @ X272 @ id )
!= X272 )
| ! [X273: term] :
( ( sub @ X273 @ id )
= X273 ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X168: term,X169: term,X170: subst > term > subst > $o] :
( ( ( X170 @ id @ ( ap @ X168 @ X169 ) @ id )
= $true )
| ( ( X170 @ id @ X169 @ id )
!= $true )
| ( ( X170 @ id @ X168 @ id )
!= $true )
| ? [X177: subst,X178: subst,X179: term,X180: subst] :
( ( ( X170 @ X178 @ X179 @ ( comp @ X177 @ X180 ) )
= $true )
& ( ( X170 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) @ X180 )
!= $true ) )
| ? [X173: subst,X174: subst,X175: term,X176: subst] :
( ( ( X170 @ ( comp @ X176 @ X173 ) @ ( sub @ X175 @ X173 ) @ X174 )
= $true )
& ( ( X170 @ X176 @ X175 @ ( comp @ X173 @ X174 ) )
!= $true ) )
| ? [X171: term,X172: term] :
( ( ( X170 @ id @ ( ap @ ( sub @ X171 @ id ) @ X172 ) @ id )
!= $true )
& ( ( X170 @ id @ X172 @ id )
= $true )
& ( ( X170 @ id @ X171 @ id )
= $true ) ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( axassoc = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 )
<=> ( axvarcons = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X92: term,X93: subst > term > subst > $o] :
( ? [X94: subst > term > term] :
( ! [X98: term] :
( ( ( X93 @ id @ X98 @ id )
!= $true )
| ( ( X93 @ id @ ( X94 @ id @ X98 ) @ id )
= $true ) )
& ! [X95: subst,X96: term,X97: subst] :
( ( X94 @ ( comp @ X95 @ X97 ) @ ( sub @ X96 @ X97 ) )
= ( sub @ ( X94 @ X95 @ X96 ) @ X97 ) )
& ( ( X93 @ id @ ( lam @ ( X94 @ sh @ one ) ) @ id )
!= $true ) )
| ( $true
= ( X93 @ id @ ( lam @ X92 ) @ id ) )
| ? [X99: term] :
( ( ( X93 @ id @ ( sub @ X92 @ ( push @ X99 @ id ) ) @ id )
!= $true )
& ( ( X93 @ id @ X99 @ id )
= $true ) ) )
| ? [X84: term,X85: subst > term > subst > $o] :
( ( ( X85 @ id @ ( lam @ ( sub @ X84 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X87: subst > term > term] :
( ? [X89: subst,X90: term,X91: subst] :
( ( X87 @ ( comp @ X91 @ X89 ) @ ( sub @ X90 @ X89 ) )
!= ( sub @ ( X87 @ X91 @ X90 ) @ X89 ) )
| ( ( X85 @ id @ ( lam @ ( X87 @ sh @ one ) ) @ id )
= $true )
| ? [X88: term] :
( ( ( X85 @ id @ X88 @ id )
= $true )
& ( $true
!= ( X85 @ id @ ( X87 @ id @ X88 ) @ id ) ) ) )
& ! [X86: term] :
( ( ( X85 @ id @ ( sub @ X84 @ ( push @ X86 @ id ) ) @ id )
= $true )
| ( ( X85 @ id @ X86 @ id )
!= $true ) ) )
| ? [X83: term] :
( ( sub @ X83 @ id )
!= X83 ) ) )
& ( pushprop = $true )
& ( ! [X292: term,X293: term] :
( ( X292 = X293 )
| ( ( lam @ X293 )
!= ( lam @ X292 ) ) )
<=> ( laminj = $true ) )
& ( termmset_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
<=> ( termmset = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X139: term,X140: term > $o,X141: subst] :
? [X142: term > $o] :
! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) ) )
& ( hoasinduction_lem1 = $true )
& ( ! [X383: term] :
( ( sub @ X383 @ id )
= X383 )
<=> ( axvarid = $true ) )
& ( pushprop_lem2v2 = $true )
& ( ! [X165: subst > term > subst > $o] :
? [X166: term > $o] :
! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( induction2lem_gthm = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X65: term] :
? [X66: subst > term > term] :
! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X452: subst > term > subst > $o,X453: term] :
( ( ( X452 @ id @ ( lam @ ( sub @ X453 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X454: term] :
( ( ( X452 @ id @ ( sub @ X453 @ ( push @ X454 @ id ) ) @ id )
!= $true )
& ( ( X452 @ id @ X454 @ id )
= $true ) )
| ? [X455: subst > term > term,X456: term] :
( ! [X461: subst,X462: term,X463: subst] :
( ( sub @ ( sub @ X456 @ ( push @ X462 @ X461 ) ) @ X463 )
= ( sub @ ( X455 @ X461 @ X462 ) @ X463 ) )
& ! [X457: subst,X458: term,X459: subst] :
( ( sub @ X456 @ ( push @ ( sub @ X458 @ X457 ) @ ( comp @ X459 @ X457 ) ) )
= ( X455 @ ( comp @ X459 @ X457 ) @ ( sub @ X458 @ X457 ) ) )
& ! [X460: term] :
( ( ( X452 @ id @ X460 @ id )
!= $true )
| ( ( X452 @ id @ ( X455 @ id @ X460 ) @ id )
= $true ) )
& ( ( X452 @ id @ ( lam @ ( X455 @ sh @ one ) ) @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( lamnotap != $true )
& ( apnotvar = $true )
& ( substmonoid_gthm = $true )
& ( axmap = $true )
& ( hoasapinj2_gthm = $true )
& ! [X413: term] :
? [X414: d_term] :
( ( d2term @ X414 )
= X413 )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X135: term] :
? [X136: subst > term > term] :
! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) ) )
& ( ( ! [X365: term,X366: subst > term > subst > $o] :
( ? [X368: subst,X369: subst,X370: subst,X371: term] :
( ( ( X366 @ ( comp @ X369 @ X370 ) @ ( sub @ X371 @ X370 ) @ X368 )
!= $true )
& ( ( X366 @ X369 @ X371 @ ( comp @ X370 @ X368 ) )
= $true ) )
| ? [X374: subst > term > term] :
( ! [X375: subst,X376: term,X377: subst] :
( ( X374 @ ( comp @ X377 @ X375 ) @ ( sub @ X376 @ X375 ) )
= ( sub @ ( X374 @ X377 @ X376 ) @ X375 ) )
& ! [X378: term] :
( ( ( X366 @ id @ X378 @ id )
!= $true )
| ( ( X366 @ id @ ( X374 @ id @ X378 ) @ id )
= $true ) )
& ( ( X366 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X367: term] :
( ( ( X366 @ id @ X367 @ id )
!= $true )
& ( ( var @ ( sub @ X367 @ id ) )
= $true ) )
| ? [X372: term,X373: term] :
( ( ( X366 @ id @ ( ap @ ( sub @ X373 @ id ) @ X372 ) @ id )
!= $true )
& ( ( X366 @ id @ X372 @ id )
= $true )
& ( ( X366 @ id @ X373 @ id )
= $true ) )
| ? [X379: subst,X380: subst,X381: term,X382: subst] :
( ( ( X366 @ X380 @ X381 @ ( comp @ X379 @ X382 ) )
!= $true )
& ( ( X366 @ ( comp @ X380 @ X379 ) @ ( sub @ X381 @ X379 ) @ X382 )
= $true ) )
| ( ( X366 @ id @ X365 @ id )
= $true ) )
| ? [X349: subst > term > subst > $o,X350: term] :
( ! [X361: subst,X362: subst,X363: term,X364: subst] :
( ( ( X349 @ X361 @ X363 @ ( comp @ X362 @ X364 ) )
!= $true )
| ( ( X349 @ ( comp @ X361 @ X362 ) @ ( sub @ X363 @ X362 ) @ X364 )
= $true ) )
& ! [X351: subst,X352: subst,X353: term,X354: subst] :
( ( ( X349 @ ( comp @ X351 @ X352 ) @ ( sub @ X353 @ X352 ) @ X354 )
!= $true )
| ( ( X349 @ X351 @ X353 @ ( comp @ X352 @ X354 ) )
= $true ) )
& ( ( X349 @ id @ ( lam @ X350 ) @ id )
!= $true )
& ! [X355: subst > term > term] :
( ( ( X349 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id )
= $true )
| ? [X357: subst,X358: subst,X359: term] :
( ( sub @ ( X355 @ X358 @ X359 ) @ X357 )
!= ( X355 @ ( comp @ X358 @ X357 ) @ ( sub @ X359 @ X357 ) ) )
| ? [X356: term] :
( ( ( X349 @ id @ X356 @ id )
= $true )
& ( ( X349 @ id @ ( X355 @ id @ X356 ) @ id )
!= $true ) ) )
& ! [X360: term] :
( ( ( X349 @ id @ ( sub @ X350 @ ( push @ X360 @ id ) ) @ id )
= $true )
| ( ( X349 @ id @ X360 @ id )
!= $true ) ) )
| ? [X336: term,X337: subst > term > subst > $o,X338: term] :
( ! [X345: subst,X346: term,X347: subst,X348: subst] :
( ( ( X337 @ ( comp @ X348 @ X347 ) @ ( sub @ X346 @ X347 ) @ X345 )
!= $true )
| ( ( X337 @ X348 @ X346 @ ( comp @ X347 @ X345 ) )
= $true ) )
& ! [X339: subst,X340: subst,X341: term,X342: subst] :
( ( ( X337 @ ( comp @ X342 @ X339 ) @ ( sub @ X341 @ X339 ) @ X340 )
= $true )
| ( ( X337 @ X342 @ X341 @ ( comp @ X339 @ X340 ) )
!= $true ) )
& ( ( X337 @ id @ X338 @ id )
= $true )
& ( ( X337 @ id @ ( ap @ X338 @ X336 ) @ id )
!= $true )
& ! [X343: term,X344: term] :
( ( ( X337 @ id @ X344 @ id )
!= $true )
| ( ( X337 @ id @ ( ap @ ( sub @ X343 @ id ) @ X344 ) @ id )
= $true )
| ( ( X337 @ id @ X343 @ id )
!= $true ) )
& ( ( X337 @ id @ X336 @ id )
= $true ) )
| ? [X329: term > $o,X330: term] :
( ( ( X329 @ X330 )
!= $true )
& ! [X332: term,X333: term] :
( ( ( X329 @ ( ap @ X332 @ X333 ) )
= $true )
| ( ( X329 @ X332 )
!= $true )
| ( ( X329 @ X333 )
!= $true ) )
& ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( ( var @ X331 )
!= $true ) )
& ! [X334: term] :
( ? [X335: term] :
( ( ( X329 @ X335 )
= $true )
& ( ( X329 @ ( sub @ X334 @ ( push @ X335 @ id ) ) )
!= $true ) )
| ( ( X329 @ ( lam @ X334 ) )
= $true ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ! [X495: term > $o,X496: term] :
( ? [X497: term,X498: term] :
( ( ( X495 @ ( ap @ X497 @ X498 ) )
!= $true )
& ( ( X495 @ X497 )
= $true )
& ( ( X495 @ X498 )
= $true ) )
| ? [X499: term] :
( ( ( var @ X499 )
= $true )
& ( ( X495 @ X499 )
!= $true ) )
| ( ( X495 @ X496 )
= $true )
| ? [X500: term] :
( ( ( X495 @ ( lam @ X500 ) )
!= $true )
& ( ( X495 @ X500 )
= $true ) ) )
<=> ( induction = $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X317: term,X318: term > $o,X319: subst] :
( ? [X324: term] :
( ( ( var @ X324 )
= $true )
& ( ( X318 @ ( sub @ X324 @ X319 ) )
!= $true ) )
| ? [X320: term] :
( ! [X321: term] :
( ( ( X318 @ ( sub @ X320 @ ( push @ X321 @ id ) ) )
= $true )
| ( ( X318 @ X321 )
!= $true ) )
& ( ( X318 @ ( lam @ X320 ) )
!= $true ) )
| ? [X322: term,X323: term] :
( ( ( X318 @ X322 )
= $true )
& ( ( X318 @ ( ap @ X323 @ X322 ) )
!= $true )
& ( ( X318 @ X323 )
= $true ) )
| ( ( X318 @ ( sub @ X317 @ X319 ) )
= $true ) )
<=> ( induction2lem = $true ) )
& ( ( hoaslaminj = $true )
<=> ! [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
!= ( lam @ ( X465 @ sh @ one ) ) )
| ( ( X465 @ X466 @ X464 )
= ( X467 @ X466 @ X464 ) )
| ? [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
!= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
| ? [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
!= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) ) )
& ( ( ( ! [X411: subst] :
( ( comp @ id @ X411 )
= X411 )
& ! [X412: subst] :
( ( comp @ X412 @ id )
= X412 ) )
| ? [X410: subst] :
( ( comp @ X410 @ id )
!= X410 )
| ? [X409: subst] :
( ( comp @ id @ X409 )
!= X409 ) )
<=> ( substmonoid_lthm = $true ) )
& ( ( pushprop_lem1 = $true )
<=> ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
= $true ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( induction2_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X257: term,X258: term,X259: term,X260: term] :
( ( X257 = X258 )
| ( ( ap @ ( sub @ X260 @ id ) @ X258 )
!= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) )
| ? [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
= ( ap @ X255 @ X254 ) )
& ( X253 != X254 ) ) ) )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasinduction = $true )
<=> ! [X274: subst > term > subst > $o,X275: term] :
( ? [X290: term,X291: term] :
( ( ( X274 @ id @ X290 @ id )
= $true )
& ( ( X274 @ id @ ( ap @ ( sub @ X291 @ id ) @ X290 ) @ id )
!= $true )
& ( ( X274 @ id @ X291 @ id )
= $true ) )
| ( ( X274 @ id @ X275 @ id )
= $true )
| ? [X285: subst > term > term] :
( ! [X286: term,X287: subst,X288: subst] :
( ( X285 @ ( comp @ X287 @ X288 ) @ ( sub @ X286 @ X288 ) )
= ( sub @ ( X285 @ X287 @ X286 ) @ X288 ) )
& ! [X289: term] :
( ( ( X274 @ id @ ( X285 @ id @ X289 ) @ id )
= $true )
| ( ( X274 @ id @ X289 @ id )
!= $true ) )
& ( ( X274 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X277: term,X278: subst,X279: subst,X280: subst] :
( ( ( X274 @ X279 @ X277 @ ( comp @ X278 @ X280 ) )
!= $true )
& ( ( X274 @ ( comp @ X279 @ X278 ) @ ( sub @ X277 @ X278 ) @ X280 )
= $true ) )
| ? [X276: term] :
( ( ( X274 @ id @ X276 @ id )
!= $true )
& ( ( var @ ( sub @ X276 @ id ) )
= $true ) )
| ? [X281: subst,X282: term,X283: subst,X284: subst] :
( ( ( X274 @ ( comp @ X283 @ X284 ) @ ( sub @ X282 @ X284 ) @ X281 )
!= $true )
& ( ( X274 @ X283 @ X282 @ ( comp @ X284 @ X281 ) )
= $true ) ) ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
= $true )
| ( ( X522 @ X521 )
!= $true )
| ? [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
!= ( X523 @ X524 ) ) ) )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& ( ! [X2: subst > term > subst > $o] :
? [X3: term > $o] :
! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( axshiftcons = $true )
<=> ! [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
= X493 ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( ( ? [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
!= X228 )
| ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ( ( X230 @ X233 )
!= $true )
| ? [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
!= ( X232 @ X234 ) )
| ( ( X232 @ one )
= $true ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( ! [X415: term,X416: term,X417: term,X418: term] :
( ( X416 = X418 )
| ( ( ap @ ( sub @ X417 @ id ) @ X418 )
!= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X502: term > $o,X503: subst > term > subst > $o,X504: term] :
( ? [X515: subst,X516: subst,X517: subst,X518: term] :
( ( ( X503 @ X516 @ X518 @ ( comp @ X515 @ X517 ) )
!= $true )
& ( ( X503 @ ( comp @ X516 @ X515 ) @ ( sub @ X518 @ X515 ) @ X517 )
= $true ) )
| ? [X519: term] :
( ( ( X502 @ ( sub @ X504 @ ( push @ X519 @ id ) ) )
!= $true )
& ( ( X502 @ X519 )
= $true ) )
| ? [X510: subst > term > term] :
( ! [X512: subst,X513: term,X514: subst] :
( ( sub @ ( X510 @ X514 @ X513 ) @ X512 )
= ( X510 @ ( comp @ X514 @ X512 ) @ ( sub @ X513 @ X512 ) ) )
& ! [X511: term] :
( ( ( X503 @ id @ ( X510 @ id @ X511 ) @ id )
= $true )
| ( ( X503 @ id @ X511 @ id )
!= $true ) )
& ( ( X503 @ id @ ( lam @ ( X510 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X506: subst,X507: subst,X508: subst,X509: term] :
( ( ( X503 @ ( comp @ X506 @ X507 ) @ ( sub @ X509 @ X507 ) @ X508 )
!= $true )
& ( ( X503 @ X506 @ X509 @ ( comp @ X507 @ X508 ) )
= $true ) )
| ( ( X502 @ ( lam @ X504 ) )
= $true )
| ? [X505: term] :
( ( X503 @ id @ X505 @ id )
!= ( X502 @ X505 ) ) )
| ? [X501: term] :
( ( sub @ X501 @ id )
!= X501 ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ! [X265: d_term] : ( d_one = X265 )
& ( hoasapinj1_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X146: subst > term > subst > $o,X147: term > $o,X148: term] :
( ? [X159: subst > term > term] :
( ( ( X146 @ id @ ( lam @ ( X159 @ sh @ one ) ) @ id )
!= $true )
& ! [X161: term,X162: subst,X163: subst] :
( ( X159 @ ( comp @ X163 @ X162 ) @ ( sub @ X161 @ X162 ) )
= ( sub @ ( X159 @ X163 @ X161 ) @ X162 ) )
& ! [X160: term] :
( ( ( X146 @ id @ X160 @ id )
!= $true )
| ( ( X146 @ id @ ( X159 @ id @ X160 ) @ id )
= $true ) ) )
| ( ( X147 @ ( lam @ X148 ) )
= $true )
| ? [X149: term] :
( ( X147 @ X149 )
!= ( X146 @ id @ X149 @ id ) )
| ? [X154: subst,X155: subst,X156: subst,X157: term] :
( ( ( X146 @ X155 @ X157 @ ( comp @ X154 @ X156 ) )
= $true )
& ( ( X146 @ ( comp @ X155 @ X154 ) @ ( sub @ X157 @ X154 ) @ X156 )
!= $true ) )
| ? [X158: term] :
( ( ( X147 @ X158 )
= $true )
& ( ( X147 @ ( sub @ X148 @ ( push @ X158 @ id ) ) )
!= $true ) )
| ? [X150: subst,X151: subst,X152: subst,X153: term] :
( ( ( X146 @ X151 @ X153 @ ( comp @ X150 @ X152 ) )
!= $true )
& ( ( X146 @ ( comp @ X151 @ X150 ) @ ( sub @ X153 @ X150 ) @ X152 )
= $true ) ) ) )
& ( hoasapnotvar = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( axidl = $true )
<=> ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 ) )
& ( ( ! [X42: subst > term > subst > $o,X43: term] :
( ? [X44: subst > term > term] :
( ( ( X42 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id )
!= $true )
& ! [X48: term] :
( ( ( X42 @ id @ ( X44 @ id @ X48 ) @ id )
= $true )
| ( ( X42 @ id @ X48 @ id )
!= $true ) )
& ! [X45: subst,X46: subst,X47: term] :
( ( sub @ ( X44 @ X45 @ X47 ) @ X46 )
= ( X44 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) ) ) )
| ? [X50: subst,X51: term,X52: subst,X53: subst] :
( ( ( X42 @ ( comp @ X53 @ X52 ) @ ( sub @ X51 @ X52 ) @ X50 )
!= $true )
& ( ( X42 @ X53 @ X51 @ ( comp @ X52 @ X50 ) )
= $true ) )
| ? [X49: term] :
( ( ( X42 @ id @ X49 @ id )
!= $true )
& ( ( var @ ( sub @ X49 @ id ) )
= $true ) )
| ( ( X42 @ id @ X43 @ id )
= $true )
| ? [X54: term,X55: term] :
( ( ( X42 @ id @ X55 @ id )
= $true )
& ( ( X42 @ id @ X54 @ id )
= $true )
& ( ( X42 @ id @ ( ap @ ( sub @ X54 @ id ) @ X55 ) @ id )
!= $true ) )
| ? [X56: subst,X57: subst,X58: subst,X59: term] :
( ( ( X42 @ X58 @ X59 @ ( comp @ X57 @ X56 ) )
!= $true )
& ( ( X42 @ ( comp @ X58 @ X57 ) @ ( sub @ X59 @ X57 ) @ X56 )
= $true ) ) )
| ? [X32: term > $o,X33: subst > term > subst > $o,X34: term] :
( ! [X35: term] :
( ( ( X32 @ ( sub @ X34 @ ( push @ X35 @ id ) ) )
= $true )
| ( ( X32 @ X35 )
!= $true ) )
& ( ( X32 @ ( lam @ X34 ) )
!= $true )
& ! [X41: term] :
( ( X33 @ id @ X41 @ id )
= ( X32 @ X41 ) )
& ! [X36: subst > term > term] :
( ( ( X33 @ id @ ( lam @ ( X36 @ sh @ one ) ) @ id )
= $true )
| ? [X38: subst,X39: term,X40: subst] :
( ( X36 @ ( comp @ X38 @ X40 ) @ ( sub @ X39 @ X40 ) )
!= ( sub @ ( X36 @ X38 @ X39 ) @ X40 ) )
| ? [X37: term] :
( ( ( X33 @ id @ X37 @ id )
= $true )
& ( ( X33 @ id @ ( X36 @ id @ X37 ) @ id )
!= $true ) ) ) )
| ? [X31: term] :
( ( sub @ X31 @ id )
!= X31 )
| ? [X24: term,X25: term > $o] :
( ! [X30: term] :
( ( ( var @ X30 )
!= $true )
| ( ( X25 @ X30 )
= $true ) )
& ( ( X25 @ X24 )
!= $true )
& ! [X26: term,X27: term] :
( ( ( X25 @ ( ap @ X27 @ X26 ) )
= $true )
| ( ( X25 @ X27 )
!= $true )
| ( ( X25 @ X26 )
!= $true ) )
& ! [X28: term] :
( ( ( X25 @ ( lam @ X28 ) )
= $true )
| ? [X29: term] :
( ( ( X25 @ ( sub @ X28 @ ( push @ X29 @ id ) ) )
!= $true )
& ( ( X25 @ X29 )
= $true ) ) ) )
| ? [X21: subst > term > subst > $o] :
! [X22: term > $o] :
? [X23: term] :
( ( X21 @ id @ X23 @ id )
!= ( X22 @ X23 ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lem2v2_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X271: d_subst] : ( d_id = X271 )
& ( lamnotvar = $true )
& ( ( induction2_lthm = $true )
<=> ( ! [X310: term,X311: term > $o] :
( ? [X313: term] :
( ! [X314: term] :
( ( ( X311 @ X314 )
!= $true )
| ( ( X311 @ ( sub @ X313 @ ( push @ X314 @ id ) ) )
= $true ) )
& ( ( X311 @ ( lam @ X313 ) )
!= $true ) )
| ? [X315: term,X316: term] :
( ( ( X311 @ X315 )
= $true )
& ( ( X311 @ ( ap @ X315 @ X316 ) )
!= $true )
& ( ( X311 @ X316 )
= $true ) )
| ? [X312: term] :
( ( ( var @ X312 )
= $true )
& ( ( X311 @ X312 )
!= $true ) )
| ( ( X311 @ X310 )
= $true ) )
| ? [X302: subst,X303: term > $o,X304: term] :
( ! [X309: term] :
( ( ( var @ X309 )
!= $true )
| ( ( X303 @ ( sub @ X309 @ X302 ) )
= $true ) )
& ( ( X303 @ ( sub @ X304 @ X302 ) )
!= $true )
& ! [X307: term] :
( ? [X308: term] :
( ( ( X303 @ ( sub @ X307 @ ( push @ X308 @ id ) ) )
!= $true )
& ( ( X303 @ X308 )
= $true ) )
| ( ( X303 @ ( lam @ X307 ) )
= $true ) )
& ! [X305: term,X306: term] :
( ( ( X303 @ ( ap @ X305 @ X306 ) )
= $true )
| ( ( X303 @ X305 )
!= $true )
| ( ( X303 @ X306 )
!= $true ) ) )
| ? [X301: term] :
( ( sub @ X301 @ id )
!= X301 ) ) )
& ( ulamvarsh = $true )
& ( ! [X144: term] :
? [X145: subst > term > term] :
( ( X145 @ sh @ one )
= ( sub @ X144 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( ! [X266: subst,X267: term > $o,X268: term] :
? [X269: term > $o] :
! [X270: term] :
( ( X269 @ X270 )
= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X419: subst > term > subst > $o,X420: term] :
( ( ( X419 @ id @ ( lam @ ( sub @ X420 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X422: subst > term > term] :
( ! [X423: term,X424: subst,X425: subst] :
( ( X422 @ ( comp @ X425 @ X424 ) @ ( sub @ X423 @ X424 ) )
= ( sub @ ( X422 @ X425 @ X423 ) @ X424 ) )
& ( ( X419 @ id @ ( lam @ ( X422 @ sh @ one ) ) @ id )
!= $true )
& ! [X426: term] :
( ( ( X419 @ id @ ( X422 @ id @ X426 ) @ id )
= $true )
| ( ( X419 @ id @ X426 @ id )
!= $true ) ) )
| ? [X421: term] :
( ( ( X419 @ id @ X421 @ id )
= $true )
& ( ( X419 @ id @ ( sub @ X420 @ ( push @ X421 @ id ) ) @ id )
!= $true ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( ! [X294: term,X295: term > $o] :
( ? [X299: term] :
( ( ( X295 @ ( lam @ X299 ) )
!= $true )
& ! [X300: term] :
( ( ( X295 @ X300 )
!= $true )
| ( ( X295 @ ( sub @ X299 @ ( push @ X300 @ id ) ) )
= $true ) ) )
| ( ( X295 @ X294 )
= $true )
| ? [X296: term] :
( ( ( var @ X296 )
= $true )
& ( ( X295 @ X296 )
!= $true ) )
| ? [X297: term,X298: term] :
( ( $true
!= ( X295 @ ( ap @ X297 @ X298 ) ) )
& ( ( X295 @ X297 )
= $true )
& ( ( X295 @ X298 )
= $true ) ) )
<=> ( induction2 = $true ) )
& ( pushprop_lthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( ( ! [X11: subst > term > term,X12: subst > term > term,X13: term,X14: subst] :
( ( ( lam @ ( X11 @ sh @ one ) )
!= ( lam @ ( X12 @ sh @ one ) ) )
| ? [X15: term,X16: subst,X17: subst] :
( ( sub @ ( X11 @ X17 @ X15 ) @ X16 )
!= ( X11 @ ( comp @ X17 @ X16 ) @ ( sub @ X15 @ X16 ) ) )
| ? [X18: subst,X19: term,X20: subst] :
( ( sub @ ( X12 @ X20 @ X19 ) @ X18 )
!= ( X12 @ ( comp @ X20 @ X18 ) @ ( sub @ X19 @ X18 ) ) )
| ( ( X11 @ X14 @ X13 )
= ( X12 @ X14 @ X13 ) ) )
| ? [X9: term,X10: term] :
( ( ( lam @ X10 )
= ( lam @ X9 ) )
& ( X9 != X10 ) )
| ? [X7: term,X8: subst] :
( ( comp @ sh @ ( push @ X7 @ X8 ) )
!= X8 )
| ? [X5: term,X6: subst] :
( ( sub @ one @ ( push @ X5 @ X6 ) )
!= X5 ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ( ! [X436: term,X437: subst > term > subst > $o] :
( ? [X443: term,X444: subst,X445: subst,X446: subst] :
( ( ( X437 @ ( comp @ X446 @ X445 ) @ ( sub @ X443 @ X445 ) @ X444 )
= $true )
& ( $true
!= ( X437 @ X446 @ X443 @ ( comp @ X445 @ X444 ) ) ) )
| ( ( X437 @ id @ ( lam @ X436 ) @ id )
= $true )
| ? [X438: subst > term > term] :
( ( ( X437 @ id @ ( lam @ ( X438 @ sh @ one ) ) @ id )
!= $true )
& ! [X439: term] :
( ( ( X437 @ id @ ( X438 @ id @ X439 ) @ id )
= $true )
| ( ( X437 @ id @ X439 @ id )
!= $true ) )
& ! [X440: term,X441: subst,X442: subst] :
( ( sub @ ( X438 @ X441 @ X440 ) @ X442 )
= ( X438 @ ( comp @ X441 @ X442 ) @ ( sub @ X440 @ X442 ) ) ) )
| ? [X448: subst,X449: subst,X450: subst,X451: term] :
( ( ( X437 @ ( comp @ X450 @ X449 ) @ ( sub @ X451 @ X449 ) @ X448 )
!= $true )
& ( ( X437 @ X450 @ X451 @ ( comp @ X449 @ X448 ) )
= $true ) )
| ? [X447: term] :
( ( ( X437 @ id @ ( sub @ X436 @ ( push @ X447 @ id ) ) @ id )
!= $true )
& ( ( X437 @ id @ X447 @ id )
= $true ) ) )
| ? [X428: term,X429: subst > term > subst > $o] :
( ! [X431: subst > term > term] :
( ? [X432: term] :
( ( ( X429 @ id @ X432 @ id )
= $true )
& ( ( X429 @ id @ ( X431 @ id @ X432 ) @ id )
!= $true ) )
| ( ( X429 @ id @ ( lam @ ( X431 @ sh @ one ) ) @ id )
= $true )
| ? [X433: term,X434: subst,X435: subst] :
( ( X431 @ ( comp @ X435 @ X434 ) @ ( sub @ X433 @ X434 ) )
!= ( sub @ ( X431 @ X435 @ X433 ) @ X434 ) ) )
& ! [X430: term] :
( ( ( X429 @ id @ X430 @ id )
!= $true )
| ( ( X429 @ id @ ( sub @ X428 @ ( push @ X430 @ id ) ) @ id )
= $true ) )
& ( ( X429 @ id @ ( lam @ ( sub @ X428 @ ( push @ one @ sh ) ) ) @ id )
!= $true ) )
| ? [X427: term] :
( ( sub @ X427 @ id )
!= X427 ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( pushprop_gthm = $true )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
| ( X193 = X194 ) )
<=> ( hoasapinj1 = $true ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X235: subst > term > subst > $o,X236: term,X237: term > $o] :
( ? [X244: term] :
( ( X237 @ X244 )
!= ( X235 @ id @ X244 @ id ) )
| ( $true
= ( X237 @ ( lam @ X236 ) ) )
| ? [X239: subst > term > term] :
( ! [X240: subst,X241: term,X242: subst] :
( ( X239 @ ( comp @ X242 @ X240 ) @ ( sub @ X241 @ X240 ) )
= ( sub @ ( X239 @ X242 @ X241 ) @ X240 ) )
& ! [X243: term] :
( ( ( X235 @ id @ X243 @ id )
!= $true )
| ( ( X235 @ id @ ( X239 @ id @ X243 ) @ id )
= $true ) )
& ( ( X235 @ id @ ( lam @ ( X239 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X238: term] :
( ( ( X237 @ X238 )
= $true )
& ( ( X237 @ ( sub @ X236 @ ( push @ X238 @ id ) ) )
!= $true ) ) ) )
& ( axapp = $true )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X219: subst > term > subst > $o,X220: term] :
( ? [X223: subst > term > term] :
( ( $true
!= ( X219 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id ) )
& ! [X225: subst,X226: term,X227: subst] :
( ( X223 @ ( comp @ X227 @ X225 ) @ ( sub @ X226 @ X225 ) )
= ( sub @ ( X223 @ X227 @ X226 ) @ X225 ) )
& ! [X224: term] :
( ( ( X219 @ id @ ( X223 @ id @ X224 ) @ id )
= $true )
| ( ( X219 @ id @ X224 @ id )
!= $true ) ) )
| ? [X221: term,X222: term] :
( ( ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id )
!= $true )
& ( ( X219 @ id @ X221 @ id )
= $true )
& ( ( X219 @ id @ X222 @ id )
= $true ) )
| ( ( X219 @ id @ X220 @ id )
= $true ) )
| ? [X209: subst > term > subst > $o,X210: term > $o,X211: term] :
( ! [X218: term] :
( ( X209 @ id @ X218 @ id )
= ( X210 @ X218 ) )
& ( ( X210 @ ( lam @ X211 ) )
!= $true )
& ! [X212: term] :
( ( ( X210 @ ( sub @ X211 @ ( push @ X212 @ id ) ) )
= $true )
| ( ( X210 @ X212 )
!= $true ) )
& ! [X213: subst > term > term] :
( ? [X215: subst,X216: subst,X217: term] :
( ( sub @ ( X213 @ X216 @ X217 ) @ X215 )
!= ( X213 @ ( comp @ X216 @ X215 ) @ ( sub @ X217 @ X215 ) ) )
| ( ( X209 @ id @ ( lam @ ( X213 @ sh @ one ) ) @ id )
= $true )
| ? [X214: term] :
( ( ( X209 @ id @ ( X213 @ id @ X214 ) @ id )
!= $true )
& ( ( X209 @ id @ X214 @ id )
= $true ) ) ) )
| ? [X208: term] :
( ( sub @ X208 @ id )
!= X208 )
| ? [X201: term > $o,X202: term] :
( ( ( X201 @ X202 )
!= $true )
& ! [X204: term,X205: term] :
( ( ( X201 @ X205 )
!= $true )
| ( ( X201 @ ( ap @ X205 @ X204 ) )
= $true )
| ( ( X201 @ X204 )
!= $true ) )
& ! [X206: term] :
( ? [X207: term] :
( ( ( X201 @ ( sub @ X206 @ ( push @ X207 @ id ) ) )
!= $true )
& ( ( X201 @ X207 )
= $true ) )
| ( ( X201 @ ( lam @ X206 ) )
= $true ) )
& ! [X203: term] :
( ( ( X201 @ X203 )
= $true )
| ( ( var @ X203 )
!= $true ) ) )
| ? [X198: subst > term > subst > $o] :
! [X199: term > $o] :
? [X200: term] :
( ( X198 @ id @ X200 @ id )
!= ( X199 @ X200 ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( ! [X261: term,X262: term,X263: term,X264: term] :
( ( X261 = X262 )
| ( ( ap @ X261 @ X263 )
!= ( ap @ X262 @ X264 ) ) )
<=> ( apinj1 = $true ) )
& ( hoasinduction_gthm = $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X73: term] :
? [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
= ( X74 @ sh @ one ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
<=> ( axidr = $true ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X405: term,X406: term,X407: term,X408: term] :
( ( X405 = X408 )
| ( ( ap @ ( sub @ X408 @ id ) @ X406 )
!= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) )
| ? [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
= ( ap @ X404 @ X402 ) )
& ( X401 != X404 ) )
| ? [X400: term] :
( ( sub @ X400 @ id )
!= X400 ) ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X108: term,X109: subst > term > subst > $o,X110: term > $o] :
( ? [X111: term] :
( ( ( X110 @ X111 )
= $true )
& ( $true
!= ( X110 @ ( sub @ X108 @ ( push @ X111 @ id ) ) ) ) )
| ( ( X110 @ ( lam @ X108 ) )
= $true )
| ? [X113: subst > term > term] :
( ( ( X109 @ id @ ( lam @ ( X113 @ sh @ one ) ) @ id )
!= $true )
& ! [X115: subst,X116: term,X117: subst] :
( ( X113 @ ( comp @ X115 @ X117 ) @ ( sub @ X116 @ X117 ) )
= ( sub @ ( X113 @ X115 @ X116 ) @ X117 ) )
& ! [X114: term] :
( ( ( X109 @ id @ X114 @ id )
!= $true )
| ( ( X109 @ id @ ( X113 @ id @ X114 ) @ id )
= $true ) ) )
| ? [X112: term] :
( ( X109 @ id @ X112 @ id )
!= ( X110 @ X112 ) ) )
| ? [X107: term] :
( ( sub @ X107 @ id )
!= X107 )
| ? [X103: term] :
! [X104: subst > term > term] :
? [X105: subst,X106: term] :
( ( sub @ X103 @ ( push @ X106 @ X105 ) )
!= ( X104 @ X105 @ X106 ) ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X183: term,X184: subst > term > subst > $o] :
( ? [X185: term,X186: term] :
( ( ( X184 @ id @ X186 @ id )
= $true )
& ( ( X184 @ id @ X185 @ id )
= $true )
& ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
!= $true ) )
| ? [X187: subst > term > term] :
( ! [X188: term] :
( ( ( X184 @ id @ ( X187 @ id @ X188 ) @ id )
= $true )
| ( ( X184 @ id @ X188 @ id )
!= $true ) )
& ( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
!= $true )
& ! [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) ) )
| ( ( X184 @ id @ X183 @ id )
= $true ) ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ! [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
!= ( ap @ X326 @ X328 ) )
| ( X325 = X328 ) )
<=> ( apinj2 = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X384: term,X385: subst > term > subst > $o] :
( ? [X386: subst,X387: subst,X388: subst,X389: term] :
( ( ( X385 @ ( comp @ X386 @ X388 ) @ ( sub @ X389 @ X388 ) @ X387 )
!= $true )
& ( ( X385 @ X386 @ X389 @ ( comp @ X388 @ X387 ) )
= $true ) )
| ? [X391: subst > term > term] :
( ! [X395: term] :
( ( ( X385 @ id @ X395 @ id )
!= $true )
| ( ( X385 @ id @ ( X391 @ id @ X395 ) @ id )
= $true ) )
& ! [X392: subst,X393: subst,X394: term] :
( ( sub @ ( X391 @ X393 @ X394 ) @ X392 )
= ( X391 @ ( comp @ X393 @ X392 ) @ ( sub @ X394 @ X392 ) ) )
& ( ( X385 @ id @ ( lam @ ( X391 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X390: term] :
( ( ( X385 @ id @ X390 @ id )
= $true )
& ( ( X385 @ id @ ( sub @ X384 @ ( push @ X390 @ id ) ) @ id )
!= $true ) )
| ? [X396: subst,X397: subst,X398: term,X399: subst] :
( ( ( X385 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X399 )
= $true )
& ( ( X385 @ X396 @ X398 @ ( comp @ X397 @ X399 ) )
!= $true ) )
| ( ( X385 @ id @ ( lam @ X384 ) @ id )
= $true ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ! [X181: d_subst,X182: d_subst] :
( ( X181 = X182 )
| ( ( d2subst @ X182 )
!= ( d2subst @ X181 ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X247: term,X248: subst > term > term,X249: term] :
( ? [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
!= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) )
| ( ( ap @ ( sub @ X247 @ id ) @ X249 )
!= ( lam @ ( X248 @ sh @ one ) ) ) ) )
& ( ( ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
!= $true )
| ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
= $true ) )
| ? [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
!= X121 )
| ? [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
!= X118 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ! [X525: term > $o,X526: term,X527: term,X528: subst > term > subst > $o] :
( ? [X532: subst,X533: subst,X534: term,X535: subst] :
( ( ( X528 @ X533 @ X534 @ ( comp @ X532 @ X535 ) )
!= $true )
& ( ( X528 @ ( comp @ X533 @ X532 ) @ ( sub @ X534 @ X532 ) @ X535 )
= $true ) )
| ( ( X525 @ X527 )
!= $true )
| ? [X536: subst,X537: subst,X538: term,X539: subst] :
( ( ( X528 @ X536 @ X538 @ ( comp @ X537 @ X539 ) )
= $true )
& ( ( X528 @ ( comp @ X536 @ X537 ) @ ( sub @ X538 @ X537 ) @ X539 )
!= $true ) )
| ? [X529: term] :
( ( X528 @ id @ X529 @ id )
!= ( X525 @ X529 ) )
| ( ( X525 @ X526 )
!= $true )
| ( ( X525 @ ( ap @ X527 @ X526 ) )
= $true )
| ? [X530: term,X531: term] :
( ( ( X528 @ id @ X531 @ id )
= $true )
& ( ( X528 @ id @ X530 @ id )
= $true )
& ( ( X528 @ id @ ( ap @ ( sub @ X531 @ id ) @ X530 ) @ id )
!= $true ) ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( pushprop_lthm_orig = $true )
& ( ( substmonoid = $true )
<=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
& ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 ) ) )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X75: subst > term > subst > $o,X76: term] :
( ? [X77: subst > term > term] :
( ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
!= $true )
& ! [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) )
& ! [X81: term] :
( ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
= $true )
| ( ( X75 @ id @ X81 @ id )
!= $true ) ) )
| ? [X82: term] :
( ( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
!= $true )
& ( ( X75 @ id @ X82 @ id )
= $true ) )
| ( ( X75 @ id @ ( lam @ X76 ) @ id )
= $true ) ) )
& ( axclos = $true )
& ( axvarshift = $true )
& ! [X245: d_term,X246: d_term] :
( ( ( d2term @ X246 )
!= ( d2term @ X245 ) )
| ( X245 = X246 ) )
& ( ulamvar1 != $true ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( ( 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 ) ) ) ) ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X127: subst > term > subst > $o,X128: term] :
( ~ ! [X130: subst > term > term] :
( ~ ! [X131: term,X132: subst,X133: subst] :
( ( X130 @ ( comp @ X132 @ X133 ) @ ( sub @ X131 @ X133 ) )
= ( sub @ ( X130 @ X132 @ X131 ) @ X133 ) )
| ~ ! [X134: term] :
( ( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
= $true )
| ( ( X127 @ id @ X134 @ id )
!= $true ) )
| ( ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X127 @ id @ ( lam @ ( sub @ X128 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X129: term] :
( ( ( X127 @ id @ ( sub @ X128 @ ( push @ X129 @ id ) ) @ id )
= $true )
| ( ( X127 @ id @ X129 @ id )
!= $true ) ) ) )
& ( axabs = $true )
& ( ( shinj = $true )
<=> ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) ) )
& ( ( ! [X474: subst,X475: term] :
( ( sub @ one @ ( push @ X475 @ X474 ) )
= X475 )
=> ( ! [X476: subst,X477: term] :
( ( comp @ sh @ ( push @ X477 @ X476 ) )
= X476 )
=> ( ! [X478: subst] :
( ( comp @ X478 @ id )
= X478 )
=> ( ! [X479: term > $o,X480: term] :
( ~ ! [X483: term,X484: term] :
( ( ( X479 @ ( ap @ X484 @ X483 ) )
= $true )
| ( ( X479 @ X483 )
!= $true )
| ( ( X479 @ X484 )
!= $true ) )
| ( ( X479 @ X480 )
= $true )
| ~ ! [X481: term] :
( ( ( X479 @ X481 )
!= $true )
| ( ( X479 @ ( lam @ X481 ) )
= $true ) )
| ~ ! [X482: term] :
( ( ( X479 @ X482 )
= $true )
| ( ( var @ X482 )
!= $true ) ) )
=> ! [X485: term,X486: subst,X487: term > $o] :
( ~ ! [X490: term] :
( ( ( var @ X490 )
!= $true )
| ( ( X487 @ ( sub @ X490 @ X486 ) )
= $true ) )
| ~ ! [X491: term,X492: term] :
( ( ( X487 @ X492 )
!= $true )
| ( ( X487 @ ( ap @ X491 @ X492 ) )
= $true )
| ( ( X487 @ X491 )
!= $true ) )
| ( ( X487 @ ( sub @ X485 @ X486 ) )
= $true )
| ~ ! [X488: term] :
( ~ ! [X489: term] :
( ( ( X487 @ X489 )
!= $true )
| ( ( X487 @ ( sub @ X488 @ ( push @ X489 @ id ) ) )
= $true ) )
| ( ( X487 @ ( lam @ X488 ) )
= $true ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ulamvarind = $true )
& ( ( ! [X272: term] :
( ( sub @ X272 @ id )
= X272 )
=> ! [X273: term] :
( ( sub @ X273 @ id )
= X273 ) )
<=> ( termmset_lthm = $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X168: term,X169: term,X170: subst > term > subst > $o] :
( ( ( X170 @ id @ X169 @ id )
!= $true )
| ~ ! [X171: term,X172: term] :
( ( ( X170 @ id @ ( ap @ ( sub @ X171 @ id ) @ X172 ) @ id )
= $true )
| ( ( X170 @ id @ X171 @ id )
!= $true )
| ( ( X170 @ id @ X172 @ id )
!= $true ) )
| ( ( X170 @ id @ ( ap @ X168 @ X169 ) @ id )
= $true )
| ( ( X170 @ id @ X168 @ id )
!= $true )
| ~ ! [X173: subst,X174: subst,X175: term,X176: subst] :
( ( ( X170 @ X176 @ X175 @ ( comp @ X173 @ X174 ) )
= $true )
| ( ( X170 @ ( comp @ X176 @ X173 ) @ ( sub @ X175 @ X173 ) @ X174 )
!= $true ) )
| ~ ! [X177: subst,X178: subst,X179: term,X180: subst] :
( ( ( X170 @ X178 @ X179 @ ( comp @ X177 @ X180 ) )
!= $true )
| ( ( X170 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) @ X180 )
= $true ) ) ) )
& ( axassoc = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 )
<=> ( axvarcons = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X83: term] :
( ( sub @ X83 @ id )
= X83 )
=> ( ! [X84: term,X85: subst > term > subst > $o] :
( ~ ! [X87: subst > term > term] :
( ~ ! [X88: term] :
( ( $true
= ( X85 @ id @ ( X87 @ id @ X88 ) @ id ) )
| ( ( X85 @ id @ X88 @ id )
!= $true ) )
| ~ ! [X89: subst,X90: term,X91: subst] :
( ( X87 @ ( comp @ X91 @ X89 ) @ ( sub @ X90 @ X89 ) )
= ( sub @ ( X87 @ X91 @ X90 ) @ X89 ) )
| ( ( X85 @ id @ ( lam @ ( X87 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X86: term] :
( ( ( X85 @ id @ ( sub @ X84 @ ( push @ X86 @ id ) ) @ id )
= $true )
| ( ( X85 @ id @ X86 @ id )
!= $true ) )
| ( ( X85 @ id @ ( lam @ ( sub @ X84 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X92: term,X93: subst > term > subst > $o] :
( ( $true
= ( X93 @ id @ ( lam @ X92 ) @ id ) )
| ~ ! [X94: subst > term > term] :
( ( ( X93 @ id @ ( lam @ ( X94 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X98: term] :
( ( ( X93 @ id @ X98 @ id )
!= $true )
| ( ( X93 @ id @ ( X94 @ id @ X98 ) @ id )
= $true ) )
| ~ ! [X95: subst,X96: term,X97: subst] :
( ( X94 @ ( comp @ X95 @ X97 ) @ ( sub @ X96 @ X97 ) )
= ( sub @ ( X94 @ X95 @ X96 ) @ X97 ) ) )
| ~ ! [X99: term] :
( ( ( X93 @ id @ ( sub @ X92 @ ( push @ X99 @ id ) ) @ id )
= $true )
| ( ( X93 @ id @ X99 @ id )
!= $true ) ) ) ) ) )
& ( pushprop = $true )
& ( ! [X292: term,X293: term] :
( ( X292 = X293 )
| ( ( lam @ X293 )
!= ( lam @ X292 ) ) )
<=> ( laminj = $true ) )
& ( termmset_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
<=> ( termmset = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ! [X139: term,X140: term > $o,X141: subst] :
~ ! [X142: term > $o] :
~ ! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ! [X383: term] :
( ( sub @ X383 @ id )
= X383 )
<=> ( axvarid = $true ) )
& ( pushprop_lem2v2 = $true )
& ( ! [X165: subst > term > subst > $o] :
~ ! [X166: term > $o] :
~ ! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( induction2lem_gthm = $true )
& ( ! [X65: term] :
~ ! [X66: subst > term > term] :
~ ! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X452: subst > term > subst > $o,X453: term] :
( ~ ! [X454: term] :
( ( ( X452 @ id @ X454 @ id )
!= $true )
| ( ( X452 @ id @ ( sub @ X453 @ ( push @ X454 @ id ) ) @ id )
= $true ) )
| ~ ! [X455: subst > term > term,X456: term] :
( ~ ! [X460: term] :
( ( ( X452 @ id @ X460 @ id )
!= $true )
| ( ( X452 @ id @ ( X455 @ id @ X460 ) @ id )
= $true ) )
| ~ ! [X457: subst,X458: term,X459: subst] :
( ( sub @ X456 @ ( push @ ( sub @ X458 @ X457 ) @ ( comp @ X459 @ X457 ) ) )
= ( X455 @ ( comp @ X459 @ X457 ) @ ( sub @ X458 @ X457 ) ) )
| ( ( X452 @ id @ ( lam @ ( X455 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X461: subst,X462: term,X463: subst] :
( ( sub @ ( sub @ X456 @ ( push @ X462 @ X461 ) ) @ X463 )
= ( sub @ ( X455 @ X461 @ X462 ) @ X463 ) ) )
| ( ( X452 @ id @ ( lam @ ( sub @ X453 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( lamnotap != $true )
& ( apnotvar = $true )
& ( substmonoid_gthm = $true )
& ( axmap = $true )
& ( hoasapinj2_gthm = $true )
& ! [X413: term] :
? [X414: d_term] :
( ( d2term @ X414 )
= X413 )
& ( hoaslamnotvar = $true )
& ( ! [X135: term] :
~ ! [X136: subst > term > term] :
~ ! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( ! [X329: term > $o,X330: term] :
( ~ ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( ( var @ X331 )
!= $true ) )
| ( ( X329 @ X330 )
= $true )
| ~ ! [X334: term] :
( ~ ! [X335: term] :
( ( ( X329 @ ( sub @ X334 @ ( push @ X335 @ id ) ) )
= $true )
| ( ( X329 @ X335 )
!= $true ) )
| ( ( X329 @ ( lam @ X334 ) )
= $true ) )
| ~ ! [X332: term,X333: term] :
( ( ( X329 @ ( ap @ X332 @ X333 ) )
= $true )
| ( ( X329 @ X332 )
!= $true )
| ( ( X329 @ X333 )
!= $true ) ) )
=> ( ! [X336: term,X337: subst > term > subst > $o,X338: term] :
( ( ( X337 @ id @ X338 @ id )
!= $true )
| ~ ! [X343: term,X344: term] :
( ( ( X337 @ id @ X344 @ id )
!= $true )
| ( ( X337 @ id @ ( ap @ ( sub @ X343 @ id ) @ X344 ) @ id )
= $true )
| ( ( X337 @ id @ X343 @ id )
!= $true ) )
| ( ( X337 @ id @ ( ap @ X338 @ X336 ) @ id )
= $true )
| ~ ! [X339: subst,X340: subst,X341: term,X342: subst] :
( ( ( X337 @ ( comp @ X342 @ X339 ) @ ( sub @ X341 @ X339 ) @ X340 )
= $true )
| ( ( X337 @ X342 @ X341 @ ( comp @ X339 @ X340 ) )
!= $true ) )
| ( ( X337 @ id @ X336 @ id )
!= $true )
| ~ ! [X345: subst,X346: term,X347: subst,X348: subst] :
( ( ( X337 @ ( comp @ X348 @ X347 ) @ ( sub @ X346 @ X347 ) @ X345 )
!= $true )
| ( ( X337 @ X348 @ X346 @ ( comp @ X347 @ X345 ) )
= $true ) ) )
=> ( ! [X349: subst > term > subst > $o,X350: term] :
( ( ( X349 @ id @ ( lam @ X350 ) @ id )
= $true )
| ~ ! [X351: subst,X352: subst,X353: term,X354: subst] :
( ( ( X349 @ ( comp @ X351 @ X352 ) @ ( sub @ X353 @ X352 ) @ X354 )
!= $true )
| ( ( X349 @ X351 @ X353 @ ( comp @ X352 @ X354 ) )
= $true ) )
| ~ ! [X360: term] :
( ( ( X349 @ id @ ( sub @ X350 @ ( push @ X360 @ id ) ) @ id )
= $true )
| ( ( X349 @ id @ X360 @ id )
!= $true ) )
| ~ ! [X355: subst > term > term] :
( ~ ! [X357: subst,X358: subst,X359: term] :
( ( sub @ ( X355 @ X358 @ X359 ) @ X357 )
= ( X355 @ ( comp @ X358 @ X357 ) @ ( sub @ X359 @ X357 ) ) )
| ( ( X349 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X356: term] :
( ( ( X349 @ id @ X356 @ id )
!= $true )
| ( ( X349 @ id @ ( X355 @ id @ X356 ) @ id )
= $true ) ) )
| ~ ! [X361: subst,X362: subst,X363: term,X364: subst] :
( ( ( X349 @ X361 @ X363 @ ( comp @ X362 @ X364 ) )
!= $true )
| ( ( X349 @ ( comp @ X361 @ X362 ) @ ( sub @ X363 @ X362 ) @ X364 )
= $true ) ) )
=> ! [X365: term,X366: subst > term > subst > $o] :
( ~ ! [X379: subst,X380: subst,X381: term,X382: subst] :
( ( ( X366 @ ( comp @ X380 @ X379 ) @ ( sub @ X381 @ X379 ) @ X382 )
!= $true )
| ( ( X366 @ X380 @ X381 @ ( comp @ X379 @ X382 ) )
= $true ) )
| ~ ! [X367: term] :
( ( ( var @ ( sub @ X367 @ id ) )
!= $true )
| ( ( X366 @ id @ X367 @ id )
= $true ) )
| ~ ! [X374: subst > term > term] :
( ~ ! [X378: term] :
( ( ( X366 @ id @ X378 @ id )
!= $true )
| ( ( X366 @ id @ ( X374 @ id @ X378 ) @ id )
= $true ) )
| ~ ! [X375: subst,X376: term,X377: subst] :
( ( X374 @ ( comp @ X377 @ X375 ) @ ( sub @ X376 @ X375 ) )
= ( sub @ ( X374 @ X377 @ X376 ) @ X375 ) )
| ( ( X366 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X366 @ id @ X365 @ id )
= $true )
| ~ ! [X372: term,X373: term] :
( ( ( X366 @ id @ X373 @ id )
!= $true )
| ( ( X366 @ id @ ( ap @ ( sub @ X373 @ id ) @ X372 ) @ id )
= $true )
| ( ( X366 @ id @ X372 @ id )
!= $true ) )
| ~ ! [X368: subst,X369: subst,X370: subst,X371: term] :
( ( ( X366 @ ( comp @ X369 @ X370 ) @ ( sub @ X371 @ X370 ) @ X368 )
= $true )
| ( ( X366 @ X369 @ X371 @ ( comp @ X370 @ X368 ) )
!= $true ) ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( induction = $true )
<=> ! [X495: term > $o,X496: term] :
( ~ ! [X499: term] :
( ( ( var @ X499 )
!= $true )
| ( ( X495 @ X499 )
= $true ) )
| ( ( X495 @ X496 )
= $true )
| ~ ! [X497: term,X498: term] :
( ( ( X495 @ ( ap @ X497 @ X498 ) )
= $true )
| ( ( X495 @ X497 )
!= $true )
| ( ( X495 @ X498 )
!= $true ) )
| ~ ! [X500: term] :
( ( ( X495 @ X500 )
!= $true )
| ( ( X495 @ ( lam @ X500 ) )
= $true ) ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X317: term,X318: term > $o,X319: subst] :
( ~ ! [X320: term] :
( ~ ! [X321: term] :
( ( ( X318 @ ( sub @ X320 @ ( push @ X321 @ id ) ) )
= $true )
| ( ( X318 @ X321 )
!= $true ) )
| ( ( X318 @ ( lam @ X320 ) )
= $true ) )
| ~ ! [X322: term,X323: term] :
( ( ( X318 @ ( ap @ X323 @ X322 ) )
= $true )
| ( ( X318 @ X322 )
!= $true )
| ( ( X318 @ X323 )
!= $true ) )
| ( ( X318 @ ( sub @ X317 @ X319 ) )
= $true )
| ~ ! [X324: term] :
( ( ( X318 @ ( sub @ X324 @ X319 ) )
= $true )
| ( ( var @ X324 )
!= $true ) ) )
<=> ( induction2lem = $true ) )
& ( ! [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
!= ( lam @ ( X465 @ sh @ one ) ) )
| ~ ! [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
| ( ( X465 @ X466 @ X464 )
= ( X467 @ X466 @ X464 ) )
| ~ ! [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( ! [X409: subst] :
( ( comp @ id @ X409 )
= X409 )
=> ( ! [X410: subst] :
( ( comp @ X410 @ id )
= X410 )
=> ( ! [X411: subst] :
( ( comp @ id @ X411 )
= X411 )
& ! [X412: subst] :
( ( comp @ X412 @ id )
= X412 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( ( pushprop_lem1 = $true )
<=> ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
= $true ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( induction2_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
!= ( ap @ X255 @ X254 ) )
| ( X253 = X254 ) )
=> ! [X257: term,X258: term,X259: term,X260: term] :
( ( X257 = X258 )
| ( ( ap @ ( sub @ X260 @ id ) @ X258 )
!= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) ) ) )
& ( pushprop_lem1_gthm = $true )
& ( ! [X274: subst > term > subst > $o,X275: term] :
( ( ( X274 @ id @ X275 @ id )
= $true )
| ~ ! [X290: term,X291: term] :
( ( ( X274 @ id @ ( ap @ ( sub @ X291 @ id ) @ X290 ) @ id )
= $true )
| ( ( X274 @ id @ X291 @ id )
!= $true )
| ( ( X274 @ id @ X290 @ id )
!= $true ) )
| ~ ! [X285: subst > term > term] :
( ( ( X274 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X286: term,X287: subst,X288: subst] :
( ( X285 @ ( comp @ X287 @ X288 ) @ ( sub @ X286 @ X288 ) )
= ( sub @ ( X285 @ X287 @ X286 ) @ X288 ) )
| ~ ! [X289: term] :
( ( ( X274 @ id @ ( X285 @ id @ X289 ) @ id )
= $true )
| ( ( X274 @ id @ X289 @ id )
!= $true ) ) )
| ~ ! [X277: term,X278: subst,X279: subst,X280: subst] :
( ( ( X274 @ X279 @ X277 @ ( comp @ X278 @ X280 ) )
= $true )
| ( ( X274 @ ( comp @ X279 @ X278 ) @ ( sub @ X277 @ X278 ) @ X280 )
!= $true ) )
| ~ ! [X276: term] :
( ( ( X274 @ id @ X276 @ id )
= $true )
| ( ( var @ ( sub @ X276 @ id ) )
!= $true ) )
| ~ ! [X281: subst,X282: term,X283: subst,X284: subst] :
( ( ( X274 @ ( comp @ X283 @ X284 ) @ ( sub @ X282 @ X284 ) @ X281 )
= $true )
| ( ( X274 @ X283 @ X282 @ ( comp @ X284 @ X281 ) )
!= $true ) ) )
<=> ( hoasinduction = $true ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
= $true )
| ( ( X522 @ X521 )
!= $true )
| ~ ! [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
= ( X523 @ X524 ) ) ) )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& ( ! [X2: subst > term > subst > $o] :
~ ! [X3: term > $o] :
~ ! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( axshiftcons = $true )
<=> ! [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
= X493 ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( ( ! [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
= X228 )
=> ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ~ ! [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
= ( X232 @ X234 ) )
| ( ( X232 @ one )
= $true )
| ( ( X230 @ X233 )
!= $true ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( ! [X415: term,X416: term,X417: term,X418: term] :
( ( X416 = X418 )
| ( ( ap @ ( sub @ X417 @ id ) @ X418 )
!= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X501: term] :
( ( sub @ X501 @ id )
= X501 )
=> ! [X502: term > $o,X503: subst > term > subst > $o,X504: term] :
( ~ ! [X515: subst,X516: subst,X517: subst,X518: term] :
( ( ( X503 @ ( comp @ X516 @ X515 ) @ ( sub @ X518 @ X515 ) @ X517 )
!= $true )
| ( ( X503 @ X516 @ X518 @ ( comp @ X515 @ X517 ) )
= $true ) )
| ~ ! [X519: term] :
( ( ( X502 @ X519 )
!= $true )
| ( ( X502 @ ( sub @ X504 @ ( push @ X519 @ id ) ) )
= $true ) )
| ( ( X502 @ ( lam @ X504 ) )
= $true )
| ~ ! [X510: subst > term > term] :
( ~ ! [X511: term] :
( ( ( X503 @ id @ ( X510 @ id @ X511 ) @ id )
= $true )
| ( ( X503 @ id @ X511 @ id )
!= $true ) )
| ( ( X503 @ id @ ( lam @ ( X510 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X512: subst,X513: term,X514: subst] :
( ( sub @ ( X510 @ X514 @ X513 ) @ X512 )
= ( X510 @ ( comp @ X514 @ X512 ) @ ( sub @ X513 @ X512 ) ) ) )
| ~ ! [X506: subst,X507: subst,X508: subst,X509: term] :
( ( ( X503 @ X506 @ X509 @ ( comp @ X507 @ X508 ) )
!= $true )
| ( ( X503 @ ( comp @ X506 @ X507 ) @ ( sub @ X509 @ X507 ) @ X508 )
= $true ) )
| ~ ! [X505: term] :
( ( X503 @ id @ X505 @ id )
= ( X502 @ X505 ) ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ! [X265: d_term] : ( d_one = X265 )
& ( hoasapinj1_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ! [X146: subst > term > subst > $o,X147: term > $o,X148: term] :
( ~ ! [X149: term] :
( ( X147 @ X149 )
= ( X146 @ id @ X149 @ id ) )
| ( ( X147 @ ( lam @ X148 ) )
= $true )
| ~ ! [X150: subst,X151: subst,X152: subst,X153: term] :
( ( ( X146 @ ( comp @ X151 @ X150 ) @ ( sub @ X153 @ X150 ) @ X152 )
!= $true )
| ( ( X146 @ X151 @ X153 @ ( comp @ X150 @ X152 ) )
= $true ) )
| ~ ! [X154: subst,X155: subst,X156: subst,X157: term] :
( ( ( X146 @ ( comp @ X155 @ X154 ) @ ( sub @ X157 @ X154 ) @ X156 )
= $true )
| ( ( X146 @ X155 @ X157 @ ( comp @ X154 @ X156 ) )
!= $true ) )
| ~ ! [X159: subst > term > term] :
( ( ( X146 @ id @ ( lam @ ( X159 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X160: term] :
( ( ( X146 @ id @ X160 @ id )
!= $true )
| ( ( X146 @ id @ ( X159 @ id @ X160 ) @ id )
= $true ) )
| ~ ! [X161: term,X162: subst,X163: subst] :
( ( X159 @ ( comp @ X163 @ X162 ) @ ( sub @ X161 @ X162 ) )
= ( sub @ ( X159 @ X163 @ X161 ) @ X162 ) ) )
| ~ ! [X158: term] :
( ( ( X147 @ X158 )
!= $true )
| ( ( X147 @ ( sub @ X148 @ ( push @ X158 @ id ) ) )
= $true ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( hoasapnotvar = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( axidl = $true )
<=> ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 ) )
& ( ( ! [X21: subst > term > subst > $o] :
~ ! [X22: term > $o] :
~ ! [X23: term] :
( ( X21 @ id @ X23 @ id )
= ( X22 @ X23 ) )
=> ( ! [X24: term,X25: term > $o] :
( ~ ! [X26: term,X27: term] :
( ( ( X25 @ ( ap @ X27 @ X26 ) )
= $true )
| ( ( X25 @ X27 )
!= $true )
| ( ( X25 @ X26 )
!= $true ) )
| ~ ! [X28: term] :
( ( ( X25 @ ( lam @ X28 ) )
= $true )
| ~ ! [X29: term] :
( ( ( X25 @ X29 )
!= $true )
| ( ( X25 @ ( sub @ X28 @ ( push @ X29 @ id ) ) )
= $true ) ) )
| ~ ! [X30: term] :
( ( ( var @ X30 )
!= $true )
| ( ( X25 @ X30 )
= $true ) )
| ( ( X25 @ X24 )
= $true ) )
=> ( ! [X31: term] :
( ( sub @ X31 @ id )
= X31 )
=> ( ! [X32: term > $o,X33: subst > term > subst > $o,X34: term] :
( ~ ! [X35: term] :
( ( ( X32 @ ( sub @ X34 @ ( push @ X35 @ id ) ) )
= $true )
| ( ( X32 @ X35 )
!= $true ) )
| ~ ! [X36: subst > term > term] :
( ( ( X33 @ id @ ( lam @ ( X36 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X38: subst,X39: term,X40: subst] :
( ( X36 @ ( comp @ X38 @ X40 ) @ ( sub @ X39 @ X40 ) )
= ( sub @ ( X36 @ X38 @ X39 ) @ X40 ) )
| ~ ! [X37: term] :
( ( ( X33 @ id @ X37 @ id )
!= $true )
| ( ( X33 @ id @ ( X36 @ id @ X37 ) @ id )
= $true ) ) )
| ( ( X32 @ ( lam @ X34 ) )
= $true )
| ~ ! [X41: term] :
( ( X33 @ id @ X41 @ id )
= ( X32 @ X41 ) ) )
=> ! [X42: subst > term > subst > $o,X43: term] :
( ~ ! [X54: term,X55: term] :
( ( ( X42 @ id @ ( ap @ ( sub @ X54 @ id ) @ X55 ) @ id )
= $true )
| ( ( X42 @ id @ X54 @ id )
!= $true )
| ( ( X42 @ id @ X55 @ id )
!= $true ) )
| ( ( X42 @ id @ X43 @ id )
= $true )
| ~ ! [X44: subst > term > term] :
( ~ ! [X45: subst,X46: subst,X47: term] :
( ( sub @ ( X44 @ X45 @ X47 ) @ X46 )
= ( X44 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) ) )
| ~ ! [X48: term] :
( ( ( X42 @ id @ ( X44 @ id @ X48 ) @ id )
= $true )
| ( ( X42 @ id @ X48 @ id )
!= $true ) )
| ( ( X42 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X49: term] :
( ( ( var @ ( sub @ X49 @ id ) )
!= $true )
| ( ( X42 @ id @ X49 @ id )
= $true ) )
| ~ ! [X50: subst,X51: term,X52: subst,X53: subst] :
( ( ( X42 @ ( comp @ X53 @ X52 ) @ ( sub @ X51 @ X52 ) @ X50 )
= $true )
| ( ( X42 @ X53 @ X51 @ ( comp @ X52 @ X50 ) )
!= $true ) )
| ~ ! [X56: subst,X57: subst,X58: subst,X59: term] :
( ( ( X42 @ ( comp @ X58 @ X57 ) @ ( sub @ X59 @ X57 ) @ X56 )
!= $true )
| ( ( X42 @ X58 @ X59 @ ( comp @ X57 @ X56 ) )
= $true ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lem2v2_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X271: d_subst] : ( d_id = X271 )
& ( lamnotvar = $true )
& ( ( induction2_lthm = $true )
<=> ( ! [X301: term] :
( ( sub @ X301 @ id )
= X301 )
=> ( ! [X302: subst,X303: term > $o,X304: term] :
( ~ ! [X309: term] :
( ( ( var @ X309 )
!= $true )
| ( ( X303 @ ( sub @ X309 @ X302 ) )
= $true ) )
| ~ ! [X305: term,X306: term] :
( ( ( X303 @ ( ap @ X305 @ X306 ) )
= $true )
| ( ( X303 @ X305 )
!= $true )
| ( ( X303 @ X306 )
!= $true ) )
| ( ( X303 @ ( sub @ X304 @ X302 ) )
= $true )
| ~ ! [X307: term] :
( ( ( X303 @ ( lam @ X307 ) )
= $true )
| ~ ! [X308: term] :
( ( ( X303 @ ( sub @ X307 @ ( push @ X308 @ id ) ) )
= $true )
| ( ( X303 @ X308 )
!= $true ) ) ) )
=> ! [X310: term,X311: term > $o] :
( ( ( X311 @ X310 )
= $true )
| ~ ! [X312: term] :
( ( ( var @ X312 )
!= $true )
| ( ( X311 @ X312 )
= $true ) )
| ~ ! [X315: term,X316: term] :
( ( ( X311 @ X315 )
!= $true )
| ( ( X311 @ ( ap @ X315 @ X316 ) )
= $true )
| ( ( X311 @ X316 )
!= $true ) )
| ~ ! [X313: term] :
( ( ( X311 @ ( lam @ X313 ) )
= $true )
| ~ ! [X314: term] :
( ( ( X311 @ X314 )
!= $true )
| ( ( X311 @ ( sub @ X313 @ ( push @ X314 @ id ) ) )
= $true ) ) ) ) ) ) )
& ( ulamvarsh = $true )
& ( ! [X144: term] :
~ ! [X145: subst > term > term] :
( ( X145 @ sh @ one )
!= ( sub @ X144 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( ! [X266: subst,X267: term > $o,X268: term] :
~ ! [X269: term > $o] :
~ ! [X270: term] :
( ( X269 @ X270 )
= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X419: subst > term > subst > $o,X420: term] :
( ~ ! [X422: subst > term > term] :
( ( ( X419 @ id @ ( lam @ ( X422 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X423: term,X424: subst,X425: subst] :
( ( X422 @ ( comp @ X425 @ X424 ) @ ( sub @ X423 @ X424 ) )
= ( sub @ ( X422 @ X425 @ X423 ) @ X424 ) )
| ~ ! [X426: term] :
( ( ( X419 @ id @ ( X422 @ id @ X426 ) @ id )
= $true )
| ( ( X419 @ id @ X426 @ id )
!= $true ) ) )
| ~ ! [X421: term] :
( ( ( X419 @ id @ ( sub @ X420 @ ( push @ X421 @ id ) ) @ id )
= $true )
| ( ( X419 @ id @ X421 @ id )
!= $true ) )
| ( ( X419 @ id @ ( lam @ ( sub @ X420 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( ( induction2 = $true )
<=> ! [X294: term,X295: term > $o] :
( ~ ! [X297: term,X298: term] :
( ( ( X295 @ X298 )
!= $true )
| ( $true
= ( X295 @ ( ap @ X297 @ X298 ) ) )
| ( ( X295 @ X297 )
!= $true ) )
| ~ ! [X296: term] :
( ( ( var @ X296 )
!= $true )
| ( ( X295 @ X296 )
= $true ) )
| ~ ! [X299: term] :
( ~ ! [X300: term] :
( ( ( X295 @ X300 )
!= $true )
| ( ( X295 @ ( sub @ X299 @ ( push @ X300 @ id ) ) )
= $true ) )
| ( ( X295 @ ( lam @ X299 ) )
= $true ) )
| ( ( X295 @ X294 )
= $true ) ) )
& ( pushprop_lthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( ( ! [X5: term,X6: subst] :
( ( sub @ one @ ( push @ X5 @ X6 ) )
= X5 )
=> ( ! [X7: term,X8: subst] :
( ( comp @ sh @ ( push @ X7 @ X8 ) )
= X8 )
=> ( ! [X9: term,X10: term] :
( ( X9 = X10 )
| ( ( lam @ X10 )
!= ( lam @ X9 ) ) )
=> ! [X11: subst > term > term,X12: subst > term > term,X13: term,X14: subst] :
( ~ ! [X15: term,X16: subst,X17: subst] :
( ( sub @ ( X11 @ X17 @ X15 ) @ X16 )
= ( X11 @ ( comp @ X17 @ X16 ) @ ( sub @ X15 @ X16 ) ) )
| ~ ! [X18: subst,X19: term,X20: subst] :
( ( sub @ ( X12 @ X20 @ X19 ) @ X18 )
= ( X12 @ ( comp @ X20 @ X18 ) @ ( sub @ X19 @ X18 ) ) )
| ( ( lam @ ( X11 @ sh @ one ) )
!= ( lam @ ( X12 @ sh @ one ) ) )
| ( ( X11 @ X14 @ X13 )
= ( X12 @ X14 @ X13 ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ( ! [X427: term] :
( ( sub @ X427 @ id )
= X427 )
=> ( ! [X428: term,X429: subst > term > subst > $o] :
( ~ ! [X431: subst > term > term] :
( ~ ! [X433: term,X434: subst,X435: subst] :
( ( X431 @ ( comp @ X435 @ X434 ) @ ( sub @ X433 @ X434 ) )
= ( sub @ ( X431 @ X435 @ X433 ) @ X434 ) )
| ( ( X429 @ id @ ( lam @ ( X431 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X432: term] :
( ( ( X429 @ id @ ( X431 @ id @ X432 ) @ id )
= $true )
| ( ( X429 @ id @ X432 @ id )
!= $true ) ) )
| ( ( X429 @ id @ ( lam @ ( sub @ X428 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X430: term] :
( ( ( X429 @ id @ X430 @ id )
!= $true )
| ( ( X429 @ id @ ( sub @ X428 @ ( push @ X430 @ id ) ) @ id )
= $true ) ) )
=> ! [X436: term,X437: subst > term > subst > $o] :
( ~ ! [X448: subst,X449: subst,X450: subst,X451: term] :
( ( ( X437 @ ( comp @ X450 @ X449 ) @ ( sub @ X451 @ X449 ) @ X448 )
= $true )
| ( ( X437 @ X450 @ X451 @ ( comp @ X449 @ X448 ) )
!= $true ) )
| ( ( X437 @ id @ ( lam @ X436 ) @ id )
= $true )
| ~ ! [X447: term] :
( ( ( X437 @ id @ ( sub @ X436 @ ( push @ X447 @ id ) ) @ id )
= $true )
| ( ( X437 @ id @ X447 @ id )
!= $true ) )
| ~ ! [X443: term,X444: subst,X445: subst,X446: subst] :
( ( ( X437 @ ( comp @ X446 @ X445 ) @ ( sub @ X443 @ X445 ) @ X444 )
!= $true )
| ( $true
= ( X437 @ X446 @ X443 @ ( comp @ X445 @ X444 ) ) ) )
| ~ ! [X438: subst > term > term] :
( ~ ! [X440: term,X441: subst,X442: subst] :
( ( sub @ ( X438 @ X441 @ X440 ) @ X442 )
= ( X438 @ ( comp @ X441 @ X442 ) @ ( sub @ X440 @ X442 ) ) )
| ~ ! [X439: term] :
( ( ( X437 @ id @ ( X438 @ id @ X439 ) @ id )
= $true )
| ( ( X437 @ id @ X439 @ id )
!= $true ) )
| ( ( X437 @ id @ ( lam @ ( X438 @ sh @ one ) ) @ id )
= $true ) ) ) ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( pushprop_gthm = $true )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
| ( X193 = X194 ) )
<=> ( hoasapinj1 = $true ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X235: subst > term > subst > $o,X236: term,X237: term > $o] :
( ~ ! [X244: term] :
( ( X237 @ X244 )
= ( X235 @ id @ X244 @ id ) )
| ( $true
= ( X237 @ ( lam @ X236 ) ) )
| ~ ! [X238: term] :
( ( ( X237 @ X238 )
!= $true )
| ( ( X237 @ ( sub @ X236 @ ( push @ X238 @ id ) ) )
= $true ) )
| ~ ! [X239: subst > term > term] :
( ~ ! [X240: subst,X241: term,X242: subst] :
( ( X239 @ ( comp @ X242 @ X240 ) @ ( sub @ X241 @ X240 ) )
= ( sub @ ( X239 @ X242 @ X241 ) @ X240 ) )
| ( ( X235 @ id @ ( lam @ ( X239 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X243: term] :
( ( ( X235 @ id @ X243 @ id )
!= $true )
| ( ( X235 @ id @ ( X239 @ id @ X243 ) @ id )
= $true ) ) ) ) )
& ( axapp = $true )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X198: subst > term > subst > $o] :
~ ! [X199: term > $o] :
~ ! [X200: term] :
( ( X198 @ id @ X200 @ id )
= ( X199 @ X200 ) )
=> ( ! [X201: term > $o,X202: term] :
( ~ ! [X203: term] :
( ( ( X201 @ X203 )
= $true )
| ( ( var @ X203 )
!= $true ) )
| ~ ! [X206: term] :
( ( ( X201 @ ( lam @ X206 ) )
= $true )
| ~ ! [X207: term] :
( ( ( X201 @ X207 )
!= $true )
| ( ( X201 @ ( sub @ X206 @ ( push @ X207 @ id ) ) )
= $true ) ) )
| ~ ! [X204: term,X205: term] :
( ( ( X201 @ X205 )
!= $true )
| ( ( X201 @ ( ap @ X205 @ X204 ) )
= $true )
| ( ( X201 @ X204 )
!= $true ) )
| ( ( X201 @ X202 )
= $true ) )
=> ( ! [X208: term] :
( ( sub @ X208 @ id )
= X208 )
=> ( ! [X209: subst > term > subst > $o,X210: term > $o,X211: term] :
( ( ( X210 @ ( lam @ X211 ) )
= $true )
| ~ ! [X213: subst > term > term] :
( ~ ! [X214: term] :
( ( ( X209 @ id @ ( X213 @ id @ X214 ) @ id )
= $true )
| ( ( X209 @ id @ X214 @ id )
!= $true ) )
| ( ( X209 @ id @ ( lam @ ( X213 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X215: subst,X216: subst,X217: term] :
( ( sub @ ( X213 @ X216 @ X217 ) @ X215 )
= ( X213 @ ( comp @ X216 @ X215 ) @ ( sub @ X217 @ X215 ) ) ) )
| ~ ! [X218: term] :
( ( X209 @ id @ X218 @ id )
= ( X210 @ X218 ) )
| ~ ! [X212: term] :
( ( ( X210 @ ( sub @ X211 @ ( push @ X212 @ id ) ) )
= $true )
| ( ( X210 @ X212 )
!= $true ) ) )
=> ! [X219: subst > term > subst > $o,X220: term] :
( ~ ! [X221: term,X222: term] :
( ( ( X219 @ id @ X222 @ id )
!= $true )
| ( ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id )
= $true )
| ( ( X219 @ id @ X221 @ id )
!= $true ) )
| ( ( X219 @ id @ X220 @ id )
= $true )
| ~ ! [X223: subst > term > term] :
( ~ ! [X225: subst,X226: term,X227: subst] :
( ( X223 @ ( comp @ X227 @ X225 ) @ ( sub @ X226 @ X225 ) )
= ( sub @ ( X223 @ X227 @ X226 ) @ X225 ) )
| ~ ! [X224: term] :
( ( ( X219 @ id @ ( X223 @ id @ X224 ) @ id )
= $true )
| ( ( X219 @ id @ X224 @ id )
!= $true ) )
| ( $true
= ( X219 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id ) ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( ! [X261: term,X262: term,X263: term,X264: term] :
( ( X261 = X262 )
| ( ( ap @ X261 @ X263 )
!= ( ap @ X262 @ X264 ) ) )
<=> ( apinj1 = $true ) )
& ( hoasinduction_gthm = $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X73: term] :
~ ! [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
!= ( X74 @ sh @ one ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
<=> ( axidr = $true ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X400: term] :
( ( sub @ X400 @ id )
= X400 )
=> ( ! [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
!= ( ap @ X404 @ X402 ) )
| ( X401 = X404 ) )
=> ! [X405: term,X406: term,X407: term,X408: term] :
( ( X405 = X408 )
| ( ( ap @ ( sub @ X408 @ id ) @ X406 )
!= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) ) ) ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X103: term] :
~ ! [X104: subst > term > term] :
~ ! [X105: subst,X106: term] :
( ( sub @ X103 @ ( push @ X106 @ X105 ) )
= ( X104 @ X105 @ X106 ) )
=> ( ! [X107: term] :
( ( sub @ X107 @ id )
= X107 )
=> ! [X108: term,X109: subst > term > subst > $o,X110: term > $o] :
( ~ ! [X112: term] :
( ( X109 @ id @ X112 @ id )
= ( X110 @ X112 ) )
| ( ( X110 @ ( lam @ X108 ) )
= $true )
| ~ ! [X113: subst > term > term] :
( ~ ! [X115: subst,X116: term,X117: subst] :
( ( X113 @ ( comp @ X115 @ X117 ) @ ( sub @ X116 @ X117 ) )
= ( sub @ ( X113 @ X115 @ X116 ) @ X117 ) )
| ( ( X109 @ id @ ( lam @ ( X113 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X114: term] :
( ( ( X109 @ id @ X114 @ id )
!= $true )
| ( ( X109 @ id @ ( X113 @ id @ X114 ) @ id )
= $true ) ) )
| ~ ! [X111: term] :
( ( ( X110 @ X111 )
!= $true )
| ( $true
= ( X110 @ ( sub @ X108 @ ( push @ X111 @ id ) ) ) ) ) ) ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X183: term,X184: subst > term > subst > $o] :
( ~ ! [X185: term,X186: term] :
( ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
= $true )
| ( ( X184 @ id @ X185 @ id )
!= $true )
| ( ( X184 @ id @ X186 @ id )
!= $true ) )
| ( ( X184 @ id @ X183 @ id )
= $true )
| ~ ! [X187: subst > term > term] :
( ~ ! [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) )
| ( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X188: term] :
( ( ( X184 @ id @ ( X187 @ id @ X188 ) @ id )
= $true )
| ( ( X184 @ id @ X188 @ id )
!= $true ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ! [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
!= ( ap @ X326 @ X328 ) )
| ( X325 = X328 ) )
<=> ( apinj2 = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( ! [X384: term,X385: subst > term > subst > $o] :
( ~ ! [X386: subst,X387: subst,X388: subst,X389: term] :
( ( ( X385 @ ( comp @ X386 @ X388 ) @ ( sub @ X389 @ X388 ) @ X387 )
= $true )
| ( ( X385 @ X386 @ X389 @ ( comp @ X388 @ X387 ) )
!= $true ) )
| ~ ! [X396: subst,X397: subst,X398: term,X399: subst] :
( ( ( X385 @ X396 @ X398 @ ( comp @ X397 @ X399 ) )
= $true )
| ( ( X385 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X399 )
!= $true ) )
| ( ( X385 @ id @ ( lam @ X384 ) @ id )
= $true )
| ~ ! [X391: subst > term > term] :
( ( ( X385 @ id @ ( lam @ ( X391 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X395: term] :
( ( ( X385 @ id @ X395 @ id )
!= $true )
| ( ( X385 @ id @ ( X391 @ id @ X395 ) @ id )
= $true ) )
| ~ ! [X392: subst,X393: subst,X394: term] :
( ( sub @ ( X391 @ X393 @ X394 ) @ X392 )
= ( X391 @ ( comp @ X393 @ X392 ) @ ( sub @ X394 @ X392 ) ) ) )
| ~ ! [X390: term] :
( ( ( X385 @ id @ ( sub @ X384 @ ( push @ X390 @ id ) ) @ id )
= $true )
| ( ( X385 @ id @ X390 @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ! [X181: d_subst,X182: d_subst] :
( ( ( d2subst @ X182 )
= ( d2subst @ X181 ) )
=> ( X181 = X182 ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X247: term,X248: subst > term > term,X249: term] :
( ( ( ap @ ( sub @ X247 @ id ) @ X249 )
!= ( lam @ ( X248 @ sh @ one ) ) )
| ~ ! [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) ) ) )
& ( ( ! [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
= X118 )
=> ( ! [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
= X121 )
=> ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
!= $true )
| ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
= $true ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X525: term > $o,X526: term,X527: term,X528: subst > term > subst > $o] :
( ( ( X525 @ X527 )
!= $true )
| ~ ! [X529: term] :
( ( X528 @ id @ X529 @ id )
= ( X525 @ X529 ) )
| ~ ! [X530: term,X531: term] :
( ( ( X528 @ id @ ( ap @ ( sub @ X531 @ id ) @ X530 ) @ id )
= $true )
| ( ( X528 @ id @ X531 @ id )
!= $true )
| ( ( X528 @ id @ X530 @ id )
!= $true ) )
| ( ( X525 @ ( ap @ X527 @ X526 ) )
= $true )
| ~ ! [X536: subst,X537: subst,X538: term,X539: subst] :
( ( ( X528 @ ( comp @ X536 @ X537 ) @ ( sub @ X538 @ X537 ) @ X539 )
= $true )
| ( ( X528 @ X536 @ X538 @ ( comp @ X537 @ X539 ) )
!= $true ) )
| ~ ! [X532: subst,X533: subst,X534: term,X535: subst] :
( ( ( X528 @ X533 @ X534 @ ( comp @ X532 @ X535 ) )
= $true )
| ( ( X528 @ ( comp @ X533 @ X532 ) @ ( sub @ X534 @ X532 ) @ X535 )
!= $true ) )
| ( ( X525 @ X526 )
!= $true ) ) )
& ( pushprop_lthm_orig = $true )
& ( ( substmonoid = $true )
<=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
& ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 ) ) )
& ( ! [X75: subst > term > subst > $o,X76: term] :
( ~ ! [X82: term] :
( ( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
= $true )
| ( ( X75 @ id @ X82 @ id )
!= $true ) )
| ~ ! [X77: subst > term > term] :
( ~ ! [X81: term] :
( ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
= $true )
| ( ( X75 @ id @ X81 @ id )
!= $true ) )
| ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) ) )
| ( ( X75 @ id @ ( lam @ X76 ) @ id )
= $true ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( axclos = $true )
& ( axvarshift = $true )
& ! [X245: d_term,X246: d_term] :
( ( ( d2term @ X246 )
= ( d2term @ X245 ) )
=> ( X245 = X246 ) )
& ( ulamvar1 != $true ) ),
inference(flattening,[status(thm)],[f10]) ).
thf(f10,plain,
( ( ( substmonoid = $true )
<=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
& ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 ) ) )
& ( lamnotap != $true )
& ( axassoc = $true )
& ( ! [X2: subst > term > subst > $o] :
~ ! [X3: term > $o] :
~ ! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( ! [X5: term,X6: subst] :
( ( sub @ one @ ( push @ X5 @ X6 ) )
= X5 )
=> ( ! [X7: term,X8: subst] :
( ( comp @ sh @ ( push @ X7 @ X8 ) )
= X8 )
=> ( ! [X9: term,X10: term] :
( ( X9 = X10 )
| ( ( lam @ X10 )
!= ( lam @ X9 ) ) )
=> ! [X11: subst > term > term,X12: subst > term > term,X13: term,X14: subst] :
( ~ ! [X15: term,X16: subst,X17: subst] :
( ( sub @ ( X11 @ X17 @ X15 ) @ X16 )
= ( X11 @ ( comp @ X17 @ X16 ) @ ( sub @ X15 @ X16 ) ) )
| ~ ! [X18: subst,X19: term,X20: subst] :
( ( sub @ ( X12 @ X20 @ X19 ) @ X18 )
= ( X12 @ ( comp @ X20 @ X18 ) @ ( sub @ X19 @ X18 ) ) )
| ( ( lam @ ( X11 @ sh @ one ) )
!= ( lam @ ( X12 @ sh @ one ) ) )
| ( ( X11 @ X14 @ X13 )
= ( X12 @ X14 @ X13 ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( pushprop_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ( ! [X21: subst > term > subst > $o] :
~ ! [X22: term > $o] :
~ ! [X23: term] :
( ( X21 @ id @ X23 @ id )
= ( X22 @ X23 ) )
=> ( ! [X24: term,X25: term > $o] :
( ~ ! [X26: term,X27: term] :
( ( ( X25 @ ( ap @ X27 @ X26 ) )
= $true )
| ( ( X25 @ X27 )
!= $true )
| ( ( X25 @ X26 )
!= $true ) )
| ~ ! [X28: term] :
( ( ( X25 @ ( lam @ X28 ) )
= $true )
| ~ ! [X29: term] :
( ( ( X25 @ ( sub @ X28 @ ( push @ X29 @ id ) ) )
= $true )
| ( ( X25 @ X29 )
!= $true ) ) )
| ~ ! [X30: term] :
( ( ( var @ X30 )
!= $true )
| ( ( X25 @ X30 )
= $true ) )
| ( ( X25 @ X24 )
= $true ) )
=> ( ! [X31: term] :
( ( sub @ X31 @ id )
= X31 )
=> ( ! [X32: term > $o,X33: subst > term > subst > $o,X34: term] :
( ~ ! [X35: term] :
( ( ( X32 @ ( sub @ X34 @ ( push @ X35 @ id ) ) )
= $true )
| ( ( X32 @ X35 )
!= $true ) )
| ~ ! [X36: subst > term > term] :
( ( ( X33 @ id @ ( lam @ ( X36 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X37: term] :
( ( ( X33 @ id @ X37 @ id )
!= $true )
| ( ( X33 @ id @ ( X36 @ id @ X37 ) @ id )
= $true ) )
| ~ ! [X38: subst,X39: term,X40: subst] :
( ( X36 @ ( comp @ X38 @ X40 ) @ ( sub @ X39 @ X40 ) )
= ( sub @ ( X36 @ X38 @ X39 ) @ X40 ) ) )
| ~ ! [X41: term] :
( ( X33 @ id @ X41 @ id )
= ( X32 @ X41 ) )
| ( ( X32 @ ( lam @ X34 ) )
= $true ) )
=> ! [X42: subst > term > subst > $o,X43: term] :
( ~ ! [X44: subst > term > term] :
( ~ ! [X45: subst,X46: subst,X47: term] :
( ( sub @ ( X44 @ X45 @ X47 ) @ X46 )
= ( X44 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) ) )
| ~ ! [X48: term] :
( ( ( X42 @ id @ ( X44 @ id @ X48 ) @ id )
= $true )
| ( ( X42 @ id @ X48 @ id )
!= $true ) )
| ( ( X42 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X42 @ id @ X43 @ id )
= $true )
| ~ ! [X49: term] :
( ( ( var @ ( sub @ X49 @ id ) )
!= $true )
| ( ( X42 @ id @ X49 @ id )
= $true ) )
| ~ ! [X50: subst,X51: term,X52: subst,X53: subst] :
( ( ( X42 @ ( comp @ X53 @ X52 ) @ ( sub @ X51 @ X52 ) @ X50 )
= $true )
| ( ( X42 @ X53 @ X51 @ ( comp @ X52 @ X50 ) )
!= $true ) )
| ~ ! [X54: term,X55: term] :
( ( ( X42 @ id @ ( ap @ ( sub @ X54 @ id ) @ X55 ) @ id )
= $true )
| ( ( X42 @ id @ X54 @ id )
!= $true )
| ( ( X42 @ id @ X55 @ id )
!= $true ) )
| ~ ! [X56: subst,X57: subst,X58: subst,X59: term] :
( ( ( X42 @ ( comp @ X58 @ X57 ) @ ( sub @ X59 @ X57 ) @ X56 )
!= $true )
| ( ( X42 @ X58 @ X59 @ ( comp @ X57 @ X56 ) )
= $true ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoasinduction_lem1_gthm = $true )
& ( ulamvarsh = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
= $true ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X65: term] :
~ ! [X66: subst > term > term] :
~ ! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( lamnotvar = $true )
& ( ( shinj = $true )
<=> ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) ) )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
<=> ( axidr = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( axidl = $true )
<=> ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X73: term] :
~ ! [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
!= ( X74 @ sh @ one ) ) )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X75: subst > term > subst > $o,X76: term] :
( ~ ! [X77: subst > term > term] :
( ~ ! [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) )
| ~ ! [X81: term] :
( ( ( X75 @ id @ X81 @ id )
!= $true )
| ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
= $true ) )
| ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X75 @ id @ ( lam @ X76 ) @ id )
= $true )
| ~ ! [X82: term] :
( ( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
= $true )
| ( ( X75 @ id @ X82 @ id )
!= $true ) ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X83: term] :
( ( sub @ X83 @ id )
= X83 )
=> ( ! [X84: term,X85: subst > term > subst > $o] :
( ~ ! [X86: term] :
( ( ( X85 @ id @ X86 @ id )
!= $true )
| ( ( X85 @ id @ ( sub @ X84 @ ( push @ X86 @ id ) ) @ id )
= $true ) )
| ~ ! [X87: subst > term > term] :
( ~ ! [X88: term] :
( ( ( X85 @ id @ X88 @ id )
!= $true )
| ( $true
= ( X85 @ id @ ( X87 @ id @ X88 ) @ id ) ) )
| ( ( X85 @ id @ ( lam @ ( X87 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X89: subst,X90: term,X91: subst] :
( ( X87 @ ( comp @ X91 @ X89 ) @ ( sub @ X90 @ X89 ) )
= ( sub @ ( X87 @ X91 @ X90 ) @ X89 ) ) )
| ( ( X85 @ id @ ( lam @ ( sub @ X84 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X92: term,X93: subst > term > subst > $o] :
( ~ ! [X94: subst > term > term] :
( ( ( X93 @ id @ ( lam @ ( X94 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X95: subst,X96: term,X97: subst] :
( ( X94 @ ( comp @ X95 @ X97 ) @ ( sub @ X96 @ X97 ) )
= ( sub @ ( X94 @ X95 @ X96 ) @ X97 ) )
| ~ ! [X98: term] :
( ( ( X93 @ id @ X98 @ id )
!= $true )
| ( ( X93 @ id @ ( X94 @ id @ X98 ) @ id )
= $true ) ) )
| ( $true
= ( X93 @ id @ ( lam @ X92 ) @ id ) )
| ~ ! [X99: term] :
( ( ( X93 @ id @ ( sub @ X92 @ ( push @ X99 @ id ) ) @ id )
= $true )
| ( ( X93 @ id @ X99 @ id )
!= $true ) ) ) ) ) )
& ( ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 )
<=> ( axvarcons = $true ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X103: term] :
~ ! [X104: subst > term > term] :
~ ! [X105: subst,X106: term] :
( ( sub @ X103 @ ( push @ X106 @ X105 ) )
= ( X104 @ X105 @ X106 ) )
=> ( ! [X107: term] :
( ( sub @ X107 @ id )
= X107 )
=> ! [X108: term,X109: subst > term > subst > $o,X110: term > $o] :
( ~ ! [X111: term] :
( ( ( X110 @ X111 )
!= $true )
| ( $true
= ( X110 @ ( sub @ X108 @ ( push @ X111 @ id ) ) ) ) )
| ( ( X110 @ ( lam @ X108 ) )
= $true )
| ~ ! [X112: term] :
( ( X109 @ id @ X112 @ id )
= ( X110 @ X112 ) )
| ~ ! [X113: subst > term > term] :
( ( ( X109 @ id @ ( lam @ ( X113 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X114: term] :
( ( ( X109 @ id @ X114 @ id )
!= $true )
| ( ( X109 @ id @ ( X113 @ id @ X114 ) @ id )
= $true ) )
| ~ ! [X115: subst,X116: term,X117: subst] :
( ( X113 @ ( comp @ X115 @ X117 ) @ ( sub @ X116 @ X117 ) )
= ( sub @ ( X113 @ X115 @ X116 ) @ X117 ) ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( ( ! [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
= X118 )
=> ( ! [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
= X121 )
=> ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
!= $true )
| ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
= $true ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoasinduction_gthm = $true )
& ( ! [X127: subst > term > subst > $o,X128: term] :
( ~ ! [X129: term] :
( ( ( X127 @ id @ ( sub @ X128 @ ( push @ X129 @ id ) ) @ id )
= $true )
| ( ( X127 @ id @ X129 @ id )
!= $true ) )
| ~ ! [X130: subst > term > term] :
( ~ ! [X131: term,X132: subst,X133: subst] :
( ( X130 @ ( comp @ X132 @ X133 ) @ ( sub @ X131 @ X133 ) )
= ( sub @ ( X130 @ X132 @ X131 ) @ X133 ) )
| ~ ! [X134: term] :
( ( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
= $true )
| ( ( X127 @ id @ X134 @ id )
!= $true ) )
| ( ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X127 @ id @ ( lam @ ( sub @ X128 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ! [X135: term] :
~ ! [X136: subst > term > term] :
~ ! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ! [X139: term,X140: term > $o,X141: subst] :
~ ! [X142: term > $o] :
~ ! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( ! [X144: term] :
~ ! [X145: subst > term > term] :
( ( X145 @ sh @ one )
!= ( sub @ X144 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X146: subst > term > subst > $o,X147: term > $o,X148: term] :
( ~ ! [X149: term] :
( ( X147 @ X149 )
= ( X146 @ id @ X149 @ id ) )
| ( ( X147 @ ( lam @ X148 ) )
= $true )
| ~ ! [X150: subst,X151: subst,X152: subst,X153: term] :
( ( ( X146 @ X151 @ X153 @ ( comp @ X150 @ X152 ) )
= $true )
| ( ( X146 @ ( comp @ X151 @ X150 ) @ ( sub @ X153 @ X150 ) @ X152 )
!= $true ) )
| ~ ! [X154: subst,X155: subst,X156: subst,X157: term] :
( ( ( X146 @ ( comp @ X155 @ X154 ) @ ( sub @ X157 @ X154 ) @ X156 )
= $true )
| ( ( X146 @ X155 @ X157 @ ( comp @ X154 @ X156 ) )
!= $true ) )
| ~ ! [X158: term] :
( ( ( X147 @ ( sub @ X148 @ ( push @ X158 @ id ) ) )
= $true )
| ( ( X147 @ X158 )
!= $true ) )
| ~ ! [X159: subst > term > term] :
( ~ ! [X160: term] :
( ( ( X146 @ id @ ( X159 @ id @ X160 ) @ id )
= $true )
| ( ( X146 @ id @ X160 @ id )
!= $true ) )
| ~ ! [X161: term,X162: subst,X163: subst] :
( ( X159 @ ( comp @ X163 @ X162 ) @ ( sub @ X161 @ X162 ) )
= ( sub @ ( X159 @ X163 @ X161 ) @ X162 ) )
| ( ( X146 @ id @ ( lam @ ( X159 @ sh @ one ) ) @ id )
= $true ) ) ) )
& ( pushprop_lem1_gthm = $true )
& ( axvarshift = $true )
& ( pushprop_lem3v2 = $true )
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
<=> ( termmset = $true ) )
& ( ! [X165: subst > term > subst > $o] :
~ ! [X166: term > $o] :
~ ! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X168: term,X169: term,X170: subst > term > subst > $o] :
( ( ( X170 @ id @ X169 @ id )
!= $true )
| ~ ! [X171: term,X172: term] :
( ( ( X170 @ id @ ( ap @ ( sub @ X171 @ id ) @ X172 ) @ id )
= $true )
| ( ( X170 @ id @ X172 @ id )
!= $true )
| ( ( X170 @ id @ X171 @ id )
!= $true ) )
| ~ ! [X173: subst,X174: subst,X175: term,X176: subst] :
( ( ( X170 @ X176 @ X175 @ ( comp @ X173 @ X174 ) )
= $true )
| ( ( X170 @ ( comp @ X176 @ X173 ) @ ( sub @ X175 @ X173 ) @ X174 )
!= $true ) )
| ( ( X170 @ id @ ( ap @ X168 @ X169 ) @ id )
= $true )
| ( ( X170 @ id @ X168 @ id )
!= $true )
| ~ ! [X177: subst,X178: subst,X179: term,X180: subst] :
( ( ( X170 @ X178 @ X179 @ ( comp @ X177 @ X180 ) )
!= $true )
| ( ( X170 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) @ X180 )
= $true ) ) ) )
& ! [X181: d_subst,X182: d_subst] :
( ( ( d2subst @ X182 )
= ( d2subst @ X181 ) )
=> ( X181 = X182 ) )
& ( substmonoid_gthm = $true )
& ( pushprop = $true )
& ( axclos = $true )
& ( ! [X183: term,X184: subst > term > subst > $o] :
( ( ( X184 @ id @ X183 @ id )
= $true )
| ~ ! [X185: term,X186: term] :
( ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
= $true )
| ( ( X184 @ id @ X186 @ id )
!= $true )
| ( ( X184 @ id @ X185 @ id )
!= $true ) )
| ~ ! [X187: subst > term > term] :
( ( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X188: term] :
( ( ( X184 @ id @ X188 @ id )
!= $true )
| ( ( X184 @ id @ ( X187 @ id @ X188 ) @ id )
= $true ) )
| ~ ! [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
| ( X193 = X194 ) )
<=> ( hoasapinj1 = $true ) )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& ( pushprop_lem1v2_gthm = $true )
& ( axabs = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ! [X198: subst > term > subst > $o] :
~ ! [X199: term > $o] :
~ ! [X200: term] :
( ( X198 @ id @ X200 @ id )
= ( X199 @ X200 ) )
=> ( ! [X201: term > $o,X202: term] :
( ( ( X201 @ X202 )
= $true )
| ~ ! [X203: term] :
( ( ( X201 @ X203 )
= $true )
| ( ( var @ X203 )
!= $true ) )
| ~ ! [X204: term,X205: term] :
( ( ( X201 @ X204 )
!= $true )
| ( ( X201 @ X205 )
!= $true )
| ( ( X201 @ ( ap @ X205 @ X204 ) )
= $true ) )
| ~ ! [X206: term] :
( ( ( X201 @ ( lam @ X206 ) )
= $true )
| ~ ! [X207: term] :
( ( ( X201 @ X207 )
!= $true )
| ( ( X201 @ ( sub @ X206 @ ( push @ X207 @ id ) ) )
= $true ) ) ) )
=> ( ! [X208: term] :
( ( sub @ X208 @ id )
= X208 )
=> ( ! [X209: subst > term > subst > $o,X210: term > $o,X211: term] :
( ( ( X210 @ ( lam @ X211 ) )
= $true )
| ~ ! [X212: term] :
( ( ( X210 @ X212 )
!= $true )
| ( ( X210 @ ( sub @ X211 @ ( push @ X212 @ id ) ) )
= $true ) )
| ~ ! [X213: subst > term > term] :
( ( ( X209 @ id @ ( lam @ ( X213 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X214: term] :
( ( ( X209 @ id @ ( X213 @ id @ X214 ) @ id )
= $true )
| ( ( X209 @ id @ X214 @ id )
!= $true ) )
| ~ ! [X215: subst,X216: subst,X217: term] :
( ( sub @ ( X213 @ X216 @ X217 ) @ X215 )
= ( X213 @ ( comp @ X216 @ X215 ) @ ( sub @ X217 @ X215 ) ) ) )
| ~ ! [X218: term] :
( ( X209 @ id @ X218 @ id )
= ( X210 @ X218 ) ) )
=> ! [X219: subst > term > subst > $o,X220: term] :
( ~ ! [X221: term,X222: term] :
( ( ( X219 @ id @ X222 @ id )
!= $true )
| ( ( X219 @ id @ X221 @ id )
!= $true )
| ( ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id )
= $true ) )
| ( ( X219 @ id @ X220 @ id )
= $true )
| ~ ! [X223: subst > term > term] :
( ~ ! [X224: term] :
( ( ( X219 @ id @ X224 @ id )
!= $true )
| ( ( X219 @ id @ ( X223 @ id @ X224 ) @ id )
= $true ) )
| ( $true
= ( X219 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id ) )
| ~ ! [X225: subst,X226: term,X227: subst] :
( ( X223 @ ( comp @ X227 @ X225 ) @ ( sub @ X226 @ X225 ) )
= ( sub @ ( X223 @ X227 @ X226 ) @ X225 ) ) ) ) ) ) ) ) )
& ( termmset_gthm = $true )
& ( axmap = $true )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
= X228 )
=> ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ~ ! [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
= ( X232 @ X234 ) )
| ( ( X232 @ one )
= $true )
| ( ( X230 @ X233 )
!= $true ) ) ) )
& ( hoasapnotvar = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( ! [X235: subst > term > subst > $o,X236: term,X237: term > $o] :
( ~ ! [X238: term] :
( ( ( X237 @ X238 )
!= $true )
| ( ( X237 @ ( sub @ X236 @ ( push @ X238 @ id ) ) )
= $true ) )
| ~ ! [X239: subst > term > term] :
( ~ ! [X240: subst,X241: term,X242: subst] :
( ( X239 @ ( comp @ X242 @ X240 ) @ ( sub @ X241 @ X240 ) )
= ( sub @ ( X239 @ X242 @ X241 ) @ X240 ) )
| ( ( X235 @ id @ ( lam @ ( X239 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X243: term] :
( ( ( X235 @ id @ ( X239 @ id @ X243 ) @ id )
= $true )
| ( ( X235 @ id @ X243 @ id )
!= $true ) ) )
| ( $true
= ( X237 @ ( lam @ X236 ) ) )
| ~ ! [X244: term] :
( ( X237 @ X244 )
= ( X235 @ id @ X244 @ id ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ! [X245: d_term,X246: d_term] :
( ( ( d2term @ X246 )
= ( d2term @ X245 ) )
=> ( X245 = X246 ) )
& ( ( hoaslamnotap = $true )
<=> ! [X247: term,X248: subst > term > term,X249: term] :
( ( ( ap @ ( sub @ X247 @ id ) @ X249 )
!= ( lam @ ( X248 @ sh @ one ) ) )
| ~ ! [X250: subst,X251: subst,X252: term] :
( ( X248 @ ( comp @ X250 @ X251 ) @ ( sub @ X252 @ X251 ) )
= ( sub @ ( X248 @ X250 @ X252 ) @ X251 ) ) ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X253: term,X254: term,X255: term,X256: term] :
( ( ( ap @ X256 @ X253 )
!= ( ap @ X255 @ X254 ) )
| ( X253 = X254 ) )
=> ! [X257: term,X258: term,X259: term,X260: term] :
( ( X257 = X258 )
| ( ( ap @ ( sub @ X260 @ id ) @ X258 )
!= ( ap @ ( sub @ X259 @ id ) @ X257 ) ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X261: term,X262: term,X263: term,X264: term] :
( ( X261 = X262 )
| ( ( ap @ X261 @ X263 )
!= ( ap @ X262 @ X264 ) ) )
<=> ( apinj1 = $true ) )
& ! [X265: d_term] : ( d_one = X265 )
& ( induction2_gthm = $true )
& ( ! [X266: subst,X267: term > $o,X268: term] :
~ ! [X269: term > $o] :
~ ! [X270: term] :
( ( X269 @ X270 )
= ( X267 @ ( sub @ X270 @ ( push @ X268 @ X266 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ( pushprop_lthm = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ! [X271: d_subst] : ( d_id = X271 )
& ( ulamvar1 != $true )
& ( ( ! [X272: term] :
( ( sub @ X272 @ id )
= X272 )
=> ! [X273: term] :
( ( sub @ X273 @ id )
= X273 ) )
<=> ( termmset_lthm = $true ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ! [X274: subst > term > subst > $o,X275: term] :
( ( ( X274 @ id @ X275 @ id )
= $true )
| ~ ! [X276: term] :
( ( ( X274 @ id @ X276 @ id )
= $true )
| ( ( var @ ( sub @ X276 @ id ) )
!= $true ) )
| ~ ! [X277: term,X278: subst,X279: subst,X280: subst] :
( ( ( X274 @ X279 @ X277 @ ( comp @ X278 @ X280 ) )
= $true )
| ( ( X274 @ ( comp @ X279 @ X278 ) @ ( sub @ X277 @ X278 ) @ X280 )
!= $true ) )
| ~ ! [X281: subst,X282: term,X283: subst,X284: subst] :
( ( ( X274 @ X283 @ X282 @ ( comp @ X284 @ X281 ) )
!= $true )
| ( ( X274 @ ( comp @ X283 @ X284 ) @ ( sub @ X282 @ X284 ) @ X281 )
= $true ) )
| ~ ! [X285: subst > term > term] :
( ~ ! [X286: term,X287: subst,X288: subst] :
( ( X285 @ ( comp @ X287 @ X288 ) @ ( sub @ X286 @ X288 ) )
= ( sub @ ( X285 @ X287 @ X286 ) @ X288 ) )
| ~ ! [X289: term] :
( ( ( X274 @ id @ X289 @ id )
!= $true )
| ( ( X274 @ id @ ( X285 @ id @ X289 ) @ id )
= $true ) )
| ( ( X274 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X290: term,X291: term] :
( ( ( X274 @ id @ X291 @ id )
!= $true )
| ( ( X274 @ id @ ( ap @ ( sub @ X291 @ id ) @ X290 ) @ id )
= $true )
| ( ( X274 @ id @ X290 @ id )
!= $true ) ) )
<=> ( hoasinduction = $true ) )
& ( ! [X292: term,X293: term] :
( ( X292 = X293 )
| ( ( lam @ X293 )
!= ( lam @ X292 ) ) )
<=> ( laminj = $true ) )
& ( ! [X294: term,X295: term > $o] :
( ~ ! [X296: term] :
( ( ( var @ X296 )
!= $true )
| ( ( X295 @ X296 )
= $true ) )
| ~ ! [X297: term,X298: term] :
( ( $true
= ( X295 @ ( ap @ X297 @ X298 ) ) )
| ( ( X295 @ X298 )
!= $true )
| ( ( X295 @ X297 )
!= $true ) )
| ~ ! [X299: term] :
( ~ ! [X300: term] :
( ( ( X295 @ X300 )
!= $true )
| ( ( X295 @ ( sub @ X299 @ ( push @ X300 @ id ) ) )
= $true ) )
| ( ( X295 @ ( lam @ X299 ) )
= $true ) )
| ( ( X295 @ X294 )
= $true ) )
<=> ( induction2 = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj2_gthm = $true )
& ( ( ! [X301: term] :
( ( sub @ X301 @ id )
= X301 )
=> ( ! [X302: subst,X303: term > $o,X304: term] :
( ~ ! [X305: term,X306: term] :
( ( ( X303 @ X305 )
!= $true )
| ( ( X303 @ X306 )
!= $true )
| ( ( X303 @ ( ap @ X305 @ X306 ) )
= $true ) )
| ~ ! [X307: term] :
( ~ ! [X308: term] :
( ( ( X303 @ X308 )
!= $true )
| ( ( X303 @ ( sub @ X307 @ ( push @ X308 @ id ) ) )
= $true ) )
| ( ( X303 @ ( lam @ X307 ) )
= $true ) )
| ( ( X303 @ ( sub @ X304 @ X302 ) )
= $true )
| ~ ! [X309: term] :
( ( ( var @ X309 )
!= $true )
| ( ( X303 @ ( sub @ X309 @ X302 ) )
= $true ) ) )
=> ! [X310: term,X311: term > $o] :
( ~ ! [X312: term] :
( ( ( var @ X312 )
!= $true )
| ( ( X311 @ X312 )
= $true ) )
| ( ( X311 @ X310 )
= $true )
| ~ ! [X313: term] :
( ~ ! [X314: term] :
( ( ( X311 @ ( sub @ X313 @ ( push @ X314 @ id ) ) )
= $true )
| ( ( X311 @ X314 )
!= $true ) )
| ( ( X311 @ ( lam @ X313 ) )
= $true ) )
| ~ ! [X315: term,X316: term] :
( ( ( X311 @ ( ap @ X315 @ X316 ) )
= $true )
| ( ( X311 @ X315 )
!= $true )
| ( ( X311 @ X316 )
!= $true ) ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( ulamvarind = $true )
& ( hoaslamnotvar_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X317: term,X318: term > $o,X319: subst] :
( ~ ! [X320: term] :
( ~ ! [X321: term] :
( ( ( X318 @ ( sub @ X320 @ ( push @ X321 @ id ) ) )
= $true )
| ( ( X318 @ X321 )
!= $true ) )
| ( ( X318 @ ( lam @ X320 ) )
= $true ) )
| ~ ! [X322: term,X323: term] :
( ( ( X318 @ ( ap @ X323 @ X322 ) )
= $true )
| ( ( X318 @ X323 )
!= $true )
| ( ( X318 @ X322 )
!= $true ) )
| ( ( X318 @ ( sub @ X317 @ X319 ) )
= $true )
| ~ ! [X324: term] :
( ( ( var @ X324 )
!= $true )
| ( ( X318 @ ( sub @ X324 @ X319 ) )
= $true ) ) )
<=> ( induction2lem = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X325: term,X326: term,X327: term,X328: term] :
( ( ( ap @ X327 @ X325 )
!= ( ap @ X326 @ X328 ) )
| ( X325 = X328 ) )
<=> ( apinj2 = $true ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X329: term > $o,X330: term] :
( ~ ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( ( var @ X331 )
!= $true ) )
| ~ ! [X332: term,X333: term] :
( ( ( X329 @ X332 )
!= $true )
| ( ( X329 @ X333 )
!= $true )
| ( ( X329 @ ( ap @ X332 @ X333 ) )
= $true ) )
| ~ ! [X334: term] :
( ~ ! [X335: term] :
( ( ( X329 @ X335 )
!= $true )
| ( ( X329 @ ( sub @ X334 @ ( push @ X335 @ id ) ) )
= $true ) )
| ( ( X329 @ ( lam @ X334 ) )
= $true ) )
| ( ( X329 @ X330 )
= $true ) )
=> ( ! [X336: term,X337: subst > term > subst > $o,X338: term] :
( ~ ! [X339: subst,X340: subst,X341: term,X342: subst] :
( ( ( X337 @ ( comp @ X342 @ X339 ) @ ( sub @ X341 @ X339 ) @ X340 )
= $true )
| ( ( X337 @ X342 @ X341 @ ( comp @ X339 @ X340 ) )
!= $true ) )
| ( ( X337 @ id @ X336 @ id )
!= $true )
| ( ( X337 @ id @ ( ap @ X338 @ X336 ) @ id )
= $true )
| ( ( X337 @ id @ X338 @ id )
!= $true )
| ~ ! [X343: term,X344: term] :
( ( ( X337 @ id @ X344 @ id )
!= $true )
| ( ( X337 @ id @ ( ap @ ( sub @ X343 @ id ) @ X344 ) @ id )
= $true )
| ( ( X337 @ id @ X343 @ id )
!= $true ) )
| ~ ! [X345: subst,X346: term,X347: subst,X348: subst] :
( ( ( X337 @ X348 @ X346 @ ( comp @ X347 @ X345 ) )
= $true )
| ( ( X337 @ ( comp @ X348 @ X347 ) @ ( sub @ X346 @ X347 ) @ X345 )
!= $true ) ) )
=> ( ! [X349: subst > term > subst > $o,X350: term] :
( ~ ! [X351: subst,X352: subst,X353: term,X354: subst] :
( ( ( X349 @ ( comp @ X351 @ X352 ) @ ( sub @ X353 @ X352 ) @ X354 )
!= $true )
| ( ( X349 @ X351 @ X353 @ ( comp @ X352 @ X354 ) )
= $true ) )
| ~ ! [X355: subst > term > term] :
( ~ ! [X356: term] :
( ( ( X349 @ id @ X356 @ id )
!= $true )
| ( ( X349 @ id @ ( X355 @ id @ X356 ) @ id )
= $true ) )
| ( ( X349 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X357: subst,X358: subst,X359: term] :
( ( sub @ ( X355 @ X358 @ X359 ) @ X357 )
= ( X355 @ ( comp @ X358 @ X357 ) @ ( sub @ X359 @ X357 ) ) ) )
| ( ( X349 @ id @ ( lam @ X350 ) @ id )
= $true )
| ~ ! [X360: term] :
( ( ( X349 @ id @ X360 @ id )
!= $true )
| ( ( X349 @ id @ ( sub @ X350 @ ( push @ X360 @ id ) ) @ id )
= $true ) )
| ~ ! [X361: subst,X362: subst,X363: term,X364: subst] :
( ( ( X349 @ X361 @ X363 @ ( comp @ X362 @ X364 ) )
!= $true )
| ( ( X349 @ ( comp @ X361 @ X362 ) @ ( sub @ X363 @ X362 ) @ X364 )
= $true ) ) )
=> ! [X365: term,X366: subst > term > subst > $o] :
( ~ ! [X367: term] :
( ( ( X366 @ id @ X367 @ id )
= $true )
| ( ( var @ ( sub @ X367 @ id ) )
!= $true ) )
| ~ ! [X368: subst,X369: subst,X370: subst,X371: term] :
( ( ( X366 @ ( comp @ X369 @ X370 ) @ ( sub @ X371 @ X370 ) @ X368 )
= $true )
| ( ( X366 @ X369 @ X371 @ ( comp @ X370 @ X368 ) )
!= $true ) )
| ~ ! [X372: term,X373: term] :
( ( ( X366 @ id @ X372 @ id )
!= $true )
| ( ( X366 @ id @ X373 @ id )
!= $true )
| ( ( X366 @ id @ ( ap @ ( sub @ X373 @ id ) @ X372 ) @ id )
= $true ) )
| ~ ! [X374: subst > term > term] :
( ~ ! [X375: subst,X376: term,X377: subst] :
( ( X374 @ ( comp @ X377 @ X375 ) @ ( sub @ X376 @ X375 ) )
= ( sub @ ( X374 @ X377 @ X376 ) @ X375 ) )
| ( ( X366 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X378: term] :
( ( ( X366 @ id @ ( X374 @ id @ X378 ) @ id )
= $true )
| ( ( X366 @ id @ X378 @ id )
!= $true ) ) )
| ( ( X366 @ id @ X365 @ id )
= $true )
| ~ ! [X379: subst,X380: subst,X381: term,X382: subst] :
( ( ( X366 @ ( comp @ X380 @ X379 ) @ ( sub @ X381 @ X379 ) @ X382 )
!= $true )
| ( ( X366 @ X380 @ X381 @ ( comp @ X379 @ X382 ) )
= $true ) ) ) ) ) ) )
& ( apnotvar = $true )
& ( ! [X383: term] :
( ( sub @ X383 @ id )
= X383 )
<=> ( axvarid = $true ) )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X384: term,X385: subst > term > subst > $o] :
( ~ ! [X386: subst,X387: subst,X388: subst,X389: term] :
( ( ( X385 @ ( comp @ X386 @ X388 ) @ ( sub @ X389 @ X388 ) @ X387 )
= $true )
| ( ( X385 @ X386 @ X389 @ ( comp @ X388 @ X387 ) )
!= $true ) )
| ~ ! [X390: term] :
( ( ( X385 @ id @ ( sub @ X384 @ ( push @ X390 @ id ) ) @ id )
= $true )
| ( ( X385 @ id @ X390 @ id )
!= $true ) )
| ( ( X385 @ id @ ( lam @ X384 ) @ id )
= $true )
| ~ ! [X391: subst > term > term] :
( ~ ! [X392: subst,X393: subst,X394: term] :
( ( sub @ ( X391 @ X393 @ X394 ) @ X392 )
= ( X391 @ ( comp @ X393 @ X392 ) @ ( sub @ X394 @ X392 ) ) )
| ~ ! [X395: term] :
( ( ( X385 @ id @ X395 @ id )
!= $true )
| ( ( X385 @ id @ ( X391 @ id @ X395 ) @ id )
= $true ) )
| ( ( X385 @ id @ ( lam @ ( X391 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X396: subst,X397: subst,X398: term,X399: subst] :
( ( ( X385 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X399 )
!= $true )
| ( ( X385 @ X396 @ X398 @ ( comp @ X397 @ X399 ) )
= $true ) ) ) )
& ( hoaslaminj_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X400: term] :
( ( sub @ X400 @ id )
= X400 )
=> ( ! [X401: term,X402: term,X403: term,X404: term] :
( ( ( ap @ X401 @ X403 )
!= ( ap @ X404 @ X402 ) )
| ( X401 = X404 ) )
=> ! [X405: term,X406: term,X407: term,X408: term] :
( ( X405 = X408 )
| ( ( ap @ ( sub @ X408 @ id ) @ X406 )
!= ( ap @ ( sub @ X405 @ id ) @ X407 ) ) ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( ! [X409: subst] :
( ( comp @ id @ X409 )
= X409 )
=> ( ! [X410: subst] :
( ( comp @ X410 @ id )
= X410 )
=> ( ! [X411: subst] :
( ( comp @ id @ X411 )
= X411 )
& ! [X412: subst] :
( ( comp @ X412 @ id )
= X412 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ! [X413: term] :
? [X414: d_term] :
( ( d2term @ X414 )
= X413 )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X415: term,X416: term,X417: term,X418: term] :
( ( X416 = X418 )
| ( ( ap @ ( sub @ X417 @ id ) @ X418 )
!= ( ap @ ( sub @ X415 @ id ) @ X416 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X419: subst > term > subst > $o,X420: term] :
( ~ ! [X421: term] :
( ( ( X419 @ id @ ( sub @ X420 @ ( push @ X421 @ id ) ) @ id )
= $true )
| ( ( X419 @ id @ X421 @ id )
!= $true ) )
| ( ( X419 @ id @ ( lam @ ( sub @ X420 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X422: subst > term > term] :
( ~ ! [X423: term,X424: subst,X425: subst] :
( ( X422 @ ( comp @ X425 @ X424 ) @ ( sub @ X423 @ X424 ) )
= ( sub @ ( X422 @ X425 @ X423 ) @ X424 ) )
| ~ ! [X426: term] :
( ( ( X419 @ id @ ( X422 @ id @ X426 ) @ id )
= $true )
| ( ( X419 @ id @ X426 @ id )
!= $true ) )
| ( ( X419 @ id @ ( lam @ ( X422 @ sh @ one ) ) @ id )
= $true ) ) ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X427: term] :
( ( sub @ X427 @ id )
= X427 )
=> ( ! [X428: term,X429: subst > term > subst > $o] :
( ~ ! [X430: term] :
( ( ( X429 @ id @ ( sub @ X428 @ ( push @ X430 @ id ) ) @ id )
= $true )
| ( ( X429 @ id @ X430 @ id )
!= $true ) )
| ~ ! [X431: subst > term > term] :
( ~ ! [X432: term] :
( ( ( X429 @ id @ X432 @ id )
!= $true )
| ( ( X429 @ id @ ( X431 @ id @ X432 ) @ id )
= $true ) )
| ( ( X429 @ id @ ( lam @ ( X431 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X433: term,X434: subst,X435: subst] :
( ( X431 @ ( comp @ X435 @ X434 ) @ ( sub @ X433 @ X434 ) )
= ( sub @ ( X431 @ X435 @ X433 ) @ X434 ) ) )
| ( ( X429 @ id @ ( lam @ ( sub @ X428 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X436: term,X437: subst > term > subst > $o] :
( ( ( X437 @ id @ ( lam @ X436 ) @ id )
= $true )
| ~ ! [X438: subst > term > term] :
( ( ( X437 @ id @ ( lam @ ( X438 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X439: term] :
( ( ( X437 @ id @ ( X438 @ id @ X439 ) @ id )
= $true )
| ( ( X437 @ id @ X439 @ id )
!= $true ) )
| ~ ! [X440: term,X441: subst,X442: subst] :
( ( sub @ ( X438 @ X441 @ X440 ) @ X442 )
= ( X438 @ ( comp @ X441 @ X442 ) @ ( sub @ X440 @ X442 ) ) ) )
| ~ ! [X443: term,X444: subst,X445: subst,X446: subst] :
( ( $true
= ( X437 @ X446 @ X443 @ ( comp @ X445 @ X444 ) ) )
| ( ( X437 @ ( comp @ X446 @ X445 ) @ ( sub @ X443 @ X445 ) @ X444 )
!= $true ) )
| ~ ! [X447: term] :
( ( ( X437 @ id @ ( sub @ X436 @ ( push @ X447 @ id ) ) @ id )
= $true )
| ( ( X437 @ id @ X447 @ id )
!= $true ) )
| ~ ! [X448: subst,X449: subst,X450: subst,X451: term] :
( ( ( X437 @ ( comp @ X450 @ X449 ) @ ( sub @ X451 @ X449 ) @ X448 )
= $true )
| ( ( X437 @ X450 @ X451 @ ( comp @ X449 @ X448 ) )
!= $true ) ) ) ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ! [X452: subst > term > subst > $o,X453: term] :
( ~ ! [X454: term] :
( ( ( X452 @ id @ ( sub @ X453 @ ( push @ X454 @ id ) ) @ id )
= $true )
| ( ( X452 @ id @ X454 @ id )
!= $true ) )
| ( ( X452 @ id @ ( lam @ ( sub @ X453 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X455: subst > term > term,X456: term] :
( ~ ! [X457: subst,X458: term,X459: subst] :
( ( sub @ X456 @ ( push @ ( sub @ X458 @ X457 ) @ ( comp @ X459 @ X457 ) ) )
= ( X455 @ ( comp @ X459 @ X457 ) @ ( sub @ X458 @ X457 ) ) )
| ( ( X452 @ id @ ( lam @ ( X455 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X460: term] :
( ( ( X452 @ id @ X460 @ id )
!= $true )
| ( ( X452 @ id @ ( X455 @ id @ X460 ) @ id )
= $true ) )
| ~ ! [X461: subst,X462: term,X463: subst] :
( ( sub @ ( sub @ X456 @ ( push @ X462 @ X461 ) ) @ X463 )
= ( sub @ ( X455 @ X461 @ X462 ) @ X463 ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X464: term,X465: subst > term > term,X466: subst,X467: subst > term > term] :
( ( ( lam @ ( X467 @ sh @ one ) )
!= ( lam @ ( X465 @ sh @ one ) ) )
| ~ ! [X471: term,X472: subst,X473: subst] :
( ( X465 @ ( comp @ X473 @ X472 ) @ ( sub @ X471 @ X472 ) )
= ( sub @ ( X465 @ X473 @ X471 ) @ X472 ) )
| ( ( X465 @ X466 @ X464 )
= ( X467 @ X466 @ X464 ) )
| ~ ! [X468: subst,X469: term,X470: subst] :
( ( X467 @ ( comp @ X470 @ X468 ) @ ( sub @ X469 @ X468 ) )
= ( sub @ ( X467 @ X470 @ X469 ) @ X468 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( ! [X474: subst,X475: term] :
( ( sub @ one @ ( push @ X475 @ X474 ) )
= X475 )
=> ( ! [X476: subst,X477: term] :
( ( comp @ sh @ ( push @ X477 @ X476 ) )
= X476 )
=> ( ! [X478: subst] :
( ( comp @ X478 @ id )
= X478 )
=> ( ! [X479: term > $o,X480: term] :
( ( ( X479 @ X480 )
= $true )
| ~ ! [X481: term] :
( ( ( X479 @ ( lam @ X481 ) )
= $true )
| ( ( X479 @ X481 )
!= $true ) )
| ~ ! [X482: term] :
( ( ( var @ X482 )
!= $true )
| ( ( X479 @ X482 )
= $true ) )
| ~ ! [X483: term,X484: term] :
( ( ( X479 @ X483 )
!= $true )
| ( ( X479 @ X484 )
!= $true )
| ( ( X479 @ ( ap @ X484 @ X483 ) )
= $true ) ) )
=> ! [X485: term,X486: subst,X487: term > $o] :
( ( ( X487 @ ( sub @ X485 @ X486 ) )
= $true )
| ~ ! [X488: term] :
( ( ( X487 @ ( lam @ X488 ) )
= $true )
| ~ ! [X489: term] :
( ( ( X487 @ X489 )
!= $true )
| ( ( X487 @ ( sub @ X488 @ ( push @ X489 @ id ) ) )
= $true ) ) )
| ~ ! [X490: term] :
( ( ( X487 @ ( sub @ X490 @ X486 ) )
= $true )
| ( ( var @ X490 )
!= $true ) )
| ~ ! [X491: term,X492: term] :
( ( ( X487 @ X491 )
!= $true )
| ( ( X487 @ ( ap @ X491 @ X492 ) )
= $true )
| ( ( X487 @ X492 )
!= $true ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( one
= ( d2term @ d_one ) )
& ( axapp = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( axshiftcons = $true )
<=> ! [X493: subst,X494: term] :
( ( comp @ sh @ ( push @ X494 @ X493 ) )
= X493 ) )
& ( 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 ) ) ) ) ) ) )
& ( ! [X495: term > $o,X496: term] :
( ~ ! [X497: term,X498: term] :
( ( ( X495 @ X497 )
!= $true )
| ( ( X495 @ ( ap @ X497 @ X498 ) )
= $true )
| ( ( X495 @ X498 )
!= $true ) )
| ~ ! [X499: term] :
( ( ( var @ X499 )
!= $true )
| ( ( X495 @ X499 )
= $true ) )
| ~ ! [X500: term] :
( ( ( X495 @ ( lam @ X500 ) )
= $true )
| ( ( X495 @ X500 )
!= $true ) )
| ( ( X495 @ X496 )
= $true ) )
<=> ( induction = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasapnotvar_gthm = $true )
& ( ( ! [X501: term] :
( ( sub @ X501 @ id )
= X501 )
=> ! [X502: term > $o,X503: subst > term > subst > $o,X504: term] :
( ~ ! [X505: term] :
( ( X503 @ id @ X505 @ id )
= ( X502 @ X505 ) )
| ( ( X502 @ ( lam @ X504 ) )
= $true )
| ~ ! [X506: subst,X507: subst,X508: subst,X509: term] :
( ( ( X503 @ X506 @ X509 @ ( comp @ X507 @ X508 ) )
!= $true )
| ( ( X503 @ ( comp @ X506 @ X507 ) @ ( sub @ X509 @ X507 ) @ X508 )
= $true ) )
| ~ ! [X510: subst > term > term] :
( ( ( X503 @ id @ ( lam @ ( X510 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X511: term] :
( ( ( X503 @ id @ X511 @ id )
!= $true )
| ( ( X503 @ id @ ( X510 @ id @ X511 ) @ id )
= $true ) )
| ~ ! [X512: subst,X513: term,X514: subst] :
( ( sub @ ( X510 @ X514 @ X513 ) @ X512 )
= ( X510 @ ( comp @ X514 @ X512 ) @ ( sub @ X513 @ X512 ) ) ) )
| ~ ! [X515: subst,X516: subst,X517: subst,X518: term] :
( ( ( X503 @ X516 @ X518 @ ( comp @ X515 @ X517 ) )
= $true )
| ( ( X503 @ ( comp @ X516 @ X515 ) @ ( sub @ X518 @ X515 ) @ X517 )
!= $true ) )
| ~ ! [X519: term] :
( ( ( X502 @ X519 )
!= $true )
| ( ( X502 @ ( sub @ X504 @ ( push @ X519 @ id ) ) )
= $true ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X520: subst,X521: term,X522: term > $o,X523: term > $o] :
( ( ( X523 @ one )
= $true )
| ( ( X522 @ X521 )
!= $true )
| ~ ! [X524: term] :
( ( X522 @ ( sub @ X524 @ ( push @ X521 @ X520 ) ) )
= ( X523 @ X524 ) ) ) )
& ( ! [X525: term > $o,X526: term,X527: term,X528: subst > term > subst > $o] :
( ( ( X525 @ X526 )
!= $true )
| ~ ! [X529: term] :
( ( X528 @ id @ X529 @ id )
= ( X525 @ X529 ) )
| ~ ! [X530: term,X531: term] :
( ( ( X528 @ id @ X530 @ id )
!= $true )
| ( ( X528 @ id @ X531 @ id )
!= $true )
| ( ( X528 @ id @ ( ap @ ( sub @ X531 @ id ) @ X530 ) @ id )
= $true ) )
| ~ ! [X532: subst,X533: subst,X534: term,X535: subst] :
( ( ( X528 @ ( comp @ X533 @ X532 ) @ ( sub @ X534 @ X532 ) @ X535 )
!= $true )
| ( ( X528 @ X533 @ X534 @ ( comp @ X532 @ X535 ) )
= $true ) )
| ( ( X525 @ ( ap @ X527 @ X526 ) )
= $true )
| ( ( X525 @ X527 )
!= $true )
| ~ ! [X536: subst,X537: subst,X538: term,X539: subst] :
( ( ( X528 @ ( comp @ X536 @ X537 ) @ ( sub @ X538 @ X537 ) @ X539 )
= $true )
| ( ( X528 @ X536 @ X538 @ ( comp @ X537 @ X539 ) )
!= $true ) ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( hoasinduction_lem1 = $true ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,plain,
( ( ( substmonoid = $true )
<=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
& ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 ) ) )
& ( lamnotap != $true )
& ( axassoc = $true )
& ( ! [X2: subst > term > subst > $o] :
~ ! [X3: term > $o] :
~ ! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( ! [X5: term,X6: subst] :
( ( sub @ one @ ( push @ X5 @ X6 ) )
= X5 )
=> ( ! [X7: term,X8: subst] :
( ( comp @ sh @ ( push @ X7 @ X8 ) )
= X8 )
=> ( ! [X9: term,X10: term] :
( ( X9 = X10 )
| ( ( lam @ X10 )
!= ( lam @ X9 ) ) )
=> ! [X11: subst > term > term,X12: subst > term > term,X13: term,X14: subst] :
( ~ ! [X15: term,X16: subst,X17: subst] :
( ( sub @ ( X11 @ X17 @ X15 ) @ X16 )
= ( X11 @ ( comp @ X17 @ X16 ) @ ( sub @ X15 @ X16 ) ) )
| ~ ! [X18: subst,X19: term,X20: subst] :
( ( sub @ ( X12 @ X20 @ X19 ) @ X18 )
= ( X12 @ ( comp @ X20 @ X18 ) @ ( sub @ X19 @ X18 ) ) )
| ( ( lam @ ( X11 @ sh @ one ) )
!= ( lam @ ( X12 @ sh @ one ) ) )
| ( ( X11 @ X14 @ X13 )
= ( X12 @ X14 @ X13 ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( pushprop_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ( ! [X21: subst > term > subst > $o] :
~ ! [X22: term > $o] :
~ ! [X23: term] :
( ( X21 @ id @ X23 @ id )
= ( X22 @ X23 ) )
=> ( ! [X24: term,X25: term > $o] :
( ~ ! [X26: term,X27: term] :
( ( ( X25 @ ( ap @ X27 @ X26 ) )
= $true )
| ( ( X25 @ X27 )
!= $true )
| ( ( X25 @ X26 )
!= $true ) )
| ~ ! [X28: term] :
( ( ( X25 @ ( lam @ X28 ) )
= $true )
| ~ ! [X29: term] :
( ( ( X25 @ ( sub @ X28 @ ( push @ X29 @ id ) ) )
= $true )
| ( ( X25 @ X29 )
!= $true ) ) )
| ~ ! [X30: term] :
( ( ( var @ X30 )
!= $true )
| ( ( X25 @ X30 )
= $true ) )
| ( ( X25 @ X24 )
= $true ) )
=> ( ! [X31: term] :
( ( sub @ X31 @ id )
= X31 )
=> ( ! [X32: term > $o,X33: subst > term > subst > $o,X34: term] :
( ~ ! [X35: term] :
( ( ( X32 @ ( sub @ X34 @ ( push @ X35 @ id ) ) )
= $true )
| ( ( X32 @ X35 )
!= $true ) )
| ~ ! [X36: subst > term > term] :
( ( ( X33 @ id @ ( lam @ ( X36 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X37: term] :
( ( ( X33 @ id @ X37 @ id )
!= $true )
| ( ( X33 @ id @ ( X36 @ id @ X37 ) @ id )
= $true ) )
| ~ ! [X38: subst,X39: term,X40: subst] :
( ( X36 @ ( comp @ X38 @ X40 ) @ ( sub @ X39 @ X40 ) )
= ( sub @ ( X36 @ X38 @ X39 ) @ X40 ) ) )
| ~ ! [X41: term] :
( ( X33 @ id @ X41 @ id )
= ( X32 @ X41 ) )
| ( ( X32 @ ( lam @ X34 ) )
= $true ) )
=> ! [X42: subst > term > subst > $o,X43: term] :
( ~ ! [X44: subst > term > term] :
( ~ ! [X45: subst,X46: subst,X47: term] :
( ( sub @ ( X44 @ X45 @ X47 ) @ X46 )
= ( X44 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) ) )
| ~ ! [X48: term] :
( ( ( X42 @ id @ ( X44 @ id @ X48 ) @ id )
= $true )
| ( ( X42 @ id @ X48 @ id )
!= $true ) )
| ( ( X42 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X42 @ id @ X43 @ id )
= $true )
| ~ ! [X49: term] :
( ( ( var @ ( sub @ X49 @ id ) )
!= $true )
| ( ( X42 @ id @ X49 @ id )
= $true ) )
| ~ ! [X50: subst,X51: term,X52: subst,X53: subst] :
( ( ( X42 @ ( comp @ X53 @ X52 ) @ ( sub @ X51 @ X52 ) @ X50 )
= $true )
| ( ( X42 @ X53 @ X51 @ ( comp @ X52 @ X50 ) )
!= $true ) )
| ~ ! [X54: term,X55: term] :
( ( ( X42 @ id @ ( ap @ ( sub @ X54 @ id ) @ X55 ) @ id )
= $true )
| ( ( X42 @ id @ X54 @ id )
!= $true )
| ( ( X42 @ id @ X55 @ id )
!= $true ) )
| ~ ! [X56: subst,X57: subst,X58: subst,X59: term] :
( ( ( X42 @ ( comp @ X58 @ X57 ) @ ( sub @ X59 @ X57 ) @ X56 )
!= $true )
| ( ( X42 @ X58 @ X59 @ ( comp @ X57 @ X56 ) )
= $true ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoasinduction_lem1_gthm = $true )
& ( ulamvarsh = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ( ( X60 @ X61 )
!= $true )
| ( ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) )
= $true ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X65: term] :
~ ! [X66: subst > term > term] :
~ ! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( lamnotvar = $true )
& ( ( shinj = $true )
<=> ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) ) )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
<=> ( axidr = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( axidl = $true )
<=> ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X73: term] :
~ ! [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
!= ( X74 @ sh @ one ) ) )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X75: subst > term > subst > $o,X76: term] :
( ~ ! [X77: subst > term > term] :
( ~ ! [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) )
| ~ ! [X81: term] :
( ( ( X75 @ id @ X81 @ id )
!= $true )
| ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
= $true ) )
| ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X75 @ id @ ( lam @ X76 ) @ id )
= $true )
| ~ ! [X82: term] :
( ( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
= $true )
| ( ( X75 @ id @ X82 @ id )
!= $true ) ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X83: term] :
( ( sub @ X83 @ id )
= X83 )
=> ( ! [X84: term,X85: subst > term > subst > $o] :
( ~ ! [X86: term] :
( ( ( X85 @ id @ X86 @ id )
!= $true )
| ( ( X85 @ id @ ( sub @ X84 @ ( push @ X86 @ id ) ) @ id )
= $true ) )
| ~ ! [X87: subst > term > term] :
( ~ ! [X88: term] :
( ( ( X85 @ id @ X88 @ id )
!= $true )
| ( $true
= ( X85 @ id @ ( X87 @ id @ X88 ) @ id ) ) )
| ( ( X85 @ id @ ( lam @ ( X87 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X89: subst,X90: term,X91: subst] :
( ( X87 @ ( comp @ X91 @ X89 ) @ ( sub @ X90 @ X89 ) )
= ( sub @ ( X87 @ X91 @ X90 ) @ X89 ) ) )
| ( ( X85 @ id @ ( lam @ ( sub @ X84 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X92: term,X93: subst > term > subst > $o] :
( ~ ! [X94: subst > term > term] :
( ( ( X93 @ id @ ( lam @ ( X94 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X95: subst,X96: term,X97: subst] :
( ( X94 @ ( comp @ X95 @ X97 ) @ ( sub @ X96 @ X97 ) )
= ( sub @ ( X94 @ X95 @ X96 ) @ X97 ) )
| ~ ! [X98: term] :
( ( ( X93 @ id @ X98 @ id )
!= $true )
| ( ( X93 @ id @ ( X94 @ id @ X98 ) @ id )
= $true ) ) )
| ( $true
= ( X93 @ id @ ( lam @ X92 ) @ id ) )
| ~ ! [X99: term] :
( ( ( X93 @ id @ ( sub @ X92 @ ( push @ X99 @ id ) ) @ id )
= $true )
| ( ( X93 @ id @ X99 @ id )
!= $true ) ) ) ) ) )
& ( ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 )
<=> ( axvarcons = $true ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X103: term] :
~ ! [X104: subst > term > term] :
~ ! [X105: subst,X106: term] :
( ( sub @ X103 @ ( push @ X106 @ X105 ) )
= ( X104 @ X105 @ X106 ) )
=> ( ! [X107: term] :
( ( sub @ X107 @ id )
= X107 )
=> ! [X108: term,X109: subst > term > subst > $o,X110: term > $o] :
( ~ ! [X111: term] :
( ( ( X110 @ X111 )
!= $true )
| ( $true
= ( X110 @ ( sub @ X108 @ ( push @ X111 @ id ) ) ) ) )
| ( ( X110 @ ( lam @ X108 ) )
= $true )
| ~ ! [X112: term] :
( ( X109 @ id @ X112 @ id )
= ( X110 @ X112 ) )
| ~ ! [X113: subst > term > term] :
( ( ( X109 @ id @ ( lam @ ( X113 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X114: term] :
( ( ( X109 @ id @ X114 @ id )
!= $true )
| ( ( X109 @ id @ ( X113 @ id @ X114 ) @ id )
= $true ) )
| ~ ! [X115: subst,X116: term,X117: subst] :
( ( X113 @ ( comp @ X115 @ X117 ) @ ( sub @ X116 @ X117 ) )
= ( sub @ ( X113 @ X115 @ X116 ) @ X117 ) ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( ( ! [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
= X118 )
=> ( ! [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
= X121 )
=> ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ( ( X122 @ X124 )
!= $true )
| ( ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) )
= $true ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoasinduction_gthm = $true )
& ( ! [X127: subst > term > subst > $o,X128: term] :
( ~ ! [X129: term] :
( ( ( X127 @ id @ ( sub @ X128 @ ( push @ X129 @ id ) ) @ id )
= $true )
| ( ( X127 @ id @ X129 @ id )
!= $true ) )
| ~ ! [X130: subst > term > term] :
( ~ ! [X131: term,X132: subst,X133: subst] :
( ( X130 @ ( comp @ X132 @ X133 ) @ ( sub @ X131 @ X133 ) )
= ( sub @ ( X130 @ X132 @ X131 ) @ X133 ) )
| ~ ! [X134: term] :
( ( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
= $true )
| ( ( X127 @ id @ X134 @ id )
!= $true ) )
| ( ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X127 @ id @ ( lam @ ( sub @ X128 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ! [X135: term] :
~ ! [X136: subst > term > term] :
~ ! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ! [X139: term,X140: term > $o,X141: subst] :
~ ! [X142: term > $o] :
~ ! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( ! [X144: term] :
~ ! [X145: subst > term > term] :
( ( X145 @ sh @ one )
!= ( sub @ X144 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X146: subst > term > subst > $o,X147: term > $o,X148: term] :
( ~ ! [X149: term] :
( ( X147 @ X149 )
= ( X146 @ id @ X149 @ id ) )
| ( ( X147 @ ( lam @ X148 ) )
= $true )
| ~ ! [X150: subst,X151: subst,X152: subst,X153: term] :
( ( ( X146 @ X151 @ X153 @ ( comp @ X150 @ X152 ) )
= $true )
| ( ( X146 @ ( comp @ X151 @ X150 ) @ ( sub @ X153 @ X150 ) @ X152 )
!= $true ) )
| ~ ! [X154: subst,X155: subst,X156: subst,X157: term] :
( ( ( X146 @ ( comp @ X155 @ X154 ) @ ( sub @ X157 @ X154 ) @ X156 )
= $true )
| ( ( X146 @ X155 @ X157 @ ( comp @ X154 @ X156 ) )
!= $true ) )
| ~ ! [X158: term] :
( ( ( X147 @ ( sub @ X148 @ ( push @ X158 @ id ) ) )
= $true )
| ( ( X147 @ X158 )
!= $true ) )
| ~ ! [X159: subst > term > term] :
( ~ ! [X160: term] :
( ( ( X146 @ id @ ( X159 @ id @ X160 ) @ id )
= $true )
| ( ( X146 @ id @ X160 @ id )
!= $true ) )
| ~ ! [X161: term,X162: subst,X163: subst] :
( ( X159 @ ( comp @ X163 @ X162 ) @ ( sub @ X161 @ X162 ) )
= ( sub @ ( X159 @ X163 @ X161 ) @ X162 ) )
| ( ( X146 @ id @ ( lam @ ( X159 @ sh @ one ) ) @ id )
= $true ) ) ) )
& ( pushprop_lem1_gthm = $true )
& ( axvarshift = $true )
& ( pushprop_lem3v2 = $true )
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
<=> ( termmset = $true ) )
& ( ! [X165: subst > term > subst > $o] :
~ ! [X166: term > $o] :
~ ! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X168: term,X169: term,X170: subst > term > subst > $o] :
( ( ( X170 @ id @ X169 @ id )
!= $true )
| ~ ! [X171: term,X172: term] :
( ( ( X170 @ id @ ( ap @ ( sub @ X171 @ id ) @ X172 ) @ id )
= $true )
| ( ( X170 @ id @ X172 @ id )
!= $true )
| ( ( X170 @ id @ X171 @ id )
!= $true ) )
| ~ ! [X173: subst,X174: subst,X175: term,X176: subst] :
( ( ( X170 @ X176 @ X175 @ ( comp @ X173 @ X174 ) )
= $true )
| ( ( X170 @ ( comp @ X176 @ X173 ) @ ( sub @ X175 @ X173 ) @ X174 )
!= $true ) )
| ( ( X170 @ id @ ( ap @ X168 @ X169 ) @ id )
= $true )
| ( ( X170 @ id @ X168 @ id )
!= $true )
| ~ ! [X177: subst,X178: subst,X179: term,X180: subst] :
( ( ( X170 @ X178 @ X179 @ ( comp @ X177 @ X180 ) )
!= $true )
| ( ( X170 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) @ X180 )
= $true ) ) ) )
& ! [X181: d_subst,X182: d_subst] :
( ( ( d2subst @ X182 )
= ( d2subst @ X181 ) )
=> ( X181 = X182 ) )
& ( substmonoid_gthm = $true )
& ( pushprop = $true )
& ( axclos = $true )
& ( ! [X183: term,X184: subst > term > subst > $o] :
( ( ( X184 @ id @ X183 @ id )
= $true )
| ~ ! [X185: term,X186: term] :
( ( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
= $true )
| ( ( X184 @ id @ X186 @ id )
!= $true )
| ( ( X184 @ id @ X185 @ id )
!= $true ) )
| ~ ! [X187: subst > term > term] :
( ( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X188: term] :
( ( ( X184 @ id @ X188 @ id )
!= $true )
| ( ( X184 @ id @ ( X187 @ id @ X188 ) @ id )
= $true ) )
| ~ ! [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) )
| ( X193 = X194 ) )
<=> ( hoasapinj1 = $true ) )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& ( pushprop_lem1v2_gthm = $true )
& ( axabs = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ! [X198: subst > term > subst > $o] :
~ ! [X199: term > $o] :
~ ! [X200: term] :
( ( X198 @ id @ X200 @ id )
= ( X199 @ X200 ) )
=> ( ! [X201: term > $o,X202: term] :
( ( ( X201 @ X202 )
= $true )
| ~ ! [X203: term] :
( ( ( X201 @ X203 )
= $true )
| ( ( var @ X203 )
!= $true ) )
| ~ ! [X204: term,X205: term] :
( ( ( X201 @ X204 )
!= $true )
| ( ( X201 @ X205 )
!= $true )
| ( ( X201 @ ( ap @ X205 @ X204 ) )
= $true ) )
| ~ ! [X206: term] :
( ( ( X201 @ ( lam @ X206 ) )
= $true )
| ~ ! [X207: term] :
( ( ( X201 @ X207 )
!= $true )
| ( ( X201 @ ( sub @ X206 @ ( push @ X207 @ id ) ) )
= $true ) ) ) )
=> ( ! [X208: term] :
( ( sub @ X208 @ id )
= X208 )
=> ( ! [X209: subst > term > subst > $o,X210: term > $o,X211: term] :
( ( ( X210 @ ( lam @ X211 ) )
= $true )
| ~ ! [X212: term] :
( ( ( X210 @ X212 )
!= $true )
| ( ( X210 @ ( sub @ X211 @ ( push @ X212 @ id ) ) )
= $true ) )
| ~ ! [X213: subst > term > term] :
( ( ( X209 @ id @ ( lam @ ( X213 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X214: term] :
( ( ( X209 @ id @ ( X213 @ id @ X214 ) @ id )
= $true )
| ( ( X209 @ id @ X214 @ id )
!= $true ) )
| ~ ! [X215: subst,X216: subst,X217: term] :
( ( sub @ ( X213 @ X216 @ X217 ) @ X215 )
= ( X213 @ ( comp @ X216 @ X215 ) @ ( sub @ X217 @ X215 ) ) ) )
| ~ ! [X218: term] :
( ( X209 @ id @ X218 @ id )
= ( X210 @ X218 ) ) )
=> ! [X219: subst > term > subst > $o,X220: term] :
( ~ ! [X221: term,X222: term] :
( ( ( X219 @ id @ X222 @ id )
!= $true )
| ( ( X219 @ id @ X221 @ id )
!= $true )
| ( ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id )
= $true ) )
| ( ( X219 @ id @ X220 @ id )
= $true )
| ~ ! [X223: subst > term > term] :
( ~ ! [X224: term] :
( ( ( X219 @ id @ X224 @ id )
!= $true )
| ( ( X219 @ id @ ( X223 @ id @ X224 ) @ id )
= $true ) )
| ( $true
= ( X219 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id ) )
| ~ ! [X225: subst,X226: term,X227: subst] :
( ( X223 @ ( comp @ X227 @ X225 ) @ ( sub @ X226 @ X225 ) )
= ( sub @ ( X223 @ X227 @ X226 ) @ X225 ) ) ) ) ) ) ) ) )
& ( termmset_gthm = $true )
& ( axmap = $true )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
= X228 )
=> ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ~ ! [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
= ( X232 @ X234 ) )
| ( ( X232 @ one )
= $true )
| ( ( X230 @ X233 )
!= $true ) ) ) )
& ( hoasapnotvar = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( ! [X238: subst > term > subst > $o,X239: term,X240: term > $o] :
( ~ ! [X241: term] :
( ( ( X240 @ X241 )
!= $true )
| ( ( X240 @ ( sub @ X239 @ ( push @ X241 @ id ) ) )
= $true ) )
| ~ ! [X242: subst > term > term] :
( ~ ! [X243: subst,X244: term,X245: subst] :
( ( sub @ ( X242 @ X245 @ X244 ) @ X243 )
= ( X242 @ ( comp @ X245 @ X243 ) @ ( sub @ X244 @ X243 ) ) )
| ( ( X238 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X246: term] :
( ( ( X238 @ id @ ( X242 @ id @ X246 ) @ id )
= $true )
| ( ( X238 @ id @ X246 @ id )
!= $true ) ) )
| ( ( X240 @ ( lam @ X239 ) )
= $true )
| ~ ! [X247: term] :
( ( X238 @ id @ X247 @ id )
= ( X240 @ X247 ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ! [X248: d_term,X249: d_term] :
( ( ( d2term @ X248 )
= ( d2term @ X249 ) )
=> ( X248 = X249 ) )
& ( ( hoaslamnotap = $true )
<=> ! [X250: term,X251: subst > term > term,X252: term] :
( ( ( lam @ ( X251 @ sh @ one ) )
!= ( ap @ ( sub @ X250 @ id ) @ X252 ) )
| ~ ! [X253: subst,X254: subst,X255: term] :
( ( sub @ ( X251 @ X253 @ X255 ) @ X254 )
= ( X251 @ ( comp @ X253 @ X254 ) @ ( sub @ X255 @ X254 ) ) ) ) )
& ( ( ! [X256: term,X257: term,X258: term,X259: term] :
( ( X256 = X257 )
| ( ( ap @ X259 @ X256 )
!= ( ap @ X258 @ X257 ) ) )
=> ! [X260: term,X261: term,X262: term,X263: term] :
( ( X260 = X261 )
| ( ( ap @ ( sub @ X262 @ id ) @ X260 )
!= ( ap @ ( sub @ X263 @ id ) @ X261 ) ) ) )
<=> ( hoasapinj2_lthm = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X264: term,X265: term,X266: term,X267: term] :
( ( X264 = X265 )
| ( ( ap @ X264 @ X266 )
!= ( ap @ X265 @ X267 ) ) )
<=> ( apinj1 = $true ) )
& ! [X268: d_term] : ( d_one = X268 )
& ( induction2_gthm = $true )
& ( ! [X269: subst,X270: term > $o,X271: term] :
~ ! [X272: term > $o] :
~ ! [X273: term] :
( ( X272 @ X273 )
= ( X270 @ ( sub @ X273 @ ( push @ X271 @ X269 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ( pushprop_lthm = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ! [X274: d_subst] : ( d_id = X274 )
& ( ulamvar1 != $true )
& ( ( termmset_lthm = $true )
<=> ( ! [X275: term] :
( ( sub @ X275 @ id )
= X275 )
=> ! [X276: term] :
( ( sub @ X276 @ id )
= X276 ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ! [X277: subst > term > subst > $o,X278: term] :
( ( ( X277 @ id @ X278 @ id )
= $true )
| ~ ! [X279: term] :
( ( ( X277 @ id @ X279 @ id )
= $true )
| ( ( var @ ( sub @ X279 @ id ) )
!= $true ) )
| ~ ! [X280: term,X281: subst,X282: subst,X283: subst] :
( ( ( X277 @ X282 @ X280 @ ( comp @ X281 @ X283 ) )
= $true )
| ( ( X277 @ ( comp @ X282 @ X281 ) @ ( sub @ X280 @ X281 ) @ X283 )
!= $true ) )
| ~ ! [X284: subst,X285: term,X286: subst,X287: subst] :
( ( ( X277 @ X286 @ X285 @ ( comp @ X287 @ X284 ) )
!= $true )
| ( ( X277 @ ( comp @ X286 @ X287 ) @ ( sub @ X285 @ X287 ) @ X284 )
= $true ) )
| ~ ! [X288: subst > term > term] :
( ~ ! [X289: term,X290: subst,X291: subst] :
( ( sub @ ( X288 @ X290 @ X289 ) @ X291 )
= ( X288 @ ( comp @ X290 @ X291 ) @ ( sub @ X289 @ X291 ) ) )
| ~ ! [X292: term] :
( ( ( X277 @ id @ X292 @ id )
!= $true )
| ( $true
= ( X277 @ id @ ( X288 @ id @ X292 ) @ id ) ) )
| ( ( X277 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X293: term,X294: term] :
( ( ( X277 @ id @ X294 @ id )
!= $true )
| ( ( X277 @ id @ ( ap @ ( sub @ X294 @ id ) @ X293 ) @ id )
= $true )
| ( ( X277 @ id @ X293 @ id )
!= $true ) ) )
<=> ( hoasinduction = $true ) )
& ( ( laminj = $true )
<=> ! [X295: term,X296: term] :
( ( X295 = X296 )
| ( ( lam @ X296 )
!= ( lam @ X295 ) ) ) )
& ( ! [X297: term,X298: term > $o] :
( ~ ! [X299: term] :
( ( ( var @ X299 )
!= $true )
| ( ( X298 @ X299 )
= $true ) )
| ~ ! [X300: term,X301: term] :
( ( ( X298 @ ( ap @ X300 @ X301 ) )
= $true )
| ( ( X298 @ X301 )
!= $true )
| ( ( X298 @ X300 )
!= $true ) )
| ~ ! [X302: term] :
( ~ ! [X303: term] :
( ( ( X298 @ X303 )
!= $true )
| ( ( X298 @ ( sub @ X302 @ ( push @ X303 @ id ) ) )
= $true ) )
| ( ( X298 @ ( lam @ X302 ) )
= $true ) )
| ( ( X298 @ X297 )
= $true ) )
<=> ( induction2 = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj2_gthm = $true )
& ( ( ! [X304: term] :
( ( sub @ X304 @ id )
= X304 )
=> ( ! [X305: subst,X306: term > $o,X307: term] :
( ~ ! [X308: term,X309: term] :
( ( ( X306 @ X308 )
!= $true )
| ( ( X306 @ X309 )
!= $true )
| ( ( X306 @ ( ap @ X308 @ X309 ) )
= $true ) )
| ~ ! [X310: term] :
( ~ ! [X311: term] :
( ( ( X306 @ X311 )
!= $true )
| ( ( X306 @ ( sub @ X310 @ ( push @ X311 @ id ) ) )
= $true ) )
| ( ( X306 @ ( lam @ X310 ) )
= $true ) )
| ( ( X306 @ ( sub @ X307 @ X305 ) )
= $true )
| ~ ! [X312: term] :
( ( ( var @ X312 )
!= $true )
| ( ( X306 @ ( sub @ X312 @ X305 ) )
= $true ) ) )
=> ! [X313: term,X314: term > $o] :
( ~ ! [X315: term] :
( ( ( var @ X315 )
!= $true )
| ( ( X314 @ X315 )
= $true ) )
| ( ( X314 @ X313 )
= $true )
| ~ ! [X316: term] :
( ~ ! [X317: term] :
( ( ( X314 @ ( sub @ X316 @ ( push @ X317 @ id ) ) )
= $true )
| ( ( X314 @ X317 )
!= $true ) )
| ( ( X314 @ ( lam @ X316 ) )
= $true ) )
| ~ ! [X318: term,X319: term] :
( ( ( X314 @ ( ap @ X318 @ X319 ) )
= $true )
| ( ( X314 @ X318 )
!= $true )
| ( ( X314 @ X319 )
!= $true ) ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( ulamvarind = $true )
& ( hoaslamnotvar_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X320: term,X321: term > $o,X322: subst] :
( ~ ! [X323: term] :
( ~ ! [X324: term] :
( ( ( X321 @ ( sub @ X323 @ ( push @ X324 @ id ) ) )
= $true )
| ( ( X321 @ X324 )
!= $true ) )
| ( ( X321 @ ( lam @ X323 ) )
= $true ) )
| ~ ! [X325: term,X326: term] :
( ( ( X321 @ ( ap @ X326 @ X325 ) )
= $true )
| ( $true
!= ( X321 @ X326 ) )
| ( ( X321 @ X325 )
!= $true ) )
| ( ( X321 @ ( sub @ X320 @ X322 ) )
= $true )
| ~ ! [X327: term] :
( ( ( var @ X327 )
!= $true )
| ( ( X321 @ ( sub @ X327 @ X322 ) )
= $true ) ) )
<=> ( induction2lem = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( ( apinj2 = $true )
<=> ! [X330: term,X331: term,X332: term,X333: term] :
( ( ( ap @ X332 @ X330 )
!= ( ap @ X331 @ X333 ) )
| ( X330 = X333 ) ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X334: term > $o,X335: term] :
( ~ ! [X336: term] :
( ( ( X334 @ X336 )
= $true )
| ( ( var @ X336 )
!= $true ) )
| ~ ! [X337: term,X338: term] :
( ( ( X334 @ X337 )
!= $true )
| ( ( X334 @ X338 )
!= $true )
| ( ( X334 @ ( ap @ X337 @ X338 ) )
= $true ) )
| ~ ! [X339: term] :
( ~ ! [X340: term] :
( ( ( X334 @ X340 )
!= $true )
| ( ( X334 @ ( sub @ X339 @ ( push @ X340 @ id ) ) )
= $true ) )
| ( ( X334 @ ( lam @ X339 ) )
= $true ) )
| ( ( X334 @ X335 )
= $true ) )
=> ( ! [X341: term,X342: subst > term > subst > $o,X343: term] :
( ~ ! [X344: subst,X345: subst,X346: term,X347: subst] :
( ( ( X342 @ ( comp @ X347 @ X344 ) @ ( sub @ X346 @ X344 ) @ X345 )
= $true )
| ( ( X342 @ X347 @ X346 @ ( comp @ X344 @ X345 ) )
!= $true ) )
| ( ( X342 @ id @ X341 @ id )
!= $true )
| ( ( X342 @ id @ ( ap @ X343 @ X341 ) @ id )
= $true )
| ( ( X342 @ id @ X343 @ id )
!= $true )
| ~ ! [X348: term,X349: term] :
( ( ( X342 @ id @ X349 @ id )
!= $true )
| ( ( X342 @ id @ ( ap @ ( sub @ X348 @ id ) @ X349 ) @ id )
= $true )
| ( ( X342 @ id @ X348 @ id )
!= $true ) )
| ~ ! [X350: subst,X351: term,X352: subst,X353: subst] :
( ( ( X342 @ X353 @ X351 @ ( comp @ X352 @ X350 ) )
= $true )
| ( ( X342 @ ( comp @ X353 @ X352 ) @ ( sub @ X351 @ X352 ) @ X350 )
!= $true ) ) )
=> ( ! [X354: subst > term > subst > $o,X355: term] :
( ~ ! [X356: subst,X357: subst,X358: term,X359: subst] :
( ( ( X354 @ ( comp @ X356 @ X357 ) @ ( sub @ X358 @ X357 ) @ X359 )
!= $true )
| ( ( X354 @ X356 @ X358 @ ( comp @ X357 @ X359 ) )
= $true ) )
| ~ ! [X360: subst > term > term] :
( ~ ! [X361: term] :
( ( ( X354 @ id @ X361 @ id )
!= $true )
| ( ( X354 @ id @ ( X360 @ id @ X361 ) @ id )
= $true ) )
| ( ( X354 @ id @ ( lam @ ( X360 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X362: subst,X363: subst,X364: term] :
( ( X360 @ ( comp @ X363 @ X362 ) @ ( sub @ X364 @ X362 ) )
= ( sub @ ( X360 @ X363 @ X364 ) @ X362 ) ) )
| ( ( X354 @ id @ ( lam @ X355 ) @ id )
= $true )
| ~ ! [X365: term] :
( ( ( X354 @ id @ X365 @ id )
!= $true )
| ( ( X354 @ id @ ( sub @ X355 @ ( push @ X365 @ id ) ) @ id )
= $true ) )
| ~ ! [X366: subst,X367: subst,X368: term,X369: subst] :
( ( ( X354 @ X366 @ X368 @ ( comp @ X367 @ X369 ) )
!= $true )
| ( ( X354 @ ( comp @ X366 @ X367 ) @ ( sub @ X368 @ X367 ) @ X369 )
= $true ) ) )
=> ! [X370: term,X371: subst > term > subst > $o] :
( ~ ! [X372: term] :
( ( ( X371 @ id @ X372 @ id )
= $true )
| ( ( var @ ( sub @ X372 @ id ) )
!= $true ) )
| ~ ! [X373: subst,X374: subst,X375: subst,X376: term] :
( ( ( X371 @ ( comp @ X374 @ X375 ) @ ( sub @ X376 @ X375 ) @ X373 )
= $true )
| ( ( X371 @ X374 @ X376 @ ( comp @ X375 @ X373 ) )
!= $true ) )
| ~ ! [X377: term,X378: term] :
( ( ( X371 @ id @ X377 @ id )
!= $true )
| ( ( X371 @ id @ X378 @ id )
!= $true )
| ( ( X371 @ id @ ( ap @ ( sub @ X378 @ id ) @ X377 ) @ id )
= $true ) )
| ~ ! [X379: subst > term > term] :
( ~ ! [X380: subst,X381: term,X382: subst] :
( ( sub @ ( X379 @ X382 @ X381 ) @ X380 )
= ( X379 @ ( comp @ X382 @ X380 ) @ ( sub @ X381 @ X380 ) ) )
| ( ( X371 @ id @ ( lam @ ( X379 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X383: term] :
( ( ( X371 @ id @ ( X379 @ id @ X383 ) @ id )
= $true )
| ( ( X371 @ id @ X383 @ id )
!= $true ) ) )
| ( ( X371 @ id @ X370 @ id )
= $true )
| ~ ! [X384: subst,X385: subst,X386: term,X387: subst] :
( ( ( X371 @ ( comp @ X385 @ X384 ) @ ( sub @ X386 @ X384 ) @ X387 )
!= $true )
| ( ( X371 @ X385 @ X386 @ ( comp @ X384 @ X387 ) )
= $true ) ) ) ) ) ) )
& ( apnotvar = $true )
& ( ( axvarid = $true )
<=> ! [X388: term] :
( ( sub @ X388 @ id )
= X388 ) )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X389: term,X390: subst > term > subst > $o] :
( ~ ! [X391: subst,X392: subst,X393: subst,X394: term] :
( ( ( X390 @ ( comp @ X391 @ X393 ) @ ( sub @ X394 @ X393 ) @ X392 )
= $true )
| ( ( X390 @ X391 @ X394 @ ( comp @ X393 @ X392 ) )
!= $true ) )
| ~ ! [X395: term] :
( ( ( X390 @ id @ ( sub @ X389 @ ( push @ X395 @ id ) ) @ id )
= $true )
| ( ( X390 @ id @ X395 @ id )
!= $true ) )
| ( ( X390 @ id @ ( lam @ X389 ) @ id )
= $true )
| ~ ! [X396: subst > term > term] :
( ~ ! [X397: subst,X398: subst,X399: term] :
( ( X396 @ ( comp @ X398 @ X397 ) @ ( sub @ X399 @ X397 ) )
= ( sub @ ( X396 @ X398 @ X399 ) @ X397 ) )
| ~ ! [X400: term] :
( ( ( X390 @ id @ X400 @ id )
!= $true )
| ( ( X390 @ id @ ( X396 @ id @ X400 ) @ id )
= $true ) )
| ( ( X390 @ id @ ( lam @ ( X396 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X401: subst,X402: subst,X403: term,X404: subst] :
( ( ( X390 @ ( comp @ X401 @ X402 ) @ ( sub @ X403 @ X402 ) @ X404 )
!= $true )
| ( ( X390 @ X401 @ X403 @ ( comp @ X402 @ X404 ) )
= $true ) ) ) )
& ( hoaslaminj_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X405: term] :
( ( sub @ X405 @ id )
= X405 )
=> ( ! [X406: term,X407: term,X408: term,X409: term] :
( ( ( ap @ X406 @ X408 )
!= ( ap @ X409 @ X407 ) )
| ( X406 = X409 ) )
=> ! [X410: term,X411: term,X412: term,X413: term] :
( ( X410 = X413 )
| ( ( ap @ ( sub @ X410 @ id ) @ X412 )
!= ( ap @ ( sub @ X413 @ id ) @ X411 ) ) ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( ! [X414: subst] :
( ( comp @ id @ X414 )
= X414 )
=> ( ! [X415: subst] :
( ( comp @ X415 @ id )
= X415 )
=> ( ! [X416: subst] :
( ( comp @ id @ X416 )
= X416 )
& ! [X417: subst] :
( ( comp @ X417 @ id )
= X417 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ! [X418: term] :
? [X419: d_term] :
( ( d2term @ X419 )
= X418 )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X420: term,X421: term,X422: term,X423: term] :
( ( X421 = X423 )
| ( ( ap @ ( sub @ X420 @ id ) @ X421 )
!= ( ap @ ( sub @ X422 @ id ) @ X423 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X424: subst > term > subst > $o,X425: term] :
( ~ ! [X426: term] :
( ( ( X424 @ id @ ( sub @ X425 @ ( push @ X426 @ id ) ) @ id )
= $true )
| ( ( X424 @ id @ X426 @ id )
!= $true ) )
| ( ( X424 @ id @ ( lam @ ( sub @ X425 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X427: subst > term > term] :
( ~ ! [X428: term,X429: subst,X430: subst] :
( ( X427 @ ( comp @ X430 @ X429 ) @ ( sub @ X428 @ X429 ) )
= ( sub @ ( X427 @ X430 @ X428 ) @ X429 ) )
| ~ ! [X431: term] :
( ( ( X424 @ id @ ( X427 @ id @ X431 ) @ id )
= $true )
| ( ( X424 @ id @ X431 @ id )
!= $true ) )
| ( ( X424 @ id @ ( lam @ ( X427 @ sh @ one ) ) @ id )
= $true ) ) ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X432: term] :
( ( sub @ X432 @ id )
= X432 )
=> ( ! [X433: term,X434: subst > term > subst > $o] :
( ~ ! [X435: term] :
( ( ( X434 @ id @ ( sub @ X433 @ ( push @ X435 @ id ) ) @ id )
= $true )
| ( ( X434 @ id @ X435 @ id )
!= $true ) )
| ~ ! [X436: subst > term > term] :
( ~ ! [X437: term] :
( ( ( X434 @ id @ X437 @ id )
!= $true )
| ( ( X434 @ id @ ( X436 @ id @ X437 ) @ id )
= $true ) )
| ( ( X434 @ id @ ( lam @ ( X436 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X438: term,X439: subst,X440: subst] :
( ( sub @ ( X436 @ X440 @ X438 ) @ X439 )
= ( X436 @ ( comp @ X440 @ X439 ) @ ( sub @ X438 @ X439 ) ) ) )
| ( ( X434 @ id @ ( lam @ ( sub @ X433 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X441: term,X442: subst > term > subst > $o] :
( ( ( X442 @ id @ ( lam @ X441 ) @ id )
= $true )
| ~ ! [X443: subst > term > term] :
( ( ( X442 @ id @ ( lam @ ( X443 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X444: term] :
( ( ( X442 @ id @ ( X443 @ id @ X444 ) @ id )
= $true )
| ( ( X442 @ id @ X444 @ id )
!= $true ) )
| ~ ! [X445: term,X446: subst,X447: subst] :
( ( X443 @ ( comp @ X446 @ X447 ) @ ( sub @ X445 @ X447 ) )
= ( sub @ ( X443 @ X446 @ X445 ) @ X447 ) ) )
| ~ ! [X448: term,X449: subst,X450: subst,X451: subst] :
( ( ( X442 @ X451 @ X448 @ ( comp @ X450 @ X449 ) )
= $true )
| ( ( X442 @ ( comp @ X451 @ X450 ) @ ( sub @ X448 @ X450 ) @ X449 )
!= $true ) )
| ~ ! [X452: term] :
( ( ( X442 @ id @ ( sub @ X441 @ ( push @ X452 @ id ) ) @ id )
= $true )
| ( ( X442 @ id @ X452 @ id )
!= $true ) )
| ~ ! [X453: subst,X454: subst,X455: subst,X456: term] :
( ( ( X442 @ ( comp @ X455 @ X454 ) @ ( sub @ X456 @ X454 ) @ X453 )
= $true )
| ( ( X442 @ X455 @ X456 @ ( comp @ X454 @ X453 ) )
!= $true ) ) ) ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ! [X457: subst > term > subst > $o,X458: term] :
( ~ ! [X459: term] :
( ( ( X457 @ id @ ( sub @ X458 @ ( push @ X459 @ id ) ) @ id )
= $true )
| ( ( X457 @ id @ X459 @ id )
!= $true ) )
| ( ( X457 @ id @ ( lam @ ( sub @ X458 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X460: subst > term > term,X461: term] :
( ~ ! [X462: subst,X463: term,X464: subst] :
( ( sub @ X461 @ ( push @ ( sub @ X463 @ X462 ) @ ( comp @ X464 @ X462 ) ) )
= ( X460 @ ( comp @ X464 @ X462 ) @ ( sub @ X463 @ X462 ) ) )
| ( ( X457 @ id @ ( lam @ ( X460 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X465: term] :
( ( ( X457 @ id @ X465 @ id )
!= $true )
| ( ( X457 @ id @ ( X460 @ id @ X465 ) @ id )
= $true ) )
| ~ ! [X466: subst,X467: term,X468: subst] :
( ( sub @ ( sub @ X461 @ ( push @ X467 @ X466 ) ) @ X468 )
= ( sub @ ( X460 @ X466 @ X467 ) @ X468 ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoaslaminj = $true )
<=> ! [X469: term,X470: subst > term > term,X471: subst,X472: subst > term > term] :
( ~ ! [X473: subst,X474: term,X475: subst] :
( ( X472 @ ( comp @ X475 @ X473 ) @ ( sub @ X474 @ X473 ) )
= ( sub @ ( X472 @ X475 @ X474 ) @ X473 ) )
| ( ( lam @ ( X470 @ sh @ one ) )
!= ( lam @ ( X472 @ sh @ one ) ) )
| ~ ! [X476: term,X477: subst,X478: subst] :
( ( sub @ ( X470 @ X478 @ X476 ) @ X477 )
= ( X470 @ ( comp @ X478 @ X477 ) @ ( sub @ X476 @ X477 ) ) )
| ( ( X470 @ X471 @ X469 )
= ( X472 @ X471 @ X469 ) ) ) )
& ( ( ! [X479: subst,X480: term] :
( ( sub @ one @ ( push @ X480 @ X479 ) )
= X480 )
=> ( ! [X481: subst,X482: term] :
( ( comp @ sh @ ( push @ X482 @ X481 ) )
= X481 )
=> ( ! [X483: subst] :
( ( comp @ X483 @ id )
= X483 )
=> ( ! [X484: term > $o,X485: term] :
( ( ( X484 @ X485 )
= $true )
| ~ ! [X486: term] :
( ( ( X484 @ ( lam @ X486 ) )
= $true )
| ( ( X484 @ X486 )
!= $true ) )
| ~ ! [X487: term] :
( ( ( var @ X487 )
!= $true )
| ( ( X484 @ X487 )
= $true ) )
| ~ ! [X488: term,X489: term] :
( ( ( X484 @ X488 )
!= $true )
| ( ( X484 @ X489 )
!= $true )
| ( ( X484 @ ( ap @ X489 @ X488 ) )
= $true ) ) )
=> ! [X490: term,X491: subst,X492: term > $o] :
( ( ( X492 @ ( sub @ X490 @ X491 ) )
= $true )
| ~ ! [X493: term] :
( ( ( X492 @ ( lam @ X493 ) )
= $true )
| ~ ! [X494: term] :
( ( ( X492 @ X494 )
!= $true )
| ( $true
= ( X492 @ ( sub @ X493 @ ( push @ X494 @ id ) ) ) ) ) )
| ~ ! [X495: term] :
( ( ( X492 @ ( sub @ X495 @ X491 ) )
= $true )
| ( ( var @ X495 )
!= $true ) )
| ~ ! [X496: term,X497: term] :
( ( ( X492 @ X496 )
!= $true )
| ( ( X492 @ ( ap @ X496 @ X497 ) )
= $true )
| ( ( X492 @ X497 )
!= $true ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( one
= ( d2term @ d_one ) )
& ( axapp = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( axshiftcons = $true )
<=> ! [X498: subst,X499: term] :
( ( comp @ sh @ ( push @ X499 @ X498 ) )
= X498 ) )
& ( 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 ) ) ) ) ) ) )
& ( ! [X505: term > $o,X506: term] :
( ~ ! [X507: term,X508: term] :
( ( ( X505 @ X507 )
!= $true )
| ( ( X505 @ ( ap @ X507 @ X508 ) )
= $true )
| ( ( X505 @ X508 )
!= $true ) )
| ~ ! [X509: term] :
( ( ( var @ X509 )
!= $true )
| ( ( X505 @ X509 )
= $true ) )
| ~ ! [X510: term] :
( ( ( X505 @ ( lam @ X510 ) )
= $true )
| ( ( X505 @ X510 )
!= $true ) )
| ( ( X505 @ X506 )
= $true ) )
<=> ( induction = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasapnotvar_gthm = $true )
& ( ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
=> ! [X512: term > $o,X513: subst > term > subst > $o,X514: term] :
( ~ ! [X515: term] :
( ( X512 @ X515 )
= ( X513 @ id @ X515 @ id ) )
| ( ( X512 @ ( lam @ X514 ) )
= $true )
| ~ ! [X516: subst,X517: subst,X518: subst,X519: term] :
( ( ( X513 @ X516 @ X519 @ ( comp @ X517 @ X518 ) )
!= $true )
| ( ( X513 @ ( comp @ X516 @ X517 ) @ ( sub @ X519 @ X517 ) @ X518 )
= $true ) )
| ~ ! [X520: subst > term > term] :
( ( ( X513 @ id @ ( lam @ ( X520 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X521: term] :
( ( ( X513 @ id @ X521 @ id )
!= $true )
| ( ( X513 @ id @ ( X520 @ id @ X521 ) @ id )
= $true ) )
| ~ ! [X522: subst,X523: term,X524: subst] :
( ( sub @ ( X520 @ X524 @ X523 ) @ X522 )
= ( X520 @ ( comp @ X524 @ X522 ) @ ( sub @ X523 @ X522 ) ) ) )
| ~ ! [X525: subst,X526: subst,X527: subst,X528: term] :
( ( ( X513 @ X526 @ X528 @ ( comp @ X525 @ X527 ) )
= $true )
| ( ( X513 @ ( comp @ X526 @ X525 ) @ ( sub @ X528 @ X525 ) @ X527 )
!= $true ) )
| ~ ! [X529: term] :
( ( ( X512 @ X529 )
!= $true )
| ( ( X512 @ ( sub @ X514 @ ( push @ X529 @ id ) ) )
= $true ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X530: subst,X531: term,X532: term > $o,X533: term > $o] :
( ( ( X533 @ one )
= $true )
| ( ( X532 @ X531 )
!= $true )
| ~ ! [X534: term] :
( ( X533 @ X534 )
= ( X532 @ ( sub @ X534 @ ( push @ X531 @ X530 ) ) ) ) ) )
& ( ! [X535: term > $o,X536: term,X537: term,X538: subst > term > subst > $o] :
( ( ( X535 @ X536 )
!= $true )
| ~ ! [X539: term] :
( ( X535 @ X539 )
= ( X538 @ id @ X539 @ id ) )
| ~ ! [X540: term,X541: term] :
( ( ( X538 @ id @ X540 @ id )
!= $true )
| ( $true
!= ( X538 @ id @ X541 @ id ) )
| ( ( X538 @ id @ ( ap @ ( sub @ X541 @ id ) @ X540 ) @ id )
= $true ) )
| ~ ! [X542: subst,X543: subst,X544: term,X545: subst] :
( ( ( X538 @ ( comp @ X543 @ X542 ) @ ( sub @ X544 @ X542 ) @ X545 )
!= $true )
| ( ( X538 @ X543 @ X544 @ ( comp @ X542 @ X545 ) )
= $true ) )
| ( ( X535 @ ( ap @ X537 @ X536 ) )
= $true )
| ( ( X535 @ X537 )
!= $true )
| ~ ! [X546: subst,X547: subst,X548: term,X549: subst] :
( ( ( X538 @ ( comp @ X546 @ X547 ) @ ( sub @ X548 @ X547 ) @ X549 )
= $true )
| ( ( X538 @ X546 @ X548 @ ( comp @ X547 @ X549 ) )
!= $true ) ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( hoasinduction_lem1 = $true ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
( ( ( ! [X0: subst] :
( ( comp @ X0 @ id )
= X0 )
& ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 ) )
<=> substmonoid )
& ~ lamnotap
& axassoc
& ( ! [X2: subst > term > subst > $o] :
~ ! [X3: term > $o] :
~ ! [X4: term] :
( ( X2 @ id @ X4 @ id )
= ( X3 @ X4 ) )
<=> hoasinduction_lem0_lthm )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoaslaminj_lthm
<=> ( ! [X5: term,X6: subst] :
( ( sub @ one @ ( push @ X5 @ X6 ) )
= X5 )
=> ( ! [X7: term,X8: subst] :
( ( comp @ sh @ ( push @ X7 @ X8 ) )
= X8 )
=> ( ! [X9: term,X10: term] :
( ( ( lam @ X10 )
!= ( lam @ X9 ) )
| ( X9 = X10 ) )
=> ! [X11: subst > term > term,X12: subst > term > term,X13: term,X14: subst] :
( ( ( lam @ ( X11 @ sh @ one ) )
!= ( lam @ ( X12 @ sh @ one ) ) )
| ( ( X11 @ X14 @ X13 )
= ( X12 @ X14 @ X13 ) )
| ~ ! [X15: term,X16: subst,X17: subst] :
( ( sub @ ( X11 @ X17 @ X15 ) @ X16 )
= ( X11 @ ( comp @ X17 @ X16 ) @ ( sub @ X15 @ X16 ) ) )
| ~ ! [X18: subst,X19: term,X20: subst] :
( ( sub @ ( X12 @ X20 @ X19 ) @ X18 )
= ( X12 @ ( comp @ X20 @ X18 ) @ ( sub @ X19 @ X18 ) ) ) ) ) ) ) )
& pushprop_gthm
& induction2lem_gthm
& ( ( ! [X21: subst > term > subst > $o] :
~ ! [X22: term > $o] :
~ ! [X23: term] :
( ( X21 @ id @ X23 @ id )
= ( X22 @ X23 ) )
=> ( ! [X24: term,X25: term > $o] :
( ~ ! [X26: term,X27: term] :
( ( X25 @ ( ap @ X27 @ X26 ) )
| ~ ( X25 @ X27 )
| ~ ( X25 @ X26 ) )
| ~ ! [X28: term] :
( ( X25 @ ( lam @ X28 ) )
| ~ ! [X29: term] :
( ( X25 @ ( sub @ X28 @ ( push @ X29 @ id ) ) )
| ~ ( X25 @ X29 ) ) )
| ~ ! [X30: term] :
( ~ ( var @ X30 )
| ( X25 @ X30 ) )
| ( X25 @ X24 ) )
=> ( ! [X31: term] :
( ( sub @ X31 @ id )
= X31 )
=> ( ! [X32: term > $o,X33: subst > term > subst > $o,X34: term] :
( ~ ! [X35: term] :
( ( X32 @ ( sub @ X34 @ ( push @ X35 @ id ) ) )
| ~ ( X32 @ X35 ) )
| ~ ! [X36: subst > term > term] :
( ( X33 @ id @ ( lam @ ( X36 @ sh @ one ) ) @ id )
| ~ ! [X37: term] :
( ~ ( X33 @ id @ X37 @ id )
| ( X33 @ id @ ( X36 @ id @ X37 ) @ id ) )
| ~ ! [X38: subst,X39: term,X40: subst] :
( ( X36 @ ( comp @ X38 @ X40 ) @ ( sub @ X39 @ X40 ) )
= ( sub @ ( X36 @ X38 @ X39 ) @ X40 ) ) )
| ~ ! [X41: term] :
( ( X33 @ id @ X41 @ id )
= ( X32 @ X41 ) )
| ( X32 @ ( lam @ X34 ) ) )
=> ! [X42: subst > term > subst > $o,X43: term] :
( ~ ! [X44: subst > term > term] :
( ~ ! [X45: subst,X46: subst,X47: term] :
( ( sub @ ( X44 @ X45 @ X47 ) @ X46 )
= ( X44 @ ( comp @ X45 @ X46 ) @ ( sub @ X47 @ X46 ) ) )
| ~ ! [X48: term] :
( ( X42 @ id @ ( X44 @ id @ X48 ) @ id )
| ~ ( X42 @ id @ X48 @ id ) )
| ( X42 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id ) )
| ( X42 @ id @ X43 @ id )
| ~ ! [X49: term] :
( ~ ( var @ ( sub @ X49 @ id ) )
| ( X42 @ id @ X49 @ id ) )
| ~ ! [X50: subst,X51: term,X52: subst,X53: subst] :
( ( X42 @ ( comp @ X53 @ X52 ) @ ( sub @ X51 @ X52 ) @ X50 )
| ~ ( X42 @ X53 @ X51 @ ( comp @ X52 @ X50 ) ) )
| ~ ! [X54: term,X55: term] :
( ( X42 @ id @ ( ap @ ( sub @ X54 @ id ) @ X55 ) @ id )
| ~ ( X42 @ id @ X54 @ id )
| ~ ( X42 @ id @ X55 @ id ) )
| ~ ! [X56: subst,X57: subst,X58: subst,X59: term] :
( ~ ( X42 @ ( comp @ X58 @ X57 ) @ ( sub @ X59 @ X57 ) @ X56 )
| ( X42 @ X58 @ X59 @ ( comp @ X57 @ X56 ) ) ) ) ) ) ) )
<=> hoasinduction_lthm_3 )
& hoasinduction_lem2_lthm
& hoasinduction_lem1_gthm
& ulamvarsh
& ( pushprop_lem1
<=> ! [X60: term > $o,X61: term,X62: term > $o,X63: term,X64: subst] :
( ~ ( X60 @ X61 )
| ( X62 @ ( sub @ X61 @ ( push @ X63 @ X64 ) ) ) ) )
& hoaslamnotap_gthm
& ( hoasinduction_lem3v2_f_lthm
<=> ! [X65: term] :
~ ! [X66: subst > term > term] :
~ ! [X67: term,X68: subst] :
( ( X66 @ X68 @ X67 )
= ( sub @ X65 @ ( push @ X67 @ X68 ) ) ) )
& lamnotvar
& ( shinj
<=> ! [X69: term,X70: term] :
( ( X69 = X70 )
| ( ( sub @ X70 @ sh )
!= ( sub @ X69 @ sh ) ) ) )
& ( ! [X71: subst] :
( ( comp @ X71 @ id )
= X71 )
<=> axidr )
& hoasinduction_lem1v2_gthm
& ( ( d2subst @ d_id )
= id )
& ( axidl
<=> ! [X72: subst] :
( ( comp @ id @ X72 )
= X72 ) )
& ( ! [X73: term] :
~ ! [X74: subst > term > term] :
( ( sub @ X73 @ ( push @ one @ sh ) )
!= ( X74 @ sh @ one ) )
<=> hoasinduction_lem3b )
& ( hoasinduction_lem3a
<=> ! [X75: subst > term > subst > $o,X76: term] :
( ~ ! [X77: subst > term > term] :
( ~ ! [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) )
| ~ ! [X81: term] :
( ~ ( X75 @ id @ X81 @ id )
| ( X75 @ id @ ( X77 @ id @ X81 ) @ id ) )
| ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id ) )
| ( X75 @ id @ ( lam @ X76 ) @ id )
| ~ ! [X82: term] :
( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
| ~ ( X75 @ id @ X82 @ id ) ) ) )
& hoasinduction_lem1_lthm
& ( hoasinduction_lem3a_lthm
<=> ( ! [X83: term] :
( ( sub @ X83 @ id )
= X83 )
=> ( ! [X84: term,X85: subst > term > subst > $o] :
( ~ ! [X86: term] :
( ~ ( X85 @ id @ X86 @ id )
| ( X85 @ id @ ( sub @ X84 @ ( push @ X86 @ id ) ) @ id ) )
| ~ ! [X87: subst > term > term] :
( ~ ! [X88: term] :
( ~ ( X85 @ id @ X88 @ id )
| ( X85 @ id @ ( X87 @ id @ X88 ) @ id ) )
| ( X85 @ id @ ( lam @ ( X87 @ sh @ one ) ) @ id )
| ~ ! [X89: subst,X90: term,X91: subst] :
( ( X87 @ ( comp @ X91 @ X89 ) @ ( sub @ X90 @ X89 ) )
= ( sub @ ( X87 @ X91 @ X90 ) @ X89 ) ) )
| ( X85 @ id @ ( lam @ ( sub @ X84 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X92: term,X93: subst > term > subst > $o] :
( ~ ! [X94: subst > term > term] :
( ( X93 @ id @ ( lam @ ( X94 @ sh @ one ) ) @ id )
| ~ ! [X95: subst,X96: term,X97: subst] :
( ( X94 @ ( comp @ X95 @ X97 ) @ ( sub @ X96 @ X97 ) )
= ( sub @ ( X94 @ X95 @ X96 ) @ X97 ) )
| ~ ! [X98: term] :
( ~ ( X93 @ id @ X98 @ id )
| ( X93 @ id @ ( X94 @ id @ X98 ) @ id ) ) )
| ( X93 @ id @ ( lam @ X92 ) @ id )
| ~ ! [X99: term] :
( ( X93 @ id @ ( sub @ X92 @ ( push @ X99 @ id ) ) @ id )
| ~ ( X93 @ id @ X99 @ id ) ) ) ) ) )
& ( axvarcons
<=> ! [X100: term,X101: subst] :
( ( sub @ one @ ( push @ X100 @ X101 ) )
= X100 ) )
& ( axscons
<=> ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 ) )
& hoaslamnotvar_gthm
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X103: term] :
~ ! [X104: subst > term > term] :
~ ! [X105: subst,X106: term] :
( ( sub @ X103 @ ( push @ X106 @ X105 ) )
= ( X104 @ X105 @ X106 ) )
=> ( ! [X107: term] :
( ( sub @ X107 @ id )
= X107 )
=> ! [X108: term,X109: subst > term > subst > $o,X110: term > $o] :
( ~ ! [X111: term] :
( ~ ( X110 @ X111 )
| ( X110 @ ( sub @ X108 @ ( push @ X111 @ id ) ) ) )
| ( X110 @ ( lam @ X108 ) )
| ~ ! [X112: term] :
( ( X109 @ id @ X112 @ id )
= ( X110 @ X112 ) )
| ~ ! [X113: subst > term > term] :
( ( X109 @ id @ ( lam @ ( X113 @ sh @ one ) ) @ id )
| ~ ! [X114: term] :
( ~ ( X109 @ id @ X114 @ id )
| ( X109 @ id @ ( X113 @ id @ X114 ) @ id ) )
| ~ ! [X115: subst,X116: term,X117: subst] :
( ( X113 @ ( comp @ X115 @ X117 ) @ ( sub @ X116 @ X117 ) )
= ( sub @ ( X113 @ X115 @ X116 ) @ X117 ) ) ) ) ) )
<=> hoasinduction_lem3v2a_lthm )
& ( ( ! [X118: term,X119: subst] :
( ( sub @ one @ ( push @ X118 @ X119 ) )
= X118 )
=> ( ! [X120: term,X121: subst] :
( ( comp @ sh @ ( push @ X120 @ X121 ) )
= X121 )
=> ! [X122: term > $o,X123: subst,X124: term,X125: term > $o,X126: term] :
( ~ ( X122 @ X124 )
| ( X125 @ ( sub @ X124 @ ( push @ X126 @ X123 ) ) ) ) ) )
<=> pushprop_lem1_lthm )
& hoasinduction_gthm
& ( ! [X127: subst > term > subst > $o,X128: term] :
( ~ ! [X129: term] :
( ( X127 @ id @ ( sub @ X128 @ ( push @ X129 @ id ) ) @ id )
| ~ ( X127 @ id @ X129 @ id ) )
| ~ ! [X130: subst > term > term] :
( ~ ! [X131: term,X132: subst,X133: subst] :
( ( X130 @ ( comp @ X132 @ X133 ) @ ( sub @ X131 @ X133 ) )
= ( sub @ ( X130 @ X132 @ X131 ) @ X133 ) )
| ~ ! [X134: term] :
( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
| ~ ( X127 @ id @ X134 @ id ) )
| ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id ) )
| ( X127 @ id @ ( lam @ ( sub @ X128 @ ( push @ one @ sh ) ) ) @ id ) )
<=> hoasinduction_lem3aa_lthm )
& ( ! [X135: term] :
~ ! [X136: subst > term > term] :
~ ! [X137: subst,X138: term] :
( ( sub @ X135 @ ( push @ X138 @ X137 ) )
= ( X136 @ X137 @ X138 ) )
<=> hoasinduction_lem3v2_f )
& ( pushprop_lem0_lthm
<=> ! [X139: term,X140: term > $o,X141: subst] :
~ ! [X142: term > $o] :
~ ! [X143: term] :
( ( X140 @ ( sub @ X143 @ ( push @ X139 @ X141 ) ) )
= ( X142 @ X143 ) ) )
& ( hoasinduction_lem3b_lthm
<=> ! [X144: term] :
~ ! [X145: subst > term > term] :
( ( X145 @ sh @ one )
!= ( sub @ X144 @ ( push @ one @ sh ) ) ) )
& hoasinduction_lem3a_gthm
& ( hoasinduction_lem3v2
<=> ! [X146: subst > term > subst > $o,X147: term > $o,X148: term] :
( ~ ! [X149: term] :
( ( X147 @ X149 )
= ( X146 @ id @ X149 @ id ) )
| ( X147 @ ( lam @ X148 ) )
| ~ ! [X150: subst,X151: subst,X152: subst,X153: term] :
( ( X146 @ X151 @ X153 @ ( comp @ X150 @ X152 ) )
| ~ ( X146 @ ( comp @ X151 @ X150 ) @ ( sub @ X153 @ X150 ) @ X152 ) )
| ~ ! [X154: subst,X155: subst,X156: subst,X157: term] :
( ( X146 @ ( comp @ X155 @ X154 ) @ ( sub @ X157 @ X154 ) @ X156 )
| ~ ( X146 @ X155 @ X157 @ ( comp @ X154 @ X156 ) ) )
| ~ ! [X158: term] :
( ( X147 @ ( sub @ X148 @ ( push @ X158 @ id ) ) )
| ~ ( X147 @ X158 ) )
| ~ ! [X159: subst > term > term] :
( ~ ! [X160: term] :
( ( X146 @ id @ ( X159 @ id @ X160 ) @ id )
| ~ ( X146 @ id @ X160 @ id ) )
| ~ ! [X161: term,X162: subst,X163: subst] :
( ( X159 @ ( comp @ X163 @ X162 ) @ ( sub @ X161 @ X162 ) )
= ( sub @ ( X159 @ X163 @ X161 ) @ X162 ) )
| ( X146 @ id @ ( lam @ ( X159 @ sh @ one ) ) @ id ) ) ) )
& pushprop_lem1_gthm
& axvarshift
& pushprop_lem3v2
& ( ! [X164: term] :
( ( sub @ X164 @ id )
= X164 )
<=> termmset )
& ( ! [X165: subst > term > subst > $o] :
~ ! [X166: term > $o] :
~ ! [X167: term] :
( ( X165 @ id @ X167 @ id )
= ( X166 @ X167 ) )
<=> hoasinduction_lem0 )
& ( hoasinduction_lem2
<=> ! [X168: term,X169: term,X170: subst > term > subst > $o] :
( ~ ( X170 @ id @ X169 @ id )
| ~ ! [X171: term,X172: term] :
( ( X170 @ id @ ( ap @ ( sub @ X171 @ id ) @ X172 ) @ id )
| ~ ( X170 @ id @ X172 @ id )
| ~ ( X170 @ id @ X171 @ id ) )
| ~ ! [X173: subst,X174: subst,X175: term,X176: subst] :
( ( X170 @ X176 @ X175 @ ( comp @ X173 @ X174 ) )
| ~ ( X170 @ ( comp @ X176 @ X173 ) @ ( sub @ X175 @ X173 ) @ X174 ) )
| ( X170 @ id @ ( ap @ X168 @ X169 ) @ id )
| ~ ( X170 @ id @ X168 @ id )
| ~ ! [X177: subst,X178: subst,X179: term,X180: subst] :
( ~ ( X170 @ X178 @ X179 @ ( comp @ X177 @ X180 ) )
| ( X170 @ ( comp @ X178 @ X177 ) @ ( sub @ X179 @ X177 ) @ X180 ) ) ) )
& ! [X181: d_subst,X182: d_subst] :
( ( ( d2subst @ X182 )
= ( d2subst @ X181 ) )
=> ( X181 = X182 ) )
& substmonoid_gthm
& pushprop
& axclos
& ( ! [X183: term,X184: subst > term > subst > $o] :
( ( X184 @ id @ X183 @ id )
| ~ ! [X185: term,X186: term] :
( ( X184 @ id @ ( ap @ ( sub @ X185 @ id ) @ X186 ) @ id )
| ~ ( X184 @ id @ X186 @ id )
| ~ ( X184 @ id @ X185 @ id ) )
| ~ ! [X187: subst > term > term] :
( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
| ~ ! [X188: term] :
( ~ ( X184 @ id @ X188 @ id )
| ( X184 @ id @ ( X187 @ id @ X188 ) @ id ) )
| ~ ! [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) ) ) )
<=> hoasinduction_no_psi_cond )
& ( ! [X192: term,X193: term,X194: term,X195: term] :
( ( X193 = X194 )
| ( ( ap @ ( sub @ X193 @ id ) @ X195 )
!= ( ap @ ( sub @ X194 @ id ) @ X192 ) ) )
<=> hoasapinj1 )
& ! [X196: subst] :
? [X197: d_subst] :
( ( d2subst @ X197 )
= X196 )
& pushprop_lem1v2_gthm
& axabs
& ( hoasinduction_no_psi_cond_lthm
<=> ( ! [X198: subst > term > subst > $o] :
~ ! [X199: term > $o] :
~ ! [X200: term] :
( ( X198 @ id @ X200 @ id )
= ( X199 @ X200 ) )
=> ( ! [X201: term > $o,X202: term] :
( ( X201 @ X202 )
| ~ ! [X203: term] :
( ( X201 @ X203 )
| ~ ( var @ X203 ) )
| ~ ! [X204: term,X205: term] :
( ~ ( X201 @ X204 )
| ~ ( X201 @ X205 )
| ( X201 @ ( ap @ X205 @ X204 ) ) )
| ~ ! [X206: term] :
( ( X201 @ ( lam @ X206 ) )
| ~ ! [X207: term] :
( ~ ( X201 @ X207 )
| ( X201 @ ( sub @ X206 @ ( push @ X207 @ id ) ) ) ) ) )
=> ( ! [X208: term] :
( ( sub @ X208 @ id )
= X208 )
=> ( ! [X209: subst > term > subst > $o,X210: term > $o,X211: term] :
( ( X210 @ ( lam @ X211 ) )
| ~ ! [X212: term] :
( ~ ( X210 @ X212 )
| ( X210 @ ( sub @ X211 @ ( push @ X212 @ id ) ) ) )
| ~ ! [X213: subst > term > term] :
( ( X209 @ id @ ( lam @ ( X213 @ sh @ one ) ) @ id )
| ~ ! [X214: term] :
( ( X209 @ id @ ( X213 @ id @ X214 ) @ id )
| ~ ( X209 @ id @ X214 @ id ) )
| ~ ! [X215: subst,X216: subst,X217: term] :
( ( sub @ ( X213 @ X216 @ X217 ) @ X215 )
= ( X213 @ ( comp @ X216 @ X215 ) @ ( sub @ X217 @ X215 ) ) ) )
| ~ ! [X218: term] :
( ( X209 @ id @ X218 @ id )
= ( X210 @ X218 ) ) )
=> ! [X219: subst > term > subst > $o,X220: term] :
( ~ ! [X221: term,X222: term] :
( ~ ( X219 @ id @ X222 @ id )
| ~ ( X219 @ id @ X221 @ id )
| ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id ) )
| ( X219 @ id @ X220 @ id )
| ~ ! [X223: subst > term > term] :
( ~ ! [X224: term] :
( ~ ( X219 @ id @ X224 @ id )
| ( X219 @ id @ ( X223 @ id @ X224 ) @ id ) )
| ( X219 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id )
| ~ ! [X225: subst,X226: term,X227: subst] :
( ( X223 @ ( comp @ X227 @ X225 ) @ ( sub @ X226 @ X225 ) )
= ( sub @ ( X223 @ X227 @ X226 ) @ X225 ) ) ) ) ) ) ) ) )
& termmset_gthm
& axmap
& ( pushprop_lem1v2_lthm
<=> ( ! [X228: term,X229: subst] :
( ( sub @ one @ ( push @ X228 @ X229 ) )
= X228 )
=> ! [X230: term > $o,X231: subst,X232: term > $o,X233: term] :
( ~ ! [X234: term] :
( ( X230 @ ( sub @ X234 @ ( push @ X233 @ X231 ) ) )
= ( X232 @ X234 ) )
| ( X232 @ one )
| ~ ( X230 @ X233 ) ) ) )
& hoasapnotvar
& ( hoasinduction_p_and_p_prime
= ( ^ [X235: subst > term > subst > $o,X236: term > $o] :
! [X237: term] :
( ( X235 @ id @ X237 @ id )
= ( X236 @ X237 ) ) ) )
& ( ! [X238: subst > term > subst > $o,X239: term,X240: term > $o] :
( ~ ! [X241: term] :
( ~ ( X240 @ X241 )
| ( X240 @ ( sub @ X239 @ ( push @ X241 @ id ) ) ) )
| ~ ! [X242: subst > term > term] :
( ~ ! [X243: subst,X244: term,X245: subst] :
( ( sub @ ( X242 @ X245 @ X244 ) @ X243 )
= ( X242 @ ( comp @ X245 @ X243 ) @ ( sub @ X244 @ X243 ) ) )
| ( X238 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id )
| ~ ! [X246: term] :
( ( X238 @ id @ ( X242 @ id @ X246 ) @ id )
| ~ ( X238 @ id @ X246 @ id ) ) )
| ( X240 @ ( lam @ X239 ) )
| ~ ! [X247: term] :
( ( X238 @ id @ X247 @ id )
= ( X240 @ X247 ) ) )
<=> hoasinduction_lem3v2a )
& hoasinduction_lem3v2_gthm
& hoasinduction_lem2v2_gthm
& ! [X248: d_term,X249: d_term] :
( ( ( d2term @ X248 )
= ( d2term @ X249 ) )
=> ( X248 = X249 ) )
& ( hoaslamnotap
<=> ! [X250: term,X251: subst > term > term,X252: term] :
( ( ( lam @ ( X251 @ sh @ one ) )
!= ( ap @ ( sub @ X250 @ id ) @ X252 ) )
| ~ ! [X253: subst,X254: subst,X255: term] :
( ( sub @ ( X251 @ X253 @ X255 ) @ X254 )
= ( X251 @ ( comp @ X253 @ X254 ) @ ( sub @ X255 @ X254 ) ) ) ) )
& ( ( ! [X256: term,X257: term,X258: term,X259: term] :
( ( X256 = X257 )
| ( ( ap @ X259 @ X256 )
!= ( ap @ X258 @ X257 ) ) )
=> ! [X260: term,X261: term,X262: term,X263: term] :
( ( X260 = X261 )
| ( ( ap @ ( sub @ X262 @ id ) @ X260 )
!= ( ap @ ( sub @ X263 @ id ) @ X261 ) ) ) )
<=> hoasapinj2_lthm )
& hoasinduction_lem2_gthm
& ( ! [X264: term,X265: term,X266: term,X267: term] :
( ( X264 = X265 )
| ( ( ap @ X264 @ X266 )
!= ( ap @ X265 @ X267 ) ) )
<=> apinj1 )
& ! [X268: d_term] : ( d_one = X268 )
& induction2_gthm
& ( ! [X269: subst,X270: term > $o,X271: term] :
~ ! [X272: term > $o] :
~ ! [X273: term] :
( ( X272 @ X273 )
= ( X270 @ ( sub @ X273 @ ( push @ X271 @ X269 ) ) ) )
<=> pushprop_lem0 )
& pushprop_lthm
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ! [X274: d_subst] : ( d_id = X274 )
& ~ ulamvar1
& ( termmset_lthm
<=> ( ! [X275: term] :
( ( sub @ X275 @ id )
= X275 )
=> ! [X276: term] :
( ( sub @ X276 @ id )
= X276 ) ) )
& hoaslamnotvar
& pushprop_lem2v2
& pushprop_lem3v2_lthm
& ( ! [X277: subst > term > subst > $o,X278: term] :
( ( X277 @ id @ X278 @ id )
| ~ ! [X279: term] :
( ( X277 @ id @ X279 @ id )
| ~ ( var @ ( sub @ X279 @ id ) ) )
| ~ ! [X280: term,X281: subst,X282: subst,X283: subst] :
( ( X277 @ X282 @ X280 @ ( comp @ X281 @ X283 ) )
| ~ ( X277 @ ( comp @ X282 @ X281 ) @ ( sub @ X280 @ X281 ) @ X283 ) )
| ~ ! [X284: subst,X285: term,X286: subst,X287: subst] :
( ~ ( X277 @ X286 @ X285 @ ( comp @ X287 @ X284 ) )
| ( X277 @ ( comp @ X286 @ X287 ) @ ( sub @ X285 @ X287 ) @ X284 ) )
| ~ ! [X288: subst > term > term] :
( ~ ! [X289: term,X290: subst,X291: subst] :
( ( sub @ ( X288 @ X290 @ X289 ) @ X291 )
= ( X288 @ ( comp @ X290 @ X291 ) @ ( sub @ X289 @ X291 ) ) )
| ~ ! [X292: term] :
( ~ ( X277 @ id @ X292 @ id )
| ( X277 @ id @ ( X288 @ id @ X292 ) @ id ) )
| ( X277 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id ) )
| ~ ! [X293: term,X294: term] :
( ~ ( X277 @ id @ X294 @ id )
| ( X277 @ id @ ( ap @ ( sub @ X294 @ id ) @ X293 ) @ id )
| ~ ( X277 @ id @ X293 @ id ) ) )
<=> hoasinduction )
& ( laminj
<=> ! [X295: term,X296: term] :
( ( X295 = X296 )
| ( ( lam @ X296 )
!= ( lam @ X295 ) ) ) )
& ( ! [X297: term,X298: term > $o] :
( ~ ! [X299: term] :
( ~ ( var @ X299 )
| ( X298 @ X299 ) )
| ~ ! [X300: term,X301: term] :
( ( X298 @ ( ap @ X300 @ X301 ) )
| ~ ( X298 @ X301 )
| ~ ( X298 @ X300 ) )
| ~ ! [X302: term] :
( ~ ! [X303: term] :
( ~ ( X298 @ X303 )
| ( X298 @ ( sub @ X302 @ ( push @ X303 @ id ) ) ) )
| ( X298 @ ( lam @ X302 ) ) )
| ( X298 @ X297 ) )
<=> induction2 )
& ( ( d2subst @ d_id )
= sh )
& hoasapinj2_gthm
& ( ( ! [X304: term] :
( ( sub @ X304 @ id )
= X304 )
=> ( ! [X305: subst,X306: term > $o,X307: term] :
( ~ ! [X308: term,X309: term] :
( ~ ( X306 @ X308 )
| ~ ( X306 @ X309 )
| ( X306 @ ( ap @ X308 @ X309 ) ) )
| ~ ! [X310: term] :
( ~ ! [X311: term] :
( ~ ( X306 @ X311 )
| ( X306 @ ( sub @ X310 @ ( push @ X311 @ id ) ) ) )
| ( X306 @ ( lam @ X310 ) ) )
| ( X306 @ ( sub @ X307 @ X305 ) )
| ~ ! [X312: term] :
( ~ ( var @ X312 )
| ( X306 @ ( sub @ X312 @ X305 ) ) ) )
=> ! [X313: term,X314: term > $o] :
( ~ ! [X315: term] :
( ~ ( var @ X315 )
| ( X314 @ X315 ) )
| ( X314 @ X313 )
| ~ ! [X316: term] :
( ~ ! [X317: term] :
( ( X314 @ ( sub @ X316 @ ( push @ X317 @ id ) ) )
| ~ ( X314 @ X317 ) )
| ( X314 @ ( lam @ X316 ) ) )
| ~ ! [X318: term,X319: term] :
( ( X314 @ ( ap @ X318 @ X319 ) )
| ~ ( X314 @ X318 )
| ~ ( X314 @ X319 ) ) ) ) )
<=> induction2_lthm )
& hoasinduction_lem3b_gthm
& ulamvarind
& hoaslamnotvar_lthm
& hoasinduction_lem3_gthm
& ( ! [X320: term,X321: term > $o,X322: subst] :
( ~ ! [X323: term] :
( ~ ! [X324: term] :
( ( X321 @ ( sub @ X323 @ ( push @ X324 @ id ) ) )
| ~ ( X321 @ X324 ) )
| ( X321 @ ( lam @ X323 ) ) )
| ~ ! [X325: term,X326: term] :
( ( X321 @ ( ap @ X326 @ X325 ) )
| ~ ( X321 @ X326 )
| ~ ( X321 @ X325 ) )
| ( X321 @ ( sub @ X320 @ X322 ) )
| ~ ! [X327: term] :
( ~ ( var @ X327 )
| ( X321 @ ( sub @ X327 @ X322 ) ) ) )
<=> induction2lem )
& ( ( ^ [X328: subst,X329: subst > term > term] : ( d2term @ d_one ) )
= hoaslam )
& hoasinduction_lem1v2
& ( apinj2
<=> ! [X330: term,X331: term,X332: term,X333: term] :
( ( ( ap @ X332 @ X330 )
!= ( ap @ X331 @ X333 ) )
| ( X330 = X333 ) ) )
& ( hoasinduction_lthm
<=> ( ! [X334: term > $o,X335: term] :
( ~ ! [X336: term] :
( ( X334 @ X336 )
| ~ ( var @ X336 ) )
| ~ ! [X337: term,X338: term] :
( ~ ( X334 @ X337 )
| ~ ( X334 @ X338 )
| ( X334 @ ( ap @ X337 @ X338 ) ) )
| ~ ! [X339: term] :
( ~ ! [X340: term] :
( ~ ( X334 @ X340 )
| ( X334 @ ( sub @ X339 @ ( push @ X340 @ id ) ) ) )
| ( X334 @ ( lam @ X339 ) ) )
| ( X334 @ X335 ) )
=> ( ! [X341: term,X342: subst > term > subst > $o,X343: term] :
( ~ ! [X344: subst,X345: subst,X346: term,X347: subst] :
( ( X342 @ ( comp @ X347 @ X344 ) @ ( sub @ X346 @ X344 ) @ X345 )
| ~ ( X342 @ X347 @ X346 @ ( comp @ X344 @ X345 ) ) )
| ~ ( X342 @ id @ X341 @ id )
| ( X342 @ id @ ( ap @ X343 @ X341 ) @ id )
| ~ ( X342 @ id @ X343 @ id )
| ~ ! [X348: term,X349: term] :
( ~ ( X342 @ id @ X349 @ id )
| ( X342 @ id @ ( ap @ ( sub @ X348 @ id ) @ X349 ) @ id )
| ~ ( X342 @ id @ X348 @ id ) )
| ~ ! [X350: subst,X351: term,X352: subst,X353: subst] :
( ( X342 @ X353 @ X351 @ ( comp @ X352 @ X350 ) )
| ~ ( X342 @ ( comp @ X353 @ X352 ) @ ( sub @ X351 @ X352 ) @ X350 ) ) )
=> ( ! [X354: subst > term > subst > $o,X355: term] :
( ~ ! [X356: subst,X357: subst,X358: term,X359: subst] :
( ~ ( X354 @ ( comp @ X356 @ X357 ) @ ( sub @ X358 @ X357 ) @ X359 )
| ( X354 @ X356 @ X358 @ ( comp @ X357 @ X359 ) ) )
| ~ ! [X360: subst > term > term] :
( ~ ! [X361: term] :
( ~ ( X354 @ id @ X361 @ id )
| ( X354 @ id @ ( X360 @ id @ X361 ) @ id ) )
| ( X354 @ id @ ( lam @ ( X360 @ sh @ one ) ) @ id )
| ~ ! [X362: subst,X363: subst,X364: term] :
( ( X360 @ ( comp @ X363 @ X362 ) @ ( sub @ X364 @ X362 ) )
= ( sub @ ( X360 @ X363 @ X364 ) @ X362 ) ) )
| ( X354 @ id @ ( lam @ X355 ) @ id )
| ~ ! [X365: term] :
( ~ ( X354 @ id @ X365 @ id )
| ( X354 @ id @ ( sub @ X355 @ ( push @ X365 @ id ) ) @ id ) )
| ~ ! [X366: subst,X367: subst,X368: term,X369: subst] :
( ~ ( X354 @ X366 @ X368 @ ( comp @ X367 @ X369 ) )
| ( X354 @ ( comp @ X366 @ X367 ) @ ( sub @ X368 @ X367 ) @ X369 ) ) )
=> ! [X370: term,X371: subst > term > subst > $o] :
( ~ ! [X372: term] :
( ( X371 @ id @ X372 @ id )
| ~ ( var @ ( sub @ X372 @ id ) ) )
| ~ ! [X373: subst,X374: subst,X375: subst,X376: term] :
( ( X371 @ ( comp @ X374 @ X375 ) @ ( sub @ X376 @ X375 ) @ X373 )
| ~ ( X371 @ X374 @ X376 @ ( comp @ X375 @ X373 ) ) )
| ~ ! [X377: term,X378: term] :
( ~ ( X371 @ id @ X377 @ id )
| ~ ( X371 @ id @ X378 @ id )
| ( X371 @ id @ ( ap @ ( sub @ X378 @ id ) @ X377 ) @ id ) )
| ~ ! [X379: subst > term > term] :
( ~ ! [X380: subst,X381: term,X382: subst] :
( ( sub @ ( X379 @ X382 @ X381 ) @ X380 )
= ( X379 @ ( comp @ X382 @ X380 ) @ ( sub @ X381 @ X380 ) ) )
| ( X371 @ id @ ( lam @ ( X379 @ sh @ one ) ) @ id )
| ~ ! [X383: term] :
( ( X371 @ id @ ( X379 @ id @ X383 ) @ id )
| ~ ( X371 @ id @ X383 @ id ) ) )
| ( X371 @ id @ X370 @ id )
| ~ ! [X384: subst,X385: subst,X386: term,X387: subst] :
( ~ ( X371 @ ( comp @ X385 @ X384 ) @ ( sub @ X386 @ X384 ) @ X387 )
| ( X371 @ X385 @ X386 @ ( comp @ X384 @ X387 ) ) ) ) ) ) ) )
& apnotvar
& ( axvarid
<=> ! [X388: term] :
( ( sub @ X388 @ id )
= X388 ) )
& ( hoasinduction_lem3
<=> ! [X389: term,X390: subst > term > subst > $o] :
( ~ ! [X391: subst,X392: subst,X393: subst,X394: term] :
( ( X390 @ ( comp @ X391 @ X393 ) @ ( sub @ X394 @ X393 ) @ X392 )
| ~ ( X390 @ X391 @ X394 @ ( comp @ X393 @ X392 ) ) )
| ~ ! [X395: term] :
( ( X390 @ id @ ( sub @ X389 @ ( push @ X395 @ id ) ) @ id )
| ~ ( X390 @ id @ X395 @ id ) )
| ( X390 @ id @ ( lam @ X389 ) @ id )
| ~ ! [X396: subst > term > term] :
( ~ ! [X397: subst,X398: subst,X399: term] :
( ( X396 @ ( comp @ X398 @ X397 ) @ ( sub @ X399 @ X397 ) )
= ( sub @ ( X396 @ X398 @ X399 ) @ X397 ) )
| ~ ! [X400: term] :
( ~ ( X390 @ id @ X400 @ id )
| ( X390 @ id @ ( X396 @ id @ X400 ) @ id ) )
| ( X390 @ id @ ( lam @ ( X396 @ sh @ one ) ) @ id ) )
| ~ ! [X401: subst,X402: subst,X403: term,X404: subst] :
( ~ ( X390 @ ( comp @ X401 @ X402 ) @ ( sub @ X403 @ X402 ) @ X404 )
| ( X390 @ X401 @ X403 @ ( comp @ X402 @ X404 ) ) ) ) )
& hoaslaminj_gthm
& hoaslamnotap_lthm
& ( hoasapinj1_lthm
<=> ( ! [X405: term] :
( ( sub @ X405 @ id )
= X405 )
=> ( ! [X406: term,X407: term,X408: term,X409: term] :
( ( ( ap @ X406 @ X408 )
!= ( ap @ X409 @ X407 ) )
| ( X406 = X409 ) )
=> ! [X410: term,X411: term,X412: term,X413: term] :
( ( X410 = X413 )
| ( ( ap @ ( sub @ X410 @ id ) @ X412 )
!= ( ap @ ( sub @ X413 @ id ) @ X411 ) ) ) ) ) )
& hoasapinj1_gthm
& ( ( ! [X414: subst] :
( ( comp @ id @ X414 )
= X414 )
=> ( ! [X415: subst] :
( ( comp @ X415 @ id )
= X415 )
=> ( ! [X416: subst] :
( ( comp @ id @ X416 )
= X416 )
& ! [X417: subst] :
( ( comp @ X417 @ id )
= X417 ) ) ) )
<=> substmonoid_lthm )
& ! [X418: term] :
? [X419: d_term] :
( ( d2term @ X419 )
= X418 )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X420: term,X421: term,X422: term,X423: term] :
( ( X421 = X423 )
| ( ( ap @ ( sub @ X420 @ id ) @ X421 )
!= ( ap @ ( sub @ X422 @ id ) @ X423 ) ) )
<=> hoasapinj2 )
& ( hoasinduction_lem3aa
<=> ! [X424: subst > term > subst > $o,X425: term] :
( ~ ! [X426: term] :
( ( X424 @ id @ ( sub @ X425 @ ( push @ X426 @ id ) ) @ id )
| ~ ( X424 @ id @ X426 @ id ) )
| ( X424 @ id @ ( lam @ ( sub @ X425 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X427: subst > term > term] :
( ~ ! [X428: term,X429: subst,X430: subst] :
( ( X427 @ ( comp @ X430 @ X429 ) @ ( sub @ X428 @ X429 ) )
= ( sub @ ( X427 @ X430 @ X428 ) @ X429 ) )
| ~ ! [X431: term] :
( ( X424 @ id @ ( X427 @ id @ X431 ) @ id )
| ~ ( X424 @ id @ X431 @ id ) )
| ( X424 @ id @ ( lam @ ( X427 @ sh @ one ) ) @ id ) ) ) )
& ( hoasinduction_lem3_lthm
<=> ( ! [X432: term] :
( ( sub @ X432 @ id )
= X432 )
=> ( ! [X433: term,X434: subst > term > subst > $o] :
( ~ ! [X435: term] :
( ( X434 @ id @ ( sub @ X433 @ ( push @ X435 @ id ) ) @ id )
| ~ ( X434 @ id @ X435 @ id ) )
| ~ ! [X436: subst > term > term] :
( ~ ! [X437: term] :
( ~ ( X434 @ id @ X437 @ id )
| ( X434 @ id @ ( X436 @ id @ X437 ) @ id ) )
| ( X434 @ id @ ( lam @ ( X436 @ sh @ one ) ) @ id )
| ~ ! [X438: term,X439: subst,X440: subst] :
( ( sub @ ( X436 @ X440 @ X438 ) @ X439 )
= ( X436 @ ( comp @ X440 @ X439 ) @ ( sub @ X438 @ X439 ) ) ) )
| ( X434 @ id @ ( lam @ ( sub @ X433 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X441: term,X442: subst > term > subst > $o] :
( ( X442 @ id @ ( lam @ X441 ) @ id )
| ~ ! [X443: subst > term > term] :
( ( X442 @ id @ ( lam @ ( X443 @ sh @ one ) ) @ id )
| ~ ! [X444: term] :
( ( X442 @ id @ ( X443 @ id @ X444 ) @ id )
| ~ ( X442 @ id @ X444 @ id ) )
| ~ ! [X445: term,X446: subst,X447: subst] :
( ( X443 @ ( comp @ X446 @ X447 ) @ ( sub @ X445 @ X447 ) )
= ( sub @ ( X443 @ X446 @ X445 ) @ X447 ) ) )
| ~ ! [X448: term,X449: subst,X450: subst,X451: subst] :
( ( X442 @ X451 @ X448 @ ( comp @ X450 @ X449 ) )
| ~ ( X442 @ ( comp @ X451 @ X450 ) @ ( sub @ X448 @ X450 ) @ X449 ) )
| ~ ! [X452: term] :
( ( X442 @ id @ ( sub @ X441 @ ( push @ X452 @ id ) ) @ id )
| ~ ( X442 @ id @ X452 @ id ) )
| ~ ! [X453: subst,X454: subst,X455: subst,X456: term] :
( ( X442 @ ( comp @ X455 @ X454 ) @ ( sub @ X456 @ X454 ) @ X453 )
| ~ ( X442 @ X455 @ X456 @ ( comp @ X454 @ X453 ) ) ) ) ) ) )
& pushprop_lem2v2_gthm
& pushprop_lem0_gthm
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ! [X457: subst > term > subst > $o,X458: term] :
( ~ ! [X459: term] :
( ( X457 @ id @ ( sub @ X458 @ ( push @ X459 @ id ) ) @ id )
| ~ ( X457 @ id @ X459 @ id ) )
| ( X457 @ id @ ( lam @ ( sub @ X458 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X460: subst > term > term,X461: term] :
( ~ ! [X462: subst,X463: term,X464: subst] :
( ( sub @ X461 @ ( push @ ( sub @ X463 @ X462 ) @ ( comp @ X464 @ X462 ) ) )
= ( X460 @ ( comp @ X464 @ X462 ) @ ( sub @ X463 @ X462 ) ) )
| ( X457 @ id @ ( lam @ ( X460 @ sh @ one ) ) @ id )
| ~ ! [X465: term] :
( ~ ( X457 @ id @ X465 @ id )
| ( X457 @ id @ ( X460 @ id @ X465 ) @ id ) )
| ~ ! [X466: subst,X467: term,X468: subst] :
( ( sub @ ( sub @ X461 @ ( push @ X467 @ X466 ) ) @ X468 )
= ( sub @ ( X460 @ X466 @ X467 ) @ X468 ) ) ) )
<=> hoasinduction_lem3aaa )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoaslaminj
<=> ! [X469: term,X470: subst > term > term,X471: subst,X472: subst > term > term] :
( ~ ! [X473: subst,X474: term,X475: subst] :
( ( X472 @ ( comp @ X475 @ X473 ) @ ( sub @ X474 @ X473 ) )
= ( sub @ ( X472 @ X475 @ X474 ) @ X473 ) )
| ( ( lam @ ( X470 @ sh @ one ) )
!= ( lam @ ( X472 @ sh @ one ) ) )
| ~ ! [X476: term,X477: subst,X478: subst] :
( ( sub @ ( X470 @ X478 @ X476 ) @ X477 )
= ( X470 @ ( comp @ X478 @ X477 ) @ ( sub @ X476 @ X477 ) ) )
| ( ( X470 @ X471 @ X469 )
= ( X472 @ X471 @ X469 ) ) ) )
& ( ( ! [X479: subst,X480: term] :
( ( sub @ one @ ( push @ X480 @ X479 ) )
= X480 )
=> ( ! [X481: subst,X482: term] :
( ( comp @ sh @ ( push @ X482 @ X481 ) )
= X481 )
=> ( ! [X483: subst] :
( ( comp @ X483 @ id )
= X483 )
=> ( ! [X484: term > $o,X485: term] :
( ( X484 @ X485 )
| ~ ! [X486: term] :
( ( X484 @ ( lam @ X486 ) )
| ~ ( X484 @ X486 ) )
| ~ ! [X487: term] :
( ~ ( var @ X487 )
| ( X484 @ X487 ) )
| ~ ! [X488: term,X489: term] :
( ~ ( X484 @ X488 )
| ~ ( X484 @ X489 )
| ( X484 @ ( ap @ X489 @ X488 ) ) ) )
=> ! [X490: term,X491: subst,X492: term > $o] :
( ( X492 @ ( sub @ X490 @ X491 ) )
| ~ ! [X493: term] :
( ( X492 @ ( lam @ X493 ) )
| ~ ! [X494: term] :
( ~ ( X492 @ X494 )
| ( X492 @ ( sub @ X493 @ ( push @ X494 @ id ) ) ) ) )
| ~ ! [X495: term] :
( ( X492 @ ( sub @ X495 @ X491 ) )
| ~ ( var @ X495 ) )
| ~ ! [X496: term,X497: term] :
( ~ ( X492 @ X496 )
| ( X492 @ ( ap @ X496 @ X497 ) )
| ~ ( X492 @ X497 ) ) ) ) ) ) )
<=> induction2lem_lthm )
& hoasapnotvar_lthm
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( one
= ( d2term @ d_one ) )
& axapp
& pushprop_lem2v2_lthm
& pushprop_lthm_orig
& ( axshiftcons
<=> ! [X498: subst,X499: term] :
( ( comp @ sh @ ( push @ X499 @ X498 ) )
= X498 ) )
& ( pushprop_p_and_p_prime
= ( ^ [X500: term,X501: subst,X502: term > $o,X503: term > $o] :
! [X504: term] :
( ( X503 @ X504 )
= ( X502 @ ( sub @ X504 @ ( push @ X500 @ X501 ) ) ) ) ) )
& ( ! [X505: term > $o,X506: term] :
( ~ ! [X507: term,X508: term] :
( ~ ( X505 @ X507 )
| ( X505 @ ( ap @ X507 @ X508 ) )
| ~ ( X505 @ X508 ) )
| ~ ! [X509: term] :
( ~ ( var @ X509 )
| ( X505 @ X509 ) )
| ~ ! [X510: term] :
( ( X505 @ ( lam @ X510 ) )
| ~ ( X505 @ X510 ) )
| ( X505 @ X506 ) )
<=> induction )
& ~ ( var @ ( d2term @ d_one ) )
& hoasapnotvar_gthm
& ( ( ! [X511: term] :
( ( sub @ X511 @ id )
= X511 )
=> ! [X512: term > $o,X513: subst > term > subst > $o,X514: term] :
( ~ ! [X515: term] :
( ( X512 @ X515 )
= ( X513 @ id @ X515 @ id ) )
| ( X512 @ ( lam @ X514 ) )
| ~ ! [X516: subst,X517: subst,X518: subst,X519: term] :
( ~ ( X513 @ X516 @ X519 @ ( comp @ X517 @ X518 ) )
| ( X513 @ ( comp @ X516 @ X517 ) @ ( sub @ X519 @ X517 ) @ X518 ) )
| ~ ! [X520: subst > term > term] :
( ( X513 @ id @ ( lam @ ( X520 @ sh @ one ) ) @ id )
| ~ ! [X521: term] :
( ~ ( X513 @ id @ X521 @ id )
| ( X513 @ id @ ( X520 @ id @ X521 ) @ id ) )
| ~ ! [X522: subst,X523: term,X524: subst] :
( ( sub @ ( X520 @ X524 @ X523 ) @ X522 )
= ( X520 @ ( comp @ X524 @ X522 ) @ ( sub @ X523 @ X522 ) ) ) )
| ~ ! [X525: subst,X526: subst,X527: subst,X528: term] :
( ( X513 @ X526 @ X528 @ ( comp @ X525 @ X527 ) )
| ~ ( X513 @ ( comp @ X526 @ X525 ) @ ( sub @ X528 @ X525 ) @ X527 ) )
| ~ ! [X529: term] :
( ~ ( X512 @ X529 )
| ( X512 @ ( sub @ X514 @ ( push @ X529 @ id ) ) ) ) ) )
<=> hoasinduction_lem3v2_lthm )
& ( pushprop_lem1v2
<=> ! [X530: subst,X531: term,X532: term > $o,X533: term > $o] :
( ( X533 @ one )
| ~ ( X532 @ X531 )
| ~ ! [X534: term] :
( ( X533 @ X534 )
= ( X532 @ ( sub @ X534 @ ( push @ X531 @ X530 ) ) ) ) ) )
& ( ! [X535: term > $o,X536: term,X537: term,X538: subst > term > subst > $o] :
( ~ ( X535 @ X536 )
| ~ ! [X539: term] :
( ( X535 @ X539 )
= ( X538 @ id @ X539 @ id ) )
| ~ ! [X540: term,X541: term] :
( ~ ( X538 @ id @ X540 @ id )
| ~ ( X538 @ id @ X541 @ id )
| ( X538 @ id @ ( ap @ ( sub @ X541 @ id ) @ X540 ) @ id ) )
| ~ ! [X542: subst,X543: subst,X544: term,X545: subst] :
( ~ ( X538 @ ( comp @ X543 @ X542 ) @ ( sub @ X544 @ X542 ) @ X545 )
| ( X538 @ X543 @ X544 @ ( comp @ X542 @ X545 ) ) )
| ( X535 @ ( ap @ X537 @ X536 ) )
| ~ ( X535 @ X537 )
| ~ ! [X546: subst,X547: subst,X548: term,X549: subst] :
( ( X538 @ ( comp @ X546 @ X547 ) @ ( sub @ X548 @ X547 ) @ X549 )
| ~ ( X538 @ X546 @ X548 @ ( comp @ X547 @ X549 ) ) ) )
<=> hoasinduction_lem2v2 )
& hoasinduction_lem1 ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) )
<=> substmonoid )
& ~ lamnotap
& axassoc
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0_lthm )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoaslaminj_lthm
<=> ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X15: term,X13: term] :
( ( ( lam @ X13 )
!= ( lam @ X15 ) )
| ( X13 = X15 ) )
=> ! [X31: subst > term > term,X18: subst > term > term,X33: term,X32: subst] :
( ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) ) ) )
& pushprop_gthm
& induction2lem_gthm
& ( ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X22: term,X10: term > $o] :
( ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) ) )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ( X10 @ X22 ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X11: term > $o,X10: subst > term > subst > $o,X29: term] :
( ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [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,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X29 ) ) )
=> ! [X10: subst > term > subst > $o,X28: term] :
( ~ ! [X18: subst > term > term] :
( ~ ! [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 ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X10 @ id @ X28 @ id )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X13: term,X26: subst,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X25: subst,X26: subst,X14: subst,X13: term] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) ) ) ) ) )
<=> hoasinduction_lthm_3 )
& hoasinduction_lem2_lthm
& hoasinduction_lem1_gthm
& ulamvarsh
& ( pushprop_lem1
<=> ! [X10: term > $o,X13: term,X26: term > $o,X15: term,X14: subst] :
( ~ ( X10 @ X13 )
| ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) ) ) )
& hoaslamnotap_gthm
& ( hoasinduction_lem3v2_f_lthm
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) ) )
& lamnotvar
& ( shinj
<=> ! [X15: term,X13: term] :
( ( X13 = X15 )
| ( ( sub @ X13 @ sh )
!= ( sub @ X15 @ sh ) ) ) )
& ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
<=> axidr )
& hoasinduction_lem1v2_gthm
& ( ( d2subst @ d_id )
= id )
& ( axidl
<=> ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) )
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) )
<=> hoasinduction_lem3b )
& ( hoasinduction_lem3a
<=> ! [X10: subst > term > subst > $o,X43: term] :
( ~ ! [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 ) )
| ( X10 @ id @ ( lam @ X43 ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) ) )
& hoasinduction_lem1_lthm
& ( hoasinduction_lem3a_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( 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 ) ) ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X43: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ( 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 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ( X10 @ id @ ( lam @ X43 ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) ) ) ) )
& ( axvarcons
<=> ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 ) )
& ( axscons
<=> ! [X14: subst] :
( ( push @ ( sub @ one @ X14 ) @ ( comp @ sh @ X14 ) )
= X14 ) )
& hoaslamnotvar_gthm
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X29: term,X10: subst > term > subst > $o,X11: term > $o] :
( ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ 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,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) ) )
<=> hoasinduction_lem3v2a_lthm )
& ( ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ! [X10: term > $o,X14: subst,X13: term,X26: term > $o,X15: term] :
( ~ ( X10 @ X13 )
| ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) ) ) ) )
<=> pushprop_lem1_lthm )
& hoasinduction_gthm
& ( ! [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] :
( ~ ! [X13: term,X14: subst,X25: 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 ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
<=> hoasinduction_lem3aa_lthm )
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
<=> hoasinduction_lem3v2_f )
& ( pushprop_lem0_lthm
<=> ! [X13: term,X10: term > $o,X14: subst] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) )
& ( hoasinduction_lem3b_lthm
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) ) )
& hoasinduction_lem3a_gthm
& ( hoasinduction_lem3v2
<=> ! [X10: subst > term > subst > $o,X11: term > $o,X48: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X26: subst,X14: subst,X25: subst,X13: term] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X26: subst,X14: subst,X25: subst,X13: term] :
( ( 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] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ 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 ) ) ) )
& pushprop_lem1_gthm
& axvarshift
& pushprop_lem3v2
& ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
<=> termmset )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0 )
& ( hoasinduction_lem2
<=> ! [X23: term,X24: term,X10: subst > term > subst > $o] :
( ~ ( X10 @ id @ X24 @ id )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ( X10 @ id @ X23 @ id )
| ~ ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) ) ) )
& ! [X6: d_subst,X7: d_subst] :
( ( ( d2subst @ X6 )
= ( d2subst @ X7 ) )
=> ( X6 = X7 ) )
& substmonoid_gthm
& pushprop
& axclos
& ( ! [X30: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ X30 @ id )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ 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 ) ) ) ) )
<=> hoasinduction_no_psi_cond )
& ( ! [X19: term,X15: term,X13: term,X20: term] :
( ( X13 = X15 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) )
<=> hoasapinj1 )
& ! [X4: subst] :
? [X5: d_subst] :
( ( d2subst @ X5 )
= X4 )
& pushprop_lem1v2_gthm
& axabs
& ( hoasinduction_no_psi_cond_lthm
<=> ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X10: term > $o,X22: term] :
( ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: subst > term > subst > $o,X11: term > $o,X29: term] :
( ( X11 @ ( lam @ X29 ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
=> ! [X10: subst > term > subst > $o,X30: term] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ( X10 @ id @ X30 @ id )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( 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 ) ) ) ) ) ) ) ) ) )
& termmset_gthm
& axmap
& ( pushprop_lem1v2_lthm
<=> ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ! [X10: term > $o,X14: subst,X11: term > $o,X13: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
| ( X11 @ one )
| ~ ( X10 @ X13 ) ) ) )
& hoasapnotvar
& ( hoasinduction_p_and_p_prime
= ( ^ [X10: subst > term > subst > $o,X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) ) )
& ( ! [X10: subst > term > subst > $o,X29: term,X11: term > $o] :
( ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X25: subst,X13: term,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 @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
<=> hoasinduction_lem3v2a )
& hoasinduction_lem3v2_gthm
& hoasinduction_lem2v2_gthm
& ! [X3: d_term,X2: d_term] :
( ( ( d2term @ X2 )
= ( d2term @ X3 ) )
=> ( X2 = X3 ) )
& ( hoaslamnotap
<=> ! [X46: term,X18: subst > term > term,X47: term] :
( ( ( lam @ ( X18 @ sh @ one ) )
!= ( ap @ ( sub @ X46 @ id ) @ X47 ) )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) )
& ( ( ! [X19: term,X20: term,X15: term,X13: term] :
( ( X19 = X20 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
=> ! [X20: term,X19: term,X15: term,X13: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) )
<=> hoasapinj2_lthm )
& hoasinduction_lem2_gthm
& ( ! [X13: term,X15: term,X19: term,X20: term] :
( ( X13 = X15 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
<=> apinj1 )
& ! [X1: d_term] : ( d_one = X1 )
& induction2_gthm
& ( ! [X14: subst,X10: term > $o,X13: term] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
<=> pushprop_lem0 )
& pushprop_lthm
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ! [X5: d_subst] : ( d_id = X5 )
& ~ ulamvar1
& ( termmset_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) ) )
& hoaslamnotvar
& pushprop_lem2v2
& pushprop_lem3v2_lthm
& ( ! [X10: subst > term > subst > $o,X28: term] :
( ( X10 @ id @ X28 @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ X13 @ id )
| ~ ( var @ ( sub @ X13 @ id ) ) )
| ~ ! [X13: term,X26: subst,X14: subst,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X25: subst,X13: term,X14: subst,X26: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [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 @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) )
<=> hoasinduction )
& ( laminj
<=> ! [X13: term,X15: term] :
( ( X13 = X15 )
| ( ( lam @ X13 )
!= ( lam @ X15 ) ) ) )
& ( ! [X22: term,X10: term > $o] :
( ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ X22 ) )
<=> induction2 )
& ( ( d2subst @ d_id )
= sh )
& hoasapinj2_gthm
& ( ( ! [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 ) ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) ) )
=> ! [X22: term,X10: term > $o] :
( ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) ) ) ) )
<=> induction2_lthm )
& hoasinduction_lem3b_gthm
& ulamvarind
& hoaslamnotvar_lthm
& hoasinduction_lem3_gthm
& ( ! [X16: term,X10: term > $o,X17: subst] :
( ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) ) )
<=> induction2lem )
& ( ( ^ [X8: subst,X9: subst > term > term] : ( d2term @ d_one ) )
= hoaslam )
& hoasinduction_lem1v2
& ( apinj2
<=> ! [X19: term,X15: term,X13: term,X20: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X19 = X20 ) ) )
& ( hoasinduction_lthm
<=> ( ! [X10: term > $o,X22: term] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ X22 ) )
=> ( ! [X24: term,X10: subst > term > subst > $o,X23: term] :
( ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ( X10 @ id @ X24 @ id )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ( X10 @ id @ X23 @ id )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X13: term,X26: subst,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) ) )
=> ( ! [X10: subst > term > subst > $o,X27: term] :
( ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ 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 @ X27 ) @ id )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) ) )
=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X13: term] :
( ( X10 @ id @ X13 @ id )
| ~ ( var @ ( sub @ X13 @ id ) ) )
| ~ ! [X25: subst,X14: subst,X26: subst,X13: term] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ 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 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ( X10 @ id @ X28 @ id )
| ~ ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) ) ) ) ) )
& apnotvar
& ( axvarid
<=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) )
& ( hoasinduction_lem3
<=> ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X14: subst,X25: subst,X26: subst,X13: term] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( lam @ X27 ) @ 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 ) )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) ) )
& hoaslaminj_gthm
& hoaslamnotap_lthm
& ( hoasapinj1_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X13: term,X20: term,X19: term,X15: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X13 = X15 ) )
=> ! [X13: term,X20: term,X19: term,X15: term] :
( ( X13 = X15 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) ) ) )
& hoasapinj1_gthm
& ( ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
& ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 ) ) ) )
<=> substmonoid_lthm )
& ! [X0: term] :
? [X1: d_term] :
( ( d2term @ X1 )
= X0 )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X13: term,X19: term,X15: term,X20: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) )
<=> hoasapinj2 )
& ( hoasinduction_lem3aa
<=> ! [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] :
( ~ ! [X13: term,X25: subst,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 ) ) ) )
& ( hoasinduction_lem3_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X27: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X13: term,X25: subst,X26: subst,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X25: subst,X26: subst,X14: subst,X13: term] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) ) ) ) )
& pushprop_lem2v2_gthm
& pushprop_lem0_gthm
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ! [X10: subst > term > subst > $o,X34: term] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X34 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X34 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X18: subst > term > term,X35: term] :
( ~ ! [X41: subst,X40: term,X39: subst] :
( ( X18 @ ( comp @ X39 @ X41 ) @ ( sub @ X40 @ X41 ) )
= ( sub @ X35 @ ( push @ ( sub @ X40 @ X41 ) @ ( comp @ X39 @ X41 ) ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X36: subst,X37: term,X38: subst] :
( ( sub @ ( X18 @ X36 @ X37 ) @ X38 )
= ( sub @ ( sub @ X35 @ ( push @ X37 @ X36 ) ) @ X38 ) ) ) )
<=> hoasinduction_lem3aaa )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoaslaminj
<=> ! [X33: term,X18: subst > term > term,X32: subst,X31: subst > term > term] :
( ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) ) ) )
& ( ( ! [X14: subst,X13: term] :
( ( 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] :
( ( X10 @ X21 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) ) ) )
=> ! [X16: term,X17: subst,X10: term > $o] :
( ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X15 @ X17 ) )
| ~ ( var @ X15 ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) ) ) ) ) ) )
<=> induction2lem_lthm )
& hoasapnotvar_lthm
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( one
= ( d2term @ d_one ) )
& axapp
& pushprop_lem2v2_lthm
& pushprop_lthm_orig
& ( axshiftcons
<=> ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 ) )
& ( 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 ) ) ) ) ) )
& ( ! [X10: term > $o,X21: term] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ( X10 @ X13 ) )
| ( X10 @ X21 ) )
<=> induction )
& ~ ( var @ ( d2term @ d_one ) )
& hoasapnotvar_gthm
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X11: term > $o,X10: subst > term > subst > $o,X48: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X14: subst,X26: subst,X25: subst,X13: term] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X26: subst,X14: subst,X25: subst,X13: term] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) ) ) ) )
<=> hoasinduction_lem3v2_lthm )
& ( pushprop_lem1v2
<=> ! [X14: subst,X13: term,X10: term > $o,X11: term > $o] :
( ( X11 @ one )
| ~ ( X10 @ X13 )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) ) )
& ( ! [X11: term > $o,X45: term,X44: term,X10: subst > term > subst > $o] :
( ~ ( X11 @ X45 )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X11 @ ( ap @ X44 @ X45 ) )
| ~ ( X11 @ X44 )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) )
<=> hoasinduction_lem2v2 )
& hoasinduction_lem1 ),
file('/export/starexec/sandbox2/tmp/tmp.Qe5lgCyrIb/DTF2THF_31149.p',alg444_1) ).
thf(f1053,plain,
! [X258: term] :
( ( d2term @ ( sK583 @ X258 ) )
= X258 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f2133,plain,
! [X1: subst > term > term] :
( ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( sK594 @ X1 ) @ ( d2subst @ d_id ) )
= $true )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(subsumption_resolution,[status(thm)],[f1745,f1898]) ).
thf(f1898,plain,
! [X0: term,X1: term] : ( X0 = X1 ),
inference(superposition,[status(thm)],[f1894,f1894]) ).
thf(f1745,plain,
! [X1: subst > term > term] :
( ( hoasinduction_no_psi_cond = $false )
| ( ( X1 @ ( comp @ ( sK595 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK595 @ X1 ) @ ( sK593 @ X1 ) ) @ ( sK592 @ X1 ) ) )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( sK594 @ X1 ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(equality_proxy_clausification,[status(thm)],[f1744]) ).
thf(f1744,plain,
! [X1: subst > term > term] :
( ( ( sK591 @ ( d2subst @ d_id ) @ ( sK594 @ X1 ) @ ( d2subst @ d_id ) )
= $true )
| ( hoasinduction_no_psi_cond = $false )
| ( ( ( sub @ ( X1 @ ( sK595 @ X1 ) @ ( sK593 @ X1 ) ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ ( sK595 @ X1 ) @ ( sK592 @ X1 ) ) @ ( sub @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) ) )
= $false )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1743]) ).
thf(f1743,plain,
! [X1: subst > term > term] :
( ( $false
= ( ^ [Y0: subst] :
( ( sub @ ( X1 @ Y0 @ ( sK593 @ X1 ) ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y0 @ ( sK592 @ X1 ) ) @ ( sub @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) ) )
@ ( sK595 @ X1 ) ) )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( sK594 @ X1 ) @ ( d2subst @ d_id ) )
= $true )
| ( hoasinduction_no_psi_cond = $false )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(sigma_clausification,[status(thm)],[f1742]) ).
thf(f1742,plain,
! [X1: subst > term > term] :
( ( ( !! @ subst
@ ^ [Y0: subst] :
( ( sub @ ( X1 @ Y0 @ ( sK593 @ X1 ) ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y0 @ ( sK592 @ X1 ) ) @ ( sub @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) ) ) )
= $false )
| ( hoasinduction_no_psi_cond = $false )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( sK594 @ X1 ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f1740]) ).
thf(f1740,plain,
! [X1: subst > term > term] :
( ( hoasinduction_no_psi_cond = $false )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( $false
= ( ( sK591 @ ( d2subst @ d_id ) @ ( sK594 @ X1 ) @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ ( sK594 @ X1 ) ) @ ( d2subst @ d_id ) ) ) )
| ( ( !! @ subst
@ ^ [Y0: subst] :
( ( sub @ ( X1 @ Y0 @ ( sK593 @ X1 ) ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y0 @ ( sK592 @ X1 ) ) @ ( sub @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) ) ) )
= $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1739]) ).
thf(f1739,plain,
! [X1: subst > term > term] :
( ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( ( !! @ subst
@ ^ [Y0: subst] :
( ( sub @ ( X1 @ Y0 @ ( sK593 @ X1 ) ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y0 @ ( sK592 @ X1 ) ) @ ( sub @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) ) ) )
= $false )
| ( hoasinduction_no_psi_cond = $false )
| ( ( ^ [Y0: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( X1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) )
@ ( sK594 @ X1 ) )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f1738]) ).
thf(f1738,plain,
! [X1: subst > term > term] :
( ( hoasinduction_no_psi_cond = $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 )
| ( ( !! @ subst
@ ^ [Y0: subst] :
( ( sub @ ( X1 @ Y0 @ ( sK593 @ X1 ) ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y0 @ ( sK592 @ X1 ) ) @ ( sub @ ( sK593 @ X1 ) @ ( sK592 @ X1 ) ) ) ) )
= $false )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1737]) ).
thf(f1737,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 )
| ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( $false
= ( ^ [Y0: term] :
( !! @ subst
@ ^ [Y1: subst] :
( ( sub @ ( X1 @ Y1 @ Y0 ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y1 @ ( sK592 @ X1 ) ) @ ( sub @ Y0 @ ( sK592 @ X1 ) ) ) ) )
@ ( sK593 @ X1 ) ) )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(sigma_clausification,[status(thm)],[f1736]) ).
thf(f1736,plain,
! [X1: subst > term > term] :
( ( ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( $false
= ( !! @ term
@ ^ [Y0: term] :
( !! @ subst
@ ^ [Y1: subst] :
( ( sub @ ( X1 @ Y1 @ Y0 ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y1 @ ( sK592 @ X1 ) ) @ ( sub @ Y0 @ ( sK592 @ X1 ) ) ) ) ) ) )
| ( hoasinduction_no_psi_cond = $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 ) ),
inference(binary_proxy_clausification,[status(thm)],[f1735]) ).
thf(f1735,plain,
! [X1: subst > term > term] :
( ( hoasinduction_no_psi_cond = $false )
| ( ( ( !! @ 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 ) ) )
= $true )
| ( $false
= ( !! @ term
@ ^ [Y0: term] :
( !! @ subst
@ ^ [Y1: subst] :
( ( sub @ ( X1 @ Y1 @ Y0 ) @ ( sK592 @ X1 ) )
= ( X1 @ ( comp @ Y1 @ ( sK592 @ X1 ) ) @ ( sub @ Y0 @ ( sK592 @ X1 ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1734]) ).
thf(f1734,plain,
! [X1: subst > term > term] :
( ( hoasinduction_no_psi_cond = $false )
| ( $false
= ( ^ [Y0: subst] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( ( sub @ ( X1 @ Y2 @ Y1 ) @ Y0 )
= ( X1 @ ( comp @ Y2 @ Y0 ) @ ( sub @ Y1 @ Y0 ) ) ) ) )
@ ( 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 ) ) ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
= $true ) ),
inference(sigma_clausification,[status(thm)],[f1733]) ).
thf(f1733,plain,
! [X1: subst > term > term] :
( ( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( ( sub @ ( X1 @ Y2 @ Y1 ) @ Y0 )
= ( X1 @ ( comp @ Y2 @ Y0 ) @ ( sub @ Y1 @ Y0 ) ) ) ) ) )
= $false )
| ( ( ( !! @ 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 ) ) )
= $true )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1732]) ).
thf(f1732,plain,
! [X1: subst > term > term] :
( ( ( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( ( sub @ ( X1 @ Y2 @ Y1 ) @ Y0 )
= ( X1 @ ( comp @ Y2 @ Y0 ) @ ( sub @ Y1 @ 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 ) ) ) )
= $true )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1731]) ).
thf(f1731,plain,
! [X1: subst > term > term] :
( ( ( ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( !! @ subst
@ ^ [Y3: subst] :
( ( sub @ ( Y0 @ Y3 @ Y2 ) @ Y1 )
= ( Y0 @ ( comp @ Y3 @ Y1 ) @ ( sub @ Y2 @ 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 )
= $true )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(pi_clausification,[status(thm)],[f1730]) ).
thf(f1730,plain,
( ( ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( !! @ subst
@ ^ [Y3: subst] :
( ( sub @ ( Y0 @ Y3 @ Y2 ) @ Y1 )
= ( Y0 @ ( comp @ Y3 @ Y1 ) @ ( sub @ Y2 @ 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 ) ) ) ) )
= $true )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1721]) ).
thf(f1721,plain,
( ( hoasinduction_no_psi_cond = $false )
| ( $false
= ( ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( !! @ subst
@ ^ [Y3: subst] :
( ( sub @ ( Y0 @ Y3 @ Y2 ) @ Y1 )
= ( Y0 @ ( comp @ Y3 @ Y1 ) @ ( sub @ Y2 @ 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] : ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1720]) ).
thf(f1720,plain,
( ( $false
= ( ( !! @ term
@ ^ [Y0: term] :
( !! @ term
@ ^ [Y1: term] :
( ( sK591 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
=> ( sK591 @ ( d2subst @ d_id ) @ ( hoasap @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) @ Y0 ) @ ( d2subst @ d_id ) ) ) ) ) )
=> ( ( !! @ ( subst > term > term )
@ ^ [Y0: subst > term > term] :
( ( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( !! @ subst
@ ^ [Y3: subst] :
( ( sub @ ( Y0 @ Y3 @ Y2 ) @ Y1 )
= ( Y0 @ ( comp @ Y3 @ Y1 ) @ ( sub @ Y2 @ 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] : ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) ) ) ) ) )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1719]) ).
thf(f1719,plain,
( ( hoasinduction_no_psi_cond = $false )
| ( ( ^ [Y0: subst > term > subst > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( ( Y0 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( hoasap @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) ) ) )
=> ( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( !! @ subst
@ ^ [Y4: subst] :
( ( sub @ ( Y1 @ Y4 @ Y3 ) @ Y2 )
= ( Y1 @ ( comp @ Y4 @ Y2 ) @ ( sub @ Y3 @ 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] : ( Y0 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) ) ) ) )
@ sK591 )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f1718]) ).
thf(f1718,plain,
( ( hoasinduction_no_psi_cond = $false )
| ( $false
= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( ( Y0 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( hoasap @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) ) ) )
=> ( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( !! @ subst
@ ^ [Y4: subst] :
( ( sub @ ( Y1 @ Y4 @ Y3 ) @ Y2 )
= ( Y1 @ ( comp @ Y4 @ Y2 ) @ ( sub @ Y3 @ 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] : ( Y0 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1716]) ).
thf(f1716,plain,
( hoasinduction_no_psi_cond
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( ( Y0 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( hoasap @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) ) ) )
=> ( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( !! @ subst
@ ^ [Y4: subst] :
( ( sub @ ( Y1 @ Y4 @ Y3 ) @ Y2 )
= ( Y1 @ ( comp @ Y4 @ Y2 ) @ ( sub @ Y3 @ 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] : ( Y0 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1262]) ).
thf(f1262,plain,
( hoasinduction_no_psi_cond
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) )
=> ( ( Y0 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) )
=> ( Y0 @ ( d2subst @ d_id ) @ ( hoasap @ ( d2subst @ d_id ) @ Y2 @ ( d2subst @ d_id ) @ Y1 ) @ ( d2subst @ d_id ) ) ) ) ) )
=> ( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( !! @ subst
@ ^ [Y4: subst] :
( ( sub @ ( Y1 @ Y4 @ Y3 ) @ Y2 )
= ( Y1 @ ( comp @ Y4 @ Y2 ) @ ( sub @ Y3 @ 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] : ( Y0 @ ( d2subst @ d_id ) @ Y1 @ ( d2subst @ d_id ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f546,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1160,f1142,f1142,f1142,f1142]) ).
thf(f1160,plain,
( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1142,plain,
( ( d2subst @ d_id )
= id ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f546,plain,
( hoasinduction_no_psi_cond
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( ( Y0 @ id @ Y1 @ id )
=> ( Y0 @ id @ ( hoasap @ id @ Y2 @ id @ Y1 ) @ id ) ) ) ) )
=> ( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( !! @ subst
@ ^ [Y4: subst] :
( ( sub @ ( Y1 @ Y4 @ Y3 ) @ Y2 )
= ( Y1 @ ( comp @ Y4 @ Y2 ) @ ( sub @ Y3 @ 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] : ( Y0 @ id @ Y1 @ id ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( hoasinduction_no_psi_cond
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( ( Y0 @ id @ Y1 @ id )
=> ( Y0 @ id @ ( hoasap @ id @ Y2 @ id @ Y1 ) @ id ) ) ) ) )
=> ( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( !! @ subst
@ ^ [Y4: subst] :
( ( sub @ ( Y1 @ Y4 @ Y3 ) @ Y2 )
= ( Y1 @ ( comp @ Y4 @ Y2 ) @ ( sub @ Y3 @ 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] : ( Y0 @ id @ Y1 @ id ) ) ) ) ) ),
inference(flattening,[status(thm)],[f8]) ).
thf(f8,plain,
( hoasinduction_no_psi_cond
!= ( !! @ ( subst > term > subst > $o )
@ ^ [Y0: subst > term > subst > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
=> ( ( Y0 @ id @ Y1 @ id )
=> ( Y0 @ id @ ( hoasap @ id @ Y2 @ id @ Y1 ) @ id ) ) ) ) )
=> ( ( !! @ ( subst > term > term )
@ ^ [Y1: subst > term > term] :
( ( !! @ subst
@ ^ [Y2: subst] :
( !! @ term
@ ^ [Y3: term] :
( !! @ subst
@ ^ [Y4: subst] :
( ( sub @ ( Y1 @ Y4 @ Y3 ) @ Y2 )
= ( Y1 @ ( comp @ Y4 @ Y2 ) @ ( sub @ Y3 @ 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] : ( Y0 @ id @ Y1 @ id ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ! [X0: subst > term > subst > $o] :
( ( ! [X1: term,X2: term] :
( ( X0 @ id @ X1 @ id )
=> ( ( X0 @ id @ X2 @ id )
=> ( X0 @ id @ ( hoasap @ id @ X1 @ id @ X2 ) @ id ) ) )
=> ( ! [X3: subst > term > term] :
( ! [X4: subst,X5: term,X6: subst] :
( ( sub @ ( X3 @ X4 @ X5 ) @ X6 )
= ( X3 @ ( comp @ X4 @ X6 ) @ ( sub @ X5 @ X6 ) ) )
=> ( ! [X7: term] :
( ( X0 @ id @ X7 @ id )
=> ( X0 @ id @ ( X3 @ id @ X7 ) @ id ) )
=> ( X0 @ id
@ ( hoaslam @ id
@ ^ [X8: subst,X9: term] : ( X3 @ X8 @ X9 ) )
@ id ) ) )
=> ! [X10: term] : ( X0 @ id @ X10 @ id ) ) )
= hoasinduction_no_psi_cond ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
~ ! [X10: subst > term > subst > $o] :
( ( ! [X13: term,X15: term] :
( ( X10 @ id @ X13 @ id )
=> ( ( X10 @ id @ X15 @ id )
=> ( X10 @ id @ ( hoasap @ id @ X13 @ id @ X15 ) @ id ) ) )
=> ( ! [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
@ ( hoaslam @ id
@ ^ [X14: subst,X13: term] : ( X18 @ X14 @ X13 ) )
@ id ) ) )
=> ! [X13: term] : ( X10 @ id @ X13 @ id ) ) )
= hoasinduction_no_psi_cond ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
! [X10: subst > term > subst > $o] :
( ( ! [X13: term,X15: term] :
( ( X10 @ id @ X13 @ id )
=> ( ( X10 @ id @ X15 @ id )
=> ( X10 @ id @ ( hoasap @ id @ X13 @ id @ X15 ) @ id ) ) )
=> ( ! [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
@ ( hoaslam @ id
@ ^ [X14: subst,X13: term] : ( X18 @ X14 @ X13 ) )
@ id ) ) )
=> ! [X13: term] : ( X10 @ id @ X13 @ id ) ) )
= hoasinduction_no_psi_cond ),
file('/export/starexec/sandbox2/tmp/tmp.Qe5lgCyrIb/DTF2THF_31149.p',hoasinduction_no_psi_cond) ).
thf(f1959,plain,
! [X0: term,X1: subst] :
( ( ( sK591 @ X1 @ X0 @ X1 )
= $false )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(superposition,[status(thm)],[f1956,f1898]) ).
thf(f1956,plain,
! [X0: subst] :
( ( ( sK591 @ X0 @ sK597 @ X0 )
= $false )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(superposition,[status(thm)],[f1750,f1951]) ).
thf(f1951,plain,
! [X159: subst] :
( ( d2subst @ d_id )
= X159 ),
inference(forward_demodulation,[status(thm)],[f1140,f1099]) ).
thf(f1099,plain,
! [X219: d_subst] : ( d_id = X219 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1140,plain,
! [X159: subst] :
( ( d2subst @ ( sK534 @ X159 ) )
= X159 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1750,plain,
( ( ( sK591 @ ( d2subst @ d_id ) @ sK597 @ ( d2subst @ d_id ) )
= $false )
| ( hoasinduction_no_psi_cond = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1749]) ).
thf(f1749,plain,
( ( hoasinduction_no_psi_cond = $false )
| ( ( ^ [Y0: term] : ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) )
@ sK597 )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f1729]) ).
thf(f1729,plain,
( ( hoasinduction_no_psi_cond = $false )
| ( $false
= ( !! @ term
@ ^ [Y0: term] : ( sK591 @ ( d2subst @ d_id ) @ Y0 @ ( d2subst @ d_id ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1721]) ).
thf(f1259,plain,
( ( ( sP58 @ sK458 )
!= $true )
| ( hoasinduction_no_psi_cond = $true ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f5107,plain,
( ( sP58 @ sK458 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f5096]) ).
thf(f5096,plain,
( ( $true != $true )
| ( ( sP58 @ sK458 )
= $true ) ),
inference(superposition,[status(thm)],[f2354,f5024]) ).
thf(f5024,plain,
! [X0: subst > term > subst > $o] :
( ( ( X0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sP58 @ X0 )
= $true ) ),
inference(subsumption_resolution,[status(thm)],[f5023,f4469]) ).
thf(f4469,plain,
! [X8: subst > term > subst > $o,X14: term] :
( ( ( X8 @ ( d2subst @ d_id ) @ X14 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4466]) ).
thf(f4466,plain,
! [X8: subst > term > subst > $o,X14: term] :
( ( ( X8 @ ( d2subst @ d_id ) @ X14 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( $true != $true ) ),
inference(backward_demodulation,[status(thm)],[f3971,f4465]) ).
thf(f4465,plain,
sP37 = $true,
inference(subsumption_resolution,[status(thm)],[f4464,f2972]) ).
thf(f2972,plain,
! [X0: term,X1: subst] :
( ( ( sK188 @ X1 @ X0 @ X1 )
!= $true )
| ( sP37 = $true ) ),
inference(superposition,[status(thm)],[f2378,f1898]) ).
thf(f2378,plain,
! [X0: subst] :
( ( ( sK188 @ X0 @ ( lam @ sK189 ) @ X0 )
!= $true )
| ( sP37 = $true ) ),
inference(superposition,[status(thm)],[f1372,f1951]) ).
thf(f1372,plain,
( ( $true
!= ( sK188 @ ( d2subst @ d_id ) @ ( lam @ sK189 ) @ ( d2subst @ d_id ) ) )
| ( sP37 = $true ) ),
inference(definition_unfolding,[status(thm)],[f717,f1142,f1142]) ).
thf(f717,plain,
( ( sP37 = $true )
| ( ( sK188 @ id @ ( lam @ sK189 ) @ id )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f217]) ).
thf(f217,plain,
( ( ( sP37 = $true )
| ( ! [X2: subst > term > term] :
( ( ( sK188 @ id @ ( lam @ ( X2 @ sh @ one ) ) @ id )
= $true )
| ( ( X2 @ ( comp @ ( sK191 @ X2 ) @ ( sK190 @ X2 ) ) @ ( sub @ ( sK192 @ X2 ) @ ( sK190 @ X2 ) ) )
!= ( sub @ ( X2 @ ( sK191 @ X2 ) @ ( sK192 @ X2 ) ) @ ( sK190 @ X2 ) ) )
| ( ( ( sK188 @ id @ ( X2 @ id @ ( sK193 @ X2 ) ) @ id )
!= $true )
& ( ( sK188 @ id @ ( sK193 @ X2 ) @ id )
= $true ) ) )
& ! [X7: term] :
( ( ( sK188 @ id @ ( sub @ sK189 @ ( push @ X7 @ id ) ) @ id )
= $true )
| ( ( sK188 @ id @ X7 @ id )
!= $true ) )
& ( ( sK188 @ id @ ( lam @ sK189 ) @ id )
!= $true ) ) )
& ( ! [X8: subst > term > subst > $o,X9: term] :
( ( ( ( X8 @ id @ ( lam @ ( sK194 @ X8 @ sh @ one ) ) @ id )
!= $true )
& ! [X11: subst,X12: subst,X13: term] :
( ( sK194 @ X8 @ ( comp @ X12 @ X11 ) @ ( sub @ X13 @ X11 ) )
= ( sub @ ( sK194 @ X8 @ X12 @ X13 ) @ X11 ) )
& ! [X14: term] :
( ( ( X8 @ id @ ( sK194 @ X8 @ id @ X14 ) @ id )
= $true )
| ( ( X8 @ id @ X14 @ id )
!= $true ) ) )
| ( ( ( X8 @ id @ ( sub @ X9 @ ( push @ ( sK195 @ X9 @ X8 ) @ id ) ) @ id )
!= $true )
& ( ( X8 @ id @ ( sK195 @ X9 @ X8 ) @ id )
= $true ) )
| ( ( X8 @ id @ ( lam @ X9 ) @ id )
= $true ) )
| ( sP37 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK188,sK189,sK190,sK191,sK192,sK193,sK194,sK195])],[f211,f216,f215,f214,f213,f212]) ).
thf(f212,plain,
( ? [X0: subst > term > subst > $o,X1: term] :
( ! [X2: subst > term > term] :
( ( ( X0 @ id @ ( lam @ ( X2 @ sh @ one ) ) @ id )
= $true )
| ? [X3: subst,X4: subst,X5: term] :
( ( X2 @ ( comp @ X4 @ X3 ) @ ( sub @ X5 @ X3 ) )
!= ( sub @ ( X2 @ X4 @ X5 ) @ X3 ) )
| ? [X6: term] :
( ( ( X0 @ id @ ( X2 @ id @ X6 ) @ id )
!= $true )
& ( ( X0 @ id @ X6 @ id )
= $true ) ) )
& ! [X7: term] :
( ( ( X0 @ id @ ( sub @ X1 @ ( push @ X7 @ id ) ) @ id )
= $true )
| ( ( X0 @ id @ X7 @ id )
!= $true ) )
& ( ( X0 @ id @ ( lam @ X1 ) @ id )
!= $true ) )
=> ( ! [X2: subst > term > term] :
( ( ( sK188 @ id @ ( lam @ ( X2 @ sh @ one ) ) @ id )
= $true )
| ? [X3: subst,X4: subst,X5: term] :
( ( X2 @ ( comp @ X4 @ X3 ) @ ( sub @ X5 @ X3 ) )
!= ( sub @ ( X2 @ X4 @ X5 ) @ X3 ) )
| ? [X6: term] :
( ( ( sK188 @ id @ ( X2 @ id @ X6 ) @ id )
!= $true )
& ( ( sK188 @ id @ X6 @ id )
= $true ) ) )
& ! [X7: term] :
( ( ( sK188 @ id @ ( sub @ sK189 @ ( push @ X7 @ id ) ) @ id )
= $true )
| ( ( sK188 @ id @ X7 @ id )
!= $true ) )
& ( ( sK188 @ id @ ( lam @ sK189 ) @ id )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f213,plain,
! [X2: subst > term > term] :
( ? [X3: subst,X4: subst,X5: term] :
( ( X2 @ ( comp @ X4 @ X3 ) @ ( sub @ X5 @ X3 ) )
!= ( sub @ ( X2 @ X4 @ X5 ) @ X3 ) )
=> ( ( X2 @ ( comp @ ( sK191 @ X2 ) @ ( sK190 @ X2 ) ) @ ( sub @ ( sK192 @ X2 ) @ ( sK190 @ X2 ) ) )
!= ( sub @ ( X2 @ ( sK191 @ X2 ) @ ( sK192 @ X2 ) ) @ ( sK190 @ X2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f214,plain,
! [X2: subst > term > term] :
( ? [X6: term] :
( ( ( sK188 @ id @ ( X2 @ id @ X6 ) @ id )
!= $true )
& ( ( sK188 @ id @ X6 @ id )
= $true ) )
=> ( ( ( sK188 @ id @ ( X2 @ id @ ( sK193 @ X2 ) ) @ id )
!= $true )
& ( ( sK188 @ id @ ( sK193 @ X2 ) @ id )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f215,plain,
! [X8: subst > term > subst > $o] :
( ? [X10: subst > term > term] :
( ( ( X8 @ id @ ( lam @ ( X10 @ sh @ one ) ) @ id )
!= $true )
& ! [X11: subst,X12: subst,X13: term] :
( ( sub @ ( X10 @ X12 @ X13 ) @ X11 )
= ( X10 @ ( comp @ X12 @ X11 ) @ ( sub @ X13 @ X11 ) ) )
& ! [X14: term] :
( ( ( X8 @ id @ ( X10 @ id @ X14 ) @ id )
= $true )
| ( ( X8 @ id @ X14 @ id )
!= $true ) ) )
=> ( ( ( X8 @ id @ ( lam @ ( sK194 @ X8 @ sh @ one ) ) @ id )
!= $true )
& ! [X13: term,X12: subst,X11: subst] :
( ( sK194 @ X8 @ ( comp @ X12 @ X11 ) @ ( sub @ X13 @ X11 ) )
= ( sub @ ( sK194 @ X8 @ X12 @ X13 ) @ X11 ) )
& ! [X14: term] :
( ( ( X8 @ id @ ( sK194 @ X8 @ id @ X14 ) @ id )
= $true )
| ( ( X8 @ id @ X14 @ id )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f216,plain,
! [X8: subst > term > subst > $o,X9: term] :
( ? [X15: term] :
( ( ( X8 @ id @ ( sub @ X9 @ ( push @ X15 @ id ) ) @ id )
!= $true )
& ( ( X8 @ id @ X15 @ id )
= $true ) )
=> ( ( ( X8 @ id @ ( sub @ X9 @ ( push @ ( sK195 @ X9 @ X8 ) @ id ) ) @ id )
!= $true )
& ( ( X8 @ id @ ( sK195 @ X9 @ X8 ) @ id )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f211,plain,
( ( ( sP37 = $true )
| ? [X0: subst > term > subst > $o,X1: term] :
( ! [X2: subst > term > term] :
( ( ( X0 @ id @ ( lam @ ( X2 @ sh @ one ) ) @ id )
= $true )
| ? [X3: subst,X4: subst,X5: term] :
( ( X2 @ ( comp @ X4 @ X3 ) @ ( sub @ X5 @ X3 ) )
!= ( sub @ ( X2 @ X4 @ X5 ) @ X3 ) )
| ? [X6: term] :
( ( ( X0 @ id @ ( X2 @ id @ X6 ) @ id )
!= $true )
& ( ( X0 @ id @ X6 @ id )
= $true ) ) )
& ! [X7: term] :
( ( ( X0 @ id @ ( sub @ X1 @ ( push @ X7 @ id ) ) @ id )
= $true )
| ( ( X0 @ id @ X7 @ id )
!= $true ) )
& ( ( X0 @ id @ ( lam @ X1 ) @ id )
!= $true ) ) )
& ( ! [X8: subst > term > subst > $o,X9: term] :
( ? [X10: subst > term > term] :
( ( ( X8 @ id @ ( lam @ ( X10 @ sh @ one ) ) @ id )
!= $true )
& ! [X11: subst,X12: subst,X13: term] :
( ( sub @ ( X10 @ X12 @ X13 ) @ X11 )
= ( X10 @ ( comp @ X12 @ X11 ) @ ( sub @ X13 @ X11 ) ) )
& ! [X14: term] :
( ( ( X8 @ id @ ( X10 @ id @ X14 ) @ id )
= $true )
| ( ( X8 @ id @ X14 @ id )
!= $true ) ) )
| ? [X15: term] :
( ( ( X8 @ id @ ( sub @ X9 @ ( push @ X15 @ id ) ) @ id )
!= $true )
& ( ( X8 @ id @ X15 @ id )
= $true ) )
| ( ( X8 @ id @ ( lam @ X9 ) @ id )
= $true ) )
| ( sP37 != $true ) ) ),
inference(rectify,[status(thm)],[f210]) ).
thf(f210,plain,
( ( ( sP37 = $true )
| ? [X75: subst > term > subst > $o,X76: term] :
( ! [X77: subst > term > term] :
( ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
= $true )
| ? [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
!= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) )
| ? [X81: term] :
( ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
!= $true )
& ( ( X75 @ id @ X81 @ id )
= $true ) ) )
& ! [X82: term] :
( ( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
= $true )
| ( ( X75 @ id @ X82 @ id )
!= $true ) )
& ( ( X75 @ id @ ( lam @ X76 ) @ id )
!= $true ) ) )
& ( ! [X75: subst > term > subst > $o,X76: term] :
( ? [X77: subst > term > term] :
( ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
!= $true )
& ! [X78: subst,X79: subst,X80: term] :
( ( sub @ ( X77 @ X79 @ X80 ) @ X78 )
= ( X77 @ ( comp @ X79 @ X78 ) @ ( sub @ X80 @ X78 ) ) )
& ! [X81: term] :
( ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
= $true )
| ( ( X75 @ id @ X81 @ id )
!= $true ) ) )
| ? [X82: term] :
( ( ( X75 @ id @ ( sub @ X76 @ ( push @ X82 @ id ) ) @ id )
!= $true )
& ( ( X75 @ id @ X82 @ id )
= $true ) )
| ( ( X75 @ id @ ( lam @ X76 ) @ id )
= $true ) )
| ( sP37 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f51]) ).
thf(f4464,plain,
! [X2: subst > term > term] :
( ( ( sK188 @ ( d2subst @ d_id ) @ ( lam @ ( X2 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) )
= $true )
| ( sP37 = $true ) ),
inference(subsumption_resolution,[status(thm)],[f4463,f2972]) ).
thf(f4463,plain,
! [X2: subst > term > term] :
( ( ( sK188 @ ( d2subst @ d_id ) @ ( sK193 @ X2 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sK188 @ ( d2subst @ d_id ) @ ( lam @ ( X2 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) )
= $true )
| ( sP37 = $true ) ),
inference(subsumption_resolution,[status(thm)],[f1370,f1898]) ).
thf(f1370,plain,
! [X2: subst > term > term] :
( ( ( sK188 @ ( d2subst @ d_id ) @ ( lam @ ( X2 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) )
= $true )
| ( ( X2 @ ( comp @ ( sK191 @ X2 ) @ ( sK190 @ X2 ) ) @ ( sub @ ( sK192 @ X2 ) @ ( sK190 @ X2 ) ) )
!= ( sub @ ( X2 @ ( sK191 @ X2 ) @ ( sK192 @ X2 ) ) @ ( sK190 @ X2 ) ) )
| ( sP37 = $true )
| ( ( sK188 @ ( d2subst @ d_id ) @ ( sK193 @ X2 ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f719,f1142,f1202,f1148,f1142,f1142,f1142]) ).
thf(f1148,plain,
( one
= ( d2term @ d_one ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1202,plain,
( ( d2subst @ d_id )
= sh ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f719,plain,
! [X2: subst > term > term] :
( ( sP37 = $true )
| ( ( sK188 @ id @ ( lam @ ( X2 @ sh @ one ) ) @ id )
= $true )
| ( ( X2 @ ( comp @ ( sK191 @ X2 ) @ ( sK190 @ X2 ) ) @ ( sub @ ( sK192 @ X2 ) @ ( sK190 @ X2 ) ) )
!= ( sub @ ( X2 @ ( sK191 @ X2 ) @ ( sK192 @ X2 ) ) @ ( sK190 @ X2 ) ) )
| ( ( sK188 @ id @ ( sK193 @ X2 ) @ id )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f217]) ).
thf(f3971,plain,
! [X8: subst > term > subst > $o,X14: term] :
( ( ( X8 @ ( d2subst @ d_id ) @ X14 @ ( d2subst @ d_id ) )
!= $true )
| ( sP37 != $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(subsumption_resolution,[status(thm)],[f3970,f3704]) ).
thf(f3704,plain,
! [X2: term,X0: subst > term > subst > $o,X1: term] :
( ( ( X0 @ ( d2subst @ d_id ) @ X2 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ X1 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f3703,f1894]) ).
thf(f3703,plain,
! [X2: term,X0: subst > term > subst > $o,X1: term] :
( ( ( X0 @ ( d2subst @ d_id ) @ X2 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ X1 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ ( ap @ ( sub @ X1 @ ( d2subst @ d_id ) ) @ X2 ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(subsumption_resolution,[status(thm)],[f1299,f3549]) ).
thf(f3549,plain,
! [X0: subst > term > subst > $o] :
( ( sP50 @ X0 )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f3548,f2752]) ).
thf(f2752,plain,
! [X0: subst > term > subst > $o] :
( ( ( sP50 @ X0 )
!= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f1301,f1894]) ).
thf(f1301,plain,
! [X0: subst > term > subst > $o] :
( ( ( X0 @ ( d2subst @ d_id ) @ ( sK106 @ X0 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sP50 @ X0 )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f612,f1142,f1142]) ).
thf(f612,plain,
! [X0: subst > term > subst > $o] :
( ( ( X0 @ id @ ( sK106 @ X0 ) @ id )
= $true )
| ( ( sP50 @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f135]) ).
thf(f135,plain,
! [X0: subst > term > subst > $o] :
( ( ( ( sP50 @ X0 )
= $true )
| ! [X1: term,X2: term] :
( ( ( X0 @ id @ ( ap @ ( sub @ X1 @ id ) @ X2 ) @ id )
= $true )
| ( $true
!= ( X0 @ id @ X1 @ id ) )
| ( ( X0 @ id @ X2 @ id )
!= $true ) ) )
& ( ( ( $true
!= ( X0 @ id @ ( ap @ ( sub @ ( sK106 @ X0 ) @ id ) @ ( sK107 @ X0 ) ) @ id ) )
& ( ( X0 @ id @ ( sK106 @ X0 ) @ id )
= $true )
& ( ( X0 @ id @ ( sK107 @ X0 ) @ id )
= $true ) )
| ( ( sP50 @ X0 )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK106,sK107])],[f133,f134]) ).
thf(f134,plain,
! [X0: subst > term > subst > $o] :
( ? [X3: term,X4: term] :
( ( ( X0 @ id @ ( ap @ ( sub @ X3 @ id ) @ X4 ) @ id )
!= $true )
& ( ( X0 @ id @ X3 @ id )
= $true )
& ( ( X0 @ id @ X4 @ id )
= $true ) )
=> ( ( $true
!= ( X0 @ id @ ( ap @ ( sub @ ( sK106 @ X0 ) @ id ) @ ( sK107 @ X0 ) ) @ id ) )
& ( ( X0 @ id @ ( sK106 @ X0 ) @ id )
= $true )
& ( ( X0 @ id @ ( sK107 @ X0 ) @ id )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f133,plain,
! [X0: subst > term > subst > $o] :
( ( ( ( sP50 @ X0 )
= $true )
| ! [X1: term,X2: term] :
( ( ( X0 @ id @ ( ap @ ( sub @ X1 @ id ) @ X2 ) @ id )
= $true )
| ( $true
!= ( X0 @ id @ X1 @ id ) )
| ( ( X0 @ id @ X2 @ id )
!= $true ) ) )
& ( ? [X3: term,X4: term] :
( ( ( X0 @ id @ ( ap @ ( sub @ X3 @ id ) @ X4 ) @ id )
!= $true )
& ( ( X0 @ id @ X3 @ id )
= $true )
& ( ( X0 @ id @ X4 @ id )
= $true ) )
| ( ( sP50 @ X0 )
!= $true ) ) ),
inference(rectify,[status(thm)],[f132]) ).
thf(f132,plain,
! [X219: subst > term > subst > $o] :
( ( ( ( sP50 @ X219 )
= $true )
| ! [X221: term,X222: term] :
( ( ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id )
= $true )
| ( ( X219 @ id @ X221 @ id )
!= $true )
| ( ( X219 @ id @ X222 @ id )
!= $true ) ) )
& ( ? [X221: term,X222: term] :
( ( ( X219 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id )
!= $true )
& ( ( X219 @ id @ X221 @ id )
= $true )
& ( ( X219 @ id @ X222 @ id )
= $true ) )
| ( ( sP50 @ X219 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f64]) ).
thf(f3548,plain,
! [X0: subst > term > subst > $o] :
( ( ( sP50 @ X0 )
!= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f1300,f1894]) ).
thf(f1300,plain,
! [X0: subst > term > subst > $o] :
( ( ( X0 @ ( d2subst @ d_id ) @ ( ap @ ( sub @ ( sK106 @ X0 ) @ ( d2subst @ d_id ) ) @ ( sK107 @ X0 ) ) @ ( d2subst @ d_id ) )
!= $true )
| ( ( sP50 @ X0 )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f613,f1142,f1142,f1142]) ).
thf(f613,plain,
! [X0: subst > term > subst > $o] :
( ( $true
!= ( X0 @ id @ ( ap @ ( sub @ ( sK106 @ X0 ) @ id ) @ ( sK107 @ X0 ) ) @ id ) )
| ( ( sP50 @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f135]) ).
thf(f1299,plain,
! [X2: term,X0: subst > term > subst > $o,X1: term] :
( ( ( sP50 @ X0 )
= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ ( ap @ ( sub @ X1 @ ( d2subst @ d_id ) ) @ X2 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ X2 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ X1 @ ( d2subst @ d_id ) )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f614,f1142,f1142,f1142,f1142,f1142,f1142,f1142]) ).
thf(f614,plain,
! [X2: term,X0: subst > term > subst > $o,X1: term] :
( ( ( sP50 @ X0 )
= $true )
| ( ( X0 @ id @ ( ap @ ( sub @ X1 @ id ) @ X2 ) @ id )
= $true )
| ( $true
!= ( X0 @ id @ X1 @ id ) )
| ( ( X0 @ id @ X2 @ id )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f135]) ).
thf(f3970,plain,
! [X8: subst > term > subst > $o,X9: term,X14: term] :
( ( ( X8 @ ( d2subst @ d_id ) @ X14 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( lam @ X9 ) @ ( d2subst @ d_id ) )
= $true )
| ( sP37 != $true ) ),
inference(subsumption_resolution,[status(thm)],[f3969,f3704]) ).
thf(f3969,plain,
! [X8: subst > term > subst > $o,X9: term,X14: term] :
( ( ( X8 @ ( d2subst @ d_id ) @ X14 @ ( d2subst @ d_id ) )
!= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true )
| ( sP37 != $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( sK194 @ X8 @ ( d2subst @ d_id ) @ X14 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( lam @ X9 ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f1378,f1894]) ).
thf(f1378,plain,
! [X8: subst > term > subst > $o,X9: term,X14: term] :
( ( ( X8 @ ( d2subst @ d_id ) @ ( sK195 @ X9 @ X8 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( sK194 @ X8 @ ( d2subst @ d_id ) @ X14 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ ( lam @ X9 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( X8 @ ( d2subst @ d_id ) @ X14 @ ( d2subst @ d_id ) )
!= $true )
| ( sP37 != $true ) ),
inference(definition_unfolding,[status(thm)],[f711,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142]) ).
thf(f711,plain,
! [X8: subst > term > subst > $o,X9: term,X14: term] :
( ( ( X8 @ id @ ( sK194 @ X8 @ id @ X14 ) @ id )
= $true )
| ( ( X8 @ id @ X14 @ id )
!= $true )
| ( ( X8 @ id @ ( sK195 @ X9 @ X8 ) @ id )
= $true )
| ( ( X8 @ id @ ( lam @ X9 ) @ id )
= $true )
| ( sP37 != $true ) ),
inference(cnf_transformation,[status(thm)],[f217]) ).
thf(f5023,plain,
! [X0: subst > term > subst > $o,X1: subst > term > term] :
( ( ( X0 @ ( d2subst @ d_id ) @ ( lam @ ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sP58 @ X0 )
= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f5022,f1894]) ).
thf(f5022,plain,
! [X0: subst > term > subst > $o,X1: subst > term > term] :
( ( ( X0 @ ( d2subst @ d_id ) @ ( sK59 @ X1 @ X0 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sP58 @ X0 )
= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ ( lam @ ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(subsumption_resolution,[status(thm)],[f1264,f1898]) ).
thf(f1264,plain,
! [X0: subst > term > subst > $o,X1: subst > term > term] :
( ( ( X1 @ ( comp @ ( sK60 @ X1 ) @ ( sK61 @ X1 ) ) @ ( sub @ ( sK62 @ X1 ) @ ( sK61 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK60 @ X1 ) @ ( sK62 @ X1 ) ) @ ( sK61 @ X1 ) ) )
| ( ( X0 @ ( d2subst @ d_id ) @ ( sK59 @ X1 @ X0 ) @ ( d2subst @ d_id ) )
= $true )
| ( ( sP58 @ X0 )
= $true )
| ( ( X0 @ ( d2subst @ d_id ) @ ( lam @ ( X1 @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) ) @ ( d2subst @ d_id ) )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f550,f1142,f1142,f1142,f1202,f1148,f1142]) ).
thf(f550,plain,
! [X0: subst > term > subst > $o,X1: subst > term > term] :
( ( ( sP58 @ X0 )
= $true )
| ( ( X0 @ id @ ( sK59 @ X1 @ X0 ) @ id )
= $true )
| ( ( X0 @ id @ ( lam @ ( X1 @ sh @ one ) ) @ id )
= $true )
| ( ( X1 @ ( comp @ ( sK60 @ X1 ) @ ( sK61 @ X1 ) ) @ ( sub @ ( sK62 @ X1 ) @ ( sK61 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK60 @ X1 ) @ ( sK62 @ X1 ) ) @ ( sK61 @ X1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f79]) ).
thf(f79,plain,
! [X0: subst > term > subst > $o] :
( ( ( ( sP58 @ X0 )
= $true )
| ! [X1: subst > term > term] :
( ( ( ( X0 @ id @ ( X1 @ id @ ( sK59 @ X1 @ X0 ) ) @ id )
!= $true )
& ( ( X0 @ id @ ( sK59 @ X1 @ X0 ) @ id )
= $true ) )
| ( ( X0 @ id @ ( lam @ ( X1 @ sh @ one ) ) @ id )
= $true )
| ( ( X1 @ ( comp @ ( sK60 @ X1 ) @ ( sK61 @ X1 ) ) @ ( sub @ ( sK62 @ X1 ) @ ( sK61 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK60 @ X1 ) @ ( sK62 @ X1 ) ) @ ( sK61 @ X1 ) ) ) ) )
& ( ( ! [X7: term] :
( ( ( X0 @ id @ ( sK63 @ X0 @ id @ X7 ) @ id )
= $true )
| ( ( X0 @ id @ X7 @ id )
!= $true ) )
& ( ( X0 @ id @ ( lam @ ( sK63 @ X0 @ sh @ one ) ) @ id )
!= $true )
& ! [X8: subst,X9: subst,X10: term] :
( ( sub @ ( sK63 @ X0 @ X8 @ X10 ) @ X9 )
= ( sK63 @ X0 @ ( comp @ X8 @ X9 ) @ ( sub @ X10 @ X9 ) ) ) )
| ( ( sP58 @ X0 )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK59,sK60,sK61,sK62,sK63])],[f75,f78,f77,f76]) ).
thf(f76,plain,
! [X0: subst > term > subst > $o,X1: subst > term > term] :
( ? [X2: term] :
( ( ( X0 @ id @ ( X1 @ id @ X2 ) @ id )
!= $true )
& ( ( X0 @ id @ X2 @ id )
= $true ) )
=> ( ( ( X0 @ id @ ( X1 @ id @ ( sK59 @ X1 @ X0 ) ) @ id )
!= $true )
& ( ( X0 @ id @ ( sK59 @ X1 @ X0 ) @ id )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f77,plain,
! [X1: subst > term > term] :
( ? [X3: subst,X4: subst,X5: term] :
( ( X1 @ ( comp @ X3 @ X4 ) @ ( sub @ X5 @ X4 ) )
!= ( sub @ ( X1 @ X3 @ X5 ) @ X4 ) )
=> ( ( X1 @ ( comp @ ( sK60 @ X1 ) @ ( sK61 @ X1 ) ) @ ( sub @ ( sK62 @ X1 ) @ ( sK61 @ X1 ) ) )
!= ( sub @ ( X1 @ ( sK60 @ X1 ) @ ( sK62 @ X1 ) ) @ ( sK61 @ X1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f78,plain,
! [X0: subst > term > subst > $o] :
( ? [X6: subst > term > term] :
( ! [X7: term] :
( ( ( X0 @ id @ ( X6 @ id @ X7 ) @ id )
= $true )
| ( ( X0 @ id @ X7 @ id )
!= $true ) )
& ( ( X0 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id )
!= $true )
& ! [X8: subst,X9: subst,X10: term] :
( ( sub @ ( X6 @ X8 @ X10 ) @ X9 )
= ( X6 @ ( comp @ X8 @ X9 ) @ ( sub @ X10 @ X9 ) ) ) )
=> ( ! [X7: term] :
( ( ( X0 @ id @ ( sK63 @ X0 @ id @ X7 ) @ id )
= $true )
| ( ( X0 @ id @ X7 @ id )
!= $true ) )
& ( ( X0 @ id @ ( lam @ ( sK63 @ X0 @ sh @ one ) ) @ id )
!= $true )
& ! [X10: term,X9: subst,X8: subst] :
( ( sub @ ( sK63 @ X0 @ X8 @ X10 ) @ X9 )
= ( sK63 @ X0 @ ( comp @ X8 @ X9 ) @ ( sub @ X10 @ X9 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f75,plain,
! [X0: subst > term > subst > $o] :
( ( ( ( sP58 @ X0 )
= $true )
| ! [X1: subst > term > term] :
( ? [X2: term] :
( ( ( X0 @ id @ ( X1 @ id @ X2 ) @ id )
!= $true )
& ( ( X0 @ id @ X2 @ id )
= $true ) )
| ( ( X0 @ id @ ( lam @ ( X1 @ sh @ one ) ) @ id )
= $true )
| ? [X3: subst,X4: subst,X5: term] :
( ( X1 @ ( comp @ X3 @ X4 ) @ ( sub @ X5 @ X4 ) )
!= ( sub @ ( X1 @ X3 @ X5 ) @ X4 ) ) ) )
& ( ? [X6: subst > term > term] :
( ! [X7: term] :
( ( ( X0 @ id @ ( X6 @ id @ X7 ) @ id )
= $true )
| ( ( X0 @ id @ X7 @ id )
!= $true ) )
& ( ( X0 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id )
!= $true )
& ! [X8: subst,X9: subst,X10: term] :
( ( sub @ ( X6 @ X8 @ X10 ) @ X9 )
= ( X6 @ ( comp @ X8 @ X9 ) @ ( sub @ X10 @ X9 ) ) ) )
| ( ( sP58 @ X0 )
!= $true ) ) ),
inference(rectify,[status(thm)],[f74]) ).
thf(f74,plain,
! [X184: subst > term > subst > $o] :
( ( ( ( sP58 @ X184 )
= $true )
| ! [X187: subst > term > term] :
( ? [X188: term] :
( ( ( X184 @ id @ ( X187 @ id @ X188 ) @ id )
!= $true )
& ( ( X184 @ id @ X188 @ id )
= $true ) )
| ( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
= $true )
| ? [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
!= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) ) ) )
& ( ? [X187: subst > term > term] :
( ! [X188: term] :
( ( ( X184 @ id @ ( X187 @ id @ X188 ) @ id )
= $true )
| ( ( X184 @ id @ X188 @ id )
!= $true ) )
& ( ( X184 @ id @ ( lam @ ( X187 @ sh @ one ) ) @ id )
!= $true )
& ! [X189: subst,X190: subst,X191: term] :
( ( X187 @ ( comp @ X189 @ X190 ) @ ( sub @ X191 @ X190 ) )
= ( sub @ ( X187 @ X189 @ X191 ) @ X190 ) ) )
| ( ( sP58 @ X184 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f72]) ).
thf(f2354,plain,
! [X0: term,X1: subst] :
( ( sK458 @ X1 @ X0 @ X1 )
!= $true ),
inference(superposition,[status(thm)],[f2350,f1898]) ).
thf(f2350,plain,
! [X0: subst] :
( ( sK458 @ X0 @ sK457 @ X0 )
!= $true ),
inference(superposition,[status(thm)],[f2344,f1951]) ).
thf(f2344,plain,
( ( sK458 @ ( d2subst @ d_id ) @ sK457 @ ( d2subst @ d_id ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2343]) ).
thf(f2343,plain,
( ( $false = $true )
| ( ( sK458 @ ( d2subst @ d_id ) @ sK457 @ ( d2subst @ d_id ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f1598,f2159]) ).
thf(f1598,plain,
( ( hoasinduction_no_psi_cond = $true )
| ( ( sK458 @ ( d2subst @ d_id ) @ sK457 @ ( d2subst @ d_id ) )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f1258,f1142,f1142]) ).
thf(f1258,plain,
( ( hoasinduction_no_psi_cond = $true )
| ( ( sK458 @ id @ sK457 @ id )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
%------------------------------------------------------------------------------
%----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
% 0.15/0.34 % Computer : n026.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue May 5 09:14:53 EDT 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.30/0.46 % Domain : Software Verification
% 0.30/0.46 % Problem : Benchmark: ALG444^1, Active target: hoasinduction_no_psi_cond
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [Kon26] Kondylidou (2026), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Kon26]
% 0.30/0.46 % Names : ALG444^1__005__axclos.verify [Kon26]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : ? v9.3.0
% 0.30/0.46 % Syntax : Number of formulae : 132 ( 1 unt; 130 typ; 0 def)
% 0.30/0.46 % Number of atoms : 255 ( 129 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 248 ( 127 avg)
% 0.30/0.46 % Number of connectives : 2617 ( 333 ~; 306 |; 129 &;1748 @)
% 0.30/0.46 % ( 56 <=>; 45 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 151 ( 76 avg)
% 0.30/0.46 % Number of types : 5 ( 4 usr)
% 0.30/0.46 % Number of type conns : 227 ( 227 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 127 ( 126 usr; 113 con; 0-4 aty)
% 0.30/0.46 % Number of variables : 561 ( 10 ^; 549 !; 2 ?; 561 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments :
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(term_type,type,
% 0.30/0.46 term: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(subst_type,type,
% 0.30/0.46 subst: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(one_decl,type,
% 0.30/0.46 one: term ).
% 0.30/0.46
% 0.30/0.46 thf(ap_decl,type,
% 0.30/0.46 ap: term > term > term ).
% 0.30/0.46
% 0.30/0.46 thf(lam_decl,type,
% 0.30/0.46 lam: term > term ).
% 0.30/0.46
% 0.30/0.46 thf(sub_decl,type,
% 0.30/0.46 sub: term > subst > term ).
% 0.30/0.46
% 0.30/0.46 thf(id_decl,type,
% 0.30/0.46 id: subst ).
% 0.30/0.46
% 0.30/0.46 thf(sh_decl,type,
% 0.30/0.46 sh: subst ).
% 0.30/0.46
% 0.30/0.46 thf(push_decl,type,
% 0.30/0.46 push: term > subst > subst ).
% 0.30/0.46
% 0.30/0.46 thf(comp_decl,type,
% 0.30/0.46 comp: subst > subst > subst ).
% 0.30/0.46
% 0.30/0.46 thf(var_decl,type,
% 0.30/0.46 var: term > $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem1v2_decl,type,
% 0.30/0.46 pushprop_lem1v2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem1_gthm_decl,type,
% 0.30/0.46 pushprop_lem1_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axmap_decl,type,
% 0.30/0.46 axmap: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem0_gthm_decl,type,
% 0.30/0.46 pushprop_lem0_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(shinj_decl,type,
% 0.30/0.46 shinj: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem1v2_decl,type,
% 0.30/0.46 hoasinduction_lem1v2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem1v2_gthm_decl,type,
% 0.30/0.46 hoasinduction_lem1v2_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasap_decl,type,
% 0.30/0.46 hoasap: subst > term > subst > term > term ).
% 0.30/0.46
% 0.30/0.46 thf(induction2lem_decl,type,
% 0.30/0.46 induction2lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3v2_f_decl,type,
% 0.30/0.46 hoasinduction_lem3v2_f: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axvarshift_decl,type,
% 0.30/0.46 axvarshift: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapinj2_decl,type,
% 0.30/0.46 hoasapinj2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapnotvar_gthm_decl,type,
% 0.30/0.46 hoasapnotvar_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapinj1_decl,type,
% 0.30/0.46 hoasapinj1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ulamvar1_decl,type,
% 0.30/0.46 ulamvar1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(induction2lem_lthm_decl,type,
% 0.30/0.46 induction2lem_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3v2_gthm_decl,type,
% 0.30/0.46 hoasinduction_lem3v2_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(apnotvar_decl,type,
% 0.30/0.46 apnotvar: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lthm_orig_decl,type,
% 0.30/0.46 pushprop_lthm_orig: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem3v2_f_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lthm_decl,type,
% 0.30/0.46 hoasinduction_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 0.30/0.46 hoasinduction_no_psi_cond_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslaminj_decl,type,
% 0.30/0.46 hoaslaminj: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3aaa_decl,type,
% 0.30/0.46 hoasinduction_lem3aaa: $o ).
% 0.30/0.46
% 0.30/0.46 thf(induction2lem_gthm_decl,type,
% 0.30/0.46 induction2lem_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3aa_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem3aa_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3_decl,type,
% 0.30/0.46 hoasinduction_lem3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem2_decl,type,
% 0.30/0.46 hoasinduction_lem2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(termmset_lthm_decl,type,
% 0.30/0.46 termmset_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem1_decl,type,
% 0.30/0.46 hoasinduction_lem1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslamnotap_lthm_decl,type,
% 0.30/0.46 hoaslamnotap_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem1v2_lthm_decl,type,
% 0.30/0.46 pushprop_lem1v2_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapnotvar_decl,type,
% 0.30/0.46 hoasapnotvar: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem0_decl,type,
% 0.30/0.46 hoasinduction_lem0: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_decl,type,
% 0.30/0.46 hoasinduction: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_gthm_decl,type,
% 0.30/0.46 hoasinduction_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axapp_decl,type,
% 0.30/0.46 axapp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslamnotvar_lthm_decl,type,
% 0.30/0.46 hoaslamnotvar_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem3v2_lthm_decl,type,
% 0.30/0.46 pushprop_lem3v2_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3b_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem3b_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ulamvarind_decl,type,
% 0.30/0.46 ulamvarind: $o ).
% 0.30/0.46
% 0.30/0.46 thf(induction_decl,type,
% 0.30/0.46 induction: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3a_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem3a_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(termmset_gthm_decl,type,
% 0.30/0.46 termmset_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3aa_decl,type,
% 0.30/0.46 hoasinduction_lem3aa: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem1v2_gthm_decl,type,
% 0.30/0.46 pushprop_lem1v2_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslamnotap_gthm_decl,type,
% 0.30/0.46 hoaslamnotap_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslamnotvar_gthm_decl,type,
% 0.30/0.46 hoaslamnotvar_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3b_gthm_decl,type,
% 0.30/0.46 hoasinduction_lem3b_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem2v2_decl,type,
% 0.30/0.46 pushprop_lem2v2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3a_gthm_decl,type,
% 0.30/0.46 hoasinduction_lem3a_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axclos_decl,type,
% 0.30/0.46 axclos: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axassoc_decl,type,
% 0.30/0.46 axassoc: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem2v2_decl,type,
% 0.30/0.46 hoasinduction_lem2v2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lthm_decl,type,
% 0.30/0.46 pushprop_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(apinj2_decl,type,
% 0.30/0.46 apinj2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(apinj1_decl,type,
% 0.30/0.46 apinj1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapinj2_lthm_decl,type,
% 0.30/0.46 hoasapinj2_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3v2a_decl,type,
% 0.30/0.46 hoasinduction_lem3v2a: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapinj1_lthm_decl,type,
% 0.30/0.46 hoasapinj1_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslaminj_lthm_decl,type,
% 0.30/0.46 hoaslaminj_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axvarcons_decl,type,
% 0.30/0.46 axvarcons: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslam_decl,type,
% 0.30/0.46 hoaslam: subst > ( subst > term > term ) > term ).
% 0.30/0.46
% 0.30/0.46 thf(axscons_decl,type,
% 0.30/0.46 axscons: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem2v2_gthm_decl,type,
% 0.30/0.46 hoasinduction_lem2v2_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axidr_decl,type,
% 0.30/0.46 axidr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem1_decl,type,
% 0.30/0.46 pushprop_lem1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(laminj_decl,type,
% 0.30/0.46 laminj: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem3_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem0_decl,type,
% 0.30/0.46 pushprop_lem0: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_gthm_decl,type,
% 0.30/0.46 pushprop_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axabs_decl,type,
% 0.30/0.46 axabs: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem3v2a_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem2_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem2_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapinj2_gthm_decl,type,
% 0.30/0.46 hoasapinj2_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_p_and_p_prime_decl,type,
% 0.30/0.46 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem1_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem1_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lamnotap_decl,type,
% 0.30/0.46 lamnotap: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapinj1_gthm_decl,type,
% 0.30/0.46 hoasapinj1_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslamnotvar_decl,type,
% 0.30/0.46 hoaslamnotvar: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axidl_decl,type,
% 0.30/0.46 axidl: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslaminj_gthm_decl,type,
% 0.30/0.46 hoaslaminj_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(induction2_lthm_decl,type,
% 0.30/0.46 induction2_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem0_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem0_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(substmonoid_lthm_decl,type,
% 0.30/0.46 substmonoid_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_decl,type,
% 0.30/0.46 pushprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3_gthm_decl,type,
% 0.30/0.46 hoasinduction_lem3_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem2_gthm_decl,type,
% 0.30/0.46 hoasinduction_lem2_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3b_decl,type,
% 0.30/0.46 hoasinduction_lem3b: $o ).
% 0.30/0.46
% 0.30/0.46 thf(substmonoid_decl,type,
% 0.30/0.46 substmonoid: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lamnotvar_decl,type,
% 0.30/0.46 lamnotvar: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3a_decl,type,
% 0.30/0.46 hoasinduction_lem3a: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem1_gthm_decl,type,
% 0.30/0.46 hoasinduction_lem1_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_no_psi_cond_decl,type,
% 0.30/0.46 hoasinduction_no_psi_cond: $o ).
% 0.30/0.46
% 0.30/0.46 thf(induction2_gthm_decl,type,
% 0.30/0.46 induction2_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem2v2_lthm_decl,type,
% 0.30/0.46 pushprop_lem2v2_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasvar_decl,type,
% 0.30/0.46 hoasvar: subst > term > subst > $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoaslamnotap_decl,type,
% 0.30/0.46 hoaslamnotap: $o ).
% 0.30/0.46
% 0.30/0.46 thf(substmonoid_gthm_decl,type,
% 0.30/0.46 substmonoid_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ulamvarsh_decl,type,
% 0.30/0.46 ulamvarsh: $o ).
% 0.30/0.46
% 0.30/0.46 thf(induction2_decl,type,
% 0.30/0.46 induction2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem3v2_decl,type,
% 0.30/0.46 pushprop_lem3v2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem2v2_gthm_decl,type,
% 0.30/0.46 pushprop_lem2v2_gthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem1_lthm_decl,type,
% 0.30/0.46 pushprop_lem1_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3v2_decl,type,
% 0.30/0.46 hoasinduction_lem3v2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(axshiftcons_decl,type,
% 0.30/0.46 axshiftcons: $o ).
% 0.30/0.46
% 0.30/0.46 thf(termmset_decl,type,
% 0.30/0.46 termmset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_lem0_lthm_decl,type,
% 0.30/0.46 pushprop_lem0_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasapnotvar_lthm_decl,type,
% 0.30/0.46 hoasapnotvar_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lem3v2_lthm_decl,type,
% 0.30/0.46 hoasinduction_lem3v2_lthm: $o ).
% 0.30/0.46
% 0.30/0.46 thf(pushprop_p_and_p_prime_decl,type,
% 0.30/0.46 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(axvarid_decl,type,
% 0.30/0.46 axvarid: $o ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_lthm_3_decl,type,
% 0.30/0.46 hoasinduction_lthm_3: $o ).
% 0.30/0.46
% 0.30/0.46 %----Types of the domains
% 0.30/0.46 thf(d_term_type,type,
% 0.30/0.46 d_term: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(d_subst_type,type,
% 0.30/0.46 d_subst: $tType ).
% 0.30/0.46
% 0.30/0.46 %----Types of the promotion functions
% 0.30/0.46 thf(d2term_decl,type,
% 0.30/0.46 d2term: d_term > term ).
% 0.30/0.46
% 0.30/0.46 thf(d2subst_decl,type,
% 0.30/0.46 d2subst: d_subst > subst ).
% 0.30/0.46
% 0.30/0.46 %----Types of the domain elements
% 0.30/0.46 thf(d_one_decl,type,
% 0.30/0.46 d_one: d_term ).
% 0.30/0.46
% 0.30/0.46 thf(d_id_decl,type,
% 0.30/0.46 d_id: d_subst ).
% 0.30/0.46
% 0.30/0.46 thf(alg444_1,axiom,
% 0.30/0.46 ( ! [T: term] :
% 0.30/0.46 ? [DT: d_term] :
% 0.30/0.46 ( T
% 0.30/0.46 = ( d2term @ DT ) )
% 0.30/0.46 & ! [DT: d_term] : ( DT = d_one )
% 0.30/0.46 & ! [DT1: d_term,DT2: d_term] :
% 0.30/0.46 ( ( ( d2term @ DT1 )
% 0.30/0.46 = ( d2term @ DT2 ) )
% 0.30/0.46 => ( DT1 = DT2 ) )
% 0.30/0.46 & ! [S: subst] :
% 0.30/0.46 ? [DS: d_subst] :
% 0.30/0.46 ( S
% 0.30/0.46 = ( d2subst @ DS ) )
% 0.30/0.46 & ! [DS: d_subst] : ( DS = d_id )
% 0.30/0.46 & ! [DS1: d_subst,DS2: d_subst] :
% 0.30/0.46 ( ( ( d2subst @ DS1 )
% 0.30/0.46 = ( d2subst @ DS2 ) )
% 0.30/0.46 => ( DS1 = DS2 ) )
% 0.30/0.46 & ( one
% 0.30/0.46 = ( d2term @ d_one ) )
% 0.30/0.46 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 0.30/0.46 = ( d2term @ d_one ) )
% 0.30/0.46 & ( ( lam @ ( d2term @ d_one ) )
% 0.30/0.46 = ( d2term @ d_one ) )
% 0.30/0.46 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.30/0.46 = ( d2term @ d_one ) )
% 0.30/0.46 & ( id
% 0.30/0.46 = ( d2subst @ d_id ) )
% 0.30/0.46 & ( sh
% 0.30/0.46 = ( d2subst @ d_id ) )
% 0.30/0.46 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.30/0.46 = ( d2subst @ d_id ) )
% 0.30/0.46 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 0.30/0.46 = ( d2subst @ d_id ) )
% 0.30/0.46 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 0.30/0.46 = ( d2term @ d_one ) )
% 0.30/0.46 & ( hoaslam
% 0.30/0.46 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 0.30/0.46 & ( hoasinduction_p_and_p_prime
% 0.30/0.46 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 0.30/0.46 ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) ) ) )
% 0.30/0.46 & ( pushprop_p_and_p_prime
% 0.30/0.46 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 0.30/0.46 ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 0.30/0.46 & ~ ( var @ ( d2term @ d_one ) )
% 0.30/0.46 & ( pushprop_lem1v2
% 0.30/0.46 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.30/0.46 | ( Q @ one ) ) )
% 0.30/0.46 & pushprop_lem1_gthm
% 0.30/0.46 & axmap
% 0.30/0.46 & pushprop_lem0_gthm
% 0.30/0.46 & ( shinj
% 0.30/0.46 <=> ! [A: term,B: term] :
% 0.30/0.46 ( ( ( sub @ A @ sh )
% 0.30/0.46 != ( sub @ B @ sh ) )
% 0.30/0.46 | ( A = B ) ) )
% 0.30/0.46 & hoasinduction_lem1v2
% 0.30/0.46 & hoasinduction_lem1v2_gthm
% 0.30/0.46 & ( induction2lem
% 0.30/0.46 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.30/0.46 ( ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ B )
% 0.30/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( var @ B )
% 0.30/0.46 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.30/0.46 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 0.30/0.46 & ( hoasinduction_lem3v2_f
% 0.30/0.46 <=> ! [B: term] :
% 0.30/0.46 ~ ! [F: subst > term > term] :
% 0.30/0.46 ~ ! [A: term,M: subst] :
% 0.30/0.46 ( ( sub @ B @ ( push @ A @ M ) )
% 0.30/0.46 = ( F @ M @ A ) ) )
% 0.30/0.46 & axvarshift
% 0.30/0.46 & ( hoasapinj2
% 0.30/0.46 <=> ! [A: term,B: term,C: term,D: term] :
% 0.30/0.46 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.30/0.46 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.30/0.46 | ( C = D ) ) )
% 0.30/0.46 & hoasapnotvar_gthm
% 0.30/0.46 & ( hoasapinj1
% 0.30/0.46 <=> ! [A: term,B: term,C: term,D: term] :
% 0.30/0.46 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.30/0.46 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.30/0.46 | ( A = B ) ) )
% 0.30/0.46 & ~ ulamvar1
% 0.30/0.46 & ( induction2lem_lthm
% 0.30/0.46 <=> ( ! [A: term,M: subst] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.30/0.46 => ( ! [A: term,M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.30/0.46 => ( ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ M @ id ) )
% 0.30/0.46 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 0.30/0.46 ( ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ A )
% 0.30/0.46 | ( P @ A ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ( P @ Bound_variable_2140 ) )
% 0.30/0.46 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.30/0.46 ( ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ B )
% 0.30/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( var @ B )
% 0.30/0.46 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.30/0.46 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 0.30/0.46 & hoasinduction_lem3v2_gthm
% 0.30/0.46 & apnotvar
% 0.30/0.46 & pushprop_lthm_orig
% 0.30/0.46 & ( hoasinduction_lem3v2_f_lthm
% 0.30/0.46 <=> ! [B: term] :
% 0.30/0.46 ~ ! [F: subst > term > term] :
% 0.30/0.46 ~ ! [A: term,M: subst] :
% 0.30/0.46 ( ( sub @ B @ ( push @ A @ M ) )
% 0.30/0.46 = ( F @ M @ A ) ) )
% 0.30/0.46 & ( hoasinduction_lthm
% 0.30/0.46 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.30/0.46 ( ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ A )
% 0.30/0.46 | ( P @ A ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ B )
% 0.30/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ( P @ Bound_variable_2367 ) )
% 0.30/0.46 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.30/0.46 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.30/0.46 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 0.30/0.46 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 0.30/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ ( sub @ A @ id ) )
% 0.30/0.46 | ( P @ id @ A @ id ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 0.30/0.46 & ( hoasinduction_no_psi_cond_lthm
% 0.30/0.46 <=> ( ! [P: subst > term > subst > $o] :
% 0.30/0.46 ~ ! [Q: term > $o] :
% 0.30/0.46 ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.30/0.46 ( ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ A )
% 0.30/0.46 | ( P @ A ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ B )
% 0.30/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ( P @ Bound_variable_2367 ) )
% 0.30/0.46 => ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( Q @ B )
% 0.30/0.46 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.30/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.30/0.46 ( ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 0.30/0.46 & ( hoaslaminj
% 0.30/0.46 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.30/0.46 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ( ( lam @ ( F @ sh @ one ) )
% 0.30/0.46 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.30/0.46 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.30/0.46 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 0.30/0.46 & ( hoasinduction_lem3aaa
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 0.30/0.46 ( ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 0.30/0.46 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 0.30/0.46 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 0.30/0.46 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 0.30/0.46 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 0.30/0.46 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 0.30/0.46 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.30/0.46 & induction2lem_gthm
% 0.30/0.46 & ( hoasinduction_lem3aa_lthm
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.30/0.46 & ( hoasinduction_lem3
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 0.30/0.46 & ( hoasinduction_lem2
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.30/0.46 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.30/0.46 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 0.30/0.46 & ( termmset_lthm
% 0.30/0.46 <=> ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) ) ) )
% 0.30/0.46 & hoasinduction_lem1
% 0.30/0.46 & hoaslamnotap_lthm
% 0.30/0.46 & ( pushprop_lem1v2_lthm
% 0.30/0.46 <=> ( ! [A: term,M: subst] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.30/0.46 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.30/0.46 | ( Q @ one ) ) ) )
% 0.30/0.46 & hoasapnotvar
% 0.30/0.46 & ( hoasinduction_lem0
% 0.30/0.46 <=> ! [P: subst > term > subst > $o] :
% 0.30/0.46 ~ ! [Q: term > $o] :
% 0.30/0.46 ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) ) )
% 0.30/0.46 & ( hoasinduction
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ ( sub @ A @ id ) )
% 0.30/0.46 | ( P @ id @ A @ id ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 0.30/0.46 & hoasinduction_gthm
% 0.30/0.46 & axapp
% 0.30/0.46 & hoaslamnotvar_lthm
% 0.30/0.46 & pushprop_lem3v2_lthm
% 0.30/0.46 & ( hoasinduction_lem3b_lthm
% 0.30/0.46 <=> ! [B: term] :
% 0.30/0.46 ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ( F @ sh @ one )
% 0.30/0.46 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.30/0.46 & ulamvarind
% 0.30/0.46 & ( induction
% 0.30/0.46 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 0.30/0.46 ( ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ A )
% 0.30/0.46 | ( P @ A ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ( P @ Bound_variable_2140 ) ) )
% 0.30/0.46 & ( hoasinduction_lem3a_lthm
% 0.30/0.46 <=> ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.30/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 0.30/0.46 & termmset_gthm
% 0.30/0.46 & ( hoasinduction_lem3aa
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.30/0.46 & pushprop_lem1v2_gthm
% 0.30/0.46 & hoaslamnotap_gthm
% 0.30/0.46 & hoaslamnotvar_gthm
% 0.30/0.46 & hoasinduction_lem3b_gthm
% 0.30/0.46 & pushprop_lem2v2
% 0.30/0.46 & hoasinduction_lem3a_gthm
% 0.30/0.46 & axclos
% 0.30/0.46 & axassoc
% 0.30/0.46 & ( hoasinduction_lem2v2
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 | ~ ( Q @ Bound_variable_2820 )
% 0.30/0.46 | ~ ( Q @ Bound_variable_2822 )
% 0.30/0.46 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 0.30/0.46 & pushprop_lthm
% 0.30/0.46 & ( apinj2
% 0.30/0.46 <=> ! [A: term,B: term,C: term,D: term] :
% 0.30/0.46 ( ( ( ap @ A @ C )
% 0.30/0.46 != ( ap @ B @ D ) )
% 0.30/0.46 | ( C = D ) ) )
% 0.30/0.46 & ( apinj1
% 0.30/0.46 <=> ! [A: term,B: term,C: term,D: term] :
% 0.30/0.46 ( ( ( ap @ A @ C )
% 0.30/0.46 != ( ap @ B @ D ) )
% 0.30/0.46 | ( A = B ) ) )
% 0.30/0.46 & ( hoasapinj2_lthm
% 0.30/0.46 <=> ( ! [A: term,B: term,C: term,D: term] :
% 0.30/0.46 ( ( ( ap @ A @ C )
% 0.30/0.46 != ( ap @ B @ D ) )
% 0.30/0.46 | ( C = D ) )
% 0.30/0.46 => ! [A: term,B: term,C: term,D: term] :
% 0.30/0.46 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.30/0.46 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.30/0.46 | ( C = D ) ) ) )
% 0.30/0.46 & ( hoasinduction_lem3v2a
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( Q @ B )
% 0.30/0.46 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 0.30/0.46 & ( hoasapinj1_lthm
% 0.30/0.46 <=> ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ( ! [A: term,B: term,C: term,D: term] :
% 0.30/0.46 ( ( ( ap @ A @ C )
% 0.30/0.46 != ( ap @ B @ D ) )
% 0.30/0.46 | ( A = B ) )
% 0.30/0.46 => ! [A: term,B: term,C: term,D: term] :
% 0.30/0.46 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.30/0.46 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.30/0.46 | ( A = B ) ) ) ) )
% 0.30/0.46 & ( hoaslaminj_lthm
% 0.30/0.46 <=> ( ! [A: term,M: subst] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.30/0.46 => ( ! [A: term,M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.30/0.46 => ( ! [A: term,B: term] :
% 0.30/0.46 ( ( ( lam @ A )
% 0.30/0.46 != ( lam @ B ) )
% 0.30/0.46 | ( A = B ) )
% 0.30/0.46 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.30/0.46 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ( ( lam @ ( F @ sh @ one ) )
% 0.30/0.46 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.30/0.46 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.30/0.46 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 0.30/0.46 & ( axvarcons
% 0.30/0.46 <=> ! [A: term,M: subst] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 0.30/0.46 & ( axscons
% 0.30/0.46 <=> ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 0.30/0.46 & hoasinduction_lem2v2_gthm
% 0.30/0.46 & ( axidr
% 0.30/0.46 <=> ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ M @ id ) ) )
% 0.30/0.46 & ( pushprop_lem1
% 0.30/0.46 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 0.30/0.46 & ( laminj
% 0.30/0.46 <=> ! [A: term,B: term] :
% 0.30/0.46 ( ( ( lam @ A )
% 0.30/0.46 != ( lam @ B ) )
% 0.30/0.46 | ( A = B ) ) )
% 0.30/0.46 & ( hoasinduction_lem3_lthm
% 0.30/0.46 <=> ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.30/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 0.30/0.46 & ( pushprop_lem0
% 0.30/0.46 <=> ! [P: term > $o,A: term,M: subst] :
% 0.30/0.46 ~ ! [Q: term > $o] :
% 0.30/0.46 ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.30/0.46 & pushprop_gthm
% 0.30/0.46 & axabs
% 0.30/0.46 & ( hoasinduction_lem3v2a_lthm
% 0.30/0.46 <=> ( ! [B: term] :
% 0.30/0.46 ~ ! [F: subst > term > term] :
% 0.30/0.46 ~ ! [A: term,M: subst] :
% 0.30/0.46 ( ( sub @ B @ ( push @ A @ M ) )
% 0.30/0.46 = ( F @ M @ A ) )
% 0.30/0.46 => ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( Q @ B )
% 0.30/0.46 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 0.30/0.46 & hoasinduction_lem2_lthm
% 0.30/0.46 & hoasapinj2_gthm
% 0.30/0.46 & hoasinduction_lem1_lthm
% 0.30/0.46 & ~ lamnotap
% 0.30/0.46 & hoasapinj1_gthm
% 0.30/0.46 & hoaslamnotvar
% 0.30/0.46 & ( axidl
% 0.30/0.46 <=> ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ id @ M ) ) )
% 0.30/0.46 & hoaslaminj_gthm
% 0.30/0.46 & ( induction2_lthm
% 0.30/0.46 <=> ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.30/0.46 ( ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ B )
% 0.30/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( var @ B )
% 0.30/0.46 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.30/0.46 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 0.30/0.46 => ! [P: term > $o,Bound_variable_2367: term] :
% 0.30/0.46 ( ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ A )
% 0.30/0.46 | ( P @ A ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ B )
% 0.30/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ( P @ Bound_variable_2367 ) ) ) ) )
% 0.30/0.46 & ( hoasinduction_lem0_lthm
% 0.30/0.46 <=> ! [P: subst > term > subst > $o] :
% 0.30/0.46 ~ ! [Q: term > $o] :
% 0.30/0.46 ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) ) )
% 0.30/0.46 & ( substmonoid_lthm
% 0.30/0.46 <=> ( ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ id @ M ) )
% 0.30/0.46 => ( ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ M @ id ) )
% 0.30/0.46 => ( ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ id @ M ) )
% 0.30/0.46 & ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ M @ id ) ) ) ) ) )
% 0.30/0.46 & pushprop
% 0.30/0.46 & hoasinduction_lem3_gthm
% 0.30/0.46 & hoasinduction_lem2_gthm
% 0.30/0.46 & ( hoasinduction_lem3b
% 0.30/0.46 <=> ! [B: term] :
% 0.30/0.46 ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ( F @ sh @ one )
% 0.30/0.46 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.30/0.46 & ( substmonoid
% 0.30/0.46 <=> ( ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ id @ M ) )
% 0.30/0.46 & ! [M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ M @ id ) ) ) )
% 0.30/0.46 & lamnotvar
% 0.30/0.46 & ( hoasinduction_lem3a
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 0.30/0.46 & hoasinduction_lem1_gthm
% 0.30/0.46 & ( hoasinduction_no_psi_cond
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.30/0.46 ( ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 0.30/0.46 & induction2_gthm
% 0.30/0.46 & pushprop_lem2v2_lthm
% 0.30/0.46 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.30/0.46 & ( hoaslamnotap
% 0.30/0.46 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ( ( lam @ ( F @ sh @ one ) )
% 0.30/0.46 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 0.30/0.46 & substmonoid_gthm
% 0.30/0.46 & ulamvarsh
% 0.30/0.46 & ( induction2
% 0.30/0.46 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 0.30/0.46 ( ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ A )
% 0.30/0.46 | ( P @ A ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ B )
% 0.30/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ( P @ Bound_variable_2367 ) ) )
% 0.30/0.46 & pushprop_lem3v2
% 0.30/0.46 & pushprop_lem2v2_gthm
% 0.30/0.46 & ( pushprop_lem1_lthm
% 0.30/0.46 <=> ( ! [A: term,M: subst] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.30/0.46 => ( ! [A: term,M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.30/0.46 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 0.30/0.46 & ( hoasinduction_lem3v2
% 0.30/0.46 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( Q @ B )
% 0.30/0.46 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 0.30/0.46 & ( axshiftcons
% 0.30/0.46 <=> ! [A: term,M: subst] :
% 0.30/0.46 ( M
% 0.30/0.46 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 0.30/0.46 & ( termmset
% 0.30/0.46 <=> ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) ) )
% 0.30/0.46 & ( pushprop_lem0_lthm
% 0.30/0.46 <=> ! [P: term > $o,A: term,M: subst] :
% 0.30/0.46 ~ ! [Q: term > $o] :
% 0.30/0.46 ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.30/0.46 & hoasapnotvar_lthm
% 0.30/0.46 & ( hoasinduction_lem3v2_lthm
% 0.30/0.46 <=> ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( Q @ B )
% 0.30/0.46 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 0.30/0.46 & ( axvarid
% 0.30/0.46 <=> ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) ) )
% 0.30/0.46 & ( hoasinduction_lthm_3
% 0.30/0.46 <=> ( ! [P: subst > term > subst > $o] :
% 0.30/0.46 ~ ! [Q: term > $o] :
% 0.30/0.46 ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.30/0.46 ( ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ A )
% 0.30/0.46 | ( P @ A ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ A )
% 0.30/0.46 | ~ ( P @ B )
% 0.30/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ! [B: term] :
% 0.30/0.46 ( ~ ( P @ B )
% 0.30/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( P @ ( lam @ A ) ) )
% 0.30/0.46 | ( P @ Bound_variable_2367 ) )
% 0.30/0.46 => ( ! [A: term] :
% 0.30/0.46 ( A
% 0.30/0.46 = ( sub @ A @ id ) )
% 0.30/0.46 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.30/0.46 ( ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ~ ! [X: term] :
% 0.30/0.46 ( ( Q @ X )
% 0.30/0.46 = ( P @ id @ X @ id ) )
% 0.30/0.46 | ~ ! [B: term] :
% 0.30/0.46 ( ~ ( Q @ B )
% 0.30/0.46 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.30/0.46 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.30/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.30/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.30/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.30/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.30/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( var @ ( sub @ A @ id ) )
% 0.30/0.46 | ( P @ id @ A @ id ) )
% 0.30/0.46 | ~ ! [A: term,B: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ~ ( P @ id @ B @ id )
% 0.30/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.30/0.46 | ~ ! [F: subst > term > term] :
% 0.30/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 | ~ ! [A: term] :
% 0.30/0.46 ( ~ ( P @ id @ A @ id )
% 0.30/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.30/0.46 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(hoasinduction_no_psi_cond,conjecture,
% 0.30/0.46 ( hoasinduction_no_psi_cond
% 0.30/0.46 = ( ! [P: subst > term > subst > $o] :
% 0.30/0.46 ( ! [A: term,B: term] :
% 0.30/0.46 ( ( P @ id @ A @ id )
% 0.30/0.46 => ( ( P @ id @ B @ id )
% 0.30/0.46 => ( P @ id @ ( hoasap @ id @ A @ id @ B ) @ id ) ) )
% 0.30/0.46 => ( ! [F: subst > term > term] :
% 0.30/0.46 ( ! [M: subst,A: term,N: subst] :
% 0.30/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.30/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.30/0.46 => ( ! [A: term] :
% 0.30/0.46 ( ( P @ id @ A @ id )
% 0.30/0.46 => ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.30/0.46 => ( P @ id
% 0.30/0.46 @ ( hoaslam @ id
% 0.30/0.46 @ ^ [M: subst,A: term] : ( F @ M @ A ) )
% 0.30/0.46 @ id ) ) )
% 0.30/0.46 => ! [A: term] : ( P @ id @ A @ id ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.42/1.41 ---- Embedded in THF ---
% 2.42/1.41 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.42/1.41 %%% Generated on Tue May 05 09:14:54 EDT 2026
% 2.42/1.41 %%% using '$$dhol' embedding, version 1.3.0.
% 2.42/1.41 %%% Logic specification used:
% 2.42/1.41 %%% thf(spec, logic, $$dhol).
% 2.42/1.41
% 2.42/1.41 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Qe5lgCyrIb/DTF2DTF31149.p
% See solution above
% 2.42/1.42 ------------------------
% 2.42/1.44 ---- Cleaned THF ---
% 2.42/1.44 thf(term_type,type,
% 2.42/1.44 term: $tType ).
% 2.42/1.44
% 2.42/1.44 thf(subst_type,type,
% 2.42/1.44 subst: $tType ).
% 2.42/1.44
% 2.42/1.44 thf(one_decl,type,
% 2.42/1.44 one: term ).
% 2.42/1.44
% 2.42/1.44 thf(ap_decl,type,
% 2.42/1.44 ap: term > term > term ).
% 2.42/1.44
% 2.42/1.44 thf(lam_decl,type,
% 2.42/1.44 lam: term > term ).
% 2.42/1.44
% 2.42/1.44 thf(sub_decl,type,
% 2.42/1.44 sub: term > subst > term ).
% 2.42/1.44
% 2.42/1.44 thf(id_decl,type,
% 2.42/1.44 id: subst ).
% 2.42/1.44
% 2.42/1.44 thf(sh_decl,type,
% 2.42/1.44 sh: subst ).
% 2.42/1.44
% 2.42/1.44 thf(push_decl,type,
% 2.42/1.44 push: term > subst > subst ).
% 2.42/1.44
% 2.42/1.44 thf(comp_decl,type,
% 2.42/1.44 comp: subst > subst > subst ).
% 2.42/1.44
% 2.42/1.44 thf(var_decl,type,
% 2.42/1.44 var: term > $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem1v2_decl,type,
% 2.42/1.44 pushprop_lem1v2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem1_gthm_decl,type,
% 2.42/1.44 pushprop_lem1_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axmap_decl,type,
% 2.42/1.44 axmap: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem0_gthm_decl,type,
% 2.42/1.44 pushprop_lem0_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(shinj_decl,type,
% 2.42/1.44 shinj: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem1v2_decl,type,
% 2.42/1.44 hoasinduction_lem1v2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem1v2_gthm_decl,type,
% 2.42/1.44 hoasinduction_lem1v2_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasap_decl,type,
% 2.42/1.44 hoasap: subst > term > subst > term > term ).
% 2.42/1.44
% 2.42/1.44 thf(induction2lem_decl,type,
% 2.42/1.44 induction2lem: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3v2_f_decl,type,
% 2.42/1.44 hoasinduction_lem3v2_f: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axvarshift_decl,type,
% 2.42/1.44 axvarshift: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapinj2_decl,type,
% 2.42/1.44 hoasapinj2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapnotvar_gthm_decl,type,
% 2.42/1.44 hoasapnotvar_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapinj1_decl,type,
% 2.42/1.44 hoasapinj1: $o ).
% 2.42/1.44
% 2.42/1.44 thf(ulamvar1_decl,type,
% 2.42/1.44 ulamvar1: $o ).
% 2.42/1.44
% 2.42/1.44 thf(induction2lem_lthm_decl,type,
% 2.42/1.44 induction2lem_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3v2_gthm_decl,type,
% 2.42/1.44 hoasinduction_lem3v2_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(apnotvar_decl,type,
% 2.42/1.44 apnotvar: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lthm_orig_decl,type,
% 2.42/1.44 pushprop_lthm_orig: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem3v2_f_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lthm_decl,type,
% 2.42/1.44 hoasinduction_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 2.42/1.44 hoasinduction_no_psi_cond_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslaminj_decl,type,
% 2.42/1.44 hoaslaminj: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3aaa_decl,type,
% 2.42/1.44 hoasinduction_lem3aaa: $o ).
% 2.42/1.44
% 2.42/1.44 thf(induction2lem_gthm_decl,type,
% 2.42/1.44 induction2lem_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3aa_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem3aa_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3_decl,type,
% 2.42/1.44 hoasinduction_lem3: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem2_decl,type,
% 2.42/1.44 hoasinduction_lem2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(termmset_lthm_decl,type,
% 2.42/1.44 termmset_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem1_decl,type,
% 2.42/1.44 hoasinduction_lem1: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslamnotap_lthm_decl,type,
% 2.42/1.44 hoaslamnotap_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem1v2_lthm_decl,type,
% 2.42/1.44 pushprop_lem1v2_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapnotvar_decl,type,
% 2.42/1.44 hoasapnotvar: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem0_decl,type,
% 2.42/1.44 hoasinduction_lem0: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_decl,type,
% 2.42/1.44 hoasinduction: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_gthm_decl,type,
% 2.42/1.44 hoasinduction_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axapp_decl,type,
% 2.42/1.44 axapp: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslamnotvar_lthm_decl,type,
% 2.42/1.44 hoaslamnotvar_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem3v2_lthm_decl,type,
% 2.42/1.44 pushprop_lem3v2_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3b_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem3b_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(ulamvarind_decl,type,
% 2.42/1.44 ulamvarind: $o ).
% 2.42/1.44
% 2.42/1.44 thf(induction_decl,type,
% 2.42/1.44 induction: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3a_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem3a_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(termmset_gthm_decl,type,
% 2.42/1.44 termmset_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3aa_decl,type,
% 2.42/1.44 hoasinduction_lem3aa: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem1v2_gthm_decl,type,
% 2.42/1.44 pushprop_lem1v2_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslamnotap_gthm_decl,type,
% 2.42/1.44 hoaslamnotap_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslamnotvar_gthm_decl,type,
% 2.42/1.44 hoaslamnotvar_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3b_gthm_decl,type,
% 2.42/1.44 hoasinduction_lem3b_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem2v2_decl,type,
% 2.42/1.44 pushprop_lem2v2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3a_gthm_decl,type,
% 2.42/1.44 hoasinduction_lem3a_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axclos_decl,type,
% 2.42/1.44 axclos: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axassoc_decl,type,
% 2.42/1.44 axassoc: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem2v2_decl,type,
% 2.42/1.44 hoasinduction_lem2v2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lthm_decl,type,
% 2.42/1.44 pushprop_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(apinj2_decl,type,
% 2.42/1.44 apinj2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(apinj1_decl,type,
% 2.42/1.44 apinj1: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapinj2_lthm_decl,type,
% 2.42/1.44 hoasapinj2_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3v2a_decl,type,
% 2.42/1.44 hoasinduction_lem3v2a: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapinj1_lthm_decl,type,
% 2.42/1.44 hoasapinj1_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslaminj_lthm_decl,type,
% 2.42/1.44 hoaslaminj_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axvarcons_decl,type,
% 2.42/1.44 axvarcons: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslam_decl,type,
% 2.42/1.44 hoaslam: subst > ( subst > term > term ) > term ).
% 2.42/1.44
% 2.42/1.44 thf(axscons_decl,type,
% 2.42/1.44 axscons: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem2v2_gthm_decl,type,
% 2.42/1.44 hoasinduction_lem2v2_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axidr_decl,type,
% 2.42/1.44 axidr: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem1_decl,type,
% 2.42/1.44 pushprop_lem1: $o ).
% 2.42/1.44
% 2.42/1.44 thf(laminj_decl,type,
% 2.42/1.44 laminj: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem3_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem0_decl,type,
% 2.42/1.44 pushprop_lem0: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_gthm_decl,type,
% 2.42/1.44 pushprop_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axabs_decl,type,
% 2.42/1.44 axabs: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem3v2a_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem2_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem2_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapinj2_gthm_decl,type,
% 2.42/1.44 hoasapinj2_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_p_and_p_prime_decl,type,
% 2.42/1.44 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem1_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem1_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(lamnotap_decl,type,
% 2.42/1.44 lamnotap: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapinj1_gthm_decl,type,
% 2.42/1.44 hoasapinj1_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslamnotvar_decl,type,
% 2.42/1.44 hoaslamnotvar: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axidl_decl,type,
% 2.42/1.44 axidl: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslaminj_gthm_decl,type,
% 2.42/1.44 hoaslaminj_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(induction2_lthm_decl,type,
% 2.42/1.44 induction2_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem0_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem0_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(substmonoid_lthm_decl,type,
% 2.42/1.44 substmonoid_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_decl,type,
% 2.42/1.44 pushprop: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3_gthm_decl,type,
% 2.42/1.44 hoasinduction_lem3_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem2_gthm_decl,type,
% 2.42/1.44 hoasinduction_lem2_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3b_decl,type,
% 2.42/1.44 hoasinduction_lem3b: $o ).
% 2.42/1.44
% 2.42/1.44 thf(substmonoid_decl,type,
% 2.42/1.44 substmonoid: $o ).
% 2.42/1.44
% 2.42/1.44 thf(lamnotvar_decl,type,
% 2.42/1.44 lamnotvar: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3a_decl,type,
% 2.42/1.44 hoasinduction_lem3a: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem1_gthm_decl,type,
% 2.42/1.44 hoasinduction_lem1_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_no_psi_cond_decl,type,
% 2.42/1.44 hoasinduction_no_psi_cond: $o ).
% 2.42/1.44
% 2.42/1.44 thf(induction2_gthm_decl,type,
% 2.42/1.44 induction2_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem2v2_lthm_decl,type,
% 2.42/1.44 pushprop_lem2v2_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasvar_decl,type,
% 2.42/1.44 hoasvar: subst > term > subst > $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoaslamnotap_decl,type,
% 2.42/1.44 hoaslamnotap: $o ).
% 2.42/1.44
% 2.42/1.44 thf(substmonoid_gthm_decl,type,
% 2.42/1.44 substmonoid_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(ulamvarsh_decl,type,
% 2.42/1.44 ulamvarsh: $o ).
% 2.42/1.44
% 2.42/1.44 thf(induction2_decl,type,
% 2.42/1.44 induction2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem3v2_decl,type,
% 2.42/1.44 pushprop_lem3v2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem2v2_gthm_decl,type,
% 2.42/1.44 pushprop_lem2v2_gthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem1_lthm_decl,type,
% 2.42/1.44 pushprop_lem1_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3v2_decl,type,
% 2.42/1.44 hoasinduction_lem3v2: $o ).
% 2.42/1.44
% 2.42/1.44 thf(axshiftcons_decl,type,
% 2.42/1.44 axshiftcons: $o ).
% 2.42/1.44
% 2.42/1.44 thf(termmset_decl,type,
% 2.42/1.44 termmset: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_lem0_lthm_decl,type,
% 2.42/1.44 pushprop_lem0_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasapnotvar_lthm_decl,type,
% 2.42/1.44 hoasapnotvar_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lem3v2_lthm_decl,type,
% 2.42/1.44 hoasinduction_lem3v2_lthm: $o ).
% 2.42/1.44
% 2.42/1.44 thf(pushprop_p_and_p_prime_decl,type,
% 2.42/1.44 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 2.42/1.44
% 2.42/1.44 thf(axvarid_decl,type,
% 2.42/1.44 axvarid: $o ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_lthm_3_decl,type,
% 2.42/1.44 hoasinduction_lthm_3: $o ).
% 2.42/1.44
% 2.42/1.44 thf(d_term_type,type,
% 2.42/1.44 d_term: $tType ).
% 2.42/1.44
% 2.42/1.44 thf(d_subst_type,type,
% 2.42/1.44 d_subst: $tType ).
% 2.42/1.44
% 2.42/1.44 thf(d2term_decl,type,
% 2.42/1.44 d2term: d_term > term ).
% 2.42/1.44
% 2.42/1.44 thf(d2subst_decl,type,
% 2.42/1.44 d2subst: d_subst > subst ).
% 2.42/1.44
% 2.42/1.44 thf(d_one_decl,type,
% 2.42/1.44 d_one: d_term ).
% 2.42/1.44
% 2.42/1.44 thf(d_id_decl,type,
% 2.42/1.44 d_id: d_subst ).
% 2.42/1.44
% 2.42/1.44 thf(alg444_1,axiom,
% 2.42/1.44 ( ! [T: term] :
% 2.42/1.44 ? [DT: d_term] :
% 2.42/1.44 ( T
% 2.42/1.44 = ( d2term @ DT ) )
% 2.42/1.44 & ! [DT: d_term] : ( DT = d_one )
% 2.42/1.44 & ! [DT1: d_term,DT2: d_term] :
% 2.42/1.44 ( ( ( d2term @ DT1 )
% 2.42/1.44 = ( d2term @ DT2 ) )
% 2.42/1.44 => ( DT1 = DT2 ) )
% 2.42/1.44 & ! [S: subst] :
% 2.42/1.44 ? [DS: d_subst] :
% 2.42/1.44 ( S
% 2.42/1.44 = ( d2subst @ DS ) )
% 2.42/1.44 & ! [DS: d_subst] : ( DS = d_id )
% 2.42/1.44 & ! [DS1: d_subst,DS2: d_subst] :
% 2.42/1.44 ( ( ( d2subst @ DS1 )
% 2.42/1.44 = ( d2subst @ DS2 ) )
% 2.42/1.44 => ( DS1 = DS2 ) )
% 2.42/1.44 & ( one
% 2.42/1.44 = ( d2term @ d_one ) )
% 2.42/1.44 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 2.42/1.44 = ( d2term @ d_one ) )
% 2.42/1.44 & ( ( lam @ ( d2term @ d_one ) )
% 2.42/1.44 = ( d2term @ d_one ) )
% 2.42/1.44 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.42/1.44 = ( d2term @ d_one ) )
% 2.42/1.44 & ( id
% 2.42/1.44 = ( d2subst @ d_id ) )
% 2.42/1.44 & ( sh
% 2.42/1.44 = ( d2subst @ d_id ) )
% 2.42/1.44 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.42/1.44 = ( d2subst @ d_id ) )
% 2.42/1.44 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 2.42/1.44 = ( d2subst @ d_id ) )
% 2.42/1.44 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 2.42/1.44 = ( d2term @ d_one ) )
% 2.42/1.44 & ( hoaslam
% 2.42/1.44 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 2.42/1.44 & ( hoasinduction_p_and_p_prime
% 2.42/1.44 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 2.42/1.44 ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) ) ) )
% 2.42/1.44 & ( pushprop_p_and_p_prime
% 2.42/1.44 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 2.42/1.44 ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 2.42/1.44 & ~ ( var @ ( d2term @ d_one ) )
% 2.42/1.44 & ( pushprop_lem1v2
% 2.42/1.44 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.42/1.44 | ( Q @ one ) ) )
% 2.42/1.44 & pushprop_lem1_gthm
% 2.42/1.44 & axmap
% 2.42/1.44 & pushprop_lem0_gthm
% 2.42/1.44 & ( shinj
% 2.42/1.44 <=> ! [A: term,B: term] :
% 2.42/1.44 ( ( ( sub @ A @ sh )
% 2.42/1.44 != ( sub @ B @ sh ) )
% 2.42/1.44 | ( A = B ) ) )
% 2.42/1.44 & hoasinduction_lem1v2
% 2.42/1.44 & hoasinduction_lem1v2_gthm
% 2.42/1.44 & ( induction2lem
% 2.42/1.44 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.42/1.44 ( ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ B )
% 2.42/1.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( var @ B )
% 2.42/1.44 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.42/1.44 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 2.42/1.44 & ( hoasinduction_lem3v2_f
% 2.42/1.44 <=> ! [B: term] :
% 2.42/1.44 ~ ! [F: subst > term > term] :
% 2.42/1.44 ~ ! [A: term,M: subst] :
% 2.42/1.44 ( ( sub @ B @ ( push @ A @ M ) )
% 2.42/1.44 = ( F @ M @ A ) ) )
% 2.42/1.44 & axvarshift
% 2.42/1.44 & ( hoasapinj2
% 2.42/1.44 <=> ! [A: term,B: term,C: term,D: term] :
% 2.42/1.44 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.42/1.44 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.42/1.44 | ( C = D ) ) )
% 2.42/1.44 & hoasapnotvar_gthm
% 2.42/1.44 & ( hoasapinj1
% 2.42/1.44 <=> ! [A: term,B: term,C: term,D: term] :
% 2.42/1.44 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.42/1.44 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.42/1.44 | ( A = B ) ) )
% 2.42/1.44 & ~ ulamvar1
% 2.42/1.44 & ( induction2lem_lthm
% 2.42/1.44 <=> ( ! [A: term,M: subst] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.42/1.44 => ( ! [A: term,M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.42/1.44 => ( ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ M @ id ) )
% 2.42/1.44 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 2.42/1.44 ( ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ A )
% 2.42/1.44 | ( P @ A ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ( P @ Bound_variable_2140 ) )
% 2.42/1.44 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.42/1.44 ( ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ B )
% 2.42/1.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( var @ B )
% 2.42/1.44 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.42/1.44 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 2.42/1.44 & hoasinduction_lem3v2_gthm
% 2.42/1.44 & apnotvar
% 2.42/1.44 & pushprop_lthm_orig
% 2.42/1.44 & ( hoasinduction_lem3v2_f_lthm
% 2.42/1.44 <=> ! [B: term] :
% 2.42/1.44 ~ ! [F: subst > term > term] :
% 2.42/1.44 ~ ! [A: term,M: subst] :
% 2.42/1.44 ( ( sub @ B @ ( push @ A @ M ) )
% 2.42/1.44 = ( F @ M @ A ) ) )
% 2.42/1.44 & ( hoasinduction_lthm
% 2.42/1.44 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.42/1.44 ( ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ A )
% 2.42/1.44 | ( P @ A ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ B )
% 2.42/1.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ( P @ Bound_variable_2367 ) )
% 2.42/1.44 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.42/1.44 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.42/1.44 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 2.42/1.44 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 2.42/1.44 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ ( sub @ A @ id ) )
% 2.42/1.44 | ( P @ id @ A @ id ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 2.42/1.44 & ( hoasinduction_no_psi_cond_lthm
% 2.42/1.44 <=> ( ! [P: subst > term > subst > $o] :
% 2.42/1.44 ~ ! [Q: term > $o] :
% 2.42/1.44 ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.42/1.44 ( ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ A )
% 2.42/1.44 | ( P @ A ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ B )
% 2.42/1.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ( P @ Bound_variable_2367 ) )
% 2.42/1.44 => ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( Q @ B )
% 2.42/1.44 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.42/1.44 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.42/1.44 ( ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 2.42/1.44 & ( hoaslaminj
% 2.42/1.44 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.42/1.44 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ( ( lam @ ( F @ sh @ one ) )
% 2.42/1.44 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.42/1.44 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.42/1.44 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 2.42/1.44 & ( hoasinduction_lem3aaa
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 2.42/1.44 ( ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 2.42/1.44 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 2.42/1.44 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 2.42/1.44 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 2.42/1.44 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 2.42/1.44 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 2.42/1.44 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.42/1.44 & induction2lem_gthm
% 2.42/1.44 & ( hoasinduction_lem3aa_lthm
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.42/1.44 & ( hoasinduction_lem3
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 2.42/1.44 & ( hoasinduction_lem2
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.42/1.44 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.42/1.44 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 2.42/1.44 & ( termmset_lthm
% 2.42/1.44 <=> ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) ) ) )
% 2.42/1.44 & hoasinduction_lem1
% 2.42/1.44 & hoaslamnotap_lthm
% 2.42/1.44 & ( pushprop_lem1v2_lthm
% 2.42/1.44 <=> ( ! [A: term,M: subst] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.42/1.44 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.42/1.44 | ( Q @ one ) ) ) )
% 2.42/1.44 & hoasapnotvar
% 2.42/1.44 & ( hoasinduction_lem0
% 2.42/1.44 <=> ! [P: subst > term > subst > $o] :
% 2.42/1.44 ~ ! [Q: term > $o] :
% 2.42/1.44 ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) ) )
% 2.42/1.44 & ( hoasinduction
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ ( sub @ A @ id ) )
% 2.42/1.44 | ( P @ id @ A @ id ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 2.42/1.44 & hoasinduction_gthm
% 2.42/1.44 & axapp
% 2.42/1.44 & hoaslamnotvar_lthm
% 2.42/1.44 & pushprop_lem3v2_lthm
% 2.42/1.44 & ( hoasinduction_lem3b_lthm
% 2.42/1.44 <=> ! [B: term] :
% 2.42/1.44 ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ( F @ sh @ one )
% 2.42/1.44 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.42/1.44 & ulamvarind
% 2.42/1.44 & ( induction
% 2.42/1.44 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 2.42/1.44 ( ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ A )
% 2.42/1.44 | ( P @ A ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ( P @ Bound_variable_2140 ) ) )
% 2.42/1.44 & ( hoasinduction_lem3a_lthm
% 2.42/1.44 <=> ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.42/1.44 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 2.42/1.44 & termmset_gthm
% 2.42/1.44 & ( hoasinduction_lem3aa
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.42/1.44 & pushprop_lem1v2_gthm
% 2.42/1.44 & hoaslamnotap_gthm
% 2.42/1.44 & hoaslamnotvar_gthm
% 2.42/1.44 & hoasinduction_lem3b_gthm
% 2.42/1.44 & pushprop_lem2v2
% 2.42/1.44 & hoasinduction_lem3a_gthm
% 2.42/1.44 & axclos
% 2.42/1.44 & axassoc
% 2.42/1.44 & ( hoasinduction_lem2v2
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 | ~ ( Q @ Bound_variable_2820 )
% 2.42/1.44 | ~ ( Q @ Bound_variable_2822 )
% 2.42/1.44 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 2.42/1.44 & pushprop_lthm
% 2.42/1.44 & ( apinj2
% 2.42/1.44 <=> ! [A: term,B: term,C: term,D: term] :
% 2.42/1.44 ( ( ( ap @ A @ C )
% 2.42/1.44 != ( ap @ B @ D ) )
% 2.42/1.44 | ( C = D ) ) )
% 2.42/1.44 & ( apinj1
% 2.42/1.44 <=> ! [A: term,B: term,C: term,D: term] :
% 2.42/1.44 ( ( ( ap @ A @ C )
% 2.42/1.44 != ( ap @ B @ D ) )
% 2.42/1.44 | ( A = B ) ) )
% 2.42/1.44 & ( hoasapinj2_lthm
% 2.42/1.44 <=> ( ! [A: term,B: term,C: term,D: term] :
% 2.42/1.44 ( ( ( ap @ A @ C )
% 2.42/1.44 != ( ap @ B @ D ) )
% 2.42/1.44 | ( C = D ) )
% 2.42/1.44 => ! [A: term,B: term,C: term,D: term] :
% 2.42/1.44 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.42/1.44 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.42/1.44 | ( C = D ) ) ) )
% 2.42/1.44 & ( hoasinduction_lem3v2a
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( Q @ B )
% 2.42/1.44 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 2.42/1.44 & ( hoasapinj1_lthm
% 2.42/1.44 <=> ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ( ! [A: term,B: term,C: term,D: term] :
% 2.42/1.44 ( ( ( ap @ A @ C )
% 2.42/1.44 != ( ap @ B @ D ) )
% 2.42/1.44 | ( A = B ) )
% 2.42/1.44 => ! [A: term,B: term,C: term,D: term] :
% 2.42/1.44 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.42/1.44 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.42/1.44 | ( A = B ) ) ) ) )
% 2.42/1.44 & ( hoaslaminj_lthm
% 2.42/1.44 <=> ( ! [A: term,M: subst] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.42/1.44 => ( ! [A: term,M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.42/1.44 => ( ! [A: term,B: term] :
% 2.42/1.44 ( ( ( lam @ A )
% 2.42/1.44 != ( lam @ B ) )
% 2.42/1.44 | ( A = B ) )
% 2.42/1.44 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.42/1.44 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ( ( lam @ ( F @ sh @ one ) )
% 2.42/1.44 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.42/1.44 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.42/1.44 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 2.42/1.44 & ( axvarcons
% 2.42/1.44 <=> ! [A: term,M: subst] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 2.42/1.44 & ( axscons
% 2.42/1.44 <=> ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 2.42/1.44 & hoasinduction_lem2v2_gthm
% 2.42/1.44 & ( axidr
% 2.42/1.44 <=> ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ M @ id ) ) )
% 2.42/1.44 & ( pushprop_lem1
% 2.42/1.44 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 2.42/1.44 & ( laminj
% 2.42/1.44 <=> ! [A: term,B: term] :
% 2.42/1.44 ( ( ( lam @ A )
% 2.42/1.44 != ( lam @ B ) )
% 2.42/1.44 | ( A = B ) ) )
% 2.42/1.44 & ( hoasinduction_lem3_lthm
% 2.42/1.44 <=> ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.42/1.44 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 2.42/1.44 & ( pushprop_lem0
% 2.42/1.44 <=> ! [P: term > $o,A: term,M: subst] :
% 2.42/1.44 ~ ! [Q: term > $o] :
% 2.42/1.44 ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.42/1.44 & pushprop_gthm
% 2.42/1.44 & axabs
% 2.42/1.44 & ( hoasinduction_lem3v2a_lthm
% 2.42/1.44 <=> ( ! [B: term] :
% 2.42/1.44 ~ ! [F: subst > term > term] :
% 2.42/1.44 ~ ! [A: term,M: subst] :
% 2.42/1.44 ( ( sub @ B @ ( push @ A @ M ) )
% 2.42/1.44 = ( F @ M @ A ) )
% 2.42/1.44 => ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( Q @ B )
% 2.42/1.44 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 2.42/1.44 & hoasinduction_lem2_lthm
% 2.42/1.44 & hoasapinj2_gthm
% 2.42/1.44 & hoasinduction_lem1_lthm
% 2.42/1.44 & ~ lamnotap
% 2.42/1.44 & hoasapinj1_gthm
% 2.42/1.44 & hoaslamnotvar
% 2.42/1.44 & ( axidl
% 2.42/1.44 <=> ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ id @ M ) ) )
% 2.42/1.44 & hoaslaminj_gthm
% 2.42/1.44 & ( induction2_lthm
% 2.42/1.44 <=> ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.42/1.44 ( ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ B )
% 2.42/1.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( var @ B )
% 2.42/1.44 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.42/1.44 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 2.42/1.44 => ! [P: term > $o,Bound_variable_2367: term] :
% 2.42/1.44 ( ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ A )
% 2.42/1.44 | ( P @ A ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ B )
% 2.42/1.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ( P @ Bound_variable_2367 ) ) ) ) )
% 2.42/1.44 & ( hoasinduction_lem0_lthm
% 2.42/1.44 <=> ! [P: subst > term > subst > $o] :
% 2.42/1.44 ~ ! [Q: term > $o] :
% 2.42/1.44 ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) ) )
% 2.42/1.44 & ( substmonoid_lthm
% 2.42/1.44 <=> ( ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ id @ M ) )
% 2.42/1.44 => ( ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ M @ id ) )
% 2.42/1.44 => ( ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ id @ M ) )
% 2.42/1.44 & ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ M @ id ) ) ) ) ) )
% 2.42/1.44 & pushprop
% 2.42/1.44 & hoasinduction_lem3_gthm
% 2.42/1.44 & hoasinduction_lem2_gthm
% 2.42/1.44 & ( hoasinduction_lem3b
% 2.42/1.44 <=> ! [B: term] :
% 2.42/1.44 ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ( F @ sh @ one )
% 2.42/1.44 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.42/1.44 & ( substmonoid
% 2.42/1.44 <=> ( ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ id @ M ) )
% 2.42/1.44 & ! [M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ M @ id ) ) ) )
% 2.42/1.44 & lamnotvar
% 2.42/1.44 & ( hoasinduction_lem3a
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 2.42/1.44 & hoasinduction_lem1_gthm
% 2.42/1.44 & ( hoasinduction_no_psi_cond
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.42/1.44 ( ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 2.42/1.44 & induction2_gthm
% 2.42/1.44 & pushprop_lem2v2_lthm
% 2.42/1.44 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.42/1.44 & ( hoaslamnotap
% 2.42/1.44 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ( ( lam @ ( F @ sh @ one ) )
% 2.42/1.44 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 2.42/1.44 & substmonoid_gthm
% 2.42/1.44 & ulamvarsh
% 2.42/1.44 & ( induction2
% 2.42/1.44 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 2.42/1.44 ( ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ A )
% 2.42/1.44 | ( P @ A ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ B )
% 2.42/1.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ( P @ Bound_variable_2367 ) ) )
% 2.42/1.44 & pushprop_lem3v2
% 2.42/1.44 & pushprop_lem2v2_gthm
% 2.42/1.44 & ( pushprop_lem1_lthm
% 2.42/1.44 <=> ( ! [A: term,M: subst] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.42/1.44 => ( ! [A: term,M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.42/1.44 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 2.42/1.44 & ( hoasinduction_lem3v2
% 2.42/1.44 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( Q @ B )
% 2.42/1.44 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 2.42/1.44 & ( axshiftcons
% 2.42/1.44 <=> ! [A: term,M: subst] :
% 2.42/1.44 ( M
% 2.42/1.44 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 2.42/1.44 & ( termmset
% 2.42/1.44 <=> ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) ) )
% 2.42/1.44 & ( pushprop_lem0_lthm
% 2.42/1.44 <=> ! [P: term > $o,A: term,M: subst] :
% 2.42/1.44 ~ ! [Q: term > $o] :
% 2.42/1.44 ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.42/1.44 & hoasapnotvar_lthm
% 2.42/1.44 & ( hoasinduction_lem3v2_lthm
% 2.42/1.44 <=> ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( Q @ B )
% 2.42/1.44 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 2.42/1.44 & ( axvarid
% 2.42/1.44 <=> ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) ) )
% 2.42/1.44 & ( hoasinduction_lthm_3
% 2.42/1.44 <=> ( ! [P: subst > term > subst > $o] :
% 2.42/1.44 ~ ! [Q: term > $o] :
% 2.42/1.44 ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.42/1.44 ( ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ A )
% 2.42/1.44 | ( P @ A ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ A )
% 2.42/1.44 | ~ ( P @ B )
% 2.42/1.44 | ( P @ ( ap @ A @ B ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ! [B: term] :
% 2.42/1.44 ( ~ ( P @ B )
% 2.42/1.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( P @ ( lam @ A ) ) )
% 2.42/1.44 | ( P @ Bound_variable_2367 ) )
% 2.42/1.44 => ( ! [A: term] :
% 2.42/1.44 ( A
% 2.42/1.44 = ( sub @ A @ id ) )
% 2.42/1.44 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.42/1.44 ( ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ~ ! [X: term] :
% 2.42/1.44 ( ( Q @ X )
% 2.42/1.44 = ( P @ id @ X @ id ) )
% 2.42/1.44 | ~ ! [B: term] :
% 2.42/1.44 ( ~ ( Q @ B )
% 2.42/1.44 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.42/1.44 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.42/1.44 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.42/1.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.42/1.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.42/1.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.42/1.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( var @ ( sub @ A @ id ) )
% 2.42/1.44 | ( P @ id @ A @ id ) )
% 2.42/1.44 | ~ ! [A: term,B: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ~ ( P @ id @ B @ id )
% 2.42/1.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.42/1.44 | ~ ! [F: subst > term > term] :
% 2.42/1.44 ( ~ ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 | ~ ! [A: term] :
% 2.42/1.44 ( ~ ( P @ id @ A @ id )
% 2.42/1.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.42/1.44 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 2.42/1.44
% 2.42/1.44 thf(hoasinduction_no_psi_cond,conjecture,
% 2.42/1.44 ( hoasinduction_no_psi_cond
% 2.42/1.44 = ( ! [P: subst > term > subst > $o] :
% 2.42/1.44 ( ! [A: term,B: term] :
% 2.42/1.44 ( ( P @ id @ A @ id )
% 2.42/1.44 => ( ( P @ id @ B @ id )
% 2.42/1.44 => ( P @ id @ ( hoasap @ id @ A @ id @ B ) @ id ) ) )
% 2.42/1.44 => ( ! [F: subst > term > term] :
% 2.42/1.44 ( ! [M: subst,A: term,N: subst] :
% 2.42/1.44 ( ( sub @ ( F @ M @ A ) @ N )
% 2.42/1.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.42/1.44 => ( ! [A: term] :
% 2.42/1.44 ( ( P @ id @ A @ id )
% 2.42/1.44 => ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.42/1.44 => ( P @ id
% 2.42/1.44 @ ( hoaslam @ id
% 2.42/1.44 @ ^ [M: subst,A: term] : ( F @ M @ A ) )
% 2.42/1.44 @ id ) ) )
% 2.42/1.44 => ! [A: term] : ( P @ id @ A @ id ) ) ) ) ) ).
% 2.42/1.44 ------------------------
% 2.42/1.46 % (31248)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/183Mi)
% 2.42/1.46 % (31249)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_31149 for (2999ds/4Mi)
% 2.42/1.46 % (31253)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_31149 for (2999ds/275Mi)
% 2.42/1.46 % (31254)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/18Mi)
% 2.42/1.46 % (31249)Instruction limit reached!
% 2.42/1.46 % (31249)------------------------------
% 2.42/1.46 % (31249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.46 % (31249)Termination reason: Unknown
% 2.42/1.46 % (31249)Termination phase: shuffling
% 2.42/1.46
% 2.42/1.46 % (31249)Memory used [KB]: 1407
% 2.42/1.46 % (31249)Time elapsed: 0.003 s
% 2.42/1.46 % (31249)Instructions burned: 6 (million)
% 2.42/1.46 % (31249)------------------------------
% 2.42/1.46 % (31249)------------------------------
% 2.42/1.46 % (31254)Instruction limit reached!
% 2.42/1.46 % (31254)------------------------------
% 2.42/1.46 % (31254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.46 % (31254)Termination reason: Unknown
% 2.42/1.46 % (31254)Termination phase: shuffling
% 2.42/1.46
% 2.42/1.46 % (31254)Memory used [KB]: 2046
% 2.42/1.46 % (31254)Time elapsed: 0.007 s
% 2.42/1.46 % (31254)Instructions burned: 18 (million)
% 2.42/1.46 % (31254)------------------------------
% 2.42/1.46 % (31254)------------------------------
% 2.42/1.46 % (31251)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/2Mi)
% 2.42/1.47 % (31250)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_31149 for (2999ds/27Mi)
% 2.42/1.47 % (31252)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_31149 for (2999ds/2Mi)
% 2.42/1.47 % (31251)Instruction limit reached!
% 2.42/1.47 % (31251)------------------------------
% 2.42/1.47 % (31251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.47 % (31251)Termination reason: Unknown
% 2.42/1.47 % (31251)Termination phase: shuffling
% 2.42/1.47
% 2.42/1.47 % (31251)Memory used [KB]: 1151
% 2.42/1.47 % (31251)Time elapsed: 0.003 s
% 2.42/1.47 % (31251)Instructions burned: 2 (million)
% 2.42/1.47 % (31251)------------------------------
% 2.42/1.47 % (31251)------------------------------
% 2.42/1.47 % (31252)Instruction limit reached!
% 2.42/1.47 % (31252)------------------------------
% 2.42/1.47 % (31252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.47 % (31252)Termination reason: Unknown
% 2.42/1.47 % (31252)Termination phase: shuffling
% 2.42/1.47
% 2.42/1.47 % (31252)Memory used [KB]: 1279
% 2.42/1.47 % (31252)Time elapsed: 0.004 s
% 2.42/1.47 % (31252)Instructions burned: 3 (million)
% 2.42/1.47 % (31252)------------------------------
% 2.42/1.47 % (31252)------------------------------
% 2.42/1.47 % (31255)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/3Mi)
% 2.42/1.47 % (31253)Refutation not found, incomplete strategy
% 2.42/1.47 % (31253)------------------------------
% 2.42/1.47 % (31253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.47 % (31253)Termination reason: Refutation not found, incomplete strategy
% 2.42/1.47
% 2.42/1.47
% 2.42/1.47 % (31253)Memory used [KB]: 6780
% 2.42/1.47 % (31253)Time elapsed: 0.014 s
% 2.42/1.47 % (31253)Instructions burned: 43 (million)
% 2.42/1.47 % (31253)------------------------------
% 2.42/1.47 % (31253)------------------------------
% 2.42/1.47 % (31255)Instruction limit reached!
% 2.42/1.47 % (31255)------------------------------
% 2.42/1.47 % (31255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.47 % (31255)Termination reason: Unknown
% 2.42/1.47 % (31255)Termination phase: shuffling
% 2.42/1.47
% 2.42/1.47 % (31255)Memory used [KB]: 1279
% 2.42/1.47 % (31255)Time elapsed: 0.002 s
% 2.42/1.47 % (31255)Instructions burned: 3 (million)
% 2.42/1.47 % (31255)------------------------------
% 2.42/1.47 % (31255)------------------------------
% 2.42/1.47 % (31256)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_31149 for (2999ds/37Mi)
% 2.42/1.48 % (31250)Instruction limit reached!
% 2.42/1.48 % (31250)------------------------------
% 2.42/1.48 % (31250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.48 % (31250)Termination reason: Unknown
% 2.42/1.48 % (31250)Termination phase: Property scanning
% 2.42/1.48
% 2.42/1.48 % (31250)Memory used [KB]: 2174
% 2.42/1.48 % (31250)Time elapsed: 0.018 s
% 2.42/1.48 % (31250)Instructions burned: 27 (million)
% 2.42/1.48 % (31250)------------------------------
% 2.42/1.48 % (31250)------------------------------
% 2.42/1.48 % (31260)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/7Mi)
% 2.42/1.48 % (31259)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_31149 for (2999ds/1041Mi)
% 2.42/1.48 % (31260)Instruction limit reached!
% 2.42/1.48 % (31260)------------------------------
% 2.42/1.48 % (31260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.48 % (31260)Termination reason: Unknown
% 2.42/1.48 % (31260)Termination phase: shuffling
% 2.42/1.48
% 2.42/1.48 % (31260)Memory used [KB]: 1407
% 2.42/1.48 % (31260)Time elapsed: 0.003 s
% 2.42/1.48 % (31260)Instructions burned: 7 (million)
% 2.42/1.48 % (31260)------------------------------
% 2.42/1.48 % (31260)------------------------------
% 2.42/1.48 % (31257)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_31149 for (2999ds/15Mi)
% 2.42/1.48 % (31258)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_31149 for (2999ds/3Mi)
% 2.42/1.49 % (31256)Instruction limit reached!
% 2.42/1.49 % (31256)------------------------------
% 2.42/1.49 % (31256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.49 % (31256)Termination reason: Unknown
% 2.42/1.49 % (31256)Termination phase: Property scanning
% 2.42/1.49
% 2.42/1.49 % (31256)Memory used [KB]: 1918
% 2.42/1.49 % (31256)Time elapsed: 0.013 s
% 2.42/1.49 % (31256)Instructions burned: 40 (million)
% 2.42/1.49 % (31256)------------------------------
% 2.42/1.49 % (31256)------------------------------
% 2.42/1.49 % (31258)Instruction limit reached!
% 2.42/1.49 % (31258)------------------------------
% 2.42/1.49 % (31258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.49 % (31258)Termination reason: Unknown
% 2.42/1.49 % (31258)Termination phase: shuffling
% 2.42/1.49
% 2.42/1.49 % (31258)Memory used [KB]: 1279
% 2.42/1.49 % (31258)Time elapsed: 0.004 s
% 2.42/1.49 % (31258)Instructions burned: 3 (million)
% 2.42/1.49 % (31258)------------------------------
% 2.42/1.49 % (31258)------------------------------
% 2.42/1.49 % (31257)Instruction limit reached!
% 2.42/1.49 % (31257)------------------------------
% 2.42/1.49 % (31257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.49 % (31257)Termination reason: Unknown
% 2.42/1.49 % (31257)Termination phase: shuffling
% 2.42/1.49
% 2.42/1.49 % (31257)Memory used [KB]: 1791
% 2.42/1.49 % (31257)Time elapsed: 0.010 s
% 2.42/1.49 % (31257)Instructions burned: 15 (million)
% 2.42/1.49 % (31257)------------------------------
% 2.42/1.49 % (31257)------------------------------
% 2.42/1.49 % (31262)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_31149 for (2999ds/3Mi)
% 2.42/1.49 % (31262)Instruction limit reached!
% 2.42/1.49 % (31262)------------------------------
% 2.42/1.49 % (31262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.42/1.49 % (31262)Termination reason: Unknown
% 2.42/1.49 % (31262)Termination phase: shuffling
% 2.42/1.49
% 2.42/1.49 % (31262)Memory used [KB]: 1279
% 2.42/1.49 % (31262)Time elapsed: 0.002 s
% 2.42/1.49 % (31262)Instructions burned: 4 (million)
% 2.42/1.49 % (31262)------------------------------
% 2.42/1.49 % (31262)------------------------------
% 2.42/1.50 % (31263)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_31149 for (2999ds/3Mi)
% 2.46/1.50 % (31263)Instruction limit reached!
% 2.46/1.50 % (31263)------------------------------
% 2.46/1.50 % (31263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.50 % (31263)Termination reason: Unknown
% 2.46/1.50 % (31263)Termination phase: shuffling
% 2.46/1.50
% 2.46/1.50 % (31263)Memory used [KB]: 1279
% 2.46/1.50 % (31263)Time elapsed: 0.003 s
% 2.46/1.50 % (31263)Instructions burned: 4 (million)
% 2.46/1.50 % (31263)------------------------------
% 2.46/1.50 % (31263)------------------------------
% 2.46/1.50 % (31261)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/16Mi)
% 2.46/1.50 % (31266)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/4Mi)
% 2.46/1.50 % (31266)Instruction limit reached!
% 2.46/1.50 % (31266)------------------------------
% 2.46/1.50 % (31266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.50 % (31266)Termination reason: Unknown
% 2.46/1.50 % (31266)Termination phase: shuffling
% 2.46/1.50
% 2.46/1.50 % (31266)Memory used [KB]: 1279
% 2.46/1.50 % (31266)Time elapsed: 0.002 s
% 2.46/1.50 % (31266)Instructions burned: 4 (million)
% 2.46/1.50 % (31266)------------------------------
% 2.46/1.50 % (31266)------------------------------
% 2.46/1.50 % (31264)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/7Mi)
% 2.46/1.51 % (31248)Instruction limit reached!
% 2.46/1.51 % (31248)------------------------------
% 2.46/1.51 % (31248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.51 % (31248)Termination reason: Unknown
% 2.46/1.51 % (31248)Termination phase: Property scanning
% 2.46/1.51
% 2.46/1.51 % (31248)Memory used [KB]: 4349
% 2.46/1.51 % (31248)Time elapsed: 0.050 s
% 2.46/1.51 % (31248)Instructions burned: 183 (million)
% 2.46/1.51 % (31248)------------------------------
% 2.46/1.51 % (31248)------------------------------
% 2.46/1.51 % (31267)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/18Mi)
% 2.46/1.51 % (31261)Instruction limit reached!
% 2.46/1.51 % (31261)------------------------------
% 2.46/1.51 % (31261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.51 % (31261)Termination reason: Unknown
% 2.46/1.51 % (31261)Termination phase: shuffling
% 2.46/1.51
% 2.46/1.51 % (31261)Memory used [KB]: 1918
% 2.46/1.51 % (31261)Time elapsed: 0.011 s
% 2.46/1.51 % (31261)Instructions burned: 17 (million)
% 2.46/1.51 % (31261)------------------------------
% 2.46/1.51 % (31261)------------------------------
% 2.46/1.51 % (31264)Instruction limit reached!
% 2.46/1.51 % (31264)------------------------------
% 2.46/1.51 % (31264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.51 % (31264)Termination reason: Unknown
% 2.46/1.51 % (31264)Termination phase: shuffling
% 2.46/1.51
% 2.46/1.51 % (31264)Memory used [KB]: 1407
% 2.46/1.51 % (31264)Time elapsed: 0.006 s
% 2.46/1.51 % (31264)Instructions burned: 7 (million)
% 2.46/1.51 % (31264)------------------------------
% 2.46/1.51 % (31264)------------------------------
% 2.46/1.51 % (31265)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/3Mi)
% 2.46/1.51 % (31265)Instruction limit reached!
% 2.46/1.51 % (31265)------------------------------
% 2.46/1.51 % (31265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.51 % (31265)Termination reason: Unknown
% 2.46/1.51 % (31265)Termination phase: shuffling
% 2.46/1.51
% 2.46/1.51 % (31265)Memory used [KB]: 1279
% 2.46/1.51 % (31265)Time elapsed: 0.004 s
% 2.46/1.51 % (31265)Instructions burned: 4 (million)
% 2.46/1.51 % (31265)------------------------------
% 2.46/1.51 % (31265)------------------------------
% 2.46/1.51 % (31268)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/710Mi)
% 2.46/1.51 % (31267)Instruction limit reached!
% 2.46/1.51 % (31267)------------------------------
% 2.46/1.51 % (31267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.51 % (31267)Termination reason: Unknown
% 2.46/1.51 % (31267)Termination phase: shuffling
% 2.46/1.51
% 2.46/1.51 % (31267)Memory used [KB]: 2046
% 2.46/1.51 % (31267)Time elapsed: 0.007 s
% 2.46/1.51 % (31267)Instructions burned: 19 (million)
% 2.46/1.51 % (31267)------------------------------
% 2.46/1.51 % (31267)------------------------------
% 2.46/1.52 % (31269)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/6Mi)
% 2.46/1.52 % (31269)Instruction limit reached!
% 2.46/1.52 % (31269)------------------------------
% 2.46/1.52 % (31269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.52 % (31269)Termination reason: Unknown
% 2.46/1.52 % (31269)Termination phase: shuffling
% 2.46/1.52
% 2.46/1.52 % (31269)Memory used [KB]: 1407
% 2.46/1.52 % (31269)Time elapsed: 0.003 s
% 2.46/1.52 % (31269)Instructions burned: 7 (million)
% 2.46/1.52 % (31269)------------------------------
% 2.46/1.52 % (31269)------------------------------
% 2.46/1.52 % (31270)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_31149 for (2999ds/902Mi)
% 2.46/1.52 % (31273)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_31149 for (2999ds/6Mi)
% 2.46/1.52 % (31273)Instruction limit reached!
% 2.46/1.52 % (31273)------------------------------
% 2.46/1.52 % (31273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.52 % (31273)Termination reason: Unknown
% 2.46/1.52 % (31273)Termination phase: shuffling
% 2.46/1.52
% 2.46/1.52 % (31273)Memory used [KB]: 1407
% 2.46/1.52 % (31273)Time elapsed: 0.003 s
% 2.46/1.52 % (31273)Instructions burned: 6 (million)
% 2.46/1.52 % (31273)------------------------------
% 2.46/1.52 % (31273)------------------------------
% 2.46/1.53 % (31271)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_31149 for (2999ds/21Mi)
% 2.46/1.53 % (31274)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/377Mi)
% 2.46/1.53 % (31272)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/5Mi)
% 2.46/1.53 % (31272)Instruction limit reached!
% 2.46/1.53 % (31272)------------------------------
% 2.46/1.53 % (31272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.53 % (31272)Termination reason: Unknown
% 2.46/1.53 % (31272)Termination phase: shuffling
% 2.46/1.53
% 2.46/1.53 % (31272)Memory used [KB]: 1279
% 2.46/1.53 % (31272)Time elapsed: 0.005 s
% 2.46/1.53 % (31272)Instructions burned: 6 (million)
% 2.46/1.53 % (31272)------------------------------
% 2.46/1.53 % (31272)------------------------------
% 2.46/1.53 % (31275)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_31149 for (2999ds/779Mi)
% 2.46/1.54 % (31271)Instruction limit reached!
% 2.46/1.54 % (31271)------------------------------
% 2.46/1.54 % (31271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.54 % (31271)Termination reason: Unknown
% 2.46/1.54 % (31271)Termination phase: shuffling
% 2.46/1.54
% 2.46/1.54 % (31271)Memory used [KB]: 2046
% 2.46/1.54 % (31271)Time elapsed: 0.015 s
% 2.46/1.54 % (31271)Instructions burned: 22 (million)
% 2.46/1.54 % (31271)------------------------------
% 2.46/1.54 % (31271)------------------------------
% 2.46/1.55 % (31276)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_31149 for (2999ds/19Mi)
% 2.46/1.56 % (31276)Instruction limit reached!
% 2.46/1.56 % (31276)------------------------------
% 2.46/1.56 % (31276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.56 % (31276)Termination reason: Unknown
% 2.46/1.56 % (31276)Termination phase: shuffling
% 2.46/1.56
% 2.46/1.56 % (31276)Memory used [KB]: 1663
% 2.46/1.56 % (31276)Time elapsed: 0.009 s
% 2.46/1.56 % (31276)Instructions burned: 20 (million)
% 2.46/1.56 % (31276)------------------------------
% 2.46/1.56 % (31276)------------------------------
% 2.46/1.56 % (31277)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_31149 for (2999ds/879Mi)
% 2.46/1.57 % (31278)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_31149 for (2998ds/17Mi)
% 2.46/1.57 % (31278)Instruction limit reached!
% 2.46/1.57 % (31278)------------------------------
% 2.46/1.57 % (31278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.57 % (31278)Termination reason: Unknown
% 2.46/1.57 % (31278)Termination phase: shuffling
% 2.46/1.57
% 2.46/1.57 % (31278)Memory used [KB]: 1918
% 2.46/1.57 % (31278)Time elapsed: 0.007 s
% 2.46/1.57 % (31278)Instructions burned: 17 (million)
% 2.46/1.57 % (31278)------------------------------
% 2.46/1.57 % (31278)------------------------------
% 2.46/1.58 % (31279)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_31149 for (2998ds/3Mi)
% 2.46/1.58 % (31279)Instruction limit reached!
% 2.46/1.58 % (31279)------------------------------
% 2.46/1.58 % (31279)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.58 % (31279)Termination reason: Unknown
% 2.46/1.58 % (31279)Termination phase: shuffling
% 2.46/1.58
% 2.46/1.58 % (31279)Memory used [KB]: 1279
% 2.46/1.58 % (31279)Time elapsed: 0.003 s
% 2.46/1.58 % (31279)Instructions burned: 4 (million)
% 2.46/1.58 % (31279)------------------------------
% 2.46/1.58 % (31279)------------------------------
% 2.46/1.59 % (31280)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_31149 for (2998ds/30Mi)
% 2.46/1.59 % (31270)Refutation not found, incomplete strategy
% 2.46/1.59 % (31270)------------------------------
% 2.46/1.59 % (31270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.59 % (31270)Termination reason: Refutation not found, incomplete strategy
% 2.46/1.59
% 2.46/1.59
% 2.46/1.59 % (31270)Memory used [KB]: 9466
% 2.46/1.59 % (31270)Time elapsed: 0.075 s
% 2.46/1.59 % (31270)Instructions burned: 233 (million)
% 2.46/1.59 % (31270)------------------------------
% 2.46/1.59 % (31270)------------------------------
% 2.46/1.60 % (31280)Instruction limit reached!
% 2.46/1.60 % (31280)------------------------------
% 2.46/1.60 % (31280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.60 % (31280)Termination reason: Unknown
% 2.46/1.60 % (31280)Termination phase: Property scanning
% 2.46/1.60
% 2.46/1.60 % (31280)Memory used [KB]: 2174
% 2.46/1.60 % (31280)Time elapsed: 0.012 s
% 2.46/1.60 % (31280)Instructions burned: 31 (million)
% 2.46/1.60 % (31280)------------------------------
% 2.46/1.60 % (31280)------------------------------
% 2.46/1.61 % (31281)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_31149 for (2998ds/127Mi)
% 2.46/1.61 % (31282)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_31149 for (2998ds/100Mi)
% 2.46/1.63 % (31274)Instruction limit reached!
% 2.46/1.63 % (31274)------------------------------
% 2.46/1.63 % (31274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.63 % (31274)Termination reason: Unknown
% 2.46/1.63 % (31274)Termination phase: Saturation
% 2.46/1.63
% 2.46/1.63 % (31274)Memory used [KB]: 10490
% 2.46/1.63 % (31274)Time elapsed: 0.108 s
% 2.46/1.63 % (31274)Instructions burned: 378 (million)
% 2.46/1.63 % (31274)------------------------------
% 2.46/1.63 % (31274)------------------------------
% 2.46/1.64 % (31282)Instruction limit reached!
% 2.46/1.64 % (31282)------------------------------
% 2.46/1.64 % (31282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.64 % (31282)Termination reason: Unknown
% 2.46/1.64 % (31282)Termination phase: Saturation
% 2.46/1.64
% 2.46/1.64 % (31282)Memory used [KB]: 6524
% 2.46/1.64 % (31282)Time elapsed: 0.030 s
% 2.46/1.64 % (31282)Instructions burned: 103 (million)
% 2.46/1.64 % (31282)------------------------------
% 2.46/1.64 % (31282)------------------------------
% 2.46/1.64 % (31283)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_31149 for (2998ds/3Mi)
% 2.46/1.64 % (31281)Instruction limit reached!
% 2.46/1.64 % (31281)------------------------------
% 2.46/1.64 % (31281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.64 % (31281)Termination reason: Unknown
% 2.46/1.64 % (31281)Termination phase: Property scanning
% 2.46/1.64
% 2.46/1.64 % (31281)Memory used [KB]: 4093
% 2.46/1.64 % (31281)Time elapsed: 0.040 s
% 2.46/1.64 % (31281)Instructions burned: 130 (million)
% 2.46/1.64 % (31281)------------------------------
% 2.46/1.64 % (31281)------------------------------
% 2.46/1.65 % (31283)Instruction limit reached!
% 2.46/1.65 % (31283)------------------------------
% 2.46/1.65 % (31283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.65 % (31283)Termination reason: Unknown
% 2.46/1.65 % (31283)Termination phase: shuffling
% 2.46/1.65
% 2.46/1.65 % (31283)Memory used [KB]: 1407
% 2.46/1.65 % (31283)Time elapsed: 0.003 s
% 2.46/1.65 % (31283)Instructions burned: 6 (million)
% 2.46/1.65 % (31283)------------------------------
% 2.46/1.65 % (31283)------------------------------
% 2.46/1.65 % (31284)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_31149 for (2998ds/20Mi)
% 2.46/1.65 % (31286)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_31149 for (2997ds/107Mi)
% 2.46/1.66 % (31285)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_31149 for (2997ds/86Mi)
% 2.46/1.66 % (31284)Instruction limit reached!
% 2.46/1.66 % (31284)------------------------------
% 2.46/1.66 % (31284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.66 % (31284)Termination reason: Unknown
% 2.46/1.66 % (31284)Termination phase: shuffling
% 2.46/1.66
% 2.46/1.66 % (31284)Memory used [KB]: 2046
% 2.46/1.66 % (31284)Time elapsed: 0.009 s
% 2.46/1.66 % (31284)Instructions burned: 21 (million)
% 2.46/1.66 % (31284)------------------------------
% 2.46/1.66 % (31284)------------------------------
% 2.46/1.67 % (31287)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_31149 for (2997ds/39Mi)
% 2.46/1.68 % (31287)Instruction limit reached!
% 2.46/1.68 % (31287)------------------------------
% 2.46/1.68 % (31287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.68 % (31287)Termination reason: Unknown
% 2.46/1.68 % (31287)Termination phase: Property scanning
% 2.46/1.68
% 2.46/1.68 % (31287)Memory used [KB]: 1918
% 2.46/1.68 % (31287)Time elapsed: 0.012 s
% 2.46/1.68 % (31287)Instructions burned: 39 (million)
% 2.46/1.68 % (31287)------------------------------
% 2.46/1.68 % (31287)------------------------------
% 2.46/1.68 % (31285)Instruction limit reached!
% 2.46/1.68 % (31285)------------------------------
% 2.46/1.68 % (31285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.68 % (31285)Termination reason: Unknown
% 2.46/1.68 % (31285)Termination phase: Preprocessing 3
% 2.46/1.68
% 2.46/1.68 % (31285)Memory used [KB]: 3965
% 2.46/1.68 % (31285)Time elapsed: 0.029 s
% 2.46/1.68 % (31285)Instructions burned: 86 (million)
% 2.46/1.68 % (31285)------------------------------
% 2.46/1.68 % (31285)------------------------------
% 2.46/1.69 % (31286)Instruction limit reached!
% 2.46/1.69 % (31286)------------------------------
% 2.46/1.69 % (31286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.69 % (31286)Termination reason: Unknown
% 2.46/1.69 % (31286)Termination phase: Property scanning
% 2.46/1.69
% 2.46/1.69 % (31286)Memory used [KB]: 4477
% 2.46/1.69 % (31286)Time elapsed: 0.032 s
% 2.46/1.69 % (31286)Instructions burned: 109 (million)
% 2.46/1.69 % (31286)------------------------------
% 2.46/1.69 % (31286)------------------------------
% 2.46/1.69 % (31288)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_31149 for (2997ds/448Mi)
% 2.46/1.69 % (31289)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_31149 for (2997ds/46Mi)
% 2.46/1.70 % (31290)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_31149 for (2997ds/98Mi)
% 2.46/1.71 % (31289)Instruction limit reached!
% 2.46/1.71 % (31289)------------------------------
% 2.46/1.71 % (31289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.71 % (31289)Termination reason: Unknown
% 2.46/1.71 % (31289)Termination phase: Preprocessing 1
% 2.46/1.71
% 2.46/1.71 % (31289)Memory used [KB]: 2558
% 2.46/1.71 % (31289)Time elapsed: 0.016 s
% 2.46/1.71 % (31289)Instructions burned: 48 (million)
% 2.46/1.71 % (31289)------------------------------
% 2.46/1.71 % (31289)------------------------------
% 2.46/1.72 % (31291)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_31149 for (2997ds/507Mi)
% 2.46/1.72 % (31290)Instruction limit reached!
% 2.46/1.72 % (31290)------------------------------
% 2.46/1.72 % (31290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.72 % (31290)Termination reason: Unknown
% 2.46/1.72 % (31290)Termination phase: Saturation
% 2.46/1.72
% 2.46/1.72 % (31290)Memory used [KB]: 2046
% 2.46/1.72 % (31290)Time elapsed: 0.026 s
% 2.46/1.72 % (31290)Instructions burned: 100 (million)
% 2.46/1.72 % (31290)------------------------------
% 2.46/1.72 % (31290)------------------------------
% 2.46/1.72 % (31275)Instruction limit reached!
% 2.46/1.72 % (31275)------------------------------
% 2.46/1.72 % (31275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.72 % (31275)Termination reason: Unknown
% 2.46/1.72 % (31275)Termination phase: Saturation
% 2.46/1.72
% 2.46/1.72 % (31275)Memory used [KB]: 6652
% 2.46/1.72 % (31275)Time elapsed: 0.188 s
% 2.46/1.72 % (31275)Instructions burned: 780 (million)
% 2.46/1.72 % (31275)------------------------------
% 2.46/1.72 % (31275)------------------------------
% 2.46/1.72 % (31268)Instruction limit reached!
% 2.46/1.72 % (31268)------------------------------
% 2.46/1.72 % (31268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.72 % (31268)Termination reason: Unknown
% 2.46/1.72 % (31268)Termination phase: Saturation
% 2.46/1.72
% 2.46/1.72 % (31268)Memory used [KB]: 11257
% 2.46/1.72 % (31268)Time elapsed: 0.211 s
% 2.46/1.72 % (31268)Instructions burned: 712 (million)
% 2.46/1.72 % (31268)------------------------------
% 2.46/1.72 % (31268)------------------------------
% 2.46/1.73 % (31292)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_31149 for (2997ds/149Mi)
% 2.46/1.73 % (31293)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_31149 for (2997ds/39Mi)
% 2.46/1.73 % (31294)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_31149 for (2997ds/6Mi)
% 2.46/1.73 % (31294)Instruction limit reached!
% 2.46/1.73 % (31294)------------------------------
% 2.46/1.73 % (31294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.73 % (31294)Termination reason: Unknown
% 2.46/1.73 % (31294)Termination phase: shuffling
% 2.46/1.73
% 2.46/1.73 % (31294)Memory used [KB]: 1407
% 2.46/1.73 % (31294)Time elapsed: 0.003 s
% 2.46/1.73 % (31294)Instructions burned: 7 (million)
% 2.46/1.73 % (31294)------------------------------
% 2.46/1.73 % (31294)------------------------------
% 2.46/1.74 % (31293)Instruction limit reached!
% 2.46/1.74 % (31293)------------------------------
% 2.46/1.74 % (31293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.74 % (31293)Termination reason: Unknown
% 2.46/1.74 % (31293)Termination phase: Preprocessing 1
% 2.46/1.74
% 2.46/1.74 % (31293)Memory used [KB]: 2302
% 2.46/1.74 % (31293)Time elapsed: 0.013 s
% 2.46/1.74 % (31293)Instructions burned: 41 (million)
% 2.46/1.74 % (31293)------------------------------
% 2.46/1.74 % (31293)------------------------------
% 2.46/1.74 % (31295)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_31149 for (2997ds/1087Mi)
% 2.46/1.75 % (31296)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_31149 for (2997ds/47Mi)
% 2.46/1.75 % (31259)Instruction limit reached!
% 2.46/1.75 % (31259)------------------------------
% 2.46/1.75 % (31259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.75 % (31259)Termination reason: Unknown
% 2.46/1.75 % (31259)Termination phase: Saturation
% 2.46/1.75
% 2.46/1.75 % (31259)Memory used [KB]: 7931
% 2.46/1.75 % (31259)Time elapsed: 0.275 s
% 2.46/1.75 % (31259)Instructions burned: 1043 (million)
% 2.46/1.75 % (31259)------------------------------
% 2.46/1.75 % (31259)------------------------------
% 2.46/1.76 % (31297)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_31149 for (2996ds/44Mi)
% 2.46/1.77 % (31296)Instruction limit reached!
% 2.46/1.77 % (31296)------------------------------
% 2.46/1.77 % (31296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.77 % (31296)Termination reason: Unknown
% 2.46/1.77 % (31296)Termination phase: Preprocessing 3
% 2.46/1.77
% 2.46/1.77 % (31296)Memory used [KB]: 2558
% 2.46/1.77 % (31296)Time elapsed: 0.015 s
% 2.46/1.77 % (31296)Instructions burned: 48 (million)
% 2.46/1.77 % (31296)------------------------------
% 2.46/1.77 % (31296)------------------------------
% 2.46/1.77 % (31292)Instruction limit reached!
% 2.46/1.77 % (31292)------------------------------
% 2.46/1.77 % (31292)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.77 % (31292)Termination reason: Unknown
% 2.46/1.77 % (31292)Termination phase: Property scanning
% 2.46/1.77
% 2.46/1.77 % (31292)Memory used [KB]: 7931
% 2.46/1.77 % (31292)Time elapsed: 0.043 s
% 2.46/1.77 % (31292)Instructions burned: 152 (million)
% 2.46/1.77 % (31292)------------------------------
% 2.46/1.77 % (31292)------------------------------
% 2.46/1.78 % (31298)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_31149 for (2996ds/432Mi)
% 2.46/1.78 % (31297)Instruction limit reached!
% 2.46/1.78 % (31297)------------------------------
% 2.46/1.78 % (31297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.78 % (31297)Termination reason: Unknown
% 2.46/1.78 % (31297)Termination phase: Property scanning
% 2.46/1.78
% 2.46/1.78 % (31297)Memory used [KB]: 1918
% 2.46/1.78 % (31297)Time elapsed: 0.013 s
% 2.46/1.78 % (31297)Instructions burned: 45 (million)
% 2.46/1.78 % (31297)------------------------------
% 2.46/1.78 % (31297)------------------------------
% 2.46/1.78 % (31299)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_31149 for (2996ds/62Mi)
% 2.46/1.79 % (31300)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_31149 for (2996ds/31Mi)
% 2.46/1.80 % (31300)Instruction limit reached!
% 2.46/1.80 % (31300)------------------------------
% 2.46/1.80 % (31300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.80 % (31300)Termination reason: Unknown
% 2.46/1.80 % (31300)Termination phase: Property scanning
% 2.46/1.80
% 2.46/1.80 % (31300)Memory used [KB]: 2174
% 2.46/1.80 % (31300)Time elapsed: 0.011 s
% 2.46/1.80 % (31300)Instructions burned: 33 (million)
% 2.46/1.80 % (31300)------------------------------
% 2.46/1.80 % (31300)------------------------------
% 2.46/1.80 % (31299)Instruction limit reached!
% 2.46/1.80 % (31299)------------------------------
% 2.46/1.80 % (31299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.80 % (31299)Termination reason: Unknown
% 2.46/1.80 % (31299)Termination phase: Property scanning
% 2.46/1.80
% 2.46/1.80 % (31299)Memory used [KB]: 1918
% 2.46/1.80 % (31299)Time elapsed: 0.017 s
% 2.46/1.80 % (31299)Instructions burned: 64 (million)
% 2.46/1.80 % (31299)------------------------------
% 2.46/1.80 % (31299)------------------------------
% 2.46/1.80 % (31291)Refutation not found, incomplete strategy
% 2.46/1.80 % (31291)------------------------------
% 2.46/1.80 % (31291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.80 % (31291)Termination reason: Refutation not found, incomplete strategy
% 2.46/1.80
% 2.46/1.80
% 2.46/1.80 % (31291)Memory used [KB]: 9850
% 2.46/1.80 % (31291)Time elapsed: 0.086 s
% 2.46/1.80 % (31291)Instructions burned: 279 (million)
% 2.46/1.80 % (31291)------------------------------
% 2.46/1.80 % (31291)------------------------------
% 2.46/1.81 % (31301)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_31149 for (2996ds/96Mi)
% 2.46/1.81 % (31302)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_31149 for (2996ds/453Mi)
% 2.46/1.82 % (31303)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_31149 for (2996ds/71Mi)
% 2.46/1.83 % (31288)Instruction limit reached!
% 2.46/1.83 % (31288)------------------------------
% 2.46/1.83 % (31288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.83 % (31288)Termination reason: Unknown
% 2.46/1.83 % (31288)Termination phase: Saturation
% 2.46/1.83
% 2.46/1.83 % (31288)Memory used [KB]: 7291
% 2.46/1.83 % (31288)Time elapsed: 0.138 s
% 2.46/1.83 % (31288)Instructions burned: 448 (million)
% 2.46/1.83 % (31288)------------------------------
% 2.46/1.83 % (31288)------------------------------
% 2.46/1.83 % (31301)Instruction limit reached!
% 2.46/1.83 % (31301)------------------------------
% 2.46/1.83 % (31301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.83 % (31301)Termination reason: Unknown
% 2.46/1.83 % (31301)Termination phase: Property scanning
% 2.46/1.83
% 2.46/1.83 % (31301)Memory used [KB]: 4221
% 2.46/1.83 % (31301)Time elapsed: 0.029 s
% 2.46/1.83 % (31301)Instructions burned: 98 (million)
% 2.46/1.83 % (31301)------------------------------
% 2.46/1.83 % (31301)------------------------------
% 2.46/1.84 % (31303)Instruction limit reached!
% 2.46/1.84 % (31303)------------------------------
% 2.46/1.84 % (31303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.84 % (31303)Termination reason: Unknown
% 2.46/1.84 % (31303)Termination phase: Preprocessing 3
% 2.46/1.84
% 2.46/1.84 % (31303)Memory used [KB]: 3454
% 2.46/1.84 % (31303)Time elapsed: 0.024 s
% 2.46/1.84 % (31303)Instructions burned: 72 (million)
% 2.46/1.84 % (31303)------------------------------
% 2.46/1.84 % (31303)------------------------------
% 2.46/1.84 % (31304)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_31149 for (2996ds/83Mi)
% 2.46/1.84 % (31305)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_31149 for (2996ds/11Mi)
% 2.46/1.85 % (31305)Instruction limit reached!
% 2.46/1.85 % (31305)------------------------------
% 2.46/1.85 % (31305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.85 % (31305)Termination reason: Unknown
% 2.46/1.85 % (31305)Termination phase: shuffling
% 2.46/1.85
% 2.46/1.85 % (31305)Memory used [KB]: 1535
% 2.46/1.85 % (31305)Time elapsed: 0.004 s
% 2.46/1.85 % (31305)Instructions burned: 11 (million)
% 2.46/1.85 % (31305)------------------------------
% 2.46/1.85 % (31305)------------------------------
% 2.46/1.85 % (31277)Instruction limit reached!
% 2.46/1.85 % (31277)------------------------------
% 2.46/1.85 % (31277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.85 % (31277)Termination reason: Unknown
% 2.46/1.85 % (31277)Termination phase: Saturation
% 2.46/1.85
% 2.46/1.85 % (31277)Memory used [KB]: 11769
% 2.46/1.85 % (31277)Time elapsed: 0.293 s
% 2.46/1.85 % (31277)Instructions burned: 879 (million)
% 2.46/1.85 % (31277)------------------------------
% 2.46/1.85 % (31277)------------------------------
% 2.46/1.85 % (31306)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_31149 for (2996ds/41Mi)
% 2.46/1.86 % (31307)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31149 for (2996ds/6Mi)
% 2.46/1.86 % (31307)Instruction limit reached!
% 2.46/1.86 % (31307)------------------------------
% 2.46/1.86 % (31307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.86 % (31307)Termination reason: Unknown
% 2.46/1.86 % (31307)Termination phase: shuffling
% 2.46/1.86
% 2.46/1.86 % (31307)Memory used [KB]: 1407
% 2.46/1.86 % (31307)Time elapsed: 0.003 s
% 2.46/1.86 % (31307)Instructions burned: 7 (million)
% 2.46/1.86 % (31307)------------------------------
% 2.46/1.86 % (31307)------------------------------
% 2.46/1.86 % (31308)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_31149 for (2996ds/150Mi)
% 2.46/1.86 % (31306)Instruction limit reached!
% 2.46/1.86 % (31306)------------------------------
% 2.46/1.86 % (31306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.86 % (31306)Termination reason: Unknown
% 2.46/1.86 % (31306)Termination phase: Property scanning
% 2.46/1.86
% 2.46/1.86 % (31306)Memory used [KB]: 1918
% 2.46/1.86 % (31306)Time elapsed: 0.012 s
% 2.46/1.86 % (31306)Instructions burned: 43 (million)
% 2.46/1.86 % (31306)------------------------------
% 2.46/1.86 % (31306)------------------------------
% 2.46/1.87 % (31304)Instruction limit reached!
% 2.46/1.87 % (31304)------------------------------
% 2.46/1.87 % (31304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.87 % (31304)Termination reason: Unknown
% 2.46/1.87 % (31304)Termination phase: Saturation
% 2.46/1.87
% 2.46/1.87 % (31304)Memory used [KB]: 7036
% 2.46/1.87 % (31304)Time elapsed: 0.028 s
% 2.46/1.87 % (31304)Instructions burned: 85 (million)
% 2.46/1.87 % (31304)------------------------------
% 2.46/1.87 % (31304)------------------------------
% 2.46/1.87 % (31309)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_31149 for (2995ds/343Mi)
% 2.46/1.87 % (31310)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_31149 for (2995ds/109Mi)
% 2.46/1.88 % (31311)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_31149 for (2995ds/253Mi)
% 2.46/1.88 % (31298)Instruction limit reached!
% 2.46/1.88 % (31298)------------------------------
% 2.46/1.88 % (31298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.88 % (31298)Termination reason: Unknown
% 2.46/1.88 % (31298)Termination phase: Saturation
% 2.46/1.88
% 2.46/1.88 % (31298)Memory used [KB]: 2430
% 2.46/1.88 % (31298)Time elapsed: 0.109 s
% 2.46/1.88 % (31298)Instructions burned: 433 (million)
% 2.46/1.88 % (31298)------------------------------
% 2.46/1.88 % (31298)------------------------------
% 2.46/1.89 % (31312)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_31149 for (2995ds/140Mi)
% 2.46/1.90 % (31310)Instruction limit reached!
% 2.46/1.90 % (31310)------------------------------
% 2.46/1.90 % (31310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.90 % (31310)Termination reason: Unknown
% 2.46/1.90 % (31310)Termination phase: Property scanning
% 2.46/1.90
% 2.46/1.90 % (31310)Memory used [KB]: 4221
% 2.46/1.90 % (31310)Time elapsed: 0.033 s
% 2.46/1.90 % (31310)Instructions burned: 111 (million)
% 2.46/1.90 % (31310)------------------------------
% 2.46/1.90 % (31310)------------------------------
% 2.46/1.90 % (31308)Instruction limit reached!
% 2.46/1.90 % (31308)------------------------------
% 2.46/1.90 % (31308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.90 % (31308)Termination reason: Unknown
% 2.46/1.90 % (31308)Termination phase: Property scanning
% 2.46/1.90
% 2.46/1.90 % (31308)Memory used [KB]: 4093
% 2.46/1.90 % (31308)Time elapsed: 0.045 s
% 2.46/1.90 % (31308)Instructions burned: 152 (million)
% 2.46/1.90 % (31308)------------------------------
% 2.46/1.90 % (31308)------------------------------
% 2.46/1.90 % (31309)Refutation not found, incomplete strategy
% 2.46/1.90 % (31309)------------------------------
% 2.46/1.90 % (31309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.90 % (31309)Termination reason: Refutation not found, incomplete strategy
% 2.46/1.90
% 2.46/1.90
% 2.46/1.90 % (31309)Memory used [KB]: 7164
% 2.46/1.90 % (31309)Time elapsed: 0.059 s
% 2.46/1.90 % (31309)Instructions burned: 122 (million)
% 2.46/1.90 % (31309)------------------------------
% 2.46/1.90 % (31309)------------------------------
% 2.46/1.91 % (31313)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_31149 for (2995ds/614Mi)
% 2.46/1.91 % (31314)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_31149 for (2995ds/202Mi)
% 2.46/1.91 % (31315)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_31149 for (2995ds/339Mi)
% 2.46/1.92 % (31302)Instruction limit reached!
% 2.46/1.92 % (31302)------------------------------
% 2.46/1.92 % (31302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.92 % (31302)Termination reason: Unknown
% 2.46/1.92 % (31302)Termination phase: Saturation
% 2.46/1.92
% 2.46/1.92 % (31302)Memory used [KB]: 6652
% 2.46/1.92 % (31302)Time elapsed: 0.109 s
% 2.46/1.92 % (31302)Instructions burned: 453 (million)
% 2.46/1.92 % (31302)------------------------------
% 2.46/1.92 % (31302)------------------------------
% 2.46/1.93 % (31316)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_31149 for (2995ds/283Mi)
% 2.46/1.93 % (31312)Instruction limit reached!
% 2.46/1.93 % (31312)------------------------------
% 2.46/1.93 % (31312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.93 % (31312)Termination reason: Unknown
% 2.46/1.93 % (31312)Termination phase: Property scanning
% 2.46/1.93
% 2.46/1.93 % (31312)Memory used [KB]: 4349
% 2.46/1.93 % (31312)Time elapsed: 0.040 s
% 2.46/1.93 % (31312)Instructions burned: 143 (million)
% 2.46/1.93 % (31312)------------------------------
% 2.46/1.93 % (31312)------------------------------
% 2.46/1.94 % (31317)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_31149 for (2995ds/134Mi)
% 2.46/1.95 % (31311)Instruction limit reached!
% 2.46/1.95 % (31311)------------------------------
% 2.46/1.95 % (31311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.95 % (31311)Termination reason: Unknown
% 2.46/1.95 % (31311)Termination phase: Saturation
% 2.46/1.95
% 2.46/1.95 % (31311)Memory used [KB]: 7164
% 2.46/1.95 % (31311)Time elapsed: 0.096 s
% 2.46/1.95 % (31311)Instructions burned: 256 (million)
% 2.46/1.95 % (31311)------------------------------
% 2.46/1.95 % (31311)------------------------------
% 2.46/1.96 % (31318)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_31149 for (2994ds/298Mi)
% 2.46/1.97 % (31314)Instruction limit reached!
% 2.46/1.97 % (31314)------------------------------
% 2.46/1.97 % (31314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.97 % (31314)Termination reason: Unknown
% 2.46/1.97 % (31314)Termination phase: Saturation
% 2.46/1.97
% 2.46/1.97 % (31314)Memory used [KB]: 4477
% 2.46/1.97 % (31314)Time elapsed: 0.055 s
% 2.46/1.97 % (31314)Instructions burned: 202 (million)
% 2.46/1.97 % (31314)------------------------------
% 2.46/1.97 % (31314)------------------------------
% 2.46/1.98 % (31319)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_31149 for (2994ds/192Mi)
% 2.46/1.98 % (31317)Instruction limit reached!
% 2.46/1.98 % (31317)------------------------------
% 2.46/1.98 % (31317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.98 % (31317)Termination reason: Unknown
% 2.46/1.98 % (31317)Termination phase: Property scanning
% 2.46/1.98
% 2.46/1.98 % (31317)Memory used [KB]: 4221
% 2.46/1.98 % (31317)Time elapsed: 0.038 s
% 2.46/1.98 % (31317)Instructions burned: 135 (million)
% 2.46/1.98 % (31317)------------------------------
% 2.46/1.98 % (31317)------------------------------
% 2.46/1.99 % (31320)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_31149 for (2994ds/258Mi)
% 2.46/1.99 % (31316)Instruction limit reached!
% 2.46/1.99 % (31316)------------------------------
% 2.46/1.99 % (31316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.99 % (31316)Termination reason: Unknown
% 2.46/1.99 % (31316)Termination phase: Saturation
% 2.46/1.99
% 2.46/2.00 % (31316)Memory used [KB]: 6652
% 2.46/2.00 % (31316)Time elapsed: 0.071 s
% 2.46/2.00 % (31316)Instructions burned: 284 (million)
% 2.46/2.00 % (31316)------------------------------
% 2.46/2.00 % (31316)------------------------------
% 2.46/2.00 % (31321)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_31149 for (2994ds/90Mi)
% 2.46/2.01 % (31315)Instruction limit reached!
% 2.46/2.01 % (31315)------------------------------
% 2.46/2.01 % (31315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/2.01 % (31315)Termination reason: Unknown
% 2.46/2.01 % (31315)Termination phase: Saturation
% 2.46/2.01
% 2.46/2.01 % (31315)Memory used [KB]: 9594
% 2.46/2.01 % (31315)Time elapsed: 0.091 s
% 2.46/2.01 % (31315)Instructions burned: 339 (million)
% 2.46/2.01 % (31315)------------------------------
% 2.46/2.01 % (31315)------------------------------
% 6.35/2.02 % (31322)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_31149 for (2994ds/231Mi)
% 6.35/2.03 % (31319)Instruction limit reached!
% 6.35/2.03 % (31319)------------------------------
% 6.35/2.03 % (31319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.35/2.03 % (31319)Termination reason: Unknown
% 6.35/2.03 % (31319)Termination phase: Property scanning
% 6.35/2.03
% 6.35/2.03 % (31319)Memory used [KB]: 4349
% 6.35/2.03 % (31319)Time elapsed: 0.052 s
% 6.35/2.03 % (31319)Instructions burned: 193 (million)
% 6.35/2.03 % (31319)------------------------------
% 6.35/2.03 % (31319)------------------------------
% 6.35/2.03 % (31321)Instruction limit reached!
% 6.35/2.03 % (31321)------------------------------
% 6.35/2.03 % (31321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.35/2.03 % (31321)Termination reason: Unknown
% 6.35/2.03 % (31321)Termination phase: Preprocessing 3
% 6.35/2.03
% 6.35/2.03 % (31321)Memory used [KB]: 4093
% 6.35/2.03 % (31321)Time elapsed: 0.028 s
% 6.35/2.03 % (31321)Instructions burned: 91 (million)
% 6.35/2.03 % (31321)------------------------------
% 6.35/2.03 % (31321)------------------------------
% 6.35/2.04 % (31323)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_31149 for (2994ds/458Mi)
% 6.35/2.04 % (31324)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_31149 for (2994ds/31Mi)
% 6.35/2.05 % (31324)Instruction limit reached!
% 6.35/2.05 % (31324)------------------------------
% 6.35/2.05 % (31324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.35/2.05 % (31324)Termination reason: Unknown
% 6.35/2.05 % (31324)Termination phase: Property scanning
% 6.35/2.05
% 6.35/2.05 % (31324)Memory used [KB]: 2174
% 6.35/2.05 % (31324)Time elapsed: 0.011 s
% 6.35/2.05 % (31324)Instructions burned: 33 (million)
% 6.35/2.05 % (31324)------------------------------
% 6.35/2.05 % (31324)------------------------------
% 6.35/2.06 % (31318)Instruction limit reached!
% 6.35/2.06 % (31318)------------------------------
% 6.35/2.06 % (31318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.35/2.06 % (31318)Termination reason: Unknown
% 6.35/2.06 % (31318)Termination phase: Saturation
% 6.35/2.06
% 6.35/2.06 % (31318)Memory used [KB]: 6780
% 6.35/2.06 % (31318)Time elapsed: 0.100 s
% 6.35/2.06 % (31318)Instructions burned: 299 (million)
% 6.35/2.06 % (31318)------------------------------
% 6.35/2.06 % (31318)------------------------------
% 6.35/2.06 % (31320)Instruction limit reached!
% 6.35/2.06 % (31320)------------------------------
% 6.35/2.06 % (31320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.35/2.06 % (31320)Termination reason: Unknown
% 6.35/2.06 % (31320)Termination phase: Property scanning
% 6.35/2.06
% 6.35/2.06 % (31320)Memory used [KB]: 7931
% 6.35/2.06 % (31320)Time elapsed: 0.069 s
% 6.35/2.06 % (31320)Instructions burned: 262 (million)
% 6.35/2.06 % (31320)------------------------------
% 6.35/2.06 % (31320)------------------------------
% 6.35/2.06 % (31325)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_31149 for (2993ds/297Mi)
% 6.35/2.07 % (31326)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_31149 for (2993ds/987Mi)
% 6.35/2.07 % (31327)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_31149 for (2993ds/182Mi)
% 6.35/2.07 % (31295)Instruction limit reached!
% 6.35/2.07 % (31295)------------------------------
% 6.35/2.07 % (31295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.35/2.07 % (31295)Termination reason: Unknown
% 6.35/2.07 % (31295)Termination phase: Saturation
% 6.35/2.07
% 6.35/2.07 % (31295)Memory used [KB]: 12537
% 6.35/2.07 % (31295)Time elapsed: 0.331 s
% 6.35/2.07 % (31295)Instructions burned: 1088 (million)
% 6.35/2.07 % (31295)------------------------------
% 6.35/2.07 % (31295)------------------------------
% 6.35/2.08 % (31328)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_31149 for (2993ds/182Mi)
% 6.35/2.09 % (31322)Instruction limit reached!
% 6.35/2.09 % (31322)------------------------------
% 6.35/2.09 % (31322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.35/2.09 % (31322)Termination reason: Unknown
% 6.35/2.09 % (31322)Termination phase: Saturation
% 6.35/2.09
% 6.35/2.09 % (31322)Memory used [KB]: 7036
% 6.35/2.09 % (31322)Time elapsed: 0.067 s
% 6.35/2.09 % (31322)Instructions burned: 233 (million)
% 6.35/2.09 % (31322)------------------------------
% 6.35/2.09 % (31322)------------------------------
% 6.35/2.09 % (31313)Instruction limit reached!
% 6.35/2.09 % (31313)------------------------------
% 6.35/2.09 % (31313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.35/2.09 % (31313)Termination reason: Unknown
% 6.35/2.09 % (31313)Termination phase: Saturation
% 6.35/2.09
% 6.35/2.09 % (31313)Memory used [KB]: 12537
% 6.35/2.09 % (31313)Time elapsed: 0.181 s
% 6.35/2.09 % (31313)Instructions burned: 615 (million)
% 6.35/2.09 % (31313)------------------------------
% 6.35/2.09 % (31313)------------------------------
% 6.35/2.10 % (31329)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_31149 for (2993ds/154Mi)
% 6.35/2.10 % (31330)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_31149 for (2993ds/79Mi)
% 6.75/2.12 % (31327)Instruction limit reached!
% 6.75/2.12 % (31327)------------------------------
% 6.75/2.12 % (31327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.12 % (31327)Termination reason: Unknown
% 6.75/2.12 % (31327)Termination phase: Property scanning
% 6.75/2.12
% 6.75/2.12 % (31327)Memory used [KB]: 7803
% 6.75/2.12 % (31327)Time elapsed: 0.050 s
% 6.75/2.12 % (31327)Instructions burned: 182 (million)
% 6.75/2.12 % (31327)------------------------------
% 6.75/2.12 % (31327)------------------------------
% 6.75/2.12 % (31330)Instruction limit reached!
% 6.75/2.12 % (31330)------------------------------
% 6.75/2.12 % (31330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.12 % (31330)Termination reason: Unknown
% 6.75/2.12 % (31330)Termination phase: Preprocessing 3
% 6.75/2.12
% 6.75/2.12 % (31330)Memory used [KB]: 3837
% 6.75/2.12 % (31330)Time elapsed: 0.024 s
% 6.75/2.12 % (31330)Instructions burned: 79 (million)
% 6.75/2.12 % (31330)------------------------------
% 6.75/2.12 % (31330)------------------------------
% 6.75/2.13 % (31331)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_31149 for (2993ds/216Mi)
% 6.75/2.13 % (31332)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_31149 for (2993ds/114Mi)
% 6.75/2.14 % (31328)Instruction limit reached!
% 6.75/2.14 % (31328)------------------------------
% 6.75/2.14 % (31328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.14 % (31328)Termination reason: Unknown
% 6.75/2.14 % (31328)Termination phase: Saturation
% 6.75/2.14
% 6.75/2.14 % (31328)Memory used [KB]: 9083
% 6.75/2.14 % (31328)Time elapsed: 0.052 s
% 6.75/2.14 % (31328)Instructions burned: 183 (million)
% 6.75/2.14 % (31328)------------------------------
% 6.75/2.14 % (31328)------------------------------
% 6.75/2.14 % (31329)Instruction limit reached!
% 6.75/2.14 % (31329)------------------------------
% 6.75/2.14 % (31329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.14 % (31329)Termination reason: Unknown
% 6.75/2.14 % (31329)Termination phase: Saturation
% 6.75/2.14
% 6.75/2.14 % (31329)Memory used [KB]: 2686
% 6.75/2.14 % (31329)Time elapsed: 0.041 s
% 6.75/2.14 % (31329)Instructions burned: 157 (million)
% 6.75/2.14 % (31329)------------------------------
% 6.75/2.14 % (31329)------------------------------
% 6.75/2.14 % (31325)Instruction limit reached!
% 6.75/2.14 % (31325)------------------------------
% 6.75/2.14 % (31325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.14 % (31325)Termination reason: Unknown
% 6.75/2.14 % (31325)Termination phase: Saturation
% 6.75/2.14
% 6.75/2.14 % (31325)Memory used [KB]: 7164
% 6.75/2.14 % (31325)Time elapsed: 0.084 s
% 6.75/2.14 % (31325)Instructions burned: 299 (million)
% 6.75/2.14 % (31325)------------------------------
% 6.75/2.14 % (31325)------------------------------
% 6.75/2.15 % (31333)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_31149 for (2993ds/922Mi)
% 6.75/2.15 % (31334)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_31149 for (2993ds/182Mi)
% 6.75/2.15 % (31335)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_31149 for (2993ds/164Mi)
% 6.75/2.17 % (31332)Instruction limit reached!
% 6.75/2.17 % (31332)------------------------------
% 6.75/2.17 % (31332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.17 % (31332)Termination reason: Unknown
% 6.75/2.17 % (31332)Termination phase: Property scanning
% 6.75/2.17
% 6.75/2.17 % (31332)Memory used [KB]: 4221
% 6.75/2.17 % (31332)Time elapsed: 0.034 s
% 6.75/2.17 % (31332)Instructions burned: 118 (million)
% 6.75/2.17 % (31332)------------------------------
% 6.75/2.17 % (31332)------------------------------
% 6.75/2.17 % (31323)Instruction limit reached!
% 6.75/2.17 % (31323)------------------------------
% 6.75/2.17 % (31323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.17 % (31323)Termination reason: Unknown
% 6.75/2.17 % (31323)Termination phase: Saturation
% 6.75/2.17
% 6.75/2.17 % (31323)Memory used [KB]: 10362
% 6.75/2.17 % (31323)Time elapsed: 0.133 s
% 6.75/2.17 % (31323)Instructions burned: 458 (million)
% 6.75/2.17 % (31323)------------------------------
% 6.75/2.17 % (31323)------------------------------
% 6.75/2.18 % (31336)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_31149 for (2992ds/50Mi)
% 6.75/2.18 % (31337)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_31149 for (2992ds/368Mi)
% 6.75/2.19 % (31331)Instruction limit reached!
% 6.75/2.19 % (31331)------------------------------
% 6.75/2.19 % (31331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.19 % (31331)Termination reason: Unknown
% 6.75/2.19 % (31331)Termination phase: Saturation
% 6.75/2.19
% 6.75/2.19 % (31331)Memory used [KB]: 6780
% 6.75/2.19 % (31331)Time elapsed: 0.062 s
% 6.75/2.19 % (31331)Instructions burned: 219 (million)
% 6.75/2.19 % (31331)------------------------------
% 6.75/2.19 % (31331)------------------------------
% 6.75/2.19 % (31336)Instruction limit reached!
% 6.75/2.19 % (31336)------------------------------
% 6.75/2.19 % (31336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.19 % (31336)Termination reason: Unknown
% 6.75/2.19 % (31336)Termination phase: Property scanning
% 6.75/2.19
% 6.75/2.19 % (31336)Memory used [KB]: 1918
% 6.75/2.19 % (31336)Time elapsed: 0.014 s
% 6.75/2.19 % (31336)Instructions burned: 50 (million)
% 6.75/2.19 % (31336)------------------------------
% 6.75/2.19 % (31336)------------------------------
% 6.75/2.19 % (31334)Instruction limit reached!
% 6.75/2.19 % (31334)------------------------------
% 6.75/2.19 % (31334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.19 % (31334)Termination reason: Unknown
% 6.75/2.19 % (31334)Termination phase: Saturation
% 6.75/2.19
% 6.75/2.19 % (31334)Memory used [KB]: 6524
% 6.75/2.19 % (31334)Time elapsed: 0.046 s
% 6.75/2.19 % (31334)Instructions burned: 186 (million)
% 6.75/2.19 % (31334)------------------------------
% 6.75/2.19 % (31334)------------------------------
% 7.38/2.20 % (31338)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_31149 for (2992ds/1021Mi)
% 7.38/2.20 % (31335)Instruction limit reached!
% 7.38/2.20 % (31335)------------------------------
% 7.38/2.20 % (31335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.20 % (31335)Termination reason: Unknown
% 7.38/2.20 % (31335)Termination phase: Saturation
% 7.38/2.20
% 7.38/2.20 % (31335)Memory used [KB]: 2302
% 7.38/2.20 % (31335)Time elapsed: 0.046 s
% 7.38/2.20 % (31335)Instructions burned: 165 (million)
% 7.38/2.20 % (31335)------------------------------
% 7.38/2.20 % (31335)------------------------------
% 7.38/2.20 % (31339)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_31149 for (2992ds/267Mi)
% 7.38/2.20 % (31340)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_31149 for (2992ds/204Mi)
% 7.38/2.21 % (31341)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_31149 for (2992ds/26Mi)
% 7.38/2.21 % (31341)Instruction limit reached!
% 7.38/2.21 % (31341)------------------------------
% 7.38/2.21 % (31341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.21 % (31341)Termination reason: Unknown
% 7.38/2.21 % (31341)Termination phase: shuffling
% 7.38/2.21
% 7.38/2.21 % (31341)Memory used [KB]: 1791
% 7.38/2.21 % (31341)Time elapsed: 0.008 s
% 7.38/2.21 % (31341)Instructions burned: 26 (million)
% 7.38/2.21 % (31341)------------------------------
% 7.38/2.21 % (31341)------------------------------
% 7.38/2.22 % (31342)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_31149 for (2992ds/53Mi)
% 7.38/2.24 % (31342)Instruction limit reached!
% 7.38/2.24 % (31342)------------------------------
% 7.38/2.24 % (31342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.24 % (31342)Termination reason: Unknown
% 7.38/2.24 % (31342)Termination phase: Property scanning
% 7.38/2.24
% 7.38/2.24 % (31342)Memory used [KB]: 1918
% 7.38/2.24 % (31342)Time elapsed: 0.015 s
% 7.38/2.24 % (31342)Instructions burned: 54 (million)
% 7.38/2.24 % (31342)------------------------------
% 7.38/2.24 % (31342)------------------------------
% 7.38/2.25 % (31343)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_31149 for (2992ds/4436Mi)
% 7.38/2.26 % (31340)Instruction limit reached!
% 7.38/2.26 % (31340)------------------------------
% 7.38/2.26 % (31340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.26 % (31340)Termination reason: Unknown
% 7.38/2.26 % (31340)Termination phase: Property scanning
% 7.38/2.26
% 7.38/2.26 % (31340)Memory used [KB]: 4605
% 7.38/2.26 % (31340)Time elapsed: 0.055 s
% 7.38/2.26 % (31340)Instructions burned: 204 (million)
% 7.38/2.26 % (31340)------------------------------
% 7.38/2.26 % (31340)------------------------------
% 7.38/2.27 % (31339)Instruction limit reached!
% 7.38/2.27 % (31339)------------------------------
% 7.38/2.27 % (31339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.27 % (31339)Termination reason: Unknown
% 7.38/2.27 % (31339)Termination phase: Saturation
% 7.38/2.27
% 7.38/2.27 % (31339)Memory used [KB]: 7419
% 7.38/2.27 % (31339)Time elapsed: 0.067 s
% 7.38/2.27 % (31339)Instructions burned: 269 (million)
% 7.38/2.27 % (31339)------------------------------
% 7.38/2.27 % (31339)------------------------------
% 7.38/2.27 % (31344)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_31149 for (2991ds/1539Mi)
% 7.38/2.28 % (31345)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_31149 for (2991ds/408Mi)
% 7.38/2.28 % (31337)Instruction limit reached!
% 7.38/2.28 % (31337)------------------------------
% 7.38/2.28 % (31337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.28 % (31337)Termination reason: Unknown
% 7.38/2.28 % (31337)Termination phase: Saturation
% 7.38/2.28
% 7.38/2.28 % (31337)Memory used [KB]: 11513
% 7.38/2.28 % (31337)Time elapsed: 0.106 s
% 7.38/2.28 % (31337)Instructions burned: 370 (million)
% 7.38/2.28 % (31337)------------------------------
% 7.38/2.28 % (31337)------------------------------
% 7.38/2.29 % (31346)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_31149 for (2991ds/5205Mi)
% 7.73/2.31 % (31326)Instruction limit reached!
% 7.73/2.31 % (31326)------------------------------
% 7.73/2.31 % (31326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.31 % (31326)Termination reason: Unknown
% 7.73/2.31 % (31326)Termination phase: Saturation
% 7.73/2.31
% 7.73/2.31 % (31326)Memory used [KB]: 7547
% 7.73/2.31 % (31326)Time elapsed: 0.242 s
% 7.73/2.31 % (31326)Instructions burned: 988 (million)
% 7.73/2.31 % (31326)------------------------------
% 7.73/2.31 % (31326)------------------------------
% 7.73/2.32 % (31347)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_31149 for (2991ds/1913Mi)
% 7.73/2.39 % (31345)Instruction limit reached!
% 7.73/2.39 % (31345)------------------------------
% 7.73/2.39 % (31345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.39 % (31345)Termination reason: Unknown
% 7.73/2.39 % (31345)Termination phase: Saturation
% 7.73/2.39
% 7.73/2.39 % (31345)Memory used [KB]: 10874
% 7.73/2.39 % (31345)Time elapsed: 0.115 s
% 7.73/2.39 % (31345)Instructions burned: 408 (million)
% 7.73/2.39 % (31345)------------------------------
% 7.73/2.39 % (31345)------------------------------
% 8.11/2.40 % (31348)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_31149 for (2990ds/4350Mi)
% 8.11/2.42 % (31333)Instruction limit reached!
% 8.11/2.42 % (31333)------------------------------
% 8.11/2.42 % (31333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.11/2.42 % (31333)Termination reason: Unknown
% 8.11/2.42 % (31333)Termination phase: Saturation
% 8.11/2.42
% 8.11/2.42 % (31333)Memory used [KB]: 12537
% 8.11/2.42 % (31333)Time elapsed: 0.279 s
% 8.11/2.42 % (31333)Instructions burned: 924 (million)
% 8.11/2.42 % (31333)------------------------------
% 8.11/2.42 % (31333)------------------------------
% 8.11/2.43 % (31349)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_31149 for (2990ds/68Mi)
% 8.11/2.45 % (31349)Instruction limit reached!
% 8.11/2.45 % (31349)------------------------------
% 8.11/2.45 % (31349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.11/2.45 % (31349)Termination reason: Unknown
% 8.11/2.45 % (31349)Termination phase: Preprocessing 3
% 8.11/2.45
% 8.11/2.45 % (31349)Memory used [KB]: 3198
% 8.11/2.45 % (31349)Time elapsed: 0.021 s
% 8.11/2.45 % (31349)Instructions burned: 68 (million)
% 8.11/2.45 % (31349)------------------------------
% 8.11/2.45 % (31349)------------------------------
% 8.11/2.46 % (31350)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_31149 for (2990ds/358Mi)
% 8.42/2.50 % (31338)Instruction limit reached!
% 8.42/2.50 % (31338)------------------------------
% 8.42/2.50 % (31338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.42/2.50 % (31338)Termination reason: Unknown
% 8.42/2.50 % (31338)Termination phase: Saturation
% 8.42/2.50
% 8.42/2.50 % (31338)Memory used [KB]: 12281
% 8.42/2.50 % (31338)Time elapsed: 0.305 s
% 8.42/2.50 % (31338)Instructions burned: 1022 (million)
% 8.42/2.50 % (31338)------------------------------
% 8.42/2.50 % (31338)------------------------------
% 8.42/2.51 % (31351)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_31149 for (2989ds/31Mi)
% 8.42/2.52 % (31351)Instruction limit reached!
% 8.42/2.52 % (31351)------------------------------
% 8.42/2.52 % (31351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.42/2.52 % (31351)Termination reason: Unknown
% 8.42/2.52 % (31351)Termination phase: Property scanning
% 8.42/2.52
% 8.42/2.52 % (31351)Memory used [KB]: 2174
% 8.42/2.52 % (31351)Time elapsed: 0.011 s
% 8.42/2.52 % (31351)Instructions burned: 31 (million)
% 8.42/2.52 % (31351)------------------------------
% 8.42/2.52 % (31351)------------------------------
% 8.42/2.53 % (31352)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_31149 for (2989ds/279Mi)
% 8.42/2.56 % (31350)Instruction limit reached!
% 8.42/2.56 % (31350)------------------------------
% 8.42/2.56 % (31350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.42/2.56 % (31350)Termination reason: Unknown
% 8.42/2.56 % (31350)Termination phase: Saturation
% 8.42/2.57
% 8.42/2.57 % (31350)Memory used [KB]: 9722
% 8.42/2.57 % (31350)Time elapsed: 0.125 s
% 8.42/2.57 % (31350)Instructions burned: 359 (million)
% 8.42/2.57 % (31350)------------------------------
% 8.42/2.57 % (31350)------------------------------
% 8.42/2.58 % (31353)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_31149 for (2988ds/258Mi)
% 8.87/2.60 % (31352)Instruction limit reached!
% 8.87/2.60 % (31352)------------------------------
% 8.87/2.60 % (31352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.87/2.60 % (31352)Termination reason: Unknown
% 8.87/2.60 % (31352)Termination phase: Saturation
% 8.87/2.60
% 8.87/2.60 % (31352)Memory used [KB]: 7675
% 8.87/2.60 % (31352)Time elapsed: 0.071 s
% 8.87/2.60 % (31352)Instructions burned: 282 (million)
% 8.87/2.60 % (31352)------------------------------
% 8.87/2.60 % (31352)------------------------------
% 8.87/2.61 % (31354)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_31149 for (2988ds/26Mi)
% 8.87/2.62 % (31354)Instruction limit reached!
% 8.87/2.62 % (31354)------------------------------
% 8.87/2.62 % (31354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.87/2.62 % (31354)Termination reason: Unknown
% 8.87/2.62 % (31354)Termination phase: shuffling
% 8.87/2.62
% 8.87/2.62 % (31354)Memory used [KB]: 1791
% 8.87/2.62 % (31354)Time elapsed: 0.009 s
% 8.87/2.62 % (31354)Instructions burned: 29 (million)
% 8.87/2.62 % (31354)------------------------------
% 8.87/2.62 % (31354)------------------------------
% 8.87/2.63 % (31355)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_31149 for (2988ds/995Mi)
% 8.87/2.65 % (31353)Instruction limit reached!
% 8.87/2.65 % (31353)------------------------------
% 8.87/2.65 % (31353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.87/2.65 % (31353)Termination reason: Unknown
% 8.87/2.65 % (31353)Termination phase: Saturation
% 8.87/2.65
% 8.87/2.65 % (31353)Memory used [KB]: 9722
% 8.87/2.65 % (31353)Time elapsed: 0.076 s
% 8.87/2.65 % (31353)Instructions burned: 260 (million)
% 8.87/2.65 % (31353)------------------------------
% 8.87/2.65 % (31353)------------------------------
% 8.87/2.66 % (31356)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_31149 for (2988ds/373Mi)
% 9.05/2.70 % (31344)Instruction limit reached!
% 9.05/2.70 % (31344)------------------------------
% 9.05/2.70 % (31344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.05/2.70 % (31344)Termination reason: Unknown
% 9.05/2.70 % (31344)Termination phase: Saturation
% 9.05/2.70
% 9.05/2.70 % (31344)Memory used [KB]: 12025
% 9.05/2.70 % (31344)Time elapsed: 0.436 s
% 9.05/2.70 % (31344)Instructions burned: 1540 (million)
% 9.05/2.70 % (31344)------------------------------
% 9.05/2.70 % (31344)------------------------------
% 9.05/2.71 % (31347)First to succeed.
% 9.05/2.71 % (31357)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_31149 for (2987ds/253Mi)
% 9.05/2.74 % (31347)Refutation found. Thanks to Tanya!
% 9.05/2.74 % SZS status Theorem for DTF2THF_31149
% 9.05/2.74 % SZS output start Proof for DTF2THF_31149
% 9.05/2.74 thf(type_def_5, type, term: $tType).
% 9.05/2.74 thf(type_def_7, type, subst: $tType).
% 9.05/2.74 thf(type_def_8, type, d_term: $tType).
% 9.05/2.74 thf(type_def_9, type, d_subst: $tType).
% 9.05/2.74 thf(func_def_0, type, term: $tType).
% 9.05/2.74 thf(func_def_1, type, subst: $tType).
% 9.05/2.74 thf(func_def_2, type, one: term).
% 9.05/2.74 thf(func_def_3, type, ap: term > term > term).
% 9.05/2.74 thf(func_def_4, type, lam: term > term).
% 9.05/2.74 thf(func_def_5, type, sub: term > subst > term).
% 9.05/2.74 thf(func_def_6, type, id: subst).
% 9.05/2.74 thf(func_def_7, type, sh: subst).
% 9.05/2.74 thf(func_def_8, type, push: term > subst > subst).
% 9.05/2.74 thf(func_def_9, type, comp: subst > subst > subst).
% 9.05/2.74 thf(func_def_10, type, var: term > $o).
% 9.05/2.74 thf(func_def_18, type, hoasap: subst > term > subst > term > term).
% 9.05/2.74 thf(func_def_73, type, hoaslam: subst > (subst > term > term) > term).
% 9.05/2.74 thf(func_def_86, type, hoasinduction_p_and_p_prime: (subst > term > subst > $o) > (term > $o) > $o).
% 9.05/2.74 thf(func_def_107, type, hoasvar: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_121, type, pushprop_p_and_p_prime: term > subst > (term > $o) > (term > $o) > $o).
% 9.05/2.74 thf(func_def_124, type, d_term: $tType).
% 9.05/2.74 thf(func_def_125, type, d_subst: $tType).
% 9.05/2.74 thf(func_def_126, type, d2term: d_term > term).
% 9.05/2.74 thf(func_def_127, type, d2subst: d_subst > subst).
% 9.05/2.74 thf(func_def_128, type, d_one: d_term).
% 9.05/2.74 thf(func_def_129, type, d_id: d_subst).
% 9.05/2.74 thf(func_def_142, type, sP0: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_144, type, sP2: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_146, type, sP4: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_148, type, sP6: (term > $o) > $o).
% 9.05/2.74 thf(func_def_149, type, sP7: (term > $o) > $o).
% 9.05/2.74 thf(func_def_152, type, sP10: (term > $o) > $o).
% 9.05/2.74 thf(func_def_153, type, sP11: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_154, type, sP12: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_155, type, sP13: (term > $o) > $o).
% 9.05/2.74 thf(func_def_160, type, sP18: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_164, type, sP22: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_165, type, sP23: (term > $o) > $o).
% 9.05/2.74 thf(func_def_166, type, sP24: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_167, type, sP25: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_168, type, sP26: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_174, type, sP32: (term > $o) > $o).
% 9.05/2.74 thf(func_def_180, type, sP38: (term > $o) > $o).
% 9.05/2.74 thf(func_def_182, type, sP40: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_183, type, sP41: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_185, type, sP43: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_187, type, sP45: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_191, type, sP49: (term > $o) > $o).
% 9.05/2.74 thf(func_def_192, type, sP50: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_196, type, sP54: (term > $o) > $o).
% 9.05/2.74 thf(func_def_197, type, sP55: (term > $o) > $o).
% 9.05/2.74 thf(func_def_200, type, sP58: (subst > term > subst > $o) > $o).
% 9.05/2.74 thf(func_def_201, type, sK59: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_202, type, sK60: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_203, type, sK61: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_204, type, sK62: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_205, type, sK63: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_206, type, sK64: (term > $o) > subst > term).
% 9.05/2.74 thf(func_def_207, type, sK65: (term > $o) > term).
% 9.05/2.74 thf(func_def_208, type, sK66: subst).
% 9.05/2.74 thf(func_def_209, type, sK67: term > $o).
% 9.05/2.74 thf(func_def_210, type, sK68: term).
% 9.05/2.74 thf(func_def_211, type, sK69: term > term).
% 9.05/2.74 thf(func_def_212, type, sK70: term).
% 9.05/2.74 thf(func_def_213, type, sK71: term).
% 9.05/2.74 thf(func_def_214, type, sK72: term > $o).
% 9.05/2.74 thf(func_def_215, type, sK73: term > term).
% 9.05/2.74 thf(func_def_216, type, sK74: (term > $o) > term).
% 9.05/2.74 thf(func_def_217, type, sK75: (term > $o) > term).
% 9.05/2.74 thf(func_def_218, type, sK76: (term > $o) > term).
% 9.05/2.74 thf(func_def_219, type, sK77: (term > $o) > term).
% 9.05/2.74 thf(func_def_220, type, sK78: (term > $o) > term).
% 9.05/2.74 thf(func_def_221, type, sK79: (term > $o) > term).
% 9.05/2.74 thf(func_def_222, type, sK80: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_223, type, sK81: term).
% 9.05/2.74 thf(func_def_224, type, sK82: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_225, type, sK83: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_226, type, sK84: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_227, type, sK85: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_228, type, sK86: (subst > term > subst > $o) > term > $o).
% 9.05/2.74 thf(func_def_229, type, sK87: term).
% 9.05/2.74 thf(func_def_230, type, sK88: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_231, type, sK89: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_232, type, sK90: (term > $o) > term).
% 9.05/2.74 thf(func_def_233, type, sK91: (term > $o) > term).
% 9.05/2.74 thf(func_def_234, type, sK92: (term > $o) > term).
% 9.05/2.74 thf(func_def_235, type, sK93: term > $o).
% 9.05/2.74 thf(func_def_236, type, sK94: term).
% 9.05/2.74 thf(func_def_237, type, sK95: term > term).
% 9.05/2.74 thf(func_def_238, type, sK96: (term > $o) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_239, type, sK97: term > (term > $o) > term).
% 9.05/2.74 thf(func_def_240, type, sK98: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_241, type, sK99: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_242, type, sK100: term > $o).
% 9.05/2.74 thf(func_def_243, type, sK101: term).
% 9.05/2.74 thf(func_def_244, type, sK102: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_245, type, sK103: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_246, type, sK104: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_247, type, sK105: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_248, type, sK106: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_249, type, sK107: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_250, type, sK108: (term > $o) > term).
% 9.05/2.74 thf(func_def_251, type, sK109: (term > $o) > term).
% 9.05/2.74 thf(func_def_252, type, sK110: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_253, type, sK111: term).
% 9.05/2.74 thf(func_def_254, type, sK112: term > (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_255, type, sK113: term > (subst > term > term) > term).
% 9.05/2.74 thf(func_def_256, type, sK114: term > (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_257, type, sK115: term > (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_258, type, sK116: term > (subst > term > term) > term).
% 9.05/2.74 thf(func_def_259, type, sK117: term > (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_260, type, sK118: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_261, type, sK119: term > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_262, type, sK120: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_263, type, sK121: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_264, type, sK122: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_265, type, sK123: term).
% 9.05/2.74 thf(func_def_266, type, sK124: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_267, type, sK125: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_268, type, sK126: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_269, type, sK127: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_270, type, sK128: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_271, type, sK129: term > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_272, type, sK130: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_273, type, sK131: term > $o).
% 9.05/2.74 thf(func_def_274, type, sK132: term).
% 9.05/2.74 thf(func_def_275, type, sK133: (term > $o) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_276, type, sK134: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_277, type, sK135: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_278, type, sK136: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_279, type, sK137: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_280, type, sK138: term > (term > $o) > term).
% 9.05/2.74 thf(func_def_281, type, sK139: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_282, type, sK140: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_283, type, sK141: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_284, type, sK142: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_285, type, sK143: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_286, type, sK144: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_287, type, sK145: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_288, type, sK146: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_289, type, sK147: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_290, type, sK148: term > $o).
% 9.05/2.74 thf(func_def_291, type, sK149: term).
% 9.05/2.74 thf(func_def_292, type, sK150: term).
% 9.05/2.74 thf(func_def_293, type, sK151: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_294, type, sK152: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_295, type, sK153: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_296, type, sK154: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_297, type, sK155: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_298, type, sK156: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_299, type, sK157: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_300, type, sK158: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_301, type, sK159: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_302, type, sK160: (subst > term > subst > $o) > (term > $o) > term).
% 9.05/2.74 thf(func_def_303, type, sK161: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_304, type, sK162: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_305, type, sK163: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_306, type, sK164: term).
% 9.05/2.74 thf(func_def_307, type, sK165: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_308, type, sK166: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_309, type, sK167: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_310, type, sK168: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_311, type, sK169: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_312, type, sK170: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_313, type, sK171: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_314, type, sK172: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_315, type, sK173: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_316, type, sK174: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_317, type, sK175: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_318, type, sK176: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_319, type, sK177: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_320, type, sK178: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_321, type, sK179: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_322, type, sK180: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_323, type, sK181: term).
% 9.05/2.74 thf(func_def_324, type, sK182: term > $o).
% 9.05/2.74 thf(func_def_325, type, sK183: term > term).
% 9.05/2.74 thf(func_def_326, type, sK184: (term > $o) > term).
% 9.05/2.74 thf(func_def_327, type, sK185: (term > $o) > term).
% 9.05/2.74 thf(func_def_328, type, sK186: (term > $o) > term).
% 9.05/2.74 thf(func_def_329, type, sK187: (term > $o) > term).
% 9.05/2.74 thf(func_def_330, type, sK188: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_331, type, sK189: term).
% 9.05/2.74 thf(func_def_332, type, sK190: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_333, type, sK191: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_334, type, sK192: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_335, type, sK193: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_336, type, sK194: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_337, type, sK195: term > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_338, type, sK196: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_339, type, sK197: term).
% 9.05/2.74 thf(func_def_340, type, sK198: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_341, type, sK199: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_342, type, sK200: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_343, type, sK201: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_344, type, sK202: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_345, type, sK203: term > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_346, type, sK204: term).
% 9.05/2.74 thf(func_def_347, type, sK205: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_348, type, sK206: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_349, type, sK207: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_350, type, sK208: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_351, type, sK209: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_352, type, sK210: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_353, type, sK211: (subst > term > subst > $o) > term > term).
% 9.05/2.74 thf(func_def_354, type, sK212: term).
% 9.05/2.74 thf(func_def_355, type, sK213: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_356, type, sK214: (subst > term > subst > $o) > term > term).
% 9.05/2.74 thf(func_def_357, type, sK215: term).
% 9.05/2.74 thf(func_def_358, type, sK216: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_359, type, sK217: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_360, type, sK218: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_361, type, sK219: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_362, type, sK220: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_363, type, sK221: term).
% 9.05/2.74 thf(func_def_364, type, sK222: term > $o).
% 9.05/2.74 thf(func_def_365, type, sK223: subst).
% 9.05/2.74 thf(func_def_366, type, sK224: term > term).
% 9.05/2.74 thf(func_def_367, type, sK225: subst > (term > $o) > term).
% 9.05/2.74 thf(func_def_368, type, sK226: (term > $o) > term).
% 9.05/2.74 thf(func_def_369, type, sK227: (term > $o) > term).
% 9.05/2.74 thf(func_def_370, type, sK228: (term > $o) > term).
% 9.05/2.74 thf(func_def_371, type, sK229: subst > term > term).
% 9.05/2.74 thf(func_def_372, type, sK230: subst > term > term).
% 9.05/2.74 thf(func_def_373, type, sK231: term).
% 9.05/2.74 thf(func_def_374, type, sK232: subst).
% 9.05/2.74 thf(func_def_375, type, sK233: term).
% 9.05/2.74 thf(func_def_376, type, sK234: term).
% 9.05/2.74 thf(func_def_377, type, sK235: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_378, type, sK236: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_379, type, sK237: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_380, type, sK238: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_381, type, sK239: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_382, type, sK240: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_383, type, sK241: term).
% 9.05/2.74 thf(func_def_384, type, sK242: subst).
% 9.05/2.74 thf(func_def_385, type, sK243: term).
% 9.05/2.74 thf(func_def_386, type, sK244: subst).
% 9.05/2.74 thf(func_def_387, type, sK245: (term > $o) > term).
% 9.05/2.74 thf(func_def_388, type, sK246: (term > $o) > term).
% 9.05/2.74 thf(func_def_389, type, sK247: term > $o).
% 9.05/2.74 thf(func_def_390, type, sK248: term).
% 9.05/2.74 thf(func_def_391, type, sK249: term > term).
% 9.05/2.74 thf(func_def_392, type, sK250: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_393, type, sK251: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_394, type, sK252: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_395, type, sK253: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_396, type, sK254: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_397, type, sK255: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_398, type, sK256: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_399, type, sK257: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_400, type, sK258: term).
% 9.05/2.74 thf(func_def_401, type, sK259: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_402, type, sK260: term).
% 9.05/2.74 thf(func_def_403, type, sK261: term).
% 9.05/2.74 thf(func_def_404, type, sK262: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_405, type, sK263: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_406, type, sK264: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_407, type, sK265: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_408, type, sK266: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_409, type, sK267: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_410, type, sK268: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_411, type, sK269: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_412, type, sK270: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_413, type, sK271: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_414, type, sK272: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_415, type, sK273: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_416, type, sK274: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_417, type, sK275: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_418, type, sK276: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_419, type, sK277: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_420, type, sK278: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_421, type, sK279: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_422, type, sK280: term > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_423, type, sK281: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_424, type, sK282: term).
% 9.05/2.74 thf(func_def_425, type, sK283: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_426, type, sK284: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_427, type, sK285: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_428, type, sK286: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_429, type, sK287: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_430, type, sK288: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_431, type, sK289: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_432, type, sK290: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_433, type, sK291: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_434, type, sK292: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_435, type, sK293: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_436, type, sK294: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_437, type, sK295: (term > $o) > term).
% 9.05/2.74 thf(func_def_438, type, sK296: (term > $o) > term).
% 9.05/2.74 thf(func_def_439, type, sK297: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_440, type, sK298: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_441, type, sK299: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_442, type, sK300: term).
% 9.05/2.74 thf(func_def_443, type, sK301: term > $o).
% 9.05/2.74 thf(func_def_444, type, sK302: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_445, type, sK303: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_446, type, sK304: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_447, type, sK305: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_448, type, sK306: (term > $o) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_449, type, sK307: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_450, type, sK308: (term > $o) > term > term).
% 9.05/2.74 thf(func_def_451, type, sK309: term).
% 9.05/2.74 thf(func_def_452, type, sK310: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_453, type, sK311: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_454, type, sK312: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_455, type, sK313: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_456, type, sK314: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_457, type, sK315: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_458, type, sK316: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_459, type, sK317: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_460, type, sK318: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_461, type, sK319: (subst > term > subst > $o) > term > term).
% 9.05/2.74 thf(func_def_462, type, sK320: term).
% 9.05/2.74 thf(func_def_463, type, sK321: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_464, type, sK322: (subst > term > subst > $o) > term > term).
% 9.05/2.74 thf(func_def_465, type, sK323: term).
% 9.05/2.74 thf(func_def_466, type, sK324: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_467, type, sK325: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_468, type, sK326: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_469, type, sK327: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_470, type, sK328: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_471, type, sK329: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_472, type, sK330: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_473, type, sK331: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_474, type, sK332: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_475, type, sK333: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_476, type, sK334: term).
% 9.05/2.74 thf(func_def_477, type, sK335: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_478, type, sK336: term > $o).
% 9.05/2.74 thf(func_def_479, type, sK337: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_480, type, sK338: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_481, type, sK339: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_482, type, sK340: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_483, type, sK341: term > subst > term > term).
% 9.05/2.74 thf(func_def_484, type, sK342: (term > $o) > term > term).
% 9.05/2.74 thf(func_def_485, type, sK343: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_486, type, sK344: (term > $o) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_487, type, sK345: term).
% 9.05/2.74 thf(func_def_488, type, sK346: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_489, type, sK347: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_490, type, sK348: term).
% 9.05/2.74 thf(func_def_491, type, sK349: (subst > term > subst > $o) > term > $o).
% 9.05/2.74 thf(func_def_492, type, sK350: (term > $o) > term).
% 9.05/2.74 thf(func_def_493, type, sK351: (term > $o) > term).
% 9.05/2.74 thf(func_def_494, type, sK352: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_495, type, sK353: (term > $o) > term).
% 9.05/2.74 thf(func_def_496, type, sK354: term).
% 9.05/2.74 thf(func_def_497, type, sK355: term > $o).
% 9.05/2.74 thf(func_def_498, type, sK356: term > term).
% 9.05/2.74 thf(func_def_499, type, sK357: term).
% 9.05/2.74 thf(func_def_500, type, sK358: term > (term > $o) > term).
% 9.05/2.74 thf(func_def_501, type, sK359: (subst > term > subst > $o) > (term > $o) > term).
% 9.05/2.74 thf(func_def_502, type, sK360: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_503, type, sK361: term > $o).
% 9.05/2.74 thf(func_def_504, type, sK362: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_505, type, sK363: term).
% 9.05/2.74 thf(func_def_506, type, sK364: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_507, type, sK365: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_508, type, sK366: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_509, type, sK367: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_510, type, sK368: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_511, type, sK369: term).
% 9.05/2.74 thf(func_def_512, type, sK370: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_513, type, sK371: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_514, type, sK372: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_515, type, sK373: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_516, type, sK374: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_517, type, sK375: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_518, type, sK376: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_519, type, sK377: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_520, type, sK378: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_521, type, sK379: (term > $o) > term).
% 9.05/2.74 thf(func_def_522, type, sK380: (term > $o) > term).
% 9.05/2.74 thf(func_def_523, type, sK381: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_524, type, sK382: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_525, type, sK383: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_526, type, sK384: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_527, type, sK385: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_528, type, sK386: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_529, type, sK387: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_530, type, sK388: (term > $o) > term).
% 9.05/2.74 thf(func_def_531, type, sK389: (term > $o) > term).
% 9.05/2.74 thf(func_def_532, type, sK390: (term > $o) > term).
% 9.05/2.74 thf(func_def_533, type, sK391: (term > $o) > term).
% 9.05/2.74 thf(func_def_534, type, sK392: subst).
% 9.05/2.74 thf(func_def_535, type, sK393: term).
% 9.05/2.74 thf(func_def_536, type, sK394: subst).
% 9.05/2.74 thf(func_def_537, type, sK395: term).
% 9.05/2.74 thf(func_def_538, type, sK396: subst).
% 9.05/2.74 thf(func_def_539, type, sK397: term > $o).
% 9.05/2.74 thf(func_def_540, type, sK398: term).
% 9.05/2.74 thf(func_def_541, type, sK399: term).
% 9.05/2.74 thf(func_def_542, type, sK400: subst).
% 9.05/2.74 thf(func_def_543, type, sK401: term > $o).
% 9.05/2.74 thf(func_def_544, type, sK402: term > term).
% 9.05/2.74 thf(func_def_545, type, sK403: (term > $o) > subst > term).
% 9.05/2.74 thf(func_def_546, type, sK404: (term > $o) > term).
% 9.05/2.74 thf(func_def_547, type, sK405: (term > $o) > term).
% 9.05/2.74 thf(func_def_548, type, sK406: (term > $o) > term).
% 9.05/2.74 thf(func_def_549, type, sK407: (term > $o) > term).
% 9.05/2.74 thf(func_def_550, type, sK408: (term > $o) > term).
% 9.05/2.74 thf(func_def_551, type, sK409: term).
% 9.05/2.74 thf(func_def_552, type, sK410: term).
% 9.05/2.74 thf(func_def_553, type, sK411: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_554, type, sK412: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_555, type, sK413: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_556, type, sK414: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_557, type, sK415: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_558, type, sK416: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_559, type, sK417: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_560, type, sK418: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_561, type, sK419: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_562, type, sK420: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_563, type, sK421: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_564, type, sK422: term).
% 9.05/2.74 thf(func_def_565, type, sK423: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_566, type, sK424: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_567, type, sK425: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_568, type, sK426: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_569, type, sK427: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_570, type, sK428: (subst > term > subst > $o) > term > term).
% 9.05/2.74 thf(func_def_571, type, sK429: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_572, type, sK430: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_573, type, sK431: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_574, type, sK432: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_575, type, sK433: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_576, type, sK434: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_577, type, sK435: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_578, type, sK436: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_579, type, sK437: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_580, type, sK438: term > $o).
% 9.05/2.74 thf(func_def_581, type, sK439: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_582, type, sK440: term).
% 9.05/2.74 thf(func_def_583, type, sK441: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_584, type, sK442: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_585, type, sK443: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_586, type, sK444: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_587, type, sK445: term > (term > $o) > term).
% 9.05/2.74 thf(func_def_588, type, sK446: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_589, type, sK447: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_590, type, sK448: (subst > term > subst > $o) > subst).
% 9.05/2.74 thf(func_def_591, type, sK449: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_592, type, sK450: (subst > term > subst > $o) > (term > $o) > term).
% 9.05/2.74 thf(func_def_593, type, sK451: term).
% 9.05/2.74 thf(func_def_594, type, sK452: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_595, type, sK453: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_596, type, sK454: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_597, type, sK455: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_598, type, sK456: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_599, type, sK457: term).
% 9.05/2.74 thf(func_def_600, type, sK458: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_601, type, sK459: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_602, type, sK460: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_603, type, sK461: term).
% 9.05/2.74 thf(func_def_604, type, sK462: term).
% 9.05/2.74 thf(func_def_605, type, sK463: (subst > term > subst > $o) > term > $o).
% 9.05/2.74 thf(func_def_606, type, sK464: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_607, type, sK465: (term > $o) > term).
% 9.05/2.74 thf(func_def_608, type, sK466: (subst > term > subst > $o) > term > $o).
% 9.05/2.74 thf(func_def_609, type, sK467: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_610, type, sK468: (term > $o) > term).
% 9.05/2.74 thf(func_def_611, type, sK469: subst).
% 9.05/2.74 thf(func_def_612, type, sK470: subst).
% 9.05/2.74 thf(func_def_613, type, sK471: subst).
% 9.05/2.74 thf(func_def_614, type, sK472: subst).
% 9.05/2.74 thf(func_def_615, type, sK473: term).
% 9.05/2.74 thf(func_def_616, type, sK474: term).
% 9.05/2.74 thf(func_def_617, type, sK475: term).
% 9.05/2.74 thf(func_def_618, type, sK476: term).
% 9.05/2.74 thf(func_def_619, type, sK477: term).
% 9.05/2.74 thf(func_def_620, type, sK478: term).
% 9.05/2.74 thf(func_def_621, type, sK479: term).
% 9.05/2.74 thf(func_def_622, type, sK480: term).
% 9.05/2.74 thf(func_def_623, type, sK481: term).
% 9.05/2.74 thf(func_def_624, type, sK482: term).
% 9.05/2.74 thf(func_def_625, type, sK483: term).
% 9.05/2.74 thf(func_def_626, type, sK484: term).
% 9.05/2.74 thf(func_def_627, type, sK485: term).
% 9.05/2.74 thf(func_def_628, type, sK486: subst > term > term).
% 9.05/2.74 thf(func_def_629, type, sK487: subst).
% 9.05/2.74 thf(func_def_630, type, sK488: subst > term > term).
% 9.05/2.74 thf(func_def_631, type, sK489: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_632, type, sK490: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_633, type, sK491: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_634, type, sK492: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_635, type, sK493: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_636, type, sK494: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_637, type, sK495: term).
% 9.05/2.74 thf(func_def_638, type, sK496: subst > term > term).
% 9.05/2.74 thf(func_def_639, type, sK497: term).
% 9.05/2.74 thf(func_def_640, type, sK498: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_641, type, sK499: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_642, type, sK500: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_643, type, sK501: term).
% 9.05/2.74 thf(func_def_644, type, sK502: term).
% 9.05/2.74 thf(func_def_645, type, sK503: term).
% 9.05/2.74 thf(func_def_646, type, sK504: term).
% 9.05/2.74 thf(func_def_647, type, sK505: term).
% 9.05/2.74 thf(func_def_648, type, sK506: term).
% 9.05/2.74 thf(func_def_649, type, sK507: term).
% 9.05/2.74 thf(func_def_650, type, sK508: term).
% 9.05/2.74 thf(func_def_651, type, sK509: term).
% 9.05/2.74 thf(func_def_652, type, sK510: term).
% 9.05/2.74 thf(func_def_653, type, sK511: term > $o).
% 9.05/2.74 thf(func_def_654, type, sK512: term).
% 9.05/2.74 thf(func_def_655, type, sK513: term > $o).
% 9.05/2.74 thf(func_def_656, type, sK514: term).
% 9.05/2.74 thf(func_def_657, type, sK515: subst).
% 9.05/2.74 thf(func_def_658, type, sK516: subst).
% 9.05/2.74 thf(func_def_659, type, sK517: subst).
% 9.05/2.74 thf(func_def_660, type, sK518: term).
% 9.05/2.74 thf(func_def_661, type, sK519: subst).
% 9.05/2.74 thf(func_def_662, type, sK520: term).
% 9.05/2.74 thf(func_def_663, type, sK521: subst).
% 9.05/2.74 thf(func_def_664, type, sK522: term > $o).
% 9.05/2.74 thf(func_def_665, type, sK523: subst).
% 9.05/2.74 thf(func_def_666, type, sK524: term).
% 9.05/2.74 thf(func_def_667, type, sK525: term > $o).
% 9.05/2.74 thf(func_def_668, type, sK526: term).
% 9.05/2.74 thf(func_def_669, type, sK527: term).
% 9.05/2.74 thf(func_def_670, type, sK528: term > $o).
% 9.05/2.74 thf(func_def_671, type, sK529: subst).
% 9.05/2.74 thf(func_def_672, type, sK530: (term > $o) > term).
% 9.05/2.74 thf(func_def_673, type, sK531: subst > (term > $o) > term > term > $o).
% 9.05/2.74 thf(func_def_674, type, sK532: term).
% 9.05/2.74 thf(func_def_675, type, sK533: term).
% 9.05/2.74 thf(func_def_676, type, sK534: subst > d_subst).
% 9.05/2.74 thf(func_def_677, type, sK535: subst).
% 9.05/2.74 thf(func_def_678, type, sK536: term).
% 9.05/2.74 thf(func_def_679, type, sK537: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_680, type, sK538: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_681, type, sK539: term > subst > term > term).
% 9.05/2.74 thf(func_def_682, type, sK540: term).
% 9.05/2.74 thf(func_def_683, type, sK541: term).
% 9.05/2.74 thf(func_def_684, type, sK542: term > subst > term > term).
% 9.05/2.74 thf(func_def_685, type, sK543: term).
% 9.05/2.74 thf(func_def_686, type, sK544: term).
% 9.05/2.74 thf(func_def_687, type, sK545: term).
% 9.05/2.74 thf(func_def_688, type, sK546: term).
% 9.05/2.74 thf(func_def_689, type, sK547: term).
% 9.05/2.74 thf(func_def_690, type, sK548: term).
% 9.05/2.74 thf(func_def_691, type, sK549: term).
% 9.05/2.74 thf(func_def_692, type, sK550: term).
% 9.05/2.74 thf(func_def_693, type, sK551: term).
% 9.05/2.74 thf(func_def_694, type, sK552: subst).
% 9.05/2.74 thf(func_def_695, type, sK553: term).
% 9.05/2.74 thf(func_def_696, type, sK554: term > $o).
% 9.05/2.74 thf(func_def_697, type, sK555: term > $o).
% 9.05/2.74 thf(func_def_698, type, sK556: (term > $o) > (term > $o) > term > subst > term).
% 9.05/2.74 thf(func_def_699, type, sK557: term).
% 9.05/2.74 thf(func_def_700, type, sK558: subst).
% 9.05/2.74 thf(func_def_701, type, sK559: term > (term > $o) > subst > (term > $o) > term).
% 9.05/2.74 thf(func_def_702, type, sK560: term > $o).
% 9.05/2.74 thf(func_def_703, type, sK561: subst).
% 9.05/2.74 thf(func_def_704, type, sK562: term > $o).
% 9.05/2.74 thf(func_def_705, type, sK563: term).
% 9.05/2.74 thf(func_def_706, type, sK564: term).
% 9.05/2.74 thf(func_def_707, type, sK565: term > subst > term > term).
% 9.05/2.74 thf(func_def_708, type, sK566: term).
% 9.05/2.74 thf(func_def_709, type, sK567: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_710, type, sK568: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_711, type, sK569: term > subst > term > term).
% 9.05/2.74 thf(func_def_712, type, sK570: subst).
% 9.05/2.74 thf(func_def_713, type, sK571: (term > $o) > term).
% 9.05/2.74 thf(func_def_714, type, sK572: (term > $o) > term).
% 9.05/2.74 thf(func_def_715, type, sK573: term > $o).
% 9.05/2.74 thf(func_def_716, type, sK574: term).
% 9.05/2.74 thf(func_def_717, type, sK575: term > (term > $o) > subst > term > $o).
% 9.05/2.74 thf(func_def_718, type, sK576: subst).
% 9.05/2.74 thf(func_def_719, type, sK577: term > $o).
% 9.05/2.74 thf(func_def_720, type, sK578: term).
% 9.05/2.74 thf(func_def_721, type, sK579: (term > $o) > term).
% 9.05/2.74 thf(func_def_722, type, sK580: term).
% 9.05/2.74 thf(func_def_723, type, sK581: term).
% 9.05/2.74 thf(func_def_724, type, sK582: subst).
% 9.05/2.74 thf(func_def_725, type, sK583: term > d_term).
% 9.05/2.74 thf(func_def_726, type, sK584: subst).
% 9.05/2.74 thf(func_def_727, type, sK585: subst).
% 9.05/2.74 thf(func_def_728, type, sK586: term).
% 9.05/2.74 thf(func_def_729, type, sK587: term).
% 9.05/2.74 thf(func_def_730, type, sK588: term).
% 9.05/2.74 thf(func_def_731, type, sK589: term).
% 9.05/2.74 thf(func_def_732, type, sK590: term).
% 9.05/2.74 thf(func_def_733, type, sK591: subst > term > subst > $o).
% 9.05/2.74 thf(func_def_734, type, sK592: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_735, type, sK593: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_736, type, sK594: (subst > term > term) > term).
% 9.05/2.74 thf(func_def_737, type, sK595: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_738, type, sK596: (subst > term > term) > subst).
% 9.05/2.74 thf(func_def_739, type, sK597: term).
% 9.05/2.74 thf(func_def_740, type, sK598: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_741, type, sK599: (subst > term > subst > $o) > term).
% 9.05/2.74 thf(func_def_742, type, sK600: (subst > term > subst > $o) > subst > term > term).
% 9.05/2.74 thf(func_def_743, type, ph601: !>[X0: $tType]:(X0)).
% 9.05/2.74 thf(f5150,plain,(
% 9.05/2.74 $false),
% 9.05/2.74 inference(subsumption_resolution,[status(thm)],[f5107,f2180])).
% 9.05/2.74 thf(f2180,plain,(
% 9.05/2.74 ((sP58 @ sK458) != $true)),
% 9.05/2.74 inference(trivial_inequality_removal,[status(thm)],[f2169])).
% 9.05/2.74 thf(f2169,plain,(
% 9.05/2.74 ($false = $true) | ((sP58 @ sK458) != $true)),
% 9.05/2.74 inference(backward_demodulation,[status(thm)],[f1259,f2159])).
% 9.05/2.74 thf(f2159,plain,(
% 9.05/2.74 (hoasinduction_no_psi_cond = $false)),
% 9.05/2.74 inference(trivial_inequality_removal,[status(thm)],[f2158])).
% 9.05/2.74 thf(f2158,plain,(
% 9.05/2.74 (hoasinduction_no_psi_cond = $false) | ($false = $true)),
% 9.05/2.74 inference(duplicate_literal_removal,[status(thm)],[f2153])).
% 9.05/2.74 thf(f2153,plain,(
% 9.05/2.74 (hoasinduction_no_psi_cond = $false) | (hoasinduction_no_psi_cond = $false) | ($false = $true)),
% 9.05/2.74 inference(superposition,[status(thm)],[f1959,f2135])).
% 9.05/2.74 thf(f2135,plain,(
% 9.05/2.74 ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | (hoasinduction_no_psi_cond = $false)),
% 9.05/2.74 inference(duplicate_literal_removal,[status(thm)],[f2134])).
% 9.05/2.74 thf(f2134,plain,(
% 9.05/2.74 (hoasinduction_no_psi_cond = $false) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)),
% 9.05/2.74 inference(forward_demodulation,[status(thm)],[f2133,f1894])).
% 9.05/2.74 thf(f1894,plain,(
% 9.05/2.74 ( ! [X258 : term] : (((d2term @ d_one) = X258)) )),
% 9.05/2.74 inference(forward_demodulation,[status(thm)],[f1053,f1212])).
% 9.05/2.74 thf(f1212,plain,(
% 9.05/2.74 ( ! [X44 : d_term] : ((d_one = X44)) )),
% 9.05/2.74 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.74 thf(f545,plain,(
% 9.05/2.74 ((hoasinduction_no_psi_cond = $true) | (! [X2 : term,X3 : term] : (((sK458 @ id @ X3 @ id) != $true) | ((sK458 @ id @ X2 @ id) != $true) | ((sK458 @ id @ (ap @ (sub @ X2 @ id) @ X3) @ id) = $true)) & ((sP58 @ sK458) != $true) & ((sK458 @ id @ sK457 @ id) != $true))) & (! [X4 : term,X5 : subst > term > subst > $o] : ((((X5 @ id @ (sK460 @ X5) @ id) = $true) & ((X5 @ id @ (sK459 @ X5) @ id) = $true) & ((X5 @ id @ (ap @ (sub @ (sK459 @ X5) @ id) @ (sK460 @ X5)) @ id) != $true)) | ((sP58 @ X5) = $true) | ((X5 @ id @ X4 @ id) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((sP57 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP57 != $true)) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP53 != $true)) & ((sP53 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) | (! [X8 : term] : ((sub @ X8 @ id) = X8) & ((sub @ sK461 @ id) != sK461))) & ((sK462 != (sub @ sK462 @ id)) | ! [X11 : term] : ((sub @ X11 @ id) = X11) | (termmset_lthm != $true)) & (! [X12 : subst > term > subst > $o] : ! [X14 : term] : ((sK463 @ X12 @ X14) = (X12 @ id @ X14 @ id)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ! [X16 : term > $o] : ((X16 @ (sK465 @ X16)) != (sK464 @ id @ (sK465 @ X16) @ id))) & (axmap = $true) & (ulamvarsh = $true) & (! [X18 : subst > term > subst > $o] : ! [X20 : term] : ((sK466 @ X18 @ X20) = (X18 @ id @ X20 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X22 : term > $o] : ((sK467 @ id @ (sK468 @ X22) @ id) != (X22 @ (sK468 @ X22)))) & (((comp @ sK469 @ id) != sK469) | ((comp @ id @ sK470) != sK470) | (! [X26 : subst] : ((comp @ id @ X26) = X26) & ! [X27 : subst] : ((comp @ X27 @ id) = X27)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X28 : subst] : ((comp @ X28 @ id) = X28) & ! [X29 : subst] : ((comp @ id @ X29) = X29) & (((comp @ id @ sK471) != sK471) | (sK472 != (comp @ sK472 @ id))))) & ((sP48 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP48 != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3aa = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa != $true)) & (hoasinduction_lem1_gthm = $true) & (substmonoid_gthm = $true) & ! [X32 : d_term,X33 : d_term] : (((d2term @ X32) != (d2term @ X33)) | (X32 = X33)) & (ulamvarind = $true) & (pushprop_lthm_orig = $true) & (! [X34 : term,X35 : term,X36 : term,X37 : term] : ((X34 = X35) | ((ap @ X34 @ X36) != (ap @ X35 @ X37))) | (apinj1 != $true)) & ((apinj1 = $true) | ((sK473 != sK474) & ((ap @ sK474 @ sK476) = (ap @ sK473 @ sK475)))) & (hoaslamnotap_gthm = $true) & ! [X42 : d_subst,X43 : d_subst] : ((X42 = X43) | ((d2subst @ X43) != (d2subst @ X42))) & (hoaslamnotvar = $true) & (pushprop_lem1v2_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2 = $true) | (sP46 != $true)) & ((sP46 = $true) | (hoasinduction_lem3v2 != $true)) & ! [X44 : d_term] : (d_one = X44) & ((sP44 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP44 != $true)) & (! [X45 : term,X46 : term,X47 : term,X48 : term] : (((ap @ X46 @ X48) != (ap @ X47 @ X45)) | (X45 = X48)) | (apinj2 != $true)) & ((apinj2 = $true) | (((ap @ sK478 @ sK480) = (ap @ sK479 @ sK477)) & (sK477 != sK480))) & (lamnotvar = $true) & (axabs = $true) & ((hoasinduction = $true) | (sP42 != $true)) & ((sP42 = $true) | (hoasinduction != $true)) & ((d2subst @ d_id) = sh) & (pushprop_lem0_gthm = $true) & (! [X53 : term,X54 : term,X55 : term,X56 : term] : (((ap @ (sub @ X55 @ id) @ X53) != (ap @ (sub @ X54 @ id) @ X56)) | (X54 = X55)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | (((ap @ (sub @ sK482 @ id) @ sK484) = (ap @ (sub @ sK483 @ id) @ sK481)) & (sK483 != sK482))) & (hoasapnotvar_gthm = $true) & ((hoaslaminj = $true) | (((lam @ (sK486 @ sh @ one)) = (lam @ (sK488 @ sh @ one))) & ((sK488 @ sK487 @ sK485) != (sK486 @ sK487 @ sK485)) & ! [X65 : term,X66 : subst,X67 : subst] : ((sK486 @ (comp @ X67 @ X66) @ (sub @ X65 @ X66)) = (sub @ (sK486 @ X67 @ X65) @ X66)) & ! [X68 : subst,X69 : term,X70 : subst] : ((sub @ (sK488 @ X70 @ X69) @ X68) = (sK488 @ (comp @ X70 @ X68) @ (sub @ X69 @ X68))))) & (! [X71 : term,X72 : subst > term > term,X73 : subst,X74 : subst > term > term] : (((lam @ (X72 @ sh @ one)) != (lam @ (X74 @ sh @ one))) | ((X72 @ X73 @ X71) = (X74 @ X73 @ X71)) | ((X72 @ (comp @ (sK491 @ X72) @ (sK490 @ X72)) @ (sub @ (sK489 @ X72) @ (sK490 @ X72))) != (sub @ (X72 @ (sK491 @ X72) @ (sK489 @ X72)) @ (sK490 @ X72))) | ((X74 @ (comp @ (sK494 @ X74) @ (sK492 @ X74)) @ (sub @ (sK493 @ X74) @ (sK492 @ X74))) != (sub @ (X74 @ (sK494 @ X74) @ (sK493 @ X74)) @ (sK492 @ X74)))) | (hoaslaminj != $true)) & ((hoaslamnotap = $true) | (! [X84 : subst,X85 : subst,X86 : term] : ((sub @ (sK496 @ X84 @ X86) @ X85) = (sK496 @ (comp @ X84 @ X85) @ (sub @ X86 @ X85))) & ((ap @ (sub @ sK495 @ id) @ sK497) = (lam @ (sK496 @ sh @ one))))) & (! [X87 : term,X88 : subst > term > term,X89 : term] : (((sub @ (X88 @ (sK498 @ X88) @ (sK500 @ X88)) @ (sK499 @ X88)) != (X88 @ (comp @ (sK498 @ X88) @ (sK499 @ X88)) @ (sub @ (sK500 @ X88) @ (sK499 @ X88)))) | ((ap @ (sub @ X87 @ id) @ X89) != (lam @ (X88 @ sh @ one)))) | (hoaslamnotap != $true)) & (hoasinduction_gthm = $true) & (pushprop_lthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) | (((sK504 != sK501) & ((ap @ (sub @ sK504 @ id) @ sK502) = (ap @ (sub @ sK501 @ id) @ sK503))) & ! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X97 @ X99) != (ap @ X100 @ X98)) | (X97 = X100)) & ! [X101 : term] : ((sub @ X101 @ id) = X101))) & (! [X102 : term,X103 : term,X104 : term,X105 : term] : ((X102 = X105) | ((ap @ (sub @ X105 @ id) @ X103) != (ap @ (sub @ X102 @ id) @ X104))) | (((ap @ sK505 @ sK507) = (ap @ sK508 @ sK506)) & (sK508 != sK505)) | ((sub @ sK509 @ id) != sK509) | (hoasapinj1_lthm != $true)) & (! [X111 : term] : ((sub @ X111 @ id) = X111) | (termmset != $true)) & ((termmset = $true) | ((sub @ sK510 @ id) != sK510)) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar = $true) & ((pushprop_lem1 = $true) | (((sK511 @ sK512) = $true) & ($true != (sK513 @ (sub @ sK512 @ (push @ sK514 @ sK515)))))) & (! [X118 : term > $o,X119 : term,X120 : term > $o,X121 : term,X122 : subst] : (((X118 @ X119) != $true) | ((X120 @ (sub @ X119 @ (push @ X121 @ X122))) = $true)) | (pushprop_lem1 != $true)) & ((sP39 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP39 != $true)) & ((hoasinduction_lem3a = $true) | (sP37 != $true)) & ((sP37 = $true) | (hoasinduction_lem3a != $true)) & (hoasinduction_lem1 = $true) & (hoaslaminj_gthm = $true) & (pushprop = $true) & (termmset_gthm = $true) & (pushprop_lem3v2 = $true) & ((substmonoid = $true) | ((comp @ id @ sK516) != sK516) | ((comp @ sK517 @ id) != sK517)) & ((! [X125 : subst] : ((comp @ id @ X125) = X125) & ! [X126 : subst] : ((comp @ X126 @ id) = X126)) | (substmonoid != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP36)) & (($true = sP36) | (hoasinduction_lem3aa_lthm != $true)) & (ulamvar1 != $true) & (((comp @ sh @ (push @ sK518 @ sK519)) != sK519) | ! [X129 : term > $o,X130 : subst,X131 : term,X132 : term > $o,X133 : term] : (((X129 @ X131) != $true) | ((X132 @ (sub @ X131 @ (push @ X133 @ X130))) = $true)) | (sK520 != (sub @ one @ (push @ sK520 @ sK521))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X136 : term,X137 : subst] : ((comp @ sh @ (push @ X136 @ X137)) = X137) & (((sK522 @ sK524) = $true) & ((sK525 @ (sub @ sK524 @ (push @ sK526 @ sK523))) != $true)) & ! [X143 : term,X144 : subst] : ((sub @ one @ (push @ X143 @ X144)) = X143))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((pushprop_lem0_lthm = $true) | ! [X148 : term > $o] : ((sK528 @ (sub @ (sK530 @ X148) @ (push @ sK527 @ sK529))) != (X148 @ (sK530 @ X148)))) & (! [X150 : term,X151 : term > $o,X152 : subst] : ! [X154 : term] : ((sK531 @ X152 @ X151 @ X150 @ X154) = (X151 @ (sub @ X154 @ (push @ X150 @ X152)))) | (pushprop_lem0_lthm != $true)) & (one = (d2term @ d_one)) & ((shinj = $true) | ((sK532 != sK533) & ((sub @ sK533 @ sh) = (sub @ sK532 @ sh)))) & (! [X157 : term,X158 : term] : ((X157 = X158) | ((sub @ X158 @ sh) != (sub @ X157 @ sh))) | (shinj != $true)) & (axvarshift = $true) & (hoasinduction_lem3_gthm = $true) & ((d2subst @ d_id) = 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)))))))))))))) & ! [X159 : subst] : ((d2subst @ (sK534 @ X159)) = X159) & ((hoasinduction_lem3a_lthm = $true) | (sP35 != $true)) & ((sP35 = $true) | (hoasinduction_lem3a_lthm != $true)) & (! [X161 : subst] : ((push @ (sub @ one @ X161) @ (comp @ sh @ X161)) = X161) | (axscons != $true)) & ((axscons = $true) | ((push @ (sub @ one @ sK535) @ (comp @ sh @ sK535)) != sK535)) & ((hoasinduction_lem3v2_f = $true) | ! [X164 : subst > term > term] : ((X164 @ (sK537 @ X164) @ (sK538 @ X164)) != (sub @ sK536 @ (push @ (sK538 @ X164) @ (sK537 @ X164))))) & (! [X167 : term] : ! [X169 : subst,X170 : term] : ((sub @ X167 @ (push @ X170 @ X169)) = (sK539 @ X167 @ X169 @ X170)) | (hoasinduction_lem3v2_f != $true)) & (! [X171 : term,X172 : term] : ((X171 = X172) | ((lam @ X172) != (lam @ X171))) | (laminj != $true)) & ((laminj = $true) | ((sK540 != sK541) & ((lam @ sK540) = (lam @ sK541)))) & (hoasinduction_lem2_gthm = $true) & (induction2lem_gthm = $true) & (! [X175 : term] : ((sub @ X175 @ (push @ one @ sh)) = (sK542 @ X175 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X178 : subst > term > term] : ((X178 @ sh @ one) != (sub @ sK543 @ (push @ one @ sh)))) & (pushprop_lem3v2_lthm = $true) & ((hoasapinj2_lthm = $true) | (((sK545 != sK544) & ((ap @ (sub @ sK546 @ id) @ sK544) = (ap @ (sub @ sK547 @ id) @ sK545))) & ! [X183 : term,X184 : term,X185 : term,X186 : term] : (((ap @ X186 @ X183) != (ap @ X185 @ X184)) | (X183 = X184)))) & (! [X187 : term,X188 : term,X189 : term,X190 : term] : ((X187 = X188) | ((ap @ (sub @ X190 @ id) @ X188) != (ap @ (sub @ X189 @ id) @ X187))) | (((ap @ sK550 @ sK549) = (ap @ sK551 @ sK548)) & (sK548 != sK549)) | (hoasapinj2_lthm != $true)) & ((sP33 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP33 != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP31)) & (($true = sP31) | (hoaslaminj_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ((sP30 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP30 != $true)) & ((pushprop_lem1v2 = $true) | (((sK555 @ one) != $true) & ((sK554 @ sK553) = $true) & ! [X199 : term] : ((sK554 @ (sub @ X199 @ (push @ sK553 @ sK552))) = (sK555 @ X199)))) & (! [X200 : subst,X201 : term,X202 : term > $o,X203 : term > $o] : (((X203 @ one) = $true) | ((X202 @ X201) != $true) | ((X203 @ (sK556 @ X203 @ X202 @ X201 @ X200)) != (X202 @ (sub @ (sK556 @ X203 @ X202 @ X201 @ X200) @ (push @ X201 @ X200))))) | (pushprop_lem1v2 != $true)) & ((hoasinduction_lem3v2a = $true) | (sP21 != $true)) & ((sP21 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP20 != $true)) & ((sP20 = $true) | (hoasinduction_lem3_lthm != $true)) & (hoasinduction_lem3v2_gthm = $true) & ((sK557 != (sub @ one @ (push @ sK557 @ sK558))) | ! [X207 : term > $o,X208 : subst,X209 : term > $o,X210 : term] : (((X207 @ X210) != $true) | ((X207 @ (sub @ (sK559 @ X210 @ X209 @ X208 @ X207) @ (push @ X210 @ X208))) != (X209 @ (sK559 @ X210 @ X209 @ X208 @ X207))) | ((X209 @ one) = $true)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X212 : term,X213 : subst] : ((sub @ one @ (push @ X212 @ X213)) = X212) & (((sK560 @ sK563) = $true) & ! [X218 : term] : ((sK562 @ X218) = (sK560 @ (sub @ X218 @ (push @ sK563 @ sK561)))) & ((sK562 @ one) != $true)))) & ! [X219 : d_subst] : (d_id = X219) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (apnotvar = $true) & ((sP17 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP17 != $true)) & (lamnotap != $true) & ((hoasinduction_lthm_3 = $true) | (sP16 != $true)) & ((sP16 = $true) | (hoasinduction_lthm_3 != $true)) & (axclos = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((hoasinduction_lem3b = $true) | ! [X221 : subst > term > term] : ((sub @ sK564 @ (push @ one @ sh)) != (X221 @ sh @ one))) & (! [X222 : term] : ((sub @ X222 @ (push @ one @ sh)) = (sK565 @ X222 @ sh @ one)) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X225 : subst > term > term] : ((X225 @ (sK568 @ X225) @ (sK567 @ X225)) != (sub @ sK566 @ (push @ (sK567 @ X225) @ (sK568 @ X225))))) & (! [X228 : term] : ! [X230 : term,X231 : subst] : ((sK569 @ X228 @ X231 @ X230) = (sub @ X228 @ (push @ X230 @ X231))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoasinduction_lem2_lthm = $true) & (! [X232 : subst] : ((comp @ X232 @ id) = X232) | (axidr != $true)) & ((axidr = $true) | (sK570 != (comp @ sK570 @ id))) & (axassoc = $true) & (! [X234 : term > $o,X235 : term] : (((sP10 @ X234) = $true) | (((var @ (sK571 @ X234)) = $true) & ((X234 @ (sK571 @ X234)) != $true)) | ((X234 @ X235) = $true) | (((X234 @ (lam @ (sK572 @ X234))) != $true) & ((X234 @ (sK572 @ X234)) = $true))) | (induction != $true)) & ((induction = $true) | (((sP10 @ sK573) != $true) & ! [X240 : term] : (((var @ X240) != $true) | ((sK573 @ X240) = $true)) & ((sK573 @ sK574) != $true) & ! [X241 : term] : (($true = (sK573 @ (lam @ X241))) | ((sK573 @ X241) != $true)))) & (hoaslamnotvar_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((sP9 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP9 != $true)) & (hoasapnotvar_lthm = $true) & (! [X242 : subst,X243 : term > $o,X244 : term] : ! [X246 : term] : ((sK575 @ X244 @ X243 @ X242 @ X246) = (X243 @ (sub @ X246 @ (push @ X244 @ X242)))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X250 : term > $o] : ((X250 @ (sK579 @ X250)) != (sK577 @ (sub @ (sK579 @ X250) @ (push @ sK578 @ sK576))))) & (! [X252 : term] : ((sub @ X252 @ id) = X252) | (axvarid != $true)) & ((axvarid = $true) | ((sub @ sK580 @ id) != sK580)) & (! [X254 : term,X255 : subst] : ((sub @ one @ (push @ X254 @ X255)) = X254) | (axvarcons != $true)) & ((axvarcons = $true) | (sK581 != (sub @ one @ (push @ sK581 @ sK582)))) & (hoasapinj1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((sP5 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP5 != $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X258 : term] : ((d2term @ (sK583 @ X258)) = X258) & ((axidl = $true) | ((comp @ id @ sK584) != sK584)) & (! [X261 : subst] : ((comp @ id @ X261) = X261) | (axidl != $true)) & (hoasinduction_lem1v2 = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((axshiftcons = $true) | ((comp @ sh @ (push @ sK586 @ sK585)) != sK585)) & (! [X264 : subst,X265 : term] : ((comp @ sh @ (push @ X265 @ X264)) = X264) | (axshiftcons != $true)) & (pushprop_gthm = $true) & (induction2_gthm = $true) & ((hoasinduction_lem3 = $true) | (sP3 != $true)) & ((sP3 = $true) | (hoasinduction_lem3 != $true)) & (! [X266 : term,X267 : term,X268 : term,X269 : term] : ((X267 = X269) | ((ap @ (sub @ X266 @ id) @ X267) != (ap @ (sub @ X268 @ id) @ X269))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ((sK590 != sK588) & ((ap @ (sub @ sK587 @ id) @ sK588) = (ap @ (sub @ sK589 @ id) @ sK590)))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (axapp = $true)),
% 9.05/2.74 inference(skolemisation,[status(esa),new_symbols(skolem,[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])],[f475,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])).
% 9.05/2.74 thf(f476,plain,(
% 9.05/2.74 ? [X0 : term,X1 : subst > term > subst > $o] : (! [X2 : term,X3 : term] : (((X1 @ id @ X3 @ id) != $true) | ((X1 @ id @ X2 @ id) != $true) | ((X1 @ id @ (ap @ (sub @ X2 @ id) @ X3) @ id) = $true)) & ($true != (sP58 @ X1)) & ((X1 @ id @ X0 @ id) != $true)) => (! [X3 : term,X2 : term] : (((sK458 @ id @ X3 @ id) != $true) | ((sK458 @ id @ X2 @ id) != $true) | ((sK458 @ id @ (ap @ (sub @ X2 @ id) @ X3) @ id) = $true)) & ((sP58 @ sK458) != $true) & ((sK458 @ id @ sK457 @ id) != $true))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f477,plain,(
% 9.05/2.74 ! [X5 : subst > term > subst > $o] : (? [X6 : term,X7 : term] : (((X5 @ id @ X7 @ id) = $true) & ((X5 @ id @ X6 @ id) = $true) & ((X5 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id) != $true)) => (((X5 @ id @ (sK460 @ X5) @ id) = $true) & ((X5 @ id @ (sK459 @ X5) @ id) = $true) & ((X5 @ id @ (ap @ (sub @ (sK459 @ X5) @ id) @ (sK460 @ X5)) @ id) != $true)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f478,plain,(
% 9.05/2.74 ? [X9 : term] : ((sub @ X9 @ id) != X9) => ((sub @ sK461 @ id) != sK461)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f479,plain,(
% 9.05/2.74 ? [X10 : term] : ((sub @ X10 @ id) != X10) => (sK462 != (sub @ sK462 @ id))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f480,plain,(
% 9.05/2.74 ! [X12 : subst > term > subst > $o] : (? [X13 : term > $o] : ! [X14 : term] : ((X13 @ X14) = (X12 @ id @ X14 @ id)) => ! [X14 : term] : ((sK463 @ X12 @ X14) = (X12 @ id @ X14 @ id)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f481,plain,(
% 9.05/2.74 ? [X15 : subst > term > subst > $o] : ! [X16 : term > $o] : ? [X17 : term] : ((X16 @ X17) != (X15 @ id @ X17 @ id)) => ! [X16 : term > $o] : ? [X17 : term] : ((X16 @ X17) != (sK464 @ id @ X17 @ id))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f482,plain,(
% 9.05/2.74 ! [X16 : term > $o] : (? [X17 : term] : ((X16 @ X17) != (sK464 @ id @ X17 @ id)) => ((X16 @ (sK465 @ X16)) != (sK464 @ id @ (sK465 @ X16) @ id)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f483,plain,(
% 9.05/2.74 ! [X18 : subst > term > subst > $o] : (? [X19 : term > $o] : ! [X20 : term] : ((X18 @ id @ X20 @ id) = (X19 @ X20)) => ! [X20 : term] : ((sK466 @ X18 @ X20) = (X18 @ id @ X20 @ id)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f484,plain,(
% 9.05/2.74 ? [X21 : subst > term > subst > $o] : ! [X22 : term > $o] : ? [X23 : term] : ((X21 @ id @ X23 @ id) != (X22 @ X23)) => ! [X22 : term > $o] : ? [X23 : term] : ((sK467 @ id @ X23 @ id) != (X22 @ X23))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f485,plain,(
% 9.05/2.74 ! [X22 : term > $o] : (? [X23 : term] : ((sK467 @ id @ X23 @ id) != (X22 @ X23)) => ((sK467 @ id @ (sK468 @ X22) @ id) != (X22 @ (sK468 @ X22))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f486,plain,(
% 9.05/2.74 ? [X24 : subst] : ((comp @ X24 @ id) != X24) => ((comp @ sK469 @ id) != sK469)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f487,plain,(
% 9.05/2.74 ? [X25 : subst] : ((comp @ id @ X25) != X25) => ((comp @ id @ sK470) != sK470)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f488,plain,(
% 9.05/2.74 ? [X30 : subst] : ((comp @ id @ X30) != X30) => ((comp @ id @ sK471) != sK471)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f489,plain,(
% 9.05/2.74 ? [X31 : subst] : ((comp @ X31 @ id) != X31) => (sK472 != (comp @ sK472 @ id))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f490,plain,(
% 9.05/2.74 ? [X38 : term,X39 : term,X40 : term,X41 : term] : ((X38 != X39) & ((ap @ X39 @ X41) = (ap @ X38 @ X40))) => ((sK473 != sK474) & ((ap @ sK474 @ sK476) = (ap @ sK473 @ sK475)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f491,plain,(
% 9.05/2.74 ? [X49 : term,X50 : term,X51 : term,X52 : term] : (((ap @ X50 @ X52) = (ap @ X51 @ X49)) & (X49 != X52)) => (((ap @ sK478 @ sK480) = (ap @ sK479 @ sK477)) & (sK477 != sK480))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f492,plain,(
% 9.05/2.74 ? [X57 : term,X58 : term,X59 : term,X60 : term] : (((ap @ (sub @ X58 @ id) @ X60) = (ap @ (sub @ X59 @ id) @ X57)) & (X58 != X59)) => (((ap @ (sub @ sK482 @ id) @ sK484) = (ap @ (sub @ sK483 @ id) @ sK481)) & (sK483 != sK482))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f493,plain,(
% 9.05/2.74 ? [X61 : term,X62 : subst > term > term,X63 : subst,X64 : subst > term > term] : (((lam @ (X64 @ sh @ one)) = (lam @ (X62 @ sh @ one))) & ((X62 @ X63 @ X61) != (X64 @ X63 @ X61)) & ! [X65 : term,X66 : subst,X67 : subst] : ((X62 @ (comp @ X67 @ X66) @ (sub @ X65 @ X66)) = (sub @ (X62 @ X67 @ X65) @ X66)) & ! [X68 : subst,X69 : term,X70 : subst] : ((sub @ (X64 @ X70 @ X69) @ X68) = (X64 @ (comp @ X70 @ X68) @ (sub @ X69 @ X68)))) => (((lam @ (sK486 @ sh @ one)) = (lam @ (sK488 @ sh @ one))) & ((sK488 @ sK487 @ sK485) != (sK486 @ sK487 @ sK485)) & ! [X67 : subst,X66 : subst,X65 : term] : ((sK486 @ (comp @ X67 @ X66) @ (sub @ X65 @ X66)) = (sub @ (sK486 @ X67 @ X65) @ X66)) & ! [X70 : subst,X69 : term,X68 : subst] : ((sub @ (sK488 @ X70 @ X69) @ X68) = (sK488 @ (comp @ X70 @ X68) @ (sub @ X69 @ X68))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f494,plain,(
% 9.05/2.74 ! [X72 : subst > term > term] : (? [X75 : term,X76 : subst,X77 : subst] : ((sub @ (X72 @ X77 @ X75) @ X76) != (X72 @ (comp @ X77 @ X76) @ (sub @ X75 @ X76))) => ((X72 @ (comp @ (sK491 @ X72) @ (sK490 @ X72)) @ (sub @ (sK489 @ X72) @ (sK490 @ X72))) != (sub @ (X72 @ (sK491 @ X72) @ (sK489 @ X72)) @ (sK490 @ X72))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f495,plain,(
% 9.05/2.74 ! [X74 : subst > term > term] : (? [X78 : subst,X79 : term,X80 : subst] : ((X74 @ (comp @ X80 @ X78) @ (sub @ X79 @ X78)) != (sub @ (X74 @ X80 @ X79) @ X78)) => ((X74 @ (comp @ (sK494 @ X74) @ (sK492 @ X74)) @ (sub @ (sK493 @ X74) @ (sK492 @ X74))) != (sub @ (X74 @ (sK494 @ X74) @ (sK493 @ X74)) @ (sK492 @ X74))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f496,plain,(
% 9.05/2.74 ? [X81 : term,X82 : subst > term > term,X83 : term] : (! [X84 : subst,X85 : subst,X86 : term] : ((sub @ (X82 @ X84 @ X86) @ X85) = (X82 @ (comp @ X84 @ X85) @ (sub @ X86 @ X85))) & ((lam @ (X82 @ sh @ one)) = (ap @ (sub @ X81 @ id) @ X83))) => (! [X86 : term,X85 : subst,X84 : subst] : ((sub @ (sK496 @ X84 @ X86) @ X85) = (sK496 @ (comp @ X84 @ X85) @ (sub @ X86 @ X85))) & ((ap @ (sub @ sK495 @ id) @ sK497) = (lam @ (sK496 @ sh @ one))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f497,plain,(
% 9.05/2.74 ! [X88 : subst > term > term] : (? [X90 : subst,X91 : subst,X92 : term] : ((sub @ (X88 @ X90 @ X92) @ X91) != (X88 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91))) => ((sub @ (X88 @ (sK498 @ X88) @ (sK500 @ X88)) @ (sK499 @ X88)) != (X88 @ (comp @ (sK498 @ X88) @ (sK499 @ X88)) @ (sub @ (sK500 @ X88) @ (sK499 @ X88)))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f498,plain,(
% 9.05/2.74 ? [X93 : term,X94 : term,X95 : term,X96 : term] : ((X93 != X96) & ((ap @ (sub @ X96 @ id) @ X94) = (ap @ (sub @ X93 @ id) @ X95))) => ((sK504 != sK501) & ((ap @ (sub @ sK504 @ id) @ sK502) = (ap @ (sub @ sK501 @ id) @ sK503)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f499,plain,(
% 9.05/2.74 ? [X106 : term,X107 : term,X108 : term,X109 : term] : (((ap @ X109 @ X107) = (ap @ X106 @ X108)) & (X106 != X109)) => (((ap @ sK505 @ sK507) = (ap @ sK508 @ sK506)) & (sK508 != sK505))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f500,plain,(
% 9.05/2.74 ? [X110 : term] : ((sub @ X110 @ id) != X110) => ((sub @ sK509 @ id) != sK509)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f501,plain,(
% 9.05/2.74 ? [X112 : term] : ((sub @ X112 @ id) != X112) => ((sub @ sK510 @ id) != sK510)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f502,plain,(
% 9.05/2.74 ? [X113 : term > $o,X114 : term,X115 : term > $o,X116 : term,X117 : subst] : (($true = (X113 @ X114)) & ((X115 @ (sub @ X114 @ (push @ X116 @ X117))) != $true)) => (((sK511 @ sK512) = $true) & ($true != (sK513 @ (sub @ sK512 @ (push @ sK514 @ sK515)))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f503,plain,(
% 9.05/2.74 ? [X123 : subst] : ((comp @ id @ X123) != X123) => ((comp @ id @ sK516) != sK516)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f504,plain,(
% 9.05/2.74 ? [X124 : subst] : ((comp @ X124 @ id) != X124) => ((comp @ sK517 @ id) != sK517)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f505,plain,(
% 9.05/2.74 ? [X127 : term,X128 : subst] : ((comp @ sh @ (push @ X127 @ X128)) != X128) => ((comp @ sh @ (push @ sK518 @ sK519)) != sK519)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f506,plain,(
% 9.05/2.74 ? [X134 : term,X135 : subst] : ((sub @ one @ (push @ X134 @ X135)) != X134) => (sK520 != (sub @ one @ (push @ sK520 @ sK521)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f507,plain,(
% 9.05/2.74 ? [X138 : term > $o,X139 : subst,X140 : term,X141 : term > $o,X142 : term] : (((X138 @ X140) = $true) & ((X141 @ (sub @ X140 @ (push @ X142 @ X139))) != $true)) => (((sK522 @ sK524) = $true) & ((sK525 @ (sub @ sK524 @ (push @ sK526 @ sK523))) != $true))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f508,plain,(
% 9.05/2.74 ? [X145 : term,X146 : term > $o,X147 : subst] : ! [X148 : term > $o] : ? [X149 : term] : ((X148 @ X149) != (X146 @ (sub @ X149 @ (push @ X145 @ X147)))) => ! [X148 : term > $o] : ? [X149 : term] : ((X148 @ X149) != (sK528 @ (sub @ X149 @ (push @ sK527 @ sK529))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f509,plain,(
% 9.05/2.74 ! [X148 : term > $o] : (? [X149 : term] : ((X148 @ X149) != (sK528 @ (sub @ X149 @ (push @ sK527 @ sK529)))) => ((sK528 @ (sub @ (sK530 @ X148) @ (push @ sK527 @ sK529))) != (X148 @ (sK530 @ X148))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f510,plain,(
% 9.05/2.74 ! [X150 : term,X151 : term > $o,X152 : subst] : (? [X153 : term > $o] : ! [X154 : term] : ((X153 @ X154) = (X151 @ (sub @ X154 @ (push @ X150 @ X152)))) => ! [X154 : term] : ((sK531 @ X152 @ X151 @ X150 @ X154) = (X151 @ (sub @ X154 @ (push @ X150 @ X152)))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f511,plain,(
% 9.05/2.74 ? [X155 : term,X156 : term] : ((X155 != X156) & ((sub @ X156 @ sh) = (sub @ X155 @ sh))) => ((sK532 != sK533) & ((sub @ sK533 @ sh) = (sub @ sK532 @ sh)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f512,plain,(
% 9.05/2.74 ! [X159 : subst] : (? [X160 : d_subst] : ((d2subst @ X160) = X159) => ((d2subst @ (sK534 @ X159)) = X159))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f513,plain,(
% 9.05/2.74 ? [X162 : subst] : ((push @ (sub @ one @ X162) @ (comp @ sh @ X162)) != X162) => ((push @ (sub @ one @ sK535) @ (comp @ sh @ sK535)) != sK535)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f514,plain,(
% 9.05/2.74 ? [X163 : term] : ! [X164 : subst > term > term] : ? [X165 : subst,X166 : term] : ((sub @ X163 @ (push @ X166 @ X165)) != (X164 @ X165 @ X166)) => ! [X164 : subst > term > term] : ? [X166 : term,X165 : subst] : ((X164 @ X165 @ X166) != (sub @ sK536 @ (push @ X166 @ X165)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f515,plain,(
% 9.05/2.74 ! [X164 : subst > term > term] : (? [X166 : term,X165 : subst] : ((X164 @ X165 @ X166) != (sub @ sK536 @ (push @ X166 @ X165))) => ((X164 @ (sK537 @ X164) @ (sK538 @ X164)) != (sub @ sK536 @ (push @ (sK538 @ X164) @ (sK537 @ X164)))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f516,plain,(
% 9.05/2.74 ! [X167 : term] : (? [X168 : subst > term > term] : ! [X169 : subst,X170 : term] : ((sub @ X167 @ (push @ X170 @ X169)) = (X168 @ X169 @ X170)) => ! [X170 : term,X169 : subst] : ((sub @ X167 @ (push @ X170 @ X169)) = (sK539 @ X167 @ X169 @ X170)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f517,plain,(
% 9.05/2.74 ? [X173 : term,X174 : term] : ((X173 != X174) & ((lam @ X174) = (lam @ X173))) => ((sK540 != sK541) & ((lam @ sK540) = (lam @ sK541)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f518,plain,(
% 9.05/2.74 ! [X175 : term] : (? [X176 : subst > term > term] : ((sub @ X175 @ (push @ one @ sh)) = (X176 @ sh @ one)) => ((sub @ X175 @ (push @ one @ sh)) = (sK542 @ X175 @ sh @ one)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f519,plain,(
% 9.05/2.74 ? [X177 : term] : ! [X178 : subst > term > term] : ((X178 @ sh @ one) != (sub @ X177 @ (push @ one @ sh))) => ! [X178 : subst > term > term] : ((X178 @ sh @ one) != (sub @ sK543 @ (push @ one @ sh)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f520,plain,(
% 9.05/2.74 ? [X179 : term,X180 : term,X181 : term,X182 : term] : ((X179 != X180) & ((ap @ (sub @ X181 @ id) @ X179) = (ap @ (sub @ X182 @ id) @ X180))) => ((sK545 != sK544) & ((ap @ (sub @ sK546 @ id) @ sK544) = (ap @ (sub @ sK547 @ id) @ sK545)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f521,plain,(
% 9.05/2.74 ? [X191 : term,X192 : term,X193 : term,X194 : term] : (((ap @ X194 @ X191) = (ap @ X193 @ X192)) & (X191 != X192)) => (((ap @ sK550 @ sK549) = (ap @ sK551 @ sK548)) & (sK548 != sK549))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f522,plain,(
% 9.05/2.74 ? [X195 : subst,X196 : term,X197 : term > $o,X198 : term > $o] : (((X198 @ one) != $true) & ((X197 @ X196) = $true) & ! [X199 : term] : ((X198 @ X199) = (X197 @ (sub @ X199 @ (push @ X196 @ X195))))) => (((sK555 @ one) != $true) & ((sK554 @ sK553) = $true) & ! [X199 : term] : ((sK554 @ (sub @ X199 @ (push @ sK553 @ sK552))) = (sK555 @ X199)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f523,plain,(
% 9.05/2.74 ! [X200 : subst,X201 : term,X202 : term > $o,X203 : term > $o] : (? [X204 : term] : ((X202 @ (sub @ X204 @ (push @ X201 @ X200))) != (X203 @ X204)) => ((X203 @ (sK556 @ X203 @ X202 @ X201 @ X200)) != (X202 @ (sub @ (sK556 @ X203 @ X202 @ X201 @ X200) @ (push @ X201 @ X200)))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f524,plain,(
% 9.05/2.74 ? [X205 : term,X206 : subst] : ((sub @ one @ (push @ X205 @ X206)) != X205) => (sK557 != (sub @ one @ (push @ sK557 @ sK558)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f525,plain,(
% 9.05/2.74 ! [X207 : term > $o,X208 : subst,X209 : term > $o,X210 : term] : (? [X211 : term] : ((X209 @ X211) != (X207 @ (sub @ X211 @ (push @ X210 @ X208)))) => ((X207 @ (sub @ (sK559 @ X210 @ X209 @ X208 @ X207) @ (push @ X210 @ X208))) != (X209 @ (sK559 @ X210 @ X209 @ X208 @ X207))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f526,plain,(
% 9.05/2.74 ? [X214 : term > $o,X215 : subst,X216 : term > $o,X217 : term] : (((X214 @ X217) = $true) & ! [X218 : term] : ((X214 @ (sub @ X218 @ (push @ X217 @ X215))) = (X216 @ X218)) & ((X216 @ one) != $true)) => (((sK560 @ sK563) = $true) & ! [X218 : term] : ((sK562 @ X218) = (sK560 @ (sub @ X218 @ (push @ sK563 @ sK561)))) & ((sK562 @ one) != $true))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f527,plain,(
% 9.05/2.74 ? [X220 : term] : ! [X221 : subst > term > term] : ((sub @ X220 @ (push @ one @ sh)) != (X221 @ sh @ one)) => ! [X221 : subst > term > term] : ((sub @ sK564 @ (push @ one @ sh)) != (X221 @ sh @ one))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f528,plain,(
% 9.05/2.74 ! [X222 : term] : (? [X223 : subst > term > term] : ((sub @ X222 @ (push @ one @ sh)) = (X223 @ sh @ one)) => ((sub @ X222 @ (push @ one @ sh)) = (sK565 @ X222 @ sh @ one)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f529,plain,(
% 9.05/2.74 ? [X224 : term] : ! [X225 : subst > term > term] : ? [X226 : term,X227 : subst] : ((X225 @ X227 @ X226) != (sub @ X224 @ (push @ X226 @ X227))) => ! [X225 : subst > term > term] : ? [X227 : subst,X226 : term] : ((sub @ sK566 @ (push @ X226 @ X227)) != (X225 @ X227 @ X226))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f530,plain,(
% 9.05/2.74 ! [X225 : subst > term > term] : (? [X227 : subst,X226 : term] : ((sub @ sK566 @ (push @ X226 @ X227)) != (X225 @ X227 @ X226)) => ((X225 @ (sK568 @ X225) @ (sK567 @ X225)) != (sub @ sK566 @ (push @ (sK567 @ X225) @ (sK568 @ X225)))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f531,plain,(
% 9.05/2.74 ! [X228 : term] : (? [X229 : subst > term > term] : ! [X230 : term,X231 : subst] : ((X229 @ X231 @ X230) = (sub @ X228 @ (push @ X230 @ X231))) => ! [X231 : subst,X230 : term] : ((sK569 @ X228 @ X231 @ X230) = (sub @ X228 @ (push @ X230 @ X231))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f532,plain,(
% 9.05/2.74 ? [X233 : subst] : ((comp @ X233 @ id) != X233) => (sK570 != (comp @ sK570 @ id))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f533,plain,(
% 9.05/2.74 ! [X234 : term > $o] : (? [X236 : term] : (($true = (var @ X236)) & ((X234 @ X236) != $true)) => (((var @ (sK571 @ X234)) = $true) & ((X234 @ (sK571 @ X234)) != $true)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f534,plain,(
% 9.05/2.74 ! [X234 : term > $o] : (? [X237 : term] : (((X234 @ (lam @ X237)) != $true) & ((X234 @ X237) = $true)) => (((X234 @ (lam @ (sK572 @ X234))) != $true) & ((X234 @ (sK572 @ X234)) = $true)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f535,plain,(
% 9.05/2.74 ? [X238 : term > $o,X239 : term] : (((sP10 @ X238) != $true) & ! [X240 : term] : (((var @ X240) != $true) | ((X238 @ X240) = $true)) & ((X238 @ X239) != $true) & ! [X241 : term] : (((X238 @ (lam @ X241)) = $true) | ((X238 @ X241) != $true))) => (((sP10 @ sK573) != $true) & ! [X240 : term] : (((var @ X240) != $true) | ((sK573 @ X240) = $true)) & ((sK573 @ sK574) != $true) & ! [X241 : term] : (($true = (sK573 @ (lam @ X241))) | ((sK573 @ X241) != $true)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f536,plain,(
% 9.05/2.74 ! [X242 : subst,X243 : term > $o,X244 : term] : (? [X245 : term > $o] : ! [X246 : term] : ((X245 @ X246) = (X243 @ (sub @ X246 @ (push @ X244 @ X242)))) => ! [X246 : term] : ((sK575 @ X244 @ X243 @ X242 @ X246) = (X243 @ (sub @ X246 @ (push @ X244 @ X242)))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f537,plain,(
% 9.05/2.74 ? [X247 : subst,X248 : term > $o,X249 : term] : ! [X250 : term > $o] : ? [X251 : term] : ((X250 @ X251) != (X248 @ (sub @ X251 @ (push @ X249 @ X247)))) => ! [X250 : term > $o] : ? [X251 : term] : ((X250 @ X251) != (sK577 @ (sub @ X251 @ (push @ sK578 @ sK576))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f538,plain,(
% 9.05/2.74 ! [X250 : term > $o] : (? [X251 : term] : ((X250 @ X251) != (sK577 @ (sub @ X251 @ (push @ sK578 @ sK576)))) => ((X250 @ (sK579 @ X250)) != (sK577 @ (sub @ (sK579 @ X250) @ (push @ sK578 @ sK576)))))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f539,plain,(
% 9.05/2.74 ? [X253 : term] : ((sub @ X253 @ id) != X253) => ((sub @ sK580 @ id) != sK580)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f540,plain,(
% 9.05/2.74 ? [X256 : term,X257 : subst] : ((sub @ one @ (push @ X256 @ X257)) != X256) => (sK581 != (sub @ one @ (push @ sK581 @ sK582)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f541,plain,(
% 9.05/2.74 ! [X258 : term] : (? [X259 : d_term] : ((d2term @ X259) = X258) => ((d2term @ (sK583 @ X258)) = X258))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f542,plain,(
% 9.05/2.74 ? [X260 : subst] : ((comp @ id @ X260) != X260) => ((comp @ id @ sK584) != sK584)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f543,plain,(
% 9.05/2.74 ? [X262 : subst,X263 : term] : ((comp @ sh @ (push @ X263 @ X262)) != X262) => ((comp @ sh @ (push @ sK586 @ sK585)) != sK585)),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f544,plain,(
% 9.05/2.74 ? [X270 : term,X271 : term,X272 : term,X273 : term] : ((X271 != X273) & ((ap @ (sub @ X270 @ id) @ X271) = (ap @ (sub @ X272 @ id) @ X273))) => ((sK590 != sK588) & ((ap @ (sub @ sK587 @ id) @ sK588) = (ap @ (sub @ sK589 @ id) @ sK590)))),
% 9.05/2.74 introduced(definition,[],[choice_axiom])).
% 9.05/2.74 thf(f475,plain,(
% 9.05/2.74 ((hoasinduction_no_psi_cond = $true) | ? [X0 : term,X1 : subst > term > subst > $o] : (! [X2 : term,X3 : term] : (((X1 @ id @ X3 @ id) != $true) | ((X1 @ id @ X2 @ id) != $true) | ((X1 @ id @ (ap @ (sub @ X2 @ id) @ X3) @ id) = $true)) & ($true != (sP58 @ X1)) & ((X1 @ id @ X0 @ id) != $true))) & (! [X4 : term,X5 : subst > term > subst > $o] : (? [X6 : term,X7 : term] : (((X5 @ id @ X7 @ id) = $true) & ((X5 @ id @ X6 @ id) = $true) & ((X5 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id) != $true)) | ((sP58 @ X5) = $true) | ((X5 @ id @ X4 @ id) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((sP57 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP57 != $true)) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP53 != $true)) & ((sP53 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) | (! [X8 : term] : ((sub @ X8 @ id) = X8) & ? [X9 : term] : ((sub @ X9 @ id) != X9))) & (? [X10 : term] : ((sub @ X10 @ id) != X10) | ! [X11 : term] : ((sub @ X11 @ id) = X11) | (termmset_lthm != $true)) & (! [X12 : subst > term > subst > $o] : ? [X13 : term > $o] : ! [X14 : term] : ((X13 @ X14) = (X12 @ id @ X14 @ id)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X15 : subst > term > subst > $o] : ! [X16 : term > $o] : ? [X17 : term] : ((X16 @ X17) != (X15 @ id @ X17 @ id))) & (axmap = $true) & (ulamvarsh = $true) & (! [X18 : subst > term > subst > $o] : ? [X19 : term > $o] : ! [X20 : term] : ((X18 @ id @ X20 @ id) = (X19 @ X20)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X21 : subst > term > subst > $o] : ! [X22 : term > $o] : ? [X23 : term] : ((X21 @ id @ X23 @ id) != (X22 @ X23))) & (? [X24 : subst] : ((comp @ X24 @ id) != X24) | ? [X25 : subst] : ((comp @ id @ X25) != X25) | (! [X26 : subst] : ((comp @ id @ X26) = X26) & ! [X27 : subst] : ((comp @ X27 @ id) = X27)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X28 : subst] : ((comp @ X28 @ id) = X28) & ! [X29 : subst] : ((comp @ id @ X29) = X29) & (? [X30 : subst] : ((comp @ id @ X30) != X30) | ? [X31 : subst] : ((comp @ X31 @ id) != X31)))) & ((sP48 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP48 != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3aa = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa != $true)) & (hoasinduction_lem1_gthm = $true) & (substmonoid_gthm = $true) & ! [X32 : d_term,X33 : d_term] : (((d2term @ X32) != (d2term @ X33)) | (X32 = X33)) & (ulamvarind = $true) & (pushprop_lthm_orig = $true) & (! [X34 : term,X35 : term,X36 : term,X37 : term] : ((X34 = X35) | ((ap @ X34 @ X36) != (ap @ X35 @ X37))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X38 : term,X39 : term,X40 : term,X41 : term] : ((X38 != X39) & ((ap @ X39 @ X41) = (ap @ X38 @ X40)))) & (hoaslamnotap_gthm = $true) & ! [X42 : d_subst,X43 : d_subst] : ((X42 = X43) | ((d2subst @ X43) != (d2subst @ X42))) & (hoaslamnotvar = $true) & (pushprop_lem1v2_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2 = $true) | (sP46 != $true)) & ((sP46 = $true) | (hoasinduction_lem3v2 != $true)) & ! [X44 : d_term] : (d_one = X44) & ((sP44 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP44 != $true)) & (! [X45 : term,X46 : term,X47 : term,X48 : term] : (((ap @ X46 @ X48) != (ap @ X47 @ X45)) | (X45 = X48)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X49 : term,X50 : term,X51 : term,X52 : term] : (((ap @ X50 @ X52) = (ap @ X51 @ X49)) & (X49 != X52))) & (lamnotvar = $true) & (axabs = $true) & ((hoasinduction = $true) | (sP42 != $true)) & ((sP42 = $true) | (hoasinduction != $true)) & ((d2subst @ d_id) = sh) & (pushprop_lem0_gthm = $true) & (! [X53 : term,X54 : term,X55 : term,X56 : term] : (((ap @ (sub @ X55 @ id) @ X53) != (ap @ (sub @ X54 @ id) @ X56)) | (X54 = X55)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X57 : term,X58 : term,X59 : term,X60 : term] : (((ap @ (sub @ X58 @ id) @ X60) = (ap @ (sub @ X59 @ id) @ X57)) & (X58 != X59))) & (hoasapnotvar_gthm = $true) & ((hoaslaminj = $true) | ? [X61 : term,X62 : subst > term > term,X63 : subst,X64 : subst > term > term] : (((lam @ (X64 @ sh @ one)) = (lam @ (X62 @ sh @ one))) & ((X62 @ X63 @ X61) != (X64 @ X63 @ X61)) & ! [X65 : term,X66 : subst,X67 : subst] : ((X62 @ (comp @ X67 @ X66) @ (sub @ X65 @ X66)) = (sub @ (X62 @ X67 @ X65) @ X66)) & ! [X68 : subst,X69 : term,X70 : subst] : ((sub @ (X64 @ X70 @ X69) @ X68) = (X64 @ (comp @ X70 @ X68) @ (sub @ X69 @ X68))))) & (! [X71 : term,X72 : subst > term > term,X73 : subst,X74 : subst > term > term] : (((lam @ (X72 @ sh @ one)) != (lam @ (X74 @ sh @ one))) | ((X72 @ X73 @ X71) = (X74 @ X73 @ X71)) | ? [X75 : term,X76 : subst,X77 : subst] : ((sub @ (X72 @ X77 @ X75) @ X76) != (X72 @ (comp @ X77 @ X76) @ (sub @ X75 @ X76))) | ? [X78 : subst,X79 : term,X80 : subst] : ((X74 @ (comp @ X80 @ X78) @ (sub @ X79 @ X78)) != (sub @ (X74 @ X80 @ X79) @ X78))) | (hoaslaminj != $true)) & ((hoaslamnotap = $true) | ? [X81 : term,X82 : subst > term > term,X83 : term] : (! [X84 : subst,X85 : subst,X86 : term] : ((sub @ (X82 @ X84 @ X86) @ X85) = (X82 @ (comp @ X84 @ X85) @ (sub @ X86 @ X85))) & ((lam @ (X82 @ sh @ one)) = (ap @ (sub @ X81 @ id) @ X83)))) & (! [X87 : term,X88 : subst > term > term,X89 : term] : (? [X90 : subst,X91 : subst,X92 : term] : ((sub @ (X88 @ X90 @ X92) @ X91) != (X88 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91))) | ((ap @ (sub @ X87 @ id) @ X89) != (lam @ (X88 @ sh @ one)))) | (hoaslamnotap != $true)) & (hoasinduction_gthm = $true) & (pushprop_lthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) | (? [X93 : term,X94 : term,X95 : term,X96 : term] : ((X93 != X96) & ((ap @ (sub @ X96 @ id) @ X94) = (ap @ (sub @ X93 @ id) @ X95))) & ! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X97 @ X99) != (ap @ X100 @ X98)) | (X97 = X100)) & ! [X101 : term] : ((sub @ X101 @ id) = X101))) & (! [X102 : term,X103 : term,X104 : term,X105 : term] : ((X102 = X105) | ((ap @ (sub @ X105 @ id) @ X103) != (ap @ (sub @ X102 @ id) @ X104))) | ? [X106 : term,X107 : term,X108 : term,X109 : term] : (((ap @ X109 @ X107) = (ap @ X106 @ X108)) & (X106 != X109)) | ? [X110 : term] : ((sub @ X110 @ id) != X110) | (hoasapinj1_lthm != $true)) & (! [X111 : term] : ((sub @ X111 @ id) = X111) | (termmset != $true)) & ((termmset = $true) | ? [X112 : term] : ((sub @ X112 @ id) != X112)) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar = $true) & ((pushprop_lem1 = $true) | ? [X113 : term > $o,X114 : term,X115 : term > $o,X116 : term,X117 : subst] : (($true = (X113 @ X114)) & ((X115 @ (sub @ X114 @ (push @ X116 @ X117))) != $true))) & (! [X118 : term > $o,X119 : term,X120 : term > $o,X121 : term,X122 : subst] : (((X118 @ X119) != $true) | ((X120 @ (sub @ X119 @ (push @ X121 @ X122))) = $true)) | (pushprop_lem1 != $true)) & ((sP39 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP39 != $true)) & ((hoasinduction_lem3a = $true) | (sP37 != $true)) & ((sP37 = $true) | (hoasinduction_lem3a != $true)) & (hoasinduction_lem1 = $true) & (hoaslaminj_gthm = $true) & (pushprop = $true) & (termmset_gthm = $true) & (pushprop_lem3v2 = $true) & ((substmonoid = $true) | ? [X123 : subst] : ((comp @ id @ X123) != X123) | ? [X124 : subst] : ((comp @ X124 @ id) != X124)) & ((! [X125 : subst] : ((comp @ id @ X125) = X125) & ! [X126 : subst] : ((comp @ X126 @ id) = X126)) | (substmonoid != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP36)) & (($true = sP36) | (hoasinduction_lem3aa_lthm != $true)) & (ulamvar1 != $true) & (? [X127 : term,X128 : subst] : ((comp @ sh @ (push @ X127 @ X128)) != X128) | ! [X129 : term > $o,X130 : subst,X131 : term,X132 : term > $o,X133 : term] : (((X129 @ X131) != $true) | ((X132 @ (sub @ X131 @ (push @ X133 @ X130))) = $true)) | ? [X134 : term,X135 : subst] : ((sub @ one @ (push @ X134 @ X135)) != X134) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X136 : term,X137 : subst] : ((comp @ sh @ (push @ X136 @ X137)) = X137) & ? [X138 : term > $o,X139 : subst,X140 : term,X141 : term > $o,X142 : term] : (((X138 @ X140) = $true) & ((X141 @ (sub @ X140 @ (push @ X142 @ X139))) != $true)) & ! [X143 : term,X144 : subst] : ((sub @ one @ (push @ X143 @ X144)) = X143))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((pushprop_lem0_lthm = $true) | ? [X145 : term,X146 : term > $o,X147 : subst] : ! [X148 : term > $o] : ? [X149 : term] : ((X148 @ X149) != (X146 @ (sub @ X149 @ (push @ X145 @ X147))))) & (! [X150 : term,X151 : term > $o,X152 : subst] : ? [X153 : term > $o] : ! [X154 : term] : ((X153 @ X154) = (X151 @ (sub @ X154 @ (push @ X150 @ X152)))) | (pushprop_lem0_lthm != $true)) & (one = (d2term @ d_one)) & ((shinj = $true) | ? [X155 : term,X156 : term] : ((X155 != X156) & ((sub @ X156 @ sh) = (sub @ X155 @ sh)))) & (! [X157 : term,X158 : term] : ((X157 = X158) | ((sub @ X158 @ sh) != (sub @ X157 @ sh))) | (shinj != $true)) & (axvarshift = $true) & (hoasinduction_lem3_gthm = $true) & ((d2subst @ d_id) = 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)))))))))))))) & ! [X159 : subst] : ? [X160 : d_subst] : ((d2subst @ X160) = X159) & ((hoasinduction_lem3a_lthm = $true) | (sP35 != $true)) & ((sP35 = $true) | (hoasinduction_lem3a_lthm != $true)) & (! [X161 : subst] : ((push @ (sub @ one @ X161) @ (comp @ sh @ X161)) = X161) | (axscons != $true)) & ((axscons = $true) | ? [X162 : subst] : ((push @ (sub @ one @ X162) @ (comp @ sh @ X162)) != X162)) & ((hoasinduction_lem3v2_f = $true) | ? [X163 : term] : ! [X164 : subst > term > term] : ? [X165 : subst,X166 : term] : ((sub @ X163 @ (push @ X166 @ X165)) != (X164 @ X165 @ X166))) & (! [X167 : term] : ? [X168 : subst > term > term] : ! [X169 : subst,X170 : term] : ((sub @ X167 @ (push @ X170 @ X169)) = (X168 @ X169 @ X170)) | (hoasinduction_lem3v2_f != $true)) & (! [X171 : term,X172 : term] : ((X171 = X172) | ((lam @ X172) != (lam @ X171))) | (laminj != $true)) & ((laminj = $true) | ? [X173 : term,X174 : term] : ((X173 != X174) & ((lam @ X174) = (lam @ X173)))) & (hoasinduction_lem2_gthm = $true) & (induction2lem_gthm = $true) & (! [X175 : term] : ? [X176 : subst > term > term] : ((sub @ X175 @ (push @ one @ sh)) = (X176 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X177 : term] : ! [X178 : subst > term > term] : ((X178 @ sh @ one) != (sub @ X177 @ (push @ one @ sh)))) & (pushprop_lem3v2_lthm = $true) & ((hoasapinj2_lthm = $true) | (? [X179 : term,X180 : term,X181 : term,X182 : term] : ((X179 != X180) & ((ap @ (sub @ X181 @ id) @ X179) = (ap @ (sub @ X182 @ id) @ X180))) & ! [X183 : term,X184 : term,X185 : term,X186 : term] : (((ap @ X186 @ X183) != (ap @ X185 @ X184)) | (X183 = X184)))) & (! [X187 : term,X188 : term,X189 : term,X190 : term] : ((X187 = X188) | ((ap @ (sub @ X190 @ id) @ X188) != (ap @ (sub @ X189 @ id) @ X187))) | ? [X191 : term,X192 : term,X193 : term,X194 : term] : (((ap @ X194 @ X191) = (ap @ X193 @ X192)) & (X191 != X192)) | (hoasapinj2_lthm != $true)) & ((sP33 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP33 != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP31)) & (($true = sP31) | (hoaslaminj_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ((sP30 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP30 != $true)) & ((pushprop_lem1v2 = $true) | ? [X195 : subst,X196 : term,X197 : term > $o,X198 : term > $o] : (((X198 @ one) != $true) & ((X197 @ X196) = $true) & ! [X199 : term] : ((X198 @ X199) = (X197 @ (sub @ X199 @ (push @ X196 @ X195)))))) & (! [X200 : subst,X201 : term,X202 : term > $o,X203 : term > $o] : (((X203 @ one) = $true) | ((X202 @ X201) != $true) | ? [X204 : term] : ((X202 @ (sub @ X204 @ (push @ X201 @ X200))) != (X203 @ X204))) | (pushprop_lem1v2 != $true)) & ((hoasinduction_lem3v2a = $true) | (sP21 != $true)) & ((sP21 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP20 != $true)) & ((sP20 = $true) | (hoasinduction_lem3_lthm != $true)) & (hoasinduction_lem3v2_gthm = $true) & (? [X205 : term,X206 : subst] : ((sub @ one @ (push @ X205 @ X206)) != X205) | ! [X207 : term > $o,X208 : subst,X209 : term > $o,X210 : term] : (((X207 @ X210) != $true) | ? [X211 : term] : ((X209 @ X211) != (X207 @ (sub @ X211 @ (push @ X210 @ X208)))) | ((X209 @ one) = $true)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X212 : term,X213 : subst] : ((sub @ one @ (push @ X212 @ X213)) = X212) & ? [X214 : term > $o,X215 : subst,X216 : term > $o,X217 : term] : (((X214 @ X217) = $true) & ! [X218 : term] : ((X214 @ (sub @ X218 @ (push @ X217 @ X215))) = (X216 @ X218)) & ((X216 @ one) != $true)))) & ! [X219 : d_subst] : (d_id = X219) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (apnotvar = $true) & ((sP17 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP17 != $true)) & (lamnotap != $true) & ((hoasinduction_lthm_3 = $true) | (sP16 != $true)) & ((sP16 = $true) | (hoasinduction_lthm_3 != $true)) & (axclos = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((hoasinduction_lem3b = $true) | ? [X220 : term] : ! [X221 : subst > term > term] : ((sub @ X220 @ (push @ one @ sh)) != (X221 @ sh @ one))) & (! [X222 : term] : ? [X223 : subst > term > term] : ((sub @ X222 @ (push @ one @ sh)) = (X223 @ sh @ one)) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X224 : term] : ! [X225 : subst > term > term] : ? [X226 : term,X227 : subst] : ((X225 @ X227 @ X226) != (sub @ X224 @ (push @ X226 @ X227)))) & (! [X228 : term] : ? [X229 : subst > term > term] : ! [X230 : term,X231 : subst] : ((X229 @ X231 @ X230) = (sub @ X228 @ (push @ X230 @ X231))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoasinduction_lem2_lthm = $true) & (! [X232 : subst] : ((comp @ X232 @ id) = X232) | (axidr != $true)) & ((axidr = $true) | ? [X233 : subst] : ((comp @ X233 @ id) != X233)) & (axassoc = $true) & (! [X234 : term > $o,X235 : term] : (((sP10 @ X234) = $true) | ? [X236 : term] : (($true = (var @ X236)) & ((X234 @ X236) != $true)) | ((X234 @ X235) = $true) | ? [X237 : term] : (((X234 @ (lam @ X237)) != $true) & ((X234 @ X237) = $true))) | (induction != $true)) & ((induction = $true) | ? [X238 : term > $o,X239 : term] : (((sP10 @ X238) != $true) & ! [X240 : term] : (((var @ X240) != $true) | ((X238 @ X240) = $true)) & ((X238 @ X239) != $true) & ! [X241 : term] : (((X238 @ (lam @ X241)) = $true) | ((X238 @ X241) != $true)))) & (hoaslamnotvar_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((sP9 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP9 != $true)) & (hoasapnotvar_lthm = $true) & (! [X242 : subst,X243 : term > $o,X244 : term] : ? [X245 : term > $o] : ! [X246 : term] : ((X245 @ X246) = (X243 @ (sub @ X246 @ (push @ X244 @ X242)))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X247 : subst,X248 : term > $o,X249 : term] : ! [X250 : term > $o] : ? [X251 : term] : ((X250 @ X251) != (X248 @ (sub @ X251 @ (push @ X249 @ X247))))) & (! [X252 : term] : ((sub @ X252 @ id) = X252) | (axvarid != $true)) & ((axvarid = $true) | ? [X253 : term] : ((sub @ X253 @ id) != X253)) & (! [X254 : term,X255 : subst] : ((sub @ one @ (push @ X254 @ X255)) = X254) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X256 : term,X257 : subst] : ((sub @ one @ (push @ X256 @ X257)) != X256)) & (hoasapinj1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((sP5 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP5 != $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X258 : term] : ? [X259 : d_term] : ((d2term @ X259) = X258) & ((axidl = $true) | ? [X260 : subst] : ((comp @ id @ X260) != X260)) & (! [X261 : subst] : ((comp @ id @ X261) = X261) | (axidl != $true)) & (hoasinduction_lem1v2 = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((axshiftcons = $true) | ? [X262 : subst,X263 : term] : ((comp @ sh @ (push @ X263 @ X262)) != X262)) & (! [X264 : subst,X265 : term] : ((comp @ sh @ (push @ X265 @ X264)) = X264) | (axshiftcons != $true)) & (pushprop_gthm = $true) & (induction2_gthm = $true) & ((hoasinduction_lem3 = $true) | (sP3 != $true)) & ((sP3 = $true) | (hoasinduction_lem3 != $true)) & (! [X266 : term,X267 : term,X268 : term,X269 : term] : ((X267 = X269) | ((ap @ (sub @ X266 @ id) @ X267) != (ap @ (sub @ X268 @ id) @ X269))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X270 : term,X271 : term,X272 : term,X273 : term] : ((X271 != X273) & ((ap @ (sub @ X270 @ id) @ X271) = (ap @ (sub @ X272 @ id) @ X273)))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (axapp = $true)),
% 9.05/2.74 inference(rectify,[status(thm)],[f474])).
% 9.05/2.74 thf(f474,plain,(
% 9.05/2.74 ((hoasinduction_no_psi_cond = $true) | ? [X183 : term,X184 : subst > term > subst > $o] : (! [X185 : term,X186 : term] : (((X184 @ id @ X186 @ id) != $true) | ((X184 @ id @ X185 @ id) != $true) | ((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) = $true)) & ((sP58 @ X184) != $true) & ((X184 @ id @ X183 @ id) != $true))) & (! [X183 : term,X184 : subst > term > subst > $o] : (? [X185 : term,X186 : term] : (((X184 @ id @ X186 @ id) = $true) & ((X184 @ id @ X185 @ id) = $true) & ((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) != $true)) | ((sP58 @ X184) = $true) | ((X184 @ id @ X183 @ id) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((sP57 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP57 != $true)) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP53 != $true)) & ((sP53 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) | (! [X272 : term] : ((sub @ X272 @ id) = X272) & ? [X273 : term] : ((sub @ X273 @ id) != X273))) & (? [X272 : term] : ((sub @ X272 @ id) != X272) | ! [X273 : term] : ((sub @ X273 @ id) = X273) | (termmset_lthm != $true)) & (! [X165 : subst > term > subst > $o] : ? [X166 : term > $o] : ! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X165 : subst > term > subst > $o] : ! [X166 : term > $o] : ? [X167 : term] : ((X165 @ id @ X167 @ id) != (X166 @ X167))) & (axmap = $true) & (ulamvarsh = $true) & (! [X2 : subst > term > subst > $o] : ? [X3 : term > $o] : ! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X2 : subst > term > subst > $o] : ! [X3 : term > $o] : ? [X4 : term] : ((X2 @ id @ X4 @ id) != (X3 @ X4))) & (? [X410 : subst] : ((comp @ X410 @ id) != X410) | ? [X409 : subst] : ((comp @ id @ X409) != X409) | (! [X411 : subst] : ((comp @ id @ X411) = X411) & ! [X412 : subst] : ((comp @ X412 @ id) = X412)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X410 : subst] : ((comp @ X410 @ id) = X410) & ! [X409 : subst] : ((comp @ id @ X409) = X409) & (? [X411 : subst] : ((comp @ id @ X411) != X411) | ? [X412 : subst] : ((comp @ X412 @ id) != X412)))) & ((sP48 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP48 != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3aa = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa != $true)) & (hoasinduction_lem1_gthm = $true) & (substmonoid_gthm = $true) & ! [X245 : d_term,X246 : d_term] : (((d2term @ X246) != (d2term @ X245)) | (X245 = X246)) & (ulamvarind = $true) & (pushprop_lthm_orig = $true) & (! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | ((ap @ X261 @ X263) != (ap @ X262 @ X264))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 != X262) & ((ap @ X261 @ X263) = (ap @ X262 @ X264)))) & (hoaslamnotap_gthm = $true) & ! [X181 : d_subst,X182 : d_subst] : ((X181 = X182) | ((d2subst @ X182) != (d2subst @ X181))) & (hoaslamnotvar = $true) & (pushprop_lem1v2_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2 = $true) | (sP46 != $true)) & ((sP46 = $true) | (hoasinduction_lem3v2 != $true)) & ! [X265 : d_term] : (d_one = X265) & ((sP44 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP44 != $true)) & (! [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) != (ap @ X326 @ X328)) | (X325 = X328)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) = (ap @ X326 @ X328)) & (X325 != X328))) & (lamnotvar = $true) & (axabs = $true) & ((hoasinduction = $true) | (sP42 != $true)) & ((sP42 = $true) | (hoasinduction != $true)) & ((d2subst @ d_id) = sh) & (pushprop_lem0_gthm = $true) & (! [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192)) | (X193 = X194)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) = (ap @ (sub @ X194 @ id) @ X192)) & (X193 != X194))) & (hoasapnotvar_gthm = $true) & ((hoaslaminj = $true) | ? [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) = (lam @ (X465 @ sh @ one))) & ((X465 @ X466 @ X464) != (X467 @ X466 @ X464)) & ! [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) = (sub @ (X465 @ X473 @ X471) @ X472)) & ! [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) = (sub @ (X467 @ X470 @ X469) @ X468)))) & (! [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) != (lam @ (X465 @ sh @ one))) | ((X465 @ X466 @ X464) = (X467 @ X466 @ X464)) | ? [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) != (sub @ (X465 @ X473 @ X471) @ X472)) | ? [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) != (sub @ (X467 @ X470 @ X469) @ X468))) | (hoaslaminj != $true)) & ((hoaslamnotap = $true) | ? [X247 : term,X248 : subst > term > term,X249 : term] : (! [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) = (sub @ (X248 @ X250 @ X252) @ X251)) & ((ap @ (sub @ X247 @ id) @ X249) = (lam @ (X248 @ sh @ one))))) & (! [X247 : term,X248 : subst > term > term,X249 : term] : (? [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) != (sub @ (X248 @ X250 @ X252) @ X251)) | ((ap @ (sub @ X247 @ id) @ X249) != (lam @ (X248 @ sh @ one)))) | (hoaslamnotap != $true)) & (hoasinduction_gthm = $true) & (pushprop_lthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) | (? [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 != X408) & ((ap @ (sub @ X408 @ id) @ X406) = (ap @ (sub @ X405 @ id) @ X407))) & ! [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) != (ap @ X404 @ X402)) | (X401 = X404)) & ! [X400 : term] : ((sub @ X400 @ id) = X400))) & (! [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 = X408) | ((ap @ (sub @ X408 @ id) @ X406) != (ap @ (sub @ X405 @ id) @ X407))) | ? [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) = (ap @ X404 @ X402)) & (X401 != X404)) | ? [X400 : term] : ((sub @ X400 @ id) != X400) | (hoasapinj1_lthm != $true)) & (! [X164 : term] : ((sub @ X164 @ id) = X164) | (termmset != $true)) & ((termmset = $true) | ? [X164 : term] : ((sub @ X164 @ id) != X164)) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar = $true) & ((pushprop_lem1 = $true) | ? [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (((X60 @ X61) = $true) & ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) != $true))) & (! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (((X60 @ X61) != $true) | ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) = $true)) | (pushprop_lem1 != $true)) & ((sP39 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP39 != $true)) & ((hoasinduction_lem3a = $true) | (sP37 != $true)) & ((sP37 = $true) | (hoasinduction_lem3a != $true)) & (hoasinduction_lem1 = $true) & (hoaslaminj_gthm = $true) & (pushprop = $true) & (termmset_gthm = $true) & (pushprop_lem3v2 = $true) & ((substmonoid = $true) | ? [X1 : subst] : ((comp @ id @ X1) != X1) | ? [X0 : subst] : ((comp @ X0 @ id) != X0)) & ((! [X1 : subst] : ((comp @ id @ X1) = X1) & ! [X0 : subst] : ((comp @ X0 @ id) = X0)) | (substmonoid != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP36)) & (($true = sP36) | (hoasinduction_lem3aa_lthm != $true)) & (ulamvar1 != $true) & (? [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) != X121) | ! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (((X122 @ X124) != $true) | ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) = $true)) | ? [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) != X118) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) = X121) & ? [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (((X122 @ X124) = $true) & ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) != $true)) & ! [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) = X118))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((pushprop_lem0_lthm = $true) | ? [X139 : term,X140 : term > $o,X141 : subst] : ! [X142 : term > $o] : ? [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) != (X142 @ X143))) & (! [X139 : term,X140 : term > $o,X141 : subst] : ? [X142 : term > $o] : ! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143)) | (pushprop_lem0_lthm != $true)) & (one = (d2term @ d_one)) & ((shinj = $true) | ? [X69 : term,X70 : term] : ((X69 != X70) & ((sub @ X70 @ sh) = (sub @ X69 @ sh)))) & (! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh))) | (shinj != $true)) & (axvarshift = $true) & (hoasinduction_lem3_gthm = $true) & ((d2subst @ d_id) = 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)))))))))))))) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & ((hoasinduction_lem3a_lthm = $true) | (sP35 != $true)) & ((sP35 = $true) | (hoasinduction_lem3a_lthm != $true)) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) | (axscons != $true)) & ((axscons = $true) | ? [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) != X102)) & ((hoasinduction_lem3v2_f = $true) | ? [X135 : term] : ! [X136 : subst > term > term] : ? [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) != (X136 @ X137 @ X138))) & (! [X135 : term] : ? [X136 : subst > term > term] : ! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138)) | (hoasinduction_lem3v2_f != $true)) & (! [X292 : term,X293 : term] : ((X292 = X293) | ((lam @ X293) != (lam @ X292))) | (laminj != $true)) & ((laminj = $true) | ? [X292 : term,X293 : term] : ((X292 != X293) & ((lam @ X293) = (lam @ X292)))) & (hoasinduction_lem2_gthm = $true) & (induction2lem_gthm = $true) & (! [X144 : term] : ? [X145 : subst > term > term] : ((X145 @ sh @ one) = (sub @ X144 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X144 : term] : ! [X145 : subst > term > term] : ((X145 @ sh @ one) != (sub @ X144 @ (push @ one @ sh)))) & (pushprop_lem3v2_lthm = $true) & ((hoasapinj2_lthm = $true) | (? [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 != X258) & ((ap @ (sub @ X260 @ id) @ X258) = (ap @ (sub @ X259 @ id) @ X257))) & ! [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) != (ap @ X255 @ X254)) | (X253 = X254)))) & (! [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 = X258) | ((ap @ (sub @ X260 @ id) @ X258) != (ap @ (sub @ X259 @ id) @ X257))) | ? [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) = (ap @ X255 @ X254)) & (X253 != X254)) | (hoasapinj2_lthm != $true)) & ((sP33 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP33 != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP31)) & (($true = sP31) | (hoaslaminj_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ((sP30 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP30 != $true)) & ((pushprop_lem1v2 = $true) | ? [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) != $true) & ((X522 @ X521) = $true) & ! [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) = (X523 @ X524)))) & (! [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) = $true) | ((X522 @ X521) != $true) | ? [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) != (X523 @ X524))) | (pushprop_lem1v2 != $true)) & ((hoasinduction_lem3v2a = $true) | (sP21 != $true)) & ((sP21 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP20 != $true)) & ((sP20 = $true) | (hoasinduction_lem3_lthm != $true)) & (hoasinduction_lem3v2_gthm = $true) & (? [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) != X228) | ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (((X230 @ X233) != $true) | ? [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) != (X232 @ X234)) | ((X232 @ one) = $true)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) = X228) & ? [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (((X230 @ X233) = $true) & ! [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) = (X232 @ X234)) & ((X232 @ one) != $true)))) & ! [X271 : d_subst] : (d_id = X271) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (apnotvar = $true) & ((sP17 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP17 != $true)) & (lamnotap != $true) & ((hoasinduction_lthm_3 = $true) | (sP16 != $true)) & ((sP16 = $true) | (hoasinduction_lthm_3 != $true)) & (axclos = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((hoasinduction_lem3b = $true) | ? [X73 : term] : ! [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) != (X74 @ sh @ one))) & (! [X73 : term] : ? [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) = (X74 @ sh @ one)) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X65 : term] : ! [X66 : subst > term > term] : ? [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) != (sub @ X65 @ (push @ X67 @ X68)))) & (! [X65 : term] : ? [X66 : subst > term > term] : ! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoasinduction_lem2_lthm = $true) & (! [X71 : subst] : ((comp @ X71 @ id) = X71) | (axidr != $true)) & ((axidr = $true) | ? [X71 : subst] : ((comp @ X71 @ id) != X71)) & (axassoc = $true) & (! [X495 : term > $o,X496 : term] : (((sP10 @ X495) = $true) | ? [X499 : term] : (((var @ X499) = $true) & ((X495 @ X499) != $true)) | ((X495 @ X496) = $true) | ? [X500 : term] : (((X495 @ (lam @ X500)) != $true) & ((X495 @ X500) = $true))) | (induction != $true)) & ((induction = $true) | ? [X495 : term > $o,X496 : term] : (((sP10 @ X495) != $true) & ! [X499 : term] : (((var @ X499) != $true) | ((X495 @ X499) = $true)) & ((X495 @ X496) != $true) & ! [X500 : term] : (((X495 @ (lam @ X500)) = $true) | ((X495 @ X500) != $true)))) & (hoaslamnotvar_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((sP9 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP9 != $true)) & (hoasapnotvar_lthm = $true) & (! [X266 : subst,X267 : term > $o,X268 : term] : ? [X269 : term > $o] : ! [X270 : term] : ((X269 @ X270) = (X267 @ (sub @ X270 @ (push @ X268 @ X266)))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X266 : subst,X267 : term > $o,X268 : term] : ! [X269 : term > $o] : ? [X270 : term] : ((X269 @ X270) != (X267 @ (sub @ X270 @ (push @ X268 @ X266))))) & (! [X383 : term] : ((sub @ X383 @ id) = X383) | (axvarid != $true)) & ((axvarid = $true) | ? [X383 : term] : ((sub @ X383 @ id) != X383)) & (! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) != X100)) & (hoasapinj1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((sP5 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP5 != $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X413 : term] : ? [X414 : d_term] : ((d2term @ X414) = X413) & ((axidl = $true) | ? [X72 : subst] : ((comp @ id @ X72) != X72)) & (! [X72 : subst] : ((comp @ id @ X72) = X72) | (axidl != $true)) & (hoasinduction_lem1v2 = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((axshiftcons = $true) | ? [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) != X493)) & (! [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) = X493) | (axshiftcons != $true)) & (pushprop_gthm = $true) & (induction2_gthm = $true) & ((hoasinduction_lem3 = $true) | (sP3 != $true)) & ((sP3 = $true) | (hoasinduction_lem3 != $true)) & (! [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 = X418) | ((ap @ (sub @ X417 @ id) @ X418) != (ap @ (sub @ X415 @ id) @ X416))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 != X418) & ((ap @ (sub @ X417 @ id) @ X418) = (ap @ (sub @ X415 @ id) @ X416)))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (axapp = $true)),
% 9.05/2.75 inference(flattening,[status(thm)],[f473])).
% 9.05/2.75 thf(f473,plain,(
% 9.05/2.75 (((hoasinduction_no_psi_cond = $true) | ? [X183 : term,X184 : subst > term > subst > $o] : (! [X185 : term,X186 : term] : (((X184 @ id @ X186 @ id) != $true) | ((X184 @ id @ X185 @ id) != $true) | ((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) = $true)) & ((sP58 @ X184) != $true) & ((X184 @ id @ X183 @ id) != $true))) & (! [X183 : term,X184 : subst > term > subst > $o] : (? [X185 : term,X186 : term] : (((X184 @ id @ X186 @ id) = $true) & ((X184 @ id @ X185 @ id) = $true) & ((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) != $true)) | ((sP58 @ X184) = $true) | ((X184 @ id @ X183 @ id) = $true)) | (hoasinduction_no_psi_cond != $true))) & (((sP57 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP57 != $true))) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & (((hoasinduction_no_psi_cond_lthm = $true) | (sP53 != $true)) & ((sP53 = $true) | (hoasinduction_no_psi_cond_lthm != $true))) & (pushprop_lem2v2_gthm = $true) & (((termmset_lthm = $true) | (! [X272 : term] : ((sub @ X272 @ id) = X272) & ? [X273 : term] : ((sub @ X273 @ id) != X273))) & ((? [X272 : term] : ((sub @ X272 @ id) != X272) | ! [X273 : term] : ((sub @ X273 @ id) = X273)) | (termmset_lthm != $true))) & ((! [X165 : subst > term > subst > $o] : ? [X166 : term > $o] : ! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X165 : subst > term > subst > $o] : ! [X166 : term > $o] : ? [X167 : term] : ((X165 @ id @ X167 @ id) != (X166 @ X167)))) & (axmap = $true) & (ulamvarsh = $true) & ((! [X2 : subst > term > subst > $o] : ? [X3 : term > $o] : ! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X2 : subst > term > subst > $o] : ! [X3 : term > $o] : ? [X4 : term] : ((X2 @ id @ X4 @ id) != (X3 @ X4)))) & (((? [X410 : subst] : ((comp @ X410 @ id) != X410) | ? [X409 : subst] : ((comp @ id @ X409) != X409) | (! [X411 : subst] : ((comp @ id @ X411) = X411) & ! [X412 : subst] : ((comp @ X412 @ id) = X412))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X410 : subst] : ((comp @ X410 @ id) = X410) & ! [X409 : subst] : ((comp @ id @ X409) = X409) & (? [X411 : subst] : ((comp @ id @ X411) != X411) | ? [X412 : subst] : ((comp @ X412 @ id) != X412))))) & (((sP48 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP48 != $true))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (((hoasinduction_lem3aa = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa != $true))) & (hoasinduction_lem1_gthm = $true) & (substmonoid_gthm = $true) & ! [X245 : d_term,X246 : d_term] : (((d2term @ X246) != (d2term @ X245)) | (X245 = X246)) & (ulamvarind = $true) & (pushprop_lthm_orig = $true) & ((! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | ((ap @ X261 @ X263) != (ap @ X262 @ X264))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 != X262) & ((ap @ X261 @ X263) = (ap @ X262 @ X264))))) & (hoaslamnotap_gthm = $true) & ! [X181 : d_subst,X182 : d_subst] : ((X181 = X182) | ((d2subst @ X182) != (d2subst @ X181))) & (hoaslamnotvar = $true) & (pushprop_lem1v2_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (hoasinduction_lem1_lthm = $true) & (((hoasinduction_lem3v2 = $true) | (sP46 != $true)) & ((sP46 = $true) | (hoasinduction_lem3v2 != $true))) & ! [X265 : d_term] : (d_one = X265) & (((sP44 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP44 != $true))) & ((! [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) != (ap @ X326 @ X328)) | (X325 = X328)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) = (ap @ X326 @ X328)) & (X325 != X328)))) & (lamnotvar = $true) & (axabs = $true) & (((hoasinduction = $true) | (sP42 != $true)) & ((sP42 = $true) | (hoasinduction != $true))) & ((d2subst @ d_id) = sh) & (pushprop_lem0_gthm = $true) & ((! [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192)) | (X193 = X194)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) = (ap @ (sub @ X194 @ id) @ X192)) & (X193 != X194)))) & (hoasapnotvar_gthm = $true) & (((hoaslaminj = $true) | ? [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) = (lam @ (X465 @ sh @ one))) & ((X465 @ X466 @ X464) != (X467 @ X466 @ X464)) & ! [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) = (sub @ (X465 @ X473 @ X471) @ X472)) & ! [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) = (sub @ (X467 @ X470 @ X469) @ X468)))) & (! [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) != (lam @ (X465 @ sh @ one))) | ((X465 @ X466 @ X464) = (X467 @ X466 @ X464)) | ? [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) != (sub @ (X465 @ X473 @ X471) @ X472)) | ? [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) != (sub @ (X467 @ X470 @ X469) @ X468))) | (hoaslaminj != $true))) & (((hoaslamnotap = $true) | ? [X247 : term,X248 : subst > term > term,X249 : term] : (! [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) = (sub @ (X248 @ X250 @ X252) @ X251)) & ((ap @ (sub @ X247 @ id) @ X249) = (lam @ (X248 @ sh @ one))))) & (! [X247 : term,X248 : subst > term > term,X249 : term] : (? [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) != (sub @ (X248 @ X250 @ X252) @ X251)) | ((ap @ (sub @ X247 @ id) @ X249) != (lam @ (X248 @ sh @ one)))) | (hoaslamnotap != $true))) & (hoasinduction_gthm = $true) & (pushprop_lthm = $true) & (pushprop_lem1_gthm = $true) & (((hoasapinj1_lthm = $true) | (? [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 != X408) & ((ap @ (sub @ X408 @ id) @ X406) = (ap @ (sub @ X405 @ id) @ X407))) & ! [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) != (ap @ X404 @ X402)) | (X401 = X404)) & ! [X400 : term] : ((sub @ X400 @ id) = X400))) & ((! [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 = X408) | ((ap @ (sub @ X408 @ id) @ X406) != (ap @ (sub @ X405 @ id) @ X407))) | ? [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) = (ap @ X404 @ X402)) & (X401 != X404)) | ? [X400 : term] : ((sub @ X400 @ id) != X400)) | (hoasapinj1_lthm != $true))) & ((! [X164 : term] : ((sub @ X164 @ id) = X164) | (termmset != $true)) & ((termmset = $true) | ? [X164 : term] : ((sub @ X164 @ id) != X164))) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar = $true) & (((pushprop_lem1 = $true) | ? [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (((X60 @ X61) = $true) & ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) != $true))) & (! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (((X60 @ X61) != $true) | ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) = $true)) | (pushprop_lem1 != $true))) & (((sP39 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP39 != $true))) & (((hoasinduction_lem3a = $true) | (sP37 != $true)) & ((sP37 = $true) | (hoasinduction_lem3a != $true))) & (hoasinduction_lem1 = $true) & (hoaslaminj_gthm = $true) & (pushprop = $true) & (termmset_gthm = $true) & (pushprop_lem3v2 = $true) & (((substmonoid = $true) | (? [X1 : subst] : ((comp @ id @ X1) != X1) | ? [X0 : subst] : ((comp @ X0 @ id) != X0))) & ((! [X1 : subst] : ((comp @ id @ X1) = X1) & ! [X0 : subst] : ((comp @ X0 @ id) = X0)) | (substmonoid != $true))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((hoasinduction_lem3aa_lthm = $true) | ($true != sP36)) & (($true = sP36) | (hoasinduction_lem3aa_lthm != $true))) & (ulamvar1 != $true) & (((? [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) != X121) | ! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (((X122 @ X124) != $true) | ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) = $true)) | ? [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) != X118)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) = X121) & ? [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (((X122 @ X124) = $true) & ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) != $true)) & ! [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) = X118)))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (((pushprop_lem0_lthm = $true) | ? [X139 : term,X140 : term > $o,X141 : subst] : ! [X142 : term > $o] : ? [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) != (X142 @ X143))) & (! [X139 : term,X140 : term > $o,X141 : subst] : ? [X142 : term > $o] : ! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143)) | (pushprop_lem0_lthm != $true))) & (one = (d2term @ d_one)) & (((shinj = $true) | ? [X69 : term,X70 : term] : ((X69 != X70) & ((sub @ X70 @ sh) = (sub @ X69 @ sh)))) & (! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh))) | (shinj != $true))) & (axvarshift = $true) & (hoasinduction_lem3_gthm = $true) & ((d2subst @ d_id) = 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)))))))))))))) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & (((hoasinduction_lem3a_lthm = $true) | (sP35 != $true)) & ((sP35 = $true) | (hoasinduction_lem3a_lthm != $true))) & ((! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) | (axscons != $true)) & ((axscons = $true) | ? [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) != X102))) & (((hoasinduction_lem3v2_f = $true) | ? [X135 : term] : ! [X136 : subst > term > term] : ? [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) != (X136 @ X137 @ X138))) & (! [X135 : term] : ? [X136 : subst > term > term] : ! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138)) | (hoasinduction_lem3v2_f != $true))) & ((! [X292 : term,X293 : term] : ((X292 = X293) | ((lam @ X293) != (lam @ X292))) | (laminj != $true)) & ((laminj = $true) | ? [X292 : term,X293 : term] : ((X292 != X293) & ((lam @ X293) = (lam @ X292))))) & (hoasinduction_lem2_gthm = $true) & (induction2lem_gthm = $true) & ((! [X144 : term] : ? [X145 : subst > term > term] : ((X145 @ sh @ one) = (sub @ X144 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X144 : term] : ! [X145 : subst > term > term] : ((X145 @ sh @ one) != (sub @ X144 @ (push @ one @ sh))))) & (pushprop_lem3v2_lthm = $true) & (((hoasapinj2_lthm = $true) | (? [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 != X258) & ((ap @ (sub @ X260 @ id) @ X258) = (ap @ (sub @ X259 @ id) @ X257))) & ! [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) != (ap @ X255 @ X254)) | (X253 = X254)))) & ((! [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 = X258) | ((ap @ (sub @ X260 @ id) @ X258) != (ap @ (sub @ X259 @ id) @ X257))) | ? [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) = (ap @ X255 @ X254)) & (X253 != X254))) | (hoasapinj2_lthm != $true))) & (((sP33 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP33 != $true))) & (((hoaslaminj_lthm = $true) | ($true != sP31)) & (($true = sP31) | (hoaslaminj_lthm != $true))) & (hoaslamnotvar_lthm = $true) & (((sP30 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP30 != $true))) & (((pushprop_lem1v2 = $true) | ? [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) != $true) & ((X522 @ X521) = $true) & ! [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) = (X523 @ X524)))) & (! [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) = $true) | ((X522 @ X521) != $true) | ? [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) != (X523 @ X524))) | (pushprop_lem1v2 != $true))) & (((hoasinduction_lem3v2a = $true) | (sP21 != $true)) & ((sP21 = $true) | (hoasinduction_lem3v2a != $true))) & (((hoasinduction_lem3_lthm = $true) | (sP20 != $true)) & ((sP20 = $true) | (hoasinduction_lem3_lthm != $true))) & (hoasinduction_lem3v2_gthm = $true) & (((? [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) != X228) | ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (((X230 @ X233) != $true) | ? [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) != (X232 @ X234)) | ((X232 @ one) = $true))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) = X228) & ? [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (((X230 @ X233) = $true) & ! [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) = (X232 @ X234)) & ((X232 @ one) != $true))))) & ! [X271 : d_subst] : (d_id = X271) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (apnotvar = $true) & (((sP17 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP17 != $true))) & (lamnotap != $true) & (((hoasinduction_lthm_3 = $true) | (sP16 != $true)) & ((sP16 = $true) | (hoasinduction_lthm_3 != $true))) & (axclos = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (((hoasinduction_lem3b = $true) | ? [X73 : term] : ! [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) != (X74 @ sh @ one))) & (! [X73 : term] : ? [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) = (X74 @ sh @ one)) | (hoasinduction_lem3b != $true))) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X65 : term] : ! [X66 : subst > term > term] : ? [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) != (sub @ X65 @ (push @ X67 @ X68)))) & (! [X65 : term] : ? [X66 : subst > term > term] : ! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68))) | (hoasinduction_lem3v2_f_lthm != $true))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoasinduction_lem2_lthm = $true) & ((! [X71 : subst] : ((comp @ X71 @ id) = X71) | (axidr != $true)) & ((axidr = $true) | ? [X71 : subst] : ((comp @ X71 @ id) != X71))) & (axassoc = $true) & ((! [X495 : term > $o,X496 : term] : (((sP10 @ X495) = $true) | ? [X499 : term] : (((var @ X499) = $true) & ((X495 @ X499) != $true)) | ((X495 @ X496) = $true) | ? [X500 : term] : (((X495 @ (lam @ X500)) != $true) & ((X495 @ X500) = $true))) | (induction != $true)) & ((induction = $true) | ? [X495 : term > $o,X496 : term] : (((sP10 @ X495) != $true) & ! [X499 : term] : (((var @ X499) != $true) | ((X495 @ X499) = $true)) & ((X495 @ X496) != $true) & ! [X500 : term] : (((X495 @ (lam @ X500)) = $true) | ((X495 @ X500) != $true))))) & (hoaslamnotvar_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & (((sP9 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP9 != $true))) & (hoasapnotvar_lthm = $true) & ((! [X266 : subst,X267 : term > $o,X268 : term] : ? [X269 : term > $o] : ! [X270 : term] : ((X269 @ X270) = (X267 @ (sub @ X270 @ (push @ X268 @ X266)))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X266 : subst,X267 : term > $o,X268 : term] : ! [X269 : term > $o] : ? [X270 : term] : ((X269 @ X270) != (X267 @ (sub @ X270 @ (push @ X268 @ X266)))))) & ((! [X383 : term] : ((sub @ X383 @ id) = X383) | (axvarid != $true)) & ((axvarid = $true) | ? [X383 : term] : ((sub @ X383 @ id) != X383))) & ((! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) != X100))) & (hoasapinj1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (((sP5 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP5 != $true))) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X413 : term] : ? [X414 : d_term] : ((d2term @ X414) = X413) & (((axidl = $true) | ? [X72 : subst] : ((comp @ id @ X72) != X72)) & (! [X72 : subst] : ((comp @ id @ X72) = X72) | (axidl != $true))) & (hoasinduction_lem1v2 = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (((axshiftcons = $true) | ? [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) != X493)) & (! [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) = X493) | (axshiftcons != $true))) & (pushprop_gthm = $true) & (induction2_gthm = $true) & (((hoasinduction_lem3 = $true) | (sP3 != $true)) & ((sP3 = $true) | (hoasinduction_lem3 != $true))) & ((! [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 = X418) | ((ap @ (sub @ X417 @ id) @ X418) != (ap @ (sub @ X415 @ id) @ X416))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 != X418) & ((ap @ (sub @ X417 @ id) @ X418) = (ap @ (sub @ X415 @ id) @ X416))))) & (hoasinduction_lem1v2_gthm = $true) & (((hoasinduction_lem3v2_lthm = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem3v2_lthm != $true))) & (axapp = $true)),
% 9.05/2.75 inference(nnf_transformation,[status(thm)],[f73])).
% 9.05/2.75 thf(f73,plain,(
% 9.05/2.75 ((hoasinduction_no_psi_cond = $true) <=> ! [X183 : term,X184 : subst > term > subst > $o] : (? [X185 : term,X186 : term] : (((X184 @ id @ X186 @ id) = $true) & ((X184 @ id @ X185 @ id) = $true) & ((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) != $true)) | ((sP58 @ X184) = $true) | ((X184 @ id @ X183 @ id) = $true))) & ((sP57 = $true) <=> (induction2_lthm = $true)) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (sP53 = $true)) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) <=> (? [X272 : term] : ((sub @ X272 @ id) != X272) | ! [X273 : term] : ((sub @ X273 @ id) = X273))) & (! [X165 : subst > term > subst > $o] : ? [X166 : term > $o] : ! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) <=> (hoasinduction_lem0 = $true)) & (axmap = $true) & (ulamvarsh = $true) & (! [X2 : subst > term > subst > $o] : ? [X3 : term > $o] : ! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) <=> (hoasinduction_lem0_lthm = $true)) & ((? [X410 : subst] : ((comp @ X410 @ id) != X410) | ? [X409 : subst] : ((comp @ id @ X409) != X409) | (! [X411 : subst] : ((comp @ id @ X411) = X411) & ! [X412 : subst] : ((comp @ X412 @ id) = X412))) <=> (substmonoid_lthm = $true)) & ((sP48 = $true) <=> (hoasinduction_lem3aaa = $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3aa = $true) <=> (sP47 = $true)) & (hoasinduction_lem1_gthm = $true) & (substmonoid_gthm = $true) & ! [X245 : d_term,X246 : d_term] : (((d2term @ X246) != (d2term @ X245)) | (X245 = X246)) & (ulamvarind = $true) & (pushprop_lthm_orig = $true) & (! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | ((ap @ X261 @ X263) != (ap @ X262 @ X264))) <=> (apinj1 = $true)) & (hoaslamnotap_gthm = $true) & ! [X181 : d_subst,X182 : d_subst] : ((X181 = X182) | ((d2subst @ X182) != (d2subst @ X181))) & (hoaslamnotvar = $true) & (pushprop_lem1v2_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> (sP46 = $true)) & ! [X265 : d_term] : (d_one = X265) & ((sP44 = $true) <=> (hoasinduction_lem2v2 = $true)) & (! [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) != (ap @ X326 @ X328)) | (X325 = X328)) <=> (apinj2 = $true)) & (lamnotvar = $true) & (axabs = $true) & ((hoasinduction = $true) <=> (sP42 = $true)) & ((d2subst @ d_id) = sh) & (pushprop_lem0_gthm = $true) & (! [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192)) | (X193 = X194)) <=> (hoasapinj1 = $true)) & (hoasapnotvar_gthm = $true) & ((hoaslaminj = $true) <=> ! [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) != (lam @ (X465 @ sh @ one))) | ((X465 @ X466 @ X464) = (X467 @ X466 @ X464)) | ? [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) != (sub @ (X465 @ X473 @ X471) @ X472)) | ? [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) != (sub @ (X467 @ X470 @ X469) @ X468)))) & ((hoaslamnotap = $true) <=> ! [X247 : term,X248 : subst > term > term,X249 : term] : (? [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) != (sub @ (X248 @ X250 @ X252) @ X251)) | ((ap @ (sub @ X247 @ id) @ X249) != (lam @ (X248 @ sh @ one))))) & (hoasinduction_gthm = $true) & (pushprop_lthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 = X408) | ((ap @ (sub @ X408 @ id) @ X406) != (ap @ (sub @ X405 @ id) @ X407))) | ? [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) = (ap @ X404 @ X402)) & (X401 != X404)) | ? [X400 : term] : ((sub @ X400 @ id) != X400))) & (! [X164 : term] : ((sub @ X164 @ id) = X164) <=> (termmset = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar = $true) & ((pushprop_lem1 = $true) <=> ! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (((X60 @ X61) != $true) | ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) = $true))) & ((sP39 = $true) <=> (induction2 = $true)) & ((hoasinduction_lem3a = $true) <=> (sP37 = $true)) & (hoasinduction_lem1 = $true) & (hoaslaminj_gthm = $true) & (pushprop = $true) & (termmset_gthm = $true) & (pushprop_lem3v2 = $true) & ((substmonoid = $true) <=> (! [X1 : subst] : ((comp @ id @ X1) = X1) & ! [X0 : subst] : ((comp @ X0 @ id) = X0))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem3aa_lthm = $true) <=> ($true = sP36)) & (ulamvar1 != $true) & ((? [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) != X121) | ! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (((X122 @ X124) != $true) | ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) = $true)) | ? [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) != X118)) <=> (pushprop_lem1_lthm = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((pushprop_lem0_lthm = $true) <=> ! [X139 : term,X140 : term > $o,X141 : subst] : ? [X142 : term > $o] : ! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143))) & (one = (d2term @ d_one)) & ((shinj = $true) <=> ! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh)))) & (axvarshift = $true) & (hoasinduction_lem3_gthm = $true) & ((d2subst @ d_id) = 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)))))))))))))) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & ((hoasinduction_lem3a_lthm = $true) <=> (sP35 = $true)) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X135 : term] : ? [X136 : subst > term > term] : ! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138))) & (! [X292 : term,X293 : term] : ((X292 = X293) | ((lam @ X293) != (lam @ X292))) <=> (laminj = $true)) & (hoasinduction_lem2_gthm = $true) & (induction2lem_gthm = $true) & (! [X144 : term] : ? [X145 : subst > term > term] : ((X145 @ sh @ one) = (sub @ X144 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 = X258) | ((ap @ (sub @ X260 @ id) @ X258) != (ap @ (sub @ X259 @ id) @ X257))) | ? [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) = (ap @ X255 @ X254)) & (X253 != X254)))) & ((sP33 = $true) <=> (induction2lem = $true)) & ((hoaslaminj_lthm = $true) <=> ($true = sP31)) & (hoaslamnotvar_lthm = $true) & ((sP30 = $true) <=> (hoasinduction_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) = $true) | ((X522 @ X521) != $true) | ? [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) != (X523 @ X524)))) & ((hoasinduction_lem3v2a = $true) <=> (sP21 = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (sP20 = $true)) & (hoasinduction_lem3v2_gthm = $true) & ((? [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) != X228) | ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (((X230 @ X233) != $true) | ? [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) != (X232 @ X234)) | ((X232 @ one) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & ! [X271 : d_subst] : (d_id = X271) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (apnotvar = $true) & ((sP17 = $true) <=> (hoasinduction_lem3v2a_lthm = $true)) & (lamnotap != $true) & ((hoasinduction_lthm_3 = $true) <=> (sP16 = $true)) & (axclos = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((hoasinduction_lem3b = $true) <=> ! [X73 : term] : ? [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) = (X74 @ sh @ one))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X65 : term] : ? [X66 : subst > term > term] : ! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68)))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoasinduction_lem2_lthm = $true) & (! [X71 : subst] : ((comp @ X71 @ id) = X71) <=> (axidr = $true)) & (axassoc = $true) & (! [X495 : term > $o,X496 : term] : (((sP10 @ X495) = $true) | ? [X499 : term] : (((var @ X499) = $true) & ((X495 @ X499) != $true)) | ((X495 @ X496) = $true) | ? [X500 : term] : (((X495 @ (lam @ X500)) != $true) & ((X495 @ X500) = $true))) <=> (induction = $true)) & (hoaslamnotvar_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((sP9 = $true) <=> (induction2lem_lthm = $true)) & (hoasapnotvar_lthm = $true) & (! [X266 : subst,X267 : term > $o,X268 : term] : ? [X269 : term > $o] : ! [X270 : term] : ((X269 @ X270) = (X267 @ (sub @ X270 @ (push @ X268 @ X266)))) <=> (pushprop_lem0 = $true)) & (! [X383 : term] : ((sub @ X383 @ id) = X383) <=> (axvarid = $true)) & (! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100) <=> (axvarcons = $true)) & (hoasapinj1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((sP5 = $true) <=> (hoasinduction_lem2 = $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X413 : term] : ? [X414 : d_term] : ((d2term @ X414) = X413) & ((axidl = $true) <=> ! [X72 : subst] : ((comp @ id @ X72) = X72)) & (hoasinduction_lem1v2 = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((axshiftcons = $true) <=> ! [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) = X493)) & (pushprop_gthm = $true) & (induction2_gthm = $true) & ((hoasinduction_lem3 = $true) <=> (sP3 = $true)) & (! [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 = X418) | ((ap @ (sub @ X417 @ id) @ X418) != (ap @ (sub @ X415 @ id) @ X416))) <=> (hoasapinj2 = $true)) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (sP1 = $true)) & (axapp = $true)),
% 9.05/2.75 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])).
% 9.05/2.75 thf(f14,plain,(
% 9.05/2.75 ! [X503 : subst > term > subst > $o] : (((sP0 @ X503) = $true) <=> ? [X510 : subst > term > term] : (! [X512 : subst,X513 : term,X514 : subst] : ((sub @ (X510 @ X514 @ X513) @ X512) = (X510 @ (comp @ X514 @ X512) @ (sub @ X513 @ X512))) & ! [X511 : term] : (((X503 @ id @ (X510 @ id @ X511) @ id) = $true) | ((X503 @ id @ X511 @ id) != $true)) & ((X503 @ id @ (lam @ (X510 @ sh @ one)) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f15,plain,(
% 9.05/2.75 (sP1 = $true) <=> (! [X502 : term > $o,X503 : subst > term > subst > $o,X504 : term] : (? [X515 : subst,X516 : subst,X517 : subst,X518 : term] : (((X503 @ X516 @ X518 @ (comp @ X515 @ X517)) != $true) & ((X503 @ (comp @ X516 @ X515) @ (sub @ X518 @ X515) @ X517) = $true)) | ? [X519 : term] : (((X502 @ (sub @ X504 @ (push @ X519 @ id))) != $true) & ((X502 @ X519) = $true)) | ((sP0 @ X503) = $true) | ? [X506 : subst,X507 : subst,X508 : subst,X509 : term] : (((X503 @ (comp @ X506 @ X507) @ (sub @ X509 @ X507) @ X508) != $true) & ((X503 @ X506 @ X509 @ (comp @ X507 @ X508)) = $true)) | ((X502 @ (lam @ X504)) = $true) | ? [X505 : term] : ((X503 @ id @ X505 @ id) != (X502 @ X505))) | ? [X501 : term] : ((sub @ X501 @ id) != X501))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f16,plain,(
% 9.05/2.75 ! [X385 : subst > term > subst > $o] : (((sP2 @ X385) = $true) <=> ? [X391 : subst > term > term] : (! [X395 : term] : (((X385 @ id @ X395 @ id) != $true) | ((X385 @ id @ (X391 @ id @ X395) @ id) = $true)) & ! [X392 : subst,X393 : subst,X394 : term] : ((sub @ (X391 @ X393 @ X394) @ X392) = (X391 @ (comp @ X393 @ X392) @ (sub @ X394 @ X392))) & ((X385 @ id @ (lam @ (X391 @ sh @ one)) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f17,plain,(
% 9.05/2.75 (sP3 = $true) <=> ! [X384 : term,X385 : subst > term > subst > $o] : (? [X386 : subst,X387 : subst,X388 : subst,X389 : term] : (((X385 @ (comp @ X386 @ X388) @ (sub @ X389 @ X388) @ X387) != $true) & ((X385 @ X386 @ X389 @ (comp @ X388 @ X387)) = $true)) | ((sP2 @ X385) = $true) | ? [X390 : term] : (((X385 @ id @ X390 @ id) = $true) & ((X385 @ id @ (sub @ X384 @ (push @ X390 @ id)) @ id) != $true)) | ? [X396 : subst,X397 : subst,X398 : term,X399 : subst] : (((X385 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X399) = $true) & ((X385 @ X396 @ X398 @ (comp @ X397 @ X399)) != $true)) | ((X385 @ id @ (lam @ X384) @ id) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f18,plain,(
% 9.05/2.75 ! [X170 : subst > term > subst > $o] : (((sP4 @ X170) = $true) <=> ? [X171 : term,X172 : term] : (((X170 @ id @ (ap @ (sub @ X171 @ id) @ X172) @ id) != $true) & ((X170 @ id @ X172 @ id) = $true) & ((X170 @ id @ X171 @ id) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f19,plain,(
% 9.05/2.75 (sP5 = $true) <=> ! [X168 : term,X169 : term,X170 : subst > term > subst > $o] : (((X170 @ id @ (ap @ X168 @ X169) @ id) = $true) | ((X170 @ id @ X169 @ id) != $true) | ((X170 @ id @ X168 @ id) != $true) | ? [X177 : subst,X178 : subst,X179 : term,X180 : subst] : (((X170 @ X178 @ X179 @ (comp @ X177 @ X180)) = $true) & ((X170 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177) @ X180) != $true)) | ? [X173 : subst,X174 : subst,X175 : term,X176 : subst] : (((X170 @ (comp @ X176 @ X173) @ (sub @ X175 @ X173) @ X174) = $true) & ((X170 @ X176 @ X175 @ (comp @ X173 @ X174)) != $true)) | ((sP4 @ X170) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f20,plain,(
% 9.05/2.75 ! [X479 : term > $o] : (((sP6 @ X479) = $true) <=> ! [X483 : term,X484 : term] : (((X479 @ (ap @ X484 @ X483)) = $true) | ((X479 @ X483) != $true) | ((X479 @ X484) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f21,plain,(
% 9.05/2.75 ! [X487 : term > $o] : (((sP7 @ X487) = $true) <=> ? [X491 : term,X492 : term] : (((X487 @ X491) = $true) & ((X487 @ X492) = $true) & ((X487 @ (ap @ X491 @ X492)) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f22,plain,(
% 9.05/2.75 (sP8 = $true) <=> ! [X485 : term,X486 : subst,X487 : term > $o] : (? [X490 : term] : (((X487 @ (sub @ X490 @ X486)) != $true) & ((var @ X490) = $true)) | ((sP7 @ X487) = $true) | ? [X488 : term] : (! [X489 : term] : (((X487 @ X489) != $true) | ((X487 @ (sub @ X488 @ (push @ X489 @ id))) = $true)) & ((X487 @ (lam @ X488)) != $true)) | ((X487 @ (sub @ X485 @ X486)) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f23,plain,(
% 9.05/2.75 (sP9 = $true) <=> ((sP8 = $true) | ? [X476 : subst,X477 : term] : ((comp @ sh @ (push @ X477 @ X476)) != X476) | ? [X474 : subst,X475 : term] : ((sub @ one @ (push @ X475 @ X474)) != X475) | ? [X478 : subst] : ((comp @ X478 @ id) != X478) | ? [X479 : term > $o,X480 : term] : (! [X482 : term] : (((X479 @ X482) = $true) | ((var @ X482) != $true)) & ((X479 @ X480) != $true) & ! [X481 : term] : (((X479 @ X481) != $true) | ((X479 @ (lam @ X481)) = $true)) & ((sP6 @ X479) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f24,plain,(
% 9.05/2.75 ! [X495 : term > $o] : (((sP10 @ X495) = $true) <=> ? [X497 : term,X498 : term] : (((X495 @ (ap @ X497 @ X498)) != $true) & ((X495 @ X497) = $true) & ((X495 @ X498) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f25,plain,(
% 9.05/2.75 ! [X42 : subst > term > subst > $o] : (((sP11 @ X42) = $true) <=> ? [X54 : term,X55 : term] : (((X42 @ id @ X55 @ id) = $true) & ((X42 @ id @ X54 @ id) = $true) & ((X42 @ id @ (ap @ (sub @ X54 @ id) @ X55) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f26,plain,(
% 9.05/2.75 ! [X42 : subst > term > subst > $o] : (((sP12 @ X42) = $true) <=> ? [X44 : subst > term > term] : (((X42 @ id @ (lam @ (X44 @ sh @ one)) @ id) != $true) & ! [X48 : term] : (((X42 @ id @ (X44 @ id @ X48) @ id) = $true) | ((X42 @ id @ X48 @ id) != $true)) & ! [X45 : subst,X46 : subst,X47 : term] : ((sub @ (X44 @ X45 @ X47) @ X46) = (X44 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46)))))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f27,plain,(
% 9.05/2.75 ! [X25 : term > $o] : (((sP13 @ X25) = $true) <=> ! [X26 : term,X27 : term] : (((X25 @ (ap @ X27 @ X26)) = $true) | ((X25 @ X27) != $true) | ((X25 @ X26) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f28,plain,(
% 9.05/2.75 (sP14 = $true) <=> ! [X42 : subst > term > subst > $o,X43 : term] : (((sP12 @ X42) = $true) | ? [X50 : subst,X51 : term,X52 : subst,X53 : subst] : (((X42 @ (comp @ X53 @ X52) @ (sub @ X51 @ X52) @ X50) != $true) & ((X42 @ X53 @ X51 @ (comp @ X52 @ X50)) = $true)) | ? [X49 : term] : (((X42 @ id @ X49 @ id) != $true) & ((var @ (sub @ X49 @ id)) = $true)) | ((X42 @ id @ X43 @ id) = $true) | ((sP11 @ X42) = $true) | ? [X56 : subst,X57 : subst,X58 : subst,X59 : term] : (((X42 @ X58 @ X59 @ (comp @ X57 @ X56)) != $true) & ((X42 @ (comp @ X58 @ X57) @ (sub @ X59 @ X57) @ X56) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f29,plain,(
% 9.05/2.75 (sP15 = $true) <=> ? [X32 : term > $o,X33 : subst > term > subst > $o,X34 : term] : (! [X35 : term] : (((X32 @ (sub @ X34 @ (push @ X35 @ id))) = $true) | ((X32 @ X35) != $true)) & ((X32 @ (lam @ X34)) != $true) & ! [X41 : term] : ((X33 @ id @ X41 @ id) = (X32 @ X41)) & ! [X36 : subst > term > term] : (((X33 @ id @ (lam @ (X36 @ sh @ one)) @ id) = $true) | ? [X38 : subst,X39 : term,X40 : subst] : ((X36 @ (comp @ X38 @ X40) @ (sub @ X39 @ X40)) != (sub @ (X36 @ X38 @ X39) @ X40)) | ? [X37 : term] : (((X33 @ id @ X37 @ id) = $true) & ((X33 @ id @ (X36 @ id @ X37) @ id) != $true))))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f30,plain,(
% 9.05/2.75 (sP16 = $true) <=> ((sP15 = $true) | ? [X21 : subst > term > subst > $o] : ! [X22 : term > $o] : ? [X23 : term] : ((X21 @ id @ X23 @ id) != (X22 @ X23)) | ? [X24 : term,X25 : term > $o] : (! [X30 : term] : (((var @ X30) != $true) | ((X25 @ X30) = $true)) & ((X25 @ X24) != $true) & ((sP13 @ X25) = $true) & ! [X28 : term] : (((X25 @ (lam @ X28)) = $true) | ? [X29 : term] : (((X25 @ (sub @ X28 @ (push @ X29 @ id))) != $true) & ((X25 @ X29) = $true)))) | (sP14 = $true) | ? [X31 : term] : ((sub @ X31 @ id) != X31))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f31,plain,(
% 9.05/2.75 (sP17 = $true) <=> (! [X108 : term,X109 : subst > term > subst > $o,X110 : term > $o] : (? [X111 : term] : (((X110 @ X111) = $true) & ($true != (X110 @ (sub @ X108 @ (push @ X111 @ id))))) | ((X110 @ (lam @ X108)) = $true) | ? [X113 : subst > term > term] : (((X109 @ id @ (lam @ (X113 @ sh @ one)) @ id) != $true) & ! [X115 : subst,X116 : term,X117 : subst] : ((X113 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117)) = (sub @ (X113 @ X115 @ X116) @ X117)) & ! [X114 : term] : (((X109 @ id @ X114 @ id) != $true) | ((X109 @ id @ (X113 @ id @ X114) @ id) = $true))) | ? [X112 : term] : ((X109 @ id @ X112 @ id) != (X110 @ X112))) | ? [X103 : term] : ! [X104 : subst > term > term] : ? [X105 : subst,X106 : term] : ((sub @ X103 @ (push @ X106 @ X105)) != (X104 @ X105 @ X106)) | ? [X107 : term] : ((sub @ X107 @ id) != X107))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f32,plain,(
% 9.05/2.75 ! [X437 : subst > term > subst > $o] : (((sP18 @ X437) = $true) <=> ? [X438 : subst > term > term] : (((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id) != $true) & ! [X439 : term] : (((X437 @ id @ (X438 @ id @ X439) @ id) = $true) | ((X437 @ id @ X439 @ id) != $true)) & ! [X440 : term,X441 : subst,X442 : subst] : ((sub @ (X438 @ X441 @ X440) @ X442) = (X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442)))))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f33,plain,(
% 9.05/2.75 (sP19 = $true) <=> ? [X428 : term,X429 : subst > term > subst > $o] : (! [X431 : subst > term > term] : (? [X432 : term] : (((X429 @ id @ X432 @ id) = $true) & ((X429 @ id @ (X431 @ id @ X432) @ id) != $true)) | ((X429 @ id @ (lam @ (X431 @ sh @ one)) @ id) = $true) | ? [X433 : term,X434 : subst,X435 : subst] : ((X431 @ (comp @ X435 @ X434) @ (sub @ X433 @ X434)) != (sub @ (X431 @ X435 @ X433) @ X434))) & ! [X430 : term] : (((X429 @ id @ X430 @ id) != $true) | ((X429 @ id @ (sub @ X428 @ (push @ X430 @ id)) @ id) = $true)) & ((X429 @ id @ (lam @ (sub @ X428 @ (push @ one @ sh))) @ id) != $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f34,plain,(
% 9.05/2.75 (sP20 = $true) <=> (! [X436 : term,X437 : subst > term > subst > $o] : (? [X443 : term,X444 : subst,X445 : subst,X446 : subst] : (((X437 @ (comp @ X446 @ X445) @ (sub @ X443 @ X445) @ X444) = $true) & ($true != (X437 @ X446 @ X443 @ (comp @ X445 @ X444)))) | ((X437 @ id @ (lam @ X436) @ id) = $true) | ((sP18 @ X437) = $true) | ? [X448 : subst,X449 : subst,X450 : subst,X451 : term] : (((X437 @ (comp @ X450 @ X449) @ (sub @ X451 @ X449) @ X448) != $true) & ((X437 @ X450 @ X451 @ (comp @ X449 @ X448)) = $true)) | ? [X447 : term] : (((X437 @ id @ (sub @ X436 @ (push @ X447 @ id)) @ id) != $true) & ((X437 @ id @ X447 @ id) = $true))) | ? [X427 : term] : ((sub @ X427 @ id) != X427) | (sP19 = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f35,plain,(
% 9.05/2.75 (sP21 = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term,X237 : term > $o] : (? [X244 : term] : ((X237 @ X244) != (X235 @ id @ X244 @ id)) | ($true = (X237 @ (lam @ X236))) | ? [X239 : subst > term > term] : (! [X240 : subst,X241 : term,X242 : subst] : ((X239 @ (comp @ X242 @ X240) @ (sub @ X241 @ X240)) = (sub @ (X239 @ X242 @ X241) @ X240)) & ! [X243 : term] : (((X235 @ id @ X243 @ id) != $true) | ((X235 @ id @ (X239 @ id @ X243) @ id) = $true)) & ((X235 @ id @ (lam @ (X239 @ sh @ one)) @ id) != $true)) | ? [X238 : term] : (((X237 @ X238) = $true) & ((X237 @ (sub @ X236 @ (push @ X238 @ id))) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f36,plain,(
% 9.05/2.75 ! [X337 : subst > term > subst > $o] : (((sP22 @ X337) = $true) <=> ! [X343 : term,X344 : term] : (((X337 @ id @ X344 @ id) != $true) | ((X337 @ id @ (ap @ (sub @ X343 @ id) @ X344) @ id) = $true) | ((X337 @ id @ X343 @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f37,plain,(
% 9.05/2.75 ! [X329 : term > $o] : (((sP23 @ X329) = $true) <=> ! [X332 : term,X333 : term] : (((X329 @ (ap @ X332 @ X333)) = $true) | ((X329 @ X332) != $true) | ((X329 @ X333) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f38,plain,(
% 9.05/2.75 ! [X366 : subst > term > subst > $o] : (((sP24 @ X366) = $true) <=> ? [X372 : term,X373 : term] : (((X366 @ id @ (ap @ (sub @ X373 @ id) @ X372) @ id) != $true) & ((X366 @ id @ X372 @ id) = $true) & ((X366 @ id @ X373 @ id) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f39,plain,(
% 9.05/2.75 ! [X366 : subst > term > subst > $o] : (((sP25 @ X366) = $true) <=> ? [X374 : subst > term > term] : (! [X375 : subst,X376 : term,X377 : subst] : ((X374 @ (comp @ X377 @ X375) @ (sub @ X376 @ X375)) = (sub @ (X374 @ X377 @ X376) @ X375)) & ! [X378 : term] : (((X366 @ id @ X378 @ id) != $true) | ((X366 @ id @ (X374 @ id @ X378) @ id) = $true)) & ((X366 @ id @ (lam @ (X374 @ sh @ one)) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f40,plain,(
% 9.05/2.75 ! [X349 : subst > term > subst > $o] : (((sP26 @ X349) = $true) <=> ! [X355 : subst > term > term] : (((X349 @ id @ (lam @ (X355 @ sh @ one)) @ id) = $true) | ? [X357 : subst,X358 : subst,X359 : term] : ((sub @ (X355 @ X358 @ X359) @ X357) != (X355 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357))) | ? [X356 : term] : (((X349 @ id @ X356 @ id) = $true) & ((X349 @ id @ (X355 @ id @ X356) @ id) != $true))))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f41,plain,(
% 9.05/2.75 (sP27 = $true) <=> ? [X349 : subst > term > subst > $o,X350 : term] : (! [X361 : subst,X362 : subst,X363 : term,X364 : subst] : (((X349 @ X361 @ X363 @ (comp @ X362 @ X364)) != $true) | ((X349 @ (comp @ X361 @ X362) @ (sub @ X363 @ X362) @ X364) = $true)) & ! [X351 : subst,X352 : subst,X353 : term,X354 : subst] : (((X349 @ (comp @ X351 @ X352) @ (sub @ X353 @ X352) @ X354) != $true) | ((X349 @ X351 @ X353 @ (comp @ X352 @ X354)) = $true)) & ((X349 @ id @ (lam @ X350) @ id) != $true) & ((sP26 @ X349) = $true) & ! [X360 : term] : (((X349 @ id @ (sub @ X350 @ (push @ X360 @ id)) @ id) = $true) | ((X349 @ id @ X360 @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f42,plain,(
% 9.05/2.75 (sP28 = $true) <=> ! [X365 : term,X366 : subst > term > subst > $o] : (? [X368 : subst,X369 : subst,X370 : subst,X371 : term] : (((X366 @ (comp @ X369 @ X370) @ (sub @ X371 @ X370) @ X368) != $true) & ((X366 @ X369 @ X371 @ (comp @ X370 @ X368)) = $true)) | ((sP25 @ X366) = $true) | ? [X367 : term] : (((X366 @ id @ X367 @ id) != $true) & ((var @ (sub @ X367 @ id)) = $true)) | ((sP24 @ X366) = $true) | ? [X379 : subst,X380 : subst,X381 : term,X382 : subst] : (((X366 @ X380 @ X381 @ (comp @ X379 @ X382)) != $true) & ((X366 @ (comp @ X380 @ X379) @ (sub @ X381 @ X379) @ X382) = $true)) | ((X366 @ id @ X365 @ id) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f43,plain,(
% 9.05/2.75 (sP29 = $true) <=> ? [X336 : term,X337 : subst > term > subst > $o,X338 : term] : (! [X345 : subst,X346 : term,X347 : subst,X348 : subst] : (((X337 @ (comp @ X348 @ X347) @ (sub @ X346 @ X347) @ X345) != $true) | ((X337 @ X348 @ X346 @ (comp @ X347 @ X345)) = $true)) & ! [X339 : subst,X340 : subst,X341 : term,X342 : subst] : (((X337 @ (comp @ X342 @ X339) @ (sub @ X341 @ X339) @ X340) = $true) | ((X337 @ X342 @ X341 @ (comp @ X339 @ X340)) != $true)) & ((X337 @ id @ X338 @ id) = $true) & ((X337 @ id @ (ap @ X338 @ X336) @ id) != $true) & ((sP22 @ X337) = $true) & ((X337 @ id @ X336 @ id) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f44,plain,(
% 9.05/2.75 (sP30 = $true) <=> ((sP27 = $true) | (sP28 = $true) | ? [X329 : term > $o,X330 : term] : (((X329 @ X330) != $true) & ((sP23 @ X329) = $true) & ! [X331 : term] : (($true = (X329 @ X331)) | ((var @ X331) != $true)) & ! [X334 : term] : (? [X335 : term] : (((X329 @ X335) = $true) & ((X329 @ (sub @ X334 @ (push @ X335 @ id))) != $true)) | ((X329 @ (lam @ X334)) = $true))) | (sP29 = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f45,plain,(
% 9.05/2.75 ($true = sP31) <=> (? [X9 : term,X10 : term] : (((lam @ X10) = (lam @ X9)) & (X9 != X10)) | ! [X11 : subst > term > term,X12 : subst > term > term,X13 : term,X14 : subst] : (((lam @ (X11 @ sh @ one)) != (lam @ (X12 @ sh @ one))) | ? [X15 : term,X16 : subst,X17 : subst] : ((sub @ (X11 @ X17 @ X15) @ X16) != (X11 @ (comp @ X17 @ X16) @ (sub @ X15 @ X16))) | ? [X18 : subst,X19 : term,X20 : subst] : ((sub @ (X12 @ X20 @ X19) @ X18) != (X12 @ (comp @ X20 @ X18) @ (sub @ X19 @ X18))) | ((X11 @ X14 @ X13) = (X12 @ X14 @ X13))) | ? [X5 : term,X6 : subst] : ((sub @ one @ (push @ X5 @ X6)) != X5) | ? [X7 : term,X8 : subst] : ((comp @ sh @ (push @ X7 @ X8)) != X8))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f46,plain,(
% 9.05/2.75 ! [X318 : term > $o] : (((sP32 @ X318) = $true) <=> ? [X322 : term,X323 : term] : (((X318 @ X322) = $true) & ((X318 @ (ap @ X323 @ X322)) != $true) & ((X318 @ X323) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f47,plain,(
% 9.05/2.75 (sP33 = $true) <=> ! [X317 : term,X318 : term > $o,X319 : subst] : (? [X324 : term] : (((var @ X324) = $true) & ((X318 @ (sub @ X324 @ X319)) != $true)) | ? [X320 : term] : (! [X321 : term] : (((X318 @ (sub @ X320 @ (push @ X321 @ id))) = $true) | ((X318 @ X321) != $true)) & ((X318 @ (lam @ X320)) != $true)) | ((sP32 @ X318) = $true) | ((X318 @ (sub @ X317 @ X319)) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f48,plain,(
% 9.05/2.75 (sP34 = $true) <=> ? [X84 : term,X85 : subst > term > subst > $o] : (((X85 @ id @ (lam @ (sub @ X84 @ (push @ one @ sh))) @ id) != $true) & ! [X87 : subst > term > term] : (? [X89 : subst,X90 : term,X91 : subst] : ((X87 @ (comp @ X91 @ X89) @ (sub @ X90 @ X89)) != (sub @ (X87 @ X91 @ X90) @ X89)) | ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id) = $true) | ? [X88 : term] : (((X85 @ id @ X88 @ id) = $true) & ($true != (X85 @ id @ (X87 @ id @ X88) @ id)))) & ! [X86 : term] : (((X85 @ id @ (sub @ X84 @ (push @ X86 @ id)) @ id) = $true) | ((X85 @ id @ X86 @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f49,plain,(
% 9.05/2.75 (sP35 = $true) <=> ((sP34 = $true) | ! [X92 : term,X93 : subst > term > subst > $o] : (? [X94 : subst > term > term] : (! [X98 : term] : (((X93 @ id @ X98 @ id) != $true) | ((X93 @ id @ (X94 @ id @ X98) @ id) = $true)) & ! [X95 : subst,X96 : term,X97 : subst] : ((X94 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97)) = (sub @ (X94 @ X95 @ X96) @ X97)) & ((X93 @ id @ (lam @ (X94 @ sh @ one)) @ id) != $true)) | ($true = (X93 @ id @ (lam @ X92) @ id)) | ? [X99 : term] : (((X93 @ id @ (sub @ X92 @ (push @ X99 @ id)) @ id) != $true) & ((X93 @ id @ X99 @ id) = $true))) | ? [X83 : term] : ((sub @ X83 @ id) != X83))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f50,plain,(
% 9.05/2.75 ($true = sP36) <=> ! [X127 : subst > term > subst > $o,X128 : term] : (((X127 @ id @ (lam @ (sub @ X128 @ (push @ one @ sh))) @ id) = $true) | ? [X130 : subst > term > term] : (! [X134 : term] : (((X127 @ id @ (X130 @ id @ X134) @ id) = $true) | ((X127 @ id @ X134 @ id) != $true)) & ((X127 @ id @ (lam @ (X130 @ sh @ one)) @ id) != $true) & ! [X131 : term,X132 : subst,X133 : subst] : ((X130 @ (comp @ X132 @ X133) @ (sub @ X131 @ X133)) = (sub @ (X130 @ X132 @ X131) @ X133))) | ? [X129 : term] : (((X127 @ id @ X129 @ id) = $true) & ((X127 @ id @ (sub @ X128 @ (push @ X129 @ id)) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f51,plain,(
% 9.05/2.75 (sP37 = $true) <=> ! [X75 : subst > term > subst > $o,X76 : term] : (? [X77 : subst > term > term] : (((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) != $true) & ! [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) = (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78))) & ! [X81 : term] : (((X75 @ id @ (X77 @ id @ X81) @ id) = $true) | ((X75 @ id @ X81 @ id) != $true))) | ? [X82 : term] : (((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) != $true) & ((X75 @ id @ X82 @ id) = $true)) | ((X75 @ id @ (lam @ X76) @ id) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f52,plain,(
% 9.05/2.75 ! [X295 : term > $o] : (((sP38 @ X295) = $true) <=> ? [X297 : term,X298 : term] : (($true != (X295 @ (ap @ X297 @ X298))) & ((X295 @ X297) = $true) & ((X295 @ X298) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f53,plain,(
% 9.05/2.75 (sP39 = $true) <=> ! [X294 : term,X295 : term > $o] : (? [X299 : term] : (((X295 @ (lam @ X299)) != $true) & ! [X300 : term] : (((X295 @ X300) != $true) | ((X295 @ (sub @ X299 @ (push @ X300 @ id))) = $true))) | ((X295 @ X294) = $true) | ? [X296 : term] : (((var @ X296) = $true) & ((X295 @ X296) != $true)) | ((sP38 @ X295) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f54,plain,(
% 9.05/2.75 ! [X274 : subst > term > subst > $o] : (((sP40 @ X274) = $true) <=> ? [X285 : subst > term > term] : (! [X286 : term,X287 : subst,X288 : subst] : ((X285 @ (comp @ X287 @ X288) @ (sub @ X286 @ X288)) = (sub @ (X285 @ X287 @ X286) @ X288)) & ! [X289 : term] : (((X274 @ id @ (X285 @ id @ X289) @ id) = $true) | ((X274 @ id @ X289 @ id) != $true)) & ((X274 @ id @ (lam @ (X285 @ sh @ one)) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f55,plain,(
% 9.05/2.75 ! [X274 : subst > term > subst > $o] : (((sP41 @ X274) = $true) <=> ? [X290 : term,X291 : term] : (((X274 @ id @ X290 @ id) = $true) & ((X274 @ id @ (ap @ (sub @ X291 @ id) @ X290) @ id) != $true) & ((X274 @ id @ X291 @ id) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f56,plain,(
% 9.05/2.75 (sP42 = $true) <=> ! [X274 : subst > term > subst > $o,X275 : term] : (((sP41 @ X274) = $true) | ((X274 @ id @ X275 @ id) = $true) | ((sP40 @ X274) = $true) | ? [X277 : term,X278 : subst,X279 : subst,X280 : subst] : (((X274 @ X279 @ X277 @ (comp @ X278 @ X280)) != $true) & ((X274 @ (comp @ X279 @ X278) @ (sub @ X277 @ X278) @ X280) = $true)) | ? [X276 : term] : (((X274 @ id @ X276 @ id) != $true) & ((var @ (sub @ X276 @ id)) = $true)) | ? [X281 : subst,X282 : term,X283 : subst,X284 : subst] : (((X274 @ (comp @ X283 @ X284) @ (sub @ X282 @ X284) @ X281) != $true) & ((X274 @ X283 @ X282 @ (comp @ X284 @ X281)) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f57,plain,(
% 9.05/2.75 ! [X528 : subst > term > subst > $o] : (((sP43 @ X528) = $true) <=> ? [X530 : term,X531 : term] : (((X528 @ id @ X531 @ id) = $true) & ((X528 @ id @ X530 @ id) = $true) & ((X528 @ id @ (ap @ (sub @ X531 @ id) @ X530) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f58,plain,(
% 9.05/2.75 (sP44 = $true) <=> ! [X525 : term > $o,X526 : term,X527 : term,X528 : subst > term > subst > $o] : (? [X532 : subst,X533 : subst,X534 : term,X535 : subst] : (((X528 @ X533 @ X534 @ (comp @ X532 @ X535)) != $true) & ((X528 @ (comp @ X533 @ X532) @ (sub @ X534 @ X532) @ X535) = $true)) | ((X525 @ X527) != $true) | ? [X536 : subst,X537 : subst,X538 : term,X539 : subst] : (((X528 @ X536 @ X538 @ (comp @ X537 @ X539)) = $true) & ((X528 @ (comp @ X536 @ X537) @ (sub @ X538 @ X537) @ X539) != $true)) | ? [X529 : term] : ((X528 @ id @ X529 @ id) != (X525 @ X529)) | ((X525 @ X526) != $true) | ((X525 @ (ap @ X527 @ X526)) = $true) | ((sP43 @ X528) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f59,plain,(
% 9.05/2.75 ! [X146 : subst > term > subst > $o] : (((sP45 @ X146) = $true) <=> ? [X159 : subst > term > term] : (((X146 @ id @ (lam @ (X159 @ sh @ one)) @ id) != $true) & ! [X161 : term,X162 : subst,X163 : subst] : ((X159 @ (comp @ X163 @ X162) @ (sub @ X161 @ X162)) = (sub @ (X159 @ X163 @ X161) @ X162)) & ! [X160 : term] : (((X146 @ id @ X160 @ id) != $true) | ((X146 @ id @ (X159 @ id @ X160) @ id) = $true))))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f60,plain,(
% 9.05/2.75 (sP46 = $true) <=> ! [X146 : subst > term > subst > $o,X147 : term > $o,X148 : term] : (((sP45 @ X146) = $true) | ((X147 @ (lam @ X148)) = $true) | ? [X149 : term] : ((X147 @ X149) != (X146 @ id @ X149 @ id)) | ? [X154 : subst,X155 : subst,X156 : subst,X157 : term] : (((X146 @ X155 @ X157 @ (comp @ X154 @ X156)) = $true) & ((X146 @ (comp @ X155 @ X154) @ (sub @ X157 @ X154) @ X156) != $true)) | ? [X158 : term] : (((X147 @ X158) = $true) & ((X147 @ (sub @ X148 @ (push @ X158 @ id))) != $true)) | ? [X150 : subst,X151 : subst,X152 : subst,X153 : term] : (((X146 @ X151 @ X153 @ (comp @ X150 @ X152)) != $true) & ((X146 @ (comp @ X151 @ X150) @ (sub @ X153 @ X150) @ X152) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f61,plain,(
% 9.05/2.75 (sP47 = $true) <=> ! [X419 : subst > term > subst > $o,X420 : term] : (((X419 @ id @ (lam @ (sub @ X420 @ (push @ one @ sh))) @ id) = $true) | ? [X422 : subst > term > term] : (! [X423 : term,X424 : subst,X425 : subst] : ((X422 @ (comp @ X425 @ X424) @ (sub @ X423 @ X424)) = (sub @ (X422 @ X425 @ X423) @ X424)) & ((X419 @ id @ (lam @ (X422 @ sh @ one)) @ id) != $true) & ! [X426 : term] : (((X419 @ id @ (X422 @ id @ X426) @ id) = $true) | ((X419 @ id @ X426 @ id) != $true))) | ? [X421 : term] : (((X419 @ id @ X421 @ id) = $true) & ((X419 @ id @ (sub @ X420 @ (push @ X421 @ id)) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f62,plain,(
% 9.05/2.75 (sP48 = $true) <=> ! [X452 : subst > term > subst > $o,X453 : term] : (((X452 @ id @ (lam @ (sub @ X453 @ (push @ one @ sh))) @ id) = $true) | ? [X454 : term] : (((X452 @ id @ (sub @ X453 @ (push @ X454 @ id)) @ id) != $true) & ((X452 @ id @ X454 @ id) = $true)) | ? [X455 : subst > term > term,X456 : term] : (! [X461 : subst,X462 : term,X463 : subst] : ((sub @ (sub @ X456 @ (push @ X462 @ X461)) @ X463) = (sub @ (X455 @ X461 @ X462) @ X463)) & ! [X457 : subst,X458 : term,X459 : subst] : ((sub @ X456 @ (push @ (sub @ X458 @ X457) @ (comp @ X459 @ X457))) = (X455 @ (comp @ X459 @ X457) @ (sub @ X458 @ X457))) & ! [X460 : term] : (((X452 @ id @ X460 @ id) != $true) | ((X452 @ id @ (X455 @ id @ X460) @ id) = $true)) & ((X452 @ id @ (lam @ (X455 @ sh @ one)) @ id) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f63,plain,(
% 9.05/2.75 ! [X201 : term > $o] : (((sP49 @ X201) = $true) <=> ! [X204 : term,X205 : term] : (((X201 @ X205) != $true) | ((X201 @ (ap @ X205 @ X204)) = $true) | ((X201 @ X204) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f64,plain,(
% 9.05/2.75 ! [X219 : subst > term > subst > $o] : (((sP50 @ X219) = $true) <=> ? [X221 : term,X222 : term] : (((X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id) != $true) & ((X219 @ id @ X221 @ id) = $true) & ((X219 @ id @ X222 @ id) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f65,plain,(
% 9.05/2.75 (sP51 = $true) <=> ? [X209 : subst > term > subst > $o,X210 : term > $o,X211 : term] : (! [X218 : term] : ((X209 @ id @ X218 @ id) = (X210 @ X218)) & ((X210 @ (lam @ X211)) != $true) & ! [X212 : term] : (((X210 @ (sub @ X211 @ (push @ X212 @ id))) = $true) | ((X210 @ X212) != $true)) & ! [X213 : subst > term > term] : (? [X215 : subst,X216 : subst,X217 : term] : ((sub @ (X213 @ X216 @ X217) @ X215) != (X213 @ (comp @ X216 @ X215) @ (sub @ X217 @ X215))) | ((X209 @ id @ (lam @ (X213 @ sh @ one)) @ id) = $true) | ? [X214 : term] : (((X209 @ id @ (X213 @ id @ X214) @ id) != $true) & ((X209 @ id @ X214 @ id) = $true))))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f66,plain,(
% 9.05/2.75 (sP52 = $true) <=> ? [X201 : term > $o,X202 : term] : (((X201 @ X202) != $true) & ((sP49 @ X201) = $true) & ! [X206 : term] : (? [X207 : term] : (((X201 @ (sub @ X206 @ (push @ X207 @ id))) != $true) & ((X201 @ X207) = $true)) | ((X201 @ (lam @ X206)) = $true)) & ! [X203 : term] : (((X201 @ X203) = $true) | ((var @ X203) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f67,plain,(
% 9.05/2.75 (sP53 = $true) <=> (? [X208 : term] : ((sub @ X208 @ id) != X208) | ! [X219 : subst > term > subst > $o,X220 : term] : (? [X223 : subst > term > term] : (($true != (X219 @ id @ (lam @ (X223 @ sh @ one)) @ id)) & ! [X225 : subst,X226 : term,X227 : subst] : ((X223 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225)) = (sub @ (X223 @ X227 @ X226) @ X225)) & ! [X224 : term] : (((X219 @ id @ (X223 @ id @ X224) @ id) = $true) | ((X219 @ id @ X224 @ id) != $true))) | ((sP50 @ X219) = $true) | ((X219 @ id @ X220 @ id) = $true)) | (sP52 = $true) | ? [X198 : subst > term > subst > $o] : ! [X199 : term > $o] : ? [X200 : term] : ((X198 @ id @ X200 @ id) != (X199 @ X200)) | (sP51 = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f68,plain,(
% 9.05/2.75 ! [X311 : term > $o] : (((sP54 @ X311) = $true) <=> ? [X315 : term,X316 : term] : (((X311 @ X315) = $true) & ((X311 @ (ap @ X315 @ X316)) != $true) & ((X311 @ X316) = $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f69,plain,(
% 9.05/2.75 ! [X303 : term > $o] : (((sP55 @ X303) = $true) <=> ! [X305 : term,X306 : term] : (((X303 @ (ap @ X305 @ X306)) = $true) | ((X303 @ X305) != $true) | ((X303 @ X306) != $true)))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f70,plain,(
% 9.05/2.75 (sP56 = $true) <=> ! [X310 : term,X311 : term > $o] : (? [X313 : term] : (! [X314 : term] : (((X311 @ X314) != $true) | ((X311 @ (sub @ X313 @ (push @ X314 @ id))) = $true)) & ((X311 @ (lam @ X313)) != $true)) | ((sP54 @ X311) = $true) | ? [X312 : term] : (((var @ X312) = $true) & ((X311 @ X312) != $true)) | ((X311 @ X310) = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f71,plain,(
% 9.05/2.75 (sP57 = $true) <=> (? [X302 : subst,X303 : term > $o,X304 : term] : (! [X309 : term] : (((var @ X309) != $true) | ((X303 @ (sub @ X309 @ X302)) = $true)) & ((X303 @ (sub @ X304 @ X302)) != $true) & ! [X307 : term] : (? [X308 : term] : (((X303 @ (sub @ X307 @ (push @ X308 @ id))) != $true) & ((X303 @ X308) = $true)) | ((X303 @ (lam @ X307)) = $true)) & ((sP55 @ X303) = $true)) | ? [X301 : term] : ((sub @ X301 @ id) != X301) | (sP56 = $true))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f72,plain,(
% 9.05/2.75 ! [X184 : subst > term > subst > $o] : (((sP58 @ X184) = $true) <=> ? [X187 : subst > term > term] : (! [X188 : term] : (((X184 @ id @ (X187 @ id @ X188) @ id) = $true) | ((X184 @ id @ X188 @ id) != $true)) & ((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) != $true) & ! [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) = (sub @ (X187 @ X189 @ X191) @ X190))))),
% 9.05/2.75 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.05/2.75 thf(f13,plain,(
% 9.05/2.75 ((hoasinduction_no_psi_cond = $true) <=> ! [X183 : term,X184 : subst > term > subst > $o] : (? [X185 : term,X186 : term] : (((X184 @ id @ X186 @ id) = $true) & ((X184 @ id @ X185 @ id) = $true) & ((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) != $true)) | ? [X187 : subst > term > term] : (! [X188 : term] : (((X184 @ id @ (X187 @ id @ X188) @ id) = $true) | ((X184 @ id @ X188 @ id) != $true)) & ((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) != $true) & ! [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) = (sub @ (X187 @ X189 @ X191) @ X190))) | ((X184 @ id @ X183 @ id) = $true))) & ((? [X302 : subst,X303 : term > $o,X304 : term] : (! [X309 : term] : (((var @ X309) != $true) | ((X303 @ (sub @ X309 @ X302)) = $true)) & ((X303 @ (sub @ X304 @ X302)) != $true) & ! [X307 : term] : (? [X308 : term] : (((X303 @ (sub @ X307 @ (push @ X308 @ id))) != $true) & ((X303 @ X308) = $true)) | ((X303 @ (lam @ X307)) = $true)) & ! [X305 : term,X306 : term] : (((X303 @ (ap @ X305 @ X306)) = $true) | ((X303 @ X305) != $true) | ((X303 @ X306) != $true))) | ? [X301 : term] : ((sub @ X301 @ id) != X301) | ! [X310 : term,X311 : term > $o] : (? [X313 : term] : (! [X314 : term] : (((X311 @ X314) != $true) | ((X311 @ (sub @ X313 @ (push @ X314 @ id))) = $true)) & ((X311 @ (lam @ X313)) != $true)) | ? [X315 : term,X316 : term] : (((X311 @ X315) = $true) & ((X311 @ (ap @ X315 @ X316)) != $true) & ((X311 @ X316) = $true)) | ? [X312 : term] : (((var @ X312) = $true) & ((X311 @ X312) != $true)) | ((X311 @ X310) = $true))) <=> (induction2_lthm = $true)) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X208 : term] : ((sub @ X208 @ id) != X208) | ! [X219 : subst > term > subst > $o,X220 : term] : (? [X223 : subst > term > term] : (($true != (X219 @ id @ (lam @ (X223 @ sh @ one)) @ id)) & ! [X225 : subst,X226 : term,X227 : subst] : ((X223 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225)) = (sub @ (X223 @ X227 @ X226) @ X225)) & ! [X224 : term] : (((X219 @ id @ (X223 @ id @ X224) @ id) = $true) | ((X219 @ id @ X224 @ id) != $true))) | ? [X221 : term,X222 : term] : (((X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id) != $true) & ((X219 @ id @ X221 @ id) = $true) & ((X219 @ id @ X222 @ id) = $true)) | ((X219 @ id @ X220 @ id) = $true)) | ? [X201 : term > $o,X202 : term] : (((X201 @ X202) != $true) & ! [X204 : term,X205 : term] : (((X201 @ X205) != $true) | ((X201 @ (ap @ X205 @ X204)) = $true) | ((X201 @ X204) != $true)) & ! [X206 : term] : (? [X207 : term] : (((X201 @ (sub @ X206 @ (push @ X207 @ id))) != $true) & ((X201 @ X207) = $true)) | ((X201 @ (lam @ X206)) = $true)) & ! [X203 : term] : (((X201 @ X203) = $true) | ((var @ X203) != $true))) | ? [X198 : subst > term > subst > $o] : ! [X199 : term > $o] : ? [X200 : term] : ((X198 @ id @ X200 @ id) != (X199 @ X200)) | ? [X209 : subst > term > subst > $o,X210 : term > $o,X211 : term] : (! [X218 : term] : ((X209 @ id @ X218 @ id) = (X210 @ X218)) & ((X210 @ (lam @ X211)) != $true) & ! [X212 : term] : (((X210 @ (sub @ X211 @ (push @ X212 @ id))) = $true) | ((X210 @ X212) != $true)) & ! [X213 : subst > term > term] : (? [X215 : subst,X216 : subst,X217 : term] : ((sub @ (X213 @ X216 @ X217) @ X215) != (X213 @ (comp @ X216 @ X215) @ (sub @ X217 @ X215))) | ((X209 @ id @ (lam @ (X213 @ sh @ one)) @ id) = $true) | ? [X214 : term] : (((X209 @ id @ (X213 @ id @ X214) @ id) != $true) & ((X209 @ id @ X214 @ id) = $true)))))) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) <=> (? [X272 : term] : ((sub @ X272 @ id) != X272) | ! [X273 : term] : ((sub @ X273 @ id) = X273))) & (! [X165 : subst > term > subst > $o] : ? [X166 : term > $o] : ! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) <=> (hoasinduction_lem0 = $true)) & (axmap = $true) & (ulamvarsh = $true) & (! [X2 : subst > term > subst > $o] : ? [X3 : term > $o] : ! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) <=> (hoasinduction_lem0_lthm = $true)) & ((? [X410 : subst] : ((comp @ X410 @ id) != X410) | ? [X409 : subst] : ((comp @ id @ X409) != X409) | (! [X411 : subst] : ((comp @ id @ X411) = X411) & ! [X412 : subst] : ((comp @ X412 @ id) = X412))) <=> (substmonoid_lthm = $true)) & (! [X452 : subst > term > subst > $o,X453 : term] : (((X452 @ id @ (lam @ (sub @ X453 @ (push @ one @ sh))) @ id) = $true) | ? [X454 : term] : (((X452 @ id @ (sub @ X453 @ (push @ X454 @ id)) @ id) != $true) & ((X452 @ id @ X454 @ id) = $true)) | ? [X455 : subst > term > term,X456 : term] : (! [X461 : subst,X462 : term,X463 : subst] : ((sub @ (sub @ X456 @ (push @ X462 @ X461)) @ X463) = (sub @ (X455 @ X461 @ X462) @ X463)) & ! [X457 : subst,X458 : term,X459 : subst] : ((sub @ X456 @ (push @ (sub @ X458 @ X457) @ (comp @ X459 @ X457))) = (X455 @ (comp @ X459 @ X457) @ (sub @ X458 @ X457))) & ! [X460 : term] : (((X452 @ id @ X460 @ id) != $true) | ((X452 @ id @ (X455 @ id @ X460) @ id) = $true)) & ((X452 @ id @ (lam @ (X455 @ sh @ one)) @ id) != $true))) <=> (hoasinduction_lem3aaa = $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3aa = $true) <=> ! [X419 : subst > term > subst > $o,X420 : term] : (((X419 @ id @ (lam @ (sub @ X420 @ (push @ one @ sh))) @ id) = $true) | ? [X422 : subst > term > term] : (! [X423 : term,X424 : subst,X425 : subst] : ((X422 @ (comp @ X425 @ X424) @ (sub @ X423 @ X424)) = (sub @ (X422 @ X425 @ X423) @ X424)) & ((X419 @ id @ (lam @ (X422 @ sh @ one)) @ id) != $true) & ! [X426 : term] : (((X419 @ id @ (X422 @ id @ X426) @ id) = $true) | ((X419 @ id @ X426 @ id) != $true))) | ? [X421 : term] : (((X419 @ id @ X421 @ id) = $true) & ((X419 @ id @ (sub @ X420 @ (push @ X421 @ id)) @ id) != $true)))) & (hoasinduction_lem1_gthm = $true) & (substmonoid_gthm = $true) & ! [X245 : d_term,X246 : d_term] : (((d2term @ X246) != (d2term @ X245)) | (X245 = X246)) & (ulamvarind = $true) & (pushprop_lthm_orig = $true) & (! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | ((ap @ X261 @ X263) != (ap @ X262 @ X264))) <=> (apinj1 = $true)) & (hoaslamnotap_gthm = $true) & ! [X181 : d_subst,X182 : d_subst] : ((X181 = X182) | ((d2subst @ X182) != (d2subst @ X181))) & (hoaslamnotvar = $true) & (pushprop_lem1v2_gthm = $true) & ((var @ (d2term @ d_one)) != $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X146 : subst > term > subst > $o,X147 : term > $o,X148 : term] : (? [X159 : subst > term > term] : (((X146 @ id @ (lam @ (X159 @ sh @ one)) @ id) != $true) & ! [X161 : term,X162 : subst,X163 : subst] : ((X159 @ (comp @ X163 @ X162) @ (sub @ X161 @ X162)) = (sub @ (X159 @ X163 @ X161) @ X162)) & ! [X160 : term] : (((X146 @ id @ X160 @ id) != $true) | ((X146 @ id @ (X159 @ id @ X160) @ id) = $true))) | ((X147 @ (lam @ X148)) = $true) | ? [X149 : term] : ((X147 @ X149) != (X146 @ id @ X149 @ id)) | ? [X154 : subst,X155 : subst,X156 : subst,X157 : term] : (((X146 @ X155 @ X157 @ (comp @ X154 @ X156)) = $true) & ((X146 @ (comp @ X155 @ X154) @ (sub @ X157 @ X154) @ X156) != $true)) | ? [X158 : term] : (((X147 @ X158) = $true) & ((X147 @ (sub @ X148 @ (push @ X158 @ id))) != $true)) | ? [X150 : subst,X151 : subst,X152 : subst,X153 : term] : (((X146 @ X151 @ X153 @ (comp @ X150 @ X152)) != $true) & ((X146 @ (comp @ X151 @ X150) @ (sub @ X153 @ X150) @ X152) = $true)))) & ! [X265 : d_term] : (d_one = X265) & (! [X525 : term > $o,X526 : term,X527 : term,X528 : subst > term > subst > $o] : (? [X532 : subst,X533 : subst,X534 : term,X535 : subst] : (((X528 @ X533 @ X534 @ (comp @ X532 @ X535)) != $true) & ((X528 @ (comp @ X533 @ X532) @ (sub @ X534 @ X532) @ X535) = $true)) | ((X525 @ X527) != $true) | ? [X536 : subst,X537 : subst,X538 : term,X539 : subst] : (((X528 @ X536 @ X538 @ (comp @ X537 @ X539)) = $true) & ((X528 @ (comp @ X536 @ X537) @ (sub @ X538 @ X537) @ X539) != $true)) | ? [X529 : term] : ((X528 @ id @ X529 @ id) != (X525 @ X529)) | ((X525 @ X526) != $true) | ((X525 @ (ap @ X527 @ X526)) = $true) | ? [X530 : term,X531 : term] : (((X528 @ id @ X531 @ id) = $true) & ((X528 @ id @ X530 @ id) = $true) & ((X528 @ id @ (ap @ (sub @ X531 @ id) @ X530) @ id) != $true))) <=> (hoasinduction_lem2v2 = $true)) & (! [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) != (ap @ X326 @ X328)) | (X325 = X328)) <=> (apinj2 = $true)) & (lamnotvar = $true) & (axabs = $true) & ((hoasinduction = $true) <=> ! [X274 : subst > term > subst > $o,X275 : term] : (? [X290 : term,X291 : term] : (((X274 @ id @ X290 @ id) = $true) & ((X274 @ id @ (ap @ (sub @ X291 @ id) @ X290) @ id) != $true) & ((X274 @ id @ X291 @ id) = $true)) | ((X274 @ id @ X275 @ id) = $true) | ? [X285 : subst > term > term] : (! [X286 : term,X287 : subst,X288 : subst] : ((X285 @ (comp @ X287 @ X288) @ (sub @ X286 @ X288)) = (sub @ (X285 @ X287 @ X286) @ X288)) & ! [X289 : term] : (((X274 @ id @ (X285 @ id @ X289) @ id) = $true) | ((X274 @ id @ X289 @ id) != $true)) & ((X274 @ id @ (lam @ (X285 @ sh @ one)) @ id) != $true)) | ? [X277 : term,X278 : subst,X279 : subst,X280 : subst] : (((X274 @ X279 @ X277 @ (comp @ X278 @ X280)) != $true) & ((X274 @ (comp @ X279 @ X278) @ (sub @ X277 @ X278) @ X280) = $true)) | ? [X276 : term] : (((X274 @ id @ X276 @ id) != $true) & ((var @ (sub @ X276 @ id)) = $true)) | ? [X281 : subst,X282 : term,X283 : subst,X284 : subst] : (((X274 @ (comp @ X283 @ X284) @ (sub @ X282 @ X284) @ X281) != $true) & ((X274 @ X283 @ X282 @ (comp @ X284 @ X281)) = $true)))) & ((d2subst @ d_id) = sh) & (pushprop_lem0_gthm = $true) & (! [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192)) | (X193 = X194)) <=> (hoasapinj1 = $true)) & (hoasapnotvar_gthm = $true) & ((hoaslaminj = $true) <=> ! [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) != (lam @ (X465 @ sh @ one))) | ((X465 @ X466 @ X464) = (X467 @ X466 @ X464)) | ? [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) != (sub @ (X465 @ X473 @ X471) @ X472)) | ? [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) != (sub @ (X467 @ X470 @ X469) @ X468)))) & ((hoaslamnotap = $true) <=> ! [X247 : term,X248 : subst > term > term,X249 : term] : (? [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) != (sub @ (X248 @ X250 @ X252) @ X251)) | ((ap @ (sub @ X247 @ id) @ X249) != (lam @ (X248 @ sh @ one))))) & (hoasinduction_gthm = $true) & (pushprop_lthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 = X408) | ((ap @ (sub @ X408 @ id) @ X406) != (ap @ (sub @ X405 @ id) @ X407))) | ? [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) = (ap @ X404 @ X402)) & (X401 != X404)) | ? [X400 : term] : ((sub @ X400 @ id) != X400))) & (! [X164 : term] : ((sub @ X164 @ id) = X164) <=> (termmset = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar = $true) & ((pushprop_lem1 = $true) <=> ! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (((X60 @ X61) != $true) | ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) = $true))) & (! [X294 : term,X295 : term > $o] : (? [X299 : term] : (((X295 @ (lam @ X299)) != $true) & ! [X300 : term] : (((X295 @ X300) != $true) | ((X295 @ (sub @ X299 @ (push @ X300 @ id))) = $true))) | ((X295 @ X294) = $true) | ? [X296 : term] : (((var @ X296) = $true) & ((X295 @ X296) != $true)) | ? [X297 : term,X298 : term] : (($true != (X295 @ (ap @ X297 @ X298))) & ((X295 @ X297) = $true) & ((X295 @ X298) = $true))) <=> (induction2 = $true)) & ((hoasinduction_lem3a = $true) <=> ! [X75 : subst > term > subst > $o,X76 : term] : (? [X77 : subst > term > term] : (((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) != $true) & ! [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) = (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78))) & ! [X81 : term] : (((X75 @ id @ (X77 @ id @ X81) @ id) = $true) | ((X75 @ id @ X81 @ id) != $true))) | ? [X82 : term] : (((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) != $true) & ((X75 @ id @ X82 @ id) = $true)) | ((X75 @ id @ (lam @ X76) @ id) = $true))) & (hoasinduction_lem1 = $true) & (hoaslaminj_gthm = $true) & (pushprop = $true) & (termmset_gthm = $true) & (pushprop_lem3v2 = $true) & ((substmonoid = $true) <=> (! [X1 : subst] : ((comp @ id @ X1) = X1) & ! [X0 : subst] : ((comp @ X0 @ id) = X0))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X127 : subst > term > subst > $o,X128 : term] : (((X127 @ id @ (lam @ (sub @ X128 @ (push @ one @ sh))) @ id) = $true) | ? [X130 : subst > term > term] : (! [X134 : term] : (((X127 @ id @ (X130 @ id @ X134) @ id) = $true) | ((X127 @ id @ X134 @ id) != $true)) & ((X127 @ id @ (lam @ (X130 @ sh @ one)) @ id) != $true) & ! [X131 : term,X132 : subst,X133 : subst] : ((X130 @ (comp @ X132 @ X133) @ (sub @ X131 @ X133)) = (sub @ (X130 @ X132 @ X131) @ X133))) | ? [X129 : term] : (((X127 @ id @ X129 @ id) = $true) & ((X127 @ id @ (sub @ X128 @ (push @ X129 @ id)) @ id) != $true)))) & (ulamvar1 != $true) & ((? [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) != X121) | ! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (((X122 @ X124) != $true) | ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) = $true)) | ? [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) != X118)) <=> (pushprop_lem1_lthm = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((pushprop_lem0_lthm = $true) <=> ! [X139 : term,X140 : term > $o,X141 : subst] : ? [X142 : term > $o] : ! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143))) & (one = (d2term @ d_one)) & ((shinj = $true) <=> ! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh)))) & (axvarshift = $true) & (hoasinduction_lem3_gthm = $true) & ((d2subst @ d_id) = 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)))))))))))))) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & ((hoasinduction_lem3a_lthm = $true) <=> (? [X84 : term,X85 : subst > term > subst > $o] : (((X85 @ id @ (lam @ (sub @ X84 @ (push @ one @ sh))) @ id) != $true) & ! [X87 : subst > term > term] : (? [X89 : subst,X90 : term,X91 : subst] : ((X87 @ (comp @ X91 @ X89) @ (sub @ X90 @ X89)) != (sub @ (X87 @ X91 @ X90) @ X89)) | ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id) = $true) | ? [X88 : term] : (((X85 @ id @ X88 @ id) = $true) & ($true != (X85 @ id @ (X87 @ id @ X88) @ id)))) & ! [X86 : term] : (((X85 @ id @ (sub @ X84 @ (push @ X86 @ id)) @ id) = $true) | ((X85 @ id @ X86 @ id) != $true))) | ! [X92 : term,X93 : subst > term > subst > $o] : (? [X94 : subst > term > term] : (! [X98 : term] : (((X93 @ id @ X98 @ id) != $true) | ((X93 @ id @ (X94 @ id @ X98) @ id) = $true)) & ! [X95 : subst,X96 : term,X97 : subst] : ((X94 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97)) = (sub @ (X94 @ X95 @ X96) @ X97)) & ((X93 @ id @ (lam @ (X94 @ sh @ one)) @ id) != $true)) | ($true = (X93 @ id @ (lam @ X92) @ id)) | ? [X99 : term] : (((X93 @ id @ (sub @ X92 @ (push @ X99 @ id)) @ id) != $true) & ((X93 @ id @ X99 @ id) = $true))) | ? [X83 : term] : ((sub @ X83 @ id) != X83))) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X135 : term] : ? [X136 : subst > term > term] : ! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138))) & (! [X292 : term,X293 : term] : ((X292 = X293) | ((lam @ X293) != (lam @ X292))) <=> (laminj = $true)) & (hoasinduction_lem2_gthm = $true) & (induction2lem_gthm = $true) & (! [X144 : term] : ? [X145 : subst > term > term] : ((X145 @ sh @ one) = (sub @ X144 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 = X258) | ((ap @ (sub @ X260 @ id) @ X258) != (ap @ (sub @ X259 @ id) @ X257))) | ? [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) = (ap @ X255 @ X254)) & (X253 != X254)))) & (! [X317 : term,X318 : term > $o,X319 : subst] : (? [X324 : term] : (((var @ X324) = $true) & ((X318 @ (sub @ X324 @ X319)) != $true)) | ? [X320 : term] : (! [X321 : term] : (((X318 @ (sub @ X320 @ (push @ X321 @ id))) = $true) | ((X318 @ X321) != $true)) & ((X318 @ (lam @ X320)) != $true)) | ? [X322 : term,X323 : term] : (((X318 @ X322) = $true) & ((X318 @ (ap @ X323 @ X322)) != $true) & ((X318 @ X323) = $true)) | ((X318 @ (sub @ X317 @ X319)) = $true)) <=> (induction2lem = $true)) & ((hoaslaminj_lthm = $true) <=> (? [X9 : term,X10 : term] : (((lam @ X10) = (lam @ X9)) & (X9 != X10)) | ! [X11 : subst > term > term,X12 : subst > term > term,X13 : term,X14 : subst] : (((lam @ (X11 @ sh @ one)) != (lam @ (X12 @ sh @ one))) | ? [X15 : term,X16 : subst,X17 : subst] : ((sub @ (X11 @ X17 @ X15) @ X16) != (X11 @ (comp @ X17 @ X16) @ (sub @ X15 @ X16))) | ? [X18 : subst,X19 : term,X20 : subst] : ((sub @ (X12 @ X20 @ X19) @ X18) != (X12 @ (comp @ X20 @ X18) @ (sub @ X19 @ X18))) | ((X11 @ X14 @ X13) = (X12 @ X14 @ X13))) | ? [X5 : term,X6 : subst] : ((sub @ one @ (push @ X5 @ X6)) != X5) | ? [X7 : term,X8 : subst] : ((comp @ sh @ (push @ X7 @ X8)) != X8))) & (hoaslamnotvar_lthm = $true) & ((? [X349 : subst > term > subst > $o,X350 : term] : (! [X361 : subst,X362 : subst,X363 : term,X364 : subst] : (((X349 @ X361 @ X363 @ (comp @ X362 @ X364)) != $true) | ((X349 @ (comp @ X361 @ X362) @ (sub @ X363 @ X362) @ X364) = $true)) & ! [X351 : subst,X352 : subst,X353 : term,X354 : subst] : (((X349 @ (comp @ X351 @ X352) @ (sub @ X353 @ X352) @ X354) != $true) | ((X349 @ X351 @ X353 @ (comp @ X352 @ X354)) = $true)) & ((X349 @ id @ (lam @ X350) @ id) != $true) & ! [X355 : subst > term > term] : (((X349 @ id @ (lam @ (X355 @ sh @ one)) @ id) = $true) | ? [X357 : subst,X358 : subst,X359 : term] : ((sub @ (X355 @ X358 @ X359) @ X357) != (X355 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357))) | ? [X356 : term] : (((X349 @ id @ X356 @ id) = $true) & ((X349 @ id @ (X355 @ id @ X356) @ id) != $true))) & ! [X360 : term] : (((X349 @ id @ (sub @ X350 @ (push @ X360 @ id)) @ id) = $true) | ((X349 @ id @ X360 @ id) != $true))) | ! [X365 : term,X366 : subst > term > subst > $o] : (? [X368 : subst,X369 : subst,X370 : subst,X371 : term] : (((X366 @ (comp @ X369 @ X370) @ (sub @ X371 @ X370) @ X368) != $true) & ((X366 @ X369 @ X371 @ (comp @ X370 @ X368)) = $true)) | ? [X374 : subst > term > term] : (! [X375 : subst,X376 : term,X377 : subst] : ((X374 @ (comp @ X377 @ X375) @ (sub @ X376 @ X375)) = (sub @ (X374 @ X377 @ X376) @ X375)) & ! [X378 : term] : (((X366 @ id @ X378 @ id) != $true) | ((X366 @ id @ (X374 @ id @ X378) @ id) = $true)) & ((X366 @ id @ (lam @ (X374 @ sh @ one)) @ id) != $true)) | ? [X367 : term] : (((X366 @ id @ X367 @ id) != $true) & ((var @ (sub @ X367 @ id)) = $true)) | ? [X372 : term,X373 : term] : (((X366 @ id @ (ap @ (sub @ X373 @ id) @ X372) @ id) != $true) & ((X366 @ id @ X372 @ id) = $true) & ((X366 @ id @ X373 @ id) = $true)) | ? [X379 : subst,X380 : subst,X381 : term,X382 : subst] : (((X366 @ X380 @ X381 @ (comp @ X379 @ X382)) != $true) & ((X366 @ (comp @ X380 @ X379) @ (sub @ X381 @ X379) @ X382) = $true)) | ((X366 @ id @ X365 @ id) = $true)) | ? [X329 : term > $o,X330 : term] : (((X329 @ X330) != $true) & ! [X332 : term,X333 : term] : (((X329 @ (ap @ X332 @ X333)) = $true) | ((X329 @ X332) != $true) | ((X329 @ X333) != $true)) & ! [X331 : term] : (($true = (X329 @ X331)) | ((var @ X331) != $true)) & ! [X334 : term] : (? [X335 : term] : (((X329 @ X335) = $true) & ((X329 @ (sub @ X334 @ (push @ X335 @ id))) != $true)) | ((X329 @ (lam @ X334)) = $true))) | ? [X336 : term,X337 : subst > term > subst > $o,X338 : term] : (! [X345 : subst,X346 : term,X347 : subst,X348 : subst] : (((X337 @ (comp @ X348 @ X347) @ (sub @ X346 @ X347) @ X345) != $true) | ((X337 @ X348 @ X346 @ (comp @ X347 @ X345)) = $true)) & ! [X339 : subst,X340 : subst,X341 : term,X342 : subst] : (((X337 @ (comp @ X342 @ X339) @ (sub @ X341 @ X339) @ X340) = $true) | ((X337 @ X342 @ X341 @ (comp @ X339 @ X340)) != $true)) & ((X337 @ id @ X338 @ id) = $true) & ((X337 @ id @ (ap @ X338 @ X336) @ id) != $true) & ! [X343 : term,X344 : term] : (((X337 @ id @ X344 @ id) != $true) | ((X337 @ id @ (ap @ (sub @ X343 @ id) @ X344) @ id) = $true) | ((X337 @ id @ X343 @ id) != $true)) & ((X337 @ id @ X336 @ id) = $true))) <=> (hoasinduction_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) = $true) | ((X522 @ X521) != $true) | ? [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) != (X523 @ X524)))) & ((hoasinduction_lem3v2a = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term,X237 : term > $o] : (? [X244 : term] : ((X237 @ X244) != (X235 @ id @ X244 @ id)) | ($true = (X237 @ (lam @ X236))) | ? [X239 : subst > term > term] : (! [X240 : subst,X241 : term,X242 : subst] : ((X239 @ (comp @ X242 @ X240) @ (sub @ X241 @ X240)) = (sub @ (X239 @ X242 @ X241) @ X240)) & ! [X243 : term] : (((X235 @ id @ X243 @ id) != $true) | ((X235 @ id @ (X239 @ id @ X243) @ id) = $true)) & ((X235 @ id @ (lam @ (X239 @ sh @ one)) @ id) != $true)) | ? [X238 : term] : (((X237 @ X238) = $true) & ((X237 @ (sub @ X236 @ (push @ X238 @ id))) != $true)))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X436 : term,X437 : subst > term > subst > $o] : (? [X443 : term,X444 : subst,X445 : subst,X446 : subst] : (((X437 @ (comp @ X446 @ X445) @ (sub @ X443 @ X445) @ X444) = $true) & ($true != (X437 @ X446 @ X443 @ (comp @ X445 @ X444)))) | ((X437 @ id @ (lam @ X436) @ id) = $true) | ? [X438 : subst > term > term] : (((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id) != $true) & ! [X439 : term] : (((X437 @ id @ (X438 @ id @ X439) @ id) = $true) | ((X437 @ id @ X439 @ id) != $true)) & ! [X440 : term,X441 : subst,X442 : subst] : ((sub @ (X438 @ X441 @ X440) @ X442) = (X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442)))) | ? [X448 : subst,X449 : subst,X450 : subst,X451 : term] : (((X437 @ (comp @ X450 @ X449) @ (sub @ X451 @ X449) @ X448) != $true) & ((X437 @ X450 @ X451 @ (comp @ X449 @ X448)) = $true)) | ? [X447 : term] : (((X437 @ id @ (sub @ X436 @ (push @ X447 @ id)) @ id) != $true) & ((X437 @ id @ X447 @ id) = $true))) | ? [X427 : term] : ((sub @ X427 @ id) != X427) | ? [X428 : term,X429 : subst > term > subst > $o] : (! [X431 : subst > term > term] : (? [X432 : term] : (((X429 @ id @ X432 @ id) = $true) & ((X429 @ id @ (X431 @ id @ X432) @ id) != $true)) | ((X429 @ id @ (lam @ (X431 @ sh @ one)) @ id) = $true) | ? [X433 : term,X434 : subst,X435 : subst] : ((X431 @ (comp @ X435 @ X434) @ (sub @ X433 @ X434)) != (sub @ (X431 @ X435 @ X433) @ X434))) & ! [X430 : term] : (((X429 @ id @ X430 @ id) != $true) | ((X429 @ id @ (sub @ X428 @ (push @ X430 @ id)) @ id) = $true)) & ((X429 @ id @ (lam @ (sub @ X428 @ (push @ one @ sh))) @ id) != $true)))) & (hoasinduction_lem3v2_gthm = $true) & ((? [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) != X228) | ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (((X230 @ X233) != $true) | ? [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) != (X232 @ X234)) | ((X232 @ one) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & ! [X271 : d_subst] : (d_id = X271) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (apnotvar = $true) & ((! [X108 : term,X109 : subst > term > subst > $o,X110 : term > $o] : (? [X111 : term] : (((X110 @ X111) = $true) & ($true != (X110 @ (sub @ X108 @ (push @ X111 @ id))))) | ((X110 @ (lam @ X108)) = $true) | ? [X113 : subst > term > term] : (((X109 @ id @ (lam @ (X113 @ sh @ one)) @ id) != $true) & ! [X115 : subst,X116 : term,X117 : subst] : ((X113 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117)) = (sub @ (X113 @ X115 @ X116) @ X117)) & ! [X114 : term] : (((X109 @ id @ X114 @ id) != $true) | ((X109 @ id @ (X113 @ id @ X114) @ id) = $true))) | ? [X112 : term] : ((X109 @ id @ X112 @ id) != (X110 @ X112))) | ? [X103 : term] : ! [X104 : subst > term > term] : ? [X105 : subst,X106 : term] : ((sub @ X103 @ (push @ X106 @ X105)) != (X104 @ X105 @ X106)) | ? [X107 : term] : ((sub @ X107 @ id) != X107)) <=> (hoasinduction_lem3v2a_lthm = $true)) & (lamnotap != $true) & ((hoasinduction_lthm_3 = $true) <=> (? [X32 : term > $o,X33 : subst > term > subst > $o,X34 : term] : (! [X35 : term] : (((X32 @ (sub @ X34 @ (push @ X35 @ id))) = $true) | ((X32 @ X35) != $true)) & ((X32 @ (lam @ X34)) != $true) & ! [X41 : term] : ((X33 @ id @ X41 @ id) = (X32 @ X41)) & ! [X36 : subst > term > term] : (((X33 @ id @ (lam @ (X36 @ sh @ one)) @ id) = $true) | ? [X38 : subst,X39 : term,X40 : subst] : ((X36 @ (comp @ X38 @ X40) @ (sub @ X39 @ X40)) != (sub @ (X36 @ X38 @ X39) @ X40)) | ? [X37 : term] : (((X33 @ id @ X37 @ id) = $true) & ((X33 @ id @ (X36 @ id @ X37) @ id) != $true)))) | ? [X21 : subst > term > subst > $o] : ! [X22 : term > $o] : ? [X23 : term] : ((X21 @ id @ X23 @ id) != (X22 @ X23)) | ? [X24 : term,X25 : term > $o] : (! [X30 : term] : (((var @ X30) != $true) | ((X25 @ X30) = $true)) & ((X25 @ X24) != $true) & ! [X26 : term,X27 : term] : (((X25 @ (ap @ X27 @ X26)) = $true) | ((X25 @ X27) != $true) | ((X25 @ X26) != $true)) & ! [X28 : term] : (((X25 @ (lam @ X28)) = $true) | ? [X29 : term] : (((X25 @ (sub @ X28 @ (push @ X29 @ id))) != $true) & ((X25 @ X29) = $true)))) | ! [X42 : subst > term > subst > $o,X43 : term] : (? [X44 : subst > term > term] : (((X42 @ id @ (lam @ (X44 @ sh @ one)) @ id) != $true) & ! [X48 : term] : (((X42 @ id @ (X44 @ id @ X48) @ id) = $true) | ((X42 @ id @ X48 @ id) != $true)) & ! [X45 : subst,X46 : subst,X47 : term] : ((sub @ (X44 @ X45 @ X47) @ X46) = (X44 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46)))) | ? [X50 : subst,X51 : term,X52 : subst,X53 : subst] : (((X42 @ (comp @ X53 @ X52) @ (sub @ X51 @ X52) @ X50) != $true) & ((X42 @ X53 @ X51 @ (comp @ X52 @ X50)) = $true)) | ? [X49 : term] : (((X42 @ id @ X49 @ id) != $true) & ((var @ (sub @ X49 @ id)) = $true)) | ((X42 @ id @ X43 @ id) = $true) | ? [X54 : term,X55 : term] : (((X42 @ id @ X55 @ id) = $true) & ((X42 @ id @ X54 @ id) = $true) & ((X42 @ id @ (ap @ (sub @ X54 @ id) @ X55) @ id) != $true)) | ? [X56 : subst,X57 : subst,X58 : subst,X59 : term] : (((X42 @ X58 @ X59 @ (comp @ X57 @ X56)) != $true) & ((X42 @ (comp @ X58 @ X57) @ (sub @ X59 @ X57) @ X56) = $true))) | ? [X31 : term] : ((sub @ X31 @ id) != X31))) & (axclos = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((hoasinduction_lem3b = $true) <=> ! [X73 : term] : ? [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) = (X74 @ sh @ one))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X65 : term] : ? [X66 : subst > term > term] : ! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68)))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoasinduction_lem2_lthm = $true) & (! [X71 : subst] : ((comp @ X71 @ id) = X71) <=> (axidr = $true)) & (axassoc = $true) & (! [X495 : term > $o,X496 : term] : (? [X497 : term,X498 : term] : (((X495 @ (ap @ X497 @ X498)) != $true) & ((X495 @ X497) = $true) & ((X495 @ X498) = $true)) | ? [X499 : term] : (((var @ X499) = $true) & ((X495 @ X499) != $true)) | ((X495 @ X496) = $true) | ? [X500 : term] : (((X495 @ (lam @ X500)) != $true) & ((X495 @ X500) = $true))) <=> (induction = $true)) & (hoaslamnotvar_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((! [X485 : term,X486 : subst,X487 : term > $o] : (? [X490 : term] : (((X487 @ (sub @ X490 @ X486)) != $true) & ((var @ X490) = $true)) | ? [X491 : term,X492 : term] : (((X487 @ X491) = $true) & ((X487 @ X492) = $true) & ((X487 @ (ap @ X491 @ X492)) != $true)) | ? [X488 : term] : (! [X489 : term] : (((X487 @ X489) != $true) | ((X487 @ (sub @ X488 @ (push @ X489 @ id))) = $true)) & ((X487 @ (lam @ X488)) != $true)) | ((X487 @ (sub @ X485 @ X486)) = $true)) | ? [X476 : subst,X477 : term] : ((comp @ sh @ (push @ X477 @ X476)) != X476) | ? [X474 : subst,X475 : term] : ((sub @ one @ (push @ X475 @ X474)) != X475) | ? [X478 : subst] : ((comp @ X478 @ id) != X478) | ? [X479 : term > $o,X480 : term] : (! [X482 : term] : (((X479 @ X482) = $true) | ((var @ X482) != $true)) & ((X479 @ X480) != $true) & ! [X481 : term] : (((X479 @ X481) != $true) | ((X479 @ (lam @ X481)) = $true)) & ! [X483 : term,X484 : term] : (((X479 @ (ap @ X484 @ X483)) = $true) | ((X479 @ X483) != $true) | ((X479 @ X484) != $true)))) <=> (induction2lem_lthm = $true)) & (hoasapnotvar_lthm = $true) & (! [X266 : subst,X267 : term > $o,X268 : term] : ? [X269 : term > $o] : ! [X270 : term] : ((X269 @ X270) = (X267 @ (sub @ X270 @ (push @ X268 @ X266)))) <=> (pushprop_lem0 = $true)) & (! [X383 : term] : ((sub @ X383 @ id) = X383) <=> (axvarid = $true)) & (! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100) <=> (axvarcons = $true)) & (hoasapinj1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X168 : term,X169 : term,X170 : subst > term > subst > $o] : (((X170 @ id @ (ap @ X168 @ X169) @ id) = $true) | ((X170 @ id @ X169 @ id) != $true) | ((X170 @ id @ X168 @ id) != $true) | ? [X177 : subst,X178 : subst,X179 : term,X180 : subst] : (((X170 @ X178 @ X179 @ (comp @ X177 @ X180)) = $true) & ((X170 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177) @ X180) != $true)) | ? [X173 : subst,X174 : subst,X175 : term,X176 : subst] : (((X170 @ (comp @ X176 @ X173) @ (sub @ X175 @ X173) @ X174) = $true) & ((X170 @ X176 @ X175 @ (comp @ X173 @ X174)) != $true)) | ? [X171 : term,X172 : term] : (((X170 @ id @ (ap @ (sub @ X171 @ id) @ X172) @ id) != $true) & ((X170 @ id @ X172 @ id) = $true) & ((X170 @ id @ X171 @ id) = $true))) <=> (hoasinduction_lem2 = $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X413 : term] : ? [X414 : d_term] : ((d2term @ X414) = X413) & ((axidl = $true) <=> ! [X72 : subst] : ((comp @ id @ X72) = X72)) & (hoasinduction_lem1v2 = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((axshiftcons = $true) <=> ! [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) = X493)) & (pushprop_gthm = $true) & (induction2_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X384 : term,X385 : subst > term > subst > $o] : (? [X386 : subst,X387 : subst,X388 : subst,X389 : term] : (((X385 @ (comp @ X386 @ X388) @ (sub @ X389 @ X388) @ X387) != $true) & ((X385 @ X386 @ X389 @ (comp @ X388 @ X387)) = $true)) | ? [X391 : subst > term > term] : (! [X395 : term] : (((X385 @ id @ X395 @ id) != $true) | ((X385 @ id @ (X391 @ id @ X395) @ id) = $true)) & ! [X392 : subst,X393 : subst,X394 : term] : ((sub @ (X391 @ X393 @ X394) @ X392) = (X391 @ (comp @ X393 @ X392) @ (sub @ X394 @ X392))) & ((X385 @ id @ (lam @ (X391 @ sh @ one)) @ id) != $true)) | ? [X390 : term] : (((X385 @ id @ X390 @ id) = $true) & ((X385 @ id @ (sub @ X384 @ (push @ X390 @ id)) @ id) != $true)) | ? [X396 : subst,X397 : subst,X398 : term,X399 : subst] : (((X385 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X399) = $true) & ((X385 @ X396 @ X398 @ (comp @ X397 @ X399)) != $true)) | ((X385 @ id @ (lam @ X384) @ id) = $true))) & (! [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 = X418) | ((ap @ (sub @ X417 @ id) @ X418) != (ap @ (sub @ X415 @ id) @ X416))) <=> (hoasapinj2 = $true)) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X502 : term > $o,X503 : subst > term > subst > $o,X504 : term] : (? [X515 : subst,X516 : subst,X517 : subst,X518 : term] : (((X503 @ X516 @ X518 @ (comp @ X515 @ X517)) != $true) & ((X503 @ (comp @ X516 @ X515) @ (sub @ X518 @ X515) @ X517) = $true)) | ? [X519 : term] : (((X502 @ (sub @ X504 @ (push @ X519 @ id))) != $true) & ((X502 @ X519) = $true)) | ? [X510 : subst > term > term] : (! [X512 : subst,X513 : term,X514 : subst] : ((sub @ (X510 @ X514 @ X513) @ X512) = (X510 @ (comp @ X514 @ X512) @ (sub @ X513 @ X512))) & ! [X511 : term] : (((X503 @ id @ (X510 @ id @ X511) @ id) = $true) | ((X503 @ id @ X511 @ id) != $true)) & ((X503 @ id @ (lam @ (X510 @ sh @ one)) @ id) != $true)) | ? [X506 : subst,X507 : subst,X508 : subst,X509 : term] : (((X503 @ (comp @ X506 @ X507) @ (sub @ X509 @ X507) @ X508) != $true) & ((X503 @ X506 @ X509 @ (comp @ X507 @ X508)) = $true)) | ((X502 @ (lam @ X504)) = $true) | ? [X505 : term] : ((X503 @ id @ X505 @ id) != (X502 @ X505))) | ? [X501 : term] : ((sub @ X501 @ id) != X501))) & (axapp = $true)),
% 9.05/2.75 inference(flattening,[status(thm)],[f12])).
% 9.05/2.75 thf(f12,plain,(
% 9.05/2.75 (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)))))))))))))) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X127 : subst > term > subst > $o,X128 : term] : (((X127 @ id @ (lam @ (sub @ X128 @ (push @ one @ sh))) @ id) = $true) | ? [X130 : subst > term > term] : (! [X134 : term] : (((X127 @ id @ (X130 @ id @ X134) @ id) = $true) | ((X127 @ id @ X134 @ id) != $true)) & ((X127 @ id @ (lam @ (X130 @ sh @ one)) @ id) != $true) & ! [X131 : term,X132 : subst,X133 : subst] : ((X130 @ (comp @ X132 @ X133) @ (sub @ X131 @ X133)) = (sub @ (X130 @ X132 @ X131) @ X133))) | ? [X129 : term] : (((X127 @ id @ X129 @ id) = $true) & ((X127 @ id @ (sub @ X128 @ (push @ X129 @ id)) @ id) != $true)))) & (axabs = $true) & ((shinj = $true) <=> ! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh)))) & (((((! [X485 : term,X486 : subst,X487 : term > $o] : (? [X490 : term] : (((X487 @ (sub @ X490 @ X486)) != $true) & ((var @ X490) = $true)) | ? [X491 : term,X492 : term] : (((X487 @ X491) = $true) & ((X487 @ X492) = $true) & ((X487 @ (ap @ X491 @ X492)) != $true)) | ? [X488 : term] : (! [X489 : term] : (((X487 @ X489) != $true) | ((X487 @ (sub @ X488 @ (push @ X489 @ id))) = $true)) & ((X487 @ (lam @ X488)) != $true)) | ((X487 @ (sub @ X485 @ X486)) = $true)) | ? [X479 : term > $o,X480 : term] : (! [X482 : term] : (((X479 @ X482) = $true) | ((var @ X482) != $true)) & ((X479 @ X480) != $true) & ! [X481 : term] : (((X479 @ X481) != $true) | ((X479 @ (lam @ X481)) = $true)) & ! [X483 : term,X484 : term] : (((X479 @ (ap @ X484 @ X483)) = $true) | ((X479 @ X483) != $true) | ((X479 @ X484) != $true)))) | ? [X478 : subst] : ((comp @ X478 @ id) != X478)) | ? [X476 : subst,X477 : term] : ((comp @ sh @ (push @ X477 @ X476)) != X476)) | ? [X474 : subst,X475 : term] : ((sub @ one @ (push @ X475 @ X474)) != X475)) <=> (induction2lem_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (ulamvarind = $true) & ((termmset_lthm = $true) <=> (? [X272 : term] : ((sub @ X272 @ id) != X272) | ! [X273 : term] : ((sub @ X273 @ id) = X273))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X168 : term,X169 : term,X170 : subst > term > subst > $o] : (((X170 @ id @ (ap @ X168 @ X169) @ id) = $true) | ((X170 @ id @ X169 @ id) != $true) | ((X170 @ id @ X168 @ id) != $true) | ? [X177 : subst,X178 : subst,X179 : term,X180 : subst] : (((X170 @ X178 @ X179 @ (comp @ X177 @ X180)) = $true) & ((X170 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177) @ X180) != $true)) | ? [X173 : subst,X174 : subst,X175 : term,X176 : subst] : (((X170 @ (comp @ X176 @ X173) @ (sub @ X175 @ X173) @ X174) = $true) & ((X170 @ X176 @ X175 @ (comp @ X173 @ X174)) != $true)) | ? [X171 : term,X172 : term] : (((X170 @ id @ (ap @ (sub @ X171 @ id) @ X172) @ id) != $true) & ((X170 @ id @ X172 @ id) = $true) & ((X170 @ id @ X171 @ id) = $true))) <=> (hoasinduction_lem2 = $true)) & (axassoc = $true) & (hoasinduction_lem3_gthm = $true) & (! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100) <=> (axvarcons = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X92 : term,X93 : subst > term > subst > $o] : (? [X94 : subst > term > term] : (! [X98 : term] : (((X93 @ id @ X98 @ id) != $true) | ((X93 @ id @ (X94 @ id @ X98) @ id) = $true)) & ! [X95 : subst,X96 : term,X97 : subst] : ((X94 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97)) = (sub @ (X94 @ X95 @ X96) @ X97)) & ((X93 @ id @ (lam @ (X94 @ sh @ one)) @ id) != $true)) | ($true = (X93 @ id @ (lam @ X92) @ id)) | ? [X99 : term] : (((X93 @ id @ (sub @ X92 @ (push @ X99 @ id)) @ id) != $true) & ((X93 @ id @ X99 @ id) = $true))) | ? [X84 : term,X85 : subst > term > subst > $o] : (((X85 @ id @ (lam @ (sub @ X84 @ (push @ one @ sh))) @ id) != $true) & ! [X87 : subst > term > term] : (? [X89 : subst,X90 : term,X91 : subst] : ((X87 @ (comp @ X91 @ X89) @ (sub @ X90 @ X89)) != (sub @ (X87 @ X91 @ X90) @ X89)) | ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id) = $true) | ? [X88 : term] : (((X85 @ id @ X88 @ id) = $true) & ($true != (X85 @ id @ (X87 @ id @ X88) @ id)))) & ! [X86 : term] : (((X85 @ id @ (sub @ X84 @ (push @ X86 @ id)) @ id) = $true) | ((X85 @ id @ X86 @ id) != $true)))) | ? [X83 : term] : ((sub @ X83 @ id) != X83))) & (pushprop = $true) & (! [X292 : term,X293 : term] : ((X292 = X293) | ((lam @ X293) != (lam @ X292))) <=> (laminj = $true)) & (termmset_gthm = $true) & (one = (d2term @ d_one)) & (! [X164 : term] : ((sub @ X164 @ id) = X164) <=> (termmset = $true)) & ((var @ (d2term @ d_one)) != $true) & ((pushprop_lem0_lthm = $true) <=> ! [X139 : term,X140 : term > $o,X141 : subst] : ? [X142 : term > $o] : ! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143))) & (hoasinduction_lem1 = $true) & (! [X383 : term] : ((sub @ X383 @ id) = X383) <=> (axvarid = $true)) & (pushprop_lem2v2 = $true) & (! [X165 : subst > term > subst > $o] : ? [X166 : term > $o] : ! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) <=> (hoasinduction_lem0 = $true)) & (induction2lem_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X65 : term] : ? [X66 : subst > term > term] : ! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68)))) & (hoasinduction_lem1_lthm = $true) & (! [X452 : subst > term > subst > $o,X453 : term] : (((X452 @ id @ (lam @ (sub @ X453 @ (push @ one @ sh))) @ id) = $true) | ? [X454 : term] : (((X452 @ id @ (sub @ X453 @ (push @ X454 @ id)) @ id) != $true) & ((X452 @ id @ X454 @ id) = $true)) | ? [X455 : subst > term > term,X456 : term] : (! [X461 : subst,X462 : term,X463 : subst] : ((sub @ (sub @ X456 @ (push @ X462 @ X461)) @ X463) = (sub @ (X455 @ X461 @ X462) @ X463)) & ! [X457 : subst,X458 : term,X459 : subst] : ((sub @ X456 @ (push @ (sub @ X458 @ X457) @ (comp @ X459 @ X457))) = (X455 @ (comp @ X459 @ X457) @ (sub @ X458 @ X457))) & ! [X460 : term] : (((X452 @ id @ X460 @ id) != $true) | ((X452 @ id @ (X455 @ id @ X460) @ id) = $true)) & ((X452 @ id @ (lam @ (X455 @ sh @ one)) @ id) != $true))) <=> (hoasinduction_lem3aaa = $true)) & (lamnotap != $true) & (apnotvar = $true) & (substmonoid_gthm = $true) & (axmap = $true) & (hoasapinj2_gthm = $true) & ! [X413 : term] : ? [X414 : d_term] : ((d2term @ X414) = X413) & (hoaslamnotvar = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X135 : term] : ? [X136 : subst > term > term] : ! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138))) & ((((! [X365 : term,X366 : subst > term > subst > $o] : (? [X368 : subst,X369 : subst,X370 : subst,X371 : term] : (((X366 @ (comp @ X369 @ X370) @ (sub @ X371 @ X370) @ X368) != $true) & ((X366 @ X369 @ X371 @ (comp @ X370 @ X368)) = $true)) | ? [X374 : subst > term > term] : (! [X375 : subst,X376 : term,X377 : subst] : ((X374 @ (comp @ X377 @ X375) @ (sub @ X376 @ X375)) = (sub @ (X374 @ X377 @ X376) @ X375)) & ! [X378 : term] : (((X366 @ id @ X378 @ id) != $true) | ((X366 @ id @ (X374 @ id @ X378) @ id) = $true)) & ((X366 @ id @ (lam @ (X374 @ sh @ one)) @ id) != $true)) | ? [X367 : term] : (((X366 @ id @ X367 @ id) != $true) & ((var @ (sub @ X367 @ id)) = $true)) | ? [X372 : term,X373 : term] : (((X366 @ id @ (ap @ (sub @ X373 @ id) @ X372) @ id) != $true) & ((X366 @ id @ X372 @ id) = $true) & ((X366 @ id @ X373 @ id) = $true)) | ? [X379 : subst,X380 : subst,X381 : term,X382 : subst] : (((X366 @ X380 @ X381 @ (comp @ X379 @ X382)) != $true) & ((X366 @ (comp @ X380 @ X379) @ (sub @ X381 @ X379) @ X382) = $true)) | ((X366 @ id @ X365 @ id) = $true)) | ? [X349 : subst > term > subst > $o,X350 : term] : (! [X361 : subst,X362 : subst,X363 : term,X364 : subst] : (((X349 @ X361 @ X363 @ (comp @ X362 @ X364)) != $true) | ((X349 @ (comp @ X361 @ X362) @ (sub @ X363 @ X362) @ X364) = $true)) & ! [X351 : subst,X352 : subst,X353 : term,X354 : subst] : (((X349 @ (comp @ X351 @ X352) @ (sub @ X353 @ X352) @ X354) != $true) | ((X349 @ X351 @ X353 @ (comp @ X352 @ X354)) = $true)) & ((X349 @ id @ (lam @ X350) @ id) != $true) & ! [X355 : subst > term > term] : (((X349 @ id @ (lam @ (X355 @ sh @ one)) @ id) = $true) | ? [X357 : subst,X358 : subst,X359 : term] : ((sub @ (X355 @ X358 @ X359) @ X357) != (X355 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357))) | ? [X356 : term] : (((X349 @ id @ X356 @ id) = $true) & ((X349 @ id @ (X355 @ id @ X356) @ id) != $true))) & ! [X360 : term] : (((X349 @ id @ (sub @ X350 @ (push @ X360 @ id)) @ id) = $true) | ((X349 @ id @ X360 @ id) != $true)))) | ? [X336 : term,X337 : subst > term > subst > $o,X338 : term] : (! [X345 : subst,X346 : term,X347 : subst,X348 : subst] : (((X337 @ (comp @ X348 @ X347) @ (sub @ X346 @ X347) @ X345) != $true) | ((X337 @ X348 @ X346 @ (comp @ X347 @ X345)) = $true)) & ! [X339 : subst,X340 : subst,X341 : term,X342 : subst] : (((X337 @ (comp @ X342 @ X339) @ (sub @ X341 @ X339) @ X340) = $true) | ((X337 @ X342 @ X341 @ (comp @ X339 @ X340)) != $true)) & ((X337 @ id @ X338 @ id) = $true) & ((X337 @ id @ (ap @ X338 @ X336) @ id) != $true) & ! [X343 : term,X344 : term] : (((X337 @ id @ X344 @ id) != $true) | ((X337 @ id @ (ap @ (sub @ X343 @ id) @ X344) @ id) = $true) | ((X337 @ id @ X343 @ id) != $true)) & ((X337 @ id @ X336 @ id) = $true))) | ? [X329 : term > $o,X330 : term] : (((X329 @ X330) != $true) & ! [X332 : term,X333 : term] : (((X329 @ (ap @ X332 @ X333)) = $true) | ((X329 @ X332) != $true) | ((X329 @ X333) != $true)) & ! [X331 : term] : (($true = (X329 @ X331)) | ((var @ X331) != $true)) & ! [X334 : term] : (? [X335 : term] : (((X329 @ X335) = $true) & ((X329 @ (sub @ X334 @ (push @ X335 @ id))) != $true)) | ((X329 @ (lam @ X334)) = $true)))) <=> (hoasinduction_lthm = $true)) & (hoasinduction_lem1_gthm = $true) & (! [X495 : term > $o,X496 : term] : (? [X497 : term,X498 : term] : (((X495 @ (ap @ X497 @ X498)) != $true) & ((X495 @ X497) = $true) & ((X495 @ X498) = $true)) | ? [X499 : term] : (((var @ X499) = $true) & ((X495 @ X499) != $true)) | ((X495 @ X496) = $true) | ? [X500 : term] : (((X495 @ (lam @ X500)) != $true) & ((X495 @ X500) = $true))) <=> (induction = $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X317 : term,X318 : term > $o,X319 : subst] : (? [X324 : term] : (((var @ X324) = $true) & ((X318 @ (sub @ X324 @ X319)) != $true)) | ? [X320 : term] : (! [X321 : term] : (((X318 @ (sub @ X320 @ (push @ X321 @ id))) = $true) | ((X318 @ X321) != $true)) & ((X318 @ (lam @ X320)) != $true)) | ? [X322 : term,X323 : term] : (((X318 @ X322) = $true) & ((X318 @ (ap @ X323 @ X322)) != $true) & ((X318 @ X323) = $true)) | ((X318 @ (sub @ X317 @ X319)) = $true)) <=> (induction2lem = $true)) & ((hoaslaminj = $true) <=> ! [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) != (lam @ (X465 @ sh @ one))) | ((X465 @ X466 @ X464) = (X467 @ X466 @ X464)) | ? [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) != (sub @ (X465 @ X473 @ X471) @ X472)) | ? [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) != (sub @ (X467 @ X470 @ X469) @ X468)))) & ((((! [X411 : subst] : ((comp @ id @ X411) = X411) & ! [X412 : subst] : ((comp @ X412 @ id) = X412)) | ? [X410 : subst] : ((comp @ X410 @ id) != X410)) | ? [X409 : subst] : ((comp @ id @ X409) != X409)) <=> (substmonoid_lthm = $true)) & ((pushprop_lem1 = $true) <=> ! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (((X60 @ X61) != $true) | ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) = $true))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (induction2_gthm = $true) & (pushprop_lem0_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 = X258) | ((ap @ (sub @ X260 @ id) @ X258) != (ap @ (sub @ X259 @ id) @ X257))) | ? [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) = (ap @ X255 @ X254)) & (X253 != X254)))) & (pushprop_lem1_gthm = $true) & ((hoasinduction = $true) <=> ! [X274 : subst > term > subst > $o,X275 : term] : (? [X290 : term,X291 : term] : (((X274 @ id @ X290 @ id) = $true) & ((X274 @ id @ (ap @ (sub @ X291 @ id) @ X290) @ id) != $true) & ((X274 @ id @ X291 @ id) = $true)) | ((X274 @ id @ X275 @ id) = $true) | ? [X285 : subst > term > term] : (! [X286 : term,X287 : subst,X288 : subst] : ((X285 @ (comp @ X287 @ X288) @ (sub @ X286 @ X288)) = (sub @ (X285 @ X287 @ X286) @ X288)) & ! [X289 : term] : (((X274 @ id @ (X285 @ id @ X289) @ id) = $true) | ((X274 @ id @ X289 @ id) != $true)) & ((X274 @ id @ (lam @ (X285 @ sh @ one)) @ id) != $true)) | ? [X277 : term,X278 : subst,X279 : subst,X280 : subst] : (((X274 @ X279 @ X277 @ (comp @ X278 @ X280)) != $true) & ((X274 @ (comp @ X279 @ X278) @ (sub @ X277 @ X278) @ X280) = $true)) | ? [X276 : term] : (((X274 @ id @ X276 @ id) != $true) & ((var @ (sub @ X276 @ id)) = $true)) | ? [X281 : subst,X282 : term,X283 : subst,X284 : subst] : (((X274 @ (comp @ X283 @ X284) @ (sub @ X282 @ X284) @ X281) != $true) & ((X274 @ X283 @ X282 @ (comp @ X284 @ X281)) = $true)))) & ((pushprop_lem1v2 = $true) <=> ! [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) = $true) | ((X522 @ X521) != $true) | ? [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) != (X523 @ X524)))) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & (! [X2 : subst > term > subst > $o] : ? [X3 : term > $o] : ! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) <=> (hoasinduction_lem0_lthm = $true)) & ((axshiftcons = $true) <=> ! [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) = X493)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((? [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) != X228) | ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (((X230 @ X233) != $true) | ? [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) != (X232 @ X234)) | ((X232 @ one) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & (! [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 = X418) | ((ap @ (sub @ X417 @ id) @ X418) != (ap @ (sub @ X415 @ id) @ X416))) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X502 : term > $o,X503 : subst > term > subst > $o,X504 : term] : (? [X515 : subst,X516 : subst,X517 : subst,X518 : term] : (((X503 @ X516 @ X518 @ (comp @ X515 @ X517)) != $true) & ((X503 @ (comp @ X516 @ X515) @ (sub @ X518 @ X515) @ X517) = $true)) | ? [X519 : term] : (((X502 @ (sub @ X504 @ (push @ X519 @ id))) != $true) & ((X502 @ X519) = $true)) | ? [X510 : subst > term > term] : (! [X512 : subst,X513 : term,X514 : subst] : ((sub @ (X510 @ X514 @ X513) @ X512) = (X510 @ (comp @ X514 @ X512) @ (sub @ X513 @ X512))) & ! [X511 : term] : (((X503 @ id @ (X510 @ id @ X511) @ id) = $true) | ((X503 @ id @ X511 @ id) != $true)) & ((X503 @ id @ (lam @ (X510 @ sh @ one)) @ id) != $true)) | ? [X506 : subst,X507 : subst,X508 : subst,X509 : term] : (((X503 @ (comp @ X506 @ X507) @ (sub @ X509 @ X507) @ X508) != $true) & ((X503 @ X506 @ X509 @ (comp @ X507 @ X508)) = $true)) | ((X502 @ (lam @ X504)) = $true) | ? [X505 : term] : ((X503 @ id @ X505 @ id) != (X502 @ X505))) | ? [X501 : term] : ((sub @ X501 @ id) != X501))) & (pushprop_lem3v2_lthm = $true) & ! [X265 : d_term] : (d_one = X265) & (hoasapinj1_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X146 : subst > term > subst > $o,X147 : term > $o,X148 : term] : (? [X159 : subst > term > term] : (((X146 @ id @ (lam @ (X159 @ sh @ one)) @ id) != $true) & ! [X161 : term,X162 : subst,X163 : subst] : ((X159 @ (comp @ X163 @ X162) @ (sub @ X161 @ X162)) = (sub @ (X159 @ X163 @ X161) @ X162)) & ! [X160 : term] : (((X146 @ id @ X160 @ id) != $true) | ((X146 @ id @ (X159 @ id @ X160) @ id) = $true))) | ((X147 @ (lam @ X148)) = $true) | ? [X149 : term] : ((X147 @ X149) != (X146 @ id @ X149 @ id)) | ? [X154 : subst,X155 : subst,X156 : subst,X157 : term] : (((X146 @ X155 @ X157 @ (comp @ X154 @ X156)) = $true) & ((X146 @ (comp @ X155 @ X154) @ (sub @ X157 @ X154) @ X156) != $true)) | ? [X158 : term] : (((X147 @ X158) = $true) & ((X147 @ (sub @ X148 @ (push @ X158 @ id))) != $true)) | ? [X150 : subst,X151 : subst,X152 : subst,X153 : term] : (((X146 @ X151 @ X153 @ (comp @ X150 @ X152)) != $true) & ((X146 @ (comp @ X151 @ X150) @ (sub @ X153 @ X150) @ X152) = $true)))) & (hoasapnotvar = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((axidl = $true) <=> ! [X72 : subst] : ((comp @ id @ X72) = X72)) & (((((! [X42 : subst > term > subst > $o,X43 : term] : (? [X44 : subst > term > term] : (((X42 @ id @ (lam @ (X44 @ sh @ one)) @ id) != $true) & ! [X48 : term] : (((X42 @ id @ (X44 @ id @ X48) @ id) = $true) | ((X42 @ id @ X48 @ id) != $true)) & ! [X45 : subst,X46 : subst,X47 : term] : ((sub @ (X44 @ X45 @ X47) @ X46) = (X44 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46)))) | ? [X50 : subst,X51 : term,X52 : subst,X53 : subst] : (((X42 @ (comp @ X53 @ X52) @ (sub @ X51 @ X52) @ X50) != $true) & ((X42 @ X53 @ X51 @ (comp @ X52 @ X50)) = $true)) | ? [X49 : term] : (((X42 @ id @ X49 @ id) != $true) & ((var @ (sub @ X49 @ id)) = $true)) | ((X42 @ id @ X43 @ id) = $true) | ? [X54 : term,X55 : term] : (((X42 @ id @ X55 @ id) = $true) & ((X42 @ id @ X54 @ id) = $true) & ((X42 @ id @ (ap @ (sub @ X54 @ id) @ X55) @ id) != $true)) | ? [X56 : subst,X57 : subst,X58 : subst,X59 : term] : (((X42 @ X58 @ X59 @ (comp @ X57 @ X56)) != $true) & ((X42 @ (comp @ X58 @ X57) @ (sub @ X59 @ X57) @ X56) = $true))) | ? [X32 : term > $o,X33 : subst > term > subst > $o,X34 : term] : (! [X35 : term] : (((X32 @ (sub @ X34 @ (push @ X35 @ id))) = $true) | ((X32 @ X35) != $true)) & ((X32 @ (lam @ X34)) != $true) & ! [X41 : term] : ((X33 @ id @ X41 @ id) = (X32 @ X41)) & ! [X36 : subst > term > term] : (((X33 @ id @ (lam @ (X36 @ sh @ one)) @ id) = $true) | ? [X38 : subst,X39 : term,X40 : subst] : ((X36 @ (comp @ X38 @ X40) @ (sub @ X39 @ X40)) != (sub @ (X36 @ X38 @ X39) @ X40)) | ? [X37 : term] : (((X33 @ id @ X37 @ id) = $true) & ((X33 @ id @ (X36 @ id @ X37) @ id) != $true))))) | ? [X31 : term] : ((sub @ X31 @ id) != X31)) | ? [X24 : term,X25 : term > $o] : (! [X30 : term] : (((var @ X30) != $true) | ((X25 @ X30) = $true)) & ((X25 @ X24) != $true) & ! [X26 : term,X27 : term] : (((X25 @ (ap @ X27 @ X26)) = $true) | ((X25 @ X27) != $true) | ((X25 @ X26) != $true)) & ! [X28 : term] : (((X25 @ (lam @ X28)) = $true) | ? [X29 : term] : (((X25 @ (sub @ X28 @ (push @ X29 @ id))) != $true) & ((X25 @ X29) = $true))))) | ? [X21 : subst > term > subst > $o] : ! [X22 : term > $o] : ? [X23 : term] : ((X21 @ id @ X23 @ id) != (X22 @ X23))) <=> (hoasinduction_lthm_3 = $true)) & ((d2subst @ d_id) = id) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X271 : d_subst] : (d_id = X271) & (lamnotvar = $true) & ((induction2_lthm = $true) <=> ((! [X310 : term,X311 : term > $o] : (? [X313 : term] : (! [X314 : term] : (((X311 @ X314) != $true) | ((X311 @ (sub @ X313 @ (push @ X314 @ id))) = $true)) & ((X311 @ (lam @ X313)) != $true)) | ? [X315 : term,X316 : term] : (((X311 @ X315) = $true) & ((X311 @ (ap @ X315 @ X316)) != $true) & ((X311 @ X316) = $true)) | ? [X312 : term] : (((var @ X312) = $true) & ((X311 @ X312) != $true)) | ((X311 @ X310) = $true)) | ? [X302 : subst,X303 : term > $o,X304 : term] : (! [X309 : term] : (((var @ X309) != $true) | ((X303 @ (sub @ X309 @ X302)) = $true)) & ((X303 @ (sub @ X304 @ X302)) != $true) & ! [X307 : term] : (? [X308 : term] : (((X303 @ (sub @ X307 @ (push @ X308 @ id))) != $true) & ((X303 @ X308) = $true)) | ((X303 @ (lam @ X307)) = $true)) & ! [X305 : term,X306 : term] : (((X303 @ (ap @ X305 @ X306)) = $true) | ((X303 @ X305) != $true) | ((X303 @ X306) != $true)))) | ? [X301 : term] : ((sub @ X301 @ id) != X301))) & (ulamvarsh = $true) & (! [X144 : term] : ? [X145 : subst > term > term] : ((X145 @ sh @ one) = (sub @ X144 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X266 : subst,X267 : term > $o,X268 : term] : ? [X269 : term > $o] : ! [X270 : term] : ((X269 @ X270) = (X267 @ (sub @ X270 @ (push @ X268 @ X266)))) <=> (pushprop_lem0 = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X419 : subst > term > subst > $o,X420 : term] : (((X419 @ id @ (lam @ (sub @ X420 @ (push @ one @ sh))) @ id) = $true) | ? [X422 : subst > term > term] : (! [X423 : term,X424 : subst,X425 : subst] : ((X422 @ (comp @ X425 @ X424) @ (sub @ X423 @ X424)) = (sub @ (X422 @ X425 @ X423) @ X424)) & ((X419 @ id @ (lam @ (X422 @ sh @ one)) @ id) != $true) & ! [X426 : term] : (((X419 @ id @ (X422 @ id @ X426) @ id) = $true) | ((X419 @ id @ X426 @ id) != $true))) | ? [X421 : term] : (((X419 @ id @ X421 @ id) = $true) & ((X419 @ id @ (sub @ X420 @ (push @ X421 @ id)) @ id) != $true)))) & ((d2subst @ d_id) = sh) & (! [X294 : term,X295 : term > $o] : (? [X299 : term] : (((X295 @ (lam @ X299)) != $true) & ! [X300 : term] : (((X295 @ X300) != $true) | ((X295 @ (sub @ X299 @ (push @ X300 @ id))) = $true))) | ((X295 @ X294) = $true) | ? [X296 : term] : (((var @ X296) = $true) & ((X295 @ X296) != $true)) | ? [X297 : term,X298 : term] : (($true != (X295 @ (ap @ X297 @ X298))) & ((X295 @ X297) = $true) & ((X295 @ X298) = $true))) <=> (induction2 = $true)) & (pushprop_lthm = $true) & (hoaslamnotap_gthm = $true) & (hoaslaminj_gthm = $true) & ((((! [X11 : subst > term > term,X12 : subst > term > term,X13 : term,X14 : subst] : (((lam @ (X11 @ sh @ one)) != (lam @ (X12 @ sh @ one))) | ? [X15 : term,X16 : subst,X17 : subst] : ((sub @ (X11 @ X17 @ X15) @ X16) != (X11 @ (comp @ X17 @ X16) @ (sub @ X15 @ X16))) | ? [X18 : subst,X19 : term,X20 : subst] : ((sub @ (X12 @ X20 @ X19) @ X18) != (X12 @ (comp @ X20 @ X18) @ (sub @ X19 @ X18))) | ((X11 @ X14 @ X13) = (X12 @ X14 @ X13))) | ? [X9 : term,X10 : term] : (((lam @ X10) = (lam @ X9)) & (X9 != X10))) | ? [X7 : term,X8 : subst] : ((comp @ sh @ (push @ X7 @ X8)) != X8)) | ? [X5 : term,X6 : subst] : ((sub @ one @ (push @ X5 @ X6)) != X5)) <=> (hoaslaminj_lthm = $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & (((! [X436 : term,X437 : subst > term > subst > $o] : (? [X443 : term,X444 : subst,X445 : subst,X446 : subst] : (((X437 @ (comp @ X446 @ X445) @ (sub @ X443 @ X445) @ X444) = $true) & ($true != (X437 @ X446 @ X443 @ (comp @ X445 @ X444)))) | ((X437 @ id @ (lam @ X436) @ id) = $true) | ? [X438 : subst > term > term] : (((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id) != $true) & ! [X439 : term] : (((X437 @ id @ (X438 @ id @ X439) @ id) = $true) | ((X437 @ id @ X439 @ id) != $true)) & ! [X440 : term,X441 : subst,X442 : subst] : ((sub @ (X438 @ X441 @ X440) @ X442) = (X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442)))) | ? [X448 : subst,X449 : subst,X450 : subst,X451 : term] : (((X437 @ (comp @ X450 @ X449) @ (sub @ X451 @ X449) @ X448) != $true) & ((X437 @ X450 @ X451 @ (comp @ X449 @ X448)) = $true)) | ? [X447 : term] : (((X437 @ id @ (sub @ X436 @ (push @ X447 @ id)) @ id) != $true) & ((X437 @ id @ X447 @ id) = $true))) | ? [X428 : term,X429 : subst > term > subst > $o] : (! [X431 : subst > term > term] : (? [X432 : term] : (((X429 @ id @ X432 @ id) = $true) & ((X429 @ id @ (X431 @ id @ X432) @ id) != $true)) | ((X429 @ id @ (lam @ (X431 @ sh @ one)) @ id) = $true) | ? [X433 : term,X434 : subst,X435 : subst] : ((X431 @ (comp @ X435 @ X434) @ (sub @ X433 @ X434)) != (sub @ (X431 @ X435 @ X433) @ X434))) & ! [X430 : term] : (((X429 @ id @ X430 @ id) != $true) | ((X429 @ id @ (sub @ X428 @ (push @ X430 @ id)) @ id) = $true)) & ((X429 @ id @ (lam @ (sub @ X428 @ (push @ one @ sh))) @ id) != $true))) | ? [X427 : term] : ((sub @ X427 @ id) != X427)) <=> (hoasinduction_lem3_lthm = $true)) & (pushprop_gthm = $true) & (! [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192)) | (X193 = X194)) <=> (hoasapinj1 = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term,X237 : term > $o] : (? [X244 : term] : ((X237 @ X244) != (X235 @ id @ X244 @ id)) | ($true = (X237 @ (lam @ X236))) | ? [X239 : subst > term > term] : (! [X240 : subst,X241 : term,X242 : subst] : ((X239 @ (comp @ X242 @ X240) @ (sub @ X241 @ X240)) = (sub @ (X239 @ X242 @ X241) @ X240)) & ! [X243 : term] : (((X235 @ id @ X243 @ id) != $true) | ((X235 @ id @ (X239 @ id @ X243) @ id) = $true)) & ((X235 @ id @ (lam @ (X239 @ sh @ one)) @ id) != $true)) | ? [X238 : term] : (((X237 @ X238) = $true) & ((X237 @ (sub @ X236 @ (push @ X238 @ id))) != $true)))) & (axapp = $true) & (hoasapnotvar_lthm = $true) & (((((! [X219 : subst > term > subst > $o,X220 : term] : (? [X223 : subst > term > term] : (($true != (X219 @ id @ (lam @ (X223 @ sh @ one)) @ id)) & ! [X225 : subst,X226 : term,X227 : subst] : ((X223 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225)) = (sub @ (X223 @ X227 @ X226) @ X225)) & ! [X224 : term] : (((X219 @ id @ (X223 @ id @ X224) @ id) = $true) | ((X219 @ id @ X224 @ id) != $true))) | ? [X221 : term,X222 : term] : (((X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id) != $true) & ((X219 @ id @ X221 @ id) = $true) & ((X219 @ id @ X222 @ id) = $true)) | ((X219 @ id @ X220 @ id) = $true)) | ? [X209 : subst > term > subst > $o,X210 : term > $o,X211 : term] : (! [X218 : term] : ((X209 @ id @ X218 @ id) = (X210 @ X218)) & ((X210 @ (lam @ X211)) != $true) & ! [X212 : term] : (((X210 @ (sub @ X211 @ (push @ X212 @ id))) = $true) | ((X210 @ X212) != $true)) & ! [X213 : subst > term > term] : (? [X215 : subst,X216 : subst,X217 : term] : ((sub @ (X213 @ X216 @ X217) @ X215) != (X213 @ (comp @ X216 @ X215) @ (sub @ X217 @ X215))) | ((X209 @ id @ (lam @ (X213 @ sh @ one)) @ id) = $true) | ? [X214 : term] : (((X209 @ id @ (X213 @ id @ X214) @ id) != $true) & ((X209 @ id @ X214 @ id) = $true))))) | ? [X208 : term] : ((sub @ X208 @ id) != X208)) | ? [X201 : term > $o,X202 : term] : (((X201 @ X202) != $true) & ! [X204 : term,X205 : term] : (((X201 @ X205) != $true) | ((X201 @ (ap @ X205 @ X204)) = $true) | ((X201 @ X204) != $true)) & ! [X206 : term] : (? [X207 : term] : (((X201 @ (sub @ X206 @ (push @ X207 @ id))) != $true) & ((X201 @ X207) = $true)) | ((X201 @ (lam @ X206)) = $true)) & ! [X203 : term] : (((X201 @ X203) = $true) | ((var @ X203) != $true)))) | ? [X198 : subst > term > subst > $o] : ! [X199 : term > $o] : ? [X200 : term] : ((X198 @ id @ X200 @ id) != (X199 @ X200))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | ((ap @ X261 @ X263) != (ap @ X262 @ X264))) <=> (apinj1 = $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3b = $true) <=> ! [X73 : term] : ? [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) = (X74 @ sh @ one))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X71 : subst] : ((comp @ X71 @ id) = X71) <=> (axidr = $true)) & ((hoasapinj1_lthm = $true) <=> ((! [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 = X408) | ((ap @ (sub @ X408 @ id) @ X406) != (ap @ (sub @ X405 @ id) @ X407))) | ? [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) = (ap @ X404 @ X402)) & (X401 != X404))) | ? [X400 : term] : ((sub @ X400 @ id) != X400))) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X108 : term,X109 : subst > term > subst > $o,X110 : term > $o] : (? [X111 : term] : (((X110 @ X111) = $true) & ($true != (X110 @ (sub @ X108 @ (push @ X111 @ id))))) | ((X110 @ (lam @ X108)) = $true) | ? [X113 : subst > term > term] : (((X109 @ id @ (lam @ (X113 @ sh @ one)) @ id) != $true) & ! [X115 : subst,X116 : term,X117 : subst] : ((X113 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117)) = (sub @ (X113 @ X115 @ X116) @ X117)) & ! [X114 : term] : (((X109 @ id @ X114 @ id) != $true) | ((X109 @ id @ (X113 @ id @ X114) @ id) = $true))) | ? [X112 : term] : ((X109 @ id @ X112 @ id) != (X110 @ X112))) | ? [X107 : term] : ((sub @ X107 @ id) != X107)) | ? [X103 : term] : ! [X104 : subst > term > term] : ? [X105 : subst,X106 : term] : ((sub @ X103 @ (push @ X106 @ X105)) != (X104 @ X105 @ X106)))) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X183 : term,X184 : subst > term > subst > $o] : (? [X185 : term,X186 : term] : (((X184 @ id @ X186 @ id) = $true) & ((X184 @ id @ X185 @ id) = $true) & ((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) != $true)) | ? [X187 : subst > term > term] : (! [X188 : term] : (((X184 @ id @ (X187 @ id @ X188) @ id) = $true) | ((X184 @ id @ X188 @ id) != $true)) & ((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) != $true) & ! [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) = (sub @ (X187 @ X189 @ X191) @ X190))) | ((X184 @ id @ X183 @ id) = $true))) & (pushprop_lem1v2_gthm = $true) & (! [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) != (ap @ X326 @ X328)) | (X325 = X328)) <=> (apinj2 = $true)) & (hoasapnotvar_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X384 : term,X385 : subst > term > subst > $o] : (? [X386 : subst,X387 : subst,X388 : subst,X389 : term] : (((X385 @ (comp @ X386 @ X388) @ (sub @ X389 @ X388) @ X387) != $true) & ((X385 @ X386 @ X389 @ (comp @ X388 @ X387)) = $true)) | ? [X391 : subst > term > term] : (! [X395 : term] : (((X385 @ id @ X395 @ id) != $true) | ((X385 @ id @ (X391 @ id @ X395) @ id) = $true)) & ! [X392 : subst,X393 : subst,X394 : term] : ((sub @ (X391 @ X393 @ X394) @ X392) = (X391 @ (comp @ X393 @ X392) @ (sub @ X394 @ X392))) & ((X385 @ id @ (lam @ (X391 @ sh @ one)) @ id) != $true)) | ? [X390 : term] : (((X385 @ id @ X390 @ id) = $true) & ((X385 @ id @ (sub @ X384 @ (push @ X390 @ id)) @ id) != $true)) | ? [X396 : subst,X397 : subst,X398 : term,X399 : subst] : (((X385 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X399) = $true) & ((X385 @ X396 @ X398 @ (comp @ X397 @ X399)) != $true)) | ((X385 @ id @ (lam @ X384) @ id) = $true))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (hoaslamnotvar_lthm = $true) & (pushprop_lem2v2_gthm = $true) & ! [X181 : d_subst,X182 : d_subst] : ((X181 = X182) | ((d2subst @ X182) != (d2subst @ X181))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoaslamnotap = $true) <=> ! [X247 : term,X248 : subst > term > term,X249 : term] : (? [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) != (sub @ (X248 @ X250 @ X252) @ X251)) | ((ap @ (sub @ X247 @ id) @ X249) != (lam @ (X248 @ sh @ one))))) & (((! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (((X122 @ X124) != $true) | ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) = $true)) | ? [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) != X121)) | ? [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) != X118)) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (! [X525 : term > $o,X526 : term,X527 : term,X528 : subst > term > subst > $o] : (? [X532 : subst,X533 : subst,X534 : term,X535 : subst] : (((X528 @ X533 @ X534 @ (comp @ X532 @ X535)) != $true) & ((X528 @ (comp @ X533 @ X532) @ (sub @ X534 @ X532) @ X535) = $true)) | ((X525 @ X527) != $true) | ? [X536 : subst,X537 : subst,X538 : term,X539 : subst] : (((X528 @ X536 @ X538 @ (comp @ X537 @ X539)) = $true) & ((X528 @ (comp @ X536 @ X537) @ (sub @ X538 @ X537) @ X539) != $true)) | ? [X529 : term] : ((X528 @ id @ X529 @ id) != (X525 @ X529)) | ((X525 @ X526) != $true) | ((X525 @ (ap @ X527 @ X526)) = $true) | ? [X530 : term,X531 : term] : (((X528 @ id @ X531 @ id) = $true) & ((X528 @ id @ X530 @ id) = $true) & ((X528 @ id @ (ap @ (sub @ X531 @ id) @ X530) @ id) != $true))) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lthm_orig = $true) & ((substmonoid = $true) <=> (! [X1 : subst] : ((comp @ id @ X1) = X1) & ! [X0 : subst] : ((comp @ X0 @ id) = X0))) & ((hoasinduction_lem3a = $true) <=> ! [X75 : subst > term > subst > $o,X76 : term] : (? [X77 : subst > term > term] : (((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) != $true) & ! [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) = (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78))) & ! [X81 : term] : (((X75 @ id @ (X77 @ id @ X81) @ id) = $true) | ((X75 @ id @ X81 @ id) != $true))) | ? [X82 : term] : (((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) != $true) & ((X75 @ id @ X82 @ id) = $true)) | ((X75 @ id @ (lam @ X76) @ id) = $true))) & (axclos = $true) & (axvarshift = $true) & ! [X245 : d_term,X246 : d_term] : (((d2term @ X246) != (d2term @ X245)) | (X245 = X246)) & (ulamvar1 != $true)),
% 9.05/2.75 inference(ennf_transformation,[status(thm)],[f11])).
% 9.05/2.75 thf(f11,plain,(
% 9.05/2.75 (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)))))))))))))) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X127 : subst > term > subst > $o,X128 : term] : (~! [X130 : subst > term > term] : (~! [X131 : term,X132 : subst,X133 : subst] : ((X130 @ (comp @ X132 @ X133) @ (sub @ X131 @ X133)) = (sub @ (X130 @ X132 @ X131) @ X133)) | ~! [X134 : term] : (((X127 @ id @ (X130 @ id @ X134) @ id) = $true) | ((X127 @ id @ X134 @ id) != $true)) | ((X127 @ id @ (lam @ (X130 @ sh @ one)) @ id) = $true)) | ((X127 @ id @ (lam @ (sub @ X128 @ (push @ one @ sh))) @ id) = $true) | ~! [X129 : term] : (((X127 @ id @ (sub @ X128 @ (push @ X129 @ id)) @ id) = $true) | ((X127 @ id @ X129 @ id) != $true)))) & (axabs = $true) & ((shinj = $true) <=> ! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh)))) & ((! [X474 : subst,X475 : term] : ((sub @ one @ (push @ X475 @ X474)) = X475) => (! [X476 : subst,X477 : term] : ((comp @ sh @ (push @ X477 @ X476)) = X476) => (! [X478 : subst] : ((comp @ X478 @ id) = X478) => (! [X479 : term > $o,X480 : term] : (~! [X483 : term,X484 : term] : (((X479 @ (ap @ X484 @ X483)) = $true) | ((X479 @ X483) != $true) | ((X479 @ X484) != $true)) | ((X479 @ X480) = $true) | ~! [X481 : term] : (((X479 @ X481) != $true) | ((X479 @ (lam @ X481)) = $true)) | ~! [X482 : term] : (((X479 @ X482) = $true) | ((var @ X482) != $true))) => ! [X485 : term,X486 : subst,X487 : term > $o] : (~! [X490 : term] : (((var @ X490) != $true) | ((X487 @ (sub @ X490 @ X486)) = $true)) | ~! [X491 : term,X492 : term] : (((X487 @ X492) != $true) | ((X487 @ (ap @ X491 @ X492)) = $true) | ((X487 @ X491) != $true)) | ((X487 @ (sub @ X485 @ X486)) = $true) | ~! [X488 : term] : (~! [X489 : term] : (((X487 @ X489) != $true) | ((X487 @ (sub @ X488 @ (push @ X489 @ id))) = $true)) | ((X487 @ (lam @ X488)) = $true))))))) <=> (induction2lem_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (ulamvarind = $true) & ((! [X272 : term] : ((sub @ X272 @ id) = X272) => ! [X273 : term] : ((sub @ X273 @ id) = X273)) <=> (termmset_lthm = $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lem2 = $true) <=> ! [X168 : term,X169 : term,X170 : subst > term > subst > $o] : (((X170 @ id @ X169 @ id) != $true) | ~! [X171 : term,X172 : term] : (((X170 @ id @ (ap @ (sub @ X171 @ id) @ X172) @ id) = $true) | ((X170 @ id @ X171 @ id) != $true) | ((X170 @ id @ X172 @ id) != $true)) | ((X170 @ id @ (ap @ X168 @ X169) @ id) = $true) | ((X170 @ id @ X168 @ id) != $true) | ~! [X173 : subst,X174 : subst,X175 : term,X176 : subst] : (((X170 @ X176 @ X175 @ (comp @ X173 @ X174)) = $true) | ((X170 @ (comp @ X176 @ X173) @ (sub @ X175 @ X173) @ X174) != $true)) | ~! [X177 : subst,X178 : subst,X179 : term,X180 : subst] : (((X170 @ X178 @ X179 @ (comp @ X177 @ X180)) != $true) | ((X170 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177) @ X180) = $true)))) & (axassoc = $true) & (hoasinduction_lem3_gthm = $true) & (! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100) <=> (axvarcons = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X83 : term] : ((sub @ X83 @ id) = X83) => (! [X84 : term,X85 : subst > term > subst > $o] : (~! [X87 : subst > term > term] : (~! [X88 : term] : (($true = (X85 @ id @ (X87 @ id @ X88) @ id)) | ((X85 @ id @ X88 @ id) != $true)) | ~! [X89 : subst,X90 : term,X91 : subst] : ((X87 @ (comp @ X91 @ X89) @ (sub @ X90 @ X89)) = (sub @ (X87 @ X91 @ X90) @ X89)) | ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id) = $true)) | ~! [X86 : term] : (((X85 @ id @ (sub @ X84 @ (push @ X86 @ id)) @ id) = $true) | ((X85 @ id @ X86 @ id) != $true)) | ((X85 @ id @ (lam @ (sub @ X84 @ (push @ one @ sh))) @ id) = $true)) => ! [X92 : term,X93 : subst > term > subst > $o] : (($true = (X93 @ id @ (lam @ X92) @ id)) | ~! [X94 : subst > term > term] : (((X93 @ id @ (lam @ (X94 @ sh @ one)) @ id) = $true) | ~! [X98 : term] : (((X93 @ id @ X98 @ id) != $true) | ((X93 @ id @ (X94 @ id @ X98) @ id) = $true)) | ~! [X95 : subst,X96 : term,X97 : subst] : ((X94 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97)) = (sub @ (X94 @ X95 @ X96) @ X97))) | ~! [X99 : term] : (((X93 @ id @ (sub @ X92 @ (push @ X99 @ id)) @ id) = $true) | ((X93 @ id @ X99 @ id) != $true)))))) & (pushprop = $true) & (! [X292 : term,X293 : term] : ((X292 = X293) | ((lam @ X293) != (lam @ X292))) <=> (laminj = $true)) & (termmset_gthm = $true) & (one = (d2term @ d_one)) & (! [X164 : term] : ((sub @ X164 @ id) = X164) <=> (termmset = $true)) & ((var @ (d2term @ d_one)) != $true) & (! [X139 : term,X140 : term > $o,X141 : subst] : ~! [X142 : term > $o] : ~! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143)) <=> (pushprop_lem0_lthm = $true)) & (hoasinduction_lem1 = $true) & (! [X383 : term] : ((sub @ X383 @ id) = X383) <=> (axvarid = $true)) & (pushprop_lem2v2 = $true) & (! [X165 : subst > term > subst > $o] : ~! [X166 : term > $o] : ~! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) <=> (hoasinduction_lem0 = $true)) & (induction2lem_gthm = $true) & (! [X65 : term] : ~! [X66 : subst > term > term] : ~! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X452 : subst > term > subst > $o,X453 : term] : (~! [X454 : term] : (((X452 @ id @ X454 @ id) != $true) | ((X452 @ id @ (sub @ X453 @ (push @ X454 @ id)) @ id) = $true)) | ~! [X455 : subst > term > term,X456 : term] : (~! [X460 : term] : (((X452 @ id @ X460 @ id) != $true) | ((X452 @ id @ (X455 @ id @ X460) @ id) = $true)) | ~! [X457 : subst,X458 : term,X459 : subst] : ((sub @ X456 @ (push @ (sub @ X458 @ X457) @ (comp @ X459 @ X457))) = (X455 @ (comp @ X459 @ X457) @ (sub @ X458 @ X457))) | ((X452 @ id @ (lam @ (X455 @ sh @ one)) @ id) = $true) | ~! [X461 : subst,X462 : term,X463 : subst] : ((sub @ (sub @ X456 @ (push @ X462 @ X461)) @ X463) = (sub @ (X455 @ X461 @ X462) @ X463))) | ((X452 @ id @ (lam @ (sub @ X453 @ (push @ one @ sh))) @ id) = $true))) & (lamnotap != $true) & (apnotvar = $true) & (substmonoid_gthm = $true) & (axmap = $true) & (hoasapinj2_gthm = $true) & ! [X413 : term] : ? [X414 : d_term] : ((d2term @ X414) = X413) & (hoaslamnotvar = $true) & (! [X135 : term] : ~! [X136 : subst > term > term] : ~! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138)) <=> (hoasinduction_lem3v2_f = $true)) & ((! [X329 : term > $o,X330 : term] : (~! [X331 : term] : (($true = (X329 @ X331)) | ((var @ X331) != $true)) | ((X329 @ X330) = $true) | ~! [X334 : term] : (~! [X335 : term] : (((X329 @ (sub @ X334 @ (push @ X335 @ id))) = $true) | ((X329 @ X335) != $true)) | ((X329 @ (lam @ X334)) = $true)) | ~! [X332 : term,X333 : term] : (((X329 @ (ap @ X332 @ X333)) = $true) | ((X329 @ X332) != $true) | ((X329 @ X333) != $true))) => (! [X336 : term,X337 : subst > term > subst > $o,X338 : term] : (((X337 @ id @ X338 @ id) != $true) | ~! [X343 : term,X344 : term] : (((X337 @ id @ X344 @ id) != $true) | ((X337 @ id @ (ap @ (sub @ X343 @ id) @ X344) @ id) = $true) | ((X337 @ id @ X343 @ id) != $true)) | ((X337 @ id @ (ap @ X338 @ X336) @ id) = $true) | ~! [X339 : subst,X340 : subst,X341 : term,X342 : subst] : (((X337 @ (comp @ X342 @ X339) @ (sub @ X341 @ X339) @ X340) = $true) | ((X337 @ X342 @ X341 @ (comp @ X339 @ X340)) != $true)) | ((X337 @ id @ X336 @ id) != $true) | ~! [X345 : subst,X346 : term,X347 : subst,X348 : subst] : (((X337 @ (comp @ X348 @ X347) @ (sub @ X346 @ X347) @ X345) != $true) | ((X337 @ X348 @ X346 @ (comp @ X347 @ X345)) = $true))) => (! [X349 : subst > term > subst > $o,X350 : term] : (((X349 @ id @ (lam @ X350) @ id) = $true) | ~! [X351 : subst,X352 : subst,X353 : term,X354 : subst] : (((X349 @ (comp @ X351 @ X352) @ (sub @ X353 @ X352) @ X354) != $true) | ((X349 @ X351 @ X353 @ (comp @ X352 @ X354)) = $true)) | ~! [X360 : term] : (((X349 @ id @ (sub @ X350 @ (push @ X360 @ id)) @ id) = $true) | ((X349 @ id @ X360 @ id) != $true)) | ~! [X355 : subst > term > term] : (~! [X357 : subst,X358 : subst,X359 : term] : ((sub @ (X355 @ X358 @ X359) @ X357) = (X355 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357))) | ((X349 @ id @ (lam @ (X355 @ sh @ one)) @ id) = $true) | ~! [X356 : term] : (((X349 @ id @ X356 @ id) != $true) | ((X349 @ id @ (X355 @ id @ X356) @ id) = $true))) | ~! [X361 : subst,X362 : subst,X363 : term,X364 : subst] : (((X349 @ X361 @ X363 @ (comp @ X362 @ X364)) != $true) | ((X349 @ (comp @ X361 @ X362) @ (sub @ X363 @ X362) @ X364) = $true))) => ! [X365 : term,X366 : subst > term > subst > $o] : (~! [X379 : subst,X380 : subst,X381 : term,X382 : subst] : (((X366 @ (comp @ X380 @ X379) @ (sub @ X381 @ X379) @ X382) != $true) | ((X366 @ X380 @ X381 @ (comp @ X379 @ X382)) = $true)) | ~! [X367 : term] : (((var @ (sub @ X367 @ id)) != $true) | ((X366 @ id @ X367 @ id) = $true)) | ~! [X374 : subst > term > term] : (~! [X378 : term] : (((X366 @ id @ X378 @ id) != $true) | ((X366 @ id @ (X374 @ id @ X378) @ id) = $true)) | ~! [X375 : subst,X376 : term,X377 : subst] : ((X374 @ (comp @ X377 @ X375) @ (sub @ X376 @ X375)) = (sub @ (X374 @ X377 @ X376) @ X375)) | ((X366 @ id @ (lam @ (X374 @ sh @ one)) @ id) = $true)) | ((X366 @ id @ X365 @ id) = $true) | ~! [X372 : term,X373 : term] : (((X366 @ id @ X373 @ id) != $true) | ((X366 @ id @ (ap @ (sub @ X373 @ id) @ X372) @ id) = $true) | ((X366 @ id @ X372 @ id) != $true)) | ~! [X368 : subst,X369 : subst,X370 : subst,X371 : term] : (((X366 @ (comp @ X369 @ X370) @ (sub @ X371 @ X370) @ X368) = $true) | ((X366 @ X369 @ X371 @ (comp @ X370 @ X368)) != $true)))))) <=> (hoasinduction_lthm = $true)) & (hoasinduction_lem1_gthm = $true) & ((induction = $true) <=> ! [X495 : term > $o,X496 : term] : (~! [X499 : term] : (((var @ X499) != $true) | ((X495 @ X499) = $true)) | ((X495 @ X496) = $true) | ~! [X497 : term,X498 : term] : (((X495 @ (ap @ X497 @ X498)) = $true) | ((X495 @ X497) != $true) | ((X495 @ X498) != $true)) | ~! [X500 : term] : (((X495 @ X500) != $true) | ((X495 @ (lam @ X500)) = $true)))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X317 : term,X318 : term > $o,X319 : subst] : (~! [X320 : term] : (~! [X321 : term] : (((X318 @ (sub @ X320 @ (push @ X321 @ id))) = $true) | ((X318 @ X321) != $true)) | ((X318 @ (lam @ X320)) = $true)) | ~! [X322 : term,X323 : term] : (((X318 @ (ap @ X323 @ X322)) = $true) | ((X318 @ X322) != $true) | ((X318 @ X323) != $true)) | ((X318 @ (sub @ X317 @ X319)) = $true) | ~! [X324 : term] : (((X318 @ (sub @ X324 @ X319)) = $true) | ((var @ X324) != $true))) <=> (induction2lem = $true)) & (! [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) != (lam @ (X465 @ sh @ one))) | ~! [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) = (sub @ (X465 @ X473 @ X471) @ X472)) | ((X465 @ X466 @ X464) = (X467 @ X466 @ X464)) | ~! [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) = (sub @ (X467 @ X470 @ X469) @ X468))) <=> (hoaslaminj = $true)) & ((! [X409 : subst] : ((comp @ id @ X409) = X409) => (! [X410 : subst] : ((comp @ X410 @ id) = X410) => (! [X411 : subst] : ((comp @ id @ X411) = X411) & ! [X412 : subst] : ((comp @ X412 @ id) = X412)))) <=> (substmonoid_lthm = $true)) & ((pushprop_lem1 = $true) <=> ! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (((X60 @ X61) != $true) | ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) = $true))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (induction2_gthm = $true) & (pushprop_lem0_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) != (ap @ X255 @ X254)) | (X253 = X254)) => ! [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 = X258) | ((ap @ (sub @ X260 @ id) @ X258) != (ap @ (sub @ X259 @ id) @ X257))))) & (pushprop_lem1_gthm = $true) & (! [X274 : subst > term > subst > $o,X275 : term] : (((X274 @ id @ X275 @ id) = $true) | ~! [X290 : term,X291 : term] : (((X274 @ id @ (ap @ (sub @ X291 @ id) @ X290) @ id) = $true) | ((X274 @ id @ X291 @ id) != $true) | ((X274 @ id @ X290 @ id) != $true)) | ~! [X285 : subst > term > term] : (((X274 @ id @ (lam @ (X285 @ sh @ one)) @ id) = $true) | ~! [X286 : term,X287 : subst,X288 : subst] : ((X285 @ (comp @ X287 @ X288) @ (sub @ X286 @ X288)) = (sub @ (X285 @ X287 @ X286) @ X288)) | ~! [X289 : term] : (((X274 @ id @ (X285 @ id @ X289) @ id) = $true) | ((X274 @ id @ X289 @ id) != $true))) | ~! [X277 : term,X278 : subst,X279 : subst,X280 : subst] : (((X274 @ X279 @ X277 @ (comp @ X278 @ X280)) = $true) | ((X274 @ (comp @ X279 @ X278) @ (sub @ X277 @ X278) @ X280) != $true)) | ~! [X276 : term] : (((X274 @ id @ X276 @ id) = $true) | ((var @ (sub @ X276 @ id)) != $true)) | ~! [X281 : subst,X282 : term,X283 : subst,X284 : subst] : (((X274 @ (comp @ X283 @ X284) @ (sub @ X282 @ X284) @ X281) = $true) | ((X274 @ X283 @ X282 @ (comp @ X284 @ X281)) != $true))) <=> (hoasinduction = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) = $true) | ((X522 @ X521) != $true) | ~! [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) = (X523 @ X524)))) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & (! [X2 : subst > term > subst > $o] : ~! [X3 : term > $o] : ~! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) <=> (hoasinduction_lem0_lthm = $true)) & ((axshiftcons = $true) <=> ! [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) = X493)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((! [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) = X228) => ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (~! [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) = (X232 @ X234)) | ((X232 @ one) = $true) | ((X230 @ X233) != $true))) <=> (pushprop_lem1v2_lthm = $true)) & (! [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 = X418) | ((ap @ (sub @ X417 @ id) @ X418) != (ap @ (sub @ X415 @ id) @ X416))) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X501 : term] : ((sub @ X501 @ id) = X501) => ! [X502 : term > $o,X503 : subst > term > subst > $o,X504 : term] : (~! [X515 : subst,X516 : subst,X517 : subst,X518 : term] : (((X503 @ (comp @ X516 @ X515) @ (sub @ X518 @ X515) @ X517) != $true) | ((X503 @ X516 @ X518 @ (comp @ X515 @ X517)) = $true)) | ~! [X519 : term] : (((X502 @ X519) != $true) | ((X502 @ (sub @ X504 @ (push @ X519 @ id))) = $true)) | ((X502 @ (lam @ X504)) = $true) | ~! [X510 : subst > term > term] : (~! [X511 : term] : (((X503 @ id @ (X510 @ id @ X511) @ id) = $true) | ((X503 @ id @ X511 @ id) != $true)) | ((X503 @ id @ (lam @ (X510 @ sh @ one)) @ id) = $true) | ~! [X512 : subst,X513 : term,X514 : subst] : ((sub @ (X510 @ X514 @ X513) @ X512) = (X510 @ (comp @ X514 @ X512) @ (sub @ X513 @ X512)))) | ~! [X506 : subst,X507 : subst,X508 : subst,X509 : term] : (((X503 @ X506 @ X509 @ (comp @ X507 @ X508)) != $true) | ((X503 @ (comp @ X506 @ X507) @ (sub @ X509 @ X507) @ X508) = $true)) | ~! [X505 : term] : ((X503 @ id @ X505 @ id) = (X502 @ X505))))) & (pushprop_lem3v2_lthm = $true) & ! [X265 : d_term] : (d_one = X265) & (hoasapinj1_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X146 : subst > term > subst > $o,X147 : term > $o,X148 : term] : (~! [X149 : term] : ((X147 @ X149) = (X146 @ id @ X149 @ id)) | ((X147 @ (lam @ X148)) = $true) | ~! [X150 : subst,X151 : subst,X152 : subst,X153 : term] : (((X146 @ (comp @ X151 @ X150) @ (sub @ X153 @ X150) @ X152) != $true) | ((X146 @ X151 @ X153 @ (comp @ X150 @ X152)) = $true)) | ~! [X154 : subst,X155 : subst,X156 : subst,X157 : term] : (((X146 @ (comp @ X155 @ X154) @ (sub @ X157 @ X154) @ X156) = $true) | ((X146 @ X155 @ X157 @ (comp @ X154 @ X156)) != $true)) | ~! [X159 : subst > term > term] : (((X146 @ id @ (lam @ (X159 @ sh @ one)) @ id) = $true) | ~! [X160 : term] : (((X146 @ id @ X160 @ id) != $true) | ((X146 @ id @ (X159 @ id @ X160) @ id) = $true)) | ~! [X161 : term,X162 : subst,X163 : subst] : ((X159 @ (comp @ X163 @ X162) @ (sub @ X161 @ X162)) = (sub @ (X159 @ X163 @ X161) @ X162))) | ~! [X158 : term] : (((X147 @ X158) != $true) | ((X147 @ (sub @ X148 @ (push @ X158 @ id))) = $true))) <=> (hoasinduction_lem3v2 = $true)) & (hoasapnotvar = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((axidl = $true) <=> ! [X72 : subst] : ((comp @ id @ X72) = X72)) & ((! [X21 : subst > term > subst > $o] : ~! [X22 : term > $o] : ~! [X23 : term] : ((X21 @ id @ X23 @ id) = (X22 @ X23)) => (! [X24 : term,X25 : term > $o] : (~! [X26 : term,X27 : term] : (((X25 @ (ap @ X27 @ X26)) = $true) | ((X25 @ X27) != $true) | ((X25 @ X26) != $true)) | ~! [X28 : term] : (((X25 @ (lam @ X28)) = $true) | ~! [X29 : term] : (((X25 @ X29) != $true) | ((X25 @ (sub @ X28 @ (push @ X29 @ id))) = $true))) | ~! [X30 : term] : (((var @ X30) != $true) | ((X25 @ X30) = $true)) | ((X25 @ X24) = $true)) => (! [X31 : term] : ((sub @ X31 @ id) = X31) => (! [X32 : term > $o,X33 : subst > term > subst > $o,X34 : term] : (~! [X35 : term] : (((X32 @ (sub @ X34 @ (push @ X35 @ id))) = $true) | ((X32 @ X35) != $true)) | ~! [X36 : subst > term > term] : (((X33 @ id @ (lam @ (X36 @ sh @ one)) @ id) = $true) | ~! [X38 : subst,X39 : term,X40 : subst] : ((X36 @ (comp @ X38 @ X40) @ (sub @ X39 @ X40)) = (sub @ (X36 @ X38 @ X39) @ X40)) | ~! [X37 : term] : (((X33 @ id @ X37 @ id) != $true) | ((X33 @ id @ (X36 @ id @ X37) @ id) = $true))) | ((X32 @ (lam @ X34)) = $true) | ~! [X41 : term] : ((X33 @ id @ X41 @ id) = (X32 @ X41))) => ! [X42 : subst > term > subst > $o,X43 : term] : (~! [X54 : term,X55 : term] : (((X42 @ id @ (ap @ (sub @ X54 @ id) @ X55) @ id) = $true) | ((X42 @ id @ X54 @ id) != $true) | ((X42 @ id @ X55 @ id) != $true)) | ((X42 @ id @ X43 @ id) = $true) | ~! [X44 : subst > term > term] : (~! [X45 : subst,X46 : subst,X47 : term] : ((sub @ (X44 @ X45 @ X47) @ X46) = (X44 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46))) | ~! [X48 : term] : (((X42 @ id @ (X44 @ id @ X48) @ id) = $true) | ((X42 @ id @ X48 @ id) != $true)) | ((X42 @ id @ (lam @ (X44 @ sh @ one)) @ id) = $true)) | ~! [X49 : term] : (((var @ (sub @ X49 @ id)) != $true) | ((X42 @ id @ X49 @ id) = $true)) | ~! [X50 : subst,X51 : term,X52 : subst,X53 : subst] : (((X42 @ (comp @ X53 @ X52) @ (sub @ X51 @ X52) @ X50) = $true) | ((X42 @ X53 @ X51 @ (comp @ X52 @ X50)) != $true)) | ~! [X56 : subst,X57 : subst,X58 : subst,X59 : term] : (((X42 @ (comp @ X58 @ X57) @ (sub @ X59 @ X57) @ X56) != $true) | ((X42 @ X58 @ X59 @ (comp @ X57 @ X56)) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & ((d2subst @ d_id) = id) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X271 : d_subst] : (d_id = X271) & (lamnotvar = $true) & ((induction2_lthm = $true) <=> (! [X301 : term] : ((sub @ X301 @ id) = X301) => (! [X302 : subst,X303 : term > $o,X304 : term] : (~! [X309 : term] : (((var @ X309) != $true) | ((X303 @ (sub @ X309 @ X302)) = $true)) | ~! [X305 : term,X306 : term] : (((X303 @ (ap @ X305 @ X306)) = $true) | ((X303 @ X305) != $true) | ((X303 @ X306) != $true)) | ((X303 @ (sub @ X304 @ X302)) = $true) | ~! [X307 : term] : (((X303 @ (lam @ X307)) = $true) | ~! [X308 : term] : (((X303 @ (sub @ X307 @ (push @ X308 @ id))) = $true) | ((X303 @ X308) != $true)))) => ! [X310 : term,X311 : term > $o] : (((X311 @ X310) = $true) | ~! [X312 : term] : (((var @ X312) != $true) | ((X311 @ X312) = $true)) | ~! [X315 : term,X316 : term] : (((X311 @ X315) != $true) | ((X311 @ (ap @ X315 @ X316)) = $true) | ((X311 @ X316) != $true)) | ~! [X313 : term] : (((X311 @ (lam @ X313)) = $true) | ~! [X314 : term] : (((X311 @ X314) != $true) | ((X311 @ (sub @ X313 @ (push @ X314 @ id))) = $true))))))) & (ulamvarsh = $true) & (! [X144 : term] : ~! [X145 : subst > term > term] : ((X145 @ sh @ one) != (sub @ X144 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X266 : subst,X267 : term > $o,X268 : term] : ~! [X269 : term > $o] : ~! [X270 : term] : ((X269 @ X270) = (X267 @ (sub @ X270 @ (push @ X268 @ X266)))) <=> (pushprop_lem0 = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X419 : subst > term > subst > $o,X420 : term] : (~! [X422 : subst > term > term] : (((X419 @ id @ (lam @ (X422 @ sh @ one)) @ id) = $true) | ~! [X423 : term,X424 : subst,X425 : subst] : ((X422 @ (comp @ X425 @ X424) @ (sub @ X423 @ X424)) = (sub @ (X422 @ X425 @ X423) @ X424)) | ~! [X426 : term] : (((X419 @ id @ (X422 @ id @ X426) @ id) = $true) | ((X419 @ id @ X426 @ id) != $true))) | ~! [X421 : term] : (((X419 @ id @ (sub @ X420 @ (push @ X421 @ id)) @ id) = $true) | ((X419 @ id @ X421 @ id) != $true)) | ((X419 @ id @ (lam @ (sub @ X420 @ (push @ one @ sh))) @ id) = $true))) & ((d2subst @ d_id) = sh) & ((induction2 = $true) <=> ! [X294 : term,X295 : term > $o] : (~! [X297 : term,X298 : term] : (((X295 @ X298) != $true) | ($true = (X295 @ (ap @ X297 @ X298))) | ((X295 @ X297) != $true)) | ~! [X296 : term] : (((var @ X296) != $true) | ((X295 @ X296) = $true)) | ~! [X299 : term] : (~! [X300 : term] : (((X295 @ X300) != $true) | ((X295 @ (sub @ X299 @ (push @ X300 @ id))) = $true)) | ((X295 @ (lam @ X299)) = $true)) | ((X295 @ X294) = $true))) & (pushprop_lthm = $true) & (hoaslamnotap_gthm = $true) & (hoaslaminj_gthm = $true) & ((! [X5 : term,X6 : subst] : ((sub @ one @ (push @ X5 @ X6)) = X5) => (! [X7 : term,X8 : subst] : ((comp @ sh @ (push @ X7 @ X8)) = X8) => (! [X9 : term,X10 : term] : ((X9 = X10) | ((lam @ X10) != (lam @ X9))) => ! [X11 : subst > term > term,X12 : subst > term > term,X13 : term,X14 : subst] : (~! [X15 : term,X16 : subst,X17 : subst] : ((sub @ (X11 @ X17 @ X15) @ X16) = (X11 @ (comp @ X17 @ X16) @ (sub @ X15 @ X16))) | ~! [X18 : subst,X19 : term,X20 : subst] : ((sub @ (X12 @ X20 @ X19) @ X18) = (X12 @ (comp @ X20 @ X18) @ (sub @ X19 @ X18))) | ((lam @ (X11 @ sh @ one)) != (lam @ (X12 @ sh @ one))) | ((X11 @ X14 @ X13) = (X12 @ X14 @ X13)))))) <=> (hoaslaminj_lthm = $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & ((! [X427 : term] : ((sub @ X427 @ id) = X427) => (! [X428 : term,X429 : subst > term > subst > $o] : (~! [X431 : subst > term > term] : (~! [X433 : term,X434 : subst,X435 : subst] : ((X431 @ (comp @ X435 @ X434) @ (sub @ X433 @ X434)) = (sub @ (X431 @ X435 @ X433) @ X434)) | ((X429 @ id @ (lam @ (X431 @ sh @ one)) @ id) = $true) | ~! [X432 : term] : (((X429 @ id @ (X431 @ id @ X432) @ id) = $true) | ((X429 @ id @ X432 @ id) != $true))) | ((X429 @ id @ (lam @ (sub @ X428 @ (push @ one @ sh))) @ id) = $true) | ~! [X430 : term] : (((X429 @ id @ X430 @ id) != $true) | ((X429 @ id @ (sub @ X428 @ (push @ X430 @ id)) @ id) = $true))) => ! [X436 : term,X437 : subst > term > subst > $o] : (~! [X448 : subst,X449 : subst,X450 : subst,X451 : term] : (((X437 @ (comp @ X450 @ X449) @ (sub @ X451 @ X449) @ X448) = $true) | ((X437 @ X450 @ X451 @ (comp @ X449 @ X448)) != $true)) | ((X437 @ id @ (lam @ X436) @ id) = $true) | ~! [X447 : term] : (((X437 @ id @ (sub @ X436 @ (push @ X447 @ id)) @ id) = $true) | ((X437 @ id @ X447 @ id) != $true)) | ~! [X443 : term,X444 : subst,X445 : subst,X446 : subst] : (((X437 @ (comp @ X446 @ X445) @ (sub @ X443 @ X445) @ X444) != $true) | ($true = (X437 @ X446 @ X443 @ (comp @ X445 @ X444)))) | ~! [X438 : subst > term > term] : (~! [X440 : term,X441 : subst,X442 : subst] : ((sub @ (X438 @ X441 @ X440) @ X442) = (X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) | ~! [X439 : term] : (((X437 @ id @ (X438 @ id @ X439) @ id) = $true) | ((X437 @ id @ X439 @ id) != $true)) | ((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & (pushprop_gthm = $true) & (! [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192)) | (X193 = X194)) <=> (hoasapinj1 = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term,X237 : term > $o] : (~! [X244 : term] : ((X237 @ X244) = (X235 @ id @ X244 @ id)) | ($true = (X237 @ (lam @ X236))) | ~! [X238 : term] : (((X237 @ X238) != $true) | ((X237 @ (sub @ X236 @ (push @ X238 @ id))) = $true)) | ~! [X239 : subst > term > term] : (~! [X240 : subst,X241 : term,X242 : subst] : ((X239 @ (comp @ X242 @ X240) @ (sub @ X241 @ X240)) = (sub @ (X239 @ X242 @ X241) @ X240)) | ((X235 @ id @ (lam @ (X239 @ sh @ one)) @ id) = $true) | ~! [X243 : term] : (((X235 @ id @ X243 @ id) != $true) | ((X235 @ id @ (X239 @ id @ X243) @ id) = $true))))) & (axapp = $true) & (hoasapnotvar_lthm = $true) & ((! [X198 : subst > term > subst > $o] : ~! [X199 : term > $o] : ~! [X200 : term] : ((X198 @ id @ X200 @ id) = (X199 @ X200)) => (! [X201 : term > $o,X202 : term] : (~! [X203 : term] : (((X201 @ X203) = $true) | ((var @ X203) != $true)) | ~! [X206 : term] : (((X201 @ (lam @ X206)) = $true) | ~! [X207 : term] : (((X201 @ X207) != $true) | ((X201 @ (sub @ X206 @ (push @ X207 @ id))) = $true))) | ~! [X204 : term,X205 : term] : (((X201 @ X205) != $true) | ((X201 @ (ap @ X205 @ X204)) = $true) | ((X201 @ X204) != $true)) | ((X201 @ X202) = $true)) => (! [X208 : term] : ((sub @ X208 @ id) = X208) => (! [X209 : subst > term > subst > $o,X210 : term > $o,X211 : term] : (((X210 @ (lam @ X211)) = $true) | ~! [X213 : subst > term > term] : (~! [X214 : term] : (((X209 @ id @ (X213 @ id @ X214) @ id) = $true) | ((X209 @ id @ X214 @ id) != $true)) | ((X209 @ id @ (lam @ (X213 @ sh @ one)) @ id) = $true) | ~! [X215 : subst,X216 : subst,X217 : term] : ((sub @ (X213 @ X216 @ X217) @ X215) = (X213 @ (comp @ X216 @ X215) @ (sub @ X217 @ X215)))) | ~! [X218 : term] : ((X209 @ id @ X218 @ id) = (X210 @ X218)) | ~! [X212 : term] : (((X210 @ (sub @ X211 @ (push @ X212 @ id))) = $true) | ((X210 @ X212) != $true))) => ! [X219 : subst > term > subst > $o,X220 : term] : (~! [X221 : term,X222 : term] : (((X219 @ id @ X222 @ id) != $true) | ((X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id) = $true) | ((X219 @ id @ X221 @ id) != $true)) | ((X219 @ id @ X220 @ id) = $true) | ~! [X223 : subst > term > term] : (~! [X225 : subst,X226 : term,X227 : subst] : ((X223 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225)) = (sub @ (X223 @ X227 @ X226) @ X225)) | ~! [X224 : term] : (((X219 @ id @ (X223 @ id @ X224) @ id) = $true) | ((X219 @ id @ X224 @ id) != $true)) | ($true = (X219 @ id @ (lam @ (X223 @ sh @ one)) @ id)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | ((ap @ X261 @ X263) != (ap @ X262 @ X264))) <=> (apinj1 = $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3b = $true) <=> ! [X73 : term] : ~! [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) != (X74 @ sh @ one))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X71 : subst] : ((comp @ X71 @ id) = X71) <=> (axidr = $true)) & ((hoasapinj1_lthm = $true) <=> (! [X400 : term] : ((sub @ X400 @ id) = X400) => (! [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) != (ap @ X404 @ X402)) | (X401 = X404)) => ! [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 = X408) | ((ap @ (sub @ X408 @ id) @ X406) != (ap @ (sub @ X405 @ id) @ X407)))))) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X103 : term] : ~! [X104 : subst > term > term] : ~! [X105 : subst,X106 : term] : ((sub @ X103 @ (push @ X106 @ X105)) = (X104 @ X105 @ X106)) => (! [X107 : term] : ((sub @ X107 @ id) = X107) => ! [X108 : term,X109 : subst > term > subst > $o,X110 : term > $o] : (~! [X112 : term] : ((X109 @ id @ X112 @ id) = (X110 @ X112)) | ((X110 @ (lam @ X108)) = $true) | ~! [X113 : subst > term > term] : (~! [X115 : subst,X116 : term,X117 : subst] : ((X113 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117)) = (sub @ (X113 @ X115 @ X116) @ X117)) | ((X109 @ id @ (lam @ (X113 @ sh @ one)) @ id) = $true) | ~! [X114 : term] : (((X109 @ id @ X114 @ id) != $true) | ((X109 @ id @ (X113 @ id @ X114) @ id) = $true))) | ~! [X111 : term] : (((X110 @ X111) != $true) | ($true = (X110 @ (sub @ X108 @ (push @ X111 @ id))))))))) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (! [X183 : term,X184 : subst > term > subst > $o] : (~! [X185 : term,X186 : term] : (((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) = $true) | ((X184 @ id @ X185 @ id) != $true) | ((X184 @ id @ X186 @ id) != $true)) | ((X184 @ id @ X183 @ id) = $true) | ~! [X187 : subst > term > term] : (~! [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) = (sub @ (X187 @ X189 @ X191) @ X190)) | ((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) = $true) | ~! [X188 : term] : (((X184 @ id @ (X187 @ id @ X188) @ id) = $true) | ((X184 @ id @ X188 @ id) != $true)))) <=> (hoasinduction_no_psi_cond = $true)) & (pushprop_lem1v2_gthm = $true) & (! [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) != (ap @ X326 @ X328)) | (X325 = X328)) <=> (apinj2 = $true)) & (hoasapnotvar_gthm = $true) & (! [X384 : term,X385 : subst > term > subst > $o] : (~! [X386 : subst,X387 : subst,X388 : subst,X389 : term] : (((X385 @ (comp @ X386 @ X388) @ (sub @ X389 @ X388) @ X387) = $true) | ((X385 @ X386 @ X389 @ (comp @ X388 @ X387)) != $true)) | ~! [X396 : subst,X397 : subst,X398 : term,X399 : subst] : (((X385 @ X396 @ X398 @ (comp @ X397 @ X399)) = $true) | ((X385 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X399) != $true)) | ((X385 @ id @ (lam @ X384) @ id) = $true) | ~! [X391 : subst > term > term] : (((X385 @ id @ (lam @ (X391 @ sh @ one)) @ id) = $true) | ~! [X395 : term] : (((X385 @ id @ X395 @ id) != $true) | ((X385 @ id @ (X391 @ id @ X395) @ id) = $true)) | ~! [X392 : subst,X393 : subst,X394 : term] : ((sub @ (X391 @ X393 @ X394) @ X392) = (X391 @ (comp @ X393 @ X392) @ (sub @ X394 @ X392)))) | ~! [X390 : term] : (((X385 @ id @ (sub @ X384 @ (push @ X390 @ id)) @ id) = $true) | ((X385 @ id @ X390 @ id) != $true))) <=> (hoasinduction_lem3 = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (hoaslamnotvar_lthm = $true) & (pushprop_lem2v2_gthm = $true) & ! [X181 : d_subst,X182 : d_subst] : (((d2subst @ X182) = (d2subst @ X181)) => (X181 = X182)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoaslamnotap = $true) <=> ! [X247 : term,X248 : subst > term > term,X249 : term] : (((ap @ (sub @ X247 @ id) @ X249) != (lam @ (X248 @ sh @ one))) | ~! [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) = (sub @ (X248 @ X250 @ X252) @ X251)))) & ((! [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) = X118) => (! [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) = X121) => ! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (((X122 @ X124) != $true) | ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X525 : term > $o,X526 : term,X527 : term,X528 : subst > term > subst > $o] : (((X525 @ X527) != $true) | ~! [X529 : term] : ((X528 @ id @ X529 @ id) = (X525 @ X529)) | ~! [X530 : term,X531 : term] : (((X528 @ id @ (ap @ (sub @ X531 @ id) @ X530) @ id) = $true) | ((X528 @ id @ X531 @ id) != $true) | ((X528 @ id @ X530 @ id) != $true)) | ((X525 @ (ap @ X527 @ X526)) = $true) | ~! [X536 : subst,X537 : subst,X538 : term,X539 : subst] : (((X528 @ (comp @ X536 @ X537) @ (sub @ X538 @ X537) @ X539) = $true) | ((X528 @ X536 @ X538 @ (comp @ X537 @ X539)) != $true)) | ~! [X532 : subst,X533 : subst,X534 : term,X535 : subst] : (((X528 @ X533 @ X534 @ (comp @ X532 @ X535)) = $true) | ((X528 @ (comp @ X533 @ X532) @ (sub @ X534 @ X532) @ X535) != $true)) | ((X525 @ X526) != $true))) & (pushprop_lthm_orig = $true) & ((substmonoid = $true) <=> (! [X1 : subst] : ((comp @ id @ X1) = X1) & ! [X0 : subst] : ((comp @ X0 @ id) = X0))) & (! [X75 : subst > term > subst > $o,X76 : term] : (~! [X82 : term] : (((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) = $true) | ((X75 @ id @ X82 @ id) != $true)) | ~! [X77 : subst > term > term] : (~! [X81 : term] : (((X75 @ id @ (X77 @ id @ X81) @ id) = $true) | ((X75 @ id @ X81 @ id) != $true)) | ((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) = $true) | ~! [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) = (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78)))) | ((X75 @ id @ (lam @ X76) @ id) = $true)) <=> (hoasinduction_lem3a = $true)) & (axclos = $true) & (axvarshift = $true) & ! [X245 : d_term,X246 : d_term] : (((d2term @ X246) = (d2term @ X245)) => (X245 = X246)) & (ulamvar1 != $true)),
% 9.05/2.75 inference(flattening,[status(thm)],[f10])).
% 9.05/2.75 thf(f10,plain,(
% 9.05/2.75 ((substmonoid = $true) <=> (! [X1 : subst] : ((comp @ id @ X1) = X1) & ! [X0 : subst] : ((comp @ X0 @ id) = X0))) & ~(lamnotap = $true) & (axassoc = $true) & (! [X2 : subst > term > subst > $o] : ~! [X3 : term > $o] : ~! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) <=> (hoasinduction_lem0_lthm = $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((! [X5 : term,X6 : subst] : ((sub @ one @ (push @ X5 @ X6)) = X5) => (! [X7 : term,X8 : subst] : ((comp @ sh @ (push @ X7 @ X8)) = X8) => (! [X9 : term,X10 : term] : ((X9 = X10) | ((lam @ X10) != (lam @ X9))) => ! [X11 : subst > term > term,X12 : subst > term > term,X13 : term,X14 : subst] : (~! [X15 : term,X16 : subst,X17 : subst] : ((sub @ (X11 @ X17 @ X15) @ X16) = (X11 @ (comp @ X17 @ X16) @ (sub @ X15 @ X16))) | ~! [X18 : subst,X19 : term,X20 : subst] : ((sub @ (X12 @ X20 @ X19) @ X18) = (X12 @ (comp @ X20 @ X18) @ (sub @ X19 @ X18))) | ((lam @ (X11 @ sh @ one)) != (lam @ (X12 @ sh @ one))) | ((X11 @ X14 @ X13) = (X12 @ X14 @ X13)))))) <=> (hoaslaminj_lthm = $true)) & (pushprop_gthm = $true) & (induction2lem_gthm = $true) & ((! [X21 : subst > term > subst > $o] : ~! [X22 : term > $o] : ~! [X23 : term] : ((X21 @ id @ X23 @ id) = (X22 @ X23)) => (! [X24 : term,X25 : term > $o] : (~! [X26 : term,X27 : term] : (((X25 @ (ap @ X27 @ X26)) = $true) | ~((X25 @ X27) = $true) | ~((X25 @ X26) = $true)) | ~! [X28 : term] : (((X25 @ (lam @ X28)) = $true) | ~! [X29 : term] : (((X25 @ (sub @ X28 @ (push @ X29 @ id))) = $true) | ~((X25 @ X29) = $true))) | ~! [X30 : term] : (~((var @ X30) = $true) | ((X25 @ X30) = $true)) | ((X25 @ X24) = $true)) => (! [X31 : term] : ((sub @ X31 @ id) = X31) => (! [X32 : term > $o,X33 : subst > term > subst > $o,X34 : term] : (~! [X35 : term] : (((X32 @ (sub @ X34 @ (push @ X35 @ id))) = $true) | ~((X32 @ X35) = $true)) | ~! [X36 : subst > term > term] : (((X33 @ id @ (lam @ (X36 @ sh @ one)) @ id) = $true) | ~! [X37 : term] : (~((X33 @ id @ X37 @ id) = $true) | ((X33 @ id @ (X36 @ id @ X37) @ id) = $true)) | ~! [X38 : subst,X39 : term,X40 : subst] : ((X36 @ (comp @ X38 @ X40) @ (sub @ X39 @ X40)) = (sub @ (X36 @ X38 @ X39) @ X40))) | ~! [X41 : term] : ((X33 @ id @ X41 @ id) = (X32 @ X41)) | ((X32 @ (lam @ X34)) = $true)) => ! [X42 : subst > term > subst > $o,X43 : term] : (~! [X44 : subst > term > term] : (~! [X45 : subst,X46 : subst,X47 : term] : ((sub @ (X44 @ X45 @ X47) @ X46) = (X44 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46))) | ~! [X48 : term] : (((X42 @ id @ (X44 @ id @ X48) @ id) = $true) | ~((X42 @ id @ X48 @ id) = $true)) | ((X42 @ id @ (lam @ (X44 @ sh @ one)) @ id) = $true)) | ((X42 @ id @ X43 @ id) = $true) | ~! [X49 : term] : (~((var @ (sub @ X49 @ id)) = $true) | ((X42 @ id @ X49 @ id) = $true)) | ~! [X50 : subst,X51 : term,X52 : subst,X53 : subst] : (((X42 @ (comp @ X53 @ X52) @ (sub @ X51 @ X52) @ X50) = $true) | ~((X42 @ X53 @ X51 @ (comp @ X52 @ X50)) = $true)) | ~! [X54 : term,X55 : term] : (((X42 @ id @ (ap @ (sub @ X54 @ id) @ X55) @ id) = $true) | ~((X42 @ id @ X54 @ id) = $true) | ~((X42 @ id @ X55 @ id) = $true)) | ~! [X56 : subst,X57 : subst,X58 : subst,X59 : term] : (~((X42 @ (comp @ X58 @ X57) @ (sub @ X59 @ X57) @ X56) = $true) | ((X42 @ X58 @ X59 @ (comp @ X57 @ X56)) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem1_gthm = $true) & (ulamvarsh = $true) & ((pushprop_lem1 = $true) <=> ! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (~((X60 @ X61) = $true) | ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) = $true))) & (hoaslamnotap_gthm = $true) & (! [X65 : term] : ~! [X66 : subst > term > term] : ~! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (lamnotvar = $true) & ((shinj = $true) <=> ! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh)))) & (! [X71 : subst] : ((comp @ X71 @ id) = X71) <=> (axidr = $true)) & (hoasinduction_lem1v2_gthm = $true) & ((d2subst @ d_id) = id) & ((axidl = $true) <=> ! [X72 : subst] : ((comp @ id @ X72) = X72)) & ((hoasinduction_lem3b = $true) <=> ! [X73 : term] : ~! [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) != (X74 @ sh @ one))) & ((hoasinduction_lem3a = $true) <=> ! [X75 : subst > term > subst > $o,X76 : term] : (~! [X77 : subst > term > term] : (~! [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) = (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78))) | ~! [X81 : term] : (~((X75 @ id @ X81 @ id) = $true) | ((X75 @ id @ (X77 @ id @ X81) @ id) = $true)) | ((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) = $true)) | ((X75 @ id @ (lam @ X76) @ id) = $true) | ~! [X82 : term] : (((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) = $true) | ~((X75 @ id @ X82 @ id) = $true)))) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X83 : term] : ((sub @ X83 @ id) = X83) => (! [X84 : term,X85 : subst > term > subst > $o] : (~! [X86 : term] : (~((X85 @ id @ X86 @ id) = $true) | ((X85 @ id @ (sub @ X84 @ (push @ X86 @ id)) @ id) = $true)) | ~! [X87 : subst > term > term] : (~! [X88 : term] : (~((X85 @ id @ X88 @ id) = $true) | ($true = (X85 @ id @ (X87 @ id @ X88) @ id))) | ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id) = $true) | ~! [X89 : subst,X90 : term,X91 : subst] : ((X87 @ (comp @ X91 @ X89) @ (sub @ X90 @ X89)) = (sub @ (X87 @ X91 @ X90) @ X89))) | ((X85 @ id @ (lam @ (sub @ X84 @ (push @ one @ sh))) @ id) = $true)) => ! [X92 : term,X93 : subst > term > subst > $o] : (~! [X94 : subst > term > term] : (((X93 @ id @ (lam @ (X94 @ sh @ one)) @ id) = $true) | ~! [X95 : subst,X96 : term,X97 : subst] : ((X94 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97)) = (sub @ (X94 @ X95 @ X96) @ X97)) | ~! [X98 : term] : (~((X93 @ id @ X98 @ id) = $true) | ((X93 @ id @ (X94 @ id @ X98) @ id) = $true))) | ($true = (X93 @ id @ (lam @ X92) @ id)) | ~! [X99 : term] : (((X93 @ id @ (sub @ X92 @ (push @ X99 @ id)) @ id) = $true) | ~((X93 @ id @ X99 @ id) = $true)))))) & (! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100) <=> (axvarcons = $true)) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & (hoaslamnotvar_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X103 : term] : ~! [X104 : subst > term > term] : ~! [X105 : subst,X106 : term] : ((sub @ X103 @ (push @ X106 @ X105)) = (X104 @ X105 @ X106)) => (! [X107 : term] : ((sub @ X107 @ id) = X107) => ! [X108 : term,X109 : subst > term > subst > $o,X110 : term > $o] : (~! [X111 : term] : (~((X110 @ X111) = $true) | ($true = (X110 @ (sub @ X108 @ (push @ X111 @ id))))) | ((X110 @ (lam @ X108)) = $true) | ~! [X112 : term] : ((X109 @ id @ X112 @ id) = (X110 @ X112)) | ~! [X113 : subst > term > term] : (((X109 @ id @ (lam @ (X113 @ sh @ one)) @ id) = $true) | ~! [X114 : term] : (~((X109 @ id @ X114 @ id) = $true) | ((X109 @ id @ (X113 @ id @ X114) @ id) = $true)) | ~! [X115 : subst,X116 : term,X117 : subst] : ((X113 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117)) = (sub @ (X113 @ X115 @ X116) @ X117)))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((! [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) = X118) => (! [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) = X121) => ! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (~((X122 @ X124) = $true) | ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_gthm = $true) & (! [X127 : subst > term > subst > $o,X128 : term] : (~! [X129 : term] : (((X127 @ id @ (sub @ X128 @ (push @ X129 @ id)) @ id) = $true) | ~((X127 @ id @ X129 @ id) = $true)) | ~! [X130 : subst > term > term] : (~! [X131 : term,X132 : subst,X133 : subst] : ((X130 @ (comp @ X132 @ X133) @ (sub @ X131 @ X133)) = (sub @ (X130 @ X132 @ X131) @ X133)) | ~! [X134 : term] : (((X127 @ id @ (X130 @ id @ X134) @ id) = $true) | ~((X127 @ id @ X134 @ id) = $true)) | ((X127 @ id @ (lam @ (X130 @ sh @ one)) @ id) = $true)) | ((X127 @ id @ (lam @ (sub @ X128 @ (push @ one @ sh))) @ id) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & (! [X135 : term] : ~! [X136 : subst > term > term] : ~! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138)) <=> (hoasinduction_lem3v2_f = $true)) & (! [X139 : term,X140 : term > $o,X141 : subst] : ~! [X142 : term > $o] : ~! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143)) <=> (pushprop_lem0_lthm = $true)) & (! [X144 : term] : ~! [X145 : subst > term > term] : ((X145 @ sh @ one) != (sub @ X144 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X146 : subst > term > subst > $o,X147 : term > $o,X148 : term] : (~! [X149 : term] : ((X147 @ X149) = (X146 @ id @ X149 @ id)) | ((X147 @ (lam @ X148)) = $true) | ~! [X150 : subst,X151 : subst,X152 : subst,X153 : term] : (((X146 @ X151 @ X153 @ (comp @ X150 @ X152)) = $true) | ~((X146 @ (comp @ X151 @ X150) @ (sub @ X153 @ X150) @ X152) = $true)) | ~! [X154 : subst,X155 : subst,X156 : subst,X157 : term] : (((X146 @ (comp @ X155 @ X154) @ (sub @ X157 @ X154) @ X156) = $true) | ~((X146 @ X155 @ X157 @ (comp @ X154 @ X156)) = $true)) | ~! [X158 : term] : (((X147 @ (sub @ X148 @ (push @ X158 @ id))) = $true) | ~((X147 @ X158) = $true)) | ~! [X159 : subst > term > term] : (~! [X160 : term] : (((X146 @ id @ (X159 @ id @ X160) @ id) = $true) | ~((X146 @ id @ X160 @ id) = $true)) | ~! [X161 : term,X162 : subst,X163 : subst] : ((X159 @ (comp @ X163 @ X162) @ (sub @ X161 @ X162)) = (sub @ (X159 @ X163 @ X161) @ X162)) | ((X146 @ id @ (lam @ (X159 @ sh @ one)) @ id) = $true)))) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (pushprop_lem3v2 = $true) & (! [X164 : term] : ((sub @ X164 @ id) = X164) <=> (termmset = $true)) & (! [X165 : subst > term > subst > $o] : ~! [X166 : term > $o] : ~! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) <=> (hoasinduction_lem0 = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X168 : term,X169 : term,X170 : subst > term > subst > $o] : (~((X170 @ id @ X169 @ id) = $true) | ~! [X171 : term,X172 : term] : (((X170 @ id @ (ap @ (sub @ X171 @ id) @ X172) @ id) = $true) | ~((X170 @ id @ X172 @ id) = $true) | ~((X170 @ id @ X171 @ id) = $true)) | ~! [X173 : subst,X174 : subst,X175 : term,X176 : subst] : (((X170 @ X176 @ X175 @ (comp @ X173 @ X174)) = $true) | ~((X170 @ (comp @ X176 @ X173) @ (sub @ X175 @ X173) @ X174) = $true)) | ((X170 @ id @ (ap @ X168 @ X169) @ id) = $true) | ~((X170 @ id @ X168 @ id) = $true) | ~! [X177 : subst,X178 : subst,X179 : term,X180 : subst] : (~((X170 @ X178 @ X179 @ (comp @ X177 @ X180)) = $true) | ((X170 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177) @ X180) = $true)))) & ! [X181 : d_subst,X182 : d_subst] : (((d2subst @ X182) = (d2subst @ X181)) => (X181 = X182)) & (substmonoid_gthm = $true) & (pushprop = $true) & (axclos = $true) & (! [X183 : term,X184 : subst > term > subst > $o] : (((X184 @ id @ X183 @ id) = $true) | ~! [X185 : term,X186 : term] : (((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) = $true) | ~((X184 @ id @ X186 @ id) = $true) | ~((X184 @ id @ X185 @ id) = $true)) | ~! [X187 : subst > term > term] : (((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) = $true) | ~! [X188 : term] : (~((X184 @ id @ X188 @ id) = $true) | ((X184 @ id @ (X187 @ id @ X188) @ id) = $true)) | ~! [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) = (sub @ (X187 @ X189 @ X191) @ X190)))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192)) | (X193 = X194)) <=> (hoasapinj1 = $true)) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & (pushprop_lem1v2_gthm = $true) & (axabs = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X198 : subst > term > subst > $o] : ~! [X199 : term > $o] : ~! [X200 : term] : ((X198 @ id @ X200 @ id) = (X199 @ X200)) => (! [X201 : term > $o,X202 : term] : (((X201 @ X202) = $true) | ~! [X203 : term] : (((X201 @ X203) = $true) | ~((var @ X203) = $true)) | ~! [X204 : term,X205 : term] : (~((X201 @ X204) = $true) | ~((X201 @ X205) = $true) | ((X201 @ (ap @ X205 @ X204)) = $true)) | ~! [X206 : term] : (((X201 @ (lam @ X206)) = $true) | ~! [X207 : term] : (~((X201 @ X207) = $true) | ((X201 @ (sub @ X206 @ (push @ X207 @ id))) = $true)))) => (! [X208 : term] : ((sub @ X208 @ id) = X208) => (! [X209 : subst > term > subst > $o,X210 : term > $o,X211 : term] : (((X210 @ (lam @ X211)) = $true) | ~! [X212 : term] : (~((X210 @ X212) = $true) | ((X210 @ (sub @ X211 @ (push @ X212 @ id))) = $true)) | ~! [X213 : subst > term > term] : (((X209 @ id @ (lam @ (X213 @ sh @ one)) @ id) = $true) | ~! [X214 : term] : (((X209 @ id @ (X213 @ id @ X214) @ id) = $true) | ~((X209 @ id @ X214 @ id) = $true)) | ~! [X215 : subst,X216 : subst,X217 : term] : ((sub @ (X213 @ X216 @ X217) @ X215) = (X213 @ (comp @ X216 @ X215) @ (sub @ X217 @ X215)))) | ~! [X218 : term] : ((X209 @ id @ X218 @ id) = (X210 @ X218))) => ! [X219 : subst > term > subst > $o,X220 : term] : (~! [X221 : term,X222 : term] : (~((X219 @ id @ X222 @ id) = $true) | ~((X219 @ id @ X221 @ id) = $true) | ((X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id) = $true)) | ((X219 @ id @ X220 @ id) = $true) | ~! [X223 : subst > term > term] : (~! [X224 : term] : (~((X219 @ id @ X224 @ id) = $true) | ((X219 @ id @ (X223 @ id @ X224) @ id) = $true)) | ($true = (X219 @ id @ (lam @ (X223 @ sh @ one)) @ id)) | ~! [X225 : subst,X226 : term,X227 : subst] : ((X223 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225)) = (sub @ (X223 @ X227 @ X226) @ X225))))))))) & (termmset_gthm = $true) & (axmap = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) = X228) => ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (~! [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) = (X232 @ X234)) | ((X232 @ one) = $true) | ~((X230 @ X233) = $true)))) & (hoasapnotvar = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X235 : subst > term > subst > $o,X236 : term,X237 : term > $o] : (~! [X238 : term] : (~((X237 @ X238) = $true) | ((X237 @ (sub @ X236 @ (push @ X238 @ id))) = $true)) | ~! [X239 : subst > term > term] : (~! [X240 : subst,X241 : term,X242 : subst] : ((X239 @ (comp @ X242 @ X240) @ (sub @ X241 @ X240)) = (sub @ (X239 @ X242 @ X241) @ X240)) | ((X235 @ id @ (lam @ (X239 @ sh @ one)) @ id) = $true) | ~! [X243 : term] : (((X235 @ id @ (X239 @ id @ X243) @ id) = $true) | ~((X235 @ id @ X243 @ id) = $true))) | ($true = (X237 @ (lam @ X236))) | ~! [X244 : term] : ((X237 @ X244) = (X235 @ id @ X244 @ id))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ! [X245 : d_term,X246 : d_term] : (((d2term @ X246) = (d2term @ X245)) => (X245 = X246)) & ((hoaslamnotap = $true) <=> ! [X247 : term,X248 : subst > term > term,X249 : term] : (((ap @ (sub @ X247 @ id) @ X249) != (lam @ (X248 @ sh @ one))) | ~! [X250 : subst,X251 : subst,X252 : term] : ((X248 @ (comp @ X250 @ X251) @ (sub @ X252 @ X251)) = (sub @ (X248 @ X250 @ X252) @ X251)))) & ((hoasapinj2_lthm = $true) <=> (! [X253 : term,X254 : term,X255 : term,X256 : term] : (((ap @ X256 @ X253) != (ap @ X255 @ X254)) | (X253 = X254)) => ! [X257 : term,X258 : term,X259 : term,X260 : term] : ((X257 = X258) | ((ap @ (sub @ X260 @ id) @ X258) != (ap @ (sub @ X259 @ id) @ X257))))) & (hoasinduction_lem2_gthm = $true) & (! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | ((ap @ X261 @ X263) != (ap @ X262 @ X264))) <=> (apinj1 = $true)) & ! [X265 : d_term] : (d_one = X265) & (induction2_gthm = $true) & (! [X266 : subst,X267 : term > $o,X268 : term] : ~! [X269 : term > $o] : ~! [X270 : term] : ((X269 @ X270) = (X267 @ (sub @ X270 @ (push @ X268 @ X266)))) <=> (pushprop_lem0 = $true)) & (pushprop_lthm = $true) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ! [X271 : d_subst] : (d_id = X271) & ~(ulamvar1 = $true) & ((! [X272 : term] : ((sub @ X272 @ id) = X272) => ! [X273 : term] : ((sub @ X273 @ id) = X273)) <=> (termmset_lthm = $true)) & (hoaslamnotvar = $true) & (pushprop_lem2v2 = $true) & (pushprop_lem3v2_lthm = $true) & (! [X274 : subst > term > subst > $o,X275 : term] : (((X274 @ id @ X275 @ id) = $true) | ~! [X276 : term] : (((X274 @ id @ X276 @ id) = $true) | ~((var @ (sub @ X276 @ id)) = $true)) | ~! [X277 : term,X278 : subst,X279 : subst,X280 : subst] : (((X274 @ X279 @ X277 @ (comp @ X278 @ X280)) = $true) | ~((X274 @ (comp @ X279 @ X278) @ (sub @ X277 @ X278) @ X280) = $true)) | ~! [X281 : subst,X282 : term,X283 : subst,X284 : subst] : (~((X274 @ X283 @ X282 @ (comp @ X284 @ X281)) = $true) | ((X274 @ (comp @ X283 @ X284) @ (sub @ X282 @ X284) @ X281) = $true)) | ~! [X285 : subst > term > term] : (~! [X286 : term,X287 : subst,X288 : subst] : ((X285 @ (comp @ X287 @ X288) @ (sub @ X286 @ X288)) = (sub @ (X285 @ X287 @ X286) @ X288)) | ~! [X289 : term] : (~((X274 @ id @ X289 @ id) = $true) | ((X274 @ id @ (X285 @ id @ X289) @ id) = $true)) | ((X274 @ id @ (lam @ (X285 @ sh @ one)) @ id) = $true)) | ~! [X290 : term,X291 : term] : (~((X274 @ id @ X291 @ id) = $true) | ((X274 @ id @ (ap @ (sub @ X291 @ id) @ X290) @ id) = $true) | ~((X274 @ id @ X290 @ id) = $true))) <=> (hoasinduction = $true)) & (! [X292 : term,X293 : term] : ((X292 = X293) | ((lam @ X293) != (lam @ X292))) <=> (laminj = $true)) & (! [X294 : term,X295 : term > $o] : (~! [X296 : term] : (~((var @ X296) = $true) | ((X295 @ X296) = $true)) | ~! [X297 : term,X298 : term] : (($true = (X295 @ (ap @ X297 @ X298))) | ~((X295 @ X298) = $true) | ~((X295 @ X297) = $true)) | ~! [X299 : term] : (~! [X300 : term] : (~((X295 @ X300) = $true) | ((X295 @ (sub @ X299 @ (push @ X300 @ id))) = $true)) | ((X295 @ (lam @ X299)) = $true)) | ((X295 @ X294) = $true)) <=> (induction2 = $true)) & ((d2subst @ d_id) = sh) & (hoasapinj2_gthm = $true) & ((! [X301 : term] : ((sub @ X301 @ id) = X301) => (! [X302 : subst,X303 : term > $o,X304 : term] : (~! [X305 : term,X306 : term] : (~((X303 @ X305) = $true) | ~((X303 @ X306) = $true) | ((X303 @ (ap @ X305 @ X306)) = $true)) | ~! [X307 : term] : (~! [X308 : term] : (~((X303 @ X308) = $true) | ((X303 @ (sub @ X307 @ (push @ X308 @ id))) = $true)) | ((X303 @ (lam @ X307)) = $true)) | ((X303 @ (sub @ X304 @ X302)) = $true) | ~! [X309 : term] : (~((var @ X309) = $true) | ((X303 @ (sub @ X309 @ X302)) = $true))) => ! [X310 : term,X311 : term > $o] : (~! [X312 : term] : (~((var @ X312) = $true) | ((X311 @ X312) = $true)) | ((X311 @ X310) = $true) | ~! [X313 : term] : (~! [X314 : term] : (((X311 @ (sub @ X313 @ (push @ X314 @ id))) = $true) | ~((X311 @ X314) = $true)) | ((X311 @ (lam @ X313)) = $true)) | ~! [X315 : term,X316 : term] : (((X311 @ (ap @ X315 @ X316)) = $true) | ~((X311 @ X315) = $true) | ~((X311 @ X316) = $true))))) <=> (induction2_lthm = $true)) & (hoasinduction_lem3b_gthm = $true) & (ulamvarind = $true) & (hoaslamnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X317 : term,X318 : term > $o,X319 : subst] : (~! [X320 : term] : (~! [X321 : term] : (((X318 @ (sub @ X320 @ (push @ X321 @ id))) = $true) | ~((X318 @ X321) = $true)) | ((X318 @ (lam @ X320)) = $true)) | ~! [X322 : term,X323 : term] : (((X318 @ (ap @ X323 @ X322)) = $true) | ~((X318 @ X323) = $true) | ~((X318 @ X322) = $true)) | ((X318 @ (sub @ X317 @ X319)) = $true) | ~! [X324 : term] : (~((var @ X324) = $true) | ((X318 @ (sub @ X324 @ X319)) = $true))) <=> (induction2lem = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem1v2 = $true) & (! [X325 : term,X326 : term,X327 : term,X328 : term] : (((ap @ X327 @ X325) != (ap @ X326 @ X328)) | (X325 = X328)) <=> (apinj2 = $true)) & ((hoasinduction_lthm = $true) <=> (! [X329 : term > $o,X330 : term] : (~! [X331 : term] : (($true = (X329 @ X331)) | ~((var @ X331) = $true)) | ~! [X332 : term,X333 : term] : (~((X329 @ X332) = $true) | ~((X329 @ X333) = $true) | ((X329 @ (ap @ X332 @ X333)) = $true)) | ~! [X334 : term] : (~! [X335 : term] : (~((X329 @ X335) = $true) | ((X329 @ (sub @ X334 @ (push @ X335 @ id))) = $true)) | ((X329 @ (lam @ X334)) = $true)) | ((X329 @ X330) = $true)) => (! [X336 : term,X337 : subst > term > subst > $o,X338 : term] : (~! [X339 : subst,X340 : subst,X341 : term,X342 : subst] : (((X337 @ (comp @ X342 @ X339) @ (sub @ X341 @ X339) @ X340) = $true) | ~((X337 @ X342 @ X341 @ (comp @ X339 @ X340)) = $true)) | ~((X337 @ id @ X336 @ id) = $true) | ((X337 @ id @ (ap @ X338 @ X336) @ id) = $true) | ~((X337 @ id @ X338 @ id) = $true) | ~! [X343 : term,X344 : term] : (~((X337 @ id @ X344 @ id) = $true) | ((X337 @ id @ (ap @ (sub @ X343 @ id) @ X344) @ id) = $true) | ~((X337 @ id @ X343 @ id) = $true)) | ~! [X345 : subst,X346 : term,X347 : subst,X348 : subst] : (((X337 @ X348 @ X346 @ (comp @ X347 @ X345)) = $true) | ~((X337 @ (comp @ X348 @ X347) @ (sub @ X346 @ X347) @ X345) = $true))) => (! [X349 : subst > term > subst > $o,X350 : term] : (~! [X351 : subst,X352 : subst,X353 : term,X354 : subst] : (~((X349 @ (comp @ X351 @ X352) @ (sub @ X353 @ X352) @ X354) = $true) | ((X349 @ X351 @ X353 @ (comp @ X352 @ X354)) = $true)) | ~! [X355 : subst > term > term] : (~! [X356 : term] : (~((X349 @ id @ X356 @ id) = $true) | ((X349 @ id @ (X355 @ id @ X356) @ id) = $true)) | ((X349 @ id @ (lam @ (X355 @ sh @ one)) @ id) = $true) | ~! [X357 : subst,X358 : subst,X359 : term] : ((sub @ (X355 @ X358 @ X359) @ X357) = (X355 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357)))) | ((X349 @ id @ (lam @ X350) @ id) = $true) | ~! [X360 : term] : (~((X349 @ id @ X360 @ id) = $true) | ((X349 @ id @ (sub @ X350 @ (push @ X360 @ id)) @ id) = $true)) | ~! [X361 : subst,X362 : subst,X363 : term,X364 : subst] : (~((X349 @ X361 @ X363 @ (comp @ X362 @ X364)) = $true) | ((X349 @ (comp @ X361 @ X362) @ (sub @ X363 @ X362) @ X364) = $true))) => ! [X365 : term,X366 : subst > term > subst > $o] : (~! [X367 : term] : (((X366 @ id @ X367 @ id) = $true) | ~((var @ (sub @ X367 @ id)) = $true)) | ~! [X368 : subst,X369 : subst,X370 : subst,X371 : term] : (((X366 @ (comp @ X369 @ X370) @ (sub @ X371 @ X370) @ X368) = $true) | ~((X366 @ X369 @ X371 @ (comp @ X370 @ X368)) = $true)) | ~! [X372 : term,X373 : term] : (~((X366 @ id @ X372 @ id) = $true) | ~((X366 @ id @ X373 @ id) = $true) | ((X366 @ id @ (ap @ (sub @ X373 @ id) @ X372) @ id) = $true)) | ~! [X374 : subst > term > term] : (~! [X375 : subst,X376 : term,X377 : subst] : ((X374 @ (comp @ X377 @ X375) @ (sub @ X376 @ X375)) = (sub @ (X374 @ X377 @ X376) @ X375)) | ((X366 @ id @ (lam @ (X374 @ sh @ one)) @ id) = $true) | ~! [X378 : term] : (((X366 @ id @ (X374 @ id @ X378) @ id) = $true) | ~((X366 @ id @ X378 @ id) = $true))) | ((X366 @ id @ X365 @ id) = $true) | ~! [X379 : subst,X380 : subst,X381 : term,X382 : subst] : (~((X366 @ (comp @ X380 @ X379) @ (sub @ X381 @ X379) @ X382) = $true) | ((X366 @ X380 @ X381 @ (comp @ X379 @ X382)) = $true))))))) & (apnotvar = $true) & (! [X383 : term] : ((sub @ X383 @ id) = X383) <=> (axvarid = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X384 : term,X385 : subst > term > subst > $o] : (~! [X386 : subst,X387 : subst,X388 : subst,X389 : term] : (((X385 @ (comp @ X386 @ X388) @ (sub @ X389 @ X388) @ X387) = $true) | ~((X385 @ X386 @ X389 @ (comp @ X388 @ X387)) = $true)) | ~! [X390 : term] : (((X385 @ id @ (sub @ X384 @ (push @ X390 @ id)) @ id) = $true) | ~((X385 @ id @ X390 @ id) = $true)) | ((X385 @ id @ (lam @ X384) @ id) = $true) | ~! [X391 : subst > term > term] : (~! [X392 : subst,X393 : subst,X394 : term] : ((sub @ (X391 @ X393 @ X394) @ X392) = (X391 @ (comp @ X393 @ X392) @ (sub @ X394 @ X392))) | ~! [X395 : term] : (~((X385 @ id @ X395 @ id) = $true) | ((X385 @ id @ (X391 @ id @ X395) @ id) = $true)) | ((X385 @ id @ (lam @ (X391 @ sh @ one)) @ id) = $true)) | ~! [X396 : subst,X397 : subst,X398 : term,X399 : subst] : (~((X385 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X399) = $true) | ((X385 @ X396 @ X398 @ (comp @ X397 @ X399)) = $true)))) & (hoaslaminj_gthm = $true) & (hoaslamnotap_lthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X400 : term] : ((sub @ X400 @ id) = X400) => (! [X401 : term,X402 : term,X403 : term,X404 : term] : (((ap @ X401 @ X403) != (ap @ X404 @ X402)) | (X401 = X404)) => ! [X405 : term,X406 : term,X407 : term,X408 : term] : ((X405 = X408) | ((ap @ (sub @ X408 @ id) @ X406) != (ap @ (sub @ X405 @ id) @ X407)))))) & (hoasapinj1_gthm = $true) & ((! [X409 : subst] : ((comp @ id @ X409) = X409) => (! [X410 : subst] : ((comp @ X410 @ id) = X410) => (! [X411 : subst] : ((comp @ id @ X411) = X411) & ! [X412 : subst] : ((comp @ X412 @ id) = X412)))) <=> (substmonoid_lthm = $true)) & ! [X413 : term] : ? [X414 : d_term] : ((d2term @ X414) = X413) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X415 : term,X416 : term,X417 : term,X418 : term] : ((X416 = X418) | ((ap @ (sub @ X417 @ id) @ X418) != (ap @ (sub @ X415 @ id) @ X416))) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X419 : subst > term > subst > $o,X420 : term] : (~! [X421 : term] : (((X419 @ id @ (sub @ X420 @ (push @ X421 @ id)) @ id) = $true) | ~((X419 @ id @ X421 @ id) = $true)) | ((X419 @ id @ (lam @ (sub @ X420 @ (push @ one @ sh))) @ id) = $true) | ~! [X422 : subst > term > term] : (~! [X423 : term,X424 : subst,X425 : subst] : ((X422 @ (comp @ X425 @ X424) @ (sub @ X423 @ X424)) = (sub @ (X422 @ X425 @ X423) @ X424)) | ~! [X426 : term] : (((X419 @ id @ (X422 @ id @ X426) @ id) = $true) | ~((X419 @ id @ X426 @ id) = $true)) | ((X419 @ id @ (lam @ (X422 @ sh @ one)) @ id) = $true)))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X427 : term] : ((sub @ X427 @ id) = X427) => (! [X428 : term,X429 : subst > term > subst > $o] : (~! [X430 : term] : (((X429 @ id @ (sub @ X428 @ (push @ X430 @ id)) @ id) = $true) | ~((X429 @ id @ X430 @ id) = $true)) | ~! [X431 : subst > term > term] : (~! [X432 : term] : (~((X429 @ id @ X432 @ id) = $true) | ((X429 @ id @ (X431 @ id @ X432) @ id) = $true)) | ((X429 @ id @ (lam @ (X431 @ sh @ one)) @ id) = $true) | ~! [X433 : term,X434 : subst,X435 : subst] : ((X431 @ (comp @ X435 @ X434) @ (sub @ X433 @ X434)) = (sub @ (X431 @ X435 @ X433) @ X434))) | ((X429 @ id @ (lam @ (sub @ X428 @ (push @ one @ sh))) @ id) = $true)) => ! [X436 : term,X437 : subst > term > subst > $o] : (((X437 @ id @ (lam @ X436) @ id) = $true) | ~! [X438 : subst > term > term] : (((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id) = $true) | ~! [X439 : term] : (((X437 @ id @ (X438 @ id @ X439) @ id) = $true) | ~((X437 @ id @ X439 @ id) = $true)) | ~! [X440 : term,X441 : subst,X442 : subst] : ((sub @ (X438 @ X441 @ X440) @ X442) = (X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442)))) | ~! [X443 : term,X444 : subst,X445 : subst,X446 : subst] : (($true = (X437 @ X446 @ X443 @ (comp @ X445 @ X444))) | ~((X437 @ (comp @ X446 @ X445) @ (sub @ X443 @ X445) @ X444) = $true)) | ~! [X447 : term] : (((X437 @ id @ (sub @ X436 @ (push @ X447 @ id)) @ id) = $true) | ~((X437 @ id @ X447 @ id) = $true)) | ~! [X448 : subst,X449 : subst,X450 : subst,X451 : term] : (((X437 @ (comp @ X450 @ X449) @ (sub @ X451 @ X449) @ X448) = $true) | ~((X437 @ X450 @ X451 @ (comp @ X449 @ X448)) = $true)))))) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem0_gthm = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (! [X452 : subst > term > subst > $o,X453 : term] : (~! [X454 : term] : (((X452 @ id @ (sub @ X453 @ (push @ X454 @ id)) @ id) = $true) | ~((X452 @ id @ X454 @ id) = $true)) | ((X452 @ id @ (lam @ (sub @ X453 @ (push @ one @ sh))) @ id) = $true) | ~! [X455 : subst > term > term,X456 : term] : (~! [X457 : subst,X458 : term,X459 : subst] : ((sub @ X456 @ (push @ (sub @ X458 @ X457) @ (comp @ X459 @ X457))) = (X455 @ (comp @ X459 @ X457) @ (sub @ X458 @ X457))) | ((X452 @ id @ (lam @ (X455 @ sh @ one)) @ id) = $true) | ~! [X460 : term] : (~((X452 @ id @ X460 @ id) = $true) | ((X452 @ id @ (X455 @ id @ X460) @ id) = $true)) | ~! [X461 : subst,X462 : term,X463 : subst] : ((sub @ (sub @ X456 @ (push @ X462 @ X461)) @ X463) = (sub @ (X455 @ X461 @ X462) @ X463)))) <=> (hoasinduction_lem3aaa = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X464 : term,X465 : subst > term > term,X466 : subst,X467 : subst > term > term] : (((lam @ (X467 @ sh @ one)) != (lam @ (X465 @ sh @ one))) | ~! [X471 : term,X472 : subst,X473 : subst] : ((X465 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472)) = (sub @ (X465 @ X473 @ X471) @ X472)) | ((X465 @ X466 @ X464) = (X467 @ X466 @ X464)) | ~! [X468 : subst,X469 : term,X470 : subst] : ((X467 @ (comp @ X470 @ X468) @ (sub @ X469 @ X468)) = (sub @ (X467 @ X470 @ X469) @ X468))) <=> (hoaslaminj = $true)) & ((! [X474 : subst,X475 : term] : ((sub @ one @ (push @ X475 @ X474)) = X475) => (! [X476 : subst,X477 : term] : ((comp @ sh @ (push @ X477 @ X476)) = X476) => (! [X478 : subst] : ((comp @ X478 @ id) = X478) => (! [X479 : term > $o,X480 : term] : (((X479 @ X480) = $true) | ~! [X481 : term] : (((X479 @ (lam @ X481)) = $true) | ~((X479 @ X481) = $true)) | ~! [X482 : term] : (~((var @ X482) = $true) | ((X479 @ X482) = $true)) | ~! [X483 : term,X484 : term] : (~((X479 @ X483) = $true) | ~((X479 @ X484) = $true) | ((X479 @ (ap @ X484 @ X483)) = $true))) => ! [X485 : term,X486 : subst,X487 : term > $o] : (((X487 @ (sub @ X485 @ X486)) = $true) | ~! [X488 : term] : (((X487 @ (lam @ X488)) = $true) | ~! [X489 : term] : (~((X487 @ X489) = $true) | ((X487 @ (sub @ X488 @ (push @ X489 @ id))) = $true))) | ~! [X490 : term] : (((X487 @ (sub @ X490 @ X486)) = $true) | ~((var @ X490) = $true)) | ~! [X491 : term,X492 : term] : (~((X487 @ X491) = $true) | ((X487 @ (ap @ X491 @ X492)) = $true) | ~((X487 @ X492) = $true))))))) <=> (induction2lem_lthm = $true)) & (hoasapnotvar_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (one = (d2term @ d_one)) & (axapp = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lthm_orig = $true) & ((axshiftcons = $true) <=> ! [X493 : subst,X494 : term] : ((comp @ sh @ (push @ X494 @ X493)) = X493)) & (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)))))))))))))) & (! [X495 : term > $o,X496 : term] : (~! [X497 : term,X498 : term] : (~((X495 @ X497) = $true) | ((X495 @ (ap @ X497 @ X498)) = $true) | ~((X495 @ X498) = $true)) | ~! [X499 : term] : (~((var @ X499) = $true) | ((X495 @ X499) = $true)) | ~! [X500 : term] : (((X495 @ (lam @ X500)) = $true) | ~((X495 @ X500) = $true)) | ((X495 @ X496) = $true)) <=> (induction = $true)) & ~((var @ (d2term @ d_one)) = $true) & (hoasapnotvar_gthm = $true) & ((! [X501 : term] : ((sub @ X501 @ id) = X501) => ! [X502 : term > $o,X503 : subst > term > subst > $o,X504 : term] : (~! [X505 : term] : ((X503 @ id @ X505 @ id) = (X502 @ X505)) | ((X502 @ (lam @ X504)) = $true) | ~! [X506 : subst,X507 : subst,X508 : subst,X509 : term] : (~((X503 @ X506 @ X509 @ (comp @ X507 @ X508)) = $true) | ((X503 @ (comp @ X506 @ X507) @ (sub @ X509 @ X507) @ X508) = $true)) | ~! [X510 : subst > term > term] : (((X503 @ id @ (lam @ (X510 @ sh @ one)) @ id) = $true) | ~! [X511 : term] : (~((X503 @ id @ X511 @ id) = $true) | ((X503 @ id @ (X510 @ id @ X511) @ id) = $true)) | ~! [X512 : subst,X513 : term,X514 : subst] : ((sub @ (X510 @ X514 @ X513) @ X512) = (X510 @ (comp @ X514 @ X512) @ (sub @ X513 @ X512)))) | ~! [X515 : subst,X516 : subst,X517 : subst,X518 : term] : (((X503 @ X516 @ X518 @ (comp @ X515 @ X517)) = $true) | ~((X503 @ (comp @ X516 @ X515) @ (sub @ X518 @ X515) @ X517) = $true)) | ~! [X519 : term] : (~((X502 @ X519) = $true) | ((X502 @ (sub @ X504 @ (push @ X519 @ id))) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X520 : subst,X521 : term,X522 : term > $o,X523 : term > $o] : (((X523 @ one) = $true) | ~((X522 @ X521) = $true) | ~! [X524 : term] : ((X522 @ (sub @ X524 @ (push @ X521 @ X520))) = (X523 @ X524)))) & (! [X525 : term > $o,X526 : term,X527 : term,X528 : subst > term > subst > $o] : (~((X525 @ X526) = $true) | ~! [X529 : term] : ((X528 @ id @ X529 @ id) = (X525 @ X529)) | ~! [X530 : term,X531 : term] : (~((X528 @ id @ X530 @ id) = $true) | ~((X528 @ id @ X531 @ id) = $true) | ((X528 @ id @ (ap @ (sub @ X531 @ id) @ X530) @ id) = $true)) | ~! [X532 : subst,X533 : subst,X534 : term,X535 : subst] : (~((X528 @ (comp @ X533 @ X532) @ (sub @ X534 @ X532) @ X535) = $true) | ((X528 @ X533 @ X534 @ (comp @ X532 @ X535)) = $true)) | ((X525 @ (ap @ X527 @ X526)) = $true) | ~((X525 @ X527) = $true) | ~! [X536 : subst,X537 : subst,X538 : term,X539 : subst] : (((X528 @ (comp @ X536 @ X537) @ (sub @ X538 @ X537) @ X539) = $true) | ~((X528 @ X536 @ X538 @ (comp @ X537 @ X539)) = $true))) <=> (hoasinduction_lem2v2 = $true)) & (hoasinduction_lem1 = $true)),
% 9.05/2.75 inference(rectify,[status(thm)],[f6])).
% 9.05/2.75 thf(f6,plain,(
% 9.05/2.75 ((substmonoid = $true) <=> (! [X1 : subst] : ((comp @ id @ X1) = X1) & ! [X0 : subst] : ((comp @ X0 @ id) = X0))) & ~(lamnotap = $true) & (axassoc = $true) & (! [X2 : subst > term > subst > $o] : ~! [X3 : term > $o] : ~! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) <=> (hoasinduction_lem0_lthm = $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((! [X5 : term,X6 : subst] : ((sub @ one @ (push @ X5 @ X6)) = X5) => (! [X7 : term,X8 : subst] : ((comp @ sh @ (push @ X7 @ X8)) = X8) => (! [X9 : term,X10 : term] : ((X9 = X10) | ((lam @ X10) != (lam @ X9))) => ! [X11 : subst > term > term,X12 : subst > term > term,X13 : term,X14 : subst] : (~! [X15 : term,X16 : subst,X17 : subst] : ((sub @ (X11 @ X17 @ X15) @ X16) = (X11 @ (comp @ X17 @ X16) @ (sub @ X15 @ X16))) | ~! [X18 : subst,X19 : term,X20 : subst] : ((sub @ (X12 @ X20 @ X19) @ X18) = (X12 @ (comp @ X20 @ X18) @ (sub @ X19 @ X18))) | ((lam @ (X11 @ sh @ one)) != (lam @ (X12 @ sh @ one))) | ((X11 @ X14 @ X13) = (X12 @ X14 @ X13)))))) <=> (hoaslaminj_lthm = $true)) & (pushprop_gthm = $true) & (induction2lem_gthm = $true) & ((! [X21 : subst > term > subst > $o] : ~! [X22 : term > $o] : ~! [X23 : term] : ((X21 @ id @ X23 @ id) = (X22 @ X23)) => (! [X24 : term,X25 : term > $o] : (~! [X26 : term,X27 : term] : (((X25 @ (ap @ X27 @ X26)) = $true) | ~((X25 @ X27) = $true) | ~((X25 @ X26) = $true)) | ~! [X28 : term] : (((X25 @ (lam @ X28)) = $true) | ~! [X29 : term] : (((X25 @ (sub @ X28 @ (push @ X29 @ id))) = $true) | ~((X25 @ X29) = $true))) | ~! [X30 : term] : (~((var @ X30) = $true) | ((X25 @ X30) = $true)) | ((X25 @ X24) = $true)) => (! [X31 : term] : ((sub @ X31 @ id) = X31) => (! [X32 : term > $o,X33 : subst > term > subst > $o,X34 : term] : (~! [X35 : term] : (((X32 @ (sub @ X34 @ (push @ X35 @ id))) = $true) | ~((X32 @ X35) = $true)) | ~! [X36 : subst > term > term] : (((X33 @ id @ (lam @ (X36 @ sh @ one)) @ id) = $true) | ~! [X37 : term] : (~((X33 @ id @ X37 @ id) = $true) | ((X33 @ id @ (X36 @ id @ X37) @ id) = $true)) | ~! [X38 : subst,X39 : term,X40 : subst] : ((X36 @ (comp @ X38 @ X40) @ (sub @ X39 @ X40)) = (sub @ (X36 @ X38 @ X39) @ X40))) | ~! [X41 : term] : ((X33 @ id @ X41 @ id) = (X32 @ X41)) | ((X32 @ (lam @ X34)) = $true)) => ! [X42 : subst > term > subst > $o,X43 : term] : (~! [X44 : subst > term > term] : (~! [X45 : subst,X46 : subst,X47 : term] : ((sub @ (X44 @ X45 @ X47) @ X46) = (X44 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46))) | ~! [X48 : term] : (((X42 @ id @ (X44 @ id @ X48) @ id) = $true) | ~((X42 @ id @ X48 @ id) = $true)) | ((X42 @ id @ (lam @ (X44 @ sh @ one)) @ id) = $true)) | ((X42 @ id @ X43 @ id) = $true) | ~! [X49 : term] : (~((var @ (sub @ X49 @ id)) = $true) | ((X42 @ id @ X49 @ id) = $true)) | ~! [X50 : subst,X51 : term,X52 : subst,X53 : subst] : (((X42 @ (comp @ X53 @ X52) @ (sub @ X51 @ X52) @ X50) = $true) | ~((X42 @ X53 @ X51 @ (comp @ X52 @ X50)) = $true)) | ~! [X54 : term,X55 : term] : (((X42 @ id @ (ap @ (sub @ X54 @ id) @ X55) @ id) = $true) | ~((X42 @ id @ X54 @ id) = $true) | ~((X42 @ id @ X55 @ id) = $true)) | ~! [X56 : subst,X57 : subst,X58 : subst,X59 : term] : (~((X42 @ (comp @ X58 @ X57) @ (sub @ X59 @ X57) @ X56) = $true) | ((X42 @ X58 @ X59 @ (comp @ X57 @ X56)) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem1_gthm = $true) & (ulamvarsh = $true) & ((pushprop_lem1 = $true) <=> ! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (~((X60 @ X61) = $true) | ((X62 @ (sub @ X61 @ (push @ X63 @ X64))) = $true))) & (hoaslamnotap_gthm = $true) & (! [X65 : term] : ~! [X66 : subst > term > term] : ~! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (lamnotvar = $true) & ((shinj = $true) <=> ! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh)))) & (! [X71 : subst] : ((comp @ X71 @ id) = X71) <=> (axidr = $true)) & (hoasinduction_lem1v2_gthm = $true) & ((d2subst @ d_id) = id) & ((axidl = $true) <=> ! [X72 : subst] : ((comp @ id @ X72) = X72)) & ((hoasinduction_lem3b = $true) <=> ! [X73 : term] : ~! [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) != (X74 @ sh @ one))) & ((hoasinduction_lem3a = $true) <=> ! [X75 : subst > term > subst > $o,X76 : term] : (~! [X77 : subst > term > term] : (~! [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) = (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78))) | ~! [X81 : term] : (~((X75 @ id @ X81 @ id) = $true) | ((X75 @ id @ (X77 @ id @ X81) @ id) = $true)) | ((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) = $true)) | ((X75 @ id @ (lam @ X76) @ id) = $true) | ~! [X82 : term] : (((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) = $true) | ~((X75 @ id @ X82 @ id) = $true)))) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X83 : term] : ((sub @ X83 @ id) = X83) => (! [X84 : term,X85 : subst > term > subst > $o] : (~! [X86 : term] : (~((X85 @ id @ X86 @ id) = $true) | ((X85 @ id @ (sub @ X84 @ (push @ X86 @ id)) @ id) = $true)) | ~! [X87 : subst > term > term] : (~! [X88 : term] : (~((X85 @ id @ X88 @ id) = $true) | ($true = (X85 @ id @ (X87 @ id @ X88) @ id))) | ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id) = $true) | ~! [X89 : subst,X90 : term,X91 : subst] : ((X87 @ (comp @ X91 @ X89) @ (sub @ X90 @ X89)) = (sub @ (X87 @ X91 @ X90) @ X89))) | ((X85 @ id @ (lam @ (sub @ X84 @ (push @ one @ sh))) @ id) = $true)) => ! [X92 : term,X93 : subst > term > subst > $o] : (~! [X94 : subst > term > term] : (((X93 @ id @ (lam @ (X94 @ sh @ one)) @ id) = $true) | ~! [X95 : subst,X96 : term,X97 : subst] : ((X94 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97)) = (sub @ (X94 @ X95 @ X96) @ X97)) | ~! [X98 : term] : (~((X93 @ id @ X98 @ id) = $true) | ((X93 @ id @ (X94 @ id @ X98) @ id) = $true))) | ($true = (X93 @ id @ (lam @ X92) @ id)) | ~! [X99 : term] : (((X93 @ id @ (sub @ X92 @ (push @ X99 @ id)) @ id) = $true) | ~((X93 @ id @ X99 @ id) = $true)))))) & (! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100) <=> (axvarcons = $true)) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & (hoaslamnotvar_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X103 : term] : ~! [X104 : subst > term > term] : ~! [X105 : subst,X106 : term] : ((sub @ X103 @ (push @ X106 @ X105)) = (X104 @ X105 @ X106)) => (! [X107 : term] : ((sub @ X107 @ id) = X107) => ! [X108 : term,X109 : subst > term > subst > $o,X110 : term > $o] : (~! [X111 : term] : (~((X110 @ X111) = $true) | ($true = (X110 @ (sub @ X108 @ (push @ X111 @ id))))) | ((X110 @ (lam @ X108)) = $true) | ~! [X112 : term] : ((X109 @ id @ X112 @ id) = (X110 @ X112)) | ~! [X113 : subst > term > term] : (((X109 @ id @ (lam @ (X113 @ sh @ one)) @ id) = $true) | ~! [X114 : term] : (~((X109 @ id @ X114 @ id) = $true) | ((X109 @ id @ (X113 @ id @ X114) @ id) = $true)) | ~! [X115 : subst,X116 : term,X117 : subst] : ((X113 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117)) = (sub @ (X113 @ X115 @ X116) @ X117)))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((! [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) = X118) => (! [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) = X121) => ! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (~((X122 @ X124) = $true) | ((X125 @ (sub @ X124 @ (push @ X126 @ X123))) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_gthm = $true) & (! [X127 : subst > term > subst > $o,X128 : term] : (~! [X129 : term] : (((X127 @ id @ (sub @ X128 @ (push @ X129 @ id)) @ id) = $true) | ~((X127 @ id @ X129 @ id) = $true)) | ~! [X130 : subst > term > term] : (~! [X131 : term,X132 : subst,X133 : subst] : ((X130 @ (comp @ X132 @ X133) @ (sub @ X131 @ X133)) = (sub @ (X130 @ X132 @ X131) @ X133)) | ~! [X134 : term] : (((X127 @ id @ (X130 @ id @ X134) @ id) = $true) | ~((X127 @ id @ X134 @ id) = $true)) | ((X127 @ id @ (lam @ (X130 @ sh @ one)) @ id) = $true)) | ((X127 @ id @ (lam @ (sub @ X128 @ (push @ one @ sh))) @ id) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & (! [X135 : term] : ~! [X136 : subst > term > term] : ~! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138)) <=> (hoasinduction_lem3v2_f = $true)) & (! [X139 : term,X140 : term > $o,X141 : subst] : ~! [X142 : term > $o] : ~! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143)) <=> (pushprop_lem0_lthm = $true)) & (! [X144 : term] : ~! [X145 : subst > term > term] : ((X145 @ sh @ one) != (sub @ X144 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X146 : subst > term > subst > $o,X147 : term > $o,X148 : term] : (~! [X149 : term] : ((X147 @ X149) = (X146 @ id @ X149 @ id)) | ((X147 @ (lam @ X148)) = $true) | ~! [X150 : subst,X151 : subst,X152 : subst,X153 : term] : (((X146 @ X151 @ X153 @ (comp @ X150 @ X152)) = $true) | ~((X146 @ (comp @ X151 @ X150) @ (sub @ X153 @ X150) @ X152) = $true)) | ~! [X154 : subst,X155 : subst,X156 : subst,X157 : term] : (((X146 @ (comp @ X155 @ X154) @ (sub @ X157 @ X154) @ X156) = $true) | ~((X146 @ X155 @ X157 @ (comp @ X154 @ X156)) = $true)) | ~! [X158 : term] : (((X147 @ (sub @ X148 @ (push @ X158 @ id))) = $true) | ~((X147 @ X158) = $true)) | ~! [X159 : subst > term > term] : (~! [X160 : term] : (((X146 @ id @ (X159 @ id @ X160) @ id) = $true) | ~((X146 @ id @ X160 @ id) = $true)) | ~! [X161 : term,X162 : subst,X163 : subst] : ((X159 @ (comp @ X163 @ X162) @ (sub @ X161 @ X162)) = (sub @ (X159 @ X163 @ X161) @ X162)) | ((X146 @ id @ (lam @ (X159 @ sh @ one)) @ id) = $true)))) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (pushprop_lem3v2 = $true) & (! [X164 : term] : ((sub @ X164 @ id) = X164) <=> (termmset = $true)) & (! [X165 : subst > term > subst > $o] : ~! [X166 : term > $o] : ~! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) <=> (hoasinduction_lem0 = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X168 : term,X169 : term,X170 : subst > term > subst > $o] : (~((X170 @ id @ X169 @ id) = $true) | ~! [X171 : term,X172 : term] : (((X170 @ id @ (ap @ (sub @ X171 @ id) @ X172) @ id) = $true) | ~((X170 @ id @ X172 @ id) = $true) | ~((X170 @ id @ X171 @ id) = $true)) | ~! [X173 : subst,X174 : subst,X175 : term,X176 : subst] : (((X170 @ X176 @ X175 @ (comp @ X173 @ X174)) = $true) | ~((X170 @ (comp @ X176 @ X173) @ (sub @ X175 @ X173) @ X174) = $true)) | ((X170 @ id @ (ap @ X168 @ X169) @ id) = $true) | ~((X170 @ id @ X168 @ id) = $true) | ~! [X177 : subst,X178 : subst,X179 : term,X180 : subst] : (~((X170 @ X178 @ X179 @ (comp @ X177 @ X180)) = $true) | ((X170 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177) @ X180) = $true)))) & ! [X181 : d_subst,X182 : d_subst] : (((d2subst @ X182) = (d2subst @ X181)) => (X181 = X182)) & (substmonoid_gthm = $true) & (pushprop = $true) & (axclos = $true) & (! [X183 : term,X184 : subst > term > subst > $o] : (((X184 @ id @ X183 @ id) = $true) | ~! [X185 : term,X186 : term] : (((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) = $true) | ~((X184 @ id @ X186 @ id) = $true) | ~((X184 @ id @ X185 @ id) = $true)) | ~! [X187 : subst > term > term] : (((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) = $true) | ~! [X188 : term] : (~((X184 @ id @ X188 @ id) = $true) | ((X184 @ id @ (X187 @ id @ X188) @ id) = $true)) | ~! [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) = (sub @ (X187 @ X189 @ X191) @ X190)))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X192 : term,X193 : term,X194 : term,X195 : term] : (((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192)) | (X193 = X194)) <=> (hoasapinj1 = $true)) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & (pushprop_lem1v2_gthm = $true) & (axabs = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X198 : subst > term > subst > $o] : ~! [X199 : term > $o] : ~! [X200 : term] : ((X198 @ id @ X200 @ id) = (X199 @ X200)) => (! [X201 : term > $o,X202 : term] : (((X201 @ X202) = $true) | ~! [X203 : term] : (((X201 @ X203) = $true) | ~((var @ X203) = $true)) | ~! [X204 : term,X205 : term] : (~((X201 @ X204) = $true) | ~((X201 @ X205) = $true) | ((X201 @ (ap @ X205 @ X204)) = $true)) | ~! [X206 : term] : (((X201 @ (lam @ X206)) = $true) | ~! [X207 : term] : (~((X201 @ X207) = $true) | ((X201 @ (sub @ X206 @ (push @ X207 @ id))) = $true)))) => (! [X208 : term] : ((sub @ X208 @ id) = X208) => (! [X209 : subst > term > subst > $o,X210 : term > $o,X211 : term] : (((X210 @ (lam @ X211)) = $true) | ~! [X212 : term] : (~((X210 @ X212) = $true) | ((X210 @ (sub @ X211 @ (push @ X212 @ id))) = $true)) | ~! [X213 : subst > term > term] : (((X209 @ id @ (lam @ (X213 @ sh @ one)) @ id) = $true) | ~! [X214 : term] : (((X209 @ id @ (X213 @ id @ X214) @ id) = $true) | ~((X209 @ id @ X214 @ id) = $true)) | ~! [X215 : subst,X216 : subst,X217 : term] : ((sub @ (X213 @ X216 @ X217) @ X215) = (X213 @ (comp @ X216 @ X215) @ (sub @ X217 @ X215)))) | ~! [X218 : term] : ((X209 @ id @ X218 @ id) = (X210 @ X218))) => ! [X219 : subst > term > subst > $o,X220 : term] : (~! [X221 : term,X222 : term] : (~((X219 @ id @ X222 @ id) = $true) | ~((X219 @ id @ X221 @ id) = $true) | ((X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id) = $true)) | ((X219 @ id @ X220 @ id) = $true) | ~! [X223 : subst > term > term] : (~! [X224 : term] : (~((X219 @ id @ X224 @ id) = $true) | ((X219 @ id @ (X223 @ id @ X224) @ id) = $true)) | ($true = (X219 @ id @ (lam @ (X223 @ sh @ one)) @ id)) | ~! [X225 : subst,X226 : term,X227 : subst] : ((X223 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225)) = (sub @ (X223 @ X227 @ X226) @ X225))))))))) & (termmset_gthm = $true) & (axmap = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) = X228) => ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (~! [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) = (X232 @ X234)) | ((X232 @ one) = $true) | ~((X230 @ X233) = $true)))) & (hoasapnotvar = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X238 : subst > term > subst > $o,X239 : term,X240 : term > $o] : (~! [X241 : term] : (~((X240 @ X241) = $true) | ((X240 @ (sub @ X239 @ (push @ X241 @ id))) = $true)) | ~! [X242 : subst > term > term] : (~! [X243 : subst,X244 : term,X245 : subst] : ((sub @ (X242 @ X245 @ X244) @ X243) = (X242 @ (comp @ X245 @ X243) @ (sub @ X244 @ X243))) | ((X238 @ id @ (lam @ (X242 @ sh @ one)) @ id) = $true) | ~! [X246 : term] : (((X238 @ id @ (X242 @ id @ X246) @ id) = $true) | ~((X238 @ id @ X246 @ id) = $true))) | ((X240 @ (lam @ X239)) = $true) | ~! [X247 : term] : ((X238 @ id @ X247 @ id) = (X240 @ X247))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ! [X248 : d_term,X249 : d_term] : (((d2term @ X248) = (d2term @ X249)) => (X248 = X249)) & ((hoaslamnotap = $true) <=> ! [X250 : term,X251 : subst > term > term,X252 : term] : (((lam @ (X251 @ sh @ one)) != (ap @ (sub @ X250 @ id) @ X252)) | ~! [X253 : subst,X254 : subst,X255 : term] : ((sub @ (X251 @ X253 @ X255) @ X254) = (X251 @ (comp @ X253 @ X254) @ (sub @ X255 @ X254))))) & ((! [X256 : term,X257 : term,X258 : term,X259 : term] : ((X256 = X257) | ((ap @ X259 @ X256) != (ap @ X258 @ X257))) => ! [X260 : term,X261 : term,X262 : term,X263 : term] : ((X260 = X261) | ((ap @ (sub @ X262 @ id) @ X260) != (ap @ (sub @ X263 @ id) @ X261)))) <=> (hoasapinj2_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (! [X264 : term,X265 : term,X266 : term,X267 : term] : ((X264 = X265) | ((ap @ X264 @ X266) != (ap @ X265 @ X267))) <=> (apinj1 = $true)) & ! [X268 : d_term] : (d_one = X268) & (induction2_gthm = $true) & (! [X269 : subst,X270 : term > $o,X271 : term] : ~! [X272 : term > $o] : ~! [X273 : term] : ((X272 @ X273) = (X270 @ (sub @ X273 @ (push @ X271 @ X269)))) <=> (pushprop_lem0 = $true)) & (pushprop_lthm = $true) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ! [X274 : d_subst] : (d_id = X274) & ~(ulamvar1 = $true) & ((termmset_lthm = $true) <=> (! [X275 : term] : ((sub @ X275 @ id) = X275) => ! [X276 : term] : ((sub @ X276 @ id) = X276))) & (hoaslamnotvar = $true) & (pushprop_lem2v2 = $true) & (pushprop_lem3v2_lthm = $true) & (! [X277 : subst > term > subst > $o,X278 : term] : (((X277 @ id @ X278 @ id) = $true) | ~! [X279 : term] : (((X277 @ id @ X279 @ id) = $true) | ~((var @ (sub @ X279 @ id)) = $true)) | ~! [X280 : term,X281 : subst,X282 : subst,X283 : subst] : (((X277 @ X282 @ X280 @ (comp @ X281 @ X283)) = $true) | ~((X277 @ (comp @ X282 @ X281) @ (sub @ X280 @ X281) @ X283) = $true)) | ~! [X284 : subst,X285 : term,X286 : subst,X287 : subst] : (~((X277 @ X286 @ X285 @ (comp @ X287 @ X284)) = $true) | ((X277 @ (comp @ X286 @ X287) @ (sub @ X285 @ X287) @ X284) = $true)) | ~! [X288 : subst > term > term] : (~! [X289 : term,X290 : subst,X291 : subst] : ((sub @ (X288 @ X290 @ X289) @ X291) = (X288 @ (comp @ X290 @ X291) @ (sub @ X289 @ X291))) | ~! [X292 : term] : (~((X277 @ id @ X292 @ id) = $true) | ($true = (X277 @ id @ (X288 @ id @ X292) @ id))) | ((X277 @ id @ (lam @ (X288 @ sh @ one)) @ id) = $true)) | ~! [X293 : term,X294 : term] : (~((X277 @ id @ X294 @ id) = $true) | ((X277 @ id @ (ap @ (sub @ X294 @ id) @ X293) @ id) = $true) | ~((X277 @ id @ X293 @ id) = $true))) <=> (hoasinduction = $true)) & ((laminj = $true) <=> ! [X295 : term,X296 : term] : ((X295 = X296) | ((lam @ X296) != (lam @ X295)))) & (! [X297 : term,X298 : term > $o] : (~! [X299 : term] : (~((var @ X299) = $true) | ((X298 @ X299) = $true)) | ~! [X300 : term,X301 : term] : (((X298 @ (ap @ X300 @ X301)) = $true) | ~((X298 @ X301) = $true) | ~((X298 @ X300) = $true)) | ~! [X302 : term] : (~! [X303 : term] : (~((X298 @ X303) = $true) | ((X298 @ (sub @ X302 @ (push @ X303 @ id))) = $true)) | ((X298 @ (lam @ X302)) = $true)) | ((X298 @ X297) = $true)) <=> (induction2 = $true)) & ((d2subst @ d_id) = sh) & (hoasapinj2_gthm = $true) & ((! [X304 : term] : ((sub @ X304 @ id) = X304) => (! [X305 : subst,X306 : term > $o,X307 : term] : (~! [X308 : term,X309 : term] : (~((X306 @ X308) = $true) | ~((X306 @ X309) = $true) | ((X306 @ (ap @ X308 @ X309)) = $true)) | ~! [X310 : term] : (~! [X311 : term] : (~((X306 @ X311) = $true) | ((X306 @ (sub @ X310 @ (push @ X311 @ id))) = $true)) | ((X306 @ (lam @ X310)) = $true)) | ((X306 @ (sub @ X307 @ X305)) = $true) | ~! [X312 : term] : (~((var @ X312) = $true) | ((X306 @ (sub @ X312 @ X305)) = $true))) => ! [X313 : term,X314 : term > $o] : (~! [X315 : term] : (~((var @ X315) = $true) | ((X314 @ X315) = $true)) | ((X314 @ X313) = $true) | ~! [X316 : term] : (~! [X317 : term] : (((X314 @ (sub @ X316 @ (push @ X317 @ id))) = $true) | ~((X314 @ X317) = $true)) | ((X314 @ (lam @ X316)) = $true)) | ~! [X318 : term,X319 : term] : (((X314 @ (ap @ X318 @ X319)) = $true) | ~((X314 @ X318) = $true) | ~((X314 @ X319) = $true))))) <=> (induction2_lthm = $true)) & (hoasinduction_lem3b_gthm = $true) & (ulamvarind = $true) & (hoaslamnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X320 : term,X321 : term > $o,X322 : subst] : (~! [X323 : term] : (~! [X324 : term] : (((X321 @ (sub @ X323 @ (push @ X324 @ id))) = $true) | ~((X321 @ X324) = $true)) | ((X321 @ (lam @ X323)) = $true)) | ~! [X325 : term,X326 : term] : (((X321 @ (ap @ X326 @ X325)) = $true) | ~($true = (X321 @ X326)) | ~((X321 @ X325) = $true)) | ((X321 @ (sub @ X320 @ X322)) = $true) | ~! [X327 : term] : (~((var @ X327) = $true) | ((X321 @ (sub @ X327 @ X322)) = $true))) <=> (induction2lem = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem1v2 = $true) & ((apinj2 = $true) <=> ! [X330 : term,X331 : term,X332 : term,X333 : term] : (((ap @ X332 @ X330) != (ap @ X331 @ X333)) | (X330 = X333))) & ((hoasinduction_lthm = $true) <=> (! [X334 : term > $o,X335 : term] : (~! [X336 : term] : (((X334 @ X336) = $true) | ~((var @ X336) = $true)) | ~! [X337 : term,X338 : term] : (~((X334 @ X337) = $true) | ~((X334 @ X338) = $true) | ((X334 @ (ap @ X337 @ X338)) = $true)) | ~! [X339 : term] : (~! [X340 : term] : (~((X334 @ X340) = $true) | ((X334 @ (sub @ X339 @ (push @ X340 @ id))) = $true)) | ((X334 @ (lam @ X339)) = $true)) | ((X334 @ X335) = $true)) => (! [X341 : term,X342 : subst > term > subst > $o,X343 : term] : (~! [X344 : subst,X345 : subst,X346 : term,X347 : subst] : (((X342 @ (comp @ X347 @ X344) @ (sub @ X346 @ X344) @ X345) = $true) | ~((X342 @ X347 @ X346 @ (comp @ X344 @ X345)) = $true)) | ~((X342 @ id @ X341 @ id) = $true) | ((X342 @ id @ (ap @ X343 @ X341) @ id) = $true) | ~((X342 @ id @ X343 @ id) = $true) | ~! [X348 : term,X349 : term] : (~((X342 @ id @ X349 @ id) = $true) | ((X342 @ id @ (ap @ (sub @ X348 @ id) @ X349) @ id) = $true) | ~((X342 @ id @ X348 @ id) = $true)) | ~! [X350 : subst,X351 : term,X352 : subst,X353 : subst] : (((X342 @ X353 @ X351 @ (comp @ X352 @ X350)) = $true) | ~((X342 @ (comp @ X353 @ X352) @ (sub @ X351 @ X352) @ X350) = $true))) => (! [X354 : subst > term > subst > $o,X355 : term] : (~! [X356 : subst,X357 : subst,X358 : term,X359 : subst] : (~((X354 @ (comp @ X356 @ X357) @ (sub @ X358 @ X357) @ X359) = $true) | ((X354 @ X356 @ X358 @ (comp @ X357 @ X359)) = $true)) | ~! [X360 : subst > term > term] : (~! [X361 : term] : (~((X354 @ id @ X361 @ id) = $true) | ((X354 @ id @ (X360 @ id @ X361) @ id) = $true)) | ((X354 @ id @ (lam @ (X360 @ sh @ one)) @ id) = $true) | ~! [X362 : subst,X363 : subst,X364 : term] : ((X360 @ (comp @ X363 @ X362) @ (sub @ X364 @ X362)) = (sub @ (X360 @ X363 @ X364) @ X362))) | ((X354 @ id @ (lam @ X355) @ id) = $true) | ~! [X365 : term] : (~((X354 @ id @ X365 @ id) = $true) | ((X354 @ id @ (sub @ X355 @ (push @ X365 @ id)) @ id) = $true)) | ~! [X366 : subst,X367 : subst,X368 : term,X369 : subst] : (~((X354 @ X366 @ X368 @ (comp @ X367 @ X369)) = $true) | ((X354 @ (comp @ X366 @ X367) @ (sub @ X368 @ X367) @ X369) = $true))) => ! [X370 : term,X371 : subst > term > subst > $o] : (~! [X372 : term] : (((X371 @ id @ X372 @ id) = $true) | ~((var @ (sub @ X372 @ id)) = $true)) | ~! [X373 : subst,X374 : subst,X375 : subst,X376 : term] : (((X371 @ (comp @ X374 @ X375) @ (sub @ X376 @ X375) @ X373) = $true) | ~((X371 @ X374 @ X376 @ (comp @ X375 @ X373)) = $true)) | ~! [X377 : term,X378 : term] : (~((X371 @ id @ X377 @ id) = $true) | ~((X371 @ id @ X378 @ id) = $true) | ((X371 @ id @ (ap @ (sub @ X378 @ id) @ X377) @ id) = $true)) | ~! [X379 : subst > term > term] : (~! [X380 : subst,X381 : term,X382 : subst] : ((sub @ (X379 @ X382 @ X381) @ X380) = (X379 @ (comp @ X382 @ X380) @ (sub @ X381 @ X380))) | ((X371 @ id @ (lam @ (X379 @ sh @ one)) @ id) = $true) | ~! [X383 : term] : (((X371 @ id @ (X379 @ id @ X383) @ id) = $true) | ~((X371 @ id @ X383 @ id) = $true))) | ((X371 @ id @ X370 @ id) = $true) | ~! [X384 : subst,X385 : subst,X386 : term,X387 : subst] : (~((X371 @ (comp @ X385 @ X384) @ (sub @ X386 @ X384) @ X387) = $true) | ((X371 @ X385 @ X386 @ (comp @ X384 @ X387)) = $true))))))) & (apnotvar = $true) & ((axvarid = $true) <=> ! [X388 : term] : ((sub @ X388 @ id) = X388)) & ((hoasinduction_lem3 = $true) <=> ! [X389 : term,X390 : subst > term > subst > $o] : (~! [X391 : subst,X392 : subst,X393 : subst,X394 : term] : (((X390 @ (comp @ X391 @ X393) @ (sub @ X394 @ X393) @ X392) = $true) | ~((X390 @ X391 @ X394 @ (comp @ X393 @ X392)) = $true)) | ~! [X395 : term] : (((X390 @ id @ (sub @ X389 @ (push @ X395 @ id)) @ id) = $true) | ~((X390 @ id @ X395 @ id) = $true)) | ((X390 @ id @ (lam @ X389) @ id) = $true) | ~! [X396 : subst > term > term] : (~! [X397 : subst,X398 : subst,X399 : term] : ((X396 @ (comp @ X398 @ X397) @ (sub @ X399 @ X397)) = (sub @ (X396 @ X398 @ X399) @ X397)) | ~! [X400 : term] : (~((X390 @ id @ X400 @ id) = $true) | ((X390 @ id @ (X396 @ id @ X400) @ id) = $true)) | ((X390 @ id @ (lam @ (X396 @ sh @ one)) @ id) = $true)) | ~! [X401 : subst,X402 : subst,X403 : term,X404 : subst] : (~((X390 @ (comp @ X401 @ X402) @ (sub @ X403 @ X402) @ X404) = $true) | ((X390 @ X401 @ X403 @ (comp @ X402 @ X404)) = $true)))) & (hoaslaminj_gthm = $true) & (hoaslamnotap_lthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X405 : term] : ((sub @ X405 @ id) = X405) => (! [X406 : term,X407 : term,X408 : term,X409 : term] : (((ap @ X406 @ X408) != (ap @ X409 @ X407)) | (X406 = X409)) => ! [X410 : term,X411 : term,X412 : term,X413 : term] : ((X410 = X413) | ((ap @ (sub @ X410 @ id) @ X412) != (ap @ (sub @ X413 @ id) @ X411)))))) & (hoasapinj1_gthm = $true) & ((! [X414 : subst] : ((comp @ id @ X414) = X414) => (! [X415 : subst] : ((comp @ X415 @ id) = X415) => (! [X416 : subst] : ((comp @ id @ X416) = X416) & ! [X417 : subst] : ((comp @ X417 @ id) = X417)))) <=> (substmonoid_lthm = $true)) & ! [X418 : term] : ? [X419 : d_term] : ((d2term @ X419) = X418) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X420 : term,X421 : term,X422 : term,X423 : term] : ((X421 = X423) | ((ap @ (sub @ X420 @ id) @ X421) != (ap @ (sub @ X422 @ id) @ X423))) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X424 : subst > term > subst > $o,X425 : term] : (~! [X426 : term] : (((X424 @ id @ (sub @ X425 @ (push @ X426 @ id)) @ id) = $true) | ~((X424 @ id @ X426 @ id) = $true)) | ((X424 @ id @ (lam @ (sub @ X425 @ (push @ one @ sh))) @ id) = $true) | ~! [X427 : subst > term > term] : (~! [X428 : term,X429 : subst,X430 : subst] : ((X427 @ (comp @ X430 @ X429) @ (sub @ X428 @ X429)) = (sub @ (X427 @ X430 @ X428) @ X429)) | ~! [X431 : term] : (((X424 @ id @ (X427 @ id @ X431) @ id) = $true) | ~((X424 @ id @ X431 @ id) = $true)) | ((X424 @ id @ (lam @ (X427 @ sh @ one)) @ id) = $true)))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X432 : term] : ((sub @ X432 @ id) = X432) => (! [X433 : term,X434 : subst > term > subst > $o] : (~! [X435 : term] : (((X434 @ id @ (sub @ X433 @ (push @ X435 @ id)) @ id) = $true) | ~((X434 @ id @ X435 @ id) = $true)) | ~! [X436 : subst > term > term] : (~! [X437 : term] : (~((X434 @ id @ X437 @ id) = $true) | ((X434 @ id @ (X436 @ id @ X437) @ id) = $true)) | ((X434 @ id @ (lam @ (X436 @ sh @ one)) @ id) = $true) | ~! [X438 : term,X439 : subst,X440 : subst] : ((sub @ (X436 @ X440 @ X438) @ X439) = (X436 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439)))) | ((X434 @ id @ (lam @ (sub @ X433 @ (push @ one @ sh))) @ id) = $true)) => ! [X441 : term,X442 : subst > term > subst > $o] : (((X442 @ id @ (lam @ X441) @ id) = $true) | ~! [X443 : subst > term > term] : (((X442 @ id @ (lam @ (X443 @ sh @ one)) @ id) = $true) | ~! [X444 : term] : (((X442 @ id @ (X443 @ id @ X444) @ id) = $true) | ~((X442 @ id @ X444 @ id) = $true)) | ~! [X445 : term,X446 : subst,X447 : subst] : ((X443 @ (comp @ X446 @ X447) @ (sub @ X445 @ X447)) = (sub @ (X443 @ X446 @ X445) @ X447))) | ~! [X448 : term,X449 : subst,X450 : subst,X451 : subst] : (((X442 @ X451 @ X448 @ (comp @ X450 @ X449)) = $true) | ~((X442 @ (comp @ X451 @ X450) @ (sub @ X448 @ X450) @ X449) = $true)) | ~! [X452 : term] : (((X442 @ id @ (sub @ X441 @ (push @ X452 @ id)) @ id) = $true) | ~((X442 @ id @ X452 @ id) = $true)) | ~! [X453 : subst,X454 : subst,X455 : subst,X456 : term] : (((X442 @ (comp @ X455 @ X454) @ (sub @ X456 @ X454) @ X453) = $true) | ~((X442 @ X455 @ X456 @ (comp @ X454 @ X453)) = $true)))))) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem0_gthm = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (! [X457 : subst > term > subst > $o,X458 : term] : (~! [X459 : term] : (((X457 @ id @ (sub @ X458 @ (push @ X459 @ id)) @ id) = $true) | ~((X457 @ id @ X459 @ id) = $true)) | ((X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id) = $true) | ~! [X460 : subst > term > term,X461 : term] : (~! [X462 : subst,X463 : term,X464 : subst] : ((sub @ X461 @ (push @ (sub @ X463 @ X462) @ (comp @ X464 @ X462))) = (X460 @ (comp @ X464 @ X462) @ (sub @ X463 @ X462))) | ((X457 @ id @ (lam @ (X460 @ sh @ one)) @ id) = $true) | ~! [X465 : term] : (~((X457 @ id @ X465 @ id) = $true) | ((X457 @ id @ (X460 @ id @ X465) @ id) = $true)) | ~! [X466 : subst,X467 : term,X468 : subst] : ((sub @ (sub @ X461 @ (push @ X467 @ X466)) @ X468) = (sub @ (X460 @ X466 @ X467) @ X468)))) <=> (hoasinduction_lem3aaa = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoaslaminj = $true) <=> ! [X469 : term,X470 : subst > term > term,X471 : subst,X472 : subst > term > term] : (~! [X473 : subst,X474 : term,X475 : subst] : ((X472 @ (comp @ X475 @ X473) @ (sub @ X474 @ X473)) = (sub @ (X472 @ X475 @ X474) @ X473)) | ((lam @ (X470 @ sh @ one)) != (lam @ (X472 @ sh @ one))) | ~! [X476 : term,X477 : subst,X478 : subst] : ((sub @ (X470 @ X478 @ X476) @ X477) = (X470 @ (comp @ X478 @ X477) @ (sub @ X476 @ X477))) | ((X470 @ X471 @ X469) = (X472 @ X471 @ X469)))) & ((! [X479 : subst,X480 : term] : ((sub @ one @ (push @ X480 @ X479)) = X480) => (! [X481 : subst,X482 : term] : ((comp @ sh @ (push @ X482 @ X481)) = X481) => (! [X483 : subst] : ((comp @ X483 @ id) = X483) => (! [X484 : term > $o,X485 : term] : (((X484 @ X485) = $true) | ~! [X486 : term] : (((X484 @ (lam @ X486)) = $true) | ~((X484 @ X486) = $true)) | ~! [X487 : term] : (~((var @ X487) = $true) | ((X484 @ X487) = $true)) | ~! [X488 : term,X489 : term] : (~((X484 @ X488) = $true) | ~((X484 @ X489) = $true) | ((X484 @ (ap @ X489 @ X488)) = $true))) => ! [X490 : term,X491 : subst,X492 : term > $o] : (((X492 @ (sub @ X490 @ X491)) = $true) | ~! [X493 : term] : (((X492 @ (lam @ X493)) = $true) | ~! [X494 : term] : (~((X492 @ X494) = $true) | ($true = (X492 @ (sub @ X493 @ (push @ X494 @ id)))))) | ~! [X495 : term] : (((X492 @ (sub @ X495 @ X491)) = $true) | ~((var @ X495) = $true)) | ~! [X496 : term,X497 : term] : (~((X492 @ X496) = $true) | ((X492 @ (ap @ X496 @ X497)) = $true) | ~((X492 @ X497) = $true))))))) <=> (induction2lem_lthm = $true)) & (hoasapnotvar_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (one = (d2term @ d_one)) & (axapp = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lthm_orig = $true) & ((axshiftcons = $true) <=> ! [X498 : subst,X499 : term] : ((comp @ sh @ (push @ X499 @ X498)) = X498)) & (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)))))))))))))) & (! [X505 : term > $o,X506 : term] : (~! [X507 : term,X508 : term] : (~((X505 @ X507) = $true) | ((X505 @ (ap @ X507 @ X508)) = $true) | ~((X505 @ X508) = $true)) | ~! [X509 : term] : (~((var @ X509) = $true) | ((X505 @ X509) = $true)) | ~! [X510 : term] : (((X505 @ (lam @ X510)) = $true) | ~((X505 @ X510) = $true)) | ((X505 @ X506) = $true)) <=> (induction = $true)) & ~((var @ (d2term @ d_one)) = $true) & (hoasapnotvar_gthm = $true) & ((! [X511 : term] : ((sub @ X511 @ id) = X511) => ! [X512 : term > $o,X513 : subst > term > subst > $o,X514 : term] : (~! [X515 : term] : ((X512 @ X515) = (X513 @ id @ X515 @ id)) | ((X512 @ (lam @ X514)) = $true) | ~! [X516 : subst,X517 : subst,X518 : subst,X519 : term] : (~((X513 @ X516 @ X519 @ (comp @ X517 @ X518)) = $true) | ((X513 @ (comp @ X516 @ X517) @ (sub @ X519 @ X517) @ X518) = $true)) | ~! [X520 : subst > term > term] : (((X513 @ id @ (lam @ (X520 @ sh @ one)) @ id) = $true) | ~! [X521 : term] : (~((X513 @ id @ X521 @ id) = $true) | ((X513 @ id @ (X520 @ id @ X521) @ id) = $true)) | ~! [X522 : subst,X523 : term,X524 : subst] : ((sub @ (X520 @ X524 @ X523) @ X522) = (X520 @ (comp @ X524 @ X522) @ (sub @ X523 @ X522)))) | ~! [X525 : subst,X526 : subst,X527 : subst,X528 : term] : (((X513 @ X526 @ X528 @ (comp @ X525 @ X527)) = $true) | ~((X513 @ (comp @ X526 @ X525) @ (sub @ X528 @ X525) @ X527) = $true)) | ~! [X529 : term] : (~((X512 @ X529) = $true) | ((X512 @ (sub @ X514 @ (push @ X529 @ id))) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X530 : subst,X531 : term,X532 : term > $o,X533 : term > $o] : (((X533 @ one) = $true) | ~((X532 @ X531) = $true) | ~! [X534 : term] : ((X533 @ X534) = (X532 @ (sub @ X534 @ (push @ X531 @ X530)))))) & (! [X535 : term > $o,X536 : term,X537 : term,X538 : subst > term > subst > $o] : (~((X535 @ X536) = $true) | ~! [X539 : term] : ((X535 @ X539) = (X538 @ id @ X539 @ id)) | ~! [X540 : term,X541 : term] : (~((X538 @ id @ X540 @ id) = $true) | ~($true = (X538 @ id @ X541 @ id)) | ((X538 @ id @ (ap @ (sub @ X541 @ id) @ X540) @ id) = $true)) | ~! [X542 : subst,X543 : subst,X544 : term,X545 : subst] : (~((X538 @ (comp @ X543 @ X542) @ (sub @ X544 @ X542) @ X545) = $true) | ((X538 @ X543 @ X544 @ (comp @ X542 @ X545)) = $true)) | ((X535 @ (ap @ X537 @ X536)) = $true) | ~((X535 @ X537) = $true) | ~! [X546 : subst,X547 : subst,X548 : term,X549 : subst] : (((X538 @ (comp @ X546 @ X547) @ (sub @ X548 @ X547) @ X549) = $true) | ~((X538 @ X546 @ X548 @ (comp @ X547 @ X549)) = $true))) <=> (hoasinduction_lem2v2 = $true)) & (hoasinduction_lem1 = $true)),
% 9.05/2.75 inference(fool_elimination,[status(thm)],[f5])).
% 9.05/2.75 thf(f5,plain,(
% 9.05/2.75 ((! [X0 : subst] : ((comp @ X0 @ id) = X0) & ! [X1 : subst] : ((comp @ id @ X1) = X1)) <=> substmonoid) & ~lamnotap & axassoc & (! [X2 : subst > term > subst > $o] : ~! [X3 : term > $o] : ~! [X4 : term] : ((X2 @ id @ X4 @ id) = (X3 @ X4)) <=> hoasinduction_lem0_lthm) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoaslaminj_lthm <=> (! [X5 : term,X6 : subst] : ((sub @ one @ (push @ X5 @ X6)) = X5) => (! [X7 : term,X8 : subst] : ((comp @ sh @ (push @ X7 @ X8)) = X8) => (! [X9 : term,X10 : term] : (((lam @ X10) != (lam @ X9)) | (X9 = X10)) => ! [X11 : subst > term > term,X12 : subst > term > term,X13 : term,X14 : subst] : (((lam @ (X11 @ sh @ one)) != (lam @ (X12 @ sh @ one))) | ((X11 @ X14 @ X13) = (X12 @ X14 @ X13)) | ~! [X15 : term,X16 : subst,X17 : subst] : ((sub @ (X11 @ X17 @ X15) @ X16) = (X11 @ (comp @ X17 @ X16) @ (sub @ X15 @ X16))) | ~! [X18 : subst,X19 : term,X20 : subst] : ((sub @ (X12 @ X20 @ X19) @ X18) = (X12 @ (comp @ X20 @ X18) @ (sub @ X19 @ X18)))))))) & pushprop_gthm & induction2lem_gthm & ((! [X21 : subst > term > subst > $o] : ~! [X22 : term > $o] : ~! [X23 : term] : ((X21 @ id @ X23 @ id) = (X22 @ X23)) => (! [X24 : term,X25 : term > $o] : (~! [X26 : term,X27 : term] : ((X25 @ (ap @ X27 @ X26)) | ~(X25 @ X27) | ~(X25 @ X26)) | ~! [X28 : term] : ((X25 @ (lam @ X28)) | ~! [X29 : term] : ((X25 @ (sub @ X28 @ (push @ X29 @ id))) | ~(X25 @ X29))) | ~! [X30 : term] : (~(var @ X30) | (X25 @ X30)) | (X25 @ X24)) => (! [X31 : term] : ((sub @ X31 @ id) = X31) => (! [X32 : term > $o,X33 : subst > term > subst > $o,X34 : term] : (~! [X35 : term] : ((X32 @ (sub @ X34 @ (push @ X35 @ id))) | ~(X32 @ X35)) | ~! [X36 : subst > term > term] : ((X33 @ id @ (lam @ (X36 @ sh @ one)) @ id) | ~! [X37 : term] : (~(X33 @ id @ X37 @ id) | (X33 @ id @ (X36 @ id @ X37) @ id)) | ~! [X38 : subst,X39 : term,X40 : subst] : ((X36 @ (comp @ X38 @ X40) @ (sub @ X39 @ X40)) = (sub @ (X36 @ X38 @ X39) @ X40))) | ~! [X41 : term] : ((X33 @ id @ X41 @ id) = (X32 @ X41)) | (X32 @ (lam @ X34))) => ! [X42 : subst > term > subst > $o,X43 : term] : (~! [X44 : subst > term > term] : (~! [X45 : subst,X46 : subst,X47 : term] : ((sub @ (X44 @ X45 @ X47) @ X46) = (X44 @ (comp @ X45 @ X46) @ (sub @ X47 @ X46))) | ~! [X48 : term] : ((X42 @ id @ (X44 @ id @ X48) @ id) | ~(X42 @ id @ X48 @ id)) | (X42 @ id @ (lam @ (X44 @ sh @ one)) @ id)) | (X42 @ id @ X43 @ id) | ~! [X49 : term] : (~(var @ (sub @ X49 @ id)) | (X42 @ id @ X49 @ id)) | ~! [X50 : subst,X51 : term,X52 : subst,X53 : subst] : ((X42 @ (comp @ X53 @ X52) @ (sub @ X51 @ X52) @ X50) | ~(X42 @ X53 @ X51 @ (comp @ X52 @ X50))) | ~! [X54 : term,X55 : term] : ((X42 @ id @ (ap @ (sub @ X54 @ id) @ X55) @ id) | ~(X42 @ id @ X54 @ id) | ~(X42 @ id @ X55 @ id)) | ~! [X56 : subst,X57 : subst,X58 : subst,X59 : term] : (~(X42 @ (comp @ X58 @ X57) @ (sub @ X59 @ X57) @ X56) | (X42 @ X58 @ X59 @ (comp @ X57 @ X56)))))))) <=> hoasinduction_lthm_3) & hoasinduction_lem2_lthm & hoasinduction_lem1_gthm & ulamvarsh & (pushprop_lem1 <=> ! [X60 : term > $o,X61 : term,X62 : term > $o,X63 : term,X64 : subst] : (~(X60 @ X61) | (X62 @ (sub @ X61 @ (push @ X63 @ X64))))) & hoaslamnotap_gthm & (hoasinduction_lem3v2_f_lthm <=> ! [X65 : term] : ~! [X66 : subst > term > term] : ~! [X67 : term,X68 : subst] : ((X66 @ X68 @ X67) = (sub @ X65 @ (push @ X67 @ X68)))) & lamnotvar & (shinj <=> ! [X69 : term,X70 : term] : ((X69 = X70) | ((sub @ X70 @ sh) != (sub @ X69 @ sh)))) & (! [X71 : subst] : ((comp @ X71 @ id) = X71) <=> axidr) & hoasinduction_lem1v2_gthm & ((d2subst @ d_id) = id) & (axidl <=> ! [X72 : subst] : ((comp @ id @ X72) = X72)) & (! [X73 : term] : ~! [X74 : subst > term > term] : ((sub @ X73 @ (push @ one @ sh)) != (X74 @ sh @ one)) <=> hoasinduction_lem3b) & (hoasinduction_lem3a <=> ! [X75 : subst > term > subst > $o,X76 : term] : (~! [X77 : subst > term > term] : (~! [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) = (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78))) | ~! [X81 : term] : (~(X75 @ id @ X81 @ id) | (X75 @ id @ (X77 @ id @ X81) @ id)) | (X75 @ id @ (lam @ (X77 @ sh @ one)) @ id)) | (X75 @ id @ (lam @ X76) @ id) | ~! [X82 : term] : ((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) | ~(X75 @ id @ X82 @ id)))) & hoasinduction_lem1_lthm & (hoasinduction_lem3a_lthm <=> (! [X83 : term] : ((sub @ X83 @ id) = X83) => (! [X84 : term,X85 : subst > term > subst > $o] : (~! [X86 : term] : (~(X85 @ id @ X86 @ id) | (X85 @ id @ (sub @ X84 @ (push @ X86 @ id)) @ id)) | ~! [X87 : subst > term > term] : (~! [X88 : term] : (~(X85 @ id @ X88 @ id) | (X85 @ id @ (X87 @ id @ X88) @ id)) | (X85 @ id @ (lam @ (X87 @ sh @ one)) @ id) | ~! [X89 : subst,X90 : term,X91 : subst] : ((X87 @ (comp @ X91 @ X89) @ (sub @ X90 @ X89)) = (sub @ (X87 @ X91 @ X90) @ X89))) | (X85 @ id @ (lam @ (sub @ X84 @ (push @ one @ sh))) @ id)) => ! [X92 : term,X93 : subst > term > subst > $o] : (~! [X94 : subst > term > term] : ((X93 @ id @ (lam @ (X94 @ sh @ one)) @ id) | ~! [X95 : subst,X96 : term,X97 : subst] : ((X94 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97)) = (sub @ (X94 @ X95 @ X96) @ X97)) | ~! [X98 : term] : (~(X93 @ id @ X98 @ id) | (X93 @ id @ (X94 @ id @ X98) @ id))) | (X93 @ id @ (lam @ X92) @ id) | ~! [X99 : term] : ((X93 @ id @ (sub @ X92 @ (push @ X99 @ id)) @ id) | ~(X93 @ id @ X99 @ id)))))) & (axvarcons <=> ! [X100 : term,X101 : subst] : ((sub @ one @ (push @ X100 @ X101)) = X100)) & (axscons <=> ! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102)) & hoaslamnotvar_gthm & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X103 : term] : ~! [X104 : subst > term > term] : ~! [X105 : subst,X106 : term] : ((sub @ X103 @ (push @ X106 @ X105)) = (X104 @ X105 @ X106)) => (! [X107 : term] : ((sub @ X107 @ id) = X107) => ! [X108 : term,X109 : subst > term > subst > $o,X110 : term > $o] : (~! [X111 : term] : (~(X110 @ X111) | (X110 @ (sub @ X108 @ (push @ X111 @ id)))) | (X110 @ (lam @ X108)) | ~! [X112 : term] : ((X109 @ id @ X112 @ id) = (X110 @ X112)) | ~! [X113 : subst > term > term] : ((X109 @ id @ (lam @ (X113 @ sh @ one)) @ id) | ~! [X114 : term] : (~(X109 @ id @ X114 @ id) | (X109 @ id @ (X113 @ id @ X114) @ id)) | ~! [X115 : subst,X116 : term,X117 : subst] : ((X113 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117)) = (sub @ (X113 @ X115 @ X116) @ X117)))))) <=> hoasinduction_lem3v2a_lthm) & ((! [X118 : term,X119 : subst] : ((sub @ one @ (push @ X118 @ X119)) = X118) => (! [X120 : term,X121 : subst] : ((comp @ sh @ (push @ X120 @ X121)) = X121) => ! [X122 : term > $o,X123 : subst,X124 : term,X125 : term > $o,X126 : term] : (~(X122 @ X124) | (X125 @ (sub @ X124 @ (push @ X126 @ X123)))))) <=> pushprop_lem1_lthm) & hoasinduction_gthm & (! [X127 : subst > term > subst > $o,X128 : term] : (~! [X129 : term] : ((X127 @ id @ (sub @ X128 @ (push @ X129 @ id)) @ id) | ~(X127 @ id @ X129 @ id)) | ~! [X130 : subst > term > term] : (~! [X131 : term,X132 : subst,X133 : subst] : ((X130 @ (comp @ X132 @ X133) @ (sub @ X131 @ X133)) = (sub @ (X130 @ X132 @ X131) @ X133)) | ~! [X134 : term] : ((X127 @ id @ (X130 @ id @ X134) @ id) | ~(X127 @ id @ X134 @ id)) | (X127 @ id @ (lam @ (X130 @ sh @ one)) @ id)) | (X127 @ id @ (lam @ (sub @ X128 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aa_lthm) & (! [X135 : term] : ~! [X136 : subst > term > term] : ~! [X137 : subst,X138 : term] : ((sub @ X135 @ (push @ X138 @ X137)) = (X136 @ X137 @ X138)) <=> hoasinduction_lem3v2_f) & (pushprop_lem0_lthm <=> ! [X139 : term,X140 : term > $o,X141 : subst] : ~! [X142 : term > $o] : ~! [X143 : term] : ((X140 @ (sub @ X143 @ (push @ X139 @ X141))) = (X142 @ X143))) & (hoasinduction_lem3b_lthm <=> ! [X144 : term] : ~! [X145 : subst > term > term] : ((X145 @ sh @ one) != (sub @ X144 @ (push @ one @ sh)))) & hoasinduction_lem3a_gthm & (hoasinduction_lem3v2 <=> ! [X146 : subst > term > subst > $o,X147 : term > $o,X148 : term] : (~! [X149 : term] : ((X147 @ X149) = (X146 @ id @ X149 @ id)) | (X147 @ (lam @ X148)) | ~! [X150 : subst,X151 : subst,X152 : subst,X153 : term] : ((X146 @ X151 @ X153 @ (comp @ X150 @ X152)) | ~(X146 @ (comp @ X151 @ X150) @ (sub @ X153 @ X150) @ X152)) | ~! [X154 : subst,X155 : subst,X156 : subst,X157 : term] : ((X146 @ (comp @ X155 @ X154) @ (sub @ X157 @ X154) @ X156) | ~(X146 @ X155 @ X157 @ (comp @ X154 @ X156))) | ~! [X158 : term] : ((X147 @ (sub @ X148 @ (push @ X158 @ id))) | ~(X147 @ X158)) | ~! [X159 : subst > term > term] : (~! [X160 : term] : ((X146 @ id @ (X159 @ id @ X160) @ id) | ~(X146 @ id @ X160 @ id)) | ~! [X161 : term,X162 : subst,X163 : subst] : ((X159 @ (comp @ X163 @ X162) @ (sub @ X161 @ X162)) = (sub @ (X159 @ X163 @ X161) @ X162)) | (X146 @ id @ (lam @ (X159 @ sh @ one)) @ id)))) & pushprop_lem1_gthm & axvarshift & pushprop_lem3v2 & (! [X164 : term] : ((sub @ X164 @ id) = X164) <=> termmset) & (! [X165 : subst > term > subst > $o] : ~! [X166 : term > $o] : ~! [X167 : term] : ((X165 @ id @ X167 @ id) = (X166 @ X167)) <=> hoasinduction_lem0) & (hoasinduction_lem2 <=> ! [X168 : term,X169 : term,X170 : subst > term > subst > $o] : (~(X170 @ id @ X169 @ id) | ~! [X171 : term,X172 : term] : ((X170 @ id @ (ap @ (sub @ X171 @ id) @ X172) @ id) | ~(X170 @ id @ X172 @ id) | ~(X170 @ id @ X171 @ id)) | ~! [X173 : subst,X174 : subst,X175 : term,X176 : subst] : ((X170 @ X176 @ X175 @ (comp @ X173 @ X174)) | ~(X170 @ (comp @ X176 @ X173) @ (sub @ X175 @ X173) @ X174)) | (X170 @ id @ (ap @ X168 @ X169) @ id) | ~(X170 @ id @ X168 @ id) | ~! [X177 : subst,X178 : subst,X179 : term,X180 : subst] : (~(X170 @ X178 @ X179 @ (comp @ X177 @ X180)) | (X170 @ (comp @ X178 @ X177) @ (sub @ X179 @ X177) @ X180)))) & ! [X181 : d_subst,X182 : d_subst] : (((d2subst @ X182) = (d2subst @ X181)) => (X181 = X182)) & substmonoid_gthm & pushprop & axclos & (! [X183 : term,X184 : subst > term > subst > $o] : ((X184 @ id @ X183 @ id) | ~! [X185 : term,X186 : term] : ((X184 @ id @ (ap @ (sub @ X185 @ id) @ X186) @ id) | ~(X184 @ id @ X186 @ id) | ~(X184 @ id @ X185 @ id)) | ~! [X187 : subst > term > term] : ((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) | ~! [X188 : term] : (~(X184 @ id @ X188 @ id) | (X184 @ id @ (X187 @ id @ X188) @ id)) | ~! [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) = (sub @ (X187 @ X189 @ X191) @ X190)))) <=> hoasinduction_no_psi_cond) & (! [X192 : term,X193 : term,X194 : term,X195 : term] : ((X193 = X194) | ((ap @ (sub @ X193 @ id) @ X195) != (ap @ (sub @ X194 @ id) @ X192))) <=> hoasapinj1) & ! [X196 : subst] : ? [X197 : d_subst] : ((d2subst @ X197) = X196) & pushprop_lem1v2_gthm & axabs & (hoasinduction_no_psi_cond_lthm <=> (! [X198 : subst > term > subst > $o] : ~! [X199 : term > $o] : ~! [X200 : term] : ((X198 @ id @ X200 @ id) = (X199 @ X200)) => (! [X201 : term > $o,X202 : term] : ((X201 @ X202) | ~! [X203 : term] : ((X201 @ X203) | ~(var @ X203)) | ~! [X204 : term,X205 : term] : (~(X201 @ X204) | ~(X201 @ X205) | (X201 @ (ap @ X205 @ X204))) | ~! [X206 : term] : ((X201 @ (lam @ X206)) | ~! [X207 : term] : (~(X201 @ X207) | (X201 @ (sub @ X206 @ (push @ X207 @ id)))))) => (! [X208 : term] : ((sub @ X208 @ id) = X208) => (! [X209 : subst > term > subst > $o,X210 : term > $o,X211 : term] : ((X210 @ (lam @ X211)) | ~! [X212 : term] : (~(X210 @ X212) | (X210 @ (sub @ X211 @ (push @ X212 @ id)))) | ~! [X213 : subst > term > term] : ((X209 @ id @ (lam @ (X213 @ sh @ one)) @ id) | ~! [X214 : term] : ((X209 @ id @ (X213 @ id @ X214) @ id) | ~(X209 @ id @ X214 @ id)) | ~! [X215 : subst,X216 : subst,X217 : term] : ((sub @ (X213 @ X216 @ X217) @ X215) = (X213 @ (comp @ X216 @ X215) @ (sub @ X217 @ X215)))) | ~! [X218 : term] : ((X209 @ id @ X218 @ id) = (X210 @ X218))) => ! [X219 : subst > term > subst > $o,X220 : term] : (~! [X221 : term,X222 : term] : (~(X219 @ id @ X222 @ id) | ~(X219 @ id @ X221 @ id) | (X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id)) | (X219 @ id @ X220 @ id) | ~! [X223 : subst > term > term] : (~! [X224 : term] : (~(X219 @ id @ X224 @ id) | (X219 @ id @ (X223 @ id @ X224) @ id)) | (X219 @ id @ (lam @ (X223 @ sh @ one)) @ id) | ~! [X225 : subst,X226 : term,X227 : subst] : ((X223 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225)) = (sub @ (X223 @ X227 @ X226) @ X225))))))))) & termmset_gthm & axmap & (pushprop_lem1v2_lthm <=> (! [X228 : term,X229 : subst] : ((sub @ one @ (push @ X228 @ X229)) = X228) => ! [X230 : term > $o,X231 : subst,X232 : term > $o,X233 : term] : (~! [X234 : term] : ((X230 @ (sub @ X234 @ (push @ X233 @ X231))) = (X232 @ X234)) | (X232 @ one) | ~(X230 @ X233)))) & hoasapnotvar & (hoasinduction_p_and_p_prime = (^[X235 : subst > term > subst > $o, X236 : term > $o] : (! [X237 : term] : ((X235 @ id @ X237 @ id) = (X236 @ X237))))) & (! [X238 : subst > term > subst > $o,X239 : term,X240 : term > $o] : (~! [X241 : term] : (~(X240 @ X241) | (X240 @ (sub @ X239 @ (push @ X241 @ id)))) | ~! [X242 : subst > term > term] : (~! [X243 : subst,X244 : term,X245 : subst] : ((sub @ (X242 @ X245 @ X244) @ X243) = (X242 @ (comp @ X245 @ X243) @ (sub @ X244 @ X243))) | (X238 @ id @ (lam @ (X242 @ sh @ one)) @ id) | ~! [X246 : term] : ((X238 @ id @ (X242 @ id @ X246) @ id) | ~(X238 @ id @ X246 @ id))) | (X240 @ (lam @ X239)) | ~! [X247 : term] : ((X238 @ id @ X247 @ id) = (X240 @ X247))) <=> hoasinduction_lem3v2a) & hoasinduction_lem3v2_gthm & hoasinduction_lem2v2_gthm & ! [X248 : d_term,X249 : d_term] : (((d2term @ X248) = (d2term @ X249)) => (X248 = X249)) & (hoaslamnotap <=> ! [X250 : term,X251 : subst > term > term,X252 : term] : (((lam @ (X251 @ sh @ one)) != (ap @ (sub @ X250 @ id) @ X252)) | ~! [X253 : subst,X254 : subst,X255 : term] : ((sub @ (X251 @ X253 @ X255) @ X254) = (X251 @ (comp @ X253 @ X254) @ (sub @ X255 @ X254))))) & ((! [X256 : term,X257 : term,X258 : term,X259 : term] : ((X256 = X257) | ((ap @ X259 @ X256) != (ap @ X258 @ X257))) => ! [X260 : term,X261 : term,X262 : term,X263 : term] : ((X260 = X261) | ((ap @ (sub @ X262 @ id) @ X260) != (ap @ (sub @ X263 @ id) @ X261)))) <=> hoasapinj2_lthm) & hoasinduction_lem2_gthm & (! [X264 : term,X265 : term,X266 : term,X267 : term] : ((X264 = X265) | ((ap @ X264 @ X266) != (ap @ X265 @ X267))) <=> apinj1) & ! [X268 : d_term] : (d_one = X268) & induction2_gthm & (! [X269 : subst,X270 : term > $o,X271 : term] : ~! [X272 : term > $o] : ~! [X273 : term] : ((X272 @ X273) = (X270 @ (sub @ X273 @ (push @ X271 @ X269)))) <=> pushprop_lem0) & pushprop_lthm & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ! [X274 : d_subst] : (d_id = X274) & ~ulamvar1 & (termmset_lthm <=> (! [X275 : term] : ((sub @ X275 @ id) = X275) => ! [X276 : term] : ((sub @ X276 @ id) = X276))) & hoaslamnotvar & pushprop_lem2v2 & pushprop_lem3v2_lthm & (! [X277 : subst > term > subst > $o,X278 : term] : ((X277 @ id @ X278 @ id) | ~! [X279 : term] : ((X277 @ id @ X279 @ id) | ~(var @ (sub @ X279 @ id))) | ~! [X280 : term,X281 : subst,X282 : subst,X283 : subst] : ((X277 @ X282 @ X280 @ (comp @ X281 @ X283)) | ~(X277 @ (comp @ X282 @ X281) @ (sub @ X280 @ X281) @ X283)) | ~! [X284 : subst,X285 : term,X286 : subst,X287 : subst] : (~(X277 @ X286 @ X285 @ (comp @ X287 @ X284)) | (X277 @ (comp @ X286 @ X287) @ (sub @ X285 @ X287) @ X284)) | ~! [X288 : subst > term > term] : (~! [X289 : term,X290 : subst,X291 : subst] : ((sub @ (X288 @ X290 @ X289) @ X291) = (X288 @ (comp @ X290 @ X291) @ (sub @ X289 @ X291))) | ~! [X292 : term] : (~(X277 @ id @ X292 @ id) | (X277 @ id @ (X288 @ id @ X292) @ id)) | (X277 @ id @ (lam @ (X288 @ sh @ one)) @ id)) | ~! [X293 : term,X294 : term] : (~(X277 @ id @ X294 @ id) | (X277 @ id @ (ap @ (sub @ X294 @ id) @ X293) @ id) | ~(X277 @ id @ X293 @ id))) <=> hoasinduction) & (laminj <=> ! [X295 : term,X296 : term] : ((X295 = X296) | ((lam @ X296) != (lam @ X295)))) & (! [X297 : term,X298 : term > $o] : (~! [X299 : term] : (~(var @ X299) | (X298 @ X299)) | ~! [X300 : term,X301 : term] : ((X298 @ (ap @ X300 @ X301)) | ~(X298 @ X301) | ~(X298 @ X300)) | ~! [X302 : term] : (~! [X303 : term] : (~(X298 @ X303) | (X298 @ (sub @ X302 @ (push @ X303 @ id)))) | (X298 @ (lam @ X302))) | (X298 @ X297)) <=> induction2) & ((d2subst @ d_id) = sh) & hoasapinj2_gthm & ((! [X304 : term] : ((sub @ X304 @ id) = X304) => (! [X305 : subst,X306 : term > $o,X307 : term] : (~! [X308 : term,X309 : term] : (~(X306 @ X308) | ~(X306 @ X309) | (X306 @ (ap @ X308 @ X309))) | ~! [X310 : term] : (~! [X311 : term] : (~(X306 @ X311) | (X306 @ (sub @ X310 @ (push @ X311 @ id)))) | (X306 @ (lam @ X310))) | (X306 @ (sub @ X307 @ X305)) | ~! [X312 : term] : (~(var @ X312) | (X306 @ (sub @ X312 @ X305)))) => ! [X313 : term,X314 : term > $o] : (~! [X315 : term] : (~(var @ X315) | (X314 @ X315)) | (X314 @ X313) | ~! [X316 : term] : (~! [X317 : term] : ((X314 @ (sub @ X316 @ (push @ X317 @ id))) | ~(X314 @ X317)) | (X314 @ (lam @ X316))) | ~! [X318 : term,X319 : term] : ((X314 @ (ap @ X318 @ X319)) | ~(X314 @ X318) | ~(X314 @ X319))))) <=> induction2_lthm) & hoasinduction_lem3b_gthm & ulamvarind & hoaslamnotvar_lthm & hoasinduction_lem3_gthm & (! [X320 : term,X321 : term > $o,X322 : subst] : (~! [X323 : term] : (~! [X324 : term] : ((X321 @ (sub @ X323 @ (push @ X324 @ id))) | ~(X321 @ X324)) | (X321 @ (lam @ X323))) | ~! [X325 : term,X326 : term] : ((X321 @ (ap @ X326 @ X325)) | ~(X321 @ X326) | ~(X321 @ X325)) | (X321 @ (sub @ X320 @ X322)) | ~! [X327 : term] : (~(var @ X327) | (X321 @ (sub @ X327 @ X322)))) <=> induction2lem) & ((^[X328 : subst, X329 : subst > term > term] : (d2term @ d_one)) = hoaslam) & hoasinduction_lem1v2 & (apinj2 <=> ! [X330 : term,X331 : term,X332 : term,X333 : term] : (((ap @ X332 @ X330) != (ap @ X331 @ X333)) | (X330 = X333))) & (hoasinduction_lthm <=> (! [X334 : term > $o,X335 : term] : (~! [X336 : term] : ((X334 @ X336) | ~(var @ X336)) | ~! [X337 : term,X338 : term] : (~(X334 @ X337) | ~(X334 @ X338) | (X334 @ (ap @ X337 @ X338))) | ~! [X339 : term] : (~! [X340 : term] : (~(X334 @ X340) | (X334 @ (sub @ X339 @ (push @ X340 @ id)))) | (X334 @ (lam @ X339))) | (X334 @ X335)) => (! [X341 : term,X342 : subst > term > subst > $o,X343 : term] : (~! [X344 : subst,X345 : subst,X346 : term,X347 : subst] : ((X342 @ (comp @ X347 @ X344) @ (sub @ X346 @ X344) @ X345) | ~(X342 @ X347 @ X346 @ (comp @ X344 @ X345))) | ~(X342 @ id @ X341 @ id) | (X342 @ id @ (ap @ X343 @ X341) @ id) | ~(X342 @ id @ X343 @ id) | ~! [X348 : term,X349 : term] : (~(X342 @ id @ X349 @ id) | (X342 @ id @ (ap @ (sub @ X348 @ id) @ X349) @ id) | ~(X342 @ id @ X348 @ id)) | ~! [X350 : subst,X351 : term,X352 : subst,X353 : subst] : ((X342 @ X353 @ X351 @ (comp @ X352 @ X350)) | ~(X342 @ (comp @ X353 @ X352) @ (sub @ X351 @ X352) @ X350))) => (! [X354 : subst > term > subst > $o,X355 : term] : (~! [X356 : subst,X357 : subst,X358 : term,X359 : subst] : (~(X354 @ (comp @ X356 @ X357) @ (sub @ X358 @ X357) @ X359) | (X354 @ X356 @ X358 @ (comp @ X357 @ X359))) | ~! [X360 : subst > term > term] : (~! [X361 : term] : (~(X354 @ id @ X361 @ id) | (X354 @ id @ (X360 @ id @ X361) @ id)) | (X354 @ id @ (lam @ (X360 @ sh @ one)) @ id) | ~! [X362 : subst,X363 : subst,X364 : term] : ((X360 @ (comp @ X363 @ X362) @ (sub @ X364 @ X362)) = (sub @ (X360 @ X363 @ X364) @ X362))) | (X354 @ id @ (lam @ X355) @ id) | ~! [X365 : term] : (~(X354 @ id @ X365 @ id) | (X354 @ id @ (sub @ X355 @ (push @ X365 @ id)) @ id)) | ~! [X366 : subst,X367 : subst,X368 : term,X369 : subst] : (~(X354 @ X366 @ X368 @ (comp @ X367 @ X369)) | (X354 @ (comp @ X366 @ X367) @ (sub @ X368 @ X367) @ X369))) => ! [X370 : term,X371 : subst > term > subst > $o] : (~! [X372 : term] : ((X371 @ id @ X372 @ id) | ~(var @ (sub @ X372 @ id))) | ~! [X373 : subst,X374 : subst,X375 : subst,X376 : term] : ((X371 @ (comp @ X374 @ X375) @ (sub @ X376 @ X375) @ X373) | ~(X371 @ X374 @ X376 @ (comp @ X375 @ X373))) | ~! [X377 : term,X378 : term] : (~(X371 @ id @ X377 @ id) | ~(X371 @ id @ X378 @ id) | (X371 @ id @ (ap @ (sub @ X378 @ id) @ X377) @ id)) | ~! [X379 : subst > term > term] : (~! [X380 : subst,X381 : term,X382 : subst] : ((sub @ (X379 @ X382 @ X381) @ X380) = (X379 @ (comp @ X382 @ X380) @ (sub @ X381 @ X380))) | (X371 @ id @ (lam @ (X379 @ sh @ one)) @ id) | ~! [X383 : term] : ((X371 @ id @ (X379 @ id @ X383) @ id) | ~(X371 @ id @ X383 @ id))) | (X371 @ id @ X370 @ id) | ~! [X384 : subst,X385 : subst,X386 : term,X387 : subst] : (~(X371 @ (comp @ X385 @ X384) @ (sub @ X386 @ X384) @ X387) | (X371 @ X385 @ X386 @ (comp @ X384 @ X387)))))))) & apnotvar & (axvarid <=> ! [X388 : term] : ((sub @ X388 @ id) = X388)) & (hoasinduction_lem3 <=> ! [X389 : term,X390 : subst > term > subst > $o] : (~! [X391 : subst,X392 : subst,X393 : subst,X394 : term] : ((X390 @ (comp @ X391 @ X393) @ (sub @ X394 @ X393) @ X392) | ~(X390 @ X391 @ X394 @ (comp @ X393 @ X392))) | ~! [X395 : term] : ((X390 @ id @ (sub @ X389 @ (push @ X395 @ id)) @ id) | ~(X390 @ id @ X395 @ id)) | (X390 @ id @ (lam @ X389) @ id) | ~! [X396 : subst > term > term] : (~! [X397 : subst,X398 : subst,X399 : term] : ((X396 @ (comp @ X398 @ X397) @ (sub @ X399 @ X397)) = (sub @ (X396 @ X398 @ X399) @ X397)) | ~! [X400 : term] : (~(X390 @ id @ X400 @ id) | (X390 @ id @ (X396 @ id @ X400) @ id)) | (X390 @ id @ (lam @ (X396 @ sh @ one)) @ id)) | ~! [X401 : subst,X402 : subst,X403 : term,X404 : subst] : (~(X390 @ (comp @ X401 @ X402) @ (sub @ X403 @ X402) @ X404) | (X390 @ X401 @ X403 @ (comp @ X402 @ X404))))) & hoaslaminj_gthm & hoaslamnotap_lthm & (hoasapinj1_lthm <=> (! [X405 : term] : ((sub @ X405 @ id) = X405) => (! [X406 : term,X407 : term,X408 : term,X409 : term] : (((ap @ X406 @ X408) != (ap @ X409 @ X407)) | (X406 = X409)) => ! [X410 : term,X411 : term,X412 : term,X413 : term] : ((X410 = X413) | ((ap @ (sub @ X410 @ id) @ X412) != (ap @ (sub @ X413 @ id) @ X411)))))) & hoasapinj1_gthm & ((! [X414 : subst] : ((comp @ id @ X414) = X414) => (! [X415 : subst] : ((comp @ X415 @ id) = X415) => (! [X416 : subst] : ((comp @ id @ X416) = X416) & ! [X417 : subst] : ((comp @ X417 @ id) = X417)))) <=> substmonoid_lthm) & ! [X418 : term] : ? [X419 : d_term] : ((d2term @ X419) = X418) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X420 : term,X421 : term,X422 : term,X423 : term] : ((X421 = X423) | ((ap @ (sub @ X420 @ id) @ X421) != (ap @ (sub @ X422 @ id) @ X423))) <=> hoasapinj2) & (hoasinduction_lem3aa <=> ! [X424 : subst > term > subst > $o,X425 : term] : (~! [X426 : term] : ((X424 @ id @ (sub @ X425 @ (push @ X426 @ id)) @ id) | ~(X424 @ id @ X426 @ id)) | (X424 @ id @ (lam @ (sub @ X425 @ (push @ one @ sh))) @ id) | ~! [X427 : subst > term > term] : (~! [X428 : term,X429 : subst,X430 : subst] : ((X427 @ (comp @ X430 @ X429) @ (sub @ X428 @ X429)) = (sub @ (X427 @ X430 @ X428) @ X429)) | ~! [X431 : term] : ((X424 @ id @ (X427 @ id @ X431) @ id) | ~(X424 @ id @ X431 @ id)) | (X424 @ id @ (lam @ (X427 @ sh @ one)) @ id)))) & (hoasinduction_lem3_lthm <=> (! [X432 : term] : ((sub @ X432 @ id) = X432) => (! [X433 : term,X434 : subst > term > subst > $o] : (~! [X435 : term] : ((X434 @ id @ (sub @ X433 @ (push @ X435 @ id)) @ id) | ~(X434 @ id @ X435 @ id)) | ~! [X436 : subst > term > term] : (~! [X437 : term] : (~(X434 @ id @ X437 @ id) | (X434 @ id @ (X436 @ id @ X437) @ id)) | (X434 @ id @ (lam @ (X436 @ sh @ one)) @ id) | ~! [X438 : term,X439 : subst,X440 : subst] : ((sub @ (X436 @ X440 @ X438) @ X439) = (X436 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439)))) | (X434 @ id @ (lam @ (sub @ X433 @ (push @ one @ sh))) @ id)) => ! [X441 : term,X442 : subst > term > subst > $o] : ((X442 @ id @ (lam @ X441) @ id) | ~! [X443 : subst > term > term] : ((X442 @ id @ (lam @ (X443 @ sh @ one)) @ id) | ~! [X444 : term] : ((X442 @ id @ (X443 @ id @ X444) @ id) | ~(X442 @ id @ X444 @ id)) | ~! [X445 : term,X446 : subst,X447 : subst] : ((X443 @ (comp @ X446 @ X447) @ (sub @ X445 @ X447)) = (sub @ (X443 @ X446 @ X445) @ X447))) | ~! [X448 : term,X449 : subst,X450 : subst,X451 : subst] : ((X442 @ X451 @ X448 @ (comp @ X450 @ X449)) | ~(X442 @ (comp @ X451 @ X450) @ (sub @ X448 @ X450) @ X449)) | ~! [X452 : term] : ((X442 @ id @ (sub @ X441 @ (push @ X452 @ id)) @ id) | ~(X442 @ id @ X452 @ id)) | ~! [X453 : subst,X454 : subst,X455 : subst,X456 : term] : ((X442 @ (comp @ X455 @ X454) @ (sub @ X456 @ X454) @ X453) | ~(X442 @ X455 @ X456 @ (comp @ X454 @ X453))))))) & pushprop_lem2v2_gthm & pushprop_lem0_gthm & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (! [X457 : subst > term > subst > $o,X458 : term] : (~! [X459 : term] : ((X457 @ id @ (sub @ X458 @ (push @ X459 @ id)) @ id) | ~(X457 @ id @ X459 @ id)) | (X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id) | ~! [X460 : subst > term > term,X461 : term] : (~! [X462 : subst,X463 : term,X464 : subst] : ((sub @ X461 @ (push @ (sub @ X463 @ X462) @ (comp @ X464 @ X462))) = (X460 @ (comp @ X464 @ X462) @ (sub @ X463 @ X462))) | (X457 @ id @ (lam @ (X460 @ sh @ one)) @ id) | ~! [X465 : term] : (~(X457 @ id @ X465 @ id) | (X457 @ id @ (X460 @ id @ X465) @ id)) | ~! [X466 : subst,X467 : term,X468 : subst] : ((sub @ (sub @ X461 @ (push @ X467 @ X466)) @ X468) = (sub @ (X460 @ X466 @ X467) @ X468)))) <=> hoasinduction_lem3aaa) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoaslaminj <=> ! [X469 : term,X470 : subst > term > term,X471 : subst,X472 : subst > term > term] : (~! [X473 : subst,X474 : term,X475 : subst] : ((X472 @ (comp @ X475 @ X473) @ (sub @ X474 @ X473)) = (sub @ (X472 @ X475 @ X474) @ X473)) | ((lam @ (X470 @ sh @ one)) != (lam @ (X472 @ sh @ one))) | ~! [X476 : term,X477 : subst,X478 : subst] : ((sub @ (X470 @ X478 @ X476) @ X477) = (X470 @ (comp @ X478 @ X477) @ (sub @ X476 @ X477))) | ((X470 @ X471 @ X469) = (X472 @ X471 @ X469)))) & ((! [X479 : subst,X480 : term] : ((sub @ one @ (push @ X480 @ X479)) = X480) => (! [X481 : subst,X482 : term] : ((comp @ sh @ (push @ X482 @ X481)) = X481) => (! [X483 : subst] : ((comp @ X483 @ id) = X483) => (! [X484 : term > $o,X485 : term] : ((X484 @ X485) | ~! [X486 : term] : ((X484 @ (lam @ X486)) | ~(X484 @ X486)) | ~! [X487 : term] : (~(var @ X487) | (X484 @ X487)) | ~! [X488 : term,X489 : term] : (~(X484 @ X488) | ~(X484 @ X489) | (X484 @ (ap @ X489 @ X488)))) => ! [X490 : term,X491 : subst,X492 : term > $o] : ((X492 @ (sub @ X490 @ X491)) | ~! [X493 : term] : ((X492 @ (lam @ X493)) | ~! [X494 : term] : (~(X492 @ X494) | (X492 @ (sub @ X493 @ (push @ X494 @ id))))) | ~! [X495 : term] : ((X492 @ (sub @ X495 @ X491)) | ~(var @ X495)) | ~! [X496 : term,X497 : term] : (~(X492 @ X496) | (X492 @ (ap @ X496 @ X497)) | ~(X492 @ X497))))))) <=> induction2lem_lthm) & hoasapnotvar_lthm & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (one = (d2term @ d_one)) & axapp & pushprop_lem2v2_lthm & pushprop_lthm_orig & (axshiftcons <=> ! [X498 : subst,X499 : term] : ((comp @ sh @ (push @ X499 @ X498)) = X498)) & (pushprop_p_and_p_prime = (^[X500 : term, X501 : subst, X502 : term > $o, X503 : term > $o] : (! [X504 : term] : ((X503 @ X504) = (X502 @ (sub @ X504 @ (push @ X500 @ X501))))))) & (! [X505 : term > $o,X506 : term] : (~! [X507 : term,X508 : term] : (~(X505 @ X507) | (X505 @ (ap @ X507 @ X508)) | ~(X505 @ X508)) | ~! [X509 : term] : (~(var @ X509) | (X505 @ X509)) | ~! [X510 : term] : ((X505 @ (lam @ X510)) | ~(X505 @ X510)) | (X505 @ X506)) <=> induction) & ~(var @ (d2term @ d_one)) & hoasapnotvar_gthm & ((! [X511 : term] : ((sub @ X511 @ id) = X511) => ! [X512 : term > $o,X513 : subst > term > subst > $o,X514 : term] : (~! [X515 : term] : ((X512 @ X515) = (X513 @ id @ X515 @ id)) | (X512 @ (lam @ X514)) | ~! [X516 : subst,X517 : subst,X518 : subst,X519 : term] : (~(X513 @ X516 @ X519 @ (comp @ X517 @ X518)) | (X513 @ (comp @ X516 @ X517) @ (sub @ X519 @ X517) @ X518)) | ~! [X520 : subst > term > term] : ((X513 @ id @ (lam @ (X520 @ sh @ one)) @ id) | ~! [X521 : term] : (~(X513 @ id @ X521 @ id) | (X513 @ id @ (X520 @ id @ X521) @ id)) | ~! [X522 : subst,X523 : term,X524 : subst] : ((sub @ (X520 @ X524 @ X523) @ X522) = (X520 @ (comp @ X524 @ X522) @ (sub @ X523 @ X522)))) | ~! [X525 : subst,X526 : subst,X527 : subst,X528 : term] : ((X513 @ X526 @ X528 @ (comp @ X525 @ X527)) | ~(X513 @ (comp @ X526 @ X525) @ (sub @ X528 @ X525) @ X527)) | ~! [X529 : term] : (~(X512 @ X529) | (X512 @ (sub @ X514 @ (push @ X529 @ id)))))) <=> hoasinduction_lem3v2_lthm) & (pushprop_lem1v2 <=> ! [X530 : subst,X531 : term,X532 : term > $o,X533 : term > $o] : ((X533 @ one) | ~(X532 @ X531) | ~! [X534 : term] : ((X533 @ X534) = (X532 @ (sub @ X534 @ (push @ X531 @ X530)))))) & (! [X535 : term > $o,X536 : term,X537 : term,X538 : subst > term > subst > $o] : (~(X535 @ X536) | ~! [X539 : term] : ((X535 @ X539) = (X538 @ id @ X539 @ id)) | ~! [X540 : term,X541 : term] : (~(X538 @ id @ X540 @ id) | ~(X538 @ id @ X541 @ id) | (X538 @ id @ (ap @ (sub @ X541 @ id) @ X540) @ id)) | ~! [X542 : subst,X543 : subst,X544 : term,X545 : subst] : (~(X538 @ (comp @ X543 @ X542) @ (sub @ X544 @ X542) @ X545) | (X538 @ X543 @ X544 @ (comp @ X542 @ X545))) | (X535 @ (ap @ X537 @ X536)) | ~(X535 @ X537) | ~! [X546 : subst,X547 : subst,X548 : term,X549 : subst] : ((X538 @ (comp @ X546 @ X547) @ (sub @ X548 @ X547) @ X549) | ~(X538 @ X546 @ X548 @ (comp @ X547 @ X549)))) <=> hoasinduction_lem2v2) & hoasinduction_lem1),
% 9.05/2.75 inference(rectify,[status(thm)],[f1])).
% 9.05/2.75 thf(f1,axiom,(
% 9.05/2.75 ((! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X14 : subst] : ((comp @ id @ X14) = X14)) <=> substmonoid) & ~lamnotap & axassoc & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0_lthm) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoaslaminj_lthm <=> (! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X15 : term,X13 : term] : (((lam @ X13) != (lam @ X15)) | (X13 = X15)) => ! [X31 : subst > term > term,X18 : subst > term > term,X33 : term,X32 : subst] : (((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))))))) & pushprop_gthm & induction2lem_gthm & ((! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X22 : term,X10 : term > $o] : (~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15))) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | (X10 @ X22)) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X11 : term > $o,X10 : subst > term > subst > $o,X29 : term] : (~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [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,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X29))) => ! [X10 : subst > term > subst > $o,X28 : term] : (~! [X18 : subst > term > term] : (~! [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)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X10 @ id @ X28 @ id) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | ~! [X25 : subst,X13 : term,X26 : subst,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X25 : subst,X26 : subst,X14 : subst,X13 : term] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25)))))))) <=> hoasinduction_lthm_3) & hoasinduction_lem2_lthm & hoasinduction_lem1_gthm & ulamvarsh & (pushprop_lem1 <=> ! [X10 : term > $o,X13 : term,X26 : term > $o,X15 : term,X14 : subst] : (~(X10 @ X13) | (X26 @ (sub @ X13 @ (push @ X15 @ X14))))) & hoaslamnotap_gthm & (hoasinduction_lem3v2_f_lthm <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13))) & lamnotvar & (shinj <=> ! [X15 : term,X13 : term] : ((X13 = X15) | ((sub @ X13 @ sh) != (sub @ X15 @ sh)))) & (! [X14 : subst] : ((comp @ X14 @ id) = X14) <=> axidr) & hoasinduction_lem1v2_gthm & ((d2subst @ d_id) = id) & (axidl <=> ! [X14 : subst] : ((comp @ id @ X14) = X14)) & (! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh))) <=> hoasinduction_lem3b) & (hoasinduction_lem3a <=> ! [X10 : subst > term > subst > $o,X43 : term] : (~! [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)) | (X10 @ id @ (lam @ X43) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)))) & hoasinduction_lem1_lthm & (hoasinduction_lem3a_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (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)))) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) => ! [X43 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : ((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))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | (X10 @ id @ (lam @ X43) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)))))) & (axvarcons <=> ! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13)) & (axscons <=> ! [X14 : subst] : ((push @ (sub @ one @ X14) @ (comp @ sh @ X14)) = X14)) & hoaslamnotvar_gthm & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X29 : term,X10 : subst > term > subst > $o,X11 : term > $o] : (~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))) | (X11 @ (lam @ X29)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ 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,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))))))) <=> hoasinduction_lem3v2a_lthm) & ((! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => ! [X10 : term > $o,X14 : subst,X13 : term,X26 : term > $o,X15 : term] : (~(X10 @ X13) | (X26 @ (sub @ X13 @ (push @ X15 @ X14)))))) <=> pushprop_lem1_lthm) & hoasinduction_gthm & (! [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] : (~! [X13 : term,X14 : subst,X25 : 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)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aa_lthm) & (! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) <=> hoasinduction_lem3v2_f) & (pushprop_lem0_lthm <=> ! [X13 : term,X10 : term > $o,X14 : subst] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14))))) & (hoasinduction_lem3b_lthm <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh)))) & hoasinduction_lem3a_gthm & (hoasinduction_lem3v2 <=> ! [X10 : subst > term > subst > $o,X11 : term > $o,X48 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X48)) | ~! [X26 : subst,X14 : subst,X25 : subst,X13 : term] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X26 : subst,X14 : subst,X25 : subst,X13 : term] : ((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] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ 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)))) & pushprop_lem1_gthm & axvarshift & pushprop_lem3v2 & (! [X13 : term] : ((sub @ X13 @ id) = X13) <=> termmset) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0) & (hoasinduction_lem2 <=> ! [X23 : term,X24 : term,X10 : subst > term > subst > $o] : (~(X10 @ id @ X24 @ id) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X10 @ id @ (ap @ X23 @ X24) @ id) | ~(X10 @ id @ X23 @ id) | ~! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)))) & ! [X6 : d_subst,X7 : d_subst] : (((d2subst @ X6) = (d2subst @ X7)) => (X6 = X7)) & substmonoid_gthm & pushprop & axclos & (! [X30 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ X30 @ id) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ 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))))) <=> hoasinduction_no_psi_cond) & (! [X19 : term,X15 : term,X13 : term,X20 : term] : ((X13 = X15) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))) <=> hoasapinj1) & ! [X4 : subst] : ? [X5 : d_subst] : ((d2subst @ X5) = X4) & pushprop_lem1v2_gthm & axabs & (hoasinduction_no_psi_cond_lthm <=> (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X10 : term > $o,X22 : term] : ((X10 @ X22) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X15 : term,X13 : term] : (~(X10 @ X15) | ~(X10 @ X13) | (X10 @ (ap @ X13 @ X15))) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : subst > term > subst > $o,X11 : term > $o,X29 : term] : ((X11 @ (lam @ X29)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))) => ! [X10 : subst > term > subst > $o,X30 : term] : (~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | (X10 @ id @ X30 @ id) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (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)))))))))) & termmset_gthm & axmap & (pushprop_lem1v2_lthm <=> (! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => ! [X10 : term > $o,X14 : subst,X11 : term > $o,X13 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) | (X11 @ one) | ~(X10 @ X13)))) & hoasapnotvar & (hoasinduction_p_and_p_prime = (^[X10 : subst > term > subst > $o, X11 : term > $o] : (! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))))) & (! [X10 : subst > term > subst > $o,X29 : term,X11 : term > $o] : (~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))) | ~! [X18 : subst > term > term] : (~! [X25 : subst,X13 : term,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 @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | (X11 @ (lam @ X29)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))) <=> hoasinduction_lem3v2a) & hoasinduction_lem3v2_gthm & hoasinduction_lem2v2_gthm & ! [X3 : d_term,X2 : d_term] : (((d2term @ X2) = (d2term @ X3)) => (X2 = X3)) & (hoaslamnotap <=> ! [X46 : term,X18 : subst > term > term,X47 : term] : (((lam @ (X18 @ sh @ one)) != (ap @ (sub @ X46 @ id) @ X47)) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))))) & ((! [X19 : term,X20 : term,X15 : term,X13 : term] : ((X19 = X20) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) => ! [X20 : term,X19 : term,X15 : term,X13 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)))) <=> hoasapinj2_lthm) & hoasinduction_lem2_gthm & (! [X13 : term,X15 : term,X19 : term,X20 : term] : ((X13 = X15) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) <=> apinj1) & ! [X1 : d_term] : (d_one = X1) & induction2_gthm & (! [X14 : subst,X10 : term > $o,X13 : term] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) <=> pushprop_lem0) & pushprop_lthm & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ! [X5 : d_subst] : (d_id = X5) & ~ulamvar1 & (termmset_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X13 : term] : ((sub @ X13 @ id) = X13))) & hoaslamnotvar & pushprop_lem2v2 & pushprop_lem3v2_lthm & (! [X10 : subst > term > subst > $o,X28 : term] : ((X10 @ id @ X28 @ id) | ~! [X13 : term] : ((X10 @ id @ X13 @ id) | ~(var @ (sub @ X13 @ id))) | ~! [X13 : term,X26 : subst,X14 : subst,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X25 : subst,X13 : term,X14 : subst,X26 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [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 @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id))) <=> hoasinduction) & (laminj <=> ! [X13 : term,X15 : term] : ((X13 = X15) | ((lam @ X13) != (lam @ X15)))) & (! [X22 : term,X10 : term > $o] : (~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X13 : term,X15 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15) | ~(X10 @ X13)) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | (X10 @ X22)) <=> induction2) & ((d2subst @ d_id) = sh) & hoasapinj2_gthm & ((! [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))) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | (X10 @ (sub @ X16 @ X17)) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17)))) => ! [X22 : term,X10 : term > $o] : (~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | (X10 @ X22) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | ~! [X13 : term,X15 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15))))) <=> induction2_lthm) & hoasinduction_lem3b_gthm & ulamvarind & hoaslamnotvar_lthm & hoasinduction_lem3_gthm & (! [X16 : term,X10 : term > $o,X17 : subst] : (~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | ~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | (X10 @ (sub @ X16 @ X17)) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17)))) <=> induction2lem) & ((^[X8 : subst, X9 : subst > term > term] : (d2term @ d_one)) = hoaslam) & hoasinduction_lem1v2 & (apinj2 <=> ! [X19 : term,X15 : term,X13 : term,X20 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X19 = X20))) & (hoasinduction_lthm <=> (! [X10 : term > $o,X22 : term] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X13 : term,X15 : term] : (~(X10 @ X13) | ~(X10 @ X15) | (X10 @ (ap @ X13 @ X15))) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | (X10 @ X22)) => (! [X24 : term,X10 : subst > term > subst > $o,X23 : term] : (~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~(X10 @ id @ X24 @ id) | (X10 @ id @ (ap @ X23 @ X24) @ id) | ~(X10 @ id @ X23 @ id) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X25 : subst,X13 : term,X26 : subst,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25))) => (! [X10 : subst > term > subst > $o,X27 : term] : (~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ 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 @ X27) @ id) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id)) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25))) => ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X13 : term] : ((X10 @ id @ X13 @ id) | ~(var @ (sub @ X13 @ id))) | ~! [X25 : subst,X14 : subst,X26 : subst,X13 : term] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ 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))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | (X10 @ id @ X28 @ id) | ~! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25)))))))) & apnotvar & (axvarid <=> ! [X13 : term] : ((sub @ X13 @ id) = X13)) & (hoasinduction_lem3 <=> ! [X27 : term,X10 : subst > term > subst > $o] : (~! [X14 : subst,X25 : subst,X26 : subst,X13 : term] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (lam @ X27) @ 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)) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))))) & hoaslaminj_gthm & hoaslamnotap_lthm & (hoasapinj1_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X13 : term,X20 : term,X19 : term,X15 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X13 = X15)) => ! [X13 : term,X20 : term,X19 : term,X15 : term] : ((X13 = X15) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)))))) & hoasapinj1_gthm & ((! [X14 : subst] : ((comp @ id @ X14) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X14 : subst] : ((comp @ id @ X14) = X14) & ! [X14 : subst] : ((comp @ X14 @ id) = X14)))) <=> substmonoid_lthm) & ! [X0 : term] : ? [X1 : d_term] : ((d2term @ X1) = X0) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X13 : term,X19 : term,X15 : term,X20 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))) <=> hoasapinj2) & (hoasinduction_lem3aa <=> ! [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] : (~! [X13 : term,X25 : subst,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)))) & (hoasinduction_lem3_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) => ! [X27 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ X27) @ id) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X13 : term,X25 : subst,X26 : subst,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X25 : subst,X26 : subst,X14 : subst,X13 : term] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))))))) & pushprop_lem2v2_gthm & pushprop_lem0_gthm & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (! [X10 : subst > term > subst > $o,X34 : term] : (~! [X15 : term] : ((X10 @ id @ (sub @ X34 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (lam @ (sub @ X34 @ (push @ one @ sh))) @ id) | ~! [X18 : subst > term > term,X35 : term] : (~! [X41 : subst,X40 : term,X39 : subst] : ((X18 @ (comp @ X39 @ X41) @ (sub @ X40 @ X41)) = (sub @ X35 @ (push @ (sub @ X40 @ X41) @ (comp @ X39 @ X41)))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X36 : subst,X37 : term,X38 : subst] : ((sub @ (X18 @ X36 @ X37) @ X38) = (sub @ (sub @ X35 @ (push @ X37 @ X36)) @ X38)))) <=> hoasinduction_lem3aaa) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoaslaminj <=> ! [X33 : term,X18 : subst > term > term,X32 : subst,X31 : subst > term > term] : (~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33)))) & ((! [X14 : subst,X13 : term] : ((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] : ((X10 @ X21) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~(X10 @ X13)) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X15 : term,X13 : term] : (~(X10 @ X15) | ~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)))) => ! [X16 : term,X17 : subst,X10 : term > $o] : ((X10 @ (sub @ X16 @ X17)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id))))) | ~! [X15 : term] : ((X10 @ (sub @ X15 @ X17)) | ~(var @ X15)) | ~! [X13 : term,X15 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15))))))) <=> induction2lem_lthm) & hoasapnotvar_lthm & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (one = (d2term @ d_one)) & axapp & pushprop_lem2v2_lthm & pushprop_lthm_orig & (axshiftcons <=> ! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14)) & (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))))))) & (! [X10 : term > $o,X21 : term] : (~! [X13 : term,X15 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~(X10 @ X13)) | (X10 @ X21)) <=> induction) & ~(var @ (d2term @ d_one)) & hoasapnotvar_gthm & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X11 : term > $o,X10 : subst > term > subst > $o,X48 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X48)) | ~! [X14 : subst,X26 : subst,X25 : subst,X13 : term] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X26 : subst,X14 : subst,X25 : subst,X13 : term] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X48 @ (push @ X15 @ id)))))) <=> hoasinduction_lem3v2_lthm) & (pushprop_lem1v2 <=> ! [X14 : subst,X13 : term,X10 : term > $o,X11 : term > $o] : ((X11 @ one) | ~(X10 @ X13) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))))) & (! [X11 : term > $o,X45 : term,X44 : term,X10 : subst > term > subst > $o] : (~(X11 @ X45) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X11 @ (ap @ X44 @ X45)) | ~(X11 @ X44) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)))) <=> hoasinduction_lem2v2) & hoasinduction_lem1),
% 9.05/2.75 file('/export/starexec/sandbox2/tmp/tmp.Qe5lgCyrIb/DTF2THF_31149.p',alg444_1)).
% 9.05/2.75 thf(f1053,plain,(
% 9.05/2.75 ( ! [X258 : term] : (((d2term @ (sK583 @ X258)) = X258)) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 thf(f2133,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : (((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ((sK591 @ (d2subst @ d_id) @ (sK594 @ X1) @ (d2subst @ d_id)) = $true) | (hoasinduction_no_psi_cond = $false)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f1745,f1898])).
% 9.05/2.75 thf(f1898,plain,(
% 9.05/2.75 ( ! [X0 : term,X1 : term] : ((X0 = X1)) )),
% 9.05/2.75 inference(superposition,[status(thm)],[f1894,f1894])).
% 9.05/2.75 thf(f1745,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : ((hoasinduction_no_psi_cond = $false) | ((X1 @ (comp @ (sK595 @ X1) @ (sK592 @ X1)) @ (sub @ (sK593 @ X1) @ (sK592 @ X1))) != (sub @ (X1 @ (sK595 @ X1) @ (sK593 @ X1)) @ (sK592 @ X1))) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ((sK591 @ (d2subst @ d_id) @ (sK594 @ X1) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(equality_proxy_clausification,[status(thm)],[f1744])).
% 9.05/2.75 thf(f1744,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : (((sK591 @ (d2subst @ d_id) @ (sK594 @ X1) @ (d2subst @ d_id)) = $true) | (hoasinduction_no_psi_cond = $false) | (((sub @ (X1 @ (sK595 @ X1) @ (sK593 @ X1)) @ (sK592 @ X1)) = (X1 @ (comp @ (sK595 @ X1) @ (sK592 @ X1)) @ (sub @ (sK593 @ X1) @ (sK592 @ X1)))) = $false) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(beta_eta_normalization,[status(thm)],[f1743])).
% 9.05/2.75 thf(f1743,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : (($false = ((^[Y0 : subst]: ((sub @ (X1 @ Y0 @ (sK593 @ X1)) @ (sK592 @ X1)) = (X1 @ (comp @ Y0 @ (sK592 @ X1)) @ (sub @ (sK593 @ X1) @ (sK592 @ X1))))) @ (sK595 @ X1))) | ((sK591 @ (d2subst @ d_id) @ (sK594 @ X1) @ (d2subst @ d_id)) = $true) | (hoasinduction_no_psi_cond = $false) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(sigma_clausification,[status(thm)],[f1742])).
% 9.05/2.75 thf(f1742,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : (((!! @ subst @ (^[Y0 : subst]: ((sub @ (X1 @ Y0 @ (sK593 @ X1)) @ (sK592 @ X1)) = (X1 @ (comp @ Y0 @ (sK592 @ X1)) @ (sub @ (sK593 @ X1) @ (sK592 @ X1)))))) = $false) | (hoasinduction_no_psi_cond = $false) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ((sK591 @ (d2subst @ d_id) @ (sK594 @ X1) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(binary_proxy_clausification,[status(thm)],[f1740])).
% 9.05/2.75 thf(f1740,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : ((hoasinduction_no_psi_cond = $false) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ($false = ((sK591 @ (d2subst @ d_id) @ (sK594 @ X1) @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ (sK594 @ X1)) @ (d2subst @ d_id)))) | ((!! @ subst @ (^[Y0 : subst]: ((sub @ (X1 @ Y0 @ (sK593 @ X1)) @ (sK592 @ X1)) = (X1 @ (comp @ Y0 @ (sK592 @ X1)) @ (sub @ (sK593 @ X1) @ (sK592 @ X1)))))) = $false)) )),
% 9.05/2.75 inference(beta_eta_normalization,[status(thm)],[f1739])).
% 9.05/2.75 thf(f1739,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : (((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ((!! @ subst @ (^[Y0 : subst]: ((sub @ (X1 @ Y0 @ (sK593 @ X1)) @ (sK592 @ X1)) = (X1 @ (comp @ Y0 @ (sK592 @ X1)) @ (sub @ (sK593 @ X1) @ (sK592 @ X1)))))) = $false) | (hoasinduction_no_psi_cond = $false) | (((^[Y0 : term]: ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (X1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id)))) @ (sK594 @ X1)) = $false)) )),
% 9.05/2.75 inference(sigma_clausification,[status(thm)],[f1738])).
% 9.05/2.75 thf(f1738,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : ((hoasinduction_no_psi_cond = $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) | ((!! @ subst @ (^[Y0 : subst]: ((sub @ (X1 @ Y0 @ (sK593 @ X1)) @ (sK592 @ X1)) = (X1 @ (comp @ Y0 @ (sK592 @ X1)) @ (sub @ (sK593 @ X1) @ (sK592 @ X1)))))) = $false) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(beta_eta_normalization,[status(thm)],[f1737])).
% 9.05/2.75 thf(f1737,plain,(
% 9.05/2.75 ( ! [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) | ((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ($false = ((^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: ((sub @ (X1 @ Y1 @ Y0) @ (sK592 @ X1)) = (X1 @ (comp @ Y1 @ (sK592 @ X1)) @ (sub @ Y0 @ (sK592 @ X1))))))) @ (sK593 @ X1))) | (hoasinduction_no_psi_cond = $false)) )),
% 9.05/2.75 inference(sigma_clausification,[status(thm)],[f1736])).
% 9.05/2.75 thf(f1736,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : (((sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ($false = (!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: ((sub @ (X1 @ Y1 @ Y0) @ (sK592 @ X1)) = (X1 @ (comp @ Y1 @ (sK592 @ X1)) @ (sub @ Y0 @ (sK592 @ X1))))))))) | (hoasinduction_no_psi_cond = $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)) )),
% 9.05/2.75 inference(binary_proxy_clausification,[status(thm)],[f1735])).
% 9.05/2.75 thf(f1735,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : ((hoasinduction_no_psi_cond = $false) | (((!! @ 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))) = $true) | ($false = (!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: ((sub @ (X1 @ Y1 @ Y0) @ (sK592 @ X1)) = (X1 @ (comp @ Y1 @ (sK592 @ X1)) @ (sub @ Y0 @ (sK592 @ X1)))))))))) )),
% 9.05/2.75 inference(beta_eta_normalization,[status(thm)],[f1734])).
% 9.05/2.75 thf(f1734,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : ((hoasinduction_no_psi_cond = $false) | ($false = ((^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: ((sub @ (X1 @ Y2 @ Y1) @ Y0) = (X1 @ (comp @ Y2 @ Y0) @ (sub @ Y1 @ Y0)))))))) @ (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))))) => (sK591 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true)) )),
% 9.05/2.75 inference(sigma_clausification,[status(thm)],[f1733])).
% 9.05/2.75 thf(f1733,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : (((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: ((sub @ (X1 @ Y2 @ Y1) @ Y0) = (X1 @ (comp @ Y2 @ Y0) @ (sub @ Y1 @ Y0))))))))) = $false) | (((!! @ 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))) = $true) | (hoasinduction_no_psi_cond = $false)) )),
% 9.05/2.75 inference(binary_proxy_clausification,[status(thm)],[f1732])).
% 9.05/2.75 thf(f1732,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : ((((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: ((sub @ (X1 @ Y2 @ Y1) @ Y0) = (X1 @ (comp @ Y2 @ Y0) @ (sub @ Y1 @ 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)))) = $true) | (hoasinduction_no_psi_cond = $false)) )),
% 9.05/2.75 inference(beta_eta_normalization,[status(thm)],[f1731])).
% 9.05/2.75 thf(f1731,plain,(
% 9.05/2.75 ( ! [X1 : subst > term > term] : ((((^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: ((sub @ (Y0 @ Y3 @ Y2) @ Y1) = (Y0 @ (comp @ Y3 @ Y1) @ (sub @ Y2 @ 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) = $true) | (hoasinduction_no_psi_cond = $false)) )),
% 9.05/2.75 inference(pi_clausification,[status(thm)],[f1730])).
% 9.05/2.75 thf(f1730,plain,(
% 9.05/2.75 ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: ((sub @ (Y0 @ Y3 @ Y2) @ Y1) = (Y0 @ (comp @ Y3 @ Y1) @ (sub @ Y2 @ 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)))))) = $true) | (hoasinduction_no_psi_cond = $false)),
% 9.05/2.75 inference(binary_proxy_clausification,[status(thm)],[f1721])).
% 9.05/2.75 thf(f1721,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond = $false) | ($false = ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: ((sub @ (Y0 @ Y3 @ Y2) @ Y1) = (Y0 @ (comp @ Y3 @ Y1) @ (sub @ Y2 @ 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]: (sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id))))))),
% 9.05/2.75 inference(binary_proxy_clausification,[status(thm)],[f1720])).
% 9.05/2.75 thf(f1720,plain,(
% 9.05/2.75 ($false = ((!! @ term @ (^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK591 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => ((sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)) => (sK591 @ (d2subst @ d_id) @ (hoasap @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id) @ Y0) @ (d2subst @ d_id)))))))) => ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: ((sub @ (Y0 @ Y3 @ Y2) @ Y1) = (Y0 @ (comp @ Y3 @ Y1) @ (sub @ Y2 @ 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]: (sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id))))))) | (hoasinduction_no_psi_cond = $false)),
% 9.05/2.75 inference(beta_eta_normalization,[status(thm)],[f1719])).
% 9.05/2.75 thf(f1719,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond = $false) | (((^[Y0 : subst > term > subst > $o]: ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => ((Y0 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (hoasap @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id)))))))) => ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y4 @ Y3) @ Y2) = (Y1 @ (comp @ Y4 @ Y2) @ (sub @ Y3 @ 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]: (Y0 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id))))))) @ sK591) = $false)),
% 9.05/2.75 inference(sigma_clausification,[status(thm)],[f1718])).
% 9.05/2.75 thf(f1718,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond = $false) | ($false = (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => ((Y0 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (hoasap @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id)))))))) => ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y4 @ Y3) @ Y2) = (Y1 @ (comp @ Y4 @ Y2) @ (sub @ Y3 @ 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]: (Y0 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)))))))))),
% 9.05/2.75 inference(binary_proxy_clausification,[status(thm)],[f1716])).
% 9.05/2.75 thf(f1716,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond != (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => ((Y0 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (hoasap @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id)))))))) => ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y4 @ Y3) @ Y2) = (Y1 @ (comp @ Y4 @ Y2) @ (sub @ Y3 @ 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]: (Y0 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)))))))))),
% 9.05/2.75 inference(beta_eta_normalization,[status(thm)],[f1262])).
% 9.05/2.75 thf(f1262,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond != (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id)) => ((Y0 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)) => (Y0 @ (d2subst @ d_id) @ (hoasap @ (d2subst @ d_id) @ Y2 @ (d2subst @ d_id) @ Y1) @ (d2subst @ d_id)))))))) => ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y4 @ Y3) @ Y2) = (Y1 @ (comp @ Y4 @ Y2) @ (sub @ Y3 @ 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]: (Y0 @ (d2subst @ d_id) @ Y1 @ (d2subst @ d_id)))))))))),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f546,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1160,f1142,f1142,f1142,f1142])).
% 9.05/2.75 thf(f1160,plain,(
% 9.05/2.75 (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one)))))),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 thf(f1142,plain,(
% 9.05/2.75 ((d2subst @ d_id) = id)),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 thf(f546,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond != (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y4 @ Y3) @ Y2) = (Y1 @ (comp @ Y4 @ Y2) @ (sub @ Y3 @ 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]: (Y0 @ id @ Y1 @ id))))))))),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f9])).
% 9.05/2.75 thf(f9,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond != (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y4 @ Y3) @ Y2) = (Y1 @ (comp @ Y4 @ Y2) @ (sub @ Y3 @ 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]: (Y0 @ id @ Y1 @ id))))))))),
% 9.05/2.75 inference(flattening,[status(thm)],[f8])).
% 9.05/2.75 thf(f8,plain,(
% 9.05/2.75 ~(hoasinduction_no_psi_cond = (!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y4 @ Y3) @ Y2) = (Y1 @ (comp @ Y4 @ Y2) @ (sub @ Y3 @ 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]: (Y0 @ id @ Y1 @ id))))))))),
% 9.05/2.75 inference(fool_elimination,[status(thm)],[f7])).
% 9.05/2.75 thf(f7,plain,(
% 9.05/2.75 ~(! [X0 : subst > term > subst > $o] : (! [X1 : term,X2 : term] : ((X0 @ id @ X1 @ id) => ((X0 @ id @ X2 @ id) => (X0 @ id @ (hoasap @ id @ X1 @ id @ X2) @ id))) => (! [X3 : subst > term > term] : (! [X4 : subst,X5 : term,X6 : subst] : ((sub @ (X3 @ X4 @ X5) @ X6) = (X3 @ (comp @ X4 @ X6) @ (sub @ X5 @ X6))) => (! [X7 : term] : ((X0 @ id @ X7 @ id) => (X0 @ id @ (X3 @ id @ X7) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X8 : subst, X9 : term] : (X3 @ X8 @ X9))) @ id))) => ! [X10 : term] : (X0 @ id @ X10 @ id))) = hoasinduction_no_psi_cond)),
% 9.05/2.75 inference(rectify,[status(thm)],[f3])).
% 9.05/2.75 thf(f3,negated_conjecture,(
% 9.05/2.75 ~(! [X10 : subst > term > subst > $o] : (! [X13 : term,X15 : term] : ((X10 @ id @ X13 @ id) => ((X10 @ id @ X15 @ id) => (X10 @ id @ (hoasap @ id @ X13 @ id @ X15) @ id))) => (! [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 @ (hoaslam @ id @ (^[X14 : subst, X13 : term] : (X18 @ X14 @ X13))) @ id))) => ! [X13 : term] : (X10 @ id @ X13 @ id))) = hoasinduction_no_psi_cond)),
% 9.05/2.75 inference(negated_conjecture,[status(cth)],[f2])).
% 9.05/2.75 thf(f2,conjecture,(
% 9.05/2.75 (! [X10 : subst > term > subst > $o] : (! [X13 : term,X15 : term] : ((X10 @ id @ X13 @ id) => ((X10 @ id @ X15 @ id) => (X10 @ id @ (hoasap @ id @ X13 @ id @ X15) @ id))) => (! [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 @ (hoaslam @ id @ (^[X14 : subst, X13 : term] : (X18 @ X14 @ X13))) @ id))) => ! [X13 : term] : (X10 @ id @ X13 @ id))) = hoasinduction_no_psi_cond)),
% 9.05/2.75 file('/export/starexec/sandbox2/tmp/tmp.Qe5lgCyrIb/DTF2THF_31149.p',hoasinduction_no_psi_cond)).
% 9.05/2.75 thf(f1959,plain,(
% 9.05/2.75 ( ! [X0 : term,X1 : subst] : (((sK591 @ X1 @ X0 @ X1) = $false) | (hoasinduction_no_psi_cond = $false)) )),
% 9.05/2.75 inference(superposition,[status(thm)],[f1956,f1898])).
% 9.05/2.75 thf(f1956,plain,(
% 9.05/2.75 ( ! [X0 : subst] : (((sK591 @ X0 @ sK597 @ X0) = $false) | (hoasinduction_no_psi_cond = $false)) )),
% 9.05/2.75 inference(superposition,[status(thm)],[f1750,f1951])).
% 9.05/2.75 thf(f1951,plain,(
% 9.05/2.75 ( ! [X159 : subst] : (((d2subst @ d_id) = X159)) )),
% 9.05/2.75 inference(forward_demodulation,[status(thm)],[f1140,f1099])).
% 9.05/2.75 thf(f1099,plain,(
% 9.05/2.75 ( ! [X219 : d_subst] : ((d_id = X219)) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 thf(f1140,plain,(
% 9.05/2.75 ( ! [X159 : subst] : (((d2subst @ (sK534 @ X159)) = X159)) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 thf(f1750,plain,(
% 9.05/2.75 ((sK591 @ (d2subst @ d_id) @ sK597 @ (d2subst @ d_id)) = $false) | (hoasinduction_no_psi_cond = $false)),
% 9.05/2.75 inference(beta_eta_normalization,[status(thm)],[f1749])).
% 9.05/2.75 thf(f1749,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond = $false) | (((^[Y0 : term]: (sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id))) @ sK597) = $false)),
% 9.05/2.75 inference(sigma_clausification,[status(thm)],[f1729])).
% 9.05/2.75 thf(f1729,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond = $false) | ($false = (!! @ term @ (^[Y0 : term]: (sK591 @ (d2subst @ d_id) @ Y0 @ (d2subst @ d_id)))))),
% 9.05/2.75 inference(binary_proxy_clausification,[status(thm)],[f1721])).
% 9.05/2.75 thf(f1259,plain,(
% 9.05/2.75 ((sP58 @ sK458) != $true) | (hoasinduction_no_psi_cond = $true)),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 thf(f5107,plain,(
% 9.05/2.75 ((sP58 @ sK458) = $true)),
% 9.05/2.75 inference(trivial_inequality_removal,[status(thm)],[f5096])).
% 9.05/2.75 thf(f5096,plain,(
% 9.05/2.75 ($true != $true) | ((sP58 @ sK458) = $true)),
% 9.05/2.75 inference(superposition,[status(thm)],[f2354,f5024])).
% 9.05/2.75 thf(f5024,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o] : (((X0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ((sP58 @ X0) = $true)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f5023,f4469])).
% 9.05/2.75 thf(f4469,plain,(
% 9.05/2.75 ( ! [X8 : subst > term > subst > $o,X14 : term] : (((X8 @ (d2subst @ d_id) @ X14 @ (d2subst @ d_id)) != $true) | ((X8 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(trivial_inequality_removal,[status(thm)],[f4466])).
% 9.05/2.75 thf(f4466,plain,(
% 9.05/2.75 ( ! [X8 : subst > term > subst > $o,X14 : term] : (((X8 @ (d2subst @ d_id) @ X14 @ (d2subst @ d_id)) != $true) | ((X8 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ($true != $true)) )),
% 9.05/2.75 inference(backward_demodulation,[status(thm)],[f3971,f4465])).
% 9.05/2.75 thf(f4465,plain,(
% 9.05/2.75 (sP37 = $true)),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f4464,f2972])).
% 9.05/2.75 thf(f2972,plain,(
% 9.05/2.75 ( ! [X0 : term,X1 : subst] : (((sK188 @ X1 @ X0 @ X1) != $true) | (sP37 = $true)) )),
% 9.05/2.75 inference(superposition,[status(thm)],[f2378,f1898])).
% 9.05/2.75 thf(f2378,plain,(
% 9.05/2.75 ( ! [X0 : subst] : (((sK188 @ X0 @ (lam @ sK189) @ X0) != $true) | (sP37 = $true)) )),
% 9.05/2.75 inference(superposition,[status(thm)],[f1372,f1951])).
% 9.05/2.75 thf(f1372,plain,(
% 9.05/2.75 ($true != (sK188 @ (d2subst @ d_id) @ (lam @ sK189) @ (d2subst @ d_id))) | (sP37 = $true)),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f717,f1142,f1142])).
% 9.05/2.75 thf(f717,plain,(
% 9.05/2.75 (sP37 = $true) | ((sK188 @ id @ (lam @ sK189) @ id) != $true)),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f217])).
% 9.05/2.75 thf(f217,plain,(
% 9.05/2.75 ((sP37 = $true) | (! [X2 : subst > term > term] : (((sK188 @ id @ (lam @ (X2 @ sh @ one)) @ id) = $true) | ((X2 @ (comp @ (sK191 @ X2) @ (sK190 @ X2)) @ (sub @ (sK192 @ X2) @ (sK190 @ X2))) != (sub @ (X2 @ (sK191 @ X2) @ (sK192 @ X2)) @ (sK190 @ X2))) | (((sK188 @ id @ (X2 @ id @ (sK193 @ X2)) @ id) != $true) & ((sK188 @ id @ (sK193 @ X2) @ id) = $true))) & ! [X7 : term] : (((sK188 @ id @ (sub @ sK189 @ (push @ X7 @ id)) @ id) = $true) | ((sK188 @ id @ X7 @ id) != $true)) & ((sK188 @ id @ (lam @ sK189) @ id) != $true))) & (! [X8 : subst > term > subst > $o,X9 : term] : ((((X8 @ id @ (lam @ (sK194 @ X8 @ sh @ one)) @ id) != $true) & ! [X11 : subst,X12 : subst,X13 : term] : ((sK194 @ X8 @ (comp @ X12 @ X11) @ (sub @ X13 @ X11)) = (sub @ (sK194 @ X8 @ X12 @ X13) @ X11)) & ! [X14 : term] : (((X8 @ id @ (sK194 @ X8 @ id @ X14) @ id) = $true) | ((X8 @ id @ X14 @ id) != $true))) | (((X8 @ id @ (sub @ X9 @ (push @ (sK195 @ X9 @ X8) @ id)) @ id) != $true) & ((X8 @ id @ (sK195 @ X9 @ X8) @ id) = $true)) | ((X8 @ id @ (lam @ X9) @ id) = $true)) | (sP37 != $true))),
% 9.05/2.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK188,sK189,sK190,sK191,sK192,sK193,sK194,sK195])],[f211,f216,f215,f214,f213,f212])).
% 9.05/2.75 thf(f212,plain,(
% 9.05/2.75 ? [X0 : subst > term > subst > $o,X1 : term] : (! [X2 : subst > term > term] : (((X0 @ id @ (lam @ (X2 @ sh @ one)) @ id) = $true) | ? [X3 : subst,X4 : subst,X5 : term] : ((X2 @ (comp @ X4 @ X3) @ (sub @ X5 @ X3)) != (sub @ (X2 @ X4 @ X5) @ X3)) | ? [X6 : term] : (((X0 @ id @ (X2 @ id @ X6) @ id) != $true) & ((X0 @ id @ X6 @ id) = $true))) & ! [X7 : term] : (((X0 @ id @ (sub @ X1 @ (push @ X7 @ id)) @ id) = $true) | ((X0 @ id @ X7 @ id) != $true)) & ((X0 @ id @ (lam @ X1) @ id) != $true)) => (! [X2 : subst > term > term] : (((sK188 @ id @ (lam @ (X2 @ sh @ one)) @ id) = $true) | ? [X3 : subst,X4 : subst,X5 : term] : ((X2 @ (comp @ X4 @ X3) @ (sub @ X5 @ X3)) != (sub @ (X2 @ X4 @ X5) @ X3)) | ? [X6 : term] : (((sK188 @ id @ (X2 @ id @ X6) @ id) != $true) & ((sK188 @ id @ X6 @ id) = $true))) & ! [X7 : term] : (((sK188 @ id @ (sub @ sK189 @ (push @ X7 @ id)) @ id) = $true) | ((sK188 @ id @ X7 @ id) != $true)) & ((sK188 @ id @ (lam @ sK189) @ id) != $true))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f213,plain,(
% 9.05/2.75 ! [X2 : subst > term > term] : (? [X3 : subst,X4 : subst,X5 : term] : ((X2 @ (comp @ X4 @ X3) @ (sub @ X5 @ X3)) != (sub @ (X2 @ X4 @ X5) @ X3)) => ((X2 @ (comp @ (sK191 @ X2) @ (sK190 @ X2)) @ (sub @ (sK192 @ X2) @ (sK190 @ X2))) != (sub @ (X2 @ (sK191 @ X2) @ (sK192 @ X2)) @ (sK190 @ X2))))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f214,plain,(
% 9.05/2.75 ! [X2 : subst > term > term] : (? [X6 : term] : (((sK188 @ id @ (X2 @ id @ X6) @ id) != $true) & ((sK188 @ id @ X6 @ id) = $true)) => (((sK188 @ id @ (X2 @ id @ (sK193 @ X2)) @ id) != $true) & ((sK188 @ id @ (sK193 @ X2) @ id) = $true)))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f215,plain,(
% 9.05/2.75 ! [X8 : subst > term > subst > $o] : (? [X10 : subst > term > term] : (((X8 @ id @ (lam @ (X10 @ sh @ one)) @ id) != $true) & ! [X11 : subst,X12 : subst,X13 : term] : ((sub @ (X10 @ X12 @ X13) @ X11) = (X10 @ (comp @ X12 @ X11) @ (sub @ X13 @ X11))) & ! [X14 : term] : (((X8 @ id @ (X10 @ id @ X14) @ id) = $true) | ((X8 @ id @ X14 @ id) != $true))) => (((X8 @ id @ (lam @ (sK194 @ X8 @ sh @ one)) @ id) != $true) & ! [X13 : term,X12 : subst,X11 : subst] : ((sK194 @ X8 @ (comp @ X12 @ X11) @ (sub @ X13 @ X11)) = (sub @ (sK194 @ X8 @ X12 @ X13) @ X11)) & ! [X14 : term] : (((X8 @ id @ (sK194 @ X8 @ id @ X14) @ id) = $true) | ((X8 @ id @ X14 @ id) != $true))))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f216,plain,(
% 9.05/2.75 ! [X8 : subst > term > subst > $o,X9 : term] : (? [X15 : term] : (((X8 @ id @ (sub @ X9 @ (push @ X15 @ id)) @ id) != $true) & ((X8 @ id @ X15 @ id) = $true)) => (((X8 @ id @ (sub @ X9 @ (push @ (sK195 @ X9 @ X8) @ id)) @ id) != $true) & ((X8 @ id @ (sK195 @ X9 @ X8) @ id) = $true)))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f211,plain,(
% 9.05/2.75 ((sP37 = $true) | ? [X0 : subst > term > subst > $o,X1 : term] : (! [X2 : subst > term > term] : (((X0 @ id @ (lam @ (X2 @ sh @ one)) @ id) = $true) | ? [X3 : subst,X4 : subst,X5 : term] : ((X2 @ (comp @ X4 @ X3) @ (sub @ X5 @ X3)) != (sub @ (X2 @ X4 @ X5) @ X3)) | ? [X6 : term] : (((X0 @ id @ (X2 @ id @ X6) @ id) != $true) & ((X0 @ id @ X6 @ id) = $true))) & ! [X7 : term] : (((X0 @ id @ (sub @ X1 @ (push @ X7 @ id)) @ id) = $true) | ((X0 @ id @ X7 @ id) != $true)) & ((X0 @ id @ (lam @ X1) @ id) != $true))) & (! [X8 : subst > term > subst > $o,X9 : term] : (? [X10 : subst > term > term] : (((X8 @ id @ (lam @ (X10 @ sh @ one)) @ id) != $true) & ! [X11 : subst,X12 : subst,X13 : term] : ((sub @ (X10 @ X12 @ X13) @ X11) = (X10 @ (comp @ X12 @ X11) @ (sub @ X13 @ X11))) & ! [X14 : term] : (((X8 @ id @ (X10 @ id @ X14) @ id) = $true) | ((X8 @ id @ X14 @ id) != $true))) | ? [X15 : term] : (((X8 @ id @ (sub @ X9 @ (push @ X15 @ id)) @ id) != $true) & ((X8 @ id @ X15 @ id) = $true)) | ((X8 @ id @ (lam @ X9) @ id) = $true)) | (sP37 != $true))),
% 9.05/2.75 inference(rectify,[status(thm)],[f210])).
% 9.05/2.75 thf(f210,plain,(
% 9.05/2.75 ((sP37 = $true) | ? [X75 : subst > term > subst > $o,X76 : term] : (! [X77 : subst > term > term] : (((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) = $true) | ? [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) != (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78))) | ? [X81 : term] : (((X75 @ id @ (X77 @ id @ X81) @ id) != $true) & ((X75 @ id @ X81 @ id) = $true))) & ! [X82 : term] : (((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) = $true) | ((X75 @ id @ X82 @ id) != $true)) & ((X75 @ id @ (lam @ X76) @ id) != $true))) & (! [X75 : subst > term > subst > $o,X76 : term] : (? [X77 : subst > term > term] : (((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) != $true) & ! [X78 : subst,X79 : subst,X80 : term] : ((sub @ (X77 @ X79 @ X80) @ X78) = (X77 @ (comp @ X79 @ X78) @ (sub @ X80 @ X78))) & ! [X81 : term] : (((X75 @ id @ (X77 @ id @ X81) @ id) = $true) | ((X75 @ id @ X81 @ id) != $true))) | ? [X82 : term] : (((X75 @ id @ (sub @ X76 @ (push @ X82 @ id)) @ id) != $true) & ((X75 @ id @ X82 @ id) = $true)) | ((X75 @ id @ (lam @ X76) @ id) = $true)) | (sP37 != $true))),
% 9.05/2.75 inference(nnf_transformation,[status(thm)],[f51])).
% 9.05/2.75 thf(f4464,plain,(
% 9.05/2.75 ( ! [X2 : subst > term > term] : (((sK188 @ (d2subst @ d_id) @ (lam @ (X2 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id)) = $true) | (sP37 = $true)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f4463,f2972])).
% 9.05/2.75 thf(f4463,plain,(
% 9.05/2.75 ( ! [X2 : subst > term > term] : (((sK188 @ (d2subst @ d_id) @ (sK193 @ X2) @ (d2subst @ d_id)) = $true) | ((sK188 @ (d2subst @ d_id) @ (lam @ (X2 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id)) = $true) | (sP37 = $true)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f1370,f1898])).
% 9.05/2.75 thf(f1370,plain,(
% 9.05/2.75 ( ! [X2 : subst > term > term] : (((sK188 @ (d2subst @ d_id) @ (lam @ (X2 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id)) = $true) | ((X2 @ (comp @ (sK191 @ X2) @ (sK190 @ X2)) @ (sub @ (sK192 @ X2) @ (sK190 @ X2))) != (sub @ (X2 @ (sK191 @ X2) @ (sK192 @ X2)) @ (sK190 @ X2))) | (sP37 = $true) | ((sK188 @ (d2subst @ d_id) @ (sK193 @ X2) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f719,f1142,f1202,f1148,f1142,f1142,f1142])).
% 9.05/2.75 thf(f1148,plain,(
% 9.05/2.75 (one = (d2term @ d_one))),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 thf(f1202,plain,(
% 9.05/2.75 ((d2subst @ d_id) = sh)),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 thf(f719,plain,(
% 9.05/2.75 ( ! [X2 : subst > term > term] : ((sP37 = $true) | ((sK188 @ id @ (lam @ (X2 @ sh @ one)) @ id) = $true) | ((X2 @ (comp @ (sK191 @ X2) @ (sK190 @ X2)) @ (sub @ (sK192 @ X2) @ (sK190 @ X2))) != (sub @ (X2 @ (sK191 @ X2) @ (sK192 @ X2)) @ (sK190 @ X2))) | ((sK188 @ id @ (sK193 @ X2) @ id) = $true)) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f217])).
% 9.05/2.75 thf(f3971,plain,(
% 9.05/2.75 ( ! [X8 : subst > term > subst > $o,X14 : term] : (((X8 @ (d2subst @ d_id) @ X14 @ (d2subst @ d_id)) != $true) | (sP37 != $true) | ((X8 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f3970,f3704])).
% 9.05/2.75 thf(f3704,plain,(
% 9.05/2.75 ( ! [X2 : term,X0 : subst > term > subst > $o,X1 : term] : (((X0 @ (d2subst @ d_id) @ X2 @ (d2subst @ d_id)) != $true) | ((X0 @ (d2subst @ d_id) @ X1 @ (d2subst @ d_id)) != $true) | ((X0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(forward_demodulation,[status(thm)],[f3703,f1894])).
% 9.05/2.75 thf(f3703,plain,(
% 9.05/2.75 ( ! [X2 : term,X0 : subst > term > subst > $o,X1 : term] : (((X0 @ (d2subst @ d_id) @ X2 @ (d2subst @ d_id)) != $true) | ((X0 @ (d2subst @ d_id) @ X1 @ (d2subst @ d_id)) != $true) | ((X0 @ (d2subst @ d_id) @ (ap @ (sub @ X1 @ (d2subst @ d_id)) @ X2) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f1299,f3549])).
% 9.05/2.75 thf(f3549,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o] : (((sP50 @ X0) != $true)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f3548,f2752])).
% 9.05/2.75 thf(f2752,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o] : (((sP50 @ X0) != $true) | ((X0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(forward_demodulation,[status(thm)],[f1301,f1894])).
% 9.05/2.75 thf(f1301,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o] : (((X0 @ (d2subst @ d_id) @ (sK106 @ X0) @ (d2subst @ d_id)) = $true) | ((sP50 @ X0) != $true)) )),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f612,f1142,f1142])).
% 9.05/2.75 thf(f612,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o] : (((X0 @ id @ (sK106 @ X0) @ id) = $true) | ((sP50 @ X0) != $true)) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f135])).
% 9.05/2.75 thf(f135,plain,(
% 9.05/2.75 ! [X0 : subst > term > subst > $o] : ((((sP50 @ X0) = $true) | ! [X1 : term,X2 : term] : (((X0 @ id @ (ap @ (sub @ X1 @ id) @ X2) @ id) = $true) | ($true != (X0 @ id @ X1 @ id)) | ((X0 @ id @ X2 @ id) != $true))) & ((($true != (X0 @ id @ (ap @ (sub @ (sK106 @ X0) @ id) @ (sK107 @ X0)) @ id)) & ((X0 @ id @ (sK106 @ X0) @ id) = $true) & ((X0 @ id @ (sK107 @ X0) @ id) = $true)) | ((sP50 @ X0) != $true)))),
% 9.05/2.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK106,sK107])],[f133,f134])).
% 9.05/2.75 thf(f134,plain,(
% 9.05/2.75 ! [X0 : subst > term > subst > $o] : (? [X3 : term,X4 : term] : (((X0 @ id @ (ap @ (sub @ X3 @ id) @ X4) @ id) != $true) & ((X0 @ id @ X3 @ id) = $true) & ((X0 @ id @ X4 @ id) = $true)) => (($true != (X0 @ id @ (ap @ (sub @ (sK106 @ X0) @ id) @ (sK107 @ X0)) @ id)) & ((X0 @ id @ (sK106 @ X0) @ id) = $true) & ((X0 @ id @ (sK107 @ X0) @ id) = $true)))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f133,plain,(
% 9.05/2.75 ! [X0 : subst > term > subst > $o] : ((((sP50 @ X0) = $true) | ! [X1 : term,X2 : term] : (((X0 @ id @ (ap @ (sub @ X1 @ id) @ X2) @ id) = $true) | ($true != (X0 @ id @ X1 @ id)) | ((X0 @ id @ X2 @ id) != $true))) & (? [X3 : term,X4 : term] : (((X0 @ id @ (ap @ (sub @ X3 @ id) @ X4) @ id) != $true) & ((X0 @ id @ X3 @ id) = $true) & ((X0 @ id @ X4 @ id) = $true)) | ((sP50 @ X0) != $true)))),
% 9.05/2.75 inference(rectify,[status(thm)],[f132])).
% 9.05/2.75 thf(f132,plain,(
% 9.05/2.75 ! [X219 : subst > term > subst > $o] : ((((sP50 @ X219) = $true) | ! [X221 : term,X222 : term] : (((X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id) = $true) | ((X219 @ id @ X221 @ id) != $true) | ((X219 @ id @ X222 @ id) != $true))) & (? [X221 : term,X222 : term] : (((X219 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id) != $true) & ((X219 @ id @ X221 @ id) = $true) & ((X219 @ id @ X222 @ id) = $true)) | ((sP50 @ X219) != $true)))),
% 9.05/2.75 inference(nnf_transformation,[status(thm)],[f64])).
% 9.05/2.75 thf(f3548,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o] : (((sP50 @ X0) != $true) | ((X0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true)) )),
% 9.05/2.75 inference(forward_demodulation,[status(thm)],[f1300,f1894])).
% 9.05/2.75 thf(f1300,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o] : (((X0 @ (d2subst @ d_id) @ (ap @ (sub @ (sK106 @ X0) @ (d2subst @ d_id)) @ (sK107 @ X0)) @ (d2subst @ d_id)) != $true) | ((sP50 @ X0) != $true)) )),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f613,f1142,f1142,f1142])).
% 9.05/2.75 thf(f613,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o] : (($true != (X0 @ id @ (ap @ (sub @ (sK106 @ X0) @ id) @ (sK107 @ X0)) @ id)) | ((sP50 @ X0) != $true)) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f135])).
% 9.05/2.75 thf(f1299,plain,(
% 9.05/2.75 ( ! [X2 : term,X0 : subst > term > subst > $o,X1 : term] : (((sP50 @ X0) = $true) | ((X0 @ (d2subst @ d_id) @ (ap @ (sub @ X1 @ (d2subst @ d_id)) @ X2) @ (d2subst @ d_id)) = $true) | ((X0 @ (d2subst @ d_id) @ X2 @ (d2subst @ d_id)) != $true) | ((X0 @ (d2subst @ d_id) @ X1 @ (d2subst @ d_id)) != $true)) )),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f614,f1142,f1142,f1142,f1142,f1142,f1142,f1142])).
% 9.05/2.75 thf(f614,plain,(
% 9.05/2.75 ( ! [X2 : term,X0 : subst > term > subst > $o,X1 : term] : (((sP50 @ X0) = $true) | ((X0 @ id @ (ap @ (sub @ X1 @ id) @ X2) @ id) = $true) | ($true != (X0 @ id @ X1 @ id)) | ((X0 @ id @ X2 @ id) != $true)) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f135])).
% 9.05/2.75 thf(f3970,plain,(
% 9.05/2.75 ( ! [X8 : subst > term > subst > $o,X9 : term,X14 : term] : (((X8 @ (d2subst @ d_id) @ X14 @ (d2subst @ d_id)) != $true) | ((X8 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | ((X8 @ (d2subst @ d_id) @ (lam @ X9) @ (d2subst @ d_id)) = $true) | (sP37 != $true)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f3969,f3704])).
% 9.05/2.75 thf(f3969,plain,(
% 9.05/2.75 ( ! [X8 : subst > term > subst > $o,X9 : term,X14 : term] : (((X8 @ (d2subst @ d_id) @ X14 @ (d2subst @ d_id)) != $true) | ((X8 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) | (sP37 != $true) | ((X8 @ (d2subst @ d_id) @ (sK194 @ X8 @ (d2subst @ d_id) @ X14) @ (d2subst @ d_id)) = $true) | ((X8 @ (d2subst @ d_id) @ (lam @ X9) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(forward_demodulation,[status(thm)],[f1378,f1894])).
% 9.05/2.75 thf(f1378,plain,(
% 9.05/2.75 ( ! [X8 : subst > term > subst > $o,X9 : term,X14 : term] : (((X8 @ (d2subst @ d_id) @ (sK195 @ X9 @ X8) @ (d2subst @ d_id)) = $true) | ((X8 @ (d2subst @ d_id) @ (sK194 @ X8 @ (d2subst @ d_id) @ X14) @ (d2subst @ d_id)) = $true) | ((X8 @ (d2subst @ d_id) @ (lam @ X9) @ (d2subst @ d_id)) = $true) | ((X8 @ (d2subst @ d_id) @ X14 @ (d2subst @ d_id)) != $true) | (sP37 != $true)) )),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f711,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142,f1142])).
% 9.05/2.75 thf(f711,plain,(
% 9.05/2.75 ( ! [X8 : subst > term > subst > $o,X9 : term,X14 : term] : (((X8 @ id @ (sK194 @ X8 @ id @ X14) @ id) = $true) | ((X8 @ id @ X14 @ id) != $true) | ((X8 @ id @ (sK195 @ X9 @ X8) @ id) = $true) | ((X8 @ id @ (lam @ X9) @ id) = $true) | (sP37 != $true)) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f217])).
% 9.05/2.75 thf(f5023,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o,X1 : subst > term > term] : (((X0 @ (d2subst @ d_id) @ (lam @ (X1 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id)) = $true) | ((sP58 @ X0) = $true) | ((X0 @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(forward_demodulation,[status(thm)],[f5022,f1894])).
% 9.05/2.75 thf(f5022,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o,X1 : subst > term > term] : (((X0 @ (d2subst @ d_id) @ (sK59 @ X1 @ X0) @ (d2subst @ d_id)) = $true) | ((sP58 @ X0) = $true) | ((X0 @ (d2subst @ d_id) @ (lam @ (X1 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(subsumption_resolution,[status(thm)],[f1264,f1898])).
% 9.05/2.75 thf(f1264,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o,X1 : subst > term > term] : (((X1 @ (comp @ (sK60 @ X1) @ (sK61 @ X1)) @ (sub @ (sK62 @ X1) @ (sK61 @ X1))) != (sub @ (X1 @ (sK60 @ X1) @ (sK62 @ X1)) @ (sK61 @ X1))) | ((X0 @ (d2subst @ d_id) @ (sK59 @ X1 @ X0) @ (d2subst @ d_id)) = $true) | ((sP58 @ X0) = $true) | ((X0 @ (d2subst @ d_id) @ (lam @ (X1 @ (d2subst @ d_id) @ (d2term @ d_one))) @ (d2subst @ d_id)) = $true)) )),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f550,f1142,f1142,f1142,f1202,f1148,f1142])).
% 9.05/2.75 thf(f550,plain,(
% 9.05/2.75 ( ! [X0 : subst > term > subst > $o,X1 : subst > term > term] : (((sP58 @ X0) = $true) | ((X0 @ id @ (sK59 @ X1 @ X0) @ id) = $true) | ((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id) = $true) | ((X1 @ (comp @ (sK60 @ X1) @ (sK61 @ X1)) @ (sub @ (sK62 @ X1) @ (sK61 @ X1))) != (sub @ (X1 @ (sK60 @ X1) @ (sK62 @ X1)) @ (sK61 @ X1)))) )),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f79])).
% 9.05/2.75 thf(f79,plain,(
% 9.05/2.75 ! [X0 : subst > term > subst > $o] : ((((sP58 @ X0) = $true) | ! [X1 : subst > term > term] : ((((X0 @ id @ (X1 @ id @ (sK59 @ X1 @ X0)) @ id) != $true) & ((X0 @ id @ (sK59 @ X1 @ X0) @ id) = $true)) | ((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id) = $true) | ((X1 @ (comp @ (sK60 @ X1) @ (sK61 @ X1)) @ (sub @ (sK62 @ X1) @ (sK61 @ X1))) != (sub @ (X1 @ (sK60 @ X1) @ (sK62 @ X1)) @ (sK61 @ X1))))) & ((! [X7 : term] : (((X0 @ id @ (sK63 @ X0 @ id @ X7) @ id) = $true) | ((X0 @ id @ X7 @ id) != $true)) & ((X0 @ id @ (lam @ (sK63 @ X0 @ sh @ one)) @ id) != $true) & ! [X8 : subst,X9 : subst,X10 : term] : ((sub @ (sK63 @ X0 @ X8 @ X10) @ X9) = (sK63 @ X0 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9)))) | ((sP58 @ X0) != $true)))),
% 9.05/2.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK59,sK60,sK61,sK62,sK63])],[f75,f78,f77,f76])).
% 9.05/2.75 thf(f76,plain,(
% 9.05/2.75 ! [X0 : subst > term > subst > $o,X1 : subst > term > term] : (? [X2 : term] : (((X0 @ id @ (X1 @ id @ X2) @ id) != $true) & ((X0 @ id @ X2 @ id) = $true)) => (((X0 @ id @ (X1 @ id @ (sK59 @ X1 @ X0)) @ id) != $true) & ((X0 @ id @ (sK59 @ X1 @ X0) @ id) = $true)))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f77,plain,(
% 9.05/2.75 ! [X1 : subst > term > term] : (? [X3 : subst,X4 : subst,X5 : term] : ((X1 @ (comp @ X3 @ X4) @ (sub @ X5 @ X4)) != (sub @ (X1 @ X3 @ X5) @ X4)) => ((X1 @ (comp @ (sK60 @ X1) @ (sK61 @ X1)) @ (sub @ (sK62 @ X1) @ (sK61 @ X1))) != (sub @ (X1 @ (sK60 @ X1) @ (sK62 @ X1)) @ (sK61 @ X1))))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f78,plain,(
% 9.05/2.75 ! [X0 : subst > term > subst > $o] : (? [X6 : subst > term > term] : (! [X7 : term] : (((X0 @ id @ (X6 @ id @ X7) @ id) = $true) | ((X0 @ id @ X7 @ id) != $true)) & ((X0 @ id @ (lam @ (X6 @ sh @ one)) @ id) != $true) & ! [X8 : subst,X9 : subst,X10 : term] : ((sub @ (X6 @ X8 @ X10) @ X9) = (X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9)))) => (! [X7 : term] : (((X0 @ id @ (sK63 @ X0 @ id @ X7) @ id) = $true) | ((X0 @ id @ X7 @ id) != $true)) & ((X0 @ id @ (lam @ (sK63 @ X0 @ sh @ one)) @ id) != $true) & ! [X10 : term,X9 : subst,X8 : subst] : ((sub @ (sK63 @ X0 @ X8 @ X10) @ X9) = (sK63 @ X0 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9)))))),
% 9.05/2.75 introduced(definition,[],[choice_axiom])).
% 9.05/2.75 thf(f75,plain,(
% 9.05/2.75 ! [X0 : subst > term > subst > $o] : ((((sP58 @ X0) = $true) | ! [X1 : subst > term > term] : (? [X2 : term] : (((X0 @ id @ (X1 @ id @ X2) @ id) != $true) & ((X0 @ id @ X2 @ id) = $true)) | ((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id) = $true) | ? [X3 : subst,X4 : subst,X5 : term] : ((X1 @ (comp @ X3 @ X4) @ (sub @ X5 @ X4)) != (sub @ (X1 @ X3 @ X5) @ X4)))) & (? [X6 : subst > term > term] : (! [X7 : term] : (((X0 @ id @ (X6 @ id @ X7) @ id) = $true) | ((X0 @ id @ X7 @ id) != $true)) & ((X0 @ id @ (lam @ (X6 @ sh @ one)) @ id) != $true) & ! [X8 : subst,X9 : subst,X10 : term] : ((sub @ (X6 @ X8 @ X10) @ X9) = (X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9)))) | ((sP58 @ X0) != $true)))),
% 9.05/2.75 inference(rectify,[status(thm)],[f74])).
% 9.05/2.75 thf(f74,plain,(
% 9.05/2.75 ! [X184 : subst > term > subst > $o] : ((((sP58 @ X184) = $true) | ! [X187 : subst > term > term] : (? [X188 : term] : (((X184 @ id @ (X187 @ id @ X188) @ id) != $true) & ((X184 @ id @ X188 @ id) = $true)) | ((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) = $true) | ? [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) != (sub @ (X187 @ X189 @ X191) @ X190)))) & (? [X187 : subst > term > term] : (! [X188 : term] : (((X184 @ id @ (X187 @ id @ X188) @ id) = $true) | ((X184 @ id @ X188 @ id) != $true)) & ((X184 @ id @ (lam @ (X187 @ sh @ one)) @ id) != $true) & ! [X189 : subst,X190 : subst,X191 : term] : ((X187 @ (comp @ X189 @ X190) @ (sub @ X191 @ X190)) = (sub @ (X187 @ X189 @ X191) @ X190))) | ((sP58 @ X184) != $true)))),
% 9.05/2.75 inference(nnf_transformation,[status(thm)],[f72])).
% 9.05/2.75 thf(f2354,plain,(
% 9.05/2.75 ( ! [X0 : term,X1 : subst] : (((sK458 @ X1 @ X0 @ X1) != $true)) )),
% 9.05/2.75 inference(superposition,[status(thm)],[f2350,f1898])).
% 9.05/2.75 thf(f2350,plain,(
% 9.05/2.75 ( ! [X0 : subst] : (((sK458 @ X0 @ sK457 @ X0) != $true)) )),
% 9.05/2.75 inference(superposition,[status(thm)],[f2344,f1951])).
% 9.05/2.75 thf(f2344,plain,(
% 9.05/2.75 ((sK458 @ (d2subst @ d_id) @ sK457 @ (d2subst @ d_id)) != $true)),
% 9.05/2.75 inference(trivial_inequality_removal,[status(thm)],[f2343])).
% 9.05/2.75 thf(f2343,plain,(
% 9.05/2.75 ($false = $true) | ((sK458 @ (d2subst @ d_id) @ sK457 @ (d2subst @ d_id)) != $true)),
% 9.05/2.75 inference(forward_demodulation,[status(thm)],[f1598,f2159])).
% 9.05/2.75 thf(f1598,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond = $true) | ((sK458 @ (d2subst @ d_id) @ sK457 @ (d2subst @ d_id)) != $true)),
% 9.05/2.75 inference(definition_unfolding,[status(thm)],[f1258,f1142,f1142])).
% 9.05/2.75 thf(f1258,plain,(
% 9.05/2.75 (hoasinduction_no_psi_cond = $true) | ((sK458 @ id @ sK457 @ id) != $true)),
% 9.05/2.75 inference(cnf_transformation,[status(thm)],[f545])).
% 9.05/2.75 % SZS output end Proof for DTF2THF_31149
% 9.05/2.75 % (31347)------------------------------
% 9.05/2.75 % (31347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.05/2.75 % (31347)Termination reason: Refutation
% 9.05/2.75
% 9.05/2.75 % (31347)Memory used [KB]: 7164
% 9.05/2.75 % (31347)Time elapsed: 0.425 s
% 9.05/2.75 % (31347)Instructions burned: 1285 (million)
% 9.05/2.75 % (31347)------------------------------
% 9.05/2.75 % (31347)------------------------------
% 9.05/2.75 % (31247)Success in time 1.312 s
%------------------------------------------------------------------------------