%------------------------------------------------------------------------------
% 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 : n007.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 06:58:46 PM UTC 2026
% Result : Theorem 2.56s 1.94s
% Output : Refutation 2.56s
% Verified :
% SZS Type : Refutation
% Derivation depth : 34
% Number of leaves : 130
% Syntax : Number of formulae : 184 ( 39 unt; 0 typ; 0 def)
% Number of atoms : 11244 (5350 equ; 0 cnn)
% Maximal formula atoms : 530 ( 61 avg)
% Number of connectives : 26513 (3044 ~;2756 |;2417 &;17363 @)
% ( 507 <=>; 332 =>; 0 <=; 0 <~>)
% Maximal formula depth : 193 ( 17 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 1989 (1989 >; 0 *; 0 +; 0 <<)
% Number of symbols : 726 ( 722 usr; 295 con; 0-4 aty)
% ( 94 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 5796 ( 205 ^;4243 !;1348 ?;5796 :)
% 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_143,type,
sP1: ( term > $o ) > $o ).
thf(func_def_145,type,
sP3: ( subst > term > subst > $o ) > $o ).
thf(func_def_147,type,
sP5: ( subst > term > subst > $o ) > $o ).
thf(func_def_148,type,
sP6: ( subst > term > subst > $o ) > $o ).
thf(func_def_149,type,
sP7: ( term > $o ) > $o ).
thf(func_def_150,type,
sP8: ( subst > term > subst > $o ) > $o ).
thf(func_def_151,type,
sP9: ( subst > term > subst > $o ) > $o ).
thf(func_def_152,type,
sP10: ( subst > term > subst > $o ) > $o ).
thf(func_def_157,type,
sP15: ( subst > term > subst > $o ) > $o ).
thf(func_def_160,type,
sP18: ( subst > term > subst > $o ) > $o ).
thf(func_def_161,type,
sP19: ( term > $o ) > $o ).
thf(func_def_167,type,
sP25: ( subst > term > subst > $o ) > $o ).
thf(func_def_172,type,
sP30: ( subst > term > subst > $o ) > $o ).
thf(func_def_174,type,
sP32: ( term > $o ) > $o ).
thf(func_def_175,type,
sP33: ( term > $o ) > $o ).
thf(func_def_178,type,
sP36: ( term > $o ) > $o ).
thf(func_def_179,type,
sP37: ( subst > term > subst > $o ) > $o ).
thf(func_def_183,type,
sP41: ( term > $o ) > $o ).
thf(func_def_185,type,
sP43: ( subst > term > subst > $o ) > $o ).
thf(func_def_186,type,
sP44: ( subst > term > subst > $o ) > $o ).
thf(func_def_188,type,
sP46: ( subst > term > subst > $o ) > $o ).
thf(func_def_189,type,
sP47: ( subst > term > subst > $o ) > $o ).
thf(func_def_190,type,
sP48: ( term > $o ) > $o ).
thf(func_def_194,type,
sP52: ( term > $o ) > $o ).
thf(func_def_195,type,
sP53: ( term > $o ) > $o ).
thf(func_def_199,type,
sP57: ( subst > term > subst > $o ) > $o ).
thf(func_def_201,type,
sK59: term ).
thf(func_def_202,type,
sK60: term ).
thf(func_def_203,type,
sK61: subst > term > subst > $o ).
thf(func_def_204,type,
sK62: ( subst > term > subst > $o ) > subst ).
thf(func_def_205,type,
sK63: ( subst > term > subst > $o ) > subst ).
thf(func_def_206,type,
sK64: ( subst > term > subst > $o ) > subst ).
thf(func_def_207,type,
sK65: ( subst > term > subst > $o ) > term ).
thf(func_def_208,type,
sK66: ( subst > term > subst > $o ) > subst ).
thf(func_def_209,type,
sK67: ( subst > term > subst > $o ) > subst ).
thf(func_def_210,type,
sK68: ( subst > term > subst > $o ) > subst ).
thf(func_def_211,type,
sK69: ( subst > term > subst > $o ) > term ).
thf(func_def_212,type,
sK70: ( subst > term > subst > $o ) > term ).
thf(func_def_213,type,
sK71: ( subst > term > subst > $o ) > term ).
thf(func_def_214,type,
sK72: term > $o ).
thf(func_def_215,type,
sK73: subst > term > subst > $o ).
thf(func_def_216,type,
sK74: term ).
thf(func_def_217,type,
sK75: ( subst > term > term ) > term ).
thf(func_def_218,type,
sK76: ( subst > term > term ) > subst ).
thf(func_def_219,type,
sK77: ( subst > term > term ) > term ).
thf(func_def_220,type,
sK78: ( subst > term > term ) > subst ).
thf(func_def_221,type,
sK79: term > subst > term > term ).
thf(func_def_222,type,
sK80: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_223,type,
sK81: term > ( term > $o ) > term ).
thf(func_def_224,type,
sK82: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_225,type,
sK83: 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: term ).
thf(func_def_229,type,
sK87: ( term > $o ) > subst > term ).
thf(func_def_230,type,
sK88: ( term > $o ) > term ).
thf(func_def_231,type,
sK89: subst ).
thf(func_def_232,type,
sK90: term ).
thf(func_def_233,type,
sK91: term > $o ).
thf(func_def_234,type,
sK92: term > term ).
thf(func_def_235,type,
sK93: term ).
thf(func_def_236,type,
sK94: term > $o ).
thf(func_def_237,type,
sK95: term ).
thf(func_def_238,type,
sK96: term > term ).
thf(func_def_239,type,
sK97: ( term > $o ) > term ).
thf(func_def_240,type,
sK98: ( term > $o ) > term ).
thf(func_def_241,type,
sK99: ( term > $o ) > term ).
thf(func_def_242,type,
sK100: ( term > $o ) > term ).
thf(func_def_243,type,
sK101: ( term > $o ) > term ).
thf(func_def_244,type,
sK102: ( term > $o ) > term ).
thf(func_def_245,type,
sK103: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_246,type,
sK104: ( term > $o ) > term ).
thf(func_def_247,type,
sK105: ( term > $o ) > term ).
thf(func_def_248,type,
sK106: subst > term > subst > $o ).
thf(func_def_249,type,
sK107: ( term > $o ) > term ).
thf(func_def_250,type,
sK108: term > $o ).
thf(func_def_251,type,
sK109: term ).
thf(func_def_252,type,
sK110: term > term ).
thf(func_def_253,type,
sK111: term ).
thf(func_def_254,type,
sK112: ( term > $o ) > term > term ).
thf(func_def_255,type,
sK113: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_256,type,
sK114: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_257,type,
sK115: term ).
thf(func_def_258,type,
sK116: term > $o ).
thf(func_def_259,type,
sK117: subst > term > subst > $o ).
thf(func_def_260,type,
sK118: ( subst > term > term ) > subst ).
thf(func_def_261,type,
sK119: ( subst > term > term ) > term ).
thf(func_def_262,type,
sK120: ( subst > term > term ) > subst ).
thf(func_def_263,type,
sK121: ( subst > term > term ) > term ).
thf(func_def_264,type,
sK122: term ).
thf(func_def_265,type,
sK123: subst > term > subst > $o ).
thf(func_def_266,type,
sK124: ( subst > term > subst > $o ) > term ).
thf(func_def_267,type,
sK125: ( subst > term > subst > $o ) > subst ).
thf(func_def_268,type,
sK126: ( subst > term > subst > $o ) > subst ).
thf(func_def_269,type,
sK127: ( subst > term > subst > $o ) > subst ).
thf(func_def_270,type,
sK128: ( subst > term > subst > $o ) > subst ).
thf(func_def_271,type,
sK129: ( subst > term > subst > $o ) > term ).
thf(func_def_272,type,
sK130: ( subst > term > subst > $o ) > subst ).
thf(func_def_273,type,
sK131: ( subst > term > subst > $o ) > subst ).
thf(func_def_274,type,
sK132: ( subst > term > subst > $o ) > term ).
thf(func_def_275,type,
sK133: ( term > $o ) > term ).
thf(func_def_276,type,
sK134: ( term > $o ) > term ).
thf(func_def_277,type,
sK135: ( subst > term > subst > $o ) > term ).
thf(func_def_278,type,
sK136: ( subst > term > subst > $o ) > term ).
thf(func_def_279,type,
sK137: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_280,type,
sK138: ( subst > term > term ) > subst ).
thf(func_def_281,type,
sK139: ( subst > term > term ) > term ).
thf(func_def_282,type,
sK140: ( subst > term > term ) > subst ).
thf(func_def_283,type,
sK141: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_284,type,
sK142: term ).
thf(func_def_285,type,
sK143: subst > term > subst > $o ).
thf(func_def_286,type,
sK144: ( subst > term > subst > $o ) > term ).
thf(func_def_287,type,
sK145: ( subst > term > subst > $o ) > term ).
thf(func_def_288,type,
sK146: ( subst > term > subst > $o ) > subst ).
thf(func_def_289,type,
sK147: ( subst > term > subst > $o ) > subst ).
thf(func_def_290,type,
sK148: ( subst > term > subst > $o ) > subst ).
thf(func_def_291,type,
sK149: ( subst > term > subst > $o ) > subst ).
thf(func_def_292,type,
sK150: ( subst > term > subst > $o ) > subst ).
thf(func_def_293,type,
sK151: ( subst > term > subst > $o ) > term ).
thf(func_def_294,type,
sK152: ( subst > term > subst > $o ) > subst ).
thf(func_def_295,type,
sK153: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_296,type,
sK154: ( subst > term > term ) > subst ).
thf(func_def_297,type,
sK155: ( subst > term > term ) > subst ).
thf(func_def_298,type,
sK156: ( subst > term > term ) > term ).
thf(func_def_299,type,
sK157: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_300,type,
sK158: ( subst > term > subst > $o ) > term ).
thf(func_def_301,type,
sK159: ( subst > term > subst > $o ) > term ).
thf(func_def_302,type,
sK160: term > $o ).
thf(func_def_303,type,
sK161: subst ).
thf(func_def_304,type,
sK162: term ).
thf(func_def_305,type,
sK163: term > term ).
thf(func_def_306,type,
sK164: subst > ( term > $o ) > term ).
thf(func_def_307,type,
sK165: ( term > $o ) > term ).
thf(func_def_308,type,
sK166: ( term > $o ) > term ).
thf(func_def_309,type,
sK167: ( term > $o ) > term ).
thf(func_def_310,type,
sK168: subst > term > subst > $o ).
thf(func_def_311,type,
sK169: term ).
thf(func_def_312,type,
sK170: ( subst > term > term ) > term ).
thf(func_def_313,type,
sK171: ( subst > term > term ) > subst ).
thf(func_def_314,type,
sK172: ( subst > term > term ) > term ).
thf(func_def_315,type,
sK173: ( subst > term > term ) > subst ).
thf(func_def_316,type,
sK174: term > ( subst > term > subst > $o ) > term ).
thf(func_def_317,type,
sK175: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_318,type,
sK176: subst > term > subst > $o ).
thf(func_def_319,type,
sK177: term ).
thf(func_def_320,type,
sK178: ( subst > term > term ) > term ).
thf(func_def_321,type,
sK179: ( subst > term > term ) > subst ).
thf(func_def_322,type,
sK180: ( subst > term > term ) > term ).
thf(func_def_323,type,
sK181: ( subst > term > term ) > subst ).
thf(func_def_324,type,
sK182: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_325,type,
sK183: term > ( subst > term > subst > $o ) > term ).
thf(func_def_326,type,
sK184: term > $o ).
thf(func_def_327,type,
sK185: subst > term > subst > $o ).
thf(func_def_328,type,
sK186: term ).
thf(func_def_329,type,
sK187: term > ( term > $o ) > term ).
thf(func_def_330,type,
sK188: ( subst > term > subst > $o ) > subst ).
thf(func_def_331,type,
sK189: ( subst > term > subst > $o ) > term ).
thf(func_def_332,type,
sK190: ( subst > term > subst > $o ) > subst ).
thf(func_def_333,type,
sK191: ( subst > term > subst > $o ) > subst ).
thf(func_def_334,type,
sK192: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_335,type,
sK193: ( subst > term > subst > $o ) > subst ).
thf(func_def_336,type,
sK194: ( subst > term > subst > $o ) > term ).
thf(func_def_337,type,
sK195: ( subst > term > subst > $o ) > subst ).
thf(func_def_338,type,
sK196: ( subst > term > subst > $o ) > subst ).
thf(func_def_339,type,
sK197: term ).
thf(func_def_340,type,
sK198: ( subst > term > term ) > subst ).
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 > term > subst > $o ) > term ).
thf(func_def_344,type,
sK202: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_345,type,
sK203: ( term > $o ) > term ).
thf(func_def_346,type,
sK204: ( term > $o ) > term ).
thf(func_def_347,type,
sK205: ( term > $o ) > term ).
thf(func_def_348,type,
sK206: ( term > $o ) > term ).
thf(func_def_349,type,
sK207: term ).
thf(func_def_350,type,
sK208: term > $o ).
thf(func_def_351,type,
sK209: subst ).
thf(func_def_352,type,
sK210: term ).
thf(func_def_353,type,
sK211: subst ).
thf(func_def_354,type,
sK212: term ).
thf(func_def_355,type,
sK213: subst ).
thf(func_def_356,type,
sK214: subst ).
thf(func_def_357,type,
sK215: term ).
thf(func_def_358,type,
sK216: term > $o ).
thf(func_def_359,type,
sK217: term > term ).
thf(func_def_360,type,
sK218: ( term > $o ) > subst > term ).
thf(func_def_361,type,
sK219: ( term > $o ) > term ).
thf(func_def_362,type,
sK220: ( term > $o ) > term ).
thf(func_def_363,type,
sK221: ( term > $o ) > term ).
thf(func_def_364,type,
sK222: ( term > $o ) > term ).
thf(func_def_365,type,
sK223: ( term > $o ) > term ).
thf(func_def_366,type,
sK224: term ).
thf(func_def_367,type,
sK225: term > $o ).
thf(func_def_368,type,
sK226: subst > term > subst > $o ).
thf(func_def_369,type,
sK227: term ).
thf(func_def_370,type,
sK228: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_371,type,
sK229: ( subst > term > subst > $o ) > term ).
thf(func_def_372,type,
sK230: ( subst > term > subst > $o ) > subst ).
thf(func_def_373,type,
sK231: ( subst > term > subst > $o ) > subst ).
thf(func_def_374,type,
sK232: ( subst > term > subst > $o ) > subst ).
thf(func_def_375,type,
sK233: ( subst > term > subst > $o ) > subst ).
thf(func_def_376,type,
sK234: ( subst > term > subst > $o ) > subst ).
thf(func_def_377,type,
sK235: ( subst > term > subst > $o ) > term ).
thf(func_def_378,type,
sK236: ( subst > term > subst > $o ) > subst ).
thf(func_def_379,type,
sK237: ( subst > term > subst > $o ) > term ).
thf(func_def_380,type,
sK238: ( subst > term > subst > $o ) > term ).
thf(func_def_381,type,
sK239: subst > term > subst > $o ).
thf(func_def_382,type,
sK240: term > $o ).
thf(func_def_383,type,
sK241: term ).
thf(func_def_384,type,
sK242: ( subst > term > term ) > term ).
thf(func_def_385,type,
sK243: ( subst > term > term ) > subst ).
thf(func_def_386,type,
sK244: ( subst > term > term ) > subst ).
thf(func_def_387,type,
sK245: ( subst > term > term ) > term ).
thf(func_def_388,type,
sK246: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_389,type,
sK247: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_390,type,
sK248: term > ( term > $o ) > term ).
thf(func_def_391,type,
sK249: term ).
thf(func_def_392,type,
sK250: subst > term > subst > $o ).
thf(func_def_393,type,
sK251: ( subst > term > term ) > term ).
thf(func_def_394,type,
sK252: ( subst > term > term ) > subst ).
thf(func_def_395,type,
sK253: ( subst > term > term ) > term ).
thf(func_def_396,type,
sK254: ( subst > term > term ) > subst ).
thf(func_def_397,type,
sK255: term ).
thf(func_def_398,type,
sK256: ( subst > term > subst > $o ) > term > term ).
thf(func_def_399,type,
sK257: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_400,type,
sK258: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_401,type,
sK259: ( subst > term > subst > $o ) > term > term ).
thf(func_def_402,type,
sK260: term ).
thf(func_def_403,type,
sK261: subst > term > subst > $o ).
thf(func_def_404,type,
sK262: ( subst > term > term ) > term ).
thf(func_def_405,type,
sK263: ( subst > term > term ) > subst ).
thf(func_def_406,type,
sK264: ( subst > term > term ) > term ).
thf(func_def_407,type,
sK265: ( subst > term > term ) > subst ).
thf(func_def_408,type,
sK266: term > $o ).
thf(func_def_409,type,
sK267: subst > term > subst > $o ).
thf(func_def_410,type,
sK268: term ).
thf(func_def_411,type,
sK269: ( subst > term > subst > $o ) > subst ).
thf(func_def_412,type,
sK270: ( subst > term > subst > $o ) > subst ).
thf(func_def_413,type,
sK271: ( subst > term > subst > $o ) > term ).
thf(func_def_414,type,
sK272: ( subst > term > subst > $o ) > subst ).
thf(func_def_415,type,
sK273: ( subst > term > subst > $o ) > ( term > $o ) > term ).
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: term > ( term > $o ) > term ).
thf(func_def_421,type,
sK279: ( subst > term > term ) > subst ).
thf(func_def_422,type,
sK280: ( subst > term > term ) > subst ).
thf(func_def_423,type,
sK281: ( subst > term > term ) > term ).
thf(func_def_424,type,
sK282: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_425,type,
sK283: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_426,type,
sK284: term ).
thf(func_def_427,type,
sK285: subst > term > subst > $o ).
thf(func_def_428,type,
sK286: term > ( subst > term > term ) > subst ).
thf(func_def_429,type,
sK287: term > ( subst > term > term ) > term ).
thf(func_def_430,type,
sK288: term > ( subst > term > term ) > subst ).
thf(func_def_431,type,
sK289: term > ( subst > term > term ) > subst ).
thf(func_def_432,type,
sK290: term > ( subst > term > term ) > term ).
thf(func_def_433,type,
sK291: term > ( subst > term > term ) > subst ).
thf(func_def_434,type,
sK292: ( subst > term > term ) > term ).
thf(func_def_435,type,
sK293: ( subst > term > subst > $o ) > term > term ).
thf(func_def_436,type,
sK294: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_437,type,
sK295: ( subst > term > subst > $o ) > term ).
thf(func_def_438,type,
sK296: subst ).
thf(func_def_439,type,
sK297: subst > term > term ).
thf(func_def_440,type,
sK298: term ).
thf(func_def_441,type,
sK299: subst > term > term ).
thf(func_def_442,type,
sK300: term ).
thf(func_def_443,type,
sK301: term ).
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: ( subst > term > term ) > subst ).
thf(func_def_449,type,
sK307: ( subst > term > term ) > subst ).
thf(func_def_450,type,
sK308: term ).
thf(func_def_451,type,
sK309: subst ).
thf(func_def_452,type,
sK310: subst ).
thf(func_def_453,type,
sK311: term ).
thf(func_def_454,type,
sK312: ( term > $o ) > term ).
thf(func_def_455,type,
sK313: ( term > $o ) > term ).
thf(func_def_456,type,
sK314: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_457,type,
sK315: term > $o ).
thf(func_def_458,type,
sK316: term ).
thf(func_def_459,type,
sK317: term > term ).
thf(func_def_460,type,
sK318: subst > term > subst > $o ).
thf(func_def_461,type,
sK319: ( term > $o ) > term ).
thf(func_def_462,type,
sK320: term ).
thf(func_def_463,type,
sK321: subst > term > subst > $o ).
thf(func_def_464,type,
sK322: term ).
thf(func_def_465,type,
sK323: ( subst > term > term ) > term ).
thf(func_def_466,type,
sK324: ( subst > term > term ) > subst ).
thf(func_def_467,type,
sK325: ( subst > term > term ) > subst ).
thf(func_def_468,type,
sK326: ( subst > term > term ) > term ).
thf(func_def_469,type,
sK327: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_470,type,
sK328: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_471,type,
sK329: ( term > $o ) > term > term ).
thf(func_def_472,type,
sK330: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_473,type,
sK331: subst > term > subst > $o ).
thf(func_def_474,type,
sK332: term ).
thf(func_def_475,type,
sK333: term > $o ).
thf(func_def_476,type,
sK334: ( subst > term > term ) > term ).
thf(func_def_477,type,
sK335: ( subst > term > term ) > subst ).
thf(func_def_478,type,
sK336: ( subst > term > term ) > term ).
thf(func_def_479,type,
sK337: ( subst > term > term ) > subst ).
thf(func_def_480,type,
sK338: ( term > $o ) > term ).
thf(func_def_481,type,
sK339: ( term > $o ) > term ).
thf(func_def_482,type,
sK340: ( subst > term > subst > $o ) > term ).
thf(func_def_483,type,
sK341: ( subst > term > subst > $o ) > term ).
thf(func_def_484,type,
sK342: term ).
thf(func_def_485,type,
sK343: subst > term > subst > $o ).
thf(func_def_486,type,
sK344: ( subst > term > subst > $o ) > subst ).
thf(func_def_487,type,
sK345: ( subst > term > subst > $o ) > subst ).
thf(func_def_488,type,
sK346: ( subst > term > subst > $o ) > subst ).
thf(func_def_489,type,
sK347: ( subst > term > subst > $o ) > term ).
thf(func_def_490,type,
sK348: ( subst > term > subst > $o ) > term > term ).
thf(func_def_491,type,
sK349: ( subst > term > subst > $o ) > term ).
thf(func_def_492,type,
sK350: ( subst > term > subst > $o ) > subst ).
thf(func_def_493,type,
sK351: ( subst > term > subst > $o ) > subst ).
thf(func_def_494,type,
sK352: ( subst > term > subst > $o ) > subst ).
thf(func_def_495,type,
sK353: term ).
thf(func_def_496,type,
sK354: ( subst > term > subst > $o ) > term > term ).
thf(func_def_497,type,
sK355: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_498,type,
sK356: term ).
thf(func_def_499,type,
sK357: subst > term > subst > $o ).
thf(func_def_500,type,
sK358: ( subst > term > term ) > subst ).
thf(func_def_501,type,
sK359: ( subst > term > term ) > subst ).
thf(func_def_502,type,
sK360: ( subst > term > term ) > term ).
thf(func_def_503,type,
sK361: ( subst > term > term ) > term ).
thf(func_def_504,type,
sK362: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_505,type,
sK363: ( subst > term > term ) > term ).
thf(func_def_506,type,
sK364: ( subst > term > term ) > subst ).
thf(func_def_507,type,
sK365: ( subst > term > term ) > subst ).
thf(func_def_508,type,
sK366: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_509,type,
sK367: ( subst > term > subst > $o ) > subst ).
thf(func_def_510,type,
sK368: ( subst > term > subst > $o ) > subst ).
thf(func_def_511,type,
sK369: ( subst > term > subst > $o ) > term ).
thf(func_def_512,type,
sK370: ( subst > term > subst > $o ) > subst ).
thf(func_def_513,type,
sK371: ( subst > term > subst > $o ) > subst ).
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: term ).
thf(func_def_518,type,
sK376: term ).
thf(func_def_519,type,
sK377: subst > term > subst > $o ).
thf(func_def_520,type,
sK378: ( term > $o ) > term ).
thf(func_def_521,type,
sK379: ( term > $o ) > term ).
thf(func_def_522,type,
sK380: term ).
thf(func_def_523,type,
sK381: term > $o ).
thf(func_def_524,type,
sK382: term > term ).
thf(func_def_525,type,
sK383: subst > term > subst > $o ).
thf(func_def_526,type,
sK384: term ).
thf(func_def_527,type,
sK385: ( subst > term > subst > $o ) > subst ).
thf(func_def_528,type,
sK386: ( subst > term > subst > $o ) > term ).
thf(func_def_529,type,
sK387: ( subst > term > subst > $o ) > subst ).
thf(func_def_530,type,
sK388: ( subst > term > subst > $o ) > subst ).
thf(func_def_531,type,
sK389: ( subst > term > subst > $o ) > term ).
thf(func_def_532,type,
sK390: ( subst > term > subst > $o ) > term ).
thf(func_def_533,type,
sK391: ( subst > term > subst > $o ) > subst ).
thf(func_def_534,type,
sK392: ( subst > term > subst > $o ) > subst ).
thf(func_def_535,type,
sK393: ( subst > term > subst > $o ) > subst ).
thf(func_def_536,type,
sK394: ( subst > term > subst > $o ) > subst ).
thf(func_def_537,type,
sK395: ( subst > term > subst > $o ) > term ).
thf(func_def_538,type,
sK396: ( subst > term > subst > $o ) > subst ).
thf(func_def_539,type,
sK397: ( subst > term > subst > $o ) > subst ).
thf(func_def_540,type,
sK398: term > ( subst > term > subst > $o ) > term ).
thf(func_def_541,type,
sK399: ( subst > term > subst > $o ) > subst ).
thf(func_def_542,type,
sK400: ( subst > term > subst > $o ) > term ).
thf(func_def_543,type,
sK401: ( subst > term > subst > $o ) > subst ).
thf(func_def_544,type,
sK402: ( subst > term > subst > $o ) > subst ).
thf(func_def_545,type,
sK403: subst > term > subst > $o ).
thf(func_def_546,type,
sK404: term ).
thf(func_def_547,type,
sK405: ( subst > term > subst > $o ) > term ).
thf(func_def_548,type,
sK406: ( subst > term > subst > $o ) > term ).
thf(func_def_549,type,
sK407: ( subst > term > term ) > term ).
thf(func_def_550,type,
sK408: ( subst > term > term ) > subst ).
thf(func_def_551,type,
sK409: ( subst > term > term ) > subst ).
thf(func_def_552,type,
sK410: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_553,type,
sK411: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_554,type,
sK412: ( subst > term > subst > $o ) > term ).
thf(func_def_555,type,
sK413: ( subst > term > subst > $o ) > term ).
thf(func_def_556,type,
sK414: ( term > $o ) > term ).
thf(func_def_557,type,
sK415: ( term > $o ) > term ).
thf(func_def_558,type,
sK416: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_559,type,
sK417: ( subst > term > term ) > term ).
thf(func_def_560,type,
sK418: ( subst > term > term ) > subst ).
thf(func_def_561,type,
sK419: ( subst > term > term ) > subst ).
thf(func_def_562,type,
sK420: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_563,type,
sK421: ( subst > term > term ) > term ).
thf(func_def_564,type,
sK422: ( subst > term > term ) > subst ).
thf(func_def_565,type,
sK423: ( subst > term > term ) > subst ).
thf(func_def_566,type,
sK424: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_567,type,
sK425: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_568,type,
sK426: term ).
thf(func_def_569,type,
sK427: subst > term > subst > $o ).
thf(func_def_570,type,
sK428: ( subst > term > subst > $o ) > subst ).
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 ) > term > term ).
thf(func_def_575,type,
sK433: ( subst > term > subst > $o ) > subst ).
thf(func_def_576,type,
sK434: ( subst > term > subst > $o ) > term ).
thf(func_def_577,type,
sK435: ( subst > term > subst > $o ) > subst ).
thf(func_def_578,type,
sK436: ( subst > term > subst > $o ) > subst ).
thf(func_def_579,type,
sK437: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_580,type,
sK438: ( subst > term > term ) > term ).
thf(func_def_581,type,
sK439: ( subst > term > term ) > subst ).
thf(func_def_582,type,
sK440: ( subst > term > term ) > subst ).
thf(func_def_583,type,
sK441: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_584,type,
sK442: term ).
thf(func_def_585,type,
sK443: term > $o ).
thf(func_def_586,type,
sK444: term > term ).
thf(func_def_587,type,
sK445: ( term > $o ) > term ).
thf(func_def_588,type,
sK446: ( term > $o ) > term ).
thf(func_def_589,type,
sK447: ( term > $o ) > term ).
thf(func_def_590,type,
sK448: ( term > $o ) > term ).
thf(func_def_591,type,
sK449: subst > term > subst > $o ).
thf(func_def_592,type,
sK450: term ).
thf(func_def_593,type,
sK451: ( subst > term > term ) > term ).
thf(func_def_594,type,
sK452: ( subst > term > term ) > term ).
thf(func_def_595,type,
sK453: ( subst > term > term ) > subst ).
thf(func_def_596,type,
sK454: ( subst > term > term ) > subst ).
thf(func_def_597,type,
sK455: 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: term ).
thf(func_def_601,type,
sK459: term ).
thf(func_def_602,type,
sK460: term ).
thf(func_def_603,type,
sK461: term ).
thf(func_def_604,type,
sK462: subst ).
thf(func_def_605,type,
sK463: subst ).
thf(func_def_606,type,
sK464: subst ).
thf(func_def_607,type,
sK465: subst ).
thf(func_def_608,type,
sK466: term ).
thf(func_def_609,type,
sK467: term > subst > term > term ).
thf(func_def_610,type,
sK468: term ).
thf(func_def_611,type,
sK469: term ).
thf(func_def_612,type,
sK470: subst > term > subst > $o ).
thf(func_def_613,type,
sK471: ( term > $o ) > term ).
thf(func_def_614,type,
sK472: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_615,type,
sK473: subst ).
thf(func_def_616,type,
sK474: term ).
thf(func_def_617,type,
sK475: subst ).
thf(func_def_618,type,
sK476: term ).
thf(func_def_619,type,
sK477: term ).
thf(func_def_620,type,
sK478: term > $o ).
thf(func_def_621,type,
sK479: term > $o ).
thf(func_def_622,type,
sK480: subst ).
thf(func_def_623,type,
sK481: term ).
thf(func_def_624,type,
sK482: term ).
thf(func_def_625,type,
sK483: term > $o ).
thf(func_def_626,type,
sK484: term ).
thf(func_def_627,type,
sK485: ( term > $o ) > term ).
thf(func_def_628,type,
sK486: ( term > $o ) > term ).
thf(func_def_629,type,
sK487: subst ).
thf(func_def_630,type,
sK488: subst ).
thf(func_def_631,type,
sK489: term ).
thf(func_def_632,type,
sK490: term ).
thf(func_def_633,type,
sK491: term > $o ).
thf(func_def_634,type,
sK492: subst ).
thf(func_def_635,type,
sK493: ( term > $o ) > term ).
thf(func_def_636,type,
sK494: subst > ( term > $o ) > term > term > $o ).
thf(func_def_637,type,
sK495: term ).
thf(func_def_638,type,
sK496: term ).
thf(func_def_639,type,
sK497: term ).
thf(func_def_640,type,
sK498: term ).
thf(func_def_641,type,
sK499: term ).
thf(func_def_642,type,
sK500: term ).
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: subst > term > term ).
thf(func_def_647,type,
sK505: term ).
thf(func_def_648,type,
sK506: term ).
thf(func_def_649,type,
sK507: ( subst > term > term ) > subst ).
thf(func_def_650,type,
sK508: ( subst > term > term ) > term ).
thf(func_def_651,type,
sK509: ( subst > term > term ) > subst ).
thf(func_def_652,type,
sK510: term ).
thf(func_def_653,type,
sK511: term ).
thf(func_def_654,type,
sK512: term ).
thf(func_def_655,type,
sK513: term ).
thf(func_def_656,type,
sK514: subst ).
thf(func_def_657,type,
sK515: subst ).
thf(func_def_658,type,
sK516: ( term > $o ) > term > subst > term > $o ).
thf(func_def_659,type,
sK517: subst ).
thf(func_def_660,type,
sK518: term ).
thf(func_def_661,type,
sK519: term > $o ).
thf(func_def_662,type,
sK520: ( term > $o ) > term ).
thf(func_def_663,type,
sK521: term ).
thf(func_def_664,type,
sK522: term > subst > term > term ).
thf(func_def_665,type,
sK523: term > $o ).
thf(func_def_666,type,
sK524: term ).
thf(func_def_667,type,
sK525: subst ).
thf(func_def_668,type,
sK526: term ).
thf(func_def_669,type,
sK527: term > $o ).
thf(func_def_670,type,
sK528: term > subst > term > term ).
thf(func_def_671,type,
sK529: term ).
thf(func_def_672,type,
sK530: ( subst > term > term ) > subst ).
thf(func_def_673,type,
sK531: ( subst > term > term ) > term ).
thf(func_def_674,type,
sK532: term > ( term > $o ) > subst > ( term > $o ) > term ).
thf(func_def_675,type,
sK533: term > $o ).
thf(func_def_676,type,
sK534: subst ).
thf(func_def_677,type,
sK535: term > $o ).
thf(func_def_678,type,
sK536: term ).
thf(func_def_679,type,
sK537: term > subst > term > term ).
thf(func_def_680,type,
sK538: term ).
thf(func_def_681,type,
sK539: ( subst > term > term ) > term ).
thf(func_def_682,type,
sK540: ( subst > term > term ) > subst ).
thf(func_def_683,type,
sK541: ( subst > term > term ) > subst ).
thf(func_def_684,type,
sK542: ( subst > term > term ) > subst ).
thf(func_def_685,type,
sK543: ( subst > term > term ) > term ).
thf(func_def_686,type,
sK544: ( subst > term > term ) > subst ).
thf(func_def_687,type,
sK545: ( subst > term > term ) > term ).
thf(func_def_688,type,
sK546: ( subst > term > term ) > subst ).
thf(func_def_689,type,
sK547: subst > term > term ).
thf(func_def_690,type,
sK548: subst ).
thf(func_def_691,type,
sK549: term ).
thf(func_def_692,type,
sK550: subst > term > term ).
thf(func_def_693,type,
sK551: term ).
thf(func_def_694,type,
sK552: term ).
thf(func_def_695,type,
sK553: subst ).
thf(func_def_696,type,
sK554: term ).
thf(func_def_697,type,
sK555: term ).
thf(func_def_698,type,
sK556: term ).
thf(func_def_699,type,
sK557: term ).
thf(func_def_700,type,
sK558: term ).
thf(func_def_701,type,
sK559: term ).
thf(func_def_702,type,
sK560: term ).
thf(func_def_703,type,
sK561: term ).
thf(func_def_704,type,
sK562: subst > term > subst > $o ).
thf(func_def_705,type,
sK563: ( term > $o ) > term ).
thf(func_def_706,type,
sK564: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_707,type,
sK565: term ).
thf(func_def_708,type,
sK566: subst > term > subst > $o ).
thf(func_def_709,type,
sK567: ( subst > term > subst > $o ) > term ).
thf(func_def_710,type,
sK568: ( subst > term > subst > $o ) > term ).
thf(func_def_711,type,
sK569: term ).
thf(func_def_712,type,
sK570: subst ).
thf(func_def_713,type,
sK571: subst > d_subst ).
thf(func_def_714,type,
sK572: subst ).
thf(func_def_715,type,
sK573: term ).
thf(func_def_716,type,
sK574: term ).
thf(func_def_717,type,
sK575: term ).
thf(func_def_718,type,
sK576: term ).
thf(func_def_719,type,
sK577: term ).
thf(func_def_720,type,
sK578: term ).
thf(func_def_721,type,
sK579: term > d_term ).
thf(func_def_722,type,
sK580: term ).
thf(func_def_723,type,
sK581: term ).
thf(func_def_724,type,
sK582: term ).
thf(func_def_725,type,
sK583: term ).
thf(func_def_726,type,
sK584: subst > ( term > $o ) > ( term > $o ) > term > term ).
thf(func_def_727,type,
sK585: term ).
thf(func_def_728,type,
sK586: subst ).
thf(func_def_729,type,
sK587: term ).
thf(func_def_730,type,
sK588: term > $o ).
thf(func_def_731,type,
sK589: term > $o ).
thf(func_def_732,type,
sK590: subst ).
thf(func_def_736,type,
sK592: term > $o ).
thf(func_def_737,type,
sK593: term ).
thf(func_def_738,type,
sK594: subst ).
thf(func_def_739,type,
sK595: term > $o ).
thf(func_def_740,type,
sK596: term ).
thf(f4413,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f4412]) ).
thf(f4412,plain,
$true = $false,
inference(backward_demodulation,[status(thm)],[f1712,f4410]) ).
thf(f4410,plain,
! [X1: term] :
( $false
= ( var @ X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f4409]) ).
thf(f4409,plain,
! [X1: term] :
( ( $false
= ( var @ X1 ) )
| ( $true = $false ) ),
inference(backward_demodulation,[status(thm)],[f1708,f4408]) ).
thf(f4408,plain,
! [X1: term] :
( $false
= ( sK595 @ X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f4407]) ).
thf(f4407,plain,
! [X1: term] :
( ( $false
= ( sK595 @ X1 ) )
| ( $true = $false ) ),
inference(forward_demodulation,[status(thm)],[f4406,f4404]) ).
thf(f4404,plain,
( $false
= ( sK592 @ ( d2term @ d_one ) ) ),
inference(forward_demodulation,[status(thm)],[f1711,f2761]) ).
thf(f2761,plain,
! [X248: term] :
( ( d2term @ d_one )
= X248 ),
inference(backward_demodulation,[status(thm)],[f1046,f1240]) ).
thf(f1240,plain,
! [X18: d_term] : ( d_one = X18 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f545,plain,
( ( hoaslamnotvar = $true )
& ( ( apinj2 = $true )
| ( ( ( ap @ sK459 @ sK457 )
= ( ap @ sK460 @ sK458 ) )
& ( sK457 != sK458 ) ) )
& ( ! [X4: term,X5: term,X6: term,X7: term] :
( ( ( ap @ X7 @ X5 )
!= ( ap @ X6 @ X4 ) )
| ( X4 = X5 ) )
| ( apinj2 != $true ) )
& ( ulamvarind = $true )
& ( axapp = $true )
& ( hoasinduction_lem1 = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( 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 ) ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X8: term] :
( ( sub @ X8 @ id )
= X8 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ( sK461
!= ( sub @ sK461 @ id ) ) )
& ( termmset_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( substmonoid_lthm = $true )
| ( ( ( sK462
!= ( comp @ sK462 @ id ) )
| ( ( comp @ id @ sK463 )
!= sK463 ) )
& ! [X12: subst] :
( ( comp @ id @ X12 )
= X12 )
& ! [X13: subst] :
( ( comp @ X13 @ id )
= X13 ) ) )
& ( ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X15: subst] :
( ( comp @ id @ X15 )
= X15 ) )
| ( ( comp @ id @ sK464 )
!= sK464 )
| ( ( comp @ sK465 @ id )
!= sK465 )
| ( substmonoid_lthm != $true ) )
& ! [X18: d_term] : ( d_one = X18 )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP58 ) )
& ( ( $true = sP58 )
| ( hoasinduction_lem2 != $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP56 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP56 ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( induction2_lthm = $true )
| ( $true != sP55 ) )
& ( ( $true = sP55 )
| ( induction2_lthm != $true ) )
& ( ( sP51 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP51 != $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoaslamnotap_gthm = $true )
& ! [X19: d_subst,X20: d_subst] :
( ( ( d2subst @ X19 )
!= ( d2subst @ X20 ) )
| ( X19 = X20 ) )
& ( ( hoasinduction_lem3b = $true )
| ! [X22: subst > term > term] :
( ( X22 @ sh @ one )
!= ( sub @ sK466 @ ( push @ one @ sh ) ) ) )
& ( ! [X23: term] :
( ( sK467 @ X23 @ sh @ one )
= ( sub @ X23 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( shinj = $true )
| ( ( ( sub @ sK469 @ sh )
= ( sub @ sK468 @ sh ) )
& ( sK469 != sK468 ) ) )
& ( ! [X27: term,X28: term] :
( ( ( sub @ X28 @ sh )
!= ( sub @ X27 @ sh ) )
| ( X27 = X28 ) )
| ( shinj != $true ) )
& ( ( $true = sP45 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP45 ) )
& ( ( induction2lem = $true )
| ( $true != sP42 ) )
& ( ( $true = sP42 )
| ( induction2lem != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ! [X30: term > $o] :
( ( sK470 @ id @ ( sK471 @ X30 ) @ id )
!= ( X30 @ ( sK471 @ X30 ) ) ) )
& ( ! [X32: subst > term > subst > $o,X34: term] :
( ( sK472 @ X32 @ X34 )
= ( X32 @ id @ X34 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP40 ) )
& ( ( $true = sP40 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( ( ( sub @ one @ ( push @ sK474 @ sK473 ) )
!= sK474 )
| ( ( comp @ sh @ ( push @ sK476 @ sK475 ) )
!= sK475 )
| ! [X39: term,X40: term > $o,X41: term > $o,X42: subst,X43: term] :
( ( $true
= ( X41 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) )
| ( $true
!= ( X40 @ X43 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X44: subst,X45: term] :
( ( sub @ one @ ( push @ X45 @ X44 ) )
= X45 )
& ! [X46: subst,X47: term] :
( ( comp @ sh @ ( push @ X47 @ X46 ) )
= X46 )
& ( $true
!= ( sK479 @ ( sub @ sK481 @ ( push @ sK477 @ sK480 ) ) ) )
& ( $true
= ( sK478 @ sK481 ) ) ) )
& ( ( sP39 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP39 != $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X53: term] :
( ( sub @ X53 @ id )
= X53 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ( sK482
!= ( sub @ sK482 @ id ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP38 ) )
& ( ( $true = sP38 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( axabs = $true )
& ( hoasapinj2_gthm = $true )
& ( lamnotvar = $true )
& ( ulamvarsh = $true )
& ! [X55: d_subst] : ( d_id = X55 )
& ( ( induction = $true )
| ( ( $true
!= ( sK483 @ sK484 ) )
& ! [X58: term] :
( ( $true
= ( sK483 @ X58 ) )
| ( $true
!= ( var @ X58 ) ) )
& ( $true
!= ( sP36 @ sK483 ) )
& ! [X59: term] :
( ( $true
= ( sK483 @ ( lam @ X59 ) ) )
| ( $true
!= ( sK483 @ X59 ) ) ) ) )
& ( ! [X60: term > $o,X61: term] :
( ( $true
= ( X60 @ X61 ) )
| ( ( $true
!= ( X60 @ ( sK485 @ X60 ) ) )
& ( ( var @ ( sK485 @ X60 ) )
= $true ) )
| ( $true
= ( sP36 @ X60 ) )
| ( ( $true
!= ( X60 @ ( lam @ ( sK486 @ X60 ) ) ) )
& ( $true
= ( X60 @ ( sK486 @ X60 ) ) ) ) )
| ( induction != $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( ( axidr = $true )
| ( sK487
!= ( comp @ sK487 @ id ) ) )
& ( ! [X65: subst] :
( ( comp @ X65 @ id )
= X65 )
| ( axidr != $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( $true = sP35 )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( $true != sP35 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP31 ) )
& ( ( $true = sP31 )
| ( hoasinduction_lem2v2 != $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_gthm = $true )
& ( induction2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem3v2a = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3v2a != $true ) )
& ( axclos = $true )
& ( lamnotap != $true )
& ( ( axvarcons = $true )
| ( sK489
!= ( sub @ one @ ( push @ sK489 @ sK488 ) ) ) )
& ( ! [X68: subst,X69: term] :
( ( sub @ one @ ( push @ X69 @ X68 ) )
= X69 )
| ( axvarcons != $true ) )
& ( axvarshift = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
| ! [X73: term > $o] :
( ( X73 @ ( sK493 @ X73 ) )
!= ( sK491 @ ( sub @ ( sK493 @ X73 ) @ ( push @ sK490 @ sK492 ) ) ) ) )
& ( ! [X75: term,X76: term > $o,X77: subst,X79: term] :
( ( X76 @ ( sub @ X79 @ ( push @ X75 @ X77 ) ) )
= ( sK494 @ X77 @ X76 @ X75 @ X79 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X80: term,X81: term,X82: term,X83: term] :
( ( ( ap @ X83 @ X82 )
!= ( ap @ X80 @ X81 ) )
| ( X80 = X83 ) )
& ( ( ap @ ( sub @ sK497 @ id ) @ sK496 )
= ( ap @ ( sub @ sK495 @ id ) @ sK498 ) )
& ( sK495 != sK497 )
& ! [X88: term] :
( ( sub @ X88 @ id )
= X88 ) ) )
& ( ( ( ( ap @ sK499 @ sK500 )
= ( ap @ sK502 @ sK501 ) )
& ( sK502 != sK499 ) )
| ! [X93: term,X94: term,X95: term,X96: term] :
( ( ( ap @ ( sub @ X95 @ id ) @ X94 )
!= ( ap @ ( sub @ X93 @ id ) @ X96 ) )
| ( X93 = X95 ) )
| ( ( sub @ sK503 @ id )
!= sK503 )
| ( hoasapinj1_lthm != $true ) )
& ( ( $true = sP28 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( $true != sP28 ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP26 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP26 != $true ) )
& ( ( hoaslamnotap = $true )
| ( ! [X101: subst,X102: term,X103: subst] :
( ( sK504 @ ( comp @ X101 @ X103 ) @ ( sub @ X102 @ X103 ) )
= ( sub @ ( sK504 @ X101 @ X102 ) @ X103 ) )
& ( ( ap @ ( sub @ sK505 @ id ) @ sK506 )
= ( lam @ ( sK504 @ sh @ one ) ) ) ) )
& ( ! [X104: subst > term > term,X105: term,X106: term] :
( ( ( X104 @ ( comp @ ( sK507 @ X104 ) @ ( sK509 @ X104 ) ) @ ( sub @ ( sK508 @ X104 ) @ ( sK509 @ X104 ) ) )
!= ( sub @ ( X104 @ ( sK507 @ X104 ) @ ( sK508 @ X104 ) ) @ ( sK509 @ X104 ) ) )
| ( ( lam @ ( X104 @ sh @ one ) )
!= ( ap @ ( sub @ X105 @ id ) @ X106 ) ) )
| ( hoaslamnotap != $true ) )
& ( pushprop = $true )
& ( pushprop_lthm = $true )
& ( ! [X110: term,X111: term,X112: term,X113: term] :
( ( ( ap @ ( sub @ X113 @ id ) @ X111 )
!= ( ap @ ( sub @ X112 @ id ) @ X110 ) )
| ( X112 = X113 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ( ( ( ap @ ( sub @ sK512 @ id ) @ sK510 )
= ( ap @ ( sub @ sK513 @ id ) @ sK511 ) )
& ( sK512 != sK513 ) ) )
& ( apnotvar = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ( $true = sP24 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP24 ) )
& ( ( $true = sP23 )
| ( hoaslaminj_lthm != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP23 ) )
& ( pushprop_lem2v2 = $true )
& ( ( substmonoid = $true )
| ( sK514
!= ( comp @ sK514 @ id ) )
| ( sK515
!= ( comp @ id @ sK515 ) ) )
& ( ( ! [X120: subst] :
( ( comp @ X120 @ id )
= X120 )
& ! [X121: subst] :
( ( comp @ id @ X121 )
= X121 ) )
| ( substmonoid != $true ) )
& ( ! [X122: subst,X123: term,X124: term > $o,X126: term] :
( ( X124 @ ( sub @ X126 @ ( push @ X123 @ X122 ) ) )
= ( sK516 @ X124 @ X123 @ X122 @ X126 ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ! [X130: term > $o] :
( ( X130 @ ( sK520 @ X130 ) )
!= ( sK519 @ ( sub @ ( sK520 @ X130 ) @ ( push @ sK518 @ sK517 ) ) ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ! [X133: subst > term > term] :
( ( sub @ sK521 @ ( push @ one @ sh ) )
!= ( X133 @ sh @ one ) ) )
& ( ! [X134: term] :
( ( sK522 @ X134 @ sh @ one )
= ( sub @ X134 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( $true = sP22 )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( $true != sP22 ) )
& ( ! [X136: term > $o,X137: term,X138: subst,X139: term,X140: term > $o] :
( ( $true
= ( X140 @ ( sub @ X137 @ ( push @ X139 @ X138 ) ) ) )
| ( $true
!= ( X136 @ X137 ) ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ( ( $true
!= ( sK527 @ ( sub @ sK524 @ ( push @ sK526 @ sK525 ) ) ) )
& ( $true
= ( sK523 @ sK524 ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X146: term,X148: subst,X149: term] :
( ( sub @ X146 @ ( push @ X149 @ X148 ) )
= ( sK528 @ X146 @ X148 @ X149 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ! [X151: subst > term > term] :
( ( sub @ sK529 @ ( push @ ( sK531 @ X151 ) @ ( sK530 @ X151 ) ) )
!= ( X151 @ ( sK530 @ X151 ) @ ( sK531 @ X151 ) ) ) )
& ( ! [X154: term > $o,X155: subst,X156: term > $o,X157: term] :
( ( $true
= ( X156 @ one ) )
| ( $true
!= ( X154 @ X157 ) )
| ( ( X154 @ ( sub @ ( sK532 @ X157 @ X156 @ X155 @ X154 ) @ ( push @ X157 @ X155 ) ) )
!= ( X156 @ ( sK532 @ X157 @ X156 @ X155 @ X154 ) ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ( ( $true
!= ( sK535 @ one ) )
& ( $true
= ( sK533 @ sK536 ) )
& ! [X163: term] :
( ( sK533 @ ( sub @ X163 @ ( push @ sK536 @ sK534 ) ) )
= ( sK535 @ X163 ) ) ) )
& ( ulamvar1 != $true )
& ( axmap = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslamnotvar_lthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X164: term,X166: term,X167: subst] :
( ( sub @ X164 @ ( push @ X166 @ X167 ) )
= ( sK537 @ X164 @ X167 @ X166 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ! [X169: subst > term > term] :
( ( X169 @ ( sK540 @ X169 ) @ ( sK539 @ X169 ) )
!= ( sub @ sK538 @ ( push @ ( sK539 @ X169 ) @ ( sK540 @ X169 ) ) ) ) )
& ( ! [X172: subst > term > term,X173: subst,X174: term,X175: subst > term > term] :
( ( ( X172 @ X173 @ X174 )
= ( X175 @ X173 @ X174 ) )
| ( ( lam @ ( X172 @ sh @ one ) )
!= ( lam @ ( X175 @ sh @ one ) ) )
| ( ( sub @ ( X172 @ ( sK541 @ X172 ) @ ( sK543 @ X172 ) ) @ ( sK542 @ X172 ) )
!= ( X172 @ ( comp @ ( sK541 @ X172 ) @ ( sK542 @ X172 ) ) @ ( sub @ ( sK543 @ X172 ) @ ( sK542 @ X172 ) ) ) )
| ( ( X175 @ ( comp @ ( sK546 @ X175 ) @ ( sK544 @ X175 ) ) @ ( sub @ ( sK545 @ X175 ) @ ( sK544 @ X175 ) ) )
!= ( sub @ ( X175 @ ( sK546 @ X175 ) @ ( sK545 @ X175 ) ) @ ( sK544 @ X175 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ( ( ( sK550 @ sK548 @ sK549 )
!= ( sK547 @ sK548 @ sK549 ) )
& ( ( lam @ ( sK547 @ sh @ one ) )
= ( lam @ ( sK550 @ sh @ one ) ) )
& ! [X186: subst,X187: subst,X188: term] :
( ( sub @ ( sK547 @ X186 @ X188 ) @ X187 )
= ( sK547 @ ( comp @ X186 @ X187 ) @ ( sub @ X188 @ X187 ) ) )
& ! [X189: subst,X190: term,X191: subst] :
( ( sK550 @ ( comp @ X191 @ X189 ) @ ( sub @ X190 @ X189 ) )
= ( sub @ ( sK550 @ X191 @ X190 ) @ X189 ) ) ) )
& ( ( termmset_lthm = $true )
| ( ( ( sub @ sK551 @ id )
!= sK551 )
& ! [X193: term] :
( ( sub @ X193 @ id )
= X193 ) ) )
& ( ! [X194: term] :
( ( sub @ X194 @ id )
= X194 )
| ( sK552
!= ( sub @ sK552 @ id ) )
| ( termmset_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP17 ) )
& ( ( $true = sP17 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( axidl = $true )
| ( ( comp @ id @ sK553 )
!= sK553 ) )
& ( ! [X197: subst] :
( ( comp @ id @ X197 )
= X197 )
| ( axidl != $true ) )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP14 ) )
& ( ( $true = sP14 )
| ( hoasinduction_lthm != $true ) )
& ( ! [X198: term,X199: term,X200: term,X201: term] :
( ( ( ap @ ( sub @ X199 @ id ) @ X201 )
!= ( ap @ ( sub @ X198 @ id ) @ X200 ) )
| ( X200 = X201 ) )
| ( ( ( ap @ sK554 @ sK557 )
= ( ap @ sK556 @ sK555 ) )
& ( sK555 != sK557 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( ( ( ap @ ( sub @ sK559 @ id ) @ sK561 )
= ( ap @ ( sub @ sK558 @ id ) @ sK560 ) )
& ( sK560 != sK561 )
& ! [X210: term,X211: term,X212: term,X213: term] :
( ( ( ap @ X210 @ X213 )
!= ( ap @ X212 @ X211 ) )
| ( X211 = X213 ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem0_lthm = $true )
| ! [X215: term > $o] :
( ( X215 @ ( sK563 @ X215 ) )
!= ( sK562 @ id @ ( sK563 @ X215 ) @ id ) ) )
& ( ! [X217: subst > term > subst > $o,X219: term] :
( ( sK564 @ X217 @ X219 )
= ( X217 @ id @ X219 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( axassoc = $true )
& ( substmonoid_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ( ! [X222: term,X223: term] :
( ( $true
= ( sK566 @ id @ ( ap @ ( sub @ X222 @ id ) @ X223 ) @ id ) )
| ( $true
!= ( sK566 @ id @ X222 @ id ) )
| ( $true
!= ( sK566 @ id @ X223 @ id ) ) )
& ( $true
!= ( sP5 @ sK566 ) )
& ( $true
!= ( sK566 @ id @ sK565 @ id ) ) ) )
& ( ! [X224: term,X225: subst > term > subst > $o] :
( ( ( $true
!= ( X225 @ id @ ( ap @ ( sub @ ( sK567 @ X225 ) @ id ) @ ( sK568 @ X225 ) ) @ id ) )
& ( $true
= ( X225 @ id @ ( sK567 @ X225 ) @ id ) )
& ( $true
= ( X225 @ id @ ( sK568 @ X225 ) @ id ) ) )
| ( $true
= ( sP5 @ X225 ) )
| ( $true
= ( X225 @ id @ X224 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( induction2lem_gthm = $true )
& ( ( axshiftcons = $true )
| ( ( comp @ sh @ ( push @ sK569 @ sK570 ) )
!= sK570 ) )
& ( ! [X230: term,X231: subst] :
( ( comp @ sh @ ( push @ X230 @ X231 ) )
= X231 )
| ( axshiftcons != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP4 ) )
& ( ( $true = sP4 )
| ( hoasinduction_lem3 != $true ) )
& ( hoasapnotvar = $true )
& ! [X232: subst] :
( ( d2subst @ ( sK571 @ X232 ) )
= X232 )
& ( ! [X234: subst] :
( ( push @ ( sub @ one @ X234 ) @ ( comp @ sh @ X234 ) )
= X234 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ( ( push @ ( sub @ one @ sK572 ) @ ( comp @ sh @ sK572 ) )
!= sK572 ) )
& ( pushprop_lthm_orig = $true )
& ( pushprop_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X236: term,X237: term,X238: term,X239: term] :
( ( X238 = X239 )
| ( ( ap @ X238 @ X236 )
!= ( ap @ X239 @ X237 ) ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ( ( sK575 != sK576 )
& ( ( ap @ sK576 @ sK574 )
= ( ap @ sK575 @ sK573 ) ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( $true = sP2 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP2 ) )
& ( ! [X244: term,X245: term] :
( ( X244 = X245 )
| ( ( lam @ X244 )
!= ( lam @ X245 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ( ( sK578 != sK577 )
& ( ( lam @ sK577 )
= ( lam @ sK578 ) ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP0 ) )
& ( ( $true = sP0 )
| ( hoasinduction_lem3aa != $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ! [X248: term] :
( ( d2term @ ( sK579 @ X248 ) )
= X248 )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X250: term,X251: term,X252: term,X253: term] :
( ( X250 = X252 )
| ( ( ap @ ( sub @ X253 @ id ) @ X252 )
!= ( ap @ ( sub @ X251 @ id ) @ X250 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ( ( sK580 != sK582 )
& ( ( ap @ ( sub @ sK581 @ id ) @ sK580 )
= ( ap @ ( sub @ sK583 @ id ) @ sK582 ) ) ) )
& ! [X258: d_term,X259: d_term] :
( ( ( d2term @ X258 )
!= ( d2term @ X259 ) )
| ( X258 = X259 ) )
& ( ! [X260: term,X261: term > $o,X262: term > $o,X263: subst] :
( ( $true
!= ( X261 @ X260 ) )
| ( ( X262 @ one )
= $true )
| ( ( X261 @ ( sub @ ( sK584 @ X263 @ X262 @ X261 @ X260 ) @ ( push @ X260 @ X263 ) ) )
!= ( X262 @ ( sK584 @ X263 @ X262 @ X261 @ X260 ) ) ) )
| ( sK585
!= ( sub @ one @ ( push @ sK585 @ sK586 ) ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ( $true
= ( sK588 @ sK587 ) )
& ( $true
!= ( sK589 @ one ) )
& ! [X271: term] :
( ( sK589 @ X271 )
= ( sK588 @ ( sub @ X271 @ ( push @ sK587 @ sK590 ) ) ) )
& ! [X272: term,X273: subst] :
( ( sub @ one @ ( push @ X272 @ X273 ) )
= X272 ) ) ) ),
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: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
= ( ap @ X3 @ X1 ) )
& ( X0 != X1 ) )
=> ( ( ( ap @ sK459 @ sK457 )
= ( ap @ sK460 @ sK458 ) )
& ( sK457 != sK458 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f477,plain,
( ? [X9: term] :
( ( sub @ X9 @ id )
!= X9 )
=> ( sK461
!= ( sub @ sK461 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f478,plain,
( ? [X10: subst] :
( ( comp @ X10 @ id )
!= X10 )
=> ( sK462
!= ( comp @ sK462 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f479,plain,
( ? [X11: subst] :
( ( comp @ id @ X11 )
!= X11 )
=> ( ( comp @ id @ sK463 )
!= sK463 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f480,plain,
( ? [X16: subst] :
( ( comp @ id @ X16 )
!= X16 )
=> ( ( comp @ id @ sK464 )
!= sK464 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f481,plain,
( ? [X17: subst] :
( ( comp @ X17 @ id )
!= X17 )
=> ( ( comp @ sK465 @ id )
!= sK465 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f482,plain,
( ? [X21: term] :
! [X22: subst > term > term] :
( ( sub @ X21 @ ( push @ one @ sh ) )
!= ( X22 @ sh @ one ) )
=> ! [X22: subst > term > term] :
( ( X22 @ sh @ one )
!= ( sub @ sK466 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f483,plain,
! [X23: term] :
( ? [X24: subst > term > term] :
( ( X24 @ sh @ one )
= ( sub @ X23 @ ( push @ one @ sh ) ) )
=> ( ( sK467 @ X23 @ sh @ one )
= ( sub @ X23 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f484,plain,
( ? [X25: term,X26: term] :
( ( ( sub @ X25 @ sh )
= ( sub @ X26 @ sh ) )
& ( X25 != X26 ) )
=> ( ( ( sub @ sK469 @ sh )
= ( sub @ sK468 @ sh ) )
& ( sK469 != sK468 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f485,plain,
( ? [X29: subst > term > subst > $o] :
! [X30: term > $o] :
? [X31: term] :
( ( X29 @ id @ X31 @ id )
!= ( X30 @ X31 ) )
=> ! [X30: term > $o] :
? [X31: term] :
( ( sK470 @ id @ X31 @ id )
!= ( X30 @ X31 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f486,plain,
! [X30: term > $o] :
( ? [X31: term] :
( ( sK470 @ id @ X31 @ id )
!= ( X30 @ X31 ) )
=> ( ( sK470 @ id @ ( sK471 @ X30 ) @ id )
!= ( X30 @ ( sK471 @ X30 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f487,plain,
! [X32: subst > term > subst > $o] :
( ? [X33: term > $o] :
! [X34: term] :
( ( X33 @ X34 )
= ( X32 @ id @ X34 @ id ) )
=> ! [X34: term] :
( ( sK472 @ X32 @ X34 )
= ( X32 @ id @ X34 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f488,plain,
( ? [X35: subst,X36: term] :
( ( sub @ one @ ( push @ X36 @ X35 ) )
!= X36 )
=> ( ( sub @ one @ ( push @ sK474 @ sK473 ) )
!= sK474 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f489,plain,
( ? [X37: subst,X38: term] :
( ( comp @ sh @ ( push @ X38 @ X37 ) )
!= X37 )
=> ( ( comp @ sh @ ( push @ sK476 @ sK475 ) )
!= sK475 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f490,plain,
( ? [X48: term,X49: term > $o,X50: term > $o,X51: subst,X52: term] :
( ( $true
!= ( X50 @ ( sub @ X52 @ ( push @ X48 @ X51 ) ) ) )
& ( $true
= ( X49 @ X52 ) ) )
=> ( ( $true
!= ( sK479 @ ( sub @ sK481 @ ( push @ sK477 @ sK480 ) ) ) )
& ( $true
= ( sK478 @ sK481 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f491,plain,
( ? [X54: term] :
( ( sub @ X54 @ id )
!= X54 )
=> ( sK482
!= ( sub @ sK482 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f492,plain,
( ? [X56: term > $o,X57: term] :
( ( $true
!= ( X56 @ X57 ) )
& ! [X58: term] :
( ( $true
= ( X56 @ X58 ) )
| ( $true
!= ( var @ X58 ) ) )
& ( $true
!= ( sP36 @ X56 ) )
& ! [X59: term] :
( ( $true
= ( X56 @ ( lam @ X59 ) ) )
| ( $true
!= ( X56 @ X59 ) ) ) )
=> ( ( $true
!= ( sK483 @ sK484 ) )
& ! [X58: term] :
( ( $true
= ( sK483 @ X58 ) )
| ( $true
!= ( var @ X58 ) ) )
& ( $true
!= ( sP36 @ sK483 ) )
& ! [X59: term] :
( ( $true
= ( sK483 @ ( lam @ X59 ) ) )
| ( $true
!= ( sK483 @ X59 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f493,plain,
! [X60: term > $o] :
( ? [X62: term] :
( ( $true
!= ( X60 @ X62 ) )
& ( $true
= ( var @ X62 ) ) )
=> ( ( $true
!= ( X60 @ ( sK485 @ X60 ) ) )
& ( ( var @ ( sK485 @ X60 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f494,plain,
! [X60: term > $o] :
( ? [X63: term] :
( ( $true
!= ( X60 @ ( lam @ X63 ) ) )
& ( ( X60 @ X63 )
= $true ) )
=> ( ( $true
!= ( X60 @ ( lam @ ( sK486 @ X60 ) ) ) )
& ( $true
= ( X60 @ ( sK486 @ X60 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f495,plain,
( ? [X64: subst] :
( ( comp @ X64 @ id )
!= X64 )
=> ( sK487
!= ( comp @ sK487 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f496,plain,
( ? [X66: subst,X67: term] :
( ( sub @ one @ ( push @ X67 @ X66 ) )
!= X67 )
=> ( sK489
!= ( sub @ one @ ( push @ sK489 @ sK488 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f497,plain,
( ? [X70: term,X71: term > $o,X72: subst] :
! [X73: term > $o] :
? [X74: term] :
( ( X73 @ X74 )
!= ( X71 @ ( sub @ X74 @ ( push @ X70 @ X72 ) ) ) )
=> ! [X73: term > $o] :
? [X74: term] :
( ( sK491 @ ( sub @ X74 @ ( push @ sK490 @ sK492 ) ) )
!= ( X73 @ X74 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f498,plain,
! [X73: term > $o] :
( ? [X74: term] :
( ( sK491 @ ( sub @ X74 @ ( push @ sK490 @ sK492 ) ) )
!= ( X73 @ X74 ) )
=> ( ( X73 @ ( sK493 @ X73 ) )
!= ( sK491 @ ( sub @ ( sK493 @ X73 ) @ ( push @ sK490 @ sK492 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f499,plain,
! [X75: term,X76: term > $o,X77: subst] :
( ? [X78: term > $o] :
! [X79: term] :
( ( X78 @ X79 )
= ( X76 @ ( sub @ X79 @ ( push @ X75 @ X77 ) ) ) )
=> ! [X79: term] :
( ( X76 @ ( sub @ X79 @ ( push @ X75 @ X77 ) ) )
= ( sK494 @ X77 @ X76 @ X75 @ X79 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f500,plain,
( ? [X84: term,X85: term,X86: term,X87: term] :
( ( ( ap @ ( sub @ X86 @ id ) @ X85 )
= ( ap @ ( sub @ X84 @ id ) @ X87 ) )
& ( X84 != X86 ) )
=> ( ( ( ap @ ( sub @ sK497 @ id ) @ sK496 )
= ( ap @ ( sub @ sK495 @ id ) @ sK498 ) )
& ( sK495 != sK497 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f501,plain,
( ? [X89: term,X90: term,X91: term,X92: term] :
( ( ( ap @ X89 @ X90 )
= ( ap @ X92 @ X91 ) )
& ( X89 != X92 ) )
=> ( ( ( ap @ sK499 @ sK500 )
= ( ap @ sK502 @ sK501 ) )
& ( sK502 != sK499 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f502,plain,
( ? [X97: term] :
( ( sub @ X97 @ id )
!= X97 )
=> ( ( sub @ sK503 @ id )
!= sK503 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f503,plain,
( ? [X98: subst > term > term,X99: term,X100: term] :
( ! [X101: subst,X102: term,X103: subst] :
( ( X98 @ ( comp @ X101 @ X103 ) @ ( sub @ X102 @ X103 ) )
= ( sub @ ( X98 @ X101 @ X102 ) @ X103 ) )
& ( ( lam @ ( X98 @ sh @ one ) )
= ( ap @ ( sub @ X99 @ id ) @ X100 ) ) )
=> ( ! [X103: subst,X102: term,X101: subst] :
( ( sK504 @ ( comp @ X101 @ X103 ) @ ( sub @ X102 @ X103 ) )
= ( sub @ ( sK504 @ X101 @ X102 ) @ X103 ) )
& ( ( ap @ ( sub @ sK505 @ id ) @ sK506 )
= ( lam @ ( sK504 @ sh @ one ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f504,plain,
! [X104: subst > term > term] :
( ? [X107: subst,X108: term,X109: subst] :
( ( X104 @ ( comp @ X107 @ X109 ) @ ( sub @ X108 @ X109 ) )
!= ( sub @ ( X104 @ X107 @ X108 ) @ X109 ) )
=> ( ( X104 @ ( comp @ ( sK507 @ X104 ) @ ( sK509 @ X104 ) ) @ ( sub @ ( sK508 @ X104 ) @ ( sK509 @ X104 ) ) )
!= ( sub @ ( X104 @ ( sK507 @ X104 ) @ ( sK508 @ X104 ) ) @ ( sK509 @ X104 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f505,plain,
( ? [X114: term,X115: term,X116: term,X117: term] :
( ( ( ap @ ( sub @ X116 @ id ) @ X114 )
= ( ap @ ( sub @ X117 @ id ) @ X115 ) )
& ( X116 != X117 ) )
=> ( ( ( ap @ ( sub @ sK512 @ id ) @ sK510 )
= ( ap @ ( sub @ sK513 @ id ) @ sK511 ) )
& ( sK512 != sK513 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f506,plain,
( ? [X118: subst] :
( ( comp @ X118 @ id )
!= X118 )
=> ( sK514
!= ( comp @ sK514 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f507,plain,
( ? [X119: subst] :
( ( comp @ id @ X119 )
!= X119 )
=> ( sK515
!= ( comp @ id @ sK515 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f508,plain,
! [X122: subst,X123: term,X124: term > $o] :
( ? [X125: term > $o] :
! [X126: term] :
( ( X124 @ ( sub @ X126 @ ( push @ X123 @ X122 ) ) )
= ( X125 @ X126 ) )
=> ! [X126: term] :
( ( X124 @ ( sub @ X126 @ ( push @ X123 @ X122 ) ) )
= ( sK516 @ X124 @ X123 @ X122 @ X126 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f509,plain,
( ? [X127: subst,X128: term,X129: term > $o] :
! [X130: term > $o] :
? [X131: term] :
( ( X129 @ ( sub @ X131 @ ( push @ X128 @ X127 ) ) )
!= ( X130 @ X131 ) )
=> ! [X130: term > $o] :
? [X131: term] :
( ( sK519 @ ( sub @ X131 @ ( push @ sK518 @ sK517 ) ) )
!= ( X130 @ X131 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f510,plain,
! [X130: term > $o] :
( ? [X131: term] :
( ( sK519 @ ( sub @ X131 @ ( push @ sK518 @ sK517 ) ) )
!= ( X130 @ X131 ) )
=> ( ( X130 @ ( sK520 @ X130 ) )
!= ( sK519 @ ( sub @ ( sK520 @ X130 ) @ ( push @ sK518 @ sK517 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f511,plain,
( ? [X132: term] :
! [X133: subst > term > term] :
( ( sub @ X132 @ ( push @ one @ sh ) )
!= ( X133 @ sh @ one ) )
=> ! [X133: subst > term > term] :
( ( sub @ sK521 @ ( push @ one @ sh ) )
!= ( X133 @ sh @ one ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f512,plain,
! [X134: term] :
( ? [X135: subst > term > term] :
( ( X135 @ sh @ one )
= ( sub @ X134 @ ( push @ one @ sh ) ) )
=> ( ( sK522 @ X134 @ sh @ one )
= ( sub @ X134 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f513,plain,
( ? [X141: term > $o,X142: term,X143: subst,X144: term,X145: term > $o] :
( ( $true
!= ( X145 @ ( sub @ X142 @ ( push @ X144 @ X143 ) ) ) )
& ( $true
= ( X141 @ X142 ) ) )
=> ( ( $true
!= ( sK527 @ ( sub @ sK524 @ ( push @ sK526 @ sK525 ) ) ) )
& ( $true
= ( sK523 @ sK524 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f514,plain,
! [X146: term] :
( ? [X147: subst > term > term] :
! [X148: subst,X149: term] :
( ( sub @ X146 @ ( push @ X149 @ X148 ) )
= ( X147 @ X148 @ X149 ) )
=> ! [X149: term,X148: subst] :
( ( sub @ X146 @ ( push @ X149 @ X148 ) )
= ( sK528 @ X146 @ X148 @ X149 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f515,plain,
( ? [X150: term] :
! [X151: subst > term > term] :
? [X152: subst,X153: term] :
( ( sub @ X150 @ ( push @ X153 @ X152 ) )
!= ( X151 @ X152 @ X153 ) )
=> ! [X151: subst > term > term] :
? [X153: term,X152: subst] :
( ( sub @ sK529 @ ( push @ X153 @ X152 ) )
!= ( X151 @ X152 @ X153 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f516,plain,
! [X151: subst > term > term] :
( ? [X153: term,X152: subst] :
( ( sub @ sK529 @ ( push @ X153 @ X152 ) )
!= ( X151 @ X152 @ X153 ) )
=> ( ( sub @ sK529 @ ( push @ ( sK531 @ X151 ) @ ( sK530 @ X151 ) ) )
!= ( X151 @ ( sK530 @ X151 ) @ ( sK531 @ X151 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f517,plain,
! [X154: term > $o,X155: subst,X156: term > $o,X157: term] :
( ? [X158: term] :
( ( X154 @ ( sub @ X158 @ ( push @ X157 @ X155 ) ) )
!= ( X156 @ X158 ) )
=> ( ( X154 @ ( sub @ ( sK532 @ X157 @ X156 @ X155 @ X154 ) @ ( push @ X157 @ X155 ) ) )
!= ( X156 @ ( sK532 @ X157 @ X156 @ X155 @ X154 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f518,plain,
( ? [X159: term > $o,X160: subst,X161: term > $o,X162: term] :
( ( $true
!= ( X161 @ one ) )
& ( $true
= ( X159 @ X162 ) )
& ! [X163: term] :
( ( X159 @ ( sub @ X163 @ ( push @ X162 @ X160 ) ) )
= ( X161 @ X163 ) ) )
=> ( ( $true
!= ( sK535 @ one ) )
& ( $true
= ( sK533 @ sK536 ) )
& ! [X163: term] :
( ( sK533 @ ( sub @ X163 @ ( push @ sK536 @ sK534 ) ) )
= ( sK535 @ X163 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f519,plain,
! [X164: term] :
( ? [X165: subst > term > term] :
! [X166: term,X167: subst] :
( ( sub @ X164 @ ( push @ X166 @ X167 ) )
= ( X165 @ X167 @ X166 ) )
=> ! [X167: subst,X166: term] :
( ( sub @ X164 @ ( push @ X166 @ X167 ) )
= ( sK537 @ X164 @ X167 @ X166 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f520,plain,
( ? [X168: term] :
! [X169: subst > term > term] :
? [X170: term,X171: subst] :
( ( X169 @ X171 @ X170 )
!= ( sub @ X168 @ ( push @ X170 @ X171 ) ) )
=> ! [X169: subst > term > term] :
? [X171: subst,X170: term] :
( ( X169 @ X171 @ X170 )
!= ( sub @ sK538 @ ( push @ X170 @ X171 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f521,plain,
! [X169: subst > term > term] :
( ? [X171: subst,X170: term] :
( ( X169 @ X171 @ X170 )
!= ( sub @ sK538 @ ( push @ X170 @ X171 ) ) )
=> ( ( X169 @ ( sK540 @ X169 ) @ ( sK539 @ X169 ) )
!= ( sub @ sK538 @ ( push @ ( sK539 @ X169 ) @ ( sK540 @ X169 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f522,plain,
! [X172: subst > term > term] :
( ? [X176: subst,X177: subst,X178: term] :
( ( X172 @ ( comp @ X176 @ X177 ) @ ( sub @ X178 @ X177 ) )
!= ( sub @ ( X172 @ X176 @ X178 ) @ X177 ) )
=> ( ( sub @ ( X172 @ ( sK541 @ X172 ) @ ( sK543 @ X172 ) ) @ ( sK542 @ X172 ) )
!= ( X172 @ ( comp @ ( sK541 @ X172 ) @ ( sK542 @ X172 ) ) @ ( sub @ ( sK543 @ X172 ) @ ( sK542 @ X172 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f523,plain,
! [X175: subst > term > term] :
( ? [X179: subst,X180: term,X181: subst] :
( ( sub @ ( X175 @ X181 @ X180 ) @ X179 )
!= ( X175 @ ( comp @ X181 @ X179 ) @ ( sub @ X180 @ X179 ) ) )
=> ( ( X175 @ ( comp @ ( sK546 @ X175 ) @ ( sK544 @ X175 ) ) @ ( sub @ ( sK545 @ X175 ) @ ( sK544 @ X175 ) ) )
!= ( sub @ ( X175 @ ( sK546 @ X175 ) @ ( sK545 @ X175 ) ) @ ( sK544 @ X175 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f524,plain,
( ? [X182: subst > term > term,X183: subst,X184: term,X185: subst > term > term] :
( ( ( X185 @ X183 @ X184 )
!= ( X182 @ X183 @ X184 ) )
& ( ( lam @ ( X182 @ sh @ one ) )
= ( lam @ ( X185 @ sh @ one ) ) )
& ! [X186: subst,X187: subst,X188: term] :
( ( X182 @ ( comp @ X186 @ X187 ) @ ( sub @ X188 @ X187 ) )
= ( sub @ ( X182 @ X186 @ X188 ) @ X187 ) )
& ! [X189: subst,X190: term,X191: subst] :
( ( X185 @ ( comp @ X191 @ X189 ) @ ( sub @ X190 @ X189 ) )
= ( sub @ ( X185 @ X191 @ X190 ) @ X189 ) ) )
=> ( ( ( sK550 @ sK548 @ sK549 )
!= ( sK547 @ sK548 @ sK549 ) )
& ( ( lam @ ( sK547 @ sh @ one ) )
= ( lam @ ( sK550 @ sh @ one ) ) )
& ! [X188: term,X187: subst,X186: subst] :
( ( sub @ ( sK547 @ X186 @ X188 ) @ X187 )
= ( sK547 @ ( comp @ X186 @ X187 ) @ ( sub @ X188 @ X187 ) ) )
& ! [X191: subst,X190: term,X189: subst] :
( ( sK550 @ ( comp @ X191 @ X189 ) @ ( sub @ X190 @ X189 ) )
= ( sub @ ( sK550 @ X191 @ X190 ) @ X189 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f525,plain,
( ? [X192: term] :
( ( sub @ X192 @ id )
!= X192 )
=> ( ( sub @ sK551 @ id )
!= sK551 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f526,plain,
( ? [X195: term] :
( ( sub @ X195 @ id )
!= X195 )
=> ( sK552
!= ( sub @ sK552 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f527,plain,
( ? [X196: subst] :
( ( comp @ id @ X196 )
!= X196 )
=> ( ( comp @ id @ sK553 )
!= sK553 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f528,plain,
( ? [X202: term,X203: term,X204: term,X205: term] :
( ( ( ap @ X204 @ X203 )
= ( ap @ X202 @ X205 ) )
& ( X203 != X205 ) )
=> ( ( ( ap @ sK554 @ sK557 )
= ( ap @ sK556 @ sK555 ) )
& ( sK555 != sK557 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f529,plain,
( ? [X206: term,X207: term,X208: term,X209: term] :
( ( ( ap @ ( sub @ X207 @ id ) @ X209 )
= ( ap @ ( sub @ X206 @ id ) @ X208 ) )
& ( X208 != X209 ) )
=> ( ( ( ap @ ( sub @ sK559 @ id ) @ sK561 )
= ( ap @ ( sub @ sK558 @ id ) @ sK560 ) )
& ( sK560 != sK561 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f530,plain,
( ? [X214: subst > term > subst > $o] :
! [X215: term > $o] :
? [X216: term] :
( ( X214 @ id @ X216 @ id )
!= ( X215 @ X216 ) )
=> ! [X215: term > $o] :
? [X216: term] :
( ( X215 @ X216 )
!= ( sK562 @ id @ X216 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f531,plain,
! [X215: term > $o] :
( ? [X216: term] :
( ( X215 @ X216 )
!= ( sK562 @ id @ X216 @ id ) )
=> ( ( X215 @ ( sK563 @ X215 ) )
!= ( sK562 @ id @ ( sK563 @ X215 ) @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f532,plain,
! [X217: subst > term > subst > $o] :
( ? [X218: term > $o] :
! [X219: term] :
( ( X218 @ X219 )
= ( X217 @ id @ X219 @ id ) )
=> ! [X219: term] :
( ( sK564 @ X217 @ X219 )
= ( X217 @ id @ X219 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f533,plain,
( ? [X220: term,X221: subst > term > subst > $o] :
( ! [X222: term,X223: term] :
( ( $true
= ( X221 @ id @ ( ap @ ( sub @ X222 @ id ) @ X223 ) @ id ) )
| ( $true
!= ( X221 @ id @ X222 @ id ) )
| ( $true
!= ( X221 @ id @ X223 @ id ) ) )
& ( $true
!= ( sP5 @ X221 ) )
& ( $true
!= ( X221 @ id @ X220 @ id ) ) )
=> ( ! [X223: term,X222: term] :
( ( $true
= ( sK566 @ id @ ( ap @ ( sub @ X222 @ id ) @ X223 ) @ id ) )
| ( $true
!= ( sK566 @ id @ X222 @ id ) )
| ( $true
!= ( sK566 @ id @ X223 @ id ) ) )
& ( $true
!= ( sP5 @ sK566 ) )
& ( $true
!= ( sK566 @ id @ sK565 @ id ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f534,plain,
! [X225: subst > term > subst > $o] :
( ? [X226: term,X227: term] :
( ( $true
!= ( X225 @ id @ ( ap @ ( sub @ X226 @ id ) @ X227 ) @ id ) )
& ( $true
= ( X225 @ id @ X226 @ id ) )
& ( $true
= ( X225 @ id @ X227 @ id ) ) )
=> ( ( $true
!= ( X225 @ id @ ( ap @ ( sub @ ( sK567 @ X225 ) @ id ) @ ( sK568 @ X225 ) ) @ id ) )
& ( $true
= ( X225 @ id @ ( sK567 @ X225 ) @ id ) )
& ( $true
= ( X225 @ id @ ( sK568 @ X225 ) @ id ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f535,plain,
( ? [X228: term,X229: subst] :
( ( comp @ sh @ ( push @ X228 @ X229 ) )
!= X229 )
=> ( ( comp @ sh @ ( push @ sK569 @ sK570 ) )
!= sK570 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f536,plain,
! [X232: subst] :
( ? [X233: d_subst] :
( ( d2subst @ X233 )
= X232 )
=> ( ( d2subst @ ( sK571 @ X232 ) )
= X232 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f537,plain,
( ? [X235: subst] :
( ( push @ ( sub @ one @ X235 ) @ ( comp @ sh @ X235 ) )
!= X235 )
=> ( ( push @ ( sub @ one @ sK572 ) @ ( comp @ sh @ sK572 ) )
!= sK572 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f538,plain,
( ? [X240: term,X241: term,X242: term,X243: term] :
( ( X242 != X243 )
& ( ( ap @ X242 @ X240 )
= ( ap @ X243 @ X241 ) ) )
=> ( ( sK575 != sK576 )
& ( ( ap @ sK576 @ sK574 )
= ( ap @ sK575 @ sK573 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f539,plain,
( ? [X246: term,X247: term] :
( ( X246 != X247 )
& ( ( lam @ X246 )
= ( lam @ X247 ) ) )
=> ( ( sK578 != sK577 )
& ( ( lam @ sK577 )
= ( lam @ sK578 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f540,plain,
! [X248: term] :
( ? [X249: d_term] :
( ( d2term @ X249 )
= X248 )
=> ( ( d2term @ ( sK579 @ X248 ) )
= X248 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f541,plain,
( ? [X254: term,X255: term,X256: term,X257: term] :
( ( X254 != X256 )
& ( ( ap @ ( sub @ X255 @ id ) @ X254 )
= ( ap @ ( sub @ X257 @ id ) @ X256 ) ) )
=> ( ( sK580 != sK582 )
& ( ( ap @ ( sub @ sK581 @ id ) @ sK580 )
= ( ap @ ( sub @ sK583 @ id ) @ sK582 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f542,plain,
! [X260: term,X261: term > $o,X262: term > $o,X263: subst] :
( ? [X264: term] :
( ( X262 @ X264 )
!= ( X261 @ ( sub @ X264 @ ( push @ X260 @ X263 ) ) ) )
=> ( ( X261 @ ( sub @ ( sK584 @ X263 @ X262 @ X261 @ X260 ) @ ( push @ X260 @ X263 ) ) )
!= ( X262 @ ( sK584 @ X263 @ X262 @ X261 @ X260 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f543,plain,
( ? [X265: term,X266: subst] :
( ( sub @ one @ ( push @ X265 @ X266 ) )
!= X265 )
=> ( sK585
!= ( sub @ one @ ( push @ sK585 @ sK586 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f544,plain,
( ? [X267: term,X268: term > $o,X269: term > $o,X270: subst] :
( ( $true
= ( X268 @ X267 ) )
& ( $true
!= ( X269 @ one ) )
& ! [X271: term] :
( ( X269 @ X271 )
= ( X268 @ ( sub @ X271 @ ( push @ X267 @ X270 ) ) ) ) )
=> ( ( $true
= ( sK588 @ sK587 ) )
& ( $true
!= ( sK589 @ one ) )
& ! [X271: term] :
( ( sK589 @ X271 )
= ( sK588 @ ( sub @ X271 @ ( push @ sK587 @ sK590 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f475,plain,
( ( hoaslamnotvar = $true )
& ( ( apinj2 = $true )
| ? [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
= ( ap @ X3 @ X1 ) )
& ( X0 != X1 ) ) )
& ( ! [X4: term,X5: term,X6: term,X7: term] :
( ( ( ap @ X7 @ X5 )
!= ( ap @ X6 @ X4 ) )
| ( X4 = X5 ) )
| ( apinj2 != $true ) )
& ( ulamvarind = $true )
& ( axapp = $true )
& ( hoasinduction_lem1 = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( 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 ) ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X8: term] :
( ( sub @ X8 @ id )
= X8 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X9: term] :
( ( sub @ X9 @ id )
!= X9 ) )
& ( termmset_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( substmonoid_lthm = $true )
| ( ( ? [X10: subst] :
( ( comp @ X10 @ id )
!= X10 )
| ? [X11: subst] :
( ( comp @ id @ X11 )
!= X11 ) )
& ! [X12: subst] :
( ( comp @ id @ X12 )
= X12 )
& ! [X13: subst] :
( ( comp @ X13 @ id )
= X13 ) ) )
& ( ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X15: subst] :
( ( comp @ id @ X15 )
= X15 ) )
| ? [X16: subst] :
( ( comp @ id @ X16 )
!= X16 )
| ? [X17: subst] :
( ( comp @ X17 @ id )
!= X17 )
| ( substmonoid_lthm != $true ) )
& ! [X18: d_term] : ( d_one = X18 )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP58 ) )
& ( ( $true = sP58 )
| ( hoasinduction_lem2 != $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP56 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP56 ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( induction2_lthm = $true )
| ( $true != sP55 ) )
& ( ( $true = sP55 )
| ( induction2_lthm != $true ) )
& ( ( sP51 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP51 != $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoaslamnotap_gthm = $true )
& ! [X19: d_subst,X20: d_subst] :
( ( ( d2subst @ X19 )
!= ( d2subst @ X20 ) )
| ( X19 = X20 ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X21: term] :
! [X22: subst > term > term] :
( ( sub @ X21 @ ( push @ one @ sh ) )
!= ( X22 @ sh @ one ) ) )
& ( ! [X23: term] :
? [X24: subst > term > term] :
( ( X24 @ sh @ one )
= ( sub @ X23 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( shinj = $true )
| ? [X25: term,X26: term] :
( ( ( sub @ X25 @ sh )
= ( sub @ X26 @ sh ) )
& ( X25 != X26 ) ) )
& ( ! [X27: term,X28: term] :
( ( ( sub @ X28 @ sh )
!= ( sub @ X27 @ sh ) )
| ( X27 = X28 ) )
| ( shinj != $true ) )
& ( ( $true = sP45 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP45 ) )
& ( ( induction2lem = $true )
| ( $true != sP42 ) )
& ( ( $true = sP42 )
| ( induction2lem != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X29: subst > term > subst > $o] :
! [X30: term > $o] :
? [X31: term] :
( ( X29 @ id @ X31 @ id )
!= ( X30 @ X31 ) ) )
& ( ! [X32: subst > term > subst > $o] :
? [X33: term > $o] :
! [X34: term] :
( ( X33 @ X34 )
= ( X32 @ id @ X34 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP40 ) )
& ( ( $true = sP40 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( ? [X35: subst,X36: term] :
( ( sub @ one @ ( push @ X36 @ X35 ) )
!= X36 )
| ? [X37: subst,X38: term] :
( ( comp @ sh @ ( push @ X38 @ X37 ) )
!= X37 )
| ! [X39: term,X40: term > $o,X41: term > $o,X42: subst,X43: term] :
( ( $true
= ( X41 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) )
| ( $true
!= ( X40 @ X43 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X44: subst,X45: term] :
( ( sub @ one @ ( push @ X45 @ X44 ) )
= X45 )
& ! [X46: subst,X47: term] :
( ( comp @ sh @ ( push @ X47 @ X46 ) )
= X46 )
& ? [X48: term,X49: term > $o,X50: term > $o,X51: subst,X52: term] :
( ( $true
!= ( X50 @ ( sub @ X52 @ ( push @ X48 @ X51 ) ) ) )
& ( $true
= ( X49 @ X52 ) ) ) ) )
& ( ( sP39 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP39 != $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X53: term] :
( ( sub @ X53 @ id )
= X53 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X54: term] :
( ( sub @ X54 @ id )
!= X54 ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP38 ) )
& ( ( $true = sP38 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( axabs = $true )
& ( hoasapinj2_gthm = $true )
& ( lamnotvar = $true )
& ( ulamvarsh = $true )
& ! [X55: d_subst] : ( d_id = X55 )
& ( ( induction = $true )
| ? [X56: term > $o,X57: term] :
( ( $true
!= ( X56 @ X57 ) )
& ! [X58: term] :
( ( $true
= ( X56 @ X58 ) )
| ( $true
!= ( var @ X58 ) ) )
& ( $true
!= ( sP36 @ X56 ) )
& ! [X59: term] :
( ( $true
= ( X56 @ ( lam @ X59 ) ) )
| ( $true
!= ( X56 @ X59 ) ) ) ) )
& ( ! [X60: term > $o,X61: term] :
( ( $true
= ( X60 @ X61 ) )
| ? [X62: term] :
( ( $true
!= ( X60 @ X62 ) )
& ( $true
= ( var @ X62 ) ) )
| ( $true
= ( sP36 @ X60 ) )
| ? [X63: term] :
( ( $true
!= ( X60 @ ( lam @ X63 ) ) )
& ( ( X60 @ X63 )
= $true ) ) )
| ( induction != $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( ( axidr = $true )
| ? [X64: subst] :
( ( comp @ X64 @ id )
!= X64 ) )
& ( ! [X65: subst] :
( ( comp @ X65 @ id )
= X65 )
| ( axidr != $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( $true = sP35 )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( $true != sP35 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP31 ) )
& ( ( $true = sP31 )
| ( hoasinduction_lem2v2 != $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_gthm = $true )
& ( induction2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem3v2a = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3v2a != $true ) )
& ( axclos = $true )
& ( lamnotap != $true )
& ( ( axvarcons = $true )
| ? [X66: subst,X67: term] :
( ( sub @ one @ ( push @ X67 @ X66 ) )
!= X67 ) )
& ( ! [X68: subst,X69: term] :
( ( sub @ one @ ( push @ X69 @ X68 ) )
= X69 )
| ( axvarcons != $true ) )
& ( axvarshift = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
| ? [X70: term,X71: term > $o,X72: subst] :
! [X73: term > $o] :
? [X74: term] :
( ( X73 @ X74 )
!= ( X71 @ ( sub @ X74 @ ( push @ X70 @ X72 ) ) ) ) )
& ( ! [X75: term,X76: term > $o,X77: subst] :
? [X78: term > $o] :
! [X79: term] :
( ( X78 @ X79 )
= ( X76 @ ( sub @ X79 @ ( push @ X75 @ X77 ) ) ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X80: term,X81: term,X82: term,X83: term] :
( ( ( ap @ X83 @ X82 )
!= ( ap @ X80 @ X81 ) )
| ( X80 = X83 ) )
& ? [X84: term,X85: term,X86: term,X87: term] :
( ( ( ap @ ( sub @ X86 @ id ) @ X85 )
= ( ap @ ( sub @ X84 @ id ) @ X87 ) )
& ( X84 != X86 ) )
& ! [X88: term] :
( ( sub @ X88 @ id )
= X88 ) ) )
& ( ? [X89: term,X90: term,X91: term,X92: term] :
( ( ( ap @ X89 @ X90 )
= ( ap @ X92 @ X91 ) )
& ( X89 != X92 ) )
| ! [X93: term,X94: term,X95: term,X96: term] :
( ( ( ap @ ( sub @ X95 @ id ) @ X94 )
!= ( ap @ ( sub @ X93 @ id ) @ X96 ) )
| ( X93 = X95 ) )
| ? [X97: term] :
( ( sub @ X97 @ id )
!= X97 )
| ( hoasapinj1_lthm != $true ) )
& ( ( $true = sP28 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( $true != sP28 ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP26 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP26 != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X98: subst > term > term,X99: term,X100: term] :
( ! [X101: subst,X102: term,X103: subst] :
( ( X98 @ ( comp @ X101 @ X103 ) @ ( sub @ X102 @ X103 ) )
= ( sub @ ( X98 @ X101 @ X102 ) @ X103 ) )
& ( ( lam @ ( X98 @ sh @ one ) )
= ( ap @ ( sub @ X99 @ id ) @ X100 ) ) ) )
& ( ! [X104: subst > term > term,X105: term,X106: term] :
( ? [X107: subst,X108: term,X109: subst] :
( ( X104 @ ( comp @ X107 @ X109 ) @ ( sub @ X108 @ X109 ) )
!= ( sub @ ( X104 @ X107 @ X108 ) @ X109 ) )
| ( ( lam @ ( X104 @ sh @ one ) )
!= ( ap @ ( sub @ X105 @ id ) @ X106 ) ) )
| ( hoaslamnotap != $true ) )
& ( pushprop = $true )
& ( pushprop_lthm = $true )
& ( ! [X110: term,X111: term,X112: term,X113: term] :
( ( ( ap @ ( sub @ X113 @ id ) @ X111 )
!= ( ap @ ( sub @ X112 @ id ) @ X110 ) )
| ( X112 = X113 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X114: term,X115: term,X116: term,X117: term] :
( ( ( ap @ ( sub @ X116 @ id ) @ X114 )
= ( ap @ ( sub @ X117 @ id ) @ X115 ) )
& ( X116 != X117 ) ) )
& ( apnotvar = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ( $true = sP24 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP24 ) )
& ( ( $true = sP23 )
| ( hoaslaminj_lthm != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP23 ) )
& ( pushprop_lem2v2 = $true )
& ( ( substmonoid = $true )
| ? [X118: subst] :
( ( comp @ X118 @ id )
!= X118 )
| ? [X119: subst] :
( ( comp @ id @ X119 )
!= X119 ) )
& ( ( ! [X120: subst] :
( ( comp @ X120 @ id )
= X120 )
& ! [X121: subst] :
( ( comp @ id @ X121 )
= X121 ) )
| ( substmonoid != $true ) )
& ( ! [X122: subst,X123: term,X124: term > $o] :
? [X125: term > $o] :
! [X126: term] :
( ( X124 @ ( sub @ X126 @ ( push @ X123 @ X122 ) ) )
= ( X125 @ X126 ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X127: subst,X128: term,X129: term > $o] :
! [X130: term > $o] :
? [X131: term] :
( ( X129 @ ( sub @ X131 @ ( push @ X128 @ X127 ) ) )
!= ( X130 @ X131 ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X132: term] :
! [X133: subst > term > term] :
( ( sub @ X132 @ ( push @ one @ sh ) )
!= ( X133 @ sh @ one ) ) )
& ( ! [X134: term] :
? [X135: subst > term > term] :
( ( X135 @ sh @ one )
= ( sub @ X134 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( $true = sP22 )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( $true != sP22 ) )
& ( ! [X136: term > $o,X137: term,X138: subst,X139: term,X140: term > $o] :
( ( $true
= ( X140 @ ( sub @ X137 @ ( push @ X139 @ X138 ) ) ) )
| ( $true
!= ( X136 @ X137 ) ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X141: term > $o,X142: term,X143: subst,X144: term,X145: term > $o] :
( ( $true
!= ( X145 @ ( sub @ X142 @ ( push @ X144 @ X143 ) ) ) )
& ( $true
= ( X141 @ X142 ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X146: term] :
? [X147: subst > term > term] :
! [X148: subst,X149: term] :
( ( sub @ X146 @ ( push @ X149 @ X148 ) )
= ( X147 @ X148 @ X149 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X150: term] :
! [X151: subst > term > term] :
? [X152: subst,X153: term] :
( ( sub @ X150 @ ( push @ X153 @ X152 ) )
!= ( X151 @ X152 @ X153 ) ) )
& ( ! [X154: term > $o,X155: subst,X156: term > $o,X157: term] :
( ( $true
= ( X156 @ one ) )
| ( $true
!= ( X154 @ X157 ) )
| ? [X158: term] :
( ( X154 @ ( sub @ X158 @ ( push @ X157 @ X155 ) ) )
!= ( X156 @ X158 ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X159: term > $o,X160: subst,X161: term > $o,X162: term] :
( ( $true
!= ( X161 @ one ) )
& ( $true
= ( X159 @ X162 ) )
& ! [X163: term] :
( ( X159 @ ( sub @ X163 @ ( push @ X162 @ X160 ) ) )
= ( X161 @ X163 ) ) ) )
& ( ulamvar1 != $true )
& ( axmap = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslamnotvar_lthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X164: term] :
? [X165: subst > term > term] :
! [X166: term,X167: subst] :
( ( sub @ X164 @ ( push @ X166 @ X167 ) )
= ( X165 @ X167 @ X166 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X168: term] :
! [X169: subst > term > term] :
? [X170: term,X171: subst] :
( ( X169 @ X171 @ X170 )
!= ( sub @ X168 @ ( push @ X170 @ X171 ) ) ) )
& ( ! [X172: subst > term > term,X173: subst,X174: term,X175: subst > term > term] :
( ( ( X172 @ X173 @ X174 )
= ( X175 @ X173 @ X174 ) )
| ( ( lam @ ( X172 @ sh @ one ) )
!= ( lam @ ( X175 @ sh @ one ) ) )
| ? [X176: subst,X177: subst,X178: term] :
( ( X172 @ ( comp @ X176 @ X177 ) @ ( sub @ X178 @ X177 ) )
!= ( sub @ ( X172 @ X176 @ X178 ) @ X177 ) )
| ? [X179: subst,X180: term,X181: subst] :
( ( sub @ ( X175 @ X181 @ X180 ) @ X179 )
!= ( X175 @ ( comp @ X181 @ X179 ) @ ( sub @ X180 @ X179 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X182: subst > term > term,X183: subst,X184: term,X185: subst > term > term] :
( ( ( X185 @ X183 @ X184 )
!= ( X182 @ X183 @ X184 ) )
& ( ( lam @ ( X182 @ sh @ one ) )
= ( lam @ ( X185 @ sh @ one ) ) )
& ! [X186: subst,X187: subst,X188: term] :
( ( X182 @ ( comp @ X186 @ X187 ) @ ( sub @ X188 @ X187 ) )
= ( sub @ ( X182 @ X186 @ X188 ) @ X187 ) )
& ! [X189: subst,X190: term,X191: subst] :
( ( X185 @ ( comp @ X191 @ X189 ) @ ( sub @ X190 @ X189 ) )
= ( sub @ ( X185 @ X191 @ X190 ) @ X189 ) ) ) )
& ( ( termmset_lthm = $true )
| ( ? [X192: term] :
( ( sub @ X192 @ id )
!= X192 )
& ! [X193: term] :
( ( sub @ X193 @ id )
= X193 ) ) )
& ( ! [X194: term] :
( ( sub @ X194 @ id )
= X194 )
| ? [X195: term] :
( ( sub @ X195 @ id )
!= X195 )
| ( termmset_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP17 ) )
& ( ( $true = sP17 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( axidl = $true )
| ? [X196: subst] :
( ( comp @ id @ X196 )
!= X196 ) )
& ( ! [X197: subst] :
( ( comp @ id @ X197 )
= X197 )
| ( axidl != $true ) )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP14 ) )
& ( ( $true = sP14 )
| ( hoasinduction_lthm != $true ) )
& ( ! [X198: term,X199: term,X200: term,X201: term] :
( ( ( ap @ ( sub @ X199 @ id ) @ X201 )
!= ( ap @ ( sub @ X198 @ id ) @ X200 ) )
| ( X200 = X201 ) )
| ? [X202: term,X203: term,X204: term,X205: term] :
( ( ( ap @ X204 @ X203 )
= ( ap @ X202 @ X205 ) )
& ( X203 != X205 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X206: term,X207: term,X208: term,X209: term] :
( ( ( ap @ ( sub @ X207 @ id ) @ X209 )
= ( ap @ ( sub @ X206 @ id ) @ X208 ) )
& ( X208 != X209 ) )
& ! [X210: term,X211: term,X212: term,X213: term] :
( ( ( ap @ X210 @ X213 )
!= ( ap @ X212 @ X211 ) )
| ( X211 = X213 ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X214: subst > term > subst > $o] :
! [X215: term > $o] :
? [X216: term] :
( ( X214 @ id @ X216 @ id )
!= ( X215 @ X216 ) ) )
& ( ! [X217: subst > term > subst > $o] :
? [X218: term > $o] :
! [X219: term] :
( ( X218 @ X219 )
= ( X217 @ id @ X219 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( axassoc = $true )
& ( substmonoid_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X220: term,X221: subst > term > subst > $o] :
( ! [X222: term,X223: term] :
( ( $true
= ( X221 @ id @ ( ap @ ( sub @ X222 @ id ) @ X223 ) @ id ) )
| ( $true
!= ( X221 @ id @ X222 @ id ) )
| ( $true
!= ( X221 @ id @ X223 @ id ) ) )
& ( $true
!= ( sP5 @ X221 ) )
& ( $true
!= ( X221 @ id @ X220 @ id ) ) ) )
& ( ! [X224: term,X225: subst > term > subst > $o] :
( ? [X226: term,X227: term] :
( ( $true
!= ( X225 @ id @ ( ap @ ( sub @ X226 @ id ) @ X227 ) @ id ) )
& ( $true
= ( X225 @ id @ X226 @ id ) )
& ( $true
= ( X225 @ id @ X227 @ id ) ) )
| ( $true
= ( sP5 @ X225 ) )
| ( $true
= ( X225 @ id @ X224 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( induction2lem_gthm = $true )
& ( ( axshiftcons = $true )
| ? [X228: term,X229: subst] :
( ( comp @ sh @ ( push @ X228 @ X229 ) )
!= X229 ) )
& ( ! [X230: term,X231: subst] :
( ( comp @ sh @ ( push @ X230 @ X231 ) )
= X231 )
| ( axshiftcons != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP4 ) )
& ( ( $true = sP4 )
| ( hoasinduction_lem3 != $true ) )
& ( hoasapnotvar = $true )
& ! [X232: subst] :
? [X233: d_subst] :
( ( d2subst @ X233 )
= X232 )
& ( ! [X234: subst] :
( ( push @ ( sub @ one @ X234 ) @ ( comp @ sh @ X234 ) )
= X234 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X235: subst] :
( ( push @ ( sub @ one @ X235 ) @ ( comp @ sh @ X235 ) )
!= X235 ) )
& ( pushprop_lthm_orig = $true )
& ( pushprop_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X236: term,X237: term,X238: term,X239: term] :
( ( X238 = X239 )
| ( ( ap @ X238 @ X236 )
!= ( ap @ X239 @ X237 ) ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X240: term,X241: term,X242: term,X243: term] :
( ( X242 != X243 )
& ( ( ap @ X242 @ X240 )
= ( ap @ X243 @ X241 ) ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( $true = sP2 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP2 ) )
& ( ! [X244: term,X245: term] :
( ( X244 = X245 )
| ( ( lam @ X244 )
!= ( lam @ X245 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X246: term,X247: term] :
( ( X246 != X247 )
& ( ( lam @ X246 )
= ( lam @ X247 ) ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP0 ) )
& ( ( $true = sP0 )
| ( hoasinduction_lem3aa != $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ! [X248: term] :
? [X249: d_term] :
( ( d2term @ X249 )
= X248 )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X250: term,X251: term,X252: term,X253: term] :
( ( X250 = X252 )
| ( ( ap @ ( sub @ X253 @ id ) @ X252 )
!= ( ap @ ( sub @ X251 @ id ) @ X250 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X254: term,X255: term,X256: term,X257: term] :
( ( X254 != X256 )
& ( ( ap @ ( sub @ X255 @ id ) @ X254 )
= ( ap @ ( sub @ X257 @ id ) @ X256 ) ) ) )
& ! [X258: d_term,X259: d_term] :
( ( ( d2term @ X258 )
!= ( d2term @ X259 ) )
| ( X258 = X259 ) )
& ( ! [X260: term,X261: term > $o,X262: term > $o,X263: subst] :
( ( $true
!= ( X261 @ X260 ) )
| ( ( X262 @ one )
= $true )
| ? [X264: term] :
( ( X262 @ X264 )
!= ( X261 @ ( sub @ X264 @ ( push @ X260 @ X263 ) ) ) ) )
| ? [X265: term,X266: subst] :
( ( sub @ one @ ( push @ X265 @ X266 ) )
!= X265 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X267: term,X268: term > $o,X269: term > $o,X270: subst] :
( ( $true
= ( X268 @ X267 ) )
& ( $true
!= ( X269 @ one ) )
& ! [X271: term] :
( ( X269 @ X271 )
= ( X268 @ ( sub @ X271 @ ( push @ X267 @ X270 ) ) ) ) )
& ! [X272: term,X273: subst] :
( ( sub @ one @ ( push @ X272 @ X273 ) )
= X272 ) ) ) ),
inference(rectify,[status(thm)],[f474]) ).
thf(f474,plain,
( ( hoaslamnotvar = $true )
& ( ( apinj2 = $true )
| ? [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
= ( ap @ X3 @ X1 ) )
& ( X0 != X1 ) ) )
& ( ! [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) )
| ( X0 = X1 ) )
| ( apinj2 != $true ) )
& ( ulamvarind = $true )
& ( axapp = $true )
& ( hoasinduction_lem1 = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( 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 ) ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X32: term] :
( ( sub @ X32 @ id )
!= X32 ) )
& ( termmset_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( substmonoid_lthm = $true )
| ( ( ? [X35: subst] :
( ( comp @ X35 @ id )
!= X35 )
| ? [X36: subst] :
( ( comp @ id @ X36 )
!= X36 ) )
& ! [X33: subst] :
( ( comp @ id @ X33 )
= X33 )
& ! [X34: subst] :
( ( comp @ X34 @ id )
= X34 ) ) )
& ( ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) )
| ? [X33: subst] :
( ( comp @ id @ X33 )
!= X33 )
| ? [X34: subst] :
( ( comp @ X34 @ id )
!= X34 )
| ( substmonoid_lthm != $true ) )
& ! [X485: d_term] : ( d_one = X485 )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP58 ) )
& ( ( $true = sP58 )
| ( hoasinduction_lem2 != $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP56 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP56 ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( induction2_lthm = $true )
| ( $true != sP55 ) )
& ( ( $true = sP55 )
| ( induction2_lthm != $true ) )
& ( ( sP51 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP51 != $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoaslamnotap_gthm = $true )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
!= ( d2subst @ X14 ) )
| ( X13 = X14 ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X537: term] :
! [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
!= ( X538 @ sh @ one ) ) )
& ( ! [X537: term] :
? [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
= ( X538 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( shinj = $true )
| ? [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
= ( sub @ X515 @ sh ) )
& ( X514 != X515 ) ) )
& ( ! [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
!= ( sub @ X515 @ sh ) )
| ( X514 = X515 ) )
| ( shinj != $true ) )
& ( ( $true = sP45 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP45 ) )
& ( ( induction2lem = $true )
| ( $true != sP42 ) )
& ( ( $true = sP42 )
| ( induction2lem != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X77: subst > term > subst > $o] :
! [X78: term > $o] :
? [X79: term] :
( ( X78 @ X79 )
!= ( X77 @ id @ X79 @ id ) ) )
& ( ! [X77: subst > term > subst > $o] :
? [X78: term > $o] :
! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP40 ) )
& ( ( $true = sP40 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( ? [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
!= X203 )
| ? [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
!= X204 )
| ! [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) )
| ( $true
!= ( X207 @ X210 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
= X203 )
& ! [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
= X204 )
& ? [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
!= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) )
& ( $true
= ( X207 @ X210 ) ) ) ) )
& ( ( sP39 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP39 != $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X318: term] :
( ( sub @ X318 @ id )
= X318 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X318: term] :
( ( sub @ X318 @ id )
!= X318 ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP38 ) )
& ( ( $true = sP38 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( axabs = $true )
& ( hoasapinj2_gthm = $true )
& ( lamnotvar = $true )
& ( ulamvarsh = $true )
& ! [X106: d_subst] : ( d_id = X106 )
& ( ( induction = $true )
| ? [X479: term > $o,X480: term] :
( ( $true
!= ( X479 @ X480 ) )
& ! [X481: term] :
( ( $true
= ( X479 @ X481 ) )
| ( $true
!= ( var @ X481 ) ) )
& ( $true
!= ( sP36 @ X479 ) )
& ! [X482: term] :
( ( $true
= ( X479 @ ( lam @ X482 ) ) )
| ( $true
!= ( X479 @ X482 ) ) ) ) )
& ( ! [X479: term > $o,X480: term] :
( ( $true
= ( X479 @ X480 ) )
| ? [X481: term] :
( ( $true
!= ( X479 @ X481 ) )
& ( $true
= ( var @ X481 ) ) )
| ( $true
= ( sP36 @ X479 ) )
| ? [X482: term] :
( ( $true
!= ( X479 @ ( lam @ X482 ) ) )
& ( $true
= ( X479 @ X482 ) ) ) )
| ( induction != $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( ( axidr = $true )
| ? [X105: subst] :
( ( comp @ X105 @ id )
!= X105 ) )
& ( ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 )
| ( axidr != $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( $true = sP35 )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( $true != sP35 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP31 ) )
& ( ( $true = sP31 )
| ( hoasinduction_lem2v2 != $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_gthm = $true )
& ( induction2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem3v2a = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3v2a != $true ) )
& ( axclos = $true )
& ( lamnotap != $true )
& ( ( axvarcons = $true )
| ? [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
!= X104 ) )
& ( ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 )
| ( axvarcons != $true ) )
& ( axvarshift = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
| ? [X532: term,X533: term > $o,X534: subst] :
! [X535: term > $o] :
? [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
!= ( X535 @ X536 ) ) )
& ( ! [X532: term,X533: term > $o,X534: subst] :
? [X535: term > $o] :
! [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
= ( X535 @ X536 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
!= ( ap @ X490 @ X489 ) )
| ( X487 = X490 ) )
& ? [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
& ( X491 != X493 ) )
& ! [X486: term] :
( ( sub @ X486 @ id )
= X486 ) ) )
& ( ? [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
= ( ap @ X490 @ X489 ) )
& ( X487 != X490 ) )
| ! [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
!= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
| ( X491 = X493 ) )
| ? [X486: term] :
( ( sub @ X486 @ id )
!= X486 )
| ( hoasapinj1_lthm != $true ) )
& ( ( $true = sP28 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( $true != sP28 ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP26 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP26 != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X117: subst > term > term,X118: term,X119: term] :
( ! [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
& ( ( lam @ ( X117 @ sh @ one ) )
= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) ) )
& ( ! [X117: subst > term > term,X118: term,X119: term] :
( ? [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
!= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) )
| ( hoaslamnotap != $true ) )
& ( pushprop = $true )
& ( pushprop_lthm = $true )
& ( ! [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
!= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
| ( X435 = X436 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
& ( X435 != X436 ) ) )
& ( apnotvar = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ( $true = sP24 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP24 ) )
& ( ( $true = sP23 )
| ( hoaslaminj_lthm != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP23 ) )
& ( pushprop_lem2v2 = $true )
& ( ( substmonoid = $true )
| ? [X449: subst] :
( ( comp @ X449 @ id )
!= X449 )
| ? [X448: subst] :
( ( comp @ id @ X448 )
!= X448 ) )
& ( ( ! [X449: subst] :
( ( comp @ X449 @ id )
= X449 )
& ! [X448: subst] :
( ( comp @ id @ X448 )
= X448 ) )
| ( substmonoid != $true ) )
& ( ! [X27: subst,X28: term,X29: term > $o] :
? [X30: term > $o] :
! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X27: subst,X28: term,X29: term > $o] :
! [X30: term > $o] :
? [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
!= ( X30 @ X31 ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X200: term] :
! [X201: subst > term > term] :
( ( X201 @ sh @ one )
!= ( sub @ X200 @ ( push @ one @ sh ) ) ) )
& ( ! [X200: term] :
? [X201: subst > term > term] :
( ( X201 @ sh @ one )
= ( sub @ X200 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( $true = sP22 )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( $true != sP22 ) )
& ( ! [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
| ( $true
!= ( X173 @ X174 ) ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
!= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
& ( $true
= ( X173 @ X174 ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X80: term] :
? [X81: subst > term > term] :
! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X80: term] :
! [X81: subst > term > term] :
? [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
!= ( X81 @ X82 @ X83 ) ) )
& ( ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
= ( X144 @ one ) )
| ( $true
!= ( X142 @ X145 ) )
| ? [X146: term] :
( ( X144 @ X146 )
!= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
!= ( X144 @ one ) )
& ( $true
= ( X142 @ X145 ) )
& ! [X146: term] :
( ( X144 @ X146 )
= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) ) ) )
& ( ulamvar1 != $true )
& ( axmap = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslamnotvar_lthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X147: term] :
! [X148: subst > term > term] :
? [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
!= ( X148 @ X150 @ X149 ) ) )
& ( ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ? [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
!= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ? [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
!= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ( ( X107 @ X108 @ X109 )
!= ( X110 @ X108 @ X109 ) )
& ( ( lam @ ( X110 @ sh @ one ) )
= ( lam @ ( X107 @ sh @ one ) ) )
& ! [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
& ! [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) ) )
& ( ( termmset_lthm = $true )
| ( ? [X26: term] :
( ( sub @ X26 @ id )
!= X26 )
& ! [X25: term] :
( ( sub @ X25 @ id )
= X25 ) ) )
& ( ! [X26: term] :
( ( sub @ X26 @ id )
= X26 )
| ? [X25: term] :
( ( sub @ X25 @ id )
!= X25 )
| ( termmset_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP17 ) )
& ( ( $true = sP17 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( axidl = $true )
| ? [X56: subst] :
( ( comp @ id @ X56 )
!= X56 ) )
& ( ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 )
| ( axidl != $true ) )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP14 ) )
& ( ( $true = sP14 )
| ( hoasinduction_lthm != $true ) )
& ( ! [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
!= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
| ( X217 = X218 ) )
| ? [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
= ( ap @ X213 @ X212 ) )
& ( X212 != X214 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
& ( X217 != X218 ) )
& ! [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
!= ( ap @ X213 @ X212 ) )
| ( X212 = X214 ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X445: subst > term > subst > $o] :
! [X446: term > $o] :
? [X447: term] :
( ( X445 @ id @ X447 @ id )
!= ( X446 @ X447 ) ) )
& ( ! [X445: subst > term > subst > $o] :
? [X446: term > $o] :
! [X447: term] :
( ( X445 @ id @ X447 @ id )
= ( X446 @ X447 ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( axassoc = $true )
& ( substmonoid_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X4: term,X5: subst > term > subst > $o] :
( ! [X11: term,X12: term] :
( ( $true
= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
| ( ( X5 @ id @ X11 @ id )
!= $true )
| ( $true
!= ( X5 @ id @ X12 @ id ) ) )
& ( $true
!= ( sP5 @ X5 ) )
& ( $true
!= ( X5 @ id @ X4 @ id ) ) ) )
& ( ! [X4: term,X5: subst > term > subst > $o] :
( ? [X11: term,X12: term] :
( ( $true
!= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
& ( ( X5 @ id @ X11 @ id )
= $true )
& ( $true
= ( X5 @ id @ X12 @ id ) ) )
| ( $true
= ( sP5 @ X5 ) )
| ( $true
= ( X5 @ id @ X4 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( induction2lem_gthm = $true )
& ( ( axshiftcons = $true )
| ? [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
!= X58 ) )
& ( ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 )
| ( axshiftcons != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP4 ) )
& ( ( $true = sP4 )
| ( hoasinduction_lem3 != $true ) )
& ( hoasapnotvar = $true )
& ! [X266: subst] :
? [X267: d_subst] :
( ( d2subst @ X267 )
= X266 )
& ( ! [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
= X539 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
!= X539 ) )
& ( pushprop_lthm_orig = $true )
& ( pushprop_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X123: term,X124: term,X125: term,X126: term] :
( ( X125 = X126 )
| ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X123: term,X124: term,X125: term,X126: term] :
( ( X125 != X126 )
& ( ( ap @ X125 @ X123 )
= ( ap @ X126 @ X124 ) ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( $true = sP2 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP2 ) )
& ( ! [X196: term,X197: term] :
( ( X196 = X197 )
| ( ( lam @ X197 )
!= ( lam @ X196 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X196: term,X197: term] :
( ( X196 != X197 )
& ( ( lam @ X197 )
= ( lam @ X196 ) ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP0 ) )
& ( ( $true = sP0 )
| ( hoasinduction_lem3aa != $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ! [X198: term] :
? [X199: d_term] :
( ( d2term @ X199 )
= X198 )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( X153 = X155 )
| ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X153: term,X154: term,X155: term,X156: term] :
( ( X153 != X155 )
& ( ( ap @ ( sub @ X156 @ id ) @ X155 )
= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) ) )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
!= ( d2term @ X152 ) )
| ( X151 = X152 ) )
& ( ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ( $true
!= ( X40 @ X39 ) )
| ( $true
= ( X41 @ one ) )
| ? [X43: term] :
( ( X41 @ X43 )
!= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) ) )
| ? [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
!= X37 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ( $true
= ( X40 @ X39 ) )
& ( $true
!= ( X41 @ one ) )
& ! [X43: term] :
( ( X41 @ X43 )
= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) ) )
& ! [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
= X37 ) ) ) ),
inference(flattening,[status(thm)],[f473]) ).
thf(f473,plain,
( ( hoaslamnotvar = $true )
& ( ( apinj2 = $true )
| ? [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
= ( ap @ X3 @ X1 ) )
& ( X0 != X1 ) ) )
& ( ! [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) )
| ( X0 = X1 ) )
| ( apinj2 != $true ) )
& ( ulamvarind = $true )
& ( axapp = $true )
& ( hoasinduction_lem1 = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( 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 ) ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X32: term] :
( ( sub @ X32 @ id )
!= X32 ) )
& ( termmset_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( substmonoid_lthm = $true )
| ( ( ? [X35: subst] :
( ( comp @ X35 @ id )
!= X35 )
| ? [X36: subst] :
( ( comp @ id @ X36 )
!= X36 ) )
& ! [X33: subst] :
( ( comp @ id @ X33 )
= X33 )
& ! [X34: subst] :
( ( comp @ X34 @ id )
= X34 ) ) )
& ( ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) )
| ? [X33: subst] :
( ( comp @ id @ X33 )
!= X33 )
| ? [X34: subst] :
( ( comp @ X34 @ id )
!= X34 )
| ( substmonoid_lthm != $true ) )
& ! [X485: d_term] : ( d_one = X485 )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP58 ) )
& ( ( $true = sP58 )
| ( hoasinduction_lem2 != $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP56 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP56 ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( induction2_lthm = $true )
| ( $true != sP55 ) )
& ( ( $true = sP55 )
| ( induction2_lthm != $true ) )
& ( ( sP51 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP51 != $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoaslamnotap_gthm = $true )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
!= ( d2subst @ X14 ) )
| ( X13 = X14 ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X537: term] :
! [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
!= ( X538 @ sh @ one ) ) )
& ( ! [X537: term] :
? [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
= ( X538 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( shinj = $true )
| ? [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
= ( sub @ X515 @ sh ) )
& ( X514 != X515 ) ) )
& ( ! [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
!= ( sub @ X515 @ sh ) )
| ( X514 = X515 ) )
| ( shinj != $true ) )
& ( ( $true = sP45 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP45 ) )
& ( ( induction2lem = $true )
| ( $true != sP42 ) )
& ( ( $true = sP42 )
| ( induction2lem != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X77: subst > term > subst > $o] :
! [X78: term > $o] :
? [X79: term] :
( ( X78 @ X79 )
!= ( X77 @ id @ X79 @ id ) ) )
& ( ! [X77: subst > term > subst > $o] :
? [X78: term > $o] :
! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP40 ) )
& ( ( $true = sP40 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( ? [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
!= X203 )
| ? [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
!= X204 )
| ! [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) )
| ( $true
!= ( X207 @ X210 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ! [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
= X203 )
& ! [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
= X204 )
& ? [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
!= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) )
& ( $true
= ( X207 @ X210 ) ) ) ) )
& ( ( sP39 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP39 != $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X318: term] :
( ( sub @ X318 @ id )
= X318 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X318: term] :
( ( sub @ X318 @ id )
!= X318 ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP38 ) )
& ( ( $true = sP38 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( axabs = $true )
& ( hoasapinj2_gthm = $true )
& ( lamnotvar = $true )
& ( ulamvarsh = $true )
& ! [X106: d_subst] : ( d_id = X106 )
& ( ( induction = $true )
| ? [X479: term > $o,X480: term] :
( ( $true
!= ( X479 @ X480 ) )
& ! [X481: term] :
( ( $true
= ( X479 @ X481 ) )
| ( $true
!= ( var @ X481 ) ) )
& ( $true
!= ( sP36 @ X479 ) )
& ! [X482: term] :
( ( $true
= ( X479 @ ( lam @ X482 ) ) )
| ( $true
!= ( X479 @ X482 ) ) ) ) )
& ( ! [X479: term > $o,X480: term] :
( ( $true
= ( X479 @ X480 ) )
| ? [X481: term] :
( ( $true
!= ( X479 @ X481 ) )
& ( $true
= ( var @ X481 ) ) )
| ( $true
= ( sP36 @ X479 ) )
| ? [X482: term] :
( ( $true
!= ( X479 @ ( lam @ X482 ) ) )
& ( $true
= ( X479 @ X482 ) ) ) )
| ( induction != $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( ( axidr = $true )
| ? [X105: subst] :
( ( comp @ X105 @ id )
!= X105 ) )
& ( ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 )
| ( axidr != $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( $true = sP35 )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( $true != sP35 ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP31 ) )
& ( ( $true = sP31 )
| ( hoasinduction_lem2v2 != $true ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_gthm = $true )
& ( induction2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem3v2a = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3v2a != $true ) )
& ( axclos = $true )
& ( lamnotap != $true )
& ( ( axvarcons = $true )
| ? [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
!= X104 ) )
& ( ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 )
| ( axvarcons != $true ) )
& ( axvarshift = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
| ? [X532: term,X533: term > $o,X534: subst] :
! [X535: term > $o] :
? [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
!= ( X535 @ X536 ) ) )
& ( ! [X532: term,X533: term > $o,X534: subst] :
? [X535: term > $o] :
! [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
= ( X535 @ X536 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
!= ( ap @ X490 @ X489 ) )
| ( X487 = X490 ) )
& ? [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
& ( X491 != X493 ) )
& ! [X486: term] :
( ( sub @ X486 @ id )
= X486 ) ) )
& ( ? [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
= ( ap @ X490 @ X489 ) )
& ( X487 != X490 ) )
| ! [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
!= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
| ( X491 = X493 ) )
| ? [X486: term] :
( ( sub @ X486 @ id )
!= X486 )
| ( hoasapinj1_lthm != $true ) )
& ( ( $true = sP28 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( $true != sP28 ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP26 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP26 != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X117: subst > term > term,X118: term,X119: term] :
( ! [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
& ( ( lam @ ( X117 @ sh @ one ) )
= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) ) )
& ( ! [X117: subst > term > term,X118: term,X119: term] :
( ? [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
!= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) )
| ( hoaslamnotap != $true ) )
& ( pushprop = $true )
& ( pushprop_lthm = $true )
& ( ! [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
!= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
| ( X435 = X436 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
& ( X435 != X436 ) ) )
& ( apnotvar = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ( $true = sP24 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP24 ) )
& ( ( $true = sP23 )
| ( hoaslaminj_lthm != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP23 ) )
& ( pushprop_lem2v2 = $true )
& ( ( substmonoid = $true )
| ? [X449: subst] :
( ( comp @ X449 @ id )
!= X449 )
| ? [X448: subst] :
( ( comp @ id @ X448 )
!= X448 ) )
& ( ( ! [X449: subst] :
( ( comp @ X449 @ id )
= X449 )
& ! [X448: subst] :
( ( comp @ id @ X448 )
= X448 ) )
| ( substmonoid != $true ) )
& ( ! [X27: subst,X28: term,X29: term > $o] :
? [X30: term > $o] :
! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X27: subst,X28: term,X29: term > $o] :
! [X30: term > $o] :
? [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
!= ( X30 @ X31 ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X200: term] :
! [X201: subst > term > term] :
( ( X201 @ sh @ one )
!= ( sub @ X200 @ ( push @ one @ sh ) ) ) )
& ( ! [X200: term] :
? [X201: subst > term > term] :
( ( X201 @ sh @ one )
= ( sub @ X200 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( $true = sP22 )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( $true != sP22 ) )
& ( ! [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
| ( $true
!= ( X173 @ X174 ) ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
!= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
& ( $true
= ( X173 @ X174 ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X80: term] :
? [X81: subst > term > term] :
! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X80: term] :
! [X81: subst > term > term] :
? [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
!= ( X81 @ X82 @ X83 ) ) )
& ( ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
= ( X144 @ one ) )
| ( $true
!= ( X142 @ X145 ) )
| ? [X146: term] :
( ( X144 @ X146 )
!= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
!= ( X144 @ one ) )
& ( $true
= ( X142 @ X145 ) )
& ! [X146: term] :
( ( X144 @ X146 )
= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) ) ) )
& ( ulamvar1 != $true )
& ( axmap = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslamnotvar_lthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X147: term] :
! [X148: subst > term > term] :
? [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
!= ( X148 @ X150 @ X149 ) ) )
& ( ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ? [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
!= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ? [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
!= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ( ( X107 @ X108 @ X109 )
!= ( X110 @ X108 @ X109 ) )
& ( ( lam @ ( X110 @ sh @ one ) )
= ( lam @ ( X107 @ sh @ one ) ) )
& ! [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
& ! [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) ) )
& ( ( termmset_lthm = $true )
| ( ? [X26: term] :
( ( sub @ X26 @ id )
!= X26 )
& ! [X25: term] :
( ( sub @ X25 @ id )
= X25 ) ) )
& ( ! [X26: term] :
( ( sub @ X26 @ id )
= X26 )
| ? [X25: term] :
( ( sub @ X25 @ id )
!= X25 )
| ( termmset_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP17 ) )
& ( ( $true = sP17 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( axidl = $true )
| ? [X56: subst] :
( ( comp @ id @ X56 )
!= X56 ) )
& ( ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 )
| ( axidl != $true ) )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP14 ) )
& ( ( $true = sP14 )
| ( hoasinduction_lthm != $true ) )
& ( ! [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
!= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
| ( X217 = X218 ) )
| ? [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
= ( ap @ X213 @ X212 ) )
& ( X212 != X214 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
& ( X217 != X218 ) )
& ! [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
!= ( ap @ X213 @ X212 ) )
| ( X212 = X214 ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X445: subst > term > subst > $o] :
! [X446: term > $o] :
? [X447: term] :
( ( X445 @ id @ X447 @ id )
!= ( X446 @ X447 ) ) )
& ( ! [X445: subst > term > subst > $o] :
? [X446: term > $o] :
! [X447: term] :
( ( X445 @ id @ X447 @ id )
= ( X446 @ X447 ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( axassoc = $true )
& ( substmonoid_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X4: term,X5: subst > term > subst > $o] :
( ! [X11: term,X12: term] :
( ( $true
= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
| ( ( X5 @ id @ X11 @ id )
!= $true )
| ( $true
!= ( X5 @ id @ X12 @ id ) ) )
& ( $true
!= ( sP5 @ X5 ) )
& ( $true
!= ( X5 @ id @ X4 @ id ) ) ) )
& ( ! [X4: term,X5: subst > term > subst > $o] :
( ? [X11: term,X12: term] :
( ( $true
!= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
& ( ( X5 @ id @ X11 @ id )
= $true )
& ( $true
= ( X5 @ id @ X12 @ id ) ) )
| ( $true
= ( sP5 @ X5 ) )
| ( $true
= ( X5 @ id @ X4 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( induction2lem_gthm = $true )
& ( ( axshiftcons = $true )
| ? [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
!= X58 ) )
& ( ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 )
| ( axshiftcons != $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP4 ) )
& ( ( $true = sP4 )
| ( hoasinduction_lem3 != $true ) )
& ( hoasapnotvar = $true )
& ! [X266: subst] :
? [X267: d_subst] :
( ( d2subst @ X267 )
= X266 )
& ( ! [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
= X539 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
!= X539 ) )
& ( pushprop_lthm_orig = $true )
& ( pushprop_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X123: term,X124: term,X125: term,X126: term] :
( ( X125 = X126 )
| ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X123: term,X124: term,X125: term,X126: term] :
( ( X125 != X126 )
& ( ( ap @ X125 @ X123 )
= ( ap @ X126 @ X124 ) ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( $true = sP2 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP2 ) )
& ( ! [X196: term,X197: term] :
( ( X196 = X197 )
| ( ( lam @ X197 )
!= ( lam @ X196 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X196: term,X197: term] :
( ( X196 != X197 )
& ( ( lam @ X197 )
= ( lam @ X196 ) ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP0 ) )
& ( ( $true = sP0 )
| ( hoasinduction_lem3aa != $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ! [X198: term] :
? [X199: d_term] :
( ( d2term @ X199 )
= X198 )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( X153 = X155 )
| ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X153: term,X154: term,X155: term,X156: term] :
( ( X153 != X155 )
& ( ( ap @ ( sub @ X156 @ id ) @ X155 )
= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) ) )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
!= ( d2term @ X152 ) )
| ( X151 = X152 ) )
& ( ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ( $true
!= ( X40 @ X39 ) )
| ( $true
= ( X41 @ one ) )
| ? [X43: term] :
( ( X41 @ X43 )
!= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) ) )
| ? [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
!= X37 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ( $true
= ( X40 @ X39 ) )
& ( $true
!= ( X41 @ one ) )
& ! [X43: term] :
( ( X41 @ X43 )
= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) ) )
& ! [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
= X37 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f73]) ).
thf(f73,plain,
( ( hoaslamnotvar = $true )
& ( ( apinj2 = $true )
<=> ! [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) )
| ( X0 = X1 ) ) )
& ( ulamvarind = $true )
& ( axapp = $true )
& ( hoasinduction_lem1 = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( 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 ) ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
<=> ( termmset = $true ) )
& ( termmset_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( substmonoid_lthm = $true )
<=> ( ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) )
| ? [X33: subst] :
( ( comp @ id @ X33 )
!= X33 )
| ? [X34: subst] :
( ( comp @ X34 @ id )
!= X34 ) ) )
& ! [X485: d_term] : ( d_one = X485 )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem2 = $true )
<=> ( $true = sP58 ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ( $true = sP56 )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( induction2_lthm = $true )
<=> ( $true = sP55 ) )
& ( ( sP51 = $true )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoaslamnotap_gthm = $true )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
!= ( d2subst @ X14 ) )
| ( X13 = X14 ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X537: term] :
? [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
= ( X538 @ sh @ one ) ) )
& ( ( shinj = $true )
<=> ! [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
!= ( sub @ X515 @ sh ) )
| ( X514 = X515 ) ) )
& ( ( $true = sP45 )
<=> ( hoasinduction = $true ) )
& ( ( induction2lem = $true )
<=> ( $true = sP42 ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X77: subst > term > subst > $o] :
? [X78: term > $o] :
! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ( $true = sP40 ) )
& ( hoaslamnotap_lthm = $true )
& ( ( ? [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
!= X203 )
| ? [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
!= X204 )
| ! [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) )
| ( $true
!= ( X207 @ X210 ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( sP39 = $true )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X318: term] :
( ( sub @ X318 @ id )
= X318 )
<=> ( axvarid = $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( $true = sP38 ) )
& ( axabs = $true )
& ( hoasapinj2_gthm = $true )
& ( lamnotvar = $true )
& ( ulamvarsh = $true )
& ! [X106: d_subst] : ( d_id = X106 )
& ( ( induction = $true )
<=> ! [X479: term > $o,X480: term] :
( ( $true
= ( X479 @ X480 ) )
| ? [X481: term] :
( ( $true
!= ( X479 @ X481 ) )
& ( $true
= ( var @ X481 ) ) )
| ( $true
= ( sP36 @ X479 ) )
| ? [X482: term] :
( ( $true
!= ( X479 @ ( lam @ X482 ) ) )
& ( $true
= ( X479 @ X482 ) ) ) ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( ( axidr = $true )
<=> ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( $true = sP35 )
<=> ( induction2lem_lthm = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ( $true = sP31 ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_gthm = $true )
& ( induction2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem3v2a = $true )
<=> ( $true = sP29 ) )
& ( axclos = $true )
& ( lamnotap != $true )
& ( ( axvarcons = $true )
<=> ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 ) )
& ( axvarshift = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X532: term,X533: term > $o,X534: subst] :
? [X535: term > $o] :
! [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
= ( X535 @ X536 ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ? [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
= ( ap @ X490 @ X489 ) )
& ( X487 != X490 ) )
| ! [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
!= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
| ( X491 = X493 ) )
| ? [X486: term] :
( ( sub @ X486 @ id )
!= X486 ) ) )
& ( ( $true = sP28 )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( sP26 = $true )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ( hoaslamnotap = $true )
<=> ! [X117: subst > term > term,X118: term,X119: term] :
( ? [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
!= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) ) )
& ( pushprop = $true )
& ( pushprop_lthm = $true )
& ( ! [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
!= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
| ( X435 = X436 ) )
<=> ( hoasapinj1 = $true ) )
& ( apnotvar = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ( $true = sP24 )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( $true = sP23 )
<=> ( hoaslaminj_lthm = $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( substmonoid = $true )
<=> ( ! [X449: subst] :
( ( comp @ X449 @ id )
= X449 )
& ! [X448: subst] :
( ( comp @ id @ X448 )
= X448 ) ) )
& ( ! [X27: subst,X28: term,X29: term > $o] :
? [X30: term > $o] :
! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) )
<=> ( pushprop_lem0 = $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X200: term] :
? [X201: subst > term > term] :
( ( X201 @ sh @ one )
= ( sub @ X200 @ ( push @ one @ sh ) ) ) )
& ( ( $true = sP22 )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( ! [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
| ( $true
!= ( X173 @ X174 ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X80: term] :
? [X81: subst > term > term] :
! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
= ( X144 @ one ) )
| ( $true
!= ( X142 @ X145 ) )
| ? [X146: term] :
( ( X144 @ X146 )
!= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( ulamvar1 != $true )
& ( axmap = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslamnotvar_lthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ? [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
!= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ? [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
!= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( termmset_lthm = $true )
<=> ( ! [X26: term] :
( ( sub @ X26 @ id )
= X26 )
| ? [X25: term] :
( ( sub @ X25 @ id )
!= X25 ) ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( $true = sP17 ) )
& ( ( axidl = $true )
<=> ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 ) )
& ( ( hoasinduction_lthm = $true )
<=> ( $true = sP14 ) )
& ( ( ! [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
!= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
| ( X217 = X218 ) )
| ? [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
= ( ap @ X213 @ X212 ) )
& ( X212 != X214 ) ) )
<=> ( hoasapinj2_lthm = $true ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X445: subst > term > subst > $o] :
? [X446: term > $o] :
! [X447: term] :
( ( X445 @ id @ X447 @ id )
= ( X446 @ X447 ) ) )
& ( axassoc = $true )
& ( substmonoid_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X4: term,X5: subst > term > subst > $o] :
( ? [X11: term,X12: term] :
( ( $true
!= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
& ( ( X5 @ id @ X11 @ id )
= $true )
& ( $true
= ( X5 @ id @ X12 @ id ) ) )
| ( $true
= ( sP5 @ X5 ) )
| ( $true
= ( X5 @ id @ X4 @ id ) ) ) )
& ( induction2lem_gthm = $true )
& ( ( axshiftcons = $true )
<=> ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ( $true = sP4 ) )
& ( hoasapnotvar = $true )
& ! [X266: subst] :
? [X267: d_subst] :
( ( d2subst @ X267 )
= X266 )
& ( ! [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
= X539 )
<=> ( axscons = $true ) )
& ( pushprop_lthm_orig = $true )
& ( pushprop_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X123: term,X124: term,X125: term,X126: term] :
( ( X125 = X126 )
| ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) ) )
<=> ( apinj1 = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ( $true = sP2 )
<=> ( induction2 = $true ) )
& ( ! [X196: term,X197: term] :
( ( X196 = X197 )
| ( ( lam @ X197 )
!= ( lam @ X196 ) ) )
<=> ( laminj = $true ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ( $true = sP0 ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ! [X198: term] :
? [X199: d_term] :
( ( d2term @ X199 )
= X198 )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( X153 = X155 )
| ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
<=> ( hoasapinj2 = $true ) )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
!= ( d2term @ X152 ) )
| ( X151 = X152 ) )
& ( ( ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ( $true
!= ( X40 @ X39 ) )
| ( $true
= ( X41 @ one ) )
| ? [X43: term] :
( ( X41 @ X43 )
!= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) ) )
| ? [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
!= X37 ) )
<=> ( pushprop_lem1v2_lthm = $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,
( ( $true = sP0 )
<=> ! [X219: subst > term > subst > $o,X220: term] :
( ( $true
= ( X219 @ id @ ( lam @ ( sub @ X220 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X226: term] :
( ( $true
= ( X219 @ id @ X226 @ id ) )
& ( $true
!= ( X219 @ id @ ( sub @ X220 @ ( push @ X226 @ id ) ) @ id ) ) )
| ? [X221: subst > term > term] :
( ! [X222: term] :
( ( $true
!= ( X219 @ id @ X222 @ id ) )
| ( $true
= ( X219 @ id @ ( X221 @ id @ X222 ) @ id ) ) )
& ! [X223: term,X224: subst,X225: subst] :
( ( X221 @ ( comp @ X225 @ X224 ) @ ( sub @ X223 @ X224 ) )
= ( sub @ ( X221 @ X225 @ X223 ) @ X224 ) )
& ( $true
!= ( X219 @ id @ ( lam @ ( X221 @ sh @ one ) ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f15,plain,
! [X320: term > $o] :
( ( $true
= ( sP1 @ X320 ) )
<=> ? [X322: term,X323: term] :
( ( $true
!= ( X320 @ ( ap @ X323 @ X322 ) ) )
& ( $true
= ( X320 @ X322 ) )
& ( $true
= ( X320 @ X323 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f16,plain,
( ( $true = sP2 )
<=> ! [X319: term,X320: term > $o] :
( ( $true
= ( X320 @ X319 ) )
| ( $true
= ( sP1 @ X320 ) )
| ? [X324: term] :
( ! [X325: term] :
( ( $true
= ( X320 @ ( sub @ X324 @ ( push @ X325 @ id ) ) ) )
| ( $true
!= ( X320 @ X325 ) ) )
& ( $true
!= ( X320 @ ( lam @ X324 ) ) ) )
| ? [X321: term] :
( ( $true
!= ( X320 @ X321 ) )
& ( $true
= ( var @ X321 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f17,plain,
! [X517: subst > term > subst > $o] :
( ( $true
= ( sP3 @ X517 ) )
<=> ? [X522: subst > term > term] :
( ( ( X517 @ id @ ( lam @ ( X522 @ sh @ one ) ) @ id )
!= $true )
& ! [X523: term] :
( ( $true
= ( X517 @ id @ ( X522 @ id @ X523 ) @ id ) )
| ( $true
!= ( X517 @ id @ X523 @ id ) ) )
& ! [X524: term,X525: subst,X526: subst] :
( ( X522 @ ( comp @ X525 @ X526 ) @ ( sub @ X524 @ X526 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X526 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f18,plain,
( ( $true = sP4 )
<=> ! [X516: term,X517: subst > term > subst > $o] :
( ( $true
= ( sP3 @ X517 ) )
| ? [X528: subst,X529: subst,X530: subst,X531: term] :
( ( ( X517 @ X529 @ X531 @ ( comp @ X530 @ X528 ) )
!= $true )
& ( $true
= ( X517 @ ( comp @ X529 @ X530 ) @ ( sub @ X531 @ X530 ) @ X528 ) ) )
| ( $true
= ( X517 @ id @ ( lam @ X516 ) @ id ) )
| ? [X527: term] :
( ( $true
= ( X517 @ id @ X527 @ id ) )
& ( $true
!= ( X517 @ id @ ( sub @ X516 @ ( push @ X527 @ id ) ) @ id ) ) )
| ? [X518: subst,X519: term,X520: subst,X521: subst] :
( ( $true
!= ( X517 @ ( comp @ X521 @ X518 ) @ ( sub @ X519 @ X518 ) @ X520 ) )
& ( $true
= ( X517 @ X521 @ X519 @ ( comp @ X518 @ X520 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f19,plain,
! [X5: subst > term > subst > $o] :
( ( $true
= ( sP5 @ X5 ) )
<=> ? [X6: subst > term > term] :
( ! [X8: term,X9: subst,X10: subst] :
( ( X6 @ ( comp @ X10 @ X9 ) @ ( sub @ X8 @ X9 ) )
= ( sub @ ( X6 @ X10 @ X8 ) @ X9 ) )
& ! [X7: term] :
( ( $true
= ( X5 @ id @ ( X6 @ id @ X7 ) @ id ) )
| ( $true
!= ( X5 @ id @ X7 @ id ) ) )
& ( $true
!= ( X5 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f20,plain,
! [X399: subst > term > subst > $o] :
( ( $true
= ( sP6 @ X399 ) )
<=> ! [X410: subst > term > term] :
( ( $true
= ( X399 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
| ? [X411: term,X412: subst,X413: subst] :
( ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) )
!= ( sub @ ( X410 @ X412 @ X411 ) @ X413 ) )
| ? [X414: term] :
( ( $true
= ( X399 @ id @ X414 @ id ) )
& ( $true
!= ( X399 @ id @ ( X410 @ id @ X414 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f21,plain,
! [X380: term > $o] :
( ( $true
= ( sP7 @ X380 ) )
<=> ! [X381: term,X382: term] :
( ( $true
!= ( X380 @ X382 ) )
| ( ( X380 @ ( ap @ X381 @ X382 ) )
= $true )
| ( $true
!= ( X380 @ X381 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f22,plain,
! [X415: subst > term > subst > $o] :
( ( $true
= ( sP8 @ X415 ) )
<=> ? [X417: term,X418: term] :
( ( ( X415 @ id @ X417 @ id )
= $true )
& ( $true
= ( X415 @ id @ X418 @ id ) )
& ( $true
!= ( X415 @ id @ ( ap @ ( sub @ X418 @ id ) @ X417 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f23,plain,
! [X415: subst > term > subst > $o] :
( ( ( sP9 @ X415 )
= $true )
<=> ? [X419: subst > term > term] :
( ! [X421: term,X422: subst,X423: subst] :
( ( sub @ ( X419 @ X422 @ X421 ) @ X423 )
= ( X419 @ ( comp @ X422 @ X423 ) @ ( sub @ X421 @ X423 ) ) )
& ( ( X415 @ id @ ( lam @ ( X419 @ sh @ one ) ) @ id )
!= $true )
& ! [X420: term] :
( ( $true
= ( X415 @ id @ ( X419 @ id @ X420 ) @ id ) )
| ( $true
!= ( X415 @ id @ X420 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f24,plain,
! [X388: subst > term > subst > $o] :
( ( $true
= ( sP10 @ X388 ) )
<=> ! [X393: term,X394: term] :
( ( $true
= ( X388 @ id @ ( ap @ ( sub @ X394 @ id ) @ X393 ) @ id ) )
| ( $true
!= ( X388 @ id @ X393 @ id ) )
| ( $true
!= ( X388 @ id @ X394 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f25,plain,
( ( sP11 = $true )
<=> ? [X399: subst > term > subst > $o,X400: term] :
( ! [X402: subst,X403: term,X404: subst,X405: subst] :
( ( $true
= ( X399 @ X404 @ X403 @ ( comp @ X402 @ X405 ) ) )
| ( $true
!= ( X399 @ ( comp @ X404 @ X402 ) @ ( sub @ X403 @ X402 ) @ X405 ) ) )
& ! [X401: term] :
( ( $true
= ( X399 @ id @ ( sub @ X400 @ ( push @ X401 @ id ) ) @ id ) )
| ( $true
!= ( X399 @ id @ X401 @ id ) ) )
& ( $true
!= ( X399 @ id @ ( lam @ X400 ) @ id ) )
& ! [X406: subst,X407: term,X408: subst,X409: subst] :
( ( $true
= ( X399 @ ( comp @ X406 @ X409 ) @ ( sub @ X407 @ X409 ) @ X408 ) )
| ( $true
!= ( X399 @ X406 @ X407 @ ( comp @ X409 @ X408 ) ) ) )
& ( $true
= ( sP6 @ X399 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f26,plain,
( ( $true = sP12 )
<=> ! [X415: subst > term > subst > $o,X416: term] :
( ( $true
= ( X415 @ id @ X416 @ id ) )
| ? [X424: subst,X425: term,X426: subst,X427: subst] :
( ( $true
= ( X415 @ X427 @ X425 @ ( comp @ X424 @ X426 ) ) )
& ( $true
!= ( X415 @ ( comp @ X427 @ X424 ) @ ( sub @ X425 @ X424 ) @ X426 ) ) )
| ( ( sP9 @ X415 )
= $true )
| ( $true
= ( sP8 @ X415 ) )
| ? [X432: term] :
( ( $true
!= ( X415 @ id @ X432 @ id ) )
& ( $true
= ( var @ ( sub @ X432 @ id ) ) ) )
| ? [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X415 @ ( comp @ X430 @ X429 ) @ ( sub @ X428 @ X429 ) @ X431 ) )
& ( $true
!= ( X415 @ X430 @ X428 @ ( comp @ X429 @ X431 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f27,plain,
( ( $true = sP13 )
<=> ? [X379: term,X380: term > $o] :
( ( $true
!= ( X380 @ X379 ) )
& ! [X383: term] :
( ? [X384: term] :
( ( $true
!= ( X380 @ ( sub @ X383 @ ( push @ X384 @ id ) ) ) )
& ( $true
= ( X380 @ X384 ) ) )
| ( $true
= ( X380 @ ( lam @ X383 ) ) ) )
& ! [X385: term] :
( ( $true
!= ( var @ X385 ) )
| ( $true
= ( X380 @ X385 ) ) )
& ( $true
= ( sP7 @ X380 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f28,plain,
( ( $true = sP14 )
<=> ( ? [X386: term,X387: term,X388: subst > term > subst > $o] :
( ! [X389: subst,X390: subst,X391: term,X392: subst] :
( ( $true
= ( X388 @ ( comp @ X390 @ X389 ) @ ( sub @ X391 @ X389 ) @ X392 ) )
| ( $true
!= ( X388 @ X390 @ X391 @ ( comp @ X389 @ X392 ) ) ) )
& ( $true
= ( sP10 @ X388 ) )
& ( $true
= ( X388 @ id @ X386 @ id ) )
& ! [X395: subst,X396: subst,X397: subst,X398: term] :
( ( $true
= ( X388 @ X396 @ X398 @ ( comp @ X397 @ X395 ) ) )
| ( $true
!= ( X388 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X395 ) ) )
& ( $true
!= ( X388 @ id @ ( ap @ X386 @ X387 ) @ id ) )
& ( $true
= ( X388 @ id @ X387 @ id ) ) )
| ( $true = sP12 )
| ( $true = sP13 )
| ( sP11 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f29,plain,
! [X295: subst > term > subst > $o] :
( ( $true
= ( sP15 @ X295 ) )
<=> ? [X301: subst > term > term] :
( ! [X305: term] :
( ( $true
!= ( X295 @ id @ X305 @ id ) )
| ( $true
= ( X295 @ id @ ( X301 @ id @ X305 ) @ id ) ) )
& ( $true
!= ( X295 @ id @ ( lam @ ( X301 @ sh @ one ) ) @ id ) )
& ! [X302: term,X303: subst,X304: subst] :
( ( X301 @ ( comp @ X303 @ X304 ) @ ( sub @ X302 @ X304 ) )
= ( sub @ ( X301 @ X303 @ X302 ) @ X304 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f30,plain,
( ( $true = sP16 )
<=> ? [X286: term,X287: subst > term > subst > $o] :
( ( $true
!= ( X287 @ id @ ( lam @ ( sub @ X286 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X293: term] :
( ( $true
= ( X287 @ id @ ( sub @ X286 @ ( push @ X293 @ id ) ) @ id ) )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) )
& ! [X288: subst > term > term] :
( ? [X290: subst,X291: subst,X292: term] :
( ( X288 @ ( comp @ X291 @ X290 ) @ ( sub @ X292 @ X290 ) )
!= ( sub @ ( X288 @ X291 @ X292 ) @ X290 ) )
| ( $true
= ( X287 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id ) )
| ? [X289: term] :
( ( $true
= ( X287 @ id @ X289 @ id ) )
& ( $true
!= ( X287 @ id @ ( X288 @ id @ X289 ) @ id ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f31,plain,
( ( $true = sP17 )
<=> ( ! [X294: term,X295: subst > term > subst > $o] :
( ? [X296: subst,X297: subst,X298: subst,X299: term] :
( ( $true
= ( X295 @ ( comp @ X298 @ X296 ) @ ( sub @ X299 @ X296 ) @ X297 ) )
& ( $true
!= ( X295 @ X298 @ X299 @ ( comp @ X296 @ X297 ) ) ) )
| ? [X300: term] :
( ( $true
!= ( X295 @ id @ ( sub @ X294 @ ( push @ X300 @ id ) ) @ id ) )
& ( $true
= ( X295 @ id @ X300 @ id ) ) )
| ( $true
= ( sP15 @ X295 ) )
| ( $true
= ( X295 @ id @ ( lam @ X294 ) @ id ) )
| ? [X306: term,X307: subst,X308: subst,X309: subst] :
( ( $true
= ( X295 @ X308 @ X306 @ ( comp @ X309 @ X307 ) ) )
& ( $true
!= ( X295 @ ( comp @ X308 @ X309 ) @ ( sub @ X306 @ X309 ) @ X307 ) ) ) )
| ? [X285: term] :
( ( sub @ X285 @ id )
!= X285 )
| ( $true = sP16 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f32,plain,
! [X347: subst > term > subst > $o] :
( ( $true
= ( sP18 @ X347 ) )
<=> ? [X349: term,X350: term] :
( ( $true
!= ( X347 @ id @ ( ap @ ( sub @ X349 @ id ) @ X350 ) @ id ) )
& ( $true
= ( X347 @ id @ X350 @ id ) )
& ( $true
= ( X347 @ id @ X349 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f33,plain,
! [X329: term > $o] :
( ( $true
= ( sP19 @ X329 ) )
<=> ! [X334: term,X335: term] :
( ( $true
!= ( X329 @ X335 ) )
| ( $true
= ( X329 @ ( ap @ X335 @ X334 ) ) )
| ( $true
!= ( X329 @ X334 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f34,plain,
( ( $true = sP20 )
<=> ? [X337: subst > term > subst > $o,X338: term,X339: term > $o] :
( ! [X346: term] :
( ( X339 @ X346 )
= ( X337 @ id @ X346 @ id ) )
& ! [X340: term] :
( ( $true
= ( X339 @ ( sub @ X338 @ ( push @ X340 @ id ) ) ) )
| ( $true
!= ( X339 @ X340 ) ) )
& ! [X341: subst > term > term] :
( ? [X342: term] :
( ( $true
= ( X337 @ id @ X342 @ id ) )
& ( ( X337 @ id @ ( X341 @ id @ X342 ) @ id )
!= $true ) )
| ( $true
= ( X337 @ id @ ( lam @ ( X341 @ sh @ one ) ) @ id ) )
| ? [X343: subst,X344: term,X345: subst] :
( ( sub @ ( X341 @ X345 @ X344 ) @ X343 )
!= ( X341 @ ( comp @ X345 @ X343 ) @ ( sub @ X344 @ X343 ) ) ) )
& ( $true
!= ( X339 @ ( lam @ X338 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f35,plain,
( ( $true = sP21 )
<=> ! [X347: subst > term > subst > $o,X348: term] :
( ( $true
= ( X347 @ id @ X348 @ id ) )
| ? [X351: subst > term > term] :
( ! [X355: term] :
( ( $true
!= ( X347 @ id @ X355 @ id ) )
| ( $true
= ( X347 @ id @ ( X351 @ id @ X355 ) @ id ) ) )
& ( $true
!= ( X347 @ id @ ( lam @ ( X351 @ sh @ one ) ) @ id ) )
& ! [X352: subst,X353: subst,X354: term] :
( ( X351 @ ( comp @ X353 @ X352 ) @ ( sub @ X354 @ X352 ) )
= ( sub @ ( X351 @ X353 @ X354 ) @ X352 ) ) )
| ( $true
= ( sP18 @ X347 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f36,plain,
( ( $true = sP22 )
<=> ( ( $true = sP20 )
| ? [X329: term > $o,X330: term] :
( ! [X332: term] :
( ? [X333: term] :
( ( $true
!= ( X329 @ ( sub @ X332 @ ( push @ X333 @ id ) ) ) )
& ( $true
= ( X329 @ X333 ) ) )
| ( $true
= ( X329 @ ( lam @ X332 ) ) ) )
& ( $true
= ( sP19 @ X329 ) )
& ( $true
!= ( X329 @ X330 ) )
& ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( $true
!= ( var @ X331 ) ) ) )
| ? [X326: subst > term > subst > $o] :
! [X327: term > $o] :
? [X328: term] :
( ( X326 @ id @ X328 @ id )
!= ( X327 @ X328 ) )
| ? [X336: term] :
( ( sub @ X336 @ id )
!= X336 )
| ( $true = sP21 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f37,plain,
( ( $true = sP23 )
<=> ( ? [X161: term,X162: term] :
( ( ( lam @ X161 )
= ( lam @ X162 ) )
& ( X161 != X162 ) )
| ! [X163: subst,X164: subst > term > term,X165: term,X166: subst > term > term] :
( ? [X167: subst,X168: term,X169: subst] :
( ( sub @ ( X164 @ X167 @ X168 ) @ X169 )
!= ( X164 @ ( comp @ X167 @ X169 ) @ ( sub @ X168 @ X169 ) ) )
| ( ( X164 @ X163 @ X165 )
= ( X166 @ X163 @ X165 ) )
| ? [X170: term,X171: subst,X172: subst] :
( ( sub @ ( X166 @ X172 @ X170 ) @ X171 )
!= ( X166 @ ( comp @ X172 @ X171 ) @ ( sub @ X170 @ X171 ) ) )
| ( ( lam @ ( X166 @ sh @ one ) )
!= ( lam @ ( X164 @ sh @ one ) ) ) )
| ? [X157: term,X158: subst] :
( ( sub @ one @ ( push @ X157 @ X158 ) )
!= X157 )
| ? [X159: subst,X160: term] :
( ( comp @ sh @ ( push @ X160 @ X159 ) )
!= X159 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f38,plain,
( ( $true = sP24 )
<=> ! [X44: term,X45: subst > term > subst > $o] :
( ( $true
= ( X45 @ id @ ( lam @ ( sub @ X44 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X46: term] :
( ( ( X45 @ id @ ( sub @ X44 @ ( push @ X46 @ id ) ) @ id )
!= $true )
& ( $true
= ( X45 @ id @ X46 @ id ) ) )
| ? [X47: subst > term > term,X48: term] :
( ! [X49: subst,X50: term,X51: subst] :
( ( sub @ ( sub @ X48 @ ( push @ X50 @ X49 ) ) @ X51 )
= ( sub @ ( X47 @ X49 @ X50 ) @ X51 ) )
& ( $true
!= ( X45 @ id @ ( lam @ ( X47 @ sh @ one ) ) @ id ) )
& ! [X52: subst,X53: term,X54: subst] :
( ( sub @ X48 @ ( push @ ( sub @ X53 @ X52 ) @ ( comp @ X54 @ X52 ) ) )
= ( X47 @ ( comp @ X54 @ X52 ) @ ( sub @ X53 @ X52 ) ) )
& ! [X55: term] :
( ( $true
!= ( X45 @ id @ X55 @ id ) )
| ( $true
= ( X45 @ id @ ( X47 @ id @ X55 ) @ id ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f39,plain,
! [X179: subst > term > subst > $o] :
( ( $true
= ( sP25 @ X179 ) )
<=> ? [X182: subst > term > term] :
( ! [X184: subst,X185: subst,X186: term] :
( ( sub @ ( X182 @ X185 @ X186 ) @ X184 )
= ( X182 @ ( comp @ X185 @ X184 ) @ ( sub @ X186 @ X184 ) ) )
& ( $true
!= ( X179 @ id @ ( lam @ ( X182 @ sh @ one ) ) @ id ) )
& ! [X183: term] :
( ( $true
= ( X179 @ id @ ( X182 @ id @ X183 ) @ id ) )
| ( $true
!= ( X179 @ id @ X183 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f40,plain,
( ( sP26 = $true )
<=> ! [X178: term > $o,X179: subst > term > subst > $o,X180: term] :
( ( $true
= ( X178 @ ( lam @ X180 ) ) )
| ( $true
= ( sP25 @ X179 ) )
| ? [X192: subst,X193: subst,X194: term,X195: subst] :
( ( $true
!= ( X179 @ ( comp @ X192 @ X193 ) @ ( sub @ X194 @ X193 ) @ X195 ) )
& ( $true
= ( X179 @ X192 @ X194 @ ( comp @ X193 @ X195 ) ) ) )
| ? [X181: term] :
( ( X179 @ id @ X181 @ id )
!= ( X178 @ X181 ) )
| ? [X188: term,X189: subst,X190: subst,X191: subst] :
( ( $true
!= ( X179 @ X189 @ X188 @ ( comp @ X190 @ X191 ) ) )
& ( $true
= ( X179 @ ( comp @ X189 @ X190 ) @ ( sub @ X188 @ X190 ) @ X191 ) ) )
| ? [X187: term] :
( ( $true
!= ( X178 @ ( sub @ X180 @ ( push @ X187 @ id ) ) ) )
& ( $true
= ( X178 @ X187 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f41,plain,
( ( $true = sP27 )
<=> ? [X269: term,X270: subst > term > subst > $o] :
( ! [X271: subst > term > term] :
( ( $true
= ( X270 @ id @ ( lam @ ( X271 @ sh @ one ) ) @ id ) )
| ? [X275: term] :
( ( $true
!= ( X270 @ id @ ( X271 @ id @ X275 ) @ id ) )
& ( $true
= ( X270 @ id @ X275 @ id ) ) )
| ? [X272: subst,X273: term,X274: subst] :
( ( X271 @ ( comp @ X272 @ X274 ) @ ( sub @ X273 @ X274 ) )
!= ( sub @ ( X271 @ X272 @ X273 ) @ X274 ) ) )
& ( $true
!= ( X270 @ id @ ( lam @ ( sub @ X269 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X276: term] :
( ( $true
= ( X270 @ id @ ( sub @ X269 @ ( push @ X276 @ id ) ) @ id ) )
| ( $true
!= ( X270 @ id @ X276 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f42,plain,
( ( $true = sP28 )
<=> ( ( $true = sP27 )
| ? [X268: term] :
( ( sub @ X268 @ id )
!= X268 )
| ! [X277: term,X278: subst > term > subst > $o] :
( ( $true
= ( X278 @ id @ ( lam @ X277 ) @ id ) )
| ? [X284: term] :
( ( $true
= ( X278 @ id @ X284 @ id ) )
& ( $true
!= ( X278 @ id @ ( sub @ X277 @ ( push @ X284 @ id ) ) @ id ) ) )
| ? [X279: subst > term > term] :
( ( $true
!= ( X278 @ id @ ( lam @ ( X279 @ sh @ one ) ) @ id ) )
& ! [X283: term] :
( ( $true
!= ( X278 @ id @ X283 @ id ) )
| ( $true
= ( X278 @ id @ ( X279 @ id @ X283 ) @ id ) ) )
& ! [X280: subst,X281: term,X282: subst] :
( ( sub @ ( X279 @ X282 @ X281 ) @ X280 )
= ( X279 @ ( comp @ X282 @ X280 ) @ ( sub @ X281 @ X280 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f43,plain,
( ( $true = sP29 )
<=> ! [X15: subst > term > subst > $o,X16: term > $o,X17: term] :
( ? [X19: subst > term > term] :
( ! [X20: term,X21: subst,X22: subst] :
( ( sub @ ( X19 @ X22 @ X20 ) @ X21 )
= ( X19 @ ( comp @ X22 @ X21 ) @ ( sub @ X20 @ X21 ) ) )
& ! [X23: term] :
( ( $true
!= ( X15 @ id @ X23 @ id ) )
| ( $true
= ( X15 @ id @ ( X19 @ id @ X23 ) @ id ) ) )
& ( $true
!= ( X15 @ id @ ( lam @ ( X19 @ sh @ one ) ) @ id ) ) )
| ? [X18: term] :
( ( X16 @ X18 )
!= ( X15 @ id @ X18 @ id ) )
| ( $true
= ( X16 @ ( lam @ X17 ) ) )
| ? [X24: term] :
( ( $true
!= ( X16 @ ( sub @ X17 @ ( push @ X24 @ id ) ) ) )
& ( $true
= ( X16 @ X24 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f44,plain,
! [X129: subst > term > subst > $o] :
( ( $true
= ( sP30 @ X129 ) )
<=> ? [X140: term,X141: term] :
( ( $true
= ( X129 @ id @ X140 @ id ) )
& ( $true
= ( X129 @ id @ X141 @ id ) )
& ( $true
!= ( X129 @ id @ ( ap @ ( sub @ X140 @ id ) @ X141 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f45,plain,
( ( $true = sP31 )
<=> ! [X127: term,X128: term > $o,X129: subst > term > subst > $o,X130: term] :
( ( $true
!= ( X128 @ X130 ) )
| ? [X139: term] :
( ( X129 @ id @ X139 @ id )
!= ( X128 @ X139 ) )
| ( $true
= ( sP30 @ X129 ) )
| ( $true
= ( X128 @ ( ap @ X130 @ X127 ) ) )
| ? [X131: term,X132: subst,X133: subst,X134: subst] :
( ( $true
= ( X129 @ X133 @ X131 @ ( comp @ X132 @ X134 ) ) )
& ( $true
!= ( X129 @ ( comp @ X133 @ X132 ) @ ( sub @ X131 @ X132 ) @ X134 ) ) )
| ? [X135: subst,X136: subst,X137: term,X138: subst] :
( ( $true
!= ( X129 @ X138 @ X137 @ ( comp @ X136 @ X135 ) ) )
& ( $true
= ( X129 @ ( comp @ X138 @ X136 ) @ ( sub @ X137 @ X136 ) @ X135 ) ) )
| ( $true
!= ( X128 @ X127 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f46,plain,
! [X97: term > $o] :
( ( $true
= ( sP32 @ X97 ) )
<=> ? [X101: term,X102: term] :
( ( $true
= ( X97 @ X102 ) )
& ( $true
!= ( X97 @ ( ap @ X102 @ X101 ) ) )
& ( $true
= ( X97 @ X101 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f47,plain,
! [X90: term > $o] :
( ( $true
= ( sP33 @ X90 ) )
<=> ! [X91: term,X92: term] :
( ( ( X90 @ X91 )
!= $true )
| ( $true
= ( X90 @ ( ap @ X91 @ X92 ) ) )
| ( $true
!= ( X90 @ X92 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f48,plain,
( ( $true = sP34 )
<=> ! [X95: subst,X96: term,X97: term > $o] :
( ( $true
= ( X97 @ ( sub @ X96 @ X95 ) ) )
| ? [X100: term] :
( ( $true
= ( var @ X100 ) )
& ( $true
!= ( X97 @ ( sub @ X100 @ X95 ) ) ) )
| ? [X98: term] :
( ! [X99: term] :
( ( $true
!= ( X97 @ X99 ) )
| ( $true
= ( X97 @ ( sub @ X98 @ ( push @ X99 @ id ) ) ) ) )
& ( $true
!= ( X97 @ ( lam @ X98 ) ) ) )
| ( $true
= ( sP32 @ X97 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f49,plain,
( ( $true = sP35 )
<=> ( ? [X89: term,X90: term > $o] :
( ( $true
= ( sP33 @ X90 ) )
& ! [X93: term] :
( ( $true
!= ( X90 @ X93 ) )
| ( $true
= ( X90 @ ( lam @ X93 ) ) ) )
& ! [X94: term] :
( ( $true
!= ( var @ X94 ) )
| ( $true
= ( X90 @ X94 ) ) )
& ( $true
!= ( X90 @ X89 ) ) )
| ? [X84: subst,X85: term] :
( ( sub @ one @ ( push @ X85 @ X84 ) )
!= X85 )
| ( $true = sP34 )
| ? [X86: subst,X87: term] :
( ( comp @ sh @ ( push @ X87 @ X86 ) )
!= X86 )
| ? [X88: subst] :
( ( comp @ X88 @ id )
!= X88 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f50,plain,
! [X479: term > $o] :
( ( $true
= ( sP36 @ X479 ) )
<=> ? [X483: term,X484: term] :
( ( $true
= ( X479 @ X484 ) )
& ( $true
= ( X479 @ X483 ) )
& ( ( X479 @ ( ap @ X483 @ X484 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f51,plain,
! [X497: subst > term > subst > $o] :
( ( $true
= ( sP37 @ X497 ) )
<=> ? [X509: subst > term > term] :
( ! [X510: subst,X511: term,X512: subst] :
( ( sub @ ( X509 @ X512 @ X511 ) @ X510 )
= ( X509 @ ( comp @ X512 @ X510 ) @ ( sub @ X511 @ X510 ) ) )
& ( $true
!= ( X497 @ id @ ( lam @ ( X509 @ sh @ one ) ) @ id ) )
& ! [X513: term] :
( ( $true
= ( X497 @ id @ ( X509 @ id @ X513 ) @ id ) )
| ( $true
!= ( X497 @ id @ X513 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f52,plain,
( ( $true = sP38 )
<=> ( ! [X496: term > $o,X497: subst > term > subst > $o,X498: term] :
( ? [X504: term] :
( ( $true
!= ( X496 @ ( sub @ X498 @ ( push @ X504 @ id ) ) ) )
& ( $true
= ( X496 @ X504 ) ) )
| ? [X499: subst,X500: term,X501: subst,X502: subst] :
( ( $true
!= ( X497 @ ( comp @ X501 @ X499 ) @ ( sub @ X500 @ X499 ) @ X502 ) )
& ( $true
= ( X497 @ X501 @ X500 @ ( comp @ X499 @ X502 ) ) ) )
| ( $true
= ( X496 @ ( lam @ X498 ) ) )
| ? [X503: term] :
( ( X496 @ X503 )
!= ( X497 @ id @ X503 @ id ) )
| ? [X505: subst,X506: term,X507: subst,X508: subst] :
( ( ( X497 @ ( comp @ X508 @ X507 ) @ ( sub @ X506 @ X507 ) @ X505 )
= $true )
& ( $true
!= ( X497 @ X508 @ X506 @ ( comp @ X507 @ X505 ) ) ) )
| ( $true
= ( sP37 @ X497 ) ) )
| ? [X495: term] :
( ( sub @ X495 @ id )
!= X495 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f53,plain,
( ( sP39 = $true )
<=> ! [X310: subst > term > subst > $o,X311: term] :
( ? [X313: subst > term > term] :
( ! [X317: term] :
( ( $true
= ( X310 @ id @ ( X313 @ id @ X317 ) @ id ) )
| ( $true
!= ( X310 @ id @ X317 @ id ) ) )
& ( $true
!= ( X310 @ id @ ( lam @ ( X313 @ sh @ one ) ) @ id ) )
& ! [X314: subst,X315: term,X316: subst] :
( ( sub @ ( X313 @ X314 @ X315 ) @ X316 )
= ( X313 @ ( comp @ X314 @ X316 ) @ ( sub @ X315 @ X316 ) ) ) )
| ? [X312: term] :
( ( $true
!= ( X310 @ id @ ( sub @ X311 @ ( push @ X312 @ id ) ) @ id ) )
& ( $true
= ( X310 @ id @ X312 @ id ) ) )
| ( $true
= ( X310 @ id @ ( lam @ X311 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f54,plain,
( ( $true = sP40 )
<=> ! [X437: subst > term > subst > $o,X438: term] :
( ? [X444: term] :
( ( $true
= ( X437 @ id @ X444 @ id ) )
& ( $true
!= ( X437 @ id @ ( sub @ X438 @ ( push @ X444 @ id ) ) @ id ) ) )
| ? [X439: subst > term > term] :
( ( $true
!= ( X437 @ id @ ( lam @ ( X439 @ sh @ one ) ) @ id ) )
& ! [X440: term] :
( ( $true
= ( X437 @ id @ ( X439 @ id @ X440 ) @ id ) )
| ( $true
!= ( X437 @ id @ X440 @ id ) ) )
& ! [X441: subst,X442: term,X443: subst] :
( ( sub @ ( X439 @ X441 @ X442 ) @ X443 )
= ( X439 @ ( comp @ X441 @ X443 ) @ ( sub @ X442 @ X443 ) ) ) )
| ( $true
= ( X437 @ id @ ( lam @ ( sub @ X438 @ ( push @ one @ sh ) ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f55,plain,
! [X356: term > $o] :
( ( $true
= ( sP41 @ X356 ) )
<=> ? [X362: term,X363: term] :
( ( $true
= ( X356 @ X362 ) )
& ( $true
= ( X356 @ X363 ) )
& ( $true
!= ( X356 @ ( ap @ X362 @ X363 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f56,plain,
( ( $true = sP42 )
<=> ! [X356: term > $o,X357: subst,X358: term] :
( ( $true
= ( X356 @ ( sub @ X358 @ X357 ) ) )
| ? [X359: term] :
( ( $true
= ( var @ X359 ) )
& ( $true
!= ( X356 @ ( sub @ X359 @ X357 ) ) ) )
| ( $true
= ( sP41 @ X356 ) )
| ? [X360: term] :
( ! [X361: term] :
( ( $true
= ( X356 @ ( sub @ X360 @ ( push @ X361 @ id ) ) ) )
| ( $true
!= ( X356 @ X361 ) ) )
& ( $true
!= ( X356 @ ( lam @ X360 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f57,plain,
! [X60: subst > term > subst > $o] :
( ( $true
= ( sP43 @ X60 ) )
<=> ? [X61: term,X62: term] :
( ( $true
= ( X60 @ id @ X61 @ id ) )
& ( $true
!= ( X60 @ id @ ( ap @ ( sub @ X61 @ id ) @ X62 ) @ id ) )
& ( $true
= ( X60 @ id @ X62 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f58,plain,
! [X60: subst > term > subst > $o] :
( ( $true
= ( sP44 @ X60 ) )
<=> ? [X63: subst > term > term] :
( ( $true
!= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
& ! [X64: term] :
( ( $true
!= ( X60 @ id @ X64 @ id ) )
| ( $true
= ( X60 @ id @ ( X63 @ id @ X64 ) @ id ) ) )
& ! [X65: subst,X66: subst,X67: term] :
( ( X63 @ ( comp @ X65 @ X66 ) @ ( sub @ X67 @ X66 ) )
= ( sub @ ( X63 @ X65 @ X67 ) @ X66 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f59,plain,
( ( $true = sP45 )
<=> ! [X59: term,X60: subst > term > subst > $o] :
( ? [X68: term] :
( ( $true
= ( var @ ( sub @ X68 @ id ) ) )
& ( $true
!= ( X60 @ id @ X68 @ id ) ) )
| ( $true
= ( sP44 @ X60 ) )
| ( $true
= ( X60 @ id @ X59 @ id ) )
| ? [X73: term,X74: subst,X75: subst,X76: subst] :
( ( $true
!= ( X60 @ ( comp @ X75 @ X74 ) @ ( sub @ X73 @ X74 ) @ X76 ) )
& ( $true
= ( X60 @ X75 @ X73 @ ( comp @ X74 @ X76 ) ) ) )
| ( $true
= ( sP43 @ X60 ) )
| ? [X69: subst,X70: subst,X71: term,X72: subst] :
( ( $true
!= ( X60 @ X69 @ X71 @ ( comp @ X72 @ X70 ) ) )
& ( $true
= ( X60 @ ( comp @ X69 @ X72 ) @ ( sub @ X71 @ X72 ) @ X70 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f60,plain,
! [X249: subst > term > subst > $o] :
( ( $true
= ( sP46 @ X249 ) )
<=> ? [X256: subst > term > term] :
( ! [X257: term] :
( ( $true
!= ( X249 @ id @ X257 @ id ) )
| ( $true
= ( X249 @ id @ ( X256 @ id @ X257 ) @ id ) ) )
& ! [X258: subst,X259: term,X260: subst] :
( ( sub @ ( X256 @ X260 @ X259 ) @ X258 )
= ( X256 @ ( comp @ X260 @ X258 ) @ ( sub @ X259 @ X258 ) ) )
& ( $true
!= ( X249 @ id @ ( lam @ ( X256 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f61,plain,
! [X249: subst > term > subst > $o] :
( ( $true
= ( sP47 @ X249 ) )
<=> ? [X250: term,X251: term] :
( ( $true
!= ( X249 @ id @ ( ap @ ( sub @ X251 @ id ) @ X250 ) @ id ) )
& ( $true
= ( X249 @ id @ X250 @ id ) )
& ( $true
= ( X249 @ id @ X251 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f62,plain,
! [X230: term > $o] :
( ( $true
= ( sP48 @ X230 ) )
<=> ! [X235: term,X236: term] :
( ( $true
!= ( X230 @ X236 ) )
| ( $true
!= ( X230 @ X235 ) )
| ( $true
= ( X230 @ ( ap @ X235 @ X236 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f63,plain,
( ( $true = sP49 )
<=> ! [X248: term,X249: subst > term > subst > $o] :
( ? [X261: term,X262: subst,X263: subst,X264: subst] :
( ( $true
!= ( X249 @ X262 @ X261 @ ( comp @ X263 @ X264 ) ) )
& ( $true
= ( X249 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) @ X264 ) ) )
| ( $true
= ( sP47 @ X249 ) )
| ( $true
= ( sP46 @ X249 ) )
| ? [X252: subst,X253: term,X254: subst,X255: subst] :
( ( $true
!= ( X249 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) @ X255 ) )
& ( $true
= ( X249 @ X252 @ X253 @ ( comp @ X254 @ X255 ) ) ) )
| ? [X265: term] :
( ( $true
= ( var @ ( sub @ X265 @ id ) ) )
& ( $true
!= ( X249 @ id @ X265 @ id ) ) )
| ( $true
= ( X249 @ id @ X248 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f64,plain,
( ( $true = sP50 )
<=> ? [X238: term,X239: term > $o,X240: subst > term > subst > $o] :
( ! [X241: term] :
( ( $true
= ( X239 @ ( sub @ X238 @ ( push @ X241 @ id ) ) ) )
| ( $true
!= ( X239 @ X241 ) ) )
& ( $true
!= ( X239 @ ( lam @ X238 ) ) )
& ! [X247: term] :
( ( X240 @ id @ X247 @ id )
= ( X239 @ X247 ) )
& ! [X242: subst > term > term] :
( ( $true
= ( X240 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id ) )
| ? [X244: subst,X245: term,X246: subst] :
( ( X242 @ ( comp @ X246 @ X244 ) @ ( sub @ X245 @ X244 ) )
!= ( sub @ ( X242 @ X246 @ X245 ) @ X244 ) )
| ? [X243: term] :
( ( $true
!= ( X240 @ id @ ( X242 @ id @ X243 ) @ id ) )
& ( $true
= ( X240 @ id @ X243 @ id ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f65,plain,
( ( sP51 = $true )
<=> ( ( $true = sP50 )
| ? [X227: subst > term > subst > $o] :
! [X228: term > $o] :
? [X229: term] :
( ( X227 @ id @ X229 @ id )
!= ( X228 @ X229 ) )
| ? [X230: term > $o,X231: term] :
( ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( $true
!= ( var @ X232 ) ) )
& ( $true
= ( sP48 @ X230 ) )
& ( $true
!= ( X230 @ X231 ) )
& ! [X233: term] :
( ? [X234: term] :
( ( $true
= ( X230 @ X234 ) )
& ( $true
!= ( X230 @ ( sub @ X233 @ ( push @ X234 @ id ) ) ) ) )
| ( $true
= ( X230 @ ( lam @ X233 ) ) ) ) )
| ? [X237: term] :
( ( sub @ X237 @ id )
!= X237 )
| ( $true = sP49 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f66,plain,
! [X459: term > $o] :
( ( ( sP52 @ X459 )
= $true )
<=> ? [X462: term,X463: term] :
( ( $true
!= ( X459 @ ( ap @ X462 @ X463 ) ) )
& ( $true
= ( X459 @ X462 ) )
& ( $true
= ( X459 @ X463 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f67,plain,
! [X453: term > $o] :
( ( $true
= ( sP53 @ X453 ) )
<=> ! [X455: term,X456: term] :
( ( $true
!= ( X453 @ X455 ) )
| ( $true
= ( X453 @ ( ap @ X455 @ X456 ) ) )
| ( ( X453 @ X456 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f68,plain,
( ( $true = sP54 )
<=> ! [X459: term > $o,X460: term] :
( ( $true
= ( X459 @ X460 ) )
| ( ( sP52 @ X459 )
= $true )
| ? [X464: term] :
( ! [X465: term] :
( ( $true
!= ( X459 @ X465 ) )
| ( $true
= ( X459 @ ( sub @ X464 @ ( push @ X465 @ id ) ) ) ) )
& ( $true
!= ( X459 @ ( lam @ X464 ) ) ) )
| ? [X461: term] :
( ( $true
= ( var @ X461 ) )
& ( $true
!= ( X459 @ X461 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f69,plain,
( ( $true = sP55 )
<=> ( ? [X451: subst,X452: term,X453: term > $o] :
( ! [X454: term] :
( ( $true
!= ( var @ X454 ) )
| ( $true
= ( X453 @ ( sub @ X454 @ X451 ) ) ) )
& ( $true
= ( sP53 @ X453 ) )
& ! [X457: term] :
( ? [X458: term] :
( ( $true
!= ( X453 @ ( sub @ X457 @ ( push @ X458 @ id ) ) ) )
& ( $true
= ( X453 @ X458 ) ) )
| ( $true
= ( X453 @ ( lam @ X457 ) ) ) )
& ( $true
!= ( X453 @ ( sub @ X452 @ X451 ) ) ) )
| ( $true = sP54 )
| ? [X450: term] :
( ( sub @ X450 @ id )
!= X450 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f70,plain,
( ( $true = sP56 )
<=> ( ! [X369: term > $o,X370: subst > term > subst > $o,X371: term] :
( ? [X374: subst > term > term] :
( ! [X375: term] :
( ( $true
!= ( X370 @ id @ X375 @ id ) )
| ( $true
= ( X370 @ id @ ( X374 @ id @ X375 ) @ id ) ) )
& ! [X376: subst,X377: term,X378: subst] :
( ( sub @ ( X374 @ X376 @ X377 ) @ X378 )
= ( X374 @ ( comp @ X376 @ X378 ) @ ( sub @ X377 @ X378 ) ) )
& ( $true
!= ( X370 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id ) ) )
| ? [X373: term] :
( ( $true
= ( X369 @ X373 ) )
& ( ( X369 @ ( sub @ X371 @ ( push @ X373 @ id ) ) )
!= $true ) )
| ( $true
= ( X369 @ ( lam @ X371 ) ) )
| ? [X372: term] :
( ( X370 @ id @ X372 @ id )
!= ( X369 @ X372 ) ) )
| ? [X364: term] :
! [X365: subst > term > term] :
? [X366: subst,X367: term] :
( ( sub @ X364 @ ( push @ X367 @ X366 ) )
!= ( X365 @ X366 @ X367 ) )
| ? [X368: term] :
( ( sub @ X368 @ id )
!= X368 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f71,plain,
! [X468: subst > term > subst > $o] :
( ( $true
= ( sP57 @ X468 ) )
<=> ? [X469: term,X470: term] :
( ( $true
= ( X468 @ id @ X470 @ id ) )
& ( $true
= ( X468 @ id @ X469 @ id ) )
& ( $true
!= ( X468 @ id @ ( ap @ ( sub @ X470 @ id ) @ X469 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f72,plain,
( ( $true = sP58 )
<=> ! [X466: term,X467: term,X468: subst > term > subst > $o] :
( ? [X471: subst,X472: subst,X473: subst,X474: term] :
( ( $true
!= ( X468 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 ) )
& ( $true
= ( X468 @ X473 @ X474 @ ( comp @ X472 @ X471 ) ) ) )
| ( $true
!= ( X468 @ id @ X467 @ id ) )
| ( $true
!= ( X468 @ id @ X466 @ id ) )
| ( $true
= ( X468 @ id @ ( ap @ X467 @ X466 ) @ id ) )
| ( $true
= ( sP57 @ X468 ) )
| ? [X475: subst,X476: subst,X477: subst,X478: term] :
( ( $true
!= ( X468 @ X476 @ X478 @ ( comp @ X475 @ X477 ) ) )
& ( $true
= ( X468 @ ( comp @ X476 @ X475 ) @ ( sub @ X478 @ X475 ) @ X477 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f13,plain,
( ( hoaslamnotvar = $true )
& ( ( apinj2 = $true )
<=> ! [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) )
| ( X0 = X1 ) ) )
& ( ulamvarind = $true )
& ( axapp = $true )
& ( hoasinduction_lem1 = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( 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 ) ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
<=> ( termmset = $true ) )
& ( termmset_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( substmonoid_lthm = $true )
<=> ( ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) )
| ? [X33: subst] :
( ( comp @ id @ X33 )
!= X33 )
| ? [X34: subst] :
( ( comp @ X34 @ id )
!= X34 ) ) )
& ! [X485: d_term] : ( d_one = X485 )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X466: term,X467: term,X468: subst > term > subst > $o] :
( ? [X471: subst,X472: subst,X473: subst,X474: term] :
( ( $true
!= ( X468 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 ) )
& ( $true
= ( X468 @ X473 @ X474 @ ( comp @ X472 @ X471 ) ) ) )
| ( $true
!= ( X468 @ id @ X467 @ id ) )
| ( $true
!= ( X468 @ id @ X466 @ id ) )
| ( $true
= ( X468 @ id @ ( ap @ X467 @ X466 ) @ id ) )
| ? [X469: term,X470: term] :
( ( $true
= ( X468 @ id @ X470 @ id ) )
& ( $true
= ( X468 @ id @ X469 @ id ) )
& ( $true
!= ( X468 @ id @ ( ap @ ( sub @ X470 @ id ) @ X469 ) @ id ) ) )
| ? [X475: subst,X476: subst,X477: subst,X478: term] :
( ( $true
!= ( X468 @ X476 @ X478 @ ( comp @ X475 @ X477 ) ) )
& ( $true
= ( X468 @ ( comp @ X476 @ X475 ) @ ( sub @ X478 @ X475 ) @ X477 ) ) ) ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ( ! [X369: term > $o,X370: subst > term > subst > $o,X371: term] :
( ? [X374: subst > term > term] :
( ! [X375: term] :
( ( $true
!= ( X370 @ id @ X375 @ id ) )
| ( $true
= ( X370 @ id @ ( X374 @ id @ X375 ) @ id ) ) )
& ! [X376: subst,X377: term,X378: subst] :
( ( sub @ ( X374 @ X376 @ X377 ) @ X378 )
= ( X374 @ ( comp @ X376 @ X378 ) @ ( sub @ X377 @ X378 ) ) )
& ( $true
!= ( X370 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id ) ) )
| ? [X373: term] :
( ( $true
= ( X369 @ X373 ) )
& ( ( X369 @ ( sub @ X371 @ ( push @ X373 @ id ) ) )
!= $true ) )
| ( $true
= ( X369 @ ( lam @ X371 ) ) )
| ? [X372: term] :
( ( X370 @ id @ X372 @ id )
!= ( X369 @ X372 ) ) )
| ? [X364: term] :
! [X365: subst > term > term] :
? [X366: subst,X367: term] :
( ( sub @ X364 @ ( push @ X367 @ X366 ) )
!= ( X365 @ X366 @ X367 ) )
| ? [X368: term] :
( ( sub @ X368 @ id )
!= X368 ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( induction2_lthm = $true )
<=> ( ? [X451: subst,X452: term,X453: term > $o] :
( ! [X454: term] :
( ( $true
!= ( var @ X454 ) )
| ( $true
= ( X453 @ ( sub @ X454 @ X451 ) ) ) )
& ! [X455: term,X456: term] :
( ( $true
!= ( X453 @ X455 ) )
| ( $true
= ( X453 @ ( ap @ X455 @ X456 ) ) )
| ( ( X453 @ X456 )
!= $true ) )
& ! [X457: term] :
( ? [X458: term] :
( ( $true
!= ( X453 @ ( sub @ X457 @ ( push @ X458 @ id ) ) ) )
& ( $true
= ( X453 @ X458 ) ) )
| ( $true
= ( X453 @ ( lam @ X457 ) ) ) )
& ( $true
!= ( X453 @ ( sub @ X452 @ X451 ) ) ) )
| ! [X459: term > $o,X460: term] :
( ( $true
= ( X459 @ X460 ) )
| ? [X462: term,X463: term] :
( ( $true
!= ( X459 @ ( ap @ X462 @ X463 ) ) )
& ( $true
= ( X459 @ X462 ) )
& ( $true
= ( X459 @ X463 ) ) )
| ? [X464: term] :
( ! [X465: term] :
( ( $true
!= ( X459 @ X465 ) )
| ( $true
= ( X459 @ ( sub @ X464 @ ( push @ X465 @ id ) ) ) ) )
& ( $true
!= ( X459 @ ( lam @ X464 ) ) ) )
| ? [X461: term] :
( ( $true
= ( var @ X461 ) )
& ( $true
!= ( X459 @ X461 ) ) ) )
| ? [X450: term] :
( ( sub @ X450 @ id )
!= X450 ) ) )
& ( ( ? [X238: term,X239: term > $o,X240: subst > term > subst > $o] :
( ! [X241: term] :
( ( $true
= ( X239 @ ( sub @ X238 @ ( push @ X241 @ id ) ) ) )
| ( $true
!= ( X239 @ X241 ) ) )
& ( $true
!= ( X239 @ ( lam @ X238 ) ) )
& ! [X247: term] :
( ( X240 @ id @ X247 @ id )
= ( X239 @ X247 ) )
& ! [X242: subst > term > term] :
( ( $true
= ( X240 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id ) )
| ? [X244: subst,X245: term,X246: subst] :
( ( X242 @ ( comp @ X246 @ X244 ) @ ( sub @ X245 @ X244 ) )
!= ( sub @ ( X242 @ X246 @ X245 ) @ X244 ) )
| ? [X243: term] :
( ( $true
!= ( X240 @ id @ ( X242 @ id @ X243 ) @ id ) )
& ( $true
= ( X240 @ id @ X243 @ id ) ) ) ) )
| ? [X227: subst > term > subst > $o] :
! [X228: term > $o] :
? [X229: term] :
( ( X227 @ id @ X229 @ id )
!= ( X228 @ X229 ) )
| ? [X230: term > $o,X231: term] :
( ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( $true
!= ( var @ X232 ) ) )
& ! [X235: term,X236: term] :
( ( $true
!= ( X230 @ X236 ) )
| ( $true
!= ( X230 @ X235 ) )
| ( $true
= ( X230 @ ( ap @ X235 @ X236 ) ) ) )
& ( $true
!= ( X230 @ X231 ) )
& ! [X233: term] :
( ? [X234: term] :
( ( $true
= ( X230 @ X234 ) )
& ( $true
!= ( X230 @ ( sub @ X233 @ ( push @ X234 @ id ) ) ) ) )
| ( $true
= ( X230 @ ( lam @ X233 ) ) ) ) )
| ? [X237: term] :
( ( sub @ X237 @ id )
!= X237 )
| ! [X248: term,X249: subst > term > subst > $o] :
( ? [X261: term,X262: subst,X263: subst,X264: subst] :
( ( $true
!= ( X249 @ X262 @ X261 @ ( comp @ X263 @ X264 ) ) )
& ( $true
= ( X249 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) @ X264 ) ) )
| ? [X250: term,X251: term] :
( ( $true
!= ( X249 @ id @ ( ap @ ( sub @ X251 @ id ) @ X250 ) @ id ) )
& ( $true
= ( X249 @ id @ X250 @ id ) )
& ( $true
= ( X249 @ id @ X251 @ id ) ) )
| ? [X256: subst > term > term] :
( ! [X257: term] :
( ( $true
!= ( X249 @ id @ X257 @ id ) )
| ( $true
= ( X249 @ id @ ( X256 @ id @ X257 ) @ id ) ) )
& ! [X258: subst,X259: term,X260: subst] :
( ( sub @ ( X256 @ X260 @ X259 ) @ X258 )
= ( X256 @ ( comp @ X260 @ X258 ) @ ( sub @ X259 @ X258 ) ) )
& ( $true
!= ( X249 @ id @ ( lam @ ( X256 @ sh @ one ) ) @ id ) ) )
| ? [X252: subst,X253: term,X254: subst,X255: subst] :
( ( $true
!= ( X249 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) @ X255 ) )
& ( $true
= ( X249 @ X252 @ X253 @ ( comp @ X254 @ X255 ) ) ) )
| ? [X265: term] :
( ( $true
= ( var @ ( sub @ X265 @ id ) ) )
& ( $true
!= ( X249 @ id @ X265 @ id ) ) )
| ( $true
= ( X249 @ id @ X248 @ id ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoaslamnotap_gthm = $true )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
!= ( d2subst @ X14 ) )
| ( X13 = X14 ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X537: term] :
? [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
= ( X538 @ sh @ one ) ) )
& ( ( shinj = $true )
<=> ! [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
!= ( sub @ X515 @ sh ) )
| ( X514 = X515 ) ) )
& ( ! [X59: term,X60: subst > term > subst > $o] :
( ? [X68: term] :
( ( $true
= ( var @ ( sub @ X68 @ id ) ) )
& ( $true
!= ( X60 @ id @ X68 @ id ) ) )
| ? [X63: subst > term > term] :
( ( $true
!= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
& ! [X64: term] :
( ( $true
!= ( X60 @ id @ X64 @ id ) )
| ( $true
= ( X60 @ id @ ( X63 @ id @ X64 ) @ id ) ) )
& ! [X65: subst,X66: subst,X67: term] :
( ( X63 @ ( comp @ X65 @ X66 ) @ ( sub @ X67 @ X66 ) )
= ( sub @ ( X63 @ X65 @ X67 ) @ X66 ) ) )
| ( $true
= ( X60 @ id @ X59 @ id ) )
| ? [X73: term,X74: subst,X75: subst,X76: subst] :
( ( $true
!= ( X60 @ ( comp @ X75 @ X74 ) @ ( sub @ X73 @ X74 ) @ X76 ) )
& ( $true
= ( X60 @ X75 @ X73 @ ( comp @ X74 @ X76 ) ) ) )
| ? [X61: term,X62: term] :
( ( $true
= ( X60 @ id @ X61 @ id ) )
& ( $true
!= ( X60 @ id @ ( ap @ ( sub @ X61 @ id ) @ X62 ) @ id ) )
& ( $true
= ( X60 @ id @ X62 @ id ) ) )
| ? [X69: subst,X70: subst,X71: term,X72: subst] :
( ( $true
!= ( X60 @ X69 @ X71 @ ( comp @ X72 @ X70 ) ) )
& ( $true
= ( X60 @ ( comp @ X69 @ X72 ) @ ( sub @ X71 @ X72 ) @ X70 ) ) ) )
<=> ( hoasinduction = $true ) )
& ( ( induction2lem = $true )
<=> ! [X356: term > $o,X357: subst,X358: term] :
( ( $true
= ( X356 @ ( sub @ X358 @ X357 ) ) )
| ? [X359: term] :
( ( $true
= ( var @ X359 ) )
& ( $true
!= ( X356 @ ( sub @ X359 @ X357 ) ) ) )
| ? [X362: term,X363: term] :
( ( $true
= ( X356 @ X362 ) )
& ( $true
= ( X356 @ X363 ) )
& ( $true
!= ( X356 @ ( ap @ X362 @ X363 ) ) ) )
| ? [X360: term] :
( ! [X361: term] :
( ( $true
= ( X356 @ ( sub @ X360 @ ( push @ X361 @ id ) ) ) )
| ( $true
!= ( X356 @ X361 ) ) )
& ( $true
!= ( X356 @ ( lam @ X360 ) ) ) ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X77: subst > term > subst > $o] :
? [X78: term > $o] :
! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X437: subst > term > subst > $o,X438: term] :
( ? [X444: term] :
( ( $true
= ( X437 @ id @ X444 @ id ) )
& ( $true
!= ( X437 @ id @ ( sub @ X438 @ ( push @ X444 @ id ) ) @ id ) ) )
| ? [X439: subst > term > term] :
( ( $true
!= ( X437 @ id @ ( lam @ ( X439 @ sh @ one ) ) @ id ) )
& ! [X440: term] :
( ( $true
= ( X437 @ id @ ( X439 @ id @ X440 ) @ id ) )
| ( $true
!= ( X437 @ id @ X440 @ id ) ) )
& ! [X441: subst,X442: term,X443: subst] :
( ( sub @ ( X439 @ X441 @ X442 ) @ X443 )
= ( X439 @ ( comp @ X441 @ X443 ) @ ( sub @ X442 @ X443 ) ) ) )
| ( $true
= ( X437 @ id @ ( lam @ ( sub @ X438 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( ? [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
!= X203 )
| ? [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
!= X204 )
| ! [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) )
| ( $true
!= ( X207 @ X210 ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ! [X310: subst > term > subst > $o,X311: term] :
( ? [X313: subst > term > term] :
( ! [X317: term] :
( ( $true
= ( X310 @ id @ ( X313 @ id @ X317 ) @ id ) )
| ( $true
!= ( X310 @ id @ X317 @ id ) ) )
& ( $true
!= ( X310 @ id @ ( lam @ ( X313 @ sh @ one ) ) @ id ) )
& ! [X314: subst,X315: term,X316: subst] :
( ( sub @ ( X313 @ X314 @ X315 ) @ X316 )
= ( X313 @ ( comp @ X314 @ X316 ) @ ( sub @ X315 @ X316 ) ) ) )
| ? [X312: term] :
( ( $true
!= ( X310 @ id @ ( sub @ X311 @ ( push @ X312 @ id ) ) @ id ) )
& ( $true
= ( X310 @ id @ X312 @ id ) ) )
| ( $true
= ( X310 @ id @ ( lam @ X311 ) @ id ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X318: term] :
( ( sub @ X318 @ id )
= X318 )
<=> ( axvarid = $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X496: term > $o,X497: subst > term > subst > $o,X498: term] :
( ? [X504: term] :
( ( $true
!= ( X496 @ ( sub @ X498 @ ( push @ X504 @ id ) ) ) )
& ( $true
= ( X496 @ X504 ) ) )
| ? [X499: subst,X500: term,X501: subst,X502: subst] :
( ( $true
!= ( X497 @ ( comp @ X501 @ X499 ) @ ( sub @ X500 @ X499 ) @ X502 ) )
& ( $true
= ( X497 @ X501 @ X500 @ ( comp @ X499 @ X502 ) ) ) )
| ( $true
= ( X496 @ ( lam @ X498 ) ) )
| ? [X503: term] :
( ( X496 @ X503 )
!= ( X497 @ id @ X503 @ id ) )
| ? [X505: subst,X506: term,X507: subst,X508: subst] :
( ( ( X497 @ ( comp @ X508 @ X507 ) @ ( sub @ X506 @ X507 ) @ X505 )
= $true )
& ( $true
!= ( X497 @ X508 @ X506 @ ( comp @ X507 @ X505 ) ) ) )
| ? [X509: subst > term > term] :
( ! [X510: subst,X511: term,X512: subst] :
( ( sub @ ( X509 @ X512 @ X511 ) @ X510 )
= ( X509 @ ( comp @ X512 @ X510 ) @ ( sub @ X511 @ X510 ) ) )
& ( $true
!= ( X497 @ id @ ( lam @ ( X509 @ sh @ one ) ) @ id ) )
& ! [X513: term] :
( ( $true
= ( X497 @ id @ ( X509 @ id @ X513 ) @ id ) )
| ( $true
!= ( X497 @ id @ X513 @ id ) ) ) ) )
| ? [X495: term] :
( ( sub @ X495 @ id )
!= X495 ) ) )
& ( axabs = $true )
& ( hoasapinj2_gthm = $true )
& ( lamnotvar = $true )
& ( ulamvarsh = $true )
& ! [X106: d_subst] : ( d_id = X106 )
& ( ( induction = $true )
<=> ! [X479: term > $o,X480: term] :
( ( $true
= ( X479 @ X480 ) )
| ? [X481: term] :
( ( $true
!= ( X479 @ X481 ) )
& ( $true
= ( var @ X481 ) ) )
| ? [X483: term,X484: term] :
( ( $true
= ( X479 @ X484 ) )
& ( $true
= ( X479 @ X483 ) )
& ( ( X479 @ ( ap @ X483 @ X484 ) )
!= $true ) )
| ? [X482: term] :
( ( $true
!= ( X479 @ ( lam @ X482 ) ) )
& ( $true
= ( X479 @ X482 ) ) ) ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( ( axidr = $true )
<=> ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( ? [X89: term,X90: term > $o] :
( ! [X91: term,X92: term] :
( ( ( X90 @ X91 )
!= $true )
| ( $true
= ( X90 @ ( ap @ X91 @ X92 ) ) )
| ( $true
!= ( X90 @ X92 ) ) )
& ! [X93: term] :
( ( $true
!= ( X90 @ X93 ) )
| ( $true
= ( X90 @ ( lam @ X93 ) ) ) )
& ! [X94: term] :
( ( $true
!= ( var @ X94 ) )
| ( $true
= ( X90 @ X94 ) ) )
& ( $true
!= ( X90 @ X89 ) ) )
| ? [X84: subst,X85: term] :
( ( sub @ one @ ( push @ X85 @ X84 ) )
!= X85 )
| ! [X95: subst,X96: term,X97: term > $o] :
( ( $true
= ( X97 @ ( sub @ X96 @ X95 ) ) )
| ? [X100: term] :
( ( $true
= ( var @ X100 ) )
& ( $true
!= ( X97 @ ( sub @ X100 @ X95 ) ) ) )
| ? [X98: term] :
( ! [X99: term] :
( ( $true
!= ( X97 @ X99 ) )
| ( $true
= ( X97 @ ( sub @ X98 @ ( push @ X99 @ id ) ) ) ) )
& ( $true
!= ( X97 @ ( lam @ X98 ) ) ) )
| ? [X101: term,X102: term] :
( ( $true
= ( X97 @ X102 ) )
& ( $true
!= ( X97 @ ( ap @ X102 @ X101 ) ) )
& ( $true
= ( X97 @ X101 ) ) ) )
| ? [X86: subst,X87: term] :
( ( comp @ sh @ ( push @ X87 @ X86 ) )
!= X86 )
| ? [X88: subst] :
( ( comp @ X88 @ id )
!= X88 ) )
<=> ( induction2lem_lthm = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X127: term,X128: term > $o,X129: subst > term > subst > $o,X130: term] :
( ( $true
!= ( X128 @ X130 ) )
| ? [X139: term] :
( ( X129 @ id @ X139 @ id )
!= ( X128 @ X139 ) )
| ? [X140: term,X141: term] :
( ( $true
= ( X129 @ id @ X140 @ id ) )
& ( $true
= ( X129 @ id @ X141 @ id ) )
& ( $true
!= ( X129 @ id @ ( ap @ ( sub @ X140 @ id ) @ X141 ) @ id ) ) )
| ( $true
= ( X128 @ ( ap @ X130 @ X127 ) ) )
| ? [X131: term,X132: subst,X133: subst,X134: subst] :
( ( $true
= ( X129 @ X133 @ X131 @ ( comp @ X132 @ X134 ) ) )
& ( $true
!= ( X129 @ ( comp @ X133 @ X132 ) @ ( sub @ X131 @ X132 ) @ X134 ) ) )
| ? [X135: subst,X136: subst,X137: term,X138: subst] :
( ( $true
!= ( X129 @ X138 @ X137 @ ( comp @ X136 @ X135 ) ) )
& ( $true
= ( X129 @ ( comp @ X138 @ X136 ) @ ( sub @ X137 @ X136 ) @ X135 ) ) )
| ( $true
!= ( X128 @ X127 ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_gthm = $true )
& ( induction2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X15: subst > term > subst > $o,X16: term > $o,X17: term] :
( ? [X19: subst > term > term] :
( ! [X20: term,X21: subst,X22: subst] :
( ( sub @ ( X19 @ X22 @ X20 ) @ X21 )
= ( X19 @ ( comp @ X22 @ X21 ) @ ( sub @ X20 @ X21 ) ) )
& ! [X23: term] :
( ( $true
!= ( X15 @ id @ X23 @ id ) )
| ( $true
= ( X15 @ id @ ( X19 @ id @ X23 ) @ id ) ) )
& ( $true
!= ( X15 @ id @ ( lam @ ( X19 @ sh @ one ) ) @ id ) ) )
| ? [X18: term] :
( ( X16 @ X18 )
!= ( X15 @ id @ X18 @ id ) )
| ( $true
= ( X16 @ ( lam @ X17 ) ) )
| ? [X24: term] :
( ( $true
!= ( X16 @ ( sub @ X17 @ ( push @ X24 @ id ) ) ) )
& ( $true
= ( X16 @ X24 ) ) ) ) )
& ( axclos = $true )
& ( lamnotap != $true )
& ( ( axvarcons = $true )
<=> ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 ) )
& ( axvarshift = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X532: term,X533: term > $o,X534: subst] :
? [X535: term > $o] :
! [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
= ( X535 @ X536 ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ? [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
= ( ap @ X490 @ X489 ) )
& ( X487 != X490 ) )
| ! [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
!= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
| ( X491 = X493 ) )
| ? [X486: term] :
( ( sub @ X486 @ id )
!= X486 ) ) )
& ( ( ? [X269: term,X270: subst > term > subst > $o] :
( ! [X271: subst > term > term] :
( ( $true
= ( X270 @ id @ ( lam @ ( X271 @ sh @ one ) ) @ id ) )
| ? [X275: term] :
( ( $true
!= ( X270 @ id @ ( X271 @ id @ X275 ) @ id ) )
& ( $true
= ( X270 @ id @ X275 @ id ) ) )
| ? [X272: subst,X273: term,X274: subst] :
( ( X271 @ ( comp @ X272 @ X274 ) @ ( sub @ X273 @ X274 ) )
!= ( sub @ ( X271 @ X272 @ X273 ) @ X274 ) ) )
& ( $true
!= ( X270 @ id @ ( lam @ ( sub @ X269 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X276: term] :
( ( $true
= ( X270 @ id @ ( sub @ X269 @ ( push @ X276 @ id ) ) @ id ) )
| ( $true
!= ( X270 @ id @ X276 @ id ) ) ) )
| ? [X268: term] :
( ( sub @ X268 @ id )
!= X268 )
| ! [X277: term,X278: subst > term > subst > $o] :
( ( $true
= ( X278 @ id @ ( lam @ X277 ) @ id ) )
| ? [X284: term] :
( ( $true
= ( X278 @ id @ X284 @ id ) )
& ( $true
!= ( X278 @ id @ ( sub @ X277 @ ( push @ X284 @ id ) ) @ id ) ) )
| ? [X279: subst > term > term] :
( ( $true
!= ( X278 @ id @ ( lam @ ( X279 @ sh @ one ) ) @ id ) )
& ! [X283: term] :
( ( $true
!= ( X278 @ id @ X283 @ id ) )
| ( $true
= ( X278 @ id @ ( X279 @ id @ X283 ) @ id ) ) )
& ! [X280: subst,X281: term,X282: subst] :
( ( sub @ ( X279 @ X282 @ X281 ) @ X280 )
= ( X279 @ ( comp @ X282 @ X280 ) @ ( sub @ X281 @ X280 ) ) ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( ! [X178: term > $o,X179: subst > term > subst > $o,X180: term] :
( ( $true
= ( X178 @ ( lam @ X180 ) ) )
| ? [X182: subst > term > term] :
( ! [X184: subst,X185: subst,X186: term] :
( ( sub @ ( X182 @ X185 @ X186 ) @ X184 )
= ( X182 @ ( comp @ X185 @ X184 ) @ ( sub @ X186 @ X184 ) ) )
& ( $true
!= ( X179 @ id @ ( lam @ ( X182 @ sh @ one ) ) @ id ) )
& ! [X183: term] :
( ( $true
= ( X179 @ id @ ( X182 @ id @ X183 ) @ id ) )
| ( $true
!= ( X179 @ id @ X183 @ id ) ) ) )
| ? [X192: subst,X193: subst,X194: term,X195: subst] :
( ( $true
!= ( X179 @ ( comp @ X192 @ X193 ) @ ( sub @ X194 @ X193 ) @ X195 ) )
& ( $true
= ( X179 @ X192 @ X194 @ ( comp @ X193 @ X195 ) ) ) )
| ? [X181: term] :
( ( X179 @ id @ X181 @ id )
!= ( X178 @ X181 ) )
| ? [X188: term,X189: subst,X190: subst,X191: subst] :
( ( $true
!= ( X179 @ X189 @ X188 @ ( comp @ X190 @ X191 ) ) )
& ( $true
= ( X179 @ ( comp @ X189 @ X190 ) @ ( sub @ X188 @ X190 ) @ X191 ) ) )
| ? [X187: term] :
( ( $true
!= ( X178 @ ( sub @ X180 @ ( push @ X187 @ id ) ) ) )
& ( $true
= ( X178 @ X187 ) ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ( hoaslamnotap = $true )
<=> ! [X117: subst > term > term,X118: term,X119: term] :
( ? [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
!= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) ) )
& ( pushprop = $true )
& ( pushprop_lthm = $true )
& ( ! [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
!= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
| ( X435 = X436 ) )
<=> ( hoasapinj1 = $true ) )
& ( apnotvar = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X44: term,X45: subst > term > subst > $o] :
( ( $true
= ( X45 @ id @ ( lam @ ( sub @ X44 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X46: term] :
( ( ( X45 @ id @ ( sub @ X44 @ ( push @ X46 @ id ) ) @ id )
!= $true )
& ( $true
= ( X45 @ id @ X46 @ id ) ) )
| ? [X47: subst > term > term,X48: term] :
( ! [X49: subst,X50: term,X51: subst] :
( ( sub @ ( sub @ X48 @ ( push @ X50 @ X49 ) ) @ X51 )
= ( sub @ ( X47 @ X49 @ X50 ) @ X51 ) )
& ( $true
!= ( X45 @ id @ ( lam @ ( X47 @ sh @ one ) ) @ id ) )
& ! [X52: subst,X53: term,X54: subst] :
( ( sub @ X48 @ ( push @ ( sub @ X53 @ X52 ) @ ( comp @ X54 @ X52 ) ) )
= ( X47 @ ( comp @ X54 @ X52 ) @ ( sub @ X53 @ X52 ) ) )
& ! [X55: term] :
( ( $true
!= ( X45 @ id @ X55 @ id ) )
| ( $true
= ( X45 @ id @ ( X47 @ id @ X55 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( ? [X161: term,X162: term] :
( ( ( lam @ X161 )
= ( lam @ X162 ) )
& ( X161 != X162 ) )
| ! [X163: subst,X164: subst > term > term,X165: term,X166: subst > term > term] :
( ? [X167: subst,X168: term,X169: subst] :
( ( sub @ ( X164 @ X167 @ X168 ) @ X169 )
!= ( X164 @ ( comp @ X167 @ X169 ) @ ( sub @ X168 @ X169 ) ) )
| ( ( X164 @ X163 @ X165 )
= ( X166 @ X163 @ X165 ) )
| ? [X170: term,X171: subst,X172: subst] :
( ( sub @ ( X166 @ X172 @ X170 ) @ X171 )
!= ( X166 @ ( comp @ X172 @ X171 ) @ ( sub @ X170 @ X171 ) ) )
| ( ( lam @ ( X166 @ sh @ one ) )
!= ( lam @ ( X164 @ sh @ one ) ) ) )
| ? [X157: term,X158: subst] :
( ( sub @ one @ ( push @ X157 @ X158 ) )
!= X157 )
| ? [X159: subst,X160: term] :
( ( comp @ sh @ ( push @ X160 @ X159 ) )
!= X159 ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( substmonoid = $true )
<=> ( ! [X449: subst] :
( ( comp @ X449 @ id )
= X449 )
& ! [X448: subst] :
( ( comp @ id @ X448 )
= X448 ) ) )
& ( ! [X27: subst,X28: term,X29: term > $o] :
? [X30: term > $o] :
! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) )
<=> ( pushprop_lem0 = $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X200: term] :
? [X201: subst > term > term] :
( ( X201 @ sh @ one )
= ( sub @ X200 @ ( push @ one @ sh ) ) ) )
& ( ( ? [X337: subst > term > subst > $o,X338: term,X339: term > $o] :
( ! [X346: term] :
( ( X339 @ X346 )
= ( X337 @ id @ X346 @ id ) )
& ! [X340: term] :
( ( $true
= ( X339 @ ( sub @ X338 @ ( push @ X340 @ id ) ) ) )
| ( $true
!= ( X339 @ X340 ) ) )
& ! [X341: subst > term > term] :
( ? [X342: term] :
( ( $true
= ( X337 @ id @ X342 @ id ) )
& ( ( X337 @ id @ ( X341 @ id @ X342 ) @ id )
!= $true ) )
| ( $true
= ( X337 @ id @ ( lam @ ( X341 @ sh @ one ) ) @ id ) )
| ? [X343: subst,X344: term,X345: subst] :
( ( sub @ ( X341 @ X345 @ X344 ) @ X343 )
!= ( X341 @ ( comp @ X345 @ X343 ) @ ( sub @ X344 @ X343 ) ) ) )
& ( $true
!= ( X339 @ ( lam @ X338 ) ) ) )
| ? [X329: term > $o,X330: term] :
( ! [X332: term] :
( ? [X333: term] :
( ( $true
!= ( X329 @ ( sub @ X332 @ ( push @ X333 @ id ) ) ) )
& ( $true
= ( X329 @ X333 ) ) )
| ( $true
= ( X329 @ ( lam @ X332 ) ) ) )
& ! [X334: term,X335: term] :
( ( $true
!= ( X329 @ X335 ) )
| ( $true
= ( X329 @ ( ap @ X335 @ X334 ) ) )
| ( $true
!= ( X329 @ X334 ) ) )
& ( $true
!= ( X329 @ X330 ) )
& ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( $true
!= ( var @ X331 ) ) ) )
| ? [X326: subst > term > subst > $o] :
! [X327: term > $o] :
? [X328: term] :
( ( X326 @ id @ X328 @ id )
!= ( X327 @ X328 ) )
| ? [X336: term] :
( ( sub @ X336 @ id )
!= X336 )
| ! [X347: subst > term > subst > $o,X348: term] :
( ( $true
= ( X347 @ id @ X348 @ id ) )
| ? [X351: subst > term > term] :
( ! [X355: term] :
( ( $true
!= ( X347 @ id @ X355 @ id ) )
| ( $true
= ( X347 @ id @ ( X351 @ id @ X355 ) @ id ) ) )
& ( $true
!= ( X347 @ id @ ( lam @ ( X351 @ sh @ one ) ) @ id ) )
& ! [X352: subst,X353: subst,X354: term] :
( ( X351 @ ( comp @ X353 @ X352 ) @ ( sub @ X354 @ X352 ) )
= ( sub @ ( X351 @ X353 @ X354 ) @ X352 ) ) )
| ? [X349: term,X350: term] :
( ( $true
!= ( X347 @ id @ ( ap @ ( sub @ X349 @ id ) @ X350 ) @ id ) )
& ( $true
= ( X347 @ id @ X350 @ id ) )
& ( $true
= ( X347 @ id @ X349 @ id ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( ! [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
| ( $true
!= ( X173 @ X174 ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X80: term] :
? [X81: subst > term > term] :
! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
= ( X144 @ one ) )
| ( $true
!= ( X142 @ X145 ) )
| ? [X146: term] :
( ( X144 @ X146 )
!= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( ulamvar1 != $true )
& ( axmap = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslamnotvar_lthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ? [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
!= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ? [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
!= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( termmset_lthm = $true )
<=> ( ! [X26: term] :
( ( sub @ X26 @ id )
= X26 )
| ? [X25: term] :
( ( sub @ X25 @ id )
!= X25 ) ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X294: term,X295: subst > term > subst > $o] :
( ? [X296: subst,X297: subst,X298: subst,X299: term] :
( ( $true
= ( X295 @ ( comp @ X298 @ X296 ) @ ( sub @ X299 @ X296 ) @ X297 ) )
& ( $true
!= ( X295 @ X298 @ X299 @ ( comp @ X296 @ X297 ) ) ) )
| ? [X300: term] :
( ( $true
!= ( X295 @ id @ ( sub @ X294 @ ( push @ X300 @ id ) ) @ id ) )
& ( $true
= ( X295 @ id @ X300 @ id ) ) )
| ? [X301: subst > term > term] :
( ! [X305: term] :
( ( $true
!= ( X295 @ id @ X305 @ id ) )
| ( $true
= ( X295 @ id @ ( X301 @ id @ X305 ) @ id ) ) )
& ( $true
!= ( X295 @ id @ ( lam @ ( X301 @ sh @ one ) ) @ id ) )
& ! [X302: term,X303: subst,X304: subst] :
( ( X301 @ ( comp @ X303 @ X304 ) @ ( sub @ X302 @ X304 ) )
= ( sub @ ( X301 @ X303 @ X302 ) @ X304 ) ) )
| ( $true
= ( X295 @ id @ ( lam @ X294 ) @ id ) )
| ? [X306: term,X307: subst,X308: subst,X309: subst] :
( ( $true
= ( X295 @ X308 @ X306 @ ( comp @ X309 @ X307 ) ) )
& ( $true
!= ( X295 @ ( comp @ X308 @ X309 ) @ ( sub @ X306 @ X309 ) @ X307 ) ) ) )
| ? [X285: term] :
( ( sub @ X285 @ id )
!= X285 )
| ? [X286: term,X287: subst > term > subst > $o] :
( ( $true
!= ( X287 @ id @ ( lam @ ( sub @ X286 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X293: term] :
( ( $true
= ( X287 @ id @ ( sub @ X286 @ ( push @ X293 @ id ) ) @ id ) )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) )
& ! [X288: subst > term > term] :
( ? [X290: subst,X291: subst,X292: term] :
( ( X288 @ ( comp @ X291 @ X290 ) @ ( sub @ X292 @ X290 ) )
!= ( sub @ ( X288 @ X291 @ X292 ) @ X290 ) )
| ( $true
= ( X287 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id ) )
| ? [X289: term] :
( ( $true
= ( X287 @ id @ X289 @ id ) )
& ( $true
!= ( X287 @ id @ ( X288 @ id @ X289 ) @ id ) ) ) ) ) ) )
& ( ( axidl = $true )
<=> ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ? [X386: term,X387: term,X388: subst > term > subst > $o] :
( ! [X389: subst,X390: subst,X391: term,X392: subst] :
( ( $true
= ( X388 @ ( comp @ X390 @ X389 ) @ ( sub @ X391 @ X389 ) @ X392 ) )
| ( $true
!= ( X388 @ X390 @ X391 @ ( comp @ X389 @ X392 ) ) ) )
& ! [X393: term,X394: term] :
( ( $true
= ( X388 @ id @ ( ap @ ( sub @ X394 @ id ) @ X393 ) @ id ) )
| ( $true
!= ( X388 @ id @ X393 @ id ) )
| ( $true
!= ( X388 @ id @ X394 @ id ) ) )
& ( $true
= ( X388 @ id @ X386 @ id ) )
& ! [X395: subst,X396: subst,X397: subst,X398: term] :
( ( $true
= ( X388 @ X396 @ X398 @ ( comp @ X397 @ X395 ) ) )
| ( $true
!= ( X388 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X395 ) ) )
& ( $true
!= ( X388 @ id @ ( ap @ X386 @ X387 ) @ id ) )
& ( $true
= ( X388 @ id @ X387 @ id ) ) )
| ! [X415: subst > term > subst > $o,X416: term] :
( ( $true
= ( X415 @ id @ X416 @ id ) )
| ? [X424: subst,X425: term,X426: subst,X427: subst] :
( ( $true
= ( X415 @ X427 @ X425 @ ( comp @ X424 @ X426 ) ) )
& ( $true
!= ( X415 @ ( comp @ X427 @ X424 ) @ ( sub @ X425 @ X424 ) @ X426 ) ) )
| ? [X419: subst > term > term] :
( ! [X421: term,X422: subst,X423: subst] :
( ( sub @ ( X419 @ X422 @ X421 ) @ X423 )
= ( X419 @ ( comp @ X422 @ X423 ) @ ( sub @ X421 @ X423 ) ) )
& ( ( X415 @ id @ ( lam @ ( X419 @ sh @ one ) ) @ id )
!= $true )
& ! [X420: term] :
( ( $true
= ( X415 @ id @ ( X419 @ id @ X420 ) @ id ) )
| ( $true
!= ( X415 @ id @ X420 @ id ) ) ) )
| ? [X417: term,X418: term] :
( ( ( X415 @ id @ X417 @ id )
= $true )
& ( $true
= ( X415 @ id @ X418 @ id ) )
& ( $true
!= ( X415 @ id @ ( ap @ ( sub @ X418 @ id ) @ X417 ) @ id ) ) )
| ? [X432: term] :
( ( $true
!= ( X415 @ id @ X432 @ id ) )
& ( $true
= ( var @ ( sub @ X432 @ id ) ) ) )
| ? [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X415 @ ( comp @ X430 @ X429 ) @ ( sub @ X428 @ X429 ) @ X431 ) )
& ( $true
!= ( X415 @ X430 @ X428 @ ( comp @ X429 @ X431 ) ) ) ) )
| ? [X379: term,X380: term > $o] :
( ( $true
!= ( X380 @ X379 ) )
& ! [X383: term] :
( ? [X384: term] :
( ( $true
!= ( X380 @ ( sub @ X383 @ ( push @ X384 @ id ) ) ) )
& ( $true
= ( X380 @ X384 ) ) )
| ( $true
= ( X380 @ ( lam @ X383 ) ) ) )
& ! [X385: term] :
( ( $true
!= ( var @ X385 ) )
| ( $true
= ( X380 @ X385 ) ) )
& ! [X381: term,X382: term] :
( ( $true
!= ( X380 @ X382 ) )
| ( ( X380 @ ( ap @ X381 @ X382 ) )
= $true )
| ( $true
!= ( X380 @ X381 ) ) ) )
| ? [X399: subst > term > subst > $o,X400: term] :
( ! [X402: subst,X403: term,X404: subst,X405: subst] :
( ( $true
= ( X399 @ X404 @ X403 @ ( comp @ X402 @ X405 ) ) )
| ( $true
!= ( X399 @ ( comp @ X404 @ X402 ) @ ( sub @ X403 @ X402 ) @ X405 ) ) )
& ! [X401: term] :
( ( $true
= ( X399 @ id @ ( sub @ X400 @ ( push @ X401 @ id ) ) @ id ) )
| ( $true
!= ( X399 @ id @ X401 @ id ) ) )
& ( $true
!= ( X399 @ id @ ( lam @ X400 ) @ id ) )
& ! [X406: subst,X407: term,X408: subst,X409: subst] :
( ( $true
= ( X399 @ ( comp @ X406 @ X409 ) @ ( sub @ X407 @ X409 ) @ X408 ) )
| ( $true
!= ( X399 @ X406 @ X407 @ ( comp @ X409 @ X408 ) ) ) )
& ! [X410: subst > term > term] :
( ( $true
= ( X399 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
| ? [X411: term,X412: subst,X413: subst] :
( ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) )
!= ( sub @ ( X410 @ X412 @ X411 ) @ X413 ) )
| ? [X414: term] :
( ( $true
= ( X399 @ id @ X414 @ id ) )
& ( $true
!= ( X399 @ id @ ( X410 @ id @ X414 ) @ id ) ) ) ) ) ) )
& ( ( ! [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
!= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
| ( X217 = X218 ) )
| ? [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
= ( ap @ X213 @ X212 ) )
& ( X212 != X214 ) ) )
<=> ( hoasapinj2_lthm = $true ) )
& ( hoasapinj1_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X445: subst > term > subst > $o] :
? [X446: term > $o] :
! [X447: term] :
( ( X445 @ id @ X447 @ id )
= ( X446 @ X447 ) ) )
& ( axassoc = $true )
& ( substmonoid_gthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X4: term,X5: subst > term > subst > $o] :
( ? [X11: term,X12: term] :
( ( $true
!= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
& ( ( X5 @ id @ X11 @ id )
= $true )
& ( $true
= ( X5 @ id @ X12 @ id ) ) )
| ? [X6: subst > term > term] :
( ! [X8: term,X9: subst,X10: subst] :
( ( X6 @ ( comp @ X10 @ X9 ) @ ( sub @ X8 @ X9 ) )
= ( sub @ ( X6 @ X10 @ X8 ) @ X9 ) )
& ! [X7: term] :
( ( $true
= ( X5 @ id @ ( X6 @ id @ X7 ) @ id ) )
| ( $true
!= ( X5 @ id @ X7 @ id ) ) )
& ( $true
!= ( X5 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X5 @ id @ X4 @ id ) ) ) )
& ( induction2lem_gthm = $true )
& ( ( axshiftcons = $true )
<=> ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X516: term,X517: subst > term > subst > $o] :
( ? [X522: subst > term > term] :
( ( ( X517 @ id @ ( lam @ ( X522 @ sh @ one ) ) @ id )
!= $true )
& ! [X523: term] :
( ( $true
= ( X517 @ id @ ( X522 @ id @ X523 ) @ id ) )
| ( $true
!= ( X517 @ id @ X523 @ id ) ) )
& ! [X524: term,X525: subst,X526: subst] :
( ( X522 @ ( comp @ X525 @ X526 ) @ ( sub @ X524 @ X526 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X526 ) ) )
| ? [X528: subst,X529: subst,X530: subst,X531: term] :
( ( ( X517 @ X529 @ X531 @ ( comp @ X530 @ X528 ) )
!= $true )
& ( $true
= ( X517 @ ( comp @ X529 @ X530 ) @ ( sub @ X531 @ X530 ) @ X528 ) ) )
| ( $true
= ( X517 @ id @ ( lam @ X516 ) @ id ) )
| ? [X527: term] :
( ( $true
= ( X517 @ id @ X527 @ id ) )
& ( $true
!= ( X517 @ id @ ( sub @ X516 @ ( push @ X527 @ id ) ) @ id ) ) )
| ? [X518: subst,X519: term,X520: subst,X521: subst] :
( ( $true
!= ( X517 @ ( comp @ X521 @ X518 ) @ ( sub @ X519 @ X518 ) @ X520 ) )
& ( $true
= ( X517 @ X521 @ X519 @ ( comp @ X518 @ X520 ) ) ) ) ) )
& ( hoasapnotvar = $true )
& ! [X266: subst] :
? [X267: d_subst] :
( ( d2subst @ X267 )
= X266 )
& ( ! [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
= X539 )
<=> ( axscons = $true ) )
& ( pushprop_lthm_orig = $true )
& ( pushprop_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X123: term,X124: term,X125: term,X126: term] :
( ( X125 = X126 )
| ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) ) )
<=> ( apinj1 = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ! [X319: term,X320: term > $o] :
( ( $true
= ( X320 @ X319 ) )
| ? [X322: term,X323: term] :
( ( $true
!= ( X320 @ ( ap @ X323 @ X322 ) ) )
& ( $true
= ( X320 @ X322 ) )
& ( $true
= ( X320 @ X323 ) ) )
| ? [X324: term] :
( ! [X325: term] :
( ( $true
= ( X320 @ ( sub @ X324 @ ( push @ X325 @ id ) ) ) )
| ( $true
!= ( X320 @ X325 ) ) )
& ( $true
!= ( X320 @ ( lam @ X324 ) ) ) )
| ? [X321: term] :
( ( $true
!= ( X320 @ X321 ) )
& ( $true
= ( var @ X321 ) ) ) )
<=> ( induction2 = $true ) )
& ( ! [X196: term,X197: term] :
( ( X196 = X197 )
| ( ( lam @ X197 )
!= ( lam @ X196 ) ) )
<=> ( laminj = $true ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X219: subst > term > subst > $o,X220: term] :
( ( $true
= ( X219 @ id @ ( lam @ ( sub @ X220 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X226: term] :
( ( $true
= ( X219 @ id @ X226 @ id ) )
& ( $true
!= ( X219 @ id @ ( sub @ X220 @ ( push @ X226 @ id ) ) @ id ) ) )
| ? [X221: subst > term > term] :
( ! [X222: term] :
( ( $true
!= ( X219 @ id @ X222 @ id ) )
| ( $true
= ( X219 @ id @ ( X221 @ id @ X222 ) @ id ) ) )
& ! [X223: term,X224: subst,X225: subst] :
( ( X221 @ ( comp @ X225 @ X224 ) @ ( sub @ X223 @ X224 ) )
= ( sub @ ( X221 @ X225 @ X223 ) @ X224 ) )
& ( $true
!= ( X219 @ id @ ( lam @ ( X221 @ sh @ one ) ) @ id ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ! [X198: term] :
? [X199: d_term] :
( ( d2term @ X199 )
= X198 )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( X153 = X155 )
| ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
<=> ( hoasapinj2 = $true ) )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
!= ( d2term @ X152 ) )
| ( X151 = X152 ) )
& ( ( ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ( $true
!= ( X40 @ X39 ) )
| ( $true
= ( X41 @ one ) )
| ? [X43: term] :
( ( X41 @ X43 )
!= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) ) )
| ? [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
!= X37 ) )
<=> ( pushprop_lem1v2_lthm = $true ) ) ),
inference(flattening,[status(thm)],[f12]) ).
thf(f12,plain,
( ( axclos = $true )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X445: subst > term > subst > $o] :
? [X446: term > $o] :
! [X447: term] :
( ( X445 @ id @ X447 @ id )
= ( X446 @ X447 ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ! [X266: subst] :
? [X267: d_subst] :
( ( d2subst @ X267 )
= X266 )
& ( ! [X59: term,X60: subst > term > subst > $o] :
( ? [X68: term] :
( ( $true
= ( var @ ( sub @ X68 @ id ) ) )
& ( $true
!= ( X60 @ id @ X68 @ id ) ) )
| ? [X63: subst > term > term] :
( ( $true
!= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
& ! [X64: term] :
( ( $true
!= ( X60 @ id @ X64 @ id ) )
| ( $true
= ( X60 @ id @ ( X63 @ id @ X64 ) @ id ) ) )
& ! [X65: subst,X66: subst,X67: term] :
( ( X63 @ ( comp @ X65 @ X66 ) @ ( sub @ X67 @ X66 ) )
= ( sub @ ( X63 @ X65 @ X67 ) @ X66 ) ) )
| ( $true
= ( X60 @ id @ X59 @ id ) )
| ? [X73: term,X74: subst,X75: subst,X76: subst] :
( ( $true
!= ( X60 @ ( comp @ X75 @ X74 ) @ ( sub @ X73 @ X74 ) @ X76 ) )
& ( $true
= ( X60 @ X75 @ X73 @ ( comp @ X74 @ X76 ) ) ) )
| ? [X61: term,X62: term] :
( ( $true
= ( X60 @ id @ X61 @ id ) )
& ( $true
!= ( X60 @ id @ ( ap @ ( sub @ X61 @ id ) @ X62 ) @ id ) )
& ( $true
= ( X60 @ id @ X62 @ id ) ) )
| ? [X69: subst,X70: subst,X71: term,X72: subst] :
( ( $true
!= ( X60 @ X69 @ X71 @ ( comp @ X72 @ X70 ) ) )
& ( $true
= ( X60 @ ( comp @ X69 @ X72 ) @ ( sub @ X71 @ X72 ) @ X70 ) ) ) )
<=> ( hoasinduction = $true ) )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
!= ( d2subst @ X14 ) )
| ( X13 = X14 ) )
& ( one
= ( d2term @ d_one ) )
& ( ( d2subst @ d_id )
= sh )
& ( ( induction2lem = $true )
<=> ! [X356: term > $o,X357: subst,X358: term] :
( ( $true
= ( X356 @ ( sub @ X358 @ X357 ) ) )
| ? [X359: term] :
( ( $true
= ( var @ X359 ) )
& ( $true
!= ( X356 @ ( sub @ X359 @ X357 ) ) ) )
| ? [X362: term,X363: term] :
( ( $true
= ( X356 @ X362 ) )
& ( $true
= ( X356 @ X363 ) )
& ( $true
!= ( X356 @ ( ap @ X362 @ X363 ) ) ) )
| ? [X360: term] :
( ! [X361: term] :
( ( $true
= ( X356 @ ( sub @ X360 @ ( push @ X361 @ id ) ) ) )
| ( $true
!= ( X356 @ X361 ) ) )
& ( $true
!= ( X356 @ ( lam @ X360 ) ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ( axidl = $true )
<=> ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X437: subst > term > subst > $o,X438: term] :
( ? [X444: term] :
( ( $true
= ( X437 @ id @ X444 @ id ) )
& ( $true
!= ( X437 @ id @ ( sub @ X438 @ ( push @ X444 @ id ) ) @ id ) ) )
| ? [X439: subst > term > term] :
( ( $true
!= ( X437 @ id @ ( lam @ ( X439 @ sh @ one ) ) @ id ) )
& ! [X440: term] :
( ( $true
= ( X437 @ id @ ( X439 @ id @ X440 ) @ id ) )
| ( $true
!= ( X437 @ id @ X440 @ id ) ) )
& ! [X441: subst,X442: term,X443: subst] :
( ( sub @ ( X439 @ X441 @ X442 ) @ X443 )
= ( X439 @ ( comp @ X441 @ X443 ) @ ( sub @ X442 @ X443 ) ) ) )
| ( $true
= ( X437 @ id @ ( lam @ ( sub @ X438 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X15: subst > term > subst > $o,X16: term > $o,X17: term] :
( ? [X19: subst > term > term] :
( ! [X20: term,X21: subst,X22: subst] :
( ( sub @ ( X19 @ X22 @ X20 ) @ X21 )
= ( X19 @ ( comp @ X22 @ X21 ) @ ( sub @ X20 @ X21 ) ) )
& ! [X23: term] :
( ( $true
!= ( X15 @ id @ X23 @ id ) )
| ( $true
= ( X15 @ id @ ( X19 @ id @ X23 ) @ id ) ) )
& ( $true
!= ( X15 @ id @ ( lam @ ( X19 @ sh @ one ) ) @ id ) ) )
| ? [X18: term] :
( ( X16 @ X18 )
!= ( X15 @ id @ X18 @ id ) )
| ( $true
= ( X16 @ ( lam @ X17 ) ) )
| ? [X24: term] :
( ( $true
!= ( X16 @ ( sub @ X17 @ ( push @ X24 @ id ) ) ) )
& ( $true
= ( X16 @ X24 ) ) ) ) )
& ( ulamvarsh = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X277: term,X278: subst > term > subst > $o] :
( ( $true
= ( X278 @ id @ ( lam @ X277 ) @ id ) )
| ? [X284: term] :
( ( $true
= ( X278 @ id @ X284 @ id ) )
& ( $true
!= ( X278 @ id @ ( sub @ X277 @ ( push @ X284 @ id ) ) @ id ) ) )
| ? [X279: subst > term > term] :
( ( $true
!= ( X278 @ id @ ( lam @ ( X279 @ sh @ one ) ) @ id ) )
& ! [X283: term] :
( ( $true
!= ( X278 @ id @ X283 @ id ) )
| ( $true
= ( X278 @ id @ ( X279 @ id @ X283 ) @ id ) ) )
& ! [X280: subst,X281: term,X282: subst] :
( ( sub @ ( X279 @ X282 @ X281 ) @ X280 )
= ( X279 @ ( comp @ X282 @ X280 ) @ ( sub @ X281 @ X280 ) ) ) ) )
| ? [X269: term,X270: subst > term > subst > $o] :
( ! [X271: subst > term > term] :
( ( $true
= ( X270 @ id @ ( lam @ ( X271 @ sh @ one ) ) @ id ) )
| ? [X275: term] :
( ( $true
!= ( X270 @ id @ ( X271 @ id @ X275 ) @ id ) )
& ( $true
= ( X270 @ id @ X275 @ id ) ) )
| ? [X272: subst,X273: term,X274: subst] :
( ( X271 @ ( comp @ X272 @ X274 ) @ ( sub @ X273 @ X274 ) )
!= ( sub @ ( X271 @ X272 @ X273 ) @ X274 ) ) )
& ( $true
!= ( X270 @ id @ ( lam @ ( sub @ X269 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X276: term] :
( ( $true
= ( X270 @ id @ ( sub @ X269 @ ( push @ X276 @ id ) ) @ id ) )
| ( $true
!= ( X270 @ id @ X276 @ id ) ) ) )
| ? [X268: term] :
( ( sub @ X268 @ id )
!= X268 ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( lamnotap != $true )
& ( axapp = $true )
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( X153 = X155 )
| ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( ! [X44: term,X45: subst > term > subst > $o] :
( ( $true
= ( X45 @ id @ ( lam @ ( sub @ X44 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X46: term] :
( ( ( X45 @ id @ ( sub @ X44 @ ( push @ X46 @ id ) ) @ id )
!= $true )
& ( $true
= ( X45 @ id @ X46 @ id ) ) )
| ? [X47: subst > term > term,X48: term] :
( ! [X49: subst,X50: term,X51: subst] :
( ( sub @ ( sub @ X48 @ ( push @ X50 @ X49 ) ) @ X51 )
= ( sub @ ( X47 @ X49 @ X50 ) @ X51 ) )
& ( $true
!= ( X45 @ id @ ( lam @ ( X47 @ sh @ one ) ) @ id ) )
& ! [X52: subst,X53: term,X54: subst] :
( ( sub @ X48 @ ( push @ ( sub @ X53 @ X52 ) @ ( comp @ X54 @ X52 ) ) )
= ( X47 @ ( comp @ X54 @ X52 ) @ ( sub @ X53 @ X52 ) ) )
& ! [X55: term] :
( ( $true
!= ( X45 @ id @ X55 @ id ) )
| ( $true
= ( X45 @ id @ ( X47 @ id @ X55 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
= ( X144 @ one ) )
| ( $true
!= ( X142 @ X145 ) )
| ? [X146: term] :
( ( X144 @ X146 )
!= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( hoasapinj2_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( lamnotvar = $true )
& ( pushprop_gthm = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( substmonoid = $true )
<=> ( ! [X449: subst] :
( ( comp @ X449 @ id )
= X449 )
& ! [X448: subst] :
( ( comp @ id @ X448 )
= X448 ) ) )
& ( ( ! [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) )
| ( $true
!= ( X207 @ X210 ) ) )
| ? [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
!= X204 )
| ? [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
!= X203 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( axvarcons = $true )
<=> ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ! [X80: term] :
? [X81: subst > term > term] :
! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ! [X123: term,X124: term,X125: term,X126: term] :
( ( X125 = X126 )
| ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) ) )
<=> ( apinj1 = $true ) )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
<=> ( termmset = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ! [X196: term,X197: term] :
( ( X196 = X197 )
| ( ( lam @ X197 )
!= ( lam @ X196 ) ) )
<=> ( laminj = $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( ! [X415: subst > term > subst > $o,X416: term] :
( ( $true
= ( X415 @ id @ X416 @ id ) )
| ? [X424: subst,X425: term,X426: subst,X427: subst] :
( ( $true
= ( X415 @ X427 @ X425 @ ( comp @ X424 @ X426 ) ) )
& ( $true
!= ( X415 @ ( comp @ X427 @ X424 ) @ ( sub @ X425 @ X424 ) @ X426 ) ) )
| ? [X419: subst > term > term] :
( ! [X421: term,X422: subst,X423: subst] :
( ( sub @ ( X419 @ X422 @ X421 ) @ X423 )
= ( X419 @ ( comp @ X422 @ X423 ) @ ( sub @ X421 @ X423 ) ) )
& ( ( X415 @ id @ ( lam @ ( X419 @ sh @ one ) ) @ id )
!= $true )
& ! [X420: term] :
( ( $true
= ( X415 @ id @ ( X419 @ id @ X420 ) @ id ) )
| ( $true
!= ( X415 @ id @ X420 @ id ) ) ) )
| ? [X417: term,X418: term] :
( ( ( X415 @ id @ X417 @ id )
= $true )
& ( $true
= ( X415 @ id @ X418 @ id ) )
& ( $true
!= ( X415 @ id @ ( ap @ ( sub @ X418 @ id ) @ X417 ) @ id ) ) )
| ? [X432: term] :
( ( $true
!= ( X415 @ id @ X432 @ id ) )
& ( $true
= ( var @ ( sub @ X432 @ id ) ) ) )
| ? [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X415 @ ( comp @ X430 @ X429 ) @ ( sub @ X428 @ X429 ) @ X431 ) )
& ( $true
!= ( X415 @ X430 @ X428 @ ( comp @ X429 @ X431 ) ) ) ) )
| ? [X399: subst > term > subst > $o,X400: term] :
( ! [X402: subst,X403: term,X404: subst,X405: subst] :
( ( $true
= ( X399 @ X404 @ X403 @ ( comp @ X402 @ X405 ) ) )
| ( $true
!= ( X399 @ ( comp @ X404 @ X402 ) @ ( sub @ X403 @ X402 ) @ X405 ) ) )
& ! [X401: term] :
( ( $true
= ( X399 @ id @ ( sub @ X400 @ ( push @ X401 @ id ) ) @ id ) )
| ( $true
!= ( X399 @ id @ X401 @ id ) ) )
& ( $true
!= ( X399 @ id @ ( lam @ X400 ) @ id ) )
& ! [X406: subst,X407: term,X408: subst,X409: subst] :
( ( $true
= ( X399 @ ( comp @ X406 @ X409 ) @ ( sub @ X407 @ X409 ) @ X408 ) )
| ( $true
!= ( X399 @ X406 @ X407 @ ( comp @ X409 @ X408 ) ) ) )
& ! [X410: subst > term > term] :
( ( $true
= ( X399 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
| ? [X411: term,X412: subst,X413: subst] :
( ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) )
!= ( sub @ ( X410 @ X412 @ X411 ) @ X413 ) )
| ? [X414: term] :
( ( $true
= ( X399 @ id @ X414 @ id ) )
& ( $true
!= ( X399 @ id @ ( X410 @ id @ X414 ) @ id ) ) ) ) )
| ? [X386: term,X387: term,X388: subst > term > subst > $o] :
( ! [X389: subst,X390: subst,X391: term,X392: subst] :
( ( $true
= ( X388 @ ( comp @ X390 @ X389 ) @ ( sub @ X391 @ X389 ) @ X392 ) )
| ( $true
!= ( X388 @ X390 @ X391 @ ( comp @ X389 @ X392 ) ) ) )
& ! [X393: term,X394: term] :
( ( $true
= ( X388 @ id @ ( ap @ ( sub @ X394 @ id ) @ X393 ) @ id ) )
| ( $true
!= ( X388 @ id @ X393 @ id ) )
| ( $true
!= ( X388 @ id @ X394 @ id ) ) )
& ( $true
= ( X388 @ id @ X386 @ id ) )
& ! [X395: subst,X396: subst,X397: subst,X398: term] :
( ( $true
= ( X388 @ X396 @ X398 @ ( comp @ X397 @ X395 ) ) )
| ( $true
!= ( X388 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X395 ) ) )
& ( $true
!= ( X388 @ id @ ( ap @ X386 @ X387 ) @ id ) )
& ( $true
= ( X388 @ id @ X387 @ id ) ) )
| ? [X379: term,X380: term > $o] :
( ( $true
!= ( X380 @ X379 ) )
& ! [X383: term] :
( ? [X384: term] :
( ( $true
!= ( X380 @ ( sub @ X383 @ ( push @ X384 @ id ) ) ) )
& ( $true
= ( X380 @ X384 ) ) )
| ( $true
= ( X380 @ ( lam @ X383 ) ) ) )
& ! [X385: term] :
( ( $true
!= ( var @ X385 ) )
| ( $true
= ( X380 @ X385 ) ) )
& ! [X381: term,X382: term] :
( ( $true
!= ( X380 @ X382 ) )
| ( ( X380 @ ( ap @ X381 @ X382 ) )
= $true )
| ( $true
!= ( X380 @ X381 ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X200: term] :
? [X201: subst > term > term] :
( ( X201 @ sh @ one )
= ( sub @ X200 @ ( push @ one @ sh ) ) ) )
& ( ( ! [X459: term > $o,X460: term] :
( ( $true
= ( X459 @ X460 ) )
| ? [X462: term,X463: term] :
( ( $true
!= ( X459 @ ( ap @ X462 @ X463 ) ) )
& ( $true
= ( X459 @ X462 ) )
& ( $true
= ( X459 @ X463 ) ) )
| ? [X464: term] :
( ! [X465: term] :
( ( $true
!= ( X459 @ X465 ) )
| ( $true
= ( X459 @ ( sub @ X464 @ ( push @ X465 @ id ) ) ) ) )
& ( $true
!= ( X459 @ ( lam @ X464 ) ) ) )
| ? [X461: term] :
( ( $true
= ( var @ X461 ) )
& ( $true
!= ( X459 @ X461 ) ) ) )
| ? [X451: subst,X452: term,X453: term > $o] :
( ! [X454: term] :
( ( $true
!= ( var @ X454 ) )
| ( $true
= ( X453 @ ( sub @ X454 @ X451 ) ) ) )
& ! [X455: term,X456: term] :
( ( $true
!= ( X453 @ X455 ) )
| ( $true
= ( X453 @ ( ap @ X455 @ X456 ) ) )
| ( ( X453 @ X456 )
!= $true ) )
& ! [X457: term] :
( ? [X458: term] :
( ( $true
!= ( X453 @ ( sub @ X457 @ ( push @ X458 @ id ) ) ) )
& ( $true
= ( X453 @ X458 ) ) )
| ( $true
= ( X453 @ ( lam @ X457 ) ) ) )
& ( $true
!= ( X453 @ ( sub @ X452 @ X451 ) ) ) )
| ? [X450: term] :
( ( sub @ X450 @ id )
!= X450 ) )
<=> ( induction2_lthm = $true ) )
& ( axassoc = $true )
& ( ( termmset_lthm = $true )
<=> ( ! [X26: term] :
( ( sub @ X26 @ id )
= X26 )
| ? [X25: term] :
( ( sub @ X25 @ id )
!= X25 ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X77: subst > term > subst > $o] :
? [X78: term > $o] :
! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
!= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
| ( X491 = X493 ) )
| ? [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
= ( ap @ X490 @ X489 ) )
& ( X487 != X490 ) )
| ? [X486: term] :
( ( sub @ X486 @ id )
!= X486 ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( axshiftcons = $true )
<=> ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 ) )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X516: term,X517: subst > term > subst > $o] :
( ? [X522: subst > term > term] :
( ( ( X517 @ id @ ( lam @ ( X522 @ sh @ one ) ) @ id )
!= $true )
& ! [X523: term] :
( ( $true
= ( X517 @ id @ ( X522 @ id @ X523 ) @ id ) )
| ( $true
!= ( X517 @ id @ X523 @ id ) ) )
& ! [X524: term,X525: subst,X526: subst] :
( ( X522 @ ( comp @ X525 @ X526 ) @ ( sub @ X524 @ X526 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X526 ) ) )
| ? [X528: subst,X529: subst,X530: subst,X531: term] :
( ( ( X517 @ X529 @ X531 @ ( comp @ X530 @ X528 ) )
!= $true )
& ( $true
= ( X517 @ ( comp @ X529 @ X530 ) @ ( sub @ X531 @ X530 ) @ X528 ) ) )
| ( $true
= ( X517 @ id @ ( lam @ X516 ) @ id ) )
| ? [X527: term] :
( ( $true
= ( X517 @ id @ X527 @ id ) )
& ( $true
!= ( X517 @ id @ ( sub @ X516 @ ( push @ X527 @ id ) ) @ id ) ) )
| ? [X518: subst,X519: term,X520: subst,X521: subst] :
( ( $true
!= ( X517 @ ( comp @ X521 @ X518 ) @ ( sub @ X519 @ X518 ) @ X520 ) )
& ( $true
= ( X517 @ X521 @ X519 @ ( comp @ X518 @ X520 ) ) ) ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lthm_orig = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X4: term,X5: subst > term > subst > $o] :
( ? [X11: term,X12: term] :
( ( $true
!= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
& ( ( X5 @ id @ X11 @ id )
= $true )
& ( $true
= ( X5 @ id @ X12 @ id ) ) )
| ? [X6: subst > term > term] :
( ! [X8: term,X9: subst,X10: subst] :
( ( X6 @ ( comp @ X10 @ X9 ) @ ( sub @ X8 @ X9 ) )
= ( sub @ ( X6 @ X10 @ X8 ) @ X9 ) )
& ! [X7: term] :
( ( $true
= ( X5 @ id @ ( X6 @ id @ X7 ) @ id ) )
| ( $true
!= ( X5 @ id @ X7 @ id ) ) )
& ( $true
!= ( X5 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X5 @ id @ X4 @ id ) ) ) )
& ( hoasinduction_gthm = $true )
& ( hoasapinj1_gthm = $true )
& ( ! [X27: subst,X28: term,X29: term > $o] :
? [X30: term > $o] :
! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) )
<=> ( pushprop_lem0 = $true ) )
& ( ( ! [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
!= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
| ( X217 = X218 ) )
| ? [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
= ( ap @ X213 @ X212 ) )
& ( X212 != X214 ) ) )
<=> ( hoasapinj2_lthm = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( hoaslamnotap = $true )
<=> ! [X117: subst > term > term,X118: term,X119: term] :
( ? [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
!= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X369: term > $o,X370: subst > term > subst > $o,X371: term] :
( ? [X374: subst > term > term] :
( ! [X375: term] :
( ( $true
!= ( X370 @ id @ X375 @ id ) )
| ( $true
= ( X370 @ id @ ( X374 @ id @ X375 ) @ id ) ) )
& ! [X376: subst,X377: term,X378: subst] :
( ( sub @ ( X374 @ X376 @ X377 ) @ X378 )
= ( X374 @ ( comp @ X376 @ X378 ) @ ( sub @ X377 @ X378 ) ) )
& ( $true
!= ( X370 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id ) ) )
| ? [X373: term] :
( ( $true
= ( X369 @ X373 ) )
& ( ( X369 @ ( sub @ X371 @ ( push @ X373 @ id ) ) )
!= $true ) )
| ( $true
= ( X369 @ ( lam @ X371 ) ) )
| ? [X372: term] :
( ( X370 @ id @ X372 @ id )
!= ( X369 @ X372 ) ) )
| ? [X368: term] :
( ( sub @ X368 @ id )
!= X368 )
| ? [X364: term] :
! [X365: subst > term > term] :
? [X366: subst,X367: term] :
( ( sub @ X364 @ ( push @ X367 @ X366 ) )
!= ( X365 @ X366 @ X367 ) ) ) )
& ( pushprop = $true )
& ! [X106: d_subst] : ( d_id = X106 )
& ( hoaslamnotvar_lthm = $true )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
!= ( d2term @ X152 ) )
| ( X151 = X152 ) )
& ( apnotvar = $true )
& ( axvarshift = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( ! [X294: term,X295: subst > term > subst > $o] :
( ? [X296: subst,X297: subst,X298: subst,X299: term] :
( ( $true
= ( X295 @ ( comp @ X298 @ X296 ) @ ( sub @ X299 @ X296 ) @ X297 ) )
& ( $true
!= ( X295 @ X298 @ X299 @ ( comp @ X296 @ X297 ) ) ) )
| ? [X300: term] :
( ( $true
!= ( X295 @ id @ ( sub @ X294 @ ( push @ X300 @ id ) ) @ id ) )
& ( $true
= ( X295 @ id @ X300 @ id ) ) )
| ? [X301: subst > term > term] :
( ! [X305: term] :
( ( $true
!= ( X295 @ id @ X305 @ id ) )
| ( $true
= ( X295 @ id @ ( X301 @ id @ X305 ) @ id ) ) )
& ( $true
!= ( X295 @ id @ ( lam @ ( X301 @ sh @ one ) ) @ id ) )
& ! [X302: term,X303: subst,X304: subst] :
( ( X301 @ ( comp @ X303 @ X304 ) @ ( sub @ X302 @ X304 ) )
= ( sub @ ( X301 @ X303 @ X302 ) @ X304 ) ) )
| ( $true
= ( X295 @ id @ ( lam @ X294 ) @ id ) )
| ? [X306: term,X307: subst,X308: subst,X309: subst] :
( ( $true
= ( X295 @ X308 @ X306 @ ( comp @ X309 @ X307 ) ) )
& ( $true
!= ( X295 @ ( comp @ X308 @ X309 ) @ ( sub @ X306 @ X309 ) @ X307 ) ) ) )
| ? [X286: term,X287: subst > term > subst > $o] :
( ( $true
!= ( X287 @ id @ ( lam @ ( sub @ X286 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X293: term] :
( ( $true
= ( X287 @ id @ ( sub @ X286 @ ( push @ X293 @ id ) ) @ id ) )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) )
& ! [X288: subst > term > term] :
( ? [X290: subst,X291: subst,X292: term] :
( ( X288 @ ( comp @ X291 @ X290 ) @ ( sub @ X292 @ X290 ) )
!= ( sub @ ( X288 @ X291 @ X292 ) @ X290 ) )
| ( $true
= ( X287 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id ) )
| ? [X289: term] :
( ( $true
= ( X287 @ id @ X289 @ id ) )
& ( $true
!= ( X287 @ id @ ( X288 @ id @ X289 ) @ id ) ) ) ) )
| ? [X285: term] :
( ( sub @ X285 @ id )
!= X285 ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( ! [X178: term > $o,X179: subst > term > subst > $o,X180: term] :
( ( $true
= ( X178 @ ( lam @ X180 ) ) )
| ? [X182: subst > term > term] :
( ! [X184: subst,X185: subst,X186: term] :
( ( sub @ ( X182 @ X185 @ X186 ) @ X184 )
= ( X182 @ ( comp @ X185 @ X184 ) @ ( sub @ X186 @ X184 ) ) )
& ( $true
!= ( X179 @ id @ ( lam @ ( X182 @ sh @ one ) ) @ id ) )
& ! [X183: term] :
( ( $true
= ( X179 @ id @ ( X182 @ id @ X183 ) @ id ) )
| ( $true
!= ( X179 @ id @ X183 @ id ) ) ) )
| ? [X192: subst,X193: subst,X194: term,X195: subst] :
( ( $true
!= ( X179 @ ( comp @ X192 @ X193 ) @ ( sub @ X194 @ X193 ) @ X195 ) )
& ( $true
= ( X179 @ X192 @ X194 @ ( comp @ X193 @ X195 ) ) ) )
| ? [X181: term] :
( ( X179 @ id @ X181 @ id )
!= ( X178 @ X181 ) )
| ? [X188: term,X189: subst,X190: subst,X191: subst] :
( ( $true
!= ( X179 @ X189 @ X188 @ ( comp @ X190 @ X191 ) ) )
& ( $true
= ( X179 @ ( comp @ X189 @ X190 ) @ ( sub @ X188 @ X190 ) @ X191 ) ) )
| ? [X187: term] :
( ( $true
!= ( X178 @ ( sub @ X180 @ ( push @ X187 @ id ) ) ) )
& ( $true
= ( X178 @ X187 ) ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( pushprop_lem3v2 = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X496: term > $o,X497: subst > term > subst > $o,X498: term] :
( ? [X504: term] :
( ( $true
!= ( X496 @ ( sub @ X498 @ ( push @ X504 @ id ) ) ) )
& ( $true
= ( X496 @ X504 ) ) )
| ? [X499: subst,X500: term,X501: subst,X502: subst] :
( ( $true
!= ( X497 @ ( comp @ X501 @ X499 ) @ ( sub @ X500 @ X499 ) @ X502 ) )
& ( $true
= ( X497 @ X501 @ X500 @ ( comp @ X499 @ X502 ) ) ) )
| ( $true
= ( X496 @ ( lam @ X498 ) ) )
| ? [X503: term] :
( ( X496 @ X503 )
!= ( X497 @ id @ X503 @ id ) )
| ? [X505: subst,X506: term,X507: subst,X508: subst] :
( ( ( X497 @ ( comp @ X508 @ X507 ) @ ( sub @ X506 @ X507 ) @ X505 )
= $true )
& ( $true
!= ( X497 @ X508 @ X506 @ ( comp @ X507 @ X505 ) ) ) )
| ? [X509: subst > term > term] :
( ! [X510: subst,X511: term,X512: subst] :
( ( sub @ ( X509 @ X512 @ X511 ) @ X510 )
= ( X509 @ ( comp @ X512 @ X510 ) @ ( sub @ X511 @ X510 ) ) )
& ( $true
!= ( X497 @ id @ ( lam @ ( X509 @ sh @ one ) ) @ id ) )
& ! [X513: term] :
( ( $true
= ( X497 @ id @ ( X509 @ id @ X513 ) @ id ) )
| ( $true
!= ( X497 @ id @ X513 @ id ) ) ) ) )
| ? [X495: term] :
( ( sub @ X495 @ id )
!= X495 ) ) )
& ! [X198: term] :
? [X199: d_term] :
( ( d2term @ X199 )
= X198 )
& ( ( ! [X347: subst > term > subst > $o,X348: term] :
( ( $true
= ( X347 @ id @ X348 @ id ) )
| ? [X351: subst > term > term] :
( ! [X355: term] :
( ( $true
!= ( X347 @ id @ X355 @ id ) )
| ( $true
= ( X347 @ id @ ( X351 @ id @ X355 ) @ id ) ) )
& ( $true
!= ( X347 @ id @ ( lam @ ( X351 @ sh @ one ) ) @ id ) )
& ! [X352: subst,X353: subst,X354: term] :
( ( X351 @ ( comp @ X353 @ X352 ) @ ( sub @ X354 @ X352 ) )
= ( sub @ ( X351 @ X353 @ X354 ) @ X352 ) ) )
| ? [X349: term,X350: term] :
( ( $true
!= ( X347 @ id @ ( ap @ ( sub @ X349 @ id ) @ X350 ) @ id ) )
& ( $true
= ( X347 @ id @ X350 @ id ) )
& ( $true
= ( X347 @ id @ X349 @ id ) ) ) )
| ? [X337: subst > term > subst > $o,X338: term,X339: term > $o] :
( ! [X346: term] :
( ( X339 @ X346 )
= ( X337 @ id @ X346 @ id ) )
& ! [X340: term] :
( ( $true
= ( X339 @ ( sub @ X338 @ ( push @ X340 @ id ) ) ) )
| ( $true
!= ( X339 @ X340 ) ) )
& ! [X341: subst > term > term] :
( ? [X342: term] :
( ( $true
= ( X337 @ id @ X342 @ id ) )
& ( ( X337 @ id @ ( X341 @ id @ X342 ) @ id )
!= $true ) )
| ( $true
= ( X337 @ id @ ( lam @ ( X341 @ sh @ one ) ) @ id ) )
| ? [X343: subst,X344: term,X345: subst] :
( ( sub @ ( X341 @ X345 @ X344 ) @ X343 )
!= ( X341 @ ( comp @ X345 @ X343 ) @ ( sub @ X344 @ X343 ) ) ) )
& ( $true
!= ( X339 @ ( lam @ X338 ) ) ) )
| ? [X336: term] :
( ( sub @ X336 @ id )
!= X336 )
| ? [X329: term > $o,X330: term] :
( ! [X332: term] :
( ? [X333: term] :
( ( $true
!= ( X329 @ ( sub @ X332 @ ( push @ X333 @ id ) ) ) )
& ( $true
= ( X329 @ X333 ) ) )
| ( $true
= ( X329 @ ( lam @ X332 ) ) ) )
& ! [X334: term,X335: term] :
( ( $true
!= ( X329 @ X335 ) )
| ( $true
= ( X329 @ ( ap @ X335 @ X334 ) ) )
| ( $true
!= ( X329 @ X334 ) ) )
& ( $true
!= ( X329 @ X330 ) )
& ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( $true
!= ( var @ X331 ) ) ) )
| ? [X326: subst > term > subst > $o] :
! [X327: term > $o] :
? [X328: term] :
( ( X326 @ id @ X328 @ id )
!= ( X327 @ X328 ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ulamvar1 != $true )
& ( ulamvarind = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( ! [X163: subst,X164: subst > term > term,X165: term,X166: subst > term > term] :
( ? [X167: subst,X168: term,X169: subst] :
( ( sub @ ( X164 @ X167 @ X168 ) @ X169 )
!= ( X164 @ ( comp @ X167 @ X169 ) @ ( sub @ X168 @ X169 ) ) )
| ( ( X164 @ X163 @ X165 )
= ( X166 @ X163 @ X165 ) )
| ? [X170: term,X171: subst,X172: subst] :
( ( sub @ ( X166 @ X172 @ X170 ) @ X171 )
!= ( X166 @ ( comp @ X172 @ X171 ) @ ( sub @ X170 @ X171 ) ) )
| ( ( lam @ ( X166 @ sh @ one ) )
!= ( lam @ ( X164 @ sh @ one ) ) ) )
| ? [X161: term,X162: term] :
( ( ( lam @ X161 )
= ( lam @ X162 ) )
& ( X161 != X162 ) )
| ? [X159: subst,X160: term] :
( ( comp @ sh @ ( push @ X160 @ X159 ) )
!= X159 )
| ? [X157: term,X158: subst] :
( ( sub @ one @ ( push @ X157 @ X158 ) )
!= X157 ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( induction2_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( axidr = $true )
<=> ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 ) )
& ( ! [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
!= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
| ( X435 = X436 ) )
<=> ( hoasapinj1 = $true ) )
& ( ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ? [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
!= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ? [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
!= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X127: term,X128: term > $o,X129: subst > term > subst > $o,X130: term] :
( ( $true
!= ( X128 @ X130 ) )
| ? [X139: term] :
( ( X129 @ id @ X139 @ id )
!= ( X128 @ X139 ) )
| ? [X140: term,X141: term] :
( ( $true
= ( X129 @ id @ X140 @ id ) )
& ( $true
= ( X129 @ id @ X141 @ id ) )
& ( $true
!= ( X129 @ id @ ( ap @ ( sub @ X140 @ id ) @ X141 ) @ id ) ) )
| ( $true
= ( X128 @ ( ap @ X130 @ X127 ) ) )
| ? [X131: term,X132: subst,X133: subst,X134: subst] :
( ( $true
= ( X129 @ X133 @ X131 @ ( comp @ X132 @ X134 ) ) )
& ( $true
!= ( X129 @ ( comp @ X133 @ X132 ) @ ( sub @ X131 @ X132 ) @ X134 ) ) )
| ? [X135: subst,X136: subst,X137: term,X138: subst] :
( ( $true
!= ( X129 @ X138 @ X137 @ ( comp @ X136 @ X135 ) ) )
& ( $true
= ( X129 @ ( comp @ X138 @ X136 ) @ ( sub @ X137 @ X136 ) @ X135 ) ) )
| ( $true
!= ( X128 @ X127 ) ) ) )
& ( ( ! [X95: subst,X96: term,X97: term > $o] :
( ( $true
= ( X97 @ ( sub @ X96 @ X95 ) ) )
| ? [X100: term] :
( ( $true
= ( var @ X100 ) )
& ( $true
!= ( X97 @ ( sub @ X100 @ X95 ) ) ) )
| ? [X98: term] :
( ! [X99: term] :
( ( $true
!= ( X97 @ X99 ) )
| ( $true
= ( X97 @ ( sub @ X98 @ ( push @ X99 @ id ) ) ) ) )
& ( $true
!= ( X97 @ ( lam @ X98 ) ) ) )
| ? [X101: term,X102: term] :
( ( $true
= ( X97 @ X102 ) )
& ( $true
!= ( X97 @ ( ap @ X102 @ X101 ) ) )
& ( $true
= ( X97 @ X101 ) ) ) )
| ? [X89: term,X90: term > $o] :
( ! [X91: term,X92: term] :
( ( ( X90 @ X91 )
!= $true )
| ( $true
= ( X90 @ ( ap @ X91 @ X92 ) ) )
| ( $true
!= ( X90 @ X92 ) ) )
& ! [X93: term] :
( ( $true
!= ( X90 @ X93 ) )
| ( $true
= ( X90 @ ( lam @ X93 ) ) ) )
& ! [X94: term] :
( ( $true
!= ( var @ X94 ) )
| ( $true
= ( X90 @ X94 ) ) )
& ( $true
!= ( X90 @ X89 ) ) )
| ? [X88: subst] :
( ( comp @ X88 @ id )
!= X88 )
| ? [X86: subst,X87: term] :
( ( comp @ sh @ ( push @ X87 @ X86 ) )
!= X86 )
| ? [X84: subst,X85: term] :
( ( sub @ one @ ( push @ X85 @ X84 ) )
!= X85 ) )
<=> ( induction2lem_lthm = $true ) )
& ( induction2lem_gthm = $true )
& ( substmonoid_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( axabs = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X219: subst > term > subst > $o,X220: term] :
( ( $true
= ( X219 @ id @ ( lam @ ( sub @ X220 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X226: term] :
( ( $true
= ( X219 @ id @ X226 @ id ) )
& ( $true
!= ( X219 @ id @ ( sub @ X220 @ ( push @ X226 @ id ) ) @ id ) ) )
| ? [X221: subst > term > term] :
( ! [X222: term] :
( ( $true
!= ( X219 @ id @ X222 @ id ) )
| ( $true
= ( X219 @ id @ ( X221 @ id @ X222 ) @ id ) ) )
& ! [X223: term,X224: subst,X225: subst] :
( ( X221 @ ( comp @ X225 @ X224 ) @ ( sub @ X223 @ X224 ) )
= ( sub @ ( X221 @ X225 @ X223 ) @ X224 ) )
& ( $true
!= ( X219 @ id @ ( lam @ ( X221 @ sh @ one ) ) @ id ) ) ) ) )
& ( ! [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
= X539 )
<=> ( axscons = $true ) )
& ( ( ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ( $true
!= ( X40 @ X39 ) )
| ( $true
= ( X41 @ one ) )
| ? [X43: term] :
( ( X41 @ X43 )
!= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) ) )
| ? [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
!= X37 ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( apinj2 = $true )
<=> ! [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) )
| ( X0 = X1 ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( axmap = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X532: term,X533: term > $o,X534: subst] :
? [X535: term > $o] :
! [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
= ( X535 @ X536 ) ) )
& ( ! [X310: subst > term > subst > $o,X311: term] :
( ? [X313: subst > term > term] :
( ! [X317: term] :
( ( $true
= ( X310 @ id @ ( X313 @ id @ X317 ) @ id ) )
| ( $true
!= ( X310 @ id @ X317 @ id ) ) )
& ( $true
!= ( X310 @ id @ ( lam @ ( X313 @ sh @ one ) ) @ id ) )
& ! [X314: subst,X315: term,X316: subst] :
( ( sub @ ( X313 @ X314 @ X315 ) @ X316 )
= ( X313 @ ( comp @ X314 @ X316 ) @ ( sub @ X315 @ X316 ) ) ) )
| ? [X312: term] :
( ( $true
!= ( X310 @ id @ ( sub @ X311 @ ( push @ X312 @ id ) ) @ id ) )
& ( $true
= ( X310 @ id @ X312 @ id ) ) )
| ( $true
= ( X310 @ id @ ( lam @ X311 ) @ id ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X537: term] :
? [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
= ( X538 @ sh @ one ) ) )
& ! [X485: d_term] : ( d_one = X485 )
& ( hoaslamnotvar_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( induction = $true )
<=> ! [X479: term > $o,X480: term] :
( ( $true
= ( X479 @ X480 ) )
| ? [X481: term] :
( ( $true
!= ( X479 @ X481 ) )
& ( $true
= ( var @ X481 ) ) )
| ? [X483: term,X484: term] :
( ( $true
= ( X479 @ X484 ) )
& ( $true
= ( X479 @ X483 ) )
& ( ( X479 @ ( ap @ X483 @ X484 ) )
!= $true ) )
| ? [X482: term] :
( ( $true
!= ( X479 @ ( lam @ X482 ) ) )
& ( $true
= ( X479 @ X482 ) ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( ! [X248: term,X249: subst > term > subst > $o] :
( ? [X261: term,X262: subst,X263: subst,X264: subst] :
( ( $true
!= ( X249 @ X262 @ X261 @ ( comp @ X263 @ X264 ) ) )
& ( $true
= ( X249 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) @ X264 ) ) )
| ? [X250: term,X251: term] :
( ( $true
!= ( X249 @ id @ ( ap @ ( sub @ X251 @ id ) @ X250 ) @ id ) )
& ( $true
= ( X249 @ id @ X250 @ id ) )
& ( $true
= ( X249 @ id @ X251 @ id ) ) )
| ? [X256: subst > term > term] :
( ! [X257: term] :
( ( $true
!= ( X249 @ id @ X257 @ id ) )
| ( $true
= ( X249 @ id @ ( X256 @ id @ X257 ) @ id ) ) )
& ! [X258: subst,X259: term,X260: subst] :
( ( sub @ ( X256 @ X260 @ X259 ) @ X258 )
= ( X256 @ ( comp @ X260 @ X258 ) @ ( sub @ X259 @ X258 ) ) )
& ( $true
!= ( X249 @ id @ ( lam @ ( X256 @ sh @ one ) ) @ id ) ) )
| ? [X252: subst,X253: term,X254: subst,X255: subst] :
( ( $true
!= ( X249 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) @ X255 ) )
& ( $true
= ( X249 @ X252 @ X253 @ ( comp @ X254 @ X255 ) ) ) )
| ? [X265: term] :
( ( $true
= ( var @ ( sub @ X265 @ id ) ) )
& ( $true
!= ( X249 @ id @ X265 @ id ) ) )
| ( $true
= ( X249 @ id @ X248 @ id ) ) )
| ? [X238: term,X239: term > $o,X240: subst > term > subst > $o] :
( ! [X241: term] :
( ( $true
= ( X239 @ ( sub @ X238 @ ( push @ X241 @ id ) ) ) )
| ( $true
!= ( X239 @ X241 ) ) )
& ( $true
!= ( X239 @ ( lam @ X238 ) ) )
& ! [X247: term] :
( ( X240 @ id @ X247 @ id )
= ( X239 @ X247 ) )
& ! [X242: subst > term > term] :
( ( $true
= ( X240 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id ) )
| ? [X244: subst,X245: term,X246: subst] :
( ( X242 @ ( comp @ X246 @ X244 ) @ ( sub @ X245 @ X244 ) )
!= ( sub @ ( X242 @ X246 @ X245 ) @ X244 ) )
| ? [X243: term] :
( ( $true
!= ( X240 @ id @ ( X242 @ id @ X243 ) @ id ) )
& ( $true
= ( X240 @ id @ X243 @ id ) ) ) ) )
| ? [X237: term] :
( ( sub @ X237 @ id )
!= X237 )
| ? [X230: term > $o,X231: term] :
( ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( $true
!= ( var @ X232 ) ) )
& ! [X235: term,X236: term] :
( ( $true
!= ( X230 @ X236 ) )
| ( $true
!= ( X230 @ X235 ) )
| ( $true
= ( X230 @ ( ap @ X235 @ X236 ) ) ) )
& ( $true
!= ( X230 @ X231 ) )
& ! [X233: term] :
( ? [X234: term] :
( ( $true
= ( X230 @ X234 ) )
& ( $true
!= ( X230 @ ( sub @ X233 @ ( push @ X234 @ id ) ) ) ) )
| ( $true
= ( X230 @ ( lam @ X233 ) ) ) ) )
| ? [X227: subst > term > subst > $o] :
! [X228: term > $o] :
? [X229: term] :
( ( X227 @ id @ X229 @ id )
!= ( X228 @ X229 ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ! [X318: term] :
( ( sub @ X318 @ id )
= X318 )
<=> ( axvarid = $true ) )
& ( ! [X319: term,X320: term > $o] :
( ( $true
= ( X320 @ X319 ) )
| ? [X322: term,X323: term] :
( ( $true
!= ( X320 @ ( ap @ X323 @ X322 ) ) )
& ( $true
= ( X320 @ X322 ) )
& ( $true
= ( X320 @ X323 ) ) )
| ? [X324: term] :
( ! [X325: term] :
( ( $true
= ( X320 @ ( sub @ X324 @ ( push @ X325 @ id ) ) ) )
| ( $true
!= ( X320 @ X325 ) ) )
& ( $true
!= ( X320 @ ( lam @ X324 ) ) ) )
| ? [X321: term] :
( ( $true
!= ( X320 @ X321 ) )
& ( $true
= ( var @ X321 ) ) ) )
<=> ( induction2 = $true ) )
& ( ( ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) )
| ? [X34: subst] :
( ( comp @ X34 @ id )
!= X34 )
| ? [X33: subst] :
( ( comp @ id @ X33 )
!= X33 ) )
<=> ( substmonoid_lthm = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( termmset_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
| ( $true
!= ( X173 @ X174 ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ( shinj = $true )
<=> ! [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
!= ( sub @ X515 @ sh ) )
| ( X514 = X515 ) ) )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X466: term,X467: term,X468: subst > term > subst > $o] :
( ? [X471: subst,X472: subst,X473: subst,X474: term] :
( ( $true
!= ( X468 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 ) )
& ( $true
= ( X468 @ X473 @ X474 @ ( comp @ X472 @ X471 ) ) ) )
| ( $true
!= ( X468 @ id @ X467 @ id ) )
| ( $true
!= ( X468 @ id @ X466 @ id ) )
| ( $true
= ( X468 @ id @ ( ap @ X467 @ X466 ) @ id ) )
| ? [X469: term,X470: term] :
( ( $true
= ( X468 @ id @ X470 @ id ) )
& ( $true
= ( X468 @ id @ X469 @ id ) )
& ( $true
!= ( X468 @ id @ ( ap @ ( sub @ X470 @ id ) @ X469 ) @ id ) ) )
| ? [X475: subst,X476: subst,X477: subst,X478: term] :
( ( $true
!= ( X468 @ X476 @ X478 @ ( comp @ X475 @ X477 ) ) )
& ( $true
= ( X468 @ ( comp @ X476 @ X475 ) @ ( sub @ X478 @ X475 ) @ X477 ) ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( ( axclos = $true )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X445: subst > term > subst > $o] :
~ ! [X446: term > $o] :
~ ! [X447: term] :
( ( X445 @ id @ X447 @ id )
= ( X446 @ X447 ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ! [X266: subst] :
? [X267: d_subst] :
( ( d2subst @ X267 )
= X266 )
& ( ! [X59: term,X60: subst > term > subst > $o] :
( ( $true
= ( X60 @ id @ X59 @ id ) )
| ~ ! [X69: subst,X70: subst,X71: term,X72: subst] :
( ( $true
!= ( X60 @ ( comp @ X69 @ X72 ) @ ( sub @ X71 @ X72 ) @ X70 ) )
| ( $true
= ( X60 @ X69 @ X71 @ ( comp @ X72 @ X70 ) ) ) )
| ~ ! [X68: term] :
( ( $true
= ( X60 @ id @ X68 @ id ) )
| ( $true
!= ( var @ ( sub @ X68 @ id ) ) ) )
| ~ ! [X61: term,X62: term] :
( ( $true
!= ( X60 @ id @ X61 @ id ) )
| ( $true
= ( X60 @ id @ ( ap @ ( sub @ X61 @ id ) @ X62 ) @ id ) )
| ( $true
!= ( X60 @ id @ X62 @ id ) ) )
| ~ ! [X73: term,X74: subst,X75: subst,X76: subst] :
( ( $true
= ( X60 @ ( comp @ X75 @ X74 ) @ ( sub @ X73 @ X74 ) @ X76 ) )
| ( $true
!= ( X60 @ X75 @ X73 @ ( comp @ X74 @ X76 ) ) ) )
| ~ ! [X63: subst > term > term] :
( ~ ! [X65: subst,X66: subst,X67: term] :
( ( X63 @ ( comp @ X65 @ X66 ) @ ( sub @ X67 @ X66 ) )
= ( sub @ ( X63 @ X65 @ X67 ) @ X66 ) )
| ~ ! [X64: term] :
( ( $true
!= ( X60 @ id @ X64 @ id ) )
| ( $true
= ( X60 @ id @ ( X63 @ id @ X64 ) @ id ) ) )
| ( $true
= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) ) ) )
<=> ( hoasinduction = $true ) )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
= ( d2subst @ X14 ) )
=> ( X13 = X14 ) )
& ( one
= ( d2term @ d_one ) )
& ( ( d2subst @ d_id )
= sh )
& ( ! [X356: term > $o,X357: subst,X358: term] :
( ~ ! [X362: term,X363: term] :
( ( $true
!= ( X356 @ X363 ) )
| ( $true
= ( X356 @ ( ap @ X362 @ X363 ) ) )
| ( $true
!= ( X356 @ X362 ) ) )
| ( $true
= ( X356 @ ( sub @ X358 @ X357 ) ) )
| ~ ! [X359: term] :
( ( $true
= ( X356 @ ( sub @ X359 @ X357 ) ) )
| ( $true
!= ( var @ X359 ) ) )
| ~ ! [X360: term] :
( ~ ! [X361: term] :
( ( $true
= ( X356 @ ( sub @ X360 @ ( push @ X361 @ id ) ) ) )
| ( $true
!= ( X356 @ X361 ) ) )
| ( $true
= ( X356 @ ( lam @ X360 ) ) ) ) )
<=> ( induction2lem = $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( axidl = $true )
<=> ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 ) )
& ( ! [X437: subst > term > subst > $o,X438: term] :
( ( $true
= ( X437 @ id @ ( lam @ ( sub @ X438 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X439: subst > term > term] :
( ( $true
= ( X437 @ id @ ( lam @ ( X439 @ sh @ one ) ) @ id ) )
| ~ ! [X440: term] :
( ( $true
= ( X437 @ id @ ( X439 @ id @ X440 ) @ id ) )
| ( $true
!= ( X437 @ id @ X440 @ id ) ) )
| ~ ! [X441: subst,X442: term,X443: subst] :
( ( sub @ ( X439 @ X441 @ X442 ) @ X443 )
= ( X439 @ ( comp @ X441 @ X443 ) @ ( sub @ X442 @ X443 ) ) ) )
| ~ ! [X444: term] :
( ( $true
!= ( X437 @ id @ X444 @ id ) )
| ( $true
= ( X437 @ id @ ( sub @ X438 @ ( push @ X444 @ id ) ) @ id ) ) ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X15: subst > term > subst > $o,X16: term > $o,X17: term] :
( ~ ! [X18: term] :
( ( X16 @ X18 )
= ( X15 @ id @ X18 @ id ) )
| ~ ! [X24: term] :
( ( $true
= ( X16 @ ( sub @ X17 @ ( push @ X24 @ id ) ) ) )
| ( $true
!= ( X16 @ X24 ) ) )
| ( $true
= ( X16 @ ( lam @ X17 ) ) )
| ~ ! [X19: subst > term > term] :
( ( $true
= ( X15 @ id @ ( lam @ ( X19 @ sh @ one ) ) @ id ) )
| ~ ! [X20: term,X21: subst,X22: subst] :
( ( sub @ ( X19 @ X22 @ X20 ) @ X21 )
= ( X19 @ ( comp @ X22 @ X21 ) @ ( sub @ X20 @ X21 ) ) )
| ~ ! [X23: term] :
( ( $true
!= ( X15 @ id @ X23 @ id ) )
| ( $true
= ( X15 @ id @ ( X19 @ id @ X23 ) @ id ) ) ) ) ) )
& ( ulamvarsh = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X268: term] :
( ( sub @ X268 @ id )
= X268 )
=> ( ! [X269: term,X270: subst > term > subst > $o] :
( ( $true
= ( X270 @ id @ ( lam @ ( sub @ X269 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X271: subst > term > term] :
( ( $true
= ( X270 @ id @ ( lam @ ( X271 @ sh @ one ) ) @ id ) )
| ~ ! [X272: subst,X273: term,X274: subst] :
( ( X271 @ ( comp @ X272 @ X274 ) @ ( sub @ X273 @ X274 ) )
= ( sub @ ( X271 @ X272 @ X273 ) @ X274 ) )
| ~ ! [X275: term] :
( ( $true
= ( X270 @ id @ ( X271 @ id @ X275 ) @ id ) )
| ( $true
!= ( X270 @ id @ X275 @ id ) ) ) )
| ~ ! [X276: term] :
( ( $true
= ( X270 @ id @ ( sub @ X269 @ ( push @ X276 @ id ) ) @ id ) )
| ( $true
!= ( X270 @ id @ X276 @ id ) ) ) )
=> ! [X277: term,X278: subst > term > subst > $o] :
( ~ ! [X279: subst > term > term] :
( ( $true
= ( X278 @ id @ ( lam @ ( X279 @ sh @ one ) ) @ id ) )
| ~ ! [X280: subst,X281: term,X282: subst] :
( ( sub @ ( X279 @ X282 @ X281 ) @ X280 )
= ( X279 @ ( comp @ X282 @ X280 ) @ ( sub @ X281 @ X280 ) ) )
| ~ ! [X283: term] :
( ( $true
!= ( X278 @ id @ X283 @ id ) )
| ( $true
= ( X278 @ id @ ( X279 @ id @ X283 ) @ id ) ) ) )
| ~ ! [X284: term] :
( ( $true
= ( X278 @ id @ ( sub @ X277 @ ( push @ X284 @ id ) ) @ id ) )
| ( $true
!= ( X278 @ id @ X284 @ id ) ) )
| ( $true
= ( X278 @ id @ ( lam @ X277 ) @ id ) ) ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( lamnotap != $true )
& ( axapp = $true )
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( X153 = X155 )
| ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X44: term,X45: subst > term > subst > $o] :
( ~ ! [X47: subst > term > term,X48: term] :
( ( $true
= ( X45 @ id @ ( lam @ ( X47 @ sh @ one ) ) @ id ) )
| ~ ! [X55: term] :
( ( $true
!= ( X45 @ id @ X55 @ id ) )
| ( $true
= ( X45 @ id @ ( X47 @ id @ X55 ) @ id ) ) )
| ~ ! [X52: subst,X53: term,X54: subst] :
( ( sub @ X48 @ ( push @ ( sub @ X53 @ X52 ) @ ( comp @ X54 @ X52 ) ) )
= ( X47 @ ( comp @ X54 @ X52 ) @ ( sub @ X53 @ X52 ) ) )
| ~ ! [X49: subst,X50: term,X51: subst] :
( ( sub @ ( sub @ X48 @ ( push @ X50 @ X49 ) ) @ X51 )
= ( sub @ ( X47 @ X49 @ X50 ) @ X51 ) ) )
| ( $true
= ( X45 @ id @ ( lam @ ( sub @ X44 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X46: term] :
( ( ( X45 @ id @ ( sub @ X44 @ ( push @ X46 @ id ) ) @ id )
= $true )
| ( $true
!= ( X45 @ id @ X46 @ id ) ) ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ~ ! [X146: term] :
( ( X144 @ X146 )
= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) )
| ( $true
= ( X144 @ one ) )
| ( $true
!= ( X142 @ X145 ) ) ) )
& ( hoasapinj2_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( lamnotvar = $true )
& ( pushprop_gthm = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( substmonoid = $true )
<=> ( ! [X449: subst] :
( ( comp @ X449 @ id )
= X449 )
& ! [X448: subst] :
( ( comp @ id @ X448 )
= X448 ) ) )
& ( ( ! [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
= X203 )
=> ( ! [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
= X204 )
=> ! [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) )
| ( $true
!= ( X207 @ X210 ) ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( axvarcons = $true )
<=> ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X80: term] :
~ ! [X81: subst > term > term] :
~ ! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) ) )
& ( ! [X123: term,X124: term,X125: term,X126: term] :
( ( X125 = X126 )
| ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) ) )
<=> ( apinj1 = $true ) )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
<=> ( termmset = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ! [X196: term,X197: term] :
( ( X196 = X197 )
| ( ( lam @ X197 )
!= ( lam @ X196 ) ) )
<=> ( laminj = $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( ! [X379: term,X380: term > $o] :
( ( $true
= ( X380 @ X379 ) )
| ~ ! [X381: term,X382: term] :
( ( $true
!= ( X380 @ X382 ) )
| ( ( X380 @ ( ap @ X381 @ X382 ) )
= $true )
| ( $true
!= ( X380 @ X381 ) ) )
| ~ ! [X383: term] :
( ( $true
= ( X380 @ ( lam @ X383 ) ) )
| ~ ! [X384: term] :
( ( $true
!= ( X380 @ X384 ) )
| ( $true
= ( X380 @ ( sub @ X383 @ ( push @ X384 @ id ) ) ) ) ) )
| ~ ! [X385: term] :
( ( $true
!= ( var @ X385 ) )
| ( $true
= ( X380 @ X385 ) ) ) )
=> ( ! [X386: term,X387: term,X388: subst > term > subst > $o] :
( ~ ! [X389: subst,X390: subst,X391: term,X392: subst] :
( ( $true
= ( X388 @ ( comp @ X390 @ X389 ) @ ( sub @ X391 @ X389 ) @ X392 ) )
| ( $true
!= ( X388 @ X390 @ X391 @ ( comp @ X389 @ X392 ) ) ) )
| ( $true
= ( X388 @ id @ ( ap @ X386 @ X387 ) @ id ) )
| ( $true
!= ( X388 @ id @ X387 @ id ) )
| ~ ! [X395: subst,X396: subst,X397: subst,X398: term] :
( ( $true
= ( X388 @ X396 @ X398 @ ( comp @ X397 @ X395 ) ) )
| ( $true
!= ( X388 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X395 ) ) )
| ( $true
!= ( X388 @ id @ X386 @ id ) )
| ~ ! [X393: term,X394: term] :
( ( $true
= ( X388 @ id @ ( ap @ ( sub @ X394 @ id ) @ X393 ) @ id ) )
| ( $true
!= ( X388 @ id @ X393 @ id ) )
| ( $true
!= ( X388 @ id @ X394 @ id ) ) ) )
=> ( ! [X399: subst > term > subst > $o,X400: term] :
( ~ ! [X410: subst > term > term] :
( ( $true
= ( X399 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
| ~ ! [X414: term] :
( ( $true
= ( X399 @ id @ ( X410 @ id @ X414 ) @ id ) )
| ( $true
!= ( X399 @ id @ X414 @ id ) ) )
| ~ ! [X411: term,X412: subst,X413: subst] :
( ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) )
= ( sub @ ( X410 @ X412 @ X411 ) @ X413 ) ) )
| ~ ! [X402: subst,X403: term,X404: subst,X405: subst] :
( ( $true
= ( X399 @ X404 @ X403 @ ( comp @ X402 @ X405 ) ) )
| ( $true
!= ( X399 @ ( comp @ X404 @ X402 ) @ ( sub @ X403 @ X402 ) @ X405 ) ) )
| ( $true
= ( X399 @ id @ ( lam @ X400 ) @ id ) )
| ~ ! [X406: subst,X407: term,X408: subst,X409: subst] :
( ( $true
= ( X399 @ ( comp @ X406 @ X409 ) @ ( sub @ X407 @ X409 ) @ X408 ) )
| ( $true
!= ( X399 @ X406 @ X407 @ ( comp @ X409 @ X408 ) ) ) )
| ~ ! [X401: term] :
( ( $true
= ( X399 @ id @ ( sub @ X400 @ ( push @ X401 @ id ) ) @ id ) )
| ( $true
!= ( X399 @ id @ X401 @ id ) ) ) )
=> ! [X415: subst > term > subst > $o,X416: term] :
( ~ ! [X432: term] :
( ( $true
= ( X415 @ id @ X432 @ id ) )
| ( $true
!= ( var @ ( sub @ X432 @ id ) ) ) )
| ~ ! [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X415 @ X430 @ X428 @ ( comp @ X429 @ X431 ) ) )
| ( $true
!= ( X415 @ ( comp @ X430 @ X429 ) @ ( sub @ X428 @ X429 ) @ X431 ) ) )
| ~ ! [X424: subst,X425: term,X426: subst,X427: subst] :
( ( $true
!= ( X415 @ X427 @ X425 @ ( comp @ X424 @ X426 ) ) )
| ( $true
= ( X415 @ ( comp @ X427 @ X424 ) @ ( sub @ X425 @ X424 ) @ X426 ) ) )
| ~ ! [X419: subst > term > term] :
( ~ ! [X421: term,X422: subst,X423: subst] :
( ( sub @ ( X419 @ X422 @ X421 ) @ X423 )
= ( X419 @ ( comp @ X422 @ X423 ) @ ( sub @ X421 @ X423 ) ) )
| ( ( X415 @ id @ ( lam @ ( X419 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X420: term] :
( ( $true
= ( X415 @ id @ ( X419 @ id @ X420 ) @ id ) )
| ( $true
!= ( X415 @ id @ X420 @ id ) ) ) )
| ( $true
= ( X415 @ id @ X416 @ id ) )
| ~ ! [X417: term,X418: term] :
( ( ( X415 @ id @ X417 @ id )
!= $true )
| ( $true
= ( X415 @ id @ ( ap @ ( sub @ X418 @ id ) @ X417 ) @ id ) )
| ( $true
!= ( X415 @ id @ X418 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X200: term] :
~ ! [X201: subst > term > term] :
( ( X201 @ sh @ one )
!= ( sub @ X200 @ ( push @ one @ sh ) ) ) )
& ( ( ! [X450: term] :
( ( sub @ X450 @ id )
= X450 )
=> ( ! [X451: subst,X452: term,X453: term > $o] :
( ( $true
= ( X453 @ ( sub @ X452 @ X451 ) ) )
| ~ ! [X454: term] :
( ( $true
!= ( var @ X454 ) )
| ( $true
= ( X453 @ ( sub @ X454 @ X451 ) ) ) )
| ~ ! [X455: term,X456: term] :
( ( $true
!= ( X453 @ X455 ) )
| ( $true
= ( X453 @ ( ap @ X455 @ X456 ) ) )
| ( ( X453 @ X456 )
!= $true ) )
| ~ ! [X457: term] :
( ~ ! [X458: term] :
( ( $true
!= ( X453 @ X458 ) )
| ( $true
= ( X453 @ ( sub @ X457 @ ( push @ X458 @ id ) ) ) ) )
| ( $true
= ( X453 @ ( lam @ X457 ) ) ) ) )
=> ! [X459: term > $o,X460: term] :
( ~ ! [X464: term] :
( ~ ! [X465: term] :
( ( $true
!= ( X459 @ X465 ) )
| ( $true
= ( X459 @ ( sub @ X464 @ ( push @ X465 @ id ) ) ) ) )
| ( $true
= ( X459 @ ( lam @ X464 ) ) ) )
| ( $true
= ( X459 @ X460 ) )
| ~ ! [X462: term,X463: term] :
( ( $true
!= ( X459 @ X463 ) )
| ( $true
!= ( X459 @ X462 ) )
| ( $true
= ( X459 @ ( ap @ X462 @ X463 ) ) ) )
| ~ ! [X461: term] :
( ( $true
!= ( var @ X461 ) )
| ( $true
= ( X459 @ X461 ) ) ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( axassoc = $true )
& ( ( ! [X25: term] :
( ( sub @ X25 @ id )
= X25 )
=> ! [X26: term] :
( ( sub @ X26 @ id )
= X26 ) )
<=> ( termmset_lthm = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X77: subst > term > subst > $o] :
~ ! [X78: term > $o] :
~ ! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X486: term] :
( ( sub @ X486 @ id )
= X486 )
=> ( ! [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
!= ( ap @ X490 @ X489 ) )
| ( X487 = X490 ) )
=> ! [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
!= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
| ( X491 = X493 ) ) ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( axshiftcons = $true )
<=> ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 ) )
& ( ! [X516: term,X517: subst > term > subst > $o] :
( ~ ! [X528: subst,X529: subst,X530: subst,X531: term] :
( ( $true
!= ( X517 @ ( comp @ X529 @ X530 ) @ ( sub @ X531 @ X530 ) @ X528 ) )
| ( ( X517 @ X529 @ X531 @ ( comp @ X530 @ X528 ) )
= $true ) )
| ~ ! [X518: subst,X519: term,X520: subst,X521: subst] :
( ( $true
= ( X517 @ ( comp @ X521 @ X518 ) @ ( sub @ X519 @ X518 ) @ X520 ) )
| ( $true
!= ( X517 @ X521 @ X519 @ ( comp @ X518 @ X520 ) ) ) )
| ( $true
= ( X517 @ id @ ( lam @ X516 ) @ id ) )
| ~ ! [X522: subst > term > term] :
( ~ ! [X524: term,X525: subst,X526: subst] :
( ( X522 @ ( comp @ X525 @ X526 ) @ ( sub @ X524 @ X526 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X526 ) )
| ( ( X517 @ id @ ( lam @ ( X522 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X523: term] :
( ( $true
= ( X517 @ id @ ( X522 @ id @ X523 ) @ id ) )
| ( $true
!= ( X517 @ id @ X523 @ id ) ) ) )
| ~ ! [X527: term] :
( ( $true
= ( X517 @ id @ ( sub @ X516 @ ( push @ X527 @ id ) ) @ id ) )
| ( $true
!= ( X517 @ id @ X527 @ id ) ) ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X4: term,X5: subst > term > subst > $o] :
( ( $true
= ( X5 @ id @ X4 @ id ) )
| ~ ! [X11: term,X12: term] :
( ( $true
= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
| ( $true
!= ( X5 @ id @ X12 @ id ) )
| ( ( X5 @ id @ X11 @ id )
!= $true ) )
| ~ ! [X6: subst > term > term] :
( ~ ! [X7: term] :
( ( $true
= ( X5 @ id @ ( X6 @ id @ X7 ) @ id ) )
| ( $true
!= ( X5 @ id @ X7 @ id ) ) )
| ( $true
= ( X5 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id ) )
| ~ ! [X8: term,X9: subst,X10: subst] :
( ( X6 @ ( comp @ X10 @ X9 ) @ ( sub @ X8 @ X9 ) )
= ( sub @ ( X6 @ X10 @ X8 ) @ X9 ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( hoasinduction_gthm = $true )
& ( hoasapinj1_gthm = $true )
& ( ( pushprop_lem0 = $true )
<=> ! [X27: subst,X28: term,X29: term > $o] :
~ ! [X30: term > $o] :
~ ! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
!= ( ap @ X213 @ X212 ) )
| ( X212 = X214 ) )
=> ! [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
!= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
| ( X217 = X218 ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X147: term] :
~ ! [X148: subst > term > term] :
~ ! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( hoaslamnotap = $true )
<=> ! [X117: subst > term > term,X118: term,X119: term] :
( ~ ! [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X364: term] :
~ ! [X365: subst > term > term] :
~ ! [X366: subst,X367: term] :
( ( sub @ X364 @ ( push @ X367 @ X366 ) )
= ( X365 @ X366 @ X367 ) )
=> ( ! [X368: term] :
( ( sub @ X368 @ id )
= X368 )
=> ! [X369: term > $o,X370: subst > term > subst > $o,X371: term] :
( ~ ! [X372: term] :
( ( X370 @ id @ X372 @ id )
= ( X369 @ X372 ) )
| ~ ! [X373: term] :
( ( $true
!= ( X369 @ X373 ) )
| ( ( X369 @ ( sub @ X371 @ ( push @ X373 @ id ) ) )
= $true ) )
| ~ ! [X374: subst > term > term] :
( ( $true
= ( X370 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id ) )
| ~ ! [X376: subst,X377: term,X378: subst] :
( ( sub @ ( X374 @ X376 @ X377 ) @ X378 )
= ( X374 @ ( comp @ X376 @ X378 ) @ ( sub @ X377 @ X378 ) ) )
| ~ ! [X375: term] :
( ( $true
!= ( X370 @ id @ X375 @ id ) )
| ( $true
= ( X370 @ id @ ( X374 @ id @ X375 ) @ id ) ) ) )
| ( $true
= ( X369 @ ( lam @ X371 ) ) ) ) ) ) )
& ( pushprop = $true )
& ! [X106: d_subst] : ( d_id = X106 )
& ( hoaslamnotvar_lthm = $true )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
= ( d2term @ X152 ) )
=> ( X151 = X152 ) )
& ( apnotvar = $true )
& ( axvarshift = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( ! [X285: term] :
( ( sub @ X285 @ id )
= X285 )
=> ( ! [X286: term,X287: subst > term > subst > $o] :
( ( $true
= ( X287 @ id @ ( lam @ ( sub @ X286 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X288: subst > term > term] :
( ~ ! [X290: subst,X291: subst,X292: term] :
( ( X288 @ ( comp @ X291 @ X290 ) @ ( sub @ X292 @ X290 ) )
= ( sub @ ( X288 @ X291 @ X292 ) @ X290 ) )
| ( $true
= ( X287 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id ) )
| ~ ! [X289: term] :
( ( $true
= ( X287 @ id @ ( X288 @ id @ X289 ) @ id ) )
| ( $true
!= ( X287 @ id @ X289 @ id ) ) ) )
| ~ ! [X293: term] :
( ( $true
= ( X287 @ id @ ( sub @ X286 @ ( push @ X293 @ id ) ) @ id ) )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) ) )
=> ! [X294: term,X295: subst > term > subst > $o] :
( ~ ! [X296: subst,X297: subst,X298: subst,X299: term] :
( ( $true
= ( X295 @ X298 @ X299 @ ( comp @ X296 @ X297 ) ) )
| ( $true
!= ( X295 @ ( comp @ X298 @ X296 ) @ ( sub @ X299 @ X296 ) @ X297 ) ) )
| ~ ! [X301: subst > term > term] :
( ( $true
= ( X295 @ id @ ( lam @ ( X301 @ sh @ one ) ) @ id ) )
| ~ ! [X302: term,X303: subst,X304: subst] :
( ( X301 @ ( comp @ X303 @ X304 ) @ ( sub @ X302 @ X304 ) )
= ( sub @ ( X301 @ X303 @ X302 ) @ X304 ) )
| ~ ! [X305: term] :
( ( $true
!= ( X295 @ id @ X305 @ id ) )
| ( $true
= ( X295 @ id @ ( X301 @ id @ X305 ) @ id ) ) ) )
| ( $true
= ( X295 @ id @ ( lam @ X294 ) @ id ) )
| ~ ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( $true
= ( X295 @ ( comp @ X308 @ X309 ) @ ( sub @ X306 @ X309 ) @ X307 ) )
| ( $true
!= ( X295 @ X308 @ X306 @ ( comp @ X309 @ X307 ) ) ) )
| ~ ! [X300: term] :
( ( $true
!= ( X295 @ id @ X300 @ id ) )
| ( $true
= ( X295 @ id @ ( sub @ X294 @ ( push @ X300 @ id ) ) @ id ) ) ) ) ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X178: term > $o,X179: subst > term > subst > $o,X180: term] :
( ~ ! [X187: term] :
( ( $true
!= ( X178 @ X187 ) )
| ( $true
= ( X178 @ ( sub @ X180 @ ( push @ X187 @ id ) ) ) ) )
| ( $true
= ( X178 @ ( lam @ X180 ) ) )
| ~ ! [X192: subst,X193: subst,X194: term,X195: subst] :
( ( $true
!= ( X179 @ X192 @ X194 @ ( comp @ X193 @ X195 ) ) )
| ( $true
= ( X179 @ ( comp @ X192 @ X193 ) @ ( sub @ X194 @ X193 ) @ X195 ) ) )
| ~ ! [X182: subst > term > term] :
( ~ ! [X183: term] :
( ( $true
= ( X179 @ id @ ( X182 @ id @ X183 ) @ id ) )
| ( $true
!= ( X179 @ id @ X183 @ id ) ) )
| ( $true
= ( X179 @ id @ ( lam @ ( X182 @ sh @ one ) ) @ id ) )
| ~ ! [X184: subst,X185: subst,X186: term] :
( ( sub @ ( X182 @ X185 @ X186 ) @ X184 )
= ( X182 @ ( comp @ X185 @ X184 ) @ ( sub @ X186 @ X184 ) ) ) )
| ~ ! [X181: term] :
( ( X179 @ id @ X181 @ id )
= ( X178 @ X181 ) )
| ~ ! [X188: term,X189: subst,X190: subst,X191: subst] :
( ( $true
= ( X179 @ X189 @ X188 @ ( comp @ X190 @ X191 ) ) )
| ( $true
!= ( X179 @ ( comp @ X189 @ X190 ) @ ( sub @ X188 @ X190 ) @ X191 ) ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X495: term] :
( ( sub @ X495 @ id )
= X495 )
=> ! [X496: term > $o,X497: subst > term > subst > $o,X498: term] :
( ~ ! [X505: subst,X506: term,X507: subst,X508: subst] :
( ( ( X497 @ ( comp @ X508 @ X507 ) @ ( sub @ X506 @ X507 ) @ X505 )
!= $true )
| ( $true
= ( X497 @ X508 @ X506 @ ( comp @ X507 @ X505 ) ) ) )
| ~ ! [X504: term] :
( ( $true
= ( X496 @ ( sub @ X498 @ ( push @ X504 @ id ) ) ) )
| ( $true
!= ( X496 @ X504 ) ) )
| ~ ! [X503: term] :
( ( X496 @ X503 )
= ( X497 @ id @ X503 @ id ) )
| ~ ! [X509: subst > term > term] :
( ( $true
= ( X497 @ id @ ( lam @ ( X509 @ sh @ one ) ) @ id ) )
| ~ ! [X513: term] :
( ( $true
= ( X497 @ id @ ( X509 @ id @ X513 ) @ id ) )
| ( $true
!= ( X497 @ id @ X513 @ id ) ) )
| ~ ! [X510: subst,X511: term,X512: subst] :
( ( sub @ ( X509 @ X512 @ X511 ) @ X510 )
= ( X509 @ ( comp @ X512 @ X510 ) @ ( sub @ X511 @ X510 ) ) ) )
| ~ ! [X499: subst,X500: term,X501: subst,X502: subst] :
( ( $true
!= ( X497 @ X501 @ X500 @ ( comp @ X499 @ X502 ) ) )
| ( $true
= ( X497 @ ( comp @ X501 @ X499 ) @ ( sub @ X500 @ X499 ) @ X502 ) ) )
| ( $true
= ( X496 @ ( lam @ X498 ) ) ) ) ) )
& ! [X198: term] :
? [X199: d_term] :
( ( d2term @ X199 )
= X198 )
& ( ( ! [X326: subst > term > subst > $o] :
~ ! [X327: term > $o] :
~ ! [X328: term] :
( ( X326 @ id @ X328 @ id )
= ( X327 @ X328 ) )
=> ( ! [X329: term > $o,X330: term] :
( ~ ! [X331: term] :
( ( $true
= ( X329 @ X331 ) )
| ( $true
!= ( var @ X331 ) ) )
| ~ ! [X334: term,X335: term] :
( ( $true
!= ( X329 @ X335 ) )
| ( $true
= ( X329 @ ( ap @ X335 @ X334 ) ) )
| ( $true
!= ( X329 @ X334 ) ) )
| ~ ! [X332: term] :
( ( $true
= ( X329 @ ( lam @ X332 ) ) )
| ~ ! [X333: term] :
( ( $true
= ( X329 @ ( sub @ X332 @ ( push @ X333 @ id ) ) ) )
| ( $true
!= ( X329 @ X333 ) ) ) )
| ( $true
= ( X329 @ X330 ) ) )
=> ( ! [X336: term] :
( ( sub @ X336 @ id )
= X336 )
=> ( ! [X337: subst > term > subst > $o,X338: term,X339: term > $o] :
( ~ ! [X346: term] :
( ( X339 @ X346 )
= ( X337 @ id @ X346 @ id ) )
| ~ ! [X340: term] :
( ( $true
= ( X339 @ ( sub @ X338 @ ( push @ X340 @ id ) ) ) )
| ( $true
!= ( X339 @ X340 ) ) )
| ( $true
= ( X339 @ ( lam @ X338 ) ) )
| ~ ! [X341: subst > term > term] :
( ~ ! [X343: subst,X344: term,X345: subst] :
( ( sub @ ( X341 @ X345 @ X344 ) @ X343 )
= ( X341 @ ( comp @ X345 @ X343 ) @ ( sub @ X344 @ X343 ) ) )
| ( $true
= ( X337 @ id @ ( lam @ ( X341 @ sh @ one ) ) @ id ) )
| ~ ! [X342: term] :
( ( $true
!= ( X337 @ id @ X342 @ id ) )
| ( ( X337 @ id @ ( X341 @ id @ X342 ) @ id )
= $true ) ) ) )
=> ! [X347: subst > term > subst > $o,X348: term] :
( ( $true
= ( X347 @ id @ X348 @ id ) )
| ~ ! [X349: term,X350: term] :
( ( $true
!= ( X347 @ id @ X350 @ id ) )
| ( $true
= ( X347 @ id @ ( ap @ ( sub @ X349 @ id ) @ X350 ) @ id ) )
| ( $true
!= ( X347 @ id @ X349 @ id ) ) )
| ~ ! [X351: subst > term > term] :
( ( $true
= ( X347 @ id @ ( lam @ ( X351 @ sh @ one ) ) @ id ) )
| ~ ! [X352: subst,X353: subst,X354: term] :
( ( X351 @ ( comp @ X353 @ X352 ) @ ( sub @ X354 @ X352 ) )
= ( sub @ ( X351 @ X353 @ X354 ) @ X352 ) )
| ~ ! [X355: term] :
( ( $true
!= ( X347 @ id @ X355 @ id ) )
| ( $true
= ( X347 @ id @ ( X351 @ id @ X355 ) @ id ) ) ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ulamvar1 != $true )
& ( ulamvarind = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( ! [X157: term,X158: subst] :
( ( sub @ one @ ( push @ X157 @ X158 ) )
= X157 )
=> ( ! [X159: subst,X160: term] :
( ( comp @ sh @ ( push @ X160 @ X159 ) )
= X159 )
=> ( ! [X161: term,X162: term] :
( ( X161 = X162 )
| ( ( lam @ X161 )
!= ( lam @ X162 ) ) )
=> ! [X163: subst,X164: subst > term > term,X165: term,X166: subst > term > term] :
( ~ ! [X167: subst,X168: term,X169: subst] :
( ( sub @ ( X164 @ X167 @ X168 ) @ X169 )
= ( X164 @ ( comp @ X167 @ X169 ) @ ( sub @ X168 @ X169 ) ) )
| ~ ! [X170: term,X171: subst,X172: subst] :
( ( sub @ ( X166 @ X172 @ X170 ) @ X171 )
= ( X166 @ ( comp @ X172 @ X171 ) @ ( sub @ X170 @ X171 ) ) )
| ( ( X164 @ X163 @ X165 )
= ( X166 @ X163 @ X165 ) )
| ( ( lam @ ( X166 @ sh @ one ) )
!= ( lam @ ( X164 @ sh @ one ) ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( induction2_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( axidr = $true )
<=> ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 ) )
& ( ! [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
!= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
| ( X435 = X436 ) )
<=> ( hoasapinj1 = $true ) )
& ( ( hoaslaminj = $true )
<=> ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ~ ! [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ~ ! [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X127: term,X128: term > $o,X129: subst > term > subst > $o,X130: term] :
( ~ ! [X131: term,X132: subst,X133: subst,X134: subst] :
( ( $true
!= ( X129 @ X133 @ X131 @ ( comp @ X132 @ X134 ) ) )
| ( $true
= ( X129 @ ( comp @ X133 @ X132 ) @ ( sub @ X131 @ X132 ) @ X134 ) ) )
| ( $true
= ( X128 @ ( ap @ X130 @ X127 ) ) )
| ( $true
!= ( X128 @ X130 ) )
| ~ ! [X140: term,X141: term] :
( ( $true
!= ( X129 @ id @ X141 @ id ) )
| ( $true
!= ( X129 @ id @ X140 @ id ) )
| ( $true
= ( X129 @ id @ ( ap @ ( sub @ X140 @ id ) @ X141 ) @ id ) ) )
| ~ ! [X135: subst,X136: subst,X137: term,X138: subst] :
( ( $true
= ( X129 @ X138 @ X137 @ ( comp @ X136 @ X135 ) ) )
| ( $true
!= ( X129 @ ( comp @ X138 @ X136 ) @ ( sub @ X137 @ X136 ) @ X135 ) ) )
| ~ ! [X139: term] :
( ( X129 @ id @ X139 @ id )
= ( X128 @ X139 ) )
| ( $true
!= ( X128 @ X127 ) ) ) )
& ( ( ! [X84: subst,X85: term] :
( ( sub @ one @ ( push @ X85 @ X84 ) )
= X85 )
=> ( ! [X86: subst,X87: term] :
( ( comp @ sh @ ( push @ X87 @ X86 ) )
= X86 )
=> ( ! [X88: subst] :
( ( comp @ X88 @ id )
= X88 )
=> ( ! [X89: term,X90: term > $o] :
( ~ ! [X94: term] :
( ( $true
!= ( var @ X94 ) )
| ( $true
= ( X90 @ X94 ) ) )
| ~ ! [X93: term] :
( ( $true
!= ( X90 @ X93 ) )
| ( $true
= ( X90 @ ( lam @ X93 ) ) ) )
| ( $true
= ( X90 @ X89 ) )
| ~ ! [X91: term,X92: term] :
( ( ( X90 @ X91 )
!= $true )
| ( $true
= ( X90 @ ( ap @ X91 @ X92 ) ) )
| ( $true
!= ( X90 @ X92 ) ) ) )
=> ! [X95: subst,X96: term,X97: term > $o] :
( ( $true
= ( X97 @ ( sub @ X96 @ X95 ) ) )
| ~ ! [X98: term] :
( ( $true
= ( X97 @ ( lam @ X98 ) ) )
| ~ ! [X99: term] :
( ( $true
!= ( X97 @ X99 ) )
| ( $true
= ( X97 @ ( sub @ X98 @ ( push @ X99 @ id ) ) ) ) ) )
| ~ ! [X100: term] :
( ( $true
!= ( var @ X100 ) )
| ( $true
= ( X97 @ ( sub @ X100 @ X95 ) ) ) )
| ~ ! [X101: term,X102: term] :
( ( $true
!= ( X97 @ X102 ) )
| ( $true
!= ( X97 @ X101 ) )
| ( $true
= ( X97 @ ( ap @ X102 @ X101 ) ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( induction2lem_gthm = $true )
& ( substmonoid_gthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( axabs = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X219: subst > term > subst > $o,X220: term] :
( ~ ! [X226: term] :
( ( $true
= ( X219 @ id @ ( sub @ X220 @ ( push @ X226 @ id ) ) @ id ) )
| ( $true
!= ( X219 @ id @ X226 @ id ) ) )
| ~ ! [X221: subst > term > term] :
( ~ ! [X222: term] :
( ( $true
!= ( X219 @ id @ X222 @ id ) )
| ( $true
= ( X219 @ id @ ( X221 @ id @ X222 ) @ id ) ) )
| ( $true
= ( X219 @ id @ ( lam @ ( X221 @ sh @ one ) ) @ id ) )
| ~ ! [X223: term,X224: subst,X225: subst] :
( ( X221 @ ( comp @ X225 @ X224 ) @ ( sub @ X223 @ X224 ) )
= ( sub @ ( X221 @ X225 @ X223 ) @ X224 ) ) )
| ( $true
= ( X219 @ id @ ( lam @ ( sub @ X220 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ! [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
= X539 )
<=> ( axscons = $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
= X37 )
=> ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ( $true
!= ( X40 @ X39 ) )
| ( $true
= ( X41 @ one ) )
| ~ ! [X43: term] :
( ( X41 @ X43 )
= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) ) ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( apinj2 = $true )
<=> ! [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) )
| ( X0 = X1 ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( axmap = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X532: term,X533: term > $o,X534: subst] :
~ ! [X535: term > $o] :
~ ! [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
= ( X535 @ X536 ) ) )
& ( ! [X310: subst > term > subst > $o,X311: term] :
( ~ ! [X312: term] :
( ( $true
= ( X310 @ id @ ( sub @ X311 @ ( push @ X312 @ id ) ) @ id ) )
| ( $true
!= ( X310 @ id @ X312 @ id ) ) )
| ( $true
= ( X310 @ id @ ( lam @ X311 ) @ id ) )
| ~ ! [X313: subst > term > term] :
( ~ ! [X317: term] :
( ( $true
= ( X310 @ id @ ( X313 @ id @ X317 ) @ id ) )
| ( $true
!= ( X310 @ id @ X317 @ id ) ) )
| ( $true
= ( X310 @ id @ ( lam @ ( X313 @ sh @ one ) ) @ id ) )
| ~ ! [X314: subst,X315: term,X316: subst] :
( ( sub @ ( X313 @ X314 @ X315 ) @ X316 )
= ( X313 @ ( comp @ X314 @ X316 ) @ ( sub @ X315 @ X316 ) ) ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X537: term] :
~ ! [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
!= ( X538 @ sh @ one ) ) )
& ! [X485: d_term] : ( d_one = X485 )
& ( hoaslamnotvar_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( induction = $true )
<=> ! [X479: term > $o,X480: term] :
( ~ ! [X482: term] :
( ( $true
!= ( X479 @ X482 ) )
| ( $true
= ( X479 @ ( lam @ X482 ) ) ) )
| ~ ! [X483: term,X484: term] :
( ( $true
!= ( X479 @ X483 ) )
| ( $true
!= ( X479 @ X484 ) )
| ( ( X479 @ ( ap @ X483 @ X484 ) )
= $true ) )
| ~ ! [X481: term] :
( ( $true
!= ( var @ X481 ) )
| ( $true
= ( X479 @ X481 ) ) )
| ( $true
= ( X479 @ X480 ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( ! [X227: subst > term > subst > $o] :
~ ! [X228: term > $o] :
~ ! [X229: term] :
( ( X227 @ id @ X229 @ id )
= ( X228 @ X229 ) )
=> ( ! [X230: term > $o,X231: term] :
( ~ ! [X235: term,X236: term] :
( ( $true
!= ( X230 @ X236 ) )
| ( $true
!= ( X230 @ X235 ) )
| ( $true
= ( X230 @ ( ap @ X235 @ X236 ) ) ) )
| ~ ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( $true
!= ( var @ X232 ) ) )
| ( $true
= ( X230 @ X231 ) )
| ~ ! [X233: term] :
( ( $true
= ( X230 @ ( lam @ X233 ) ) )
| ~ ! [X234: term] :
( ( $true
= ( X230 @ ( sub @ X233 @ ( push @ X234 @ id ) ) ) )
| ( $true
!= ( X230 @ X234 ) ) ) ) )
=> ( ! [X237: term] :
( ( sub @ X237 @ id )
= X237 )
=> ( ! [X238: term,X239: term > $o,X240: subst > term > subst > $o] :
( ~ ! [X241: term] :
( ( $true
= ( X239 @ ( sub @ X238 @ ( push @ X241 @ id ) ) ) )
| ( $true
!= ( X239 @ X241 ) ) )
| ( $true
= ( X239 @ ( lam @ X238 ) ) )
| ~ ! [X242: subst > term > term] :
( ~ ! [X244: subst,X245: term,X246: subst] :
( ( X242 @ ( comp @ X246 @ X244 ) @ ( sub @ X245 @ X244 ) )
= ( sub @ ( X242 @ X246 @ X245 ) @ X244 ) )
| ( $true
= ( X240 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id ) )
| ~ ! [X243: term] :
( ( $true
!= ( X240 @ id @ X243 @ id ) )
| ( $true
= ( X240 @ id @ ( X242 @ id @ X243 ) @ id ) ) ) )
| ~ ! [X247: term] :
( ( X240 @ id @ X247 @ id )
= ( X239 @ X247 ) ) )
=> ! [X248: term,X249: subst > term > subst > $o] :
( ~ ! [X252: subst,X253: term,X254: subst,X255: subst] :
( ( $true
!= ( X249 @ X252 @ X253 @ ( comp @ X254 @ X255 ) ) )
| ( $true
= ( X249 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) @ X255 ) ) )
| ~ ! [X256: subst > term > term] :
( ~ ! [X257: term] :
( ( $true
!= ( X249 @ id @ X257 @ id ) )
| ( $true
= ( X249 @ id @ ( X256 @ id @ X257 ) @ id ) ) )
| ( $true
= ( X249 @ id @ ( lam @ ( X256 @ sh @ one ) ) @ id ) )
| ~ ! [X258: subst,X259: term,X260: subst] :
( ( sub @ ( X256 @ X260 @ X259 ) @ X258 )
= ( X256 @ ( comp @ X260 @ X258 ) @ ( sub @ X259 @ X258 ) ) ) )
| ~ ! [X250: term,X251: term] :
( ( $true
!= ( X249 @ id @ X250 @ id ) )
| ( $true
!= ( X249 @ id @ X251 @ id ) )
| ( $true
= ( X249 @ id @ ( ap @ ( sub @ X251 @ id ) @ X250 ) @ id ) ) )
| ( $true
= ( X249 @ id @ X248 @ id ) )
| ~ ! [X265: term] :
( ( $true
!= ( var @ ( sub @ X265 @ id ) ) )
| ( $true
= ( X249 @ id @ X265 @ id ) ) )
| ~ ! [X261: term,X262: subst,X263: subst,X264: subst] :
( ( $true
!= ( X249 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) @ X264 ) )
| ( $true
= ( X249 @ X262 @ X261 @ ( comp @ X263 @ X264 ) ) ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ! [X318: term] :
( ( sub @ X318 @ id )
= X318 )
<=> ( axvarid = $true ) )
& ( ( induction2 = $true )
<=> ! [X319: term,X320: term > $o] :
( ( $true
= ( X320 @ X319 ) )
| ~ ! [X322: term,X323: term] :
( ( $true
!= ( X320 @ X323 ) )
| ( $true
!= ( X320 @ X322 ) )
| ( $true
= ( X320 @ ( ap @ X323 @ X322 ) ) ) )
| ~ ! [X321: term] :
( ( $true
= ( X320 @ X321 ) )
| ( $true
!= ( var @ X321 ) ) )
| ~ ! [X324: term] :
( ~ ! [X325: term] :
( ( $true
= ( X320 @ ( sub @ X324 @ ( push @ X325 @ id ) ) ) )
| ( $true
!= ( X320 @ X325 ) ) )
| ( $true
= ( X320 @ ( lam @ X324 ) ) ) ) ) )
& ( ( ! [X33: subst] :
( ( comp @ id @ X33 )
= X33 )
=> ( ! [X34: subst] :
( ( comp @ X34 @ id )
= X34 )
=> ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( termmset_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
| ( $true
!= ( X173 @ X174 ) ) )
<=> ( pushprop_lem1 = $true ) )
& ( ( shinj = $true )
<=> ! [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
!= ( sub @ X515 @ sh ) )
| ( X514 = X515 ) ) )
& ( ! [X466: term,X467: term,X468: subst > term > subst > $o] :
( ~ ! [X469: term,X470: term] :
( ( $true
= ( X468 @ id @ ( ap @ ( sub @ X470 @ id ) @ X469 ) @ id ) )
| ( $true
!= ( X468 @ id @ X470 @ id ) )
| ( $true
!= ( X468 @ id @ X469 @ id ) ) )
| ( $true
= ( X468 @ id @ ( ap @ X467 @ X466 ) @ id ) )
| ~ ! [X471: subst,X472: subst,X473: subst,X474: term] :
( ( $true
= ( X468 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 ) )
| ( $true
!= ( X468 @ X473 @ X474 @ ( comp @ X472 @ X471 ) ) ) )
| ( $true
!= ( X468 @ id @ X467 @ id ) )
| ( $true
!= ( X468 @ id @ X466 @ id ) )
| ~ ! [X475: subst,X476: subst,X477: subst,X478: term] :
( ( $true
= ( X468 @ X476 @ X478 @ ( comp @ X475 @ X477 ) ) )
| ( $true
!= ( X468 @ ( comp @ X476 @ X475 ) @ ( sub @ X478 @ X475 ) @ X477 ) ) ) )
<=> ( hoasinduction_lem2 = $true ) ) ),
inference(flattening,[status(thm)],[f10]) ).
thf(f10,plain,
( ( ( apinj2 = $true )
<=> ! [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) )
| ( X0 = X1 ) ) )
& ( ! [X4: term,X5: subst > term > subst > $o] :
( ( $true
= ( X5 @ id @ X4 @ id ) )
| ~ ! [X6: subst > term > term] :
( ~ ! [X7: term] :
( ( $true
!= ( X5 @ id @ X7 @ id ) )
| ( $true
= ( X5 @ id @ ( X6 @ id @ X7 ) @ id ) ) )
| ~ ! [X8: term,X9: subst,X10: subst] :
( ( X6 @ ( comp @ X10 @ X9 ) @ ( sub @ X8 @ X9 ) )
= ( sub @ ( X6 @ X10 @ X8 ) @ X9 ) )
| ( $true
= ( X5 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id ) ) )
| ~ ! [X11: term,X12: term] :
( ( $true
!= ( X5 @ id @ X12 @ id ) )
| ( $true
= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
| ( ( X5 @ id @ X11 @ id )
!= $true ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
= ( d2subst @ X14 ) )
=> ( X13 = X14 ) )
& ( one
= ( d2term @ d_one ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X15: subst > term > subst > $o,X16: term > $o,X17: term] :
( ~ ! [X18: term] :
( ( X16 @ X18 )
= ( X15 @ id @ X18 @ id ) )
| ~ ! [X19: subst > term > term] :
( ~ ! [X20: term,X21: subst,X22: subst] :
( ( sub @ ( X19 @ X22 @ X20 ) @ X21 )
= ( X19 @ ( comp @ X22 @ X21 ) @ ( sub @ X20 @ X21 ) ) )
| ( $true
= ( X15 @ id @ ( lam @ ( X19 @ sh @ one ) ) @ id ) )
| ~ ! [X23: term] :
( ( $true
= ( X15 @ id @ ( X19 @ id @ X23 ) @ id ) )
| ( $true
!= ( X15 @ id @ X23 @ id ) ) ) )
| ~ ! [X24: term] :
( ( $true
= ( X16 @ ( sub @ X17 @ ( push @ X24 @ id ) ) ) )
| ( $true
!= ( X16 @ X24 ) ) )
| ( $true
= ( X16 @ ( lam @ X17 ) ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( ! [X25: term] :
( ( sub @ X25 @ id )
= X25 )
=> ! [X26: term] :
( ( sub @ X26 @ id )
= X26 ) )
<=> ( termmset_lthm = $true ) )
& ( ( pushprop_lem0 = $true )
<=> ! [X27: subst,X28: term,X29: term > $o] :
~ ! [X30: term > $o] :
~ ! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) ) )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
<=> ( termmset = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ( ! [X33: subst] :
( ( comp @ id @ X33 )
= X33 )
=> ( ! [X34: subst] :
( ( comp @ X34 @ id )
= X34 )
=> ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
= X37 )
=> ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ~ ! [X43: term] :
( ( X41 @ X43 )
= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) )
| ( $true
!= ( X40 @ X39 ) )
| ( $true
= ( X41 @ one ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( lamnotap != $true )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X44: term,X45: subst > term > subst > $o] :
( ( $true
= ( X45 @ id @ ( lam @ ( sub @ X44 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X46: term] :
( ( ( X45 @ id @ ( sub @ X44 @ ( push @ X46 @ id ) ) @ id )
= $true )
| ( $true
!= ( X45 @ id @ X46 @ id ) ) )
| ~ ! [X47: subst > term > term,X48: term] :
( ~ ! [X49: subst,X50: term,X51: subst] :
( ( sub @ ( sub @ X48 @ ( push @ X50 @ X49 ) ) @ X51 )
= ( sub @ ( X47 @ X49 @ X50 ) @ X51 ) )
| ~ ! [X52: subst,X53: term,X54: subst] :
( ( sub @ X48 @ ( push @ ( sub @ X53 @ X52 ) @ ( comp @ X54 @ X52 ) ) )
= ( X47 @ ( comp @ X54 @ X52 ) @ ( sub @ X53 @ X52 ) ) )
| ( $true
= ( X45 @ id @ ( lam @ ( X47 @ sh @ one ) ) @ id ) )
| ~ ! [X55: term] :
( ( $true
= ( X45 @ id @ ( X47 @ id @ X55 ) @ id ) )
| ( $true
!= ( X45 @ id @ X55 @ id ) ) ) ) ) )
& ( ( axidl = $true )
<=> ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 ) )
& ( hoasinduction_gthm = $true )
& ( ( axshiftcons = $true )
<=> ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 ) )
& ( axapp = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction = $true )
<=> ! [X59: term,X60: subst > term > subst > $o] :
( ~ ! [X61: term,X62: term] :
( ( $true
= ( X60 @ id @ ( ap @ ( sub @ X61 @ id ) @ X62 ) @ id ) )
| ( $true
!= ( X60 @ id @ X61 @ id ) )
| ( $true
!= ( X60 @ id @ X62 @ id ) ) )
| ~ ! [X63: subst > term > term] :
( ~ ! [X64: term] :
( ( $true
!= ( X60 @ id @ X64 @ id ) )
| ( $true
= ( X60 @ id @ ( X63 @ id @ X64 ) @ id ) ) )
| ( $true
= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
| ~ ! [X65: subst,X66: subst,X67: term] :
( ( X63 @ ( comp @ X65 @ X66 ) @ ( sub @ X67 @ X66 ) )
= ( sub @ ( X63 @ X65 @ X67 ) @ X66 ) ) )
| ~ ! [X68: term] :
( ( $true
!= ( var @ ( sub @ X68 @ id ) ) )
| ( $true
= ( X60 @ id @ X68 @ id ) ) )
| ~ ! [X69: subst,X70: subst,X71: term,X72: subst] :
( ( $true
!= ( X60 @ ( comp @ X69 @ X72 ) @ ( sub @ X71 @ X72 ) @ X70 ) )
| ( $true
= ( X60 @ X69 @ X71 @ ( comp @ X72 @ X70 ) ) ) )
| ~ ! [X73: term,X74: subst,X75: subst,X76: subst] :
( ( $true
= ( X60 @ ( comp @ X75 @ X74 ) @ ( sub @ X73 @ X74 ) @ X76 ) )
| ( $true
!= ( X60 @ X75 @ X73 @ ( comp @ X74 @ X76 ) ) ) )
| ( $true
= ( X60 @ id @ X59 @ id ) ) ) )
& ( ulamvar1 != $true )
& ( ( d2subst @ d_id )
= sh )
& ( hoaslamnotvar_lthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X77: subst > term > subst > $o] :
~ ! [X78: term > $o] :
~ ! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X80: term] :
~ ! [X81: subst > term > term] :
~ ! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) ) )
& ( induction2lem_gthm = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ( induction2lem_lthm = $true )
<=> ( ! [X84: subst,X85: term] :
( ( sub @ one @ ( push @ X85 @ X84 ) )
= X85 )
=> ( ! [X86: subst,X87: term] :
( ( comp @ sh @ ( push @ X87 @ X86 ) )
= X86 )
=> ( ! [X88: subst] :
( ( comp @ X88 @ id )
= X88 )
=> ( ! [X89: term,X90: term > $o] :
( ~ ! [X91: term,X92: term] :
( ( $true
!= ( X90 @ X92 ) )
| ( $true
= ( X90 @ ( ap @ X91 @ X92 ) ) )
| ( ( X90 @ X91 )
!= $true ) )
| ~ ! [X93: term] :
( ( $true
!= ( X90 @ X93 ) )
| ( $true
= ( X90 @ ( lam @ X93 ) ) ) )
| ( $true
= ( X90 @ X89 ) )
| ~ ! [X94: term] :
( ( $true
!= ( var @ X94 ) )
| ( $true
= ( X90 @ X94 ) ) ) )
=> ! [X95: subst,X96: term,X97: term > $o] :
( ~ ! [X98: term] :
( ~ ! [X99: term] :
( ( $true
= ( X97 @ ( sub @ X98 @ ( push @ X99 @ id ) ) ) )
| ( $true
!= ( X97 @ X99 ) ) )
| ( $true
= ( X97 @ ( lam @ X98 ) ) ) )
| ~ ! [X100: term] :
( ( $true
!= ( var @ X100 ) )
| ( $true
= ( X97 @ ( sub @ X100 @ X95 ) ) ) )
| ~ ! [X101: term,X102: term] :
( ( $true
!= ( X97 @ X102 ) )
| ( $true
= ( X97 @ ( ap @ X102 @ X101 ) ) )
| ( $true
!= ( X97 @ X101 ) ) )
| ( $true
= ( X97 @ ( sub @ X96 @ X95 ) ) ) ) ) ) ) ) )
& ( ( axvarcons = $true )
<=> ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 ) )
& ( hoasapinj2_gthm = $true )
& ( ( axidr = $true )
<=> ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 ) )
& ! [X106: d_subst] : ( d_id = X106 )
& ( ( hoaslaminj = $true )
<=> ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ~ ! [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ~ ! [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X117: subst > term > term,X118: term,X119: term] :
( ~ ! [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) ) )
& ( ! [X123: term,X124: term,X125: term,X126: term] :
( ( X125 = X126 )
| ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) ) )
<=> ( apinj1 = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X127: term,X128: term > $o,X129: subst > term > subst > $o,X130: term] :
( ( $true
!= ( X128 @ X130 ) )
| ( $true
= ( X128 @ ( ap @ X130 @ X127 ) ) )
| ~ ! [X131: term,X132: subst,X133: subst,X134: subst] :
( ( $true
= ( X129 @ ( comp @ X133 @ X132 ) @ ( sub @ X131 @ X132 ) @ X134 ) )
| ( $true
!= ( X129 @ X133 @ X131 @ ( comp @ X132 @ X134 ) ) ) )
| ( $true
!= ( X128 @ X127 ) )
| ~ ! [X135: subst,X136: subst,X137: term,X138: subst] :
( ( $true
!= ( X129 @ ( comp @ X138 @ X136 ) @ ( sub @ X137 @ X136 ) @ X135 ) )
| ( $true
= ( X129 @ X138 @ X137 @ ( comp @ X136 @ X135 ) ) ) )
| ~ ! [X139: term] :
( ( X129 @ id @ X139 @ id )
= ( X128 @ X139 ) )
| ~ ! [X140: term,X141: term] :
( ( $true
= ( X129 @ id @ ( ap @ ( sub @ X140 @ id ) @ X141 ) @ id ) )
| ( $true
!= ( X129 @ id @ X140 @ id ) )
| ( $true
!= ( X129 @ id @ X141 @ id ) ) ) ) )
& ( pushprop_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar_lthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
= ( X144 @ one ) )
| ~ ! [X146: term] :
( ( X144 @ X146 )
= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) )
| ( $true
!= ( X142 @ X145 ) ) ) )
& ( ! [X147: term] :
~ ! [X148: subst > term > term] :
~ ! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
= ( d2term @ X152 ) )
=> ( X151 = X152 ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( X153 = X155 )
| ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( ! [X157: term,X158: subst] :
( ( sub @ one @ ( push @ X157 @ X158 ) )
= X157 )
=> ( ! [X159: subst,X160: term] :
( ( comp @ sh @ ( push @ X160 @ X159 ) )
= X159 )
=> ( ! [X161: term,X162: term] :
( ( X161 = X162 )
| ( ( lam @ X161 )
!= ( lam @ X162 ) ) )
=> ! [X163: subst,X164: subst > term > term,X165: term,X166: subst > term > term] :
( ~ ! [X167: subst,X168: term,X169: subst] :
( ( sub @ ( X164 @ X167 @ X168 ) @ X169 )
= ( X164 @ ( comp @ X167 @ X169 ) @ ( sub @ X168 @ X169 ) ) )
| ~ ! [X170: term,X171: subst,X172: subst] :
( ( sub @ ( X166 @ X172 @ X170 ) @ X171 )
= ( X166 @ ( comp @ X172 @ X171 ) @ ( sub @ X170 @ X171 ) ) )
| ( ( X164 @ X163 @ X165 )
= ( X166 @ X163 @ X165 ) )
| ( ( lam @ ( X166 @ sh @ one ) )
!= ( lam @ ( X164 @ sh @ one ) ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( pushprop_lem1 = $true )
<=> ! [X173: term > $o,X174: term,X175: subst,X176: term,X177: term > $o] :
( ( $true
= ( X177 @ ( sub @ X174 @ ( push @ X176 @ X175 ) ) ) )
| ( $true
!= ( X173 @ X174 ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X178: term > $o,X179: subst > term > subst > $o,X180: term] :
( ~ ! [X181: term] :
( ( X179 @ id @ X181 @ id )
= ( X178 @ X181 ) )
| ~ ! [X182: subst > term > term] :
( ( $true
= ( X179 @ id @ ( lam @ ( X182 @ sh @ one ) ) @ id ) )
| ~ ! [X183: term] :
( ( $true
= ( X179 @ id @ ( X182 @ id @ X183 ) @ id ) )
| ( $true
!= ( X179 @ id @ X183 @ id ) ) )
| ~ ! [X184: subst,X185: subst,X186: term] :
( ( sub @ ( X182 @ X185 @ X186 ) @ X184 )
= ( X182 @ ( comp @ X185 @ X184 ) @ ( sub @ X186 @ X184 ) ) ) )
| ~ ! [X187: term] :
( ( $true
= ( X178 @ ( sub @ X180 @ ( push @ X187 @ id ) ) ) )
| ( $true
!= ( X178 @ X187 ) ) )
| ~ ! [X188: term,X189: subst,X190: subst,X191: subst] :
( ( $true
= ( X179 @ X189 @ X188 @ ( comp @ X190 @ X191 ) ) )
| ( $true
!= ( X179 @ ( comp @ X189 @ X190 ) @ ( sub @ X188 @ X190 ) @ X191 ) ) )
| ~ ! [X192: subst,X193: subst,X194: term,X195: subst] :
( ( $true
= ( X179 @ ( comp @ X192 @ X193 ) @ ( sub @ X194 @ X193 ) @ X195 ) )
| ( $true
!= ( X179 @ X192 @ X194 @ ( comp @ X193 @ X195 ) ) ) )
| ( $true
= ( X178 @ ( lam @ X180 ) ) ) ) )
& ( ! [X196: term,X197: term] :
( ( X196 = X197 )
| ( ( lam @ X197 )
!= ( lam @ X196 ) ) )
<=> ( laminj = $true ) )
& ! [X198: term] :
? [X199: d_term] :
( ( d2term @ X199 )
= X198 )
& ( pushprop_lem3v2 = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X200: term] :
~ ! [X201: subst > term > term] :
( ( X201 @ sh @ one )
!= ( sub @ X200 @ ( push @ one @ sh ) ) ) )
& ( hoasapnotvar_gthm = $true )
& ( axmap = $true )
& ( axclos = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoaslamnotvar = $true )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X202: subst,X203: term] :
( ( sub @ one @ ( push @ X203 @ X202 ) )
= X203 )
=> ( ! [X204: subst,X205: term] :
( ( comp @ sh @ ( push @ X205 @ X204 ) )
= X204 )
=> ! [X206: term,X207: term > $o,X208: term > $o,X209: subst,X210: term] :
( ( $true
!= ( X207 @ X210 ) )
| ( $true
= ( X208 @ ( sub @ X210 @ ( push @ X206 @ X209 ) ) ) ) ) ) ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X211 @ X214 )
!= ( ap @ X213 @ X212 ) )
| ( X212 = X214 ) )
=> ! [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ ( sub @ X215 @ id ) @ X217 )
!= ( ap @ ( sub @ X216 @ id ) @ X218 ) )
| ( X217 = X218 ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X219: subst > term > subst > $o,X220: term] :
( ~ ! [X221: subst > term > term] :
( ~ ! [X222: term] :
( ( $true
!= ( X219 @ id @ X222 @ id ) )
| ( $true
= ( X219 @ id @ ( X221 @ id @ X222 ) @ id ) ) )
| ( $true
= ( X219 @ id @ ( lam @ ( X221 @ sh @ one ) ) @ id ) )
| ~ ! [X223: term,X224: subst,X225: subst] :
( ( X221 @ ( comp @ X225 @ X224 ) @ ( sub @ X223 @ X224 ) )
= ( sub @ ( X221 @ X225 @ X223 ) @ X224 ) ) )
| ~ ! [X226: term] :
( ( $true
= ( X219 @ id @ ( sub @ X220 @ ( push @ X226 @ id ) ) @ id ) )
| ( $true
!= ( X219 @ id @ X226 @ id ) ) )
| ( $true
= ( X219 @ id @ ( lam @ ( sub @ X220 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ! [X227: subst > term > subst > $o] :
~ ! [X228: term > $o] :
~ ! [X229: term] :
( ( X227 @ id @ X229 @ id )
= ( X228 @ X229 ) )
=> ( ! [X230: term > $o,X231: term] :
( ~ ! [X232: term] :
( ( $true
= ( X230 @ X232 ) )
| ( $true
!= ( var @ X232 ) ) )
| ( $true
= ( X230 @ X231 ) )
| ~ ! [X233: term] :
( ~ ! [X234: term] :
( ( $true
= ( X230 @ ( sub @ X233 @ ( push @ X234 @ id ) ) ) )
| ( $true
!= ( X230 @ X234 ) ) )
| ( $true
= ( X230 @ ( lam @ X233 ) ) ) )
| ~ ! [X235: term,X236: term] :
( ( $true
= ( X230 @ ( ap @ X235 @ X236 ) ) )
| ( $true
!= ( X230 @ X236 ) )
| ( $true
!= ( X230 @ X235 ) ) ) )
=> ( ! [X237: term] :
( ( sub @ X237 @ id )
= X237 )
=> ( ! [X238: term,X239: term > $o,X240: subst > term > subst > $o] :
( ~ ! [X241: term] :
( ( $true
!= ( X239 @ X241 ) )
| ( $true
= ( X239 @ ( sub @ X238 @ ( push @ X241 @ id ) ) ) ) )
| ~ ! [X242: subst > term > term] :
( ( $true
= ( X240 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id ) )
| ~ ! [X243: term] :
( ( $true
!= ( X240 @ id @ X243 @ id ) )
| ( $true
= ( X240 @ id @ ( X242 @ id @ X243 ) @ id ) ) )
| ~ ! [X244: subst,X245: term,X246: subst] :
( ( X242 @ ( comp @ X246 @ X244 ) @ ( sub @ X245 @ X244 ) )
= ( sub @ ( X242 @ X246 @ X245 ) @ X244 ) ) )
| ~ ! [X247: term] :
( ( X240 @ id @ X247 @ id )
= ( X239 @ X247 ) )
| ( $true
= ( X239 @ ( lam @ X238 ) ) ) )
=> ! [X248: term,X249: subst > term > subst > $o] :
( ~ ! [X250: term,X251: term] :
( ( $true
!= ( X249 @ id @ X251 @ id ) )
| ( $true
!= ( X249 @ id @ X250 @ id ) )
| ( $true
= ( X249 @ id @ ( ap @ ( sub @ X251 @ id ) @ X250 ) @ id ) ) )
| ~ ! [X252: subst,X253: term,X254: subst,X255: subst] :
( ( $true
!= ( X249 @ X252 @ X253 @ ( comp @ X254 @ X255 ) ) )
| ( $true
= ( X249 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) @ X255 ) ) )
| ~ ! [X256: subst > term > term] :
( ~ ! [X257: term] :
( ( $true
!= ( X249 @ id @ X257 @ id ) )
| ( $true
= ( X249 @ id @ ( X256 @ id @ X257 ) @ id ) ) )
| ~ ! [X258: subst,X259: term,X260: subst] :
( ( sub @ ( X256 @ X260 @ X259 ) @ X258 )
= ( X256 @ ( comp @ X260 @ X258 ) @ ( sub @ X259 @ X258 ) ) )
| ( $true
= ( X249 @ id @ ( lam @ ( X256 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X249 @ id @ X248 @ id ) )
| ~ ! [X261: term,X262: subst,X263: subst,X264: subst] :
( ( $true
!= ( X249 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) @ X264 ) )
| ( $true
= ( X249 @ X262 @ X261 @ ( comp @ X263 @ X264 ) ) ) )
| ~ ! [X265: term] :
( ( $true
!= ( var @ ( sub @ X265 @ id ) ) )
| ( $true
= ( X249 @ id @ X265 @ id ) ) ) ) ) ) ) ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ! [X266: subst] :
? [X267: d_subst] :
( ( d2subst @ X267 )
= X266 )
& ( ( ! [X268: term] :
( ( sub @ X268 @ id )
= X268 )
=> ( ! [X269: term,X270: subst > term > subst > $o] :
( ~ ! [X271: subst > term > term] :
( ~ ! [X272: subst,X273: term,X274: subst] :
( ( X271 @ ( comp @ X272 @ X274 ) @ ( sub @ X273 @ X274 ) )
= ( sub @ ( X271 @ X272 @ X273 ) @ X274 ) )
| ( $true
= ( X270 @ id @ ( lam @ ( X271 @ sh @ one ) ) @ id ) )
| ~ ! [X275: term] :
( ( $true
!= ( X270 @ id @ X275 @ id ) )
| ( $true
= ( X270 @ id @ ( X271 @ id @ X275 ) @ id ) ) ) )
| ( $true
= ( X270 @ id @ ( lam @ ( sub @ X269 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X276: term] :
( ( $true
= ( X270 @ id @ ( sub @ X269 @ ( push @ X276 @ id ) ) @ id ) )
| ( $true
!= ( X270 @ id @ X276 @ id ) ) ) )
=> ! [X277: term,X278: subst > term > subst > $o] :
( ~ ! [X279: subst > term > term] :
( ( $true
= ( X278 @ id @ ( lam @ ( X279 @ sh @ one ) ) @ id ) )
| ~ ! [X280: subst,X281: term,X282: subst] :
( ( sub @ ( X279 @ X282 @ X281 ) @ X280 )
= ( X279 @ ( comp @ X282 @ X280 ) @ ( sub @ X281 @ X280 ) ) )
| ~ ! [X283: term] :
( ( $true
= ( X278 @ id @ ( X279 @ id @ X283 ) @ id ) )
| ( $true
!= ( X278 @ id @ X283 @ id ) ) ) )
| ~ ! [X284: term] :
( ( $true
= ( X278 @ id @ ( sub @ X277 @ ( push @ X284 @ id ) ) @ id ) )
| ( $true
!= ( X278 @ id @ X284 @ id ) ) )
| ( $true
= ( X278 @ id @ ( lam @ X277 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( pushprop_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X285: term] :
( ( sub @ X285 @ id )
= X285 )
=> ( ! [X286: term,X287: subst > term > subst > $o] :
( ~ ! [X288: subst > term > term] :
( ~ ! [X289: term] :
( ( $true
= ( X287 @ id @ ( X288 @ id @ X289 ) @ id ) )
| ( $true
!= ( X287 @ id @ X289 @ id ) ) )
| ~ ! [X290: subst,X291: subst,X292: term] :
( ( X288 @ ( comp @ X291 @ X290 ) @ ( sub @ X292 @ X290 ) )
= ( sub @ ( X288 @ X291 @ X292 ) @ X290 ) )
| ( $true
= ( X287 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X287 @ id @ ( lam @ ( sub @ X286 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X293: term] :
( ( $true
= ( X287 @ id @ ( sub @ X286 @ ( push @ X293 @ id ) ) @ id ) )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) ) )
=> ! [X294: term,X295: subst > term > subst > $o] :
( ( $true
= ( X295 @ id @ ( lam @ X294 ) @ id ) )
| ~ ! [X296: subst,X297: subst,X298: subst,X299: term] :
( ( $true
!= ( X295 @ ( comp @ X298 @ X296 ) @ ( sub @ X299 @ X296 ) @ X297 ) )
| ( $true
= ( X295 @ X298 @ X299 @ ( comp @ X296 @ X297 ) ) ) )
| ~ ! [X300: term] :
( ( $true
= ( X295 @ id @ ( sub @ X294 @ ( push @ X300 @ id ) ) @ id ) )
| ( $true
!= ( X295 @ id @ X300 @ id ) ) )
| ~ ! [X301: subst > term > term] :
( ~ ! [X302: term,X303: subst,X304: subst] :
( ( X301 @ ( comp @ X303 @ X304 ) @ ( sub @ X302 @ X304 ) )
= ( sub @ ( X301 @ X303 @ X302 ) @ X304 ) )
| ~ ! [X305: term] :
( ( $true
!= ( X295 @ id @ X305 @ id ) )
| ( $true
= ( X295 @ id @ ( X301 @ id @ X305 ) @ id ) ) )
| ( $true
= ( X295 @ id @ ( lam @ ( X301 @ sh @ one ) ) @ id ) ) )
| ~ ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( $true
!= ( X295 @ X308 @ X306 @ ( comp @ X309 @ X307 ) ) )
| ( $true
= ( X295 @ ( comp @ X308 @ X309 ) @ ( sub @ X306 @ X309 ) @ X307 ) ) ) ) ) ) )
& ( termmset_gthm = $true )
& ( axassoc = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ! [X310: subst > term > subst > $o,X311: term] :
( ~ ! [X312: term] :
( ( $true
!= ( X310 @ id @ X312 @ id ) )
| ( $true
= ( X310 @ id @ ( sub @ X311 @ ( push @ X312 @ id ) ) @ id ) ) )
| ~ ! [X313: subst > term > term] :
( ( $true
= ( X310 @ id @ ( lam @ ( X313 @ sh @ one ) ) @ id ) )
| ~ ! [X314: subst,X315: term,X316: subst] :
( ( sub @ ( X313 @ X314 @ X315 ) @ X316 )
= ( X313 @ ( comp @ X314 @ X316 ) @ ( sub @ X315 @ X316 ) ) )
| ~ ! [X317: term] :
( ( $true
!= ( X310 @ id @ X317 @ id ) )
| ( $true
= ( X310 @ id @ ( X313 @ id @ X317 ) @ id ) ) ) )
| ( $true
= ( X310 @ id @ ( lam @ X311 ) @ id ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ! [X318: term] :
( ( sub @ X318 @ id )
= X318 )
<=> ( axvarid = $true ) )
& ( ulamvarind = $true )
& ( ( induction2 = $true )
<=> ! [X319: term,X320: term > $o] :
( ~ ! [X321: term] :
( ( $true
= ( X320 @ X321 ) )
| ( $true
!= ( var @ X321 ) ) )
| ~ ! [X322: term,X323: term] :
( ( $true
!= ( X320 @ X322 ) )
| ( $true
= ( X320 @ ( ap @ X323 @ X322 ) ) )
| ( $true
!= ( X320 @ X323 ) ) )
| ( $true
= ( X320 @ X319 ) )
| ~ ! [X324: term] :
( ~ ! [X325: term] :
( ( $true
= ( X320 @ ( sub @ X324 @ ( push @ X325 @ id ) ) ) )
| ( $true
!= ( X320 @ X325 ) ) )
| ( $true
= ( X320 @ ( lam @ X324 ) ) ) ) ) )
& ( axabs = $true )
& ( ( ! [X326: subst > term > subst > $o] :
~ ! [X327: term > $o] :
~ ! [X328: term] :
( ( X326 @ id @ X328 @ id )
= ( X327 @ X328 ) )
=> ( ! [X329: term > $o,X330: term] :
( ~ ! [X331: term] :
( ( $true
!= ( var @ X331 ) )
| ( $true
= ( X329 @ X331 ) ) )
| ( $true
= ( X329 @ X330 ) )
| ~ ! [X332: term] :
( ~ ! [X333: term] :
( ( $true
!= ( X329 @ X333 ) )
| ( $true
= ( X329 @ ( sub @ X332 @ ( push @ X333 @ id ) ) ) ) )
| ( $true
= ( X329 @ ( lam @ X332 ) ) ) )
| ~ ! [X334: term,X335: term] :
( ( $true
!= ( X329 @ X335 ) )
| ( $true
!= ( X329 @ X334 ) )
| ( $true
= ( X329 @ ( ap @ X335 @ X334 ) ) ) ) )
=> ( ! [X336: term] :
( ( sub @ X336 @ id )
= X336 )
=> ( ! [X337: subst > term > subst > $o,X338: term,X339: term > $o] :
( ~ ! [X340: term] :
( ( $true
= ( X339 @ ( sub @ X338 @ ( push @ X340 @ id ) ) ) )
| ( $true
!= ( X339 @ X340 ) ) )
| ~ ! [X341: subst > term > term] :
( ~ ! [X342: term] :
( ( $true
!= ( X337 @ id @ X342 @ id ) )
| ( ( X337 @ id @ ( X341 @ id @ X342 ) @ id )
= $true ) )
| ~ ! [X343: subst,X344: term,X345: subst] :
( ( sub @ ( X341 @ X345 @ X344 ) @ X343 )
= ( X341 @ ( comp @ X345 @ X343 ) @ ( sub @ X344 @ X343 ) ) )
| ( $true
= ( X337 @ id @ ( lam @ ( X341 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X339 @ ( lam @ X338 ) ) )
| ~ ! [X346: term] :
( ( X339 @ X346 )
= ( X337 @ id @ X346 @ id ) ) )
=> ! [X347: subst > term > subst > $o,X348: term] :
( ~ ! [X349: term,X350: term] :
( ( $true
!= ( X347 @ id @ X350 @ id ) )
| ( $true
!= ( X347 @ id @ X349 @ id ) )
| ( $true
= ( X347 @ id @ ( ap @ ( sub @ X349 @ id ) @ X350 ) @ id ) ) )
| ~ ! [X351: subst > term > term] :
( ( $true
= ( X347 @ id @ ( lam @ ( X351 @ sh @ one ) ) @ id ) )
| ~ ! [X352: subst,X353: subst,X354: term] :
( ( X351 @ ( comp @ X353 @ X352 ) @ ( sub @ X354 @ X352 ) )
= ( sub @ ( X351 @ X353 @ X354 ) @ X352 ) )
| ~ ! [X355: term] :
( ( $true
= ( X347 @ id @ ( X351 @ id @ X355 ) @ id ) )
| ( $true
!= ( X347 @ id @ X355 @ id ) ) ) )
| ( $true
= ( X347 @ id @ X348 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( pushprop = $true )
& ( ! [X356: term > $o,X357: subst,X358: term] :
( ~ ! [X359: term] :
( ( $true
!= ( var @ X359 ) )
| ( $true
= ( X356 @ ( sub @ X359 @ X357 ) ) ) )
| ( $true
= ( X356 @ ( sub @ X358 @ X357 ) ) )
| ~ ! [X360: term] :
( ( $true
= ( X356 @ ( lam @ X360 ) ) )
| ~ ! [X361: term] :
( ( $true
!= ( X356 @ X361 ) )
| ( $true
= ( X356 @ ( sub @ X360 @ ( push @ X361 @ id ) ) ) ) ) )
| ~ ! [X362: term,X363: term] :
( ( $true
= ( X356 @ ( ap @ X362 @ X363 ) ) )
| ( $true
!= ( X356 @ X362 ) )
| ( $true
!= ( X356 @ X363 ) ) ) )
<=> ( induction2lem = $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X364: term] :
~ ! [X365: subst > term > term] :
~ ! [X366: subst,X367: term] :
( ( sub @ X364 @ ( push @ X367 @ X366 ) )
= ( X365 @ X366 @ X367 ) )
=> ( ! [X368: term] :
( ( sub @ X368 @ id )
= X368 )
=> ! [X369: term > $o,X370: subst > term > subst > $o,X371: term] :
( ~ ! [X372: term] :
( ( X370 @ id @ X372 @ id )
= ( X369 @ X372 ) )
| ~ ! [X373: term] :
( ( ( X369 @ ( sub @ X371 @ ( push @ X373 @ id ) ) )
= $true )
| ( $true
!= ( X369 @ X373 ) ) )
| ( $true
= ( X369 @ ( lam @ X371 ) ) )
| ~ ! [X374: subst > term > term] :
( ( $true
= ( X370 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id ) )
| ~ ! [X375: term] :
( ( $true
= ( X370 @ id @ ( X374 @ id @ X375 ) @ id ) )
| ( $true
!= ( X370 @ id @ X375 @ id ) ) )
| ~ ! [X376: subst,X377: term,X378: subst] :
( ( sub @ ( X374 @ X376 @ X377 ) @ X378 )
= ( X374 @ ( comp @ X376 @ X378 ) @ ( sub @ X377 @ X378 ) ) ) ) ) ) ) )
& ( induction2_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X379: term,X380: term > $o] :
( ~ ! [X381: term,X382: term] :
( ( $true
!= ( X380 @ X381 ) )
| ( ( X380 @ ( ap @ X381 @ X382 ) )
= $true )
| ( $true
!= ( X380 @ X382 ) ) )
| ( $true
= ( X380 @ X379 ) )
| ~ ! [X383: term] :
( ( $true
= ( X380 @ ( lam @ X383 ) ) )
| ~ ! [X384: term] :
( ( $true
= ( X380 @ ( sub @ X383 @ ( push @ X384 @ id ) ) ) )
| ( $true
!= ( X380 @ X384 ) ) ) )
| ~ ! [X385: term] :
( ( $true
= ( X380 @ X385 ) )
| ( $true
!= ( var @ X385 ) ) ) )
=> ( ! [X386: term,X387: term,X388: subst > term > subst > $o] :
( ( $true
= ( X388 @ id @ ( ap @ X386 @ X387 ) @ id ) )
| ( $true
!= ( X388 @ id @ X387 @ id ) )
| ( $true
!= ( X388 @ id @ X386 @ id ) )
| ~ ! [X389: subst,X390: subst,X391: term,X392: subst] :
( ( $true
!= ( X388 @ X390 @ X391 @ ( comp @ X389 @ X392 ) ) )
| ( $true
= ( X388 @ ( comp @ X390 @ X389 ) @ ( sub @ X391 @ X389 ) @ X392 ) ) )
| ~ ! [X393: term,X394: term] :
( ( $true
= ( X388 @ id @ ( ap @ ( sub @ X394 @ id ) @ X393 ) @ id ) )
| ( $true
!= ( X388 @ id @ X394 @ id ) )
| ( $true
!= ( X388 @ id @ X393 @ id ) ) )
| ~ ! [X395: subst,X396: subst,X397: subst,X398: term] :
( ( $true
!= ( X388 @ ( comp @ X396 @ X397 ) @ ( sub @ X398 @ X397 ) @ X395 ) )
| ( $true
= ( X388 @ X396 @ X398 @ ( comp @ X397 @ X395 ) ) ) ) )
=> ( ! [X399: subst > term > subst > $o,X400: term] :
( ( $true
= ( X399 @ id @ ( lam @ X400 ) @ id ) )
| ~ ! [X401: term] :
( ( $true
= ( X399 @ id @ ( sub @ X400 @ ( push @ X401 @ id ) ) @ id ) )
| ( $true
!= ( X399 @ id @ X401 @ id ) ) )
| ~ ! [X402: subst,X403: term,X404: subst,X405: subst] :
( ( $true
= ( X399 @ X404 @ X403 @ ( comp @ X402 @ X405 ) ) )
| ( $true
!= ( X399 @ ( comp @ X404 @ X402 ) @ ( sub @ X403 @ X402 ) @ X405 ) ) )
| ~ ! [X406: subst,X407: term,X408: subst,X409: subst] :
( ( $true
!= ( X399 @ X406 @ X407 @ ( comp @ X409 @ X408 ) ) )
| ( $true
= ( X399 @ ( comp @ X406 @ X409 ) @ ( sub @ X407 @ X409 ) @ X408 ) ) )
| ~ ! [X410: subst > term > term] :
( ~ ! [X411: term,X412: subst,X413: subst] :
( ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) )
= ( sub @ ( X410 @ X412 @ X411 ) @ X413 ) )
| ( $true
= ( X399 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
| ~ ! [X414: term] :
( ( $true
!= ( X399 @ id @ X414 @ id ) )
| ( $true
= ( X399 @ id @ ( X410 @ id @ X414 ) @ id ) ) ) ) )
=> ! [X415: subst > term > subst > $o,X416: term] :
( ~ ! [X417: term,X418: term] :
( ( $true
= ( X415 @ id @ ( ap @ ( sub @ X418 @ id ) @ X417 ) @ id ) )
| ( ( X415 @ id @ X417 @ id )
!= $true )
| ( $true
!= ( X415 @ id @ X418 @ id ) ) )
| ~ ! [X419: subst > term > term] :
( ( ( X415 @ id @ ( lam @ ( X419 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X420: term] :
( ( $true
= ( X415 @ id @ ( X419 @ id @ X420 ) @ id ) )
| ( $true
!= ( X415 @ id @ X420 @ id ) ) )
| ~ ! [X421: term,X422: subst,X423: subst] :
( ( sub @ ( X419 @ X422 @ X421 ) @ X423 )
= ( X419 @ ( comp @ X422 @ X423 ) @ ( sub @ X421 @ X423 ) ) ) )
| ( $true
= ( X415 @ id @ X416 @ id ) )
| ~ ! [X424: subst,X425: term,X426: subst,X427: subst] :
( ( $true
= ( X415 @ ( comp @ X427 @ X424 ) @ ( sub @ X425 @ X424 ) @ X426 ) )
| ( $true
!= ( X415 @ X427 @ X425 @ ( comp @ X424 @ X426 ) ) ) )
| ~ ! [X428: term,X429: subst,X430: subst,X431: subst] :
( ( $true
= ( X415 @ X430 @ X428 @ ( comp @ X429 @ X431 ) ) )
| ( $true
!= ( X415 @ ( comp @ X430 @ X429 ) @ ( sub @ X428 @ X429 ) @ X431 ) ) )
| ~ ! [X432: term] :
( ( $true
= ( X415 @ id @ X432 @ id ) )
| ( $true
!= ( var @ ( sub @ X432 @ id ) ) ) ) ) ) ) ) )
& ( ! [X433: term,X434: term,X435: term,X436: term] :
( ( ( ap @ ( sub @ X435 @ id ) @ X433 )
!= ( ap @ ( sub @ X436 @ id ) @ X434 ) )
| ( X435 = X436 ) )
<=> ( hoasapinj1 = $true ) )
& ( pushprop_lem1_gthm = $true )
& ( apnotvar = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X437: subst > term > subst > $o,X438: term] :
( ~ ! [X439: subst > term > term] :
( ~ ! [X440: term] :
( ( $true
= ( X437 @ id @ ( X439 @ id @ X440 ) @ id ) )
| ( $true
!= ( X437 @ id @ X440 @ id ) ) )
| ( $true
= ( X437 @ id @ ( lam @ ( X439 @ sh @ one ) ) @ id ) )
| ~ ! [X441: subst,X442: term,X443: subst] :
( ( sub @ ( X439 @ X441 @ X442 ) @ X443 )
= ( X439 @ ( comp @ X441 @ X443 ) @ ( sub @ X442 @ X443 ) ) ) )
| ( $true
= ( X437 @ id @ ( lam @ ( sub @ X438 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X444: term] :
( ( $true
= ( X437 @ id @ ( sub @ X438 @ ( push @ X444 @ id ) ) @ id ) )
| ( $true
!= ( X437 @ id @ X444 @ id ) ) ) ) )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X445: subst > term > subst > $o] :
~ ! [X446: term > $o] :
~ ! [X447: term] :
( ( X445 @ id @ X447 @ id )
= ( X446 @ X447 ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( substmonoid = $true )
<=> ( ! [X449: subst] :
( ( comp @ X449 @ id )
= X449 )
& ! [X448: subst] :
( ( comp @ id @ X448 )
= X448 ) ) )
& ( ( induction2_lthm = $true )
<=> ( ! [X450: term] :
( ( sub @ X450 @ id )
= X450 )
=> ( ! [X451: subst,X452: term,X453: term > $o] :
( ~ ! [X454: term] :
( ( $true
!= ( var @ X454 ) )
| ( $true
= ( X453 @ ( sub @ X454 @ X451 ) ) ) )
| ~ ! [X455: term,X456: term] :
( ( ( X453 @ X456 )
!= $true )
| ( $true
= ( X453 @ ( ap @ X455 @ X456 ) ) )
| ( $true
!= ( X453 @ X455 ) ) )
| ~ ! [X457: term] :
( ~ ! [X458: term] :
( ( $true
= ( X453 @ ( sub @ X457 @ ( push @ X458 @ id ) ) ) )
| ( $true
!= ( X453 @ X458 ) ) )
| ( $true
= ( X453 @ ( lam @ X457 ) ) ) )
| ( $true
= ( X453 @ ( sub @ X452 @ X451 ) ) ) )
=> ! [X459: term > $o,X460: term] :
( ~ ! [X461: term] :
( ( $true
!= ( var @ X461 ) )
| ( $true
= ( X459 @ X461 ) ) )
| ~ ! [X462: term,X463: term] :
( ( $true
!= ( X459 @ X462 ) )
| ( $true
!= ( X459 @ X463 ) )
| ( $true
= ( X459 @ ( ap @ X462 @ X463 ) ) ) )
| ( $true
= ( X459 @ X460 ) )
| ~ ! [X464: term] :
( ( $true
= ( X459 @ ( lam @ X464 ) ) )
| ~ ! [X465: term] :
( ( $true
= ( X459 @ ( sub @ X464 @ ( push @ X465 @ id ) ) ) )
| ( $true
!= ( X459 @ X465 ) ) ) ) ) ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X466: term,X467: term,X468: subst > term > subst > $o] :
( ( $true
= ( X468 @ id @ ( ap @ X467 @ X466 ) @ id ) )
| ( $true
!= ( X468 @ id @ X467 @ id ) )
| ~ ! [X469: term,X470: term] :
( ( $true
!= ( X468 @ id @ X470 @ id ) )
| ( $true
= ( X468 @ id @ ( ap @ ( sub @ X470 @ id ) @ X469 ) @ id ) )
| ( $true
!= ( X468 @ id @ X469 @ id ) ) )
| ~ ! [X471: subst,X472: subst,X473: subst,X474: term] :
( ( $true
!= ( X468 @ X473 @ X474 @ ( comp @ X472 @ X471 ) ) )
| ( $true
= ( X468 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 ) ) )
| ~ ! [X475: subst,X476: subst,X477: subst,X478: term] :
( ( $true
= ( X468 @ X476 @ X478 @ ( comp @ X475 @ X477 ) ) )
| ( $true
!= ( X468 @ ( comp @ X476 @ X475 ) @ ( sub @ X478 @ X475 ) @ X477 ) ) )
| ( $true
!= ( X468 @ id @ X466 @ id ) ) ) )
& ( ! [X479: term > $o,X480: term] :
( ~ ! [X481: term] :
( ( $true
= ( X479 @ X481 ) )
| ( $true
!= ( var @ X481 ) ) )
| ~ ! [X482: term] :
( ( $true
!= ( X479 @ X482 ) )
| ( $true
= ( X479 @ ( lam @ X482 ) ) ) )
| ( $true
= ( X479 @ X480 ) )
| ~ ! [X483: term,X484: term] :
( ( $true
!= ( X479 @ X484 ) )
| ( ( X479 @ ( ap @ X483 @ X484 ) )
= $true )
| ( $true
!= ( X479 @ X483 ) ) ) )
<=> ( induction = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X485: d_term] : ( d_one = X485 )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X486: term] :
( ( sub @ X486 @ id )
= X486 )
=> ( ! [X487: term,X488: term,X489: term,X490: term] :
( ( ( ap @ X487 @ X488 )
!= ( ap @ X490 @ X489 ) )
| ( X487 = X490 ) )
=> ! [X491: term,X492: term,X493: term,X494: term] :
( ( ( ap @ ( sub @ X493 @ id ) @ X492 )
!= ( ap @ ( sub @ X491 @ id ) @ X494 ) )
| ( X491 = X493 ) ) ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( axvarshift = $true )
& ( ( ! [X495: term] :
( ( sub @ X495 @ id )
= X495 )
=> ! [X496: term > $o,X497: subst > term > subst > $o,X498: term] :
( ~ ! [X499: subst,X500: term,X501: subst,X502: subst] :
( ( $true
= ( X497 @ ( comp @ X501 @ X499 ) @ ( sub @ X500 @ X499 ) @ X502 ) )
| ( $true
!= ( X497 @ X501 @ X500 @ ( comp @ X499 @ X502 ) ) ) )
| ~ ! [X503: term] :
( ( X496 @ X503 )
= ( X497 @ id @ X503 @ id ) )
| ~ ! [X504: term] :
( ( $true
= ( X496 @ ( sub @ X498 @ ( push @ X504 @ id ) ) ) )
| ( $true
!= ( X496 @ X504 ) ) )
| ~ ! [X505: subst,X506: term,X507: subst,X508: subst] :
( ( ( X497 @ ( comp @ X508 @ X507 ) @ ( sub @ X506 @ X507 ) @ X505 )
!= $true )
| ( $true
= ( X497 @ X508 @ X506 @ ( comp @ X507 @ X505 ) ) ) )
| ( $true
= ( X496 @ ( lam @ X498 ) ) )
| ~ ! [X509: subst > term > term] :
( ( $true
= ( X497 @ id @ ( lam @ ( X509 @ sh @ one ) ) @ id ) )
| ~ ! [X510: subst,X511: term,X512: subst] :
( ( sub @ ( X509 @ X512 @ X511 ) @ X510 )
= ( X509 @ ( comp @ X512 @ X510 ) @ ( sub @ X511 @ X510 ) ) )
| ~ ! [X513: term] :
( ( $true
!= ( X497 @ id @ X513 @ id ) )
| ( $true
= ( X497 @ id @ ( X509 @ id @ X513 ) @ id ) ) ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( ( shinj = $true )
<=> ! [X514: term,X515: term] :
( ( ( sub @ X514 @ sh )
!= ( sub @ X515 @ sh ) )
| ( X514 = X515 ) ) )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X516: term,X517: subst > term > subst > $o] :
( ~ ! [X518: subst,X519: term,X520: subst,X521: subst] :
( ( $true
= ( X517 @ ( comp @ X521 @ X518 ) @ ( sub @ X519 @ X518 ) @ X520 ) )
| ( $true
!= ( X517 @ X521 @ X519 @ ( comp @ X518 @ X520 ) ) ) )
| ~ ! [X522: subst > term > term] :
( ( ( X517 @ id @ ( lam @ ( X522 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X523: term] :
( ( $true
= ( X517 @ id @ ( X522 @ id @ X523 ) @ id ) )
| ( $true
!= ( X517 @ id @ X523 @ id ) ) )
| ~ ! [X524: term,X525: subst,X526: subst] :
( ( X522 @ ( comp @ X525 @ X526 ) @ ( sub @ X524 @ X526 ) )
= ( sub @ ( X522 @ X525 @ X524 ) @ X526 ) ) )
| ( $true
= ( X517 @ id @ ( lam @ X516 ) @ id ) )
| ~ ! [X527: term] :
( ( $true
= ( X517 @ id @ ( sub @ X516 @ ( push @ X527 @ id ) ) @ id ) )
| ( $true
!= ( X517 @ id @ X527 @ id ) ) )
| ~ ! [X528: subst,X529: subst,X530: subst,X531: term] :
( ( $true
!= ( X517 @ ( comp @ X529 @ X530 ) @ ( sub @ X531 @ X530 ) @ X528 ) )
| ( ( X517 @ X529 @ X531 @ ( comp @ X530 @ X528 ) )
= $true ) ) ) )
& ( substmonoid_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X532: term,X533: term > $o,X534: subst] :
~ ! [X535: term > $o] :
~ ! [X536: term] :
( ( X533 @ ( sub @ X536 @ ( push @ X532 @ X534 ) ) )
= ( X535 @ X536 ) ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X537: term] :
~ ! [X538: subst > term > term] :
( ( sub @ X537 @ ( push @ one @ sh ) )
!= ( X538 @ sh @ one ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X539: subst] :
( ( push @ ( sub @ one @ X539 ) @ ( comp @ sh @ X539 ) )
= X539 )
<=> ( axscons = $true ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem2_lthm = $true ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,plain,
( ( ( apinj2 = $true )
<=> ! [X0: term,X1: term,X2: term,X3: term] :
( ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) )
| ( X0 = X1 ) ) )
& ( ! [X4: term,X5: subst > term > subst > $o] :
( ( $true
= ( X5 @ id @ X4 @ id ) )
| ~ ! [X6: subst > term > term] :
( ~ ! [X7: term] :
( ( $true
!= ( X5 @ id @ X7 @ id ) )
| ( $true
= ( X5 @ id @ ( X6 @ id @ X7 ) @ id ) ) )
| ~ ! [X8: term,X9: subst,X10: subst] :
( ( X6 @ ( comp @ X10 @ X9 ) @ ( sub @ X8 @ X9 ) )
= ( sub @ ( X6 @ X10 @ X8 ) @ X9 ) )
| ( $true
= ( X5 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id ) ) )
| ~ ! [X11: term,X12: term] :
( ( $true
!= ( X5 @ id @ X12 @ id ) )
| ( $true
= ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id ) )
| ( ( X5 @ id @ X11 @ id )
!= $true ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
= ( d2subst @ X14 ) )
=> ( X13 = X14 ) )
& ( one
= ( d2term @ d_one ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X15: subst > term > subst > $o,X16: term > $o,X17: term] :
( ~ ! [X18: term] :
( ( X16 @ X18 )
= ( X15 @ id @ X18 @ id ) )
| ~ ! [X19: subst > term > term] :
( ~ ! [X20: term,X21: subst,X22: subst] :
( ( sub @ ( X19 @ X22 @ X20 ) @ X21 )
= ( X19 @ ( comp @ X22 @ X21 ) @ ( sub @ X20 @ X21 ) ) )
| ( $true
= ( X15 @ id @ ( lam @ ( X19 @ sh @ one ) ) @ id ) )
| ~ ! [X23: term] :
( ( $true
= ( X15 @ id @ ( X19 @ id @ X23 ) @ id ) )
| ( $true
!= ( X15 @ id @ X23 @ id ) ) ) )
| ~ ! [X24: term] :
( ( $true
= ( X16 @ ( sub @ X17 @ ( push @ X24 @ id ) ) ) )
| ( $true
!= ( X16 @ X24 ) ) )
| ( $true
= ( X16 @ ( lam @ X17 ) ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( ! [X25: term] :
( ( sub @ X25 @ id )
= X25 )
=> ! [X26: term] :
( ( sub @ X26 @ id )
= X26 ) )
<=> ( termmset_lthm = $true ) )
& ( ( pushprop_lem0 = $true )
<=> ! [X27: subst,X28: term,X29: term > $o] :
~ ! [X30: term > $o] :
~ ! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) ) )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
<=> ( termmset = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ( ! [X33: subst] :
( ( comp @ id @ X33 )
= X33 )
=> ( ! [X34: subst] :
( ( comp @ X34 @ id )
= X34 )
=> ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
= X37 )
=> ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ~ ! [X43: term] :
( ( X41 @ X43 )
= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) )
| ( $true
!= ( X40 @ X39 ) )
| ( $true
= ( X41 @ one ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( lamnotap != $true )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X44: term,X45: subst > term > subst > $o] :
( ( $true
= ( X45 @ id @ ( lam @ ( sub @ X44 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X46: term] :
( ( ( X45 @ id @ ( sub @ X44 @ ( push @ X46 @ id ) ) @ id )
= $true )
| ( $true
!= ( X45 @ id @ X46 @ id ) ) )
| ~ ! [X47: subst > term > term,X48: term] :
( ~ ! [X49: subst,X50: term,X51: subst] :
( ( sub @ ( sub @ X48 @ ( push @ X50 @ X49 ) ) @ X51 )
= ( sub @ ( X47 @ X49 @ X50 ) @ X51 ) )
| ~ ! [X52: subst,X53: term,X54: subst] :
( ( sub @ X48 @ ( push @ ( sub @ X53 @ X52 ) @ ( comp @ X54 @ X52 ) ) )
= ( X47 @ ( comp @ X54 @ X52 ) @ ( sub @ X53 @ X52 ) ) )
| ( $true
= ( X45 @ id @ ( lam @ ( X47 @ sh @ one ) ) @ id ) )
| ~ ! [X55: term] :
( ( $true
= ( X45 @ id @ ( X47 @ id @ X55 ) @ id ) )
| ( $true
!= ( X45 @ id @ X55 @ id ) ) ) ) ) )
& ( ( axidl = $true )
<=> ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 ) )
& ( hoasinduction_gthm = $true )
& ( ( axshiftcons = $true )
<=> ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 ) )
& ( axapp = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction = $true )
<=> ! [X59: term,X60: subst > term > subst > $o] :
( ~ ! [X61: term,X62: term] :
( ( $true
= ( X60 @ id @ ( ap @ ( sub @ X61 @ id ) @ X62 ) @ id ) )
| ( $true
!= ( X60 @ id @ X61 @ id ) )
| ( $true
!= ( X60 @ id @ X62 @ id ) ) )
| ~ ! [X63: subst > term > term] :
( ~ ! [X64: term] :
( ( $true
!= ( X60 @ id @ X64 @ id ) )
| ( $true
= ( X60 @ id @ ( X63 @ id @ X64 ) @ id ) ) )
| ( $true
= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
| ~ ! [X65: subst,X66: subst,X67: term] :
( ( X63 @ ( comp @ X65 @ X66 ) @ ( sub @ X67 @ X66 ) )
= ( sub @ ( X63 @ X65 @ X67 ) @ X66 ) ) )
| ~ ! [X68: term] :
( ( $true
!= ( var @ ( sub @ X68 @ id ) ) )
| ( $true
= ( X60 @ id @ X68 @ id ) ) )
| ~ ! [X69: subst,X70: subst,X71: term,X72: subst] :
( ( $true
!= ( X60 @ ( comp @ X69 @ X72 ) @ ( sub @ X71 @ X72 ) @ X70 ) )
| ( $true
= ( X60 @ X69 @ X71 @ ( comp @ X72 @ X70 ) ) ) )
| ~ ! [X73: term,X74: subst,X75: subst,X76: subst] :
( ( $true
= ( X60 @ ( comp @ X75 @ X74 ) @ ( sub @ X73 @ X74 ) @ X76 ) )
| ( $true
!= ( X60 @ X75 @ X73 @ ( comp @ X74 @ X76 ) ) ) )
| ( $true
= ( X60 @ id @ X59 @ id ) ) ) )
& ( ulamvar1 != $true )
& ( ( d2subst @ d_id )
= sh )
& ( hoaslamnotvar_lthm = $true )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X77: subst > term > subst > $o] :
~ ! [X78: term > $o] :
~ ! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) ) )
& ( hoasapinj1_gthm = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X80: term] :
~ ! [X81: subst > term > term] :
~ ! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) ) )
& ( induction2lem_gthm = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ( induction2lem_lthm = $true )
<=> ( ! [X84: subst,X85: term] :
( ( sub @ one @ ( push @ X85 @ X84 ) )
= X85 )
=> ( ! [X86: subst,X87: term] :
( ( comp @ sh @ ( push @ X87 @ X86 ) )
= X86 )
=> ( ! [X88: subst] :
( ( comp @ X88 @ id )
= X88 )
=> ( ! [X89: term,X90: term > $o] :
( ~ ! [X91: term,X92: term] :
( ( $true
!= ( X90 @ X92 ) )
| ( $true
= ( X90 @ ( ap @ X91 @ X92 ) ) )
| ( ( X90 @ X91 )
!= $true ) )
| ~ ! [X93: term] :
( ( $true
!= ( X90 @ X93 ) )
| ( $true
= ( X90 @ ( lam @ X93 ) ) ) )
| ( $true
= ( X90 @ X89 ) )
| ~ ! [X94: term] :
( ( $true
!= ( var @ X94 ) )
| ( $true
= ( X90 @ X94 ) ) ) )
=> ! [X95: subst,X96: term,X97: term > $o] :
( ~ ! [X98: term] :
( ~ ! [X99: term] :
( ( $true
= ( X97 @ ( sub @ X98 @ ( push @ X99 @ id ) ) ) )
| ( $true
!= ( X97 @ X99 ) ) )
| ( $true
= ( X97 @ ( lam @ X98 ) ) ) )
| ~ ! [X100: term] :
( ( $true
!= ( var @ X100 ) )
| ( $true
= ( X97 @ ( sub @ X100 @ X95 ) ) ) )
| ~ ! [X101: term,X102: term] :
( ( $true
!= ( X97 @ X102 ) )
| ( $true
= ( X97 @ ( ap @ X102 @ X101 ) ) )
| ( $true
!= ( X97 @ X101 ) ) )
| ( $true
= ( X97 @ ( sub @ X96 @ X95 ) ) ) ) ) ) ) ) )
& ( ( axvarcons = $true )
<=> ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 ) )
& ( hoasapinj2_gthm = $true )
& ( ( axidr = $true )
<=> ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 ) )
& ! [X106: d_subst] : ( d_id = X106 )
& ( ( hoaslaminj = $true )
<=> ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ~ ! [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ~ ! [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X117: subst > term > term,X118: term,X119: term] :
( ~ ! [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) ) )
& ( ! [X123: term,X124: term,X125: term,X126: term] :
( ( X125 = X126 )
| ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) ) )
<=> ( apinj1 = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X127: term,X128: term > $o,X129: subst > term > subst > $o,X130: term] :
( ( $true
!= ( X128 @ X130 ) )
| ( $true
= ( X128 @ ( ap @ X130 @ X127 ) ) )
| ~ ! [X131: term,X132: subst,X133: subst,X134: subst] :
( ( $true
= ( X129 @ ( comp @ X133 @ X132 ) @ ( sub @ X131 @ X132 ) @ X134 ) )
| ( $true
!= ( X129 @ X133 @ X131 @ ( comp @ X132 @ X134 ) ) ) )
| ( $true
!= ( X128 @ X127 ) )
| ~ ! [X135: subst,X136: subst,X137: term,X138: subst] :
( ( $true
!= ( X129 @ ( comp @ X138 @ X136 ) @ ( sub @ X137 @ X136 ) @ X135 ) )
| ( $true
= ( X129 @ X138 @ X137 @ ( comp @ X136 @ X135 ) ) ) )
| ~ ! [X139: term] :
( ( X129 @ id @ X139 @ id )
= ( X128 @ X139 ) )
| ~ ! [X140: term,X141: term] :
( ( $true
= ( X129 @ id @ ( ap @ ( sub @ X140 @ id ) @ X141 ) @ id ) )
| ( $true
!= ( X129 @ id @ X140 @ id ) )
| ( $true
!= ( X129 @ id @ X141 @ id ) ) ) ) )
& ( pushprop_lthm = $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar_lthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( $true
= ( X144 @ one ) )
| ~ ! [X146: term] :
( ( X144 @ X146 )
= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) )
| ( $true
!= ( X142 @ X145 ) ) ) )
& ( ! [X147: term] :
~ ! [X148: subst > term > term] :
~ ! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
= ( d2term @ X152 ) )
=> ( X151 = X152 ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( X153 = X155 )
| ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
<=> ( hoasapinj2 = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( ! [X157: term,X158: subst] :
( ( sub @ one @ ( push @ X157 @ X158 ) )
= X157 )
=> ( ! [X159: subst,X160: term] :
( ( comp @ sh @ ( push @ X160 @ X159 ) )
= X159 )
=> ( ! [X161: term,X162: term] :
( ( X161 = X162 )
| ( ( lam @ X161 )
!= ( lam @ X162 ) ) )
=> ! [X163: subst,X164: subst > term > term,X165: term,X166: subst > term > term] :
( ~ ! [X167: subst,X168: term,X169: subst] :
( ( sub @ ( X164 @ X167 @ X168 ) @ X169 )
= ( X164 @ ( comp @ X167 @ X169 ) @ ( sub @ X168 @ X169 ) ) )
| ~ ! [X170: term,X171: subst,X172: subst] :
( ( sub @ ( X166 @ X172 @ X170 ) @ X171 )
= ( X166 @ ( comp @ X172 @ X171 ) @ ( sub @ X170 @ X171 ) ) )
| ( ( X164 @ X163 @ X165 )
= ( X166 @ X163 @ X165 ) )
| ( ( lam @ ( X166 @ sh @ one ) )
!= ( lam @ ( X164 @ sh @ one ) ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( pushprop_lem1 = $true )
<=> ! [X175: term > $o,X176: term,X177: subst,X178: term,X179: term > $o] :
( ( $true
= ( X179 @ ( sub @ X176 @ ( push @ X178 @ X177 ) ) ) )
| ( $true
!= ( X175 @ X176 ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X180: term > $o,X181: subst > term > subst > $o,X182: term] :
( ~ ! [X183: term] :
( ( X180 @ X183 )
= ( X181 @ id @ X183 @ id ) )
| ~ ! [X184: subst > term > term] :
( ( $true
= ( X181 @ id @ ( lam @ ( X184 @ sh @ one ) ) @ id ) )
| ~ ! [X185: term] :
( ( $true
= ( X181 @ id @ ( X184 @ id @ X185 ) @ id ) )
| ( $true
!= ( X181 @ id @ X185 @ id ) ) )
| ~ ! [X186: subst,X187: subst,X188: term] :
( ( sub @ ( X184 @ X187 @ X188 ) @ X186 )
= ( X184 @ ( comp @ X187 @ X186 ) @ ( sub @ X188 @ X186 ) ) ) )
| ~ ! [X189: term] :
( ( ( X180 @ ( sub @ X182 @ ( push @ X189 @ id ) ) )
= $true )
| ( $true
!= ( X180 @ X189 ) ) )
| ~ ! [X190: term,X191: subst,X192: subst,X193: subst] :
( ( $true
= ( X181 @ X191 @ X190 @ ( comp @ X192 @ X193 ) ) )
| ( $true
!= ( X181 @ ( comp @ X191 @ X192 ) @ ( sub @ X190 @ X192 ) @ X193 ) ) )
| ~ ! [X194: subst,X195: subst,X196: term,X197: subst] :
( ( $true
= ( X181 @ ( comp @ X194 @ X195 ) @ ( sub @ X196 @ X195 ) @ X197 ) )
| ( $true
!= ( X181 @ X194 @ X196 @ ( comp @ X195 @ X197 ) ) ) )
| ( $true
= ( X180 @ ( lam @ X182 ) ) ) ) )
& ( ( laminj = $true )
<=> ! [X198: term,X199: term] :
( ( ( lam @ X199 )
!= ( lam @ X198 ) )
| ( X198 = X199 ) ) )
& ! [X200: term] :
? [X201: d_term] :
( ( d2term @ X201 )
= X200 )
& ( pushprop_lem3v2 = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X202: term] :
~ ! [X203: subst > term > term] :
( ( sub @ X202 @ ( push @ one @ sh ) )
!= ( X203 @ sh @ one ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( axmap = $true )
& ( axclos = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoaslamnotvar = $true )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X204: subst,X205: term] :
( ( sub @ one @ ( push @ X205 @ X204 ) )
= X205 )
=> ( ! [X206: subst,X207: term] :
( ( comp @ sh @ ( push @ X207 @ X206 ) )
= X206 )
=> ! [X208: term,X209: term > $o,X210: term > $o,X211: subst,X212: term] :
( ( $true
!= ( X209 @ X212 ) )
| ( $true
= ( X210 @ ( sub @ X212 @ ( push @ X208 @ X211 ) ) ) ) ) ) ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X213: term,X214: term,X215: term,X216: term] :
( ( X214 = X216 )
| ( ( ap @ X215 @ X214 )
!= ( ap @ X213 @ X216 ) ) )
=> ! [X217: term,X218: term,X219: term,X220: term] :
( ( X219 = X220 )
| ( ( ap @ ( sub @ X218 @ id ) @ X220 )
!= ( ap @ ( sub @ X217 @ id ) @ X219 ) ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X221: subst > term > subst > $o,X222: term] :
( ~ ! [X223: subst > term > term] :
( ~ ! [X224: term] :
( ( $true
!= ( X221 @ id @ X224 @ id ) )
| ( $true
= ( X221 @ id @ ( X223 @ id @ X224 ) @ id ) ) )
| ( ( X221 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X225: term,X226: subst,X227: subst] :
( ( X223 @ ( comp @ X227 @ X226 ) @ ( sub @ X225 @ X226 ) )
= ( sub @ ( X223 @ X227 @ X225 ) @ X226 ) ) )
| ~ ! [X228: term] :
( ( $true
= ( X221 @ id @ ( sub @ X222 @ ( push @ X228 @ id ) ) @ id ) )
| ( $true
!= ( X221 @ id @ X228 @ id ) ) )
| ( $true
= ( X221 @ id @ ( lam @ ( sub @ X222 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ! [X229: subst > term > subst > $o] :
~ ! [X230: term > $o] :
~ ! [X231: term] :
( ( X229 @ id @ X231 @ id )
= ( X230 @ X231 ) )
=> ( ! [X232: term > $o,X233: term] :
( ~ ! [X234: term] :
( ( $true
= ( X232 @ X234 ) )
| ( $true
!= ( var @ X234 ) ) )
| ( $true
= ( X232 @ X233 ) )
| ~ ! [X235: term] :
( ~ ! [X236: term] :
( ( $true
= ( X232 @ ( sub @ X235 @ ( push @ X236 @ id ) ) ) )
| ( $true
!= ( X232 @ X236 ) ) )
| ( $true
= ( X232 @ ( lam @ X235 ) ) ) )
| ~ ! [X237: term,X238: term] :
( ( $true
= ( X232 @ ( ap @ X237 @ X238 ) ) )
| ( $true
!= ( X232 @ X238 ) )
| ( $true
!= ( X232 @ X237 ) ) ) )
=> ( ! [X239: term] :
( ( sub @ X239 @ id )
= X239 )
=> ( ! [X240: term,X241: term > $o,X242: subst > term > subst > $o] :
( ~ ! [X243: term] :
( ( $true
!= ( X241 @ X243 ) )
| ( $true
= ( X241 @ ( sub @ X240 @ ( push @ X243 @ id ) ) ) ) )
| ~ ! [X244: subst > term > term] :
( ( $true
= ( X242 @ id @ ( lam @ ( X244 @ sh @ one ) ) @ id ) )
| ~ ! [X245: term] :
( ( $true
!= ( X242 @ id @ X245 @ id ) )
| ( $true
= ( X242 @ id @ ( X244 @ id @ X245 ) @ id ) ) )
| ~ ! [X246: subst,X247: term,X248: subst] :
( ( sub @ ( X244 @ X248 @ X247 ) @ X246 )
= ( X244 @ ( comp @ X248 @ X246 ) @ ( sub @ X247 @ X246 ) ) ) )
| ~ ! [X249: term] :
( ( X241 @ X249 )
= ( X242 @ id @ X249 @ id ) )
| ( $true
= ( X241 @ ( lam @ X240 ) ) ) )
=> ! [X250: term,X251: subst > term > subst > $o] :
( ~ ! [X252: term,X253: term] :
( ( $true
!= ( X251 @ id @ X253 @ id ) )
| ( $true
!= ( X251 @ id @ X252 @ id ) )
| ( $true
= ( X251 @ id @ ( ap @ ( sub @ X253 @ id ) @ X252 ) @ id ) ) )
| ~ ! [X254: subst,X255: term,X256: subst,X257: subst] :
( ( $true
!= ( X251 @ X254 @ X255 @ ( comp @ X256 @ X257 ) ) )
| ( $true
= ( X251 @ ( comp @ X254 @ X256 ) @ ( sub @ X255 @ X256 ) @ X257 ) ) )
| ~ ! [X258: subst > term > term] :
( ~ ! [X259: term] :
( ( $true
!= ( X251 @ id @ X259 @ id ) )
| ( $true
= ( X251 @ id @ ( X258 @ id @ X259 ) @ id ) ) )
| ~ ! [X260: subst,X261: term,X262: subst] :
( ( sub @ ( X258 @ X262 @ X261 ) @ X260 )
= ( X258 @ ( comp @ X262 @ X260 ) @ ( sub @ X261 @ X260 ) ) )
| ( ( X251 @ id @ ( lam @ ( X258 @ sh @ one ) ) @ id )
= $true ) )
| ( $true
= ( X251 @ id @ X250 @ id ) )
| ~ ! [X263: term,X264: subst,X265: subst,X266: subst] :
( ( $true
!= ( X251 @ ( comp @ X264 @ X265 ) @ ( sub @ X263 @ X265 ) @ X266 ) )
| ( $true
= ( X251 @ X264 @ X263 @ ( comp @ X265 @ X266 ) ) ) )
| ~ ! [X267: term] :
( ( $true
!= ( var @ ( sub @ X267 @ id ) ) )
| ( $true
= ( X251 @ id @ X267 @ id ) ) ) ) ) ) ) ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ! [X271: subst] :
? [X272: d_subst] :
( ( d2subst @ X272 )
= X271 )
& ( ( ! [X273: term] :
( ( sub @ X273 @ id )
= X273 )
=> ( ! [X274: term,X275: subst > term > subst > $o] :
( ~ ! [X276: subst > term > term] :
( ~ ! [X277: subst,X278: term,X279: subst] :
( ( sub @ ( X276 @ X277 @ X278 ) @ X279 )
= ( X276 @ ( comp @ X277 @ X279 ) @ ( sub @ X278 @ X279 ) ) )
| ( $true
= ( X275 @ id @ ( lam @ ( X276 @ sh @ one ) ) @ id ) )
| ~ ! [X280: term] :
( ( $true
!= ( X275 @ id @ X280 @ id ) )
| ( $true
= ( X275 @ id @ ( X276 @ id @ X280 ) @ id ) ) ) )
| ( ( X275 @ id @ ( lam @ ( sub @ X274 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X281: term] :
( ( $true
= ( X275 @ id @ ( sub @ X274 @ ( push @ X281 @ id ) ) @ id ) )
| ( $true
!= ( X275 @ id @ X281 @ id ) ) ) )
=> ! [X282: term,X283: subst > term > subst > $o] :
( ~ ! [X284: subst > term > term] :
( ( $true
= ( X283 @ id @ ( lam @ ( X284 @ sh @ one ) ) @ id ) )
| ~ ! [X285: subst,X286: term,X287: subst] :
( ( sub @ ( X284 @ X287 @ X286 ) @ X285 )
= ( X284 @ ( comp @ X287 @ X285 ) @ ( sub @ X286 @ X285 ) ) )
| ~ ! [X288: term] :
( ( $true
= ( X283 @ id @ ( X284 @ id @ X288 ) @ id ) )
| ( $true
!= ( X283 @ id @ X288 @ id ) ) ) )
| ~ ! [X289: term] :
( ( $true
= ( X283 @ id @ ( sub @ X282 @ ( push @ X289 @ id ) ) @ id ) )
| ( $true
!= ( X283 @ id @ X289 @ id ) ) )
| ( $true
= ( X283 @ id @ ( lam @ X282 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( pushprop_gthm = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X290: term] :
( ( sub @ X290 @ id )
= X290 )
=> ( ! [X291: term,X292: subst > term > subst > $o] :
( ~ ! [X293: subst > term > term] :
( ~ ! [X294: term] :
( ( $true
= ( X292 @ id @ ( X293 @ id @ X294 ) @ id ) )
| ( $true
!= ( X292 @ id @ X294 @ id ) ) )
| ~ ! [X295: subst,X296: subst,X297: term] :
( ( X293 @ ( comp @ X296 @ X295 ) @ ( sub @ X297 @ X295 ) )
= ( sub @ ( X293 @ X296 @ X297 ) @ X295 ) )
| ( $true
= ( X292 @ id @ ( lam @ ( X293 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X292 @ id @ ( lam @ ( sub @ X291 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X298: term] :
( ( $true
= ( X292 @ id @ ( sub @ X291 @ ( push @ X298 @ id ) ) @ id ) )
| ( $true
!= ( X292 @ id @ X298 @ id ) ) ) )
=> ! [X299: term,X300: subst > term > subst > $o] :
( ( $true
= ( X300 @ id @ ( lam @ X299 ) @ id ) )
| ~ ! [X301: subst,X302: subst,X303: subst,X304: term] :
( ( $true
!= ( X300 @ ( comp @ X303 @ X301 ) @ ( sub @ X304 @ X301 ) @ X302 ) )
| ( $true
= ( X300 @ X303 @ X304 @ ( comp @ X301 @ X302 ) ) ) )
| ~ ! [X305: term] :
( ( ( X300 @ id @ ( sub @ X299 @ ( push @ X305 @ id ) ) @ id )
= $true )
| ( $true
!= ( X300 @ id @ X305 @ id ) ) )
| ~ ! [X306: subst > term > term] :
( ~ ! [X307: term,X308: subst,X309: subst] :
( ( X306 @ ( comp @ X308 @ X309 ) @ ( sub @ X307 @ X309 ) )
= ( sub @ ( X306 @ X308 @ X307 ) @ X309 ) )
| ~ ! [X310: term] :
( ( $true
!= ( X300 @ id @ X310 @ id ) )
| ( $true
= ( X300 @ id @ ( X306 @ id @ X310 ) @ id ) ) )
| ( $true
= ( X300 @ id @ ( lam @ ( X306 @ sh @ one ) ) @ id ) ) )
| ~ ! [X311: term,X312: subst,X313: subst,X314: subst] :
( ( $true
!= ( X300 @ X313 @ X311 @ ( comp @ X314 @ X312 ) ) )
| ( ( X300 @ ( comp @ X313 @ X314 ) @ ( sub @ X311 @ X314 ) @ X312 )
= $true ) ) ) ) ) )
& ( termmset_gthm = $true )
& ( axassoc = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ! [X315: subst > term > subst > $o,X316: term] :
( ~ ! [X317: term] :
( ( $true
!= ( X315 @ id @ X317 @ id ) )
| ( $true
= ( X315 @ id @ ( sub @ X316 @ ( push @ X317 @ id ) ) @ id ) ) )
| ~ ! [X318: subst > term > term] :
( ( $true
= ( X315 @ id @ ( lam @ ( X318 @ sh @ one ) ) @ id ) )
| ~ ! [X319: subst,X320: term,X321: subst] :
( ( sub @ ( X318 @ X319 @ X320 ) @ X321 )
= ( X318 @ ( comp @ X319 @ X321 ) @ ( sub @ X320 @ X321 ) ) )
| ~ ! [X322: term] :
( ( $true
!= ( X315 @ id @ X322 @ id ) )
| ( $true
= ( X315 @ id @ ( X318 @ id @ X322 ) @ id ) ) ) )
| ( $true
= ( X315 @ id @ ( lam @ X316 ) @ id ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ! [X323: term] :
( ( sub @ X323 @ id )
= X323 )
<=> ( axvarid = $true ) )
& ( ulamvarind = $true )
& ( ( induction2 = $true )
<=> ! [X324: term,X325: term > $o] :
( ~ ! [X326: term] :
( ( $true
= ( X325 @ X326 ) )
| ( $true
!= ( var @ X326 ) ) )
| ~ ! [X327: term,X328: term] :
( ( $true
!= ( X325 @ X327 ) )
| ( $true
= ( X325 @ ( ap @ X328 @ X327 ) ) )
| ( $true
!= ( X325 @ X328 ) ) )
| ( $true
= ( X325 @ X324 ) )
| ~ ! [X329: term] :
( ~ ! [X330: term] :
( ( $true
= ( X325 @ ( sub @ X329 @ ( push @ X330 @ id ) ) ) )
| ( $true
!= ( X325 @ X330 ) ) )
| ( $true
= ( X325 @ ( lam @ X329 ) ) ) ) ) )
& ( axabs = $true )
& ( ( ! [X331: subst > term > subst > $o] :
~ ! [X332: term > $o] :
~ ! [X333: term] :
( ( X331 @ id @ X333 @ id )
= ( X332 @ X333 ) )
=> ( ! [X334: term > $o,X335: term] :
( ~ ! [X336: term] :
( ( $true
!= ( var @ X336 ) )
| ( $true
= ( X334 @ X336 ) ) )
| ( $true
= ( X334 @ X335 ) )
| ~ ! [X337: term] :
( ~ ! [X338: term] :
( ( $true
!= ( X334 @ X338 ) )
| ( $true
= ( X334 @ ( sub @ X337 @ ( push @ X338 @ id ) ) ) ) )
| ( $true
= ( X334 @ ( lam @ X337 ) ) ) )
| ~ ! [X339: term,X340: term] :
( ( $true
!= ( X334 @ X340 ) )
| ( $true
!= ( X334 @ X339 ) )
| ( $true
= ( X334 @ ( ap @ X340 @ X339 ) ) ) ) )
=> ( ! [X341: term] :
( ( sub @ X341 @ id )
= X341 )
=> ( ! [X342: subst > term > subst > $o,X343: term,X344: term > $o] :
( ~ ! [X345: term] :
( ( $true
= ( X344 @ ( sub @ X343 @ ( push @ X345 @ id ) ) ) )
| ( $true
!= ( X344 @ X345 ) ) )
| ~ ! [X346: subst > term > term] :
( ~ ! [X347: term] :
( ( $true
!= ( X342 @ id @ X347 @ id ) )
| ( $true
= ( X342 @ id @ ( X346 @ id @ X347 ) @ id ) ) )
| ~ ! [X348: subst,X349: term,X350: subst] :
( ( sub @ ( X346 @ X350 @ X349 ) @ X348 )
= ( X346 @ ( comp @ X350 @ X348 ) @ ( sub @ X349 @ X348 ) ) )
| ( $true
= ( X342 @ id @ ( lam @ ( X346 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X344 @ ( lam @ X343 ) ) )
| ~ ! [X351: term] :
( ( X342 @ id @ X351 @ id )
= ( X344 @ X351 ) ) )
=> ! [X352: subst > term > subst > $o,X353: term] :
( ~ ! [X354: term,X355: term] :
( ( $true
!= ( X352 @ id @ X355 @ id ) )
| ( $true
!= ( X352 @ id @ X354 @ id ) )
| ( $true
= ( X352 @ id @ ( ap @ ( sub @ X354 @ id ) @ X355 ) @ id ) ) )
| ~ ! [X356: subst > term > term] :
( ( $true
= ( X352 @ id @ ( lam @ ( X356 @ sh @ one ) ) @ id ) )
| ~ ! [X357: subst,X358: subst,X359: term] :
( ( X356 @ ( comp @ X358 @ X357 ) @ ( sub @ X359 @ X357 ) )
= ( sub @ ( X356 @ X358 @ X359 ) @ X357 ) )
| ~ ! [X360: term] :
( ( $true
= ( X352 @ id @ ( X356 @ id @ X360 ) @ id ) )
| ( $true
!= ( X352 @ id @ X360 @ id ) ) ) )
| ( $true
= ( X352 @ id @ X353 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( pushprop = $true )
& ( ! [X361: term > $o,X362: subst,X363: term] :
( ~ ! [X364: term] :
( ( $true
!= ( var @ X364 ) )
| ( ( X361 @ ( sub @ X364 @ X362 ) )
= $true ) )
| ( $true
= ( X361 @ ( sub @ X363 @ X362 ) ) )
| ~ ! [X365: term] :
( ( $true
= ( X361 @ ( lam @ X365 ) ) )
| ~ ! [X366: term] :
( ( ( X361 @ X366 )
!= $true )
| ( $true
= ( X361 @ ( sub @ X365 @ ( push @ X366 @ id ) ) ) ) ) )
| ~ ! [X367: term,X368: term] :
( ( $true
= ( X361 @ ( ap @ X367 @ X368 ) ) )
| ( $true
!= ( X361 @ X367 ) )
| ( $true
!= ( X361 @ X368 ) ) ) )
<=> ( induction2lem = $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X369: term] :
~ ! [X370: subst > term > term] :
~ ! [X371: subst,X372: term] :
( ( X370 @ X371 @ X372 )
= ( sub @ X369 @ ( push @ X372 @ X371 ) ) )
=> ( ! [X373: term] :
( ( sub @ X373 @ id )
= X373 )
=> ! [X374: term > $o,X375: subst > term > subst > $o,X376: term] :
( ~ ! [X377: term] :
( ( X374 @ X377 )
= ( X375 @ id @ X377 @ id ) )
| ~ ! [X378: term] :
( ( $true
= ( X374 @ ( sub @ X376 @ ( push @ X378 @ id ) ) ) )
| ( $true
!= ( X374 @ X378 ) ) )
| ( $true
= ( X374 @ ( lam @ X376 ) ) )
| ~ ! [X379: subst > term > term] :
( ( $true
= ( X375 @ id @ ( lam @ ( X379 @ sh @ one ) ) @ id ) )
| ~ ! [X380: term] :
( ( $true
= ( X375 @ id @ ( X379 @ id @ X380 ) @ id ) )
| ( $true
!= ( X375 @ id @ X380 @ id ) ) )
| ~ ! [X381: subst,X382: term,X383: subst] :
( ( sub @ ( X379 @ X381 @ X382 ) @ X383 )
= ( X379 @ ( comp @ X381 @ X383 ) @ ( sub @ X382 @ X383 ) ) ) ) ) ) ) )
& ( induction2_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X384: term,X385: term > $o] :
( ~ ! [X386: term,X387: term] :
( ( $true
!= ( X385 @ X386 ) )
| ( $true
= ( X385 @ ( ap @ X386 @ X387 ) ) )
| ( $true
!= ( X385 @ X387 ) ) )
| ( $true
= ( X385 @ X384 ) )
| ~ ! [X388: term] :
( ( $true
= ( X385 @ ( lam @ X388 ) ) )
| ~ ! [X389: term] :
( ( $true
= ( X385 @ ( sub @ X388 @ ( push @ X389 @ id ) ) ) )
| ( $true
!= ( X385 @ X389 ) ) ) )
| ~ ! [X390: term] :
( ( $true
= ( X385 @ X390 ) )
| ( $true
!= ( var @ X390 ) ) ) )
=> ( ! [X391: term,X392: term,X393: subst > term > subst > $o] :
( ( ( X393 @ id @ ( ap @ X391 @ X392 ) @ id )
= $true )
| ( $true
!= ( X393 @ id @ X392 @ id ) )
| ( $true
!= ( X393 @ id @ X391 @ id ) )
| ~ ! [X394: subst,X395: subst,X396: term,X397: subst] :
( ( $true
!= ( X393 @ X395 @ X396 @ ( comp @ X394 @ X397 ) ) )
| ( $true
= ( X393 @ ( comp @ X395 @ X394 ) @ ( sub @ X396 @ X394 ) @ X397 ) ) )
| ~ ! [X398: term,X399: term] :
( ( $true
= ( X393 @ id @ ( ap @ ( sub @ X399 @ id ) @ X398 ) @ id ) )
| ( $true
!= ( X393 @ id @ X399 @ id ) )
| ( $true
!= ( X393 @ id @ X398 @ id ) ) )
| ~ ! [X400: subst,X401: subst,X402: subst,X403: term] :
( ( $true
!= ( X393 @ ( comp @ X401 @ X402 ) @ ( sub @ X403 @ X402 ) @ X400 ) )
| ( $true
= ( X393 @ X401 @ X403 @ ( comp @ X402 @ X400 ) ) ) ) )
=> ( ! [X404: subst > term > subst > $o,X405: term] :
( ( $true
= ( X404 @ id @ ( lam @ X405 ) @ id ) )
| ~ ! [X406: term] :
( ( $true
= ( X404 @ id @ ( sub @ X405 @ ( push @ X406 @ id ) ) @ id ) )
| ( $true
!= ( X404 @ id @ X406 @ id ) ) )
| ~ ! [X407: subst,X408: term,X409: subst,X410: subst] :
( ( $true
= ( X404 @ X409 @ X408 @ ( comp @ X407 @ X410 ) ) )
| ( $true
!= ( X404 @ ( comp @ X409 @ X407 ) @ ( sub @ X408 @ X407 ) @ X410 ) ) )
| ~ ! [X411: subst,X412: term,X413: subst,X414: subst] :
( ( ( X404 @ X411 @ X412 @ ( comp @ X414 @ X413 ) )
!= $true )
| ( $true
= ( X404 @ ( comp @ X411 @ X414 ) @ ( sub @ X412 @ X414 ) @ X413 ) ) )
| ~ ! [X415: subst > term > term] :
( ~ ! [X416: term,X417: subst,X418: subst] :
( ( sub @ ( X415 @ X417 @ X416 ) @ X418 )
= ( X415 @ ( comp @ X417 @ X418 ) @ ( sub @ X416 @ X418 ) ) )
| ( $true
= ( X404 @ id @ ( lam @ ( X415 @ sh @ one ) ) @ id ) )
| ~ ! [X419: term] :
( ( $true
!= ( X404 @ id @ X419 @ id ) )
| ( $true
= ( X404 @ id @ ( X415 @ id @ X419 ) @ id ) ) ) ) )
=> ! [X420: subst > term > subst > $o,X421: term] :
( ~ ! [X422: term,X423: term] :
( ( $true
= ( X420 @ id @ ( ap @ ( sub @ X423 @ id ) @ X422 ) @ id ) )
| ( $true
!= ( X420 @ id @ X422 @ id ) )
| ( $true
!= ( X420 @ id @ X423 @ id ) ) )
| ~ ! [X424: subst > term > term] :
( ( $true
= ( X420 @ id @ ( lam @ ( X424 @ sh @ one ) ) @ id ) )
| ~ ! [X425: term] :
( ( $true
= ( X420 @ id @ ( X424 @ id @ X425 ) @ id ) )
| ( $true
!= ( X420 @ id @ X425 @ id ) ) )
| ~ ! [X426: term,X427: subst,X428: subst] :
( ( X424 @ ( comp @ X427 @ X428 ) @ ( sub @ X426 @ X428 ) )
= ( sub @ ( X424 @ X427 @ X426 ) @ X428 ) ) )
| ( $true
= ( X420 @ id @ X421 @ id ) )
| ~ ! [X429: subst,X430: term,X431: subst,X432: subst] :
( ( $true
= ( X420 @ ( comp @ X432 @ X429 ) @ ( sub @ X430 @ X429 ) @ X431 ) )
| ( $true
!= ( X420 @ X432 @ X430 @ ( comp @ X429 @ X431 ) ) ) )
| ~ ! [X433: term,X434: subst,X435: subst,X436: subst] :
( ( $true
= ( X420 @ X435 @ X433 @ ( comp @ X434 @ X436 ) ) )
| ( $true
!= ( X420 @ ( comp @ X435 @ X434 ) @ ( sub @ X433 @ X434 ) @ X436 ) ) )
| ~ ! [X437: term] :
( ( $true
= ( X420 @ id @ X437 @ id ) )
| ( $true
!= ( var @ ( sub @ X437 @ id ) ) ) ) ) ) ) ) )
& ( ( hoasapinj1 = $true )
<=> ! [X438: term,X439: term,X440: term,X441: term] :
( ( X440 = X441 )
| ( ( ap @ ( sub @ X441 @ id ) @ X439 )
!= ( ap @ ( sub @ X440 @ id ) @ X438 ) ) ) )
& ( pushprop_lem1_gthm = $true )
& ( apnotvar = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X447: subst > term > subst > $o,X448: term] :
( ~ ! [X449: subst > term > term] :
( ~ ! [X450: term] :
( ( $true
= ( X447 @ id @ ( X449 @ id @ X450 ) @ id ) )
| ( $true
!= ( X447 @ id @ X450 @ id ) ) )
| ( $true
= ( X447 @ id @ ( lam @ ( X449 @ sh @ one ) ) @ id ) )
| ~ ! [X451: subst,X452: term,X453: subst] :
( ( sub @ ( X449 @ X451 @ X452 ) @ X453 )
= ( X449 @ ( comp @ X451 @ X453 ) @ ( sub @ X452 @ X453 ) ) ) )
| ( $true
= ( X447 @ id @ ( lam @ ( sub @ X448 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X454: term] :
( ( $true
= ( X447 @ id @ ( sub @ X448 @ ( push @ X454 @ id ) ) @ id ) )
| ( $true
!= ( X447 @ id @ X454 @ id ) ) ) ) )
& ( ! [X455: subst > term > subst > $o] :
~ ! [X456: term > $o] :
~ ! [X457: term] :
( ( X455 @ id @ X457 @ id )
= ( X456 @ X457 ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( ! [X458: subst] :
( ( comp @ id @ X458 )
= X458 )
& ! [X459: subst] :
( ( comp @ X459 @ id )
= X459 ) )
<=> ( substmonoid = $true ) )
& ( ( induction2_lthm = $true )
<=> ( ! [X460: term] :
( ( sub @ X460 @ id )
= X460 )
=> ( ! [X461: subst,X462: term,X463: term > $o] :
( ~ ! [X464: term] :
( ( $true
!= ( var @ X464 ) )
| ( $true
= ( X463 @ ( sub @ X464 @ X461 ) ) ) )
| ~ ! [X465: term,X466: term] :
( ( $true
!= ( X463 @ X466 ) )
| ( $true
= ( X463 @ ( ap @ X465 @ X466 ) ) )
| ( $true
!= ( X463 @ X465 ) ) )
| ~ ! [X467: term] :
( ~ ! [X468: term] :
( ( $true
= ( X463 @ ( sub @ X467 @ ( push @ X468 @ id ) ) ) )
| ( $true
!= ( X463 @ X468 ) ) )
| ( ( X463 @ ( lam @ X467 ) )
= $true ) )
| ( $true
= ( X463 @ ( sub @ X462 @ X461 ) ) ) )
=> ! [X469: term > $o,X470: term] :
( ~ ! [X471: term] :
( ( $true
!= ( var @ X471 ) )
| ( $true
= ( X469 @ X471 ) ) )
| ~ ! [X472: term,X473: term] :
( ( $true
!= ( X469 @ X472 ) )
| ( $true
!= ( X469 @ X473 ) )
| ( $true
= ( X469 @ ( ap @ X472 @ X473 ) ) ) )
| ( $true
= ( X469 @ X470 ) )
| ~ ! [X474: term] :
( ( ( X469 @ ( lam @ X474 ) )
= $true )
| ~ ! [X475: term] :
( ( $true
= ( X469 @ ( sub @ X474 @ ( push @ X475 @ id ) ) ) )
| ( $true
!= ( X469 @ X475 ) ) ) ) ) ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X476: term,X477: term,X478: subst > term > subst > $o] :
( ( $true
= ( X478 @ id @ ( ap @ X477 @ X476 ) @ id ) )
| ( $true
!= ( X478 @ id @ X477 @ id ) )
| ~ ! [X479: term,X480: term] :
( ( $true
!= ( X478 @ id @ X480 @ id ) )
| ( $true
= ( X478 @ id @ ( ap @ ( sub @ X480 @ id ) @ X479 ) @ id ) )
| ( $true
!= ( X478 @ id @ X479 @ id ) ) )
| ~ ! [X481: subst,X482: subst,X483: subst,X484: term] :
( ( $true
!= ( X478 @ X483 @ X484 @ ( comp @ X482 @ X481 ) ) )
| ( $true
= ( X478 @ ( comp @ X483 @ X482 ) @ ( sub @ X484 @ X482 ) @ X481 ) ) )
| ~ ! [X485: subst,X486: subst,X487: subst,X488: term] :
( ( $true
= ( X478 @ X486 @ X488 @ ( comp @ X485 @ X487 ) ) )
| ( $true
!= ( X478 @ ( comp @ X486 @ X485 ) @ ( sub @ X488 @ X485 ) @ X487 ) ) )
| ( $true
!= ( X478 @ id @ X476 @ id ) ) ) )
& ( ! [X489: term > $o,X490: term] :
( ~ ! [X491: term] :
( ( $true
= ( X489 @ X491 ) )
| ( $true
!= ( var @ X491 ) ) )
| ~ ! [X492: term] :
( ( $true
!= ( X489 @ X492 ) )
| ( $true
= ( X489 @ ( lam @ X492 ) ) ) )
| ( $true
= ( X489 @ X490 ) )
| ~ ! [X493: term,X494: term] :
( ( $true
!= ( X489 @ X494 ) )
| ( $true
= ( X489 @ ( ap @ X493 @ X494 ) ) )
| ( $true
!= ( X489 @ X493 ) ) ) )
<=> ( induction = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X495: d_term] : ( d_one = X495 )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X496: term] :
( ( sub @ X496 @ id )
= X496 )
=> ( ! [X497: term,X498: term,X499: term,X500: term] :
( ( ( ap @ X500 @ X499 )
!= ( ap @ X497 @ X498 ) )
| ( X497 = X500 ) )
=> ! [X501: term,X502: term,X503: term,X504: term] :
( ( ( ap @ ( sub @ X503 @ id ) @ X502 )
!= ( ap @ ( sub @ X501 @ id ) @ X504 ) )
| ( X501 = X503 ) ) ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( axvarshift = $true )
& ( ( ! [X505: term] :
( ( sub @ X505 @ id )
= X505 )
=> ! [X506: term > $o,X507: subst > term > subst > $o,X508: term] :
( ~ ! [X509: subst,X510: term,X511: subst,X512: subst] :
( ( $true
= ( X507 @ ( comp @ X511 @ X509 ) @ ( sub @ X510 @ X509 ) @ X512 ) )
| ( $true
!= ( X507 @ X511 @ X510 @ ( comp @ X509 @ X512 ) ) ) )
| ~ ! [X513: term] :
( ( X506 @ X513 )
= ( X507 @ id @ X513 @ id ) )
| ~ ! [X514: term] :
( ( $true
= ( X506 @ ( sub @ X508 @ ( push @ X514 @ id ) ) ) )
| ( $true
!= ( X506 @ X514 ) ) )
| ~ ! [X515: subst,X516: term,X517: subst,X518: subst] :
( ( $true
!= ( X507 @ ( comp @ X518 @ X517 ) @ ( sub @ X516 @ X517 ) @ X515 ) )
| ( ( X507 @ X518 @ X516 @ ( comp @ X517 @ X515 ) )
= $true ) )
| ( ( X506 @ ( lam @ X508 ) )
= $true )
| ~ ! [X519: subst > term > term] :
( ( ( X507 @ id @ ( lam @ ( X519 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X520: subst,X521: term,X522: subst] :
( ( X519 @ ( comp @ X522 @ X520 ) @ ( sub @ X521 @ X520 ) )
= ( sub @ ( X519 @ X522 @ X521 ) @ X520 ) )
| ~ ! [X523: term] :
( ( $true
!= ( X507 @ id @ X523 @ id ) )
| ( $true
= ( X507 @ id @ ( X519 @ id @ X523 ) @ id ) ) ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( ( shinj = $true )
<=> ! [X524: term,X525: term] :
( ( ( sub @ X525 @ sh )
!= ( sub @ X524 @ sh ) )
| ( X524 = X525 ) ) )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X526: term,X527: subst > term > subst > $o] :
( ~ ! [X528: subst,X529: term,X530: subst,X531: subst] :
( ( $true
= ( X527 @ ( comp @ X531 @ X528 ) @ ( sub @ X529 @ X528 ) @ X530 ) )
| ( ( X527 @ X531 @ X529 @ ( comp @ X528 @ X530 ) )
!= $true ) )
| ~ ! [X532: subst > term > term] :
( ( $true
= ( X527 @ id @ ( lam @ ( X532 @ sh @ one ) ) @ id ) )
| ~ ! [X533: term] :
( ( $true
= ( X527 @ id @ ( X532 @ id @ X533 ) @ id ) )
| ( $true
!= ( X527 @ id @ X533 @ id ) ) )
| ~ ! [X534: term,X535: subst,X536: subst] :
( ( X532 @ ( comp @ X535 @ X536 ) @ ( sub @ X534 @ X536 ) )
= ( sub @ ( X532 @ X535 @ X534 ) @ X536 ) ) )
| ( $true
= ( X527 @ id @ ( lam @ X526 ) @ id ) )
| ~ ! [X537: term] :
( ( ( X527 @ id @ ( sub @ X526 @ ( push @ X537 @ id ) ) @ id )
= $true )
| ( $true
!= ( X527 @ id @ X537 @ id ) ) )
| ~ ! [X538: subst,X539: subst,X540: subst,X541: term] :
( ( $true
!= ( X527 @ ( comp @ X539 @ X540 ) @ ( sub @ X541 @ X540 ) @ X538 ) )
| ( $true
= ( X527 @ X539 @ X541 @ ( comp @ X540 @ X538 ) ) ) ) ) )
& ( substmonoid_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X542: term,X543: term > $o,X544: subst] :
~ ! [X545: term > $o] :
~ ! [X546: term] :
( ( X543 @ ( sub @ X546 @ ( push @ X542 @ X544 ) ) )
= ( X545 @ X546 ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X547: term] :
~ ! [X548: subst > term > term] :
( ( X548 @ sh @ one )
!= ( sub @ X547 @ ( push @ one @ sh ) ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X549: subst] :
( ( push @ ( sub @ one @ X549 ) @ ( comp @ sh @ X549 ) )
= X549 )
<=> ( axscons = $true ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem2_lthm = $true ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
( ( ! [X0: term,X1: term,X2: term,X3: term] :
( ( X0 = X1 )
| ( ( ap @ X2 @ X0 )
!= ( ap @ X3 @ X1 ) ) )
<=> apinj2 )
& ( ! [X4: term,X5: subst > term > subst > $o] :
( ( X5 @ id @ X4 @ id )
| ~ ! [X6: subst > term > term] :
( ~ ! [X7: term] :
( ~ ( X5 @ id @ X7 @ id )
| ( X5 @ id @ ( X6 @ id @ X7 ) @ id ) )
| ~ ! [X8: term,X9: subst,X10: subst] :
( ( X6 @ ( comp @ X10 @ X9 ) @ ( sub @ X8 @ X9 ) )
= ( sub @ ( X6 @ X10 @ X8 ) @ X9 ) )
| ( X5 @ id @ ( lam @ ( X6 @ sh @ one ) ) @ id ) )
| ~ ! [X11: term,X12: term] :
( ~ ( X5 @ id @ X12 @ id )
| ( X5 @ id @ ( ap @ ( sub @ X11 @ id ) @ X12 ) @ id )
| ~ ( X5 @ id @ X11 @ id ) ) )
<=> hoasinduction_no_psi_cond )
& ! [X13: d_subst,X14: d_subst] :
( ( ( d2subst @ X13 )
= ( d2subst @ X14 ) )
=> ( X13 = X14 ) )
& ( one
= ( d2term @ d_one ) )
& ( hoasinduction_lem3v2a
<=> ! [X15: subst > term > subst > $o,X16: term > $o,X17: term] :
( ~ ! [X18: term] :
( ( X16 @ X18 )
= ( X15 @ id @ X18 @ id ) )
| ~ ! [X19: subst > term > term] :
( ~ ! [X20: term,X21: subst,X22: subst] :
( ( sub @ ( X19 @ X22 @ X20 ) @ X21 )
= ( X19 @ ( comp @ X22 @ X21 ) @ ( sub @ X20 @ X21 ) ) )
| ( X15 @ id @ ( lam @ ( X19 @ sh @ one ) ) @ id )
| ~ ! [X23: term] :
( ( X15 @ id @ ( X19 @ id @ X23 ) @ id )
| ~ ( X15 @ id @ X23 @ id ) ) )
| ~ ! [X24: term] :
( ( X16 @ ( sub @ X17 @ ( push @ X24 @ id ) ) )
| ~ ( X16 @ X24 ) )
| ( X16 @ ( lam @ X17 ) ) ) )
& hoasinduction_lem3a_gthm
& ( termmset_lthm
<=> ( ! [X25: term] :
( ( sub @ X25 @ id )
= X25 )
=> ! [X26: term] :
( ( sub @ X26 @ id )
= X26 ) ) )
& ( ! [X27: subst,X28: term,X29: term > $o] :
~ ! [X30: term > $o] :
~ ! [X31: term] :
( ( X29 @ ( sub @ X31 @ ( push @ X28 @ X27 ) ) )
= ( X30 @ X31 ) )
<=> pushprop_lem0 )
& ( ! [X32: term] :
( ( sub @ X32 @ id )
= X32 )
<=> termmset )
& hoasinduction_lem1
& ( substmonoid_lthm
<=> ( ! [X33: subst] :
( ( comp @ id @ X33 )
= X33 )
=> ( ! [X34: subst] :
( ( comp @ X34 @ id )
= X34 )
=> ( ! [X35: subst] :
( ( comp @ X35 @ id )
= X35 )
& ! [X36: subst] :
( ( comp @ id @ X36 )
= X36 ) ) ) ) )
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ( pushprop_lem1v2_lthm
<=> ( ! [X37: term,X38: subst] :
( ( sub @ one @ ( push @ X37 @ X38 ) )
= X37 )
=> ! [X39: term,X40: term > $o,X41: term > $o,X42: subst] :
( ~ ! [X43: term] :
( ( X41 @ X43 )
= ( X40 @ ( sub @ X43 @ ( push @ X39 @ X42 ) ) ) )
| ~ ( X40 @ X39 )
| ( X41 @ one ) ) ) )
& pushprop_lem2v2
& ~ lamnotap
& ( hoasinduction_lem3aaa
<=> ! [X44: term,X45: subst > term > subst > $o] :
( ( X45 @ id @ ( lam @ ( sub @ X44 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X46: term] :
( ( X45 @ id @ ( sub @ X44 @ ( push @ X46 @ id ) ) @ id )
| ~ ( X45 @ id @ X46 @ id ) )
| ~ ! [X47: subst > term > term,X48: term] :
( ~ ! [X49: subst,X50: term,X51: subst] :
( ( sub @ ( sub @ X48 @ ( push @ X50 @ X49 ) ) @ X51 )
= ( sub @ ( X47 @ X49 @ X50 ) @ X51 ) )
| ~ ! [X52: subst,X53: term,X54: subst] :
( ( sub @ X48 @ ( push @ ( sub @ X53 @ X52 ) @ ( comp @ X54 @ X52 ) ) )
= ( X47 @ ( comp @ X54 @ X52 ) @ ( sub @ X53 @ X52 ) ) )
| ( X45 @ id @ ( lam @ ( X47 @ sh @ one ) ) @ id )
| ~ ! [X55: term] :
( ( X45 @ id @ ( X47 @ id @ X55 ) @ id )
| ~ ( X45 @ id @ X55 @ id ) ) ) ) )
& ( axidl
<=> ! [X56: subst] :
( ( comp @ id @ X56 )
= X56 ) )
& hoasinduction_gthm
& ( ! [X57: term,X58: subst] :
( ( comp @ sh @ ( push @ X57 @ X58 ) )
= X58 )
<=> axshiftcons )
& axapp
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( hoasinduction
<=> ! [X59: term,X60: subst > term > subst > $o] :
( ~ ! [X61: term,X62: term] :
( ( X60 @ id @ ( ap @ ( sub @ X61 @ id ) @ X62 ) @ id )
| ~ ( X60 @ id @ X61 @ id )
| ~ ( X60 @ id @ X62 @ id ) )
| ~ ! [X63: subst > term > term] :
( ~ ! [X64: term] :
( ~ ( X60 @ id @ X64 @ id )
| ( X60 @ id @ ( X63 @ id @ X64 ) @ id ) )
| ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
| ~ ! [X65: subst,X66: subst,X67: term] :
( ( X63 @ ( comp @ X65 @ X66 ) @ ( sub @ X67 @ X66 ) )
= ( sub @ ( X63 @ X65 @ X67 ) @ X66 ) ) )
| ~ ! [X68: term] :
( ~ ( var @ ( sub @ X68 @ id ) )
| ( X60 @ id @ X68 @ id ) )
| ~ ! [X69: subst,X70: subst,X71: term,X72: subst] :
( ~ ( X60 @ ( comp @ X69 @ X72 ) @ ( sub @ X71 @ X72 ) @ X70 )
| ( X60 @ X69 @ X71 @ ( comp @ X72 @ X70 ) ) )
| ~ ! [X73: term,X74: subst,X75: subst,X76: subst] :
( ( X60 @ ( comp @ X75 @ X74 ) @ ( sub @ X73 @ X74 ) @ X76 )
| ~ ( X60 @ X75 @ X73 @ ( comp @ X74 @ X76 ) ) )
| ( X60 @ id @ X59 @ id ) ) )
& ~ ulamvar1
& ( ( d2subst @ d_id )
= sh )
& hoaslamnotvar_lthm
& hoaslamnotap_lthm
& ( hoasinduction_lem0
<=> ! [X77: subst > term > subst > $o] :
~ ! [X78: term > $o] :
~ ! [X79: term] :
( ( X78 @ X79 )
= ( X77 @ id @ X79 @ id ) ) )
& hoasapinj1_gthm
& ( hoasinduction_lem3v2_f
<=> ! [X80: term] :
~ ! [X81: subst > term > term] :
~ ! [X82: subst,X83: term] :
( ( sub @ X80 @ ( push @ X83 @ X82 ) )
= ( X81 @ X82 @ X83 ) ) )
& induction2lem_gthm
& hoasinduction_lem2_gthm
& ( induction2lem_lthm
<=> ( ! [X84: subst,X85: term] :
( ( sub @ one @ ( push @ X85 @ X84 ) )
= X85 )
=> ( ! [X86: subst,X87: term] :
( ( comp @ sh @ ( push @ X87 @ X86 ) )
= X86 )
=> ( ! [X88: subst] :
( ( comp @ X88 @ id )
= X88 )
=> ( ! [X89: term,X90: term > $o] :
( ~ ! [X91: term,X92: term] :
( ~ ( X90 @ X92 )
| ( X90 @ ( ap @ X91 @ X92 ) )
| ~ ( X90 @ X91 ) )
| ~ ! [X93: term] :
( ~ ( X90 @ X93 )
| ( X90 @ ( lam @ X93 ) ) )
| ( X90 @ X89 )
| ~ ! [X94: term] :
( ~ ( var @ X94 )
| ( X90 @ X94 ) ) )
=> ! [X95: subst,X96: term,X97: term > $o] :
( ~ ! [X98: term] :
( ~ ! [X99: term] :
( ( X97 @ ( sub @ X98 @ ( push @ X99 @ id ) ) )
| ~ ( X97 @ X99 ) )
| ( X97 @ ( lam @ X98 ) ) )
| ~ ! [X100: term] :
( ~ ( var @ X100 )
| ( X97 @ ( sub @ X100 @ X95 ) ) )
| ~ ! [X101: term,X102: term] :
( ~ ( X97 @ X102 )
| ( X97 @ ( ap @ X102 @ X101 ) )
| ~ ( X97 @ X101 ) )
| ( X97 @ ( sub @ X96 @ X95 ) ) ) ) ) ) ) )
& ( ! [X103: subst,X104: term] :
( ( sub @ one @ ( push @ X104 @ X103 ) )
= X104 )
<=> axvarcons )
& hoasapinj2_gthm
& ( axidr
<=> ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 ) )
& ! [X106: d_subst] : ( d_id = X106 )
& ( ! [X107: subst > term > term,X108: subst,X109: term,X110: subst > term > term] :
( ( ( X107 @ X108 @ X109 )
= ( X110 @ X108 @ X109 ) )
| ~ ! [X111: subst,X112: subst,X113: term] :
( ( X107 @ ( comp @ X111 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X107 @ X111 @ X113 ) @ X112 ) )
| ~ ! [X114: subst,X115: term,X116: subst] :
( ( X110 @ ( comp @ X116 @ X114 ) @ ( sub @ X115 @ X114 ) )
= ( sub @ ( X110 @ X116 @ X115 ) @ X114 ) )
| ( ( lam @ ( X110 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) ) )
<=> hoaslaminj )
& ( ! [X117: subst > term > term,X118: term,X119: term] :
( ~ ! [X120: subst,X121: term,X122: subst] :
( ( X117 @ ( comp @ X120 @ X122 ) @ ( sub @ X121 @ X122 ) )
= ( sub @ ( X117 @ X120 @ X121 ) @ X122 ) )
| ( ( lam @ ( X117 @ sh @ one ) )
!= ( ap @ ( sub @ X118 @ id ) @ X119 ) ) )
<=> hoaslamnotap )
& ( apinj1
<=> ! [X123: term,X124: term,X125: term,X126: term] :
( ( ( ap @ X125 @ X123 )
!= ( ap @ X126 @ X124 ) )
| ( X125 = X126 ) ) )
& hoasinduction_lem3v2_gthm
& ( hoasinduction_lem2v2
<=> ! [X127: term,X128: term > $o,X129: subst > term > subst > $o,X130: term] :
( ~ ( X128 @ X130 )
| ( X128 @ ( ap @ X130 @ X127 ) )
| ~ ! [X131: term,X132: subst,X133: subst,X134: subst] :
( ( X129 @ ( comp @ X133 @ X132 ) @ ( sub @ X131 @ X132 ) @ X134 )
| ~ ( X129 @ X133 @ X131 @ ( comp @ X132 @ X134 ) ) )
| ~ ( X128 @ X127 )
| ~ ! [X135: subst,X136: subst,X137: term,X138: subst] :
( ~ ( X129 @ ( comp @ X138 @ X136 ) @ ( sub @ X137 @ X136 ) @ X135 )
| ( X129 @ X138 @ X137 @ ( comp @ X136 @ X135 ) ) )
| ~ ! [X139: term] :
( ( X129 @ id @ X139 @ id )
= ( X128 @ X139 ) )
| ~ ! [X140: term,X141: term] :
( ( X129 @ id @ ( ap @ ( sub @ X140 @ id ) @ X141 ) @ id )
| ~ ( X129 @ id @ X140 @ id )
| ~ ( X129 @ id @ X141 @ id ) ) ) )
& pushprop_lthm
& hoaslaminj_gthm
& hoasapnotvar_lthm
& ( pushprop_lem1v2
<=> ! [X142: term > $o,X143: subst,X144: term > $o,X145: term] :
( ( X144 @ one )
| ~ ! [X146: term] :
( ( X144 @ X146 )
= ( X142 @ ( sub @ X146 @ ( push @ X145 @ X143 ) ) ) )
| ~ ( X142 @ X145 ) ) )
& ( ! [X147: term] :
~ ! [X148: subst > term > term] :
~ ! [X149: term,X150: subst] :
( ( sub @ X147 @ ( push @ X149 @ X150 ) )
= ( X148 @ X150 @ X149 ) )
<=> hoasinduction_lem3v2_f_lthm )
& ! [X151: d_term,X152: d_term] :
( ( ( d2term @ X151 )
= ( d2term @ X152 ) )
=> ( X151 = X152 ) )
& pushprop_lthm_orig
& ( ! [X153: term,X154: term,X155: term,X156: term] :
( ( ( ap @ ( sub @ X156 @ id ) @ X155 )
!= ( ap @ ( sub @ X154 @ id ) @ X153 ) )
| ( X153 = X155 ) )
<=> hoasapinj2 )
& hoasinduction_lem1v2
& hoaslamnotvar_gthm
& ( hoaslaminj_lthm
<=> ( ! [X157: term,X158: subst] :
( ( sub @ one @ ( push @ X157 @ X158 ) )
= X157 )
=> ( ! [X159: subst,X160: term] :
( ( comp @ sh @ ( push @ X160 @ X159 ) )
= X159 )
=> ( ! [X161: term,X162: term] :
( ( X161 = X162 )
| ( ( lam @ X161 )
!= ( lam @ X162 ) ) )
=> ! [X163: subst,X164: subst > term > term,X165: term,X166: subst > term > term] :
( ( ( lam @ ( X166 @ sh @ one ) )
!= ( lam @ ( X164 @ sh @ one ) ) )
| ~ ! [X167: subst,X168: term,X169: subst] :
( ( sub @ ( X164 @ X167 @ X168 ) @ X169 )
= ( X164 @ ( comp @ X167 @ X169 ) @ ( sub @ X168 @ X169 ) ) )
| ( ( X164 @ X163 @ X165 )
= ( X166 @ X163 @ X165 ) )
| ~ ! [X170: term,X171: subst,X172: subst] :
( ( sub @ ( X166 @ X172 @ X170 ) @ X171 )
= ( X166 @ ( comp @ X172 @ X171 ) @ ( sub @ X170 @ X171 ) ) ) ) ) ) ) )
& ( ( ^ [X173: subst,X174: subst > term > term] : ( d2term @ d_one ) )
= hoaslam )
& ( pushprop_lem1
<=> ! [X175: term > $o,X176: term,X177: subst,X178: term,X179: term > $o] :
( ( X179 @ ( sub @ X176 @ ( push @ X178 @ X177 ) ) )
| ~ ( X175 @ X176 ) ) )
& pushprop_lem3v2_lthm
& ( hoasinduction_lem3v2
<=> ! [X180: term > $o,X181: subst > term > subst > $o,X182: term] :
( ~ ! [X183: term] :
( ( X180 @ X183 )
= ( X181 @ id @ X183 @ id ) )
| ~ ! [X184: subst > term > term] :
( ( X181 @ id @ ( lam @ ( X184 @ sh @ one ) ) @ id )
| ~ ! [X185: term] :
( ( X181 @ id @ ( X184 @ id @ X185 ) @ id )
| ~ ( X181 @ id @ X185 @ id ) )
| ~ ! [X186: subst,X187: subst,X188: term] :
( ( sub @ ( X184 @ X187 @ X188 ) @ X186 )
= ( X184 @ ( comp @ X187 @ X186 ) @ ( sub @ X188 @ X186 ) ) ) )
| ~ ! [X189: term] :
( ( X180 @ ( sub @ X182 @ ( push @ X189 @ id ) ) )
| ~ ( X180 @ X189 ) )
| ~ ! [X190: term,X191: subst,X192: subst,X193: subst] :
( ( X181 @ X191 @ X190 @ ( comp @ X192 @ X193 ) )
| ~ ( X181 @ ( comp @ X191 @ X192 ) @ ( sub @ X190 @ X192 ) @ X193 ) )
| ~ ! [X194: subst,X195: subst,X196: term,X197: subst] :
( ( X181 @ ( comp @ X194 @ X195 ) @ ( sub @ X196 @ X195 ) @ X197 )
| ~ ( X181 @ X194 @ X196 @ ( comp @ X195 @ X197 ) ) )
| ( X180 @ ( lam @ X182 ) ) ) )
& ( laminj
<=> ! [X198: term,X199: term] :
( ( ( lam @ X199 )
!= ( lam @ X198 ) )
| ( X198 = X199 ) ) )
& ! [X200: term] :
? [X201: d_term] :
( ( d2term @ X201 )
= X200 )
& pushprop_lem3v2
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X202: term] :
~ ! [X203: subst > term > term] :
( ( sub @ X202 @ ( push @ one @ sh ) )
!= ( X203 @ sh @ one ) )
<=> hoasinduction_lem3b_lthm )
& hoasapnotvar_gthm
& axmap
& axclos
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& hoaslamnotvar
& ( pushprop_lem1_lthm
<=> ( ! [X204: subst,X205: term] :
( ( sub @ one @ ( push @ X205 @ X204 ) )
= X205 )
=> ( ! [X206: subst,X207: term] :
( ( comp @ sh @ ( push @ X207 @ X206 ) )
= X206 )
=> ! [X208: term,X209: term > $o,X210: term > $o,X211: subst,X212: term] :
( ~ ( X209 @ X212 )
| ( X210 @ ( sub @ X212 @ ( push @ X208 @ X211 ) ) ) ) ) ) )
& ( hoasapinj2_lthm
<=> ( ! [X213: term,X214: term,X215: term,X216: term] :
( ( X214 = X216 )
| ( ( ap @ X215 @ X214 )
!= ( ap @ X213 @ X216 ) ) )
=> ! [X217: term,X218: term,X219: term,X220: term] :
( ( X219 = X220 )
| ( ( ap @ ( sub @ X218 @ id ) @ X220 )
!= ( ap @ ( sub @ X217 @ id ) @ X219 ) ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3aa
<=> ! [X221: subst > term > subst > $o,X222: term] :
( ~ ! [X223: subst > term > term] :
( ~ ! [X224: term] :
( ~ ( X221 @ id @ X224 @ id )
| ( X221 @ id @ ( X223 @ id @ X224 ) @ id ) )
| ( X221 @ id @ ( lam @ ( X223 @ sh @ one ) ) @ id )
| ~ ! [X225: term,X226: subst,X227: subst] :
( ( X223 @ ( comp @ X227 @ X226 ) @ ( sub @ X225 @ X226 ) )
= ( sub @ ( X223 @ X227 @ X225 ) @ X226 ) ) )
| ~ ! [X228: term] :
( ( X221 @ id @ ( sub @ X222 @ ( push @ X228 @ id ) ) @ id )
| ~ ( X221 @ id @ X228 @ id ) )
| ( X221 @ id @ ( lam @ ( sub @ X222 @ ( push @ one @ sh ) ) ) @ id ) ) )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lthm_3
<=> ( ! [X229: subst > term > subst > $o] :
~ ! [X230: term > $o] :
~ ! [X231: term] :
( ( X229 @ id @ X231 @ id )
= ( X230 @ X231 ) )
=> ( ! [X232: term > $o,X233: term] :
( ~ ! [X234: term] :
( ( X232 @ X234 )
| ~ ( var @ X234 ) )
| ( X232 @ X233 )
| ~ ! [X235: term] :
( ~ ! [X236: term] :
( ( X232 @ ( sub @ X235 @ ( push @ X236 @ id ) ) )
| ~ ( X232 @ X236 ) )
| ( X232 @ ( lam @ X235 ) ) )
| ~ ! [X237: term,X238: term] :
( ( X232 @ ( ap @ X237 @ X238 ) )
| ~ ( X232 @ X238 )
| ~ ( X232 @ X237 ) ) )
=> ( ! [X239: term] :
( ( sub @ X239 @ id )
= X239 )
=> ( ! [X240: term,X241: term > $o,X242: subst > term > subst > $o] :
( ~ ! [X243: term] :
( ~ ( X241 @ X243 )
| ( X241 @ ( sub @ X240 @ ( push @ X243 @ id ) ) ) )
| ~ ! [X244: subst > term > term] :
( ( X242 @ id @ ( lam @ ( X244 @ sh @ one ) ) @ id )
| ~ ! [X245: term] :
( ~ ( X242 @ id @ X245 @ id )
| ( X242 @ id @ ( X244 @ id @ X245 ) @ id ) )
| ~ ! [X246: subst,X247: term,X248: subst] :
( ( sub @ ( X244 @ X248 @ X247 ) @ X246 )
= ( X244 @ ( comp @ X248 @ X246 ) @ ( sub @ X247 @ X246 ) ) ) )
| ~ ! [X249: term] :
( ( X241 @ X249 )
= ( X242 @ id @ X249 @ id ) )
| ( X241 @ ( lam @ X240 ) ) )
=> ! [X250: term,X251: subst > term > subst > $o] :
( ~ ! [X252: term,X253: term] :
( ~ ( X251 @ id @ X253 @ id )
| ~ ( X251 @ id @ X252 @ id )
| ( X251 @ id @ ( ap @ ( sub @ X253 @ id ) @ X252 ) @ id ) )
| ~ ! [X254: subst,X255: term,X256: subst,X257: subst] :
( ~ ( X251 @ X254 @ X255 @ ( comp @ X256 @ X257 ) )
| ( X251 @ ( comp @ X254 @ X256 ) @ ( sub @ X255 @ X256 ) @ X257 ) )
| ~ ! [X258: subst > term > term] :
( ~ ! [X259: term] :
( ~ ( X251 @ id @ X259 @ id )
| ( X251 @ id @ ( X258 @ id @ X259 ) @ id ) )
| ~ ! [X260: subst,X261: term,X262: subst] :
( ( sub @ ( X258 @ X262 @ X261 ) @ X260 )
= ( X258 @ ( comp @ X262 @ X260 ) @ ( sub @ X261 @ X260 ) ) )
| ( X251 @ id @ ( lam @ ( X258 @ sh @ one ) ) @ id ) )
| ( X251 @ id @ X250 @ id )
| ~ ! [X263: term,X264: subst,X265: subst,X266: subst] :
( ~ ( X251 @ ( comp @ X264 @ X265 ) @ ( sub @ X263 @ X265 ) @ X266 )
| ( X251 @ X264 @ X263 @ ( comp @ X265 @ X266 ) ) )
| ~ ! [X267: term] :
( ~ ( var @ ( sub @ X267 @ id ) )
| ( X251 @ id @ X267 @ id ) ) ) ) ) ) ) )
& ( ( ^ [X268: subst > term > subst > $o,X269: term > $o] :
! [X270: term] :
( ( X269 @ X270 )
= ( X268 @ id @ X270 @ id ) ) )
= hoasinduction_p_and_p_prime )
& ! [X271: subst] :
? [X272: d_subst] :
( ( d2subst @ X272 )
= X271 )
& ( ( ! [X273: term] :
( ( sub @ X273 @ id )
= X273 )
=> ( ! [X274: term,X275: subst > term > subst > $o] :
( ~ ! [X276: subst > term > term] :
( ~ ! [X277: subst,X278: term,X279: subst] :
( ( sub @ ( X276 @ X277 @ X278 ) @ X279 )
= ( X276 @ ( comp @ X277 @ X279 ) @ ( sub @ X278 @ X279 ) ) )
| ( X275 @ id @ ( lam @ ( X276 @ sh @ one ) ) @ id )
| ~ ! [X280: term] :
( ~ ( X275 @ id @ X280 @ id )
| ( X275 @ id @ ( X276 @ id @ X280 ) @ id ) ) )
| ( X275 @ id @ ( lam @ ( sub @ X274 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X281: term] :
( ( X275 @ id @ ( sub @ X274 @ ( push @ X281 @ id ) ) @ id )
| ~ ( X275 @ id @ X281 @ id ) ) )
=> ! [X282: term,X283: subst > term > subst > $o] :
( ~ ! [X284: subst > term > term] :
( ( X283 @ id @ ( lam @ ( X284 @ sh @ one ) ) @ id )
| ~ ! [X285: subst,X286: term,X287: subst] :
( ( sub @ ( X284 @ X287 @ X286 ) @ X285 )
= ( X284 @ ( comp @ X287 @ X285 ) @ ( sub @ X286 @ X285 ) ) )
| ~ ! [X288: term] :
( ( X283 @ id @ ( X284 @ id @ X288 ) @ id )
| ~ ( X283 @ id @ X288 @ id ) ) )
| ~ ! [X289: term] :
( ( X283 @ id @ ( sub @ X282 @ ( push @ X289 @ id ) ) @ id )
| ~ ( X283 @ id @ X289 @ id ) )
| ( X283 @ id @ ( lam @ X282 ) @ id ) ) ) )
<=> hoasinduction_lem3a_lthm )
& pushprop_gthm
& hoasapnotvar
& ( hoasinduction_lem3_lthm
<=> ( ! [X290: term] :
( ( sub @ X290 @ id )
= X290 )
=> ( ! [X291: term,X292: subst > term > subst > $o] :
( ~ ! [X293: subst > term > term] :
( ~ ! [X294: term] :
( ( X292 @ id @ ( X293 @ id @ X294 ) @ id )
| ~ ( X292 @ id @ X294 @ id ) )
| ~ ! [X295: subst,X296: subst,X297: term] :
( ( X293 @ ( comp @ X296 @ X295 ) @ ( sub @ X297 @ X295 ) )
= ( sub @ ( X293 @ X296 @ X297 ) @ X295 ) )
| ( X292 @ id @ ( lam @ ( X293 @ sh @ one ) ) @ id ) )
| ( X292 @ id @ ( lam @ ( sub @ X291 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X298: term] :
( ( X292 @ id @ ( sub @ X291 @ ( push @ X298 @ id ) ) @ id )
| ~ ( X292 @ id @ X298 @ id ) ) )
=> ! [X299: term,X300: subst > term > subst > $o] :
( ( X300 @ id @ ( lam @ X299 ) @ id )
| ~ ! [X301: subst,X302: subst,X303: subst,X304: term] :
( ~ ( X300 @ ( comp @ X303 @ X301 ) @ ( sub @ X304 @ X301 ) @ X302 )
| ( X300 @ X303 @ X304 @ ( comp @ X301 @ X302 ) ) )
| ~ ! [X305: term] :
( ( X300 @ id @ ( sub @ X299 @ ( push @ X305 @ id ) ) @ id )
| ~ ( X300 @ id @ X305 @ id ) )
| ~ ! [X306: subst > term > term] :
( ~ ! [X307: term,X308: subst,X309: subst] :
( ( X306 @ ( comp @ X308 @ X309 ) @ ( sub @ X307 @ X309 ) )
= ( sub @ ( X306 @ X308 @ X307 ) @ X309 ) )
| ~ ! [X310: term] :
( ~ ( X300 @ id @ X310 @ id )
| ( X300 @ id @ ( X306 @ id @ X310 ) @ id ) )
| ( X300 @ id @ ( lam @ ( X306 @ sh @ one ) ) @ id ) )
| ~ ! [X311: term,X312: subst,X313: subst,X314: subst] :
( ~ ( X300 @ X313 @ X311 @ ( comp @ X314 @ X312 ) )
| ( X300 @ ( comp @ X313 @ X314 ) @ ( sub @ X311 @ X314 ) @ X312 ) ) ) ) ) )
& termmset_gthm
& axassoc
& hoasinduction_lem1v2_gthm
& hoasinduction_lem1_lthm
& pushprop_lem2v2_lthm
& ( ! [X315: subst > term > subst > $o,X316: term] :
( ~ ! [X317: term] :
( ~ ( X315 @ id @ X317 @ id )
| ( X315 @ id @ ( sub @ X316 @ ( push @ X317 @ id ) ) @ id ) )
| ~ ! [X318: subst > term > term] :
( ( X315 @ id @ ( lam @ ( X318 @ sh @ one ) ) @ id )
| ~ ! [X319: subst,X320: term,X321: subst] :
( ( sub @ ( X318 @ X319 @ X320 ) @ X321 )
= ( X318 @ ( comp @ X319 @ X321 ) @ ( sub @ X320 @ X321 ) ) )
| ~ ! [X322: term] :
( ~ ( X315 @ id @ X322 @ id )
| ( X315 @ id @ ( X318 @ id @ X322 ) @ id ) ) )
| ( X315 @ id @ ( lam @ X316 ) @ id ) )
<=> hoasinduction_lem3a )
& hoasinduction_lem2v2_gthm
& ( ! [X323: term] :
( ( sub @ X323 @ id )
= X323 )
<=> axvarid )
& ulamvarind
& ( induction2
<=> ! [X324: term,X325: term > $o] :
( ~ ! [X326: term] :
( ( X325 @ X326 )
| ~ ( var @ X326 ) )
| ~ ! [X327: term,X328: term] :
( ~ ( X325 @ X327 )
| ( X325 @ ( ap @ X328 @ X327 ) )
| ~ ( X325 @ X328 ) )
| ( X325 @ X324 )
| ~ ! [X329: term] :
( ~ ! [X330: term] :
( ( X325 @ ( sub @ X329 @ ( push @ X330 @ id ) ) )
| ~ ( X325 @ X330 ) )
| ( X325 @ ( lam @ X329 ) ) ) ) )
& axabs
& ( ( ! [X331: subst > term > subst > $o] :
~ ! [X332: term > $o] :
~ ! [X333: term] :
( ( X331 @ id @ X333 @ id )
= ( X332 @ X333 ) )
=> ( ! [X334: term > $o,X335: term] :
( ~ ! [X336: term] :
( ~ ( var @ X336 )
| ( X334 @ X336 ) )
| ( X334 @ X335 )
| ~ ! [X337: term] :
( ~ ! [X338: term] :
( ~ ( X334 @ X338 )
| ( X334 @ ( sub @ X337 @ ( push @ X338 @ id ) ) ) )
| ( X334 @ ( lam @ X337 ) ) )
| ~ ! [X339: term,X340: term] :
( ~ ( X334 @ X340 )
| ~ ( X334 @ X339 )
| ( X334 @ ( ap @ X340 @ X339 ) ) ) )
=> ( ! [X341: term] :
( ( sub @ X341 @ id )
= X341 )
=> ( ! [X342: subst > term > subst > $o,X343: term,X344: term > $o] :
( ~ ! [X345: term] :
( ( X344 @ ( sub @ X343 @ ( push @ X345 @ id ) ) )
| ~ ( X344 @ X345 ) )
| ~ ! [X346: subst > term > term] :
( ~ ! [X347: term] :
( ~ ( X342 @ id @ X347 @ id )
| ( X342 @ id @ ( X346 @ id @ X347 ) @ id ) )
| ~ ! [X348: subst,X349: term,X350: subst] :
( ( sub @ ( X346 @ X350 @ X349 ) @ X348 )
= ( X346 @ ( comp @ X350 @ X348 ) @ ( sub @ X349 @ X348 ) ) )
| ( X342 @ id @ ( lam @ ( X346 @ sh @ one ) ) @ id ) )
| ( X344 @ ( lam @ X343 ) )
| ~ ! [X351: term] :
( ( X342 @ id @ X351 @ id )
= ( X344 @ X351 ) ) )
=> ! [X352: subst > term > subst > $o,X353: term] :
( ~ ! [X354: term,X355: term] :
( ~ ( X352 @ id @ X355 @ id )
| ~ ( X352 @ id @ X354 @ id )
| ( X352 @ id @ ( ap @ ( sub @ X354 @ id ) @ X355 ) @ id ) )
| ~ ! [X356: subst > term > term] :
( ( X352 @ id @ ( lam @ ( X356 @ sh @ one ) ) @ id )
| ~ ! [X357: subst,X358: subst,X359: term] :
( ( X356 @ ( comp @ X358 @ X357 ) @ ( sub @ X359 @ X357 ) )
= ( sub @ ( X356 @ X358 @ X359 ) @ X357 ) )
| ~ ! [X360: term] :
( ( X352 @ id @ ( X356 @ id @ X360 ) @ id )
| ~ ( X352 @ id @ X360 @ id ) ) )
| ( X352 @ id @ X353 @ id ) ) ) ) ) )
<=> hoasinduction_no_psi_cond_lthm )
& pushprop
& ( ! [X361: term > $o,X362: subst,X363: term] :
( ~ ! [X364: term] :
( ~ ( var @ X364 )
| ( X361 @ ( sub @ X364 @ X362 ) ) )
| ( X361 @ ( sub @ X363 @ X362 ) )
| ~ ! [X365: term] :
( ( X361 @ ( lam @ X365 ) )
| ~ ! [X366: term] :
( ~ ( X361 @ X366 )
| ( X361 @ ( sub @ X365 @ ( push @ X366 @ id ) ) ) ) )
| ~ ! [X367: term,X368: term] :
( ( X361 @ ( ap @ X367 @ X368 ) )
| ~ ( X361 @ X367 )
| ~ ( X361 @ X368 ) ) )
<=> induction2lem )
& ( hoasinduction_lem3v2a_lthm
<=> ( ! [X369: term] :
~ ! [X370: subst > term > term] :
~ ! [X371: subst,X372: term] :
( ( X370 @ X371 @ X372 )
= ( sub @ X369 @ ( push @ X372 @ X371 ) ) )
=> ( ! [X373: term] :
( ( sub @ X373 @ id )
= X373 )
=> ! [X374: term > $o,X375: subst > term > subst > $o,X376: term] :
( ~ ! [X377: term] :
( ( X374 @ X377 )
= ( X375 @ id @ X377 @ id ) )
| ~ ! [X378: term] :
( ( X374 @ ( sub @ X376 @ ( push @ X378 @ id ) ) )
| ~ ( X374 @ X378 ) )
| ( X374 @ ( lam @ X376 ) )
| ~ ! [X379: subst > term > term] :
( ( X375 @ id @ ( lam @ ( X379 @ sh @ one ) ) @ id )
| ~ ! [X380: term] :
( ( X375 @ id @ ( X379 @ id @ X380 ) @ id )
| ~ ( X375 @ id @ X380 @ id ) )
| ~ ! [X381: subst,X382: term,X383: subst] :
( ( sub @ ( X379 @ X381 @ X382 ) @ X383 )
= ( X379 @ ( comp @ X381 @ X383 ) @ ( sub @ X382 @ X383 ) ) ) ) ) ) ) )
& induction2_gthm
& hoaslamnotap_gthm
& ( hoasinduction_lthm
<=> ( ! [X384: term,X385: term > $o] :
( ~ ! [X386: term,X387: term] :
( ~ ( X385 @ X386 )
| ( X385 @ ( ap @ X386 @ X387 ) )
| ~ ( X385 @ X387 ) )
| ( X385 @ X384 )
| ~ ! [X388: term] :
( ( X385 @ ( lam @ X388 ) )
| ~ ! [X389: term] :
( ( X385 @ ( sub @ X388 @ ( push @ X389 @ id ) ) )
| ~ ( X385 @ X389 ) ) )
| ~ ! [X390: term] :
( ( X385 @ X390 )
| ~ ( var @ X390 ) ) )
=> ( ! [X391: term,X392: term,X393: subst > term > subst > $o] :
( ( X393 @ id @ ( ap @ X391 @ X392 ) @ id )
| ~ ( X393 @ id @ X392 @ id )
| ~ ( X393 @ id @ X391 @ id )
| ~ ! [X394: subst,X395: subst,X396: term,X397: subst] :
( ~ ( X393 @ X395 @ X396 @ ( comp @ X394 @ X397 ) )
| ( X393 @ ( comp @ X395 @ X394 ) @ ( sub @ X396 @ X394 ) @ X397 ) )
| ~ ! [X398: term,X399: term] :
( ( X393 @ id @ ( ap @ ( sub @ X399 @ id ) @ X398 ) @ id )
| ~ ( X393 @ id @ X399 @ id )
| ~ ( X393 @ id @ X398 @ id ) )
| ~ ! [X400: subst,X401: subst,X402: subst,X403: term] :
( ~ ( X393 @ ( comp @ X401 @ X402 ) @ ( sub @ X403 @ X402 ) @ X400 )
| ( X393 @ X401 @ X403 @ ( comp @ X402 @ X400 ) ) ) )
=> ( ! [X404: subst > term > subst > $o,X405: term] :
( ( X404 @ id @ ( lam @ X405 ) @ id )
| ~ ! [X406: term] :
( ( X404 @ id @ ( sub @ X405 @ ( push @ X406 @ id ) ) @ id )
| ~ ( X404 @ id @ X406 @ id ) )
| ~ ! [X407: subst,X408: term,X409: subst,X410: subst] :
( ( X404 @ X409 @ X408 @ ( comp @ X407 @ X410 ) )
| ~ ( X404 @ ( comp @ X409 @ X407 ) @ ( sub @ X408 @ X407 ) @ X410 ) )
| ~ ! [X411: subst,X412: term,X413: subst,X414: subst] :
( ~ ( X404 @ X411 @ X412 @ ( comp @ X414 @ X413 ) )
| ( X404 @ ( comp @ X411 @ X414 ) @ ( sub @ X412 @ X414 ) @ X413 ) )
| ~ ! [X415: subst > term > term] :
( ~ ! [X416: term,X417: subst,X418: subst] :
( ( sub @ ( X415 @ X417 @ X416 ) @ X418 )
= ( X415 @ ( comp @ X417 @ X418 ) @ ( sub @ X416 @ X418 ) ) )
| ( X404 @ id @ ( lam @ ( X415 @ sh @ one ) ) @ id )
| ~ ! [X419: term] :
( ~ ( X404 @ id @ X419 @ id )
| ( X404 @ id @ ( X415 @ id @ X419 ) @ id ) ) ) )
=> ! [X420: subst > term > subst > $o,X421: term] :
( ~ ! [X422: term,X423: term] :
( ( X420 @ id @ ( ap @ ( sub @ X423 @ id ) @ X422 ) @ id )
| ~ ( X420 @ id @ X422 @ id )
| ~ ( X420 @ id @ X423 @ id ) )
| ~ ! [X424: subst > term > term] :
( ( X420 @ id @ ( lam @ ( X424 @ sh @ one ) ) @ id )
| ~ ! [X425: term] :
( ( X420 @ id @ ( X424 @ id @ X425 ) @ id )
| ~ ( X420 @ id @ X425 @ id ) )
| ~ ! [X426: term,X427: subst,X428: subst] :
( ( X424 @ ( comp @ X427 @ X428 ) @ ( sub @ X426 @ X428 ) )
= ( sub @ ( X424 @ X427 @ X426 ) @ X428 ) ) )
| ( X420 @ id @ X421 @ id )
| ~ ! [X429: subst,X430: term,X431: subst,X432: subst] :
( ( X420 @ ( comp @ X432 @ X429 ) @ ( sub @ X430 @ X429 ) @ X431 )
| ~ ( X420 @ X432 @ X430 @ ( comp @ X429 @ X431 ) ) )
| ~ ! [X433: term,X434: subst,X435: subst,X436: subst] :
( ( X420 @ X435 @ X433 @ ( comp @ X434 @ X436 ) )
| ~ ( X420 @ ( comp @ X435 @ X434 ) @ ( sub @ X433 @ X434 ) @ X436 ) )
| ~ ! [X437: term] :
( ( X420 @ id @ X437 @ id )
| ~ ( var @ ( sub @ X437 @ id ) ) ) ) ) ) ) )
& ( hoasapinj1
<=> ! [X438: term,X439: term,X440: term,X441: term] :
( ( X440 = X441 )
| ( ( ap @ ( sub @ X441 @ id ) @ X439 )
!= ( ap @ ( sub @ X440 @ id ) @ X438 ) ) ) )
& pushprop_lem1_gthm
& apnotvar
& ( pushprop_p_and_p_prime
= ( ^ [X442: term,X443: subst,X444: term > $o,X445: term > $o] :
! [X446: term] :
( ( X445 @ X446 )
= ( X444 @ ( sub @ X446 @ ( push @ X442 @ X443 ) ) ) ) ) )
& lamnotvar
& ( hoasinduction_lem3aa_lthm
<=> ! [X447: subst > term > subst > $o,X448: term] :
( ~ ! [X449: subst > term > term] :
( ~ ! [X450: term] :
( ( X447 @ id @ ( X449 @ id @ X450 ) @ id )
| ~ ( X447 @ id @ X450 @ id ) )
| ( X447 @ id @ ( lam @ ( X449 @ sh @ one ) ) @ id )
| ~ ! [X451: subst,X452: term,X453: subst] :
( ( sub @ ( X449 @ X451 @ X452 ) @ X453 )
= ( X449 @ ( comp @ X451 @ X453 ) @ ( sub @ X452 @ X453 ) ) ) )
| ( X447 @ id @ ( lam @ ( sub @ X448 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X454: term] :
( ( X447 @ id @ ( sub @ X448 @ ( push @ X454 @ id ) ) @ id )
| ~ ( X447 @ id @ X454 @ id ) ) ) )
& ( ! [X455: subst > term > subst > $o] :
~ ! [X456: term > $o] :
~ ! [X457: term] :
( ( X455 @ id @ X457 @ id )
= ( X456 @ X457 ) )
<=> hoasinduction_lem0_lthm )
& hoasinduction_lem3_gthm
& ( ( ! [X458: subst] :
( ( comp @ id @ X458 )
= X458 )
& ! [X459: subst] :
( ( comp @ X459 @ id )
= X459 ) )
<=> substmonoid )
& ( induction2_lthm
<=> ( ! [X460: term] :
( ( sub @ X460 @ id )
= X460 )
=> ( ! [X461: subst,X462: term,X463: term > $o] :
( ~ ! [X464: term] :
( ~ ( var @ X464 )
| ( X463 @ ( sub @ X464 @ X461 ) ) )
| ~ ! [X465: term,X466: term] :
( ~ ( X463 @ X466 )
| ( X463 @ ( ap @ X465 @ X466 ) )
| ~ ( X463 @ X465 ) )
| ~ ! [X467: term] :
( ~ ! [X468: term] :
( ( X463 @ ( sub @ X467 @ ( push @ X468 @ id ) ) )
| ~ ( X463 @ X468 ) )
| ( X463 @ ( lam @ X467 ) ) )
| ( X463 @ ( sub @ X462 @ X461 ) ) )
=> ! [X469: term > $o,X470: term] :
( ~ ! [X471: term] :
( ~ ( var @ X471 )
| ( X469 @ X471 ) )
| ~ ! [X472: term,X473: term] :
( ~ ( X469 @ X472 )
| ~ ( X469 @ X473 )
| ( X469 @ ( ap @ X472 @ X473 ) ) )
| ( X469 @ X470 )
| ~ ! [X474: term] :
( ( X469 @ ( lam @ X474 ) )
| ~ ! [X475: term] :
( ( X469 @ ( sub @ X474 @ ( push @ X475 @ id ) ) )
| ~ ( X469 @ X475 ) ) ) ) ) ) )
& ~ ( var @ ( d2term @ d_one ) )
& pushprop_lem1v2_gthm
& pushprop_lem0_gthm
& ( hoasinduction_lem2
<=> ! [X476: term,X477: term,X478: subst > term > subst > $o] :
( ( X478 @ id @ ( ap @ X477 @ X476 ) @ id )
| ~ ( X478 @ id @ X477 @ id )
| ~ ! [X479: term,X480: term] :
( ~ ( X478 @ id @ X480 @ id )
| ( X478 @ id @ ( ap @ ( sub @ X480 @ id ) @ X479 ) @ id )
| ~ ( X478 @ id @ X479 @ id ) )
| ~ ! [X481: subst,X482: subst,X483: subst,X484: term] :
( ~ ( X478 @ X483 @ X484 @ ( comp @ X482 @ X481 ) )
| ( X478 @ ( comp @ X483 @ X482 ) @ ( sub @ X484 @ X482 ) @ X481 ) )
| ~ ! [X485: subst,X486: subst,X487: subst,X488: term] :
( ( X478 @ X486 @ X488 @ ( comp @ X485 @ X487 ) )
| ~ ( X478 @ ( comp @ X486 @ X485 ) @ ( sub @ X488 @ X485 ) @ X487 ) )
| ~ ( X478 @ id @ X476 @ id ) ) )
& ( ! [X489: term > $o,X490: term] :
( ~ ! [X491: term] :
( ( X489 @ X491 )
| ~ ( var @ X491 ) )
| ~ ! [X492: term] :
( ~ ( X489 @ X492 )
| ( X489 @ ( lam @ X492 ) ) )
| ( X489 @ X490 )
| ~ ! [X493: term,X494: term] :
( ~ ( X489 @ X494 )
| ( X489 @ ( ap @ X493 @ X494 ) )
| ~ ( X489 @ X493 ) ) )
<=> induction )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X495: d_term] : ( d_one = X495 )
& ( hoasapinj1_lthm
<=> ( ! [X496: term] :
( ( sub @ X496 @ id )
= X496 )
=> ( ! [X497: term,X498: term,X499: term,X500: term] :
( ( ( ap @ X500 @ X499 )
!= ( ap @ X497 @ X498 ) )
| ( X497 = X500 ) )
=> ! [X501: term,X502: term,X503: term,X504: term] :
( ( ( ap @ ( sub @ X503 @ id ) @ X502 )
!= ( ap @ ( sub @ X501 @ id ) @ X504 ) )
| ( X501 = X503 ) ) ) ) )
& hoasinduction_lem1_gthm
& hoasinduction_lem3b_gthm
& axvarshift
& ( ( ! [X505: term] :
( ( sub @ X505 @ id )
= X505 )
=> ! [X506: term > $o,X507: subst > term > subst > $o,X508: term] :
( ~ ! [X509: subst,X510: term,X511: subst,X512: subst] :
( ( X507 @ ( comp @ X511 @ X509 ) @ ( sub @ X510 @ X509 ) @ X512 )
| ~ ( X507 @ X511 @ X510 @ ( comp @ X509 @ X512 ) ) )
| ~ ! [X513: term] :
( ( X506 @ X513 )
= ( X507 @ id @ X513 @ id ) )
| ~ ! [X514: term] :
( ( X506 @ ( sub @ X508 @ ( push @ X514 @ id ) ) )
| ~ ( X506 @ X514 ) )
| ~ ! [X515: subst,X516: term,X517: subst,X518: subst] :
( ~ ( X507 @ ( comp @ X518 @ X517 ) @ ( sub @ X516 @ X517 ) @ X515 )
| ( X507 @ X518 @ X516 @ ( comp @ X517 @ X515 ) ) )
| ( X506 @ ( lam @ X508 ) )
| ~ ! [X519: subst > term > term] :
( ( X507 @ id @ ( lam @ ( X519 @ sh @ one ) ) @ id )
| ~ ! [X520: subst,X521: term,X522: subst] :
( ( X519 @ ( comp @ X522 @ X520 ) @ ( sub @ X521 @ X520 ) )
= ( sub @ ( X519 @ X522 @ X521 ) @ X520 ) )
| ~ ! [X523: term] :
( ~ ( X507 @ id @ X523 @ id )
| ( X507 @ id @ ( X519 @ id @ X523 ) @ id ) ) ) ) )
<=> hoasinduction_lem3v2_lthm )
& ( shinj
<=> ! [X524: term,X525: term] :
( ( ( sub @ X525 @ sh )
!= ( sub @ X524 @ sh ) )
| ( X524 = X525 ) ) )
& ( hoasinduction_lem3
<=> ! [X526: term,X527: subst > term > subst > $o] :
( ~ ! [X528: subst,X529: term,X530: subst,X531: subst] :
( ( X527 @ ( comp @ X531 @ X528 ) @ ( sub @ X529 @ X528 ) @ X530 )
| ~ ( X527 @ X531 @ X529 @ ( comp @ X528 @ X530 ) ) )
| ~ ! [X532: subst > term > term] :
( ( X527 @ id @ ( lam @ ( X532 @ sh @ one ) ) @ id )
| ~ ! [X533: term] :
( ( X527 @ id @ ( X532 @ id @ X533 ) @ id )
| ~ ( X527 @ id @ X533 @ id ) )
| ~ ! [X534: term,X535: subst,X536: subst] :
( ( X532 @ ( comp @ X535 @ X536 ) @ ( sub @ X534 @ X536 ) )
= ( sub @ ( X532 @ X535 @ X534 ) @ X536 ) ) )
| ( X527 @ id @ ( lam @ X526 ) @ id )
| ~ ! [X537: term] :
( ( X527 @ id @ ( sub @ X526 @ ( push @ X537 @ id ) ) @ id )
| ~ ( X527 @ id @ X537 @ id ) )
| ~ ! [X538: subst,X539: subst,X540: subst,X541: term] :
( ~ ( X527 @ ( comp @ X539 @ X540 ) @ ( sub @ X541 @ X540 ) @ X538 )
| ( X527 @ X539 @ X541 @ ( comp @ X540 @ X538 ) ) ) ) )
& substmonoid_gthm
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X542: term,X543: term > $o,X544: subst] :
~ ! [X545: term > $o] :
~ ! [X546: term] :
( ( X543 @ ( sub @ X546 @ ( push @ X542 @ X544 ) ) )
= ( X545 @ X546 ) )
<=> pushprop_lem0_lthm )
& ( hoasinduction_lem3b
<=> ! [X547: term] :
~ ! [X548: subst > term > term] :
( ( X548 @ sh @ one )
!= ( sub @ X547 @ ( push @ one @ sh ) ) ) )
& pushprop_lem2v2_gthm
& ( ! [X549: subst] :
( ( push @ ( sub @ one @ X549 ) @ ( comp @ sh @ X549 ) )
= X549 )
<=> axscons )
& ulamvarsh
& hoasinduction_lem2_lthm ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ! [X19: term,X20: term,X13: term,X15: term] :
( ( X19 = X20 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
<=> apinj2 )
& ( ! [X30: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ X30 @ id )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
<=> hoasinduction_no_psi_cond )
& ! [X6: d_subst,X7: d_subst] :
( ( ( d2subst @ X6 )
= ( d2subst @ X7 ) )
=> ( X6 = X7 ) )
& ( one
= ( d2term @ d_one ) )
& ( hoasinduction_lem3v2a
<=> ! [X10: subst > term > subst > $o,X11: term > $o,X29: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ( X11 @ ( lam @ X29 ) ) ) )
& hoasinduction_lem3a_gthm
& ( termmset_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) ) )
& ( ! [X14: subst,X13: term,X10: term > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
<=> pushprop_lem0 )
& ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
<=> termmset )
& hoasinduction_lem1
& ( substmonoid_lthm
<=> ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) ) ) ) )
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ( pushprop_lem1v2_lthm
<=> ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ! [X13: term,X10: term > $o,X11: term > $o,X14: subst] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
| ~ ( X10 @ X13 )
| ( X11 @ one ) ) ) )
& pushprop_lem2v2
& ~ lamnotap
& ( hoasinduction_lem3aaa
<=> ! [X34: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ ( sub @ X34 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X34 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term,X35: term] :
( ~ ! [X36: subst,X37: term,X38: subst] :
( ( sub @ ( X18 @ X36 @ X37 ) @ X38 )
= ( sub @ ( sub @ X35 @ ( push @ X37 @ X36 ) ) @ X38 ) )
| ~ ! [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 @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) ) ) )
& ( axidl
<=> ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) )
& hoasinduction_gthm
& ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
<=> axshiftcons )
& axapp
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( hoasinduction
<=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ 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 )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ~ ! [X14: subst,X25: subst,X13: term,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term,X26: subst,X14: subst,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ X28 @ id ) ) )
& ~ ulamvar1
& ( ( d2subst @ d_id )
= sh )
& hoaslamnotvar_lthm
& hoaslamnotap_lthm
& ( hoasinduction_lem0
<=> ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
& hoasapinj1_gthm
& ( hoasinduction_lem3v2_f
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) ) )
& induction2lem_gthm
& hoasinduction_lem2_gthm
& ( induction2lem_lthm
<=> ( ! [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 )
=> ( ! [X21: term,X10: term > $o] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ X21 )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) ) )
=> ! [X17: subst,X16: term,X10: term > $o] :
( ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( sub @ X16 @ X17 ) ) ) ) ) ) ) )
& ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
<=> axvarcons )
& hoasapinj2_gthm
& ( axidr
<=> ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 ) )
& ! [X5: d_subst] : ( d_id = X5 )
& ( ! [X18: subst > term > term,X32: subst,X33: term,X31: subst > term > term] :
( ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [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 ) ) ) )
<=> hoaslaminj )
& ( ! [X18: subst > term > term,X46: term,X47: term] :
( ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( ap @ ( sub @ X46 @ id ) @ X47 ) ) )
<=> hoaslamnotap )
& ( apinj1
<=> ! [X20: term,X19: term,X15: term,X13: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X13 = X15 ) ) )
& hoasinduction_lem3v2_gthm
& ( hoasinduction_lem2v2
<=> ! [X45: term,X11: term > $o,X10: subst > term > subst > $o,X44: term] :
( ~ ( X11 @ X44 )
| ( X11 @ ( ap @ X44 @ X45 ) )
| ~ ! [X13: term,X26: subst,X14: subst,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ( X11 @ X45 )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) ) ) )
& pushprop_lthm
& hoaslaminj_gthm
& hoasapnotvar_lthm
& ( pushprop_lem1v2
<=> ! [X10: term > $o,X14: subst,X11: term > $o,X13: term] :
( ( X11 @ one )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
| ~ ( X10 @ X13 ) ) )
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
<=> hoasinduction_lem3v2_f_lthm )
& ! [X2: d_term,X3: d_term] :
( ( ( d2term @ X2 )
= ( d2term @ X3 ) )
=> ( X2 = X3 ) )
& pushprop_lthm_orig
& ( ! [X20: term,X15: term,X19: term,X13: term] :
( ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) )
| ( X19 = X20 ) )
<=> hoasapinj2 )
& hoasinduction_lem1v2
& hoaslamnotvar_gthm
& ( hoaslaminj_lthm
<=> ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X15: term,X13: term] :
( ( X13 = X15 )
| ( ( lam @ X13 )
!= ( lam @ X15 ) ) )
=> ! [X32: subst,X18: subst > term > term,X33: term,X31: subst > term > term] :
( ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) )
| ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) ) ) )
& ( ( ^ [X8: subst,X9: subst > term > term] : ( d2term @ d_one ) )
= hoaslam )
& ( pushprop_lem1
<=> ! [X10: term > $o,X13: term,X14: subst,X15: term,X26: term > $o] :
( ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) )
| ~ ( X10 @ X13 ) ) )
& pushprop_lem3v2_lthm
& ( hoasinduction_lem3v2
<=> ! [X11: term > $o,X10: subst > term > subst > $o,X48: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ 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 ) ) ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X13: term,X14: subst,X26: subst,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X11 @ ( lam @ X48 ) ) ) )
& ( laminj
<=> ! [X13: term,X15: term] :
( ( ( lam @ X13 )
!= ( lam @ X15 ) )
| ( X13 = X15 ) ) )
& ! [X0: term] :
? [X1: d_term] :
( ( d2term @ X1 )
= X0 )
& pushprop_lem3v2
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) )
<=> hoasinduction_lem3b_lthm )
& hoasapnotvar_gthm
& axmap
& axclos
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& hoaslamnotvar
& ( pushprop_lem1_lthm
<=> ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ! [X15: term,X10: term > $o,X26: term > $o,X14: subst,X13: term] :
( ~ ( X10 @ X13 )
| ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) ) ) ) ) )
& ( hoasapinj2_lthm
<=> ( ! [X13: term,X20: term,X15: term,X19: term] :
( ( X19 = X20 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
=> ! [X13: term,X15: term,X19: term,X20: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3aa
<=> ! [X10: subst > term > subst > $o,X42: term] :
( ~ ! [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 ) ) ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) ) )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lthm_3
<=> ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X10: term > $o,X22: term] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ 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 @ X15 )
| ~ ( X10 @ X13 ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X29: term,X11: term > $o,X10: subst > term > subst > $o] :
( ~ ! [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 @ 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 ) ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X29 ) ) )
=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ! [X14: subst,X13: term,X26: subst,X25: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [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 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X10 @ id @ X28 @ id )
| ~ ! [X13: term,X14: subst,X26: subst,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) ) ) ) ) ) ) )
& ( ( ^ [X10: subst > term > subst > $o,X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
= hoasinduction_p_and_p_prime )
& ! [X4: subst] :
? [X5: d_subst] :
( ( d2subst @ X5 )
= X4 )
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) )
=> ! [X43: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( lam @ X43 ) @ id ) ) ) )
<=> hoasinduction_lem3a_lthm )
& pushprop_gthm
& hoasapnotvar
& ( hoasinduction_lem3_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) )
=> ! [X27: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X26: subst,X25: subst,X14: 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 ) )
| ~ ! [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 ) )
| ~ ! [X13: term,X25: subst,X14: subst,X26: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) ) ) ) ) )
& termmset_gthm
& axassoc
& hoasinduction_lem1v2_gthm
& hoasinduction_lem1_lthm
& pushprop_lem2v2_lthm
& ( ! [X10: subst > term > subst > $o,X43: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [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 ) )
<=> hoasinduction_lem3a )
& hoasinduction_lem2v2_gthm
& ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
<=> axvarid )
& ulamvarind
& ( induction2
<=> ! [X22: term,X10: term > $o] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) ) ) )
& axabs
& ( ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X10: term > $o,X22: term] :
( ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: subst > term > subst > $o,X29: term,X11: term > $o] :
( ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [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 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [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 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ( X10 @ id @ X30 @ id ) ) ) ) ) )
<=> hoasinduction_no_psi_cond_lthm )
& pushprop
& ( ! [X10: term > $o,X17: subst,X16: term] :
( ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) ) )
<=> induction2lem )
& ( hoasinduction_lem3v2a_lthm
<=> ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X11: term > $o,X10: subst > term > subst > $o,X29: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) ) ) )
& induction2_gthm
& hoaslamnotap_gthm
& ( hoasinduction_lthm
<=> ( ! [X22: term,X10: term > $o] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) ) )
=> ( ! [X23: term,X24: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ( X10 @ id @ 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 ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X25: subst,X14: subst,X26: subst,X13: term] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) )
=> ( ! [X10: subst > term > subst > $o,X27: term] :
( ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X26: subst,X13: term,X14: subst,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X14: subst,X13: term,X25: 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 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) ) )
=> ! [X10: subst > term > subst > $o,X28: term] :
( ~ ! [X15: term,X13: 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 @ ( 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 ) ) ) )
| ( X10 @ id @ X28 @ id )
| ~ ! [X26: subst,X13: term,X25: subst,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term,X26: subst,X14: subst,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term] :
( ( X10 @ id @ X13 @ id )
| ~ ( var @ ( sub @ X13 @ id ) ) ) ) ) ) ) )
& ( hoasapinj1
<=> ! [X19: term,X20: term,X13: term,X15: term] :
( ( X13 = X15 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) )
& pushprop_lem1_gthm
& apnotvar
& ( 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 ) ) ) ) ) )
& lamnotvar
& ( hoasinduction_lem3aa_lthm
<=> ! [X10: subst > term > subst > $o,X42: term] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) ) )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0_lthm )
& hoasinduction_lem3_gthm
& ( ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
& ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 ) )
<=> substmonoid )
& ( induction2_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X17: subst,X16: term,X10: term > $o] :
( ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) ) )
=> ! [X10: term > $o,X22: term] :
( ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) ) ) ) ) ) )
& ~ ( var @ ( d2term @ d_one ) )
& pushprop_lem1v2_gthm
& pushprop_lem0_gthm
& ( hoasinduction_lem2
<=> ! [X24: term,X23: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ( X10 @ id @ X23 @ id )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X25: subst,X26: subst,X14: 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 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ( X10 @ id @ X24 @ id ) ) )
& ( ! [X10: term > $o,X21: term] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ X21 )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) ) )
<=> induction )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X1: d_term] : ( d_one = X1 )
& ( hoasapinj1_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X15: term,X20: term,X19: term,X13: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X13 = X15 ) )
=> ! [X13: term,X20: term,X15: term,X19: term] :
( ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) )
| ( X13 = X15 ) ) ) ) )
& hoasinduction_lem1_gthm
& hoasinduction_lem3b_gthm
& axvarshift
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X11: term > $o,X10: subst > term > subst > $o,X48: term] :
( ~ ! [X26: subst,X13: term,X14: subst,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X25: subst,X13: term,X26: subst,X14: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) ) ) )
<=> hoasinduction_lem3v2_lthm )
& ( shinj
<=> ! [X15: term,X13: term] :
( ( ( sub @ X13 @ sh )
!= ( sub @ X15 @ sh ) )
| ( X13 = X15 ) ) )
& ( hoasinduction_lem3
<=> ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X26: subst,X13: term,X25: subst,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [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 ) ) ) )
| ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X25: subst,X14: subst,X26: subst,X13: term] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) ) )
& substmonoid_gthm
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X13: term,X10: term > $o,X14: subst] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
<=> pushprop_lem0_lthm )
& ( hoasinduction_lem3b
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) ) )
& pushprop_lem2v2_gthm
& ( ! [X14: subst] :
( ( push @ ( sub @ one @ X14 ) @ ( comp @ sh @ X14 ) )
= X14 )
<=> axscons )
& ulamvarsh
& hoasinduction_lem2_lthm ),
file('/export/starexec/sandbox2/tmp/tmp.ae6moAB9X6/DTF2THF_11154.p',alg444_1) ).
thf(f1046,plain,
! [X248: term] :
( ( d2term @ ( sK579 @ X248 ) )
= X248 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1711,plain,
( $false
= ( sK592 @ ( sub @ sK596 @ ( push @ sK593 @ sK594 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1710]) ).
thf(f1710,plain,
( $false
= ( ( var @ sK596 )
=> ( sK592 @ ( sub @ sK596 @ ( push @ sK593 @ sK594 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1709]) ).
thf(f1709,plain,
( $false
= ( ^ [Y0: term] :
( ( var @ Y0 )
=> ( sK592 @ ( sub @ Y0 @ ( push @ sK593 @ sK594 ) ) ) )
@ sK596 ) ),
inference(sigma_clausification,[status(thm)],[f1704]) ).
thf(f1704,plain,
( $false
= ( !! @ term
@ ^ [Y0: term] :
( ( var @ Y0 )
=> ( sK592 @ ( sub @ Y0 @ ( push @ sK593 @ sK594 ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1697]) ).
thf(f1697,plain,
( $false
= ( ( !! @ term
@ ^ [Y0: term] :
( ( var @ Y0 )
=> ( sK595 @ Y0 ) ) )
=> ( !! @ term
@ ^ [Y0: term] :
( ( var @ Y0 )
=> ( sK592 @ ( sub @ Y0 @ ( push @ sK593 @ sK594 ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1696]) ).
thf(f1696,plain,
( ( ( !! @ term
@ ^ [Y0: term] :
( ( sK595 @ Y0 )
= ( sK592 @ ( sub @ Y0 @ ( push @ sK593 @ sK594 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y0: term] :
( ( var @ Y0 )
=> ( sK595 @ Y0 ) ) )
=> ( !! @ term
@ ^ [Y0: term] :
( ( var @ Y0 )
=> ( sK592 @ ( sub @ Y0 @ ( push @ sK593 @ sK594 ) ) ) ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f1695]) ).
thf(f1695,plain,
( $false
= ( ^ [Y0: term > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( ( Y0 @ Y1 )
= ( sK592 @ ( sub @ Y1 @ ( push @ sK593 @ sK594 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( var @ Y1 )
=> ( Y0 @ Y1 ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( var @ Y1 )
=> ( sK592 @ ( sub @ Y1 @ ( push @ sK593 @ sK594 ) ) ) ) ) ) )
@ sK595 ) ),
inference(sigma_clausification,[status(thm)],[f1694]) ).
thf(f1694,plain,
( $false
= ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( ( !! @ term
@ ^ [Y1: term] :
( ( Y0 @ Y1 )
= ( sK592 @ ( sub @ Y1 @ ( push @ sK593 @ sK594 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( var @ Y1 )
=> ( Y0 @ Y1 ) ) )
=> ( !! @ term
@ ^ [Y1: term] :
( ( var @ Y1 )
=> ( sK592 @ ( sub @ Y1 @ ( push @ sK593 @ sK594 ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1693]) ).
thf(f1693,plain,
( $false
= ( ^ [Y0: subst] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( sK592 @ ( sub @ Y2 @ ( push @ sK593 @ Y0 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( var @ Y2 )
=> ( Y1 @ Y2 ) ) )
=> ( !! @ term
@ ^ [Y2: term] :
( ( var @ Y2 )
=> ( sK592 @ ( sub @ Y2 @ ( push @ sK593 @ Y0 ) ) ) ) ) ) ) )
@ sK594 ) ),
inference(sigma_clausification,[status(thm)],[f1692]) ).
thf(f1692,plain,
( $false
= ( !! @ subst
@ ^ [Y0: subst] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( ( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( sK592 @ ( sub @ Y2 @ ( push @ sK593 @ Y0 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( var @ Y2 )
=> ( Y1 @ Y2 ) ) )
=> ( !! @ term
@ ^ [Y2: term] :
( ( var @ Y2 )
=> ( sK592 @ ( sub @ Y2 @ ( push @ sK593 @ Y0 ) ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1691]) ).
thf(f1691,plain,
( $false
= ( ^ [Y0: term] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( ( !! @ term
@ ^ [Y3: term] :
( ( Y2 @ Y3 )
= ( sK592 @ ( sub @ Y3 @ ( push @ Y0 @ Y1 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y3: term] :
( ( var @ Y3 )
=> ( Y2 @ Y3 ) ) )
=> ( !! @ term
@ ^ [Y3: term] :
( ( var @ Y3 )
=> ( sK592 @ ( sub @ Y3 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) ) ) )
@ sK593 ) ),
inference(sigma_clausification,[status(thm)],[f1690]) ).
thf(f1690,plain,
( $false
= ( !! @ term
@ ^ [Y0: term] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( ( !! @ term
@ ^ [Y3: term] :
( ( Y2 @ Y3 )
= ( sK592 @ ( sub @ Y3 @ ( push @ Y0 @ Y1 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y3: term] :
( ( var @ Y3 )
=> ( Y2 @ Y3 ) ) )
=> ( !! @ term
@ ^ [Y3: term] :
( ( var @ Y3 )
=> ( sK592 @ ( sub @ Y3 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1689]) ).
thf(f1689,plain,
( $false
= ( ^ [Y0: term > $o] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ ( term > $o )
@ ^ [Y3: term > $o] :
( ( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y0 @ ( sub @ Y4 @ ( push @ Y1 @ Y2 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y3 @ Y4 ) ) )
=> ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y0 @ ( sub @ Y4 @ ( push @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) )
@ sK592 ) ),
inference(sigma_clausification,[status(thm)],[f1688]) ).
thf(f1688,plain,
( $true
!= ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ ( term > $o )
@ ^ [Y3: term > $o] :
( ( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y0 @ ( sub @ Y4 @ ( push @ Y1 @ Y2 ) ) ) ) )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y3 @ Y4 ) ) )
=> ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y0 @ ( sub @ Y4 @ ( push @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1663]) ).
thf(f1663,plain,
( $true
!= ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ ( term > $o )
@ ^ [Y3: term > $o] :
( ( ^ [Y4: term,Y5: subst,Y6: term > $o,Y7: term > $o] :
( !! @ term
@ ^ [Y8: term] :
( ( Y7 @ Y8 )
= ( Y6 @ ( sub @ Y8 @ ( push @ Y4 @ Y5 ) ) ) ) )
@ Y1
@ Y2
@ Y0
@ Y3 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y3 @ Y4 ) ) )
=> ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y0 @ ( sub @ Y4 @ ( push @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1260,f1174,f1251]) ).
thf(f1251,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 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1174,plain,
pushprop_lem3v2 = $true,
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1260,plain,
( pushprop_lem3v2
!= ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ ( term > $o )
@ ^ [Y3: term > $o] :
( ( pushprop_p_and_p_prime @ Y1 @ Y2 @ Y0 @ Y3 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y3 @ Y4 ) ) )
=> ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y0 @ ( sub @ Y4 @ ( push @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( pushprop_lem3v2
!= ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ ( term > $o )
@ ^ [Y3: term > $o] :
( ( pushprop_p_and_p_prime @ Y1 @ Y2 @ Y0 @ Y3 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y3 @ Y4 ) ) )
=> ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y0 @ ( sub @ Y4 @ ( push @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f8]) ).
thf(f8,plain,
( pushprop_lem3v2
!= ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ term
@ ^ [Y1: term] :
( !! @ subst
@ ^ [Y2: subst] :
( !! @ ( term > $o )
@ ^ [Y3: term > $o] :
( ( pushprop_p_and_p_prime @ Y1 @ Y2 @ Y0 @ Y3 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y3 @ Y4 ) ) )
=> ( !! @ term
@ ^ [Y4: term] :
( ( var @ Y4 )
=> ( Y0 @ ( sub @ Y4 @ ( push @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ! [X0: term > $o,X1: subst,X2: term,X3: term > $o] :
( ( ( pushprop_p_and_p_prime @ X2 @ X1 @ X3 @ X0 )
=> ( ! [X4: term] :
( ( var @ X4 )
=> ( X0 @ X4 ) )
=> ! [X5: term] :
( ( var @ X5 )
=> ( X3 @ ( sub @ X5 @ ( push @ X2 @ X1 ) ) ) ) ) )
= pushprop_lem3v2 ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
~ ! [X11: term > $o,X14: subst,X13: term,X10: term > $o] :
( ( ( pushprop_p_and_p_prime @ X13 @ X14 @ X10 @ X11 )
=> ( ! [X15: term] :
( ( var @ X15 )
=> ( X11 @ X15 ) )
=> ! [X15: term] :
( ( var @ X15 )
=> ( X10 @ ( sub @ X15 @ ( push @ X13 @ X14 ) ) ) ) ) )
= pushprop_lem3v2 ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
! [X11: term > $o,X14: subst,X13: term,X10: term > $o] :
( ( ( pushprop_p_and_p_prime @ X13 @ X14 @ X10 @ X11 )
=> ( ! [X15: term] :
( ( var @ X15 )
=> ( X11 @ X15 ) )
=> ! [X15: term] :
( ( var @ X15 )
=> ( X10 @ ( sub @ X15 @ ( push @ X13 @ X14 ) ) ) ) ) )
= pushprop_lem3v2 ),
file('/export/starexec/sandbox2/tmp/tmp.ae6moAB9X6/DTF2THF_11154.p',pushprop_lem3v2) ).
thf(f4406,plain,
! [X1: term] :
( ( $false
= ( sK595 @ X1 ) )
| ( $true
= ( sK592 @ ( d2term @ d_one ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1703,f2761]) ).
thf(f1703,plain,
! [X1: term] :
( ( $true
= ( sK592 @ ( sub @ X1 @ ( push @ sK593 @ sK594 ) ) ) )
| ( $false
= ( sK595 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1701]) ).
thf(f1701,plain,
! [X1: term] :
( ( sK595 @ X1 )
= ( sK592 @ ( sub @ X1 @ ( push @ sK593 @ sK594 ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1700]) ).
thf(f1700,plain,
! [X1: term] :
( $true
= ( ( sK595 @ X1 )
= ( sK592 @ ( sub @ X1 @ ( push @ sK593 @ sK594 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1699]) ).
thf(f1699,plain,
! [X1: term] :
( $true
= ( ^ [Y0: term] :
( ( sK595 @ Y0 )
= ( sK592 @ ( sub @ Y0 @ ( push @ sK593 @ sK594 ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1698]) ).
thf(f1698,plain,
( $true
= ( !! @ term
@ ^ [Y0: term] :
( ( sK595 @ Y0 )
= ( sK592 @ ( sub @ Y0 @ ( push @ sK593 @ sK594 ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1696]) ).
thf(f1708,plain,
! [X1: term] :
( ( $true
= ( sK595 @ X1 ) )
| ( $false
= ( var @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1707]) ).
thf(f1707,plain,
! [X1: term] :
( ( ( var @ X1 )
=> ( sK595 @ X1 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1706]) ).
thf(f1706,plain,
! [X1: term] :
( $true
= ( ^ [Y0: term] :
( ( var @ Y0 )
=> ( sK595 @ Y0 ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1705]) ).
thf(f1705,plain,
( $true
= ( !! @ term
@ ^ [Y0: term] :
( ( var @ Y0 )
=> ( sK595 @ Y0 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1697]) ).
thf(f1712,plain,
( $true
= ( var @ sK596 ) ),
inference(binary_proxy_clausification,[status(thm)],[f1710]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n007.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue May 5 09:00:23 EDT 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.47 thf(spec,logic,$$dhol).
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % File : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.19/0.47 % Domain : Software Verification
% 0.19/0.47 % Problem : Benchmark: ALG444^1, Active target: pushprop_lem3v2
% 0.19/0.47 % Version : Especial.
% 0.19/0.47 % English :
% 0.19/0.47
% 0.19/0.47 % Refs : [Kon26] Kondylidou (2026), Email to Geoff Sutcliffe
% 0.19/0.47 % Source : [Kon26]
% 0.19/0.47 % Names : ALG444^1__005__axclos.verify [Kon26]
% 0.19/0.47
% 0.19/0.47 % Status : Theorem
% 0.19/0.47 % Rating : ? v9.3.0
% 0.19/0.47 % Syntax : Number of formulae : 132 ( 1 unt; 130 typ; 0 def)
% 0.19/0.47 % Number of atoms : 257 ( 128 equ; 0 cnn)
% 0.19/0.47 % Maximal formula atoms : 248 ( 128 avg)
% 0.19/0.47 % Number of connectives : 2585 ( 333 ~; 306 |; 129 &;1719 @)
% 0.19/0.47 % ( 56 <=>; 42 =>; 0 <=; 0 <~>)
% 0.19/0.47 % Maximal formula depth : 151 ( 76 avg)
% 0.19/0.47 % Number of types : 5 ( 4 usr)
% 0.19/0.47 % Number of type conns : 224 ( 224 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 127 ( 126 usr; 113 con; 0-4 aty)
% 0.19/0.47 % Number of variables : 556 ( 8 ^; 546 !; 2 ?; 556 :)
% 0.19/0.47 % SPC : TH0_THM_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments :
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 thf(term_type,type,
% 0.19/0.47 term: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(subst_type,type,
% 0.19/0.47 subst: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(one_decl,type,
% 0.19/0.47 one: term ).
% 0.19/0.47
% 0.19/0.47 thf(ap_decl,type,
% 0.19/0.47 ap: term > term > term ).
% 0.19/0.47
% 0.19/0.47 thf(lam_decl,type,
% 0.19/0.47 lam: term > term ).
% 0.19/0.47
% 0.19/0.47 thf(sub_decl,type,
% 0.19/0.47 sub: term > subst > term ).
% 0.19/0.47
% 0.19/0.47 thf(id_decl,type,
% 0.19/0.47 id: subst ).
% 0.19/0.47
% 0.19/0.47 thf(sh_decl,type,
% 0.19/0.47 sh: subst ).
% 0.19/0.47
% 0.19/0.47 thf(push_decl,type,
% 0.19/0.47 push: term > subst > subst ).
% 0.19/0.47
% 0.19/0.47 thf(comp_decl,type,
% 0.19/0.47 comp: subst > subst > subst ).
% 0.19/0.47
% 0.19/0.47 thf(var_decl,type,
% 0.19/0.47 var: term > $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem1v2_decl,type,
% 0.19/0.47 pushprop_lem1v2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem1_gthm_decl,type,
% 0.19/0.47 pushprop_lem1_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axmap_decl,type,
% 0.19/0.47 axmap: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem0_gthm_decl,type,
% 0.19/0.47 pushprop_lem0_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(shinj_decl,type,
% 0.19/0.47 shinj: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem1v2_decl,type,
% 0.19/0.47 hoasinduction_lem1v2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem1v2_gthm_decl,type,
% 0.19/0.47 hoasinduction_lem1v2_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasap_decl,type,
% 0.19/0.47 hoasap: subst > term > subst > term > term ).
% 0.19/0.47
% 0.19/0.47 thf(induction2lem_decl,type,
% 0.19/0.47 induction2lem: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3v2_f_decl,type,
% 0.19/0.47 hoasinduction_lem3v2_f: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axvarshift_decl,type,
% 0.19/0.47 axvarshift: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapinj2_decl,type,
% 0.19/0.47 hoasapinj2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapnotvar_gthm_decl,type,
% 0.19/0.47 hoasapnotvar_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapinj1_decl,type,
% 0.19/0.47 hoasapinj1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ulamvar1_decl,type,
% 0.19/0.47 ulamvar1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(induction2lem_lthm_decl,type,
% 0.19/0.47 induction2lem_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3v2_gthm_decl,type,
% 0.19/0.47 hoasinduction_lem3v2_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(apnotvar_decl,type,
% 0.19/0.47 apnotvar: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lthm_orig_decl,type,
% 0.19/0.47 pushprop_lthm_orig: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem3v2_f_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lthm_decl,type,
% 0.19/0.47 hoasinduction_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 0.19/0.47 hoasinduction_no_psi_cond_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslaminj_decl,type,
% 0.19/0.47 hoaslaminj: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3aaa_decl,type,
% 0.19/0.47 hoasinduction_lem3aaa: $o ).
% 0.19/0.47
% 0.19/0.47 thf(induction2lem_gthm_decl,type,
% 0.19/0.47 induction2lem_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3aa_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem3aa_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3_decl,type,
% 0.19/0.47 hoasinduction_lem3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem2_decl,type,
% 0.19/0.47 hoasinduction_lem2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(termmset_lthm_decl,type,
% 0.19/0.47 termmset_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem1_decl,type,
% 0.19/0.47 hoasinduction_lem1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslamnotap_lthm_decl,type,
% 0.19/0.47 hoaslamnotap_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem1v2_lthm_decl,type,
% 0.19/0.47 pushprop_lem1v2_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapnotvar_decl,type,
% 0.19/0.47 hoasapnotvar: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem0_decl,type,
% 0.19/0.47 hoasinduction_lem0: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_decl,type,
% 0.19/0.47 hoasinduction: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_gthm_decl,type,
% 0.19/0.47 hoasinduction_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axapp_decl,type,
% 0.19/0.47 axapp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslamnotvar_lthm_decl,type,
% 0.19/0.47 hoaslamnotvar_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem3v2_lthm_decl,type,
% 0.19/0.47 pushprop_lem3v2_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3b_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem3b_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ulamvarind_decl,type,
% 0.19/0.47 ulamvarind: $o ).
% 0.19/0.47
% 0.19/0.47 thf(induction_decl,type,
% 0.19/0.47 induction: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3a_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem3a_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(termmset_gthm_decl,type,
% 0.19/0.47 termmset_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3aa_decl,type,
% 0.19/0.47 hoasinduction_lem3aa: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem1v2_gthm_decl,type,
% 0.19/0.47 pushprop_lem1v2_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslamnotap_gthm_decl,type,
% 0.19/0.47 hoaslamnotap_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslamnotvar_gthm_decl,type,
% 0.19/0.47 hoaslamnotvar_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3b_gthm_decl,type,
% 0.19/0.47 hoasinduction_lem3b_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem2v2_decl,type,
% 0.19/0.47 pushprop_lem2v2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3a_gthm_decl,type,
% 0.19/0.47 hoasinduction_lem3a_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axclos_decl,type,
% 0.19/0.47 axclos: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axassoc_decl,type,
% 0.19/0.47 axassoc: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem2v2_decl,type,
% 0.19/0.47 hoasinduction_lem2v2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lthm_decl,type,
% 0.19/0.47 pushprop_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(apinj2_decl,type,
% 0.19/0.47 apinj2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(apinj1_decl,type,
% 0.19/0.47 apinj1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapinj2_lthm_decl,type,
% 0.19/0.47 hoasapinj2_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3v2a_decl,type,
% 0.19/0.47 hoasinduction_lem3v2a: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapinj1_lthm_decl,type,
% 0.19/0.47 hoasapinj1_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslaminj_lthm_decl,type,
% 0.19/0.47 hoaslaminj_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axvarcons_decl,type,
% 0.19/0.47 axvarcons: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslam_decl,type,
% 0.19/0.47 hoaslam: subst > ( subst > term > term ) > term ).
% 0.19/0.47
% 0.19/0.47 thf(axscons_decl,type,
% 0.19/0.47 axscons: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem2v2_gthm_decl,type,
% 0.19/0.47 hoasinduction_lem2v2_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axidr_decl,type,
% 0.19/0.47 axidr: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem1_decl,type,
% 0.19/0.47 pushprop_lem1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(laminj_decl,type,
% 0.19/0.47 laminj: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem3_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem0_decl,type,
% 0.19/0.47 pushprop_lem0: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_gthm_decl,type,
% 0.19/0.47 pushprop_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axabs_decl,type,
% 0.19/0.47 axabs: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem3v2a_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem2_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem2_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapinj2_gthm_decl,type,
% 0.19/0.47 hoasapinj2_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_p_and_p_prime_decl,type,
% 0.19/0.47 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem1_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem1_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(lamnotap_decl,type,
% 0.19/0.47 lamnotap: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapinj1_gthm_decl,type,
% 0.19/0.47 hoasapinj1_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslamnotvar_decl,type,
% 0.19/0.47 hoaslamnotvar: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axidl_decl,type,
% 0.19/0.47 axidl: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslaminj_gthm_decl,type,
% 0.19/0.47 hoaslaminj_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(induction2_lthm_decl,type,
% 0.19/0.47 induction2_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem0_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem0_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(substmonoid_lthm_decl,type,
% 0.19/0.47 substmonoid_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_decl,type,
% 0.19/0.47 pushprop: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3_gthm_decl,type,
% 0.19/0.47 hoasinduction_lem3_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem2_gthm_decl,type,
% 0.19/0.47 hoasinduction_lem2_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3b_decl,type,
% 0.19/0.47 hoasinduction_lem3b: $o ).
% 0.19/0.47
% 0.19/0.47 thf(substmonoid_decl,type,
% 0.19/0.47 substmonoid: $o ).
% 0.19/0.47
% 0.19/0.47 thf(lamnotvar_decl,type,
% 0.19/0.47 lamnotvar: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3a_decl,type,
% 0.19/0.47 hoasinduction_lem3a: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem1_gthm_decl,type,
% 0.19/0.47 hoasinduction_lem1_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_no_psi_cond_decl,type,
% 0.19/0.47 hoasinduction_no_psi_cond: $o ).
% 0.19/0.47
% 0.19/0.47 thf(induction2_gthm_decl,type,
% 0.19/0.47 induction2_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem2v2_lthm_decl,type,
% 0.19/0.47 pushprop_lem2v2_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasvar_decl,type,
% 0.19/0.47 hoasvar: subst > term > subst > $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoaslamnotap_decl,type,
% 0.19/0.47 hoaslamnotap: $o ).
% 0.19/0.47
% 0.19/0.47 thf(substmonoid_gthm_decl,type,
% 0.19/0.47 substmonoid_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ulamvarsh_decl,type,
% 0.19/0.47 ulamvarsh: $o ).
% 0.19/0.47
% 0.19/0.47 thf(induction2_decl,type,
% 0.19/0.47 induction2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem3v2_decl,type,
% 0.19/0.47 pushprop_lem3v2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem2v2_gthm_decl,type,
% 0.19/0.47 pushprop_lem2v2_gthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem1_lthm_decl,type,
% 0.19/0.47 pushprop_lem1_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3v2_decl,type,
% 0.19/0.47 hoasinduction_lem3v2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(axshiftcons_decl,type,
% 0.19/0.47 axshiftcons: $o ).
% 0.19/0.47
% 0.19/0.47 thf(termmset_decl,type,
% 0.19/0.47 termmset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem0_lthm_decl,type,
% 0.19/0.47 pushprop_lem0_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasapnotvar_lthm_decl,type,
% 0.19/0.47 hoasapnotvar_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lem3v2_lthm_decl,type,
% 0.19/0.47 hoasinduction_lem3v2_lthm: $o ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_p_and_p_prime_decl,type,
% 0.19/0.47 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 0.19/0.47
% 0.19/0.47 thf(axvarid_decl,type,
% 0.19/0.47 axvarid: $o ).
% 0.19/0.47
% 0.19/0.47 thf(hoasinduction_lthm_3_decl,type,
% 0.19/0.47 hoasinduction_lthm_3: $o ).
% 0.19/0.47
% 0.19/0.47 %----Types of the domains
% 0.19/0.47 thf(d_term_type,type,
% 0.19/0.47 d_term: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(d_subst_type,type,
% 0.19/0.47 d_subst: $tType ).
% 0.19/0.47
% 0.19/0.47 %----Types of the promotion functions
% 0.19/0.47 thf(d2term_decl,type,
% 0.19/0.47 d2term: d_term > term ).
% 0.19/0.47
% 0.19/0.47 thf(d2subst_decl,type,
% 0.19/0.47 d2subst: d_subst > subst ).
% 0.19/0.47
% 0.19/0.47 %----Types of the domain elements
% 0.19/0.47 thf(d_one_decl,type,
% 0.19/0.47 d_one: d_term ).
% 0.19/0.47
% 0.19/0.47 thf(d_id_decl,type,
% 0.19/0.47 d_id: d_subst ).
% 0.19/0.47
% 0.19/0.47 thf(alg444_1,axiom,
% 0.19/0.47 ( ! [T: term] :
% 0.19/0.47 ? [DT: d_term] :
% 0.19/0.47 ( T
% 0.19/0.47 = ( d2term @ DT ) )
% 0.19/0.47 & ! [DT: d_term] : ( DT = d_one )
% 0.19/0.47 & ! [DT1: d_term,DT2: d_term] :
% 0.19/0.47 ( ( ( d2term @ DT1 )
% 0.19/0.47 = ( d2term @ DT2 ) )
% 0.19/0.47 => ( DT1 = DT2 ) )
% 0.19/0.47 & ! [S: subst] :
% 0.19/0.47 ? [DS: d_subst] :
% 0.19/0.47 ( S
% 0.19/0.47 = ( d2subst @ DS ) )
% 0.19/0.47 & ! [DS: d_subst] : ( DS = d_id )
% 0.19/0.47 & ! [DS1: d_subst,DS2: d_subst] :
% 0.19/0.47 ( ( ( d2subst @ DS1 )
% 0.19/0.47 = ( d2subst @ DS2 ) )
% 0.19/0.47 => ( DS1 = DS2 ) )
% 0.19/0.47 & ( one
% 0.19/0.47 = ( d2term @ d_one ) )
% 0.19/0.47 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 0.19/0.47 = ( d2term @ d_one ) )
% 0.19/0.47 & ( ( lam @ ( d2term @ d_one ) )
% 0.19/0.47 = ( d2term @ d_one ) )
% 0.19/0.47 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.19/0.47 = ( d2term @ d_one ) )
% 0.19/0.47 & ( id
% 0.19/0.47 = ( d2subst @ d_id ) )
% 0.19/0.47 & ( sh
% 0.19/0.47 = ( d2subst @ d_id ) )
% 0.19/0.47 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.19/0.47 = ( d2subst @ d_id ) )
% 0.19/0.47 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 0.19/0.47 = ( d2subst @ d_id ) )
% 0.19/0.47 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 0.19/0.47 = ( d2term @ d_one ) )
% 0.19/0.47 & ( hoaslam
% 0.19/0.47 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 0.19/0.47 & ( hoasinduction_p_and_p_prime
% 0.19/0.47 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 0.19/0.47 ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) ) ) )
% 0.19/0.47 & ( pushprop_p_and_p_prime
% 0.19/0.47 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 0.19/0.47 ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 0.19/0.47 & ~ ( var @ ( d2term @ d_one ) )
% 0.19/0.47 & ( pushprop_lem1v2
% 0.19/0.47 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.19/0.47 | ( Q @ one ) ) )
% 0.19/0.47 & pushprop_lem1_gthm
% 0.19/0.47 & axmap
% 0.19/0.47 & pushprop_lem0_gthm
% 0.19/0.47 & ( shinj
% 0.19/0.47 <=> ! [A: term,B: term] :
% 0.19/0.47 ( ( ( sub @ A @ sh )
% 0.19/0.47 != ( sub @ B @ sh ) )
% 0.19/0.47 | ( A = B ) ) )
% 0.19/0.47 & hoasinduction_lem1v2
% 0.19/0.47 & hoasinduction_lem1v2_gthm
% 0.19/0.47 & ( induction2lem
% 0.19/0.47 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.19/0.47 ( ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ B )
% 0.19/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( var @ B )
% 0.19/0.47 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.19/0.47 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 0.19/0.47 & ( hoasinduction_lem3v2_f
% 0.19/0.47 <=> ! [B: term] :
% 0.19/0.47 ~ ! [F: subst > term > term] :
% 0.19/0.47 ~ ! [A: term,M: subst] :
% 0.19/0.47 ( ( sub @ B @ ( push @ A @ M ) )
% 0.19/0.47 = ( F @ M @ A ) ) )
% 0.19/0.47 & axvarshift
% 0.19/0.47 & ( hoasapinj2
% 0.19/0.47 <=> ! [A: term,B: term,C: term,D: term] :
% 0.19/0.47 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.19/0.47 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.19/0.47 | ( C = D ) ) )
% 0.19/0.47 & hoasapnotvar_gthm
% 0.19/0.47 & ( hoasapinj1
% 0.19/0.47 <=> ! [A: term,B: term,C: term,D: term] :
% 0.19/0.47 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.19/0.47 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.19/0.47 | ( A = B ) ) )
% 0.19/0.47 & ~ ulamvar1
% 0.19/0.47 & ( induction2lem_lthm
% 0.19/0.47 <=> ( ! [A: term,M: subst] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.19/0.47 => ( ! [A: term,M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.19/0.47 => ( ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ M @ id ) )
% 0.19/0.47 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 0.19/0.47 ( ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ A )
% 0.19/0.47 | ( P @ A ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ( P @ Bound_variable_2140 ) )
% 0.19/0.47 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.19/0.47 ( ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ B )
% 0.19/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( var @ B )
% 0.19/0.47 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.19/0.47 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 0.19/0.47 & hoasinduction_lem3v2_gthm
% 0.19/0.47 & apnotvar
% 0.19/0.47 & pushprop_lthm_orig
% 0.19/0.47 & ( hoasinduction_lem3v2_f_lthm
% 0.19/0.47 <=> ! [B: term] :
% 0.19/0.47 ~ ! [F: subst > term > term] :
% 0.19/0.47 ~ ! [A: term,M: subst] :
% 0.19/0.47 ( ( sub @ B @ ( push @ A @ M ) )
% 0.19/0.47 = ( F @ M @ A ) ) )
% 0.19/0.47 & ( hoasinduction_lthm
% 0.19/0.47 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.19/0.47 ( ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ A )
% 0.19/0.47 | ( P @ A ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ B )
% 0.19/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ( P @ Bound_variable_2367 ) )
% 0.19/0.47 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.19/0.47 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.19/0.47 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 0.19/0.47 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 0.19/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ ( sub @ A @ id ) )
% 0.19/0.47 | ( P @ id @ A @ id ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 0.19/0.47 & ( hoasinduction_no_psi_cond_lthm
% 0.19/0.47 <=> ( ! [P: subst > term > subst > $o] :
% 0.19/0.47 ~ ! [Q: term > $o] :
% 0.19/0.47 ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.19/0.47 ( ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ A )
% 0.19/0.47 | ( P @ A ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ B )
% 0.19/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ( P @ Bound_variable_2367 ) )
% 0.19/0.47 => ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( Q @ B )
% 0.19/0.47 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.19/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.19/0.47 ( ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 0.19/0.47 & ( hoaslaminj
% 0.19/0.47 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.19/0.47 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ( ( lam @ ( F @ sh @ one ) )
% 0.19/0.47 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.19/0.47 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.19/0.47 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 0.19/0.47 & ( hoasinduction_lem3aaa
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 0.19/0.47 ( ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 0.19/0.47 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 0.19/0.47 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 0.19/0.47 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 0.19/0.47 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 0.19/0.47 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 0.19/0.47 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.19/0.47 & induction2lem_gthm
% 0.19/0.47 & ( hoasinduction_lem3aa_lthm
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.19/0.47 & ( hoasinduction_lem3
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 0.19/0.47 & ( hoasinduction_lem2
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.19/0.47 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.19/0.47 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 0.19/0.47 & ( termmset_lthm
% 0.19/0.47 <=> ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) ) ) )
% 0.19/0.47 & hoasinduction_lem1
% 0.19/0.47 & hoaslamnotap_lthm
% 0.19/0.47 & ( pushprop_lem1v2_lthm
% 0.19/0.47 <=> ( ! [A: term,M: subst] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.19/0.47 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.19/0.47 | ( Q @ one ) ) ) )
% 0.19/0.47 & hoasapnotvar
% 0.19/0.47 & ( hoasinduction_lem0
% 0.19/0.47 <=> ! [P: subst > term > subst > $o] :
% 0.19/0.47 ~ ! [Q: term > $o] :
% 0.19/0.47 ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) ) )
% 0.19/0.47 & ( hoasinduction
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ ( sub @ A @ id ) )
% 0.19/0.47 | ( P @ id @ A @ id ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 0.19/0.47 & hoasinduction_gthm
% 0.19/0.47 & axapp
% 0.19/0.47 & hoaslamnotvar_lthm
% 0.19/0.47 & pushprop_lem3v2_lthm
% 0.19/0.47 & ( hoasinduction_lem3b_lthm
% 0.19/0.47 <=> ! [B: term] :
% 0.19/0.47 ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ( F @ sh @ one )
% 0.19/0.47 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.19/0.47 & ulamvarind
% 0.19/0.47 & ( induction
% 0.19/0.47 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 0.19/0.47 ( ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ A )
% 0.19/0.47 | ( P @ A ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ( P @ Bound_variable_2140 ) ) )
% 0.19/0.47 & ( hoasinduction_lem3a_lthm
% 0.19/0.47 <=> ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.19/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 0.19/0.47 & termmset_gthm
% 0.19/0.47 & ( hoasinduction_lem3aa
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.19/0.47 & pushprop_lem1v2_gthm
% 0.19/0.47 & hoaslamnotap_gthm
% 0.19/0.47 & hoaslamnotvar_gthm
% 0.19/0.47 & hoasinduction_lem3b_gthm
% 0.19/0.47 & pushprop_lem2v2
% 0.19/0.47 & hoasinduction_lem3a_gthm
% 0.19/0.47 & axclos
% 0.19/0.47 & axassoc
% 0.19/0.47 & ( hoasinduction_lem2v2
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 | ~ ( Q @ Bound_variable_2820 )
% 0.19/0.47 | ~ ( Q @ Bound_variable_2822 )
% 0.19/0.47 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 0.19/0.47 & pushprop_lthm
% 0.19/0.47 & ( apinj2
% 0.19/0.47 <=> ! [A: term,B: term,C: term,D: term] :
% 0.19/0.47 ( ( ( ap @ A @ C )
% 0.19/0.47 != ( ap @ B @ D ) )
% 0.19/0.47 | ( C = D ) ) )
% 0.19/0.47 & ( apinj1
% 0.19/0.47 <=> ! [A: term,B: term,C: term,D: term] :
% 0.19/0.47 ( ( ( ap @ A @ C )
% 0.19/0.47 != ( ap @ B @ D ) )
% 0.19/0.47 | ( A = B ) ) )
% 0.19/0.47 & ( hoasapinj2_lthm
% 0.19/0.47 <=> ( ! [A: term,B: term,C: term,D: term] :
% 0.19/0.47 ( ( ( ap @ A @ C )
% 0.19/0.47 != ( ap @ B @ D ) )
% 0.19/0.47 | ( C = D ) )
% 0.19/0.47 => ! [A: term,B: term,C: term,D: term] :
% 0.19/0.47 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.19/0.47 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.19/0.47 | ( C = D ) ) ) )
% 0.19/0.47 & ( hoasinduction_lem3v2a
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( Q @ B )
% 0.19/0.47 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 0.19/0.47 & ( hoasapinj1_lthm
% 0.19/0.47 <=> ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ( ! [A: term,B: term,C: term,D: term] :
% 0.19/0.47 ( ( ( ap @ A @ C )
% 0.19/0.47 != ( ap @ B @ D ) )
% 0.19/0.47 | ( A = B ) )
% 0.19/0.47 => ! [A: term,B: term,C: term,D: term] :
% 0.19/0.47 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.19/0.47 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.19/0.47 | ( A = B ) ) ) ) )
% 0.19/0.47 & ( hoaslaminj_lthm
% 0.19/0.47 <=> ( ! [A: term,M: subst] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.19/0.47 => ( ! [A: term,M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.19/0.47 => ( ! [A: term,B: term] :
% 0.19/0.47 ( ( ( lam @ A )
% 0.19/0.47 != ( lam @ B ) )
% 0.19/0.47 | ( A = B ) )
% 0.19/0.47 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.19/0.47 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ( ( lam @ ( F @ sh @ one ) )
% 0.19/0.47 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.19/0.47 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.19/0.47 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 0.19/0.47 & ( axvarcons
% 0.19/0.47 <=> ! [A: term,M: subst] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 0.19/0.47 & ( axscons
% 0.19/0.47 <=> ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 0.19/0.47 & hoasinduction_lem2v2_gthm
% 0.19/0.47 & ( axidr
% 0.19/0.47 <=> ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ M @ id ) ) )
% 0.19/0.47 & ( pushprop_lem1
% 0.19/0.47 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 0.19/0.47 & ( laminj
% 0.19/0.47 <=> ! [A: term,B: term] :
% 0.19/0.47 ( ( ( lam @ A )
% 0.19/0.47 != ( lam @ B ) )
% 0.19/0.47 | ( A = B ) ) )
% 0.19/0.47 & ( hoasinduction_lem3_lthm
% 0.19/0.47 <=> ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.19/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 0.19/0.47 & ( pushprop_lem0
% 0.19/0.47 <=> ! [P: term > $o,A: term,M: subst] :
% 0.19/0.47 ~ ! [Q: term > $o] :
% 0.19/0.47 ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.19/0.47 & pushprop_gthm
% 0.19/0.47 & axabs
% 0.19/0.47 & ( hoasinduction_lem3v2a_lthm
% 0.19/0.47 <=> ( ! [B: term] :
% 0.19/0.47 ~ ! [F: subst > term > term] :
% 0.19/0.47 ~ ! [A: term,M: subst] :
% 0.19/0.47 ( ( sub @ B @ ( push @ A @ M ) )
% 0.19/0.47 = ( F @ M @ A ) )
% 0.19/0.47 => ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( Q @ B )
% 0.19/0.47 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 0.19/0.47 & hoasinduction_lem2_lthm
% 0.19/0.47 & hoasapinj2_gthm
% 0.19/0.47 & hoasinduction_lem1_lthm
% 0.19/0.47 & ~ lamnotap
% 0.19/0.47 & hoasapinj1_gthm
% 0.19/0.47 & hoaslamnotvar
% 0.19/0.47 & ( axidl
% 0.19/0.47 <=> ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ id @ M ) ) )
% 0.19/0.47 & hoaslaminj_gthm
% 0.19/0.47 & ( induction2_lthm
% 0.19/0.47 <=> ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.19/0.47 ( ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ B )
% 0.19/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( var @ B )
% 0.19/0.47 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.19/0.47 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 0.19/0.47 => ! [P: term > $o,Bound_variable_2367: term] :
% 0.19/0.47 ( ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ A )
% 0.19/0.47 | ( P @ A ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ B )
% 0.19/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ( P @ Bound_variable_2367 ) ) ) ) )
% 0.19/0.47 & ( hoasinduction_lem0_lthm
% 0.19/0.47 <=> ! [P: subst > term > subst > $o] :
% 0.19/0.47 ~ ! [Q: term > $o] :
% 0.19/0.47 ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) ) )
% 0.19/0.47 & ( substmonoid_lthm
% 0.19/0.47 <=> ( ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ id @ M ) )
% 0.19/0.47 => ( ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ M @ id ) )
% 0.19/0.47 => ( ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ id @ M ) )
% 0.19/0.47 & ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ M @ id ) ) ) ) ) )
% 0.19/0.47 & pushprop
% 0.19/0.47 & hoasinduction_lem3_gthm
% 0.19/0.47 & hoasinduction_lem2_gthm
% 0.19/0.47 & ( hoasinduction_lem3b
% 0.19/0.47 <=> ! [B: term] :
% 0.19/0.47 ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ( F @ sh @ one )
% 0.19/0.47 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.19/0.47 & ( substmonoid
% 0.19/0.47 <=> ( ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ id @ M ) )
% 0.19/0.47 & ! [M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ M @ id ) ) ) )
% 0.19/0.47 & lamnotvar
% 0.19/0.47 & ( hoasinduction_lem3a
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 0.19/0.47 & hoasinduction_lem1_gthm
% 0.19/0.47 & ( hoasinduction_no_psi_cond
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.19/0.47 ( ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 0.19/0.47 & induction2_gthm
% 0.19/0.47 & pushprop_lem2v2_lthm
% 0.19/0.47 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.19/0.47 & ( hoaslamnotap
% 0.19/0.47 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ( ( lam @ ( F @ sh @ one ) )
% 0.19/0.47 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 0.19/0.47 & substmonoid_gthm
% 0.19/0.47 & ulamvarsh
% 0.19/0.47 & ( induction2
% 0.19/0.47 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 0.19/0.47 ( ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ A )
% 0.19/0.47 | ( P @ A ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ B )
% 0.19/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ( P @ Bound_variable_2367 ) ) )
% 0.19/0.47 & pushprop_lem3v2
% 0.19/0.47 & pushprop_lem2v2_gthm
% 0.19/0.47 & ( pushprop_lem1_lthm
% 0.19/0.47 <=> ( ! [A: term,M: subst] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.19/0.47 => ( ! [A: term,M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.19/0.47 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 0.19/0.47 & ( hoasinduction_lem3v2
% 0.19/0.47 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( Q @ B )
% 0.19/0.47 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 0.19/0.47 & ( axshiftcons
% 0.19/0.47 <=> ! [A: term,M: subst] :
% 0.19/0.47 ( M
% 0.19/0.47 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 0.19/0.47 & ( termmset
% 0.19/0.47 <=> ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) ) )
% 0.19/0.47 & ( pushprop_lem0_lthm
% 0.19/0.47 <=> ! [P: term > $o,A: term,M: subst] :
% 0.19/0.47 ~ ! [Q: term > $o] :
% 0.19/0.47 ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.19/0.47 & hoasapnotvar_lthm
% 0.19/0.47 & ( hoasinduction_lem3v2_lthm
% 0.19/0.47 <=> ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( Q @ B )
% 0.19/0.47 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 0.19/0.47 & ( axvarid
% 0.19/0.47 <=> ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) ) )
% 0.19/0.47 & ( hoasinduction_lthm_3
% 0.19/0.47 <=> ( ! [P: subst > term > subst > $o] :
% 0.19/0.47 ~ ! [Q: term > $o] :
% 0.19/0.47 ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.19/0.47 ( ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ A )
% 0.19/0.47 | ( P @ A ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ A )
% 0.19/0.47 | ~ ( P @ B )
% 0.19/0.47 | ( P @ ( ap @ A @ B ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ! [B: term] :
% 0.19/0.47 ( ~ ( P @ B )
% 0.19/0.47 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( P @ ( lam @ A ) ) )
% 0.19/0.47 | ( P @ Bound_variable_2367 ) )
% 0.19/0.47 => ( ! [A: term] :
% 0.19/0.47 ( A
% 0.19/0.47 = ( sub @ A @ id ) )
% 0.19/0.47 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.19/0.47 ( ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ~ ! [X: term] :
% 0.19/0.47 ( ( Q @ X )
% 0.19/0.47 = ( P @ id @ X @ id ) )
% 0.19/0.47 | ~ ! [B: term] :
% 0.19/0.47 ( ~ ( Q @ B )
% 0.19/0.47 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.19/0.47 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.19/0.47 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.19/0.47 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.19/0.47 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.19/0.47 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.19/0.47 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( var @ ( sub @ A @ id ) )
% 0.19/0.47 | ( P @ id @ A @ id ) )
% 0.19/0.47 | ~ ! [A: term,B: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ~ ( P @ id @ B @ id )
% 0.19/0.47 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.19/0.47 | ~ ! [F: subst > term > term] :
% 0.19/0.47 ( ~ ! [M: subst,A: term,N: subst] :
% 0.19/0.47 ( ( sub @ ( F @ M @ A ) @ N )
% 0.19/0.47 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.19/0.47 | ~ ! [A: term] :
% 0.19/0.47 ( ~ ( P @ id @ A @ id )
% 0.19/0.47 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.19/0.47 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.19/0.47 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(pushprop_lem3v2,conjecture,
% 0.19/0.47 ( pushprop_lem3v2
% 0.19/0.47 = ( ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.19/0.47 ( ( pushprop_p_and_p_prime @ A @ M @ P @ Q )
% 0.19/0.47 => ( ! [B: term] :
% 0.19/0.47 ( ( var @ B )
% 0.19/0.47 => ( Q @ B ) )
% 0.19/0.47 => ! [B: term] :
% 0.19/0.47 ( ( var @ B )
% 0.19/0.47 => ( P @ ( sub @ B @ ( push @ A @ M ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 2.35/1.47 ---- Embedded in THF ---
% 2.35/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.35/1.47 %%% Generated on Tue May 05 09:00:24 EDT 2026
% 2.35/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.35/1.47 %%% Logic specification used:
% 2.35/1.47 %%% thf(spec, logic, $$dhol).
% 2.35/1.47
% 2.35/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ae6moAB9X6/DTF2DTF11154.p
% See solution above
% 2.35/1.47 ------------------------
% 2.35/1.50 ---- Cleaned THF ---
% 2.56/1.50 thf(term_type,type,
% 2.56/1.50 term: $tType ).
% 2.56/1.50
% 2.56/1.50 thf(subst_type,type,
% 2.56/1.50 subst: $tType ).
% 2.56/1.50
% 2.56/1.50 thf(one_decl,type,
% 2.56/1.50 one: term ).
% 2.56/1.50
% 2.56/1.50 thf(ap_decl,type,
% 2.56/1.50 ap: term > term > term ).
% 2.56/1.50
% 2.56/1.50 thf(lam_decl,type,
% 2.56/1.50 lam: term > term ).
% 2.56/1.50
% 2.56/1.50 thf(sub_decl,type,
% 2.56/1.50 sub: term > subst > term ).
% 2.56/1.50
% 2.56/1.50 thf(id_decl,type,
% 2.56/1.50 id: subst ).
% 2.56/1.50
% 2.56/1.50 thf(sh_decl,type,
% 2.56/1.50 sh: subst ).
% 2.56/1.50
% 2.56/1.50 thf(push_decl,type,
% 2.56/1.50 push: term > subst > subst ).
% 2.56/1.50
% 2.56/1.50 thf(comp_decl,type,
% 2.56/1.50 comp: subst > subst > subst ).
% 2.56/1.50
% 2.56/1.50 thf(var_decl,type,
% 2.56/1.50 var: term > $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem1v2_decl,type,
% 2.56/1.50 pushprop_lem1v2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem1_gthm_decl,type,
% 2.56/1.50 pushprop_lem1_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axmap_decl,type,
% 2.56/1.50 axmap: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem0_gthm_decl,type,
% 2.56/1.50 pushprop_lem0_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(shinj_decl,type,
% 2.56/1.50 shinj: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem1v2_decl,type,
% 2.56/1.50 hoasinduction_lem1v2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem1v2_gthm_decl,type,
% 2.56/1.50 hoasinduction_lem1v2_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasap_decl,type,
% 2.56/1.50 hoasap: subst > term > subst > term > term ).
% 2.56/1.50
% 2.56/1.50 thf(induction2lem_decl,type,
% 2.56/1.50 induction2lem: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3v2_f_decl,type,
% 2.56/1.50 hoasinduction_lem3v2_f: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axvarshift_decl,type,
% 2.56/1.50 axvarshift: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapinj2_decl,type,
% 2.56/1.50 hoasapinj2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapnotvar_gthm_decl,type,
% 2.56/1.50 hoasapnotvar_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapinj1_decl,type,
% 2.56/1.50 hoasapinj1: $o ).
% 2.56/1.50
% 2.56/1.50 thf(ulamvar1_decl,type,
% 2.56/1.50 ulamvar1: $o ).
% 2.56/1.50
% 2.56/1.50 thf(induction2lem_lthm_decl,type,
% 2.56/1.50 induction2lem_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3v2_gthm_decl,type,
% 2.56/1.50 hoasinduction_lem3v2_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(apnotvar_decl,type,
% 2.56/1.50 apnotvar: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lthm_orig_decl,type,
% 2.56/1.50 pushprop_lthm_orig: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem3v2_f_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lthm_decl,type,
% 2.56/1.50 hoasinduction_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 2.56/1.50 hoasinduction_no_psi_cond_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslaminj_decl,type,
% 2.56/1.50 hoaslaminj: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3aaa_decl,type,
% 2.56/1.50 hoasinduction_lem3aaa: $o ).
% 2.56/1.50
% 2.56/1.50 thf(induction2lem_gthm_decl,type,
% 2.56/1.50 induction2lem_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3aa_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem3aa_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3_decl,type,
% 2.56/1.50 hoasinduction_lem3: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem2_decl,type,
% 2.56/1.50 hoasinduction_lem2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(termmset_lthm_decl,type,
% 2.56/1.50 termmset_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem1_decl,type,
% 2.56/1.50 hoasinduction_lem1: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslamnotap_lthm_decl,type,
% 2.56/1.50 hoaslamnotap_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem1v2_lthm_decl,type,
% 2.56/1.50 pushprop_lem1v2_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapnotvar_decl,type,
% 2.56/1.50 hoasapnotvar: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem0_decl,type,
% 2.56/1.50 hoasinduction_lem0: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_decl,type,
% 2.56/1.50 hoasinduction: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_gthm_decl,type,
% 2.56/1.50 hoasinduction_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axapp_decl,type,
% 2.56/1.50 axapp: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslamnotvar_lthm_decl,type,
% 2.56/1.50 hoaslamnotvar_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem3v2_lthm_decl,type,
% 2.56/1.50 pushprop_lem3v2_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3b_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem3b_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(ulamvarind_decl,type,
% 2.56/1.50 ulamvarind: $o ).
% 2.56/1.50
% 2.56/1.50 thf(induction_decl,type,
% 2.56/1.50 induction: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3a_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem3a_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(termmset_gthm_decl,type,
% 2.56/1.50 termmset_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3aa_decl,type,
% 2.56/1.50 hoasinduction_lem3aa: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem1v2_gthm_decl,type,
% 2.56/1.50 pushprop_lem1v2_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslamnotap_gthm_decl,type,
% 2.56/1.50 hoaslamnotap_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslamnotvar_gthm_decl,type,
% 2.56/1.50 hoaslamnotvar_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3b_gthm_decl,type,
% 2.56/1.50 hoasinduction_lem3b_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem2v2_decl,type,
% 2.56/1.50 pushprop_lem2v2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3a_gthm_decl,type,
% 2.56/1.50 hoasinduction_lem3a_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axclos_decl,type,
% 2.56/1.50 axclos: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axassoc_decl,type,
% 2.56/1.50 axassoc: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem2v2_decl,type,
% 2.56/1.50 hoasinduction_lem2v2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lthm_decl,type,
% 2.56/1.50 pushprop_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(apinj2_decl,type,
% 2.56/1.50 apinj2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(apinj1_decl,type,
% 2.56/1.50 apinj1: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapinj2_lthm_decl,type,
% 2.56/1.50 hoasapinj2_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3v2a_decl,type,
% 2.56/1.50 hoasinduction_lem3v2a: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapinj1_lthm_decl,type,
% 2.56/1.50 hoasapinj1_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslaminj_lthm_decl,type,
% 2.56/1.50 hoaslaminj_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axvarcons_decl,type,
% 2.56/1.50 axvarcons: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslam_decl,type,
% 2.56/1.50 hoaslam: subst > ( subst > term > term ) > term ).
% 2.56/1.50
% 2.56/1.50 thf(axscons_decl,type,
% 2.56/1.50 axscons: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem2v2_gthm_decl,type,
% 2.56/1.50 hoasinduction_lem2v2_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axidr_decl,type,
% 2.56/1.50 axidr: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem1_decl,type,
% 2.56/1.50 pushprop_lem1: $o ).
% 2.56/1.50
% 2.56/1.50 thf(laminj_decl,type,
% 2.56/1.50 laminj: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem3_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem0_decl,type,
% 2.56/1.50 pushprop_lem0: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_gthm_decl,type,
% 2.56/1.50 pushprop_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axabs_decl,type,
% 2.56/1.50 axabs: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem3v2a_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem2_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem2_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapinj2_gthm_decl,type,
% 2.56/1.50 hoasapinj2_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_p_and_p_prime_decl,type,
% 2.56/1.50 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem1_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem1_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(lamnotap_decl,type,
% 2.56/1.50 lamnotap: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapinj1_gthm_decl,type,
% 2.56/1.50 hoasapinj1_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslamnotvar_decl,type,
% 2.56/1.50 hoaslamnotvar: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axidl_decl,type,
% 2.56/1.50 axidl: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslaminj_gthm_decl,type,
% 2.56/1.50 hoaslaminj_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(induction2_lthm_decl,type,
% 2.56/1.50 induction2_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem0_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem0_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(substmonoid_lthm_decl,type,
% 2.56/1.50 substmonoid_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_decl,type,
% 2.56/1.50 pushprop: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3_gthm_decl,type,
% 2.56/1.50 hoasinduction_lem3_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem2_gthm_decl,type,
% 2.56/1.50 hoasinduction_lem2_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3b_decl,type,
% 2.56/1.50 hoasinduction_lem3b: $o ).
% 2.56/1.50
% 2.56/1.50 thf(substmonoid_decl,type,
% 2.56/1.50 substmonoid: $o ).
% 2.56/1.50
% 2.56/1.50 thf(lamnotvar_decl,type,
% 2.56/1.50 lamnotvar: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3a_decl,type,
% 2.56/1.50 hoasinduction_lem3a: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem1_gthm_decl,type,
% 2.56/1.50 hoasinduction_lem1_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_no_psi_cond_decl,type,
% 2.56/1.50 hoasinduction_no_psi_cond: $o ).
% 2.56/1.50
% 2.56/1.50 thf(induction2_gthm_decl,type,
% 2.56/1.50 induction2_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem2v2_lthm_decl,type,
% 2.56/1.50 pushprop_lem2v2_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasvar_decl,type,
% 2.56/1.50 hoasvar: subst > term > subst > $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoaslamnotap_decl,type,
% 2.56/1.50 hoaslamnotap: $o ).
% 2.56/1.50
% 2.56/1.50 thf(substmonoid_gthm_decl,type,
% 2.56/1.50 substmonoid_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(ulamvarsh_decl,type,
% 2.56/1.50 ulamvarsh: $o ).
% 2.56/1.50
% 2.56/1.50 thf(induction2_decl,type,
% 2.56/1.50 induction2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem3v2_decl,type,
% 2.56/1.50 pushprop_lem3v2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem2v2_gthm_decl,type,
% 2.56/1.50 pushprop_lem2v2_gthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem1_lthm_decl,type,
% 2.56/1.50 pushprop_lem1_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3v2_decl,type,
% 2.56/1.50 hoasinduction_lem3v2: $o ).
% 2.56/1.50
% 2.56/1.50 thf(axshiftcons_decl,type,
% 2.56/1.50 axshiftcons: $o ).
% 2.56/1.50
% 2.56/1.50 thf(termmset_decl,type,
% 2.56/1.50 termmset: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem0_lthm_decl,type,
% 2.56/1.50 pushprop_lem0_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasapnotvar_lthm_decl,type,
% 2.56/1.50 hoasapnotvar_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lem3v2_lthm_decl,type,
% 2.56/1.50 hoasinduction_lem3v2_lthm: $o ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_p_and_p_prime_decl,type,
% 2.56/1.50 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 2.56/1.50
% 2.56/1.50 thf(axvarid_decl,type,
% 2.56/1.50 axvarid: $o ).
% 2.56/1.50
% 2.56/1.50 thf(hoasinduction_lthm_3_decl,type,
% 2.56/1.50 hoasinduction_lthm_3: $o ).
% 2.56/1.50
% 2.56/1.50 thf(d_term_type,type,
% 2.56/1.50 d_term: $tType ).
% 2.56/1.50
% 2.56/1.50 thf(d_subst_type,type,
% 2.56/1.50 d_subst: $tType ).
% 2.56/1.50
% 2.56/1.50 thf(d2term_decl,type,
% 2.56/1.50 d2term: d_term > term ).
% 2.56/1.50
% 2.56/1.50 thf(d2subst_decl,type,
% 2.56/1.50 d2subst: d_subst > subst ).
% 2.56/1.50
% 2.56/1.50 thf(d_one_decl,type,
% 2.56/1.50 d_one: d_term ).
% 2.56/1.50
% 2.56/1.50 thf(d_id_decl,type,
% 2.56/1.50 d_id: d_subst ).
% 2.56/1.50
% 2.56/1.50 thf(alg444_1,axiom,
% 2.56/1.50 ( ! [T: term] :
% 2.56/1.50 ? [DT: d_term] :
% 2.56/1.50 ( T
% 2.56/1.50 = ( d2term @ DT ) )
% 2.56/1.50 & ! [DT: d_term] : ( DT = d_one )
% 2.56/1.50 & ! [DT1: d_term,DT2: d_term] :
% 2.56/1.50 ( ( ( d2term @ DT1 )
% 2.56/1.50 = ( d2term @ DT2 ) )
% 2.56/1.50 => ( DT1 = DT2 ) )
% 2.56/1.50 & ! [S: subst] :
% 2.56/1.50 ? [DS: d_subst] :
% 2.56/1.50 ( S
% 2.56/1.50 = ( d2subst @ DS ) )
% 2.56/1.50 & ! [DS: d_subst] : ( DS = d_id )
% 2.56/1.50 & ! [DS1: d_subst,DS2: d_subst] :
% 2.56/1.50 ( ( ( d2subst @ DS1 )
% 2.56/1.50 = ( d2subst @ DS2 ) )
% 2.56/1.50 => ( DS1 = DS2 ) )
% 2.56/1.50 & ( one
% 2.56/1.50 = ( d2term @ d_one ) )
% 2.56/1.50 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 2.56/1.50 = ( d2term @ d_one ) )
% 2.56/1.50 & ( ( lam @ ( d2term @ d_one ) )
% 2.56/1.50 = ( d2term @ d_one ) )
% 2.56/1.50 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.56/1.50 = ( d2term @ d_one ) )
% 2.56/1.50 & ( id
% 2.56/1.50 = ( d2subst @ d_id ) )
% 2.56/1.50 & ( sh
% 2.56/1.50 = ( d2subst @ d_id ) )
% 2.56/1.50 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.56/1.50 = ( d2subst @ d_id ) )
% 2.56/1.50 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 2.56/1.50 = ( d2subst @ d_id ) )
% 2.56/1.50 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 2.56/1.50 = ( d2term @ d_one ) )
% 2.56/1.50 & ( hoaslam
% 2.56/1.50 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 2.56/1.50 & ( hoasinduction_p_and_p_prime
% 2.56/1.50 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 2.56/1.50 ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) ) ) )
% 2.56/1.50 & ( pushprop_p_and_p_prime
% 2.56/1.50 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 2.56/1.50 ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 2.56/1.50 & ~ ( var @ ( d2term @ d_one ) )
% 2.56/1.50 & ( pushprop_lem1v2
% 2.56/1.50 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.56/1.50 | ( Q @ one ) ) )
% 2.56/1.50 & pushprop_lem1_gthm
% 2.56/1.50 & axmap
% 2.56/1.50 & pushprop_lem0_gthm
% 2.56/1.50 & ( shinj
% 2.56/1.50 <=> ! [A: term,B: term] :
% 2.56/1.50 ( ( ( sub @ A @ sh )
% 2.56/1.50 != ( sub @ B @ sh ) )
% 2.56/1.50 | ( A = B ) ) )
% 2.56/1.50 & hoasinduction_lem1v2
% 2.56/1.50 & hoasinduction_lem1v2_gthm
% 2.56/1.50 & ( induction2lem
% 2.56/1.50 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.56/1.50 ( ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ B )
% 2.56/1.50 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( var @ B )
% 2.56/1.50 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.56/1.50 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 2.56/1.50 & ( hoasinduction_lem3v2_f
% 2.56/1.50 <=> ! [B: term] :
% 2.56/1.50 ~ ! [F: subst > term > term] :
% 2.56/1.50 ~ ! [A: term,M: subst] :
% 2.56/1.50 ( ( sub @ B @ ( push @ A @ M ) )
% 2.56/1.50 = ( F @ M @ A ) ) )
% 2.56/1.50 & axvarshift
% 2.56/1.50 & ( hoasapinj2
% 2.56/1.50 <=> ! [A: term,B: term,C: term,D: term] :
% 2.56/1.50 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.56/1.50 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.56/1.50 | ( C = D ) ) )
% 2.56/1.50 & hoasapnotvar_gthm
% 2.56/1.50 & ( hoasapinj1
% 2.56/1.50 <=> ! [A: term,B: term,C: term,D: term] :
% 2.56/1.50 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.56/1.50 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.56/1.50 | ( A = B ) ) )
% 2.56/1.50 & ~ ulamvar1
% 2.56/1.50 & ( induction2lem_lthm
% 2.56/1.50 <=> ( ! [A: term,M: subst] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.56/1.50 => ( ! [A: term,M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.56/1.50 => ( ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ M @ id ) )
% 2.56/1.50 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 2.56/1.50 ( ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ A )
% 2.56/1.50 | ( P @ A ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ( P @ Bound_variable_2140 ) )
% 2.56/1.50 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.56/1.50 ( ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ B )
% 2.56/1.50 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( var @ B )
% 2.56/1.50 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.56/1.50 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 2.56/1.50 & hoasinduction_lem3v2_gthm
% 2.56/1.50 & apnotvar
% 2.56/1.50 & pushprop_lthm_orig
% 2.56/1.50 & ( hoasinduction_lem3v2_f_lthm
% 2.56/1.50 <=> ! [B: term] :
% 2.56/1.50 ~ ! [F: subst > term > term] :
% 2.56/1.50 ~ ! [A: term,M: subst] :
% 2.56/1.50 ( ( sub @ B @ ( push @ A @ M ) )
% 2.56/1.50 = ( F @ M @ A ) ) )
% 2.56/1.50 & ( hoasinduction_lthm
% 2.56/1.50 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.56/1.50 ( ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ A )
% 2.56/1.50 | ( P @ A ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ B )
% 2.56/1.50 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ( P @ Bound_variable_2367 ) )
% 2.56/1.50 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.56/1.50 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.56/1.50 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 2.56/1.50 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 2.56/1.50 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ ( sub @ A @ id ) )
% 2.56/1.50 | ( P @ id @ A @ id ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 2.56/1.50 & ( hoasinduction_no_psi_cond_lthm
% 2.56/1.50 <=> ( ! [P: subst > term > subst > $o] :
% 2.56/1.50 ~ ! [Q: term > $o] :
% 2.56/1.50 ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.56/1.50 ( ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ A )
% 2.56/1.50 | ( P @ A ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ B )
% 2.56/1.50 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ( P @ Bound_variable_2367 ) )
% 2.56/1.50 => ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( Q @ B )
% 2.56/1.50 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.56/1.50 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.56/1.50 ( ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 2.56/1.50 & ( hoaslaminj
% 2.56/1.50 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.56/1.50 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ( ( lam @ ( F @ sh @ one ) )
% 2.56/1.50 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.56/1.50 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.56/1.50 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 2.56/1.50 & ( hoasinduction_lem3aaa
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 2.56/1.50 ( ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 2.56/1.50 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 2.56/1.50 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 2.56/1.50 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 2.56/1.50 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 2.56/1.50 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 2.56/1.50 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.56/1.50 & induction2lem_gthm
% 2.56/1.50 & ( hoasinduction_lem3aa_lthm
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.56/1.50 & ( hoasinduction_lem3
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 2.56/1.50 & ( hoasinduction_lem2
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.56/1.50 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.56/1.50 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 2.56/1.50 & ( termmset_lthm
% 2.56/1.50 <=> ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) ) ) )
% 2.56/1.50 & hoasinduction_lem1
% 2.56/1.50 & hoaslamnotap_lthm
% 2.56/1.50 & ( pushprop_lem1v2_lthm
% 2.56/1.50 <=> ( ! [A: term,M: subst] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.56/1.50 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.56/1.50 | ( Q @ one ) ) ) )
% 2.56/1.50 & hoasapnotvar
% 2.56/1.50 & ( hoasinduction_lem0
% 2.56/1.50 <=> ! [P: subst > term > subst > $o] :
% 2.56/1.50 ~ ! [Q: term > $o] :
% 2.56/1.50 ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) ) )
% 2.56/1.50 & ( hoasinduction
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ ( sub @ A @ id ) )
% 2.56/1.50 | ( P @ id @ A @ id ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 2.56/1.50 & hoasinduction_gthm
% 2.56/1.50 & axapp
% 2.56/1.50 & hoaslamnotvar_lthm
% 2.56/1.50 & pushprop_lem3v2_lthm
% 2.56/1.50 & ( hoasinduction_lem3b_lthm
% 2.56/1.50 <=> ! [B: term] :
% 2.56/1.50 ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ( F @ sh @ one )
% 2.56/1.50 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.56/1.50 & ulamvarind
% 2.56/1.50 & ( induction
% 2.56/1.50 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 2.56/1.50 ( ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ A )
% 2.56/1.50 | ( P @ A ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ( P @ Bound_variable_2140 ) ) )
% 2.56/1.50 & ( hoasinduction_lem3a_lthm
% 2.56/1.50 <=> ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.56/1.50 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 2.56/1.50 & termmset_gthm
% 2.56/1.50 & ( hoasinduction_lem3aa
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.56/1.50 & pushprop_lem1v2_gthm
% 2.56/1.50 & hoaslamnotap_gthm
% 2.56/1.50 & hoaslamnotvar_gthm
% 2.56/1.50 & hoasinduction_lem3b_gthm
% 2.56/1.50 & pushprop_lem2v2
% 2.56/1.50 & hoasinduction_lem3a_gthm
% 2.56/1.50 & axclos
% 2.56/1.50 & axassoc
% 2.56/1.50 & ( hoasinduction_lem2v2
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 | ~ ( Q @ Bound_variable_2820 )
% 2.56/1.50 | ~ ( Q @ Bound_variable_2822 )
% 2.56/1.50 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 2.56/1.50 & pushprop_lthm
% 2.56/1.50 & ( apinj2
% 2.56/1.50 <=> ! [A: term,B: term,C: term,D: term] :
% 2.56/1.50 ( ( ( ap @ A @ C )
% 2.56/1.50 != ( ap @ B @ D ) )
% 2.56/1.50 | ( C = D ) ) )
% 2.56/1.50 & ( apinj1
% 2.56/1.50 <=> ! [A: term,B: term,C: term,D: term] :
% 2.56/1.50 ( ( ( ap @ A @ C )
% 2.56/1.50 != ( ap @ B @ D ) )
% 2.56/1.50 | ( A = B ) ) )
% 2.56/1.50 & ( hoasapinj2_lthm
% 2.56/1.50 <=> ( ! [A: term,B: term,C: term,D: term] :
% 2.56/1.50 ( ( ( ap @ A @ C )
% 2.56/1.50 != ( ap @ B @ D ) )
% 2.56/1.50 | ( C = D ) )
% 2.56/1.50 => ! [A: term,B: term,C: term,D: term] :
% 2.56/1.50 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.56/1.50 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.56/1.50 | ( C = D ) ) ) )
% 2.56/1.50 & ( hoasinduction_lem3v2a
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( Q @ B )
% 2.56/1.50 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 2.56/1.50 & ( hoasapinj1_lthm
% 2.56/1.50 <=> ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ( ! [A: term,B: term,C: term,D: term] :
% 2.56/1.50 ( ( ( ap @ A @ C )
% 2.56/1.50 != ( ap @ B @ D ) )
% 2.56/1.50 | ( A = B ) )
% 2.56/1.50 => ! [A: term,B: term,C: term,D: term] :
% 2.56/1.50 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.56/1.50 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.56/1.50 | ( A = B ) ) ) ) )
% 2.56/1.50 & ( hoaslaminj_lthm
% 2.56/1.50 <=> ( ! [A: term,M: subst] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.56/1.50 => ( ! [A: term,M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.56/1.50 => ( ! [A: term,B: term] :
% 2.56/1.50 ( ( ( lam @ A )
% 2.56/1.50 != ( lam @ B ) )
% 2.56/1.50 | ( A = B ) )
% 2.56/1.50 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.56/1.50 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ( ( lam @ ( F @ sh @ one ) )
% 2.56/1.50 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.56/1.50 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.56/1.50 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 2.56/1.50 & ( axvarcons
% 2.56/1.50 <=> ! [A: term,M: subst] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 2.56/1.50 & ( axscons
% 2.56/1.50 <=> ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 2.56/1.50 & hoasinduction_lem2v2_gthm
% 2.56/1.50 & ( axidr
% 2.56/1.50 <=> ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ M @ id ) ) )
% 2.56/1.50 & ( pushprop_lem1
% 2.56/1.50 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 2.56/1.50 & ( laminj
% 2.56/1.50 <=> ! [A: term,B: term] :
% 2.56/1.50 ( ( ( lam @ A )
% 2.56/1.50 != ( lam @ B ) )
% 2.56/1.50 | ( A = B ) ) )
% 2.56/1.50 & ( hoasinduction_lem3_lthm
% 2.56/1.50 <=> ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.56/1.50 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 2.56/1.50 & ( pushprop_lem0
% 2.56/1.50 <=> ! [P: term > $o,A: term,M: subst] :
% 2.56/1.50 ~ ! [Q: term > $o] :
% 2.56/1.50 ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.56/1.50 & pushprop_gthm
% 2.56/1.50 & axabs
% 2.56/1.50 & ( hoasinduction_lem3v2a_lthm
% 2.56/1.50 <=> ( ! [B: term] :
% 2.56/1.50 ~ ! [F: subst > term > term] :
% 2.56/1.50 ~ ! [A: term,M: subst] :
% 2.56/1.50 ( ( sub @ B @ ( push @ A @ M ) )
% 2.56/1.50 = ( F @ M @ A ) )
% 2.56/1.50 => ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( Q @ B )
% 2.56/1.50 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 2.56/1.50 & hoasinduction_lem2_lthm
% 2.56/1.50 & hoasapinj2_gthm
% 2.56/1.50 & hoasinduction_lem1_lthm
% 2.56/1.50 & ~ lamnotap
% 2.56/1.50 & hoasapinj1_gthm
% 2.56/1.50 & hoaslamnotvar
% 2.56/1.50 & ( axidl
% 2.56/1.50 <=> ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ id @ M ) ) )
% 2.56/1.50 & hoaslaminj_gthm
% 2.56/1.50 & ( induction2_lthm
% 2.56/1.50 <=> ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.56/1.50 ( ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ B )
% 2.56/1.50 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( var @ B )
% 2.56/1.50 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.56/1.50 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 2.56/1.50 => ! [P: term > $o,Bound_variable_2367: term] :
% 2.56/1.50 ( ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ A )
% 2.56/1.50 | ( P @ A ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ B )
% 2.56/1.50 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ( P @ Bound_variable_2367 ) ) ) ) )
% 2.56/1.50 & ( hoasinduction_lem0_lthm
% 2.56/1.50 <=> ! [P: subst > term > subst > $o] :
% 2.56/1.50 ~ ! [Q: term > $o] :
% 2.56/1.50 ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) ) )
% 2.56/1.50 & ( substmonoid_lthm
% 2.56/1.50 <=> ( ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ id @ M ) )
% 2.56/1.50 => ( ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ M @ id ) )
% 2.56/1.50 => ( ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ id @ M ) )
% 2.56/1.50 & ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ M @ id ) ) ) ) ) )
% 2.56/1.50 & pushprop
% 2.56/1.50 & hoasinduction_lem3_gthm
% 2.56/1.50 & hoasinduction_lem2_gthm
% 2.56/1.50 & ( hoasinduction_lem3b
% 2.56/1.50 <=> ! [B: term] :
% 2.56/1.50 ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ( F @ sh @ one )
% 2.56/1.50 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.56/1.50 & ( substmonoid
% 2.56/1.50 <=> ( ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ id @ M ) )
% 2.56/1.50 & ! [M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ M @ id ) ) ) )
% 2.56/1.50 & lamnotvar
% 2.56/1.50 & ( hoasinduction_lem3a
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 2.56/1.50 & hoasinduction_lem1_gthm
% 2.56/1.50 & ( hoasinduction_no_psi_cond
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.56/1.50 ( ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 2.56/1.50 & induction2_gthm
% 2.56/1.50 & pushprop_lem2v2_lthm
% 2.56/1.50 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.56/1.50 & ( hoaslamnotap
% 2.56/1.50 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ( ( lam @ ( F @ sh @ one ) )
% 2.56/1.50 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 2.56/1.50 & substmonoid_gthm
% 2.56/1.50 & ulamvarsh
% 2.56/1.50 & ( induction2
% 2.56/1.50 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 2.56/1.50 ( ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ A )
% 2.56/1.50 | ( P @ A ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ B )
% 2.56/1.50 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ( P @ Bound_variable_2367 ) ) )
% 2.56/1.50 & pushprop_lem3v2
% 2.56/1.50 & pushprop_lem2v2_gthm
% 2.56/1.50 & ( pushprop_lem1_lthm
% 2.56/1.50 <=> ( ! [A: term,M: subst] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.56/1.50 => ( ! [A: term,M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.56/1.50 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 2.56/1.50 & ( hoasinduction_lem3v2
% 2.56/1.50 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( Q @ B )
% 2.56/1.50 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 2.56/1.50 & ( axshiftcons
% 2.56/1.50 <=> ! [A: term,M: subst] :
% 2.56/1.50 ( M
% 2.56/1.50 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 2.56/1.50 & ( termmset
% 2.56/1.50 <=> ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) ) )
% 2.56/1.50 & ( pushprop_lem0_lthm
% 2.56/1.50 <=> ! [P: term > $o,A: term,M: subst] :
% 2.56/1.50 ~ ! [Q: term > $o] :
% 2.56/1.50 ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.56/1.50 & hoasapnotvar_lthm
% 2.56/1.50 & ( hoasinduction_lem3v2_lthm
% 2.56/1.50 <=> ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( Q @ B )
% 2.56/1.50 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 2.56/1.50 & ( axvarid
% 2.56/1.50 <=> ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) ) )
% 2.56/1.50 & ( hoasinduction_lthm_3
% 2.56/1.50 <=> ( ! [P: subst > term > subst > $o] :
% 2.56/1.50 ~ ! [Q: term > $o] :
% 2.56/1.50 ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.56/1.50 ( ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ A )
% 2.56/1.50 | ( P @ A ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ A )
% 2.56/1.50 | ~ ( P @ B )
% 2.56/1.50 | ( P @ ( ap @ A @ B ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ! [B: term] :
% 2.56/1.50 ( ~ ( P @ B )
% 2.56/1.50 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( P @ ( lam @ A ) ) )
% 2.56/1.50 | ( P @ Bound_variable_2367 ) )
% 2.56/1.50 => ( ! [A: term] :
% 2.56/1.50 ( A
% 2.56/1.50 = ( sub @ A @ id ) )
% 2.56/1.50 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.56/1.50 ( ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ~ ! [X: term] :
% 2.56/1.50 ( ( Q @ X )
% 2.56/1.50 = ( P @ id @ X @ id ) )
% 2.56/1.50 | ~ ! [B: term] :
% 2.56/1.50 ( ~ ( Q @ B )
% 2.56/1.50 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.56/1.50 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.56/1.50 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.56/1.50 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.56/1.50 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.56/1.50 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.56/1.50 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( var @ ( sub @ A @ id ) )
% 2.56/1.50 | ( P @ id @ A @ id ) )
% 2.56/1.50 | ~ ! [A: term,B: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ~ ( P @ id @ B @ id )
% 2.56/1.50 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.56/1.50 | ~ ! [F: subst > term > term] :
% 2.56/1.50 ( ~ ! [M: subst,A: term,N: subst] :
% 2.56/1.50 ( ( sub @ ( F @ M @ A ) @ N )
% 2.56/1.50 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.56/1.50 | ~ ! [A: term] :
% 2.56/1.50 ( ~ ( P @ id @ A @ id )
% 2.56/1.50 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.56/1.50 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.56/1.50 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 2.56/1.50
% 2.56/1.50 thf(pushprop_lem3v2,conjecture,
% 2.56/1.50 ( pushprop_lem3v2
% 2.56/1.50 = ( ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.56/1.50 ( ( pushprop_p_and_p_prime @ A @ M @ P @ Q )
% 2.56/1.50 => ( ! [B: term] :
% 2.56/1.50 ( ( var @ B )
% 2.56/1.50 => ( Q @ B ) )
% 2.56/1.50 => ! [B: term] :
% 2.56/1.50 ( ( var @ B )
% 2.56/1.50 => ( P @ ( sub @ B @ ( push @ A @ M ) ) ) ) ) ) ) ) ).
% 2.56/1.50 ------------------------
% 2.56/1.52 % (11507)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/183Mi)
% 2.56/1.52 % (11514)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/18Mi)
% 2.56/1.52 % (11514)Instruction limit reached!
% 2.56/1.52 % (11514)------------------------------
% 2.56/1.52 % (11514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.52 % (11514)Termination reason: Unknown
% 2.56/1.52 % (11514)Termination phase: shuffling
% 2.56/1.52
% 2.56/1.52 % (11514)Memory used [KB]: 1918
% 2.56/1.52 % (11514)Time elapsed: 0.007 s
% 2.56/1.52 % (11514)Instructions burned: 18 (million)
% 2.56/1.52 % (11514)------------------------------
% 2.56/1.52 % (11514)------------------------------
% 2.56/1.52 % (11509)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_11154 for (2999ds/4Mi)
% 2.56/1.52 % (11510)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_11154 for (2999ds/27Mi)
% 2.56/1.52 % (11512)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_11154 for (2999ds/2Mi)
% 2.56/1.52 % (11511)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/2Mi)
% 2.56/1.52 % (11513)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_11154 for (2999ds/275Mi)
% 2.56/1.52 % (11512)Instruction limit reached!
% 2.56/1.52 % (11512)------------------------------
% 2.56/1.52 % (11512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.52 % (11512)Termination reason: Unknown
% 2.56/1.52 % (11512)Termination phase: shuffling
% 2.56/1.52
% 2.56/1.52 % (11512)Memory used [KB]: 1151
% 2.56/1.52 % (11512)Time elapsed: 0.003 s
% 2.56/1.52 % (11512)Instructions burned: 2 (million)
% 2.56/1.52 % (11512)------------------------------
% 2.56/1.52 % (11512)------------------------------
% 2.56/1.53 % (11511)Instruction limit reached!
% 2.56/1.53 % (11511)------------------------------
% 2.56/1.53 % (11511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (11511)Termination reason: Unknown
% 2.56/1.53 % (11511)Termination phase: shuffling
% 2.56/1.53
% 2.56/1.53 % (11511)Memory used [KB]: 1279
% 2.56/1.53 % (11511)Time elapsed: 0.004 s
% 2.56/1.53 % (11511)Instructions burned: 3 (million)
% 2.56/1.53 % (11511)------------------------------
% 2.56/1.53 % (11511)------------------------------
% 2.56/1.53 % (11509)Instruction limit reached!
% 2.56/1.53 % (11509)------------------------------
% 2.56/1.53 % (11509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (11509)Termination reason: Unknown
% 2.56/1.53 % (11509)Termination phase: shuffling
% 2.56/1.53
% 2.56/1.53 % (11509)Memory used [KB]: 1279
% 2.56/1.53 % (11509)Time elapsed: 0.005 s
% 2.56/1.53 % (11509)Instructions burned: 5 (million)
% 2.56/1.53 % (11509)------------------------------
% 2.56/1.53 % (11509)------------------------------
% 2.56/1.53 % (11515)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/3Mi)
% 2.56/1.53 % (11515)Instruction limit reached!
% 2.56/1.53 % (11515)------------------------------
% 2.56/1.53 % (11515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (11515)Termination reason: Unknown
% 2.56/1.53 % (11515)Termination phase: shuffling
% 2.56/1.53
% 2.56/1.53 % (11515)Memory used [KB]: 1279
% 2.56/1.53 % (11515)Time elapsed: 0.003 s
% 2.56/1.53 % (11515)Instructions burned: 4 (million)
% 2.56/1.53 % (11515)------------------------------
% 2.56/1.53 % (11515)------------------------------
% 2.56/1.54 % (11510)Instruction limit reached!
% 2.56/1.54 % (11510)------------------------------
% 2.56/1.54 % (11510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.54 % (11510)Termination reason: Unknown
% 2.56/1.54 % (11510)Termination phase: Property scanning
% 2.56/1.54
% 2.56/1.54 % (11510)Memory used [KB]: 2174
% 2.56/1.54 % (11510)Time elapsed: 0.018 s
% 2.56/1.54 % (11510)Instructions burned: 27 (million)
% 2.56/1.54 % (11510)------------------------------
% 2.56/1.54 % (11510)------------------------------
% 2.56/1.54 % (11516)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_11154 for (2999ds/37Mi)
% 2.56/1.54 % (11518)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_11154 for (2999ds/3Mi)
% 2.56/1.54 % (11517)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_11154 for (2999ds/15Mi)
% 2.56/1.54 % (11518)Instruction limit reached!
% 2.56/1.54 % (11518)------------------------------
% 2.56/1.54 % (11518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.54 % (11518)Termination reason: Unknown
% 2.56/1.54 % (11518)Termination phase: shuffling
% 2.56/1.54
% 2.56/1.54 % (11518)Memory used [KB]: 1279
% 2.56/1.54 % (11518)Time elapsed: 0.003 s
% 2.56/1.54 % (11518)Instructions burned: 4 (million)
% 2.56/1.54 % (11518)------------------------------
% 2.56/1.54 % (11518)------------------------------
% 2.56/1.54 % (11519)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_11154 for (2999ds/1041Mi)
% 2.56/1.55 % (11513)Refutation not found, incomplete strategy
% 2.56/1.55 % (11513)------------------------------
% 2.56/1.55 % (11513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.55 % (11513)Termination reason: Refutation not found, incomplete strategy
% 2.56/1.55
% 2.56/1.55
% 2.56/1.55 % (11513)Memory used [KB]: 6652
% 2.56/1.55 % (11513)Time elapsed: 0.024 s
% 2.56/1.55 % (11513)Instructions burned: 38 (million)
% 2.56/1.55 % (11513)------------------------------
% 2.56/1.55 % (11513)------------------------------
% 2.56/1.55 % (11517)Instruction limit reached!
% 2.56/1.55 % (11517)------------------------------
% 2.56/1.55 % (11517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.55 % (11517)Termination reason: Unknown
% 2.56/1.55 % (11517)Termination phase: shuffling
% 2.56/1.55
% 2.56/1.55 % (11517)Memory used [KB]: 1918
% 2.56/1.55 % (11517)Time elapsed: 0.011 s
% 2.56/1.55 % (11517)Instructions burned: 16 (million)
% 2.56/1.55 % (11517)------------------------------
% 2.56/1.55 % (11517)------------------------------
% 2.56/1.56 % (11521)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/16Mi)
% 2.56/1.56 % (11520)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/7Mi)
% 2.56/1.56 % (11516)Instruction limit reached!
% 2.56/1.56 % (11516)------------------------------
% 2.56/1.56 % (11516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (11516)Termination reason: Unknown
% 2.56/1.56 % (11516)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (11516)Memory used [KB]: 1918
% 2.56/1.56 % (11516)Time elapsed: 0.020 s
% 2.56/1.56 % (11516)Instructions burned: 37 (million)
% 2.56/1.56 % (11516)------------------------------
% 2.56/1.56 % (11516)------------------------------
% 2.56/1.56 % (11520)Instruction limit reached!
% 2.56/1.56 % (11520)------------------------------
% 2.56/1.56 % (11520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (11520)Termination reason: Unknown
% 2.56/1.56 % (11520)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (11520)Memory used [KB]: 1407
% 2.56/1.56 % (11520)Time elapsed: 0.006 s
% 2.56/1.56 % (11520)Instructions burned: 8 (million)
% 2.56/1.56 % (11520)------------------------------
% 2.56/1.56 % (11520)------------------------------
% 2.56/1.56 % (11521)Instruction limit reached!
% 2.56/1.56 % (11521)------------------------------
% 2.56/1.56 % (11521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (11521)Termination reason: Unknown
% 2.56/1.56 % (11521)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (11521)Memory used [KB]: 1918
% 2.56/1.56 % (11521)Time elapsed: 0.007 s
% 2.56/1.56 % (11521)Instructions burned: 16 (million)
% 2.56/1.56 % (11521)------------------------------
% 2.56/1.56 % (11521)------------------------------
% 2.56/1.56 % (11522)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_11154 for (2999ds/3Mi)
% 2.56/1.56 % (11522)Instruction limit reached!
% 2.56/1.56 % (11522)------------------------------
% 2.56/1.56 % (11522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (11522)Termination reason: Unknown
% 2.56/1.56 % (11522)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (11522)Memory used [KB]: 1279
% 2.56/1.56 % (11522)Time elapsed: 0.003 s
% 2.56/1.56 % (11522)Instructions burned: 3 (million)
% 2.56/1.56 % (11522)------------------------------
% 2.56/1.56 % (11522)------------------------------
% 2.56/1.57 % (11523)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_11154 for (2999ds/3Mi)
% 2.56/1.57 % (11507)Instruction limit reached!
% 2.56/1.57 % (11507)------------------------------
% 2.56/1.57 % (11507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.57 % (11507)Termination reason: Unknown
% 2.56/1.57 % (11507)Termination phase: Property scanning
% 2.56/1.57
% 2.56/1.57 % (11507)Memory used [KB]: 4605
% 2.56/1.57 % (11507)Time elapsed: 0.055 s
% 2.56/1.57 % (11507)Instructions burned: 186 (million)
% 2.56/1.57 % (11507)------------------------------
% 2.56/1.57 % (11507)------------------------------
% 2.56/1.57 % (11525)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/3Mi)
% 2.56/1.57 % (11523)Instruction limit reached!
% 2.56/1.57 % (11523)------------------------------
% 2.56/1.57 % (11523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.57 % (11523)Termination reason: Unknown
% 2.56/1.57 % (11523)Termination phase: shuffling
% 2.56/1.57
% 2.56/1.57 % (11523)Memory used [KB]: 1279
% 2.56/1.57 % (11523)Time elapsed: 0.004 s
% 2.56/1.57 % (11523)Instructions burned: 4 (million)
% 2.56/1.57 % (11523)------------------------------
% 2.56/1.57 % (11523)------------------------------
% 2.56/1.57 % (11525)Instruction limit reached!
% 2.56/1.57 % (11525)------------------------------
% 2.56/1.57 % (11525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.57 % (11525)Termination reason: Unknown
% 2.56/1.57 % (11525)Termination phase: shuffling
% 2.56/1.57
% 2.56/1.57 % (11525)Memory used [KB]: 1279
% 2.56/1.57 % (11525)Time elapsed: 0.003 s
% 2.56/1.57 % (11525)Instructions burned: 4 (million)
% 2.56/1.57 % (11525)------------------------------
% 2.56/1.57 % (11525)------------------------------
% 2.56/1.57 % (11527)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/18Mi)
% 2.56/1.58 % (11524)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/7Mi)
% 2.56/1.58 % (11526)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/4Mi)
% 2.56/1.58 % (11526)Instruction limit reached!
% 2.56/1.58 % (11526)------------------------------
% 2.56/1.58 % (11526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.58 % (11526)Termination reason: Unknown
% 2.56/1.58 % (11526)Termination phase: shuffling
% 2.56/1.58
% 2.56/1.58 % (11526)Memory used [KB]: 1279
% 2.56/1.58 % (11526)Time elapsed: 0.004 s
% 2.56/1.58 % (11526)Instructions burned: 5 (million)
% 2.56/1.58 % (11526)------------------------------
% 2.56/1.58 % (11526)------------------------------
% 2.56/1.58 % (11528)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/710Mi)
% 2.56/1.58 % (11527)Instruction limit reached!
% 2.56/1.58 % (11527)------------------------------
% 2.56/1.58 % (11527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.58 % (11527)Termination reason: Unknown
% 2.56/1.58 % (11527)Termination phase: shuffling
% 2.56/1.58 % (11524)Instruction limit reached!
% 2.56/1.58 % (11524)------------------------------
% 2.56/1.58 % (11524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.58
% 2.56/1.58 % (11527)Memory used [KB]: 2046
% 2.56/1.58 % (11524)Termination reason: Unknown
% 2.56/1.58 % (11527)Time elapsed: 0.008 s
% 2.56/1.58 % (11527)Instructions burned: 20 (million)
% 2.56/1.58 % (11527)------------------------------
% 2.56/1.58 % (11527)------------------------------
% 2.56/1.58 % (11524)Termination phase: shuffling
% 2.56/1.58
% 2.56/1.58 % (11524)Memory used [KB]: 1407
% 2.56/1.58 % (11524)Time elapsed: 0.006 s
% 2.56/1.58 % (11524)Instructions burned: 7 (million)
% 2.56/1.58 % (11524)------------------------------
% 2.56/1.58 % (11524)------------------------------
% 2.56/1.58 % (11530)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_11154 for (2999ds/902Mi)
% 2.56/1.59 % (11529)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/6Mi)
% 2.56/1.59 % (11532)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/5Mi)
% 2.56/1.59 % (11529)Instruction limit reached!
% 2.56/1.59 % (11529)------------------------------
% 2.56/1.59 % (11529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.59 % (11529)Termination reason: Unknown
% 2.56/1.59 % (11529)Termination phase: shuffling
% 2.56/1.59
% 2.56/1.59 % (11529)Memory used [KB]: 1407
% 2.56/1.59 % (11529)Time elapsed: 0.006 s
% 2.56/1.59 % (11529)Instructions burned: 7 (million)
% 2.56/1.59 % (11529)------------------------------
% 2.56/1.59 % (11529)------------------------------
% 2.56/1.59 % (11532)Instruction limit reached!
% 2.56/1.59 % (11532)------------------------------
% 2.56/1.59 % (11532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.59 % (11532)Termination reason: Unknown
% 2.56/1.59 % (11532)Termination phase: shuffling
% 2.56/1.59
% 2.56/1.59 % (11532)Memory used [KB]: 1407
% 2.56/1.59 % (11532)Time elapsed: 0.004 s
% 2.56/1.59 % (11532)Instructions burned: 7 (million)
% 2.56/1.59 % (11532)------------------------------
% 2.56/1.59 % (11532)------------------------------
% 2.56/1.59 % (11531)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_11154 for (2999ds/21Mi)
% 2.56/1.60 % (11533)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_11154 for (2999ds/6Mi)
% 2.56/1.60 % (11533)Instruction limit reached!
% 2.56/1.60 % (11533)------------------------------
% 2.56/1.60 % (11533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.60 % (11531)Instruction limit reached!
% 2.56/1.60 % (11531)------------------------------
% 2.56/1.60 % (11531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.60 % (11531)Termination reason: Unknown
% 2.56/1.60 % (11533)Termination reason: Unknown
% 2.56/1.60 % (11531)Termination phase: shuffling
% 2.56/1.60
% 2.56/1.60 % (11531)Memory used [KB]: 2046
% 2.56/1.60 % (11533)Termination phase: shuffling
% 2.56/1.60 % (11531)Time elapsed: 0.009 s
% 2.56/1.60
% 2.56/1.60 % (11531)Instructions burned: 22 (million)
% 2.56/1.60 % (11531)------------------------------
% 2.56/1.60 % (11531)------------------------------
% 2.56/1.60 % (11533)Memory used [KB]: 1407
% 2.56/1.60 % (11533)Time elapsed: 0.005 s
% 2.56/1.60 % (11533)Instructions burned: 6 (million)
% 2.56/1.60 % (11533)------------------------------
% 2.56/1.60 % (11533)------------------------------
% 2.56/1.60 % (11535)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_11154 for (2999ds/779Mi)
% 2.56/1.61 % (11534)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/377Mi)
% 2.56/1.61 % (11536)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_11154 for (2999ds/19Mi)
% 2.56/1.62 % (11536)Instruction limit reached!
% 2.56/1.62 % (11536)------------------------------
% 2.56/1.62 % (11536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.62 % (11536)Termination reason: Unknown
% 2.56/1.62 % (11536)Termination phase: shuffling
% 2.56/1.62
% 2.56/1.62 % (11536)Memory used [KB]: 1663
% 2.56/1.62 % (11536)Time elapsed: 0.008 s
% 2.56/1.62 % (11536)Instructions burned: 22 (million)
% 2.56/1.62 % (11536)------------------------------
% 2.56/1.62 % (11536)------------------------------
% 2.56/1.62 % (11537)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_11154 for (2999ds/879Mi)
% 2.56/1.63 % (11538)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_11154 for (2998ds/17Mi)
% 2.56/1.63 % (11538)Instruction limit reached!
% 2.56/1.63 % (11538)------------------------------
% 2.56/1.63 % (11538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.63 % (11538)Termination reason: Unknown
% 2.56/1.63 % (11538)Termination phase: shuffling
% 2.56/1.63
% 2.56/1.63 % (11538)Memory used [KB]: 2046
% 2.56/1.63 % (11538)Time elapsed: 0.007 s
% 2.56/1.63 % (11538)Instructions burned: 18 (million)
% 2.56/1.63 % (11538)------------------------------
% 2.56/1.63 % (11538)------------------------------
% 2.56/1.65 % (11539)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_11154 for (2998ds/3Mi)
% 2.56/1.65 % (11539)Instruction limit reached!
% 2.56/1.65 % (11539)------------------------------
% 2.56/1.65 % (11539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.65 % (11539)Termination reason: Unknown
% 2.56/1.65 % (11539)Termination phase: shuffling
% 2.56/1.65
% 2.56/1.65 % (11539)Memory used [KB]: 1279
% 2.56/1.65 % (11539)Time elapsed: 0.003 s
% 2.56/1.65 % (11539)Instructions burned: 4 (million)
% 2.56/1.65 % (11539)------------------------------
% 2.56/1.65 % (11539)------------------------------
% 2.56/1.66 % (11540)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_11154 for (2998ds/30Mi)
% 2.56/1.67 % (11540)Instruction limit reached!
% 2.56/1.67 % (11540)------------------------------
% 2.56/1.67 % (11540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.67 % (11540)Termination reason: Unknown
% 2.56/1.67 % (11540)Termination phase: Property scanning
% 2.56/1.67
% 2.56/1.67 % (11540)Memory used [KB]: 2174
% 2.56/1.67 % (11540)Time elapsed: 0.012 s
% 2.56/1.67 % (11540)Instructions burned: 31 (million)
% 2.56/1.67 % (11540)------------------------------
% 2.56/1.67 % (11540)------------------------------
% 2.56/1.68 % (11541)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_11154 for (2998ds/127Mi)
% 2.56/1.70 % (11530)Refutation not found, incomplete strategy
% 2.56/1.70 % (11530)------------------------------
% 2.56/1.70 % (11530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.70 % (11530)Termination reason: Refutation not found, incomplete strategy
% 2.56/1.70
% 2.56/1.70
% 2.56/1.70 % (11530)Memory used [KB]: 10234
% 2.56/1.70 % (11530)Time elapsed: 0.119 s
% 2.56/1.70 % (11530)Instructions burned: 372 (million)
% 2.56/1.70 % (11530)------------------------------
% 2.56/1.70 % (11530)------------------------------
% 2.56/1.71 % (11542)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_11154 for (2998ds/100Mi)
% 2.56/1.72 % (11541)Instruction limit reached!
% 2.56/1.72 % (11541)------------------------------
% 2.56/1.72 % (11541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.72 % (11541)Termination reason: Unknown
% 2.56/1.72 % (11541)Termination phase: Property scanning
% 2.56/1.72
% 2.56/1.72 % (11541)Memory used [KB]: 4093
% 2.56/1.72 % (11541)Time elapsed: 0.040 s
% 2.56/1.72 % (11541)Instructions burned: 130 (million)
% 2.56/1.72 % (11541)------------------------------
% 2.56/1.72 % (11541)------------------------------
% 2.56/1.73 % (11543)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_11154 for (2997ds/3Mi)
% 2.56/1.73 % (11543)Instruction limit reached!
% 2.56/1.73 % (11543)------------------------------
% 2.56/1.73 % (11543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.73 % (11543)Termination reason: Unknown
% 2.56/1.73 % (11543)Termination phase: shuffling
% 2.56/1.73
% 2.56/1.73 % (11543)Memory used [KB]: 1279
% 2.56/1.73 % (11543)Time elapsed: 0.002 s
% 2.56/1.73 % (11543)Instructions burned: 4 (million)
% 2.56/1.73 % (11543)------------------------------
% 2.56/1.73 % (11543)------------------------------
% 2.56/1.74 % (11544)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_11154 for (2997ds/20Mi)
% 2.56/1.74 % (11534)Instruction limit reached!
% 2.56/1.74 % (11534)------------------------------
% 2.56/1.74 % (11534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.74 % (11534)Termination reason: Unknown
% 2.56/1.74 % (11534)Termination phase: Saturation
% 2.56/1.74
% 2.56/1.74 % (11534)Memory used [KB]: 10490
% 2.56/1.74 % (11534)Time elapsed: 0.131 s
% 2.56/1.74 % (11534)Instructions burned: 377 (million)
% 2.56/1.74 % (11534)------------------------------
% 2.56/1.74 % (11534)------------------------------
% 2.56/1.74 % (11542)Instruction limit reached!
% 2.56/1.74 % (11542)------------------------------
% 2.56/1.74 % (11542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.74 % (11542)Termination reason: Unknown
% 2.56/1.74 % (11542)Termination phase: Saturation
% 2.56/1.74
% 2.56/1.74 % (11542)Memory used [KB]: 6524
% 2.56/1.74 % (11542)Time elapsed: 0.029 s
% 2.56/1.74 % (11542)Instructions burned: 103 (million)
% 2.56/1.74 % (11542)------------------------------
% 2.56/1.74 % (11542)------------------------------
% 2.56/1.75 % (11544)Instruction limit reached!
% 2.56/1.75 % (11544)------------------------------
% 2.56/1.75 % (11544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.75 % (11544)Termination reason: Unknown
% 2.56/1.75 % (11544)Termination phase: shuffling
% 2.56/1.75
% 2.56/1.75 % (11544)Memory used [KB]: 2046
% 2.56/1.75 % (11544)Time elapsed: 0.009 s
% 2.56/1.75 % (11544)Instructions burned: 21 (million)
% 2.56/1.75 % (11544)------------------------------
% 2.56/1.75 % (11544)------------------------------
% 2.56/1.75 % (11545)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_11154 for (2997ds/86Mi)
% 2.56/1.75 % (11546)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_11154 for (2997ds/107Mi)
% 2.56/1.76 % (11547)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_11154 for (2997ds/39Mi)
% 2.56/1.77 % (11547)Instruction limit reached!
% 2.56/1.77 % (11547)------------------------------
% 2.56/1.77 % (11547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.77 % (11547)Termination reason: Unknown
% 2.56/1.77 % (11547)Termination phase: Property scanning
% 2.56/1.77
% 2.56/1.77 % (11547)Memory used [KB]: 1918
% 2.56/1.77 % (11547)Time elapsed: 0.013 s
% 2.56/1.77 % (11547)Instructions burned: 40 (million)
% 2.56/1.77 % (11547)------------------------------
% 2.56/1.77 % (11547)------------------------------
% 2.56/1.78 % (11548)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_11154 for (2997ds/448Mi)
% 2.56/1.78 % (11545)Instruction limit reached!
% 2.56/1.78 % (11545)------------------------------
% 2.56/1.78 % (11545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.78 % (11545)Termination reason: Unknown
% 2.56/1.78 % (11545)Termination phase: Preprocessing 3
% 2.56/1.78
% 2.56/1.78 % (11545)Memory used [KB]: 3965
% 2.56/1.78 % (11545)Time elapsed: 0.028 s
% 2.56/1.78 % (11545)Instructions burned: 86 (million)
% 2.56/1.78 % (11545)------------------------------
% 2.56/1.78 % (11545)------------------------------
% 2.56/1.78 % (11546)Instruction limit reached!
% 2.56/1.78 % (11546)------------------------------
% 2.56/1.78 % (11546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.78 % (11546)Termination reason: Unknown
% 2.56/1.78 % (11546)Termination phase: Property scanning
% 2.56/1.78
% 2.56/1.78 % (11546)Memory used [KB]: 4477
% 2.56/1.78 % (11546)Time elapsed: 0.035 s
% 2.56/1.78 % (11546)Instructions burned: 107 (million)
% 2.56/1.78 % (11546)------------------------------
% 2.56/1.78 % (11546)------------------------------
% 2.56/1.79 % (11549)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_11154 for (2997ds/46Mi)
% 2.56/1.79 % (11550)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_11154 for (2997ds/98Mi)
% 2.56/1.80 % (11528)Instruction limit reached!
% 2.56/1.80 % (11528)------------------------------
% 2.56/1.80 % (11528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.80 % (11528)Termination reason: Unknown
% 2.56/1.80 % (11528)Termination phase: Saturation
% 2.56/1.80
% 2.56/1.80 % (11528)Memory used [KB]: 11257
% 2.56/1.80 % (11528)Time elapsed: 0.219 s
% 2.56/1.80 % (11528)Instructions burned: 711 (million)
% 2.56/1.80 % (11528)------------------------------
% 2.56/1.80 % (11528)------------------------------
% 2.56/1.80 % (11548)Refutation not found, incomplete strategy
% 2.56/1.80 % (11548)------------------------------
% 2.56/1.80 % (11548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.80 % (11548)Termination reason: Refutation not found, incomplete strategy
% 2.56/1.80
% 2.56/1.80
% 2.56/1.80 % (11548)Memory used [KB]: 6524
% 2.56/1.80 % (11548)Time elapsed: 0.023 s
% 2.56/1.80 % (11548)Instructions burned: 81 (million)
% 2.56/1.80 % (11548)------------------------------
% 2.56/1.80 % (11548)------------------------------
% 2.56/1.80 % (11549)Instruction limit reached!
% 2.56/1.80 % (11549)------------------------------
% 2.56/1.80 % (11549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.80 % (11549)Termination reason: Unknown
% 2.56/1.80 % (11549)Termination phase: Naming
% 2.56/1.80
% 2.56/1.80 % (11549)Memory used [KB]: 2430
% 2.56/1.80 % (11549)Time elapsed: 0.017 s
% 2.56/1.80 % (11549)Instructions burned: 48 (million)
% 2.56/1.80 % (11549)------------------------------
% 2.56/1.80 % (11549)------------------------------
% 2.56/1.81 % (11535)Instruction limit reached!
% 2.56/1.81 % (11535)------------------------------
% 2.56/1.81 % (11535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.81 % (11535)Termination reason: Unknown
% 2.56/1.81 % (11535)Termination phase: Saturation
% 2.56/1.81
% 2.56/1.81 % (11535)Memory used [KB]: 6780
% 2.56/1.81 % (11535)Time elapsed: 0.206 s
% 2.56/1.81 % (11535)Instructions burned: 779 (million)
% 2.56/1.81 % (11535)------------------------------
% 2.56/1.81 % (11535)------------------------------
% 2.56/1.81 % (11551)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_11154 for (2996ds/507Mi)
% 2.56/1.81 % (11552)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_11154 for (2996ds/149Mi)
% 2.56/1.81 % (11519)Instruction limit reached!
% 2.56/1.81 % (11519)------------------------------
% 2.56/1.81 % (11519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.81 % (11519)Termination reason: Unknown
% 2.56/1.81 % (11519)Termination phase: Saturation
% 2.56/1.81
% 2.56/1.81 % (11519)Memory used [KB]: 8315
% 2.56/1.81 % (11519)Time elapsed: 0.270 s
% 2.56/1.81 % (11519)Instructions burned: 1043 (million)
% 2.56/1.81 % (11519)------------------------------
% 2.56/1.81 % (11519)------------------------------
% 2.56/1.81 % (11553)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_11154 for (2996ds/39Mi)
% 2.56/1.82 % (11554)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_11154 for (2996ds/6Mi)
% 2.56/1.82 % (11554)Instruction limit reached!
% 2.56/1.82 % (11554)------------------------------
% 2.56/1.82 % (11554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.82 % (11554)Termination reason: Unknown
% 2.56/1.82 % (11554)Termination phase: shuffling
% 2.56/1.82
% 2.56/1.82 % (11554)Memory used [KB]: 1407
% 2.56/1.82 % (11554)Time elapsed: 0.003 s
% 2.56/1.82 % (11554)Instructions burned: 7 (million)
% 2.56/1.82 % (11554)------------------------------
% 2.56/1.82 % (11554)------------------------------
% 2.56/1.82 % (11550)Instruction limit reached!
% 2.56/1.82 % (11550)------------------------------
% 2.56/1.82 % (11550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.82 % (11550)Termination reason: Unknown
% 2.56/1.82 % (11550)Termination phase: Saturation
% 2.56/1.82
% 2.56/1.82 % (11550)Memory used [KB]: 2046
% 2.56/1.82 % (11550)Time elapsed: 0.029 s
% 2.56/1.82 % (11550)Instructions burned: 101 (million)
% 2.56/1.82 % (11550)------------------------------
% 2.56/1.82 % (11550)------------------------------
% 2.56/1.82 % (11555)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_11154 for (2996ds/1087Mi)
% 2.56/1.83 % (11553)Instruction limit reached!
% 2.56/1.83 % (11553)------------------------------
% 2.56/1.83 % (11553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.83 % (11553)Termination reason: Unknown
% 2.56/1.83 % (11553)Termination phase: Preprocessing 2
% 2.56/1.83
% 2.56/1.83 % (11553)Memory used [KB]: 2302
% 2.56/1.83 % (11553)Time elapsed: 0.015 s
% 2.56/1.83 % (11553)Instructions burned: 41 (million)
% 2.56/1.83 % (11553)------------------------------
% 2.56/1.83 % (11553)------------------------------
% 2.56/1.83 % (11556)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_11154 for (2996ds/47Mi)
% 2.56/1.83 % (11557)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_11154 for (2996ds/44Mi)
% 2.56/1.84 % (11558)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_11154 for (2996ds/432Mi)
% 2.56/1.84 % (11557)Instruction limit reached!
% 2.56/1.84 % (11557)------------------------------
% 2.56/1.84 % (11557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.84 % (11557)Termination reason: Unknown
% 2.56/1.84 % (11557)Termination phase: Property scanning
% 2.56/1.84
% 2.56/1.84 % (11557)Memory used [KB]: 1918
% 2.56/1.84 % (11557)Time elapsed: 0.015 s
% 2.56/1.84 % (11557)Instructions burned: 46 (million)
% 2.56/1.84 % (11557)------------------------------
% 2.56/1.84 % (11557)------------------------------
% 2.56/1.85 % (11556)Instruction limit reached!
% 2.56/1.85 % (11556)------------------------------
% 2.56/1.85 % (11556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.85 % (11556)Termination reason: Unknown
% 2.56/1.85 % (11556)Termination phase: Naming
% 2.56/1.85
% 2.56/1.85 % (11556)Memory used [KB]: 2686
% 2.56/1.85 % (11556)Time elapsed: 0.017 s
% 2.56/1.85 % (11556)Instructions burned: 47 (million)
% 2.56/1.85 % (11556)------------------------------
% 2.56/1.85 % (11556)------------------------------
% 2.56/1.86 % (11559)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_11154 for (2996ds/62Mi)
% 2.56/1.86 % (11552)Instruction limit reached!
% 2.56/1.86 % (11552)------------------------------
% 2.56/1.86 % (11552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.86 % (11552)Termination reason: Unknown
% 2.56/1.86 % (11552)Termination phase: Property scanning
% 2.56/1.86
% 2.56/1.86 % (11552)Memory used [KB]: 7803
% 2.56/1.86 % (11552)Time elapsed: 0.048 s
% 2.56/1.86 % (11552)Instructions burned: 151 (million)
% 2.56/1.86 % (11552)------------------------------
% 2.56/1.86 % (11552)------------------------------
% 2.56/1.86 % (11560)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_11154 for (2996ds/31Mi)
% 2.56/1.87 % (11560)Instruction limit reached!
% 2.56/1.87 % (11560)------------------------------
% 2.56/1.87 % (11560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.87 % (11560)Termination reason: Unknown
% 2.56/1.87 % (11560)Termination phase: Property scanning
% 2.56/1.87
% 2.56/1.87 % (11560)Memory used [KB]: 2174
% 2.56/1.87 % (11560)Time elapsed: 0.011 s
% 2.56/1.87 % (11560)Instructions burned: 31 (million)
% 2.56/1.87 % (11560)------------------------------
% 2.56/1.87 % (11560)------------------------------
% 2.56/1.87 % (11561)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_11154 for (2996ds/96Mi)
% 2.56/1.87 % (11559)Instruction limit reached!
% 2.56/1.87 % (11559)------------------------------
% 2.56/1.87 % (11559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.87 % (11559)Termination reason: Unknown
% 2.56/1.87 % (11559)Termination phase: Property scanning
% 2.56/1.87
% 2.56/1.87 % (11559)Memory used [KB]: 1918
% 2.56/1.87 % (11559)Time elapsed: 0.018 s
% 2.56/1.87 % (11559)Instructions burned: 62 (million)
% 2.56/1.87 % (11559)------------------------------
% 2.56/1.87 % (11559)------------------------------
% 2.56/1.88 % (11562)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_11154 for (2996ds/453Mi)
% 2.56/1.88 % (11563)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_11154 for (2996ds/71Mi)
% 2.56/1.89 % (11551)Refutation not found, incomplete strategy
% 2.56/1.89 % (11551)------------------------------
% 2.56/1.89 % (11551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.89 % (11551)Termination reason: Refutation not found, incomplete strategy
% 2.56/1.89
% 2.56/1.89
% 2.56/1.89 % (11551)Memory used [KB]: 9850
% 2.56/1.89 % (11551)Time elapsed: 0.078 s
% 2.56/1.89 % (11551)Instructions burned: 273 (million)
% 2.56/1.89 % (11551)------------------------------
% 2.56/1.89 % (11551)------------------------------
% 2.56/1.90 % (11564)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_11154 for (2996ds/83Mi)
% 2.56/1.90 % (11561)Instruction limit reached!
% 2.56/1.90 % (11561)------------------------------
% 2.56/1.90 % (11561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.90 % (11561)Termination reason: Unknown
% 2.56/1.90 % (11561)Termination phase: Property scanning
% 2.56/1.90
% 2.56/1.90 % (11561)Memory used [KB]: 4221
% 2.56/1.90 % (11561)Time elapsed: 0.032 s
% 2.56/1.90 % (11561)Instructions burned: 99 (million)
% 2.56/1.90 % (11561)------------------------------
% 2.56/1.90 % (11561)------------------------------
% 2.56/1.90 % (11537)First to succeed.
% 2.56/1.90 % (11563)Instruction limit reached!
% 2.56/1.90 % (11563)------------------------------
% 2.56/1.90 % (11563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.90 % (11563)Termination reason: Unknown
% 2.56/1.90 % (11563)Termination phase: Preprocessing 3
% 2.56/1.90
% 2.56/1.90 % (11563)Memory used [KB]: 3454
% 2.56/1.90 % (11563)Time elapsed: 0.023 s
% 2.56/1.90 % (11563)Instructions burned: 74 (million)
% 2.56/1.90 % (11563)------------------------------
% 2.56/1.90 % (11563)------------------------------
% 2.56/1.91 % (11565)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_11154 for (2996ds/11Mi)
% 2.56/1.91 % (11564)Refutation not found, incomplete strategy
% 2.56/1.91 % (11564)------------------------------
% 2.56/1.91 % (11564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.91 % (11564)Termination reason: Refutation not found, incomplete strategy
% 2.56/1.91
% 2.56/1.91
% 2.56/1.91 % (11564)Memory used [KB]: 6652
% 2.56/1.91 % (11564)Time elapsed: 0.014 s
% 2.56/1.91 % (11564)Instructions burned: 43 (million)
% 2.56/1.91 % (11564)------------------------------
% 2.56/1.91 % (11564)------------------------------
% 2.56/1.91 % (11565)Instruction limit reached!
% 2.56/1.91 % (11565)------------------------------
% 2.56/1.91 % (11565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.91 % (11565)Termination reason: Unknown
% 2.56/1.91 % (11565)Termination phase: shuffling
% 2.56/1.91
% 2.56/1.91 % (11565)Memory used [KB]: 1535
% 2.56/1.91 % (11565)Time elapsed: 0.005 s
% 2.56/1.91 % (11565)Instructions burned: 11 (million)
% 2.56/1.91 % (11565)------------------------------
% 2.56/1.91 % (11565)------------------------------
% 2.56/1.91 % (11566)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_11154 for (2996ds/41Mi)
% 2.56/1.92 % (11567)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11154 for (2995ds/6Mi)
% 2.56/1.92 % (11567)Instruction limit reached!
% 2.56/1.92 % (11567)------------------------------
% 2.56/1.92 % (11567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.92 % (11567)Termination reason: Unknown
% 2.56/1.92 % (11567)Termination phase: shuffling
% 2.56/1.92
% 2.56/1.92 % (11567)Memory used [KB]: 1407
% 2.56/1.92 % (11567)Time elapsed: 0.003 s
% 2.56/1.92 % (11567)Instructions burned: 7 (million)
% 2.56/1.92 % (11567)------------------------------
% 2.56/1.92 % (11567)------------------------------
% 2.56/1.92 % (11568)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_11154 for (2995ds/150Mi)
% 2.56/1.92 % (11566)Instruction limit reached!
% 2.56/1.92 % (11566)------------------------------
% 2.56/1.92 % (11566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.92 % (11566)Termination reason: Unknown
% 2.56/1.92 % (11566)Termination phase: Property scanning
% 2.56/1.92
% 2.56/1.92 % (11566)Memory used [KB]: 1918
% 2.56/1.92 % (11566)Time elapsed: 0.012 s
% 2.56/1.92 % (11566)Instructions burned: 42 (million)
% 2.56/1.92 % (11566)------------------------------
% 2.56/1.92 % (11566)------------------------------
% 2.56/1.93 % (11569)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_11154 for (2995ds/343Mi)
% 2.56/1.93 % (11570)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_11154 for (2995ds/109Mi)
% 2.56/1.94 % (11537)Refutation found. Thanks to Tanya!
% 2.56/1.94 % SZS status Theorem for DTF2THF_11154
% 2.56/1.94 % SZS output start Proof for DTF2THF_11154
% 2.56/1.94 thf(type_def_5, type, term: $tType).
% 2.56/1.94 thf(type_def_7, type, subst: $tType).
% 2.56/1.94 thf(type_def_8, type, d_term: $tType).
% 2.56/1.94 thf(type_def_9, type, d_subst: $tType).
% 2.56/1.94 thf(func_def_0, type, term: $tType).
% 2.56/1.94 thf(func_def_1, type, subst: $tType).
% 2.56/1.94 thf(func_def_2, type, one: term).
% 2.56/1.94 thf(func_def_3, type, ap: term > term > term).
% 2.56/1.94 thf(func_def_4, type, lam: term > term).
% 2.56/1.94 thf(func_def_5, type, sub: term > subst > term).
% 2.56/1.94 thf(func_def_6, type, id: subst).
% 2.56/1.94 thf(func_def_7, type, sh: subst).
% 2.56/1.94 thf(func_def_8, type, push: term > subst > subst).
% 2.56/1.94 thf(func_def_9, type, comp: subst > subst > subst).
% 2.56/1.94 thf(func_def_10, type, var: term > $o).
% 2.56/1.94 thf(func_def_18, type, hoasap: subst > term > subst > term > term).
% 2.56/1.94 thf(func_def_73, type, hoaslam: subst > (subst > term > term) > term).
% 2.56/1.94 thf(func_def_86, type, hoasinduction_p_and_p_prime: (subst > term > subst > $o) > (term > $o) > $o).
% 2.56/1.94 thf(func_def_107, type, hoasvar: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_121, type, pushprop_p_and_p_prime: term > subst > (term > $o) > (term > $o) > $o).
% 2.56/1.94 thf(func_def_124, type, d_term: $tType).
% 2.56/1.94 thf(func_def_125, type, d_subst: $tType).
% 2.56/1.94 thf(func_def_126, type, d2term: d_term > term).
% 2.56/1.94 thf(func_def_127, type, d2subst: d_subst > subst).
% 2.56/1.94 thf(func_def_128, type, d_one: d_term).
% 2.56/1.94 thf(func_def_129, type, d_id: d_subst).
% 2.56/1.94 thf(func_def_143, type, sP1: (term > $o) > $o).
% 2.56/1.94 thf(func_def_145, type, sP3: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_147, type, sP5: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_148, type, sP6: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_149, type, sP7: (term > $o) > $o).
% 2.56/1.94 thf(func_def_150, type, sP8: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_151, type, sP9: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_152, type, sP10: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_157, type, sP15: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_160, type, sP18: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_161, type, sP19: (term > $o) > $o).
% 2.56/1.94 thf(func_def_167, type, sP25: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_172, type, sP30: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_174, type, sP32: (term > $o) > $o).
% 2.56/1.94 thf(func_def_175, type, sP33: (term > $o) > $o).
% 2.56/1.94 thf(func_def_178, type, sP36: (term > $o) > $o).
% 2.56/1.94 thf(func_def_179, type, sP37: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_183, type, sP41: (term > $o) > $o).
% 2.56/1.94 thf(func_def_185, type, sP43: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_186, type, sP44: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_188, type, sP46: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_189, type, sP47: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_190, type, sP48: (term > $o) > $o).
% 2.56/1.94 thf(func_def_194, type, sP52: (term > $o) > $o).
% 2.56/1.94 thf(func_def_195, type, sP53: (term > $o) > $o).
% 2.56/1.94 thf(func_def_199, type, sP57: (subst > term > subst > $o) > $o).
% 2.56/1.94 thf(func_def_201, type, sK59: term).
% 2.56/1.94 thf(func_def_202, type, sK60: term).
% 2.56/1.94 thf(func_def_203, type, sK61: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_204, type, sK62: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_205, type, sK63: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_206, type, sK64: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_207, type, sK65: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_208, type, sK66: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_209, type, sK67: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_210, type, sK68: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_211, type, sK69: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_212, type, sK70: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_213, type, sK71: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_214, type, sK72: term > $o).
% 2.56/1.94 thf(func_def_215, type, sK73: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_216, type, sK74: term).
% 2.56/1.94 thf(func_def_217, type, sK75: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_218, type, sK76: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_219, type, sK77: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_220, type, sK78: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_221, type, sK79: term > subst > term > term).
% 2.56/1.94 thf(func_def_222, type, sK80: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_223, type, sK81: term > (term > $o) > term).
% 2.56/1.94 thf(func_def_224, type, sK82: (subst > term > subst > $o) > (term > $o) > term).
% 2.56/1.94 thf(func_def_225, type, sK83: term).
% 2.56/1.94 thf(func_def_226, type, sK84: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_227, type, sK85: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_228, type, sK86: term).
% 2.56/1.94 thf(func_def_229, type, sK87: (term > $o) > subst > term).
% 2.56/1.94 thf(func_def_230, type, sK88: (term > $o) > term).
% 2.56/1.94 thf(func_def_231, type, sK89: subst).
% 2.56/1.94 thf(func_def_232, type, sK90: term).
% 2.56/1.94 thf(func_def_233, type, sK91: term > $o).
% 2.56/1.94 thf(func_def_234, type, sK92: term > term).
% 2.56/1.94 thf(func_def_235, type, sK93: term).
% 2.56/1.94 thf(func_def_236, type, sK94: term > $o).
% 2.56/1.94 thf(func_def_237, type, sK95: term).
% 2.56/1.94 thf(func_def_238, type, sK96: term > term).
% 2.56/1.94 thf(func_def_239, type, sK97: (term > $o) > term).
% 2.56/1.94 thf(func_def_240, type, sK98: (term > $o) > term).
% 2.56/1.94 thf(func_def_241, type, sK99: (term > $o) > term).
% 2.56/1.94 thf(func_def_242, type, sK100: (term > $o) > term).
% 2.56/1.94 thf(func_def_243, type, sK101: (term > $o) > term).
% 2.56/1.94 thf(func_def_244, type, sK102: (term > $o) > term).
% 2.56/1.94 thf(func_def_245, type, sK103: (subst > term > subst > $o) > term > $o).
% 2.56/1.94 thf(func_def_246, type, sK104: (term > $o) > term).
% 2.56/1.94 thf(func_def_247, type, sK105: (term > $o) > term).
% 2.56/1.94 thf(func_def_248, type, sK106: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_249, type, sK107: (term > $o) > term).
% 2.56/1.94 thf(func_def_250, type, sK108: term > $o).
% 2.56/1.94 thf(func_def_251, type, sK109: term).
% 2.56/1.94 thf(func_def_252, type, sK110: term > term).
% 2.56/1.94 thf(func_def_253, type, sK111: term).
% 2.56/1.94 thf(func_def_254, type, sK112: (term > $o) > term > term).
% 2.56/1.94 thf(func_def_255, type, sK113: (subst > term > subst > $o) > (term > $o) > term).
% 2.56/1.94 thf(func_def_256, type, sK114: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_257, type, sK115: term).
% 2.56/1.94 thf(func_def_258, type, sK116: term > $o).
% 2.56/1.94 thf(func_def_259, type, sK117: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_260, type, sK118: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_261, type, sK119: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_262, type, sK120: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_263, type, sK121: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_264, type, sK122: term).
% 2.56/1.94 thf(func_def_265, type, sK123: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_266, type, sK124: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_267, type, sK125: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_268, type, sK126: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_269, type, sK127: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_270, type, sK128: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_271, type, sK129: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_272, type, sK130: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_273, type, sK131: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_274, type, sK132: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_275, type, sK133: (term > $o) > term).
% 2.56/1.94 thf(func_def_276, type, sK134: (term > $o) > term).
% 2.56/1.94 thf(func_def_277, type, sK135: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_278, type, sK136: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_279, type, sK137: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_280, type, sK138: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_281, type, sK139: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_282, type, sK140: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_283, type, sK141: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_284, type, sK142: term).
% 2.56/1.94 thf(func_def_285, type, sK143: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_286, type, sK144: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_287, type, sK145: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_288, type, sK146: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_289, type, sK147: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_290, type, sK148: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_291, type, sK149: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_292, type, sK150: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_293, type, sK151: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_294, type, sK152: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_295, type, sK153: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_296, type, sK154: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_297, type, sK155: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_298, type, sK156: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_299, type, sK157: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_300, type, sK158: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_301, type, sK159: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_302, type, sK160: term > $o).
% 2.56/1.94 thf(func_def_303, type, sK161: subst).
% 2.56/1.94 thf(func_def_304, type, sK162: term).
% 2.56/1.94 thf(func_def_305, type, sK163: term > term).
% 2.56/1.94 thf(func_def_306, type, sK164: subst > (term > $o) > term).
% 2.56/1.94 thf(func_def_307, type, sK165: (term > $o) > term).
% 2.56/1.94 thf(func_def_308, type, sK166: (term > $o) > term).
% 2.56/1.94 thf(func_def_309, type, sK167: (term > $o) > term).
% 2.56/1.94 thf(func_def_310, type, sK168: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_311, type, sK169: term).
% 2.56/1.94 thf(func_def_312, type, sK170: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_313, type, sK171: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_314, type, sK172: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_315, type, sK173: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_316, type, sK174: term > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_317, type, sK175: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_318, type, sK176: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_319, type, sK177: term).
% 2.56/1.94 thf(func_def_320, type, sK178: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_321, type, sK179: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_322, type, sK180: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_323, type, sK181: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_324, type, sK182: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_325, type, sK183: term > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_326, type, sK184: term > $o).
% 2.56/1.94 thf(func_def_327, type, sK185: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_328, type, sK186: term).
% 2.56/1.94 thf(func_def_329, type, sK187: term > (term > $o) > term).
% 2.56/1.94 thf(func_def_330, type, sK188: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_331, type, sK189: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_332, type, sK190: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_333, type, sK191: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_334, type, sK192: (subst > term > subst > $o) > (term > $o) > term).
% 2.56/1.94 thf(func_def_335, type, sK193: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_336, type, sK194: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_337, type, sK195: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_338, type, sK196: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_339, type, sK197: term).
% 2.56/1.94 thf(func_def_340, type, sK198: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_341, type, sK199: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_342, type, sK200: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_343, type, sK201: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_344, type, sK202: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_345, type, sK203: (term > $o) > term).
% 2.56/1.94 thf(func_def_346, type, sK204: (term > $o) > term).
% 2.56/1.94 thf(func_def_347, type, sK205: (term > $o) > term).
% 2.56/1.94 thf(func_def_348, type, sK206: (term > $o) > term).
% 2.56/1.94 thf(func_def_349, type, sK207: term).
% 2.56/1.94 thf(func_def_350, type, sK208: term > $o).
% 2.56/1.94 thf(func_def_351, type, sK209: subst).
% 2.56/1.94 thf(func_def_352, type, sK210: term).
% 2.56/1.94 thf(func_def_353, type, sK211: subst).
% 2.56/1.94 thf(func_def_354, type, sK212: term).
% 2.56/1.94 thf(func_def_355, type, sK213: subst).
% 2.56/1.94 thf(func_def_356, type, sK214: subst).
% 2.56/1.94 thf(func_def_357, type, sK215: term).
% 2.56/1.94 thf(func_def_358, type, sK216: term > $o).
% 2.56/1.94 thf(func_def_359, type, sK217: term > term).
% 2.56/1.94 thf(func_def_360, type, sK218: (term > $o) > subst > term).
% 2.56/1.94 thf(func_def_361, type, sK219: (term > $o) > term).
% 2.56/1.94 thf(func_def_362, type, sK220: (term > $o) > term).
% 2.56/1.94 thf(func_def_363, type, sK221: (term > $o) > term).
% 2.56/1.94 thf(func_def_364, type, sK222: (term > $o) > term).
% 2.56/1.94 thf(func_def_365, type, sK223: (term > $o) > term).
% 2.56/1.94 thf(func_def_366, type, sK224: term).
% 2.56/1.94 thf(func_def_367, type, sK225: term > $o).
% 2.56/1.94 thf(func_def_368, type, sK226: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_369, type, sK227: term).
% 2.56/1.94 thf(func_def_370, type, sK228: (subst > term > subst > $o) > (term > $o) > term).
% 2.56/1.94 thf(func_def_371, type, sK229: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_372, type, sK230: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_373, type, sK231: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_374, type, sK232: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_375, type, sK233: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_376, type, sK234: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_377, type, sK235: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_378, type, sK236: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_379, type, sK237: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_380, type, sK238: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_381, type, sK239: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_382, type, sK240: term > $o).
% 2.56/1.94 thf(func_def_383, type, sK241: term).
% 2.56/1.94 thf(func_def_384, type, sK242: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_385, type, sK243: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_386, type, sK244: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_387, type, sK245: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_388, type, sK246: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_389, type, sK247: (term > $o) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_390, type, sK248: term > (term > $o) > term).
% 2.56/1.94 thf(func_def_391, type, sK249: term).
% 2.56/1.94 thf(func_def_392, type, sK250: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_393, type, sK251: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_394, type, sK252: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_395, type, sK253: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_396, type, sK254: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_397, type, sK255: term).
% 2.56/1.94 thf(func_def_398, type, sK256: (subst > term > subst > $o) > term > term).
% 2.56/1.94 thf(func_def_399, type, sK257: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_400, type, sK258: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_401, type, sK259: (subst > term > subst > $o) > term > term).
% 2.56/1.94 thf(func_def_402, type, sK260: term).
% 2.56/1.94 thf(func_def_403, type, sK261: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_404, type, sK262: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_405, type, sK263: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_406, type, sK264: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_407, type, sK265: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_408, type, sK266: term > $o).
% 2.56/1.94 thf(func_def_409, type, sK267: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_410, type, sK268: term).
% 2.56/1.94 thf(func_def_411, type, sK269: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_412, type, sK270: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_413, type, sK271: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_414, type, sK272: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_415, type, sK273: (subst > term > subst > $o) > (term > $o) > term).
% 2.56/1.94 thf(func_def_416, type, sK274: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_417, type, sK275: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_418, type, sK276: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_419, type, sK277: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_420, type, sK278: term > (term > $o) > term).
% 2.56/1.94 thf(func_def_421, type, sK279: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_422, type, sK280: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_423, type, sK281: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_424, type, sK282: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_425, type, sK283: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_426, type, sK284: term).
% 2.56/1.94 thf(func_def_427, type, sK285: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_428, type, sK286: term > (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_429, type, sK287: term > (subst > term > term) > term).
% 2.56/1.94 thf(func_def_430, type, sK288: term > (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_431, type, sK289: term > (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_432, type, sK290: term > (subst > term > term) > term).
% 2.56/1.94 thf(func_def_433, type, sK291: term > (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_434, type, sK292: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_435, type, sK293: (subst > term > subst > $o) > term > term).
% 2.56/1.94 thf(func_def_436, type, sK294: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_437, type, sK295: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_438, type, sK296: subst).
% 2.56/1.94 thf(func_def_439, type, sK297: subst > term > term).
% 2.56/1.94 thf(func_def_440, type, sK298: term).
% 2.56/1.94 thf(func_def_441, type, sK299: subst > term > term).
% 2.56/1.94 thf(func_def_442, type, sK300: term).
% 2.56/1.94 thf(func_def_443, type, sK301: term).
% 2.56/1.94 thf(func_def_444, type, sK302: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_445, type, sK303: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_446, type, sK304: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_447, type, sK305: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_448, type, sK306: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_449, type, sK307: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_450, type, sK308: term).
% 2.56/1.94 thf(func_def_451, type, sK309: subst).
% 2.56/1.94 thf(func_def_452, type, sK310: subst).
% 2.56/1.94 thf(func_def_453, type, sK311: term).
% 2.56/1.94 thf(func_def_454, type, sK312: (term > $o) > term).
% 2.56/1.94 thf(func_def_455, type, sK313: (term > $o) > term).
% 2.56/1.94 thf(func_def_456, type, sK314: (subst > term > subst > $o) > term > $o).
% 2.56/1.94 thf(func_def_457, type, sK315: term > $o).
% 2.56/1.94 thf(func_def_458, type, sK316: term).
% 2.56/1.94 thf(func_def_459, type, sK317: term > term).
% 2.56/1.94 thf(func_def_460, type, sK318: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_461, type, sK319: (term > $o) > term).
% 2.56/1.94 thf(func_def_462, type, sK320: term).
% 2.56/1.94 thf(func_def_463, type, sK321: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_464, type, sK322: term).
% 2.56/1.94 thf(func_def_465, type, sK323: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_466, type, sK324: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_467, type, sK325: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_468, type, sK326: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_469, type, sK327: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_470, type, sK328: (term > $o) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_471, type, sK329: (term > $o) > term > term).
% 2.56/1.94 thf(func_def_472, type, sK330: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_473, type, sK331: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_474, type, sK332: term).
% 2.56/1.94 thf(func_def_475, type, sK333: term > $o).
% 2.56/1.94 thf(func_def_476, type, sK334: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_477, type, sK335: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_478, type, sK336: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_479, type, sK337: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_480, type, sK338: (term > $o) > term).
% 2.56/1.94 thf(func_def_481, type, sK339: (term > $o) > term).
% 2.56/1.94 thf(func_def_482, type, sK340: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_483, type, sK341: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_484, type, sK342: term).
% 2.56/1.94 thf(func_def_485, type, sK343: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_486, type, sK344: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_487, type, sK345: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_488, type, sK346: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_489, type, sK347: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_490, type, sK348: (subst > term > subst > $o) > term > term).
% 2.56/1.94 thf(func_def_491, type, sK349: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_492, type, sK350: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_493, type, sK351: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_494, type, sK352: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_495, type, sK353: term).
% 2.56/1.94 thf(func_def_496, type, sK354: (subst > term > subst > $o) > term > term).
% 2.56/1.94 thf(func_def_497, type, sK355: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_498, type, sK356: term).
% 2.56/1.94 thf(func_def_499, type, sK357: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_500, type, sK358: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_501, type, sK359: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_502, type, sK360: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_503, type, sK361: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_504, type, sK362: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_505, type, sK363: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_506, type, sK364: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_507, type, sK365: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_508, type, sK366: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_509, type, sK367: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_510, type, sK368: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_511, type, sK369: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_512, type, sK370: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_513, type, sK371: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_514, type, sK372: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_515, type, sK373: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_516, type, sK374: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_517, type, sK375: term).
% 2.56/1.94 thf(func_def_518, type, sK376: term).
% 2.56/1.94 thf(func_def_519, type, sK377: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_520, type, sK378: (term > $o) > term).
% 2.56/1.94 thf(func_def_521, type, sK379: (term > $o) > term).
% 2.56/1.94 thf(func_def_522, type, sK380: term).
% 2.56/1.94 thf(func_def_523, type, sK381: term > $o).
% 2.56/1.94 thf(func_def_524, type, sK382: term > term).
% 2.56/1.94 thf(func_def_525, type, sK383: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_526, type, sK384: term).
% 2.56/1.94 thf(func_def_527, type, sK385: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_528, type, sK386: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_529, type, sK387: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_530, type, sK388: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_531, type, sK389: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_532, type, sK390: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_533, type, sK391: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_534, type, sK392: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_535, type, sK393: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_536, type, sK394: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_537, type, sK395: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_538, type, sK396: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_539, type, sK397: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_540, type, sK398: term > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_541, type, sK399: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_542, type, sK400: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_543, type, sK401: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_544, type, sK402: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_545, type, sK403: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_546, type, sK404: term).
% 2.56/1.94 thf(func_def_547, type, sK405: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_548, type, sK406: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_549, type, sK407: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_550, type, sK408: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_551, type, sK409: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_552, type, sK410: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_553, type, sK411: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_554, type, sK412: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_555, type, sK413: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_556, type, sK414: (term > $o) > term).
% 2.56/1.94 thf(func_def_557, type, sK415: (term > $o) > term).
% 2.56/1.94 thf(func_def_558, type, sK416: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_559, type, sK417: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_560, type, sK418: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_561, type, sK419: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_562, type, sK420: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_563, type, sK421: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_564, type, sK422: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_565, type, sK423: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_566, type, sK424: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_567, type, sK425: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_568, type, sK426: term).
% 2.56/1.94 thf(func_def_569, type, sK427: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_570, type, sK428: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_571, type, sK429: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_572, type, sK430: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_573, type, sK431: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_574, type, sK432: (subst > term > subst > $o) > term > term).
% 2.56/1.94 thf(func_def_575, type, sK433: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_576, type, sK434: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_577, type, sK435: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_578, type, sK436: (subst > term > subst > $o) > subst).
% 2.56/1.94 thf(func_def_579, type, sK437: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_580, type, sK438: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_581, type, sK439: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_582, type, sK440: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_583, type, sK441: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_584, type, sK442: term).
% 2.56/1.94 thf(func_def_585, type, sK443: term > $o).
% 2.56/1.94 thf(func_def_586, type, sK444: term > term).
% 2.56/1.94 thf(func_def_587, type, sK445: (term > $o) > term).
% 2.56/1.94 thf(func_def_588, type, sK446: (term > $o) > term).
% 2.56/1.94 thf(func_def_589, type, sK447: (term > $o) > term).
% 2.56/1.94 thf(func_def_590, type, sK448: (term > $o) > term).
% 2.56/1.94 thf(func_def_591, type, sK449: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_592, type, sK450: term).
% 2.56/1.94 thf(func_def_593, type, sK451: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_594, type, sK452: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_595, type, sK453: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_596, type, sK454: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_597, type, sK455: term > (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_598, type, sK456: (subst > term > subst > $o) > subst > term > term).
% 2.56/1.94 thf(func_def_599, type, sK457: term).
% 2.56/1.94 thf(func_def_600, type, sK458: term).
% 2.56/1.94 thf(func_def_601, type, sK459: term).
% 2.56/1.94 thf(func_def_602, type, sK460: term).
% 2.56/1.94 thf(func_def_603, type, sK461: term).
% 2.56/1.94 thf(func_def_604, type, sK462: subst).
% 2.56/1.94 thf(func_def_605, type, sK463: subst).
% 2.56/1.94 thf(func_def_606, type, sK464: subst).
% 2.56/1.94 thf(func_def_607, type, sK465: subst).
% 2.56/1.94 thf(func_def_608, type, sK466: term).
% 2.56/1.94 thf(func_def_609, type, sK467: term > subst > term > term).
% 2.56/1.94 thf(func_def_610, type, sK468: term).
% 2.56/1.94 thf(func_def_611, type, sK469: term).
% 2.56/1.94 thf(func_def_612, type, sK470: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_613, type, sK471: (term > $o) > term).
% 2.56/1.94 thf(func_def_614, type, sK472: (subst > term > subst > $o) > term > $o).
% 2.56/1.94 thf(func_def_615, type, sK473: subst).
% 2.56/1.94 thf(func_def_616, type, sK474: term).
% 2.56/1.94 thf(func_def_617, type, sK475: subst).
% 2.56/1.94 thf(func_def_618, type, sK476: term).
% 2.56/1.94 thf(func_def_619, type, sK477: term).
% 2.56/1.94 thf(func_def_620, type, sK478: term > $o).
% 2.56/1.94 thf(func_def_621, type, sK479: term > $o).
% 2.56/1.94 thf(func_def_622, type, sK480: subst).
% 2.56/1.94 thf(func_def_623, type, sK481: term).
% 2.56/1.94 thf(func_def_624, type, sK482: term).
% 2.56/1.94 thf(func_def_625, type, sK483: term > $o).
% 2.56/1.94 thf(func_def_626, type, sK484: term).
% 2.56/1.94 thf(func_def_627, type, sK485: (term > $o) > term).
% 2.56/1.94 thf(func_def_628, type, sK486: (term > $o) > term).
% 2.56/1.94 thf(func_def_629, type, sK487: subst).
% 2.56/1.94 thf(func_def_630, type, sK488: subst).
% 2.56/1.94 thf(func_def_631, type, sK489: term).
% 2.56/1.94 thf(func_def_632, type, sK490: term).
% 2.56/1.94 thf(func_def_633, type, sK491: term > $o).
% 2.56/1.94 thf(func_def_634, type, sK492: subst).
% 2.56/1.94 thf(func_def_635, type, sK493: (term > $o) > term).
% 2.56/1.94 thf(func_def_636, type, sK494: subst > (term > $o) > term > term > $o).
% 2.56/1.94 thf(func_def_637, type, sK495: term).
% 2.56/1.94 thf(func_def_638, type, sK496: term).
% 2.56/1.94 thf(func_def_639, type, sK497: term).
% 2.56/1.94 thf(func_def_640, type, sK498: term).
% 2.56/1.94 thf(func_def_641, type, sK499: term).
% 2.56/1.94 thf(func_def_642, type, sK500: term).
% 2.56/1.94 thf(func_def_643, type, sK501: term).
% 2.56/1.94 thf(func_def_644, type, sK502: term).
% 2.56/1.94 thf(func_def_645, type, sK503: term).
% 2.56/1.94 thf(func_def_646, type, sK504: subst > term > term).
% 2.56/1.94 thf(func_def_647, type, sK505: term).
% 2.56/1.94 thf(func_def_648, type, sK506: term).
% 2.56/1.94 thf(func_def_649, type, sK507: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_650, type, sK508: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_651, type, sK509: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_652, type, sK510: term).
% 2.56/1.94 thf(func_def_653, type, sK511: term).
% 2.56/1.94 thf(func_def_654, type, sK512: term).
% 2.56/1.94 thf(func_def_655, type, sK513: term).
% 2.56/1.94 thf(func_def_656, type, sK514: subst).
% 2.56/1.94 thf(func_def_657, type, sK515: subst).
% 2.56/1.94 thf(func_def_658, type, sK516: (term > $o) > term > subst > term > $o).
% 2.56/1.94 thf(func_def_659, type, sK517: subst).
% 2.56/1.94 thf(func_def_660, type, sK518: term).
% 2.56/1.94 thf(func_def_661, type, sK519: term > $o).
% 2.56/1.94 thf(func_def_662, type, sK520: (term > $o) > term).
% 2.56/1.94 thf(func_def_663, type, sK521: term).
% 2.56/1.94 thf(func_def_664, type, sK522: term > subst > term > term).
% 2.56/1.94 thf(func_def_665, type, sK523: term > $o).
% 2.56/1.94 thf(func_def_666, type, sK524: term).
% 2.56/1.94 thf(func_def_667, type, sK525: subst).
% 2.56/1.94 thf(func_def_668, type, sK526: term).
% 2.56/1.94 thf(func_def_669, type, sK527: term > $o).
% 2.56/1.94 thf(func_def_670, type, sK528: term > subst > term > term).
% 2.56/1.94 thf(func_def_671, type, sK529: term).
% 2.56/1.94 thf(func_def_672, type, sK530: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_673, type, sK531: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_674, type, sK532: term > (term > $o) > subst > (term > $o) > term).
% 2.56/1.94 thf(func_def_675, type, sK533: term > $o).
% 2.56/1.94 thf(func_def_676, type, sK534: subst).
% 2.56/1.94 thf(func_def_677, type, sK535: term > $o).
% 2.56/1.94 thf(func_def_678, type, sK536: term).
% 2.56/1.94 thf(func_def_679, type, sK537: term > subst > term > term).
% 2.56/1.94 thf(func_def_680, type, sK538: term).
% 2.56/1.94 thf(func_def_681, type, sK539: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_682, type, sK540: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_683, type, sK541: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_684, type, sK542: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_685, type, sK543: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_686, type, sK544: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_687, type, sK545: (subst > term > term) > term).
% 2.56/1.94 thf(func_def_688, type, sK546: (subst > term > term) > subst).
% 2.56/1.94 thf(func_def_689, type, sK547: subst > term > term).
% 2.56/1.94 thf(func_def_690, type, sK548: subst).
% 2.56/1.94 thf(func_def_691, type, sK549: term).
% 2.56/1.94 thf(func_def_692, type, sK550: subst > term > term).
% 2.56/1.94 thf(func_def_693, type, sK551: term).
% 2.56/1.94 thf(func_def_694, type, sK552: term).
% 2.56/1.94 thf(func_def_695, type, sK553: subst).
% 2.56/1.94 thf(func_def_696, type, sK554: term).
% 2.56/1.94 thf(func_def_697, type, sK555: term).
% 2.56/1.94 thf(func_def_698, type, sK556: term).
% 2.56/1.94 thf(func_def_699, type, sK557: term).
% 2.56/1.94 thf(func_def_700, type, sK558: term).
% 2.56/1.94 thf(func_def_701, type, sK559: term).
% 2.56/1.94 thf(func_def_702, type, sK560: term).
% 2.56/1.94 thf(func_def_703, type, sK561: term).
% 2.56/1.94 thf(func_def_704, type, sK562: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_705, type, sK563: (term > $o) > term).
% 2.56/1.94 thf(func_def_706, type, sK564: (subst > term > subst > $o) > term > $o).
% 2.56/1.94 thf(func_def_707, type, sK565: term).
% 2.56/1.94 thf(func_def_708, type, sK566: subst > term > subst > $o).
% 2.56/1.94 thf(func_def_709, type, sK567: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_710, type, sK568: (subst > term > subst > $o) > term).
% 2.56/1.94 thf(func_def_711, type, sK569: term).
% 2.56/1.94 thf(func_def_712, type, sK570: subst).
% 2.56/1.94 thf(func_def_713, type, sK571: subst > d_subst).
% 2.56/1.94 thf(func_def_714, type, sK572: subst).
% 2.56/1.94 thf(func_def_715, type, sK573: term).
% 2.56/1.94 thf(func_def_716, type, sK574: term).
% 2.56/1.94 thf(func_def_717, type, sK575: term).
% 2.56/1.94 thf(func_def_718, type, sK576: term).
% 2.56/1.94 thf(func_def_719, type, sK577: term).
% 2.56/1.94 thf(func_def_720, type, sK578: term).
% 2.56/1.94 thf(func_def_721, type, sK579: term > d_term).
% 2.56/1.94 thf(func_def_722, type, sK580: term).
% 2.56/1.94 thf(func_def_723, type, sK581: term).
% 2.56/1.94 thf(func_def_724, type, sK582: term).
% 2.56/1.94 thf(func_def_725, type, sK583: term).
% 2.56/1.94 thf(func_def_726, type, sK584: subst > (term > $o) > (term > $o) > term > term).
% 2.56/1.94 thf(func_def_727, type, sK585: term).
% 2.56/1.94 thf(func_def_728, type, sK586: subst).
% 2.56/1.94 thf(func_def_729, type, sK587: term).
% 2.56/1.94 thf(func_def_730, type, sK588: term > $o).
% 2.56/1.94 thf(func_def_731, type, sK589: term > $o).
% 2.56/1.94 thf(func_def_732, type, sK590: subst).
% 2.56/1.94 thf(func_def_736, type, sK592: term > $o).
% 2.56/1.94 thf(func_def_737, type, sK593: term).
% 2.56/1.94 thf(func_def_738, type, sK594: subst).
% 2.56/1.94 thf(func_def_739, type, sK595: term > $o).
% 2.56/1.94 thf(func_def_740, type, sK596: term).
% 2.56/1.94 thf(f4413,plain,(
% 2.56/1.94 $false),
% 2.56/1.94 inference(trivial_inequality_removal,[status(thm)],[f4412])).
% 2.56/1.94 thf(f4412,plain,(
% 2.56/1.94 ($true = $false)),
% 2.56/1.94 inference(backward_demodulation,[status(thm)],[f1712,f4410])).
% 2.56/1.94 thf(f4410,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($false = (var @ X1))) )),
% 2.56/1.94 inference(trivial_inequality_removal,[status(thm)],[f4409])).
% 2.56/1.94 thf(f4409,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($false = (var @ X1)) | ($true = $false)) )),
% 2.56/1.94 inference(backward_demodulation,[status(thm)],[f1708,f4408])).
% 2.56/1.94 thf(f4408,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($false = (sK595 @ X1))) )),
% 2.56/1.94 inference(trivial_inequality_removal,[status(thm)],[f4407])).
% 2.56/1.94 thf(f4407,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($false = (sK595 @ X1)) | ($true = $false)) )),
% 2.56/1.94 inference(forward_demodulation,[status(thm)],[f4406,f4404])).
% 2.56/1.94 thf(f4404,plain,(
% 2.56/1.94 ($false = (sK592 @ (d2term @ d_one)))),
% 2.56/1.94 inference(forward_demodulation,[status(thm)],[f1711,f2761])).
% 2.56/1.94 thf(f2761,plain,(
% 2.56/1.94 ( ! [X248 : term] : (((d2term @ d_one) = X248)) )),
% 2.56/1.94 inference(backward_demodulation,[status(thm)],[f1046,f1240])).
% 2.56/1.94 thf(f1240,plain,(
% 2.56/1.94 ( ! [X18 : d_term] : ((d_one = X18)) )),
% 2.56/1.94 inference(cnf_transformation,[status(thm)],[f545])).
% 2.56/1.94 thf(f545,plain,(
% 2.56/1.94 (hoaslamnotvar = $true) & ((apinj2 = $true) | (((ap @ sK459 @ sK457) = (ap @ sK460 @ sK458)) & (sK457 != sK458))) & (! [X4 : term,X5 : term,X6 : term,X7 : term] : (((ap @ X7 @ X5) != (ap @ X6 @ X4)) | (X4 = X5)) | (apinj2 != $true)) & (ulamvarind = $true) & (axapp = $true) & (hoasinduction_lem1 = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (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)))))))))))))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X8 : term] : ((sub @ X8 @ id) = X8) | (termmset != $true)) & ((termmset = $true) | (sK461 != (sub @ sK461 @ id))) & (termmset_gthm = $true) & (pushprop_lem0_gthm = $true) & ((substmonoid_lthm = $true) | (((sK462 != (comp @ sK462 @ id)) | ((comp @ id @ sK463) != sK463)) & ! [X12 : subst] : ((comp @ id @ X12) = X12) & ! [X13 : subst] : ((comp @ X13 @ id) = X13))) & ((! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X15 : subst] : ((comp @ id @ X15) = X15)) | ((comp @ id @ sK464) != sK464) | ((comp @ sK465 @ id) != sK465) | (substmonoid_lthm != $true)) & ! [X18 : d_term] : (d_one = X18) & ((d2subst @ d_id) = id) & ((hoasinduction_lem2 = $true) | ($true != sP58)) & (($true = sP58) | (hoasinduction_lem2 != $true)) & (pushprop_lem1v2_gthm = $true) & (($true = sP56) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP56)) & (pushprop_lem3v2_lthm = $true) & ((induction2_lthm = $true) | ($true != sP55)) & (($true = sP55) | (induction2_lthm != $true)) & ((sP51 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP51 != $true)) & ((d2subst @ d_id) = sh) & (hoaslamnotap_gthm = $true) & ! [X19 : d_subst,X20 : d_subst] : (((d2subst @ X19) != (d2subst @ X20)) | (X19 = X20)) & ((hoasinduction_lem3b = $true) | ! [X22 : subst > term > term] : ((X22 @ sh @ one) != (sub @ sK466 @ (push @ one @ sh)))) & (! [X23 : term] : ((sK467 @ X23 @ sh @ one) = (sub @ X23 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((shinj = $true) | (((sub @ sK469 @ sh) = (sub @ sK468 @ sh)) & (sK469 != sK468))) & (! [X27 : term,X28 : term] : (((sub @ X28 @ sh) != (sub @ X27 @ sh)) | (X27 = X28)) | (shinj != $true)) & (($true = sP45) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP45)) & ((induction2lem = $true) | ($true != sP42)) & (($true = sP42) | (induction2lem != $true)) & ((hoasinduction_lem0 = $true) | ! [X30 : term > $o] : ((sK470 @ id @ (sK471 @ X30) @ id) != (X30 @ (sK471 @ X30)))) & (! [X32 : subst > term > subst > $o] : ! [X34 : term] : ((sK472 @ X32 @ X34) = (X32 @ id @ X34 @ id)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP40)) & (($true = sP40) | (hoasinduction_lem3aa_lthm != $true)) & (hoaslamnotap_lthm = $true) & (((sub @ one @ (push @ sK474 @ sK473)) != sK474) | ((comp @ sh @ (push @ sK476 @ sK475)) != sK475) | ! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst,X43 : term] : (($true = (X41 @ (sub @ X43 @ (push @ X39 @ X42)))) | ($true != (X40 @ X43))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X44 : subst,X45 : term] : ((sub @ one @ (push @ X45 @ X44)) = X45) & ! [X46 : subst,X47 : term] : ((comp @ sh @ (push @ X47 @ X46)) = X46) & (($true != (sK479 @ (sub @ sK481 @ (push @ sK477 @ sK480)))) & ($true = (sK478 @ sK481))))) & ((sP39 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP39 != $true)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X53 : term] : ((sub @ X53 @ id) = X53) | (axvarid != $true)) & ((axvarid = $true) | (sK482 != (sub @ sK482 @ id))) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3v2_lthm != $true)) & (axabs = $true) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & (ulamvarsh = $true) & ! [X55 : d_subst] : (d_id = X55) & ((induction = $true) | (($true != (sK483 @ sK484)) & ! [X58 : term] : (($true = (sK483 @ X58)) | ($true != (var @ X58))) & ($true != (sP36 @ sK483)) & ! [X59 : term] : (($true = (sK483 @ (lam @ X59))) | ($true != (sK483 @ X59))))) & (! [X60 : term > $o,X61 : term] : (($true = (X60 @ X61)) | (($true != (X60 @ (sK485 @ X60))) & ((var @ (sK485 @ X60)) = $true)) | ($true = (sP36 @ X60)) | (($true != (X60 @ (lam @ (sK486 @ X60)))) & ($true = (X60 @ (sK486 @ X60))))) | (induction != $true)) & (pushprop_lem2v2_lthm = $true) & (hoaslaminj_gthm = $true) & ((axidr = $true) | (sK487 != (comp @ sK487 @ id))) & (! [X65 : subst] : ((comp @ X65 @ id) = X65) | (axidr != $true)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & (($true = sP35) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP35)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem2v2 = $true) | ($true != sP31)) & (($true = sP31) | (hoasinduction_lem2v2 != $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_gthm = $true) & (induction2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2a != $true)) & (axclos = $true) & (lamnotap != $true) & ((axvarcons = $true) | (sK489 != (sub @ one @ (push @ sK489 @ sK488)))) & (! [X68 : subst,X69 : term] : ((sub @ one @ (push @ X69 @ X68)) = X69) | (axvarcons != $true)) & (axvarshift = $true) & (pushprop_lem1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lem2v2_gthm = $true) & ((pushprop_lem0_lthm = $true) | ! [X73 : term > $o] : ((X73 @ (sK493 @ X73)) != (sK491 @ (sub @ (sK493 @ X73) @ (push @ sK490 @ sK492))))) & (! [X75 : term,X76 : term > $o,X77 : subst] : ! [X79 : term] : ((X76 @ (sub @ X79 @ (push @ X75 @ X77))) = (sK494 @ X77 @ X76 @ X75 @ X79)) | (pushprop_lem0_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X80 : term,X81 : term,X82 : term,X83 : term] : (((ap @ X83 @ X82) != (ap @ X80 @ X81)) | (X80 = X83)) & (((ap @ (sub @ sK497 @ id) @ sK496) = (ap @ (sub @ sK495 @ id) @ sK498)) & (sK495 != sK497)) & ! [X88 : term] : ((sub @ X88 @ id) = X88))) & ((((ap @ sK499 @ sK500) = (ap @ sK502 @ sK501)) & (sK502 != sK499)) | ! [X93 : term,X94 : term,X95 : term,X96 : term] : (((ap @ (sub @ X95 @ id) @ X94) != (ap @ (sub @ X93 @ id) @ X96)) | (X93 = X95)) | ((sub @ sK503 @ id) != sK503) | (hoasapinj1_lthm != $true)) & (($true = sP28) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP28)) & (hoaslamnotvar_gthm = $true) & ((sP26 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP26 != $true)) & ((hoaslamnotap = $true) | (! [X101 : subst,X102 : term,X103 : subst] : ((sK504 @ (comp @ X101 @ X103) @ (sub @ X102 @ X103)) = (sub @ (sK504 @ X101 @ X102) @ X103)) & ((ap @ (sub @ sK505 @ id) @ sK506) = (lam @ (sK504 @ sh @ one))))) & (! [X104 : subst > term > term,X105 : term,X106 : term] : (((X104 @ (comp @ (sK507 @ X104) @ (sK509 @ X104)) @ (sub @ (sK508 @ X104) @ (sK509 @ X104))) != (sub @ (X104 @ (sK507 @ X104) @ (sK508 @ X104)) @ (sK509 @ X104))) | ((lam @ (X104 @ sh @ one)) != (ap @ (sub @ X105 @ id) @ X106))) | (hoaslamnotap != $true)) & (pushprop = $true) & (pushprop_lthm = $true) & (! [X110 : term,X111 : term,X112 : term,X113 : term] : (((ap @ (sub @ X113 @ id) @ X111) != (ap @ (sub @ X112 @ id) @ X110)) | (X112 = X113)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | (((ap @ (sub @ sK512 @ id) @ sK510) = (ap @ (sub @ sK513 @ id) @ sK511)) & (sK512 != sK513))) & (apnotvar = $true) & (hoasinduction_lem2_gthm = $true) & (($true = sP24) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP24)) & (($true = sP23) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP23)) & (pushprop_lem2v2 = $true) & ((substmonoid = $true) | (sK514 != (comp @ sK514 @ id)) | (sK515 != (comp @ id @ sK515))) & ((! [X120 : subst] : ((comp @ X120 @ id) = X120) & ! [X121 : subst] : ((comp @ id @ X121) = X121)) | (substmonoid != $true)) & (! [X122 : subst,X123 : term,X124 : term > $o] : ! [X126 : term] : ((X124 @ (sub @ X126 @ (push @ X123 @ X122))) = (sK516 @ X124 @ X123 @ X122 @ X126)) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X130 : term > $o] : ((X130 @ (sK520 @ X130)) != (sK519 @ (sub @ (sK520 @ X130) @ (push @ sK518 @ sK517))))) & ((hoasinduction_lem3b_lthm = $true) | ! [X133 : subst > term > term] : ((sub @ sK521 @ (push @ one @ sh)) != (X133 @ sh @ one))) & (! [X134 : term] : ((sK522 @ X134 @ sh @ one) = (sub @ X134 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & (($true = sP22) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP22)) & (! [X136 : term > $o,X137 : term,X138 : subst,X139 : term,X140 : term > $o] : (($true = (X140 @ (sub @ X137 @ (push @ X139 @ X138)))) | ($true != (X136 @ X137))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | (($true != (sK527 @ (sub @ sK524 @ (push @ sK526 @ sK525)))) & ($true = (sK523 @ sK524)))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X146 : term] : ! [X148 : subst,X149 : term] : ((sub @ X146 @ (push @ X149 @ X148)) = (sK528 @ X146 @ X148 @ X149)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X151 : subst > term > term] : ((sub @ sK529 @ (push @ (sK531 @ X151) @ (sK530 @ X151))) != (X151 @ (sK530 @ X151) @ (sK531 @ X151)))) & (! [X154 : term > $o,X155 : subst,X156 : term > $o,X157 : term] : (($true = (X156 @ one)) | ($true != (X154 @ X157)) | ((X154 @ (sub @ (sK532 @ X157 @ X156 @ X155 @ X154) @ (push @ X157 @ X155))) != (X156 @ (sK532 @ X157 @ X156 @ X155 @ X154)))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | (($true != (sK535 @ one)) & ($true = (sK533 @ sK536)) & ! [X163 : term] : ((sK533 @ (sub @ X163 @ (push @ sK536 @ sK534))) = (sK535 @ X163)))) & (ulamvar1 != $true) & (axmap = $true) & ((var @ (d2term @ d_one)) != $true) & (hoaslamnotvar_lthm = $true) & (one = (d2term @ d_one)) & (! [X164 : term] : ! [X166 : term,X167 : subst] : ((sub @ X164 @ (push @ X166 @ X167)) = (sK537 @ X164 @ X167 @ X166)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X169 : subst > term > term] : ((X169 @ (sK540 @ X169) @ (sK539 @ X169)) != (sub @ sK538 @ (push @ (sK539 @ X169) @ (sK540 @ X169))))) & (! [X172 : subst > term > term,X173 : subst,X174 : term,X175 : subst > term > term] : (((X172 @ X173 @ X174) = (X175 @ X173 @ X174)) | ((lam @ (X172 @ sh @ one)) != (lam @ (X175 @ sh @ one))) | ((sub @ (X172 @ (sK541 @ X172) @ (sK543 @ X172)) @ (sK542 @ X172)) != (X172 @ (comp @ (sK541 @ X172) @ (sK542 @ X172)) @ (sub @ (sK543 @ X172) @ (sK542 @ X172)))) | ((X175 @ (comp @ (sK546 @ X175) @ (sK544 @ X175)) @ (sub @ (sK545 @ X175) @ (sK544 @ X175))) != (sub @ (X175 @ (sK546 @ X175) @ (sK545 @ X175)) @ (sK544 @ X175)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | (((sK550 @ sK548 @ sK549) != (sK547 @ sK548 @ sK549)) & ((lam @ (sK547 @ sh @ one)) = (lam @ (sK550 @ sh @ one))) & ! [X186 : subst,X187 : subst,X188 : term] : ((sub @ (sK547 @ X186 @ X188) @ X187) = (sK547 @ (comp @ X186 @ X187) @ (sub @ X188 @ X187))) & ! [X189 : subst,X190 : term,X191 : subst] : ((sK550 @ (comp @ X191 @ X189) @ (sub @ X190 @ X189)) = (sub @ (sK550 @ X191 @ X190) @ X189)))) & ((termmset_lthm = $true) | (((sub @ sK551 @ id) != sK551) & ! [X193 : term] : ((sub @ X193 @ id) = X193))) & (! [X194 : term] : ((sub @ X194 @ id) = X194) | (sK552 != (sub @ sK552 @ id)) | (termmset_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP17)) & (($true = sP17) | (hoasinduction_lem3_lthm != $true)) & ((axidl = $true) | ((comp @ id @ sK553) != sK553)) & (! [X197 : subst] : ((comp @ id @ X197) = X197) | (axidl != $true)) & ((hoasinduction_lthm = $true) | ($true != sP14)) & (($true = sP14) | (hoasinduction_lthm != $true)) & (! [X198 : term,X199 : term,X200 : term,X201 : term] : (((ap @ (sub @ X199 @ id) @ X201) != (ap @ (sub @ X198 @ id) @ X200)) | (X200 = X201)) | (((ap @ sK554 @ sK557) = (ap @ sK556 @ sK555)) & (sK555 != sK557)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | ((((ap @ (sub @ sK559 @ id) @ sK561) = (ap @ (sub @ sK558 @ id) @ sK560)) & (sK560 != sK561)) & ! [X210 : term,X211 : term,X212 : term,X213 : term] : (((ap @ X210 @ X213) != (ap @ X212 @ X211)) | (X211 = X213)))) & (hoasapinj1_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ! [X215 : term > $o] : ((X215 @ (sK563 @ X215)) != (sK562 @ id @ (sK563 @ X215) @ id))) & (! [X217 : subst > term > subst > $o] : ! [X219 : term] : ((sK564 @ X217 @ X219) = (X217 @ id @ X219 @ id)) | (hoasinduction_lem0_lthm != $true)) & (axassoc = $true) & (substmonoid_gthm = $true) & ((hoasinduction_no_psi_cond = $true) | (! [X222 : term,X223 : term] : (($true = (sK566 @ id @ (ap @ (sub @ X222 @ id) @ X223) @ id)) | ($true != (sK566 @ id @ X222 @ id)) | ($true != (sK566 @ id @ X223 @ id))) & ($true != (sP5 @ sK566)) & ($true != (sK566 @ id @ sK565 @ id)))) & (! [X224 : term,X225 : subst > term > subst > $o] : ((($true != (X225 @ id @ (ap @ (sub @ (sK567 @ X225) @ id) @ (sK568 @ X225)) @ id)) & ($true = (X225 @ id @ (sK567 @ X225) @ id)) & ($true = (X225 @ id @ (sK568 @ X225) @ id))) | ($true = (sP5 @ X225)) | ($true = (X225 @ id @ X224 @ id))) | (hoasinduction_no_psi_cond != $true)) & (induction2lem_gthm = $true) & ((axshiftcons = $true) | ((comp @ sh @ (push @ sK569 @ sK570)) != sK570)) & (! [X230 : term,X231 : subst] : ((comp @ sh @ (push @ X230 @ X231)) = X231) | (axshiftcons != $true)) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3 = $true) | ($true != sP4)) & (($true = sP4) | (hoasinduction_lem3 != $true)) & (hoasapnotvar = $true) & ! [X232 : subst] : ((d2subst @ (sK571 @ X232)) = X232) & (! [X234 : subst] : ((push @ (sub @ one @ X234) @ (comp @ sh @ X234)) = X234) | (axscons != $true)) & ((axscons = $true) | ((push @ (sub @ one @ sK572) @ (comp @ sh @ sK572)) != sK572)) & (pushprop_lthm_orig = $true) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X238 = X239) | ((ap @ X238 @ X236) != (ap @ X239 @ X237))) | (apinj1 != $true)) & ((apinj1 = $true) | ((sK575 != sK576) & ((ap @ sK576 @ sK574) = (ap @ sK575 @ sK573)))) & (hoasinduction_lem3b_gthm = $true) & (hoasapnotvar_gthm = $true) & (($true = sP2) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP2)) & (! [X244 : term,X245 : term] : ((X244 = X245) | ((lam @ X244) != (lam @ X245))) | (laminj != $true)) & ((laminj = $true) | ((sK578 != sK577) & ((lam @ sK577) = (lam @ sK578)))) & ((hoasinduction_lem3aa = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3aa != $true)) & (hoasinduction_lem2v2_gthm = $true) & ! [X248 : term] : ((d2term @ (sK579 @ X248)) = X248) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X250 : term,X251 : term,X252 : term,X253 : term] : ((X250 = X252) | ((ap @ (sub @ X253 @ id) @ X252) != (ap @ (sub @ X251 @ id) @ X250))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ((sK580 != sK582) & ((ap @ (sub @ sK581 @ id) @ sK580) = (ap @ (sub @ sK583 @ id) @ sK582)))) & ! [X258 : d_term,X259 : d_term] : (((d2term @ X258) != (d2term @ X259)) | (X258 = X259)) & (! [X260 : term,X261 : term > $o,X262 : term > $o,X263 : subst] : (($true != (X261 @ X260)) | ((X262 @ one) = $true) | ((X261 @ (sub @ (sK584 @ X263 @ X262 @ X261 @ X260) @ (push @ X260 @ X263))) != (X262 @ (sK584 @ X263 @ X262 @ X261 @ X260)))) | (sK585 != (sub @ one @ (push @ sK585 @ sK586))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | ((($true = (sK588 @ sK587)) & ($true != (sK589 @ one)) & ! [X271 : term] : ((sK589 @ X271) = (sK588 @ (sub @ X271 @ (push @ sK587 @ sK590))))) & ! [X272 : term,X273 : subst] : ((sub @ one @ (push @ X272 @ X273)) = X272)))),
% 2.56/1.94 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])).
% 2.56/1.94 thf(f476,plain,(
% 2.56/1.94 ? [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) = (ap @ X3 @ X1)) & (X0 != X1)) => (((ap @ sK459 @ sK457) = (ap @ sK460 @ sK458)) & (sK457 != sK458))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f477,plain,(
% 2.56/1.94 ? [X9 : term] : ((sub @ X9 @ id) != X9) => (sK461 != (sub @ sK461 @ id))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f478,plain,(
% 2.56/1.94 ? [X10 : subst] : ((comp @ X10 @ id) != X10) => (sK462 != (comp @ sK462 @ id))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f479,plain,(
% 2.56/1.94 ? [X11 : subst] : ((comp @ id @ X11) != X11) => ((comp @ id @ sK463) != sK463)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f480,plain,(
% 2.56/1.94 ? [X16 : subst] : ((comp @ id @ X16) != X16) => ((comp @ id @ sK464) != sK464)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f481,plain,(
% 2.56/1.94 ? [X17 : subst] : ((comp @ X17 @ id) != X17) => ((comp @ sK465 @ id) != sK465)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f482,plain,(
% 2.56/1.94 ? [X21 : term] : ! [X22 : subst > term > term] : ((sub @ X21 @ (push @ one @ sh)) != (X22 @ sh @ one)) => ! [X22 : subst > term > term] : ((X22 @ sh @ one) != (sub @ sK466 @ (push @ one @ sh)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f483,plain,(
% 2.56/1.94 ! [X23 : term] : (? [X24 : subst > term > term] : ((X24 @ sh @ one) = (sub @ X23 @ (push @ one @ sh))) => ((sK467 @ X23 @ sh @ one) = (sub @ X23 @ (push @ one @ sh))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f484,plain,(
% 2.56/1.94 ? [X25 : term,X26 : term] : (((sub @ X25 @ sh) = (sub @ X26 @ sh)) & (X25 != X26)) => (((sub @ sK469 @ sh) = (sub @ sK468 @ sh)) & (sK469 != sK468))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f485,plain,(
% 2.56/1.94 ? [X29 : subst > term > subst > $o] : ! [X30 : term > $o] : ? [X31 : term] : ((X29 @ id @ X31 @ id) != (X30 @ X31)) => ! [X30 : term > $o] : ? [X31 : term] : ((sK470 @ id @ X31 @ id) != (X30 @ X31))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f486,plain,(
% 2.56/1.94 ! [X30 : term > $o] : (? [X31 : term] : ((sK470 @ id @ X31 @ id) != (X30 @ X31)) => ((sK470 @ id @ (sK471 @ X30) @ id) != (X30 @ (sK471 @ X30))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f487,plain,(
% 2.56/1.94 ! [X32 : subst > term > subst > $o] : (? [X33 : term > $o] : ! [X34 : term] : ((X33 @ X34) = (X32 @ id @ X34 @ id)) => ! [X34 : term] : ((sK472 @ X32 @ X34) = (X32 @ id @ X34 @ id)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f488,plain,(
% 2.56/1.94 ? [X35 : subst,X36 : term] : ((sub @ one @ (push @ X36 @ X35)) != X36) => ((sub @ one @ (push @ sK474 @ sK473)) != sK474)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f489,plain,(
% 2.56/1.94 ? [X37 : subst,X38 : term] : ((comp @ sh @ (push @ X38 @ X37)) != X37) => ((comp @ sh @ (push @ sK476 @ sK475)) != sK475)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f490,plain,(
% 2.56/1.94 ? [X48 : term,X49 : term > $o,X50 : term > $o,X51 : subst,X52 : term] : (($true != (X50 @ (sub @ X52 @ (push @ X48 @ X51)))) & ($true = (X49 @ X52))) => (($true != (sK479 @ (sub @ sK481 @ (push @ sK477 @ sK480)))) & ($true = (sK478 @ sK481)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f491,plain,(
% 2.56/1.94 ? [X54 : term] : ((sub @ X54 @ id) != X54) => (sK482 != (sub @ sK482 @ id))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f492,plain,(
% 2.56/1.94 ? [X56 : term > $o,X57 : term] : (($true != (X56 @ X57)) & ! [X58 : term] : (($true = (X56 @ X58)) | ($true != (var @ X58))) & ($true != (sP36 @ X56)) & ! [X59 : term] : (($true = (X56 @ (lam @ X59))) | ($true != (X56 @ X59)))) => (($true != (sK483 @ sK484)) & ! [X58 : term] : (($true = (sK483 @ X58)) | ($true != (var @ X58))) & ($true != (sP36 @ sK483)) & ! [X59 : term] : (($true = (sK483 @ (lam @ X59))) | ($true != (sK483 @ X59))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f493,plain,(
% 2.56/1.94 ! [X60 : term > $o] : (? [X62 : term] : (($true != (X60 @ X62)) & ($true = (var @ X62))) => (($true != (X60 @ (sK485 @ X60))) & ((var @ (sK485 @ X60)) = $true)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f494,plain,(
% 2.56/1.94 ! [X60 : term > $o] : (? [X63 : term] : (($true != (X60 @ (lam @ X63))) & ((X60 @ X63) = $true)) => (($true != (X60 @ (lam @ (sK486 @ X60)))) & ($true = (X60 @ (sK486 @ X60)))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f495,plain,(
% 2.56/1.94 ? [X64 : subst] : ((comp @ X64 @ id) != X64) => (sK487 != (comp @ sK487 @ id))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f496,plain,(
% 2.56/1.94 ? [X66 : subst,X67 : term] : ((sub @ one @ (push @ X67 @ X66)) != X67) => (sK489 != (sub @ one @ (push @ sK489 @ sK488)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f497,plain,(
% 2.56/1.94 ? [X70 : term,X71 : term > $o,X72 : subst] : ! [X73 : term > $o] : ? [X74 : term] : ((X73 @ X74) != (X71 @ (sub @ X74 @ (push @ X70 @ X72)))) => ! [X73 : term > $o] : ? [X74 : term] : ((sK491 @ (sub @ X74 @ (push @ sK490 @ sK492))) != (X73 @ X74))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f498,plain,(
% 2.56/1.94 ! [X73 : term > $o] : (? [X74 : term] : ((sK491 @ (sub @ X74 @ (push @ sK490 @ sK492))) != (X73 @ X74)) => ((X73 @ (sK493 @ X73)) != (sK491 @ (sub @ (sK493 @ X73) @ (push @ sK490 @ sK492)))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f499,plain,(
% 2.56/1.94 ! [X75 : term,X76 : term > $o,X77 : subst] : (? [X78 : term > $o] : ! [X79 : term] : ((X78 @ X79) = (X76 @ (sub @ X79 @ (push @ X75 @ X77)))) => ! [X79 : term] : ((X76 @ (sub @ X79 @ (push @ X75 @ X77))) = (sK494 @ X77 @ X76 @ X75 @ X79)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f500,plain,(
% 2.56/1.94 ? [X84 : term,X85 : term,X86 : term,X87 : term] : (((ap @ (sub @ X86 @ id) @ X85) = (ap @ (sub @ X84 @ id) @ X87)) & (X84 != X86)) => (((ap @ (sub @ sK497 @ id) @ sK496) = (ap @ (sub @ sK495 @ id) @ sK498)) & (sK495 != sK497))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f501,plain,(
% 2.56/1.94 ? [X89 : term,X90 : term,X91 : term,X92 : term] : (((ap @ X89 @ X90) = (ap @ X92 @ X91)) & (X89 != X92)) => (((ap @ sK499 @ sK500) = (ap @ sK502 @ sK501)) & (sK502 != sK499))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f502,plain,(
% 2.56/1.94 ? [X97 : term] : ((sub @ X97 @ id) != X97) => ((sub @ sK503 @ id) != sK503)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f503,plain,(
% 2.56/1.94 ? [X98 : subst > term > term,X99 : term,X100 : term] : (! [X101 : subst,X102 : term,X103 : subst] : ((X98 @ (comp @ X101 @ X103) @ (sub @ X102 @ X103)) = (sub @ (X98 @ X101 @ X102) @ X103)) & ((lam @ (X98 @ sh @ one)) = (ap @ (sub @ X99 @ id) @ X100))) => (! [X103 : subst,X102 : term,X101 : subst] : ((sK504 @ (comp @ X101 @ X103) @ (sub @ X102 @ X103)) = (sub @ (sK504 @ X101 @ X102) @ X103)) & ((ap @ (sub @ sK505 @ id) @ sK506) = (lam @ (sK504 @ sh @ one))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f504,plain,(
% 2.56/1.94 ! [X104 : subst > term > term] : (? [X107 : subst,X108 : term,X109 : subst] : ((X104 @ (comp @ X107 @ X109) @ (sub @ X108 @ X109)) != (sub @ (X104 @ X107 @ X108) @ X109)) => ((X104 @ (comp @ (sK507 @ X104) @ (sK509 @ X104)) @ (sub @ (sK508 @ X104) @ (sK509 @ X104))) != (sub @ (X104 @ (sK507 @ X104) @ (sK508 @ X104)) @ (sK509 @ X104))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f505,plain,(
% 2.56/1.94 ? [X114 : term,X115 : term,X116 : term,X117 : term] : (((ap @ (sub @ X116 @ id) @ X114) = (ap @ (sub @ X117 @ id) @ X115)) & (X116 != X117)) => (((ap @ (sub @ sK512 @ id) @ sK510) = (ap @ (sub @ sK513 @ id) @ sK511)) & (sK512 != sK513))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f506,plain,(
% 2.56/1.94 ? [X118 : subst] : ((comp @ X118 @ id) != X118) => (sK514 != (comp @ sK514 @ id))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f507,plain,(
% 2.56/1.94 ? [X119 : subst] : ((comp @ id @ X119) != X119) => (sK515 != (comp @ id @ sK515))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f508,plain,(
% 2.56/1.94 ! [X122 : subst,X123 : term,X124 : term > $o] : (? [X125 : term > $o] : ! [X126 : term] : ((X124 @ (sub @ X126 @ (push @ X123 @ X122))) = (X125 @ X126)) => ! [X126 : term] : ((X124 @ (sub @ X126 @ (push @ X123 @ X122))) = (sK516 @ X124 @ X123 @ X122 @ X126)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f509,plain,(
% 2.56/1.94 ? [X127 : subst,X128 : term,X129 : term > $o] : ! [X130 : term > $o] : ? [X131 : term] : ((X129 @ (sub @ X131 @ (push @ X128 @ X127))) != (X130 @ X131)) => ! [X130 : term > $o] : ? [X131 : term] : ((sK519 @ (sub @ X131 @ (push @ sK518 @ sK517))) != (X130 @ X131))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f510,plain,(
% 2.56/1.94 ! [X130 : term > $o] : (? [X131 : term] : ((sK519 @ (sub @ X131 @ (push @ sK518 @ sK517))) != (X130 @ X131)) => ((X130 @ (sK520 @ X130)) != (sK519 @ (sub @ (sK520 @ X130) @ (push @ sK518 @ sK517)))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f511,plain,(
% 2.56/1.94 ? [X132 : term] : ! [X133 : subst > term > term] : ((sub @ X132 @ (push @ one @ sh)) != (X133 @ sh @ one)) => ! [X133 : subst > term > term] : ((sub @ sK521 @ (push @ one @ sh)) != (X133 @ sh @ one))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f512,plain,(
% 2.56/1.94 ! [X134 : term] : (? [X135 : subst > term > term] : ((X135 @ sh @ one) = (sub @ X134 @ (push @ one @ sh))) => ((sK522 @ X134 @ sh @ one) = (sub @ X134 @ (push @ one @ sh))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f513,plain,(
% 2.56/1.94 ? [X141 : term > $o,X142 : term,X143 : subst,X144 : term,X145 : term > $o] : (($true != (X145 @ (sub @ X142 @ (push @ X144 @ X143)))) & ($true = (X141 @ X142))) => (($true != (sK527 @ (sub @ sK524 @ (push @ sK526 @ sK525)))) & ($true = (sK523 @ sK524)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f514,plain,(
% 2.56/1.94 ! [X146 : term] : (? [X147 : subst > term > term] : ! [X148 : subst,X149 : term] : ((sub @ X146 @ (push @ X149 @ X148)) = (X147 @ X148 @ X149)) => ! [X149 : term,X148 : subst] : ((sub @ X146 @ (push @ X149 @ X148)) = (sK528 @ X146 @ X148 @ X149)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f515,plain,(
% 2.56/1.94 ? [X150 : term] : ! [X151 : subst > term > term] : ? [X152 : subst,X153 : term] : ((sub @ X150 @ (push @ X153 @ X152)) != (X151 @ X152 @ X153)) => ! [X151 : subst > term > term] : ? [X153 : term,X152 : subst] : ((sub @ sK529 @ (push @ X153 @ X152)) != (X151 @ X152 @ X153))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f516,plain,(
% 2.56/1.94 ! [X151 : subst > term > term] : (? [X153 : term,X152 : subst] : ((sub @ sK529 @ (push @ X153 @ X152)) != (X151 @ X152 @ X153)) => ((sub @ sK529 @ (push @ (sK531 @ X151) @ (sK530 @ X151))) != (X151 @ (sK530 @ X151) @ (sK531 @ X151))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f517,plain,(
% 2.56/1.94 ! [X154 : term > $o,X155 : subst,X156 : term > $o,X157 : term] : (? [X158 : term] : ((X154 @ (sub @ X158 @ (push @ X157 @ X155))) != (X156 @ X158)) => ((X154 @ (sub @ (sK532 @ X157 @ X156 @ X155 @ X154) @ (push @ X157 @ X155))) != (X156 @ (sK532 @ X157 @ X156 @ X155 @ X154))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f518,plain,(
% 2.56/1.94 ? [X159 : term > $o,X160 : subst,X161 : term > $o,X162 : term] : (($true != (X161 @ one)) & ($true = (X159 @ X162)) & ! [X163 : term] : ((X159 @ (sub @ X163 @ (push @ X162 @ X160))) = (X161 @ X163))) => (($true != (sK535 @ one)) & ($true = (sK533 @ sK536)) & ! [X163 : term] : ((sK533 @ (sub @ X163 @ (push @ sK536 @ sK534))) = (sK535 @ X163)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f519,plain,(
% 2.56/1.94 ! [X164 : term] : (? [X165 : subst > term > term] : ! [X166 : term,X167 : subst] : ((sub @ X164 @ (push @ X166 @ X167)) = (X165 @ X167 @ X166)) => ! [X167 : subst,X166 : term] : ((sub @ X164 @ (push @ X166 @ X167)) = (sK537 @ X164 @ X167 @ X166)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f520,plain,(
% 2.56/1.94 ? [X168 : term] : ! [X169 : subst > term > term] : ? [X170 : term,X171 : subst] : ((X169 @ X171 @ X170) != (sub @ X168 @ (push @ X170 @ X171))) => ! [X169 : subst > term > term] : ? [X171 : subst,X170 : term] : ((X169 @ X171 @ X170) != (sub @ sK538 @ (push @ X170 @ X171)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f521,plain,(
% 2.56/1.94 ! [X169 : subst > term > term] : (? [X171 : subst,X170 : term] : ((X169 @ X171 @ X170) != (sub @ sK538 @ (push @ X170 @ X171))) => ((X169 @ (sK540 @ X169) @ (sK539 @ X169)) != (sub @ sK538 @ (push @ (sK539 @ X169) @ (sK540 @ X169)))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f522,plain,(
% 2.56/1.94 ! [X172 : subst > term > term] : (? [X176 : subst,X177 : subst,X178 : term] : ((X172 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177)) != (sub @ (X172 @ X176 @ X178) @ X177)) => ((sub @ (X172 @ (sK541 @ X172) @ (sK543 @ X172)) @ (sK542 @ X172)) != (X172 @ (comp @ (sK541 @ X172) @ (sK542 @ X172)) @ (sub @ (sK543 @ X172) @ (sK542 @ X172)))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f523,plain,(
% 2.56/1.94 ! [X175 : subst > term > term] : (? [X179 : subst,X180 : term,X181 : subst] : ((sub @ (X175 @ X181 @ X180) @ X179) != (X175 @ (comp @ X181 @ X179) @ (sub @ X180 @ X179))) => ((X175 @ (comp @ (sK546 @ X175) @ (sK544 @ X175)) @ (sub @ (sK545 @ X175) @ (sK544 @ X175))) != (sub @ (X175 @ (sK546 @ X175) @ (sK545 @ X175)) @ (sK544 @ X175))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f524,plain,(
% 2.56/1.94 ? [X182 : subst > term > term,X183 : subst,X184 : term,X185 : subst > term > term] : (((X185 @ X183 @ X184) != (X182 @ X183 @ X184)) & ((lam @ (X182 @ sh @ one)) = (lam @ (X185 @ sh @ one))) & ! [X186 : subst,X187 : subst,X188 : term] : ((X182 @ (comp @ X186 @ X187) @ (sub @ X188 @ X187)) = (sub @ (X182 @ X186 @ X188) @ X187)) & ! [X189 : subst,X190 : term,X191 : subst] : ((X185 @ (comp @ X191 @ X189) @ (sub @ X190 @ X189)) = (sub @ (X185 @ X191 @ X190) @ X189))) => (((sK550 @ sK548 @ sK549) != (sK547 @ sK548 @ sK549)) & ((lam @ (sK547 @ sh @ one)) = (lam @ (sK550 @ sh @ one))) & ! [X188 : term,X187 : subst,X186 : subst] : ((sub @ (sK547 @ X186 @ X188) @ X187) = (sK547 @ (comp @ X186 @ X187) @ (sub @ X188 @ X187))) & ! [X191 : subst,X190 : term,X189 : subst] : ((sK550 @ (comp @ X191 @ X189) @ (sub @ X190 @ X189)) = (sub @ (sK550 @ X191 @ X190) @ X189)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f525,plain,(
% 2.56/1.94 ? [X192 : term] : ((sub @ X192 @ id) != X192) => ((sub @ sK551 @ id) != sK551)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f526,plain,(
% 2.56/1.94 ? [X195 : term] : ((sub @ X195 @ id) != X195) => (sK552 != (sub @ sK552 @ id))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f527,plain,(
% 2.56/1.94 ? [X196 : subst] : ((comp @ id @ X196) != X196) => ((comp @ id @ sK553) != sK553)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f528,plain,(
% 2.56/1.94 ? [X202 : term,X203 : term,X204 : term,X205 : term] : (((ap @ X204 @ X203) = (ap @ X202 @ X205)) & (X203 != X205)) => (((ap @ sK554 @ sK557) = (ap @ sK556 @ sK555)) & (sK555 != sK557))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f529,plain,(
% 2.56/1.94 ? [X206 : term,X207 : term,X208 : term,X209 : term] : (((ap @ (sub @ X207 @ id) @ X209) = (ap @ (sub @ X206 @ id) @ X208)) & (X208 != X209)) => (((ap @ (sub @ sK559 @ id) @ sK561) = (ap @ (sub @ sK558 @ id) @ sK560)) & (sK560 != sK561))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f530,plain,(
% 2.56/1.94 ? [X214 : subst > term > subst > $o] : ! [X215 : term > $o] : ? [X216 : term] : ((X214 @ id @ X216 @ id) != (X215 @ X216)) => ! [X215 : term > $o] : ? [X216 : term] : ((X215 @ X216) != (sK562 @ id @ X216 @ id))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f531,plain,(
% 2.56/1.94 ! [X215 : term > $o] : (? [X216 : term] : ((X215 @ X216) != (sK562 @ id @ X216 @ id)) => ((X215 @ (sK563 @ X215)) != (sK562 @ id @ (sK563 @ X215) @ id)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f532,plain,(
% 2.56/1.94 ! [X217 : subst > term > subst > $o] : (? [X218 : term > $o] : ! [X219 : term] : ((X218 @ X219) = (X217 @ id @ X219 @ id)) => ! [X219 : term] : ((sK564 @ X217 @ X219) = (X217 @ id @ X219 @ id)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f533,plain,(
% 2.56/1.94 ? [X220 : term,X221 : subst > term > subst > $o] : (! [X222 : term,X223 : term] : (($true = (X221 @ id @ (ap @ (sub @ X222 @ id) @ X223) @ id)) | ($true != (X221 @ id @ X222 @ id)) | ($true != (X221 @ id @ X223 @ id))) & ($true != (sP5 @ X221)) & ($true != (X221 @ id @ X220 @ id))) => (! [X223 : term,X222 : term] : (($true = (sK566 @ id @ (ap @ (sub @ X222 @ id) @ X223) @ id)) | ($true != (sK566 @ id @ X222 @ id)) | ($true != (sK566 @ id @ X223 @ id))) & ($true != (sP5 @ sK566)) & ($true != (sK566 @ id @ sK565 @ id)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f534,plain,(
% 2.56/1.94 ! [X225 : subst > term > subst > $o] : (? [X226 : term,X227 : term] : (($true != (X225 @ id @ (ap @ (sub @ X226 @ id) @ X227) @ id)) & ($true = (X225 @ id @ X226 @ id)) & ($true = (X225 @ id @ X227 @ id))) => (($true != (X225 @ id @ (ap @ (sub @ (sK567 @ X225) @ id) @ (sK568 @ X225)) @ id)) & ($true = (X225 @ id @ (sK567 @ X225) @ id)) & ($true = (X225 @ id @ (sK568 @ X225) @ id))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f535,plain,(
% 2.56/1.94 ? [X228 : term,X229 : subst] : ((comp @ sh @ (push @ X228 @ X229)) != X229) => ((comp @ sh @ (push @ sK569 @ sK570)) != sK570)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f536,plain,(
% 2.56/1.94 ! [X232 : subst] : (? [X233 : d_subst] : ((d2subst @ X233) = X232) => ((d2subst @ (sK571 @ X232)) = X232))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f537,plain,(
% 2.56/1.94 ? [X235 : subst] : ((push @ (sub @ one @ X235) @ (comp @ sh @ X235)) != X235) => ((push @ (sub @ one @ sK572) @ (comp @ sh @ sK572)) != sK572)),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f538,plain,(
% 2.56/1.94 ? [X240 : term,X241 : term,X242 : term,X243 : term] : ((X242 != X243) & ((ap @ X242 @ X240) = (ap @ X243 @ X241))) => ((sK575 != sK576) & ((ap @ sK576 @ sK574) = (ap @ sK575 @ sK573)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f539,plain,(
% 2.56/1.94 ? [X246 : term,X247 : term] : ((X246 != X247) & ((lam @ X246) = (lam @ X247))) => ((sK578 != sK577) & ((lam @ sK577) = (lam @ sK578)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f540,plain,(
% 2.56/1.94 ! [X248 : term] : (? [X249 : d_term] : ((d2term @ X249) = X248) => ((d2term @ (sK579 @ X248)) = X248))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f541,plain,(
% 2.56/1.94 ? [X254 : term,X255 : term,X256 : term,X257 : term] : ((X254 != X256) & ((ap @ (sub @ X255 @ id) @ X254) = (ap @ (sub @ X257 @ id) @ X256))) => ((sK580 != sK582) & ((ap @ (sub @ sK581 @ id) @ sK580) = (ap @ (sub @ sK583 @ id) @ sK582)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f542,plain,(
% 2.56/1.94 ! [X260 : term,X261 : term > $o,X262 : term > $o,X263 : subst] : (? [X264 : term] : ((X262 @ X264) != (X261 @ (sub @ X264 @ (push @ X260 @ X263)))) => ((X261 @ (sub @ (sK584 @ X263 @ X262 @ X261 @ X260) @ (push @ X260 @ X263))) != (X262 @ (sK584 @ X263 @ X262 @ X261 @ X260))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f543,plain,(
% 2.56/1.94 ? [X265 : term,X266 : subst] : ((sub @ one @ (push @ X265 @ X266)) != X265) => (sK585 != (sub @ one @ (push @ sK585 @ sK586)))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f544,plain,(
% 2.56/1.94 ? [X267 : term,X268 : term > $o,X269 : term > $o,X270 : subst] : (($true = (X268 @ X267)) & ($true != (X269 @ one)) & ! [X271 : term] : ((X269 @ X271) = (X268 @ (sub @ X271 @ (push @ X267 @ X270))))) => (($true = (sK588 @ sK587)) & ($true != (sK589 @ one)) & ! [X271 : term] : ((sK589 @ X271) = (sK588 @ (sub @ X271 @ (push @ sK587 @ sK590)))))),
% 2.56/1.94 introduced(definition,[],[choice_axiom])).
% 2.56/1.94 thf(f475,plain,(
% 2.56/1.94 (hoaslamnotvar = $true) & ((apinj2 = $true) | ? [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) = (ap @ X3 @ X1)) & (X0 != X1))) & (! [X4 : term,X5 : term,X6 : term,X7 : term] : (((ap @ X7 @ X5) != (ap @ X6 @ X4)) | (X4 = X5)) | (apinj2 != $true)) & (ulamvarind = $true) & (axapp = $true) & (hoasinduction_lem1 = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (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)))))))))))))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X8 : term] : ((sub @ X8 @ id) = X8) | (termmset != $true)) & ((termmset = $true) | ? [X9 : term] : ((sub @ X9 @ id) != X9)) & (termmset_gthm = $true) & (pushprop_lem0_gthm = $true) & ((substmonoid_lthm = $true) | ((? [X10 : subst] : ((comp @ X10 @ id) != X10) | ? [X11 : subst] : ((comp @ id @ X11) != X11)) & ! [X12 : subst] : ((comp @ id @ X12) = X12) & ! [X13 : subst] : ((comp @ X13 @ id) = X13))) & ((! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X15 : subst] : ((comp @ id @ X15) = X15)) | ? [X16 : subst] : ((comp @ id @ X16) != X16) | ? [X17 : subst] : ((comp @ X17 @ id) != X17) | (substmonoid_lthm != $true)) & ! [X18 : d_term] : (d_one = X18) & ((d2subst @ d_id) = id) & ((hoasinduction_lem2 = $true) | ($true != sP58)) & (($true = sP58) | (hoasinduction_lem2 != $true)) & (pushprop_lem1v2_gthm = $true) & (($true = sP56) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP56)) & (pushprop_lem3v2_lthm = $true) & ((induction2_lthm = $true) | ($true != sP55)) & (($true = sP55) | (induction2_lthm != $true)) & ((sP51 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP51 != $true)) & ((d2subst @ d_id) = sh) & (hoaslamnotap_gthm = $true) & ! [X19 : d_subst,X20 : d_subst] : (((d2subst @ X19) != (d2subst @ X20)) | (X19 = X20)) & ((hoasinduction_lem3b = $true) | ? [X21 : term] : ! [X22 : subst > term > term] : ((sub @ X21 @ (push @ one @ sh)) != (X22 @ sh @ one))) & (! [X23 : term] : ? [X24 : subst > term > term] : ((X24 @ sh @ one) = (sub @ X23 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((shinj = $true) | ? [X25 : term,X26 : term] : (((sub @ X25 @ sh) = (sub @ X26 @ sh)) & (X25 != X26))) & (! [X27 : term,X28 : term] : (((sub @ X28 @ sh) != (sub @ X27 @ sh)) | (X27 = X28)) | (shinj != $true)) & (($true = sP45) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP45)) & ((induction2lem = $true) | ($true != sP42)) & (($true = sP42) | (induction2lem != $true)) & ((hoasinduction_lem0 = $true) | ? [X29 : subst > term > subst > $o] : ! [X30 : term > $o] : ? [X31 : term] : ((X29 @ id @ X31 @ id) != (X30 @ X31))) & (! [X32 : subst > term > subst > $o] : ? [X33 : term > $o] : ! [X34 : term] : ((X33 @ X34) = (X32 @ id @ X34 @ id)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP40)) & (($true = sP40) | (hoasinduction_lem3aa_lthm != $true)) & (hoaslamnotap_lthm = $true) & (? [X35 : subst,X36 : term] : ((sub @ one @ (push @ X36 @ X35)) != X36) | ? [X37 : subst,X38 : term] : ((comp @ sh @ (push @ X38 @ X37)) != X37) | ! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst,X43 : term] : (($true = (X41 @ (sub @ X43 @ (push @ X39 @ X42)))) | ($true != (X40 @ X43))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X44 : subst,X45 : term] : ((sub @ one @ (push @ X45 @ X44)) = X45) & ! [X46 : subst,X47 : term] : ((comp @ sh @ (push @ X47 @ X46)) = X46) & ? [X48 : term,X49 : term > $o,X50 : term > $o,X51 : subst,X52 : term] : (($true != (X50 @ (sub @ X52 @ (push @ X48 @ X51)))) & ($true = (X49 @ X52))))) & ((sP39 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP39 != $true)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X53 : term] : ((sub @ X53 @ id) = X53) | (axvarid != $true)) & ((axvarid = $true) | ? [X54 : term] : ((sub @ X54 @ id) != X54)) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3v2_lthm != $true)) & (axabs = $true) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & (ulamvarsh = $true) & ! [X55 : d_subst] : (d_id = X55) & ((induction = $true) | ? [X56 : term > $o,X57 : term] : (($true != (X56 @ X57)) & ! [X58 : term] : (($true = (X56 @ X58)) | ($true != (var @ X58))) & ($true != (sP36 @ X56)) & ! [X59 : term] : (($true = (X56 @ (lam @ X59))) | ($true != (X56 @ X59))))) & (! [X60 : term > $o,X61 : term] : (($true = (X60 @ X61)) | ? [X62 : term] : (($true != (X60 @ X62)) & ($true = (var @ X62))) | ($true = (sP36 @ X60)) | ? [X63 : term] : (($true != (X60 @ (lam @ X63))) & ((X60 @ X63) = $true))) | (induction != $true)) & (pushprop_lem2v2_lthm = $true) & (hoaslaminj_gthm = $true) & ((axidr = $true) | ? [X64 : subst] : ((comp @ X64 @ id) != X64)) & (! [X65 : subst] : ((comp @ X65 @ id) = X65) | (axidr != $true)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & (($true = sP35) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP35)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem2v2 = $true) | ($true != sP31)) & (($true = sP31) | (hoasinduction_lem2v2 != $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_gthm = $true) & (induction2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2a != $true)) & (axclos = $true) & (lamnotap != $true) & ((axvarcons = $true) | ? [X66 : subst,X67 : term] : ((sub @ one @ (push @ X67 @ X66)) != X67)) & (! [X68 : subst,X69 : term] : ((sub @ one @ (push @ X69 @ X68)) = X69) | (axvarcons != $true)) & (axvarshift = $true) & (pushprop_lem1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lem2v2_gthm = $true) & ((pushprop_lem0_lthm = $true) | ? [X70 : term,X71 : term > $o,X72 : subst] : ! [X73 : term > $o] : ? [X74 : term] : ((X73 @ X74) != (X71 @ (sub @ X74 @ (push @ X70 @ X72))))) & (! [X75 : term,X76 : term > $o,X77 : subst] : ? [X78 : term > $o] : ! [X79 : term] : ((X78 @ X79) = (X76 @ (sub @ X79 @ (push @ X75 @ X77)))) | (pushprop_lem0_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X80 : term,X81 : term,X82 : term,X83 : term] : (((ap @ X83 @ X82) != (ap @ X80 @ X81)) | (X80 = X83)) & ? [X84 : term,X85 : term,X86 : term,X87 : term] : (((ap @ (sub @ X86 @ id) @ X85) = (ap @ (sub @ X84 @ id) @ X87)) & (X84 != X86)) & ! [X88 : term] : ((sub @ X88 @ id) = X88))) & (? [X89 : term,X90 : term,X91 : term,X92 : term] : (((ap @ X89 @ X90) = (ap @ X92 @ X91)) & (X89 != X92)) | ! [X93 : term,X94 : term,X95 : term,X96 : term] : (((ap @ (sub @ X95 @ id) @ X94) != (ap @ (sub @ X93 @ id) @ X96)) | (X93 = X95)) | ? [X97 : term] : ((sub @ X97 @ id) != X97) | (hoasapinj1_lthm != $true)) & (($true = sP28) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP28)) & (hoaslamnotvar_gthm = $true) & ((sP26 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP26 != $true)) & ((hoaslamnotap = $true) | ? [X98 : subst > term > term,X99 : term,X100 : term] : (! [X101 : subst,X102 : term,X103 : subst] : ((X98 @ (comp @ X101 @ X103) @ (sub @ X102 @ X103)) = (sub @ (X98 @ X101 @ X102) @ X103)) & ((lam @ (X98 @ sh @ one)) = (ap @ (sub @ X99 @ id) @ X100)))) & (! [X104 : subst > term > term,X105 : term,X106 : term] : (? [X107 : subst,X108 : term,X109 : subst] : ((X104 @ (comp @ X107 @ X109) @ (sub @ X108 @ X109)) != (sub @ (X104 @ X107 @ X108) @ X109)) | ((lam @ (X104 @ sh @ one)) != (ap @ (sub @ X105 @ id) @ X106))) | (hoaslamnotap != $true)) & (pushprop = $true) & (pushprop_lthm = $true) & (! [X110 : term,X111 : term,X112 : term,X113 : term] : (((ap @ (sub @ X113 @ id) @ X111) != (ap @ (sub @ X112 @ id) @ X110)) | (X112 = X113)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X114 : term,X115 : term,X116 : term,X117 : term] : (((ap @ (sub @ X116 @ id) @ X114) = (ap @ (sub @ X117 @ id) @ X115)) & (X116 != X117))) & (apnotvar = $true) & (hoasinduction_lem2_gthm = $true) & (($true = sP24) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP24)) & (($true = sP23) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP23)) & (pushprop_lem2v2 = $true) & ((substmonoid = $true) | ? [X118 : subst] : ((comp @ X118 @ id) != X118) | ? [X119 : subst] : ((comp @ id @ X119) != X119)) & ((! [X120 : subst] : ((comp @ X120 @ id) = X120) & ! [X121 : subst] : ((comp @ id @ X121) = X121)) | (substmonoid != $true)) & (! [X122 : subst,X123 : term,X124 : term > $o] : ? [X125 : term > $o] : ! [X126 : term] : ((X124 @ (sub @ X126 @ (push @ X123 @ X122))) = (X125 @ X126)) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X127 : subst,X128 : term,X129 : term > $o] : ! [X130 : term > $o] : ? [X131 : term] : ((X129 @ (sub @ X131 @ (push @ X128 @ X127))) != (X130 @ X131))) & ((hoasinduction_lem3b_lthm = $true) | ? [X132 : term] : ! [X133 : subst > term > term] : ((sub @ X132 @ (push @ one @ sh)) != (X133 @ sh @ one))) & (! [X134 : term] : ? [X135 : subst > term > term] : ((X135 @ sh @ one) = (sub @ X134 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & (($true = sP22) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP22)) & (! [X136 : term > $o,X137 : term,X138 : subst,X139 : term,X140 : term > $o] : (($true = (X140 @ (sub @ X137 @ (push @ X139 @ X138)))) | ($true != (X136 @ X137))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X141 : term > $o,X142 : term,X143 : subst,X144 : term,X145 : term > $o] : (($true != (X145 @ (sub @ X142 @ (push @ X144 @ X143)))) & ($true = (X141 @ X142)))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X146 : term] : ? [X147 : subst > term > term] : ! [X148 : subst,X149 : term] : ((sub @ X146 @ (push @ X149 @ X148)) = (X147 @ X148 @ X149)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X150 : term] : ! [X151 : subst > term > term] : ? [X152 : subst,X153 : term] : ((sub @ X150 @ (push @ X153 @ X152)) != (X151 @ X152 @ X153))) & (! [X154 : term > $o,X155 : subst,X156 : term > $o,X157 : term] : (($true = (X156 @ one)) | ($true != (X154 @ X157)) | ? [X158 : term] : ((X154 @ (sub @ X158 @ (push @ X157 @ X155))) != (X156 @ X158))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X159 : term > $o,X160 : subst,X161 : term > $o,X162 : term] : (($true != (X161 @ one)) & ($true = (X159 @ X162)) & ! [X163 : term] : ((X159 @ (sub @ X163 @ (push @ X162 @ X160))) = (X161 @ X163)))) & (ulamvar1 != $true) & (axmap = $true) & ((var @ (d2term @ d_one)) != $true) & (hoaslamnotvar_lthm = $true) & (one = (d2term @ d_one)) & (! [X164 : term] : ? [X165 : subst > term > term] : ! [X166 : term,X167 : subst] : ((sub @ X164 @ (push @ X166 @ X167)) = (X165 @ X167 @ X166)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X168 : term] : ! [X169 : subst > term > term] : ? [X170 : term,X171 : subst] : ((X169 @ X171 @ X170) != (sub @ X168 @ (push @ X170 @ X171)))) & (! [X172 : subst > term > term,X173 : subst,X174 : term,X175 : subst > term > term] : (((X172 @ X173 @ X174) = (X175 @ X173 @ X174)) | ((lam @ (X172 @ sh @ one)) != (lam @ (X175 @ sh @ one))) | ? [X176 : subst,X177 : subst,X178 : term] : ((X172 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177)) != (sub @ (X172 @ X176 @ X178) @ X177)) | ? [X179 : subst,X180 : term,X181 : subst] : ((sub @ (X175 @ X181 @ X180) @ X179) != (X175 @ (comp @ X181 @ X179) @ (sub @ X180 @ X179)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X182 : subst > term > term,X183 : subst,X184 : term,X185 : subst > term > term] : (((X185 @ X183 @ X184) != (X182 @ X183 @ X184)) & ((lam @ (X182 @ sh @ one)) = (lam @ (X185 @ sh @ one))) & ! [X186 : subst,X187 : subst,X188 : term] : ((X182 @ (comp @ X186 @ X187) @ (sub @ X188 @ X187)) = (sub @ (X182 @ X186 @ X188) @ X187)) & ! [X189 : subst,X190 : term,X191 : subst] : ((X185 @ (comp @ X191 @ X189) @ (sub @ X190 @ X189)) = (sub @ (X185 @ X191 @ X190) @ X189)))) & ((termmset_lthm = $true) | (? [X192 : term] : ((sub @ X192 @ id) != X192) & ! [X193 : term] : ((sub @ X193 @ id) = X193))) & (! [X194 : term] : ((sub @ X194 @ id) = X194) | ? [X195 : term] : ((sub @ X195 @ id) != X195) | (termmset_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP17)) & (($true = sP17) | (hoasinduction_lem3_lthm != $true)) & ((axidl = $true) | ? [X196 : subst] : ((comp @ id @ X196) != X196)) & (! [X197 : subst] : ((comp @ id @ X197) = X197) | (axidl != $true)) & ((hoasinduction_lthm = $true) | ($true != sP14)) & (($true = sP14) | (hoasinduction_lthm != $true)) & (! [X198 : term,X199 : term,X200 : term,X201 : term] : (((ap @ (sub @ X199 @ id) @ X201) != (ap @ (sub @ X198 @ id) @ X200)) | (X200 = X201)) | ? [X202 : term,X203 : term,X204 : term,X205 : term] : (((ap @ X204 @ X203) = (ap @ X202 @ X205)) & (X203 != X205)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X206 : term,X207 : term,X208 : term,X209 : term] : (((ap @ (sub @ X207 @ id) @ X209) = (ap @ (sub @ X206 @ id) @ X208)) & (X208 != X209)) & ! [X210 : term,X211 : term,X212 : term,X213 : term] : (((ap @ X210 @ X213) != (ap @ X212 @ X211)) | (X211 = X213)))) & (hoasapinj1_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X214 : subst > term > subst > $o] : ! [X215 : term > $o] : ? [X216 : term] : ((X214 @ id @ X216 @ id) != (X215 @ X216))) & (! [X217 : subst > term > subst > $o] : ? [X218 : term > $o] : ! [X219 : term] : ((X218 @ X219) = (X217 @ id @ X219 @ id)) | (hoasinduction_lem0_lthm != $true)) & (axassoc = $true) & (substmonoid_gthm = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X220 : term,X221 : subst > term > subst > $o] : (! [X222 : term,X223 : term] : (($true = (X221 @ id @ (ap @ (sub @ X222 @ id) @ X223) @ id)) | ($true != (X221 @ id @ X222 @ id)) | ($true != (X221 @ id @ X223 @ id))) & ($true != (sP5 @ X221)) & ($true != (X221 @ id @ X220 @ id)))) & (! [X224 : term,X225 : subst > term > subst > $o] : (? [X226 : term,X227 : term] : (($true != (X225 @ id @ (ap @ (sub @ X226 @ id) @ X227) @ id)) & ($true = (X225 @ id @ X226 @ id)) & ($true = (X225 @ id @ X227 @ id))) | ($true = (sP5 @ X225)) | ($true = (X225 @ id @ X224 @ id))) | (hoasinduction_no_psi_cond != $true)) & (induction2lem_gthm = $true) & ((axshiftcons = $true) | ? [X228 : term,X229 : subst] : ((comp @ sh @ (push @ X228 @ X229)) != X229)) & (! [X230 : term,X231 : subst] : ((comp @ sh @ (push @ X230 @ X231)) = X231) | (axshiftcons != $true)) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3 = $true) | ($true != sP4)) & (($true = sP4) | (hoasinduction_lem3 != $true)) & (hoasapnotvar = $true) & ! [X232 : subst] : ? [X233 : d_subst] : ((d2subst @ X233) = X232) & (! [X234 : subst] : ((push @ (sub @ one @ X234) @ (comp @ sh @ X234)) = X234) | (axscons != $true)) & ((axscons = $true) | ? [X235 : subst] : ((push @ (sub @ one @ X235) @ (comp @ sh @ X235)) != X235)) & (pushprop_lthm_orig = $true) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X236 : term,X237 : term,X238 : term,X239 : term] : ((X238 = X239) | ((ap @ X238 @ X236) != (ap @ X239 @ X237))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X240 : term,X241 : term,X242 : term,X243 : term] : ((X242 != X243) & ((ap @ X242 @ X240) = (ap @ X243 @ X241)))) & (hoasinduction_lem3b_gthm = $true) & (hoasapnotvar_gthm = $true) & (($true = sP2) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP2)) & (! [X244 : term,X245 : term] : ((X244 = X245) | ((lam @ X244) != (lam @ X245))) | (laminj != $true)) & ((laminj = $true) | ? [X246 : term,X247 : term] : ((X246 != X247) & ((lam @ X246) = (lam @ X247)))) & ((hoasinduction_lem3aa = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3aa != $true)) & (hoasinduction_lem2v2_gthm = $true) & ! [X248 : term] : ? [X249 : d_term] : ((d2term @ X249) = X248) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X250 : term,X251 : term,X252 : term,X253 : term] : ((X250 = X252) | ((ap @ (sub @ X253 @ id) @ X252) != (ap @ (sub @ X251 @ id) @ X250))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X254 : term,X255 : term,X256 : term,X257 : term] : ((X254 != X256) & ((ap @ (sub @ X255 @ id) @ X254) = (ap @ (sub @ X257 @ id) @ X256)))) & ! [X258 : d_term,X259 : d_term] : (((d2term @ X258) != (d2term @ X259)) | (X258 = X259)) & (! [X260 : term,X261 : term > $o,X262 : term > $o,X263 : subst] : (($true != (X261 @ X260)) | ((X262 @ one) = $true) | ? [X264 : term] : ((X262 @ X264) != (X261 @ (sub @ X264 @ (push @ X260 @ X263))))) | ? [X265 : term,X266 : subst] : ((sub @ one @ (push @ X265 @ X266)) != X265) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X267 : term,X268 : term > $o,X269 : term > $o,X270 : subst] : (($true = (X268 @ X267)) & ($true != (X269 @ one)) & ! [X271 : term] : ((X269 @ X271) = (X268 @ (sub @ X271 @ (push @ X267 @ X270))))) & ! [X272 : term,X273 : subst] : ((sub @ one @ (push @ X272 @ X273)) = X272)))),
% 2.56/1.94 inference(rectify,[status(thm)],[f474])).
% 2.56/1.94 thf(f474,plain,(
% 2.56/1.94 (hoaslamnotvar = $true) & ((apinj2 = $true) | ? [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) = (ap @ X3 @ X1)) & (X0 != X1))) & (! [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) != (ap @ X3 @ X1)) | (X0 = X1)) | (apinj2 != $true)) & (ulamvarind = $true) & (axapp = $true) & (hoasinduction_lem1 = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (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)))))))))))))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X32 : term] : ((sub @ X32 @ id) = X32) | (termmset != $true)) & ((termmset = $true) | ? [X32 : term] : ((sub @ X32 @ id) != X32)) & (termmset_gthm = $true) & (pushprop_lem0_gthm = $true) & ((substmonoid_lthm = $true) | ((? [X35 : subst] : ((comp @ X35 @ id) != X35) | ? [X36 : subst] : ((comp @ id @ X36) != X36)) & ! [X33 : subst] : ((comp @ id @ X33) = X33) & ! [X34 : subst] : ((comp @ X34 @ id) = X34))) & ((! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36)) | ? [X33 : subst] : ((comp @ id @ X33) != X33) | ? [X34 : subst] : ((comp @ X34 @ id) != X34) | (substmonoid_lthm != $true)) & ! [X485 : d_term] : (d_one = X485) & ((d2subst @ d_id) = id) & ((hoasinduction_lem2 = $true) | ($true != sP58)) & (($true = sP58) | (hoasinduction_lem2 != $true)) & (pushprop_lem1v2_gthm = $true) & (($true = sP56) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP56)) & (pushprop_lem3v2_lthm = $true) & ((induction2_lthm = $true) | ($true != sP55)) & (($true = sP55) | (induction2_lthm != $true)) & ((sP51 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP51 != $true)) & ((d2subst @ d_id) = sh) & (hoaslamnotap_gthm = $true) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) != (d2subst @ X14)) | (X13 = X14)) & ((hoasinduction_lem3b = $true) | ? [X537 : term] : ! [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) != (X538 @ sh @ one))) & (! [X537 : term] : ? [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) = (X538 @ sh @ one)) | (hoasinduction_lem3b != $true)) & ((shinj = $true) | ? [X514 : term,X515 : term] : (((sub @ X514 @ sh) = (sub @ X515 @ sh)) & (X514 != X515))) & (! [X514 : term,X515 : term] : (((sub @ X514 @ sh) != (sub @ X515 @ sh)) | (X514 = X515)) | (shinj != $true)) & (($true = sP45) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP45)) & ((induction2lem = $true) | ($true != sP42)) & (($true = sP42) | (induction2lem != $true)) & ((hoasinduction_lem0 = $true) | ? [X77 : subst > term > subst > $o] : ! [X78 : term > $o] : ? [X79 : term] : ((X78 @ X79) != (X77 @ id @ X79 @ id))) & (! [X77 : subst > term > subst > $o] : ? [X78 : term > $o] : ! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP40)) & (($true = sP40) | (hoasinduction_lem3aa_lthm != $true)) & (hoaslamnotap_lthm = $true) & (? [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) != X203) | ? [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) != X204) | ! [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (($true = (X208 @ (sub @ X210 @ (push @ X206 @ X209)))) | ($true != (X207 @ X210))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) = X203) & ! [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) = X204) & ? [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (($true != (X208 @ (sub @ X210 @ (push @ X206 @ X209)))) & ($true = (X207 @ X210))))) & ((sP39 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP39 != $true)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X318 : term] : ((sub @ X318 @ id) = X318) | (axvarid != $true)) & ((axvarid = $true) | ? [X318 : term] : ((sub @ X318 @ id) != X318)) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3v2_lthm != $true)) & (axabs = $true) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & (ulamvarsh = $true) & ! [X106 : d_subst] : (d_id = X106) & ((induction = $true) | ? [X479 : term > $o,X480 : term] : (($true != (X479 @ X480)) & ! [X481 : term] : (($true = (X479 @ X481)) | ($true != (var @ X481))) & ($true != (sP36 @ X479)) & ! [X482 : term] : (($true = (X479 @ (lam @ X482))) | ($true != (X479 @ X482))))) & (! [X479 : term > $o,X480 : term] : (($true = (X479 @ X480)) | ? [X481 : term] : (($true != (X479 @ X481)) & ($true = (var @ X481))) | ($true = (sP36 @ X479)) | ? [X482 : term] : (($true != (X479 @ (lam @ X482))) & ($true = (X479 @ X482)))) | (induction != $true)) & (pushprop_lem2v2_lthm = $true) & (hoaslaminj_gthm = $true) & ((axidr = $true) | ? [X105 : subst] : ((comp @ X105 @ id) != X105)) & (! [X105 : subst] : ((comp @ X105 @ id) = X105) | (axidr != $true)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & (($true = sP35) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP35)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem2v2 = $true) | ($true != sP31)) & (($true = sP31) | (hoasinduction_lem2v2 != $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_gthm = $true) & (induction2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2a != $true)) & (axclos = $true) & (lamnotap != $true) & ((axvarcons = $true) | ? [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) != X104)) & (! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104) | (axvarcons != $true)) & (axvarshift = $true) & (pushprop_lem1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lem2v2_gthm = $true) & ((pushprop_lem0_lthm = $true) | ? [X532 : term,X533 : term > $o,X534 : subst] : ! [X535 : term > $o] : ? [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) != (X535 @ X536))) & (! [X532 : term,X533 : term > $o,X534 : subst] : ? [X535 : term > $o] : ! [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) = (X535 @ X536)) | (pushprop_lem0_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) != (ap @ X490 @ X489)) | (X487 = X490)) & ? [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) = (ap @ (sub @ X491 @ id) @ X494)) & (X491 != X493)) & ! [X486 : term] : ((sub @ X486 @ id) = X486))) & (? [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) = (ap @ X490 @ X489)) & (X487 != X490)) | ! [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) != (ap @ (sub @ X491 @ id) @ X494)) | (X491 = X493)) | ? [X486 : term] : ((sub @ X486 @ id) != X486) | (hoasapinj1_lthm != $true)) & (($true = sP28) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP28)) & (hoaslamnotvar_gthm = $true) & ((sP26 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP26 != $true)) & ((hoaslamnotap = $true) | ? [X117 : subst > term > term,X118 : term,X119 : term] : (! [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) = (sub @ (X117 @ X120 @ X121) @ X122)) & ((lam @ (X117 @ sh @ one)) = (ap @ (sub @ X118 @ id) @ X119)))) & (! [X117 : subst > term > term,X118 : term,X119 : term] : (? [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) != (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119))) | (hoaslamnotap != $true)) & (pushprop = $true) & (pushprop_lthm = $true) & (! [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) != (ap @ (sub @ X436 @ id) @ X434)) | (X435 = X436)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) = (ap @ (sub @ X436 @ id) @ X434)) & (X435 != X436))) & (apnotvar = $true) & (hoasinduction_lem2_gthm = $true) & (($true = sP24) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP24)) & (($true = sP23) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP23)) & (pushprop_lem2v2 = $true) & ((substmonoid = $true) | ? [X449 : subst] : ((comp @ X449 @ id) != X449) | ? [X448 : subst] : ((comp @ id @ X448) != X448)) & ((! [X449 : subst] : ((comp @ X449 @ id) = X449) & ! [X448 : subst] : ((comp @ id @ X448) = X448)) | (substmonoid != $true)) & (! [X27 : subst,X28 : term,X29 : term > $o] : ? [X30 : term > $o] : ! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31)) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X27 : subst,X28 : term,X29 : term > $o] : ! [X30 : term > $o] : ? [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) != (X30 @ X31))) & ((hoasinduction_lem3b_lthm = $true) | ? [X200 : term] : ! [X201 : subst > term > term] : ((X201 @ sh @ one) != (sub @ X200 @ (push @ one @ sh)))) & (! [X200 : term] : ? [X201 : subst > term > term] : ((X201 @ sh @ one) = (sub @ X200 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & (($true = sP22) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP22)) & (! [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true = (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) | ($true != (X173 @ X174))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true != (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) & ($true = (X173 @ X174)))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X80 : term] : ? [X81 : subst > term > term] : ! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X80 : term] : ! [X81 : subst > term > term] : ? [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) != (X81 @ X82 @ X83))) & (! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true = (X144 @ one)) | ($true != (X142 @ X145)) | ? [X146 : term] : ((X144 @ X146) != (X142 @ (sub @ X146 @ (push @ X145 @ X143))))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true != (X144 @ one)) & ($true = (X142 @ X145)) & ! [X146 : term] : ((X144 @ X146) = (X142 @ (sub @ X146 @ (push @ X145 @ X143)))))) & (ulamvar1 != $true) & (axmap = $true) & ((var @ (d2term @ d_one)) != $true) & (hoaslamnotvar_lthm = $true) & (one = (d2term @ d_one)) & (! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X147 : term] : ! [X148 : subst > term > term] : ? [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) != (X148 @ X150 @ X149))) & (! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ? [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) != (sub @ (X107 @ X111 @ X113) @ X112)) | ? [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) != (sub @ (X110 @ X116 @ X115) @ X114))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (((X107 @ X108 @ X109) != (X110 @ X108 @ X109)) & ((lam @ (X110 @ sh @ one)) = (lam @ (X107 @ sh @ one))) & ! [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X107 @ X111 @ X113) @ X112)) & ! [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) = (sub @ (X110 @ X116 @ X115) @ X114)))) & ((termmset_lthm = $true) | (? [X26 : term] : ((sub @ X26 @ id) != X26) & ! [X25 : term] : ((sub @ X25 @ id) = X25))) & (! [X26 : term] : ((sub @ X26 @ id) = X26) | ? [X25 : term] : ((sub @ X25 @ id) != X25) | (termmset_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP17)) & (($true = sP17) | (hoasinduction_lem3_lthm != $true)) & ((axidl = $true) | ? [X56 : subst] : ((comp @ id @ X56) != X56)) & (! [X56 : subst] : ((comp @ id @ X56) = X56) | (axidl != $true)) & ((hoasinduction_lthm = $true) | ($true != sP14)) & (($true = sP14) | (hoasinduction_lthm != $true)) & (! [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) != (ap @ (sub @ X216 @ id) @ X218)) | (X217 = X218)) | ? [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) = (ap @ X213 @ X212)) & (X212 != X214)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) = (ap @ (sub @ X216 @ id) @ X218)) & (X217 != X218)) & ! [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) != (ap @ X213 @ X212)) | (X212 = X214)))) & (hoasapinj1_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X445 : subst > term > subst > $o] : ! [X446 : term > $o] : ? [X447 : term] : ((X445 @ id @ X447 @ id) != (X446 @ X447))) & (! [X445 : subst > term > subst > $o] : ? [X446 : term > $o] : ! [X447 : term] : ((X445 @ id @ X447 @ id) = (X446 @ X447)) | (hoasinduction_lem0_lthm != $true)) & (axassoc = $true) & (substmonoid_gthm = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X4 : term,X5 : subst > term > subst > $o] : (! [X11 : term,X12 : term] : (($true = (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) | ((X5 @ id @ X11 @ id) != $true) | ($true != (X5 @ id @ X12 @ id))) & ($true != (sP5 @ X5)) & ($true != (X5 @ id @ X4 @ id)))) & (! [X4 : term,X5 : subst > term > subst > $o] : (? [X11 : term,X12 : term] : (($true != (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) & ((X5 @ id @ X11 @ id) = $true) & ($true = (X5 @ id @ X12 @ id))) | ($true = (sP5 @ X5)) | ($true = (X5 @ id @ X4 @ id))) | (hoasinduction_no_psi_cond != $true)) & (induction2lem_gthm = $true) & ((axshiftcons = $true) | ? [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) != X58)) & (! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58) | (axshiftcons != $true)) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3 = $true) | ($true != sP4)) & (($true = sP4) | (hoasinduction_lem3 != $true)) & (hoasapnotvar = $true) & ! [X266 : subst] : ? [X267 : d_subst] : ((d2subst @ X267) = X266) & (! [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) = X539) | (axscons != $true)) & ((axscons = $true) | ? [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) != X539)) & (pushprop_lthm_orig = $true) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 = X126) | ((ap @ X125 @ X123) != (ap @ X126 @ X124))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 != X126) & ((ap @ X125 @ X123) = (ap @ X126 @ X124)))) & (hoasinduction_lem3b_gthm = $true) & (hoasapnotvar_gthm = $true) & (($true = sP2) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP2)) & (! [X196 : term,X197 : term] : ((X196 = X197) | ((lam @ X197) != (lam @ X196))) | (laminj != $true)) & ((laminj = $true) | ? [X196 : term,X197 : term] : ((X196 != X197) & ((lam @ X197) = (lam @ X196)))) & ((hoasinduction_lem3aa = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3aa != $true)) & (hoasinduction_lem2v2_gthm = $true) & ! [X198 : term] : ? [X199 : d_term] : ((d2term @ X199) = X198) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 = X155) | ((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 != X155) & ((ap @ (sub @ X156 @ id) @ X155) = (ap @ (sub @ X154 @ id) @ X153)))) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) != (d2term @ X152)) | (X151 = X152)) & (! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (($true != (X40 @ X39)) | ($true = (X41 @ one)) | ? [X43 : term] : ((X41 @ X43) != (X40 @ (sub @ X43 @ (push @ X39 @ X42))))) | ? [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) != X37) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (($true = (X40 @ X39)) & ($true != (X41 @ one)) & ! [X43 : term] : ((X41 @ X43) = (X40 @ (sub @ X43 @ (push @ X39 @ X42))))) & ! [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) = X37)))),
% 2.56/1.94 inference(flattening,[status(thm)],[f473])).
% 2.56/1.94 thf(f473,plain,(
% 2.56/1.94 (hoaslamnotvar = $true) & (((apinj2 = $true) | ? [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) = (ap @ X3 @ X1)) & (X0 != X1))) & (! [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) != (ap @ X3 @ X1)) | (X0 = X1)) | (apinj2 != $true))) & (ulamvarind = $true) & (axapp = $true) & (hoasinduction_lem1 = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (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)))))))))))))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X32 : term] : ((sub @ X32 @ id) = X32) | (termmset != $true)) & ((termmset = $true) | ? [X32 : term] : ((sub @ X32 @ id) != X32))) & (termmset_gthm = $true) & (pushprop_lem0_gthm = $true) & (((substmonoid_lthm = $true) | ((? [X35 : subst] : ((comp @ X35 @ id) != X35) | ? [X36 : subst] : ((comp @ id @ X36) != X36)) & ! [X33 : subst] : ((comp @ id @ X33) = X33) & ! [X34 : subst] : ((comp @ X34 @ id) = X34))) & (((! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36)) | ? [X33 : subst] : ((comp @ id @ X33) != X33) | ? [X34 : subst] : ((comp @ X34 @ id) != X34)) | (substmonoid_lthm != $true))) & ! [X485 : d_term] : (d_one = X485) & ((d2subst @ d_id) = id) & (((hoasinduction_lem2 = $true) | ($true != sP58)) & (($true = sP58) | (hoasinduction_lem2 != $true))) & (pushprop_lem1v2_gthm = $true) & ((($true = sP56) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP56))) & (pushprop_lem3v2_lthm = $true) & (((induction2_lthm = $true) | ($true != sP55)) & (($true = sP55) | (induction2_lthm != $true))) & (((sP51 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP51 != $true))) & ((d2subst @ d_id) = sh) & (hoaslamnotap_gthm = $true) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) != (d2subst @ X14)) | (X13 = X14)) & (((hoasinduction_lem3b = $true) | ? [X537 : term] : ! [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) != (X538 @ sh @ one))) & (! [X537 : term] : ? [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) = (X538 @ sh @ one)) | (hoasinduction_lem3b != $true))) & (((shinj = $true) | ? [X514 : term,X515 : term] : (((sub @ X514 @ sh) = (sub @ X515 @ sh)) & (X514 != X515))) & (! [X514 : term,X515 : term] : (((sub @ X514 @ sh) != (sub @ X515 @ sh)) | (X514 = X515)) | (shinj != $true))) & ((($true = sP45) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP45))) & (((induction2lem = $true) | ($true != sP42)) & (($true = sP42) | (induction2lem != $true))) & (((hoasinduction_lem0 = $true) | ? [X77 : subst > term > subst > $o] : ! [X78 : term > $o] : ? [X79 : term] : ((X78 @ X79) != (X77 @ id @ X79 @ id))) & (! [X77 : subst > term > subst > $o] : ? [X78 : term > $o] : ! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id)) | (hoasinduction_lem0 != $true))) & (((hoasinduction_lem3aa_lthm = $true) | ($true != sP40)) & (($true = sP40) | (hoasinduction_lem3aa_lthm != $true))) & (hoaslamnotap_lthm = $true) & (((? [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) != X203) | ? [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) != X204) | ! [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (($true = (X208 @ (sub @ X210 @ (push @ X206 @ X209)))) | ($true != (X207 @ X210)))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) = X203) & ! [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) = X204) & ? [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (($true != (X208 @ (sub @ X210 @ (push @ X206 @ X209)))) & ($true = (X207 @ X210)))))) & (((sP39 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP39 != $true))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((! [X318 : term] : ((sub @ X318 @ id) = X318) | (axvarid != $true)) & ((axvarid = $true) | ? [X318 : term] : ((sub @ X318 @ id) != X318))) & (hoasinduction_lem1_lthm = $true) & (((hoasinduction_lem3v2_lthm = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3v2_lthm != $true))) & (axabs = $true) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & (ulamvarsh = $true) & ! [X106 : d_subst] : (d_id = X106) & (((induction = $true) | ? [X479 : term > $o,X480 : term] : (($true != (X479 @ X480)) & ! [X481 : term] : (($true = (X479 @ X481)) | ($true != (var @ X481))) & ($true != (sP36 @ X479)) & ! [X482 : term] : (($true = (X479 @ (lam @ X482))) | ($true != (X479 @ X482))))) & (! [X479 : term > $o,X480 : term] : (($true = (X479 @ X480)) | ? [X481 : term] : (($true != (X479 @ X481)) & ($true = (var @ X481))) | ($true = (sP36 @ X479)) | ? [X482 : term] : (($true != (X479 @ (lam @ X482))) & ($true = (X479 @ X482)))) | (induction != $true))) & (pushprop_lem2v2_lthm = $true) & (hoaslaminj_gthm = $true) & (((axidr = $true) | ? [X105 : subst] : ((comp @ X105 @ id) != X105)) & (! [X105 : subst] : ((comp @ X105 @ id) = X105) | (axidr != $true))) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & ((($true = sP35) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP35))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (((hoasinduction_lem2v2 = $true) | ($true != sP31)) & (($true = sP31) | (hoasinduction_lem2v2 != $true))) & (pushprop_lem3v2 = $true) & (hoasinduction_gthm = $true) & (induction2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((hoasinduction_lem3v2a = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2a != $true))) & (axclos = $true) & (lamnotap != $true) & (((axvarcons = $true) | ? [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) != X104)) & (! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104) | (axvarcons != $true))) & (axvarshift = $true) & (pushprop_lem1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lem2v2_gthm = $true) & (((pushprop_lem0_lthm = $true) | ? [X532 : term,X533 : term > $o,X534 : subst] : ! [X535 : term > $o] : ? [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) != (X535 @ X536))) & (! [X532 : term,X533 : term > $o,X534 : subst] : ? [X535 : term > $o] : ! [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) = (X535 @ X536)) | (pushprop_lem0_lthm != $true))) & (((hoasapinj1_lthm = $true) | (! [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) != (ap @ X490 @ X489)) | (X487 = X490)) & ? [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) = (ap @ (sub @ X491 @ id) @ X494)) & (X491 != X493)) & ! [X486 : term] : ((sub @ X486 @ id) = X486))) & ((? [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) = (ap @ X490 @ X489)) & (X487 != X490)) | ! [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) != (ap @ (sub @ X491 @ id) @ X494)) | (X491 = X493)) | ? [X486 : term] : ((sub @ X486 @ id) != X486)) | (hoasapinj1_lthm != $true))) & ((($true = sP28) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP28))) & (hoaslamnotvar_gthm = $true) & (((sP26 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP26 != $true))) & (((hoaslamnotap = $true) | ? [X117 : subst > term > term,X118 : term,X119 : term] : (! [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) = (sub @ (X117 @ X120 @ X121) @ X122)) & ((lam @ (X117 @ sh @ one)) = (ap @ (sub @ X118 @ id) @ X119)))) & (! [X117 : subst > term > term,X118 : term,X119 : term] : (? [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) != (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119))) | (hoaslamnotap != $true))) & (pushprop = $true) & (pushprop_lthm = $true) & ((! [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) != (ap @ (sub @ X436 @ id) @ X434)) | (X435 = X436)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) = (ap @ (sub @ X436 @ id) @ X434)) & (X435 != X436)))) & (apnotvar = $true) & (hoasinduction_lem2_gthm = $true) & ((($true = sP24) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP24))) & ((($true = sP23) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP23))) & (pushprop_lem2v2 = $true) & (((substmonoid = $true) | (? [X449 : subst] : ((comp @ X449 @ id) != X449) | ? [X448 : subst] : ((comp @ id @ X448) != X448))) & ((! [X449 : subst] : ((comp @ X449 @ id) = X449) & ! [X448 : subst] : ((comp @ id @ X448) = X448)) | (substmonoid != $true))) & ((! [X27 : subst,X28 : term,X29 : term > $o] : ? [X30 : term > $o] : ! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31)) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X27 : subst,X28 : term,X29 : term > $o] : ! [X30 : term > $o] : ? [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) != (X30 @ X31)))) & (((hoasinduction_lem3b_lthm = $true) | ? [X200 : term] : ! [X201 : subst > term > term] : ((X201 @ sh @ one) != (sub @ X200 @ (push @ one @ sh)))) & (! [X200 : term] : ? [X201 : subst > term > term] : ((X201 @ sh @ one) = (sub @ X200 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true))) & ((($true = sP22) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP22))) & ((! [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true = (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) | ($true != (X173 @ X174))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true != (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) & ($true = (X173 @ X174))))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X80 : term] : ? [X81 : subst > term > term] : ! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X80 : term] : ! [X81 : subst > term > term] : ? [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) != (X81 @ X82 @ X83)))) & ((! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true = (X144 @ one)) | ($true != (X142 @ X145)) | ? [X146 : term] : ((X144 @ X146) != (X142 @ (sub @ X146 @ (push @ X145 @ X143))))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true != (X144 @ one)) & ($true = (X142 @ X145)) & ! [X146 : term] : ((X144 @ X146) = (X142 @ (sub @ X146 @ (push @ X145 @ X143))))))) & (ulamvar1 != $true) & (axmap = $true) & ((var @ (d2term @ d_one)) != $true) & (hoaslamnotvar_lthm = $true) & (one = (d2term @ d_one)) & ((! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X147 : term] : ! [X148 : subst > term > term] : ? [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) != (X148 @ X150 @ X149)))) & ((! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ? [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) != (sub @ (X107 @ X111 @ X113) @ X112)) | ? [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) != (sub @ (X110 @ X116 @ X115) @ X114))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (((X107 @ X108 @ X109) != (X110 @ X108 @ X109)) & ((lam @ (X110 @ sh @ one)) = (lam @ (X107 @ sh @ one))) & ! [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X107 @ X111 @ X113) @ X112)) & ! [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) = (sub @ (X110 @ X116 @ X115) @ X114))))) & (((termmset_lthm = $true) | (? [X26 : term] : ((sub @ X26 @ id) != X26) & ! [X25 : term] : ((sub @ X25 @ id) = X25))) & ((! [X26 : term] : ((sub @ X26 @ id) = X26) | ? [X25 : term] : ((sub @ X25 @ id) != X25)) | (termmset_lthm != $true))) & (((hoasinduction_lem3_lthm = $true) | ($true != sP17)) & (($true = sP17) | (hoasinduction_lem3_lthm != $true))) & (((axidl = $true) | ? [X56 : subst] : ((comp @ id @ X56) != X56)) & (! [X56 : subst] : ((comp @ id @ X56) = X56) | (axidl != $true))) & (((hoasinduction_lthm = $true) | ($true != sP14)) & (($true = sP14) | (hoasinduction_lthm != $true))) & (((! [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) != (ap @ (sub @ X216 @ id) @ X218)) | (X217 = X218)) | ? [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) = (ap @ X213 @ X212)) & (X212 != X214))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) = (ap @ (sub @ X216 @ id) @ X218)) & (X217 != X218)) & ! [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) != (ap @ X213 @ X212)) | (X212 = X214))))) & (hoasapinj1_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem1v2_gthm = $true) & (((hoasinduction_lem0_lthm = $true) | ? [X445 : subst > term > subst > $o] : ! [X446 : term > $o] : ? [X447 : term] : ((X445 @ id @ X447 @ id) != (X446 @ X447))) & (! [X445 : subst > term > subst > $o] : ? [X446 : term > $o] : ! [X447 : term] : ((X445 @ id @ X447 @ id) = (X446 @ X447)) | (hoasinduction_lem0_lthm != $true))) & (axassoc = $true) & (substmonoid_gthm = $true) & (((hoasinduction_no_psi_cond = $true) | ? [X4 : term,X5 : subst > term > subst > $o] : (! [X11 : term,X12 : term] : (($true = (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) | ((X5 @ id @ X11 @ id) != $true) | ($true != (X5 @ id @ X12 @ id))) & ($true != (sP5 @ X5)) & ($true != (X5 @ id @ X4 @ id)))) & (! [X4 : term,X5 : subst > term > subst > $o] : (? [X11 : term,X12 : term] : (($true != (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) & ((X5 @ id @ X11 @ id) = $true) & ($true = (X5 @ id @ X12 @ id))) | ($true = (sP5 @ X5)) | ($true = (X5 @ id @ X4 @ id))) | (hoasinduction_no_psi_cond != $true))) & (induction2lem_gthm = $true) & (((axshiftcons = $true) | ? [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) != X58)) & (! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58) | (axshiftcons != $true))) & (hoasinduction_lem1_gthm = $true) & (((hoasinduction_lem3 = $true) | ($true != sP4)) & (($true = sP4) | (hoasinduction_lem3 != $true))) & (hoasapnotvar = $true) & ! [X266 : subst] : ? [X267 : d_subst] : ((d2subst @ X267) = X266) & ((! [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) = X539) | (axscons != $true)) & ((axscons = $true) | ? [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) != X539))) & (pushprop_lthm_orig = $true) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & ((! [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 = X126) | ((ap @ X125 @ X123) != (ap @ X126 @ X124))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 != X126) & ((ap @ X125 @ X123) = (ap @ X126 @ X124))))) & (hoasinduction_lem3b_gthm = $true) & (hoasapnotvar_gthm = $true) & ((($true = sP2) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP2))) & ((! [X196 : term,X197 : term] : ((X196 = X197) | ((lam @ X197) != (lam @ X196))) | (laminj != $true)) & ((laminj = $true) | ? [X196 : term,X197 : term] : ((X196 != X197) & ((lam @ X197) = (lam @ X196))))) & (((hoasinduction_lem3aa = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3aa != $true))) & (hoasinduction_lem2v2_gthm = $true) & ! [X198 : term] : ? [X199 : d_term] : ((d2term @ X199) = X198) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & ((! [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 = X155) | ((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 != X155) & ((ap @ (sub @ X156 @ id) @ X155) = (ap @ (sub @ X154 @ id) @ X153))))) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) != (d2term @ X152)) | (X151 = X152)) & (((! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (($true != (X40 @ X39)) | ($true = (X41 @ one)) | ? [X43 : term] : ((X41 @ X43) != (X40 @ (sub @ X43 @ (push @ X39 @ X42))))) | ? [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) != X37)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (($true = (X40 @ X39)) & ($true != (X41 @ one)) & ! [X43 : term] : ((X41 @ X43) = (X40 @ (sub @ X43 @ (push @ X39 @ X42))))) & ! [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) = X37))))),
% 2.56/1.94 inference(nnf_transformation,[status(thm)],[f73])).
% 2.56/1.94 thf(f73,plain,(
% 2.56/1.94 (hoaslamnotvar = $true) & ((apinj2 = $true) <=> ! [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) != (ap @ X3 @ X1)) | (X0 = X1))) & (ulamvarind = $true) & (axapp = $true) & (hoasinduction_lem1 = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (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)))))))))))))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X32 : term] : ((sub @ X32 @ id) = X32) <=> (termmset = $true)) & (termmset_gthm = $true) & (pushprop_lem0_gthm = $true) & ((substmonoid_lthm = $true) <=> ((! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36)) | ? [X33 : subst] : ((comp @ id @ X33) != X33) | ? [X34 : subst] : ((comp @ X34 @ id) != X34))) & ! [X485 : d_term] : (d_one = X485) & ((d2subst @ d_id) = id) & ((hoasinduction_lem2 = $true) <=> ($true = sP58)) & (pushprop_lem1v2_gthm = $true) & (($true = sP56) <=> (hoasinduction_lem3v2a_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((induction2_lthm = $true) <=> ($true = sP55)) & ((sP51 = $true) <=> (hoasinduction_lthm_3 = $true)) & ((d2subst @ d_id) = sh) & (hoaslamnotap_gthm = $true) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) != (d2subst @ X14)) | (X13 = X14)) & ((hoasinduction_lem3b = $true) <=> ! [X537 : term] : ? [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) = (X538 @ sh @ one))) & ((shinj = $true) <=> ! [X514 : term,X515 : term] : (((sub @ X514 @ sh) != (sub @ X515 @ sh)) | (X514 = X515))) & (($true = sP45) <=> (hoasinduction = $true)) & ((induction2lem = $true) <=> ($true = sP42)) & ((hoasinduction_lem0 = $true) <=> ! [X77 : subst > term > subst > $o] : ? [X78 : term > $o] : ! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id))) & ((hoasinduction_lem3aa_lthm = $true) <=> ($true = sP40)) & (hoaslamnotap_lthm = $true) & ((? [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) != X203) | ? [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) != X204) | ! [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (($true = (X208 @ (sub @ X210 @ (push @ X206 @ X209)))) | ($true != (X207 @ X210)))) <=> (pushprop_lem1_lthm = $true)) & ((sP39 = $true) <=> (hoasinduction_lem3a = $true)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X318 : term] : ((sub @ X318 @ id) = X318) <=> (axvarid = $true)) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> ($true = sP38)) & (axabs = $true) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & (ulamvarsh = $true) & ! [X106 : d_subst] : (d_id = X106) & ((induction = $true) <=> ! [X479 : term > $o,X480 : term] : (($true = (X479 @ X480)) | ? [X481 : term] : (($true != (X479 @ X481)) & ($true = (var @ X481))) | ($true = (sP36 @ X479)) | ? [X482 : term] : (($true != (X479 @ (lam @ X482))) & ($true = (X479 @ X482))))) & (pushprop_lem2v2_lthm = $true) & (hoaslaminj_gthm = $true) & ((axidr = $true) <=> ! [X105 : subst] : ((comp @ X105 @ id) = X105)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & (($true = sP35) <=> (induction2lem_lthm = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem2v2 = $true) <=> ($true = sP31)) & (pushprop_lem3v2 = $true) & (hoasinduction_gthm = $true) & (induction2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ($true = sP29)) & (axclos = $true) & (lamnotap != $true) & ((axvarcons = $true) <=> ! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104)) & (axvarshift = $true) & (pushprop_lem1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lem2v2_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X532 : term,X533 : term > $o,X534 : subst] : ? [X535 : term > $o] : ! [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) = (X535 @ X536))) & ((hoasapinj1_lthm = $true) <=> (? [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) = (ap @ X490 @ X489)) & (X487 != X490)) | ! [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) != (ap @ (sub @ X491 @ id) @ X494)) | (X491 = X493)) | ? [X486 : term] : ((sub @ X486 @ id) != X486))) & (($true = sP28) <=> (hoasinduction_lem3a_lthm = $true)) & (hoaslamnotvar_gthm = $true) & ((sP26 = $true) <=> (hoasinduction_lem3v2 = $true)) & ((hoaslamnotap = $true) <=> ! [X117 : subst > term > term,X118 : term,X119 : term] : (? [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) != (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119)))) & (pushprop = $true) & (pushprop_lthm = $true) & (! [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) != (ap @ (sub @ X436 @ id) @ X434)) | (X435 = X436)) <=> (hoasapinj1 = $true)) & (apnotvar = $true) & (hoasinduction_lem2_gthm = $true) & (($true = sP24) <=> (hoasinduction_lem3aaa = $true)) & (($true = sP23) <=> (hoaslaminj_lthm = $true)) & (pushprop_lem2v2 = $true) & ((substmonoid = $true) <=> (! [X449 : subst] : ((comp @ X449 @ id) = X449) & ! [X448 : subst] : ((comp @ id @ X448) = X448))) & (! [X27 : subst,X28 : term,X29 : term > $o] : ? [X30 : term > $o] : ! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31)) <=> (pushprop_lem0 = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X200 : term] : ? [X201 : subst > term > term] : ((X201 @ sh @ one) = (sub @ X200 @ (push @ one @ sh)))) & (($true = sP22) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (! [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true = (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) | ($true != (X173 @ X174))) <=> (pushprop_lem1 = $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X80 : term] : ? [X81 : subst > term > term] : ! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83)) <=> (hoasinduction_lem3v2_f = $true)) & (! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true = (X144 @ one)) | ($true != (X142 @ X145)) | ? [X146 : term] : ((X144 @ X146) != (X142 @ (sub @ X146 @ (push @ X145 @ X143))))) <=> (pushprop_lem1v2 = $true)) & (ulamvar1 != $true) & (axmap = $true) & ((var @ (d2term @ d_one)) != $true) & (hoaslamnotvar_lthm = $true) & (one = (d2term @ d_one)) & (! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ? [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) != (sub @ (X107 @ X111 @ X113) @ X112)) | ? [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) != (sub @ (X110 @ X116 @ X115) @ X114))) <=> (hoaslaminj = $true)) & ((termmset_lthm = $true) <=> (! [X26 : term] : ((sub @ X26 @ id) = X26) | ? [X25 : term] : ((sub @ X25 @ id) != X25))) & ((hoasinduction_lem3_lthm = $true) <=> ($true = sP17)) & ((axidl = $true) <=> ! [X56 : subst] : ((comp @ id @ X56) = X56)) & ((hoasinduction_lthm = $true) <=> ($true = sP14)) & ((! [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) != (ap @ (sub @ X216 @ id) @ X218)) | (X217 = X218)) | ? [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) = (ap @ X213 @ X212)) & (X212 != X214))) <=> (hoasapinj2_lthm = $true)) & (hoasapinj1_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X445 : subst > term > subst > $o] : ? [X446 : term > $o] : ! [X447 : term] : ((X445 @ id @ X447 @ id) = (X446 @ X447))) & (axassoc = $true) & (substmonoid_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X4 : term,X5 : subst > term > subst > $o] : (? [X11 : term,X12 : term] : (($true != (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) & ((X5 @ id @ X11 @ id) = $true) & ($true = (X5 @ id @ X12 @ id))) | ($true = (sP5 @ X5)) | ($true = (X5 @ id @ X4 @ id)))) & (induction2lem_gthm = $true) & ((axshiftcons = $true) <=> ! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58)) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ($true = sP4)) & (hoasapnotvar = $true) & ! [X266 : subst] : ? [X267 : d_subst] : ((d2subst @ X267) = X266) & (! [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) = X539) <=> (axscons = $true)) & (pushprop_lthm_orig = $true) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 = X126) | ((ap @ X125 @ X123) != (ap @ X126 @ X124))) <=> (apinj1 = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoasapnotvar_gthm = $true) & (($true = sP2) <=> (induction2 = $true)) & (! [X196 : term,X197 : term] : ((X196 = X197) | ((lam @ X197) != (lam @ X196))) <=> (laminj = $true)) & ((hoasinduction_lem3aa = $true) <=> ($true = sP0)) & (hoasinduction_lem2v2_gthm = $true) & ! [X198 : term] : ? [X199 : d_term] : ((d2term @ X199) = X198) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 = X155) | ((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153))) <=> (hoasapinj2 = $true)) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) != (d2term @ X152)) | (X151 = X152)) & ((! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (($true != (X40 @ X39)) | ($true = (X41 @ one)) | ? [X43 : term] : ((X41 @ X43) != (X40 @ (sub @ X43 @ (push @ X39 @ X42))))) | ? [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) != X37)) <=> (pushprop_lem1v2_lthm = $true))),
% 2.56/1.94 inference(definition_folding,[status(thm)],[f13,f72,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14])).
% 2.56/1.94 thf(f14,plain,(
% 2.56/1.94 ($true = sP0) <=> ! [X219 : subst > term > subst > $o,X220 : term] : (($true = (X219 @ id @ (lam @ (sub @ X220 @ (push @ one @ sh))) @ id)) | ? [X226 : term] : (($true = (X219 @ id @ X226 @ id)) & ($true != (X219 @ id @ (sub @ X220 @ (push @ X226 @ id)) @ id))) | ? [X221 : subst > term > term] : (! [X222 : term] : (($true != (X219 @ id @ X222 @ id)) | ($true = (X219 @ id @ (X221 @ id @ X222) @ id))) & ! [X223 : term,X224 : subst,X225 : subst] : ((X221 @ (comp @ X225 @ X224) @ (sub @ X223 @ X224)) = (sub @ (X221 @ X225 @ X223) @ X224)) & ($true != (X219 @ id @ (lam @ (X221 @ sh @ one)) @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f15,plain,(
% 2.56/1.94 ! [X320 : term > $o] : (($true = (sP1 @ X320)) <=> ? [X322 : term,X323 : term] : (($true != (X320 @ (ap @ X323 @ X322))) & ($true = (X320 @ X322)) & ($true = (X320 @ X323))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f16,plain,(
% 2.56/1.94 ($true = sP2) <=> ! [X319 : term,X320 : term > $o] : (($true = (X320 @ X319)) | ($true = (sP1 @ X320)) | ? [X324 : term] : (! [X325 : term] : (($true = (X320 @ (sub @ X324 @ (push @ X325 @ id)))) | ($true != (X320 @ X325))) & ($true != (X320 @ (lam @ X324)))) | ? [X321 : term] : (($true != (X320 @ X321)) & ($true = (var @ X321))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f17,plain,(
% 2.56/1.94 ! [X517 : subst > term > subst > $o] : (($true = (sP3 @ X517)) <=> ? [X522 : subst > term > term] : (((X517 @ id @ (lam @ (X522 @ sh @ one)) @ id) != $true) & ! [X523 : term] : (($true = (X517 @ id @ (X522 @ id @ X523) @ id)) | ($true != (X517 @ id @ X523 @ id))) & ! [X524 : term,X525 : subst,X526 : subst] : ((X522 @ (comp @ X525 @ X526) @ (sub @ X524 @ X526)) = (sub @ (X522 @ X525 @ X524) @ X526))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f18,plain,(
% 2.56/1.94 ($true = sP4) <=> ! [X516 : term,X517 : subst > term > subst > $o] : (($true = (sP3 @ X517)) | ? [X528 : subst,X529 : subst,X530 : subst,X531 : term] : (((X517 @ X529 @ X531 @ (comp @ X530 @ X528)) != $true) & ($true = (X517 @ (comp @ X529 @ X530) @ (sub @ X531 @ X530) @ X528))) | ($true = (X517 @ id @ (lam @ X516) @ id)) | ? [X527 : term] : (($true = (X517 @ id @ X527 @ id)) & ($true != (X517 @ id @ (sub @ X516 @ (push @ X527 @ id)) @ id))) | ? [X518 : subst,X519 : term,X520 : subst,X521 : subst] : (($true != (X517 @ (comp @ X521 @ X518) @ (sub @ X519 @ X518) @ X520)) & ($true = (X517 @ X521 @ X519 @ (comp @ X518 @ X520)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f19,plain,(
% 2.56/1.94 ! [X5 : subst > term > subst > $o] : (($true = (sP5 @ X5)) <=> ? [X6 : subst > term > term] : (! [X8 : term,X9 : subst,X10 : subst] : ((X6 @ (comp @ X10 @ X9) @ (sub @ X8 @ X9)) = (sub @ (X6 @ X10 @ X8) @ X9)) & ! [X7 : term] : (($true = (X5 @ id @ (X6 @ id @ X7) @ id)) | ($true != (X5 @ id @ X7 @ id))) & ($true != (X5 @ id @ (lam @ (X6 @ sh @ one)) @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f20,plain,(
% 2.56/1.94 ! [X399 : subst > term > subst > $o] : (($true = (sP6 @ X399)) <=> ! [X410 : subst > term > term] : (($true = (X399 @ id @ (lam @ (X410 @ sh @ one)) @ id)) | ? [X411 : term,X412 : subst,X413 : subst] : ((X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413)) != (sub @ (X410 @ X412 @ X411) @ X413)) | ? [X414 : term] : (($true = (X399 @ id @ X414 @ id)) & ($true != (X399 @ id @ (X410 @ id @ X414) @ id)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f21,plain,(
% 2.56/1.94 ! [X380 : term > $o] : (($true = (sP7 @ X380)) <=> ! [X381 : term,X382 : term] : (($true != (X380 @ X382)) | ((X380 @ (ap @ X381 @ X382)) = $true) | ($true != (X380 @ X381))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f22,plain,(
% 2.56/1.94 ! [X415 : subst > term > subst > $o] : (($true = (sP8 @ X415)) <=> ? [X417 : term,X418 : term] : (((X415 @ id @ X417 @ id) = $true) & ($true = (X415 @ id @ X418 @ id)) & ($true != (X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f23,plain,(
% 2.56/1.94 ! [X415 : subst > term > subst > $o] : (((sP9 @ X415) = $true) <=> ? [X419 : subst > term > term] : (! [X421 : term,X422 : subst,X423 : subst] : ((sub @ (X419 @ X422 @ X421) @ X423) = (X419 @ (comp @ X422 @ X423) @ (sub @ X421 @ X423))) & ((X415 @ id @ (lam @ (X419 @ sh @ one)) @ id) != $true) & ! [X420 : term] : (($true = (X415 @ id @ (X419 @ id @ X420) @ id)) | ($true != (X415 @ id @ X420 @ id)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f24,plain,(
% 2.56/1.94 ! [X388 : subst > term > subst > $o] : (($true = (sP10 @ X388)) <=> ! [X393 : term,X394 : term] : (($true = (X388 @ id @ (ap @ (sub @ X394 @ id) @ X393) @ id)) | ($true != (X388 @ id @ X393 @ id)) | ($true != (X388 @ id @ X394 @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f25,plain,(
% 2.56/1.94 (sP11 = $true) <=> ? [X399 : subst > term > subst > $o,X400 : term] : (! [X402 : subst,X403 : term,X404 : subst,X405 : subst] : (($true = (X399 @ X404 @ X403 @ (comp @ X402 @ X405))) | ($true != (X399 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) & ! [X401 : term] : (($true = (X399 @ id @ (sub @ X400 @ (push @ X401 @ id)) @ id)) | ($true != (X399 @ id @ X401 @ id))) & ($true != (X399 @ id @ (lam @ X400) @ id)) & ! [X406 : subst,X407 : term,X408 : subst,X409 : subst] : (($true = (X399 @ (comp @ X406 @ X409) @ (sub @ X407 @ X409) @ X408)) | ($true != (X399 @ X406 @ X407 @ (comp @ X409 @ X408)))) & ($true = (sP6 @ X399)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f26,plain,(
% 2.56/1.94 ($true = sP12) <=> ! [X415 : subst > term > subst > $o,X416 : term] : (($true = (X415 @ id @ X416 @ id)) | ? [X424 : subst,X425 : term,X426 : subst,X427 : subst] : (($true = (X415 @ X427 @ X425 @ (comp @ X424 @ X426))) & ($true != (X415 @ (comp @ X427 @ X424) @ (sub @ X425 @ X424) @ X426))) | ((sP9 @ X415) = $true) | ($true = (sP8 @ X415)) | ? [X432 : term] : (($true != (X415 @ id @ X432 @ id)) & ($true = (var @ (sub @ X432 @ id)))) | ? [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X415 @ (comp @ X430 @ X429) @ (sub @ X428 @ X429) @ X431)) & ($true != (X415 @ X430 @ X428 @ (comp @ X429 @ X431)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f27,plain,(
% 2.56/1.94 ($true = sP13) <=> ? [X379 : term,X380 : term > $o] : (($true != (X380 @ X379)) & ! [X383 : term] : (? [X384 : term] : (($true != (X380 @ (sub @ X383 @ (push @ X384 @ id)))) & ($true = (X380 @ X384))) | ($true = (X380 @ (lam @ X383)))) & ! [X385 : term] : (($true != (var @ X385)) | ($true = (X380 @ X385))) & ($true = (sP7 @ X380)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f28,plain,(
% 2.56/1.94 ($true = sP14) <=> (? [X386 : term,X387 : term,X388 : subst > term > subst > $o] : (! [X389 : subst,X390 : subst,X391 : term,X392 : subst] : (($true = (X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389) @ X392)) | ($true != (X388 @ X390 @ X391 @ (comp @ X389 @ X392)))) & ($true = (sP10 @ X388)) & ($true = (X388 @ id @ X386 @ id)) & ! [X395 : subst,X396 : subst,X397 : subst,X398 : term] : (($true = (X388 @ X396 @ X398 @ (comp @ X397 @ X395))) | ($true != (X388 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X395))) & ($true != (X388 @ id @ (ap @ X386 @ X387) @ id)) & ($true = (X388 @ id @ X387 @ id))) | ($true = sP12) | ($true = sP13) | (sP11 = $true))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f29,plain,(
% 2.56/1.94 ! [X295 : subst > term > subst > $o] : (($true = (sP15 @ X295)) <=> ? [X301 : subst > term > term] : (! [X305 : term] : (($true != (X295 @ id @ X305 @ id)) | ($true = (X295 @ id @ (X301 @ id @ X305) @ id))) & ($true != (X295 @ id @ (lam @ (X301 @ sh @ one)) @ id)) & ! [X302 : term,X303 : subst,X304 : subst] : ((X301 @ (comp @ X303 @ X304) @ (sub @ X302 @ X304)) = (sub @ (X301 @ X303 @ X302) @ X304))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f30,plain,(
% 2.56/1.94 ($true = sP16) <=> ? [X286 : term,X287 : subst > term > subst > $o] : (($true != (X287 @ id @ (lam @ (sub @ X286 @ (push @ one @ sh))) @ id)) & ! [X293 : term] : (($true = (X287 @ id @ (sub @ X286 @ (push @ X293 @ id)) @ id)) | ($true != (X287 @ id @ X293 @ id))) & ! [X288 : subst > term > term] : (? [X290 : subst,X291 : subst,X292 : term] : ((X288 @ (comp @ X291 @ X290) @ (sub @ X292 @ X290)) != (sub @ (X288 @ X291 @ X292) @ X290)) | ($true = (X287 @ id @ (lam @ (X288 @ sh @ one)) @ id)) | ? [X289 : term] : (($true = (X287 @ id @ X289 @ id)) & ($true != (X287 @ id @ (X288 @ id @ X289) @ id)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f31,plain,(
% 2.56/1.94 ($true = sP17) <=> (! [X294 : term,X295 : subst > term > subst > $o] : (? [X296 : subst,X297 : subst,X298 : subst,X299 : term] : (($true = (X295 @ (comp @ X298 @ X296) @ (sub @ X299 @ X296) @ X297)) & ($true != (X295 @ X298 @ X299 @ (comp @ X296 @ X297)))) | ? [X300 : term] : (($true != (X295 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id)) & ($true = (X295 @ id @ X300 @ id))) | ($true = (sP15 @ X295)) | ($true = (X295 @ id @ (lam @ X294) @ id)) | ? [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (($true = (X295 @ X308 @ X306 @ (comp @ X309 @ X307))) & ($true != (X295 @ (comp @ X308 @ X309) @ (sub @ X306 @ X309) @ X307)))) | ? [X285 : term] : ((sub @ X285 @ id) != X285) | ($true = sP16))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f32,plain,(
% 2.56/1.94 ! [X347 : subst > term > subst > $o] : (($true = (sP18 @ X347)) <=> ? [X349 : term,X350 : term] : (($true != (X347 @ id @ (ap @ (sub @ X349 @ id) @ X350) @ id)) & ($true = (X347 @ id @ X350 @ id)) & ($true = (X347 @ id @ X349 @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f33,plain,(
% 2.56/1.94 ! [X329 : term > $o] : (($true = (sP19 @ X329)) <=> ! [X334 : term,X335 : term] : (($true != (X329 @ X335)) | ($true = (X329 @ (ap @ X335 @ X334))) | ($true != (X329 @ X334))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f34,plain,(
% 2.56/1.94 ($true = sP20) <=> ? [X337 : subst > term > subst > $o,X338 : term,X339 : term > $o] : (! [X346 : term] : ((X339 @ X346) = (X337 @ id @ X346 @ id)) & ! [X340 : term] : (($true = (X339 @ (sub @ X338 @ (push @ X340 @ id)))) | ($true != (X339 @ X340))) & ! [X341 : subst > term > term] : (? [X342 : term] : (($true = (X337 @ id @ X342 @ id)) & ((X337 @ id @ (X341 @ id @ X342) @ id) != $true)) | ($true = (X337 @ id @ (lam @ (X341 @ sh @ one)) @ id)) | ? [X343 : subst,X344 : term,X345 : subst] : ((sub @ (X341 @ X345 @ X344) @ X343) != (X341 @ (comp @ X345 @ X343) @ (sub @ X344 @ X343)))) & ($true != (X339 @ (lam @ X338))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f35,plain,(
% 2.56/1.94 ($true = sP21) <=> ! [X347 : subst > term > subst > $o,X348 : term] : (($true = (X347 @ id @ X348 @ id)) | ? [X351 : subst > term > term] : (! [X355 : term] : (($true != (X347 @ id @ X355 @ id)) | ($true = (X347 @ id @ (X351 @ id @ X355) @ id))) & ($true != (X347 @ id @ (lam @ (X351 @ sh @ one)) @ id)) & ! [X352 : subst,X353 : subst,X354 : term] : ((X351 @ (comp @ X353 @ X352) @ (sub @ X354 @ X352)) = (sub @ (X351 @ X353 @ X354) @ X352))) | ($true = (sP18 @ X347)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f36,plain,(
% 2.56/1.94 ($true = sP22) <=> (($true = sP20) | ? [X329 : term > $o,X330 : term] : (! [X332 : term] : (? [X333 : term] : (($true != (X329 @ (sub @ X332 @ (push @ X333 @ id)))) & ($true = (X329 @ X333))) | ($true = (X329 @ (lam @ X332)))) & ($true = (sP19 @ X329)) & ($true != (X329 @ X330)) & ! [X331 : term] : (($true = (X329 @ X331)) | ($true != (var @ X331)))) | ? [X326 : subst > term > subst > $o] : ! [X327 : term > $o] : ? [X328 : term] : ((X326 @ id @ X328 @ id) != (X327 @ X328)) | ? [X336 : term] : ((sub @ X336 @ id) != X336) | ($true = sP21))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f37,plain,(
% 2.56/1.94 ($true = sP23) <=> (? [X161 : term,X162 : term] : (((lam @ X161) = (lam @ X162)) & (X161 != X162)) | ! [X163 : subst,X164 : subst > term > term,X165 : term,X166 : subst > term > term] : (? [X167 : subst,X168 : term,X169 : subst] : ((sub @ (X164 @ X167 @ X168) @ X169) != (X164 @ (comp @ X167 @ X169) @ (sub @ X168 @ X169))) | ((X164 @ X163 @ X165) = (X166 @ X163 @ X165)) | ? [X170 : term,X171 : subst,X172 : subst] : ((sub @ (X166 @ X172 @ X170) @ X171) != (X166 @ (comp @ X172 @ X171) @ (sub @ X170 @ X171))) | ((lam @ (X166 @ sh @ one)) != (lam @ (X164 @ sh @ one)))) | ? [X157 : term,X158 : subst] : ((sub @ one @ (push @ X157 @ X158)) != X157) | ? [X159 : subst,X160 : term] : ((comp @ sh @ (push @ X160 @ X159)) != X159))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f38,plain,(
% 2.56/1.94 ($true = sP24) <=> ! [X44 : term,X45 : subst > term > subst > $o] : (($true = (X45 @ id @ (lam @ (sub @ X44 @ (push @ one @ sh))) @ id)) | ? [X46 : term] : (((X45 @ id @ (sub @ X44 @ (push @ X46 @ id)) @ id) != $true) & ($true = (X45 @ id @ X46 @ id))) | ? [X47 : subst > term > term,X48 : term] : (! [X49 : subst,X50 : term,X51 : subst] : ((sub @ (sub @ X48 @ (push @ X50 @ X49)) @ X51) = (sub @ (X47 @ X49 @ X50) @ X51)) & ($true != (X45 @ id @ (lam @ (X47 @ sh @ one)) @ id)) & ! [X52 : subst,X53 : term,X54 : subst] : ((sub @ X48 @ (push @ (sub @ X53 @ X52) @ (comp @ X54 @ X52))) = (X47 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) & ! [X55 : term] : (($true != (X45 @ id @ X55 @ id)) | ($true = (X45 @ id @ (X47 @ id @ X55) @ id)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f39,plain,(
% 2.56/1.94 ! [X179 : subst > term > subst > $o] : (($true = (sP25 @ X179)) <=> ? [X182 : subst > term > term] : (! [X184 : subst,X185 : subst,X186 : term] : ((sub @ (X182 @ X185 @ X186) @ X184) = (X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184))) & ($true != (X179 @ id @ (lam @ (X182 @ sh @ one)) @ id)) & ! [X183 : term] : (($true = (X179 @ id @ (X182 @ id @ X183) @ id)) | ($true != (X179 @ id @ X183 @ id)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f40,plain,(
% 2.56/1.94 (sP26 = $true) <=> ! [X178 : term > $o,X179 : subst > term > subst > $o,X180 : term] : (($true = (X178 @ (lam @ X180))) | ($true = (sP25 @ X179)) | ? [X192 : subst,X193 : subst,X194 : term,X195 : subst] : (($true != (X179 @ (comp @ X192 @ X193) @ (sub @ X194 @ X193) @ X195)) & ($true = (X179 @ X192 @ X194 @ (comp @ X193 @ X195)))) | ? [X181 : term] : ((X179 @ id @ X181 @ id) != (X178 @ X181)) | ? [X188 : term,X189 : subst,X190 : subst,X191 : subst] : (($true != (X179 @ X189 @ X188 @ (comp @ X190 @ X191))) & ($true = (X179 @ (comp @ X189 @ X190) @ (sub @ X188 @ X190) @ X191))) | ? [X187 : term] : (($true != (X178 @ (sub @ X180 @ (push @ X187 @ id)))) & ($true = (X178 @ X187))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f41,plain,(
% 2.56/1.94 ($true = sP27) <=> ? [X269 : term,X270 : subst > term > subst > $o] : (! [X271 : subst > term > term] : (($true = (X270 @ id @ (lam @ (X271 @ sh @ one)) @ id)) | ? [X275 : term] : (($true != (X270 @ id @ (X271 @ id @ X275) @ id)) & ($true = (X270 @ id @ X275 @ id))) | ? [X272 : subst,X273 : term,X274 : subst] : ((X271 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)) != (sub @ (X271 @ X272 @ X273) @ X274))) & ($true != (X270 @ id @ (lam @ (sub @ X269 @ (push @ one @ sh))) @ id)) & ! [X276 : term] : (($true = (X270 @ id @ (sub @ X269 @ (push @ X276 @ id)) @ id)) | ($true != (X270 @ id @ X276 @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f42,plain,(
% 2.56/1.94 ($true = sP28) <=> (($true = sP27) | ? [X268 : term] : ((sub @ X268 @ id) != X268) | ! [X277 : term,X278 : subst > term > subst > $o] : (($true = (X278 @ id @ (lam @ X277) @ id)) | ? [X284 : term] : (($true = (X278 @ id @ X284 @ id)) & ($true != (X278 @ id @ (sub @ X277 @ (push @ X284 @ id)) @ id))) | ? [X279 : subst > term > term] : (($true != (X278 @ id @ (lam @ (X279 @ sh @ one)) @ id)) & ! [X283 : term] : (($true != (X278 @ id @ X283 @ id)) | ($true = (X278 @ id @ (X279 @ id @ X283) @ id))) & ! [X280 : subst,X281 : term,X282 : subst] : ((sub @ (X279 @ X282 @ X281) @ X280) = (X279 @ (comp @ X282 @ X280) @ (sub @ X281 @ X280))))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f43,plain,(
% 2.56/1.94 ($true = sP29) <=> ! [X15 : subst > term > subst > $o,X16 : term > $o,X17 : term] : (? [X19 : subst > term > term] : (! [X20 : term,X21 : subst,X22 : subst] : ((sub @ (X19 @ X22 @ X20) @ X21) = (X19 @ (comp @ X22 @ X21) @ (sub @ X20 @ X21))) & ! [X23 : term] : (($true != (X15 @ id @ X23 @ id)) | ($true = (X15 @ id @ (X19 @ id @ X23) @ id))) & ($true != (X15 @ id @ (lam @ (X19 @ sh @ one)) @ id))) | ? [X18 : term] : ((X16 @ X18) != (X15 @ id @ X18 @ id)) | ($true = (X16 @ (lam @ X17))) | ? [X24 : term] : (($true != (X16 @ (sub @ X17 @ (push @ X24 @ id)))) & ($true = (X16 @ X24))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f44,plain,(
% 2.56/1.94 ! [X129 : subst > term > subst > $o] : (($true = (sP30 @ X129)) <=> ? [X140 : term,X141 : term] : (($true = (X129 @ id @ X140 @ id)) & ($true = (X129 @ id @ X141 @ id)) & ($true != (X129 @ id @ (ap @ (sub @ X140 @ id) @ X141) @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f45,plain,(
% 2.56/1.94 ($true = sP31) <=> ! [X127 : term,X128 : term > $o,X129 : subst > term > subst > $o,X130 : term] : (($true != (X128 @ X130)) | ? [X139 : term] : ((X129 @ id @ X139 @ id) != (X128 @ X139)) | ($true = (sP30 @ X129)) | ($true = (X128 @ (ap @ X130 @ X127))) | ? [X131 : term,X132 : subst,X133 : subst,X134 : subst] : (($true = (X129 @ X133 @ X131 @ (comp @ X132 @ X134))) & ($true != (X129 @ (comp @ X133 @ X132) @ (sub @ X131 @ X132) @ X134))) | ? [X135 : subst,X136 : subst,X137 : term,X138 : subst] : (($true != (X129 @ X138 @ X137 @ (comp @ X136 @ X135))) & ($true = (X129 @ (comp @ X138 @ X136) @ (sub @ X137 @ X136) @ X135))) | ($true != (X128 @ X127)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f46,plain,(
% 2.56/1.94 ! [X97 : term > $o] : (($true = (sP32 @ X97)) <=> ? [X101 : term,X102 : term] : (($true = (X97 @ X102)) & ($true != (X97 @ (ap @ X102 @ X101))) & ($true = (X97 @ X101))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f47,plain,(
% 2.56/1.94 ! [X90 : term > $o] : (($true = (sP33 @ X90)) <=> ! [X91 : term,X92 : term] : (((X90 @ X91) != $true) | ($true = (X90 @ (ap @ X91 @ X92))) | ($true != (X90 @ X92))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f48,plain,(
% 2.56/1.94 ($true = sP34) <=> ! [X95 : subst,X96 : term,X97 : term > $o] : (($true = (X97 @ (sub @ X96 @ X95))) | ? [X100 : term] : (($true = (var @ X100)) & ($true != (X97 @ (sub @ X100 @ X95)))) | ? [X98 : term] : (! [X99 : term] : (($true != (X97 @ X99)) | ($true = (X97 @ (sub @ X98 @ (push @ X99 @ id))))) & ($true != (X97 @ (lam @ X98)))) | ($true = (sP32 @ X97)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f49,plain,(
% 2.56/1.94 ($true = sP35) <=> (? [X89 : term,X90 : term > $o] : (($true = (sP33 @ X90)) & ! [X93 : term] : (($true != (X90 @ X93)) | ($true = (X90 @ (lam @ X93)))) & ! [X94 : term] : (($true != (var @ X94)) | ($true = (X90 @ X94))) & ($true != (X90 @ X89))) | ? [X84 : subst,X85 : term] : ((sub @ one @ (push @ X85 @ X84)) != X85) | ($true = sP34) | ? [X86 : subst,X87 : term] : ((comp @ sh @ (push @ X87 @ X86)) != X86) | ? [X88 : subst] : ((comp @ X88 @ id) != X88))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f50,plain,(
% 2.56/1.94 ! [X479 : term > $o] : (($true = (sP36 @ X479)) <=> ? [X483 : term,X484 : term] : (($true = (X479 @ X484)) & ($true = (X479 @ X483)) & ((X479 @ (ap @ X483 @ X484)) != $true)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f51,plain,(
% 2.56/1.94 ! [X497 : subst > term > subst > $o] : (($true = (sP37 @ X497)) <=> ? [X509 : subst > term > term] : (! [X510 : subst,X511 : term,X512 : subst] : ((sub @ (X509 @ X512 @ X511) @ X510) = (X509 @ (comp @ X512 @ X510) @ (sub @ X511 @ X510))) & ($true != (X497 @ id @ (lam @ (X509 @ sh @ one)) @ id)) & ! [X513 : term] : (($true = (X497 @ id @ (X509 @ id @ X513) @ id)) | ($true != (X497 @ id @ X513 @ id)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f52,plain,(
% 2.56/1.94 ($true = sP38) <=> (! [X496 : term > $o,X497 : subst > term > subst > $o,X498 : term] : (? [X504 : term] : (($true != (X496 @ (sub @ X498 @ (push @ X504 @ id)))) & ($true = (X496 @ X504))) | ? [X499 : subst,X500 : term,X501 : subst,X502 : subst] : (($true != (X497 @ (comp @ X501 @ X499) @ (sub @ X500 @ X499) @ X502)) & ($true = (X497 @ X501 @ X500 @ (comp @ X499 @ X502)))) | ($true = (X496 @ (lam @ X498))) | ? [X503 : term] : ((X496 @ X503) != (X497 @ id @ X503 @ id)) | ? [X505 : subst,X506 : term,X507 : subst,X508 : subst] : (((X497 @ (comp @ X508 @ X507) @ (sub @ X506 @ X507) @ X505) = $true) & ($true != (X497 @ X508 @ X506 @ (comp @ X507 @ X505)))) | ($true = (sP37 @ X497))) | ? [X495 : term] : ((sub @ X495 @ id) != X495))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f53,plain,(
% 2.56/1.94 (sP39 = $true) <=> ! [X310 : subst > term > subst > $o,X311 : term] : (? [X313 : subst > term > term] : (! [X317 : term] : (($true = (X310 @ id @ (X313 @ id @ X317) @ id)) | ($true != (X310 @ id @ X317 @ id))) & ($true != (X310 @ id @ (lam @ (X313 @ sh @ one)) @ id)) & ! [X314 : subst,X315 : term,X316 : subst] : ((sub @ (X313 @ X314 @ X315) @ X316) = (X313 @ (comp @ X314 @ X316) @ (sub @ X315 @ X316)))) | ? [X312 : term] : (($true != (X310 @ id @ (sub @ X311 @ (push @ X312 @ id)) @ id)) & ($true = (X310 @ id @ X312 @ id))) | ($true = (X310 @ id @ (lam @ X311) @ id)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f54,plain,(
% 2.56/1.94 ($true = sP40) <=> ! [X437 : subst > term > subst > $o,X438 : term] : (? [X444 : term] : (($true = (X437 @ id @ X444 @ id)) & ($true != (X437 @ id @ (sub @ X438 @ (push @ X444 @ id)) @ id))) | ? [X439 : subst > term > term] : (($true != (X437 @ id @ (lam @ (X439 @ sh @ one)) @ id)) & ! [X440 : term] : (($true = (X437 @ id @ (X439 @ id @ X440) @ id)) | ($true != (X437 @ id @ X440 @ id))) & ! [X441 : subst,X442 : term,X443 : subst] : ((sub @ (X439 @ X441 @ X442) @ X443) = (X439 @ (comp @ X441 @ X443) @ (sub @ X442 @ X443)))) | ($true = (X437 @ id @ (lam @ (sub @ X438 @ (push @ one @ sh))) @ id)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f55,plain,(
% 2.56/1.94 ! [X356 : term > $o] : (($true = (sP41 @ X356)) <=> ? [X362 : term,X363 : term] : (($true = (X356 @ X362)) & ($true = (X356 @ X363)) & ($true != (X356 @ (ap @ X362 @ X363)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f56,plain,(
% 2.56/1.94 ($true = sP42) <=> ! [X356 : term > $o,X357 : subst,X358 : term] : (($true = (X356 @ (sub @ X358 @ X357))) | ? [X359 : term] : (($true = (var @ X359)) & ($true != (X356 @ (sub @ X359 @ X357)))) | ($true = (sP41 @ X356)) | ? [X360 : term] : (! [X361 : term] : (($true = (X356 @ (sub @ X360 @ (push @ X361 @ id)))) | ($true != (X356 @ X361))) & ($true != (X356 @ (lam @ X360)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f57,plain,(
% 2.56/1.94 ! [X60 : subst > term > subst > $o] : (($true = (sP43 @ X60)) <=> ? [X61 : term,X62 : term] : (($true = (X60 @ id @ X61 @ id)) & ($true != (X60 @ id @ (ap @ (sub @ X61 @ id) @ X62) @ id)) & ($true = (X60 @ id @ X62 @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f58,plain,(
% 2.56/1.94 ! [X60 : subst > term > subst > $o] : (($true = (sP44 @ X60)) <=> ? [X63 : subst > term > term] : (($true != (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) & ! [X64 : term] : (($true != (X60 @ id @ X64 @ id)) | ($true = (X60 @ id @ (X63 @ id @ X64) @ id))) & ! [X65 : subst,X66 : subst,X67 : term] : ((X63 @ (comp @ X65 @ X66) @ (sub @ X67 @ X66)) = (sub @ (X63 @ X65 @ X67) @ X66))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f59,plain,(
% 2.56/1.94 ($true = sP45) <=> ! [X59 : term,X60 : subst > term > subst > $o] : (? [X68 : term] : (($true = (var @ (sub @ X68 @ id))) & ($true != (X60 @ id @ X68 @ id))) | ($true = (sP44 @ X60)) | ($true = (X60 @ id @ X59 @ id)) | ? [X73 : term,X74 : subst,X75 : subst,X76 : subst] : (($true != (X60 @ (comp @ X75 @ X74) @ (sub @ X73 @ X74) @ X76)) & ($true = (X60 @ X75 @ X73 @ (comp @ X74 @ X76)))) | ($true = (sP43 @ X60)) | ? [X69 : subst,X70 : subst,X71 : term,X72 : subst] : (($true != (X60 @ X69 @ X71 @ (comp @ X72 @ X70))) & ($true = (X60 @ (comp @ X69 @ X72) @ (sub @ X71 @ X72) @ X70))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f60,plain,(
% 2.56/1.94 ! [X249 : subst > term > subst > $o] : (($true = (sP46 @ X249)) <=> ? [X256 : subst > term > term] : (! [X257 : term] : (($true != (X249 @ id @ X257 @ id)) | ($true = (X249 @ id @ (X256 @ id @ X257) @ id))) & ! [X258 : subst,X259 : term,X260 : subst] : ((sub @ (X256 @ X260 @ X259) @ X258) = (X256 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258))) & ($true != (X249 @ id @ (lam @ (X256 @ sh @ one)) @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f61,plain,(
% 2.56/1.94 ! [X249 : subst > term > subst > $o] : (($true = (sP47 @ X249)) <=> ? [X250 : term,X251 : term] : (($true != (X249 @ id @ (ap @ (sub @ X251 @ id) @ X250) @ id)) & ($true = (X249 @ id @ X250 @ id)) & ($true = (X249 @ id @ X251 @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f62,plain,(
% 2.56/1.94 ! [X230 : term > $o] : (($true = (sP48 @ X230)) <=> ! [X235 : term,X236 : term] : (($true != (X230 @ X236)) | ($true != (X230 @ X235)) | ($true = (X230 @ (ap @ X235 @ X236)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f63,plain,(
% 2.56/1.94 ($true = sP49) <=> ! [X248 : term,X249 : subst > term > subst > $o] : (? [X261 : term,X262 : subst,X263 : subst,X264 : subst] : (($true != (X249 @ X262 @ X261 @ (comp @ X263 @ X264))) & ($true = (X249 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263) @ X264))) | ($true = (sP47 @ X249)) | ($true = (sP46 @ X249)) | ? [X252 : subst,X253 : term,X254 : subst,X255 : subst] : (($true != (X249 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254) @ X255)) & ($true = (X249 @ X252 @ X253 @ (comp @ X254 @ X255)))) | ? [X265 : term] : (($true = (var @ (sub @ X265 @ id))) & ($true != (X249 @ id @ X265 @ id))) | ($true = (X249 @ id @ X248 @ id)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f64,plain,(
% 2.56/1.94 ($true = sP50) <=> ? [X238 : term,X239 : term > $o,X240 : subst > term > subst > $o] : (! [X241 : term] : (($true = (X239 @ (sub @ X238 @ (push @ X241 @ id)))) | ($true != (X239 @ X241))) & ($true != (X239 @ (lam @ X238))) & ! [X247 : term] : ((X240 @ id @ X247 @ id) = (X239 @ X247)) & ! [X242 : subst > term > term] : (($true = (X240 @ id @ (lam @ (X242 @ sh @ one)) @ id)) | ? [X244 : subst,X245 : term,X246 : subst] : ((X242 @ (comp @ X246 @ X244) @ (sub @ X245 @ X244)) != (sub @ (X242 @ X246 @ X245) @ X244)) | ? [X243 : term] : (($true != (X240 @ id @ (X242 @ id @ X243) @ id)) & ($true = (X240 @ id @ X243 @ id)))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f65,plain,(
% 2.56/1.94 (sP51 = $true) <=> (($true = sP50) | ? [X227 : subst > term > subst > $o] : ! [X228 : term > $o] : ? [X229 : term] : ((X227 @ id @ X229 @ id) != (X228 @ X229)) | ? [X230 : term > $o,X231 : term] : (! [X232 : term] : (($true = (X230 @ X232)) | ($true != (var @ X232))) & ($true = (sP48 @ X230)) & ($true != (X230 @ X231)) & ! [X233 : term] : (? [X234 : term] : (($true = (X230 @ X234)) & ($true != (X230 @ (sub @ X233 @ (push @ X234 @ id))))) | ($true = (X230 @ (lam @ X233))))) | ? [X237 : term] : ((sub @ X237 @ id) != X237) | ($true = sP49))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f66,plain,(
% 2.56/1.94 ! [X459 : term > $o] : (((sP52 @ X459) = $true) <=> ? [X462 : term,X463 : term] : (($true != (X459 @ (ap @ X462 @ X463))) & ($true = (X459 @ X462)) & ($true = (X459 @ X463))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f67,plain,(
% 2.56/1.94 ! [X453 : term > $o] : (($true = (sP53 @ X453)) <=> ! [X455 : term,X456 : term] : (($true != (X453 @ X455)) | ($true = (X453 @ (ap @ X455 @ X456))) | ((X453 @ X456) != $true)))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f68,plain,(
% 2.56/1.94 ($true = sP54) <=> ! [X459 : term > $o,X460 : term] : (($true = (X459 @ X460)) | ((sP52 @ X459) = $true) | ? [X464 : term] : (! [X465 : term] : (($true != (X459 @ X465)) | ($true = (X459 @ (sub @ X464 @ (push @ X465 @ id))))) & ($true != (X459 @ (lam @ X464)))) | ? [X461 : term] : (($true = (var @ X461)) & ($true != (X459 @ X461))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f69,plain,(
% 2.56/1.94 ($true = sP55) <=> (? [X451 : subst,X452 : term,X453 : term > $o] : (! [X454 : term] : (($true != (var @ X454)) | ($true = (X453 @ (sub @ X454 @ X451)))) & ($true = (sP53 @ X453)) & ! [X457 : term] : (? [X458 : term] : (($true != (X453 @ (sub @ X457 @ (push @ X458 @ id)))) & ($true = (X453 @ X458))) | ($true = (X453 @ (lam @ X457)))) & ($true != (X453 @ (sub @ X452 @ X451)))) | ($true = sP54) | ? [X450 : term] : ((sub @ X450 @ id) != X450))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f70,plain,(
% 2.56/1.94 ($true = sP56) <=> (! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term] : (? [X374 : subst > term > term] : (! [X375 : term] : (($true != (X370 @ id @ X375 @ id)) | ($true = (X370 @ id @ (X374 @ id @ X375) @ id))) & ! [X376 : subst,X377 : term,X378 : subst] : ((sub @ (X374 @ X376 @ X377) @ X378) = (X374 @ (comp @ X376 @ X378) @ (sub @ X377 @ X378))) & ($true != (X370 @ id @ (lam @ (X374 @ sh @ one)) @ id))) | ? [X373 : term] : (($true = (X369 @ X373)) & ((X369 @ (sub @ X371 @ (push @ X373 @ id))) != $true)) | ($true = (X369 @ (lam @ X371))) | ? [X372 : term] : ((X370 @ id @ X372 @ id) != (X369 @ X372))) | ? [X364 : term] : ! [X365 : subst > term > term] : ? [X366 : subst,X367 : term] : ((sub @ X364 @ (push @ X367 @ X366)) != (X365 @ X366 @ X367)) | ? [X368 : term] : ((sub @ X368 @ id) != X368))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f71,plain,(
% 2.56/1.94 ! [X468 : subst > term > subst > $o] : (($true = (sP57 @ X468)) <=> ? [X469 : term,X470 : term] : (($true = (X468 @ id @ X470 @ id)) & ($true = (X468 @ id @ X469 @ id)) & ($true != (X468 @ id @ (ap @ (sub @ X470 @ id) @ X469) @ id))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f72,plain,(
% 2.56/1.94 ($true = sP58) <=> ! [X466 : term,X467 : term,X468 : subst > term > subst > $o] : (? [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (($true != (X468 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471)) & ($true = (X468 @ X473 @ X474 @ (comp @ X472 @ X471)))) | ($true != (X468 @ id @ X467 @ id)) | ($true != (X468 @ id @ X466 @ id)) | ($true = (X468 @ id @ (ap @ X467 @ X466) @ id)) | ($true = (sP57 @ X468)) | ? [X475 : subst,X476 : subst,X477 : subst,X478 : term] : (($true != (X468 @ X476 @ X478 @ (comp @ X475 @ X477))) & ($true = (X468 @ (comp @ X476 @ X475) @ (sub @ X478 @ X475) @ X477))))),
% 2.56/1.94 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.56/1.94 thf(f13,plain,(
% 2.56/1.94 (hoaslamnotvar = $true) & ((apinj2 = $true) <=> ! [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) != (ap @ X3 @ X1)) | (X0 = X1))) & (ulamvarind = $true) & (axapp = $true) & (hoasinduction_lem1 = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (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)))))))))))))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X32 : term] : ((sub @ X32 @ id) = X32) <=> (termmset = $true)) & (termmset_gthm = $true) & (pushprop_lem0_gthm = $true) & ((substmonoid_lthm = $true) <=> ((! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36)) | ? [X33 : subst] : ((comp @ id @ X33) != X33) | ? [X34 : subst] : ((comp @ X34 @ id) != X34))) & ! [X485 : d_term] : (d_one = X485) & ((d2subst @ d_id) = id) & ((hoasinduction_lem2 = $true) <=> ! [X466 : term,X467 : term,X468 : subst > term > subst > $o] : (? [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (($true != (X468 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471)) & ($true = (X468 @ X473 @ X474 @ (comp @ X472 @ X471)))) | ($true != (X468 @ id @ X467 @ id)) | ($true != (X468 @ id @ X466 @ id)) | ($true = (X468 @ id @ (ap @ X467 @ X466) @ id)) | ? [X469 : term,X470 : term] : (($true = (X468 @ id @ X470 @ id)) & ($true = (X468 @ id @ X469 @ id)) & ($true != (X468 @ id @ (ap @ (sub @ X470 @ id) @ X469) @ id))) | ? [X475 : subst,X476 : subst,X477 : subst,X478 : term] : (($true != (X468 @ X476 @ X478 @ (comp @ X475 @ X477))) & ($true = (X468 @ (comp @ X476 @ X475) @ (sub @ X478 @ X475) @ X477))))) & (pushprop_lem1v2_gthm = $true) & ((! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term] : (? [X374 : subst > term > term] : (! [X375 : term] : (($true != (X370 @ id @ X375 @ id)) | ($true = (X370 @ id @ (X374 @ id @ X375) @ id))) & ! [X376 : subst,X377 : term,X378 : subst] : ((sub @ (X374 @ X376 @ X377) @ X378) = (X374 @ (comp @ X376 @ X378) @ (sub @ X377 @ X378))) & ($true != (X370 @ id @ (lam @ (X374 @ sh @ one)) @ id))) | ? [X373 : term] : (($true = (X369 @ X373)) & ((X369 @ (sub @ X371 @ (push @ X373 @ id))) != $true)) | ($true = (X369 @ (lam @ X371))) | ? [X372 : term] : ((X370 @ id @ X372 @ id) != (X369 @ X372))) | ? [X364 : term] : ! [X365 : subst > term > term] : ? [X366 : subst,X367 : term] : ((sub @ X364 @ (push @ X367 @ X366)) != (X365 @ X366 @ X367)) | ? [X368 : term] : ((sub @ X368 @ id) != X368)) <=> (hoasinduction_lem3v2a_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((induction2_lthm = $true) <=> (? [X451 : subst,X452 : term,X453 : term > $o] : (! [X454 : term] : (($true != (var @ X454)) | ($true = (X453 @ (sub @ X454 @ X451)))) & ! [X455 : term,X456 : term] : (($true != (X453 @ X455)) | ($true = (X453 @ (ap @ X455 @ X456))) | ((X453 @ X456) != $true)) & ! [X457 : term] : (? [X458 : term] : (($true != (X453 @ (sub @ X457 @ (push @ X458 @ id)))) & ($true = (X453 @ X458))) | ($true = (X453 @ (lam @ X457)))) & ($true != (X453 @ (sub @ X452 @ X451)))) | ! [X459 : term > $o,X460 : term] : (($true = (X459 @ X460)) | ? [X462 : term,X463 : term] : (($true != (X459 @ (ap @ X462 @ X463))) & ($true = (X459 @ X462)) & ($true = (X459 @ X463))) | ? [X464 : term] : (! [X465 : term] : (($true != (X459 @ X465)) | ($true = (X459 @ (sub @ X464 @ (push @ X465 @ id))))) & ($true != (X459 @ (lam @ X464)))) | ? [X461 : term] : (($true = (var @ X461)) & ($true != (X459 @ X461)))) | ? [X450 : term] : ((sub @ X450 @ id) != X450))) & ((? [X238 : term,X239 : term > $o,X240 : subst > term > subst > $o] : (! [X241 : term] : (($true = (X239 @ (sub @ X238 @ (push @ X241 @ id)))) | ($true != (X239 @ X241))) & ($true != (X239 @ (lam @ X238))) & ! [X247 : term] : ((X240 @ id @ X247 @ id) = (X239 @ X247)) & ! [X242 : subst > term > term] : (($true = (X240 @ id @ (lam @ (X242 @ sh @ one)) @ id)) | ? [X244 : subst,X245 : term,X246 : subst] : ((X242 @ (comp @ X246 @ X244) @ (sub @ X245 @ X244)) != (sub @ (X242 @ X246 @ X245) @ X244)) | ? [X243 : term] : (($true != (X240 @ id @ (X242 @ id @ X243) @ id)) & ($true = (X240 @ id @ X243 @ id))))) | ? [X227 : subst > term > subst > $o] : ! [X228 : term > $o] : ? [X229 : term] : ((X227 @ id @ X229 @ id) != (X228 @ X229)) | ? [X230 : term > $o,X231 : term] : (! [X232 : term] : (($true = (X230 @ X232)) | ($true != (var @ X232))) & ! [X235 : term,X236 : term] : (($true != (X230 @ X236)) | ($true != (X230 @ X235)) | ($true = (X230 @ (ap @ X235 @ X236)))) & ($true != (X230 @ X231)) & ! [X233 : term] : (? [X234 : term] : (($true = (X230 @ X234)) & ($true != (X230 @ (sub @ X233 @ (push @ X234 @ id))))) | ($true = (X230 @ (lam @ X233))))) | ? [X237 : term] : ((sub @ X237 @ id) != X237) | ! [X248 : term,X249 : subst > term > subst > $o] : (? [X261 : term,X262 : subst,X263 : subst,X264 : subst] : (($true != (X249 @ X262 @ X261 @ (comp @ X263 @ X264))) & ($true = (X249 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263) @ X264))) | ? [X250 : term,X251 : term] : (($true != (X249 @ id @ (ap @ (sub @ X251 @ id) @ X250) @ id)) & ($true = (X249 @ id @ X250 @ id)) & ($true = (X249 @ id @ X251 @ id))) | ? [X256 : subst > term > term] : (! [X257 : term] : (($true != (X249 @ id @ X257 @ id)) | ($true = (X249 @ id @ (X256 @ id @ X257) @ id))) & ! [X258 : subst,X259 : term,X260 : subst] : ((sub @ (X256 @ X260 @ X259) @ X258) = (X256 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258))) & ($true != (X249 @ id @ (lam @ (X256 @ sh @ one)) @ id))) | ? [X252 : subst,X253 : term,X254 : subst,X255 : subst] : (($true != (X249 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254) @ X255)) & ($true = (X249 @ X252 @ X253 @ (comp @ X254 @ X255)))) | ? [X265 : term] : (($true = (var @ (sub @ X265 @ id))) & ($true != (X249 @ id @ X265 @ id))) | ($true = (X249 @ id @ X248 @ id)))) <=> (hoasinduction_lthm_3 = $true)) & ((d2subst @ d_id) = sh) & (hoaslamnotap_gthm = $true) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) != (d2subst @ X14)) | (X13 = X14)) & ((hoasinduction_lem3b = $true) <=> ! [X537 : term] : ? [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) = (X538 @ sh @ one))) & ((shinj = $true) <=> ! [X514 : term,X515 : term] : (((sub @ X514 @ sh) != (sub @ X515 @ sh)) | (X514 = X515))) & (! [X59 : term,X60 : subst > term > subst > $o] : (? [X68 : term] : (($true = (var @ (sub @ X68 @ id))) & ($true != (X60 @ id @ X68 @ id))) | ? [X63 : subst > term > term] : (($true != (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) & ! [X64 : term] : (($true != (X60 @ id @ X64 @ id)) | ($true = (X60 @ id @ (X63 @ id @ X64) @ id))) & ! [X65 : subst,X66 : subst,X67 : term] : ((X63 @ (comp @ X65 @ X66) @ (sub @ X67 @ X66)) = (sub @ (X63 @ X65 @ X67) @ X66))) | ($true = (X60 @ id @ X59 @ id)) | ? [X73 : term,X74 : subst,X75 : subst,X76 : subst] : (($true != (X60 @ (comp @ X75 @ X74) @ (sub @ X73 @ X74) @ X76)) & ($true = (X60 @ X75 @ X73 @ (comp @ X74 @ X76)))) | ? [X61 : term,X62 : term] : (($true = (X60 @ id @ X61 @ id)) & ($true != (X60 @ id @ (ap @ (sub @ X61 @ id) @ X62) @ id)) & ($true = (X60 @ id @ X62 @ id))) | ? [X69 : subst,X70 : subst,X71 : term,X72 : subst] : (($true != (X60 @ X69 @ X71 @ (comp @ X72 @ X70))) & ($true = (X60 @ (comp @ X69 @ X72) @ (sub @ X71 @ X72) @ X70)))) <=> (hoasinduction = $true)) & ((induction2lem = $true) <=> ! [X356 : term > $o,X357 : subst,X358 : term] : (($true = (X356 @ (sub @ X358 @ X357))) | ? [X359 : term] : (($true = (var @ X359)) & ($true != (X356 @ (sub @ X359 @ X357)))) | ? [X362 : term,X363 : term] : (($true = (X356 @ X362)) & ($true = (X356 @ X363)) & ($true != (X356 @ (ap @ X362 @ X363)))) | ? [X360 : term] : (! [X361 : term] : (($true = (X356 @ (sub @ X360 @ (push @ X361 @ id)))) | ($true != (X356 @ X361))) & ($true != (X356 @ (lam @ X360)))))) & ((hoasinduction_lem0 = $true) <=> ! [X77 : subst > term > subst > $o] : ? [X78 : term > $o] : ! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X437 : subst > term > subst > $o,X438 : term] : (? [X444 : term] : (($true = (X437 @ id @ X444 @ id)) & ($true != (X437 @ id @ (sub @ X438 @ (push @ X444 @ id)) @ id))) | ? [X439 : subst > term > term] : (($true != (X437 @ id @ (lam @ (X439 @ sh @ one)) @ id)) & ! [X440 : term] : (($true = (X437 @ id @ (X439 @ id @ X440) @ id)) | ($true != (X437 @ id @ X440 @ id))) & ! [X441 : subst,X442 : term,X443 : subst] : ((sub @ (X439 @ X441 @ X442) @ X443) = (X439 @ (comp @ X441 @ X443) @ (sub @ X442 @ X443)))) | ($true = (X437 @ id @ (lam @ (sub @ X438 @ (push @ one @ sh))) @ id)))) & (hoaslamnotap_lthm = $true) & ((? [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) != X203) | ? [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) != X204) | ! [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (($true = (X208 @ (sub @ X210 @ (push @ X206 @ X209)))) | ($true != (X207 @ X210)))) <=> (pushprop_lem1_lthm = $true)) & (! [X310 : subst > term > subst > $o,X311 : term] : (? [X313 : subst > term > term] : (! [X317 : term] : (($true = (X310 @ id @ (X313 @ id @ X317) @ id)) | ($true != (X310 @ id @ X317 @ id))) & ($true != (X310 @ id @ (lam @ (X313 @ sh @ one)) @ id)) & ! [X314 : subst,X315 : term,X316 : subst] : ((sub @ (X313 @ X314 @ X315) @ X316) = (X313 @ (comp @ X314 @ X316) @ (sub @ X315 @ X316)))) | ? [X312 : term] : (($true != (X310 @ id @ (sub @ X311 @ (push @ X312 @ id)) @ id)) & ($true = (X310 @ id @ X312 @ id))) | ($true = (X310 @ id @ (lam @ X311) @ id))) <=> (hoasinduction_lem3a = $true)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X318 : term] : ((sub @ X318 @ id) = X318) <=> (axvarid = $true)) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X496 : term > $o,X497 : subst > term > subst > $o,X498 : term] : (? [X504 : term] : (($true != (X496 @ (sub @ X498 @ (push @ X504 @ id)))) & ($true = (X496 @ X504))) | ? [X499 : subst,X500 : term,X501 : subst,X502 : subst] : (($true != (X497 @ (comp @ X501 @ X499) @ (sub @ X500 @ X499) @ X502)) & ($true = (X497 @ X501 @ X500 @ (comp @ X499 @ X502)))) | ($true = (X496 @ (lam @ X498))) | ? [X503 : term] : ((X496 @ X503) != (X497 @ id @ X503 @ id)) | ? [X505 : subst,X506 : term,X507 : subst,X508 : subst] : (((X497 @ (comp @ X508 @ X507) @ (sub @ X506 @ X507) @ X505) = $true) & ($true != (X497 @ X508 @ X506 @ (comp @ X507 @ X505)))) | ? [X509 : subst > term > term] : (! [X510 : subst,X511 : term,X512 : subst] : ((sub @ (X509 @ X512 @ X511) @ X510) = (X509 @ (comp @ X512 @ X510) @ (sub @ X511 @ X510))) & ($true != (X497 @ id @ (lam @ (X509 @ sh @ one)) @ id)) & ! [X513 : term] : (($true = (X497 @ id @ (X509 @ id @ X513) @ id)) | ($true != (X497 @ id @ X513 @ id))))) | ? [X495 : term] : ((sub @ X495 @ id) != X495))) & (axabs = $true) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & (ulamvarsh = $true) & ! [X106 : d_subst] : (d_id = X106) & ((induction = $true) <=> ! [X479 : term > $o,X480 : term] : (($true = (X479 @ X480)) | ? [X481 : term] : (($true != (X479 @ X481)) & ($true = (var @ X481))) | ? [X483 : term,X484 : term] : (($true = (X479 @ X484)) & ($true = (X479 @ X483)) & ((X479 @ (ap @ X483 @ X484)) != $true)) | ? [X482 : term] : (($true != (X479 @ (lam @ X482))) & ($true = (X479 @ X482))))) & (pushprop_lem2v2_lthm = $true) & (hoaslaminj_gthm = $true) & ((axidr = $true) <=> ! [X105 : subst] : ((comp @ X105 @ id) = X105)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & ((? [X89 : term,X90 : term > $o] : (! [X91 : term,X92 : term] : (((X90 @ X91) != $true) | ($true = (X90 @ (ap @ X91 @ X92))) | ($true != (X90 @ X92))) & ! [X93 : term] : (($true != (X90 @ X93)) | ($true = (X90 @ (lam @ X93)))) & ! [X94 : term] : (($true != (var @ X94)) | ($true = (X90 @ X94))) & ($true != (X90 @ X89))) | ? [X84 : subst,X85 : term] : ((sub @ one @ (push @ X85 @ X84)) != X85) | ! [X95 : subst,X96 : term,X97 : term > $o] : (($true = (X97 @ (sub @ X96 @ X95))) | ? [X100 : term] : (($true = (var @ X100)) & ($true != (X97 @ (sub @ X100 @ X95)))) | ? [X98 : term] : (! [X99 : term] : (($true != (X97 @ X99)) | ($true = (X97 @ (sub @ X98 @ (push @ X99 @ id))))) & ($true != (X97 @ (lam @ X98)))) | ? [X101 : term,X102 : term] : (($true = (X97 @ X102)) & ($true != (X97 @ (ap @ X102 @ X101))) & ($true = (X97 @ X101)))) | ? [X86 : subst,X87 : term] : ((comp @ sh @ (push @ X87 @ X86)) != X86) | ? [X88 : subst] : ((comp @ X88 @ id) != X88)) <=> (induction2lem_lthm = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem2v2 = $true) <=> ! [X127 : term,X128 : term > $o,X129 : subst > term > subst > $o,X130 : term] : (($true != (X128 @ X130)) | ? [X139 : term] : ((X129 @ id @ X139 @ id) != (X128 @ X139)) | ? [X140 : term,X141 : term] : (($true = (X129 @ id @ X140 @ id)) & ($true = (X129 @ id @ X141 @ id)) & ($true != (X129 @ id @ (ap @ (sub @ X140 @ id) @ X141) @ id))) | ($true = (X128 @ (ap @ X130 @ X127))) | ? [X131 : term,X132 : subst,X133 : subst,X134 : subst] : (($true = (X129 @ X133 @ X131 @ (comp @ X132 @ X134))) & ($true != (X129 @ (comp @ X133 @ X132) @ (sub @ X131 @ X132) @ X134))) | ? [X135 : subst,X136 : subst,X137 : term,X138 : subst] : (($true != (X129 @ X138 @ X137 @ (comp @ X136 @ X135))) & ($true = (X129 @ (comp @ X138 @ X136) @ (sub @ X137 @ X136) @ X135))) | ($true != (X128 @ X127)))) & (pushprop_lem3v2 = $true) & (hoasinduction_gthm = $true) & (induction2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X15 : subst > term > subst > $o,X16 : term > $o,X17 : term] : (? [X19 : subst > term > term] : (! [X20 : term,X21 : subst,X22 : subst] : ((sub @ (X19 @ X22 @ X20) @ X21) = (X19 @ (comp @ X22 @ X21) @ (sub @ X20 @ X21))) & ! [X23 : term] : (($true != (X15 @ id @ X23 @ id)) | ($true = (X15 @ id @ (X19 @ id @ X23) @ id))) & ($true != (X15 @ id @ (lam @ (X19 @ sh @ one)) @ id))) | ? [X18 : term] : ((X16 @ X18) != (X15 @ id @ X18 @ id)) | ($true = (X16 @ (lam @ X17))) | ? [X24 : term] : (($true != (X16 @ (sub @ X17 @ (push @ X24 @ id)))) & ($true = (X16 @ X24))))) & (axclos = $true) & (lamnotap != $true) & ((axvarcons = $true) <=> ! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104)) & (axvarshift = $true) & (pushprop_lem1_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lem2v2_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X532 : term,X533 : term > $o,X534 : subst] : ? [X535 : term > $o] : ! [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) = (X535 @ X536))) & ((hoasapinj1_lthm = $true) <=> (? [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) = (ap @ X490 @ X489)) & (X487 != X490)) | ! [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) != (ap @ (sub @ X491 @ id) @ X494)) | (X491 = X493)) | ? [X486 : term] : ((sub @ X486 @ id) != X486))) & ((? [X269 : term,X270 : subst > term > subst > $o] : (! [X271 : subst > term > term] : (($true = (X270 @ id @ (lam @ (X271 @ sh @ one)) @ id)) | ? [X275 : term] : (($true != (X270 @ id @ (X271 @ id @ X275) @ id)) & ($true = (X270 @ id @ X275 @ id))) | ? [X272 : subst,X273 : term,X274 : subst] : ((X271 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)) != (sub @ (X271 @ X272 @ X273) @ X274))) & ($true != (X270 @ id @ (lam @ (sub @ X269 @ (push @ one @ sh))) @ id)) & ! [X276 : term] : (($true = (X270 @ id @ (sub @ X269 @ (push @ X276 @ id)) @ id)) | ($true != (X270 @ id @ X276 @ id)))) | ? [X268 : term] : ((sub @ X268 @ id) != X268) | ! [X277 : term,X278 : subst > term > subst > $o] : (($true = (X278 @ id @ (lam @ X277) @ id)) | ? [X284 : term] : (($true = (X278 @ id @ X284 @ id)) & ($true != (X278 @ id @ (sub @ X277 @ (push @ X284 @ id)) @ id))) | ? [X279 : subst > term > term] : (($true != (X278 @ id @ (lam @ (X279 @ sh @ one)) @ id)) & ! [X283 : term] : (($true != (X278 @ id @ X283 @ id)) | ($true = (X278 @ id @ (X279 @ id @ X283) @ id))) & ! [X280 : subst,X281 : term,X282 : subst] : ((sub @ (X279 @ X282 @ X281) @ X280) = (X279 @ (comp @ X282 @ X280) @ (sub @ X281 @ X280)))))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoaslamnotvar_gthm = $true) & (! [X178 : term > $o,X179 : subst > term > subst > $o,X180 : term] : (($true = (X178 @ (lam @ X180))) | ? [X182 : subst > term > term] : (! [X184 : subst,X185 : subst,X186 : term] : ((sub @ (X182 @ X185 @ X186) @ X184) = (X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184))) & ($true != (X179 @ id @ (lam @ (X182 @ sh @ one)) @ id)) & ! [X183 : term] : (($true = (X179 @ id @ (X182 @ id @ X183) @ id)) | ($true != (X179 @ id @ X183 @ id)))) | ? [X192 : subst,X193 : subst,X194 : term,X195 : subst] : (($true != (X179 @ (comp @ X192 @ X193) @ (sub @ X194 @ X193) @ X195)) & ($true = (X179 @ X192 @ X194 @ (comp @ X193 @ X195)))) | ? [X181 : term] : ((X179 @ id @ X181 @ id) != (X178 @ X181)) | ? [X188 : term,X189 : subst,X190 : subst,X191 : subst] : (($true != (X179 @ X189 @ X188 @ (comp @ X190 @ X191))) & ($true = (X179 @ (comp @ X189 @ X190) @ (sub @ X188 @ X190) @ X191))) | ? [X187 : term] : (($true != (X178 @ (sub @ X180 @ (push @ X187 @ id)))) & ($true = (X178 @ X187)))) <=> (hoasinduction_lem3v2 = $true)) & ((hoaslamnotap = $true) <=> ! [X117 : subst > term > term,X118 : term,X119 : term] : (? [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) != (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119)))) & (pushprop = $true) & (pushprop_lthm = $true) & (! [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) != (ap @ (sub @ X436 @ id) @ X434)) | (X435 = X436)) <=> (hoasapinj1 = $true)) & (apnotvar = $true) & (hoasinduction_lem2_gthm = $true) & (! [X44 : term,X45 : subst > term > subst > $o] : (($true = (X45 @ id @ (lam @ (sub @ X44 @ (push @ one @ sh))) @ id)) | ? [X46 : term] : (((X45 @ id @ (sub @ X44 @ (push @ X46 @ id)) @ id) != $true) & ($true = (X45 @ id @ X46 @ id))) | ? [X47 : subst > term > term,X48 : term] : (! [X49 : subst,X50 : term,X51 : subst] : ((sub @ (sub @ X48 @ (push @ X50 @ X49)) @ X51) = (sub @ (X47 @ X49 @ X50) @ X51)) & ($true != (X45 @ id @ (lam @ (X47 @ sh @ one)) @ id)) & ! [X52 : subst,X53 : term,X54 : subst] : ((sub @ X48 @ (push @ (sub @ X53 @ X52) @ (comp @ X54 @ X52))) = (X47 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) & ! [X55 : term] : (($true != (X45 @ id @ X55 @ id)) | ($true = (X45 @ id @ (X47 @ id @ X55) @ id))))) <=> (hoasinduction_lem3aaa = $true)) & ((? [X161 : term,X162 : term] : (((lam @ X161) = (lam @ X162)) & (X161 != X162)) | ! [X163 : subst,X164 : subst > term > term,X165 : term,X166 : subst > term > term] : (? [X167 : subst,X168 : term,X169 : subst] : ((sub @ (X164 @ X167 @ X168) @ X169) != (X164 @ (comp @ X167 @ X169) @ (sub @ X168 @ X169))) | ((X164 @ X163 @ X165) = (X166 @ X163 @ X165)) | ? [X170 : term,X171 : subst,X172 : subst] : ((sub @ (X166 @ X172 @ X170) @ X171) != (X166 @ (comp @ X172 @ X171) @ (sub @ X170 @ X171))) | ((lam @ (X166 @ sh @ one)) != (lam @ (X164 @ sh @ one)))) | ? [X157 : term,X158 : subst] : ((sub @ one @ (push @ X157 @ X158)) != X157) | ? [X159 : subst,X160 : term] : ((comp @ sh @ (push @ X160 @ X159)) != X159)) <=> (hoaslaminj_lthm = $true)) & (pushprop_lem2v2 = $true) & ((substmonoid = $true) <=> (! [X449 : subst] : ((comp @ X449 @ id) = X449) & ! [X448 : subst] : ((comp @ id @ X448) = X448))) & (! [X27 : subst,X28 : term,X29 : term > $o] : ? [X30 : term > $o] : ! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31)) <=> (pushprop_lem0 = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X200 : term] : ? [X201 : subst > term > term] : ((X201 @ sh @ one) = (sub @ X200 @ (push @ one @ sh)))) & ((? [X337 : subst > term > subst > $o,X338 : term,X339 : term > $o] : (! [X346 : term] : ((X339 @ X346) = (X337 @ id @ X346 @ id)) & ! [X340 : term] : (($true = (X339 @ (sub @ X338 @ (push @ X340 @ id)))) | ($true != (X339 @ X340))) & ! [X341 : subst > term > term] : (? [X342 : term] : (($true = (X337 @ id @ X342 @ id)) & ((X337 @ id @ (X341 @ id @ X342) @ id) != $true)) | ($true = (X337 @ id @ (lam @ (X341 @ sh @ one)) @ id)) | ? [X343 : subst,X344 : term,X345 : subst] : ((sub @ (X341 @ X345 @ X344) @ X343) != (X341 @ (comp @ X345 @ X343) @ (sub @ X344 @ X343)))) & ($true != (X339 @ (lam @ X338)))) | ? [X329 : term > $o,X330 : term] : (! [X332 : term] : (? [X333 : term] : (($true != (X329 @ (sub @ X332 @ (push @ X333 @ id)))) & ($true = (X329 @ X333))) | ($true = (X329 @ (lam @ X332)))) & ! [X334 : term,X335 : term] : (($true != (X329 @ X335)) | ($true = (X329 @ (ap @ X335 @ X334))) | ($true != (X329 @ X334))) & ($true != (X329 @ X330)) & ! [X331 : term] : (($true = (X329 @ X331)) | ($true != (var @ X331)))) | ? [X326 : subst > term > subst > $o] : ! [X327 : term > $o] : ? [X328 : term] : ((X326 @ id @ X328 @ id) != (X327 @ X328)) | ? [X336 : term] : ((sub @ X336 @ id) != X336) | ! [X347 : subst > term > subst > $o,X348 : term] : (($true = (X347 @ id @ X348 @ id)) | ? [X351 : subst > term > term] : (! [X355 : term] : (($true != (X347 @ id @ X355 @ id)) | ($true = (X347 @ id @ (X351 @ id @ X355) @ id))) & ($true != (X347 @ id @ (lam @ (X351 @ sh @ one)) @ id)) & ! [X352 : subst,X353 : subst,X354 : term] : ((X351 @ (comp @ X353 @ X352) @ (sub @ X354 @ X352)) = (sub @ (X351 @ X353 @ X354) @ X352))) | ? [X349 : term,X350 : term] : (($true != (X347 @ id @ (ap @ (sub @ X349 @ id) @ X350) @ id)) & ($true = (X347 @ id @ X350 @ id)) & ($true = (X347 @ id @ X349 @ id))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (! [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true = (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) | ($true != (X173 @ X174))) <=> (pushprop_lem1 = $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X80 : term] : ? [X81 : subst > term > term] : ! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83)) <=> (hoasinduction_lem3v2_f = $true)) & (! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true = (X144 @ one)) | ($true != (X142 @ X145)) | ? [X146 : term] : ((X144 @ X146) != (X142 @ (sub @ X146 @ (push @ X145 @ X143))))) <=> (pushprop_lem1v2 = $true)) & (ulamvar1 != $true) & (axmap = $true) & ((var @ (d2term @ d_one)) != $true) & (hoaslamnotvar_lthm = $true) & (one = (d2term @ d_one)) & (! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ? [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) != (sub @ (X107 @ X111 @ X113) @ X112)) | ? [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) != (sub @ (X110 @ X116 @ X115) @ X114))) <=> (hoaslaminj = $true)) & ((termmset_lthm = $true) <=> (! [X26 : term] : ((sub @ X26 @ id) = X26) | ? [X25 : term] : ((sub @ X25 @ id) != X25))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X294 : term,X295 : subst > term > subst > $o] : (? [X296 : subst,X297 : subst,X298 : subst,X299 : term] : (($true = (X295 @ (comp @ X298 @ X296) @ (sub @ X299 @ X296) @ X297)) & ($true != (X295 @ X298 @ X299 @ (comp @ X296 @ X297)))) | ? [X300 : term] : (($true != (X295 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id)) & ($true = (X295 @ id @ X300 @ id))) | ? [X301 : subst > term > term] : (! [X305 : term] : (($true != (X295 @ id @ X305 @ id)) | ($true = (X295 @ id @ (X301 @ id @ X305) @ id))) & ($true != (X295 @ id @ (lam @ (X301 @ sh @ one)) @ id)) & ! [X302 : term,X303 : subst,X304 : subst] : ((X301 @ (comp @ X303 @ X304) @ (sub @ X302 @ X304)) = (sub @ (X301 @ X303 @ X302) @ X304))) | ($true = (X295 @ id @ (lam @ X294) @ id)) | ? [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (($true = (X295 @ X308 @ X306 @ (comp @ X309 @ X307))) & ($true != (X295 @ (comp @ X308 @ X309) @ (sub @ X306 @ X309) @ X307)))) | ? [X285 : term] : ((sub @ X285 @ id) != X285) | ? [X286 : term,X287 : subst > term > subst > $o] : (($true != (X287 @ id @ (lam @ (sub @ X286 @ (push @ one @ sh))) @ id)) & ! [X293 : term] : (($true = (X287 @ id @ (sub @ X286 @ (push @ X293 @ id)) @ id)) | ($true != (X287 @ id @ X293 @ id))) & ! [X288 : subst > term > term] : (? [X290 : subst,X291 : subst,X292 : term] : ((X288 @ (comp @ X291 @ X290) @ (sub @ X292 @ X290)) != (sub @ (X288 @ X291 @ X292) @ X290)) | ($true = (X287 @ id @ (lam @ (X288 @ sh @ one)) @ id)) | ? [X289 : term] : (($true = (X287 @ id @ X289 @ id)) & ($true != (X287 @ id @ (X288 @ id @ X289) @ id))))))) & ((axidl = $true) <=> ! [X56 : subst] : ((comp @ id @ X56) = X56)) & ((hoasinduction_lthm = $true) <=> (? [X386 : term,X387 : term,X388 : subst > term > subst > $o] : (! [X389 : subst,X390 : subst,X391 : term,X392 : subst] : (($true = (X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389) @ X392)) | ($true != (X388 @ X390 @ X391 @ (comp @ X389 @ X392)))) & ! [X393 : term,X394 : term] : (($true = (X388 @ id @ (ap @ (sub @ X394 @ id) @ X393) @ id)) | ($true != (X388 @ id @ X393 @ id)) | ($true != (X388 @ id @ X394 @ id))) & ($true = (X388 @ id @ X386 @ id)) & ! [X395 : subst,X396 : subst,X397 : subst,X398 : term] : (($true = (X388 @ X396 @ X398 @ (comp @ X397 @ X395))) | ($true != (X388 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X395))) & ($true != (X388 @ id @ (ap @ X386 @ X387) @ id)) & ($true = (X388 @ id @ X387 @ id))) | ! [X415 : subst > term > subst > $o,X416 : term] : (($true = (X415 @ id @ X416 @ id)) | ? [X424 : subst,X425 : term,X426 : subst,X427 : subst] : (($true = (X415 @ X427 @ X425 @ (comp @ X424 @ X426))) & ($true != (X415 @ (comp @ X427 @ X424) @ (sub @ X425 @ X424) @ X426))) | ? [X419 : subst > term > term] : (! [X421 : term,X422 : subst,X423 : subst] : ((sub @ (X419 @ X422 @ X421) @ X423) = (X419 @ (comp @ X422 @ X423) @ (sub @ X421 @ X423))) & ((X415 @ id @ (lam @ (X419 @ sh @ one)) @ id) != $true) & ! [X420 : term] : (($true = (X415 @ id @ (X419 @ id @ X420) @ id)) | ($true != (X415 @ id @ X420 @ id)))) | ? [X417 : term,X418 : term] : (((X415 @ id @ X417 @ id) = $true) & ($true = (X415 @ id @ X418 @ id)) & ($true != (X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id))) | ? [X432 : term] : (($true != (X415 @ id @ X432 @ id)) & ($true = (var @ (sub @ X432 @ id)))) | ? [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X415 @ (comp @ X430 @ X429) @ (sub @ X428 @ X429) @ X431)) & ($true != (X415 @ X430 @ X428 @ (comp @ X429 @ X431))))) | ? [X379 : term,X380 : term > $o] : (($true != (X380 @ X379)) & ! [X383 : term] : (? [X384 : term] : (($true != (X380 @ (sub @ X383 @ (push @ X384 @ id)))) & ($true = (X380 @ X384))) | ($true = (X380 @ (lam @ X383)))) & ! [X385 : term] : (($true != (var @ X385)) | ($true = (X380 @ X385))) & ! [X381 : term,X382 : term] : (($true != (X380 @ X382)) | ((X380 @ (ap @ X381 @ X382)) = $true) | ($true != (X380 @ X381)))) | ? [X399 : subst > term > subst > $o,X400 : term] : (! [X402 : subst,X403 : term,X404 : subst,X405 : subst] : (($true = (X399 @ X404 @ X403 @ (comp @ X402 @ X405))) | ($true != (X399 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) & ! [X401 : term] : (($true = (X399 @ id @ (sub @ X400 @ (push @ X401 @ id)) @ id)) | ($true != (X399 @ id @ X401 @ id))) & ($true != (X399 @ id @ (lam @ X400) @ id)) & ! [X406 : subst,X407 : term,X408 : subst,X409 : subst] : (($true = (X399 @ (comp @ X406 @ X409) @ (sub @ X407 @ X409) @ X408)) | ($true != (X399 @ X406 @ X407 @ (comp @ X409 @ X408)))) & ! [X410 : subst > term > term] : (($true = (X399 @ id @ (lam @ (X410 @ sh @ one)) @ id)) | ? [X411 : term,X412 : subst,X413 : subst] : ((X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413)) != (sub @ (X410 @ X412 @ X411) @ X413)) | ? [X414 : term] : (($true = (X399 @ id @ X414 @ id)) & ($true != (X399 @ id @ (X410 @ id @ X414) @ id))))))) & ((! [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) != (ap @ (sub @ X216 @ id) @ X218)) | (X217 = X218)) | ? [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) = (ap @ X213 @ X212)) & (X212 != X214))) <=> (hoasapinj2_lthm = $true)) & (hoasapinj1_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X445 : subst > term > subst > $o] : ? [X446 : term > $o] : ! [X447 : term] : ((X445 @ id @ X447 @ id) = (X446 @ X447))) & (axassoc = $true) & (substmonoid_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X4 : term,X5 : subst > term > subst > $o] : (? [X11 : term,X12 : term] : (($true != (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) & ((X5 @ id @ X11 @ id) = $true) & ($true = (X5 @ id @ X12 @ id))) | ? [X6 : subst > term > term] : (! [X8 : term,X9 : subst,X10 : subst] : ((X6 @ (comp @ X10 @ X9) @ (sub @ X8 @ X9)) = (sub @ (X6 @ X10 @ X8) @ X9)) & ! [X7 : term] : (($true = (X5 @ id @ (X6 @ id @ X7) @ id)) | ($true != (X5 @ id @ X7 @ id))) & ($true != (X5 @ id @ (lam @ (X6 @ sh @ one)) @ id))) | ($true = (X5 @ id @ X4 @ id)))) & (induction2lem_gthm = $true) & ((axshiftcons = $true) <=> ! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58)) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X516 : term,X517 : subst > term > subst > $o] : (? [X522 : subst > term > term] : (((X517 @ id @ (lam @ (X522 @ sh @ one)) @ id) != $true) & ! [X523 : term] : (($true = (X517 @ id @ (X522 @ id @ X523) @ id)) | ($true != (X517 @ id @ X523 @ id))) & ! [X524 : term,X525 : subst,X526 : subst] : ((X522 @ (comp @ X525 @ X526) @ (sub @ X524 @ X526)) = (sub @ (X522 @ X525 @ X524) @ X526))) | ? [X528 : subst,X529 : subst,X530 : subst,X531 : term] : (((X517 @ X529 @ X531 @ (comp @ X530 @ X528)) != $true) & ($true = (X517 @ (comp @ X529 @ X530) @ (sub @ X531 @ X530) @ X528))) | ($true = (X517 @ id @ (lam @ X516) @ id)) | ? [X527 : term] : (($true = (X517 @ id @ X527 @ id)) & ($true != (X517 @ id @ (sub @ X516 @ (push @ X527 @ id)) @ id))) | ? [X518 : subst,X519 : term,X520 : subst,X521 : subst] : (($true != (X517 @ (comp @ X521 @ X518) @ (sub @ X519 @ X518) @ X520)) & ($true = (X517 @ X521 @ X519 @ (comp @ X518 @ X520)))))) & (hoasapnotvar = $true) & ! [X266 : subst] : ? [X267 : d_subst] : ((d2subst @ X267) = X266) & (! [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) = X539) <=> (axscons = $true)) & (pushprop_lthm_orig = $true) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 = X126) | ((ap @ X125 @ X123) != (ap @ X126 @ X124))) <=> (apinj1 = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoasapnotvar_gthm = $true) & (! [X319 : term,X320 : term > $o] : (($true = (X320 @ X319)) | ? [X322 : term,X323 : term] : (($true != (X320 @ (ap @ X323 @ X322))) & ($true = (X320 @ X322)) & ($true = (X320 @ X323))) | ? [X324 : term] : (! [X325 : term] : (($true = (X320 @ (sub @ X324 @ (push @ X325 @ id)))) | ($true != (X320 @ X325))) & ($true != (X320 @ (lam @ X324)))) | ? [X321 : term] : (($true != (X320 @ X321)) & ($true = (var @ X321)))) <=> (induction2 = $true)) & (! [X196 : term,X197 : term] : ((X196 = X197) | ((lam @ X197) != (lam @ X196))) <=> (laminj = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X219 : subst > term > subst > $o,X220 : term] : (($true = (X219 @ id @ (lam @ (sub @ X220 @ (push @ one @ sh))) @ id)) | ? [X226 : term] : (($true = (X219 @ id @ X226 @ id)) & ($true != (X219 @ id @ (sub @ X220 @ (push @ X226 @ id)) @ id))) | ? [X221 : subst > term > term] : (! [X222 : term] : (($true != (X219 @ id @ X222 @ id)) | ($true = (X219 @ id @ (X221 @ id @ X222) @ id))) & ! [X223 : term,X224 : subst,X225 : subst] : ((X221 @ (comp @ X225 @ X224) @ (sub @ X223 @ X224)) = (sub @ (X221 @ X225 @ X223) @ X224)) & ($true != (X219 @ id @ (lam @ (X221 @ sh @ one)) @ id))))) & (hoasinduction_lem2v2_gthm = $true) & ! [X198 : term] : ? [X199 : d_term] : ((d2term @ X199) = X198) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 = X155) | ((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153))) <=> (hoasapinj2 = $true)) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) != (d2term @ X152)) | (X151 = X152)) & ((! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (($true != (X40 @ X39)) | ($true = (X41 @ one)) | ? [X43 : term] : ((X41 @ X43) != (X40 @ (sub @ X43 @ (push @ X39 @ X42))))) | ? [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) != X37)) <=> (pushprop_lem1v2_lthm = $true))),
% 2.56/1.94 inference(flattening,[status(thm)],[f12])).
% 2.56/1.94 thf(f12,plain,(
% 2.56/1.94 (axclos = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X445 : subst > term > subst > $o] : ? [X446 : term > $o] : ! [X447 : term] : ((X445 @ id @ X447 @ id) = (X446 @ X447))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ! [X266 : subst] : ? [X267 : d_subst] : ((d2subst @ X267) = X266) & (! [X59 : term,X60 : subst > term > subst > $o] : (? [X68 : term] : (($true = (var @ (sub @ X68 @ id))) & ($true != (X60 @ id @ X68 @ id))) | ? [X63 : subst > term > term] : (($true != (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) & ! [X64 : term] : (($true != (X60 @ id @ X64 @ id)) | ($true = (X60 @ id @ (X63 @ id @ X64) @ id))) & ! [X65 : subst,X66 : subst,X67 : term] : ((X63 @ (comp @ X65 @ X66) @ (sub @ X67 @ X66)) = (sub @ (X63 @ X65 @ X67) @ X66))) | ($true = (X60 @ id @ X59 @ id)) | ? [X73 : term,X74 : subst,X75 : subst,X76 : subst] : (($true != (X60 @ (comp @ X75 @ X74) @ (sub @ X73 @ X74) @ X76)) & ($true = (X60 @ X75 @ X73 @ (comp @ X74 @ X76)))) | ? [X61 : term,X62 : term] : (($true = (X60 @ id @ X61 @ id)) & ($true != (X60 @ id @ (ap @ (sub @ X61 @ id) @ X62) @ id)) & ($true = (X60 @ id @ X62 @ id))) | ? [X69 : subst,X70 : subst,X71 : term,X72 : subst] : (($true != (X60 @ X69 @ X71 @ (comp @ X72 @ X70))) & ($true = (X60 @ (comp @ X69 @ X72) @ (sub @ X71 @ X72) @ X70)))) <=> (hoasinduction = $true)) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) != (d2subst @ X14)) | (X13 = X14)) & (one = (d2term @ d_one)) & ((d2subst @ d_id) = sh) & ((induction2lem = $true) <=> ! [X356 : term > $o,X357 : subst,X358 : term] : (($true = (X356 @ (sub @ X358 @ X357))) | ? [X359 : term] : (($true = (var @ X359)) & ($true != (X356 @ (sub @ X359 @ X357)))) | ? [X362 : term,X363 : term] : (($true = (X356 @ X362)) & ($true = (X356 @ X363)) & ($true != (X356 @ (ap @ X362 @ X363)))) | ? [X360 : term] : (! [X361 : term] : (($true = (X356 @ (sub @ X360 @ (push @ X361 @ id)))) | ($true != (X356 @ X361))) & ($true != (X356 @ (lam @ X360)))))) & (pushprop_lem2v2 = $true) & ((axidl = $true) <=> ! [X56 : subst] : ((comp @ id @ X56) = X56)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X437 : subst > term > subst > $o,X438 : term] : (? [X444 : term] : (($true = (X437 @ id @ X444 @ id)) & ($true != (X437 @ id @ (sub @ X438 @ (push @ X444 @ id)) @ id))) | ? [X439 : subst > term > term] : (($true != (X437 @ id @ (lam @ (X439 @ sh @ one)) @ id)) & ! [X440 : term] : (($true = (X437 @ id @ (X439 @ id @ X440) @ id)) | ($true != (X437 @ id @ X440 @ id))) & ! [X441 : subst,X442 : term,X443 : subst] : ((sub @ (X439 @ X441 @ X442) @ X443) = (X439 @ (comp @ X441 @ X443) @ (sub @ X442 @ X443)))) | ($true = (X437 @ id @ (lam @ (sub @ X438 @ (push @ one @ sh))) @ id)))) & ((var @ (d2term @ d_one)) != $true) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X15 : subst > term > subst > $o,X16 : term > $o,X17 : term] : (? [X19 : subst > term > term] : (! [X20 : term,X21 : subst,X22 : subst] : ((sub @ (X19 @ X22 @ X20) @ X21) = (X19 @ (comp @ X22 @ X21) @ (sub @ X20 @ X21))) & ! [X23 : term] : (($true != (X15 @ id @ X23 @ id)) | ($true = (X15 @ id @ (X19 @ id @ X23) @ id))) & ($true != (X15 @ id @ (lam @ (X19 @ sh @ one)) @ id))) | ? [X18 : term] : ((X16 @ X18) != (X15 @ id @ X18 @ id)) | ($true = (X16 @ (lam @ X17))) | ? [X24 : term] : (($true != (X16 @ (sub @ X17 @ (push @ X24 @ id)))) & ($true = (X16 @ X24))))) & (ulamvarsh = $true) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X277 : term,X278 : subst > term > subst > $o] : (($true = (X278 @ id @ (lam @ X277) @ id)) | ? [X284 : term] : (($true = (X278 @ id @ X284 @ id)) & ($true != (X278 @ id @ (sub @ X277 @ (push @ X284 @ id)) @ id))) | ? [X279 : subst > term > term] : (($true != (X278 @ id @ (lam @ (X279 @ sh @ one)) @ id)) & ! [X283 : term] : (($true != (X278 @ id @ X283 @ id)) | ($true = (X278 @ id @ (X279 @ id @ X283) @ id))) & ! [X280 : subst,X281 : term,X282 : subst] : ((sub @ (X279 @ X282 @ X281) @ X280) = (X279 @ (comp @ X282 @ X280) @ (sub @ X281 @ X280))))) | ? [X269 : term,X270 : subst > term > subst > $o] : (! [X271 : subst > term > term] : (($true = (X270 @ id @ (lam @ (X271 @ sh @ one)) @ id)) | ? [X275 : term] : (($true != (X270 @ id @ (X271 @ id @ X275) @ id)) & ($true = (X270 @ id @ X275 @ id))) | ? [X272 : subst,X273 : term,X274 : subst] : ((X271 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)) != (sub @ (X271 @ X272 @ X273) @ X274))) & ($true != (X270 @ id @ (lam @ (sub @ X269 @ (push @ one @ sh))) @ id)) & ! [X276 : term] : (($true = (X270 @ id @ (sub @ X269 @ (push @ X276 @ id)) @ id)) | ($true != (X270 @ id @ X276 @ id))))) | ? [X268 : term] : ((sub @ X268 @ id) != X268))) & (hoasinduction_lem2_gthm = $true) & (lamnotap != $true) & (axapp = $true) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 = X155) | ((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153))) <=> (hoasapinj2 = $true)) & (! [X44 : term,X45 : subst > term > subst > $o] : (($true = (X45 @ id @ (lam @ (sub @ X44 @ (push @ one @ sh))) @ id)) | ? [X46 : term] : (((X45 @ id @ (sub @ X44 @ (push @ X46 @ id)) @ id) != $true) & ($true = (X45 @ id @ X46 @ id))) | ? [X47 : subst > term > term,X48 : term] : (! [X49 : subst,X50 : term,X51 : subst] : ((sub @ (sub @ X48 @ (push @ X50 @ X49)) @ X51) = (sub @ (X47 @ X49 @ X50) @ X51)) & ($true != (X45 @ id @ (lam @ (X47 @ sh @ one)) @ id)) & ! [X52 : subst,X53 : term,X54 : subst] : ((sub @ X48 @ (push @ (sub @ X53 @ X52) @ (comp @ X54 @ X52))) = (X47 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) & ! [X55 : term] : (($true != (X45 @ id @ X55 @ id)) | ($true = (X45 @ id @ (X47 @ id @ X55) @ id))))) <=> (hoasinduction_lem3aaa = $true)) & (hoasinduction_lem1_lthm = $true) & (! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true = (X144 @ one)) | ($true != (X142 @ X145)) | ? [X146 : term] : ((X144 @ X146) != (X142 @ (sub @ X146 @ (push @ X145 @ X143))))) <=> (pushprop_lem1v2 = $true)) & (hoasapinj2_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (lamnotvar = $true) & (pushprop_gthm = $true) & ((d2subst @ d_id) = id) & ((substmonoid = $true) <=> (! [X449 : subst] : ((comp @ X449 @ id) = X449) & ! [X448 : subst] : ((comp @ id @ X448) = X448))) & (((! [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (($true = (X208 @ (sub @ X210 @ (push @ X206 @ X209)))) | ($true != (X207 @ X210))) | ? [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) != X204)) | ? [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) != X203)) <=> (pushprop_lem1_lthm = $true)) & ((axvarcons = $true) <=> ! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104)) & (hoasinduction_lem3a_gthm = $true) & (! [X80 : term] : ? [X81 : subst > term > term] : ! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83)) <=> (hoasinduction_lem3v2_f = $true)) & (! [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 = X126) | ((ap @ X125 @ X123) != (ap @ X126 @ X124))) <=> (apinj1 = $true)) & (! [X32 : term] : ((sub @ X32 @ id) = X32) <=> (termmset = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X196 : term,X197 : term] : ((X196 = X197) | ((lam @ X197) != (lam @ X196))) <=> (laminj = $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem2v2_gthm = $true) & ((((! [X415 : subst > term > subst > $o,X416 : term] : (($true = (X415 @ id @ X416 @ id)) | ? [X424 : subst,X425 : term,X426 : subst,X427 : subst] : (($true = (X415 @ X427 @ X425 @ (comp @ X424 @ X426))) & ($true != (X415 @ (comp @ X427 @ X424) @ (sub @ X425 @ X424) @ X426))) | ? [X419 : subst > term > term] : (! [X421 : term,X422 : subst,X423 : subst] : ((sub @ (X419 @ X422 @ X421) @ X423) = (X419 @ (comp @ X422 @ X423) @ (sub @ X421 @ X423))) & ((X415 @ id @ (lam @ (X419 @ sh @ one)) @ id) != $true) & ! [X420 : term] : (($true = (X415 @ id @ (X419 @ id @ X420) @ id)) | ($true != (X415 @ id @ X420 @ id)))) | ? [X417 : term,X418 : term] : (((X415 @ id @ X417 @ id) = $true) & ($true = (X415 @ id @ X418 @ id)) & ($true != (X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id))) | ? [X432 : term] : (($true != (X415 @ id @ X432 @ id)) & ($true = (var @ (sub @ X432 @ id)))) | ? [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X415 @ (comp @ X430 @ X429) @ (sub @ X428 @ X429) @ X431)) & ($true != (X415 @ X430 @ X428 @ (comp @ X429 @ X431))))) | ? [X399 : subst > term > subst > $o,X400 : term] : (! [X402 : subst,X403 : term,X404 : subst,X405 : subst] : (($true = (X399 @ X404 @ X403 @ (comp @ X402 @ X405))) | ($true != (X399 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) & ! [X401 : term] : (($true = (X399 @ id @ (sub @ X400 @ (push @ X401 @ id)) @ id)) | ($true != (X399 @ id @ X401 @ id))) & ($true != (X399 @ id @ (lam @ X400) @ id)) & ! [X406 : subst,X407 : term,X408 : subst,X409 : subst] : (($true = (X399 @ (comp @ X406 @ X409) @ (sub @ X407 @ X409) @ X408)) | ($true != (X399 @ X406 @ X407 @ (comp @ X409 @ X408)))) & ! [X410 : subst > term > term] : (($true = (X399 @ id @ (lam @ (X410 @ sh @ one)) @ id)) | ? [X411 : term,X412 : subst,X413 : subst] : ((X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413)) != (sub @ (X410 @ X412 @ X411) @ X413)) | ? [X414 : term] : (($true = (X399 @ id @ X414 @ id)) & ($true != (X399 @ id @ (X410 @ id @ X414) @ id)))))) | ? [X386 : term,X387 : term,X388 : subst > term > subst > $o] : (! [X389 : subst,X390 : subst,X391 : term,X392 : subst] : (($true = (X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389) @ X392)) | ($true != (X388 @ X390 @ X391 @ (comp @ X389 @ X392)))) & ! [X393 : term,X394 : term] : (($true = (X388 @ id @ (ap @ (sub @ X394 @ id) @ X393) @ id)) | ($true != (X388 @ id @ X393 @ id)) | ($true != (X388 @ id @ X394 @ id))) & ($true = (X388 @ id @ X386 @ id)) & ! [X395 : subst,X396 : subst,X397 : subst,X398 : term] : (($true = (X388 @ X396 @ X398 @ (comp @ X397 @ X395))) | ($true != (X388 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X395))) & ($true != (X388 @ id @ (ap @ X386 @ X387) @ id)) & ($true = (X388 @ id @ X387 @ id)))) | ? [X379 : term,X380 : term > $o] : (($true != (X380 @ X379)) & ! [X383 : term] : (? [X384 : term] : (($true != (X380 @ (sub @ X383 @ (push @ X384 @ id)))) & ($true = (X380 @ X384))) | ($true = (X380 @ (lam @ X383)))) & ! [X385 : term] : (($true != (var @ X385)) | ($true = (X380 @ X385))) & ! [X381 : term,X382 : term] : (($true != (X380 @ X382)) | ((X380 @ (ap @ X381 @ X382)) = $true) | ($true != (X380 @ X381))))) <=> (hoasinduction_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X200 : term] : ? [X201 : subst > term > term] : ((X201 @ sh @ one) = (sub @ X200 @ (push @ one @ sh)))) & (((! [X459 : term > $o,X460 : term] : (($true = (X459 @ X460)) | ? [X462 : term,X463 : term] : (($true != (X459 @ (ap @ X462 @ X463))) & ($true = (X459 @ X462)) & ($true = (X459 @ X463))) | ? [X464 : term] : (! [X465 : term] : (($true != (X459 @ X465)) | ($true = (X459 @ (sub @ X464 @ (push @ X465 @ id))))) & ($true != (X459 @ (lam @ X464)))) | ? [X461 : term] : (($true = (var @ X461)) & ($true != (X459 @ X461)))) | ? [X451 : subst,X452 : term,X453 : term > $o] : (! [X454 : term] : (($true != (var @ X454)) | ($true = (X453 @ (sub @ X454 @ X451)))) & ! [X455 : term,X456 : term] : (($true != (X453 @ X455)) | ($true = (X453 @ (ap @ X455 @ X456))) | ((X453 @ X456) != $true)) & ! [X457 : term] : (? [X458 : term] : (($true != (X453 @ (sub @ X457 @ (push @ X458 @ id)))) & ($true = (X453 @ X458))) | ($true = (X453 @ (lam @ X457)))) & ($true != (X453 @ (sub @ X452 @ X451))))) | ? [X450 : term] : ((sub @ X450 @ id) != X450)) <=> (induction2_lthm = $true)) & (axassoc = $true) & ((termmset_lthm = $true) <=> (! [X26 : term] : ((sub @ X26 @ id) = X26) | ? [X25 : term] : ((sub @ X25 @ id) != X25))) & ((hoasinduction_lem0 = $true) <=> ! [X77 : subst > term > subst > $o] : ? [X78 : term > $o] : ! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id))) & ((hoasapinj1_lthm = $true) <=> ((! [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) != (ap @ (sub @ X491 @ id) @ X494)) | (X491 = X493)) | ? [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) = (ap @ X490 @ X489)) & (X487 != X490))) | ? [X486 : term] : ((sub @ X486 @ id) != X486))) & (hoasinduction_lem1v2_gthm = $true) & ((axshiftcons = $true) <=> ! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58)) & ((hoasinduction_lem3 = $true) <=> ! [X516 : term,X517 : subst > term > subst > $o] : (? [X522 : subst > term > term] : (((X517 @ id @ (lam @ (X522 @ sh @ one)) @ id) != $true) & ! [X523 : term] : (($true = (X517 @ id @ (X522 @ id @ X523) @ id)) | ($true != (X517 @ id @ X523 @ id))) & ! [X524 : term,X525 : subst,X526 : subst] : ((X522 @ (comp @ X525 @ X526) @ (sub @ X524 @ X526)) = (sub @ (X522 @ X525 @ X524) @ X526))) | ? [X528 : subst,X529 : subst,X530 : subst,X531 : term] : (((X517 @ X529 @ X531 @ (comp @ X530 @ X528)) != $true) & ($true = (X517 @ (comp @ X529 @ X530) @ (sub @ X531 @ X530) @ X528))) | ($true = (X517 @ id @ (lam @ X516) @ id)) | ? [X527 : term] : (($true = (X517 @ id @ X527 @ id)) & ($true != (X517 @ id @ (sub @ X516 @ (push @ X527 @ id)) @ id))) | ? [X518 : subst,X519 : term,X520 : subst,X521 : subst] : (($true != (X517 @ (comp @ X521 @ X518) @ (sub @ X519 @ X518) @ X520)) & ($true = (X517 @ X521 @ X519 @ (comp @ X518 @ X520)))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lthm_orig = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X4 : term,X5 : subst > term > subst > $o] : (? [X11 : term,X12 : term] : (($true != (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) & ((X5 @ id @ X11 @ id) = $true) & ($true = (X5 @ id @ X12 @ id))) | ? [X6 : subst > term > term] : (! [X8 : term,X9 : subst,X10 : subst] : ((X6 @ (comp @ X10 @ X9) @ (sub @ X8 @ X9)) = (sub @ (X6 @ X10 @ X8) @ X9)) & ! [X7 : term] : (($true = (X5 @ id @ (X6 @ id @ X7) @ id)) | ($true != (X5 @ id @ X7 @ id))) & ($true != (X5 @ id @ (lam @ (X6 @ sh @ one)) @ id))) | ($true = (X5 @ id @ X4 @ id)))) & (hoasinduction_gthm = $true) & (hoasapinj1_gthm = $true) & (! [X27 : subst,X28 : term,X29 : term > $o] : ? [X30 : term > $o] : ! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31)) <=> (pushprop_lem0 = $true)) & ((! [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) != (ap @ (sub @ X216 @ id) @ X218)) | (X217 = X218)) | ? [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) = (ap @ X213 @ X212)) & (X212 != X214))) <=> (hoasapinj2_lthm = $true)) & (hoasapnotvar_lthm = $true) & (! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoaslamnotap = $true) <=> ! [X117 : subst > term > term,X118 : term,X119 : term] : (? [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) != (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119)))) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term] : (? [X374 : subst > term > term] : (! [X375 : term] : (($true != (X370 @ id @ X375 @ id)) | ($true = (X370 @ id @ (X374 @ id @ X375) @ id))) & ! [X376 : subst,X377 : term,X378 : subst] : ((sub @ (X374 @ X376 @ X377) @ X378) = (X374 @ (comp @ X376 @ X378) @ (sub @ X377 @ X378))) & ($true != (X370 @ id @ (lam @ (X374 @ sh @ one)) @ id))) | ? [X373 : term] : (($true = (X369 @ X373)) & ((X369 @ (sub @ X371 @ (push @ X373 @ id))) != $true)) | ($true = (X369 @ (lam @ X371))) | ? [X372 : term] : ((X370 @ id @ X372 @ id) != (X369 @ X372))) | ? [X368 : term] : ((sub @ X368 @ id) != X368)) | ? [X364 : term] : ! [X365 : subst > term > term] : ? [X366 : subst,X367 : term] : ((sub @ X364 @ (push @ X367 @ X366)) != (X365 @ X366 @ X367)))) & (pushprop = $true) & ! [X106 : d_subst] : (d_id = X106) & (hoaslamnotvar_lthm = $true) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) != (d2term @ X152)) | (X151 = X152)) & (apnotvar = $true) & (axvarshift = $true) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (((! [X294 : term,X295 : subst > term > subst > $o] : (? [X296 : subst,X297 : subst,X298 : subst,X299 : term] : (($true = (X295 @ (comp @ X298 @ X296) @ (sub @ X299 @ X296) @ X297)) & ($true != (X295 @ X298 @ X299 @ (comp @ X296 @ X297)))) | ? [X300 : term] : (($true != (X295 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id)) & ($true = (X295 @ id @ X300 @ id))) | ? [X301 : subst > term > term] : (! [X305 : term] : (($true != (X295 @ id @ X305 @ id)) | ($true = (X295 @ id @ (X301 @ id @ X305) @ id))) & ($true != (X295 @ id @ (lam @ (X301 @ sh @ one)) @ id)) & ! [X302 : term,X303 : subst,X304 : subst] : ((X301 @ (comp @ X303 @ X304) @ (sub @ X302 @ X304)) = (sub @ (X301 @ X303 @ X302) @ X304))) | ($true = (X295 @ id @ (lam @ X294) @ id)) | ? [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (($true = (X295 @ X308 @ X306 @ (comp @ X309 @ X307))) & ($true != (X295 @ (comp @ X308 @ X309) @ (sub @ X306 @ X309) @ X307)))) | ? [X286 : term,X287 : subst > term > subst > $o] : (($true != (X287 @ id @ (lam @ (sub @ X286 @ (push @ one @ sh))) @ id)) & ! [X293 : term] : (($true = (X287 @ id @ (sub @ X286 @ (push @ X293 @ id)) @ id)) | ($true != (X287 @ id @ X293 @ id))) & ! [X288 : subst > term > term] : (? [X290 : subst,X291 : subst,X292 : term] : ((X288 @ (comp @ X291 @ X290) @ (sub @ X292 @ X290)) != (sub @ (X288 @ X291 @ X292) @ X290)) | ($true = (X287 @ id @ (lam @ (X288 @ sh @ one)) @ id)) | ? [X289 : term] : (($true = (X287 @ id @ X289 @ id)) & ($true != (X287 @ id @ (X288 @ id @ X289) @ id)))))) | ? [X285 : term] : ((sub @ X285 @ id) != X285)) <=> (hoasinduction_lem3_lthm = $true)) & (! [X178 : term > $o,X179 : subst > term > subst > $o,X180 : term] : (($true = (X178 @ (lam @ X180))) | ? [X182 : subst > term > term] : (! [X184 : subst,X185 : subst,X186 : term] : ((sub @ (X182 @ X185 @ X186) @ X184) = (X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184))) & ($true != (X179 @ id @ (lam @ (X182 @ sh @ one)) @ id)) & ! [X183 : term] : (($true = (X179 @ id @ (X182 @ id @ X183) @ id)) | ($true != (X179 @ id @ X183 @ id)))) | ? [X192 : subst,X193 : subst,X194 : term,X195 : subst] : (($true != (X179 @ (comp @ X192 @ X193) @ (sub @ X194 @ X193) @ X195)) & ($true = (X179 @ X192 @ X194 @ (comp @ X193 @ X195)))) | ? [X181 : term] : ((X179 @ id @ X181 @ id) != (X178 @ X181)) | ? [X188 : term,X189 : subst,X190 : subst,X191 : subst] : (($true != (X179 @ X189 @ X188 @ (comp @ X190 @ X191))) & ($true = (X179 @ (comp @ X189 @ X190) @ (sub @ X188 @ X190) @ X191))) | ? [X187 : term] : (($true != (X178 @ (sub @ X180 @ (push @ X187 @ id)))) & ($true = (X178 @ X187)))) <=> (hoasinduction_lem3v2 = $true)) & (pushprop_lem3v2 = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X496 : term > $o,X497 : subst > term > subst > $o,X498 : term] : (? [X504 : term] : (($true != (X496 @ (sub @ X498 @ (push @ X504 @ id)))) & ($true = (X496 @ X504))) | ? [X499 : subst,X500 : term,X501 : subst,X502 : subst] : (($true != (X497 @ (comp @ X501 @ X499) @ (sub @ X500 @ X499) @ X502)) & ($true = (X497 @ X501 @ X500 @ (comp @ X499 @ X502)))) | ($true = (X496 @ (lam @ X498))) | ? [X503 : term] : ((X496 @ X503) != (X497 @ id @ X503 @ id)) | ? [X505 : subst,X506 : term,X507 : subst,X508 : subst] : (((X497 @ (comp @ X508 @ X507) @ (sub @ X506 @ X507) @ X505) = $true) & ($true != (X497 @ X508 @ X506 @ (comp @ X507 @ X505)))) | ? [X509 : subst > term > term] : (! [X510 : subst,X511 : term,X512 : subst] : ((sub @ (X509 @ X512 @ X511) @ X510) = (X509 @ (comp @ X512 @ X510) @ (sub @ X511 @ X510))) & ($true != (X497 @ id @ (lam @ (X509 @ sh @ one)) @ id)) & ! [X513 : term] : (($true = (X497 @ id @ (X509 @ id @ X513) @ id)) | ($true != (X497 @ id @ X513 @ id))))) | ? [X495 : term] : ((sub @ X495 @ id) != X495))) & ! [X198 : term] : ? [X199 : d_term] : ((d2term @ X199) = X198) & (((((! [X347 : subst > term > subst > $o,X348 : term] : (($true = (X347 @ id @ X348 @ id)) | ? [X351 : subst > term > term] : (! [X355 : term] : (($true != (X347 @ id @ X355 @ id)) | ($true = (X347 @ id @ (X351 @ id @ X355) @ id))) & ($true != (X347 @ id @ (lam @ (X351 @ sh @ one)) @ id)) & ! [X352 : subst,X353 : subst,X354 : term] : ((X351 @ (comp @ X353 @ X352) @ (sub @ X354 @ X352)) = (sub @ (X351 @ X353 @ X354) @ X352))) | ? [X349 : term,X350 : term] : (($true != (X347 @ id @ (ap @ (sub @ X349 @ id) @ X350) @ id)) & ($true = (X347 @ id @ X350 @ id)) & ($true = (X347 @ id @ X349 @ id)))) | ? [X337 : subst > term > subst > $o,X338 : term,X339 : term > $o] : (! [X346 : term] : ((X339 @ X346) = (X337 @ id @ X346 @ id)) & ! [X340 : term] : (($true = (X339 @ (sub @ X338 @ (push @ X340 @ id)))) | ($true != (X339 @ X340))) & ! [X341 : subst > term > term] : (? [X342 : term] : (($true = (X337 @ id @ X342 @ id)) & ((X337 @ id @ (X341 @ id @ X342) @ id) != $true)) | ($true = (X337 @ id @ (lam @ (X341 @ sh @ one)) @ id)) | ? [X343 : subst,X344 : term,X345 : subst] : ((sub @ (X341 @ X345 @ X344) @ X343) != (X341 @ (comp @ X345 @ X343) @ (sub @ X344 @ X343)))) & ($true != (X339 @ (lam @ X338))))) | ? [X336 : term] : ((sub @ X336 @ id) != X336)) | ? [X329 : term > $o,X330 : term] : (! [X332 : term] : (? [X333 : term] : (($true != (X329 @ (sub @ X332 @ (push @ X333 @ id)))) & ($true = (X329 @ X333))) | ($true = (X329 @ (lam @ X332)))) & ! [X334 : term,X335 : term] : (($true != (X329 @ X335)) | ($true = (X329 @ (ap @ X335 @ X334))) | ($true != (X329 @ X334))) & ($true != (X329 @ X330)) & ! [X331 : term] : (($true = (X329 @ X331)) | ($true != (var @ X331))))) | ? [X326 : subst > term > subst > $o] : ! [X327 : term > $o] : ? [X328 : term] : ((X326 @ id @ X328 @ id) != (X327 @ X328))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoasinduction_lem1 = $true) & (ulamvar1 != $true) & (ulamvarind = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotap_gthm = $true) & ((((! [X163 : subst,X164 : subst > term > term,X165 : term,X166 : subst > term > term] : (? [X167 : subst,X168 : term,X169 : subst] : ((sub @ (X164 @ X167 @ X168) @ X169) != (X164 @ (comp @ X167 @ X169) @ (sub @ X168 @ X169))) | ((X164 @ X163 @ X165) = (X166 @ X163 @ X165)) | ? [X170 : term,X171 : subst,X172 : subst] : ((sub @ (X166 @ X172 @ X170) @ X171) != (X166 @ (comp @ X172 @ X171) @ (sub @ X170 @ X171))) | ((lam @ (X166 @ sh @ one)) != (lam @ (X164 @ sh @ one)))) | ? [X161 : term,X162 : term] : (((lam @ X161) = (lam @ X162)) & (X161 != X162))) | ? [X159 : subst,X160 : term] : ((comp @ sh @ (push @ X160 @ X159)) != X159)) | ? [X157 : term,X158 : subst] : ((sub @ one @ (push @ X157 @ X158)) != X157)) <=> (hoaslaminj_lthm = $true)) & (hoasinduction_lem1v2 = $true) & (induction2_gthm = $true) & (hoasapnotvar = $true) & ((axidr = $true) <=> ! [X105 : subst] : ((comp @ X105 @ id) = X105)) & (! [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) != (ap @ (sub @ X436 @ id) @ X434)) | (X435 = X436)) <=> (hoasapinj1 = $true)) & (! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ? [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) != (sub @ (X107 @ X111 @ X113) @ X112)) | ? [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) != (sub @ (X110 @ X116 @ X115) @ X114))) <=> (hoaslaminj = $true)) & ((hoasinduction_lem2v2 = $true) <=> ! [X127 : term,X128 : term > $o,X129 : subst > term > subst > $o,X130 : term] : (($true != (X128 @ X130)) | ? [X139 : term] : ((X129 @ id @ X139 @ id) != (X128 @ X139)) | ? [X140 : term,X141 : term] : (($true = (X129 @ id @ X140 @ id)) & ($true = (X129 @ id @ X141 @ id)) & ($true != (X129 @ id @ (ap @ (sub @ X140 @ id) @ X141) @ id))) | ($true = (X128 @ (ap @ X130 @ X127))) | ? [X131 : term,X132 : subst,X133 : subst,X134 : subst] : (($true = (X129 @ X133 @ X131 @ (comp @ X132 @ X134))) & ($true != (X129 @ (comp @ X133 @ X132) @ (sub @ X131 @ X132) @ X134))) | ? [X135 : subst,X136 : subst,X137 : term,X138 : subst] : (($true != (X129 @ X138 @ X137 @ (comp @ X136 @ X135))) & ($true = (X129 @ (comp @ X138 @ X136) @ (sub @ X137 @ X136) @ X135))) | ($true != (X128 @ X127)))) & (((((! [X95 : subst,X96 : term,X97 : term > $o] : (($true = (X97 @ (sub @ X96 @ X95))) | ? [X100 : term] : (($true = (var @ X100)) & ($true != (X97 @ (sub @ X100 @ X95)))) | ? [X98 : term] : (! [X99 : term] : (($true != (X97 @ X99)) | ($true = (X97 @ (sub @ X98 @ (push @ X99 @ id))))) & ($true != (X97 @ (lam @ X98)))) | ? [X101 : term,X102 : term] : (($true = (X97 @ X102)) & ($true != (X97 @ (ap @ X102 @ X101))) & ($true = (X97 @ X101)))) | ? [X89 : term,X90 : term > $o] : (! [X91 : term,X92 : term] : (((X90 @ X91) != $true) | ($true = (X90 @ (ap @ X91 @ X92))) | ($true != (X90 @ X92))) & ! [X93 : term] : (($true != (X90 @ X93)) | ($true = (X90 @ (lam @ X93)))) & ! [X94 : term] : (($true != (var @ X94)) | ($true = (X90 @ X94))) & ($true != (X90 @ X89)))) | ? [X88 : subst] : ((comp @ X88 @ id) != X88)) | ? [X86 : subst,X87 : term] : ((comp @ sh @ (push @ X87 @ X86)) != X86)) | ? [X84 : subst,X85 : term] : ((sub @ one @ (push @ X85 @ X84)) != X85)) <=> (induction2lem_lthm = $true)) & (induction2lem_gthm = $true) & (substmonoid_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoaslaminj_gthm = $true) & (axabs = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X219 : subst > term > subst > $o,X220 : term] : (($true = (X219 @ id @ (lam @ (sub @ X220 @ (push @ one @ sh))) @ id)) | ? [X226 : term] : (($true = (X219 @ id @ X226 @ id)) & ($true != (X219 @ id @ (sub @ X220 @ (push @ X226 @ id)) @ id))) | ? [X221 : subst > term > term] : (! [X222 : term] : (($true != (X219 @ id @ X222 @ id)) | ($true = (X219 @ id @ (X221 @ id @ X222) @ id))) & ! [X223 : term,X224 : subst,X225 : subst] : ((X221 @ (comp @ X225 @ X224) @ (sub @ X223 @ X224)) = (sub @ (X221 @ X225 @ X223) @ X224)) & ($true != (X219 @ id @ (lam @ (X221 @ sh @ one)) @ id))))) & (! [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) = X539) <=> (axscons = $true)) & ((! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (($true != (X40 @ X39)) | ($true = (X41 @ one)) | ? [X43 : term] : ((X41 @ X43) != (X40 @ (sub @ X43 @ (push @ X39 @ X42))))) | ? [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) != X37)) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslamnotvar = $true) & (pushprop_lthm = $true) & (pushprop_lem0_gthm = $true) & ((apinj2 = $true) <=> ! [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) != (ap @ X3 @ X1)) | (X0 = X1))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (axmap = $true) & (pushprop_lem1_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X532 : term,X533 : term > $o,X534 : subst] : ? [X535 : term > $o] : ! [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) = (X535 @ X536))) & (! [X310 : subst > term > subst > $o,X311 : term] : (? [X313 : subst > term > term] : (! [X317 : term] : (($true = (X310 @ id @ (X313 @ id @ X317) @ id)) | ($true != (X310 @ id @ X317 @ id))) & ($true != (X310 @ id @ (lam @ (X313 @ sh @ one)) @ id)) & ! [X314 : subst,X315 : term,X316 : subst] : ((sub @ (X313 @ X314 @ X315) @ X316) = (X313 @ (comp @ X314 @ X316) @ (sub @ X315 @ X316)))) | ? [X312 : term] : (($true != (X310 @ id @ (sub @ X311 @ (push @ X312 @ id)) @ id)) & ($true = (X310 @ id @ X312 @ id))) | ($true = (X310 @ id @ (lam @ X311) @ id))) <=> (hoasinduction_lem3a = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X537 : term] : ? [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) = (X538 @ sh @ one))) & ! [X485 : d_term] : (d_one = X485) & (hoaslamnotvar_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((induction = $true) <=> ! [X479 : term > $o,X480 : term] : (($true = (X479 @ X480)) | ? [X481 : term] : (($true != (X479 @ X481)) & ($true = (var @ X481))) | ? [X483 : term,X484 : term] : (($true = (X479 @ X484)) & ($true = (X479 @ X483)) & ((X479 @ (ap @ X483 @ X484)) != $true)) | ? [X482 : term] : (($true != (X479 @ (lam @ X482))) & ($true = (X479 @ X482))))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (((((! [X248 : term,X249 : subst > term > subst > $o] : (? [X261 : term,X262 : subst,X263 : subst,X264 : subst] : (($true != (X249 @ X262 @ X261 @ (comp @ X263 @ X264))) & ($true = (X249 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263) @ X264))) | ? [X250 : term,X251 : term] : (($true != (X249 @ id @ (ap @ (sub @ X251 @ id) @ X250) @ id)) & ($true = (X249 @ id @ X250 @ id)) & ($true = (X249 @ id @ X251 @ id))) | ? [X256 : subst > term > term] : (! [X257 : term] : (($true != (X249 @ id @ X257 @ id)) | ($true = (X249 @ id @ (X256 @ id @ X257) @ id))) & ! [X258 : subst,X259 : term,X260 : subst] : ((sub @ (X256 @ X260 @ X259) @ X258) = (X256 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258))) & ($true != (X249 @ id @ (lam @ (X256 @ sh @ one)) @ id))) | ? [X252 : subst,X253 : term,X254 : subst,X255 : subst] : (($true != (X249 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254) @ X255)) & ($true = (X249 @ X252 @ X253 @ (comp @ X254 @ X255)))) | ? [X265 : term] : (($true = (var @ (sub @ X265 @ id))) & ($true != (X249 @ id @ X265 @ id))) | ($true = (X249 @ id @ X248 @ id))) | ? [X238 : term,X239 : term > $o,X240 : subst > term > subst > $o] : (! [X241 : term] : (($true = (X239 @ (sub @ X238 @ (push @ X241 @ id)))) | ($true != (X239 @ X241))) & ($true != (X239 @ (lam @ X238))) & ! [X247 : term] : ((X240 @ id @ X247 @ id) = (X239 @ X247)) & ! [X242 : subst > term > term] : (($true = (X240 @ id @ (lam @ (X242 @ sh @ one)) @ id)) | ? [X244 : subst,X245 : term,X246 : subst] : ((X242 @ (comp @ X246 @ X244) @ (sub @ X245 @ X244)) != (sub @ (X242 @ X246 @ X245) @ X244)) | ? [X243 : term] : (($true != (X240 @ id @ (X242 @ id @ X243) @ id)) & ($true = (X240 @ id @ X243 @ id)))))) | ? [X237 : term] : ((sub @ X237 @ id) != X237)) | ? [X230 : term > $o,X231 : term] : (! [X232 : term] : (($true = (X230 @ X232)) | ($true != (var @ X232))) & ! [X235 : term,X236 : term] : (($true != (X230 @ X236)) | ($true != (X230 @ X235)) | ($true = (X230 @ (ap @ X235 @ X236)))) & ($true != (X230 @ X231)) & ! [X233 : term] : (? [X234 : term] : (($true = (X230 @ X234)) & ($true != (X230 @ (sub @ X233 @ (push @ X234 @ id))))) | ($true = (X230 @ (lam @ X233)))))) | ? [X227 : subst > term > subst > $o] : ! [X228 : term > $o] : ? [X229 : term] : ((X227 @ id @ X229 @ id) != (X228 @ X229))) <=> (hoasinduction_lthm_3 = $true)) & (! [X318 : term] : ((sub @ X318 @ id) = X318) <=> (axvarid = $true)) & (! [X319 : term,X320 : term > $o] : (($true = (X320 @ X319)) | ? [X322 : term,X323 : term] : (($true != (X320 @ (ap @ X323 @ X322))) & ($true = (X320 @ X322)) & ($true = (X320 @ X323))) | ? [X324 : term] : (! [X325 : term] : (($true = (X320 @ (sub @ X324 @ (push @ X325 @ id)))) | ($true != (X320 @ X325))) & ($true != (X320 @ (lam @ X324)))) | ? [X321 : term] : (($true != (X320 @ X321)) & ($true = (var @ X321)))) <=> (induction2 = $true)) & ((((! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36)) | ? [X34 : subst] : ((comp @ X34 @ id) != X34)) | ? [X33 : subst] : ((comp @ id @ X33) != X33)) <=> (substmonoid_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & (termmset_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true = (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) | ($true != (X173 @ X174))) <=> (pushprop_lem1 = $true)) & ((shinj = $true) <=> ! [X514 : term,X515 : term] : (((sub @ X514 @ sh) != (sub @ X515 @ sh)) | (X514 = X515))) & ((hoasinduction_lem2 = $true) <=> ! [X466 : term,X467 : term,X468 : subst > term > subst > $o] : (? [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (($true != (X468 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471)) & ($true = (X468 @ X473 @ X474 @ (comp @ X472 @ X471)))) | ($true != (X468 @ id @ X467 @ id)) | ($true != (X468 @ id @ X466 @ id)) | ($true = (X468 @ id @ (ap @ X467 @ X466) @ id)) | ? [X469 : term,X470 : term] : (($true = (X468 @ id @ X470 @ id)) & ($true = (X468 @ id @ X469 @ id)) & ($true != (X468 @ id @ (ap @ (sub @ X470 @ id) @ X469) @ id))) | ? [X475 : subst,X476 : subst,X477 : subst,X478 : term] : (($true != (X468 @ X476 @ X478 @ (comp @ X475 @ X477))) & ($true = (X468 @ (comp @ X476 @ X475) @ (sub @ X478 @ X475) @ X477)))))),
% 2.56/1.94 inference(ennf_transformation,[status(thm)],[f11])).
% 2.56/1.94 thf(f11,plain,(
% 2.56/1.94 (axclos = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X445 : subst > term > subst > $o] : ~! [X446 : term > $o] : ~! [X447 : term] : ((X445 @ id @ X447 @ id) = (X446 @ X447))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ! [X266 : subst] : ? [X267 : d_subst] : ((d2subst @ X267) = X266) & (! [X59 : term,X60 : subst > term > subst > $o] : (($true = (X60 @ id @ X59 @ id)) | ~! [X69 : subst,X70 : subst,X71 : term,X72 : subst] : (($true != (X60 @ (comp @ X69 @ X72) @ (sub @ X71 @ X72) @ X70)) | ($true = (X60 @ X69 @ X71 @ (comp @ X72 @ X70)))) | ~! [X68 : term] : (($true = (X60 @ id @ X68 @ id)) | ($true != (var @ (sub @ X68 @ id)))) | ~! [X61 : term,X62 : term] : (($true != (X60 @ id @ X61 @ id)) | ($true = (X60 @ id @ (ap @ (sub @ X61 @ id) @ X62) @ id)) | ($true != (X60 @ id @ X62 @ id))) | ~! [X73 : term,X74 : subst,X75 : subst,X76 : subst] : (($true = (X60 @ (comp @ X75 @ X74) @ (sub @ X73 @ X74) @ X76)) | ($true != (X60 @ X75 @ X73 @ (comp @ X74 @ X76)))) | ~! [X63 : subst > term > term] : (~! [X65 : subst,X66 : subst,X67 : term] : ((X63 @ (comp @ X65 @ X66) @ (sub @ X67 @ X66)) = (sub @ (X63 @ X65 @ X67) @ X66)) | ~! [X64 : term] : (($true != (X60 @ id @ X64 @ id)) | ($true = (X60 @ id @ (X63 @ id @ X64) @ id))) | ($true = (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)))) <=> (hoasinduction = $true)) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) = (d2subst @ X14)) => (X13 = X14)) & (one = (d2term @ d_one)) & ((d2subst @ d_id) = sh) & (! [X356 : term > $o,X357 : subst,X358 : term] : (~! [X362 : term,X363 : term] : (($true != (X356 @ X363)) | ($true = (X356 @ (ap @ X362 @ X363))) | ($true != (X356 @ X362))) | ($true = (X356 @ (sub @ X358 @ X357))) | ~! [X359 : term] : (($true = (X356 @ (sub @ X359 @ X357))) | ($true != (var @ X359))) | ~! [X360 : term] : (~! [X361 : term] : (($true = (X356 @ (sub @ X360 @ (push @ X361 @ id)))) | ($true != (X356 @ X361))) | ($true = (X356 @ (lam @ X360))))) <=> (induction2lem = $true)) & (pushprop_lem2v2 = $true) & ((axidl = $true) <=> ! [X56 : subst] : ((comp @ id @ X56) = X56)) & (! [X437 : subst > term > subst > $o,X438 : term] : (($true = (X437 @ id @ (lam @ (sub @ X438 @ (push @ one @ sh))) @ id)) | ~! [X439 : subst > term > term] : (($true = (X437 @ id @ (lam @ (X439 @ sh @ one)) @ id)) | ~! [X440 : term] : (($true = (X437 @ id @ (X439 @ id @ X440) @ id)) | ($true != (X437 @ id @ X440 @ id))) | ~! [X441 : subst,X442 : term,X443 : subst] : ((sub @ (X439 @ X441 @ X442) @ X443) = (X439 @ (comp @ X441 @ X443) @ (sub @ X442 @ X443)))) | ~! [X444 : term] : (($true != (X437 @ id @ X444 @ id)) | ($true = (X437 @ id @ (sub @ X438 @ (push @ X444 @ id)) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((var @ (d2term @ d_one)) != $true) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X15 : subst > term > subst > $o,X16 : term > $o,X17 : term] : (~! [X18 : term] : ((X16 @ X18) = (X15 @ id @ X18 @ id)) | ~! [X24 : term] : (($true = (X16 @ (sub @ X17 @ (push @ X24 @ id)))) | ($true != (X16 @ X24))) | ($true = (X16 @ (lam @ X17))) | ~! [X19 : subst > term > term] : (($true = (X15 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~! [X20 : term,X21 : subst,X22 : subst] : ((sub @ (X19 @ X22 @ X20) @ X21) = (X19 @ (comp @ X22 @ X21) @ (sub @ X20 @ X21))) | ~! [X23 : term] : (($true != (X15 @ id @ X23 @ id)) | ($true = (X15 @ id @ (X19 @ id @ X23) @ id)))))) & (ulamvarsh = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X268 : term] : ((sub @ X268 @ id) = X268) => (! [X269 : term,X270 : subst > term > subst > $o] : (($true = (X270 @ id @ (lam @ (sub @ X269 @ (push @ one @ sh))) @ id)) | ~! [X271 : subst > term > term] : (($true = (X270 @ id @ (lam @ (X271 @ sh @ one)) @ id)) | ~! [X272 : subst,X273 : term,X274 : subst] : ((X271 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)) = (sub @ (X271 @ X272 @ X273) @ X274)) | ~! [X275 : term] : (($true = (X270 @ id @ (X271 @ id @ X275) @ id)) | ($true != (X270 @ id @ X275 @ id)))) | ~! [X276 : term] : (($true = (X270 @ id @ (sub @ X269 @ (push @ X276 @ id)) @ id)) | ($true != (X270 @ id @ X276 @ id)))) => ! [X277 : term,X278 : subst > term > subst > $o] : (~! [X279 : subst > term > term] : (($true = (X278 @ id @ (lam @ (X279 @ sh @ one)) @ id)) | ~! [X280 : subst,X281 : term,X282 : subst] : ((sub @ (X279 @ X282 @ X281) @ X280) = (X279 @ (comp @ X282 @ X280) @ (sub @ X281 @ X280))) | ~! [X283 : term] : (($true != (X278 @ id @ X283 @ id)) | ($true = (X278 @ id @ (X279 @ id @ X283) @ id)))) | ~! [X284 : term] : (($true = (X278 @ id @ (sub @ X277 @ (push @ X284 @ id)) @ id)) | ($true != (X278 @ id @ X284 @ id))) | ($true = (X278 @ id @ (lam @ X277) @ id)))))) & (hoasinduction_lem2_gthm = $true) & (lamnotap != $true) & (axapp = $true) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 = X155) | ((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153))) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X44 : term,X45 : subst > term > subst > $o] : (~! [X47 : subst > term > term,X48 : term] : (($true = (X45 @ id @ (lam @ (X47 @ sh @ one)) @ id)) | ~! [X55 : term] : (($true != (X45 @ id @ X55 @ id)) | ($true = (X45 @ id @ (X47 @ id @ X55) @ id))) | ~! [X52 : subst,X53 : term,X54 : subst] : ((sub @ X48 @ (push @ (sub @ X53 @ X52) @ (comp @ X54 @ X52))) = (X47 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) | ~! [X49 : subst,X50 : term,X51 : subst] : ((sub @ (sub @ X48 @ (push @ X50 @ X49)) @ X51) = (sub @ (X47 @ X49 @ X50) @ X51))) | ($true = (X45 @ id @ (lam @ (sub @ X44 @ (push @ one @ sh))) @ id)) | ~! [X46 : term] : (((X45 @ id @ (sub @ X44 @ (push @ X46 @ id)) @ id) = $true) | ($true != (X45 @ id @ X46 @ id))))) & (hoasinduction_lem1_lthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (~! [X146 : term] : ((X144 @ X146) = (X142 @ (sub @ X146 @ (push @ X145 @ X143)))) | ($true = (X144 @ one)) | ($true != (X142 @ X145)))) & (hoasapinj2_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (lamnotvar = $true) & (pushprop_gthm = $true) & ((d2subst @ d_id) = id) & ((substmonoid = $true) <=> (! [X449 : subst] : ((comp @ X449 @ id) = X449) & ! [X448 : subst] : ((comp @ id @ X448) = X448))) & ((! [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) = X203) => (! [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) = X204) => ! [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (($true = (X208 @ (sub @ X210 @ (push @ X206 @ X209)))) | ($true != (X207 @ X210))))) <=> (pushprop_lem1_lthm = $true)) & ((axvarcons = $true) <=> ! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X80 : term] : ~! [X81 : subst > term > term] : ~! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83))) & (! [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 = X126) | ((ap @ X125 @ X123) != (ap @ X126 @ X124))) <=> (apinj1 = $true)) & (! [X32 : term] : ((sub @ X32 @ id) = X32) <=> (termmset = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X196 : term,X197 : term] : ((X196 = X197) | ((lam @ X197) != (lam @ X196))) <=> (laminj = $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem2v2_gthm = $true) & ((! [X379 : term,X380 : term > $o] : (($true = (X380 @ X379)) | ~! [X381 : term,X382 : term] : (($true != (X380 @ X382)) | ((X380 @ (ap @ X381 @ X382)) = $true) | ($true != (X380 @ X381))) | ~! [X383 : term] : (($true = (X380 @ (lam @ X383))) | ~! [X384 : term] : (($true != (X380 @ X384)) | ($true = (X380 @ (sub @ X383 @ (push @ X384 @ id)))))) | ~! [X385 : term] : (($true != (var @ X385)) | ($true = (X380 @ X385)))) => (! [X386 : term,X387 : term,X388 : subst > term > subst > $o] : (~! [X389 : subst,X390 : subst,X391 : term,X392 : subst] : (($true = (X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389) @ X392)) | ($true != (X388 @ X390 @ X391 @ (comp @ X389 @ X392)))) | ($true = (X388 @ id @ (ap @ X386 @ X387) @ id)) | ($true != (X388 @ id @ X387 @ id)) | ~! [X395 : subst,X396 : subst,X397 : subst,X398 : term] : (($true = (X388 @ X396 @ X398 @ (comp @ X397 @ X395))) | ($true != (X388 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X395))) | ($true != (X388 @ id @ X386 @ id)) | ~! [X393 : term,X394 : term] : (($true = (X388 @ id @ (ap @ (sub @ X394 @ id) @ X393) @ id)) | ($true != (X388 @ id @ X393 @ id)) | ($true != (X388 @ id @ X394 @ id)))) => (! [X399 : subst > term > subst > $o,X400 : term] : (~! [X410 : subst > term > term] : (($true = (X399 @ id @ (lam @ (X410 @ sh @ one)) @ id)) | ~! [X414 : term] : (($true = (X399 @ id @ (X410 @ id @ X414) @ id)) | ($true != (X399 @ id @ X414 @ id))) | ~! [X411 : term,X412 : subst,X413 : subst] : ((X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413)) = (sub @ (X410 @ X412 @ X411) @ X413))) | ~! [X402 : subst,X403 : term,X404 : subst,X405 : subst] : (($true = (X399 @ X404 @ X403 @ (comp @ X402 @ X405))) | ($true != (X399 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) | ($true = (X399 @ id @ (lam @ X400) @ id)) | ~! [X406 : subst,X407 : term,X408 : subst,X409 : subst] : (($true = (X399 @ (comp @ X406 @ X409) @ (sub @ X407 @ X409) @ X408)) | ($true != (X399 @ X406 @ X407 @ (comp @ X409 @ X408)))) | ~! [X401 : term] : (($true = (X399 @ id @ (sub @ X400 @ (push @ X401 @ id)) @ id)) | ($true != (X399 @ id @ X401 @ id)))) => ! [X415 : subst > term > subst > $o,X416 : term] : (~! [X432 : term] : (($true = (X415 @ id @ X432 @ id)) | ($true != (var @ (sub @ X432 @ id)))) | ~! [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X415 @ X430 @ X428 @ (comp @ X429 @ X431))) | ($true != (X415 @ (comp @ X430 @ X429) @ (sub @ X428 @ X429) @ X431))) | ~! [X424 : subst,X425 : term,X426 : subst,X427 : subst] : (($true != (X415 @ X427 @ X425 @ (comp @ X424 @ X426))) | ($true = (X415 @ (comp @ X427 @ X424) @ (sub @ X425 @ X424) @ X426))) | ~! [X419 : subst > term > term] : (~! [X421 : term,X422 : subst,X423 : subst] : ((sub @ (X419 @ X422 @ X421) @ X423) = (X419 @ (comp @ X422 @ X423) @ (sub @ X421 @ X423))) | ((X415 @ id @ (lam @ (X419 @ sh @ one)) @ id) = $true) | ~! [X420 : term] : (($true = (X415 @ id @ (X419 @ id @ X420) @ id)) | ($true != (X415 @ id @ X420 @ id)))) | ($true = (X415 @ id @ X416 @ id)) | ~! [X417 : term,X418 : term] : (((X415 @ id @ X417 @ id) != $true) | ($true = (X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id)) | ($true != (X415 @ id @ X418 @ id))))))) <=> (hoasinduction_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X200 : term] : ~! [X201 : subst > term > term] : ((X201 @ sh @ one) != (sub @ X200 @ (push @ one @ sh)))) & ((! [X450 : term] : ((sub @ X450 @ id) = X450) => (! [X451 : subst,X452 : term,X453 : term > $o] : (($true = (X453 @ (sub @ X452 @ X451))) | ~! [X454 : term] : (($true != (var @ X454)) | ($true = (X453 @ (sub @ X454 @ X451)))) | ~! [X455 : term,X456 : term] : (($true != (X453 @ X455)) | ($true = (X453 @ (ap @ X455 @ X456))) | ((X453 @ X456) != $true)) | ~! [X457 : term] : (~! [X458 : term] : (($true != (X453 @ X458)) | ($true = (X453 @ (sub @ X457 @ (push @ X458 @ id))))) | ($true = (X453 @ (lam @ X457))))) => ! [X459 : term > $o,X460 : term] : (~! [X464 : term] : (~! [X465 : term] : (($true != (X459 @ X465)) | ($true = (X459 @ (sub @ X464 @ (push @ X465 @ id))))) | ($true = (X459 @ (lam @ X464)))) | ($true = (X459 @ X460)) | ~! [X462 : term,X463 : term] : (($true != (X459 @ X463)) | ($true != (X459 @ X462)) | ($true = (X459 @ (ap @ X462 @ X463)))) | ~! [X461 : term] : (($true != (var @ X461)) | ($true = (X459 @ X461)))))) <=> (induction2_lthm = $true)) & (axassoc = $true) & ((! [X25 : term] : ((sub @ X25 @ id) = X25) => ! [X26 : term] : ((sub @ X26 @ id) = X26)) <=> (termmset_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X77 : subst > term > subst > $o] : ~! [X78 : term > $o] : ~! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id))) & ((hoasapinj1_lthm = $true) <=> (! [X486 : term] : ((sub @ X486 @ id) = X486) => (! [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) != (ap @ X490 @ X489)) | (X487 = X490)) => ! [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) != (ap @ (sub @ X491 @ id) @ X494)) | (X491 = X493))))) & (hoasinduction_lem1v2_gthm = $true) & ((axshiftcons = $true) <=> ! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58)) & (! [X516 : term,X517 : subst > term > subst > $o] : (~! [X528 : subst,X529 : subst,X530 : subst,X531 : term] : (($true != (X517 @ (comp @ X529 @ X530) @ (sub @ X531 @ X530) @ X528)) | ((X517 @ X529 @ X531 @ (comp @ X530 @ X528)) = $true)) | ~! [X518 : subst,X519 : term,X520 : subst,X521 : subst] : (($true = (X517 @ (comp @ X521 @ X518) @ (sub @ X519 @ X518) @ X520)) | ($true != (X517 @ X521 @ X519 @ (comp @ X518 @ X520)))) | ($true = (X517 @ id @ (lam @ X516) @ id)) | ~! [X522 : subst > term > term] : (~! [X524 : term,X525 : subst,X526 : subst] : ((X522 @ (comp @ X525 @ X526) @ (sub @ X524 @ X526)) = (sub @ (X522 @ X525 @ X524) @ X526)) | ((X517 @ id @ (lam @ (X522 @ sh @ one)) @ id) = $true) | ~! [X523 : term] : (($true = (X517 @ id @ (X522 @ id @ X523) @ id)) | ($true != (X517 @ id @ X523 @ id)))) | ~! [X527 : term] : (($true = (X517 @ id @ (sub @ X516 @ (push @ X527 @ id)) @ id)) | ($true != (X517 @ id @ X527 @ id)))) <=> (hoasinduction_lem3 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lthm_orig = $true) & (! [X4 : term,X5 : subst > term > subst > $o] : (($true = (X5 @ id @ X4 @ id)) | ~! [X11 : term,X12 : term] : (($true = (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) | ($true != (X5 @ id @ X12 @ id)) | ((X5 @ id @ X11 @ id) != $true)) | ~! [X6 : subst > term > term] : (~! [X7 : term] : (($true = (X5 @ id @ (X6 @ id @ X7) @ id)) | ($true != (X5 @ id @ X7 @ id))) | ($true = (X5 @ id @ (lam @ (X6 @ sh @ one)) @ id)) | ~! [X8 : term,X9 : subst,X10 : subst] : ((X6 @ (comp @ X10 @ X9) @ (sub @ X8 @ X9)) = (sub @ (X6 @ X10 @ X8) @ X9)))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasinduction_gthm = $true) & (hoasapinj1_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X27 : subst,X28 : term,X29 : term > $o] : ~! [X30 : term > $o] : ~! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31))) & ((hoasapinj2_lthm = $true) <=> (! [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) != (ap @ X213 @ X212)) | (X212 = X214)) => ! [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) != (ap @ (sub @ X216 @ id) @ X218)) | (X217 = X218)))) & (hoasapnotvar_lthm = $true) & (! [X147 : term] : ~! [X148 : subst > term > term] : ~! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoaslamnotap = $true) <=> ! [X117 : subst > term > term,X118 : term,X119 : term] : (~! [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) = (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119)))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X364 : term] : ~! [X365 : subst > term > term] : ~! [X366 : subst,X367 : term] : ((sub @ X364 @ (push @ X367 @ X366)) = (X365 @ X366 @ X367)) => (! [X368 : term] : ((sub @ X368 @ id) = X368) => ! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term] : (~! [X372 : term] : ((X370 @ id @ X372 @ id) = (X369 @ X372)) | ~! [X373 : term] : (($true != (X369 @ X373)) | ((X369 @ (sub @ X371 @ (push @ X373 @ id))) = $true)) | ~! [X374 : subst > term > term] : (($true = (X370 @ id @ (lam @ (X374 @ sh @ one)) @ id)) | ~! [X376 : subst,X377 : term,X378 : subst] : ((sub @ (X374 @ X376 @ X377) @ X378) = (X374 @ (comp @ X376 @ X378) @ (sub @ X377 @ X378))) | ~! [X375 : term] : (($true != (X370 @ id @ X375 @ id)) | ($true = (X370 @ id @ (X374 @ id @ X375) @ id)))) | ($true = (X369 @ (lam @ X371))))))) & (pushprop = $true) & ! [X106 : d_subst] : (d_id = X106) & (hoaslamnotvar_lthm = $true) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) = (d2term @ X152)) => (X151 = X152)) & (apnotvar = $true) & (axvarshift = $true) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem3_gthm = $true) & ((! [X285 : term] : ((sub @ X285 @ id) = X285) => (! [X286 : term,X287 : subst > term > subst > $o] : (($true = (X287 @ id @ (lam @ (sub @ X286 @ (push @ one @ sh))) @ id)) | ~! [X288 : subst > term > term] : (~! [X290 : subst,X291 : subst,X292 : term] : ((X288 @ (comp @ X291 @ X290) @ (sub @ X292 @ X290)) = (sub @ (X288 @ X291 @ X292) @ X290)) | ($true = (X287 @ id @ (lam @ (X288 @ sh @ one)) @ id)) | ~! [X289 : term] : (($true = (X287 @ id @ (X288 @ id @ X289) @ id)) | ($true != (X287 @ id @ X289 @ id)))) | ~! [X293 : term] : (($true = (X287 @ id @ (sub @ X286 @ (push @ X293 @ id)) @ id)) | ($true != (X287 @ id @ X293 @ id)))) => ! [X294 : term,X295 : subst > term > subst > $o] : (~! [X296 : subst,X297 : subst,X298 : subst,X299 : term] : (($true = (X295 @ X298 @ X299 @ (comp @ X296 @ X297))) | ($true != (X295 @ (comp @ X298 @ X296) @ (sub @ X299 @ X296) @ X297))) | ~! [X301 : subst > term > term] : (($true = (X295 @ id @ (lam @ (X301 @ sh @ one)) @ id)) | ~! [X302 : term,X303 : subst,X304 : subst] : ((X301 @ (comp @ X303 @ X304) @ (sub @ X302 @ X304)) = (sub @ (X301 @ X303 @ X302) @ X304)) | ~! [X305 : term] : (($true != (X295 @ id @ X305 @ id)) | ($true = (X295 @ id @ (X301 @ id @ X305) @ id)))) | ($true = (X295 @ id @ (lam @ X294) @ id)) | ~! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (($true = (X295 @ (comp @ X308 @ X309) @ (sub @ X306 @ X309) @ X307)) | ($true != (X295 @ X308 @ X306 @ (comp @ X309 @ X307)))) | ~! [X300 : term] : (($true != (X295 @ id @ X300 @ id)) | ($true = (X295 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id)))))) <=> (hoasinduction_lem3_lthm = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X178 : term > $o,X179 : subst > term > subst > $o,X180 : term] : (~! [X187 : term] : (($true != (X178 @ X187)) | ($true = (X178 @ (sub @ X180 @ (push @ X187 @ id))))) | ($true = (X178 @ (lam @ X180))) | ~! [X192 : subst,X193 : subst,X194 : term,X195 : subst] : (($true != (X179 @ X192 @ X194 @ (comp @ X193 @ X195))) | ($true = (X179 @ (comp @ X192 @ X193) @ (sub @ X194 @ X193) @ X195))) | ~! [X182 : subst > term > term] : (~! [X183 : term] : (($true = (X179 @ id @ (X182 @ id @ X183) @ id)) | ($true != (X179 @ id @ X183 @ id))) | ($true = (X179 @ id @ (lam @ (X182 @ sh @ one)) @ id)) | ~! [X184 : subst,X185 : subst,X186 : term] : ((sub @ (X182 @ X185 @ X186) @ X184) = (X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184)))) | ~! [X181 : term] : ((X179 @ id @ X181 @ id) = (X178 @ X181)) | ~! [X188 : term,X189 : subst,X190 : subst,X191 : subst] : (($true = (X179 @ X189 @ X188 @ (comp @ X190 @ X191))) | ($true != (X179 @ (comp @ X189 @ X190) @ (sub @ X188 @ X190) @ X191))))) & (pushprop_lem3v2 = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X495 : term] : ((sub @ X495 @ id) = X495) => ! [X496 : term > $o,X497 : subst > term > subst > $o,X498 : term] : (~! [X505 : subst,X506 : term,X507 : subst,X508 : subst] : (((X497 @ (comp @ X508 @ X507) @ (sub @ X506 @ X507) @ X505) != $true) | ($true = (X497 @ X508 @ X506 @ (comp @ X507 @ X505)))) | ~! [X504 : term] : (($true = (X496 @ (sub @ X498 @ (push @ X504 @ id)))) | ($true != (X496 @ X504))) | ~! [X503 : term] : ((X496 @ X503) = (X497 @ id @ X503 @ id)) | ~! [X509 : subst > term > term] : (($true = (X497 @ id @ (lam @ (X509 @ sh @ one)) @ id)) | ~! [X513 : term] : (($true = (X497 @ id @ (X509 @ id @ X513) @ id)) | ($true != (X497 @ id @ X513 @ id))) | ~! [X510 : subst,X511 : term,X512 : subst] : ((sub @ (X509 @ X512 @ X511) @ X510) = (X509 @ (comp @ X512 @ X510) @ (sub @ X511 @ X510)))) | ~! [X499 : subst,X500 : term,X501 : subst,X502 : subst] : (($true != (X497 @ X501 @ X500 @ (comp @ X499 @ X502))) | ($true = (X497 @ (comp @ X501 @ X499) @ (sub @ X500 @ X499) @ X502))) | ($true = (X496 @ (lam @ X498)))))) & ! [X198 : term] : ? [X199 : d_term] : ((d2term @ X199) = X198) & ((! [X326 : subst > term > subst > $o] : ~! [X327 : term > $o] : ~! [X328 : term] : ((X326 @ id @ X328 @ id) = (X327 @ X328)) => (! [X329 : term > $o,X330 : term] : (~! [X331 : term] : (($true = (X329 @ X331)) | ($true != (var @ X331))) | ~! [X334 : term,X335 : term] : (($true != (X329 @ X335)) | ($true = (X329 @ (ap @ X335 @ X334))) | ($true != (X329 @ X334))) | ~! [X332 : term] : (($true = (X329 @ (lam @ X332))) | ~! [X333 : term] : (($true = (X329 @ (sub @ X332 @ (push @ X333 @ id)))) | ($true != (X329 @ X333)))) | ($true = (X329 @ X330))) => (! [X336 : term] : ((sub @ X336 @ id) = X336) => (! [X337 : subst > term > subst > $o,X338 : term,X339 : term > $o] : (~! [X346 : term] : ((X339 @ X346) = (X337 @ id @ X346 @ id)) | ~! [X340 : term] : (($true = (X339 @ (sub @ X338 @ (push @ X340 @ id)))) | ($true != (X339 @ X340))) | ($true = (X339 @ (lam @ X338))) | ~! [X341 : subst > term > term] : (~! [X343 : subst,X344 : term,X345 : subst] : ((sub @ (X341 @ X345 @ X344) @ X343) = (X341 @ (comp @ X345 @ X343) @ (sub @ X344 @ X343))) | ($true = (X337 @ id @ (lam @ (X341 @ sh @ one)) @ id)) | ~! [X342 : term] : (($true != (X337 @ id @ X342 @ id)) | ((X337 @ id @ (X341 @ id @ X342) @ id) = $true)))) => ! [X347 : subst > term > subst > $o,X348 : term] : (($true = (X347 @ id @ X348 @ id)) | ~! [X349 : term,X350 : term] : (($true != (X347 @ id @ X350 @ id)) | ($true = (X347 @ id @ (ap @ (sub @ X349 @ id) @ X350) @ id)) | ($true != (X347 @ id @ X349 @ id))) | ~! [X351 : subst > term > term] : (($true = (X347 @ id @ (lam @ (X351 @ sh @ one)) @ id)) | ~! [X352 : subst,X353 : subst,X354 : term] : ((X351 @ (comp @ X353 @ X352) @ (sub @ X354 @ X352)) = (sub @ (X351 @ X353 @ X354) @ X352)) | ~! [X355 : term] : (($true != (X347 @ id @ X355 @ id)) | ($true = (X347 @ id @ (X351 @ id @ X355) @ id))))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoasinduction_lem1 = $true) & (ulamvar1 != $true) & (ulamvarind = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotap_gthm = $true) & ((! [X157 : term,X158 : subst] : ((sub @ one @ (push @ X157 @ X158)) = X157) => (! [X159 : subst,X160 : term] : ((comp @ sh @ (push @ X160 @ X159)) = X159) => (! [X161 : term,X162 : term] : ((X161 = X162) | ((lam @ X161) != (lam @ X162))) => ! [X163 : subst,X164 : subst > term > term,X165 : term,X166 : subst > term > term] : (~! [X167 : subst,X168 : term,X169 : subst] : ((sub @ (X164 @ X167 @ X168) @ X169) = (X164 @ (comp @ X167 @ X169) @ (sub @ X168 @ X169))) | ~! [X170 : term,X171 : subst,X172 : subst] : ((sub @ (X166 @ X172 @ X170) @ X171) = (X166 @ (comp @ X172 @ X171) @ (sub @ X170 @ X171))) | ((X164 @ X163 @ X165) = (X166 @ X163 @ X165)) | ((lam @ (X166 @ sh @ one)) != (lam @ (X164 @ sh @ one))))))) <=> (hoaslaminj_lthm = $true)) & (hoasinduction_lem1v2 = $true) & (induction2_gthm = $true) & (hoasapnotvar = $true) & ((axidr = $true) <=> ! [X105 : subst] : ((comp @ X105 @ id) = X105)) & (! [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) != (ap @ (sub @ X436 @ id) @ X434)) | (X435 = X436)) <=> (hoasapinj1 = $true)) & ((hoaslaminj = $true) <=> ! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (~! [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X107 @ X111 @ X113) @ X112)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ~! [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) = (sub @ (X110 @ X116 @ X115) @ X114)))) & ((hoasinduction_lem2v2 = $true) <=> ! [X127 : term,X128 : term > $o,X129 : subst > term > subst > $o,X130 : term] : (~! [X131 : term,X132 : subst,X133 : subst,X134 : subst] : (($true != (X129 @ X133 @ X131 @ (comp @ X132 @ X134))) | ($true = (X129 @ (comp @ X133 @ X132) @ (sub @ X131 @ X132) @ X134))) | ($true = (X128 @ (ap @ X130 @ X127))) | ($true != (X128 @ X130)) | ~! [X140 : term,X141 : term] : (($true != (X129 @ id @ X141 @ id)) | ($true != (X129 @ id @ X140 @ id)) | ($true = (X129 @ id @ (ap @ (sub @ X140 @ id) @ X141) @ id))) | ~! [X135 : subst,X136 : subst,X137 : term,X138 : subst] : (($true = (X129 @ X138 @ X137 @ (comp @ X136 @ X135))) | ($true != (X129 @ (comp @ X138 @ X136) @ (sub @ X137 @ X136) @ X135))) | ~! [X139 : term] : ((X129 @ id @ X139 @ id) = (X128 @ X139)) | ($true != (X128 @ X127)))) & ((! [X84 : subst,X85 : term] : ((sub @ one @ (push @ X85 @ X84)) = X85) => (! [X86 : subst,X87 : term] : ((comp @ sh @ (push @ X87 @ X86)) = X86) => (! [X88 : subst] : ((comp @ X88 @ id) = X88) => (! [X89 : term,X90 : term > $o] : (~! [X94 : term] : (($true != (var @ X94)) | ($true = (X90 @ X94))) | ~! [X93 : term] : (($true != (X90 @ X93)) | ($true = (X90 @ (lam @ X93)))) | ($true = (X90 @ X89)) | ~! [X91 : term,X92 : term] : (((X90 @ X91) != $true) | ($true = (X90 @ (ap @ X91 @ X92))) | ($true != (X90 @ X92)))) => ! [X95 : subst,X96 : term,X97 : term > $o] : (($true = (X97 @ (sub @ X96 @ X95))) | ~! [X98 : term] : (($true = (X97 @ (lam @ X98))) | ~! [X99 : term] : (($true != (X97 @ X99)) | ($true = (X97 @ (sub @ X98 @ (push @ X99 @ id)))))) | ~! [X100 : term] : (($true != (var @ X100)) | ($true = (X97 @ (sub @ X100 @ X95)))) | ~! [X101 : term,X102 : term] : (($true != (X97 @ X102)) | ($true != (X97 @ X101)) | ($true = (X97 @ (ap @ X102 @ X101))))))))) <=> (induction2lem_lthm = $true)) & (induction2lem_gthm = $true) & (substmonoid_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoaslaminj_gthm = $true) & (axabs = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X219 : subst > term > subst > $o,X220 : term] : (~! [X226 : term] : (($true = (X219 @ id @ (sub @ X220 @ (push @ X226 @ id)) @ id)) | ($true != (X219 @ id @ X226 @ id))) | ~! [X221 : subst > term > term] : (~! [X222 : term] : (($true != (X219 @ id @ X222 @ id)) | ($true = (X219 @ id @ (X221 @ id @ X222) @ id))) | ($true = (X219 @ id @ (lam @ (X221 @ sh @ one)) @ id)) | ~! [X223 : term,X224 : subst,X225 : subst] : ((X221 @ (comp @ X225 @ X224) @ (sub @ X223 @ X224)) = (sub @ (X221 @ X225 @ X223) @ X224))) | ($true = (X219 @ id @ (lam @ (sub @ X220 @ (push @ one @ sh))) @ id)))) & (! [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) = X539) <=> (axscons = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) = X37) => ! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (($true != (X40 @ X39)) | ($true = (X41 @ one)) | ~! [X43 : term] : ((X41 @ X43) = (X40 @ (sub @ X43 @ (push @ X39 @ X42))))))) & (hoaslamnotvar = $true) & (pushprop_lthm = $true) & (pushprop_lem0_gthm = $true) & ((apinj2 = $true) <=> ! [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) != (ap @ X3 @ X1)) | (X0 = X1))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (axmap = $true) & (pushprop_lem1_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X532 : term,X533 : term > $o,X534 : subst] : ~! [X535 : term > $o] : ~! [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) = (X535 @ X536))) & (! [X310 : subst > term > subst > $o,X311 : term] : (~! [X312 : term] : (($true = (X310 @ id @ (sub @ X311 @ (push @ X312 @ id)) @ id)) | ($true != (X310 @ id @ X312 @ id))) | ($true = (X310 @ id @ (lam @ X311) @ id)) | ~! [X313 : subst > term > term] : (~! [X317 : term] : (($true = (X310 @ id @ (X313 @ id @ X317) @ id)) | ($true != (X310 @ id @ X317 @ id))) | ($true = (X310 @ id @ (lam @ (X313 @ sh @ one)) @ id)) | ~! [X314 : subst,X315 : term,X316 : subst] : ((sub @ (X313 @ X314 @ X315) @ X316) = (X313 @ (comp @ X314 @ X316) @ (sub @ X315 @ X316))))) <=> (hoasinduction_lem3a = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X537 : term] : ~! [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) != (X538 @ sh @ one))) & ! [X485 : d_term] : (d_one = X485) & (hoaslamnotvar_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((induction = $true) <=> ! [X479 : term > $o,X480 : term] : (~! [X482 : term] : (($true != (X479 @ X482)) | ($true = (X479 @ (lam @ X482)))) | ~! [X483 : term,X484 : term] : (($true != (X479 @ X483)) | ($true != (X479 @ X484)) | ((X479 @ (ap @ X483 @ X484)) = $true)) | ~! [X481 : term] : (($true != (var @ X481)) | ($true = (X479 @ X481))) | ($true = (X479 @ X480)))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((! [X227 : subst > term > subst > $o] : ~! [X228 : term > $o] : ~! [X229 : term] : ((X227 @ id @ X229 @ id) = (X228 @ X229)) => (! [X230 : term > $o,X231 : term] : (~! [X235 : term,X236 : term] : (($true != (X230 @ X236)) | ($true != (X230 @ X235)) | ($true = (X230 @ (ap @ X235 @ X236)))) | ~! [X232 : term] : (($true = (X230 @ X232)) | ($true != (var @ X232))) | ($true = (X230 @ X231)) | ~! [X233 : term] : (($true = (X230 @ (lam @ X233))) | ~! [X234 : term] : (($true = (X230 @ (sub @ X233 @ (push @ X234 @ id)))) | ($true != (X230 @ X234))))) => (! [X237 : term] : ((sub @ X237 @ id) = X237) => (! [X238 : term,X239 : term > $o,X240 : subst > term > subst > $o] : (~! [X241 : term] : (($true = (X239 @ (sub @ X238 @ (push @ X241 @ id)))) | ($true != (X239 @ X241))) | ($true = (X239 @ (lam @ X238))) | ~! [X242 : subst > term > term] : (~! [X244 : subst,X245 : term,X246 : subst] : ((X242 @ (comp @ X246 @ X244) @ (sub @ X245 @ X244)) = (sub @ (X242 @ X246 @ X245) @ X244)) | ($true = (X240 @ id @ (lam @ (X242 @ sh @ one)) @ id)) | ~! [X243 : term] : (($true != (X240 @ id @ X243 @ id)) | ($true = (X240 @ id @ (X242 @ id @ X243) @ id)))) | ~! [X247 : term] : ((X240 @ id @ X247 @ id) = (X239 @ X247))) => ! [X248 : term,X249 : subst > term > subst > $o] : (~! [X252 : subst,X253 : term,X254 : subst,X255 : subst] : (($true != (X249 @ X252 @ X253 @ (comp @ X254 @ X255))) | ($true = (X249 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254) @ X255))) | ~! [X256 : subst > term > term] : (~! [X257 : term] : (($true != (X249 @ id @ X257 @ id)) | ($true = (X249 @ id @ (X256 @ id @ X257) @ id))) | ($true = (X249 @ id @ (lam @ (X256 @ sh @ one)) @ id)) | ~! [X258 : subst,X259 : term,X260 : subst] : ((sub @ (X256 @ X260 @ X259) @ X258) = (X256 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258)))) | ~! [X250 : term,X251 : term] : (($true != (X249 @ id @ X250 @ id)) | ($true != (X249 @ id @ X251 @ id)) | ($true = (X249 @ id @ (ap @ (sub @ X251 @ id) @ X250) @ id))) | ($true = (X249 @ id @ X248 @ id)) | ~! [X265 : term] : (($true != (var @ (sub @ X265 @ id))) | ($true = (X249 @ id @ X265 @ id))) | ~! [X261 : term,X262 : subst,X263 : subst,X264 : subst] : (($true != (X249 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263) @ X264)) | ($true = (X249 @ X262 @ X261 @ (comp @ X263 @ X264))))))))) <=> (hoasinduction_lthm_3 = $true)) & (! [X318 : term] : ((sub @ X318 @ id) = X318) <=> (axvarid = $true)) & ((induction2 = $true) <=> ! [X319 : term,X320 : term > $o] : (($true = (X320 @ X319)) | ~! [X322 : term,X323 : term] : (($true != (X320 @ X323)) | ($true != (X320 @ X322)) | ($true = (X320 @ (ap @ X323 @ X322)))) | ~! [X321 : term] : (($true = (X320 @ X321)) | ($true != (var @ X321))) | ~! [X324 : term] : (~! [X325 : term] : (($true = (X320 @ (sub @ X324 @ (push @ X325 @ id)))) | ($true != (X320 @ X325))) | ($true = (X320 @ (lam @ X324)))))) & ((! [X33 : subst] : ((comp @ id @ X33) = X33) => (! [X34 : subst] : ((comp @ X34 @ id) = X34) => (! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36)))) <=> (substmonoid_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & (termmset_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true = (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) | ($true != (X173 @ X174))) <=> (pushprop_lem1 = $true)) & ((shinj = $true) <=> ! [X514 : term,X515 : term] : (((sub @ X514 @ sh) != (sub @ X515 @ sh)) | (X514 = X515))) & (! [X466 : term,X467 : term,X468 : subst > term > subst > $o] : (~! [X469 : term,X470 : term] : (($true = (X468 @ id @ (ap @ (sub @ X470 @ id) @ X469) @ id)) | ($true != (X468 @ id @ X470 @ id)) | ($true != (X468 @ id @ X469 @ id))) | ($true = (X468 @ id @ (ap @ X467 @ X466) @ id)) | ~! [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (($true = (X468 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471)) | ($true != (X468 @ X473 @ X474 @ (comp @ X472 @ X471)))) | ($true != (X468 @ id @ X467 @ id)) | ($true != (X468 @ id @ X466 @ id)) | ~! [X475 : subst,X476 : subst,X477 : subst,X478 : term] : (($true = (X468 @ X476 @ X478 @ (comp @ X475 @ X477))) | ($true != (X468 @ (comp @ X476 @ X475) @ (sub @ X478 @ X475) @ X477)))) <=> (hoasinduction_lem2 = $true))),
% 2.56/1.94 inference(flattening,[status(thm)],[f10])).
% 2.56/1.94 thf(f10,plain,(
% 2.56/1.94 ((apinj2 = $true) <=> ! [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) != (ap @ X3 @ X1)) | (X0 = X1))) & (! [X4 : term,X5 : subst > term > subst > $o] : (($true = (X5 @ id @ X4 @ id)) | ~! [X6 : subst > term > term] : (~! [X7 : term] : (~($true = (X5 @ id @ X7 @ id)) | ($true = (X5 @ id @ (X6 @ id @ X7) @ id))) | ~! [X8 : term,X9 : subst,X10 : subst] : ((X6 @ (comp @ X10 @ X9) @ (sub @ X8 @ X9)) = (sub @ (X6 @ X10 @ X8) @ X9)) | ($true = (X5 @ id @ (lam @ (X6 @ sh @ one)) @ id))) | ~! [X11 : term,X12 : term] : (~($true = (X5 @ id @ X12 @ id)) | ($true = (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) | ~((X5 @ id @ X11 @ id) = $true))) <=> (hoasinduction_no_psi_cond = $true)) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) = (d2subst @ X14)) => (X13 = X14)) & (one = (d2term @ d_one)) & ((hoasinduction_lem3v2a = $true) <=> ! [X15 : subst > term > subst > $o,X16 : term > $o,X17 : term] : (~! [X18 : term] : ((X16 @ X18) = (X15 @ id @ X18 @ id)) | ~! [X19 : subst > term > term] : (~! [X20 : term,X21 : subst,X22 : subst] : ((sub @ (X19 @ X22 @ X20) @ X21) = (X19 @ (comp @ X22 @ X21) @ (sub @ X20 @ X21))) | ($true = (X15 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~! [X23 : term] : (($true = (X15 @ id @ (X19 @ id @ X23) @ id)) | ~($true = (X15 @ id @ X23 @ id)))) | ~! [X24 : term] : (($true = (X16 @ (sub @ X17 @ (push @ X24 @ id)))) | ~($true = (X16 @ X24))) | ($true = (X16 @ (lam @ X17))))) & (hoasinduction_lem3a_gthm = $true) & ((! [X25 : term] : ((sub @ X25 @ id) = X25) => ! [X26 : term] : ((sub @ X26 @ id) = X26)) <=> (termmset_lthm = $true)) & ((pushprop_lem0 = $true) <=> ! [X27 : subst,X28 : term,X29 : term > $o] : ~! [X30 : term > $o] : ~! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31))) & (! [X32 : term] : ((sub @ X32 @ id) = X32) <=> (termmset = $true)) & (hoasinduction_lem1 = $true) & ((! [X33 : subst] : ((comp @ id @ X33) = X33) => (! [X34 : subst] : ((comp @ X34 @ id) = X34) => (! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36)))) <=> (substmonoid_lthm = $true)) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) = X37) => ! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (~! [X43 : term] : ((X41 @ X43) = (X40 @ (sub @ X43 @ (push @ X39 @ X42)))) | ~($true = (X40 @ X39)) | ($true = (X41 @ one))))) & (pushprop_lem2v2 = $true) & ~(lamnotap = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X44 : term,X45 : subst > term > subst > $o] : (($true = (X45 @ id @ (lam @ (sub @ X44 @ (push @ one @ sh))) @ id)) | ~! [X46 : term] : (((X45 @ id @ (sub @ X44 @ (push @ X46 @ id)) @ id) = $true) | ~($true = (X45 @ id @ X46 @ id))) | ~! [X47 : subst > term > term,X48 : term] : (~! [X49 : subst,X50 : term,X51 : subst] : ((sub @ (sub @ X48 @ (push @ X50 @ X49)) @ X51) = (sub @ (X47 @ X49 @ X50) @ X51)) | ~! [X52 : subst,X53 : term,X54 : subst] : ((sub @ X48 @ (push @ (sub @ X53 @ X52) @ (comp @ X54 @ X52))) = (X47 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) | ($true = (X45 @ id @ (lam @ (X47 @ sh @ one)) @ id)) | ~! [X55 : term] : (($true = (X45 @ id @ (X47 @ id @ X55) @ id)) | ~($true = (X45 @ id @ X55 @ id)))))) & ((axidl = $true) <=> ! [X56 : subst] : ((comp @ id @ X56) = X56)) & (hoasinduction_gthm = $true) & ((axshiftcons = $true) <=> ! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58)) & (axapp = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction = $true) <=> ! [X59 : term,X60 : subst > term > subst > $o] : (~! [X61 : term,X62 : term] : (($true = (X60 @ id @ (ap @ (sub @ X61 @ id) @ X62) @ id)) | ~($true = (X60 @ id @ X61 @ id)) | ~($true = (X60 @ id @ X62 @ id))) | ~! [X63 : subst > term > term] : (~! [X64 : term] : (~($true = (X60 @ id @ X64 @ id)) | ($true = (X60 @ id @ (X63 @ id @ X64) @ id))) | ($true = (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) | ~! [X65 : subst,X66 : subst,X67 : term] : ((X63 @ (comp @ X65 @ X66) @ (sub @ X67 @ X66)) = (sub @ (X63 @ X65 @ X67) @ X66))) | ~! [X68 : term] : (~($true = (var @ (sub @ X68 @ id))) | ($true = (X60 @ id @ X68 @ id))) | ~! [X69 : subst,X70 : subst,X71 : term,X72 : subst] : (~($true = (X60 @ (comp @ X69 @ X72) @ (sub @ X71 @ X72) @ X70)) | ($true = (X60 @ X69 @ X71 @ (comp @ X72 @ X70)))) | ~! [X73 : term,X74 : subst,X75 : subst,X76 : subst] : (($true = (X60 @ (comp @ X75 @ X74) @ (sub @ X73 @ X74) @ X76)) | ~($true = (X60 @ X75 @ X73 @ (comp @ X74 @ X76)))) | ($true = (X60 @ id @ X59 @ id)))) & ~(ulamvar1 = $true) & ((d2subst @ d_id) = sh) & (hoaslamnotvar_lthm = $true) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem0 = $true) <=> ! [X77 : subst > term > subst > $o] : ~! [X78 : term > $o] : ~! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id))) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X80 : term] : ~! [X81 : subst > term > term] : ~! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83))) & (induction2lem_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((induction2lem_lthm = $true) <=> (! [X84 : subst,X85 : term] : ((sub @ one @ (push @ X85 @ X84)) = X85) => (! [X86 : subst,X87 : term] : ((comp @ sh @ (push @ X87 @ X86)) = X86) => (! [X88 : subst] : ((comp @ X88 @ id) = X88) => (! [X89 : term,X90 : term > $o] : (~! [X91 : term,X92 : term] : (~($true = (X90 @ X92)) | ($true = (X90 @ (ap @ X91 @ X92))) | ~((X90 @ X91) = $true)) | ~! [X93 : term] : (~($true = (X90 @ X93)) | ($true = (X90 @ (lam @ X93)))) | ($true = (X90 @ X89)) | ~! [X94 : term] : (~($true = (var @ X94)) | ($true = (X90 @ X94)))) => ! [X95 : subst,X96 : term,X97 : term > $o] : (~! [X98 : term] : (~! [X99 : term] : (($true = (X97 @ (sub @ X98 @ (push @ X99 @ id)))) | ~($true = (X97 @ X99))) | ($true = (X97 @ (lam @ X98)))) | ~! [X100 : term] : (~($true = (var @ X100)) | ($true = (X97 @ (sub @ X100 @ X95)))) | ~! [X101 : term,X102 : term] : (~($true = (X97 @ X102)) | ($true = (X97 @ (ap @ X102 @ X101))) | ~($true = (X97 @ X101))) | ($true = (X97 @ (sub @ X96 @ X95))))))))) & ((axvarcons = $true) <=> ! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104)) & (hoasapinj2_gthm = $true) & ((axidr = $true) <=> ! [X105 : subst] : ((comp @ X105 @ id) = X105)) & ! [X106 : d_subst] : (d_id = X106) & ((hoaslaminj = $true) <=> ! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (~! [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X107 @ X111 @ X113) @ X112)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ~! [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) = (sub @ (X110 @ X116 @ X115) @ X114)))) & ((hoaslamnotap = $true) <=> ! [X117 : subst > term > term,X118 : term,X119 : term] : (~! [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) = (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119)))) & (! [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 = X126) | ((ap @ X125 @ X123) != (ap @ X126 @ X124))) <=> (apinj1 = $true)) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X127 : term,X128 : term > $o,X129 : subst > term > subst > $o,X130 : term] : (~($true = (X128 @ X130)) | ($true = (X128 @ (ap @ X130 @ X127))) | ~! [X131 : term,X132 : subst,X133 : subst,X134 : subst] : (($true = (X129 @ (comp @ X133 @ X132) @ (sub @ X131 @ X132) @ X134)) | ~($true = (X129 @ X133 @ X131 @ (comp @ X132 @ X134)))) | ~($true = (X128 @ X127)) | ~! [X135 : subst,X136 : subst,X137 : term,X138 : subst] : (~($true = (X129 @ (comp @ X138 @ X136) @ (sub @ X137 @ X136) @ X135)) | ($true = (X129 @ X138 @ X137 @ (comp @ X136 @ X135)))) | ~! [X139 : term] : ((X129 @ id @ X139 @ id) = (X128 @ X139)) | ~! [X140 : term,X141 : term] : (($true = (X129 @ id @ (ap @ (sub @ X140 @ id) @ X141) @ id)) | ~($true = (X129 @ id @ X140 @ id)) | ~($true = (X129 @ id @ X141 @ id))))) & (pushprop_lthm = $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true = (X144 @ one)) | ~! [X146 : term] : ((X144 @ X146) = (X142 @ (sub @ X146 @ (push @ X145 @ X143)))) | ~($true = (X142 @ X145)))) & (! [X147 : term] : ~! [X148 : subst > term > term] : ~! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) = (d2term @ X152)) => (X151 = X152)) & (pushprop_lthm_orig = $true) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 = X155) | ((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153))) <=> (hoasapinj2 = $true)) & (hoasinduction_lem1v2 = $true) & (hoaslamnotvar_gthm = $true) & ((! [X157 : term,X158 : subst] : ((sub @ one @ (push @ X157 @ X158)) = X157) => (! [X159 : subst,X160 : term] : ((comp @ sh @ (push @ X160 @ X159)) = X159) => (! [X161 : term,X162 : term] : ((X161 = X162) | ((lam @ X161) != (lam @ X162))) => ! [X163 : subst,X164 : subst > term > term,X165 : term,X166 : subst > term > term] : (~! [X167 : subst,X168 : term,X169 : subst] : ((sub @ (X164 @ X167 @ X168) @ X169) = (X164 @ (comp @ X167 @ X169) @ (sub @ X168 @ X169))) | ~! [X170 : term,X171 : subst,X172 : subst] : ((sub @ (X166 @ X172 @ X170) @ X171) = (X166 @ (comp @ X172 @ X171) @ (sub @ X170 @ X171))) | ((X164 @ X163 @ X165) = (X166 @ X163 @ X165)) | ((lam @ (X166 @ sh @ one)) != (lam @ (X164 @ sh @ one))))))) <=> (hoaslaminj_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem1 = $true) <=> ! [X173 : term > $o,X174 : term,X175 : subst,X176 : term,X177 : term > $o] : (($true = (X177 @ (sub @ X174 @ (push @ X176 @ X175)))) | ~($true = (X173 @ X174)))) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X178 : term > $o,X179 : subst > term > subst > $o,X180 : term] : (~! [X181 : term] : ((X179 @ id @ X181 @ id) = (X178 @ X181)) | ~! [X182 : subst > term > term] : (($true = (X179 @ id @ (lam @ (X182 @ sh @ one)) @ id)) | ~! [X183 : term] : (($true = (X179 @ id @ (X182 @ id @ X183) @ id)) | ~($true = (X179 @ id @ X183 @ id))) | ~! [X184 : subst,X185 : subst,X186 : term] : ((sub @ (X182 @ X185 @ X186) @ X184) = (X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184)))) | ~! [X187 : term] : (($true = (X178 @ (sub @ X180 @ (push @ X187 @ id)))) | ~($true = (X178 @ X187))) | ~! [X188 : term,X189 : subst,X190 : subst,X191 : subst] : (($true = (X179 @ X189 @ X188 @ (comp @ X190 @ X191))) | ~($true = (X179 @ (comp @ X189 @ X190) @ (sub @ X188 @ X190) @ X191))) | ~! [X192 : subst,X193 : subst,X194 : term,X195 : subst] : (($true = (X179 @ (comp @ X192 @ X193) @ (sub @ X194 @ X193) @ X195)) | ~($true = (X179 @ X192 @ X194 @ (comp @ X193 @ X195)))) | ($true = (X178 @ (lam @ X180))))) & (! [X196 : term,X197 : term] : ((X196 = X197) | ((lam @ X197) != (lam @ X196))) <=> (laminj = $true)) & ! [X198 : term] : ? [X199 : d_term] : ((d2term @ X199) = X198) & (pushprop_lem3v2 = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X200 : term] : ~! [X201 : subst > term > term] : ((X201 @ sh @ one) != (sub @ X200 @ (push @ one @ sh)))) & (hoasapnotvar_gthm = $true) & (axmap = $true) & (axclos = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoaslamnotvar = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X202 : subst,X203 : term] : ((sub @ one @ (push @ X203 @ X202)) = X203) => (! [X204 : subst,X205 : term] : ((comp @ sh @ (push @ X205 @ X204)) = X204) => ! [X206 : term,X207 : term > $o,X208 : term > $o,X209 : subst,X210 : term] : (~($true = (X207 @ X210)) | ($true = (X208 @ (sub @ X210 @ (push @ X206 @ X209)))))))) & ((hoasapinj2_lthm = $true) <=> (! [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X211 @ X214) != (ap @ X213 @ X212)) | (X212 = X214)) => ! [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ (sub @ X215 @ id) @ X217) != (ap @ (sub @ X216 @ id) @ X218)) | (X217 = X218)))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem3aa = $true) <=> ! [X219 : subst > term > subst > $o,X220 : term] : (~! [X221 : subst > term > term] : (~! [X222 : term] : (~($true = (X219 @ id @ X222 @ id)) | ($true = (X219 @ id @ (X221 @ id @ X222) @ id))) | ($true = (X219 @ id @ (lam @ (X221 @ sh @ one)) @ id)) | ~! [X223 : term,X224 : subst,X225 : subst] : ((X221 @ (comp @ X225 @ X224) @ (sub @ X223 @ X224)) = (sub @ (X221 @ X225 @ X223) @ X224))) | ~! [X226 : term] : (($true = (X219 @ id @ (sub @ X220 @ (push @ X226 @ id)) @ id)) | ~($true = (X219 @ id @ X226 @ id))) | ($true = (X219 @ id @ (lam @ (sub @ X220 @ (push @ one @ sh))) @ id)))) & ((d2subst @ d_id) = id) & ((hoasinduction_lthm_3 = $true) <=> (! [X227 : subst > term > subst > $o] : ~! [X228 : term > $o] : ~! [X229 : term] : ((X227 @ id @ X229 @ id) = (X228 @ X229)) => (! [X230 : term > $o,X231 : term] : (~! [X232 : term] : (($true = (X230 @ X232)) | ~($true = (var @ X232))) | ($true = (X230 @ X231)) | ~! [X233 : term] : (~! [X234 : term] : (($true = (X230 @ (sub @ X233 @ (push @ X234 @ id)))) | ~($true = (X230 @ X234))) | ($true = (X230 @ (lam @ X233)))) | ~! [X235 : term,X236 : term] : (($true = (X230 @ (ap @ X235 @ X236))) | ~($true = (X230 @ X236)) | ~($true = (X230 @ X235)))) => (! [X237 : term] : ((sub @ X237 @ id) = X237) => (! [X238 : term,X239 : term > $o,X240 : subst > term > subst > $o] : (~! [X241 : term] : (~($true = (X239 @ X241)) | ($true = (X239 @ (sub @ X238 @ (push @ X241 @ id))))) | ~! [X242 : subst > term > term] : (($true = (X240 @ id @ (lam @ (X242 @ sh @ one)) @ id)) | ~! [X243 : term] : (~($true = (X240 @ id @ X243 @ id)) | ($true = (X240 @ id @ (X242 @ id @ X243) @ id))) | ~! [X244 : subst,X245 : term,X246 : subst] : ((X242 @ (comp @ X246 @ X244) @ (sub @ X245 @ X244)) = (sub @ (X242 @ X246 @ X245) @ X244))) | ~! [X247 : term] : ((X240 @ id @ X247 @ id) = (X239 @ X247)) | ($true = (X239 @ (lam @ X238)))) => ! [X248 : term,X249 : subst > term > subst > $o] : (~! [X250 : term,X251 : term] : (~($true = (X249 @ id @ X251 @ id)) | ~($true = (X249 @ id @ X250 @ id)) | ($true = (X249 @ id @ (ap @ (sub @ X251 @ id) @ X250) @ id))) | ~! [X252 : subst,X253 : term,X254 : subst,X255 : subst] : (~($true = (X249 @ X252 @ X253 @ (comp @ X254 @ X255))) | ($true = (X249 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254) @ X255))) | ~! [X256 : subst > term > term] : (~! [X257 : term] : (~($true = (X249 @ id @ X257 @ id)) | ($true = (X249 @ id @ (X256 @ id @ X257) @ id))) | ~! [X258 : subst,X259 : term,X260 : subst] : ((sub @ (X256 @ X260 @ X259) @ X258) = (X256 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258))) | ($true = (X249 @ id @ (lam @ (X256 @ sh @ one)) @ id))) | ($true = (X249 @ id @ X248 @ id)) | ~! [X261 : term,X262 : subst,X263 : subst,X264 : subst] : (~($true = (X249 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263) @ X264)) | ($true = (X249 @ X262 @ X261 @ (comp @ X263 @ X264)))) | ~! [X265 : term] : (~($true = (var @ (sub @ X265 @ id))) | ($true = (X249 @ id @ X265 @ id))))))))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X266 : subst] : ? [X267 : d_subst] : ((d2subst @ X267) = X266) & ((! [X268 : term] : ((sub @ X268 @ id) = X268) => (! [X269 : term,X270 : subst > term > subst > $o] : (~! [X271 : subst > term > term] : (~! [X272 : subst,X273 : term,X274 : subst] : ((X271 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)) = (sub @ (X271 @ X272 @ X273) @ X274)) | ($true = (X270 @ id @ (lam @ (X271 @ sh @ one)) @ id)) | ~! [X275 : term] : (~($true = (X270 @ id @ X275 @ id)) | ($true = (X270 @ id @ (X271 @ id @ X275) @ id)))) | ($true = (X270 @ id @ (lam @ (sub @ X269 @ (push @ one @ sh))) @ id)) | ~! [X276 : term] : (($true = (X270 @ id @ (sub @ X269 @ (push @ X276 @ id)) @ id)) | ~($true = (X270 @ id @ X276 @ id)))) => ! [X277 : term,X278 : subst > term > subst > $o] : (~! [X279 : subst > term > term] : (($true = (X278 @ id @ (lam @ (X279 @ sh @ one)) @ id)) | ~! [X280 : subst,X281 : term,X282 : subst] : ((sub @ (X279 @ X282 @ X281) @ X280) = (X279 @ (comp @ X282 @ X280) @ (sub @ X281 @ X280))) | ~! [X283 : term] : (($true = (X278 @ id @ (X279 @ id @ X283) @ id)) | ~($true = (X278 @ id @ X283 @ id)))) | ~! [X284 : term] : (($true = (X278 @ id @ (sub @ X277 @ (push @ X284 @ id)) @ id)) | ~($true = (X278 @ id @ X284 @ id))) | ($true = (X278 @ id @ (lam @ X277) @ id))))) <=> (hoasinduction_lem3a_lthm = $true)) & (pushprop_gthm = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X285 : term] : ((sub @ X285 @ id) = X285) => (! [X286 : term,X287 : subst > term > subst > $o] : (~! [X288 : subst > term > term] : (~! [X289 : term] : (($true = (X287 @ id @ (X288 @ id @ X289) @ id)) | ~($true = (X287 @ id @ X289 @ id))) | ~! [X290 : subst,X291 : subst,X292 : term] : ((X288 @ (comp @ X291 @ X290) @ (sub @ X292 @ X290)) = (sub @ (X288 @ X291 @ X292) @ X290)) | ($true = (X287 @ id @ (lam @ (X288 @ sh @ one)) @ id))) | ($true = (X287 @ id @ (lam @ (sub @ X286 @ (push @ one @ sh))) @ id)) | ~! [X293 : term] : (($true = (X287 @ id @ (sub @ X286 @ (push @ X293 @ id)) @ id)) | ~($true = (X287 @ id @ X293 @ id)))) => ! [X294 : term,X295 : subst > term > subst > $o] : (($true = (X295 @ id @ (lam @ X294) @ id)) | ~! [X296 : subst,X297 : subst,X298 : subst,X299 : term] : (~($true = (X295 @ (comp @ X298 @ X296) @ (sub @ X299 @ X296) @ X297)) | ($true = (X295 @ X298 @ X299 @ (comp @ X296 @ X297)))) | ~! [X300 : term] : (($true = (X295 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id)) | ~($true = (X295 @ id @ X300 @ id))) | ~! [X301 : subst > term > term] : (~! [X302 : term,X303 : subst,X304 : subst] : ((X301 @ (comp @ X303 @ X304) @ (sub @ X302 @ X304)) = (sub @ (X301 @ X303 @ X302) @ X304)) | ~! [X305 : term] : (~($true = (X295 @ id @ X305 @ id)) | ($true = (X295 @ id @ (X301 @ id @ X305) @ id))) | ($true = (X295 @ id @ (lam @ (X301 @ sh @ one)) @ id))) | ~! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (~($true = (X295 @ X308 @ X306 @ (comp @ X309 @ X307))) | ($true = (X295 @ (comp @ X308 @ X309) @ (sub @ X306 @ X309) @ X307))))))) & (termmset_gthm = $true) & (axassoc = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (! [X310 : subst > term > subst > $o,X311 : term] : (~! [X312 : term] : (~($true = (X310 @ id @ X312 @ id)) | ($true = (X310 @ id @ (sub @ X311 @ (push @ X312 @ id)) @ id))) | ~! [X313 : subst > term > term] : (($true = (X310 @ id @ (lam @ (X313 @ sh @ one)) @ id)) | ~! [X314 : subst,X315 : term,X316 : subst] : ((sub @ (X313 @ X314 @ X315) @ X316) = (X313 @ (comp @ X314 @ X316) @ (sub @ X315 @ X316))) | ~! [X317 : term] : (~($true = (X310 @ id @ X317 @ id)) | ($true = (X310 @ id @ (X313 @ id @ X317) @ id)))) | ($true = (X310 @ id @ (lam @ X311) @ id))) <=> (hoasinduction_lem3a = $true)) & (hoasinduction_lem2v2_gthm = $true) & (! [X318 : term] : ((sub @ X318 @ id) = X318) <=> (axvarid = $true)) & (ulamvarind = $true) & ((induction2 = $true) <=> ! [X319 : term,X320 : term > $o] : (~! [X321 : term] : (($true = (X320 @ X321)) | ~($true = (var @ X321))) | ~! [X322 : term,X323 : term] : (~($true = (X320 @ X322)) | ($true = (X320 @ (ap @ X323 @ X322))) | ~($true = (X320 @ X323))) | ($true = (X320 @ X319)) | ~! [X324 : term] : (~! [X325 : term] : (($true = (X320 @ (sub @ X324 @ (push @ X325 @ id)))) | ~($true = (X320 @ X325))) | ($true = (X320 @ (lam @ X324)))))) & (axabs = $true) & ((! [X326 : subst > term > subst > $o] : ~! [X327 : term > $o] : ~! [X328 : term] : ((X326 @ id @ X328 @ id) = (X327 @ X328)) => (! [X329 : term > $o,X330 : term] : (~! [X331 : term] : (~($true = (var @ X331)) | ($true = (X329 @ X331))) | ($true = (X329 @ X330)) | ~! [X332 : term] : (~! [X333 : term] : (~($true = (X329 @ X333)) | ($true = (X329 @ (sub @ X332 @ (push @ X333 @ id))))) | ($true = (X329 @ (lam @ X332)))) | ~! [X334 : term,X335 : term] : (~($true = (X329 @ X335)) | ~($true = (X329 @ X334)) | ($true = (X329 @ (ap @ X335 @ X334))))) => (! [X336 : term] : ((sub @ X336 @ id) = X336) => (! [X337 : subst > term > subst > $o,X338 : term,X339 : term > $o] : (~! [X340 : term] : (($true = (X339 @ (sub @ X338 @ (push @ X340 @ id)))) | ~($true = (X339 @ X340))) | ~! [X341 : subst > term > term] : (~! [X342 : term] : (~($true = (X337 @ id @ X342 @ id)) | ((X337 @ id @ (X341 @ id @ X342) @ id) = $true)) | ~! [X343 : subst,X344 : term,X345 : subst] : ((sub @ (X341 @ X345 @ X344) @ X343) = (X341 @ (comp @ X345 @ X343) @ (sub @ X344 @ X343))) | ($true = (X337 @ id @ (lam @ (X341 @ sh @ one)) @ id))) | ($true = (X339 @ (lam @ X338))) | ~! [X346 : term] : ((X339 @ X346) = (X337 @ id @ X346 @ id))) => ! [X347 : subst > term > subst > $o,X348 : term] : (~! [X349 : term,X350 : term] : (~($true = (X347 @ id @ X350 @ id)) | ~($true = (X347 @ id @ X349 @ id)) | ($true = (X347 @ id @ (ap @ (sub @ X349 @ id) @ X350) @ id))) | ~! [X351 : subst > term > term] : (($true = (X347 @ id @ (lam @ (X351 @ sh @ one)) @ id)) | ~! [X352 : subst,X353 : subst,X354 : term] : ((X351 @ (comp @ X353 @ X352) @ (sub @ X354 @ X352)) = (sub @ (X351 @ X353 @ X354) @ X352)) | ~! [X355 : term] : (($true = (X347 @ id @ (X351 @ id @ X355) @ id)) | ~($true = (X347 @ id @ X355 @ id)))) | ($true = (X347 @ id @ X348 @ id))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop = $true) & (! [X356 : term > $o,X357 : subst,X358 : term] : (~! [X359 : term] : (~($true = (var @ X359)) | ($true = (X356 @ (sub @ X359 @ X357)))) | ($true = (X356 @ (sub @ X358 @ X357))) | ~! [X360 : term] : (($true = (X356 @ (lam @ X360))) | ~! [X361 : term] : (~($true = (X356 @ X361)) | ($true = (X356 @ (sub @ X360 @ (push @ X361 @ id)))))) | ~! [X362 : term,X363 : term] : (($true = (X356 @ (ap @ X362 @ X363))) | ~($true = (X356 @ X362)) | ~($true = (X356 @ X363)))) <=> (induction2lem = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X364 : term] : ~! [X365 : subst > term > term] : ~! [X366 : subst,X367 : term] : ((sub @ X364 @ (push @ X367 @ X366)) = (X365 @ X366 @ X367)) => (! [X368 : term] : ((sub @ X368 @ id) = X368) => ! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term] : (~! [X372 : term] : ((X370 @ id @ X372 @ id) = (X369 @ X372)) | ~! [X373 : term] : (((X369 @ (sub @ X371 @ (push @ X373 @ id))) = $true) | ~($true = (X369 @ X373))) | ($true = (X369 @ (lam @ X371))) | ~! [X374 : subst > term > term] : (($true = (X370 @ id @ (lam @ (X374 @ sh @ one)) @ id)) | ~! [X375 : term] : (($true = (X370 @ id @ (X374 @ id @ X375) @ id)) | ~($true = (X370 @ id @ X375 @ id))) | ~! [X376 : subst,X377 : term,X378 : subst] : ((sub @ (X374 @ X376 @ X377) @ X378) = (X374 @ (comp @ X376 @ X378) @ (sub @ X377 @ X378)))))))) & (induction2_gthm = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X379 : term,X380 : term > $o] : (~! [X381 : term,X382 : term] : (~($true = (X380 @ X381)) | ((X380 @ (ap @ X381 @ X382)) = $true) | ~($true = (X380 @ X382))) | ($true = (X380 @ X379)) | ~! [X383 : term] : (($true = (X380 @ (lam @ X383))) | ~! [X384 : term] : (($true = (X380 @ (sub @ X383 @ (push @ X384 @ id)))) | ~($true = (X380 @ X384)))) | ~! [X385 : term] : (($true = (X380 @ X385)) | ~($true = (var @ X385)))) => (! [X386 : term,X387 : term,X388 : subst > term > subst > $o] : (($true = (X388 @ id @ (ap @ X386 @ X387) @ id)) | ~($true = (X388 @ id @ X387 @ id)) | ~($true = (X388 @ id @ X386 @ id)) | ~! [X389 : subst,X390 : subst,X391 : term,X392 : subst] : (~($true = (X388 @ X390 @ X391 @ (comp @ X389 @ X392))) | ($true = (X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389) @ X392))) | ~! [X393 : term,X394 : term] : (($true = (X388 @ id @ (ap @ (sub @ X394 @ id) @ X393) @ id)) | ~($true = (X388 @ id @ X394 @ id)) | ~($true = (X388 @ id @ X393 @ id))) | ~! [X395 : subst,X396 : subst,X397 : subst,X398 : term] : (~($true = (X388 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397) @ X395)) | ($true = (X388 @ X396 @ X398 @ (comp @ X397 @ X395))))) => (! [X399 : subst > term > subst > $o,X400 : term] : (($true = (X399 @ id @ (lam @ X400) @ id)) | ~! [X401 : term] : (($true = (X399 @ id @ (sub @ X400 @ (push @ X401 @ id)) @ id)) | ~($true = (X399 @ id @ X401 @ id))) | ~! [X402 : subst,X403 : term,X404 : subst,X405 : subst] : (($true = (X399 @ X404 @ X403 @ (comp @ X402 @ X405))) | ~($true = (X399 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) | ~! [X406 : subst,X407 : term,X408 : subst,X409 : subst] : (~($true = (X399 @ X406 @ X407 @ (comp @ X409 @ X408))) | ($true = (X399 @ (comp @ X406 @ X409) @ (sub @ X407 @ X409) @ X408))) | ~! [X410 : subst > term > term] : (~! [X411 : term,X412 : subst,X413 : subst] : ((X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413)) = (sub @ (X410 @ X412 @ X411) @ X413)) | ($true = (X399 @ id @ (lam @ (X410 @ sh @ one)) @ id)) | ~! [X414 : term] : (~($true = (X399 @ id @ X414 @ id)) | ($true = (X399 @ id @ (X410 @ id @ X414) @ id))))) => ! [X415 : subst > term > subst > $o,X416 : term] : (~! [X417 : term,X418 : term] : (($true = (X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id)) | ~((X415 @ id @ X417 @ id) = $true) | ~($true = (X415 @ id @ X418 @ id))) | ~! [X419 : subst > term > term] : (((X415 @ id @ (lam @ (X419 @ sh @ one)) @ id) = $true) | ~! [X420 : term] : (($true = (X415 @ id @ (X419 @ id @ X420) @ id)) | ~($true = (X415 @ id @ X420 @ id))) | ~! [X421 : term,X422 : subst,X423 : subst] : ((sub @ (X419 @ X422 @ X421) @ X423) = (X419 @ (comp @ X422 @ X423) @ (sub @ X421 @ X423)))) | ($true = (X415 @ id @ X416 @ id)) | ~! [X424 : subst,X425 : term,X426 : subst,X427 : subst] : (($true = (X415 @ (comp @ X427 @ X424) @ (sub @ X425 @ X424) @ X426)) | ~($true = (X415 @ X427 @ X425 @ (comp @ X424 @ X426)))) | ~! [X428 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = (X415 @ X430 @ X428 @ (comp @ X429 @ X431))) | ~($true = (X415 @ (comp @ X430 @ X429) @ (sub @ X428 @ X429) @ X431))) | ~! [X432 : term] : (($true = (X415 @ id @ X432 @ id)) | ~($true = (var @ (sub @ X432 @ id))))))))) & (! [X433 : term,X434 : term,X435 : term,X436 : term] : (((ap @ (sub @ X435 @ id) @ X433) != (ap @ (sub @ X436 @ id) @ X434)) | (X435 = X436)) <=> (hoasapinj1 = $true)) & (pushprop_lem1_gthm = $true) & (apnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (lamnotvar = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X437 : subst > term > subst > $o,X438 : term] : (~! [X439 : subst > term > term] : (~! [X440 : term] : (($true = (X437 @ id @ (X439 @ id @ X440) @ id)) | ~($true = (X437 @ id @ X440 @ id))) | ($true = (X437 @ id @ (lam @ (X439 @ sh @ one)) @ id)) | ~! [X441 : subst,X442 : term,X443 : subst] : ((sub @ (X439 @ X441 @ X442) @ X443) = (X439 @ (comp @ X441 @ X443) @ (sub @ X442 @ X443)))) | ($true = (X437 @ id @ (lam @ (sub @ X438 @ (push @ one @ sh))) @ id)) | ~! [X444 : term] : (($true = (X437 @ id @ (sub @ X438 @ (push @ X444 @ id)) @ id)) | ~($true = (X437 @ id @ X444 @ id))))) & ((hoasinduction_lem0_lthm = $true) <=> ! [X445 : subst > term > subst > $o] : ~! [X446 : term > $o] : ~! [X447 : term] : ((X445 @ id @ X447 @ id) = (X446 @ X447))) & (hoasinduction_lem3_gthm = $true) & ((substmonoid = $true) <=> (! [X449 : subst] : ((comp @ X449 @ id) = X449) & ! [X448 : subst] : ((comp @ id @ X448) = X448))) & ((induction2_lthm = $true) <=> (! [X450 : term] : ((sub @ X450 @ id) = X450) => (! [X451 : subst,X452 : term,X453 : term > $o] : (~! [X454 : term] : (~($true = (var @ X454)) | ($true = (X453 @ (sub @ X454 @ X451)))) | ~! [X455 : term,X456 : term] : (~((X453 @ X456) = $true) | ($true = (X453 @ (ap @ X455 @ X456))) | ~($true = (X453 @ X455))) | ~! [X457 : term] : (~! [X458 : term] : (($true = (X453 @ (sub @ X457 @ (push @ X458 @ id)))) | ~($true = (X453 @ X458))) | ($true = (X453 @ (lam @ X457)))) | ($true = (X453 @ (sub @ X452 @ X451)))) => ! [X459 : term > $o,X460 : term] : (~! [X461 : term] : (~($true = (var @ X461)) | ($true = (X459 @ X461))) | ~! [X462 : term,X463 : term] : (~($true = (X459 @ X462)) | ~($true = (X459 @ X463)) | ($true = (X459 @ (ap @ X462 @ X463)))) | ($true = (X459 @ X460)) | ~! [X464 : term] : (($true = (X459 @ (lam @ X464))) | ~! [X465 : term] : (($true = (X459 @ (sub @ X464 @ (push @ X465 @ id)))) | ~($true = (X459 @ X465)))))))) & ~((var @ (d2term @ d_one)) = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X466 : term,X467 : term,X468 : subst > term > subst > $o] : (($true = (X468 @ id @ (ap @ X467 @ X466) @ id)) | ~($true = (X468 @ id @ X467 @ id)) | ~! [X469 : term,X470 : term] : (~($true = (X468 @ id @ X470 @ id)) | ($true = (X468 @ id @ (ap @ (sub @ X470 @ id) @ X469) @ id)) | ~($true = (X468 @ id @ X469 @ id))) | ~! [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (~($true = (X468 @ X473 @ X474 @ (comp @ X472 @ X471))) | ($true = (X468 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471))) | ~! [X475 : subst,X476 : subst,X477 : subst,X478 : term] : (($true = (X468 @ X476 @ X478 @ (comp @ X475 @ X477))) | ~($true = (X468 @ (comp @ X476 @ X475) @ (sub @ X478 @ X475) @ X477))) | ~($true = (X468 @ id @ X466 @ id)))) & (! [X479 : term > $o,X480 : term] : (~! [X481 : term] : (($true = (X479 @ X481)) | ~($true = (var @ X481))) | ~! [X482 : term] : (~($true = (X479 @ X482)) | ($true = (X479 @ (lam @ X482)))) | ($true = (X479 @ X480)) | ~! [X483 : term,X484 : term] : (~($true = (X479 @ X484)) | ((X479 @ (ap @ X483 @ X484)) = $true) | ~($true = (X479 @ X483)))) <=> (induction = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X485 : d_term] : (d_one = X485) & ((hoasapinj1_lthm = $true) <=> (! [X486 : term] : ((sub @ X486 @ id) = X486) => (! [X487 : term,X488 : term,X489 : term,X490 : term] : (((ap @ X487 @ X488) != (ap @ X490 @ X489)) | (X487 = X490)) => ! [X491 : term,X492 : term,X493 : term,X494 : term] : (((ap @ (sub @ X493 @ id) @ X492) != (ap @ (sub @ X491 @ id) @ X494)) | (X491 = X493))))) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axvarshift = $true) & ((! [X495 : term] : ((sub @ X495 @ id) = X495) => ! [X496 : term > $o,X497 : subst > term > subst > $o,X498 : term] : (~! [X499 : subst,X500 : term,X501 : subst,X502 : subst] : (($true = (X497 @ (comp @ X501 @ X499) @ (sub @ X500 @ X499) @ X502)) | ~($true = (X497 @ X501 @ X500 @ (comp @ X499 @ X502)))) | ~! [X503 : term] : ((X496 @ X503) = (X497 @ id @ X503 @ id)) | ~! [X504 : term] : (($true = (X496 @ (sub @ X498 @ (push @ X504 @ id)))) | ~($true = (X496 @ X504))) | ~! [X505 : subst,X506 : term,X507 : subst,X508 : subst] : (~((X497 @ (comp @ X508 @ X507) @ (sub @ X506 @ X507) @ X505) = $true) | ($true = (X497 @ X508 @ X506 @ (comp @ X507 @ X505)))) | ($true = (X496 @ (lam @ X498))) | ~! [X509 : subst > term > term] : (($true = (X497 @ id @ (lam @ (X509 @ sh @ one)) @ id)) | ~! [X510 : subst,X511 : term,X512 : subst] : ((sub @ (X509 @ X512 @ X511) @ X510) = (X509 @ (comp @ X512 @ X510) @ (sub @ X511 @ X510))) | ~! [X513 : term] : (~($true = (X497 @ id @ X513 @ id)) | ($true = (X497 @ id @ (X509 @ id @ X513) @ id)))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((shinj = $true) <=> ! [X514 : term,X515 : term] : (((sub @ X514 @ sh) != (sub @ X515 @ sh)) | (X514 = X515))) & ((hoasinduction_lem3 = $true) <=> ! [X516 : term,X517 : subst > term > subst > $o] : (~! [X518 : subst,X519 : term,X520 : subst,X521 : subst] : (($true = (X517 @ (comp @ X521 @ X518) @ (sub @ X519 @ X518) @ X520)) | ~($true = (X517 @ X521 @ X519 @ (comp @ X518 @ X520)))) | ~! [X522 : subst > term > term] : (((X517 @ id @ (lam @ (X522 @ sh @ one)) @ id) = $true) | ~! [X523 : term] : (($true = (X517 @ id @ (X522 @ id @ X523) @ id)) | ~($true = (X517 @ id @ X523 @ id))) | ~! [X524 : term,X525 : subst,X526 : subst] : ((X522 @ (comp @ X525 @ X526) @ (sub @ X524 @ X526)) = (sub @ (X522 @ X525 @ X524) @ X526))) | ($true = (X517 @ id @ (lam @ X516) @ id)) | ~! [X527 : term] : (($true = (X517 @ id @ (sub @ X516 @ (push @ X527 @ id)) @ id)) | ~($true = (X517 @ id @ X527 @ id))) | ~! [X528 : subst,X529 : subst,X530 : subst,X531 : term] : (~($true = (X517 @ (comp @ X529 @ X530) @ (sub @ X531 @ X530) @ X528)) | ((X517 @ X529 @ X531 @ (comp @ X530 @ X528)) = $true)))) & (substmonoid_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((pushprop_lem0_lthm = $true) <=> ! [X532 : term,X533 : term > $o,X534 : subst] : ~! [X535 : term > $o] : ~! [X536 : term] : ((X533 @ (sub @ X536 @ (push @ X532 @ X534))) = (X535 @ X536))) & ((hoasinduction_lem3b = $true) <=> ! [X537 : term] : ~! [X538 : subst > term > term] : ((sub @ X537 @ (push @ one @ sh)) != (X538 @ sh @ one))) & (pushprop_lem2v2_gthm = $true) & (! [X539 : subst] : ((push @ (sub @ one @ X539) @ (comp @ sh @ X539)) = X539) <=> (axscons = $true)) & (ulamvarsh = $true) & (hoasinduction_lem2_lthm = $true)),
% 2.56/1.94 inference(rectify,[status(thm)],[f6])).
% 2.56/1.94 thf(f6,plain,(
% 2.56/1.94 ((apinj2 = $true) <=> ! [X0 : term,X1 : term,X2 : term,X3 : term] : (((ap @ X2 @ X0) != (ap @ X3 @ X1)) | (X0 = X1))) & (! [X4 : term,X5 : subst > term > subst > $o] : (($true = (X5 @ id @ X4 @ id)) | ~! [X6 : subst > term > term] : (~! [X7 : term] : (~($true = (X5 @ id @ X7 @ id)) | ($true = (X5 @ id @ (X6 @ id @ X7) @ id))) | ~! [X8 : term,X9 : subst,X10 : subst] : ((X6 @ (comp @ X10 @ X9) @ (sub @ X8 @ X9)) = (sub @ (X6 @ X10 @ X8) @ X9)) | ($true = (X5 @ id @ (lam @ (X6 @ sh @ one)) @ id))) | ~! [X11 : term,X12 : term] : (~($true = (X5 @ id @ X12 @ id)) | ($true = (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id)) | ~((X5 @ id @ X11 @ id) = $true))) <=> (hoasinduction_no_psi_cond = $true)) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) = (d2subst @ X14)) => (X13 = X14)) & (one = (d2term @ d_one)) & ((hoasinduction_lem3v2a = $true) <=> ! [X15 : subst > term > subst > $o,X16 : term > $o,X17 : term] : (~! [X18 : term] : ((X16 @ X18) = (X15 @ id @ X18 @ id)) | ~! [X19 : subst > term > term] : (~! [X20 : term,X21 : subst,X22 : subst] : ((sub @ (X19 @ X22 @ X20) @ X21) = (X19 @ (comp @ X22 @ X21) @ (sub @ X20 @ X21))) | ($true = (X15 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~! [X23 : term] : (($true = (X15 @ id @ (X19 @ id @ X23) @ id)) | ~($true = (X15 @ id @ X23 @ id)))) | ~! [X24 : term] : (($true = (X16 @ (sub @ X17 @ (push @ X24 @ id)))) | ~($true = (X16 @ X24))) | ($true = (X16 @ (lam @ X17))))) & (hoasinduction_lem3a_gthm = $true) & ((! [X25 : term] : ((sub @ X25 @ id) = X25) => ! [X26 : term] : ((sub @ X26 @ id) = X26)) <=> (termmset_lthm = $true)) & ((pushprop_lem0 = $true) <=> ! [X27 : subst,X28 : term,X29 : term > $o] : ~! [X30 : term > $o] : ~! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31))) & (! [X32 : term] : ((sub @ X32 @ id) = X32) <=> (termmset = $true)) & (hoasinduction_lem1 = $true) & ((! [X33 : subst] : ((comp @ id @ X33) = X33) => (! [X34 : subst] : ((comp @ X34 @ id) = X34) => (! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36)))) <=> (substmonoid_lthm = $true)) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) = X37) => ! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (~! [X43 : term] : ((X41 @ X43) = (X40 @ (sub @ X43 @ (push @ X39 @ X42)))) | ~($true = (X40 @ X39)) | ($true = (X41 @ one))))) & (pushprop_lem2v2 = $true) & ~(lamnotap = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X44 : term,X45 : subst > term > subst > $o] : (($true = (X45 @ id @ (lam @ (sub @ X44 @ (push @ one @ sh))) @ id)) | ~! [X46 : term] : (((X45 @ id @ (sub @ X44 @ (push @ X46 @ id)) @ id) = $true) | ~($true = (X45 @ id @ X46 @ id))) | ~! [X47 : subst > term > term,X48 : term] : (~! [X49 : subst,X50 : term,X51 : subst] : ((sub @ (sub @ X48 @ (push @ X50 @ X49)) @ X51) = (sub @ (X47 @ X49 @ X50) @ X51)) | ~! [X52 : subst,X53 : term,X54 : subst] : ((sub @ X48 @ (push @ (sub @ X53 @ X52) @ (comp @ X54 @ X52))) = (X47 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) | ($true = (X45 @ id @ (lam @ (X47 @ sh @ one)) @ id)) | ~! [X55 : term] : (($true = (X45 @ id @ (X47 @ id @ X55) @ id)) | ~($true = (X45 @ id @ X55 @ id)))))) & ((axidl = $true) <=> ! [X56 : subst] : ((comp @ id @ X56) = X56)) & (hoasinduction_gthm = $true) & ((axshiftcons = $true) <=> ! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58)) & (axapp = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction = $true) <=> ! [X59 : term,X60 : subst > term > subst > $o] : (~! [X61 : term,X62 : term] : (($true = (X60 @ id @ (ap @ (sub @ X61 @ id) @ X62) @ id)) | ~($true = (X60 @ id @ X61 @ id)) | ~($true = (X60 @ id @ X62 @ id))) | ~! [X63 : subst > term > term] : (~! [X64 : term] : (~($true = (X60 @ id @ X64 @ id)) | ($true = (X60 @ id @ (X63 @ id @ X64) @ id))) | ($true = (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) | ~! [X65 : subst,X66 : subst,X67 : term] : ((X63 @ (comp @ X65 @ X66) @ (sub @ X67 @ X66)) = (sub @ (X63 @ X65 @ X67) @ X66))) | ~! [X68 : term] : (~($true = (var @ (sub @ X68 @ id))) | ($true = (X60 @ id @ X68 @ id))) | ~! [X69 : subst,X70 : subst,X71 : term,X72 : subst] : (~($true = (X60 @ (comp @ X69 @ X72) @ (sub @ X71 @ X72) @ X70)) | ($true = (X60 @ X69 @ X71 @ (comp @ X72 @ X70)))) | ~! [X73 : term,X74 : subst,X75 : subst,X76 : subst] : (($true = (X60 @ (comp @ X75 @ X74) @ (sub @ X73 @ X74) @ X76)) | ~($true = (X60 @ X75 @ X73 @ (comp @ X74 @ X76)))) | ($true = (X60 @ id @ X59 @ id)))) & ~(ulamvar1 = $true) & ((d2subst @ d_id) = sh) & (hoaslamnotvar_lthm = $true) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem0 = $true) <=> ! [X77 : subst > term > subst > $o] : ~! [X78 : term > $o] : ~! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id))) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X80 : term] : ~! [X81 : subst > term > term] : ~! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83))) & (induction2lem_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((induction2lem_lthm = $true) <=> (! [X84 : subst,X85 : term] : ((sub @ one @ (push @ X85 @ X84)) = X85) => (! [X86 : subst,X87 : term] : ((comp @ sh @ (push @ X87 @ X86)) = X86) => (! [X88 : subst] : ((comp @ X88 @ id) = X88) => (! [X89 : term,X90 : term > $o] : (~! [X91 : term,X92 : term] : (~($true = (X90 @ X92)) | ($true = (X90 @ (ap @ X91 @ X92))) | ~((X90 @ X91) = $true)) | ~! [X93 : term] : (~($true = (X90 @ X93)) | ($true = (X90 @ (lam @ X93)))) | ($true = (X90 @ X89)) | ~! [X94 : term] : (~($true = (var @ X94)) | ($true = (X90 @ X94)))) => ! [X95 : subst,X96 : term,X97 : term > $o] : (~! [X98 : term] : (~! [X99 : term] : (($true = (X97 @ (sub @ X98 @ (push @ X99 @ id)))) | ~($true = (X97 @ X99))) | ($true = (X97 @ (lam @ X98)))) | ~! [X100 : term] : (~($true = (var @ X100)) | ($true = (X97 @ (sub @ X100 @ X95)))) | ~! [X101 : term,X102 : term] : (~($true = (X97 @ X102)) | ($true = (X97 @ (ap @ X102 @ X101))) | ~($true = (X97 @ X101))) | ($true = (X97 @ (sub @ X96 @ X95))))))))) & ((axvarcons = $true) <=> ! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104)) & (hoasapinj2_gthm = $true) & ((axidr = $true) <=> ! [X105 : subst] : ((comp @ X105 @ id) = X105)) & ! [X106 : d_subst] : (d_id = X106) & ((hoaslaminj = $true) <=> ! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (~! [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X107 @ X111 @ X113) @ X112)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ~! [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) = (sub @ (X110 @ X116 @ X115) @ X114)))) & ((hoaslamnotap = $true) <=> ! [X117 : subst > term > term,X118 : term,X119 : term] : (~! [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) = (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119)))) & (! [X123 : term,X124 : term,X125 : term,X126 : term] : ((X125 = X126) | ((ap @ X125 @ X123) != (ap @ X126 @ X124))) <=> (apinj1 = $true)) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X127 : term,X128 : term > $o,X129 : subst > term > subst > $o,X130 : term] : (~($true = (X128 @ X130)) | ($true = (X128 @ (ap @ X130 @ X127))) | ~! [X131 : term,X132 : subst,X133 : subst,X134 : subst] : (($true = (X129 @ (comp @ X133 @ X132) @ (sub @ X131 @ X132) @ X134)) | ~($true = (X129 @ X133 @ X131 @ (comp @ X132 @ X134)))) | ~($true = (X128 @ X127)) | ~! [X135 : subst,X136 : subst,X137 : term,X138 : subst] : (~($true = (X129 @ (comp @ X138 @ X136) @ (sub @ X137 @ X136) @ X135)) | ($true = (X129 @ X138 @ X137 @ (comp @ X136 @ X135)))) | ~! [X139 : term] : ((X129 @ id @ X139 @ id) = (X128 @ X139)) | ~! [X140 : term,X141 : term] : (($true = (X129 @ id @ (ap @ (sub @ X140 @ id) @ X141) @ id)) | ~($true = (X129 @ id @ X140 @ id)) | ~($true = (X129 @ id @ X141 @ id))))) & (pushprop_lthm = $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : (($true = (X144 @ one)) | ~! [X146 : term] : ((X144 @ X146) = (X142 @ (sub @ X146 @ (push @ X145 @ X143)))) | ~($true = (X142 @ X145)))) & (! [X147 : term] : ~! [X148 : subst > term > term] : ~! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) = (d2term @ X152)) => (X151 = X152)) & (pushprop_lthm_orig = $true) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 = X155) | ((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153))) <=> (hoasapinj2 = $true)) & (hoasinduction_lem1v2 = $true) & (hoaslamnotvar_gthm = $true) & ((! [X157 : term,X158 : subst] : ((sub @ one @ (push @ X157 @ X158)) = X157) => (! [X159 : subst,X160 : term] : ((comp @ sh @ (push @ X160 @ X159)) = X159) => (! [X161 : term,X162 : term] : ((X161 = X162) | ((lam @ X161) != (lam @ X162))) => ! [X163 : subst,X164 : subst > term > term,X165 : term,X166 : subst > term > term] : (~! [X167 : subst,X168 : term,X169 : subst] : ((sub @ (X164 @ X167 @ X168) @ X169) = (X164 @ (comp @ X167 @ X169) @ (sub @ X168 @ X169))) | ~! [X170 : term,X171 : subst,X172 : subst] : ((sub @ (X166 @ X172 @ X170) @ X171) = (X166 @ (comp @ X172 @ X171) @ (sub @ X170 @ X171))) | ((X164 @ X163 @ X165) = (X166 @ X163 @ X165)) | ((lam @ (X166 @ sh @ one)) != (lam @ (X164 @ sh @ one))))))) <=> (hoaslaminj_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem1 = $true) <=> ! [X175 : term > $o,X176 : term,X177 : subst,X178 : term,X179 : term > $o] : (($true = (X179 @ (sub @ X176 @ (push @ X178 @ X177)))) | ~($true = (X175 @ X176)))) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X180 : term > $o,X181 : subst > term > subst > $o,X182 : term] : (~! [X183 : term] : ((X180 @ X183) = (X181 @ id @ X183 @ id)) | ~! [X184 : subst > term > term] : (($true = (X181 @ id @ (lam @ (X184 @ sh @ one)) @ id)) | ~! [X185 : term] : (($true = (X181 @ id @ (X184 @ id @ X185) @ id)) | ~($true = (X181 @ id @ X185 @ id))) | ~! [X186 : subst,X187 : subst,X188 : term] : ((sub @ (X184 @ X187 @ X188) @ X186) = (X184 @ (comp @ X187 @ X186) @ (sub @ X188 @ X186)))) | ~! [X189 : term] : (((X180 @ (sub @ X182 @ (push @ X189 @ id))) = $true) | ~($true = (X180 @ X189))) | ~! [X190 : term,X191 : subst,X192 : subst,X193 : subst] : (($true = (X181 @ X191 @ X190 @ (comp @ X192 @ X193))) | ~($true = (X181 @ (comp @ X191 @ X192) @ (sub @ X190 @ X192) @ X193))) | ~! [X194 : subst,X195 : subst,X196 : term,X197 : subst] : (($true = (X181 @ (comp @ X194 @ X195) @ (sub @ X196 @ X195) @ X197)) | ~($true = (X181 @ X194 @ X196 @ (comp @ X195 @ X197)))) | ($true = (X180 @ (lam @ X182))))) & ((laminj = $true) <=> ! [X198 : term,X199 : term] : (((lam @ X199) != (lam @ X198)) | (X198 = X199))) & ! [X200 : term] : ? [X201 : d_term] : ((d2term @ X201) = X200) & (pushprop_lem3v2 = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X202 : term] : ~! [X203 : subst > term > term] : ((sub @ X202 @ (push @ one @ sh)) != (X203 @ sh @ one)) <=> (hoasinduction_lem3b_lthm = $true)) & (hoasapnotvar_gthm = $true) & (axmap = $true) & (axclos = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoaslamnotvar = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X204 : subst,X205 : term] : ((sub @ one @ (push @ X205 @ X204)) = X205) => (! [X206 : subst,X207 : term] : ((comp @ sh @ (push @ X207 @ X206)) = X206) => ! [X208 : term,X209 : term > $o,X210 : term > $o,X211 : subst,X212 : term] : (~($true = (X209 @ X212)) | ($true = (X210 @ (sub @ X212 @ (push @ X208 @ X211)))))))) & ((hoasapinj2_lthm = $true) <=> (! [X213 : term,X214 : term,X215 : term,X216 : term] : ((X214 = X216) | ((ap @ X215 @ X214) != (ap @ X213 @ X216))) => ! [X217 : term,X218 : term,X219 : term,X220 : term] : ((X219 = X220) | ((ap @ (sub @ X218 @ id) @ X220) != (ap @ (sub @ X217 @ id) @ X219))))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem3aa = $true) <=> ! [X221 : subst > term > subst > $o,X222 : term] : (~! [X223 : subst > term > term] : (~! [X224 : term] : (~($true = (X221 @ id @ X224 @ id)) | ($true = (X221 @ id @ (X223 @ id @ X224) @ id))) | ((X221 @ id @ (lam @ (X223 @ sh @ one)) @ id) = $true) | ~! [X225 : term,X226 : subst,X227 : subst] : ((X223 @ (comp @ X227 @ X226) @ (sub @ X225 @ X226)) = (sub @ (X223 @ X227 @ X225) @ X226))) | ~! [X228 : term] : (($true = (X221 @ id @ (sub @ X222 @ (push @ X228 @ id)) @ id)) | ~($true = (X221 @ id @ X228 @ id))) | ($true = (X221 @ id @ (lam @ (sub @ X222 @ (push @ one @ sh))) @ id)))) & ((d2subst @ d_id) = id) & ((hoasinduction_lthm_3 = $true) <=> (! [X229 : subst > term > subst > $o] : ~! [X230 : term > $o] : ~! [X231 : term] : ((X229 @ id @ X231 @ id) = (X230 @ X231)) => (! [X232 : term > $o,X233 : term] : (~! [X234 : term] : (($true = (X232 @ X234)) | ~($true = (var @ X234))) | ($true = (X232 @ X233)) | ~! [X235 : term] : (~! [X236 : term] : (($true = (X232 @ (sub @ X235 @ (push @ X236 @ id)))) | ~($true = (X232 @ X236))) | ($true = (X232 @ (lam @ X235)))) | ~! [X237 : term,X238 : term] : (($true = (X232 @ (ap @ X237 @ X238))) | ~($true = (X232 @ X238)) | ~($true = (X232 @ X237)))) => (! [X239 : term] : ((sub @ X239 @ id) = X239) => (! [X240 : term,X241 : term > $o,X242 : subst > term > subst > $o] : (~! [X243 : term] : (~($true = (X241 @ X243)) | ($true = (X241 @ (sub @ X240 @ (push @ X243 @ id))))) | ~! [X244 : subst > term > term] : (($true = (X242 @ id @ (lam @ (X244 @ sh @ one)) @ id)) | ~! [X245 : term] : (~($true = (X242 @ id @ X245 @ id)) | ($true = (X242 @ id @ (X244 @ id @ X245) @ id))) | ~! [X246 : subst,X247 : term,X248 : subst] : ((sub @ (X244 @ X248 @ X247) @ X246) = (X244 @ (comp @ X248 @ X246) @ (sub @ X247 @ X246)))) | ~! [X249 : term] : ((X241 @ X249) = (X242 @ id @ X249 @ id)) | ($true = (X241 @ (lam @ X240)))) => ! [X250 : term,X251 : subst > term > subst > $o] : (~! [X252 : term,X253 : term] : (~($true = (X251 @ id @ X253 @ id)) | ~($true = (X251 @ id @ X252 @ id)) | ($true = (X251 @ id @ (ap @ (sub @ X253 @ id) @ X252) @ id))) | ~! [X254 : subst,X255 : term,X256 : subst,X257 : subst] : (~($true = (X251 @ X254 @ X255 @ (comp @ X256 @ X257))) | ($true = (X251 @ (comp @ X254 @ X256) @ (sub @ X255 @ X256) @ X257))) | ~! [X258 : subst > term > term] : (~! [X259 : term] : (~($true = (X251 @ id @ X259 @ id)) | ($true = (X251 @ id @ (X258 @ id @ X259) @ id))) | ~! [X260 : subst,X261 : term,X262 : subst] : ((sub @ (X258 @ X262 @ X261) @ X260) = (X258 @ (comp @ X262 @ X260) @ (sub @ X261 @ X260))) | ((X251 @ id @ (lam @ (X258 @ sh @ one)) @ id) = $true)) | ($true = (X251 @ id @ X250 @ id)) | ~! [X263 : term,X264 : subst,X265 : subst,X266 : subst] : (~($true = (X251 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265) @ X266)) | ($true = (X251 @ X264 @ X263 @ (comp @ X265 @ X266)))) | ~! [X267 : term] : (~($true = (var @ (sub @ X267 @ id))) | ($true = (X251 @ id @ X267 @ id))))))))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X271 : subst] : ? [X272 : d_subst] : ((d2subst @ X272) = X271) & ((! [X273 : term] : ((sub @ X273 @ id) = X273) => (! [X274 : term,X275 : subst > term > subst > $o] : (~! [X276 : subst > term > term] : (~! [X277 : subst,X278 : term,X279 : subst] : ((sub @ (X276 @ X277 @ X278) @ X279) = (X276 @ (comp @ X277 @ X279) @ (sub @ X278 @ X279))) | ($true = (X275 @ id @ (lam @ (X276 @ sh @ one)) @ id)) | ~! [X280 : term] : (~($true = (X275 @ id @ X280 @ id)) | ($true = (X275 @ id @ (X276 @ id @ X280) @ id)))) | ((X275 @ id @ (lam @ (sub @ X274 @ (push @ one @ sh))) @ id) = $true) | ~! [X281 : term] : (($true = (X275 @ id @ (sub @ X274 @ (push @ X281 @ id)) @ id)) | ~($true = (X275 @ id @ X281 @ id)))) => ! [X282 : term,X283 : subst > term > subst > $o] : (~! [X284 : subst > term > term] : (($true = (X283 @ id @ (lam @ (X284 @ sh @ one)) @ id)) | ~! [X285 : subst,X286 : term,X287 : subst] : ((sub @ (X284 @ X287 @ X286) @ X285) = (X284 @ (comp @ X287 @ X285) @ (sub @ X286 @ X285))) | ~! [X288 : term] : (($true = (X283 @ id @ (X284 @ id @ X288) @ id)) | ~($true = (X283 @ id @ X288 @ id)))) | ~! [X289 : term] : (($true = (X283 @ id @ (sub @ X282 @ (push @ X289 @ id)) @ id)) | ~($true = (X283 @ id @ X289 @ id))) | ($true = (X283 @ id @ (lam @ X282) @ id))))) <=> (hoasinduction_lem3a_lthm = $true)) & (pushprop_gthm = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X290 : term] : ((sub @ X290 @ id) = X290) => (! [X291 : term,X292 : subst > term > subst > $o] : (~! [X293 : subst > term > term] : (~! [X294 : term] : (($true = (X292 @ id @ (X293 @ id @ X294) @ id)) | ~($true = (X292 @ id @ X294 @ id))) | ~! [X295 : subst,X296 : subst,X297 : term] : ((X293 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)) = (sub @ (X293 @ X296 @ X297) @ X295)) | ($true = (X292 @ id @ (lam @ (X293 @ sh @ one)) @ id))) | ($true = (X292 @ id @ (lam @ (sub @ X291 @ (push @ one @ sh))) @ id)) | ~! [X298 : term] : (($true = (X292 @ id @ (sub @ X291 @ (push @ X298 @ id)) @ id)) | ~($true = (X292 @ id @ X298 @ id)))) => ! [X299 : term,X300 : subst > term > subst > $o] : (($true = (X300 @ id @ (lam @ X299) @ id)) | ~! [X301 : subst,X302 : subst,X303 : subst,X304 : term] : (~($true = (X300 @ (comp @ X303 @ X301) @ (sub @ X304 @ X301) @ X302)) | ($true = (X300 @ X303 @ X304 @ (comp @ X301 @ X302)))) | ~! [X305 : term] : (((X300 @ id @ (sub @ X299 @ (push @ X305 @ id)) @ id) = $true) | ~($true = (X300 @ id @ X305 @ id))) | ~! [X306 : subst > term > term] : (~! [X307 : term,X308 : subst,X309 : subst] : ((X306 @ (comp @ X308 @ X309) @ (sub @ X307 @ X309)) = (sub @ (X306 @ X308 @ X307) @ X309)) | ~! [X310 : term] : (~($true = (X300 @ id @ X310 @ id)) | ($true = (X300 @ id @ (X306 @ id @ X310) @ id))) | ($true = (X300 @ id @ (lam @ (X306 @ sh @ one)) @ id))) | ~! [X311 : term,X312 : subst,X313 : subst,X314 : subst] : (~($true = (X300 @ X313 @ X311 @ (comp @ X314 @ X312))) | ((X300 @ (comp @ X313 @ X314) @ (sub @ X311 @ X314) @ X312) = $true)))))) & (termmset_gthm = $true) & (axassoc = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (! [X315 : subst > term > subst > $o,X316 : term] : (~! [X317 : term] : (~($true = (X315 @ id @ X317 @ id)) | ($true = (X315 @ id @ (sub @ X316 @ (push @ X317 @ id)) @ id))) | ~! [X318 : subst > term > term] : (($true = (X315 @ id @ (lam @ (X318 @ sh @ one)) @ id)) | ~! [X319 : subst,X320 : term,X321 : subst] : ((sub @ (X318 @ X319 @ X320) @ X321) = (X318 @ (comp @ X319 @ X321) @ (sub @ X320 @ X321))) | ~! [X322 : term] : (~($true = (X315 @ id @ X322 @ id)) | ($true = (X315 @ id @ (X318 @ id @ X322) @ id)))) | ($true = (X315 @ id @ (lam @ X316) @ id))) <=> (hoasinduction_lem3a = $true)) & (hoasinduction_lem2v2_gthm = $true) & (! [X323 : term] : ((sub @ X323 @ id) = X323) <=> (axvarid = $true)) & (ulamvarind = $true) & ((induction2 = $true) <=> ! [X324 : term,X325 : term > $o] : (~! [X326 : term] : (($true = (X325 @ X326)) | ~($true = (var @ X326))) | ~! [X327 : term,X328 : term] : (~($true = (X325 @ X327)) | ($true = (X325 @ (ap @ X328 @ X327))) | ~($true = (X325 @ X328))) | ($true = (X325 @ X324)) | ~! [X329 : term] : (~! [X330 : term] : (($true = (X325 @ (sub @ X329 @ (push @ X330 @ id)))) | ~($true = (X325 @ X330))) | ($true = (X325 @ (lam @ X329)))))) & (axabs = $true) & ((! [X331 : subst > term > subst > $o] : ~! [X332 : term > $o] : ~! [X333 : term] : ((X331 @ id @ X333 @ id) = (X332 @ X333)) => (! [X334 : term > $o,X335 : term] : (~! [X336 : term] : (~($true = (var @ X336)) | ($true = (X334 @ X336))) | ($true = (X334 @ X335)) | ~! [X337 : term] : (~! [X338 : term] : (~($true = (X334 @ X338)) | ($true = (X334 @ (sub @ X337 @ (push @ X338 @ id))))) | ($true = (X334 @ (lam @ X337)))) | ~! [X339 : term,X340 : term] : (~($true = (X334 @ X340)) | ~($true = (X334 @ X339)) | ($true = (X334 @ (ap @ X340 @ X339))))) => (! [X341 : term] : ((sub @ X341 @ id) = X341) => (! [X342 : subst > term > subst > $o,X343 : term,X344 : term > $o] : (~! [X345 : term] : (($true = (X344 @ (sub @ X343 @ (push @ X345 @ id)))) | ~($true = (X344 @ X345))) | ~! [X346 : subst > term > term] : (~! [X347 : term] : (~($true = (X342 @ id @ X347 @ id)) | ($true = (X342 @ id @ (X346 @ id @ X347) @ id))) | ~! [X348 : subst,X349 : term,X350 : subst] : ((sub @ (X346 @ X350 @ X349) @ X348) = (X346 @ (comp @ X350 @ X348) @ (sub @ X349 @ X348))) | ($true = (X342 @ id @ (lam @ (X346 @ sh @ one)) @ id))) | ($true = (X344 @ (lam @ X343))) | ~! [X351 : term] : ((X342 @ id @ X351 @ id) = (X344 @ X351))) => ! [X352 : subst > term > subst > $o,X353 : term] : (~! [X354 : term,X355 : term] : (~($true = (X352 @ id @ X355 @ id)) | ~($true = (X352 @ id @ X354 @ id)) | ($true = (X352 @ id @ (ap @ (sub @ X354 @ id) @ X355) @ id))) | ~! [X356 : subst > term > term] : (($true = (X352 @ id @ (lam @ (X356 @ sh @ one)) @ id)) | ~! [X357 : subst,X358 : subst,X359 : term] : ((X356 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357)) = (sub @ (X356 @ X358 @ X359) @ X357)) | ~! [X360 : term] : (($true = (X352 @ id @ (X356 @ id @ X360) @ id)) | ~($true = (X352 @ id @ X360 @ id)))) | ($true = (X352 @ id @ X353 @ id))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop = $true) & (! [X361 : term > $o,X362 : subst,X363 : term] : (~! [X364 : term] : (~($true = (var @ X364)) | ((X361 @ (sub @ X364 @ X362)) = $true)) | ($true = (X361 @ (sub @ X363 @ X362))) | ~! [X365 : term] : (($true = (X361 @ (lam @ X365))) | ~! [X366 : term] : (~((X361 @ X366) = $true) | ($true = (X361 @ (sub @ X365 @ (push @ X366 @ id)))))) | ~! [X367 : term,X368 : term] : (($true = (X361 @ (ap @ X367 @ X368))) | ~($true = (X361 @ X367)) | ~($true = (X361 @ X368)))) <=> (induction2lem = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X369 : term] : ~! [X370 : subst > term > term] : ~! [X371 : subst,X372 : term] : ((X370 @ X371 @ X372) = (sub @ X369 @ (push @ X372 @ X371))) => (! [X373 : term] : ((sub @ X373 @ id) = X373) => ! [X374 : term > $o,X375 : subst > term > subst > $o,X376 : term] : (~! [X377 : term] : ((X374 @ X377) = (X375 @ id @ X377 @ id)) | ~! [X378 : term] : (($true = (X374 @ (sub @ X376 @ (push @ X378 @ id)))) | ~($true = (X374 @ X378))) | ($true = (X374 @ (lam @ X376))) | ~! [X379 : subst > term > term] : (($true = (X375 @ id @ (lam @ (X379 @ sh @ one)) @ id)) | ~! [X380 : term] : (($true = (X375 @ id @ (X379 @ id @ X380) @ id)) | ~($true = (X375 @ id @ X380 @ id))) | ~! [X381 : subst,X382 : term,X383 : subst] : ((sub @ (X379 @ X381 @ X382) @ X383) = (X379 @ (comp @ X381 @ X383) @ (sub @ X382 @ X383)))))))) & (induction2_gthm = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X384 : term,X385 : term > $o] : (~! [X386 : term,X387 : term] : (~($true = (X385 @ X386)) | ($true = (X385 @ (ap @ X386 @ X387))) | ~($true = (X385 @ X387))) | ($true = (X385 @ X384)) | ~! [X388 : term] : (($true = (X385 @ (lam @ X388))) | ~! [X389 : term] : (($true = (X385 @ (sub @ X388 @ (push @ X389 @ id)))) | ~($true = (X385 @ X389)))) | ~! [X390 : term] : (($true = (X385 @ X390)) | ~($true = (var @ X390)))) => (! [X391 : term,X392 : term,X393 : subst > term > subst > $o] : (((X393 @ id @ (ap @ X391 @ X392) @ id) = $true) | ~($true = (X393 @ id @ X392 @ id)) | ~($true = (X393 @ id @ X391 @ id)) | ~! [X394 : subst,X395 : subst,X396 : term,X397 : subst] : (~($true = (X393 @ X395 @ X396 @ (comp @ X394 @ X397))) | ($true = (X393 @ (comp @ X395 @ X394) @ (sub @ X396 @ X394) @ X397))) | ~! [X398 : term,X399 : term] : (($true = (X393 @ id @ (ap @ (sub @ X399 @ id) @ X398) @ id)) | ~($true = (X393 @ id @ X399 @ id)) | ~($true = (X393 @ id @ X398 @ id))) | ~! [X400 : subst,X401 : subst,X402 : subst,X403 : term] : (~($true = (X393 @ (comp @ X401 @ X402) @ (sub @ X403 @ X402) @ X400)) | ($true = (X393 @ X401 @ X403 @ (comp @ X402 @ X400))))) => (! [X404 : subst > term > subst > $o,X405 : term] : (($true = (X404 @ id @ (lam @ X405) @ id)) | ~! [X406 : term] : (($true = (X404 @ id @ (sub @ X405 @ (push @ X406 @ id)) @ id)) | ~($true = (X404 @ id @ X406 @ id))) | ~! [X407 : subst,X408 : term,X409 : subst,X410 : subst] : (($true = (X404 @ X409 @ X408 @ (comp @ X407 @ X410))) | ~($true = (X404 @ (comp @ X409 @ X407) @ (sub @ X408 @ X407) @ X410))) | ~! [X411 : subst,X412 : term,X413 : subst,X414 : subst] : (~((X404 @ X411 @ X412 @ (comp @ X414 @ X413)) = $true) | ($true = (X404 @ (comp @ X411 @ X414) @ (sub @ X412 @ X414) @ X413))) | ~! [X415 : subst > term > term] : (~! [X416 : term,X417 : subst,X418 : subst] : ((sub @ (X415 @ X417 @ X416) @ X418) = (X415 @ (comp @ X417 @ X418) @ (sub @ X416 @ X418))) | ($true = (X404 @ id @ (lam @ (X415 @ sh @ one)) @ id)) | ~! [X419 : term] : (~($true = (X404 @ id @ X419 @ id)) | ($true = (X404 @ id @ (X415 @ id @ X419) @ id))))) => ! [X420 : subst > term > subst > $o,X421 : term] : (~! [X422 : term,X423 : term] : (($true = (X420 @ id @ (ap @ (sub @ X423 @ id) @ X422) @ id)) | ~($true = (X420 @ id @ X422 @ id)) | ~($true = (X420 @ id @ X423 @ id))) | ~! [X424 : subst > term > term] : (($true = (X420 @ id @ (lam @ (X424 @ sh @ one)) @ id)) | ~! [X425 : term] : (($true = (X420 @ id @ (X424 @ id @ X425) @ id)) | ~($true = (X420 @ id @ X425 @ id))) | ~! [X426 : term,X427 : subst,X428 : subst] : ((X424 @ (comp @ X427 @ X428) @ (sub @ X426 @ X428)) = (sub @ (X424 @ X427 @ X426) @ X428))) | ($true = (X420 @ id @ X421 @ id)) | ~! [X429 : subst,X430 : term,X431 : subst,X432 : subst] : (($true = (X420 @ (comp @ X432 @ X429) @ (sub @ X430 @ X429) @ X431)) | ~($true = (X420 @ X432 @ X430 @ (comp @ X429 @ X431)))) | ~! [X433 : term,X434 : subst,X435 : subst,X436 : subst] : (($true = (X420 @ X435 @ X433 @ (comp @ X434 @ X436))) | ~($true = (X420 @ (comp @ X435 @ X434) @ (sub @ X433 @ X434) @ X436))) | ~! [X437 : term] : (($true = (X420 @ id @ X437 @ id)) | ~($true = (var @ (sub @ X437 @ id))))))))) & ((hoasapinj1 = $true) <=> ! [X438 : term,X439 : term,X440 : term,X441 : term] : ((X440 = X441) | ((ap @ (sub @ X441 @ id) @ X439) != (ap @ (sub @ X440 @ id) @ X438)))) & (pushprop_lem1_gthm = $true) & (apnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (lamnotvar = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X447 : subst > term > subst > $o,X448 : term] : (~! [X449 : subst > term > term] : (~! [X450 : term] : (($true = (X447 @ id @ (X449 @ id @ X450) @ id)) | ~($true = (X447 @ id @ X450 @ id))) | ($true = (X447 @ id @ (lam @ (X449 @ sh @ one)) @ id)) | ~! [X451 : subst,X452 : term,X453 : subst] : ((sub @ (X449 @ X451 @ X452) @ X453) = (X449 @ (comp @ X451 @ X453) @ (sub @ X452 @ X453)))) | ($true = (X447 @ id @ (lam @ (sub @ X448 @ (push @ one @ sh))) @ id)) | ~! [X454 : term] : (($true = (X447 @ id @ (sub @ X448 @ (push @ X454 @ id)) @ id)) | ~($true = (X447 @ id @ X454 @ id))))) & (! [X455 : subst > term > subst > $o] : ~! [X456 : term > $o] : ~! [X457 : term] : ((X455 @ id @ X457 @ id) = (X456 @ X457)) <=> (hoasinduction_lem0_lthm = $true)) & (hoasinduction_lem3_gthm = $true) & ((! [X458 : subst] : ((comp @ id @ X458) = X458) & ! [X459 : subst] : ((comp @ X459 @ id) = X459)) <=> (substmonoid = $true)) & ((induction2_lthm = $true) <=> (! [X460 : term] : ((sub @ X460 @ id) = X460) => (! [X461 : subst,X462 : term,X463 : term > $o] : (~! [X464 : term] : (~($true = (var @ X464)) | ($true = (X463 @ (sub @ X464 @ X461)))) | ~! [X465 : term,X466 : term] : (~($true = (X463 @ X466)) | ($true = (X463 @ (ap @ X465 @ X466))) | ~($true = (X463 @ X465))) | ~! [X467 : term] : (~! [X468 : term] : (($true = (X463 @ (sub @ X467 @ (push @ X468 @ id)))) | ~($true = (X463 @ X468))) | ((X463 @ (lam @ X467)) = $true)) | ($true = (X463 @ (sub @ X462 @ X461)))) => ! [X469 : term > $o,X470 : term] : (~! [X471 : term] : (~($true = (var @ X471)) | ($true = (X469 @ X471))) | ~! [X472 : term,X473 : term] : (~($true = (X469 @ X472)) | ~($true = (X469 @ X473)) | ($true = (X469 @ (ap @ X472 @ X473)))) | ($true = (X469 @ X470)) | ~! [X474 : term] : (((X469 @ (lam @ X474)) = $true) | ~! [X475 : term] : (($true = (X469 @ (sub @ X474 @ (push @ X475 @ id)))) | ~($true = (X469 @ X475)))))))) & ~((var @ (d2term @ d_one)) = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X476 : term,X477 : term,X478 : subst > term > subst > $o] : (($true = (X478 @ id @ (ap @ X477 @ X476) @ id)) | ~($true = (X478 @ id @ X477 @ id)) | ~! [X479 : term,X480 : term] : (~($true = (X478 @ id @ X480 @ id)) | ($true = (X478 @ id @ (ap @ (sub @ X480 @ id) @ X479) @ id)) | ~($true = (X478 @ id @ X479 @ id))) | ~! [X481 : subst,X482 : subst,X483 : subst,X484 : term] : (~($true = (X478 @ X483 @ X484 @ (comp @ X482 @ X481))) | ($true = (X478 @ (comp @ X483 @ X482) @ (sub @ X484 @ X482) @ X481))) | ~! [X485 : subst,X486 : subst,X487 : subst,X488 : term] : (($true = (X478 @ X486 @ X488 @ (comp @ X485 @ X487))) | ~($true = (X478 @ (comp @ X486 @ X485) @ (sub @ X488 @ X485) @ X487))) | ~($true = (X478 @ id @ X476 @ id)))) & (! [X489 : term > $o,X490 : term] : (~! [X491 : term] : (($true = (X489 @ X491)) | ~($true = (var @ X491))) | ~! [X492 : term] : (~($true = (X489 @ X492)) | ($true = (X489 @ (lam @ X492)))) | ($true = (X489 @ X490)) | ~! [X493 : term,X494 : term] : (~($true = (X489 @ X494)) | ($true = (X489 @ (ap @ X493 @ X494))) | ~($true = (X489 @ X493)))) <=> (induction = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X495 : d_term] : (d_one = X495) & ((hoasapinj1_lthm = $true) <=> (! [X496 : term] : ((sub @ X496 @ id) = X496) => (! [X497 : term,X498 : term,X499 : term,X500 : term] : (((ap @ X500 @ X499) != (ap @ X497 @ X498)) | (X497 = X500)) => ! [X501 : term,X502 : term,X503 : term,X504 : term] : (((ap @ (sub @ X503 @ id) @ X502) != (ap @ (sub @ X501 @ id) @ X504)) | (X501 = X503))))) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axvarshift = $true) & ((! [X505 : term] : ((sub @ X505 @ id) = X505) => ! [X506 : term > $o,X507 : subst > term > subst > $o,X508 : term] : (~! [X509 : subst,X510 : term,X511 : subst,X512 : subst] : (($true = (X507 @ (comp @ X511 @ X509) @ (sub @ X510 @ X509) @ X512)) | ~($true = (X507 @ X511 @ X510 @ (comp @ X509 @ X512)))) | ~! [X513 : term] : ((X506 @ X513) = (X507 @ id @ X513 @ id)) | ~! [X514 : term] : (($true = (X506 @ (sub @ X508 @ (push @ X514 @ id)))) | ~($true = (X506 @ X514))) | ~! [X515 : subst,X516 : term,X517 : subst,X518 : subst] : (~($true = (X507 @ (comp @ X518 @ X517) @ (sub @ X516 @ X517) @ X515)) | ((X507 @ X518 @ X516 @ (comp @ X517 @ X515)) = $true)) | ((X506 @ (lam @ X508)) = $true) | ~! [X519 : subst > term > term] : (((X507 @ id @ (lam @ (X519 @ sh @ one)) @ id) = $true) | ~! [X520 : subst,X521 : term,X522 : subst] : ((X519 @ (comp @ X522 @ X520) @ (sub @ X521 @ X520)) = (sub @ (X519 @ X522 @ X521) @ X520)) | ~! [X523 : term] : (~($true = (X507 @ id @ X523 @ id)) | ($true = (X507 @ id @ (X519 @ id @ X523) @ id)))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((shinj = $true) <=> ! [X524 : term,X525 : term] : (((sub @ X525 @ sh) != (sub @ X524 @ sh)) | (X524 = X525))) & ((hoasinduction_lem3 = $true) <=> ! [X526 : term,X527 : subst > term > subst > $o] : (~! [X528 : subst,X529 : term,X530 : subst,X531 : subst] : (($true = (X527 @ (comp @ X531 @ X528) @ (sub @ X529 @ X528) @ X530)) | ~((X527 @ X531 @ X529 @ (comp @ X528 @ X530)) = $true)) | ~! [X532 : subst > term > term] : (($true = (X527 @ id @ (lam @ (X532 @ sh @ one)) @ id)) | ~! [X533 : term] : (($true = (X527 @ id @ (X532 @ id @ X533) @ id)) | ~($true = (X527 @ id @ X533 @ id))) | ~! [X534 : term,X535 : subst,X536 : subst] : ((X532 @ (comp @ X535 @ X536) @ (sub @ X534 @ X536)) = (sub @ (X532 @ X535 @ X534) @ X536))) | ($true = (X527 @ id @ (lam @ X526) @ id)) | ~! [X537 : term] : (((X527 @ id @ (sub @ X526 @ (push @ X537 @ id)) @ id) = $true) | ~($true = (X527 @ id @ X537 @ id))) | ~! [X538 : subst,X539 : subst,X540 : subst,X541 : term] : (~($true = (X527 @ (comp @ X539 @ X540) @ (sub @ X541 @ X540) @ X538)) | ($true = (X527 @ X539 @ X541 @ (comp @ X540 @ X538)))))) & (substmonoid_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X542 : term,X543 : term > $o,X544 : subst] : ~! [X545 : term > $o] : ~! [X546 : term] : ((X543 @ (sub @ X546 @ (push @ X542 @ X544))) = (X545 @ X546)) <=> (pushprop_lem0_lthm = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X547 : term] : ~! [X548 : subst > term > term] : ((X548 @ sh @ one) != (sub @ X547 @ (push @ one @ sh)))) & (pushprop_lem2v2_gthm = $true) & (! [X549 : subst] : ((push @ (sub @ one @ X549) @ (comp @ sh @ X549)) = X549) <=> (axscons = $true)) & (ulamvarsh = $true) & (hoasinduction_lem2_lthm = $true)),
% 2.56/1.94 inference(fool_elimination,[status(thm)],[f5])).
% 2.56/1.94 thf(f5,plain,(
% 2.56/1.94 (! [X0 : term,X1 : term,X2 : term,X3 : term] : ((X0 = X1) | ((ap @ X2 @ X0) != (ap @ X3 @ X1))) <=> apinj2) & (! [X4 : term,X5 : subst > term > subst > $o] : ((X5 @ id @ X4 @ id) | ~! [X6 : subst > term > term] : (~! [X7 : term] : (~(X5 @ id @ X7 @ id) | (X5 @ id @ (X6 @ id @ X7) @ id)) | ~! [X8 : term,X9 : subst,X10 : subst] : ((X6 @ (comp @ X10 @ X9) @ (sub @ X8 @ X9)) = (sub @ (X6 @ X10 @ X8) @ X9)) | (X5 @ id @ (lam @ (X6 @ sh @ one)) @ id)) | ~! [X11 : term,X12 : term] : (~(X5 @ id @ X12 @ id) | (X5 @ id @ (ap @ (sub @ X11 @ id) @ X12) @ id) | ~(X5 @ id @ X11 @ id))) <=> hoasinduction_no_psi_cond) & ! [X13 : d_subst,X14 : d_subst] : (((d2subst @ X13) = (d2subst @ X14)) => (X13 = X14)) & (one = (d2term @ d_one)) & (hoasinduction_lem3v2a <=> ! [X15 : subst > term > subst > $o,X16 : term > $o,X17 : term] : (~! [X18 : term] : ((X16 @ X18) = (X15 @ id @ X18 @ id)) | ~! [X19 : subst > term > term] : (~! [X20 : term,X21 : subst,X22 : subst] : ((sub @ (X19 @ X22 @ X20) @ X21) = (X19 @ (comp @ X22 @ X21) @ (sub @ X20 @ X21))) | (X15 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~! [X23 : term] : ((X15 @ id @ (X19 @ id @ X23) @ id) | ~(X15 @ id @ X23 @ id))) | ~! [X24 : term] : ((X16 @ (sub @ X17 @ (push @ X24 @ id))) | ~(X16 @ X24)) | (X16 @ (lam @ X17)))) & hoasinduction_lem3a_gthm & (termmset_lthm <=> (! [X25 : term] : ((sub @ X25 @ id) = X25) => ! [X26 : term] : ((sub @ X26 @ id) = X26))) & (! [X27 : subst,X28 : term,X29 : term > $o] : ~! [X30 : term > $o] : ~! [X31 : term] : ((X29 @ (sub @ X31 @ (push @ X28 @ X27))) = (X30 @ X31)) <=> pushprop_lem0) & (! [X32 : term] : ((sub @ X32 @ id) = X32) <=> termmset) & hoasinduction_lem1 & (substmonoid_lthm <=> (! [X33 : subst] : ((comp @ id @ X33) = X33) => (! [X34 : subst] : ((comp @ X34 @ id) = X34) => (! [X35 : subst] : ((comp @ X35 @ id) = X35) & ! [X36 : subst] : ((comp @ id @ X36) = X36))))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (pushprop_lem1v2_lthm <=> (! [X37 : term,X38 : subst] : ((sub @ one @ (push @ X37 @ X38)) = X37) => ! [X39 : term,X40 : term > $o,X41 : term > $o,X42 : subst] : (~! [X43 : term] : ((X41 @ X43) = (X40 @ (sub @ X43 @ (push @ X39 @ X42)))) | ~(X40 @ X39) | (X41 @ one)))) & pushprop_lem2v2 & ~lamnotap & (hoasinduction_lem3aaa <=> ! [X44 : term,X45 : subst > term > subst > $o] : ((X45 @ id @ (lam @ (sub @ X44 @ (push @ one @ sh))) @ id) | ~! [X46 : term] : ((X45 @ id @ (sub @ X44 @ (push @ X46 @ id)) @ id) | ~(X45 @ id @ X46 @ id)) | ~! [X47 : subst > term > term,X48 : term] : (~! [X49 : subst,X50 : term,X51 : subst] : ((sub @ (sub @ X48 @ (push @ X50 @ X49)) @ X51) = (sub @ (X47 @ X49 @ X50) @ X51)) | ~! [X52 : subst,X53 : term,X54 : subst] : ((sub @ X48 @ (push @ (sub @ X53 @ X52) @ (comp @ X54 @ X52))) = (X47 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) | (X45 @ id @ (lam @ (X47 @ sh @ one)) @ id) | ~! [X55 : term] : ((X45 @ id @ (X47 @ id @ X55) @ id) | ~(X45 @ id @ X55 @ id))))) & (axidl <=> ! [X56 : subst] : ((comp @ id @ X56) = X56)) & hoasinduction_gthm & (! [X57 : term,X58 : subst] : ((comp @ sh @ (push @ X57 @ X58)) = X58) <=> axshiftcons) & axapp & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (hoasinduction <=> ! [X59 : term,X60 : subst > term > subst > $o] : (~! [X61 : term,X62 : term] : ((X60 @ id @ (ap @ (sub @ X61 @ id) @ X62) @ id) | ~(X60 @ id @ X61 @ id) | ~(X60 @ id @ X62 @ id)) | ~! [X63 : subst > term > term] : (~! [X64 : term] : (~(X60 @ id @ X64 @ id) | (X60 @ id @ (X63 @ id @ X64) @ id)) | (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id) | ~! [X65 : subst,X66 : subst,X67 : term] : ((X63 @ (comp @ X65 @ X66) @ (sub @ X67 @ X66)) = (sub @ (X63 @ X65 @ X67) @ X66))) | ~! [X68 : term] : (~(var @ (sub @ X68 @ id)) | (X60 @ id @ X68 @ id)) | ~! [X69 : subst,X70 : subst,X71 : term,X72 : subst] : (~(X60 @ (comp @ X69 @ X72) @ (sub @ X71 @ X72) @ X70) | (X60 @ X69 @ X71 @ (comp @ X72 @ X70))) | ~! [X73 : term,X74 : subst,X75 : subst,X76 : subst] : ((X60 @ (comp @ X75 @ X74) @ (sub @ X73 @ X74) @ X76) | ~(X60 @ X75 @ X73 @ (comp @ X74 @ X76))) | (X60 @ id @ X59 @ id))) & ~ulamvar1 & ((d2subst @ d_id) = sh) & hoaslamnotvar_lthm & hoaslamnotap_lthm & (hoasinduction_lem0 <=> ! [X77 : subst > term > subst > $o] : ~! [X78 : term > $o] : ~! [X79 : term] : ((X78 @ X79) = (X77 @ id @ X79 @ id))) & hoasapinj1_gthm & (hoasinduction_lem3v2_f <=> ! [X80 : term] : ~! [X81 : subst > term > term] : ~! [X82 : subst,X83 : term] : ((sub @ X80 @ (push @ X83 @ X82)) = (X81 @ X82 @ X83))) & induction2lem_gthm & hoasinduction_lem2_gthm & (induction2lem_lthm <=> (! [X84 : subst,X85 : term] : ((sub @ one @ (push @ X85 @ X84)) = X85) => (! [X86 : subst,X87 : term] : ((comp @ sh @ (push @ X87 @ X86)) = X86) => (! [X88 : subst] : ((comp @ X88 @ id) = X88) => (! [X89 : term,X90 : term > $o] : (~! [X91 : term,X92 : term] : (~(X90 @ X92) | (X90 @ (ap @ X91 @ X92)) | ~(X90 @ X91)) | ~! [X93 : term] : (~(X90 @ X93) | (X90 @ (lam @ X93))) | (X90 @ X89) | ~! [X94 : term] : (~(var @ X94) | (X90 @ X94))) => ! [X95 : subst,X96 : term,X97 : term > $o] : (~! [X98 : term] : (~! [X99 : term] : ((X97 @ (sub @ X98 @ (push @ X99 @ id))) | ~(X97 @ X99)) | (X97 @ (lam @ X98))) | ~! [X100 : term] : (~(var @ X100) | (X97 @ (sub @ X100 @ X95))) | ~! [X101 : term,X102 : term] : (~(X97 @ X102) | (X97 @ (ap @ X102 @ X101)) | ~(X97 @ X101)) | (X97 @ (sub @ X96 @ X95)))))))) & (! [X103 : subst,X104 : term] : ((sub @ one @ (push @ X104 @ X103)) = X104) <=> axvarcons) & hoasapinj2_gthm & (axidr <=> ! [X105 : subst] : ((comp @ X105 @ id) = X105)) & ! [X106 : d_subst] : (d_id = X106) & (! [X107 : subst > term > term,X108 : subst,X109 : term,X110 : subst > term > term] : (((X107 @ X108 @ X109) = (X110 @ X108 @ X109)) | ~! [X111 : subst,X112 : subst,X113 : term] : ((X107 @ (comp @ X111 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X107 @ X111 @ X113) @ X112)) | ~! [X114 : subst,X115 : term,X116 : subst] : ((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)) = (sub @ (X110 @ X116 @ X115) @ X114)) | ((lam @ (X110 @ sh @ one)) != (lam @ (X107 @ sh @ one)))) <=> hoaslaminj) & (! [X117 : subst > term > term,X118 : term,X119 : term] : (~! [X120 : subst,X121 : term,X122 : subst] : ((X117 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122)) = (sub @ (X117 @ X120 @ X121) @ X122)) | ((lam @ (X117 @ sh @ one)) != (ap @ (sub @ X118 @ id) @ X119))) <=> hoaslamnotap) & (apinj1 <=> ! [X123 : term,X124 : term,X125 : term,X126 : term] : (((ap @ X125 @ X123) != (ap @ X126 @ X124)) | (X125 = X126))) & hoasinduction_lem3v2_gthm & (hoasinduction_lem2v2 <=> ! [X127 : term,X128 : term > $o,X129 : subst > term > subst > $o,X130 : term] : (~(X128 @ X130) | (X128 @ (ap @ X130 @ X127)) | ~! [X131 : term,X132 : subst,X133 : subst,X134 : subst] : ((X129 @ (comp @ X133 @ X132) @ (sub @ X131 @ X132) @ X134) | ~(X129 @ X133 @ X131 @ (comp @ X132 @ X134))) | ~(X128 @ X127) | ~! [X135 : subst,X136 : subst,X137 : term,X138 : subst] : (~(X129 @ (comp @ X138 @ X136) @ (sub @ X137 @ X136) @ X135) | (X129 @ X138 @ X137 @ (comp @ X136 @ X135))) | ~! [X139 : term] : ((X129 @ id @ X139 @ id) = (X128 @ X139)) | ~! [X140 : term,X141 : term] : ((X129 @ id @ (ap @ (sub @ X140 @ id) @ X141) @ id) | ~(X129 @ id @ X140 @ id) | ~(X129 @ id @ X141 @ id)))) & pushprop_lthm & hoaslaminj_gthm & hoasapnotvar_lthm & (pushprop_lem1v2 <=> ! [X142 : term > $o,X143 : subst,X144 : term > $o,X145 : term] : ((X144 @ one) | ~! [X146 : term] : ((X144 @ X146) = (X142 @ (sub @ X146 @ (push @ X145 @ X143)))) | ~(X142 @ X145))) & (! [X147 : term] : ~! [X148 : subst > term > term] : ~! [X149 : term,X150 : subst] : ((sub @ X147 @ (push @ X149 @ X150)) = (X148 @ X150 @ X149)) <=> hoasinduction_lem3v2_f_lthm) & ! [X151 : d_term,X152 : d_term] : (((d2term @ X151) = (d2term @ X152)) => (X151 = X152)) & pushprop_lthm_orig & (! [X153 : term,X154 : term,X155 : term,X156 : term] : (((ap @ (sub @ X156 @ id) @ X155) != (ap @ (sub @ X154 @ id) @ X153)) | (X153 = X155)) <=> hoasapinj2) & hoasinduction_lem1v2 & hoaslamnotvar_gthm & (hoaslaminj_lthm <=> (! [X157 : term,X158 : subst] : ((sub @ one @ (push @ X157 @ X158)) = X157) => (! [X159 : subst,X160 : term] : ((comp @ sh @ (push @ X160 @ X159)) = X159) => (! [X161 : term,X162 : term] : ((X161 = X162) | ((lam @ X161) != (lam @ X162))) => ! [X163 : subst,X164 : subst > term > term,X165 : term,X166 : subst > term > term] : (((lam @ (X166 @ sh @ one)) != (lam @ (X164 @ sh @ one))) | ~! [X167 : subst,X168 : term,X169 : subst] : ((sub @ (X164 @ X167 @ X168) @ X169) = (X164 @ (comp @ X167 @ X169) @ (sub @ X168 @ X169))) | ((X164 @ X163 @ X165) = (X166 @ X163 @ X165)) | ~! [X170 : term,X171 : subst,X172 : subst] : ((sub @ (X166 @ X172 @ X170) @ X171) = (X166 @ (comp @ X172 @ X171) @ (sub @ X170 @ X171)))))))) & ((^[X173 : subst, X174 : subst > term > term] : (d2term @ d_one)) = hoaslam) & (pushprop_lem1 <=> ! [X175 : term > $o,X176 : term,X177 : subst,X178 : term,X179 : term > $o] : ((X179 @ (sub @ X176 @ (push @ X178 @ X177))) | ~(X175 @ X176))) & pushprop_lem3v2_lthm & (hoasinduction_lem3v2 <=> ! [X180 : term > $o,X181 : subst > term > subst > $o,X182 : term] : (~! [X183 : term] : ((X180 @ X183) = (X181 @ id @ X183 @ id)) | ~! [X184 : subst > term > term] : ((X181 @ id @ (lam @ (X184 @ sh @ one)) @ id) | ~! [X185 : term] : ((X181 @ id @ (X184 @ id @ X185) @ id) | ~(X181 @ id @ X185 @ id)) | ~! [X186 : subst,X187 : subst,X188 : term] : ((sub @ (X184 @ X187 @ X188) @ X186) = (X184 @ (comp @ X187 @ X186) @ (sub @ X188 @ X186)))) | ~! [X189 : term] : ((X180 @ (sub @ X182 @ (push @ X189 @ id))) | ~(X180 @ X189)) | ~! [X190 : term,X191 : subst,X192 : subst,X193 : subst] : ((X181 @ X191 @ X190 @ (comp @ X192 @ X193)) | ~(X181 @ (comp @ X191 @ X192) @ (sub @ X190 @ X192) @ X193)) | ~! [X194 : subst,X195 : subst,X196 : term,X197 : subst] : ((X181 @ (comp @ X194 @ X195) @ (sub @ X196 @ X195) @ X197) | ~(X181 @ X194 @ X196 @ (comp @ X195 @ X197))) | (X180 @ (lam @ X182)))) & (laminj <=> ! [X198 : term,X199 : term] : (((lam @ X199) != (lam @ X198)) | (X198 = X199))) & ! [X200 : term] : ? [X201 : d_term] : ((d2term @ X201) = X200) & pushprop_lem3v2 & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X202 : term] : ~! [X203 : subst > term > term] : ((sub @ X202 @ (push @ one @ sh)) != (X203 @ sh @ one)) <=> hoasinduction_lem3b_lthm) & hoasapnotvar_gthm & axmap & axclos & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & hoaslamnotvar & (pushprop_lem1_lthm <=> (! [X204 : subst,X205 : term] : ((sub @ one @ (push @ X205 @ X204)) = X205) => (! [X206 : subst,X207 : term] : ((comp @ sh @ (push @ X207 @ X206)) = X206) => ! [X208 : term,X209 : term > $o,X210 : term > $o,X211 : subst,X212 : term] : (~(X209 @ X212) | (X210 @ (sub @ X212 @ (push @ X208 @ X211))))))) & (hoasapinj2_lthm <=> (! [X213 : term,X214 : term,X215 : term,X216 : term] : ((X214 = X216) | ((ap @ X215 @ X214) != (ap @ X213 @ X216))) => ! [X217 : term,X218 : term,X219 : term,X220 : term] : ((X219 = X220) | ((ap @ (sub @ X218 @ id) @ X220) != (ap @ (sub @ X217 @ id) @ X219))))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (hoasinduction_lem3aa <=> ! [X221 : subst > term > subst > $o,X222 : term] : (~! [X223 : subst > term > term] : (~! [X224 : term] : (~(X221 @ id @ X224 @ id) | (X221 @ id @ (X223 @ id @ X224) @ id)) | (X221 @ id @ (lam @ (X223 @ sh @ one)) @ id) | ~! [X225 : term,X226 : subst,X227 : subst] : ((X223 @ (comp @ X227 @ X226) @ (sub @ X225 @ X226)) = (sub @ (X223 @ X227 @ X225) @ X226))) | ~! [X228 : term] : ((X221 @ id @ (sub @ X222 @ (push @ X228 @ id)) @ id) | ~(X221 @ id @ X228 @ id)) | (X221 @ id @ (lam @ (sub @ X222 @ (push @ one @ sh))) @ id))) & ((d2subst @ d_id) = id) & (hoasinduction_lthm_3 <=> (! [X229 : subst > term > subst > $o] : ~! [X230 : term > $o] : ~! [X231 : term] : ((X229 @ id @ X231 @ id) = (X230 @ X231)) => (! [X232 : term > $o,X233 : term] : (~! [X234 : term] : ((X232 @ X234) | ~(var @ X234)) | (X232 @ X233) | ~! [X235 : term] : (~! [X236 : term] : ((X232 @ (sub @ X235 @ (push @ X236 @ id))) | ~(X232 @ X236)) | (X232 @ (lam @ X235))) | ~! [X237 : term,X238 : term] : ((X232 @ (ap @ X237 @ X238)) | ~(X232 @ X238) | ~(X232 @ X237))) => (! [X239 : term] : ((sub @ X239 @ id) = X239) => (! [X240 : term,X241 : term > $o,X242 : subst > term > subst > $o] : (~! [X243 : term] : (~(X241 @ X243) | (X241 @ (sub @ X240 @ (push @ X243 @ id)))) | ~! [X244 : subst > term > term] : ((X242 @ id @ (lam @ (X244 @ sh @ one)) @ id) | ~! [X245 : term] : (~(X242 @ id @ X245 @ id) | (X242 @ id @ (X244 @ id @ X245) @ id)) | ~! [X246 : subst,X247 : term,X248 : subst] : ((sub @ (X244 @ X248 @ X247) @ X246) = (X244 @ (comp @ X248 @ X246) @ (sub @ X247 @ X246)))) | ~! [X249 : term] : ((X241 @ X249) = (X242 @ id @ X249 @ id)) | (X241 @ (lam @ X240))) => ! [X250 : term,X251 : subst > term > subst > $o] : (~! [X252 : term,X253 : term] : (~(X251 @ id @ X253 @ id) | ~(X251 @ id @ X252 @ id) | (X251 @ id @ (ap @ (sub @ X253 @ id) @ X252) @ id)) | ~! [X254 : subst,X255 : term,X256 : subst,X257 : subst] : (~(X251 @ X254 @ X255 @ (comp @ X256 @ X257)) | (X251 @ (comp @ X254 @ X256) @ (sub @ X255 @ X256) @ X257)) | ~! [X258 : subst > term > term] : (~! [X259 : term] : (~(X251 @ id @ X259 @ id) | (X251 @ id @ (X258 @ id @ X259) @ id)) | ~! [X260 : subst,X261 : term,X262 : subst] : ((sub @ (X258 @ X262 @ X261) @ X260) = (X258 @ (comp @ X262 @ X260) @ (sub @ X261 @ X260))) | (X251 @ id @ (lam @ (X258 @ sh @ one)) @ id)) | (X251 @ id @ X250 @ id) | ~! [X263 : term,X264 : subst,X265 : subst,X266 : subst] : (~(X251 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265) @ X266) | (X251 @ X264 @ X263 @ (comp @ X265 @ X266))) | ~! [X267 : term] : (~(var @ (sub @ X267 @ id)) | (X251 @ id @ X267 @ id)))))))) & ((^[X268 : subst > term > subst > $o, X269 : term > $o] : (! [X270 : term] : ((X269 @ X270) = (X268 @ id @ X270 @ id)))) = hoasinduction_p_and_p_prime) & ! [X271 : subst] : ? [X272 : d_subst] : ((d2subst @ X272) = X271) & ((! [X273 : term] : ((sub @ X273 @ id) = X273) => (! [X274 : term,X275 : subst > term > subst > $o] : (~! [X276 : subst > term > term] : (~! [X277 : subst,X278 : term,X279 : subst] : ((sub @ (X276 @ X277 @ X278) @ X279) = (X276 @ (comp @ X277 @ X279) @ (sub @ X278 @ X279))) | (X275 @ id @ (lam @ (X276 @ sh @ one)) @ id) | ~! [X280 : term] : (~(X275 @ id @ X280 @ id) | (X275 @ id @ (X276 @ id @ X280) @ id))) | (X275 @ id @ (lam @ (sub @ X274 @ (push @ one @ sh))) @ id) | ~! [X281 : term] : ((X275 @ id @ (sub @ X274 @ (push @ X281 @ id)) @ id) | ~(X275 @ id @ X281 @ id))) => ! [X282 : term,X283 : subst > term > subst > $o] : (~! [X284 : subst > term > term] : ((X283 @ id @ (lam @ (X284 @ sh @ one)) @ id) | ~! [X285 : subst,X286 : term,X287 : subst] : ((sub @ (X284 @ X287 @ X286) @ X285) = (X284 @ (comp @ X287 @ X285) @ (sub @ X286 @ X285))) | ~! [X288 : term] : ((X283 @ id @ (X284 @ id @ X288) @ id) | ~(X283 @ id @ X288 @ id))) | ~! [X289 : term] : ((X283 @ id @ (sub @ X282 @ (push @ X289 @ id)) @ id) | ~(X283 @ id @ X289 @ id)) | (X283 @ id @ (lam @ X282) @ id)))) <=> hoasinduction_lem3a_lthm) & pushprop_gthm & hoasapnotvar & (hoasinduction_lem3_lthm <=> (! [X290 : term] : ((sub @ X290 @ id) = X290) => (! [X291 : term,X292 : subst > term > subst > $o] : (~! [X293 : subst > term > term] : (~! [X294 : term] : ((X292 @ id @ (X293 @ id @ X294) @ id) | ~(X292 @ id @ X294 @ id)) | ~! [X295 : subst,X296 : subst,X297 : term] : ((X293 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)) = (sub @ (X293 @ X296 @ X297) @ X295)) | (X292 @ id @ (lam @ (X293 @ sh @ one)) @ id)) | (X292 @ id @ (lam @ (sub @ X291 @ (push @ one @ sh))) @ id) | ~! [X298 : term] : ((X292 @ id @ (sub @ X291 @ (push @ X298 @ id)) @ id) | ~(X292 @ id @ X298 @ id))) => ! [X299 : term,X300 : subst > term > subst > $o] : ((X300 @ id @ (lam @ X299) @ id) | ~! [X301 : subst,X302 : subst,X303 : subst,X304 : term] : (~(X300 @ (comp @ X303 @ X301) @ (sub @ X304 @ X301) @ X302) | (X300 @ X303 @ X304 @ (comp @ X301 @ X302))) | ~! [X305 : term] : ((X300 @ id @ (sub @ X299 @ (push @ X305 @ id)) @ id) | ~(X300 @ id @ X305 @ id)) | ~! [X306 : subst > term > term] : (~! [X307 : term,X308 : subst,X309 : subst] : ((X306 @ (comp @ X308 @ X309) @ (sub @ X307 @ X309)) = (sub @ (X306 @ X308 @ X307) @ X309)) | ~! [X310 : term] : (~(X300 @ id @ X310 @ id) | (X300 @ id @ (X306 @ id @ X310) @ id)) | (X300 @ id @ (lam @ (X306 @ sh @ one)) @ id)) | ~! [X311 : term,X312 : subst,X313 : subst,X314 : subst] : (~(X300 @ X313 @ X311 @ (comp @ X314 @ X312)) | (X300 @ (comp @ X313 @ X314) @ (sub @ X311 @ X314) @ X312)))))) & termmset_gthm & axassoc & hoasinduction_lem1v2_gthm & hoasinduction_lem1_lthm & pushprop_lem2v2_lthm & (! [X315 : subst > term > subst > $o,X316 : term] : (~! [X317 : term] : (~(X315 @ id @ X317 @ id) | (X315 @ id @ (sub @ X316 @ (push @ X317 @ id)) @ id)) | ~! [X318 : subst > term > term] : ((X315 @ id @ (lam @ (X318 @ sh @ one)) @ id) | ~! [X319 : subst,X320 : term,X321 : subst] : ((sub @ (X318 @ X319 @ X320) @ X321) = (X318 @ (comp @ X319 @ X321) @ (sub @ X320 @ X321))) | ~! [X322 : term] : (~(X315 @ id @ X322 @ id) | (X315 @ id @ (X318 @ id @ X322) @ id))) | (X315 @ id @ (lam @ X316) @ id)) <=> hoasinduction_lem3a) & hoasinduction_lem2v2_gthm & (! [X323 : term] : ((sub @ X323 @ id) = X323) <=> axvarid) & ulamvarind & (induction2 <=> ! [X324 : term,X325 : term > $o] : (~! [X326 : term] : ((X325 @ X326) | ~(var @ X326)) | ~! [X327 : term,X328 : term] : (~(X325 @ X327) | (X325 @ (ap @ X328 @ X327)) | ~(X325 @ X328)) | (X325 @ X324) | ~! [X329 : term] : (~! [X330 : term] : ((X325 @ (sub @ X329 @ (push @ X330 @ id))) | ~(X325 @ X330)) | (X325 @ (lam @ X329))))) & axabs & ((! [X331 : subst > term > subst > $o] : ~! [X332 : term > $o] : ~! [X333 : term] : ((X331 @ id @ X333 @ id) = (X332 @ X333)) => (! [X334 : term > $o,X335 : term] : (~! [X336 : term] : (~(var @ X336) | (X334 @ X336)) | (X334 @ X335) | ~! [X337 : term] : (~! [X338 : term] : (~(X334 @ X338) | (X334 @ (sub @ X337 @ (push @ X338 @ id)))) | (X334 @ (lam @ X337))) | ~! [X339 : term,X340 : term] : (~(X334 @ X340) | ~(X334 @ X339) | (X334 @ (ap @ X340 @ X339)))) => (! [X341 : term] : ((sub @ X341 @ id) = X341) => (! [X342 : subst > term > subst > $o,X343 : term,X344 : term > $o] : (~! [X345 : term] : ((X344 @ (sub @ X343 @ (push @ X345 @ id))) | ~(X344 @ X345)) | ~! [X346 : subst > term > term] : (~! [X347 : term] : (~(X342 @ id @ X347 @ id) | (X342 @ id @ (X346 @ id @ X347) @ id)) | ~! [X348 : subst,X349 : term,X350 : subst] : ((sub @ (X346 @ X350 @ X349) @ X348) = (X346 @ (comp @ X350 @ X348) @ (sub @ X349 @ X348))) | (X342 @ id @ (lam @ (X346 @ sh @ one)) @ id)) | (X344 @ (lam @ X343)) | ~! [X351 : term] : ((X342 @ id @ X351 @ id) = (X344 @ X351))) => ! [X352 : subst > term > subst > $o,X353 : term] : (~! [X354 : term,X355 : term] : (~(X352 @ id @ X355 @ id) | ~(X352 @ id @ X354 @ id) | (X352 @ id @ (ap @ (sub @ X354 @ id) @ X355) @ id)) | ~! [X356 : subst > term > term] : ((X352 @ id @ (lam @ (X356 @ sh @ one)) @ id) | ~! [X357 : subst,X358 : subst,X359 : term] : ((X356 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357)) = (sub @ (X356 @ X358 @ X359) @ X357)) | ~! [X360 : term] : ((X352 @ id @ (X356 @ id @ X360) @ id) | ~(X352 @ id @ X360 @ id))) | (X352 @ id @ X353 @ id)))))) <=> hoasinduction_no_psi_cond_lthm) & pushprop & (! [X361 : term > $o,X362 : subst,X363 : term] : (~! [X364 : term] : (~(var @ X364) | (X361 @ (sub @ X364 @ X362))) | (X361 @ (sub @ X363 @ X362)) | ~! [X365 : term] : ((X361 @ (lam @ X365)) | ~! [X366 : term] : (~(X361 @ X366) | (X361 @ (sub @ X365 @ (push @ X366 @ id))))) | ~! [X367 : term,X368 : term] : ((X361 @ (ap @ X367 @ X368)) | ~(X361 @ X367) | ~(X361 @ X368))) <=> induction2lem) & (hoasinduction_lem3v2a_lthm <=> (! [X369 : term] : ~! [X370 : subst > term > term] : ~! [X371 : subst,X372 : term] : ((X370 @ X371 @ X372) = (sub @ X369 @ (push @ X372 @ X371))) => (! [X373 : term] : ((sub @ X373 @ id) = X373) => ! [X374 : term > $o,X375 : subst > term > subst > $o,X376 : term] : (~! [X377 : term] : ((X374 @ X377) = (X375 @ id @ X377 @ id)) | ~! [X378 : term] : ((X374 @ (sub @ X376 @ (push @ X378 @ id))) | ~(X374 @ X378)) | (X374 @ (lam @ X376)) | ~! [X379 : subst > term > term] : ((X375 @ id @ (lam @ (X379 @ sh @ one)) @ id) | ~! [X380 : term] : ((X375 @ id @ (X379 @ id @ X380) @ id) | ~(X375 @ id @ X380 @ id)) | ~! [X381 : subst,X382 : term,X383 : subst] : ((sub @ (X379 @ X381 @ X382) @ X383) = (X379 @ (comp @ X381 @ X383) @ (sub @ X382 @ X383)))))))) & induction2_gthm & hoaslamnotap_gthm & (hoasinduction_lthm <=> (! [X384 : term,X385 : term > $o] : (~! [X386 : term,X387 : term] : (~(X385 @ X386) | (X385 @ (ap @ X386 @ X387)) | ~(X385 @ X387)) | (X385 @ X384) | ~! [X388 : term] : ((X385 @ (lam @ X388)) | ~! [X389 : term] : ((X385 @ (sub @ X388 @ (push @ X389 @ id))) | ~(X385 @ X389))) | ~! [X390 : term] : ((X385 @ X390) | ~(var @ X390))) => (! [X391 : term,X392 : term,X393 : subst > term > subst > $o] : ((X393 @ id @ (ap @ X391 @ X392) @ id) | ~(X393 @ id @ X392 @ id) | ~(X393 @ id @ X391 @ id) | ~! [X394 : subst,X395 : subst,X396 : term,X397 : subst] : (~(X393 @ X395 @ X396 @ (comp @ X394 @ X397)) | (X393 @ (comp @ X395 @ X394) @ (sub @ X396 @ X394) @ X397)) | ~! [X398 : term,X399 : term] : ((X393 @ id @ (ap @ (sub @ X399 @ id) @ X398) @ id) | ~(X393 @ id @ X399 @ id) | ~(X393 @ id @ X398 @ id)) | ~! [X400 : subst,X401 : subst,X402 : subst,X403 : term] : (~(X393 @ (comp @ X401 @ X402) @ (sub @ X403 @ X402) @ X400) | (X393 @ X401 @ X403 @ (comp @ X402 @ X400)))) => (! [X404 : subst > term > subst > $o,X405 : term] : ((X404 @ id @ (lam @ X405) @ id) | ~! [X406 : term] : ((X404 @ id @ (sub @ X405 @ (push @ X406 @ id)) @ id) | ~(X404 @ id @ X406 @ id)) | ~! [X407 : subst,X408 : term,X409 : subst,X410 : subst] : ((X404 @ X409 @ X408 @ (comp @ X407 @ X410)) | ~(X404 @ (comp @ X409 @ X407) @ (sub @ X408 @ X407) @ X410)) | ~! [X411 : subst,X412 : term,X413 : subst,X414 : subst] : (~(X404 @ X411 @ X412 @ (comp @ X414 @ X413)) | (X404 @ (comp @ X411 @ X414) @ (sub @ X412 @ X414) @ X413)) | ~! [X415 : subst > term > term] : (~! [X416 : term,X417 : subst,X418 : subst] : ((sub @ (X415 @ X417 @ X416) @ X418) = (X415 @ (comp @ X417 @ X418) @ (sub @ X416 @ X418))) | (X404 @ id @ (lam @ (X415 @ sh @ one)) @ id) | ~! [X419 : term] : (~(X404 @ id @ X419 @ id) | (X404 @ id @ (X415 @ id @ X419) @ id)))) => ! [X420 : subst > term > subst > $o,X421 : term] : (~! [X422 : term,X423 : term] : ((X420 @ id @ (ap @ (sub @ X423 @ id) @ X422) @ id) | ~(X420 @ id @ X422 @ id) | ~(X420 @ id @ X423 @ id)) | ~! [X424 : subst > term > term] : ((X420 @ id @ (lam @ (X424 @ sh @ one)) @ id) | ~! [X425 : term] : ((X420 @ id @ (X424 @ id @ X425) @ id) | ~(X420 @ id @ X425 @ id)) | ~! [X426 : term,X427 : subst,X428 : subst] : ((X424 @ (comp @ X427 @ X428) @ (sub @ X426 @ X428)) = (sub @ (X424 @ X427 @ X426) @ X428))) | (X420 @ id @ X421 @ id) | ~! [X429 : subst,X430 : term,X431 : subst,X432 : subst] : ((X420 @ (comp @ X432 @ X429) @ (sub @ X430 @ X429) @ X431) | ~(X420 @ X432 @ X430 @ (comp @ X429 @ X431))) | ~! [X433 : term,X434 : subst,X435 : subst,X436 : subst] : ((X420 @ X435 @ X433 @ (comp @ X434 @ X436)) | ~(X420 @ (comp @ X435 @ X434) @ (sub @ X433 @ X434) @ X436)) | ~! [X437 : term] : ((X420 @ id @ X437 @ id) | ~(var @ (sub @ X437 @ id)))))))) & (hoasapinj1 <=> ! [X438 : term,X439 : term,X440 : term,X441 : term] : ((X440 = X441) | ((ap @ (sub @ X441 @ id) @ X439) != (ap @ (sub @ X440 @ id) @ X438)))) & pushprop_lem1_gthm & apnotvar & (pushprop_p_and_p_prime = (^[X442 : term, X443 : subst, X444 : term > $o, X445 : term > $o] : (! [X446 : term] : ((X445 @ X446) = (X444 @ (sub @ X446 @ (push @ X442 @ X443))))))) & lamnotvar & (hoasinduction_lem3aa_lthm <=> ! [X447 : subst > term > subst > $o,X448 : term] : (~! [X449 : subst > term > term] : (~! [X450 : term] : ((X447 @ id @ (X449 @ id @ X450) @ id) | ~(X447 @ id @ X450 @ id)) | (X447 @ id @ (lam @ (X449 @ sh @ one)) @ id) | ~! [X451 : subst,X452 : term,X453 : subst] : ((sub @ (X449 @ X451 @ X452) @ X453) = (X449 @ (comp @ X451 @ X453) @ (sub @ X452 @ X453)))) | (X447 @ id @ (lam @ (sub @ X448 @ (push @ one @ sh))) @ id) | ~! [X454 : term] : ((X447 @ id @ (sub @ X448 @ (push @ X454 @ id)) @ id) | ~(X447 @ id @ X454 @ id)))) & (! [X455 : subst > term > subst > $o] : ~! [X456 : term > $o] : ~! [X457 : term] : ((X455 @ id @ X457 @ id) = (X456 @ X457)) <=> hoasinduction_lem0_lthm) & hoasinduction_lem3_gthm & ((! [X458 : subst] : ((comp @ id @ X458) = X458) & ! [X459 : subst] : ((comp @ X459 @ id) = X459)) <=> substmonoid) & (induction2_lthm <=> (! [X460 : term] : ((sub @ X460 @ id) = X460) => (! [X461 : subst,X462 : term,X463 : term > $o] : (~! [X464 : term] : (~(var @ X464) | (X463 @ (sub @ X464 @ X461))) | ~! [X465 : term,X466 : term] : (~(X463 @ X466) | (X463 @ (ap @ X465 @ X466)) | ~(X463 @ X465)) | ~! [X467 : term] : (~! [X468 : term] : ((X463 @ (sub @ X467 @ (push @ X468 @ id))) | ~(X463 @ X468)) | (X463 @ (lam @ X467))) | (X463 @ (sub @ X462 @ X461))) => ! [X469 : term > $o,X470 : term] : (~! [X471 : term] : (~(var @ X471) | (X469 @ X471)) | ~! [X472 : term,X473 : term] : (~(X469 @ X472) | ~(X469 @ X473) | (X469 @ (ap @ X472 @ X473))) | (X469 @ X470) | ~! [X474 : term] : ((X469 @ (lam @ X474)) | ~! [X475 : term] : ((X469 @ (sub @ X474 @ (push @ X475 @ id))) | ~(X469 @ X475))))))) & ~(var @ (d2term @ d_one)) & pushprop_lem1v2_gthm & pushprop_lem0_gthm & (hoasinduction_lem2 <=> ! [X476 : term,X477 : term,X478 : subst > term > subst > $o] : ((X478 @ id @ (ap @ X477 @ X476) @ id) | ~(X478 @ id @ X477 @ id) | ~! [X479 : term,X480 : term] : (~(X478 @ id @ X480 @ id) | (X478 @ id @ (ap @ (sub @ X480 @ id) @ X479) @ id) | ~(X478 @ id @ X479 @ id)) | ~! [X481 : subst,X482 : subst,X483 : subst,X484 : term] : (~(X478 @ X483 @ X484 @ (comp @ X482 @ X481)) | (X478 @ (comp @ X483 @ X482) @ (sub @ X484 @ X482) @ X481)) | ~! [X485 : subst,X486 : subst,X487 : subst,X488 : term] : ((X478 @ X486 @ X488 @ (comp @ X485 @ X487)) | ~(X478 @ (comp @ X486 @ X485) @ (sub @ X488 @ X485) @ X487)) | ~(X478 @ id @ X476 @ id))) & (! [X489 : term > $o,X490 : term] : (~! [X491 : term] : ((X489 @ X491) | ~(var @ X491)) | ~! [X492 : term] : (~(X489 @ X492) | (X489 @ (lam @ X492))) | (X489 @ X490) | ~! [X493 : term,X494 : term] : (~(X489 @ X494) | (X489 @ (ap @ X493 @ X494)) | ~(X489 @ X493))) <=> induction) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X495 : d_term] : (d_one = X495) & (hoasapinj1_lthm <=> (! [X496 : term] : ((sub @ X496 @ id) = X496) => (! [X497 : term,X498 : term,X499 : term,X500 : term] : (((ap @ X500 @ X499) != (ap @ X497 @ X498)) | (X497 = X500)) => ! [X501 : term,X502 : term,X503 : term,X504 : term] : (((ap @ (sub @ X503 @ id) @ X502) != (ap @ (sub @ X501 @ id) @ X504)) | (X501 = X503))))) & hoasinduction_lem1_gthm & hoasinduction_lem3b_gthm & axvarshift & ((! [X505 : term] : ((sub @ X505 @ id) = X505) => ! [X506 : term > $o,X507 : subst > term > subst > $o,X508 : term] : (~! [X509 : subst,X510 : term,X511 : subst,X512 : subst] : ((X507 @ (comp @ X511 @ X509) @ (sub @ X510 @ X509) @ X512) | ~(X507 @ X511 @ X510 @ (comp @ X509 @ X512))) | ~! [X513 : term] : ((X506 @ X513) = (X507 @ id @ X513 @ id)) | ~! [X514 : term] : ((X506 @ (sub @ X508 @ (push @ X514 @ id))) | ~(X506 @ X514)) | ~! [X515 : subst,X516 : term,X517 : subst,X518 : subst] : (~(X507 @ (comp @ X518 @ X517) @ (sub @ X516 @ X517) @ X515) | (X507 @ X518 @ X516 @ (comp @ X517 @ X515))) | (X506 @ (lam @ X508)) | ~! [X519 : subst > term > term] : ((X507 @ id @ (lam @ (X519 @ sh @ one)) @ id) | ~! [X520 : subst,X521 : term,X522 : subst] : ((X519 @ (comp @ X522 @ X520) @ (sub @ X521 @ X520)) = (sub @ (X519 @ X522 @ X521) @ X520)) | ~! [X523 : term] : (~(X507 @ id @ X523 @ id) | (X507 @ id @ (X519 @ id @ X523) @ id))))) <=> hoasinduction_lem3v2_lthm) & (shinj <=> ! [X524 : term,X525 : term] : (((sub @ X525 @ sh) != (sub @ X524 @ sh)) | (X524 = X525))) & (hoasinduction_lem3 <=> ! [X526 : term,X527 : subst > term > subst > $o] : (~! [X528 : subst,X529 : term,X530 : subst,X531 : subst] : ((X527 @ (comp @ X531 @ X528) @ (sub @ X529 @ X528) @ X530) | ~(X527 @ X531 @ X529 @ (comp @ X528 @ X530))) | ~! [X532 : subst > term > term] : ((X527 @ id @ (lam @ (X532 @ sh @ one)) @ id) | ~! [X533 : term] : ((X527 @ id @ (X532 @ id @ X533) @ id) | ~(X527 @ id @ X533 @ id)) | ~! [X534 : term,X535 : subst,X536 : subst] : ((X532 @ (comp @ X535 @ X536) @ (sub @ X534 @ X536)) = (sub @ (X532 @ X535 @ X534) @ X536))) | (X527 @ id @ (lam @ X526) @ id) | ~! [X537 : term] : ((X527 @ id @ (sub @ X526 @ (push @ X537 @ id)) @ id) | ~(X527 @ id @ X537 @ id)) | ~! [X538 : subst,X539 : subst,X540 : subst,X541 : term] : (~(X527 @ (comp @ X539 @ X540) @ (sub @ X541 @ X540) @ X538) | (X527 @ X539 @ X541 @ (comp @ X540 @ X538))))) & substmonoid_gthm & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X542 : term,X543 : term > $o,X544 : subst] : ~! [X545 : term > $o] : ~! [X546 : term] : ((X543 @ (sub @ X546 @ (push @ X542 @ X544))) = (X545 @ X546)) <=> pushprop_lem0_lthm) & (hoasinduction_lem3b <=> ! [X547 : term] : ~! [X548 : subst > term > term] : ((X548 @ sh @ one) != (sub @ X547 @ (push @ one @ sh)))) & pushprop_lem2v2_gthm & (! [X549 : subst] : ((push @ (sub @ one @ X549) @ (comp @ sh @ X549)) = X549) <=> axscons) & ulamvarsh & hoasinduction_lem2_lthm),
% 2.56/1.94 inference(rectify,[status(thm)],[f1])).
% 2.56/1.94 thf(f1,axiom,(
% 2.56/1.94 (! [X19 : term,X20 : term,X13 : term,X15 : term] : ((X19 = X20) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) <=> apinj2) & (! [X30 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ X30 @ id) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id))) <=> hoasinduction_no_psi_cond) & ! [X6 : d_subst,X7 : d_subst] : (((d2subst @ X6) = (d2subst @ X7)) => (X6 = X7)) & (one = (d2term @ d_one)) & (hoasinduction_lem3v2a <=> ! [X10 : subst > term > subst > $o,X11 : term > $o,X29 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | (X11 @ (lam @ X29)))) & hoasinduction_lem3a_gthm & (termmset_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X13 : term] : ((sub @ X13 @ id) = X13))) & (! [X14 : subst,X13 : term,X10 : term > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) <=> pushprop_lem0) & (! [X13 : term] : ((sub @ X13 @ id) = X13) <=> termmset) & hoasinduction_lem1 & (substmonoid_lthm <=> (! [X14 : subst] : ((comp @ id @ X14) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X14 : subst] : ((comp @ id @ X14) = X14))))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (pushprop_lem1v2_lthm <=> (! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => ! [X13 : term,X10 : term > $o,X11 : term > $o,X14 : subst] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) | ~(X10 @ X13) | (X11 @ one)))) & pushprop_lem2v2 & ~lamnotap & (hoasinduction_lem3aaa <=> ! [X34 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ (sub @ X34 @ (push @ one @ sh))) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X34 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term,X35 : term] : (~! [X36 : subst,X37 : term,X38 : subst] : ((sub @ (X18 @ X36 @ X37) @ X38) = (sub @ (sub @ X35 @ (push @ X37 @ X36)) @ X38)) | ~! [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 @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))))) & (axidl <=> ! [X14 : subst] : ((comp @ id @ X14) = X14)) & hoasinduction_gthm & (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) <=> axshiftcons) & axapp & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (hoasinduction <=> ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ 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) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | ~! [X14 : subst,X25 : subst,X13 : term,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term,X26 : subst,X14 : subst,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ X28 @ id))) & ~ulamvar1 & ((d2subst @ d_id) = sh) & hoaslamnotvar_lthm & hoaslamnotap_lthm & (hoasinduction_lem0 <=> ! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))) & hoasapinj1_gthm & (hoasinduction_lem3v2_f <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13))) & induction2lem_gthm & hoasinduction_lem2_gthm & (induction2lem_lthm <=> (! [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) => (! [X21 : term,X10 : term > $o] : (~! [X13 : term,X15 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13)) | ~! [X13 : term] : (~(X10 @ X13) | (X10 @ (lam @ X13))) | (X10 @ X21) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13))) => ! [X17 : subst,X16 : term,X10 : term > $o] : (~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | ~! [X15 : term,X13 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | (X10 @ (sub @ X16 @ X17)))))))) & (! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) <=> axvarcons) & hoasapinj2_gthm & (axidr <=> ! [X14 : subst] : ((comp @ X14 @ id) = X14)) & ! [X5 : d_subst] : (d_id = X5) & (! [X18 : subst > term > term,X32 : subst,X33 : term,X31 : subst > term > term] : (((X18 @ X32 @ X33) = (X31 @ X32 @ X33)) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [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)))) <=> hoaslaminj) & (! [X18 : subst > term > term,X46 : term,X47 : term] : (~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (ap @ (sub @ X46 @ id) @ X47))) <=> hoaslamnotap) & (apinj1 <=> ! [X20 : term,X19 : term,X15 : term,X13 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X13 = X15))) & hoasinduction_lem3v2_gthm & (hoasinduction_lem2v2 <=> ! [X45 : term,X11 : term > $o,X10 : subst > term > subst > $o,X44 : term] : (~(X11 @ X44) | (X11 @ (ap @ X44 @ X45)) | ~! [X13 : term,X26 : subst,X14 : subst,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~(X11 @ X45) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id)))) & pushprop_lthm & hoaslaminj_gthm & hoasapnotvar_lthm & (pushprop_lem1v2 <=> ! [X10 : term > $o,X14 : subst,X11 : term > $o,X13 : term] : ((X11 @ one) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) | ~(X10 @ X13))) & (! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) <=> hoasinduction_lem3v2_f_lthm) & ! [X2 : d_term,X3 : d_term] : (((d2term @ X2) = (d2term @ X3)) => (X2 = X3)) & pushprop_lthm_orig & (! [X20 : term,X15 : term,X19 : term,X13 : term] : (((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)) | (X19 = X20)) <=> hoasapinj2) & hoasinduction_lem1v2 & hoaslamnotvar_gthm & (hoaslaminj_lthm <=> (! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X15 : term,X13 : term] : ((X13 = X15) | ((lam @ X13) != (lam @ X15))) => ! [X32 : subst,X18 : subst > term > term,X33 : term,X31 : subst > term > term] : (((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))) | ~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))))))) & ((^[X8 : subst, X9 : subst > term > term] : (d2term @ d_one)) = hoaslam) & (pushprop_lem1 <=> ! [X10 : term > $o,X13 : term,X14 : subst,X15 : term,X26 : term > $o] : ((X26 @ (sub @ X13 @ (push @ X15 @ X14))) | ~(X10 @ X13))) & pushprop_lem3v2_lthm & (hoasinduction_lem3v2 <=> ! [X11 : term > $o,X10 : subst > term > subst > $o,X48 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ 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)))) | ~! [X15 : term] : ((X11 @ (sub @ X48 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X13 : term,X14 : subst,X26 : subst,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X11 @ (lam @ X48)))) & (laminj <=> ! [X13 : term,X15 : term] : (((lam @ X13) != (lam @ X15)) | (X13 = X15))) & ! [X0 : term] : ? [X1 : d_term] : ((d2term @ X1) = X0) & pushprop_lem3v2 & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh))) <=> hoasinduction_lem3b_lthm) & hoasapnotvar_gthm & axmap & axclos & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & hoaslamnotvar & (pushprop_lem1_lthm <=> (! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => ! [X15 : term,X10 : term > $o,X26 : term > $o,X14 : subst,X13 : term] : (~(X10 @ X13) | (X26 @ (sub @ X13 @ (push @ X15 @ X14))))))) & (hoasapinj2_lthm <=> (! [X13 : term,X20 : term,X15 : term,X19 : term] : ((X19 = X20) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) => ! [X13 : term,X15 : term,X19 : term,X20 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (hoasinduction_lem3aa <=> ! [X10 : subst > term > subst > $o,X42 : term] : (~! [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)))) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id))) & ((d2subst @ d_id) = id) & (hoasinduction_lthm_3 <=> (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X10 : term > $o,X22 : term] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ 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 @ X15) | ~(X10 @ X13))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X29 : term,X11 : term > $o,X10 : subst > term > subst > $o] : (~! [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 @ 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)))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X29))) => ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X15 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~! [X14 : subst,X13 : term,X26 : subst,X25 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X18 : subst > term > term] : (~! [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))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X10 @ id @ X28 @ id) | ~! [X13 : term,X14 : subst,X26 : subst,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)))))))) & ((^[X10 : subst > term > subst > $o, X11 : term > $o] : (! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)))) = hoasinduction_p_and_p_prime) & ! [X4 : subst] : ? [X5 : d_subst] : ((d2subst @ X5) = X4) & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : (~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id))) => ! [X43 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X15 : term] : ((X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (lam @ X43) @ id)))) <=> hoasinduction_lem3a_lthm) & pushprop_gthm & hoasapnotvar & (hoasinduction_lem3_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id))) => ! [X27 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ X27) @ id) | ~! [X26 : subst,X25 : subst,X14 : 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)) | ~! [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)) | ~! [X13 : term,X25 : subst,X14 : subst,X26 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)))))) & termmset_gthm & axassoc & hoasinduction_lem1v2_gthm & hoasinduction_lem1_lthm & pushprop_lem2v2_lthm & (! [X10 : subst > term > subst > $o,X43 : term] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id)) | ~! [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)) <=> hoasinduction_lem3a) & hoasinduction_lem2v2_gthm & (! [X13 : term] : ((sub @ X13 @ id) = X13) <=> axvarid) & ulamvarind & (induction2 <=> ! [X22 : term,X10 : term > $o] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X15 : term,X13 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13)) | (X10 @ X22) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))))) & axabs & ((! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X10 : term > $o,X22 : term] : (~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | (X10 @ X22) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | ~! [X15 : term,X13 : term] : (~(X10 @ X13) | ~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : subst > term > subst > $o,X29 : term,X11 : term > $o] : (~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X18 : subst > term > term] : (~! [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))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X11 @ (lam @ X29)) | ~! [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)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | (X10 @ id @ X30 @ id)))))) <=> hoasinduction_no_psi_cond_lthm) & pushprop & (! [X10 : term > $o,X17 : subst,X16 : term] : (~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | (X10 @ (sub @ X16 @ X17)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id))))) | ~! [X13 : term,X15 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15))) <=> induction2lem) & (hoasinduction_lem3v2a_lthm <=> (! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X11 : term > $o,X10 : subst > term > subst > $o,X29 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | (X11 @ (lam @ X29)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))))))) & induction2_gthm & hoaslamnotap_gthm & (hoasinduction_lthm <=> (! [X22 : term,X10 : term > $o] : (~! [X13 : term,X15 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | (X10 @ X22) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15))) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13))) => (! [X23 : term,X24 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (ap @ X23 @ X24) @ id) | ~(X10 @ id @ 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)) | ~! [X15 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X25 : subst,X14 : subst,X26 : subst,X13 : term] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25)))) => (! [X10 : subst > term > subst > $o,X27 : term] : ((X10 @ id @ (lam @ X27) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X26 : subst,X13 : term,X14 : subst,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X14 : subst,X13 : term,X25 : 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))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)))) => ! [X10 : subst > term > subst > $o,X28 : term] : (~! [X15 : term,X13 : 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 @ (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)))) | (X10 @ id @ X28 @ id) | ~! [X26 : subst,X13 : term,X25 : subst,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term,X26 : subst,X14 : subst,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term] : ((X10 @ id @ X13 @ id) | ~(var @ (sub @ X13 @ id)))))))) & (hoasapinj1 <=> ! [X19 : term,X20 : term,X13 : term,X15 : term] : ((X13 = X15) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)))) & pushprop_lem1_gthm & apnotvar & (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))))))) & lamnotvar & (hoasinduction_lem3aa_lthm <=> ! [X10 : subst > term > subst > $o,X42 : term] : (~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)))) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0_lthm) & hoasinduction_lem3_gthm & ((! [X14 : subst] : ((comp @ id @ X14) = X14) & ! [X14 : subst] : ((comp @ X14 @ id) = X14)) <=> substmonoid) & (induction2_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X17 : subst,X16 : term,X10 : term > $o] : (~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | ~! [X13 : term,X15 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13)) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | (X10 @ (sub @ X16 @ X17))) => ! [X10 : term > $o,X22 : term] : (~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X13 : term,X15 : term] : (~(X10 @ X13) | ~(X10 @ X15) | (X10 @ (ap @ X13 @ X15))) | (X10 @ X22) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15))))))) & ~(var @ (d2term @ d_one)) & pushprop_lem1v2_gthm & pushprop_lem0_gthm & (hoasinduction_lem2 <=> ! [X24 : term,X23 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (ap @ X23 @ X24) @ id) | ~(X10 @ id @ X23 @ id) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X25 : subst,X26 : subst,X14 : 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 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~(X10 @ id @ X24 @ id))) & (! [X10 : term > $o,X21 : term] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X13 : term] : (~(X10 @ X13) | (X10 @ (lam @ X13))) | (X10 @ X21) | ~! [X13 : term,X15 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13))) <=> induction) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X1 : d_term] : (d_one = X1) & (hoasapinj1_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X15 : term,X20 : term,X19 : term,X13 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X13 = X15)) => ! [X13 : term,X20 : term,X15 : term,X19 : term] : (((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)) | (X13 = X15))))) & hoasinduction_lem1_gthm & hoasinduction_lem3b_gthm & axvarshift & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X11 : term > $o,X10 : subst > term > subst > $o,X48 : term] : (~! [X26 : subst,X13 : term,X14 : subst,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term] : ((X11 @ (sub @ X48 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X25 : subst,X13 : term,X26 : subst,X14 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X11 @ (lam @ X48)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))))) <=> hoasinduction_lem3v2_lthm) & (shinj <=> ! [X15 : term,X13 : term] : (((sub @ X13 @ sh) != (sub @ X15 @ sh)) | (X13 = X15))) & (hoasinduction_lem3 <=> ! [X27 : term,X10 : subst > term > subst > $o] : (~! [X26 : subst,X13 : term,X25 : subst,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [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)))) | (X10 @ id @ (lam @ X27) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X25 : subst,X14 : subst,X26 : subst,X13 : term] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))))) & substmonoid_gthm & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X13 : term,X10 : term > $o,X14 : subst] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) <=> pushprop_lem0_lthm) & (hoasinduction_lem3b <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh)))) & pushprop_lem2v2_gthm & (! [X14 : subst] : ((push @ (sub @ one @ X14) @ (comp @ sh @ X14)) = X14) <=> axscons) & ulamvarsh & hoasinduction_lem2_lthm),
% 2.56/1.94 file('/export/starexec/sandbox2/tmp/tmp.ae6moAB9X6/DTF2THF_11154.p',alg444_1)).
% 2.56/1.94 thf(f1046,plain,(
% 2.56/1.94 ( ! [X248 : term] : (((d2term @ (sK579 @ X248)) = X248)) )),
% 2.56/1.94 inference(cnf_transformation,[status(thm)],[f545])).
% 2.56/1.94 thf(f1711,plain,(
% 2.56/1.94 ($false = (sK592 @ (sub @ sK596 @ (push @ sK593 @ sK594))))),
% 2.56/1.94 inference(binary_proxy_clausification,[status(thm)],[f1710])).
% 2.56/1.94 thf(f1710,plain,(
% 2.56/1.94 ($false = ((var @ sK596) => (sK592 @ (sub @ sK596 @ (push @ sK593 @ sK594)))))),
% 2.56/1.94 inference(beta_eta_normalization,[status(thm)],[f1709])).
% 2.56/1.94 thf(f1709,plain,(
% 2.56/1.94 ($false = ((^[Y0 : term]: ((var @ Y0) => (sK592 @ (sub @ Y0 @ (push @ sK593 @ sK594))))) @ sK596))),
% 2.56/1.94 inference(sigma_clausification,[status(thm)],[f1704])).
% 2.56/1.94 thf(f1704,plain,(
% 2.56/1.94 ($false = (!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK592 @ (sub @ Y0 @ (push @ sK593 @ sK594)))))))),
% 2.56/1.94 inference(binary_proxy_clausification,[status(thm)],[f1697])).
% 2.56/1.94 thf(f1697,plain,(
% 2.56/1.94 ($false = ((!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK595 @ Y0)))) => (!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK592 @ (sub @ Y0 @ (push @ sK593 @ sK594))))))))),
% 2.56/1.94 inference(binary_proxy_clausification,[status(thm)],[f1696])).
% 2.56/1.94 thf(f1696,plain,(
% 2.56/1.94 (((!! @ term @ (^[Y0 : term]: ((sK595 @ Y0) = (sK592 @ (sub @ Y0 @ (push @ sK593 @ sK594)))))) => ((!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK595 @ Y0)))) => (!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK592 @ (sub @ Y0 @ (push @ sK593 @ sK594)))))))) = $false)),
% 2.56/1.94 inference(beta_eta_normalization,[status(thm)],[f1695])).
% 2.56/1.94 thf(f1695,plain,(
% 2.56/1.94 ($false = ((^[Y0 : term > $o]: ((!! @ term @ (^[Y1 : term]: ((Y0 @ Y1) = (sK592 @ (sub @ Y1 @ (push @ sK593 @ sK594)))))) => ((!! @ term @ (^[Y1 : term]: ((var @ Y1) => (Y0 @ Y1)))) => (!! @ term @ (^[Y1 : term]: ((var @ Y1) => (sK592 @ (sub @ Y1 @ (push @ sK593 @ sK594))))))))) @ sK595))),
% 2.56/1.94 inference(sigma_clausification,[status(thm)],[f1694])).
% 2.56/1.94 thf(f1694,plain,(
% 2.56/1.94 ($false = (!! @ (term > $o) @ (^[Y0 : term > $o]: ((!! @ term @ (^[Y1 : term]: ((Y0 @ Y1) = (sK592 @ (sub @ Y1 @ (push @ sK593 @ sK594)))))) => ((!! @ term @ (^[Y1 : term]: ((var @ Y1) => (Y0 @ Y1)))) => (!! @ term @ (^[Y1 : term]: ((var @ Y1) => (sK592 @ (sub @ Y1 @ (push @ sK593 @ sK594)))))))))))),
% 2.56/1.94 inference(beta_eta_normalization,[status(thm)],[f1693])).
% 2.56/1.94 thf(f1693,plain,(
% 2.56/1.94 ($false = ((^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (sK592 @ (sub @ Y2 @ (push @ sK593 @ Y0)))))) => ((!! @ term @ (^[Y2 : term]: ((var @ Y2) => (Y1 @ Y2)))) => (!! @ term @ (^[Y2 : term]: ((var @ Y2) => (sK592 @ (sub @ Y2 @ (push @ sK593 @ Y0))))))))))) @ sK594))),
% 2.56/1.94 inference(sigma_clausification,[status(thm)],[f1692])).
% 2.56/1.94 thf(f1692,plain,(
% 2.56/1.94 ($false = (!! @ subst @ (^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (sK592 @ (sub @ Y2 @ (push @ sK593 @ Y0)))))) => ((!! @ term @ (^[Y2 : term]: ((var @ Y2) => (Y1 @ Y2)))) => (!! @ term @ (^[Y2 : term]: ((var @ Y2) => (sK592 @ (sub @ Y2 @ (push @ sK593 @ Y0)))))))))))))),
% 2.56/1.94 inference(beta_eta_normalization,[status(thm)],[f1691])).
% 2.56/1.94 thf(f1691,plain,(
% 2.56/1.94 ($false = ((^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ (term > $o) @ (^[Y2 : term > $o]: ((!! @ term @ (^[Y3 : term]: ((Y2 @ Y3) = (sK592 @ (sub @ Y3 @ (push @ Y0 @ Y1)))))) => ((!! @ term @ (^[Y3 : term]: ((var @ Y3) => (Y2 @ Y3)))) => (!! @ term @ (^[Y3 : term]: ((var @ Y3) => (sK592 @ (sub @ Y3 @ (push @ Y0 @ Y1))))))))))))) @ sK593))),
% 2.56/1.94 inference(sigma_clausification,[status(thm)],[f1690])).
% 2.56/1.94 thf(f1690,plain,(
% 2.56/1.94 ($false = (!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ (term > $o) @ (^[Y2 : term > $o]: ((!! @ term @ (^[Y3 : term]: ((Y2 @ Y3) = (sK592 @ (sub @ Y3 @ (push @ Y0 @ Y1)))))) => ((!! @ term @ (^[Y3 : term]: ((var @ Y3) => (Y2 @ Y3)))) => (!! @ term @ (^[Y3 : term]: ((var @ Y3) => (sK592 @ (sub @ Y3 @ (push @ Y0 @ Y1)))))))))))))))),
% 2.56/1.94 inference(beta_eta_normalization,[status(thm)],[f1689])).
% 2.56/1.94 thf(f1689,plain,(
% 2.56/1.94 ($false = ((^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y0 @ (sub @ Y4 @ (push @ Y1 @ Y2)))))) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y3 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y0 @ (sub @ Y4 @ (push @ Y1 @ Y2))))))))))))))) @ sK592))),
% 2.56/1.94 inference(sigma_clausification,[status(thm)],[f1688])).
% 2.56/1.94 thf(f1688,plain,(
% 2.56/1.94 ($true != (!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y0 @ (sub @ Y4 @ (push @ Y1 @ Y2)))))) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y3 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y0 @ (sub @ Y4 @ (push @ Y1 @ Y2)))))))))))))))))),
% 2.56/1.94 inference(beta_eta_normalization,[status(thm)],[f1663])).
% 2.56/1.94 thf(f1663,plain,(
% 2.56/1.94 ($true != (!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y7 @ Y8) = (Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))))))))))))) @ Y1 @ Y2 @ Y0 @ Y3) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y3 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y0 @ (sub @ Y4 @ (push @ Y1 @ Y2)))))))))))))))))),
% 2.56/1.94 inference(definition_unfolding,[status(thm)],[f1260,f1174,f1251])).
% 2.56/1.94 thf(f1251,plain,(
% 2.56/1.94 (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))))))))))))))),
% 2.56/1.94 inference(cnf_transformation,[status(thm)],[f545])).
% 2.56/1.94 thf(f1174,plain,(
% 2.56/1.94 (pushprop_lem3v2 = $true)),
% 2.56/1.94 inference(cnf_transformation,[status(thm)],[f545])).
% 2.56/1.94 thf(f1260,plain,(
% 2.56/1.94 (pushprop_lem3v2 != (!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((pushprop_p_and_p_prime @ Y1 @ Y2 @ Y0 @ Y3) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y3 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y0 @ (sub @ Y4 @ (push @ Y1 @ Y2)))))))))))))))))),
% 2.56/1.94 inference(cnf_transformation,[status(thm)],[f9])).
% 2.56/1.94 thf(f9,plain,(
% 2.56/1.94 (pushprop_lem3v2 != (!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((pushprop_p_and_p_prime @ Y1 @ Y2 @ Y0 @ Y3) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y3 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y0 @ (sub @ Y4 @ (push @ Y1 @ Y2)))))))))))))))))),
% 2.56/1.94 inference(flattening,[status(thm)],[f8])).
% 2.56/1.94 thf(f8,plain,(
% 2.56/1.94 ~(pushprop_lem3v2 = (!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((pushprop_p_and_p_prime @ Y1 @ Y2 @ Y0 @ Y3) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y3 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y0 @ (sub @ Y4 @ (push @ Y1 @ Y2)))))))))))))))))),
% 2.56/1.94 inference(fool_elimination,[status(thm)],[f7])).
% 2.56/1.94 thf(f7,plain,(
% 2.56/1.94 ~(! [X0 : term > $o,X1 : subst,X2 : term,X3 : term > $o] : ((pushprop_p_and_p_prime @ X2 @ X1 @ X3 @ X0) => (! [X4 : term] : ((var @ X4) => (X0 @ X4)) => ! [X5 : term] : ((var @ X5) => (X3 @ (sub @ X5 @ (push @ X2 @ X1)))))) = pushprop_lem3v2)),
% 2.56/1.94 inference(rectify,[status(thm)],[f3])).
% 2.56/1.94 thf(f3,negated_conjecture,(
% 2.56/1.94 ~(! [X11 : term > $o,X14 : subst,X13 : term,X10 : term > $o] : ((pushprop_p_and_p_prime @ X13 @ X14 @ X10 @ X11) => (! [X15 : term] : ((var @ X15) => (X11 @ X15)) => ! [X15 : term] : ((var @ X15) => (X10 @ (sub @ X15 @ (push @ X13 @ X14)))))) = pushprop_lem3v2)),
% 2.56/1.94 inference(negated_conjecture,[status(cth)],[f2])).
% 2.56/1.94 thf(f2,conjecture,(
% 2.56/1.94 (! [X11 : term > $o,X14 : subst,X13 : term,X10 : term > $o] : ((pushprop_p_and_p_prime @ X13 @ X14 @ X10 @ X11) => (! [X15 : term] : ((var @ X15) => (X11 @ X15)) => ! [X15 : term] : ((var @ X15) => (X10 @ (sub @ X15 @ (push @ X13 @ X14)))))) = pushprop_lem3v2)),
% 2.56/1.94 file('/export/starexec/sandbox2/tmp/tmp.ae6moAB9X6/DTF2THF_11154.p',pushprop_lem3v2)).
% 2.56/1.94 thf(f4406,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($false = (sK595 @ X1)) | ($true = (sK592 @ (d2term @ d_one)))) )),
% 2.56/1.94 inference(forward_demodulation,[status(thm)],[f1703,f2761])).
% 2.56/1.94 thf(f1703,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($true = (sK592 @ (sub @ X1 @ (push @ sK593 @ sK594)))) | ($false = (sK595 @ X1))) )),
% 2.56/1.94 inference(binary_proxy_clausification,[status(thm)],[f1701])).
% 2.56/1.94 thf(f1701,plain,(
% 2.56/1.94 ( ! [X1 : term] : (((sK595 @ X1) = (sK592 @ (sub @ X1 @ (push @ sK593 @ sK594))))) )),
% 2.56/1.94 inference(equality_proxy_clausification,[status(thm)],[f1700])).
% 2.56/1.94 thf(f1700,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($true = ((sK595 @ X1) = (sK592 @ (sub @ X1 @ (push @ sK593 @ sK594)))))) )),
% 2.56/1.94 inference(beta_eta_normalization,[status(thm)],[f1699])).
% 2.56/1.94 thf(f1699,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($true = ((^[Y0 : term]: ((sK595 @ Y0) = (sK592 @ (sub @ Y0 @ (push @ sK593 @ sK594))))) @ X1))) )),
% 2.56/1.94 inference(pi_clausification,[status(thm)],[f1698])).
% 2.56/1.94 thf(f1698,plain,(
% 2.56/1.94 ($true = (!! @ term @ (^[Y0 : term]: ((sK595 @ Y0) = (sK592 @ (sub @ Y0 @ (push @ sK593 @ sK594)))))))),
% 2.56/1.94 inference(binary_proxy_clausification,[status(thm)],[f1696])).
% 2.56/1.94 thf(f1708,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($true = (sK595 @ X1)) | ($false = (var @ X1))) )),
% 2.56/1.94 inference(binary_proxy_clausification,[status(thm)],[f1707])).
% 2.56/1.94 thf(f1707,plain,(
% 2.56/1.94 ( ! [X1 : term] : ((((var @ X1) => (sK595 @ X1)) = $true)) )),
% 2.56/1.94 inference(beta_eta_normalization,[status(thm)],[f1706])).
% 2.56/1.94 thf(f1706,plain,(
% 2.56/1.94 ( ! [X1 : term] : (($true = ((^[Y0 : term]: ((var @ Y0) => (sK595 @ Y0))) @ X1))) )),
% 2.56/1.94 inference(pi_clausification,[status(thm)],[f1705])).
% 2.56/1.94 thf(f1705,plain,(
% 2.56/1.94 ($true = (!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK595 @ Y0)))))),
% 2.56/1.94 inference(binary_proxy_clausification,[status(thm)],[f1697])).
% 2.56/1.94 thf(f1712,plain,(
% 2.56/1.94 ($true = (var @ sK596))),
% 2.56/1.94 inference(binary_proxy_clausification,[status(thm)],[f1710])).
% 2.56/1.94 % SZS output end Proof for DTF2THF_11154
% 2.56/1.94 % (11537)------------------------------
% 2.56/1.94 % (11537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.94 % (11537)Termination reason: Refutation
% 2.56/1.94
% 2.56/1.94 % (11537)Memory used [KB]: 12409
% 2.56/1.94 % (11537)Time elapsed: 0.319 s
% 2.56/1.94 % (11537)Instructions burned: 835 (million)
% 2.56/1.94 % (11537)------------------------------
% 2.56/1.94 % (11537)------------------------------
% 2.56/1.94 % (11506)Success in time 0.433 s
%------------------------------------------------------------------------------