%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n021.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 06:58:43 PM UTC 2026
% Result : Theorem 2.65s 1.76s
% Output : Refutation 2.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 130
% Syntax : Number of formulae : 160 ( 20 unt; 0 typ; 0 def)
% Number of atoms : 11032 (5320 equ; 0 cnn)
% Maximal formula atoms : 530 ( 68 avg)
% Number of connectives : 26080 (3045 ~;2751 |;2417 &;17063 @)
% ( 507 <=>; 259 =>; 0 <=; 0 <~>)
% Maximal formula depth : 191 ( 19 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 1945 (1945 >; 0 *; 0 +; 0 <<)
% Number of symbols : 724 ( 720 usr; 295 con; 0-4 aty)
% ( 38 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 5707 ( 137 ^;4222 !;1348 ?;5707 :)
% 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_141,type,
sP0: ( subst > term > subst > $o ) > $o ).
thf(func_def_143,type,
sP2: ( subst > term > subst > $o ) > $o ).
thf(func_def_144,type,
sP3: ( term > $o ) > $o ).
thf(func_def_148,type,
sP7: ( subst > term > subst > $o ) > $o ).
thf(func_def_149,type,
sP8: ( term > $o ) > $o ).
thf(func_def_150,type,
sP9: ( subst > term > subst > $o ) > $o ).
thf(func_def_151,type,
sP10: ( subst > term > subst > $o ) > $o ).
thf(func_def_152,type,
sP11: ( subst > term > subst > $o ) > $o ).
thf(func_def_157,type,
sP16: ( subst > term > subst > $o ) > $o ).
thf(func_def_161,type,
sP20: ( subst > term > subst > $o ) > $o ).
thf(func_def_163,type,
sP22: ( term > $o ) > $o ).
thf(func_def_166,type,
sP25: ( term > $o ) > $o ).
thf(func_def_167,type,
sP26: ( subst > term > subst > $o ) > $o ).
thf(func_def_168,type,
sP27: ( subst > term > subst > $o ) > $o ).
thf(func_def_172,type,
sP31: ( subst > term > subst > $o ) > $o ).
thf(func_def_173,type,
sP32: ( subst > term > subst > $o ) > $o ).
thf(func_def_175,type,
sP34: ( subst > term > subst > $o ) > $o ).
thf(func_def_176,type,
sP35: ( term > $o ) > $o ).
thf(func_def_177,type,
sP36: ( term > $o ) > $o ).
thf(func_def_181,type,
sP40: ( term > $o ) > $o ).
thf(func_def_182,type,
sP41: ( term > $o ) > $o ).
thf(func_def_186,type,
sP45: ( subst > term > subst > $o ) > $o ).
thf(func_def_188,type,
sP47: ( subst > term > subst > $o ) > $o ).
thf(func_def_191,type,
sP50: ( subst > term > subst > $o ) > $o ).
thf(func_def_197,type,
sP56: ( term > $o ) > $o ).
thf(func_def_198,type,
sP57: ( term > $o ) > $o ).
thf(func_def_200,type,
sK59: subst ).
thf(func_def_201,type,
sK60: term > $o ).
thf(func_def_202,type,
sK61: term ).
thf(func_def_203,type,
sK62: term > term ).
thf(func_def_204,type,
sK63: ( term > $o ) > term ).
thf(func_def_205,type,
sK64: ( term > $o ) > subst > term ).
thf(func_def_206,type,
sK65: ( term > $o ) > term ).
thf(func_def_207,type,
sK66: ( term > $o ) > term ).
thf(func_def_208,type,
sK67: ( term > $o ) > term ).
thf(func_def_209,type,
sK68: ( term > $o ) > term ).
thf(func_def_210,type,
sK69: subst > term > subst > $o ).
thf(func_def_211,type,
sK70: term ).
thf(func_def_212,type,
sK71: ( subst > term > term ) > term ).
thf(func_def_213,type,
sK72: ( subst > term > term ) > term ).
thf(func_def_214,type,
sK73: ( subst > term > term ) > subst ).
thf(func_def_215,type,
sK74: ( subst > term > term ) > subst ).
thf(func_def_216,type,
sK75: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_217,type,
sK76: term > ( subst > term > subst > $o ) > term ).
thf(func_def_218,type,
sK77: term ).
thf(func_def_219,type,
sK78: subst > term > subst > $o ).
thf(func_def_220,type,
sK79: term > $o ).
thf(func_def_221,type,
sK80: ( subst > term > term ) > subst ).
thf(func_def_222,type,
sK81: ( subst > term > term ) > subst ).
thf(func_def_223,type,
sK82: ( subst > term > term ) > term ).
thf(func_def_224,type,
sK83: ( subst > term > term ) > term ).
thf(func_def_225,type,
sK84: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_226,type,
sK85: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_227,type,
sK86: ( term > $o ) > term > term ).
thf(func_def_228,type,
sK87: subst > term > subst > $o ).
thf(func_def_229,type,
sK88: term ).
thf(func_def_230,type,
sK89: ( subst > term > term ) > term ).
thf(func_def_231,type,
sK90: ( subst > term > term ) > term ).
thf(func_def_232,type,
sK91: ( subst > term > term ) > subst ).
thf(func_def_233,type,
sK92: ( subst > term > term ) > subst ).
thf(func_def_234,type,
sK93: term ).
thf(func_def_235,type,
sK94: term > ( subst > term > subst > $o ) > term ).
thf(func_def_236,type,
sK95: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_237,type,
sK96: term > ( subst > term > subst > $o ) > term ).
thf(func_def_238,type,
sK97: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_239,type,
sK98: subst > term > subst > $o ).
thf(func_def_240,type,
sK99: term ).
thf(func_def_241,type,
sK100: ( subst > term > term ) > term ).
thf(func_def_242,type,
sK101: ( subst > term > term ) > subst ).
thf(func_def_243,type,
sK102: ( subst > term > term ) > term ).
thf(func_def_244,type,
sK103: ( subst > term > term ) > subst ).
thf(func_def_245,type,
sK104: term ).
thf(func_def_246,type,
sK105: subst > term > subst > $o ).
thf(func_def_247,type,
sK106: term > $o ).
thf(func_def_248,type,
sK107: ( subst > term > subst > $o ) > subst ).
thf(func_def_249,type,
sK108: ( subst > term > subst > $o ) > subst ).
thf(func_def_250,type,
sK109: ( subst > term > subst > $o ) > subst ).
thf(func_def_251,type,
sK110: ( subst > term > subst > $o ) > term ).
thf(func_def_252,type,
sK111: ( term > $o ) > term > term ).
thf(func_def_253,type,
sK112: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_254,type,
sK113: ( subst > term > subst > $o ) > term ).
thf(func_def_255,type,
sK114: ( subst > term > subst > $o ) > subst ).
thf(func_def_256,type,
sK115: ( subst > term > subst > $o ) > subst ).
thf(func_def_257,type,
sK116: ( subst > term > subst > $o ) > subst ).
thf(func_def_258,type,
sK117: term ).
thf(func_def_259,type,
sK118: ( subst > term > term ) > term ).
thf(func_def_260,type,
sK119: ( subst > term > term ) > subst ).
thf(func_def_261,type,
sK120: ( subst > term > term ) > subst ).
thf(func_def_262,type,
sK121: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_263,type,
sK122: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_264,type,
sK123: term ).
thf(func_def_265,type,
sK124: subst > term > subst > $o ).
thf(func_def_266,type,
sK125: ( subst > term > term ) > term ).
thf(func_def_267,type,
sK126: ( subst > term > term ) > subst ).
thf(func_def_268,type,
sK127: ( subst > term > term ) > subst ).
thf(func_def_269,type,
sK128: ( subst > term > term ) > term ).
thf(func_def_270,type,
sK129: ( subst > term > subst > $o ) > term > term ).
thf(func_def_271,type,
sK130: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_272,type,
sK131: term ).
thf(func_def_273,type,
sK132: subst > term > subst > $o ).
thf(func_def_274,type,
sK133: term > $o ).
thf(func_def_275,type,
sK134: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_276,type,
sK135: ( term > $o ) > term > term ).
thf(func_def_277,type,
sK136: ( subst > term > subst > $o ) > subst ).
thf(func_def_278,type,
sK137: ( subst > term > subst > $o ) > term ).
thf(func_def_279,type,
sK138: ( subst > term > subst > $o ) > subst ).
thf(func_def_280,type,
sK139: ( subst > term > subst > $o ) > subst ).
thf(func_def_281,type,
sK140: ( subst > term > subst > $o ) > subst ).
thf(func_def_282,type,
sK141: ( subst > term > subst > $o ) > subst ).
thf(func_def_283,type,
sK142: ( subst > term > subst > $o ) > subst ).
thf(func_def_284,type,
sK143: ( subst > term > subst > $o ) > term ).
thf(func_def_285,type,
sK144: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_286,type,
sK145: ( subst > term > term ) > subst ).
thf(func_def_287,type,
sK146: ( subst > term > term ) > subst ).
thf(func_def_288,type,
sK147: ( subst > term > term ) > term ).
thf(func_def_289,type,
sK148: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_290,type,
sK149: term ).
thf(func_def_291,type,
sK150: subst > term > subst > $o ).
thf(func_def_292,type,
sK151: ( subst > term > subst > $o ) > term > term ).
thf(func_def_293,type,
sK152: ( subst > term > subst > $o ) > subst ).
thf(func_def_294,type,
sK153: ( subst > term > subst > $o ) > subst ).
thf(func_def_295,type,
sK154: ( subst > term > subst > $o ) > term ).
thf(func_def_296,type,
sK155: ( subst > term > subst > $o ) > subst ).
thf(func_def_297,type,
sK156: ( subst > term > subst > $o ) > subst ).
thf(func_def_298,type,
sK157: ( subst > term > subst > $o ) > subst ).
thf(func_def_299,type,
sK158: ( subst > term > subst > $o ) > subst ).
thf(func_def_300,type,
sK159: ( subst > term > subst > $o ) > term ).
thf(func_def_301,type,
sK160: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_302,type,
sK161: ( subst > term > term ) > subst ).
thf(func_def_303,type,
sK162: ( subst > term > term ) > subst ).
thf(func_def_304,type,
sK163: ( subst > term > term ) > term ).
thf(func_def_305,type,
sK164: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_306,type,
sK165: term ).
thf(func_def_307,type,
sK166: subst > term > term ).
thf(func_def_308,type,
sK167: subst > term > term ).
thf(func_def_309,type,
sK168: subst ).
thf(func_def_310,type,
sK169: term ).
thf(func_def_311,type,
sK170: term ).
thf(func_def_312,type,
sK171: term ).
thf(func_def_313,type,
sK172: subst ).
thf(func_def_314,type,
sK173: term ).
thf(func_def_315,type,
sK174: subst ).
thf(func_def_316,type,
sK175: ( subst > term > term ) > term ).
thf(func_def_317,type,
sK176: ( subst > term > term ) > subst ).
thf(func_def_318,type,
sK177: ( subst > term > term ) > subst ).
thf(func_def_319,type,
sK178: ( subst > term > term ) > subst ).
thf(func_def_320,type,
sK179: ( subst > term > term ) > subst ).
thf(func_def_321,type,
sK180: ( subst > term > term ) > term ).
thf(func_def_322,type,
sK181: ( term > $o ) > term ).
thf(func_def_323,type,
sK182: ( term > $o ) > term ).
thf(func_def_324,type,
sK183: term ).
thf(func_def_325,type,
sK184: term > $o ).
thf(func_def_326,type,
sK185: subst ).
thf(func_def_327,type,
sK186: term ).
thf(func_def_328,type,
sK187: subst ).
thf(func_def_329,type,
sK188: subst ).
thf(func_def_330,type,
sK189: term ).
thf(func_def_331,type,
sK190: term ).
thf(func_def_332,type,
sK191: term > $o ).
thf(func_def_333,type,
sK192: subst ).
thf(func_def_334,type,
sK193: term > term ).
thf(func_def_335,type,
sK194: ( term > $o ) > term ).
thf(func_def_336,type,
sK195: subst > ( term > $o ) > term ).
thf(func_def_337,type,
sK196: ( term > $o ) > term ).
thf(func_def_338,type,
sK197: ( term > $o ) > term ).
thf(func_def_339,type,
sK198: ( term > $o ) > term ).
thf(func_def_340,type,
sK199: ( term > $o ) > term ).
thf(func_def_341,type,
sK200: term ).
thf(func_def_342,type,
sK201: subst > term > subst > $o ).
thf(func_def_343,type,
sK202: ( subst > term > term ) > term > subst ).
thf(func_def_344,type,
sK203: ( subst > term > term ) > term > term ).
thf(func_def_345,type,
sK204: ( subst > term > term ) > term > subst ).
thf(func_def_346,type,
sK205: ( subst > term > term ) > term ).
thf(func_def_347,type,
sK206: ( subst > term > term ) > term > term ).
thf(func_def_348,type,
sK207: ( subst > term > term ) > term > subst ).
thf(func_def_349,type,
sK208: ( subst > term > term ) > term > subst ).
thf(func_def_350,type,
sK209: ( subst > term > subst > $o ) > term > term ).
thf(func_def_351,type,
sK210: ( subst > term > subst > $o ) > term ).
thf(func_def_352,type,
sK211: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_353,type,
sK212: ( term > $o ) > term ).
thf(func_def_354,type,
sK213: subst > ( term > $o ) > term ).
thf(func_def_355,type,
sK214: term > $o ).
thf(func_def_356,type,
sK215: term ).
thf(func_def_357,type,
sK216: subst ).
thf(func_def_358,type,
sK217: term > term ).
thf(func_def_359,type,
sK218: term ).
thf(func_def_360,type,
sK219: term > $o ).
thf(func_def_361,type,
sK220: term ).
thf(func_def_362,type,
sK221: term > term ).
thf(func_def_363,type,
sK222: ( term > $o ) > term ).
thf(func_def_364,type,
sK223: ( term > $o ) > term ).
thf(func_def_365,type,
sK224: ( term > $o ) > term ).
thf(func_def_366,type,
sK225: ( term > $o ) > term ).
thf(func_def_367,type,
sK226: ( term > $o ) > term ).
thf(func_def_368,type,
sK227: ( term > $o ) > term ).
thf(func_def_369,type,
sK228: ( subst > term > term ) > subst ).
thf(func_def_370,type,
sK229: ( subst > term > term ) > term ).
thf(func_def_371,type,
sK230: ( subst > term > term ) > subst ).
thf(func_def_372,type,
sK231: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_373,type,
sK232: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_374,type,
sK233: term ).
thf(func_def_375,type,
sK234: subst > term > subst > $o ).
thf(func_def_376,type,
sK235: ( subst > term > subst > $o ) > subst ).
thf(func_def_377,type,
sK236: ( subst > term > subst > $o ) > subst ).
thf(func_def_378,type,
sK237: ( subst > term > subst > $o ) > term ).
thf(func_def_379,type,
sK238: ( subst > term > subst > $o ) > subst ).
thf(func_def_380,type,
sK239: ( subst > term > subst > $o ) > term ).
thf(func_def_381,type,
sK240: ( subst > term > subst > $o ) > subst ).
thf(func_def_382,type,
sK241: ( subst > term > subst > $o ) > subst ).
thf(func_def_383,type,
sK242: ( subst > term > subst > $o ) > subst ).
thf(func_def_384,type,
sK243: ( subst > term > subst > $o ) > term ).
thf(func_def_385,type,
sK244: ( subst > term > subst > $o ) > term ).
thf(func_def_386,type,
sK245: ( subst > term > subst > $o ) > term ).
thf(func_def_387,type,
sK246: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_388,type,
sK247: ( subst > term > term ) > subst ).
thf(func_def_389,type,
sK248: ( subst > term > term ) > term ).
thf(func_def_390,type,
sK249: ( subst > term > term ) > subst ).
thf(func_def_391,type,
sK250: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_392,type,
sK251: ( term > $o ) > term ).
thf(func_def_393,type,
sK252: ( term > $o ) > term ).
thf(func_def_394,type,
sK253: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_395,type,
sK254: term ).
thf(func_def_396,type,
sK255: term ).
thf(func_def_397,type,
sK256: term > $o ).
thf(func_def_398,type,
sK257: term > term ).
thf(func_def_399,type,
sK258: subst > term > subst > $o ).
thf(func_def_400,type,
sK259: ( term > $o ) > term ).
thf(func_def_401,type,
sK260: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_402,type,
sK261: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_403,type,
sK262: term > ( term > $o ) > term ).
thf(func_def_404,type,
sK263: term > $o ).
thf(func_def_405,type,
sK264: term ).
thf(func_def_406,type,
sK265: subst > term > subst > $o ).
thf(func_def_407,type,
sK266: ( subst > term > term ) > term ).
thf(func_def_408,type,
sK267: ( subst > term > term ) > term ).
thf(func_def_409,type,
sK268: ( subst > term > term ) > subst ).
thf(func_def_410,type,
sK269: ( subst > term > term ) > subst ).
thf(func_def_411,type,
sK270: term ).
thf(func_def_412,type,
sK271: subst > term > subst > $o ).
thf(func_def_413,type,
sK272: ( subst > term > subst > $o ) > subst ).
thf(func_def_414,type,
sK273: ( subst > term > subst > $o ) > term ).
thf(func_def_415,type,
sK274: ( subst > term > subst > $o ) > subst ).
thf(func_def_416,type,
sK275: ( subst > term > subst > $o ) > subst ).
thf(func_def_417,type,
sK276: ( subst > term > subst > $o ) > term ).
thf(func_def_418,type,
sK277: ( subst > term > subst > $o ) > subst ).
thf(func_def_419,type,
sK278: ( subst > term > subst > $o ) > subst ).
thf(func_def_420,type,
sK279: ( subst > term > subst > $o ) > term ).
thf(func_def_421,type,
sK280: ( subst > term > subst > $o ) > subst ).
thf(func_def_422,type,
sK281: ( subst > term > subst > $o ) > term ).
thf(func_def_423,type,
sK282: ( subst > term > subst > $o ) > term ).
thf(func_def_424,type,
sK283: ( subst > term > term ) > term ).
thf(func_def_425,type,
sK284: ( subst > term > term ) > subst ).
thf(func_def_426,type,
sK285: ( subst > term > term ) > subst ).
thf(func_def_427,type,
sK286: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_428,type,
sK287: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_429,type,
sK288: ( term > $o ) > term ).
thf(func_def_430,type,
sK289: ( term > $o ) > term ).
thf(func_def_431,type,
sK290: term ).
thf(func_def_432,type,
sK291: subst > term > subst > $o ).
thf(func_def_433,type,
sK292: ( subst > term > term ) > subst ).
thf(func_def_434,type,
sK293: ( subst > term > term ) > term ).
thf(func_def_435,type,
sK294: ( subst > term > term ) > subst ).
thf(func_def_436,type,
sK295: ( subst > term > term ) > term ).
thf(func_def_437,type,
sK296: ( subst > term > subst > $o ) > term > term ).
thf(func_def_438,type,
sK297: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_439,type,
sK298: term > $o ).
thf(func_def_440,type,
sK299: term ).
thf(func_def_441,type,
sK300: term > term ).
thf(func_def_442,type,
sK301: ( term > $o ) > term ).
thf(func_def_443,type,
sK302: ( term > $o ) > term ).
thf(func_def_444,type,
sK303: ( term > $o ) > term ).
thf(func_def_445,type,
sK304: ( term > $o ) > term ).
thf(func_def_446,type,
sK305: term > $o ).
thf(func_def_447,type,
sK306: subst > term > subst > $o ).
thf(func_def_448,type,
sK307: term ).
thf(func_def_449,type,
sK308: term ).
thf(func_def_450,type,
sK309: ( subst > term > subst > $o ) > subst ).
thf(func_def_451,type,
sK310: ( subst > term > subst > $o ) > subst ).
thf(func_def_452,type,
sK311: ( subst > term > subst > $o ) > term ).
thf(func_def_453,type,
sK312: ( subst > term > subst > $o ) > subst ).
thf(func_def_454,type,
sK313: ( subst > term > subst > $o ) > term ).
thf(func_def_455,type,
sK314: ( subst > term > subst > $o ) > subst ).
thf(func_def_456,type,
sK315: ( subst > term > subst > $o ) > subst ).
thf(func_def_457,type,
sK316: ( subst > term > subst > $o ) > subst ).
thf(func_def_458,type,
sK317: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_459,type,
sK318: ( subst > term > subst > $o ) > term ).
thf(func_def_460,type,
sK319: ( subst > term > subst > $o ) > term ).
thf(func_def_461,type,
sK320: term ).
thf(func_def_462,type,
sK321: subst > term > subst > $o ).
thf(func_def_463,type,
sK322: term > $o ).
thf(func_def_464,type,
sK323: ( subst > term > term ) > term ).
thf(func_def_465,type,
sK324: ( subst > term > term ) > subst ).
thf(func_def_466,type,
sK325: ( subst > term > term ) > subst ).
thf(func_def_467,type,
sK326: ( subst > term > term ) > term ).
thf(func_def_468,type,
sK327: term > subst > term > term ).
thf(func_def_469,type,
sK328: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_470,type,
sK329: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_471,type,
sK330: ( term > $o ) > term > term ).
thf(func_def_472,type,
sK331: term ).
thf(func_def_473,type,
sK332: ( subst > term > term ) > term ).
thf(func_def_474,type,
sK333: ( subst > term > term ) > subst ).
thf(func_def_475,type,
sK334: term ).
thf(func_def_476,type,
sK335: term ).
thf(func_def_477,type,
sK336: subst > term > subst > $o ).
thf(func_def_478,type,
sK337: term ).
thf(func_def_479,type,
sK338: ( subst > term > subst > $o ) > subst ).
thf(func_def_480,type,
sK339: ( subst > term > subst > $o ) > subst ).
thf(func_def_481,type,
sK340: ( subst > term > subst > $o ) > term ).
thf(func_def_482,type,
sK341: ( subst > term > subst > $o ) > subst ).
thf(func_def_483,type,
sK342: ( subst > term > subst > $o ) > term > term ).
thf(func_def_484,type,
sK343: ( subst > term > subst > $o ) > subst ).
thf(func_def_485,type,
sK344: ( subst > term > subst > $o ) > subst ).
thf(func_def_486,type,
sK345: ( subst > term > subst > $o ) > subst ).
thf(func_def_487,type,
sK346: ( subst > term > subst > $o ) > term ).
thf(func_def_488,type,
sK347: ( subst > term > subst > $o ) > term > term ).
thf(func_def_489,type,
sK348: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_490,type,
sK349: term ).
thf(func_def_491,type,
sK350: subst > term > subst > $o ).
thf(func_def_492,type,
sK351: ( subst > term > term ) > subst ).
thf(func_def_493,type,
sK352: ( subst > term > term ) > subst ).
thf(func_def_494,type,
sK353: ( subst > term > term ) > term ).
thf(func_def_495,type,
sK354: ( subst > term > term ) > term ).
thf(func_def_496,type,
sK355: ( subst > term > term ) > term ).
thf(func_def_497,type,
sK356: ( subst > term > term ) > subst ).
thf(func_def_498,type,
sK357: ( subst > term > term ) > subst ).
thf(func_def_499,type,
sK358: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_500,type,
sK359: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_501,type,
sK360: ( term > $o ) > term ).
thf(func_def_502,type,
sK361: ( term > $o ) > term ).
thf(func_def_503,type,
sK362: term ).
thf(func_def_504,type,
sK363: term > $o ).
thf(func_def_505,type,
sK364: term > term ).
thf(func_def_506,type,
sK365: ( subst > term > subst > $o ) > subst ).
thf(func_def_507,type,
sK366: ( subst > term > subst > $o ) > subst ).
thf(func_def_508,type,
sK367: ( subst > term > subst > $o ) > term ).
thf(func_def_509,type,
sK368: ( subst > term > subst > $o ) > subst ).
thf(func_def_510,type,
sK369: ( subst > term > subst > $o ) > subst ).
thf(func_def_511,type,
sK370: ( subst > term > subst > $o ) > subst ).
thf(func_def_512,type,
sK371: ( subst > term > subst > $o ) > term ).
thf(func_def_513,type,
sK372: ( subst > term > subst > $o ) > subst ).
thf(func_def_514,type,
sK373: term ).
thf(func_def_515,type,
sK374: subst > term > subst > $o ).
thf(func_def_516,type,
sK375: term ).
thf(func_def_517,type,
sK376: term ).
thf(func_def_518,type,
sK377: subst > term > subst > $o ).
thf(func_def_519,type,
sK378: ( subst > term > subst > $o ) > subst ).
thf(func_def_520,type,
sK379: ( subst > term > subst > $o ) > subst ).
thf(func_def_521,type,
sK380: ( subst > term > subst > $o ) > subst ).
thf(func_def_522,type,
sK381: ( subst > term > subst > $o ) > term ).
thf(func_def_523,type,
sK382: ( subst > term > subst > $o ) > subst ).
thf(func_def_524,type,
sK383: ( subst > term > subst > $o ) > term ).
thf(func_def_525,type,
sK384: ( subst > term > subst > $o ) > subst ).
thf(func_def_526,type,
sK385: ( subst > term > subst > $o ) > subst ).
thf(func_def_527,type,
sK386: ( subst > term > subst > $o ) > term ).
thf(func_def_528,type,
sK387: ( subst > term > subst > $o ) > subst ).
thf(func_def_529,type,
sK388: ( subst > term > subst > $o ) > subst ).
thf(func_def_530,type,
sK389: ( subst > term > subst > $o ) > term ).
thf(func_def_531,type,
sK390: ( subst > term > subst > $o ) > subst ).
thf(func_def_532,type,
sK391: ( subst > term > subst > $o ) > term > term ).
thf(func_def_533,type,
sK392: ( subst > term > subst > $o ) > term ).
thf(func_def_534,type,
sK393: ( subst > term > subst > $o ) > subst ).
thf(func_def_535,type,
sK394: ( subst > term > subst > $o ) > subst ).
thf(func_def_536,type,
sK395: ( subst > term > subst > $o ) > subst ).
thf(func_def_537,type,
sK396: term ).
thf(func_def_538,type,
sK397: subst > term > subst > $o ).
thf(func_def_539,type,
sK398: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_540,type,
sK399: ( subst > term > term ) > term ).
thf(func_def_541,type,
sK400: ( subst > term > term ) > subst ).
thf(func_def_542,type,
sK401: ( subst > term > term ) > subst ).
thf(func_def_543,type,
sK402: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_544,type,
sK403: ( subst > term > subst > $o ) > term ).
thf(func_def_545,type,
sK404: ( subst > term > subst > $o ) > term ).
thf(func_def_546,type,
sK405: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_547,type,
sK406: ( subst > term > term ) > term ).
thf(func_def_548,type,
sK407: ( subst > term > term ) > subst ).
thf(func_def_549,type,
sK408: ( subst > term > term ) > subst ).
thf(func_def_550,type,
sK409: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_551,type,
sK410: ( term > $o ) > term ).
thf(func_def_552,type,
sK411: ( term > $o ) > term ).
thf(func_def_553,type,
sK412: ( subst > term > subst > $o ) > term ).
thf(func_def_554,type,
sK413: ( subst > term > subst > $o ) > term ).
thf(func_def_555,type,
sK414: ( term > $o ) > term ).
thf(func_def_556,type,
sK415: ( term > $o ) > term ).
thf(func_def_557,type,
sK416: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_558,type,
sK417: term > $o ).
thf(func_def_559,type,
sK418: term ).
thf(func_def_560,type,
sK419: term > term ).
thf(func_def_561,type,
sK420: subst > term > subst > $o ).
thf(func_def_562,type,
sK421: ( term > $o ) > term ).
thf(func_def_563,type,
sK422: term ).
thf(func_def_564,type,
sK423: subst > term > subst > $o ).
thf(func_def_565,type,
sK424: term ).
thf(func_def_566,type,
sK425: ( subst > term > term ) > term ).
thf(func_def_567,type,
sK426: ( subst > term > term ) > subst ).
thf(func_def_568,type,
sK427: ( subst > term > term ) > term ).
thf(func_def_569,type,
sK428: ( subst > term > term ) > subst ).
thf(func_def_570,type,
sK429: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_571,type,
sK430: term > ( term > $o ) > term ).
thf(func_def_572,type,
sK431: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_573,type,
sK432: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_574,type,
sK433: subst > term > subst > $o ).
thf(func_def_575,type,
sK434: term > $o ).
thf(func_def_576,type,
sK435: term ).
thf(func_def_577,type,
sK436: ( subst > term > term ) > subst ).
thf(func_def_578,type,
sK437: ( subst > term > term ) > subst ).
thf(func_def_579,type,
sK438: ( subst > term > term ) > term ).
thf(func_def_580,type,
sK439: ( subst > term > term ) > term ).
thf(func_def_581,type,
sK440: ( term > $o ) > term ).
thf(func_def_582,type,
sK441: ( term > $o ) > term ).
thf(func_def_583,type,
sK442: ( subst > term > subst > $o ) > term ).
thf(func_def_584,type,
sK443: ( subst > term > subst > $o ) > term ).
thf(func_def_585,type,
sK444: subst > term > subst > $o ).
thf(func_def_586,type,
sK445: term ).
thf(func_def_587,type,
sK446: term ).
thf(func_def_588,type,
sK447: ( subst > term > subst > $o ) > subst ).
thf(func_def_589,type,
sK448: ( subst > term > subst > $o ) > subst ).
thf(func_def_590,type,
sK449: ( subst > term > subst > $o ) > term ).
thf(func_def_591,type,
sK450: ( subst > term > subst > $o ) > subst ).
thf(func_def_592,type,
sK451: ( subst > term > subst > $o ) > subst ).
thf(func_def_593,type,
sK452: ( subst > term > subst > $o ) > subst ).
thf(func_def_594,type,
sK453: ( subst > term > subst > $o ) > term ).
thf(func_def_595,type,
sK454: ( subst > term > subst > $o ) > subst ).
thf(func_def_596,type,
sK455: ( subst > term > subst > $o ) > term ).
thf(func_def_597,type,
sK456: ( subst > term > subst > $o ) > term ).
thf(func_def_598,type,
sK457: term ).
thf(func_def_599,type,
sK458: subst ).
thf(func_def_600,type,
sK459: term > $o ).
thf(func_def_601,type,
sK460: ( term > $o ) > term ).
thf(func_def_602,type,
sK461: ( term > $o ) > subst > term > term > $o ).
thf(func_def_603,type,
sK462: subst ).
thf(func_def_604,type,
sK463: term ).
thf(func_def_605,type,
sK464: term > $o ).
thf(func_def_606,type,
sK465: term ).
thf(func_def_607,type,
sK466: ( term > $o ) > term ).
thf(func_def_608,type,
sK467: ( term > $o ) > term ).
thf(func_def_609,type,
sK468: term > $o ).
thf(func_def_610,type,
sK469: term ).
thf(func_def_611,type,
sK470: term ).
thf(func_def_612,type,
sK471: subst ).
thf(func_def_613,type,
sK472: term > $o ).
thf(func_def_614,type,
sK473: term ).
thf(func_def_615,type,
sK474: term ).
thf(func_def_616,type,
sK475: term ).
thf(func_def_617,type,
sK476: term ).
thf(func_def_618,type,
sK477: subst > d_subst ).
thf(func_def_619,type,
sK478: term ).
thf(func_def_620,type,
sK479: term ).
thf(func_def_621,type,
sK480: term ).
thf(func_def_622,type,
sK481: term ).
thf(func_def_623,type,
sK482: term ).
thf(func_def_624,type,
sK483: term ).
thf(func_def_625,type,
sK484: term ).
thf(func_def_626,type,
sK485: term ).
thf(func_def_627,type,
sK486: term ).
thf(func_def_628,type,
sK487: term > d_term ).
thf(func_def_629,type,
sK488: term ).
thf(func_def_630,type,
sK489: term ).
thf(func_def_631,type,
sK490: subst > term > subst > $o ).
thf(func_def_632,type,
sK491: ( term > $o ) > term ).
thf(func_def_633,type,
sK492: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_634,type,
sK493: term > subst > term > term ).
thf(func_def_635,type,
sK494: term ).
thf(func_def_636,type,
sK495: ( subst > term > term ) > subst ).
thf(func_def_637,type,
sK496: ( subst > term > term ) > term ).
thf(func_def_638,type,
sK497: term ).
thf(func_def_639,type,
sK498: ( subst > term > term ) > subst ).
thf(func_def_640,type,
sK499: ( subst > term > term ) > term ).
thf(func_def_641,type,
sK500: term > subst > term > term ).
thf(func_def_642,type,
sK501: subst ).
thf(func_def_643,type,
sK502: term ).
thf(func_def_644,type,
sK503: term ).
thf(func_def_645,type,
sK504: term ).
thf(func_def_646,type,
sK505: term ).
thf(func_def_647,type,
sK506: term ).
thf(func_def_648,type,
sK507: term > subst > term > term ).
thf(func_def_649,type,
sK508: subst ).
thf(func_def_650,type,
sK509: subst ).
thf(func_def_651,type,
sK510: subst ).
thf(func_def_652,type,
sK511: subst ).
thf(func_def_653,type,
sK512: term ).
thf(func_def_654,type,
sK513: subst ).
thf(func_def_655,type,
sK514: subst ).
thf(func_def_656,type,
sK515: subst > term > subst > $o ).
thf(func_def_657,type,
sK516: term ).
thf(func_def_658,type,
sK517: ( subst > term > subst > $o ) > term ).
thf(func_def_659,type,
sK518: ( subst > term > subst > $o ) > term ).
thf(func_def_660,type,
sK519: term ).
thf(func_def_661,type,
sK520: term ).
thf(func_def_662,type,
sK521: subst ).
thf(func_def_663,type,
sK522: term ).
thf(func_def_664,type,
sK523: term ).
thf(func_def_665,type,
sK524: term ).
thf(func_def_666,type,
sK525: term ).
thf(func_def_667,type,
sK526: term ).
thf(func_def_668,type,
sK527: term ).
thf(func_def_669,type,
sK528: subst > term > term ).
thf(func_def_670,type,
sK529: ( subst > term > term ) > subst ).
thf(func_def_671,type,
sK530: ( subst > term > term ) > subst ).
thf(func_def_672,type,
sK531: ( subst > term > term ) > term ).
thf(func_def_673,type,
sK532: term ).
thf(func_def_674,type,
sK533: term ).
thf(func_def_675,type,
sK534: term ).
thf(func_def_676,type,
sK535: term ).
thf(func_def_677,type,
sK536: term ).
thf(func_def_678,type,
sK537: term ).
thf(func_def_679,type,
sK538: term ).
thf(func_def_680,type,
sK539: term ).
thf(func_def_681,type,
sK540: term ).
thf(func_def_682,type,
sK541: term ).
thf(func_def_683,type,
sK542: term ).
thf(func_def_684,type,
sK543: term ).
thf(func_def_685,type,
sK544: term > subst > term > term ).
thf(func_def_686,type,
sK545: term ).
thf(func_def_687,type,
sK546: subst ).
thf(func_def_688,type,
sK547: term > $o ).
thf(func_def_689,type,
sK548: term > $o ).
thf(func_def_690,type,
sK549: term ).
thf(func_def_691,type,
sK550: subst ).
thf(func_def_692,type,
sK551: term ).
thf(func_def_693,type,
sK552: term > ( term > $o ) > ( term > $o ) > subst > term ).
thf(func_def_694,type,
sK553: term > ( term > $o ) > subst > ( term > $o ) > term ).
thf(func_def_695,type,
sK554: term > $o ).
thf(func_def_696,type,
sK555: subst ).
thf(func_def_697,type,
sK556: term > $o ).
thf(func_def_698,type,
sK557: term ).
thf(func_def_699,type,
sK558: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_700,type,
sK559: subst > term > subst > $o ).
thf(func_def_701,type,
sK560: ( term > $o ) > term ).
thf(func_def_702,type,
sK561: term ).
thf(func_def_703,type,
sK562: term ).
thf(func_def_704,type,
sK563: term ).
thf(func_def_705,type,
sK564: term ).
thf(func_def_706,type,
sK565: subst ).
thf(func_def_707,type,
sK566: subst ).
thf(func_def_708,type,
sK567: subst > term > term ).
thf(func_def_709,type,
sK568: subst > term > term ).
thf(func_def_710,type,
sK569: subst ).
thf(func_def_711,type,
sK570: term ).
thf(func_def_712,type,
sK571: ( subst > term > term ) > term ).
thf(func_def_713,type,
sK572: ( subst > term > term ) > subst ).
thf(func_def_714,type,
sK573: ( subst > term > term ) > subst ).
thf(func_def_715,type,
sK574: ( subst > term > term ) > term ).
thf(func_def_716,type,
sK575: ( subst > term > term ) > subst ).
thf(func_def_717,type,
sK576: ( subst > term > term ) > subst ).
thf(func_def_718,type,
sK577: subst ).
thf(func_def_719,type,
sK578: term ).
thf(func_def_720,type,
sK579: subst ).
thf(func_def_721,type,
sK580: term ).
thf(func_def_722,type,
sK581: term > $o ).
thf(func_def_723,type,
sK582: term > $o ).
thf(func_def_724,type,
sK583: term ).
thf(func_def_725,type,
sK584: subst ).
thf(func_def_726,type,
sK585: term ).
thf(func_def_727,type,
sK586: term > $o ).
thf(func_def_728,type,
sK587: term ).
thf(func_def_729,type,
sK588: subst ).
thf(func_def_730,type,
sK589: ( term > $o ) > term ).
thf(func_def_731,type,
sK590: subst > term > ( term > $o ) > term > $o ).
thf(func_def_733,type,
sK592: subst ).
thf(func_def_734,type,
sK593: subst ).
thf(func_def_735,type,
sK594: term ).
thf(f2675,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2674,f2587]) ).
thf(f2587,plain,
! [X62: term] :
( ( d2term @ d_one )
= X62 ),
inference(forward_demodulation,[status(thm)],[f1212,f1038]) ).
thf(f1038,plain,
! [X263: d_term] : ( d_one = X263 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f545,plain,
( ( ( pushprop_lem0_lthm = $true )
| ! [X3: term > $o] :
( ( sK459 @ ( sub @ ( sK460 @ X3 ) @ ( push @ sK457 @ sK458 ) ) )
!= ( X3 @ ( sK460 @ X3 ) ) ) )
& ( ! [X5: term,X6: subst,X7: term > $o,X9: term] :
( ( X7 @ ( sub @ X9 @ ( push @ X5 @ X6 ) ) )
= ( sK461 @ X7 @ X6 @ X5 @ X9 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ! [X10: subst] :
( ( comp @ X10 @ id )
= X10 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ( sK462
!= ( comp @ sK462 @ id ) ) )
& ! [X12: d_subst,X13: d_subst] :
( ( ( d2subst @ X13 )
!= ( d2subst @ X12 ) )
| ( X12 = X13 ) )
& ( hoasinduction_lem2_lthm = $true )
& ( axclos = $true )
& ( ( $true = sP58 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP58 ) )
& ( ! [X14: term] :
( ( sub @ X14 @ id )
= X14 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ( sK463
!= ( sub @ sK463 @ id ) ) )
& ( substmonoid_gthm = $true )
& ( ( induction = $true )
| ( ( ( sP56 @ sK464 )
!= $true )
& ! [X18: term] :
( ( ( var @ X18 )
!= $true )
| ( $true
= ( sK464 @ X18 ) ) )
& ( ( sK464 @ sK465 )
!= $true )
& ! [X19: term] :
( ( $true
= ( sK464 @ ( lam @ X19 ) ) )
| ( ( sK464 @ X19 )
!= $true ) ) ) )
& ( ! [X20: term > $o,X21: term] :
( ( ( sP56 @ X20 )
= $true )
| ( ( $true
= ( var @ ( sK466 @ X20 ) ) )
& ( ( X20 @ ( sK466 @ X20 ) )
!= $true ) )
| ( ( X20 @ X21 )
= $true )
| ( ( ( X20 @ ( lam @ ( sK467 @ X20 ) ) )
!= $true )
& ( ( X20 @ ( sK467 @ X20 ) )
= $true ) ) )
| ( induction != $true ) )
& ( ( pushprop_lem1 = $true )
| ( ( $true
!= ( sK468 @ ( sub @ sK469 @ ( push @ sK470 @ sK471 ) ) ) )
& ( $true
= ( sK472 @ sK469 ) ) ) )
& ( ! [X29: term > $o,X30: term,X31: term,X32: subst,X33: term > $o] :
( ( ( X29 @ ( sub @ X30 @ ( push @ X31 @ X32 ) ) )
= $true )
| ( ( X33 @ X30 )
!= $true ) )
| ( pushprop_lem1 != $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( axassoc = $true )
& ( ! [X34: term,X35: term,X36: term,X37: term] :
( ( ( ap @ X37 @ X34 )
!= ( ap @ X35 @ X36 ) )
| ( X35 = X37 ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ( ( ( ap @ sK474 @ sK475 )
= ( ap @ sK476 @ sK473 ) )
& ( sK476 != sK474 ) ) )
& ( hoasapinj2_gthm = $true )
& ! [X42: subst] :
( ( d2subst @ ( sK477 @ X42 ) )
= X42 )
& ( ( ( sK481 != sK480 )
& ( ( ap @ sK481 @ sK478 )
= ( ap @ sK480 @ sK479 ) ) )
| ( ( sub @ sK482 @ id )
!= sK482 )
| ! [X49: term,X50: term,X51: term,X52: term] :
( ( X49 = X50 )
| ( ( ap @ ( sub @ X49 @ id ) @ X51 )
!= ( ap @ ( sub @ X50 @ id ) @ X52 ) ) )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X53: term,X54: term,X55: term,X56: term] :
( ( X55 = X56 )
| ( ( ap @ X55 @ X54 )
!= ( ap @ X56 @ X53 ) ) )
& ! [X57: term] :
( ( sub @ X57 @ id )
= X57 )
& ( sK484 != sK483 )
& ( ( ap @ ( sub @ sK484 @ id ) @ sK486 )
= ( ap @ ( sub @ sK483 @ id ) @ sK485 ) ) ) )
& ( hoasapnotvar = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP55 ) )
& ( ( $true = sP55 )
| ( hoasinduction_lem3aa != $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar_lthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( sP54 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP54 != $true ) )
& ( termmset_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ! [X62: term] :
( ( d2term @ ( sK487 @ X62 ) )
= X62 )
& ( ( shinj = $true )
| ( ( sK488 != sK489 )
& ( ( sub @ sK488 @ sh )
= ( sub @ sK489 @ sh ) ) ) )
& ( ! [X66: term,X67: term] :
( ( X66 = X67 )
| ( ( sub @ X66 @ sh )
!= ( sub @ X67 @ sh ) ) )
| ( shinj != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ! [X69: term > $o] :
( ( sK490 @ id @ ( sK491 @ X69 ) @ id )
!= ( X69 @ ( sK491 @ X69 ) ) ) )
& ( ! [X71: subst > term > subst > $o,X73: term] :
( ( sK492 @ X71 @ X73 )
= ( X71 @ id @ X73 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( induction2_gthm = $true )
& ( ! [X74: term,X76: subst,X77: term] :
( ( sub @ X74 @ ( push @ X77 @ X76 ) )
= ( sK493 @ X74 @ X76 @ X77 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ! [X79: subst > term > term] :
( ( X79 @ ( sK495 @ X79 ) @ ( sK496 @ X79 ) )
!= ( sub @ sK494 @ ( push @ ( sK496 @ X79 ) @ ( sK495 @ X79 ) ) ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ! [X83: subst > term > term] :
( ( sub @ sK497 @ ( push @ ( sK499 @ X83 ) @ ( sK498 @ X83 ) ) )
!= ( X83 @ ( sK498 @ X83 ) @ ( sK499 @ X83 ) ) ) )
& ( ! [X86: term,X88: subst,X89: term] :
( ( sub @ X86 @ ( push @ X89 @ X88 ) )
= ( sK500 @ X86 @ X88 @ X89 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP53 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP53 != $true ) )
& ! [X90: d_subst] : ( d_id = X90 )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X91: subst,X92: term] :
( ( sub @ one @ ( push @ X92 @ X91 ) )
= X92 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ( ( sub @ one @ ( push @ sK502 @ sK501 ) )
!= sK502 ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP51 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP51 ) )
& ( axapp = $true )
& ( induction2lem_gthm = $true )
& ( ulamvarsh = $true )
& ( axvarshift = $true )
& ( ulamvarind = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( termmset = $true )
| ( ( sub @ sK503 @ id )
!= sK503 ) )
& ( ! [X96: term] :
( ( sub @ X96 @ id )
= X96 )
| ( termmset != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( sP49 != $true ) )
& ( ( sP49 = $true )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( apnotvar = $true )
& ( ! [X97: term] :
( ( sub @ X97 @ id )
= X97 )
| ( sK504
!= ( sub @ sK504 @ id ) )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ( ( sub @ sK505 @ id )
!= sK505 )
& ! [X100: term] :
( ( sub @ X100 @ id )
= X100 ) ) )
& ( ( sP48 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP48 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP46 ) )
& ( ( $true = sP46 )
| ( hoasinduction_lem3 != $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP44 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( sP44 != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasinduction_lem3b = $true )
| ! [X102: subst > term > term] :
( ( X102 @ sh @ one )
!= ( sub @ sK506 @ ( push @ one @ sh ) ) ) )
& ( ! [X103: term] :
( ( sK507 @ X103 @ sh @ one )
= ( sub @ X103 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( axmap = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop = $true )
& ( ( substmonoid_lthm = $true )
| ( ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 )
& ( ( ( comp @ sK508 @ id )
!= sK508 )
| ( sK509
!= ( comp @ id @ sK509 ) ) )
& ! [X108: subst] :
( ( comp @ id @ X108 )
= X108 ) ) )
& ( ( ( comp @ sK510 @ id )
!= sK510 )
| ( ! [X110: subst] :
( ( comp @ X110 @ id )
= X110 )
& ! [X111: subst] :
( ( comp @ id @ X111 )
= X111 ) )
| ( sK511
!= ( comp @ id @ sK511 ) )
| ( substmonoid_lthm != $true ) )
& ( axabs = $true )
& ( lamnotvar = $true )
& ( ! [X113: term,X114: subst] :
( ( comp @ sh @ ( push @ X113 @ X114 ) )
= X114 )
| ( axshiftcons != $true ) )
& ( ( axshiftcons = $true )
| ( sK513
!= ( comp @ sh @ ( push @ sK512 @ sK513 ) ) ) )
& ( ( sP43 = $true )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( sP43 != $true ) )
& ( ( $true = sP39 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP39 ) )
& ( ( sP38 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP38 != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X117: subst] :
( ( push @ ( sub @ one @ X117 ) @ ( comp @ sh @ X117 ) )
= X117 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ( sK514
!= ( push @ ( sub @ one @ sK514 ) @ ( comp @ sh @ sK514 ) ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( pushprop_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ( ( ( sK515 @ id @ sK516 @ id )
!= $true )
& ! [X121: term,X122: term] :
( ( ( sK515 @ id @ ( ap @ ( sub @ X121 @ id ) @ X122 ) @ id )
= $true )
| ( ( sK515 @ id @ X121 @ id )
!= $true )
| ( ( sK515 @ id @ X122 @ id )
!= $true ) )
& ( ( sP34 @ sK515 )
!= $true ) ) )
& ( ! [X123: subst > term > subst > $o,X124: term] :
( ( $true
= ( X123 @ id @ X124 @ id ) )
| ( ( ( X123 @ id @ ( ap @ ( sub @ ( sK517 @ X123 ) @ id ) @ ( sK518 @ X123 ) ) @ id )
!= $true )
& ( ( X123 @ id @ ( sK517 @ X123 ) @ id )
= $true )
& ( ( X123 @ id @ ( sK518 @ X123 ) @ id )
= $true ) )
| ( $true
= ( sP34 @ X123 ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction = $true )
| ( sP33 != $true ) )
& ( ( sP33 = $true )
| ( hoasinduction != $true ) )
& ( ( laminj = $true )
| ( ( sK520 != sK519 )
& ( ( lam @ sK520 )
= ( lam @ sK519 ) ) ) )
& ( ! [X129: term,X130: term] :
( ( X129 = X130 )
| ( ( lam @ X129 )
!= ( lam @ X130 ) ) )
| ( laminj != $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( axidl = $true )
| ( ( comp @ id @ sK521 )
!= sK521 ) )
& ( ! [X132: subst] :
( ( comp @ id @ X132 )
= X132 )
| ( axidl != $true ) )
& ( ( hoasapinj2 = $true )
| ( ( sK525 != sK524 )
& ( ( ap @ ( sub @ sK523 @ id ) @ sK525 )
= ( ap @ ( sub @ sK522 @ id ) @ sK524 ) ) ) )
& ( ! [X137: term,X138: term,X139: term,X140: term] :
( ( X139 = X140 )
| ( ( ap @ ( sub @ X138 @ id ) @ X140 )
!= ( ap @ ( sub @ X137 @ id ) @ X139 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoaslamnotap = $true )
| ( ! [X144: subst,X145: subst,X146: term] :
( ( sK528 @ ( comp @ X145 @ X144 ) @ ( sub @ X146 @ X144 ) )
= ( sub @ ( sK528 @ X145 @ X146 ) @ X144 ) )
& ( ( lam @ ( sK528 @ sh @ one ) )
= ( ap @ ( sub @ sK526 @ id ) @ sK527 ) ) ) )
& ( ! [X147: term,X148: term,X149: subst > term > term] :
( ( ( X149 @ ( comp @ ( sK530 @ X149 ) @ ( sK529 @ X149 ) ) @ ( sub @ ( sK531 @ X149 ) @ ( sK529 @ X149 ) ) )
!= ( sub @ ( X149 @ ( sK530 @ X149 ) @ ( sK531 @ X149 ) ) @ ( sK529 @ X149 ) ) )
| ( ( ap @ ( sub @ X147 @ id ) @ X148 )
!= ( lam @ ( X149 @ sh @ one ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP30 != $true ) )
& ( ( sP30 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( sP24 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP24 != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( ( sK532 != sK534 )
& ( ( ap @ ( sub @ sK533 @ id ) @ sK532 )
= ( ap @ ( sub @ sK535 @ id ) @ sK534 ) )
& ! [X157: term,X158: term,X159: term,X160: term] :
( ( ( ap @ X157 @ X159 )
!= ( ap @ X160 @ X158 ) )
| ( X158 = X159 ) ) ) )
& ( ! [X161: term,X162: term,X163: term,X164: term] :
( ( X161 = X163 )
| ( ( ap @ ( sub @ X162 @ id ) @ X161 )
!= ( ap @ ( sub @ X164 @ id ) @ X163 ) ) )
| ( ( ( ap @ sK539 @ sK537 )
= ( ap @ sK536 @ sK538 ) )
& ( sK537 != sK538 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( hoasapinj1 = $true )
| ( ( sK543 != sK540 )
& ( ( ap @ ( sub @ sK540 @ id ) @ sK541 )
= ( ap @ ( sub @ sK543 @ id ) @ sK542 ) ) ) )
& ( ! [X173: term,X174: term,X175: term,X176: term] :
( ( X173 = X176 )
| ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X176 @ id ) @ X175 ) ) )
| ( hoasapinj1 != $true ) )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ! [X177: term] :
( ( sub @ X177 @ ( push @ one @ sh ) )
= ( sK544 @ X177 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ! [X180: subst > term > term] :
( ( X180 @ sh @ one )
!= ( sub @ sK545 @ ( push @ one @ sh ) ) ) )
& ( ( sP23 = $true )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( sP23 != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ! [X181: subst,X182: term] :
( ( sub @ one @ ( push @ X182 @ X181 ) )
= X182 )
& ! [X187: term] :
( ( sK548 @ ( sub @ X187 @ ( push @ sK549 @ sK546 ) ) )
= ( sK547 @ X187 ) )
& ( ( sK547 @ one )
!= $true )
& ( $true
= ( sK548 @ sK549 ) ) ) )
& ( ( ( sub @ one @ ( push @ sK551 @ sK550 ) )
!= sK551 )
| ! [X190: subst,X191: term > $o,X192: term > $o,X193: term] :
( ( ( X191 @ ( sK552 @ X193 @ X192 @ X191 @ X190 ) )
!= ( X192 @ ( sub @ ( sK552 @ X193 @ X192 @ X191 @ X190 ) @ ( push @ X193 @ X190 ) ) ) )
| ( ( X191 @ one )
= $true )
| ( $true
!= ( X192 @ X193 ) ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ! [X195: term > $o,X196: subst,X197: term > $o,X198: term] :
( ( ( X197 @ X198 )
!= $true )
| ( ( X197 @ ( sub @ ( sK553 @ X198 @ X197 @ X196 @ X195 ) @ ( push @ X198 @ X196 ) ) )
!= ( X195 @ ( sK553 @ X198 @ X197 @ X196 @ X195 ) ) )
| ( $true
= ( X195 @ one ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ( ( $true
= ( sK556 @ sK557 ) )
& ! [X204: term] :
( ( sK556 @ ( sub @ X204 @ ( push @ sK557 @ sK555 ) ) )
= ( sK554 @ X204 ) )
& ( ( sK554 @ one )
!= $true ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( sP21 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP21 != $true ) )
& ( pushprop_gthm = $true )
& ( ! [X205: subst > term > subst > $o,X207: term] :
( ( sK558 @ X205 @ X207 )
= ( X205 @ id @ X207 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ! [X209: term > $o] :
( ( sK559 @ id @ ( sK560 @ X209 ) @ id )
!= ( X209 @ ( sK560 @ X209 ) ) ) )
& ( ( $true = sP19 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP19 ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj1_gthm = $true )
& ( ! [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X214 @ X212 )
!= ( ap @ X213 @ X211 ) )
| ( X211 = X212 ) )
| ( apinj2 != $true ) )
& ( ( apinj2 = $true )
| ( ( ( ap @ sK564 @ sK562 )
= ( ap @ sK563 @ sK561 ) )
& ( sK561 != sK562 ) ) )
& ( ( substmonoid = $true )
| ( sK565
!= ( comp @ id @ sK565 ) )
| ( sK566
!= ( comp @ sK566 @ id ) ) )
& ( ( ! [X221: subst] :
( ( comp @ id @ X221 )
= X221 )
& ! [X222: subst] :
( ( comp @ X222 @ id )
= X222 ) )
| ( substmonoid != $true ) )
& ( ( $true = sP18 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP18 ) )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP15 ) )
& ( ( $true = sP15 )
| ( hoasinduction_lthm != $true ) )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP6 != $true ) )
& ( ( sP6 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoaslaminj = $true )
| ( ( ( lam @ ( sK568 @ sh @ one ) )
= ( lam @ ( sK567 @ sh @ one ) ) )
& ! [X227: term,X228: subst,X229: subst] :
( ( sub @ ( sK567 @ X229 @ X227 ) @ X228 )
= ( sK567 @ ( comp @ X229 @ X228 ) @ ( sub @ X227 @ X228 ) ) )
& ( ( sK567 @ sK569 @ sK570 )
!= ( sK568 @ sK569 @ sK570 ) )
& ! [X230: term,X231: subst,X232: subst] :
( ( sK568 @ ( comp @ X231 @ X232 ) @ ( sub @ X230 @ X232 ) )
= ( sub @ ( sK568 @ X231 @ X230 ) @ X232 ) ) ) )
& ( ! [X233: subst > term > term,X234: subst > term > term,X235: subst,X236: term] :
( ( ( lam @ ( X233 @ sh @ one ) )
!= ( lam @ ( X234 @ sh @ one ) ) )
| ( ( sub @ ( X233 @ ( sK573 @ X233 ) @ ( sK571 @ X233 ) ) @ ( sK572 @ X233 ) )
!= ( X233 @ ( comp @ ( sK573 @ X233 ) @ ( sK572 @ X233 ) ) @ ( sub @ ( sK571 @ X233 ) @ ( sK572 @ X233 ) ) ) )
| ( ( X233 @ X235 @ X236 )
= ( X234 @ X235 @ X236 ) )
| ( ( X234 @ ( comp @ ( sK575 @ X234 ) @ ( sK576 @ X234 ) ) @ ( sub @ ( sK574 @ X234 ) @ ( sK576 @ X234 ) ) )
!= ( sub @ ( X234 @ ( sK575 @ X234 ) @ ( sK574 @ X234 ) ) @ ( sK576 @ X234 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ! [X243: term > $o,X244: term > $o,X245: term,X246: subst,X247: term] :
( ( ( X244 @ ( sub @ X247 @ ( push @ X245 @ X246 ) ) )
= $true )
| ( ( X243 @ X247 )
!= $true ) )
| ( sK577
!= ( comp @ sh @ ( push @ sK578 @ sK577 ) ) )
| ( sK580
!= ( sub @ one @ ( push @ sK580 @ sK579 ) ) )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ( $true
!= ( sK582 @ ( sub @ sK585 @ ( push @ sK583 @ sK584 ) ) ) )
& ( $true
= ( sK581 @ sK585 ) )
& ! [X257: subst,X258: term] :
( ( comp @ sh @ ( push @ X258 @ X257 ) )
= X257 )
& ! [X259: subst,X260: term] :
( ( sub @ one @ ( push @ X260 @ X259 ) )
= X260 ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ulamvar1 != $true )
& ( hoasinduction_lem2_gthm = $true )
& ! [X261: d_term,X262: d_term] :
( ( ( d2term @ X262 )
!= ( d2term @ X261 ) )
| ( X261 = X262 ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
| ( sP1 != $true ) )
& ( ( sP1 = $true )
| ( hoasinduction_lem2 != $true ) )
& ( pushprop_lthm_orig = $true )
& ! [X263: d_term] : ( d_one = X263 )
& ( ( pushprop_lem0 = $true )
| ! [X267: term > $o] :
( ( sK586 @ ( sub @ ( sK589 @ X267 ) @ ( push @ sK587 @ sK588 ) ) )
!= ( X267 @ ( sK589 @ X267 ) ) ) )
& ( ! [X269: term > $o,X270: term,X271: subst,X273: term] :
( ( X269 @ ( sub @ X273 @ ( push @ X270 @ X271 ) ) )
= ( sK590 @ X271 @ X270 @ X269 @ X273 ) )
| ( pushprop_lem0 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK457,sK458,sK459,sK460,sK461,sK462,sK463,sK464,sK465,sK466,sK467,sK468,sK469,sK470,sK471,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,sK487,sK488,sK489,sK490,sK491,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,sK509,sK510,sK511,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,sK521,sK522,sK523,sK524,sK525,sK526,sK527,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,sK536,sK537,sK538,sK539,sK540,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,sK562,sK563,sK564,sK565,sK566,sK567,sK568,sK569,sK570,sK571,sK572,sK573,sK574,sK575,sK576,sK577,sK578,sK579,sK580,sK581,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590])],[f475,f544,f543,f542,f541,f540,f539,f538,f537,f536,f535,f534,f533,f532,f531,f530,f529,f528,f527,f526,f525,f524,f523,f522,f521,f520,f519,f518,f517,f516,f515,f514,f513,f512,f511,f510,f509,f508,f507,f506,f505,f504,f503,f502,f501,f500,f499,f498,f497,f496,f495,f494,f493,f492,f491,f490,f489,f488,f487,f486,f485,f484,f483,f482,f481,f480,f479,f478,f477,f476]) ).
thf(f476,plain,
( ? [X0: term,X1: subst,X2: term > $o] :
! [X3: term > $o] :
? [X4: term] :
( ( X3 @ X4 )
!= ( X2 @ ( sub @ X4 @ ( push @ X0 @ X1 ) ) ) )
=> ! [X3: term > $o] :
? [X4: term] :
( ( sK459 @ ( sub @ X4 @ ( push @ sK457 @ sK458 ) ) )
!= ( X3 @ X4 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f477,plain,
! [X3: term > $o] :
( ? [X4: term] :
( ( sK459 @ ( sub @ X4 @ ( push @ sK457 @ sK458 ) ) )
!= ( X3 @ X4 ) )
=> ( ( sK459 @ ( sub @ ( sK460 @ X3 ) @ ( push @ sK457 @ sK458 ) ) )
!= ( X3 @ ( sK460 @ X3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f478,plain,
! [X5: term,X6: subst,X7: term > $o] :
( ? [X8: term > $o] :
! [X9: term] :
( ( X7 @ ( sub @ X9 @ ( push @ X5 @ X6 ) ) )
= ( X8 @ X9 ) )
=> ! [X9: term] :
( ( X7 @ ( sub @ X9 @ ( push @ X5 @ X6 ) ) )
= ( sK461 @ X7 @ X6 @ X5 @ X9 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f479,plain,
( ? [X11: subst] :
( ( comp @ X11 @ id )
!= X11 )
=> ( sK462
!= ( comp @ sK462 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f480,plain,
( ? [X15: term] :
( ( sub @ X15 @ id )
!= X15 )
=> ( sK463
!= ( sub @ sK463 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f481,plain,
( ? [X16: term > $o,X17: term] :
( ( ( sP56 @ X16 )
!= $true )
& ! [X18: term] :
( ( ( var @ X18 )
!= $true )
| ( $true
= ( X16 @ X18 ) ) )
& ( $true
!= ( X16 @ X17 ) )
& ! [X19: term] :
( ( ( X16 @ ( lam @ X19 ) )
= $true )
| ( $true
!= ( X16 @ X19 ) ) ) )
=> ( ( ( sP56 @ sK464 )
!= $true )
& ! [X18: term] :
( ( ( var @ X18 )
!= $true )
| ( $true
= ( sK464 @ X18 ) ) )
& ( ( sK464 @ sK465 )
!= $true )
& ! [X19: term] :
( ( $true
= ( sK464 @ ( lam @ X19 ) ) )
| ( ( sK464 @ X19 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f482,plain,
! [X20: term > $o] :
( ? [X22: term] :
( ( $true
= ( var @ X22 ) )
& ( ( X20 @ X22 )
!= $true ) )
=> ( ( $true
= ( var @ ( sK466 @ X20 ) ) )
& ( ( X20 @ ( sK466 @ X20 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f483,plain,
! [X20: term > $o] :
( ? [X23: term] :
( ( $true
!= ( X20 @ ( lam @ X23 ) ) )
& ( $true
= ( X20 @ X23 ) ) )
=> ( ( ( X20 @ ( lam @ ( sK467 @ X20 ) ) )
!= $true )
& ( ( X20 @ ( sK467 @ X20 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f484,plain,
( ? [X24: term > $o,X25: term,X26: term,X27: subst,X28: term > $o] :
( ( ( X24 @ ( sub @ X25 @ ( push @ X26 @ X27 ) ) )
!= $true )
& ( $true
= ( X28 @ X25 ) ) )
=> ( ( $true
!= ( sK468 @ ( sub @ sK469 @ ( push @ sK470 @ sK471 ) ) ) )
& ( $true
= ( sK472 @ sK469 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f485,plain,
( ? [X38: term,X39: term,X40: term,X41: term] :
( ( ( ap @ X41 @ X38 )
= ( ap @ X39 @ X40 ) )
& ( X39 != X41 ) )
=> ( ( ( ap @ sK474 @ sK475 )
= ( ap @ sK476 @ sK473 ) )
& ( sK476 != sK474 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f486,plain,
! [X42: subst] :
( ? [X43: d_subst] :
( ( d2subst @ X43 )
= X42 )
=> ( ( d2subst @ ( sK477 @ X42 ) )
= X42 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f487,plain,
( ? [X44: term,X45: term,X46: term,X47: term] :
( ( X46 != X47 )
& ( ( ap @ X46 @ X45 )
= ( ap @ X47 @ X44 ) ) )
=> ( ( sK481 != sK480 )
& ( ( ap @ sK481 @ sK478 )
= ( ap @ sK480 @ sK479 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f488,plain,
( ? [X48: term] :
( ( sub @ X48 @ id )
!= X48 )
=> ( ( sub @ sK482 @ id )
!= sK482 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f489,plain,
( ? [X58: term,X59: term,X60: term,X61: term] :
( ( X58 != X59 )
& ( ( ap @ ( sub @ X59 @ id ) @ X61 )
= ( ap @ ( sub @ X58 @ id ) @ X60 ) ) )
=> ( ( sK484 != sK483 )
& ( ( ap @ ( sub @ sK484 @ id ) @ sK486 )
= ( ap @ ( sub @ sK483 @ id ) @ sK485 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f490,plain,
! [X62: term] :
( ? [X63: d_term] :
( ( d2term @ X63 )
= X62 )
=> ( ( d2term @ ( sK487 @ X62 ) )
= X62 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f491,plain,
( ? [X64: term,X65: term] :
( ( X64 != X65 )
& ( ( sub @ X65 @ sh )
= ( sub @ X64 @ sh ) ) )
=> ( ( sK488 != sK489 )
& ( ( sub @ sK488 @ sh )
= ( sub @ sK489 @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f492,plain,
( ? [X68: subst > term > subst > $o] :
! [X69: term > $o] :
? [X70: term] :
( ( X68 @ id @ X70 @ id )
!= ( X69 @ X70 ) )
=> ! [X69: term > $o] :
? [X70: term] :
( ( sK490 @ id @ X70 @ id )
!= ( X69 @ X70 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f493,plain,
! [X69: term > $o] :
( ? [X70: term] :
( ( sK490 @ id @ X70 @ id )
!= ( X69 @ X70 ) )
=> ( ( sK490 @ id @ ( sK491 @ X69 ) @ id )
!= ( X69 @ ( sK491 @ X69 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f494,plain,
! [X71: subst > term > subst > $o] :
( ? [X72: term > $o] :
! [X73: term] :
( ( X72 @ X73 )
= ( X71 @ id @ X73 @ id ) )
=> ! [X73: term] :
( ( sK492 @ X71 @ X73 )
= ( X71 @ id @ X73 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f495,plain,
! [X74: term] :
( ? [X75: subst > term > term] :
! [X76: subst,X77: term] :
( ( X75 @ X76 @ X77 )
= ( sub @ X74 @ ( push @ X77 @ X76 ) ) )
=> ! [X77: term,X76: subst] :
( ( sub @ X74 @ ( push @ X77 @ X76 ) )
= ( sK493 @ X74 @ X76 @ X77 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f496,plain,
( ? [X78: term] :
! [X79: subst > term > term] :
? [X80: subst,X81: term] :
( ( sub @ X78 @ ( push @ X81 @ X80 ) )
!= ( X79 @ X80 @ X81 ) )
=> ! [X79: subst > term > term] :
? [X81: term,X80: subst] :
( ( sub @ sK494 @ ( push @ X81 @ X80 ) )
!= ( X79 @ X80 @ X81 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f497,plain,
! [X79: subst > term > term] :
( ? [X81: term,X80: subst] :
( ( sub @ sK494 @ ( push @ X81 @ X80 ) )
!= ( X79 @ X80 @ X81 ) )
=> ( ( X79 @ ( sK495 @ X79 ) @ ( sK496 @ X79 ) )
!= ( sub @ sK494 @ ( push @ ( sK496 @ X79 ) @ ( sK495 @ X79 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f498,plain,
( ? [X82: term] :
! [X83: subst > term > term] :
? [X84: subst,X85: term] :
( ( X83 @ X84 @ X85 )
!= ( sub @ X82 @ ( push @ X85 @ X84 ) ) )
=> ! [X83: subst > term > term] :
? [X85: term,X84: subst] :
( ( X83 @ X84 @ X85 )
!= ( sub @ sK497 @ ( push @ X85 @ X84 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f499,plain,
! [X83: subst > term > term] :
( ? [X85: term,X84: subst] :
( ( X83 @ X84 @ X85 )
!= ( sub @ sK497 @ ( push @ X85 @ X84 ) ) )
=> ( ( sub @ sK497 @ ( push @ ( sK499 @ X83 ) @ ( sK498 @ X83 ) ) )
!= ( X83 @ ( sK498 @ X83 ) @ ( sK499 @ X83 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f500,plain,
! [X86: term] :
( ? [X87: subst > term > term] :
! [X88: subst,X89: term] :
( ( sub @ X86 @ ( push @ X89 @ X88 ) )
= ( X87 @ X88 @ X89 ) )
=> ! [X89: term,X88: subst] :
( ( sub @ X86 @ ( push @ X89 @ X88 ) )
= ( sK500 @ X86 @ X88 @ X89 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f501,plain,
( ? [X93: subst,X94: term] :
( ( sub @ one @ ( push @ X94 @ X93 ) )
!= X94 )
=> ( ( sub @ one @ ( push @ sK502 @ sK501 ) )
!= sK502 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f502,plain,
( ? [X95: term] :
( ( sub @ X95 @ id )
!= X95 )
=> ( ( sub @ sK503 @ id )
!= sK503 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f503,plain,
( ? [X98: term] :
( ( sub @ X98 @ id )
!= X98 )
=> ( sK504
!= ( sub @ sK504 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f504,plain,
( ? [X99: term] :
( ( sub @ X99 @ id )
!= X99 )
=> ( ( sub @ sK505 @ id )
!= sK505 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f505,plain,
( ? [X101: term] :
! [X102: subst > term > term] :
( ( sub @ X101 @ ( push @ one @ sh ) )
!= ( X102 @ sh @ one ) )
=> ! [X102: subst > term > term] :
( ( X102 @ sh @ one )
!= ( sub @ sK506 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f506,plain,
! [X103: term] :
( ? [X104: subst > term > term] :
( ( X104 @ sh @ one )
= ( sub @ X103 @ ( push @ one @ sh ) ) )
=> ( ( sK507 @ X103 @ sh @ one )
= ( sub @ X103 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f507,plain,
( ? [X106: subst] :
( ( comp @ X106 @ id )
!= X106 )
=> ( ( comp @ sK508 @ id )
!= sK508 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f508,plain,
( ? [X107: subst] :
( ( comp @ id @ X107 )
!= X107 )
=> ( sK509
!= ( comp @ id @ sK509 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f509,plain,
( ? [X109: subst] :
( ( comp @ X109 @ id )
!= X109 )
=> ( ( comp @ sK510 @ id )
!= sK510 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f510,plain,
( ? [X112: subst] :
( ( comp @ id @ X112 )
!= X112 )
=> ( sK511
!= ( comp @ id @ sK511 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f511,plain,
( ? [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
!= X116 )
=> ( sK513
!= ( comp @ sh @ ( push @ sK512 @ sK513 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f512,plain,
( ? [X118: subst] :
( ( push @ ( sub @ one @ X118 ) @ ( comp @ sh @ X118 ) )
!= X118 )
=> ( sK514
!= ( push @ ( sub @ one @ sK514 ) @ ( comp @ sh @ sK514 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f513,plain,
( ? [X119: subst > term > subst > $o,X120: term] :
( ( $true
!= ( X119 @ id @ X120 @ id ) )
& ! [X121: term,X122: term] :
( ( ( X119 @ id @ ( ap @ ( sub @ X121 @ id ) @ X122 ) @ id )
= $true )
| ( ( X119 @ id @ X121 @ id )
!= $true )
| ( ( X119 @ id @ X122 @ id )
!= $true ) )
& ( $true
!= ( sP34 @ X119 ) ) )
=> ( ( ( sK515 @ id @ sK516 @ id )
!= $true )
& ! [X122: term,X121: term] :
( ( ( sK515 @ id @ ( ap @ ( sub @ X121 @ id ) @ X122 ) @ id )
= $true )
| ( ( sK515 @ id @ X121 @ id )
!= $true )
| ( ( sK515 @ id @ X122 @ id )
!= $true ) )
& ( ( sP34 @ sK515 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f514,plain,
! [X123: subst > term > subst > $o] :
( ? [X125: term,X126: term] :
( ( ( X123 @ id @ ( ap @ ( sub @ X125 @ id ) @ X126 ) @ id )
!= $true )
& ( $true
= ( X123 @ id @ X125 @ id ) )
& ( ( X123 @ id @ X126 @ id )
= $true ) )
=> ( ( ( X123 @ id @ ( ap @ ( sub @ ( sK517 @ X123 ) @ id ) @ ( sK518 @ X123 ) ) @ id )
!= $true )
& ( ( X123 @ id @ ( sK517 @ X123 ) @ id )
= $true )
& ( ( X123 @ id @ ( sK518 @ X123 ) @ id )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f515,plain,
( ? [X127: term,X128: term] :
( ( X127 != X128 )
& ( ( lam @ X127 )
= ( lam @ X128 ) ) )
=> ( ( sK520 != sK519 )
& ( ( lam @ sK520 )
= ( lam @ sK519 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f516,plain,
( ? [X131: subst] :
( ( comp @ id @ X131 )
!= X131 )
=> ( ( comp @ id @ sK521 )
!= sK521 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f517,plain,
( ? [X133: term,X134: term,X135: term,X136: term] :
( ( X135 != X136 )
& ( ( ap @ ( sub @ X134 @ id ) @ X136 )
= ( ap @ ( sub @ X133 @ id ) @ X135 ) ) )
=> ( ( sK525 != sK524 )
& ( ( ap @ ( sub @ sK523 @ id ) @ sK525 )
= ( ap @ ( sub @ sK522 @ id ) @ sK524 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f518,plain,
( ? [X141: term,X142: term,X143: subst > term > term] :
( ! [X144: subst,X145: subst,X146: term] :
( ( sub @ ( X143 @ X145 @ X146 ) @ X144 )
= ( X143 @ ( comp @ X145 @ X144 ) @ ( sub @ X146 @ X144 ) ) )
& ( ( lam @ ( X143 @ sh @ one ) )
= ( ap @ ( sub @ X141 @ id ) @ X142 ) ) )
=> ( ! [X146: term,X145: subst,X144: subst] :
( ( sK528 @ ( comp @ X145 @ X144 ) @ ( sub @ X146 @ X144 ) )
= ( sub @ ( sK528 @ X145 @ X146 ) @ X144 ) )
& ( ( lam @ ( sK528 @ sh @ one ) )
= ( ap @ ( sub @ sK526 @ id ) @ sK527 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f519,plain,
! [X149: subst > term > term] :
( ? [X150: subst,X151: subst,X152: term] :
( ( sub @ ( X149 @ X151 @ X152 ) @ X150 )
!= ( X149 @ ( comp @ X151 @ X150 ) @ ( sub @ X152 @ X150 ) ) )
=> ( ( X149 @ ( comp @ ( sK530 @ X149 ) @ ( sK529 @ X149 ) ) @ ( sub @ ( sK531 @ X149 ) @ ( sK529 @ X149 ) ) )
!= ( sub @ ( X149 @ ( sK530 @ X149 ) @ ( sK531 @ X149 ) ) @ ( sK529 @ X149 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f520,plain,
( ? [X153: term,X154: term,X155: term,X156: term] :
( ( X153 != X155 )
& ( ( ap @ ( sub @ X156 @ id ) @ X155 )
= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
=> ( ( sK532 != sK534 )
& ( ( ap @ ( sub @ sK533 @ id ) @ sK532 )
= ( ap @ ( sub @ sK535 @ id ) @ sK534 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f521,plain,
( ? [X165: term,X166: term,X167: term,X168: term] :
( ( ( ap @ X168 @ X166 )
= ( ap @ X165 @ X167 ) )
& ( X166 != X167 ) )
=> ( ( ( ap @ sK539 @ sK537 )
= ( ap @ sK536 @ sK538 ) )
& ( sK537 != sK538 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f522,plain,
( ? [X169: term,X170: term,X171: term,X172: term] :
( ( X169 != X172 )
& ( ( ap @ ( sub @ X172 @ id ) @ X171 )
= ( ap @ ( sub @ X169 @ id ) @ X170 ) ) )
=> ( ( sK543 != sK540 )
& ( ( ap @ ( sub @ sK540 @ id ) @ sK541 )
= ( ap @ ( sub @ sK543 @ id ) @ sK542 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f523,plain,
! [X177: term] :
( ? [X178: subst > term > term] :
( ( sub @ X177 @ ( push @ one @ sh ) )
= ( X178 @ sh @ one ) )
=> ( ( sub @ X177 @ ( push @ one @ sh ) )
= ( sK544 @ X177 @ sh @ one ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f524,plain,
( ? [X179: term] :
! [X180: subst > term > term] :
( ( sub @ X179 @ ( push @ one @ sh ) )
!= ( X180 @ sh @ one ) )
=> ! [X180: subst > term > term] :
( ( X180 @ sh @ one )
!= ( sub @ sK545 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f525,plain,
( ? [X183: subst,X184: term > $o,X185: term > $o,X186: term] :
( ! [X187: term] :
( ( X184 @ X187 )
= ( X185 @ ( sub @ X187 @ ( push @ X186 @ X183 ) ) ) )
& ( ( X184 @ one )
!= $true )
& ( $true
= ( X185 @ X186 ) ) )
=> ( ! [X187: term] :
( ( sK548 @ ( sub @ X187 @ ( push @ sK549 @ sK546 ) ) )
= ( sK547 @ X187 ) )
& ( ( sK547 @ one )
!= $true )
& ( $true
= ( sK548 @ sK549 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f526,plain,
( ? [X188: subst,X189: term] :
( ( sub @ one @ ( push @ X189 @ X188 ) )
!= X189 )
=> ( ( sub @ one @ ( push @ sK551 @ sK550 ) )
!= sK551 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f527,plain,
! [X190: subst,X191: term > $o,X192: term > $o,X193: term] :
( ? [X194: term] :
( ( X191 @ X194 )
!= ( X192 @ ( sub @ X194 @ ( push @ X193 @ X190 ) ) ) )
=> ( ( X191 @ ( sK552 @ X193 @ X192 @ X191 @ X190 ) )
!= ( X192 @ ( sub @ ( sK552 @ X193 @ X192 @ X191 @ X190 ) @ ( push @ X193 @ X190 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f528,plain,
! [X195: term > $o,X196: subst,X197: term > $o,X198: term] :
( ? [X199: term] :
( ( X197 @ ( sub @ X199 @ ( push @ X198 @ X196 ) ) )
!= ( X195 @ X199 ) )
=> ( ( X197 @ ( sub @ ( sK553 @ X198 @ X197 @ X196 @ X195 ) @ ( push @ X198 @ X196 ) ) )
!= ( X195 @ ( sK553 @ X198 @ X197 @ X196 @ X195 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f529,plain,
( ? [X200: term > $o,X201: subst,X202: term > $o,X203: term] :
( ( ( X202 @ X203 )
= $true )
& ! [X204: term] :
( ( X200 @ X204 )
= ( X202 @ ( sub @ X204 @ ( push @ X203 @ X201 ) ) ) )
& ( ( X200 @ one )
!= $true ) )
=> ( ( $true
= ( sK556 @ sK557 ) )
& ! [X204: term] :
( ( sK556 @ ( sub @ X204 @ ( push @ sK557 @ sK555 ) ) )
= ( sK554 @ X204 ) )
& ( ( sK554 @ one )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f530,plain,
! [X205: subst > term > subst > $o] :
( ? [X206: term > $o] :
! [X207: term] :
( ( X206 @ X207 )
= ( X205 @ id @ X207 @ id ) )
=> ! [X207: term] :
( ( sK558 @ X205 @ X207 )
= ( X205 @ id @ X207 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f531,plain,
( ? [X208: subst > term > subst > $o] :
! [X209: term > $o] :
? [X210: term] :
( ( X208 @ id @ X210 @ id )
!= ( X209 @ X210 ) )
=> ! [X209: term > $o] :
? [X210: term] :
( ( sK559 @ id @ X210 @ id )
!= ( X209 @ X210 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f532,plain,
! [X209: term > $o] :
( ? [X210: term] :
( ( sK559 @ id @ X210 @ id )
!= ( X209 @ X210 ) )
=> ( ( sK559 @ id @ ( sK560 @ X209 ) @ id )
!= ( X209 @ ( sK560 @ X209 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f533,plain,
( ? [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ X217 @ X215 )
= ( ap @ X218 @ X216 ) )
& ( X215 != X216 ) )
=> ( ( ( ap @ sK564 @ sK562 )
= ( ap @ sK563 @ sK561 ) )
& ( sK561 != sK562 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f534,plain,
( ? [X219: subst] :
( ( comp @ id @ X219 )
!= X219 )
=> ( sK565
!= ( comp @ id @ sK565 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f535,plain,
( ? [X220: subst] :
( ( comp @ X220 @ id )
!= X220 )
=> ( sK566
!= ( comp @ sK566 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f536,plain,
( ? [X223: subst > term > term,X224: subst > term > term,X225: subst,X226: term] :
( ( ( lam @ ( X224 @ sh @ one ) )
= ( lam @ ( X223 @ sh @ one ) ) )
& ! [X227: term,X228: subst,X229: subst] :
( ( X223 @ ( comp @ X229 @ X228 ) @ ( sub @ X227 @ X228 ) )
= ( sub @ ( X223 @ X229 @ X227 ) @ X228 ) )
& ( ( X224 @ X225 @ X226 )
!= ( X223 @ X225 @ X226 ) )
& ! [X230: term,X231: subst,X232: subst] :
( ( sub @ ( X224 @ X231 @ X230 ) @ X232 )
= ( X224 @ ( comp @ X231 @ X232 ) @ ( sub @ X230 @ X232 ) ) ) )
=> ( ( ( lam @ ( sK568 @ sh @ one ) )
= ( lam @ ( sK567 @ sh @ one ) ) )
& ! [X229: subst,X228: subst,X227: term] :
( ( sub @ ( sK567 @ X229 @ X227 ) @ X228 )
= ( sK567 @ ( comp @ X229 @ X228 ) @ ( sub @ X227 @ X228 ) ) )
& ( ( sK567 @ sK569 @ sK570 )
!= ( sK568 @ sK569 @ sK570 ) )
& ! [X232: subst,X231: subst,X230: term] :
( ( sK568 @ ( comp @ X231 @ X232 ) @ ( sub @ X230 @ X232 ) )
= ( sub @ ( sK568 @ X231 @ X230 ) @ X232 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f537,plain,
! [X233: subst > term > term] :
( ? [X237: term,X238: subst,X239: subst] :
( ( sub @ ( X233 @ X239 @ X237 ) @ X238 )
!= ( X233 @ ( comp @ X239 @ X238 ) @ ( sub @ X237 @ X238 ) ) )
=> ( ( sub @ ( X233 @ ( sK573 @ X233 ) @ ( sK571 @ X233 ) ) @ ( sK572 @ X233 ) )
!= ( X233 @ ( comp @ ( sK573 @ X233 ) @ ( sK572 @ X233 ) ) @ ( sub @ ( sK571 @ X233 ) @ ( sK572 @ X233 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f538,plain,
! [X234: subst > term > term] :
( ? [X240: term,X241: subst,X242: subst] :
( ( sub @ ( X234 @ X241 @ X240 ) @ X242 )
!= ( X234 @ ( comp @ X241 @ X242 ) @ ( sub @ X240 @ X242 ) ) )
=> ( ( X234 @ ( comp @ ( sK575 @ X234 ) @ ( sK576 @ X234 ) ) @ ( sub @ ( sK574 @ X234 ) @ ( sK576 @ X234 ) ) )
!= ( sub @ ( X234 @ ( sK575 @ X234 ) @ ( sK574 @ X234 ) ) @ ( sK576 @ X234 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f539,plain,
( ? [X248: subst,X249: term] :
( ( comp @ sh @ ( push @ X249 @ X248 ) )
!= X248 )
=> ( sK577
!= ( comp @ sh @ ( push @ sK578 @ sK577 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f540,plain,
( ? [X250: subst,X251: term] :
( ( sub @ one @ ( push @ X251 @ X250 ) )
!= X251 )
=> ( sK580
!= ( sub @ one @ ( push @ sK580 @ sK579 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f541,plain,
( ? [X252: term > $o,X253: term > $o,X254: term,X255: subst,X256: term] :
( ( ( X253 @ ( sub @ X256 @ ( push @ X254 @ X255 ) ) )
!= $true )
& ( ( X252 @ X256 )
= $true ) )
=> ( ( $true
!= ( sK582 @ ( sub @ sK585 @ ( push @ sK583 @ sK584 ) ) ) )
& ( $true
= ( sK581 @ sK585 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f542,plain,
( ? [X264: term > $o,X265: term,X266: subst] :
! [X267: term > $o] :
? [X268: term] :
( ( X267 @ X268 )
!= ( X264 @ ( sub @ X268 @ ( push @ X265 @ X266 ) ) ) )
=> ! [X267: term > $o] :
? [X268: term] :
( ( X267 @ X268 )
!= ( sK586 @ ( sub @ X268 @ ( push @ sK587 @ sK588 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f543,plain,
! [X267: term > $o] :
( ? [X268: term] :
( ( X267 @ X268 )
!= ( sK586 @ ( sub @ X268 @ ( push @ sK587 @ sK588 ) ) ) )
=> ( ( sK586 @ ( sub @ ( sK589 @ X267 ) @ ( push @ sK587 @ sK588 ) ) )
!= ( X267 @ ( sK589 @ X267 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f544,plain,
! [X269: term > $o,X270: term,X271: subst] :
( ? [X272: term > $o] :
! [X273: term] :
( ( X269 @ ( sub @ X273 @ ( push @ X270 @ X271 ) ) )
= ( X272 @ X273 ) )
=> ! [X273: term] :
( ( X269 @ ( sub @ X273 @ ( push @ X270 @ X271 ) ) )
= ( sK590 @ X271 @ X270 @ X269 @ X273 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f475,plain,
( ( ( pushprop_lem0_lthm = $true )
| ? [X0: term,X1: subst,X2: term > $o] :
! [X3: term > $o] :
? [X4: term] :
( ( X3 @ X4 )
!= ( X2 @ ( sub @ X4 @ ( push @ X0 @ X1 ) ) ) ) )
& ( ! [X5: term,X6: subst,X7: term > $o] :
? [X8: term > $o] :
! [X9: term] :
( ( X7 @ ( sub @ X9 @ ( push @ X5 @ X6 ) ) )
= ( X8 @ X9 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ! [X10: subst] :
( ( comp @ X10 @ id )
= X10 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X11: subst] :
( ( comp @ X11 @ id )
!= X11 ) )
& ! [X12: d_subst,X13: d_subst] :
( ( ( d2subst @ X13 )
!= ( d2subst @ X12 ) )
| ( X12 = X13 ) )
& ( hoasinduction_lem2_lthm = $true )
& ( axclos = $true )
& ( ( $true = sP58 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP58 ) )
& ( ! [X14: term] :
( ( sub @ X14 @ id )
= X14 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X15: term] :
( ( sub @ X15 @ id )
!= X15 ) )
& ( substmonoid_gthm = $true )
& ( ( induction = $true )
| ? [X16: term > $o,X17: term] :
( ( ( sP56 @ X16 )
!= $true )
& ! [X18: term] :
( ( ( var @ X18 )
!= $true )
| ( $true
= ( X16 @ X18 ) ) )
& ( $true
!= ( X16 @ X17 ) )
& ! [X19: term] :
( ( ( X16 @ ( lam @ X19 ) )
= $true )
| ( $true
!= ( X16 @ X19 ) ) ) ) )
& ( ! [X20: term > $o,X21: term] :
( ( ( sP56 @ X20 )
= $true )
| ? [X22: term] :
( ( $true
= ( var @ X22 ) )
& ( ( X20 @ X22 )
!= $true ) )
| ( ( X20 @ X21 )
= $true )
| ? [X23: term] :
( ( $true
!= ( X20 @ ( lam @ X23 ) ) )
& ( $true
= ( X20 @ X23 ) ) ) )
| ( induction != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X24: term > $o,X25: term,X26: term,X27: subst,X28: term > $o] :
( ( ( X24 @ ( sub @ X25 @ ( push @ X26 @ X27 ) ) )
!= $true )
& ( $true
= ( X28 @ X25 ) ) ) )
& ( ! [X29: term > $o,X30: term,X31: term,X32: subst,X33: term > $o] :
( ( ( X29 @ ( sub @ X30 @ ( push @ X31 @ X32 ) ) )
= $true )
| ( ( X33 @ X30 )
!= $true ) )
| ( pushprop_lem1 != $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( axassoc = $true )
& ( ! [X34: term,X35: term,X36: term,X37: term] :
( ( ( ap @ X37 @ X34 )
!= ( ap @ X35 @ X36 ) )
| ( X35 = X37 ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X38: term,X39: term,X40: term,X41: term] :
( ( ( ap @ X41 @ X38 )
= ( ap @ X39 @ X40 ) )
& ( X39 != X41 ) ) )
& ( hoasapinj2_gthm = $true )
& ! [X42: subst] :
? [X43: d_subst] :
( ( d2subst @ X43 )
= X42 )
& ( ? [X44: term,X45: term,X46: term,X47: term] :
( ( X46 != X47 )
& ( ( ap @ X46 @ X45 )
= ( ap @ X47 @ X44 ) ) )
| ? [X48: term] :
( ( sub @ X48 @ id )
!= X48 )
| ! [X49: term,X50: term,X51: term,X52: term] :
( ( X49 = X50 )
| ( ( ap @ ( sub @ X49 @ id ) @ X51 )
!= ( ap @ ( sub @ X50 @ id ) @ X52 ) ) )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X53: term,X54: term,X55: term,X56: term] :
( ( X55 = X56 )
| ( ( ap @ X55 @ X54 )
!= ( ap @ X56 @ X53 ) ) )
& ! [X57: term] :
( ( sub @ X57 @ id )
= X57 )
& ? [X58: term,X59: term,X60: term,X61: term] :
( ( X58 != X59 )
& ( ( ap @ ( sub @ X59 @ id ) @ X61 )
= ( ap @ ( sub @ X58 @ id ) @ X60 ) ) ) ) )
& ( hoasapnotvar = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP55 ) )
& ( ( $true = sP55 )
| ( hoasinduction_lem3aa != $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar_lthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( sP54 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP54 != $true ) )
& ( termmset_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ! [X62: term] :
? [X63: d_term] :
( ( d2term @ X63 )
= X62 )
& ( ( shinj = $true )
| ? [X64: term,X65: term] :
( ( X64 != X65 )
& ( ( sub @ X65 @ sh )
= ( sub @ X64 @ sh ) ) ) )
& ( ! [X66: term,X67: term] :
( ( X66 = X67 )
| ( ( sub @ X66 @ sh )
!= ( sub @ X67 @ sh ) ) )
| ( shinj != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X68: subst > term > subst > $o] :
! [X69: term > $o] :
? [X70: term] :
( ( X68 @ id @ X70 @ id )
!= ( X69 @ X70 ) ) )
& ( ! [X71: subst > term > subst > $o] :
? [X72: term > $o] :
! [X73: term] :
( ( X72 @ X73 )
= ( X71 @ id @ X73 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( induction2_gthm = $true )
& ( ! [X74: term] :
? [X75: subst > term > term] :
! [X76: subst,X77: term] :
( ( X75 @ X76 @ X77 )
= ( sub @ X74 @ ( push @ X77 @ X76 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X78: term] :
! [X79: subst > term > term] :
? [X80: subst,X81: term] :
( ( sub @ X78 @ ( push @ X81 @ X80 ) )
!= ( X79 @ X80 @ X81 ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X82: term] :
! [X83: subst > term > term] :
? [X84: subst,X85: term] :
( ( X83 @ X84 @ X85 )
!= ( sub @ X82 @ ( push @ X85 @ X84 ) ) ) )
& ( ! [X86: term] :
? [X87: subst > term > term] :
! [X88: subst,X89: term] :
( ( sub @ X86 @ ( push @ X89 @ X88 ) )
= ( X87 @ X88 @ X89 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP53 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP53 != $true ) )
& ! [X90: d_subst] : ( d_id = X90 )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X91: subst,X92: term] :
( ( sub @ one @ ( push @ X92 @ X91 ) )
= X92 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X93: subst,X94: term] :
( ( sub @ one @ ( push @ X94 @ X93 ) )
!= X94 ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP51 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP51 ) )
& ( axapp = $true )
& ( induction2lem_gthm = $true )
& ( ulamvarsh = $true )
& ( axvarshift = $true )
& ( ulamvarind = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( termmset = $true )
| ? [X95: term] :
( ( sub @ X95 @ id )
!= X95 ) )
& ( ! [X96: term] :
( ( sub @ X96 @ id )
= X96 )
| ( termmset != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( sP49 != $true ) )
& ( ( sP49 = $true )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( apnotvar = $true )
& ( ! [X97: term] :
( ( sub @ X97 @ id )
= X97 )
| ? [X98: term] :
( ( sub @ X98 @ id )
!= X98 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ? [X99: term] :
( ( sub @ X99 @ id )
!= X99 )
& ! [X100: term] :
( ( sub @ X100 @ id )
= X100 ) ) )
& ( ( sP48 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP48 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP46 ) )
& ( ( $true = sP46 )
| ( hoasinduction_lem3 != $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP44 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( sP44 != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasinduction_lem3b = $true )
| ? [X101: term] :
! [X102: subst > term > term] :
( ( sub @ X101 @ ( push @ one @ sh ) )
!= ( X102 @ sh @ one ) ) )
& ( ! [X103: term] :
? [X104: subst > term > term] :
( ( X104 @ sh @ one )
= ( sub @ X103 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( axmap = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop = $true )
& ( ( substmonoid_lthm = $true )
| ( ! [X105: subst] :
( ( comp @ X105 @ id )
= X105 )
& ( ? [X106: subst] :
( ( comp @ X106 @ id )
!= X106 )
| ? [X107: subst] :
( ( comp @ id @ X107 )
!= X107 ) )
& ! [X108: subst] :
( ( comp @ id @ X108 )
= X108 ) ) )
& ( ? [X109: subst] :
( ( comp @ X109 @ id )
!= X109 )
| ( ! [X110: subst] :
( ( comp @ X110 @ id )
= X110 )
& ! [X111: subst] :
( ( comp @ id @ X111 )
= X111 ) )
| ? [X112: subst] :
( ( comp @ id @ X112 )
!= X112 )
| ( substmonoid_lthm != $true ) )
& ( axabs = $true )
& ( lamnotvar = $true )
& ( ! [X113: term,X114: subst] :
( ( comp @ sh @ ( push @ X113 @ X114 ) )
= X114 )
| ( axshiftcons != $true ) )
& ( ( axshiftcons = $true )
| ? [X115: term,X116: subst] :
( ( comp @ sh @ ( push @ X115 @ X116 ) )
!= X116 ) )
& ( ( sP43 = $true )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( sP43 != $true ) )
& ( ( $true = sP39 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP39 ) )
& ( ( sP38 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP38 != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X117: subst] :
( ( push @ ( sub @ one @ X117 ) @ ( comp @ sh @ X117 ) )
= X117 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X118: subst] :
( ( push @ ( sub @ one @ X118 ) @ ( comp @ sh @ X118 ) )
!= X118 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( pushprop_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X119: subst > term > subst > $o,X120: term] :
( ( $true
!= ( X119 @ id @ X120 @ id ) )
& ! [X121: term,X122: term] :
( ( ( X119 @ id @ ( ap @ ( sub @ X121 @ id ) @ X122 ) @ id )
= $true )
| ( ( X119 @ id @ X121 @ id )
!= $true )
| ( ( X119 @ id @ X122 @ id )
!= $true ) )
& ( $true
!= ( sP34 @ X119 ) ) ) )
& ( ! [X123: subst > term > subst > $o,X124: term] :
( ( $true
= ( X123 @ id @ X124 @ id ) )
| ? [X125: term,X126: term] :
( ( ( X123 @ id @ ( ap @ ( sub @ X125 @ id ) @ X126 ) @ id )
!= $true )
& ( $true
= ( X123 @ id @ X125 @ id ) )
& ( ( X123 @ id @ X126 @ id )
= $true ) )
| ( $true
= ( sP34 @ X123 ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction = $true )
| ( sP33 != $true ) )
& ( ( sP33 = $true )
| ( hoasinduction != $true ) )
& ( ( laminj = $true )
| ? [X127: term,X128: term] :
( ( X127 != X128 )
& ( ( lam @ X127 )
= ( lam @ X128 ) ) ) )
& ( ! [X129: term,X130: term] :
( ( X129 = X130 )
| ( ( lam @ X129 )
!= ( lam @ X130 ) ) )
| ( laminj != $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( axidl = $true )
| ? [X131: subst] :
( ( comp @ id @ X131 )
!= X131 ) )
& ( ! [X132: subst] :
( ( comp @ id @ X132 )
= X132 )
| ( axidl != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X133: term,X134: term,X135: term,X136: term] :
( ( X135 != X136 )
& ( ( ap @ ( sub @ X134 @ id ) @ X136 )
= ( ap @ ( sub @ X133 @ id ) @ X135 ) ) ) )
& ( ! [X137: term,X138: term,X139: term,X140: term] :
( ( X139 = X140 )
| ( ( ap @ ( sub @ X138 @ id ) @ X140 )
!= ( ap @ ( sub @ X137 @ id ) @ X139 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X141: term,X142: term,X143: subst > term > term] :
( ! [X144: subst,X145: subst,X146: term] :
( ( sub @ ( X143 @ X145 @ X146 ) @ X144 )
= ( X143 @ ( comp @ X145 @ X144 ) @ ( sub @ X146 @ X144 ) ) )
& ( ( lam @ ( X143 @ sh @ one ) )
= ( ap @ ( sub @ X141 @ id ) @ X142 ) ) ) )
& ( ! [X147: term,X148: term,X149: subst > term > term] :
( ? [X150: subst,X151: subst,X152: term] :
( ( sub @ ( X149 @ X151 @ X152 ) @ X150 )
!= ( X149 @ ( comp @ X151 @ X150 ) @ ( sub @ X152 @ X150 ) ) )
| ( ( ap @ ( sub @ X147 @ id ) @ X148 )
!= ( lam @ ( X149 @ sh @ one ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP30 != $true ) )
& ( ( sP30 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( sP24 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP24 != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X153: term,X154: term,X155: term,X156: term] :
( ( X153 != X155 )
& ( ( ap @ ( sub @ X156 @ id ) @ X155 )
= ( ap @ ( sub @ X154 @ id ) @ X153 ) ) )
& ! [X157: term,X158: term,X159: term,X160: term] :
( ( ( ap @ X157 @ X159 )
!= ( ap @ X160 @ X158 ) )
| ( X158 = X159 ) ) ) )
& ( ! [X161: term,X162: term,X163: term,X164: term] :
( ( X161 = X163 )
| ( ( ap @ ( sub @ X162 @ id ) @ X161 )
!= ( ap @ ( sub @ X164 @ id ) @ X163 ) ) )
| ? [X165: term,X166: term,X167: term,X168: term] :
( ( ( ap @ X168 @ X166 )
= ( ap @ X165 @ X167 ) )
& ( X166 != X167 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X169: term,X170: term,X171: term,X172: term] :
( ( X169 != X172 )
& ( ( ap @ ( sub @ X172 @ id ) @ X171 )
= ( ap @ ( sub @ X169 @ id ) @ X170 ) ) ) )
& ( ! [X173: term,X174: term,X175: term,X176: term] :
( ( X173 = X176 )
| ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X176 @ id ) @ X175 ) ) )
| ( hoasapinj1 != $true ) )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ! [X177: term] :
? [X178: subst > term > term] :
( ( sub @ X177 @ ( push @ one @ sh ) )
= ( X178 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X179: term] :
! [X180: subst > term > term] :
( ( sub @ X179 @ ( push @ one @ sh ) )
!= ( X180 @ sh @ one ) ) )
& ( ( sP23 = $true )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( sP23 != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ! [X181: subst,X182: term] :
( ( sub @ one @ ( push @ X182 @ X181 ) )
= X182 )
& ? [X183: subst,X184: term > $o,X185: term > $o,X186: term] :
( ! [X187: term] :
( ( X184 @ X187 )
= ( X185 @ ( sub @ X187 @ ( push @ X186 @ X183 ) ) ) )
& ( ( X184 @ one )
!= $true )
& ( $true
= ( X185 @ X186 ) ) ) ) )
& ( ? [X188: subst,X189: term] :
( ( sub @ one @ ( push @ X189 @ X188 ) )
!= X189 )
| ! [X190: subst,X191: term > $o,X192: term > $o,X193: term] :
( ? [X194: term] :
( ( X191 @ X194 )
!= ( X192 @ ( sub @ X194 @ ( push @ X193 @ X190 ) ) ) )
| ( ( X191 @ one )
= $true )
| ( $true
!= ( X192 @ X193 ) ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ! [X195: term > $o,X196: subst,X197: term > $o,X198: term] :
( ( ( X197 @ X198 )
!= $true )
| ? [X199: term] :
( ( X197 @ ( sub @ X199 @ ( push @ X198 @ X196 ) ) )
!= ( X195 @ X199 ) )
| ( $true
= ( X195 @ one ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X200: term > $o,X201: subst,X202: term > $o,X203: term] :
( ( ( X202 @ X203 )
= $true )
& ! [X204: term] :
( ( X200 @ X204 )
= ( X202 @ ( sub @ X204 @ ( push @ X203 @ X201 ) ) ) )
& ( ( X200 @ one )
!= $true ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( sP21 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP21 != $true ) )
& ( pushprop_gthm = $true )
& ( ! [X205: subst > term > subst > $o] :
? [X206: term > $o] :
! [X207: term] :
( ( X206 @ X207 )
= ( X205 @ id @ X207 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X208: subst > term > subst > $o] :
! [X209: term > $o] :
? [X210: term] :
( ( X208 @ id @ X210 @ id )
!= ( X209 @ X210 ) ) )
& ( ( $true = sP19 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP19 ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj1_gthm = $true )
& ( ! [X211: term,X212: term,X213: term,X214: term] :
( ( ( ap @ X214 @ X212 )
!= ( ap @ X213 @ X211 ) )
| ( X211 = X212 ) )
| ( apinj2 != $true ) )
& ( ( apinj2 = $true )
| ? [X215: term,X216: term,X217: term,X218: term] :
( ( ( ap @ X217 @ X215 )
= ( ap @ X218 @ X216 ) )
& ( X215 != X216 ) ) )
& ( ( substmonoid = $true )
| ? [X219: subst] :
( ( comp @ id @ X219 )
!= X219 )
| ? [X220: subst] :
( ( comp @ X220 @ id )
!= X220 ) )
& ( ( ! [X221: subst] :
( ( comp @ id @ X221 )
= X221 )
& ! [X222: subst] :
( ( comp @ X222 @ id )
= X222 ) )
| ( substmonoid != $true ) )
& ( ( $true = sP18 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP18 ) )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP15 ) )
& ( ( $true = sP15 )
| ( hoasinduction_lthm != $true ) )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP6 != $true ) )
& ( ( sP6 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoaslaminj = $true )
| ? [X223: subst > term > term,X224: subst > term > term,X225: subst,X226: term] :
( ( ( lam @ ( X224 @ sh @ one ) )
= ( lam @ ( X223 @ sh @ one ) ) )
& ! [X227: term,X228: subst,X229: subst] :
( ( X223 @ ( comp @ X229 @ X228 ) @ ( sub @ X227 @ X228 ) )
= ( sub @ ( X223 @ X229 @ X227 ) @ X228 ) )
& ( ( X224 @ X225 @ X226 )
!= ( X223 @ X225 @ X226 ) )
& ! [X230: term,X231: subst,X232: subst] :
( ( sub @ ( X224 @ X231 @ X230 ) @ X232 )
= ( X224 @ ( comp @ X231 @ X232 ) @ ( sub @ X230 @ X232 ) ) ) ) )
& ( ! [X233: subst > term > term,X234: subst > term > term,X235: subst,X236: term] :
( ( ( lam @ ( X233 @ sh @ one ) )
!= ( lam @ ( X234 @ sh @ one ) ) )
| ? [X237: term,X238: subst,X239: subst] :
( ( sub @ ( X233 @ X239 @ X237 ) @ X238 )
!= ( X233 @ ( comp @ X239 @ X238 ) @ ( sub @ X237 @ X238 ) ) )
| ( ( X233 @ X235 @ X236 )
= ( X234 @ X235 @ X236 ) )
| ? [X240: term,X241: subst,X242: subst] :
( ( sub @ ( X234 @ X241 @ X240 ) @ X242 )
!= ( X234 @ ( comp @ X241 @ X242 ) @ ( sub @ X240 @ X242 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ! [X243: term > $o,X244: term > $o,X245: term,X246: subst,X247: term] :
( ( ( X244 @ ( sub @ X247 @ ( push @ X245 @ X246 ) ) )
= $true )
| ( ( X243 @ X247 )
!= $true ) )
| ? [X248: subst,X249: term] :
( ( comp @ sh @ ( push @ X249 @ X248 ) )
!= X248 )
| ? [X250: subst,X251: term] :
( ( sub @ one @ ( push @ X251 @ X250 ) )
!= X251 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ? [X252: term > $o,X253: term > $o,X254: term,X255: subst,X256: term] :
( ( ( X253 @ ( sub @ X256 @ ( push @ X254 @ X255 ) ) )
!= $true )
& ( ( X252 @ X256 )
= $true ) )
& ! [X257: subst,X258: term] :
( ( comp @ sh @ ( push @ X258 @ X257 ) )
= X257 )
& ! [X259: subst,X260: term] :
( ( sub @ one @ ( push @ X260 @ X259 ) )
= X260 ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ulamvar1 != $true )
& ( hoasinduction_lem2_gthm = $true )
& ! [X261: d_term,X262: d_term] :
( ( ( d2term @ X262 )
!= ( d2term @ X261 ) )
| ( X261 = X262 ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
| ( sP1 != $true ) )
& ( ( sP1 = $true )
| ( hoasinduction_lem2 != $true ) )
& ( pushprop_lthm_orig = $true )
& ! [X263: d_term] : ( d_one = X263 )
& ( ( pushprop_lem0 = $true )
| ? [X264: term > $o,X265: term,X266: subst] :
! [X267: term > $o] :
? [X268: term] :
( ( X267 @ X268 )
!= ( X264 @ ( sub @ X268 @ ( push @ X265 @ X266 ) ) ) ) )
& ( ! [X269: term > $o,X270: term,X271: subst] :
? [X272: term > $o] :
! [X273: term] :
( ( X269 @ ( sub @ X273 @ ( push @ X270 @ X271 ) ) )
= ( X272 @ X273 ) )
| ( pushprop_lem0 != $true ) ) ),
inference(rectify,[status(thm)],[f474]) ).
thf(f474,plain,
( ( ( pushprop_lem0_lthm = $true )
| ? [X116: term,X117: subst,X118: term > $o] :
! [X119: term > $o] :
? [X120: term] :
( ( X119 @ X120 )
!= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) ) )
& ( ! [X116: term,X117: subst,X118: term > $o] :
? [X119: term > $o] :
! [X120: term] :
( ( X119 @ X120 )
= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ! [X466: subst] :
( ( comp @ X466 @ id )
= X466 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X466: subst] :
( ( comp @ X466 @ id )
!= X466 ) )
& ! [X112: d_subst,X113: d_subst] :
( ( ( d2subst @ X112 )
!= ( d2subst @ X113 ) )
| ( X112 = X113 ) )
& ( hoasinduction_lem2_lthm = $true )
& ( axclos = $true )
& ( ( $true = sP58 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP58 ) )
& ( ! [X174: term] :
( ( sub @ X174 @ id )
= X174 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X174: term] :
( ( sub @ X174 @ id )
!= X174 ) )
& ( substmonoid_gthm = $true )
& ( ( induction = $true )
| ? [X403: term > $o,X404: term] :
( ( ( sP56 @ X403 )
!= $true )
& ! [X407: term] :
( ( $true
!= ( var @ X407 ) )
| ( $true
= ( X403 @ X407 ) ) )
& ( ( X403 @ X404 )
!= $true )
& ! [X408: term] :
( ( ( X403 @ ( lam @ X408 ) )
= $true )
| ( ( X403 @ X408 )
!= $true ) ) ) )
& ( ! [X403: term > $o,X404: term] :
( ( ( sP56 @ X403 )
= $true )
| ? [X407: term] :
( ( $true
= ( var @ X407 ) )
& ( $true
!= ( X403 @ X407 ) ) )
| ( ( X403 @ X404 )
= $true )
| ? [X408: term] :
( ( ( X403 @ ( lam @ X408 ) )
!= $true )
& ( ( X403 @ X408 )
= $true ) ) )
| ( induction != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
!= $true )
& ( $true
= ( X137 @ X134 ) ) ) )
& ( ! [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
= $true )
| ( $true
!= ( X137 @ X134 ) ) )
| ( pushprop_lem1 != $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( axassoc = $true )
& ( ! [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
!= ( ap @ X276 @ X277 ) )
| ( X276 = X278 ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
= ( ap @ X276 @ X277 ) )
& ( X276 != X278 ) ) )
& ( hoasapinj2_gthm = $true )
& ! [X464: subst] :
? [X465: d_subst] :
( ( d2subst @ X465 )
= X464 )
& ( ? [X104: term,X105: term,X106: term,X107: term] :
( ( X106 != X107 )
& ( ( ap @ X106 @ X105 )
= ( ap @ X107 @ X104 ) ) )
| ? [X103: term] :
( ( sub @ X103 @ id )
!= X103 )
| ! [X108: term,X109: term,X110: term,X111: term] :
( ( X108 = X109 )
| ( ( ap @ ( sub @ X109 @ id ) @ X111 )
!= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X104: term,X105: term,X106: term,X107: term] :
( ( X106 = X107 )
| ( ( ap @ X106 @ X105 )
!= ( ap @ X107 @ X104 ) ) )
& ! [X103: term] :
( ( sub @ X103 @ id )
= X103 )
& ? [X108: term,X109: term,X110: term,X111: term] :
( ( X108 != X109 )
& ( ( ap @ ( sub @ X109 @ id ) @ X111 )
= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) ) ) )
& ( hoasapnotvar = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP55 ) )
& ( ( $true = sP55 )
| ( hoasinduction_lem3aa != $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar_lthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( sP54 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP54 != $true ) )
& ( termmset_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ! [X114: term] :
? [X115: d_term] :
( ( d2term @ X115 )
= X114 )
& ( ( shinj = $true )
| ? [X35: term,X36: term] :
( ( X35 != X36 )
& ( ( sub @ X36 @ sh )
= ( sub @ X35 @ sh ) ) ) )
& ( ! [X35: term,X36: term] :
( ( X35 = X36 )
| ( ( sub @ X36 @ sh )
!= ( sub @ X35 @ sh ) ) )
| ( shinj != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X333: subst > term > subst > $o] :
! [X334: term > $o] :
? [X335: term] :
( ( X334 @ X335 )
!= ( X333 @ id @ X335 @ id ) ) )
& ( ! [X333: subst > term > subst > $o] :
? [X334: term > $o] :
! [X335: term] :
( ( X334 @ X335 )
= ( X333 @ id @ X335 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( induction2_gthm = $true )
& ( ! [X178: term] :
? [X179: subst > term > term] :
! [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
= ( X179 @ X180 @ X181 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X178: term] :
! [X179: subst > term > term] :
? [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
!= ( X179 @ X180 @ X181 ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X384: term] :
! [X385: subst > term > term] :
? [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
!= ( X385 @ X386 @ X387 ) ) )
& ( ! [X384: term] :
? [X385: subst > term > term] :
! [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
= ( X385 @ X386 @ X387 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP53 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP53 != $true ) )
& ! [X177: d_subst] : ( d_id = X177 )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
= X176 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
!= X176 ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP51 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP51 ) )
& ( axapp = $true )
& ( induction2lem_gthm = $true )
& ( ulamvarsh = $true )
& ( axvarshift = $true )
& ( ulamvarind = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( termmset = $true )
| ? [X247: term] :
( ( sub @ X247 @ id )
!= X247 ) )
& ( ! [X247: term] :
( ( sub @ X247 @ id )
= X247 )
| ( termmset != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( sP49 != $true ) )
& ( ( sP49 = $true )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( apnotvar = $true )
& ( ! [X150: term] :
( ( sub @ X150 @ id )
= X150 )
| ? [X149: term] :
( ( sub @ X149 @ id )
!= X149 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ? [X150: term] :
( ( sub @ X150 @ id )
!= X150 )
& ! [X149: term] :
( ( sub @ X149 @ id )
= X149 ) ) )
& ( ( sP48 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP48 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP46 ) )
& ( ( $true = sP46 )
| ( hoasinduction_lem3 != $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP44 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( sP44 != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasinduction_lem3b = $true )
| ? [X8: term] :
! [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
!= ( X9 @ sh @ one ) ) )
& ( ! [X8: term] :
? [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
= ( X9 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( axmap = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop = $true )
& ( ( substmonoid_lthm = $true )
| ( ! [X537: subst] :
( ( comp @ X537 @ id )
= X537 )
& ( ? [X538: subst] :
( ( comp @ X538 @ id )
!= X538 )
| ? [X539: subst] :
( ( comp @ id @ X539 )
!= X539 ) )
& ! [X536: subst] :
( ( comp @ id @ X536 )
= X536 ) ) )
& ( ? [X537: subst] :
( ( comp @ X537 @ id )
!= X537 )
| ( ! [X538: subst] :
( ( comp @ X538 @ id )
= X538 )
& ! [X539: subst] :
( ( comp @ id @ X539 )
= X539 ) )
| ? [X536: subst] :
( ( comp @ id @ X536 )
!= X536 )
| ( substmonoid_lthm != $true ) )
& ( axabs = $true )
& ( lamnotvar = $true )
& ( ! [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
= X240 )
| ( axshiftcons != $true ) )
& ( ( axshiftcons = $true )
| ? [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
!= X240 ) )
& ( ( sP43 = $true )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( sP43 != $true ) )
& ( ( $true = sP39 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP39 ) )
& ( ( sP38 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP38 != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
!= X102 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( pushprop_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X138: subst > term > subst > $o,X139: term] :
( ( ( X138 @ id @ X139 @ id )
!= $true )
& ! [X145: term,X146: term] :
( ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
= $true )
| ( $true
!= ( X138 @ id @ X145 @ id ) )
| ( ( X138 @ id @ X146 @ id )
!= $true ) )
& ( $true
!= ( sP34 @ X138 ) ) ) )
& ( ! [X138: subst > term > subst > $o,X139: term] :
( ( ( X138 @ id @ X139 @ id )
= $true )
| ? [X145: term,X146: term] :
( ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
!= $true )
& ( $true
= ( X138 @ id @ X145 @ id ) )
& ( ( X138 @ id @ X146 @ id )
= $true ) )
| ( $true
= ( sP34 @ X138 ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction = $true )
| ( sP33 != $true ) )
& ( ( sP33 = $true )
| ( hoasinduction != $true ) )
& ( ( laminj = $true )
| ? [X147: term,X148: term] :
( ( X147 != X148 )
& ( ( lam @ X147 )
= ( lam @ X148 ) ) ) )
& ( ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) )
| ( laminj != $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( axidl = $true )
| ? [X471: subst] :
( ( comp @ id @ X471 )
!= X471 ) )
& ( ! [X471: subst] :
( ( comp @ id @ X471 )
= X471 )
| ( axidl != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X98: term,X99: term,X100: term,X101: term] :
( ( X100 != X101 )
& ( ( ap @ ( sub @ X98 @ id ) @ X100 )
= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) ) )
& ( ! [X98: term,X99: term,X100: term,X101: term] :
( ( X100 = X101 )
| ( ( ap @ ( sub @ X98 @ id ) @ X100 )
!= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X336: term,X337: term,X338: subst > term > term] :
( ! [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
& ( ( ap @ ( sub @ X336 @ id ) @ X337 )
= ( lam @ ( X338 @ sh @ one ) ) ) ) )
& ( ! [X336: term,X337: term,X338: subst > term > term] :
( ? [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
!= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
| ( ( ap @ ( sub @ X336 @ id ) @ X337 )
!= ( lam @ ( X338 @ sh @ one ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP30 != $true ) )
& ( ( sP30 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( sP24 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP24 != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X532: term,X533: term,X534: term,X535: term] :
( ( X532 != X534 )
& ( ( ap @ ( sub @ X533 @ id ) @ X532 )
= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) )
& ! [X528: term,X529: term,X530: term,X531: term] :
( ( ( ap @ X531 @ X529 )
!= ( ap @ X528 @ X530 ) )
| ( X529 = X530 ) ) ) )
& ( ! [X532: term,X533: term,X534: term,X535: term] :
( ( X532 = X534 )
| ( ( ap @ ( sub @ X533 @ id ) @ X532 )
!= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) )
| ? [X528: term,X529: term,X530: term,X531: term] :
( ( ( ap @ X531 @ X529 )
= ( ap @ X528 @ X530 ) )
& ( X529 != X530 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X13: term,X14: term,X15: term,X16: term] :
( ( X13 != X16 )
& ( ( ap @ ( sub @ X13 @ id ) @ X14 )
= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) ) )
& ( ! [X13: term,X14: term,X15: term,X16: term] :
( ( X13 = X16 )
| ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) )
| ( hoasapinj1 != $true ) )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ! [X467: term] :
? [X468: subst > term > term] :
( ( X468 @ sh @ one )
= ( sub @ X467 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X467: term] :
! [X468: subst > term > term] :
( ( X468 @ sh @ one )
!= ( sub @ X467 @ ( push @ one @ sh ) ) ) )
& ( ( sP23 = $true )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( sP23 != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ! [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
= X168 )
& ? [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ! [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
= ( X170 @ X173 ) )
& ( ( X170 @ one )
!= $true )
& ( ( X171 @ X172 )
= $true ) ) ) )
& ( ? [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
!= X168 )
| ! [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ? [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
!= ( X170 @ X173 ) )
| ( ( X170 @ one )
= $true )
| ( ( X171 @ X172 )
!= $true ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ! [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ( ( X474 @ X475 )
!= $true )
| ? [X476: term] :
( ( X472 @ X476 )
!= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
| ( ( X472 @ one )
= $true ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ( ( X474 @ X475 )
= $true )
& ! [X476: term] :
( ( X472 @ X476 )
= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
& ( ( X472 @ one )
!= $true ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( sP21 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP21 != $true ) )
& ( pushprop_gthm = $true )
& ( ! [X10: subst > term > subst > $o] :
? [X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X10: subst > term > subst > $o] :
! [X11: term > $o] :
? [X12: term] :
( ( X11 @ X12 )
!= ( X10 @ id @ X12 @ id ) ) )
& ( ( $true = sP19 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP19 ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj1_gthm = $true )
& ( ! [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
!= ( ap @ X243 @ X241 ) )
| ( X241 = X242 ) )
| ( apinj2 != $true ) )
& ( ( apinj2 = $true )
| ? [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
= ( ap @ X243 @ X241 ) )
& ( X241 != X242 ) ) )
& ( ( substmonoid = $true )
| ? [X246: subst] :
( ( comp @ id @ X246 )
!= X246 )
| ? [X245: subst] :
( ( comp @ X245 @ id )
!= X245 ) )
& ( ( ! [X246: subst] :
( ( comp @ id @ X246 )
= X246 )
& ! [X245: subst] :
( ( comp @ X245 @ id )
= X245 ) )
| ( substmonoid != $true ) )
& ( ( $true = sP18 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP18 ) )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP15 ) )
& ( ( $true = sP15 )
| ( hoasinduction_lthm != $true ) )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP6 != $true ) )
& ( ( sP6 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoaslaminj = $true )
| ? [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ( ( lam @ ( X191 @ sh @ one ) )
= ( lam @ ( X190 @ sh @ one ) ) )
& ! [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
& ( ( X191 @ X192 @ X193 )
!= ( X190 @ X192 @ X193 ) )
& ! [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) ) ) )
& ( ! [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ( ( lam @ ( X191 @ sh @ one ) )
!= ( lam @ ( X190 @ sh @ one ) ) )
| ? [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
!= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
| ( ( X191 @ X192 @ X193 )
= ( X190 @ X192 @ X193 ) )
| ? [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
!= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ! [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) )
| ( $true
!= ( X523 @ X527 ) ) )
| ? [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
!= X521 )
| ? [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
!= X520 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ? [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
!= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) )
& ( $true
= ( X523 @ X527 ) ) )
& ! [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
= X521 )
& ! [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
= X520 ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ulamvar1 != $true )
& ( hoasinduction_lem2_gthm = $true )
& ! [X469: d_term,X470: d_term] :
( ( ( d2term @ X469 )
!= ( d2term @ X470 ) )
| ( X469 = X470 ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
| ( sP1 != $true ) )
& ( ( sP1 = $true )
| ( hoasinduction_lem2 != $true ) )
& ( pushprop_lthm_orig = $true )
& ! [X450: d_term] : ( d_one = X450 )
& ( ( pushprop_lem0 = $true )
| ? [X37: term > $o,X38: term,X39: subst] :
! [X40: term > $o] :
? [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
!= ( X40 @ X41 ) ) )
& ( ! [X37: term > $o,X38: term,X39: subst] :
? [X40: term > $o] :
! [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
= ( X40 @ X41 ) )
| ( pushprop_lem0 != $true ) ) ),
inference(flattening,[status(thm)],[f473]) ).
thf(f473,plain,
( ( ( pushprop_lem0_lthm = $true )
| ? [X116: term,X117: subst,X118: term > $o] :
! [X119: term > $o] :
? [X120: term] :
( ( X119 @ X120 )
!= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) ) )
& ( ! [X116: term,X117: subst,X118: term > $o] :
? [X119: term > $o] :
! [X120: term] :
( ( X119 @ X120 )
= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) )
| ( pushprop_lem0_lthm != $true ) )
& ( ! [X466: subst] :
( ( comp @ X466 @ id )
= X466 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X466: subst] :
( ( comp @ X466 @ id )
!= X466 ) )
& ! [X112: d_subst,X113: d_subst] :
( ( ( d2subst @ X112 )
!= ( d2subst @ X113 ) )
| ( X112 = X113 ) )
& ( hoasinduction_lem2_lthm = $true )
& ( axclos = $true )
& ( ( $true = sP58 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP58 ) )
& ( ! [X174: term] :
( ( sub @ X174 @ id )
= X174 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X174: term] :
( ( sub @ X174 @ id )
!= X174 ) )
& ( substmonoid_gthm = $true )
& ( ( induction = $true )
| ? [X403: term > $o,X404: term] :
( ( ( sP56 @ X403 )
!= $true )
& ! [X407: term] :
( ( $true
!= ( var @ X407 ) )
| ( $true
= ( X403 @ X407 ) ) )
& ( ( X403 @ X404 )
!= $true )
& ! [X408: term] :
( ( ( X403 @ ( lam @ X408 ) )
= $true )
| ( ( X403 @ X408 )
!= $true ) ) ) )
& ( ! [X403: term > $o,X404: term] :
( ( ( sP56 @ X403 )
= $true )
| ? [X407: term] :
( ( $true
= ( var @ X407 ) )
& ( $true
!= ( X403 @ X407 ) ) )
| ( ( X403 @ X404 )
= $true )
| ? [X408: term] :
( ( ( X403 @ ( lam @ X408 ) )
!= $true )
& ( ( X403 @ X408 )
= $true ) ) )
| ( induction != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
!= $true )
& ( $true
= ( X137 @ X134 ) ) ) )
& ( ! [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
= $true )
| ( $true
!= ( X137 @ X134 ) ) )
| ( pushprop_lem1 != $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( axassoc = $true )
& ( ! [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
!= ( ap @ X276 @ X277 ) )
| ( X276 = X278 ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
= ( ap @ X276 @ X277 ) )
& ( X276 != X278 ) ) )
& ( hoasapinj2_gthm = $true )
& ! [X464: subst] :
? [X465: d_subst] :
( ( d2subst @ X465 )
= X464 )
& ( ? [X104: term,X105: term,X106: term,X107: term] :
( ( X106 != X107 )
& ( ( ap @ X106 @ X105 )
= ( ap @ X107 @ X104 ) ) )
| ? [X103: term] :
( ( sub @ X103 @ id )
!= X103 )
| ! [X108: term,X109: term,X110: term,X111: term] :
( ( X108 = X109 )
| ( ( ap @ ( sub @ X109 @ id ) @ X111 )
!= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X104: term,X105: term,X106: term,X107: term] :
( ( X106 = X107 )
| ( ( ap @ X106 @ X105 )
!= ( ap @ X107 @ X104 ) ) )
& ! [X103: term] :
( ( sub @ X103 @ id )
= X103 )
& ? [X108: term,X109: term,X110: term,X111: term] :
( ( X108 != X109 )
& ( ( ap @ ( sub @ X109 @ id ) @ X111 )
= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) ) ) )
& ( hoasapnotvar = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP55 ) )
& ( ( $true = sP55 )
| ( hoasinduction_lem3aa != $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar_lthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( sP54 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP54 != $true ) )
& ( termmset_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ! [X114: term] :
? [X115: d_term] :
( ( d2term @ X115 )
= X114 )
& ( ( shinj = $true )
| ? [X35: term,X36: term] :
( ( X35 != X36 )
& ( ( sub @ X36 @ sh )
= ( sub @ X35 @ sh ) ) ) )
& ( ! [X35: term,X36: term] :
( ( X35 = X36 )
| ( ( sub @ X36 @ sh )
!= ( sub @ X35 @ sh ) ) )
| ( shinj != $true ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X333: subst > term > subst > $o] :
! [X334: term > $o] :
? [X335: term] :
( ( X334 @ X335 )
!= ( X333 @ id @ X335 @ id ) ) )
& ( ! [X333: subst > term > subst > $o] :
? [X334: term > $o] :
! [X335: term] :
( ( X334 @ X335 )
= ( X333 @ id @ X335 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( induction2_gthm = $true )
& ( ! [X178: term] :
? [X179: subst > term > term] :
! [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
= ( X179 @ X180 @ X181 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X178: term] :
! [X179: subst > term > term] :
? [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
!= ( X179 @ X180 @ X181 ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X384: term] :
! [X385: subst > term > term] :
? [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
!= ( X385 @ X386 @ X387 ) ) )
& ( ! [X384: term] :
? [X385: subst > term > term] :
! [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
= ( X385 @ X386 @ X387 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP53 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP53 != $true ) )
& ! [X177: d_subst] : ( d_id = X177 )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
= X176 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
!= X176 ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP51 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP51 ) )
& ( axapp = $true )
& ( induction2lem_gthm = $true )
& ( ulamvarsh = $true )
& ( axvarshift = $true )
& ( ulamvarind = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( termmset = $true )
| ? [X247: term] :
( ( sub @ X247 @ id )
!= X247 ) )
& ( ! [X247: term] :
( ( sub @ X247 @ id )
= X247 )
| ( termmset != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( sP49 != $true ) )
& ( ( sP49 = $true )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( apnotvar = $true )
& ( ! [X150: term] :
( ( sub @ X150 @ id )
= X150 )
| ? [X149: term] :
( ( sub @ X149 @ id )
!= X149 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ? [X150: term] :
( ( sub @ X150 @ id )
!= X150 )
& ! [X149: term] :
( ( sub @ X149 @ id )
= X149 ) ) )
& ( ( sP48 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP48 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP46 ) )
& ( ( $true = sP46 )
| ( hoasinduction_lem3 != $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP44 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( ( hoaslaminj_lthm = $true )
| ( sP44 != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasinduction_lem3b = $true )
| ? [X8: term] :
! [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
!= ( X9 @ sh @ one ) ) )
& ( ! [X8: term] :
? [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
= ( X9 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( axmap = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop = $true )
& ( ( substmonoid_lthm = $true )
| ( ! [X537: subst] :
( ( comp @ X537 @ id )
= X537 )
& ( ? [X538: subst] :
( ( comp @ X538 @ id )
!= X538 )
| ? [X539: subst] :
( ( comp @ id @ X539 )
!= X539 ) )
& ! [X536: subst] :
( ( comp @ id @ X536 )
= X536 ) ) )
& ( ? [X537: subst] :
( ( comp @ X537 @ id )
!= X537 )
| ( ! [X538: subst] :
( ( comp @ X538 @ id )
= X538 )
& ! [X539: subst] :
( ( comp @ id @ X539 )
= X539 ) )
| ? [X536: subst] :
( ( comp @ id @ X536 )
!= X536 )
| ( substmonoid_lthm != $true ) )
& ( axabs = $true )
& ( lamnotvar = $true )
& ( ! [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
= X240 )
| ( axshiftcons != $true ) )
& ( ( axshiftcons = $true )
| ? [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
!= X240 ) )
& ( ( sP43 = $true )
| ( induction2lem_lthm != $true ) )
& ( ( induction2lem_lthm = $true )
| ( sP43 != $true ) )
& ( ( $true = sP39 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP39 ) )
& ( ( sP38 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP38 != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
!= X102 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( pushprop_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X138: subst > term > subst > $o,X139: term] :
( ( ( X138 @ id @ X139 @ id )
!= $true )
& ! [X145: term,X146: term] :
( ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
= $true )
| ( $true
!= ( X138 @ id @ X145 @ id ) )
| ( ( X138 @ id @ X146 @ id )
!= $true ) )
& ( $true
!= ( sP34 @ X138 ) ) ) )
& ( ! [X138: subst > term > subst > $o,X139: term] :
( ( ( X138 @ id @ X139 @ id )
= $true )
| ? [X145: term,X146: term] :
( ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
!= $true )
& ( $true
= ( X138 @ id @ X145 @ id ) )
& ( ( X138 @ id @ X146 @ id )
= $true ) )
| ( $true
= ( sP34 @ X138 ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( lamnotap != $true )
& ( ( hoasinduction = $true )
| ( sP33 != $true ) )
& ( ( sP33 = $true )
| ( hoasinduction != $true ) )
& ( ( laminj = $true )
| ? [X147: term,X148: term] :
( ( X147 != X148 )
& ( ( lam @ X147 )
= ( lam @ X148 ) ) ) )
& ( ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) )
| ( laminj != $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( axidl = $true )
| ? [X471: subst] :
( ( comp @ id @ X471 )
!= X471 ) )
& ( ! [X471: subst] :
( ( comp @ id @ X471 )
= X471 )
| ( axidl != $true ) )
& ( ( hoasapinj2 = $true )
| ? [X98: term,X99: term,X100: term,X101: term] :
( ( X100 != X101 )
& ( ( ap @ ( sub @ X98 @ id ) @ X100 )
= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) ) )
& ( ! [X98: term,X99: term,X100: term,X101: term] :
( ( X100 = X101 )
| ( ( ap @ ( sub @ X98 @ id ) @ X100 )
!= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) )
| ( hoasapinj2 != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X336: term,X337: term,X338: subst > term > term] :
( ! [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
& ( ( ap @ ( sub @ X336 @ id ) @ X337 )
= ( lam @ ( X338 @ sh @ one ) ) ) ) )
& ( ! [X336: term,X337: term,X338: subst > term > term] :
( ? [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
!= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
| ( ( ap @ ( sub @ X336 @ id ) @ X337 )
!= ( lam @ ( X338 @ sh @ one ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( sP30 != $true ) )
& ( ( sP30 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( sP24 = $true )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( sP24 != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( ? [X532: term,X533: term,X534: term,X535: term] :
( ( X532 != X534 )
& ( ( ap @ ( sub @ X533 @ id ) @ X532 )
= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) )
& ! [X528: term,X529: term,X530: term,X531: term] :
( ( ( ap @ X531 @ X529 )
!= ( ap @ X528 @ X530 ) )
| ( X529 = X530 ) ) ) )
& ( ! [X532: term,X533: term,X534: term,X535: term] :
( ( X532 = X534 )
| ( ( ap @ ( sub @ X533 @ id ) @ X532 )
!= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) )
| ? [X528: term,X529: term,X530: term,X531: term] :
( ( ( ap @ X531 @ X529 )
= ( ap @ X528 @ X530 ) )
& ( X529 != X530 ) )
| ( hoasapinj2_lthm != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X13: term,X14: term,X15: term,X16: term] :
( ( X13 != X16 )
& ( ( ap @ ( sub @ X13 @ id ) @ X14 )
= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) ) )
& ( ! [X13: term,X14: term,X15: term,X16: term] :
( ( X13 = X16 )
| ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) )
| ( hoasapinj1 != $true ) )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ! [X467: term] :
? [X468: subst > term > term] :
( ( X468 @ sh @ one )
= ( sub @ X467 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X467: term] :
! [X468: subst > term > term] :
( ( X468 @ sh @ one )
!= ( sub @ X467 @ ( push @ one @ sh ) ) ) )
& ( ( sP23 = $true )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( sP23 != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ! [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
= X168 )
& ? [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ! [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
= ( X170 @ X173 ) )
& ( ( X170 @ one )
!= $true )
& ( ( X171 @ X172 )
= $true ) ) ) )
& ( ? [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
!= X168 )
| ! [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ? [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
!= ( X170 @ X173 ) )
| ( ( X170 @ one )
= $true )
| ( ( X171 @ X172 )
!= $true ) )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ! [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ( ( X474 @ X475 )
!= $true )
| ? [X476: term] :
( ( X472 @ X476 )
!= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
| ( ( X472 @ one )
= $true ) )
| ( pushprop_lem1v2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ( ( X474 @ X475 )
= $true )
& ! [X476: term] :
( ( X472 @ X476 )
= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
& ( ( X472 @ one )
!= $true ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( sP21 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP21 != $true ) )
& ( pushprop_gthm = $true )
& ( ! [X10: subst > term > subst > $o] :
? [X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X10: subst > term > subst > $o] :
! [X11: term > $o] :
? [X12: term] :
( ( X11 @ X12 )
!= ( X10 @ id @ X12 @ id ) ) )
& ( ( $true = sP19 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP19 ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj1_gthm = $true )
& ( ! [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
!= ( ap @ X243 @ X241 ) )
| ( X241 = X242 ) )
| ( apinj2 != $true ) )
& ( ( apinj2 = $true )
| ? [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
= ( ap @ X243 @ X241 ) )
& ( X241 != X242 ) ) )
& ( ( substmonoid = $true )
| ? [X246: subst] :
( ( comp @ id @ X246 )
!= X246 )
| ? [X245: subst] :
( ( comp @ X245 @ id )
!= X245 ) )
& ( ( ! [X246: subst] :
( ( comp @ id @ X246 )
= X246 )
& ! [X245: subst] :
( ( comp @ X245 @ id )
= X245 ) )
| ( substmonoid != $true ) )
& ( ( $true = sP18 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP18 ) )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP15 ) )
& ( ( $true = sP15 )
| ( hoasinduction_lthm != $true ) )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP6 != $true ) )
& ( ( sP6 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoaslaminj = $true )
| ? [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ( ( lam @ ( X191 @ sh @ one ) )
= ( lam @ ( X190 @ sh @ one ) ) )
& ! [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
& ( ( X191 @ X192 @ X193 )
!= ( X190 @ X192 @ X193 ) )
& ! [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) ) ) )
& ( ! [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ( ( lam @ ( X191 @ sh @ one ) )
!= ( lam @ ( X190 @ sh @ one ) ) )
| ? [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
!= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
| ( ( X191 @ X192 @ X193 )
= ( X190 @ X192 @ X193 ) )
| ? [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
!= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ! [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) )
| ( $true
!= ( X523 @ X527 ) ) )
| ? [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
!= X521 )
| ? [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
!= X520 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ? [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
!= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) )
& ( $true
= ( X523 @ X527 ) ) )
& ! [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
= X521 )
& ! [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
= X520 ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ulamvar1 != $true )
& ( hoasinduction_lem2_gthm = $true )
& ! [X469: d_term,X470: d_term] :
( ( ( d2term @ X469 )
!= ( d2term @ X470 ) )
| ( X469 = X470 ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
| ( sP1 != $true ) )
& ( ( sP1 = $true )
| ( hoasinduction_lem2 != $true ) )
& ( pushprop_lthm_orig = $true )
& ! [X450: d_term] : ( d_one = X450 )
& ( ( pushprop_lem0 = $true )
| ? [X37: term > $o,X38: term,X39: subst] :
! [X40: term > $o] :
? [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
!= ( X40 @ X41 ) ) )
& ( ! [X37: term > $o,X38: term,X39: subst] :
? [X40: term > $o] :
! [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
= ( X40 @ X41 ) )
| ( pushprop_lem0 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f73]) ).
thf(f73,plain,
( ( ( pushprop_lem0_lthm = $true )
<=> ! [X116: term,X117: subst,X118: term > $o] :
? [X119: term > $o] :
! [X120: term] :
( ( X119 @ X120 )
= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) ) )
& ( ! [X466: subst] :
( ( comp @ X466 @ id )
= X466 )
<=> ( axidr = $true ) )
& ! [X112: d_subst,X113: d_subst] :
( ( ( d2subst @ X112 )
!= ( d2subst @ X113 ) )
| ( X112 = X113 ) )
& ( hoasinduction_lem2_lthm = $true )
& ( axclos = $true )
& ( ( $true = sP58 )
<=> ( induction2lem = $true ) )
& ( ! [X174: term] :
( ( sub @ X174 @ id )
= X174 )
<=> ( axvarid = $true ) )
& ( substmonoid_gthm = $true )
& ( ( induction = $true )
<=> ! [X403: term > $o,X404: term] :
( ( ( sP56 @ X403 )
= $true )
| ? [X407: term] :
( ( $true
= ( var @ X407 ) )
& ( $true
!= ( X403 @ X407 ) ) )
| ( ( X403 @ X404 )
= $true )
| ? [X408: term] :
( ( ( X403 @ ( lam @ X408 ) )
!= $true )
& ( ( X403 @ X408 )
= $true ) ) ) )
& ( ( pushprop_lem1 = $true )
<=> ! [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
= $true )
| ( $true
!= ( X137 @ X134 ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( axassoc = $true )
& ( ! [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
!= ( ap @ X276 @ X277 ) )
| ( X276 = X278 ) )
<=> ( apinj1 = $true ) )
& ( hoasapinj2_gthm = $true )
& ! [X464: subst] :
? [X465: d_subst] :
( ( d2subst @ X465 )
= X464 )
& ( ( ? [X104: term,X105: term,X106: term,X107: term] :
( ( X106 != X107 )
& ( ( ap @ X106 @ X105 )
= ( ap @ X107 @ X104 ) ) )
| ? [X103: term] :
( ( sub @ X103 @ id )
!= X103 )
| ! [X108: term,X109: term,X110: term,X111: term] :
( ( X108 = X109 )
| ( ( ap @ ( sub @ X109 @ id ) @ X111 )
!= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( hoasapnotvar = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ( $true = sP55 ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar_lthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( sP54 = $true )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( termmset_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ! [X114: term] :
? [X115: d_term] :
( ( d2term @ X115 )
= X114 )
& ( ( shinj = $true )
<=> ! [X35: term,X36: term] :
( ( X35 = X36 )
| ( ( sub @ X36 @ sh )
!= ( sub @ X35 @ sh ) ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X333: subst > term > subst > $o] :
? [X334: term > $o] :
! [X335: term] :
( ( X334 @ X335 )
= ( X333 @ id @ X335 @ id ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( induction2_gthm = $true )
& ( ! [X178: term] :
? [X179: subst > term > term] :
! [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
= ( X179 @ X180 @ X181 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X384: term] :
? [X385: subst > term > term] :
! [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
= ( X385 @ X386 @ X387 ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( sP53 = $true )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ! [X177: d_subst] : ( d_id = X177 )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
= X176 )
<=> ( axvarcons = $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP51 )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( axapp = $true )
& ( induction2lem_gthm = $true )
& ( ulamvarsh = $true )
& ( axvarshift = $true )
& ( ulamvarind = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( termmset = $true )
<=> ! [X247: term] :
( ( sub @ X247 @ id )
= X247 ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ( sP49 = $true ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( apnotvar = $true )
& ( ( ! [X150: term] :
( ( sub @ X150 @ id )
= X150 )
| ? [X149: term] :
( ( sub @ X149 @ id )
!= X149 ) )
<=> ( termmset_lthm = $true ) )
& ( ( sP48 = $true )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ( hoasinduction_lem3 = $true )
<=> ( $true = sP46 ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP44 = $true )
<=> ( hoaslaminj_lthm = $true ) )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X8: term] :
? [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
= ( X9 @ sh @ one ) ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( axmap = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop = $true )
& ( ( substmonoid_lthm = $true )
<=> ( ? [X537: subst] :
( ( comp @ X537 @ id )
!= X537 )
| ( ! [X538: subst] :
( ( comp @ X538 @ id )
= X538 )
& ! [X539: subst] :
( ( comp @ id @ X539 )
= X539 ) )
| ? [X536: subst] :
( ( comp @ id @ X536 )
!= X536 ) ) )
& ( axabs = $true )
& ( lamnotvar = $true )
& ( ! [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
= X240 )
<=> ( axshiftcons = $true ) )
& ( ( sP43 = $true )
<=> ( induction2lem_lthm = $true ) )
& ( ( $true = sP39 )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( sP38 = $true )
<=> ( induction2_lthm = $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( pushprop_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X138: subst > term > subst > $o,X139: term] :
( ( ( X138 @ id @ X139 @ id )
= $true )
| ? [X145: term,X146: term] :
( ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
!= $true )
& ( $true
= ( X138 @ id @ X145 @ id ) )
& ( ( X138 @ id @ X146 @ id )
= $true ) )
| ( $true
= ( sP34 @ X138 ) ) ) )
& ( lamnotap != $true )
& ( ( hoasinduction = $true )
<=> ( sP33 = $true ) )
& ( ( laminj = $true )
<=> ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( axidl = $true )
<=> ! [X471: subst] :
( ( comp @ id @ X471 )
= X471 ) )
& ( ( hoasapinj2 = $true )
<=> ! [X98: term,X99: term,X100: term,X101: term] :
( ( X100 = X101 )
| ( ( ap @ ( sub @ X98 @ id ) @ X100 )
!= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X336: term,X337: term,X338: subst > term > term] :
( ? [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
!= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
| ( ( ap @ ( sub @ X336 @ id ) @ X337 )
!= ( lam @ ( X338 @ sh @ one ) ) ) ) )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( sP30 = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( sP24 = $true )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X532: term,X533: term,X534: term,X535: term] :
( ( X532 = X534 )
| ( ( ap @ ( sub @ X533 @ id ) @ X532 )
!= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) )
| ? [X528: term,X529: term,X530: term,X531: term] :
( ( ( ap @ X531 @ X529 )
= ( ap @ X528 @ X530 ) )
& ( X529 != X530 ) ) ) )
& ( ( hoasapinj1 = $true )
<=> ! [X13: term,X14: term,X15: term,X16: term] :
( ( X13 = X16 )
| ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ! [X467: term] :
? [X468: subst > term > term] :
( ( X468 @ sh @ one )
= ( sub @ X467 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( ( sP23 = $true )
<=> ( induction2 = $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ? [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
!= X168 )
| ! [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ? [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
!= ( X170 @ X173 ) )
| ( ( X170 @ one )
= $true )
| ( ( X171 @ X172 )
!= $true ) ) ) )
& ( ! [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ( ( X474 @ X475 )
!= $true )
| ? [X476: term] :
( ( X472 @ X476 )
!= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
| ( ( X472 @ one )
= $true ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( ( sP21 = $true )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( pushprop_gthm = $true )
& ( ! [X10: subst > term > subst > $o] :
? [X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( $true = sP19 )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj1_gthm = $true )
& ( ! [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
!= ( ap @ X243 @ X241 ) )
| ( X241 = X242 ) )
<=> ( apinj2 = $true ) )
& ( ( substmonoid = $true )
<=> ( ! [X246: subst] :
( ( comp @ id @ X246 )
= X246 )
& ! [X245: subst] :
( ( comp @ X245 @ id )
= X245 ) ) )
& ( ( $true = sP18 )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( ( hoasinduction_lthm = $true )
<=> ( $true = sP15 ) )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( sP6 = $true ) )
& ( ( hoaslaminj = $true )
<=> ! [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ( ( lam @ ( X191 @ sh @ one ) )
!= ( lam @ ( X190 @ sh @ one ) ) )
| ? [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
!= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
| ( ( X191 @ X192 @ X193 )
= ( X190 @ X192 @ X193 ) )
| ? [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
!= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) ) ) )
& ( ( ! [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) )
| ( $true
!= ( X523 @ X527 ) ) )
| ? [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
!= X521 )
| ? [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
!= X520 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ulamvar1 != $true )
& ( hoasinduction_lem2_gthm = $true )
& ! [X469: d_term,X470: d_term] :
( ( ( d2term @ X469 )
!= ( d2term @ X470 ) )
| ( X469 = X470 ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ( sP1 = $true ) )
& ( pushprop_lthm_orig = $true )
& ! [X450: d_term] : ( d_one = X450 )
& ( ( pushprop_lem0 = $true )
<=> ! [X37: term > $o,X38: term,X39: subst] :
? [X40: term > $o] :
! [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
= ( X40 @ X41 ) ) ) ),
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,
! [X451: subst > term > subst > $o] :
( ( ( sP0 @ X451 )
= $true )
<=> ? [X458: term,X459: term] :
( ( $true
!= ( X451 @ id @ ( ap @ ( sub @ X458 @ id ) @ X459 ) @ id ) )
& ( $true
= ( X451 @ id @ X459 @ id ) )
& ( ( X451 @ id @ X458 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f15,plain,
( ( sP1 = $true )
<=> ! [X451: subst > term > subst > $o,X452: term,X453: term] :
( ? [X460: subst,X461: subst,X462: term,X463: subst] :
( ( ( X451 @ X460 @ X462 @ ( comp @ X463 @ X461 ) )
!= $true )
& ( ( X451 @ ( comp @ X460 @ X463 ) @ ( sub @ X462 @ X463 ) @ X461 )
= $true ) )
| ( ( X451 @ id @ X453 @ id )
!= $true )
| ( ( sP0 @ X451 )
= $true )
| ? [X454: subst,X455: subst,X456: term,X457: subst] :
( ( ( X451 @ X457 @ X456 @ ( comp @ X455 @ X454 ) )
= $true )
& ( $true
!= ( X451 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 ) ) )
| ( ( X451 @ id @ ( ap @ X453 @ X452 ) @ id )
= $true )
| ( ( X451 @ id @ X452 @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f16,plain,
! [X70: subst > term > subst > $o] :
( ( ( sP2 @ X70 )
= $true )
<=> ? [X77: term,X78: term] :
( ( $true
= ( X70 @ id @ X77 @ id ) )
& ( ( X70 @ id @ ( ap @ ( sub @ X78 @ id ) @ X77 ) @ id )
!= $true )
& ( ( X70 @ id @ X78 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f17,plain,
! [X52: term > $o] :
( ( ( sP3 @ X52 )
= $true )
<=> ! [X54: term,X55: term] :
( ( $true
!= ( X52 @ X54 ) )
| ( ( X52 @ ( ap @ X55 @ X54 ) )
= $true )
| ( ( X52 @ X55 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f18,plain,
( ( sP4 = $true )
<=> ? [X60: subst > term > subst > $o,X61: term > $o,X62: term] :
( ! [X68: term] :
( ( $true
= ( X61 @ ( sub @ X62 @ ( push @ X68 @ id ) ) ) )
| ( ( X61 @ X68 )
!= $true ) )
& ! [X69: term] :
( ( X60 @ id @ X69 @ id )
= ( X61 @ X69 ) )
& ! [X63: subst > term > term] :
( ( $true
= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
| ? [X65: subst,X66: subst,X67: term] :
( ( sub @ ( X63 @ X66 @ X67 ) @ X65 )
!= ( X63 @ ( comp @ X66 @ X65 ) @ ( sub @ X67 @ X65 ) ) )
| ? [X64: term] :
( ( ( X60 @ id @ X64 @ id )
= $true )
& ( ( X60 @ id @ ( X63 @ id @ X64 ) @ id )
!= $true ) ) )
& ( ( X61 @ ( lam @ X62 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f19,plain,
( ( sP5 = $true )
<=> ! [X70: subst > term > subst > $o,X71: term] :
( ? [X72: subst > term > term] :
( ! [X76: term] :
( ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true )
| ( $true
!= ( X70 @ id @ X76 @ id ) ) )
& ! [X73: subst,X74: term,X75: subst] :
( ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) )
= ( sub @ ( X72 @ X75 @ X74 ) @ X73 ) )
& ( ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( sP2 @ X70 )
= $true )
| ( $true
= ( X70 @ id @ X71 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f20,plain,
( ( sP6 = $true )
<=> ( ? [X52: term > $o,X53: term] :
( ( ( sP3 @ X52 )
= $true )
& ! [X57: term] :
( ? [X58: term] :
( ( $true
= ( X52 @ X58 ) )
& ( ( X52 @ ( sub @ X57 @ ( push @ X58 @ id ) ) )
!= $true ) )
| ( ( X52 @ ( lam @ X57 ) )
= $true ) )
& ! [X56: term] :
( ( ( var @ X56 )
!= $true )
| ( $true
= ( X52 @ X56 ) ) )
& ( ( X52 @ X53 )
!= $true ) )
| ( sP5 = $true )
| ? [X49: subst > term > subst > $o] :
! [X50: term > $o] :
? [X51: term] :
( ( X49 @ id @ X51 @ id )
!= ( X50 @ X51 ) )
| ? [X59: term] :
( ( sub @ X59 @ id )
!= X59 )
| ( sP4 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f21,plain,
! [X287: subst > term > subst > $o] :
( ( ( sP7 @ X287 )
= $true )
<=> ! [X293: term,X294: term] :
( ( ( X287 @ id @ X294 @ id )
!= $true )
| ( ( X287 @ id @ ( ap @ ( sub @ X293 @ id ) @ X294 ) @ id )
= $true )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f22,plain,
! [X280: term > $o] :
( ( $true
= ( sP8 @ X280 ) )
<=> ! [X284: term,X285: term] :
( ( $true
!= ( X280 @ X285 ) )
| ( ( X280 @ ( ap @ X285 @ X284 ) )
= $true )
| ( $true
!= ( X280 @ X284 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f23,plain,
! [X300: subst > term > subst > $o] :
( ( $true
= ( sP9 @ X300 ) )
<=> ! [X310: subst > term > term] :
( ? [X312: term,X313: subst,X314: subst] :
( ( X310 @ ( comp @ X314 @ X313 ) @ ( sub @ X312 @ X313 ) )
!= ( sub @ ( X310 @ X314 @ X312 ) @ X313 ) )
| ( ( X300 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id )
= $true )
| ? [X311: term] :
( ( ( X300 @ id @ ( X310 @ id @ X311 ) @ id )
!= $true )
& ( $true
= ( X300 @ id @ X311 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f24,plain,
! [X316: subst > term > subst > $o] :
( ( $true
= ( sP10 @ X316 ) )
<=> ? [X317: term,X318: term] :
( ( ( X316 @ id @ X318 @ id )
= $true )
& ( ( X316 @ id @ X317 @ id )
= $true )
& ( $true
!= ( X316 @ id @ ( ap @ ( sub @ X318 @ id ) @ X317 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f25,plain,
! [X316: subst > term > subst > $o] :
( ( ( sP11 @ X316 )
= $true )
<=> ? [X319: subst > term > term] :
( ! [X323: term] :
( ( ( X316 @ id @ ( X319 @ id @ X323 ) @ id )
= $true )
| ( ( X316 @ id @ X323 @ id )
!= $true ) )
& ! [X320: term,X321: subst,X322: subst] :
( ( sub @ ( X319 @ X321 @ X320 ) @ X322 )
= ( X319 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) ) )
& ( $true
!= ( X316 @ id @ ( lam @ ( X319 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f26,plain,
( ( sP12 = $true )
<=> ? [X299: term,X300: subst > term > subst > $o] :
( ! [X302: subst,X303: subst,X304: term,X305: subst] :
( ( ( X300 @ ( comp @ X305 @ X302 ) @ ( sub @ X304 @ X302 ) @ X303 )
= $true )
| ( ( X300 @ X305 @ X304 @ ( comp @ X302 @ X303 ) )
!= $true ) )
& ( ( X300 @ id @ ( lam @ X299 ) @ id )
!= $true )
& ! [X301: term] :
( ( $true
!= ( X300 @ id @ X301 @ id ) )
| ( ( X300 @ id @ ( sub @ X299 @ ( push @ X301 @ id ) ) @ id )
= $true ) )
& ( $true
= ( sP9 @ X300 ) )
& ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( ( X300 @ ( comp @ X307 @ X308 ) @ ( sub @ X306 @ X308 ) @ X309 )
!= $true )
| ( $true
= ( X300 @ X307 @ X306 @ ( comp @ X308 @ X309 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f27,plain,
( ( sP13 = $true )
<=> ! [X315: term,X316: subst > term > subst > $o] :
( ? [X328: subst,X329: subst,X330: subst,X331: term] :
( ( ( X316 @ X328 @ X331 @ ( comp @ X329 @ X330 ) )
!= $true )
& ( ( X316 @ ( comp @ X328 @ X329 ) @ ( sub @ X331 @ X329 ) @ X330 )
= $true ) )
| ( ( sP11 @ X316 )
= $true )
| ? [X324: subst,X325: term,X326: subst,X327: subst] :
( ( $true
= ( X316 @ X324 @ X325 @ ( comp @ X327 @ X326 ) ) )
& ( ( X316 @ ( comp @ X324 @ X327 ) @ ( sub @ X325 @ X327 ) @ X326 )
!= $true ) )
| ( ( X316 @ id @ X315 @ id )
= $true )
| ? [X332: term] :
( ( ( var @ ( sub @ X332 @ id ) )
= $true )
& ( ( X316 @ id @ X332 @ id )
!= $true ) )
| ( $true
= ( sP10 @ X316 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f28,plain,
( ( sP14 = $true )
<=> ? [X286: term,X287: subst > term > subst > $o,X288: term] :
( ! [X295: subst,X296: subst,X297: term,X298: subst] :
( ( $true
= ( X287 @ X295 @ X297 @ ( comp @ X296 @ X298 ) ) )
| ( ( X287 @ ( comp @ X295 @ X296 ) @ ( sub @ X297 @ X296 ) @ X298 )
!= $true ) )
& ( $true
= ( X287 @ id @ X286 @ id ) )
& ( ( X287 @ id @ X288 @ id )
= $true )
& ! [X289: subst,X290: subst,X291: term,X292: subst] :
( ( ( X287 @ ( comp @ X289 @ X290 ) @ ( sub @ X291 @ X290 ) @ X292 )
= $true )
| ( $true
!= ( X287 @ X289 @ X291 @ ( comp @ X290 @ X292 ) ) ) )
& ( $true
!= ( X287 @ id @ ( ap @ X286 @ X288 ) @ id ) )
& ( ( sP7 @ X287 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f29,plain,
( ( $true = sP15 )
<=> ( ( sP13 = $true )
| ( sP12 = $true )
| ? [X279: term,X280: term > $o] :
( ( ( X280 @ X279 )
!= $true )
& ! [X282: term] :
( ? [X283: term] :
( ( $true
!= ( X280 @ ( sub @ X282 @ ( push @ X283 @ id ) ) ) )
& ( ( X280 @ X283 )
= $true ) )
| ( ( X280 @ ( lam @ X282 ) )
= $true ) )
& ( $true
= ( sP8 @ X280 ) )
& ! [X281: term] :
( ( ( X280 @ X281 )
= $true )
| ( $true
!= ( var @ X281 ) ) ) )
| ( sP14 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f30,plain,
! [X419: subst > term > subst > $o] :
( ( $true
= ( sP16 @ X419 ) )
<=> ? [X428: subst > term > term] :
( ( $true
!= ( X419 @ id @ ( lam @ ( X428 @ sh @ one ) ) @ id ) )
& ! [X429: term,X430: subst,X431: subst] :
( ( sub @ ( X428 @ X431 @ X429 ) @ X430 )
= ( X428 @ ( comp @ X431 @ X430 ) @ ( sub @ X429 @ X430 ) ) )
& ! [X432: term] :
( ( $true
= ( X419 @ id @ ( X428 @ id @ X432 ) @ id ) )
| ( ( X419 @ id @ X432 @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f31,plain,
( ( sP17 = $true )
<=> ? [X410: term,X411: subst > term > subst > $o] :
( ! [X417: term] :
( ( $true
= ( X411 @ id @ ( sub @ X410 @ ( push @ X417 @ id ) ) @ id ) )
| ( ( X411 @ id @ X417 @ id )
!= $true ) )
& ( ( X411 @ id @ ( lam @ ( sub @ X410 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X412: subst > term > term] :
( ? [X414: subst,X415: subst,X416: term] :
( ( X412 @ ( comp @ X414 @ X415 ) @ ( sub @ X416 @ X415 ) )
!= ( sub @ ( X412 @ X414 @ X416 ) @ X415 ) )
| ? [X413: term] :
( ( ( X411 @ id @ ( X412 @ id @ X413 ) @ id )
!= $true )
& ( $true
= ( X411 @ id @ X413 @ id ) ) )
| ( ( X411 @ id @ ( lam @ ( X412 @ sh @ one ) ) @ id )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f32,plain,
( ( $true = sP18 )
<=> ( ? [X409: term] :
( ( sub @ X409 @ id )
!= X409 )
| ! [X418: term,X419: subst > term > subst > $o] :
( ? [X420: subst,X421: subst,X422: term,X423: subst] :
( ( ( X419 @ X423 @ X422 @ ( comp @ X421 @ X420 ) )
!= $true )
& ( ( X419 @ ( comp @ X423 @ X421 ) @ ( sub @ X422 @ X421 ) @ X420 )
= $true ) )
| ? [X433: term] :
( ( ( X419 @ id @ ( sub @ X418 @ ( push @ X433 @ id ) ) @ id )
!= $true )
& ( $true
= ( X419 @ id @ X433 @ id ) ) )
| ( $true
= ( X419 @ id @ ( lam @ X418 ) @ id ) )
| ( $true
= ( sP16 @ X419 ) )
| ? [X424: subst,X425: subst,X426: subst,X427: term] :
( ( $true
!= ( X419 @ ( comp @ X426 @ X425 ) @ ( sub @ X427 @ X425 ) @ X424 ) )
& ( ( X419 @ X426 @ X427 @ ( comp @ X425 @ X424 ) )
= $true ) ) )
| ( sP17 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f33,plain,
( ( $true = sP19 )
<=> ( ! [X393: term,X394: subst > term > subst > $o,X395: term > $o] :
( ? [X402: term] :
( ( X395 @ X402 )
!= ( X394 @ id @ X402 @ id ) )
| ? [X397: subst > term > term] :
( ! [X399: term,X400: subst,X401: subst] :
( ( sub @ ( X397 @ X400 @ X399 ) @ X401 )
= ( X397 @ ( comp @ X400 @ X401 ) @ ( sub @ X399 @ X401 ) ) )
& ! [X398: term] :
( ( ( X394 @ id @ X398 @ id )
!= $true )
| ( ( X394 @ id @ ( X397 @ id @ X398 ) @ id )
= $true ) )
& ( ( X394 @ id @ ( lam @ ( X397 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X395 @ ( lam @ X393 ) )
= $true )
| ? [X396: term] :
( ( ( X395 @ ( sub @ X393 @ ( push @ X396 @ id ) ) )
!= $true )
& ( ( X395 @ X396 )
= $true ) ) )
| ? [X388: term] :
! [X389: subst > term > term] :
? [X390: term,X391: subst] :
( ( sub @ X388 @ ( push @ X390 @ X391 ) )
!= ( X389 @ X391 @ X390 ) )
| ? [X392: term] :
( ( sub @ X392 @ id )
!= X392 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f34,plain,
! [X370: subst > term > subst > $o] :
( ( ( sP20 @ X370 )
= $true )
<=> ? [X382: term,X383: term] :
( ( ( X370 @ id @ ( ap @ ( sub @ X383 @ id ) @ X382 ) @ id )
!= $true )
& ( ( X370 @ id @ X383 @ id )
= $true )
& ( ( X370 @ id @ X382 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f35,plain,
( ( sP21 = $true )
<=> ! [X369: term > $o,X370: subst > term > subst > $o,X371: term,X372: term] :
( ( ( sP20 @ X370 )
= $true )
| ( $true
!= ( X369 @ X371 ) )
| ( ( X369 @ X372 )
!= $true )
| ( ( X369 @ ( ap @ X372 @ X371 ) )
= $true )
| ? [X373: subst,X374: subst,X375: term,X376: subst] :
( ( ( X370 @ ( comp @ X374 @ X373 ) @ ( sub @ X375 @ X373 ) @ X376 )
!= $true )
& ( ( X370 @ X374 @ X375 @ ( comp @ X373 @ X376 ) )
= $true ) )
| ? [X378: term,X379: subst,X380: subst,X381: subst] :
( ( $true
!= ( X370 @ X380 @ X378 @ ( comp @ X381 @ X379 ) ) )
& ( ( X370 @ ( comp @ X380 @ X381 ) @ ( sub @ X378 @ X381 ) @ X379 )
= $true ) )
| ? [X377: term] :
( ( X369 @ X377 )
!= ( X370 @ id @ X377 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f36,plain,
! [X42: term > $o] :
( ( ( sP22 @ X42 )
= $true )
<=> ? [X47: term,X48: term] :
( ( $true
!= ( X42 @ ( ap @ X47 @ X48 ) ) )
& ( $true
= ( X42 @ X47 ) )
& ( ( X42 @ X48 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f37,plain,
( ( sP23 = $true )
<=> ! [X42: term > $o,X43: term] :
( ? [X44: term] :
( ( ( var @ X44 )
= $true )
& ( ( X42 @ X44 )
!= $true ) )
| ( ( sP22 @ X42 )
= $true )
| ? [X45: term] :
( ( ( X42 @ ( lam @ X45 ) )
!= $true )
& ! [X46: term] :
( ( $true
!= ( X42 @ X46 ) )
| ( $true
= ( X42 @ ( sub @ X45 @ ( push @ X46 @ id ) ) ) ) ) )
| ( ( X42 @ X43 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f38,plain,
( ( sP24 = $true )
<=> ! [X511: term,X512: subst > term > subst > $o] :
( ? [X518: term] :
( ( ( X512 @ id @ ( sub @ X511 @ ( push @ X518 @ id ) ) @ id )
!= $true )
& ( ( X512 @ id @ X518 @ id )
= $true ) )
| ( ( X512 @ id @ ( lam @ X511 ) @ id )
= $true )
| ? [X513: subst > term > term] :
( ( ( X512 @ id @ ( lam @ ( X513 @ sh @ one ) ) @ id )
!= $true )
& ! [X514: subst,X515: term,X516: subst] :
( ( X513 @ ( comp @ X514 @ X516 ) @ ( sub @ X515 @ X516 ) )
= ( sub @ ( X513 @ X514 @ X515 ) @ X516 ) )
& ! [X517: term] :
( ( $true
= ( X512 @ id @ ( X513 @ id @ X517 ) @ id ) )
| ( ( X512 @ id @ X517 @ id )
!= $true ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f39,plain,
! [X204: term > $o] :
( ( $true
= ( sP25 @ X204 ) )
<=> ! [X205: term,X206: term] :
( ( $true
!= ( X204 @ X206 ) )
| ( ( X204 @ X205 )
!= $true )
| ( ( X204 @ ( ap @ X206 @ X205 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f40,plain,
! [X222: subst > term > subst > $o] :
( ( ( sP26 @ X222 )
= $true )
<=> ? [X234: subst > term > term] :
( ( ( X222 @ id @ ( lam @ ( X234 @ sh @ one ) ) @ id )
!= $true )
& ! [X236: term,X237: subst,X238: subst] :
( ( X234 @ ( comp @ X238 @ X237 ) @ ( sub @ X236 @ X237 ) )
= ( sub @ ( X234 @ X238 @ X236 ) @ X237 ) )
& ! [X235: term] :
( ( $true
= ( X222 @ id @ ( X234 @ id @ X235 ) @ id ) )
| ( ( X222 @ id @ X235 @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f41,plain,
! [X222: subst > term > subst > $o] :
( ( ( sP27 @ X222 )
= $true )
<=> ? [X227: term,X228: term] :
( ( ( X222 @ id @ X227 @ id )
= $true )
& ( ( X222 @ id @ X228 @ id )
= $true )
& ( ( X222 @ id @ ( ap @ ( sub @ X228 @ id ) @ X227 ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f42,plain,
( ( sP28 = $true )
<=> ! [X221: term,X222: subst > term > subst > $o] :
( ? [X223: subst,X224: term,X225: subst,X226: subst] :
( ( $true
= ( X222 @ X225 @ X224 @ ( comp @ X226 @ X223 ) ) )
& ( ( X222 @ ( comp @ X225 @ X226 ) @ ( sub @ X224 @ X226 ) @ X223 )
!= $true ) )
| ? [X229: term] :
( ( $true
!= ( X222 @ id @ X229 @ id ) )
& ( ( var @ ( sub @ X229 @ id ) )
= $true ) )
| ( ( X222 @ id @ X221 @ id )
= $true )
| ( ( sP27 @ X222 )
= $true )
| ( ( sP26 @ X222 )
= $true )
| ? [X230: subst,X231: subst,X232: term,X233: subst] :
( ( ( X222 @ X233 @ X232 @ ( comp @ X230 @ X231 ) )
!= $true )
& ( $true
= ( X222 @ ( comp @ X233 @ X230 ) @ ( sub @ X232 @ X230 ) @ X231 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f43,plain,
( ( $true = sP29 )
<=> ? [X211: term > $o,X212: term,X213: subst > term > subst > $o] :
( ! [X216: subst > term > term] :
( ( ( X213 @ id @ ( lam @ ( X216 @ sh @ one ) ) @ id )
= $true )
| ? [X220: term] :
( ( ( X213 @ id @ X220 @ id )
= $true )
& ( ( X213 @ id @ ( X216 @ id @ X220 ) @ id )
!= $true ) )
| ? [X217: term,X218: subst,X219: subst] :
( ( sub @ ( X216 @ X219 @ X217 ) @ X218 )
!= ( X216 @ ( comp @ X219 @ X218 ) @ ( sub @ X217 @ X218 ) ) ) )
& ! [X214: term] :
( ( X213 @ id @ X214 @ id )
= ( X211 @ X214 ) )
& ( $true
!= ( X211 @ ( lam @ X212 ) ) )
& ! [X215: term] :
( ( $true
!= ( X211 @ X215 ) )
| ( $true
= ( X211 @ ( sub @ X212 @ ( push @ X215 @ id ) ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f44,plain,
( ( sP30 = $true )
<=> ( ( $true = sP29 )
| ( sP28 = $true )
| ? [X210: term] :
( ( sub @ X210 @ id )
!= X210 )
| ? [X203: term,X204: term > $o] :
( ( $true
!= ( X204 @ X203 ) )
& ! [X207: term] :
( ? [X208: term] :
( ( $true
!= ( X204 @ ( sub @ X207 @ ( push @ X208 @ id ) ) ) )
& ( $true
= ( X204 @ X208 ) ) )
| ( $true
= ( X204 @ ( lam @ X207 ) ) ) )
& ! [X209: term] :
( ( ( X204 @ X209 )
= $true )
| ( ( var @ X209 )
!= $true ) )
& ( $true
= ( sP25 @ X204 ) ) )
| ? [X200: subst > term > subst > $o] :
! [X201: term > $o] :
? [X202: term] :
( ( X201 @ X202 )
!= ( X200 @ id @ X202 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f45,plain,
! [X494: subst > term > subst > $o] :
( ( $true
= ( sP31 @ X494 ) )
<=> ? [X501: subst > term > term] :
( ( $true
!= ( X494 @ id @ ( lam @ ( X501 @ sh @ one ) ) @ id ) )
& ! [X505: term] :
( ( ( X494 @ id @ X505 @ id )
!= $true )
| ( ( X494 @ id @ ( X501 @ id @ X505 ) @ id )
= $true ) )
& ! [X502: subst,X503: term,X504: subst] :
( ( X501 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X501 @ X502 @ X503 ) @ X504 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f46,plain,
! [X494: subst > term > subst > $o] :
( ( ( sP32 @ X494 )
= $true )
<=> ? [X499: term,X500: term] :
( ( ( X494 @ id @ X500 @ id )
= $true )
& ( ( X494 @ id @ ( ap @ ( sub @ X500 @ id ) @ X499 ) @ id )
!= $true )
& ( ( X494 @ id @ X499 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f47,plain,
( ( sP33 = $true )
<=> ! [X493: term,X494: subst > term > subst > $o] :
( ( $true
= ( X494 @ id @ X493 @ id ) )
| ( ( sP32 @ X494 )
= $true )
| ? [X507: subst,X508: subst,X509: term,X510: subst] :
( ( ( X494 @ X510 @ X509 @ ( comp @ X508 @ X507 ) )
= $true )
& ( $true
!= ( X494 @ ( comp @ X510 @ X508 ) @ ( sub @ X509 @ X508 ) @ X507 ) ) )
| ( $true
= ( sP31 @ X494 ) )
| ? [X495: term,X496: subst,X497: subst,X498: subst] :
( ( $true
= ( X494 @ ( comp @ X498 @ X497 ) @ ( sub @ X495 @ X497 ) @ X496 ) )
& ( ( X494 @ X498 @ X495 @ ( comp @ X497 @ X496 ) )
!= $true ) )
| ? [X506: term] :
( ( ( var @ ( sub @ X506 @ id ) )
= $true )
& ( $true
!= ( X494 @ id @ X506 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f48,plain,
! [X138: subst > term > subst > $o] :
( ( $true
= ( sP34 @ X138 ) )
<=> ? [X140: subst > term > term] :
( ( ( X138 @ id @ ( lam @ ( X140 @ sh @ one ) ) @ id )
!= $true )
& ! [X142: subst,X143: term,X144: subst] :
( ( X140 @ ( comp @ X142 @ X144 ) @ ( sub @ X143 @ X144 ) )
= ( sub @ ( X140 @ X142 @ X143 ) @ X144 ) )
& ! [X141: term] :
( ( ( X138 @ id @ X141 @ id )
!= $true )
| ( $true
= ( X138 @ id @ ( X140 @ id @ X141 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f49,plain,
! [X152: term > $o] :
( ( ( sP35 @ X152 )
= $true )
<=> ! [X155: term,X156: term] :
( ( $true
= ( X152 @ ( ap @ X155 @ X156 ) ) )
| ( ( X152 @ X156 )
!= $true )
| ( $true
!= ( X152 @ X155 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f50,plain,
! [X160: term > $o] :
( ( ( sP36 @ X160 )
= $true )
<=> ? [X162: term,X163: term] :
( ( ( X160 @ X163 )
= $true )
& ( $true
!= ( X160 @ ( ap @ X163 @ X162 ) ) )
& ( $true
= ( X160 @ X162 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f51,plain,
( ( sP37 = $true )
<=> ! [X160: term > $o,X161: term] :
( ( ( sP36 @ X160 )
= $true )
| ? [X164: term] :
( ( ( var @ X164 )
= $true )
& ( ( X160 @ X164 )
!= $true ) )
| ( ( X160 @ X161 )
= $true )
| ? [X165: term] :
( ( $true
!= ( X160 @ ( lam @ X165 ) ) )
& ! [X166: term] :
( ( $true
!= ( X160 @ X166 ) )
| ( ( X160 @ ( sub @ X165 @ ( push @ X166 @ id ) ) )
= $true ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f52,plain,
( ( sP38 = $true )
<=> ( ( sP37 = $true )
| ? [X152: term > $o,X153: term,X154: subst] :
( ! [X158: term] :
( ? [X159: term] :
( ( $true
!= ( X152 @ ( sub @ X158 @ ( push @ X159 @ id ) ) ) )
& ( ( X152 @ X159 )
= $true ) )
| ( ( X152 @ ( lam @ X158 ) )
= $true ) )
& ! [X157: term] :
( ( ( X152 @ ( sub @ X157 @ X154 ) )
= $true )
| ( ( var @ X157 )
!= $true ) )
& ( ( X152 @ ( sub @ X153 @ X154 ) )
!= $true )
& ( ( sP35 @ X152 )
= $true ) )
| ? [X151: term] :
( ( sub @ X151 @ id )
!= X151 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f53,plain,
( ( $true = sP39 )
<=> ! [X121: term,X122: subst > term > subst > $o] :
( ? [X123: term] :
( ( ( X122 @ id @ X123 @ id )
= $true )
& ( ( X122 @ id @ ( sub @ X121 @ ( push @ X123 @ id ) ) @ id )
!= $true ) )
| ( $true
= ( X122 @ id @ ( lam @ ( sub @ X121 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X124: term,X125: subst > term > term] :
( ! [X126: subst,X127: term,X128: subst] :
( ( sub @ ( sub @ X124 @ ( push @ X127 @ X128 ) ) @ X126 )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
& ( ( X122 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id )
!= $true )
& ! [X129: term] :
( ( ( X122 @ id @ X129 @ id )
!= $true )
| ( ( X122 @ id @ ( X125 @ id @ X129 ) @ id )
= $true ) )
& ! [X130: term,X131: subst,X132: subst] :
( ( sub @ X124 @ ( push @ ( sub @ X130 @ X132 ) @ ( comp @ X131 @ X132 ) ) )
= ( X125 @ ( comp @ X131 @ X132 ) @ ( sub @ X130 @ X132 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f54,plain,
! [X362: term > $o] :
( ( $true
= ( sP40 @ X362 ) )
<=> ? [X367: term,X368: term] :
( ( ( X362 @ X367 )
= $true )
& ( ( X362 @ X368 )
= $true )
& ( $true
!= ( X362 @ ( ap @ X368 @ X367 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f55,plain,
! [X356: term > $o] :
( ( $true
= ( sP41 @ X356 ) )
<=> ! [X359: term,X360: term] :
( ( $true
!= ( X356 @ X360 ) )
| ( $true
= ( X356 @ ( ap @ X360 @ X359 ) ) )
| ( $true
!= ( X356 @ X359 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f56,plain,
( ( $true = sP42 )
<=> ! [X361: term,X362: term > $o,X363: subst] :
( ? [X365: term] :
( ! [X366: term] :
( ( $true
= ( X362 @ ( sub @ X365 @ ( push @ X366 @ id ) ) ) )
| ( ( X362 @ X366 )
!= $true ) )
& ( $true
!= ( X362 @ ( lam @ X365 ) ) ) )
| ? [X364: term] :
( ( ( var @ X364 )
= $true )
& ( $true
!= ( X362 @ ( sub @ X364 @ X363 ) ) ) )
| ( $true
= ( sP40 @ X362 ) )
| ( ( X362 @ ( sub @ X361 @ X363 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f57,plain,
( ( sP43 = $true )
<=> ( ? [X355: term,X356: term > $o] :
( ( ( X356 @ X355 )
!= $true )
& ! [X358: term] :
( ( $true
= ( X356 @ ( lam @ X358 ) ) )
| ( ( X356 @ X358 )
!= $true ) )
& ! [X357: term] :
( ( ( var @ X357 )
!= $true )
| ( $true
= ( X356 @ X357 ) ) )
& ( $true
= ( sP41 @ X356 ) ) )
| ? [X354: subst] :
( ( comp @ X354 @ id )
!= X354 )
| ? [X352: term,X353: subst] :
( ( comp @ sh @ ( push @ X352 @ X353 ) )
!= X353 )
| ? [X350: subst,X351: term] :
( ( sub @ one @ ( push @ X351 @ X350 ) )
!= X351 )
| ( $true = sP42 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f58,plain,
( ( sP44 = $true )
<=> ( ? [X438: term,X439: term] :
( ( X438 != X439 )
& ( ( lam @ X438 )
= ( lam @ X439 ) ) )
| ? [X436: term,X437: subst] :
( ( comp @ sh @ ( push @ X436 @ X437 ) )
!= X437 )
| ? [X434: term,X435: subst] :
( ( sub @ one @ ( push @ X434 @ X435 ) )
!= X434 )
| ! [X440: term,X441: subst > term > term,X442: subst > term > term,X443: subst] :
( ? [X444: term,X445: subst,X446: subst] :
( ( X441 @ ( comp @ X446 @ X445 ) @ ( sub @ X444 @ X445 ) )
!= ( sub @ ( X441 @ X446 @ X444 ) @ X445 ) )
| ( ( X441 @ X443 @ X440 )
= ( X442 @ X443 @ X440 ) )
| ( ( lam @ ( X441 @ sh @ one ) )
!= ( lam @ ( X442 @ sh @ one ) ) )
| ? [X447: subst,X448: subst,X449: term] :
( ( X442 @ ( comp @ X448 @ X447 ) @ ( sub @ X449 @ X447 ) )
!= ( sub @ ( X442 @ X448 @ X449 ) @ X447 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f59,plain,
! [X478: subst > term > subst > $o] :
( ( ( sP45 @ X478 )
= $true )
<=> ? [X484: subst > term > term] :
( ! [X488: term] :
( ( ( X478 @ id @ ( X484 @ id @ X488 ) @ id )
= $true )
| ( $true
!= ( X478 @ id @ X488 @ id ) ) )
& ! [X485: subst,X486: subst,X487: term] :
( ( sub @ ( X484 @ X486 @ X487 ) @ X485 )
= ( X484 @ ( comp @ X486 @ X485 ) @ ( sub @ X487 @ X485 ) ) )
& ( $true
!= ( X478 @ id @ ( lam @ ( X484 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f60,plain,
( ( $true = sP46 )
<=> ! [X477: term,X478: subst > term > subst > $o] :
( ( ( X478 @ id @ ( lam @ X477 ) @ id )
= $true )
| ? [X483: term] :
( ( $true
!= ( X478 @ id @ ( sub @ X477 @ ( push @ X483 @ id ) ) @ id ) )
& ( ( X478 @ id @ X483 @ id )
= $true ) )
| ? [X489: subst,X490: subst,X491: term,X492: subst] :
( ( ( X478 @ ( comp @ X492 @ X490 ) @ ( sub @ X491 @ X490 ) @ X489 )
= $true )
& ( ( X478 @ X492 @ X491 @ ( comp @ X490 @ X489 ) )
!= $true ) )
| ? [X479: subst,X480: subst,X481: subst,X482: term] :
( ( ( X478 @ ( comp @ X481 @ X479 ) @ ( sub @ X482 @ X479 ) @ X480 )
!= $true )
& ( ( X478 @ X481 @ X482 @ ( comp @ X479 @ X480 ) )
= $true ) )
| ( ( sP45 @ X478 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f61,plain,
! [X18: subst > term > subst > $o] :
( ( ( sP47 @ X18 )
= $true )
<=> ? [X22: subst > term > term] :
( ! [X26: term] :
( ( $true
!= ( X18 @ id @ X26 @ id ) )
| ( ( X18 @ id @ ( X22 @ id @ X26 ) @ id )
= $true ) )
& ! [X23: subst,X24: subst,X25: term] :
( ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( X22 @ X24 @ X25 ) @ X23 ) )
& ( $true
!= ( X18 @ id @ ( lam @ ( X22 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f62,plain,
( ( sP48 = $true )
<=> ! [X17: term,X18: subst > term > subst > $o,X19: term > $o] :
( ( ( sP47 @ X18 )
= $true )
| ? [X21: term] :
( ( X19 @ X21 )
!= ( X18 @ id @ X21 @ id ) )
| ? [X20: term] :
( ( $true
= ( X19 @ X20 ) )
& ( ( X19 @ ( sub @ X17 @ ( push @ X20 @ id ) ) )
!= $true ) )
| ( ( X19 @ ( lam @ X17 ) )
= $true )
| ? [X27: subst,X28: term,X29: subst,X30: subst] :
( ( $true
= ( X18 @ X27 @ X28 @ ( comp @ X29 @ X30 ) ) )
& ( ( X18 @ ( comp @ X27 @ X29 ) @ ( sub @ X28 @ X29 ) @ X30 )
!= $true ) )
| ? [X31: subst,X32: subst,X33: subst,X34: term] :
( ( $true
= ( X18 @ ( comp @ X33 @ X32 ) @ ( sub @ X34 @ X32 ) @ X31 ) )
& ( ( X18 @ X33 @ X34 @ ( comp @ X32 @ X31 ) )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f63,plain,
( ( sP49 = $true )
<=> ! [X342: term,X343: subst > term > subst > $o] :
( ? [X344: term] :
( ( ( X343 @ id @ X344 @ id )
= $true )
& ( ( X343 @ id @ ( sub @ X342 @ ( push @ X344 @ id ) ) @ id )
!= $true ) )
| ? [X345: subst > term > term] :
( ! [X346: term,X347: subst,X348: subst] :
( ( sub @ ( X345 @ X347 @ X346 ) @ X348 )
= ( X345 @ ( comp @ X347 @ X348 ) @ ( sub @ X346 @ X348 ) ) )
& ! [X349: term] :
( ( ( X343 @ id @ X349 @ id )
!= $true )
| ( $true
= ( X343 @ id @ ( X345 @ id @ X349 ) @ id ) ) )
& ( ( X343 @ id @ ( lam @ ( X345 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X343 @ id @ ( lam @ ( sub @ X342 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f64,plain,
! [X81: subst > term > subst > $o] :
( ( ( sP50 @ X81 )
= $true )
<=> ? [X84: subst > term > term] :
( ! [X86: term,X87: subst,X88: subst] :
( ( sub @ ( X84 @ X88 @ X86 ) @ X87 )
= ( X84 @ ( comp @ X88 @ X87 ) @ ( sub @ X86 @ X87 ) ) )
& ! [X85: term] :
( ( ( X81 @ id @ X85 @ id )
!= $true )
| ( ( X81 @ id @ ( X84 @ id @ X85 ) @ id )
= $true ) )
& ( ( X81 @ id @ ( lam @ ( X84 @ sh @ one ) ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f65,plain,
( ( $true = sP51 )
<=> ( ! [X80: term,X81: subst > term > subst > $o,X82: term > $o] :
( ( ( X82 @ ( lam @ X80 ) )
= $true )
| ? [X89: subst,X90: subst,X91: subst,X92: term] :
( ( $true
= ( X81 @ X90 @ X92 @ ( comp @ X91 @ X89 ) ) )
& ( ( X81 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) @ X89 )
!= $true ) )
| ? [X97: term] :
( ( ( X82 @ ( sub @ X80 @ ( push @ X97 @ id ) ) )
!= $true )
& ( ( X82 @ X97 )
= $true ) )
| ? [X83: term] :
( ( X82 @ X83 )
!= ( X81 @ id @ X83 @ id ) )
| ? [X93: term,X94: subst,X95: subst,X96: subst] :
( ( $true
!= ( X81 @ X94 @ X93 @ ( comp @ X95 @ X96 ) ) )
& ( ( X81 @ ( comp @ X94 @ X95 ) @ ( sub @ X93 @ X95 ) @ X96 )
= $true ) )
| ( ( sP50 @ X81 )
= $true ) )
| ? [X79: term] :
( ( sub @ X79 @ id )
!= X79 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f66,plain,
( ( $true = sP52 )
<=> ? [X249: subst > term > subst > $o,X250: term] :
( ! [X256: term] :
( ( ( X249 @ id @ ( sub @ X250 @ ( push @ X256 @ id ) ) @ id )
= $true )
| ( ( X249 @ id @ X256 @ id )
!= $true ) )
& ( ( X249 @ id @ ( lam @ ( sub @ X250 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X251: subst > term > term] :
( ? [X255: term] :
( ( ( X249 @ id @ ( X251 @ id @ X255 ) @ id )
!= $true )
& ( ( X249 @ id @ X255 @ id )
= $true ) )
| ( $true
= ( X249 @ id @ ( lam @ ( X251 @ sh @ one ) ) @ id ) )
| ? [X252: subst,X253: term,X254: subst] :
( ( sub @ ( X251 @ X252 @ X253 ) @ X254 )
!= ( X251 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f67,plain,
( ( sP53 = $true )
<=> ( ? [X248: term] :
( ( sub @ X248 @ id )
!= X248 )
| ! [X257: subst > term > subst > $o,X258: term] :
( ( ( X257 @ id @ ( lam @ X258 ) @ id )
= $true )
| ? [X259: term] :
( ( ( X257 @ id @ X259 @ id )
= $true )
& ( ( X257 @ id @ ( sub @ X258 @ ( push @ X259 @ id ) ) @ id )
!= $true ) )
| ? [X260: subst > term > term] :
( ! [X264: term] :
( ( ( X257 @ id @ ( X260 @ id @ X264 ) @ id )
= $true )
| ( ( X257 @ id @ X264 @ id )
!= $true ) )
& ! [X261: term,X262: subst,X263: subst] :
( ( X260 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) )
= ( sub @ ( X260 @ X262 @ X261 ) @ X263 ) )
& ( ( X257 @ id @ ( lam @ ( X260 @ sh @ one ) ) @ id )
!= $true ) ) )
| ( $true = sP52 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f68,plain,
( ( sP54 = $true )
<=> ! [X265: term,X266: subst > term > subst > $o,X267: term > $o] :
( ? [X269: term] :
( ( X267 @ X269 )
!= ( X266 @ id @ X269 @ id ) )
| ( $true
= ( X267 @ ( lam @ X265 ) ) )
| ? [X270: subst > term > term] :
( ! [X272: subst,X273: subst,X274: term] :
( ( sub @ ( X270 @ X272 @ X274 ) @ X273 )
= ( X270 @ ( comp @ X272 @ X273 ) @ ( sub @ X274 @ X273 ) ) )
& ! [X271: term] :
( ( ( X266 @ id @ X271 @ id )
!= $true )
| ( $true
= ( X266 @ id @ ( X270 @ id @ X271 ) @ id ) ) )
& ( ( X266 @ id @ ( lam @ ( X270 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X268: term] :
( ( ( X267 @ X268 )
= $true )
& ( $true
!= ( X267 @ ( sub @ X265 @ ( push @ X268 @ id ) ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f69,plain,
( ( $true = sP55 )
<=> ! [X182: subst > term > subst > $o,X183: term] :
( ? [X185: subst > term > term] :
( ! [X186: term] :
( ( $true
!= ( X182 @ id @ X186 @ id ) )
| ( ( X182 @ id @ ( X185 @ id @ X186 ) @ id )
= $true ) )
& ! [X187: term,X188: subst,X189: subst] :
( ( X185 @ ( comp @ X189 @ X188 ) @ ( sub @ X187 @ X188 ) )
= ( sub @ ( X185 @ X189 @ X187 ) @ X188 ) )
& ( ( X182 @ id @ ( lam @ ( X185 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X182 @ id @ ( lam @ ( sub @ X183 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X184: term] :
( ( ( X182 @ id @ X184 @ id )
= $true )
& ( ( X182 @ id @ ( sub @ X183 @ ( push @ X184 @ id ) ) @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f70,plain,
! [X403: term > $o] :
( ( ( sP56 @ X403 )
= $true )
<=> ? [X405: term,X406: term] :
( ( ( X403 @ X405 )
= $true )
& ( ( X403 @ ( ap @ X405 @ X406 ) )
!= $true )
& ( $true
= ( X403 @ X406 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f71,plain,
! [X1: term > $o] :
( ( $true
= ( sP57 @ X1 ) )
<=> ? [X3: term,X4: term] :
( ( ( X1 @ X3 )
= $true )
& ( ( X1 @ ( ap @ X3 @ X4 ) )
!= $true )
& ( ( X1 @ X4 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f72,plain,
( ( $true = sP58 )
<=> ! [X0: subst,X1: term > $o,X2: term] :
( ? [X6: term] :
( ! [X7: term] :
( ( $true
!= ( X1 @ X7 ) )
| ( $true
= ( X1 @ ( sub @ X6 @ ( push @ X7 @ id ) ) ) ) )
& ( ( X1 @ ( lam @ X6 ) )
!= $true ) )
| ( ( X1 @ ( sub @ X2 @ X0 ) )
= $true )
| ? [X5: term] :
( ( ( var @ X5 )
= $true )
& ( ( X1 @ ( sub @ X5 @ X0 ) )
!= $true ) )
| ( $true
= ( sP57 @ X1 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f13,plain,
( ( ( pushprop_lem0_lthm = $true )
<=> ! [X116: term,X117: subst,X118: term > $o] :
? [X119: term > $o] :
! [X120: term] :
( ( X119 @ X120 )
= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) ) )
& ( ! [X466: subst] :
( ( comp @ X466 @ id )
= X466 )
<=> ( axidr = $true ) )
& ! [X112: d_subst,X113: d_subst] :
( ( ( d2subst @ X112 )
!= ( d2subst @ X113 ) )
| ( X112 = X113 ) )
& ( hoasinduction_lem2_lthm = $true )
& ( axclos = $true )
& ( ! [X0: subst,X1: term > $o,X2: term] :
( ? [X6: term] :
( ! [X7: term] :
( ( $true
!= ( X1 @ X7 ) )
| ( $true
= ( X1 @ ( sub @ X6 @ ( push @ X7 @ id ) ) ) ) )
& ( ( X1 @ ( lam @ X6 ) )
!= $true ) )
| ( ( X1 @ ( sub @ X2 @ X0 ) )
= $true )
| ? [X5: term] :
( ( ( var @ X5 )
= $true )
& ( ( X1 @ ( sub @ X5 @ X0 ) )
!= $true ) )
| ? [X3: term,X4: term] :
( ( ( X1 @ X3 )
= $true )
& ( ( X1 @ ( ap @ X3 @ X4 ) )
!= $true )
& ( ( X1 @ X4 )
= $true ) ) )
<=> ( induction2lem = $true ) )
& ( ! [X174: term] :
( ( sub @ X174 @ id )
= X174 )
<=> ( axvarid = $true ) )
& ( substmonoid_gthm = $true )
& ( ( induction = $true )
<=> ! [X403: term > $o,X404: term] :
( ? [X405: term,X406: term] :
( ( ( X403 @ X405 )
= $true )
& ( ( X403 @ ( ap @ X405 @ X406 ) )
!= $true )
& ( $true
= ( X403 @ X406 ) ) )
| ? [X407: term] :
( ( $true
= ( var @ X407 ) )
& ( $true
!= ( X403 @ X407 ) ) )
| ( ( X403 @ X404 )
= $true )
| ? [X408: term] :
( ( ( X403 @ ( lam @ X408 ) )
!= $true )
& ( ( X403 @ X408 )
= $true ) ) ) )
& ( ( pushprop_lem1 = $true )
<=> ! [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
= $true )
| ( $true
!= ( X137 @ X134 ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( axassoc = $true )
& ( ! [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
!= ( ap @ X276 @ X277 ) )
| ( X276 = X278 ) )
<=> ( apinj1 = $true ) )
& ( hoasapinj2_gthm = $true )
& ! [X464: subst] :
? [X465: d_subst] :
( ( d2subst @ X465 )
= X464 )
& ( ( ? [X104: term,X105: term,X106: term,X107: term] :
( ( X106 != X107 )
& ( ( ap @ X106 @ X105 )
= ( ap @ X107 @ X104 ) ) )
| ? [X103: term] :
( ( sub @ X103 @ id )
!= X103 )
| ! [X108: term,X109: term,X110: term,X111: term] :
( ( X108 = X109 )
| ( ( ap @ ( sub @ X109 @ id ) @ X111 )
!= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( hoasapnotvar = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X182: subst > term > subst > $o,X183: term] :
( ? [X185: subst > term > term] :
( ! [X186: term] :
( ( $true
!= ( X182 @ id @ X186 @ id ) )
| ( ( X182 @ id @ ( X185 @ id @ X186 ) @ id )
= $true ) )
& ! [X187: term,X188: subst,X189: subst] :
( ( X185 @ ( comp @ X189 @ X188 ) @ ( sub @ X187 @ X188 ) )
= ( sub @ ( X185 @ X189 @ X187 ) @ X188 ) )
& ( ( X182 @ id @ ( lam @ ( X185 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X182 @ id @ ( lam @ ( sub @ X183 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X184: term] :
( ( ( X182 @ id @ X184 @ id )
= $true )
& ( ( X182 @ id @ ( sub @ X183 @ ( push @ X184 @ id ) ) @ id )
!= $true ) ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslaminj_gthm = $true )
& ( hoasapnotvar_lthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ! [X265: term,X266: subst > term > subst > $o,X267: term > $o] :
( ? [X269: term] :
( ( X267 @ X269 )
!= ( X266 @ id @ X269 @ id ) )
| ( $true
= ( X267 @ ( lam @ X265 ) ) )
| ? [X270: subst > term > term] :
( ! [X272: subst,X273: subst,X274: term] :
( ( sub @ ( X270 @ X272 @ X274 ) @ X273 )
= ( X270 @ ( comp @ X272 @ X273 ) @ ( sub @ X274 @ X273 ) ) )
& ! [X271: term] :
( ( ( X266 @ id @ X271 @ id )
!= $true )
| ( $true
= ( X266 @ id @ ( X270 @ id @ X271 ) @ id ) ) )
& ( ( X266 @ id @ ( lam @ ( X270 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X268: term] :
( ( ( X267 @ X268 )
= $true )
& ( $true
!= ( X267 @ ( sub @ X265 @ ( push @ X268 @ id ) ) ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( termmset_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ! [X114: term] :
? [X115: d_term] :
( ( d2term @ X115 )
= X114 )
& ( ( shinj = $true )
<=> ! [X35: term,X36: term] :
( ( X35 = X36 )
| ( ( sub @ X36 @ sh )
!= ( sub @ X35 @ sh ) ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X333: subst > term > subst > $o] :
? [X334: term > $o] :
! [X335: term] :
( ( X334 @ X335 )
= ( X333 @ id @ X335 @ id ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( induction2_gthm = $true )
& ( ! [X178: term] :
? [X179: subst > term > term] :
! [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
= ( X179 @ X180 @ X181 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X384: term] :
? [X385: subst > term > term] :
! [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
= ( X385 @ X386 @ X387 ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( ? [X248: term] :
( ( sub @ X248 @ id )
!= X248 )
| ! [X257: subst > term > subst > $o,X258: term] :
( ( ( X257 @ id @ ( lam @ X258 ) @ id )
= $true )
| ? [X259: term] :
( ( ( X257 @ id @ X259 @ id )
= $true )
& ( ( X257 @ id @ ( sub @ X258 @ ( push @ X259 @ id ) ) @ id )
!= $true ) )
| ? [X260: subst > term > term] :
( ! [X264: term] :
( ( ( X257 @ id @ ( X260 @ id @ X264 ) @ id )
= $true )
| ( ( X257 @ id @ X264 @ id )
!= $true ) )
& ! [X261: term,X262: subst,X263: subst] :
( ( X260 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) )
= ( sub @ ( X260 @ X262 @ X261 ) @ X263 ) )
& ( ( X257 @ id @ ( lam @ ( X260 @ sh @ one ) ) @ id )
!= $true ) ) )
| ? [X249: subst > term > subst > $o,X250: term] :
( ! [X256: term] :
( ( ( X249 @ id @ ( sub @ X250 @ ( push @ X256 @ id ) ) @ id )
= $true )
| ( ( X249 @ id @ X256 @ id )
!= $true ) )
& ( ( X249 @ id @ ( lam @ ( sub @ X250 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X251: subst > term > term] :
( ? [X255: term] :
( ( ( X249 @ id @ ( X251 @ id @ X255 ) @ id )
!= $true )
& ( ( X249 @ id @ X255 @ id )
= $true ) )
| ( $true
= ( X249 @ id @ ( lam @ ( X251 @ sh @ one ) ) @ id ) )
| ? [X252: subst,X253: term,X254: subst] :
( ( sub @ ( X251 @ X252 @ X253 ) @ X254 )
!= ( X251 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) ) ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ! [X177: d_subst] : ( d_id = X177 )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
= X176 )
<=> ( axvarcons = $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X80: term,X81: subst > term > subst > $o,X82: term > $o] :
( ( ( X82 @ ( lam @ X80 ) )
= $true )
| ? [X89: subst,X90: subst,X91: subst,X92: term] :
( ( $true
= ( X81 @ X90 @ X92 @ ( comp @ X91 @ X89 ) ) )
& ( ( X81 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) @ X89 )
!= $true ) )
| ? [X97: term] :
( ( ( X82 @ ( sub @ X80 @ ( push @ X97 @ id ) ) )
!= $true )
& ( ( X82 @ X97 )
= $true ) )
| ? [X83: term] :
( ( X82 @ X83 )
!= ( X81 @ id @ X83 @ id ) )
| ? [X93: term,X94: subst,X95: subst,X96: subst] :
( ( $true
!= ( X81 @ X94 @ X93 @ ( comp @ X95 @ X96 ) ) )
& ( ( X81 @ ( comp @ X94 @ X95 ) @ ( sub @ X93 @ X95 ) @ X96 )
= $true ) )
| ? [X84: subst > term > term] :
( ! [X86: term,X87: subst,X88: subst] :
( ( sub @ ( X84 @ X88 @ X86 ) @ X87 )
= ( X84 @ ( comp @ X88 @ X87 ) @ ( sub @ X86 @ X87 ) ) )
& ! [X85: term] :
( ( ( X81 @ id @ X85 @ id )
!= $true )
| ( ( X81 @ id @ ( X84 @ id @ X85 ) @ id )
= $true ) )
& ( ( X81 @ id @ ( lam @ ( X84 @ sh @ one ) ) @ id )
!= $true ) ) )
| ? [X79: term] :
( ( sub @ X79 @ id )
!= X79 ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( axapp = $true )
& ( induction2lem_gthm = $true )
& ( ulamvarsh = $true )
& ( axvarshift = $true )
& ( ulamvarind = $true )
& ( ( d2subst @ d_id )
= id )
& ( ( termmset = $true )
<=> ! [X247: term] :
( ( sub @ X247 @ id )
= X247 ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X342: term,X343: subst > term > subst > $o] :
( ? [X344: term] :
( ( ( X343 @ id @ X344 @ id )
= $true )
& ( ( X343 @ id @ ( sub @ X342 @ ( push @ X344 @ id ) ) @ id )
!= $true ) )
| ? [X345: subst > term > term] :
( ! [X346: term,X347: subst,X348: subst] :
( ( sub @ ( X345 @ X347 @ X346 ) @ X348 )
= ( X345 @ ( comp @ X347 @ X348 ) @ ( sub @ X346 @ X348 ) ) )
& ! [X349: term] :
( ( ( X343 @ id @ X349 @ id )
!= $true )
| ( $true
= ( X343 @ id @ ( X345 @ id @ X349 ) @ id ) ) )
& ( ( X343 @ id @ ( lam @ ( X345 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X343 @ id @ ( lam @ ( sub @ X342 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( apnotvar = $true )
& ( ( ! [X150: term] :
( ( sub @ X150 @ id )
= X150 )
| ? [X149: term] :
( ( sub @ X149 @ id )
!= X149 ) )
<=> ( termmset_lthm = $true ) )
& ( ! [X17: term,X18: subst > term > subst > $o,X19: term > $o] :
( ? [X22: subst > term > term] :
( ! [X26: term] :
( ( $true
!= ( X18 @ id @ X26 @ id ) )
| ( ( X18 @ id @ ( X22 @ id @ X26 ) @ id )
= $true ) )
& ! [X23: subst,X24: subst,X25: term] :
( ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( X22 @ X24 @ X25 ) @ X23 ) )
& ( $true
!= ( X18 @ id @ ( lam @ ( X22 @ sh @ one ) ) @ id ) ) )
| ? [X21: term] :
( ( X19 @ X21 )
!= ( X18 @ id @ X21 @ id ) )
| ? [X20: term] :
( ( $true
= ( X19 @ X20 ) )
& ( ( X19 @ ( sub @ X17 @ ( push @ X20 @ id ) ) )
!= $true ) )
| ( ( X19 @ ( lam @ X17 ) )
= $true )
| ? [X27: subst,X28: term,X29: subst,X30: subst] :
( ( $true
= ( X18 @ X27 @ X28 @ ( comp @ X29 @ X30 ) ) )
& ( ( X18 @ ( comp @ X27 @ X29 ) @ ( sub @ X28 @ X29 ) @ X30 )
!= $true ) )
| ? [X31: subst,X32: subst,X33: subst,X34: term] :
( ( $true
= ( X18 @ ( comp @ X33 @ X32 ) @ ( sub @ X34 @ X32 ) @ X31 ) )
& ( ( X18 @ X33 @ X34 @ ( comp @ X32 @ X31 ) )
!= $true ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X477: term,X478: subst > term > subst > $o] :
( ( ( X478 @ id @ ( lam @ X477 ) @ id )
= $true )
| ? [X483: term] :
( ( $true
!= ( X478 @ id @ ( sub @ X477 @ ( push @ X483 @ id ) ) @ id ) )
& ( ( X478 @ id @ X483 @ id )
= $true ) )
| ? [X489: subst,X490: subst,X491: term,X492: subst] :
( ( ( X478 @ ( comp @ X492 @ X490 ) @ ( sub @ X491 @ X490 ) @ X489 )
= $true )
& ( ( X478 @ X492 @ X491 @ ( comp @ X490 @ X489 ) )
!= $true ) )
| ? [X479: subst,X480: subst,X481: subst,X482: term] :
( ( ( X478 @ ( comp @ X481 @ X479 ) @ ( sub @ X482 @ X479 ) @ X480 )
!= $true )
& ( ( X478 @ X481 @ X482 @ ( comp @ X479 @ X480 ) )
= $true ) )
| ? [X484: subst > term > term] :
( ! [X488: term] :
( ( ( X478 @ id @ ( X484 @ id @ X488 ) @ id )
= $true )
| ( $true
!= ( X478 @ id @ X488 @ id ) ) )
& ! [X485: subst,X486: subst,X487: term] :
( ( sub @ ( X484 @ X486 @ X487 ) @ X485 )
= ( X484 @ ( comp @ X486 @ X485 ) @ ( sub @ X487 @ X485 ) ) )
& ( $true
!= ( X478 @ id @ ( lam @ ( X484 @ sh @ one ) ) @ id ) ) ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( ? [X438: term,X439: term] :
( ( X438 != X439 )
& ( ( lam @ X438 )
= ( lam @ X439 ) ) )
| ? [X436: term,X437: subst] :
( ( comp @ sh @ ( push @ X436 @ X437 ) )
!= X437 )
| ? [X434: term,X435: subst] :
( ( sub @ one @ ( push @ X434 @ X435 ) )
!= X434 )
| ! [X440: term,X441: subst > term > term,X442: subst > term > term,X443: subst] :
( ? [X444: term,X445: subst,X446: subst] :
( ( X441 @ ( comp @ X446 @ X445 ) @ ( sub @ X444 @ X445 ) )
!= ( sub @ ( X441 @ X446 @ X444 ) @ X445 ) )
| ( ( X441 @ X443 @ X440 )
= ( X442 @ X443 @ X440 ) )
| ( ( lam @ ( X441 @ sh @ one ) )
!= ( lam @ ( X442 @ sh @ one ) ) )
| ? [X447: subst,X448: subst,X449: term] :
( ( X442 @ ( comp @ X448 @ X447 ) @ ( sub @ X449 @ X447 ) )
!= ( sub @ ( X442 @ X448 @ X449 ) @ X447 ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( pushprop_lem0_gthm = $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X8: term] :
? [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
= ( X9 @ sh @ one ) ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( axmap = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop = $true )
& ( ( substmonoid_lthm = $true )
<=> ( ? [X537: subst] :
( ( comp @ X537 @ id )
!= X537 )
| ( ! [X538: subst] :
( ( comp @ X538 @ id )
= X538 )
& ! [X539: subst] :
( ( comp @ id @ X539 )
= X539 ) )
| ? [X536: subst] :
( ( comp @ id @ X536 )
!= X536 ) ) )
& ( axabs = $true )
& ( lamnotvar = $true )
& ( ! [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
= X240 )
<=> ( axshiftcons = $true ) )
& ( ( ? [X355: term,X356: term > $o] :
( ( ( X356 @ X355 )
!= $true )
& ! [X358: term] :
( ( $true
= ( X356 @ ( lam @ X358 ) ) )
| ( ( X356 @ X358 )
!= $true ) )
& ! [X357: term] :
( ( ( var @ X357 )
!= $true )
| ( $true
= ( X356 @ X357 ) ) )
& ! [X359: term,X360: term] :
( ( $true
!= ( X356 @ X360 ) )
| ( $true
= ( X356 @ ( ap @ X360 @ X359 ) ) )
| ( $true
!= ( X356 @ X359 ) ) ) )
| ? [X354: subst] :
( ( comp @ X354 @ id )
!= X354 )
| ? [X352: term,X353: subst] :
( ( comp @ sh @ ( push @ X352 @ X353 ) )
!= X353 )
| ? [X350: subst,X351: term] :
( ( sub @ one @ ( push @ X351 @ X350 ) )
!= X351 )
| ! [X361: term,X362: term > $o,X363: subst] :
( ? [X365: term] :
( ! [X366: term] :
( ( $true
= ( X362 @ ( sub @ X365 @ ( push @ X366 @ id ) ) ) )
| ( ( X362 @ X366 )
!= $true ) )
& ( $true
!= ( X362 @ ( lam @ X365 ) ) ) )
| ? [X364: term] :
( ( ( var @ X364 )
= $true )
& ( $true
!= ( X362 @ ( sub @ X364 @ X363 ) ) ) )
| ? [X367: term,X368: term] :
( ( ( X362 @ X367 )
= $true )
& ( ( X362 @ X368 )
= $true )
& ( $true
!= ( X362 @ ( ap @ X368 @ X367 ) ) ) )
| ( ( X362 @ ( sub @ X361 @ X363 ) )
= $true ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( ! [X121: term,X122: subst > term > subst > $o] :
( ? [X123: term] :
( ( ( X122 @ id @ X123 @ id )
= $true )
& ( ( X122 @ id @ ( sub @ X121 @ ( push @ X123 @ id ) ) @ id )
!= $true ) )
| ( $true
= ( X122 @ id @ ( lam @ ( sub @ X121 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X124: term,X125: subst > term > term] :
( ! [X126: subst,X127: term,X128: subst] :
( ( sub @ ( sub @ X124 @ ( push @ X127 @ X128 ) ) @ X126 )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
& ( ( X122 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id )
!= $true )
& ! [X129: term] :
( ( ( X122 @ id @ X129 @ id )
!= $true )
| ( ( X122 @ id @ ( X125 @ id @ X129 ) @ id )
= $true ) )
& ! [X130: term,X131: subst,X132: subst] :
( ( sub @ X124 @ ( push @ ( sub @ X130 @ X132 ) @ ( comp @ X131 @ X132 ) ) )
= ( X125 @ ( comp @ X131 @ X132 ) @ ( sub @ X130 @ X132 ) ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( ! [X160: term > $o,X161: term] :
( ? [X162: term,X163: term] :
( ( ( X160 @ X163 )
= $true )
& ( $true
!= ( X160 @ ( ap @ X163 @ X162 ) ) )
& ( $true
= ( X160 @ X162 ) ) )
| ? [X164: term] :
( ( ( var @ X164 )
= $true )
& ( ( X160 @ X164 )
!= $true ) )
| ( ( X160 @ X161 )
= $true )
| ? [X165: term] :
( ( $true
!= ( X160 @ ( lam @ X165 ) ) )
& ! [X166: term] :
( ( $true
!= ( X160 @ X166 ) )
| ( ( X160 @ ( sub @ X165 @ ( push @ X166 @ id ) ) )
= $true ) ) ) )
| ? [X152: term > $o,X153: term,X154: subst] :
( ! [X158: term] :
( ? [X159: term] :
( ( $true
!= ( X152 @ ( sub @ X158 @ ( push @ X159 @ id ) ) ) )
& ( ( X152 @ X159 )
= $true ) )
| ( ( X152 @ ( lam @ X158 ) )
= $true ) )
& ! [X157: term] :
( ( ( X152 @ ( sub @ X157 @ X154 ) )
= $true )
| ( ( var @ X157 )
!= $true ) )
& ( ( X152 @ ( sub @ X153 @ X154 ) )
!= $true )
& ! [X155: term,X156: term] :
( ( $true
= ( X152 @ ( ap @ X155 @ X156 ) ) )
| ( ( X152 @ X156 )
!= $true )
| ( $true
!= ( X152 @ X155 ) ) ) )
| ? [X151: term] :
( ( sub @ X151 @ id )
!= X151 ) )
<=> ( induction2_lthm = $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( pushprop_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X138: subst > term > subst > $o,X139: term] :
( ( ( X138 @ id @ X139 @ id )
= $true )
| ? [X145: term,X146: term] :
( ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
!= $true )
& ( $true
= ( X138 @ id @ X145 @ id ) )
& ( ( X138 @ id @ X146 @ id )
= $true ) )
| ? [X140: subst > term > term] :
( ( ( X138 @ id @ ( lam @ ( X140 @ sh @ one ) ) @ id )
!= $true )
& ! [X142: subst,X143: term,X144: subst] :
( ( X140 @ ( comp @ X142 @ X144 ) @ ( sub @ X143 @ X144 ) )
= ( sub @ ( X140 @ X142 @ X143 ) @ X144 ) )
& ! [X141: term] :
( ( ( X138 @ id @ X141 @ id )
!= $true )
| ( $true
= ( X138 @ id @ ( X140 @ id @ X141 ) @ id ) ) ) ) ) )
& ( lamnotap != $true )
& ( ( hoasinduction = $true )
<=> ! [X493: term,X494: subst > term > subst > $o] :
( ( $true
= ( X494 @ id @ X493 @ id ) )
| ? [X499: term,X500: term] :
( ( ( X494 @ id @ X500 @ id )
= $true )
& ( ( X494 @ id @ ( ap @ ( sub @ X500 @ id ) @ X499 ) @ id )
!= $true )
& ( ( X494 @ id @ X499 @ id )
= $true ) )
| ? [X507: subst,X508: subst,X509: term,X510: subst] :
( ( ( X494 @ X510 @ X509 @ ( comp @ X508 @ X507 ) )
= $true )
& ( $true
!= ( X494 @ ( comp @ X510 @ X508 ) @ ( sub @ X509 @ X508 ) @ X507 ) ) )
| ? [X501: subst > term > term] :
( ( $true
!= ( X494 @ id @ ( lam @ ( X501 @ sh @ one ) ) @ id ) )
& ! [X505: term] :
( ( ( X494 @ id @ X505 @ id )
!= $true )
| ( ( X494 @ id @ ( X501 @ id @ X505 ) @ id )
= $true ) )
& ! [X502: subst,X503: term,X504: subst] :
( ( X501 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X501 @ X502 @ X503 ) @ X504 ) ) )
| ? [X495: term,X496: subst,X497: subst,X498: subst] :
( ( $true
= ( X494 @ ( comp @ X498 @ X497 ) @ ( sub @ X495 @ X497 ) @ X496 ) )
& ( ( X494 @ X498 @ X495 @ ( comp @ X497 @ X496 ) )
!= $true ) )
| ? [X506: term] :
( ( ( var @ ( sub @ X506 @ id ) )
= $true )
& ( $true
!= ( X494 @ id @ X506 @ id ) ) ) ) )
& ( ( laminj = $true )
<=> ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( axidl = $true )
<=> ! [X471: subst] :
( ( comp @ id @ X471 )
= X471 ) )
& ( ( hoasapinj2 = $true )
<=> ! [X98: term,X99: term,X100: term,X101: term] :
( ( X100 = X101 )
| ( ( ap @ ( sub @ X98 @ id ) @ X100 )
!= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X336: term,X337: term,X338: subst > term > term] :
( ? [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
!= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
| ( ( ap @ ( sub @ X336 @ id ) @ X337 )
!= ( lam @ ( X338 @ sh @ one ) ) ) ) )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ? [X211: term > $o,X212: term,X213: subst > term > subst > $o] :
( ! [X216: subst > term > term] :
( ( ( X213 @ id @ ( lam @ ( X216 @ sh @ one ) ) @ id )
= $true )
| ? [X220: term] :
( ( ( X213 @ id @ X220 @ id )
= $true )
& ( ( X213 @ id @ ( X216 @ id @ X220 ) @ id )
!= $true ) )
| ? [X217: term,X218: subst,X219: subst] :
( ( sub @ ( X216 @ X219 @ X217 ) @ X218 )
!= ( X216 @ ( comp @ X219 @ X218 ) @ ( sub @ X217 @ X218 ) ) ) )
& ! [X214: term] :
( ( X213 @ id @ X214 @ id )
= ( X211 @ X214 ) )
& ( $true
!= ( X211 @ ( lam @ X212 ) ) )
& ! [X215: term] :
( ( $true
!= ( X211 @ X215 ) )
| ( $true
= ( X211 @ ( sub @ X212 @ ( push @ X215 @ id ) ) ) ) ) )
| ! [X221: term,X222: subst > term > subst > $o] :
( ? [X223: subst,X224: term,X225: subst,X226: subst] :
( ( $true
= ( X222 @ X225 @ X224 @ ( comp @ X226 @ X223 ) ) )
& ( ( X222 @ ( comp @ X225 @ X226 ) @ ( sub @ X224 @ X226 ) @ X223 )
!= $true ) )
| ? [X229: term] :
( ( $true
!= ( X222 @ id @ X229 @ id ) )
& ( ( var @ ( sub @ X229 @ id ) )
= $true ) )
| ( ( X222 @ id @ X221 @ id )
= $true )
| ? [X227: term,X228: term] :
( ( ( X222 @ id @ X227 @ id )
= $true )
& ( ( X222 @ id @ X228 @ id )
= $true )
& ( ( X222 @ id @ ( ap @ ( sub @ X228 @ id ) @ X227 ) @ id )
!= $true ) )
| ? [X234: subst > term > term] :
( ( ( X222 @ id @ ( lam @ ( X234 @ sh @ one ) ) @ id )
!= $true )
& ! [X236: term,X237: subst,X238: subst] :
( ( X234 @ ( comp @ X238 @ X237 ) @ ( sub @ X236 @ X237 ) )
= ( sub @ ( X234 @ X238 @ X236 ) @ X237 ) )
& ! [X235: term] :
( ( $true
= ( X222 @ id @ ( X234 @ id @ X235 ) @ id ) )
| ( ( X222 @ id @ X235 @ id )
!= $true ) ) )
| ? [X230: subst,X231: subst,X232: term,X233: subst] :
( ( ( X222 @ X233 @ X232 @ ( comp @ X230 @ X231 ) )
!= $true )
& ( $true
= ( X222 @ ( comp @ X233 @ X230 ) @ ( sub @ X232 @ X230 ) @ X231 ) ) ) )
| ? [X210: term] :
( ( sub @ X210 @ id )
!= X210 )
| ? [X203: term,X204: term > $o] :
( ( $true
!= ( X204 @ X203 ) )
& ! [X207: term] :
( ? [X208: term] :
( ( $true
!= ( X204 @ ( sub @ X207 @ ( push @ X208 @ id ) ) ) )
& ( $true
= ( X204 @ X208 ) ) )
| ( $true
= ( X204 @ ( lam @ X207 ) ) ) )
& ! [X209: term] :
( ( ( X204 @ X209 )
= $true )
| ( ( var @ X209 )
!= $true ) )
& ! [X205: term,X206: term] :
( ( $true
!= ( X204 @ X206 ) )
| ( ( X204 @ X205 )
!= $true )
| ( ( X204 @ ( ap @ X206 @ X205 ) )
= $true ) ) )
| ? [X200: subst > term > subst > $o] :
! [X201: term > $o] :
? [X202: term] :
( ( X201 @ X202 )
!= ( X200 @ id @ X202 @ id ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ! [X511: term,X512: subst > term > subst > $o] :
( ? [X518: term] :
( ( ( X512 @ id @ ( sub @ X511 @ ( push @ X518 @ id ) ) @ id )
!= $true )
& ( ( X512 @ id @ X518 @ id )
= $true ) )
| ( ( X512 @ id @ ( lam @ X511 ) @ id )
= $true )
| ? [X513: subst > term > term] :
( ( ( X512 @ id @ ( lam @ ( X513 @ sh @ one ) ) @ id )
!= $true )
& ! [X514: subst,X515: term,X516: subst] :
( ( X513 @ ( comp @ X514 @ X516 ) @ ( sub @ X515 @ X516 ) )
= ( sub @ ( X513 @ X514 @ X515 ) @ X516 ) )
& ! [X517: term] :
( ( $true
= ( X512 @ id @ ( X513 @ id @ X517 ) @ id ) )
| ( ( X512 @ id @ X517 @ id )
!= $true ) ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X532: term,X533: term,X534: term,X535: term] :
( ( X532 = X534 )
| ( ( ap @ ( sub @ X533 @ id ) @ X532 )
!= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) )
| ? [X528: term,X529: term,X530: term,X531: term] :
( ( ( ap @ X531 @ X529 )
= ( ap @ X528 @ X530 ) )
& ( X529 != X530 ) ) ) )
& ( ( hoasapinj1 = $true )
<=> ! [X13: term,X14: term,X15: term,X16: term] :
( ( X13 = X16 )
| ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( ! [X467: term] :
? [X468: subst > term > term] :
( ( X468 @ sh @ one )
= ( sub @ X467 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( ! [X42: term > $o,X43: term] :
( ? [X44: term] :
( ( ( var @ X44 )
= $true )
& ( ( X42 @ X44 )
!= $true ) )
| ? [X47: term,X48: term] :
( ( $true
!= ( X42 @ ( ap @ X47 @ X48 ) ) )
& ( $true
= ( X42 @ X47 ) )
& ( ( X42 @ X48 )
= $true ) )
| ? [X45: term] :
( ( ( X42 @ ( lam @ X45 ) )
!= $true )
& ! [X46: term] :
( ( $true
!= ( X42 @ X46 ) )
| ( $true
= ( X42 @ ( sub @ X45 @ ( push @ X46 @ id ) ) ) ) ) )
| ( ( X42 @ X43 )
= $true ) )
<=> ( induction2 = $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ? [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
!= X168 )
| ! [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ? [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
!= ( X170 @ X173 ) )
| ( ( X170 @ one )
= $true )
| ( ( X171 @ X172 )
!= $true ) ) ) )
& ( ! [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ( ( X474 @ X475 )
!= $true )
| ? [X476: term] :
( ( X472 @ X476 )
!= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
| ( ( X472 @ one )
= $true ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( ! [X369: term > $o,X370: subst > term > subst > $o,X371: term,X372: term] :
( ? [X382: term,X383: term] :
( ( ( X370 @ id @ ( ap @ ( sub @ X383 @ id ) @ X382 ) @ id )
!= $true )
& ( ( X370 @ id @ X383 @ id )
= $true )
& ( ( X370 @ id @ X382 @ id )
= $true ) )
| ( $true
!= ( X369 @ X371 ) )
| ( ( X369 @ X372 )
!= $true )
| ( ( X369 @ ( ap @ X372 @ X371 ) )
= $true )
| ? [X373: subst,X374: subst,X375: term,X376: subst] :
( ( ( X370 @ ( comp @ X374 @ X373 ) @ ( sub @ X375 @ X373 ) @ X376 )
!= $true )
& ( ( X370 @ X374 @ X375 @ ( comp @ X373 @ X376 ) )
= $true ) )
| ? [X378: term,X379: subst,X380: subst,X381: subst] :
( ( $true
!= ( X370 @ X380 @ X378 @ ( comp @ X381 @ X379 ) ) )
& ( ( X370 @ ( comp @ X380 @ X381 ) @ ( sub @ X378 @ X381 ) @ X379 )
= $true ) )
| ? [X377: term] :
( ( X369 @ X377 )
!= ( X370 @ id @ X377 @ id ) ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( pushprop_gthm = $true )
& ( ! [X10: subst > term > subst > $o] :
? [X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( ! [X393: term,X394: subst > term > subst > $o,X395: term > $o] :
( ? [X402: term] :
( ( X395 @ X402 )
!= ( X394 @ id @ X402 @ id ) )
| ? [X397: subst > term > term] :
( ! [X399: term,X400: subst,X401: subst] :
( ( sub @ ( X397 @ X400 @ X399 ) @ X401 )
= ( X397 @ ( comp @ X400 @ X401 ) @ ( sub @ X399 @ X401 ) ) )
& ! [X398: term] :
( ( ( X394 @ id @ X398 @ id )
!= $true )
| ( ( X394 @ id @ ( X397 @ id @ X398 ) @ id )
= $true ) )
& ( ( X394 @ id @ ( lam @ ( X397 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X395 @ ( lam @ X393 ) )
= $true )
| ? [X396: term] :
( ( ( X395 @ ( sub @ X393 @ ( push @ X396 @ id ) ) )
!= $true )
& ( ( X395 @ X396 )
= $true ) ) )
| ? [X388: term] :
! [X389: subst > term > term] :
? [X390: term,X391: subst] :
( ( sub @ X388 @ ( push @ X390 @ X391 ) )
!= ( X389 @ X391 @ X390 ) )
| ? [X392: term] :
( ( sub @ X392 @ id )
!= X392 ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj1_gthm = $true )
& ( ! [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
!= ( ap @ X243 @ X241 ) )
| ( X241 = X242 ) )
<=> ( apinj2 = $true ) )
& ( ( substmonoid = $true )
<=> ( ! [X246: subst] :
( ( comp @ id @ X246 )
= X246 )
& ! [X245: subst] :
( ( comp @ X245 @ id )
= X245 ) ) )
& ( ( ? [X409: term] :
( ( sub @ X409 @ id )
!= X409 )
| ! [X418: term,X419: subst > term > subst > $o] :
( ? [X420: subst,X421: subst,X422: term,X423: subst] :
( ( ( X419 @ X423 @ X422 @ ( comp @ X421 @ X420 ) )
!= $true )
& ( ( X419 @ ( comp @ X423 @ X421 ) @ ( sub @ X422 @ X421 ) @ X420 )
= $true ) )
| ? [X433: term] :
( ( ( X419 @ id @ ( sub @ X418 @ ( push @ X433 @ id ) ) @ id )
!= $true )
& ( $true
= ( X419 @ id @ X433 @ id ) ) )
| ( $true
= ( X419 @ id @ ( lam @ X418 ) @ id ) )
| ? [X428: subst > term > term] :
( ( $true
!= ( X419 @ id @ ( lam @ ( X428 @ sh @ one ) ) @ id ) )
& ! [X429: term,X430: subst,X431: subst] :
( ( sub @ ( X428 @ X431 @ X429 ) @ X430 )
= ( X428 @ ( comp @ X431 @ X430 ) @ ( sub @ X429 @ X430 ) ) )
& ! [X432: term] :
( ( $true
= ( X419 @ id @ ( X428 @ id @ X432 ) @ id ) )
| ( ( X419 @ id @ X432 @ id )
!= $true ) ) )
| ? [X424: subst,X425: subst,X426: subst,X427: term] :
( ( $true
!= ( X419 @ ( comp @ X426 @ X425 ) @ ( sub @ X427 @ X425 ) @ X424 ) )
& ( ( X419 @ X426 @ X427 @ ( comp @ X425 @ X424 ) )
= $true ) ) )
| ? [X410: term,X411: subst > term > subst > $o] :
( ! [X417: term] :
( ( $true
= ( X411 @ id @ ( sub @ X410 @ ( push @ X417 @ id ) ) @ id ) )
| ( ( X411 @ id @ X417 @ id )
!= $true ) )
& ( ( X411 @ id @ ( lam @ ( sub @ X410 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X412: subst > term > term] :
( ? [X414: subst,X415: subst,X416: term] :
( ( X412 @ ( comp @ X414 @ X415 ) @ ( sub @ X416 @ X415 ) )
!= ( sub @ ( X412 @ X414 @ X416 ) @ X415 ) )
| ? [X413: term] :
( ( ( X411 @ id @ ( X412 @ id @ X413 ) @ id )
!= $true )
& ( $true
= ( X411 @ id @ X413 @ id ) ) )
| ( ( X411 @ id @ ( lam @ ( X412 @ sh @ one ) ) @ id )
= $true ) ) ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X315: term,X316: subst > term > subst > $o] :
( ? [X328: subst,X329: subst,X330: subst,X331: term] :
( ( ( X316 @ X328 @ X331 @ ( comp @ X329 @ X330 ) )
!= $true )
& ( ( X316 @ ( comp @ X328 @ X329 ) @ ( sub @ X331 @ X329 ) @ X330 )
= $true ) )
| ? [X319: subst > term > term] :
( ! [X323: term] :
( ( ( X316 @ id @ ( X319 @ id @ X323 ) @ id )
= $true )
| ( ( X316 @ id @ X323 @ id )
!= $true ) )
& ! [X320: term,X321: subst,X322: subst] :
( ( sub @ ( X319 @ X321 @ X320 ) @ X322 )
= ( X319 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) ) )
& ( $true
!= ( X316 @ id @ ( lam @ ( X319 @ sh @ one ) ) @ id ) ) )
| ? [X324: subst,X325: term,X326: subst,X327: subst] :
( ( $true
= ( X316 @ X324 @ X325 @ ( comp @ X327 @ X326 ) ) )
& ( ( X316 @ ( comp @ X324 @ X327 ) @ ( sub @ X325 @ X327 ) @ X326 )
!= $true ) )
| ( ( X316 @ id @ X315 @ id )
= $true )
| ? [X332: term] :
( ( ( var @ ( sub @ X332 @ id ) )
= $true )
& ( ( X316 @ id @ X332 @ id )
!= $true ) )
| ? [X317: term,X318: term] :
( ( ( X316 @ id @ X318 @ id )
= $true )
& ( ( X316 @ id @ X317 @ id )
= $true )
& ( $true
!= ( X316 @ id @ ( ap @ ( sub @ X318 @ id ) @ X317 ) @ id ) ) ) )
| ? [X299: term,X300: subst > term > subst > $o] :
( ! [X302: subst,X303: subst,X304: term,X305: subst] :
( ( ( X300 @ ( comp @ X305 @ X302 ) @ ( sub @ X304 @ X302 ) @ X303 )
= $true )
| ( ( X300 @ X305 @ X304 @ ( comp @ X302 @ X303 ) )
!= $true ) )
& ( ( X300 @ id @ ( lam @ X299 ) @ id )
!= $true )
& ! [X301: term] :
( ( $true
!= ( X300 @ id @ X301 @ id ) )
| ( ( X300 @ id @ ( sub @ X299 @ ( push @ X301 @ id ) ) @ id )
= $true ) )
& ! [X310: subst > term > term] :
( ? [X312: term,X313: subst,X314: subst] :
( ( X310 @ ( comp @ X314 @ X313 ) @ ( sub @ X312 @ X313 ) )
!= ( sub @ ( X310 @ X314 @ X312 ) @ X313 ) )
| ( ( X300 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id )
= $true )
| ? [X311: term] :
( ( ( X300 @ id @ ( X310 @ id @ X311 ) @ id )
!= $true )
& ( $true
= ( X300 @ id @ X311 @ id ) ) ) )
& ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( ( X300 @ ( comp @ X307 @ X308 ) @ ( sub @ X306 @ X308 ) @ X309 )
!= $true )
| ( $true
= ( X300 @ X307 @ X306 @ ( comp @ X308 @ X309 ) ) ) ) )
| ? [X279: term,X280: term > $o] :
( ( ( X280 @ X279 )
!= $true )
& ! [X282: term] :
( ? [X283: term] :
( ( $true
!= ( X280 @ ( sub @ X282 @ ( push @ X283 @ id ) ) ) )
& ( ( X280 @ X283 )
= $true ) )
| ( ( X280 @ ( lam @ X282 ) )
= $true ) )
& ! [X284: term,X285: term] :
( ( $true
!= ( X280 @ X285 ) )
| ( ( X280 @ ( ap @ X285 @ X284 ) )
= $true )
| ( $true
!= ( X280 @ X284 ) ) )
& ! [X281: term] :
( ( ( X280 @ X281 )
= $true )
| ( $true
!= ( var @ X281 ) ) ) )
| ? [X286: term,X287: subst > term > subst > $o,X288: term] :
( ! [X295: subst,X296: subst,X297: term,X298: subst] :
( ( $true
= ( X287 @ X295 @ X297 @ ( comp @ X296 @ X298 ) ) )
| ( ( X287 @ ( comp @ X295 @ X296 ) @ ( sub @ X297 @ X296 ) @ X298 )
!= $true ) )
& ( $true
= ( X287 @ id @ X286 @ id ) )
& ( ( X287 @ id @ X288 @ id )
= $true )
& ! [X289: subst,X290: subst,X291: term,X292: subst] :
( ( ( X287 @ ( comp @ X289 @ X290 ) @ ( sub @ X291 @ X290 ) @ X292 )
= $true )
| ( $true
!= ( X287 @ X289 @ X291 @ ( comp @ X290 @ X292 ) ) ) )
& ( $true
!= ( X287 @ id @ ( ap @ X286 @ X288 ) @ id ) )
& ! [X293: term,X294: term] :
( ( ( X287 @ id @ X294 @ id )
!= $true )
| ( ( X287 @ id @ ( ap @ ( sub @ X293 @ id ) @ X294 ) @ id )
= $true )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) ) ) ) )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ? [X52: term > $o,X53: term] :
( ! [X54: term,X55: term] :
( ( $true
!= ( X52 @ X54 ) )
| ( ( X52 @ ( ap @ X55 @ X54 ) )
= $true )
| ( ( X52 @ X55 )
!= $true ) )
& ! [X57: term] :
( ? [X58: term] :
( ( $true
= ( X52 @ X58 ) )
& ( ( X52 @ ( sub @ X57 @ ( push @ X58 @ id ) ) )
!= $true ) )
| ( ( X52 @ ( lam @ X57 ) )
= $true ) )
& ! [X56: term] :
( ( ( var @ X56 )
!= $true )
| ( $true
= ( X52 @ X56 ) ) )
& ( ( X52 @ X53 )
!= $true ) )
| ! [X70: subst > term > subst > $o,X71: term] :
( ? [X72: subst > term > term] :
( ! [X76: term] :
( ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true )
| ( $true
!= ( X70 @ id @ X76 @ id ) ) )
& ! [X73: subst,X74: term,X75: subst] :
( ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) )
= ( sub @ ( X72 @ X75 @ X74 ) @ X73 ) )
& ( ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X77: term,X78: term] :
( ( $true
= ( X70 @ id @ X77 @ id ) )
& ( ( X70 @ id @ ( ap @ ( sub @ X78 @ id ) @ X77 ) @ id )
!= $true )
& ( ( X70 @ id @ X78 @ id )
= $true ) )
| ( $true
= ( X70 @ id @ X71 @ id ) ) )
| ? [X49: subst > term > subst > $o] :
! [X50: term > $o] :
? [X51: term] :
( ( X49 @ id @ X51 @ id )
!= ( X50 @ X51 ) )
| ? [X59: term] :
( ( sub @ X59 @ id )
!= X59 )
| ? [X60: subst > term > subst > $o,X61: term > $o,X62: term] :
( ! [X68: term] :
( ( $true
= ( X61 @ ( sub @ X62 @ ( push @ X68 @ id ) ) ) )
| ( ( X61 @ X68 )
!= $true ) )
& ! [X69: term] :
( ( X60 @ id @ X69 @ id )
= ( X61 @ X69 ) )
& ! [X63: subst > term > term] :
( ( $true
= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
| ? [X65: subst,X66: subst,X67: term] :
( ( sub @ ( X63 @ X66 @ X67 ) @ X65 )
!= ( X63 @ ( comp @ X66 @ X65 ) @ ( sub @ X67 @ X65 ) ) )
| ? [X64: term] :
( ( ( X60 @ id @ X64 @ id )
= $true )
& ( ( X60 @ id @ ( X63 @ id @ X64 ) @ id )
!= $true ) ) )
& ( ( X61 @ ( lam @ X62 ) )
!= $true ) ) ) )
& ( ( hoaslaminj = $true )
<=> ! [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ( ( lam @ ( X191 @ sh @ one ) )
!= ( lam @ ( X190 @ sh @ one ) ) )
| ? [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
!= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
| ( ( X191 @ X192 @ X193 )
= ( X190 @ X192 @ X193 ) )
| ? [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
!= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) ) ) )
& ( ( ! [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) )
| ( $true
!= ( X523 @ X527 ) ) )
| ? [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
!= X521 )
| ? [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
!= X520 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ulamvar1 != $true )
& ( hoasinduction_lem2_gthm = $true )
& ! [X469: d_term,X470: d_term] :
( ( ( d2term @ X469 )
!= ( d2term @ X470 ) )
| ( X469 = X470 ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X451: subst > term > subst > $o,X452: term,X453: term] :
( ? [X460: subst,X461: subst,X462: term,X463: subst] :
( ( ( X451 @ X460 @ X462 @ ( comp @ X463 @ X461 ) )
!= $true )
& ( ( X451 @ ( comp @ X460 @ X463 ) @ ( sub @ X462 @ X463 ) @ X461 )
= $true ) )
| ( ( X451 @ id @ X453 @ id )
!= $true )
| ? [X458: term,X459: term] :
( ( $true
!= ( X451 @ id @ ( ap @ ( sub @ X458 @ id ) @ X459 ) @ id ) )
& ( $true
= ( X451 @ id @ X459 @ id ) )
& ( ( X451 @ id @ X458 @ id )
= $true ) )
| ? [X454: subst,X455: subst,X456: term,X457: subst] :
( ( ( X451 @ X457 @ X456 @ ( comp @ X455 @ X454 ) )
= $true )
& ( $true
!= ( X451 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 ) ) )
| ( ( X451 @ id @ ( ap @ X453 @ X452 ) @ id )
= $true )
| ( ( X451 @ id @ X452 @ id )
!= $true ) ) )
& ( pushprop_lthm_orig = $true )
& ! [X450: d_term] : ( d_one = X450 )
& ( ( pushprop_lem0 = $true )
<=> ! [X37: term > $o,X38: term,X39: subst] :
? [X40: term > $o] :
! [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
= ( X40 @ X41 ) ) ) ),
inference(flattening,[status(thm)],[f12]) ).
thf(f12,plain,
( ( ( hoasapinj1 = $true )
<=> ! [X13: term,X14: term,X15: term,X16: term] :
( ( X13 = X16 )
| ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) ) )
& ( ( ! [X70: subst > term > subst > $o,X71: term] :
( ? [X72: subst > term > term] :
( ! [X76: term] :
( ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true )
| ( $true
!= ( X70 @ id @ X76 @ id ) ) )
& ! [X73: subst,X74: term,X75: subst] :
( ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) )
= ( sub @ ( X72 @ X75 @ X74 ) @ X73 ) )
& ( ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X77: term,X78: term] :
( ( $true
= ( X70 @ id @ X77 @ id ) )
& ( ( X70 @ id @ ( ap @ ( sub @ X78 @ id ) @ X77 ) @ id )
!= $true )
& ( ( X70 @ id @ X78 @ id )
= $true ) )
| ( $true
= ( X70 @ id @ X71 @ id ) ) )
| ? [X60: subst > term > subst > $o,X61: term > $o,X62: term] :
( ! [X68: term] :
( ( $true
= ( X61 @ ( sub @ X62 @ ( push @ X68 @ id ) ) ) )
| ( ( X61 @ X68 )
!= $true ) )
& ! [X69: term] :
( ( X60 @ id @ X69 @ id )
= ( X61 @ X69 ) )
& ! [X63: subst > term > term] :
( ( $true
= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
| ? [X65: subst,X66: subst,X67: term] :
( ( sub @ ( X63 @ X66 @ X67 ) @ X65 )
!= ( X63 @ ( comp @ X66 @ X65 ) @ ( sub @ X67 @ X65 ) ) )
| ? [X64: term] :
( ( ( X60 @ id @ X64 @ id )
= $true )
& ( ( X60 @ id @ ( X63 @ id @ X64 ) @ id )
!= $true ) ) )
& ( ( X61 @ ( lam @ X62 ) )
!= $true ) )
| ? [X59: term] :
( ( sub @ X59 @ id )
!= X59 )
| ? [X52: term > $o,X53: term] :
( ! [X54: term,X55: term] :
( ( $true
!= ( X52 @ X54 ) )
| ( ( X52 @ ( ap @ X55 @ X54 ) )
= $true )
| ( ( X52 @ X55 )
!= $true ) )
& ! [X57: term] :
( ? [X58: term] :
( ( $true
= ( X52 @ X58 ) )
& ( ( X52 @ ( sub @ X57 @ ( push @ X58 @ id ) ) )
!= $true ) )
| ( ( X52 @ ( lam @ X57 ) )
= $true ) )
& ! [X56: term] :
( ( ( var @ X56 )
!= $true )
| ( $true
= ( X52 @ X56 ) ) )
& ( ( X52 @ X53 )
!= $true ) )
| ? [X49: subst > term > subst > $o] :
! [X50: term > $o] :
? [X51: term] :
( ( X49 @ id @ X51 @ id )
!= ( X50 @ X51 ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X333: subst > term > subst > $o] :
? [X334: term > $o] :
! [X335: term] :
( ( X334 @ X335 )
= ( X333 @ id @ X335 @ id ) ) )
& ( axclos = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X477: term,X478: subst > term > subst > $o] :
( ( ( X478 @ id @ ( lam @ X477 ) @ id )
= $true )
| ? [X483: term] :
( ( $true
!= ( X478 @ id @ ( sub @ X477 @ ( push @ X483 @ id ) ) @ id ) )
& ( ( X478 @ id @ X483 @ id )
= $true ) )
| ? [X489: subst,X490: subst,X491: term,X492: subst] :
( ( ( X478 @ ( comp @ X492 @ X490 ) @ ( sub @ X491 @ X490 ) @ X489 )
= $true )
& ( ( X478 @ X492 @ X491 @ ( comp @ X490 @ X489 ) )
!= $true ) )
| ? [X479: subst,X480: subst,X481: subst,X482: term] :
( ( ( X478 @ ( comp @ X481 @ X479 ) @ ( sub @ X482 @ X479 ) @ X480 )
!= $true )
& ( ( X478 @ X481 @ X482 @ ( comp @ X479 @ X480 ) )
= $true ) )
| ? [X484: subst > term > term] :
( ! [X488: term] :
( ( ( X478 @ id @ ( X484 @ id @ X488 ) @ id )
= $true )
| ( $true
!= ( X478 @ id @ X488 @ id ) ) )
& ! [X485: subst,X486: subst,X487: term] :
( ( sub @ ( X484 @ X486 @ X487 ) @ X485 )
= ( X484 @ ( comp @ X486 @ X485 ) @ ( sub @ X487 @ X485 ) ) )
& ( $true
!= ( X478 @ id @ ( lam @ ( X484 @ sh @ one ) ) @ id ) ) ) ) )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X138: subst > term > subst > $o,X139: term] :
( ( ( X138 @ id @ X139 @ id )
= $true )
| ? [X145: term,X146: term] :
( ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
!= $true )
& ( $true
= ( X138 @ id @ X145 @ id ) )
& ( ( X138 @ id @ X146 @ id )
= $true ) )
| ? [X140: subst > term > term] :
( ( ( X138 @ id @ ( lam @ ( X140 @ sh @ one ) ) @ id )
!= $true )
& ! [X142: subst,X143: term,X144: subst] :
( ( X140 @ ( comp @ X142 @ X144 ) @ ( sub @ X143 @ X144 ) )
= ( sub @ ( X140 @ X142 @ X143 ) @ X144 ) )
& ! [X141: term] :
( ( ( X138 @ id @ X141 @ id )
!= $true )
| ( $true
= ( X138 @ id @ ( X140 @ id @ X141 ) @ id ) ) ) ) ) )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X418: term,X419: subst > term > subst > $o] :
( ? [X420: subst,X421: subst,X422: term,X423: subst] :
( ( ( X419 @ X423 @ X422 @ ( comp @ X421 @ X420 ) )
!= $true )
& ( ( X419 @ ( comp @ X423 @ X421 ) @ ( sub @ X422 @ X421 ) @ X420 )
= $true ) )
| ? [X433: term] :
( ( ( X419 @ id @ ( sub @ X418 @ ( push @ X433 @ id ) ) @ id )
!= $true )
& ( $true
= ( X419 @ id @ X433 @ id ) ) )
| ( $true
= ( X419 @ id @ ( lam @ X418 ) @ id ) )
| ? [X428: subst > term > term] :
( ( $true
!= ( X419 @ id @ ( lam @ ( X428 @ sh @ one ) ) @ id ) )
& ! [X429: term,X430: subst,X431: subst] :
( ( sub @ ( X428 @ X431 @ X429 ) @ X430 )
= ( X428 @ ( comp @ X431 @ X430 ) @ ( sub @ X429 @ X430 ) ) )
& ! [X432: term] :
( ( $true
= ( X419 @ id @ ( X428 @ id @ X432 ) @ id ) )
| ( ( X419 @ id @ X432 @ id )
!= $true ) ) )
| ? [X424: subst,X425: subst,X426: subst,X427: term] :
( ( $true
!= ( X419 @ ( comp @ X426 @ X425 ) @ ( sub @ X427 @ X425 ) @ X424 ) )
& ( ( X419 @ X426 @ X427 @ ( comp @ X425 @ X424 ) )
= $true ) ) )
| ? [X410: term,X411: subst > term > subst > $o] :
( ! [X417: term] :
( ( $true
= ( X411 @ id @ ( sub @ X410 @ ( push @ X417 @ id ) ) @ id ) )
| ( ( X411 @ id @ X417 @ id )
!= $true ) )
& ( ( X411 @ id @ ( lam @ ( sub @ X410 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X412: subst > term > term] :
( ? [X414: subst,X415: subst,X416: term] :
( ( X412 @ ( comp @ X414 @ X415 ) @ ( sub @ X416 @ X415 ) )
!= ( sub @ ( X412 @ X414 @ X416 ) @ X415 ) )
| ? [X413: term] :
( ( ( X411 @ id @ ( X412 @ id @ X413 ) @ id )
!= $true )
& ( $true
= ( X411 @ id @ X413 @ id ) ) )
| ( ( X411 @ id @ ( lam @ ( X412 @ sh @ one ) ) @ id )
= $true ) ) )
| ? [X409: term] :
( ( sub @ X409 @ id )
!= X409 ) ) )
& ( hoaslaminj_gthm = $true )
& ! [X112: d_subst,X113: d_subst] :
( ( ( d2subst @ X112 )
!= ( d2subst @ X113 ) )
| ( X112 = X113 ) )
& ! [X114: term] :
? [X115: d_term] :
( ( d2term @ X115 )
= X114 )
& ( hoasapnotvar = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoaslaminj = $true )
<=> ! [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ( ( lam @ ( X191 @ sh @ one ) )
!= ( lam @ ( X190 @ sh @ one ) ) )
| ? [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
!= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
| ( ( X191 @ X192 @ X193 )
= ( X190 @ X192 @ X193 ) )
| ? [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
!= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) ) ) )
& ( ( d2subst @ d_id )
= id )
& ( ( ! [X221: term,X222: subst > term > subst > $o] :
( ? [X223: subst,X224: term,X225: subst,X226: subst] :
( ( $true
= ( X222 @ X225 @ X224 @ ( comp @ X226 @ X223 ) ) )
& ( ( X222 @ ( comp @ X225 @ X226 ) @ ( sub @ X224 @ X226 ) @ X223 )
!= $true ) )
| ? [X229: term] :
( ( $true
!= ( X222 @ id @ X229 @ id ) )
& ( ( var @ ( sub @ X229 @ id ) )
= $true ) )
| ( ( X222 @ id @ X221 @ id )
= $true )
| ? [X227: term,X228: term] :
( ( ( X222 @ id @ X227 @ id )
= $true )
& ( ( X222 @ id @ X228 @ id )
= $true )
& ( ( X222 @ id @ ( ap @ ( sub @ X228 @ id ) @ X227 ) @ id )
!= $true ) )
| ? [X234: subst > term > term] :
( ( ( X222 @ id @ ( lam @ ( X234 @ sh @ one ) ) @ id )
!= $true )
& ! [X236: term,X237: subst,X238: subst] :
( ( X234 @ ( comp @ X238 @ X237 ) @ ( sub @ X236 @ X237 ) )
= ( sub @ ( X234 @ X238 @ X236 ) @ X237 ) )
& ! [X235: term] :
( ( $true
= ( X222 @ id @ ( X234 @ id @ X235 ) @ id ) )
| ( ( X222 @ id @ X235 @ id )
!= $true ) ) )
| ? [X230: subst,X231: subst,X232: term,X233: subst] :
( ( ( X222 @ X233 @ X232 @ ( comp @ X230 @ X231 ) )
!= $true )
& ( $true
= ( X222 @ ( comp @ X233 @ X230 ) @ ( sub @ X232 @ X230 ) @ X231 ) ) ) )
| ? [X211: term > $o,X212: term,X213: subst > term > subst > $o] :
( ! [X216: subst > term > term] :
( ( ( X213 @ id @ ( lam @ ( X216 @ sh @ one ) ) @ id )
= $true )
| ? [X220: term] :
( ( ( X213 @ id @ X220 @ id )
= $true )
& ( ( X213 @ id @ ( X216 @ id @ X220 ) @ id )
!= $true ) )
| ? [X217: term,X218: subst,X219: subst] :
( ( sub @ ( X216 @ X219 @ X217 ) @ X218 )
!= ( X216 @ ( comp @ X219 @ X218 ) @ ( sub @ X217 @ X218 ) ) ) )
& ! [X214: term] :
( ( X213 @ id @ X214 @ id )
= ( X211 @ X214 ) )
& ( $true
!= ( X211 @ ( lam @ X212 ) ) )
& ! [X215: term] :
( ( $true
!= ( X211 @ X215 ) )
| ( $true
= ( X211 @ ( sub @ X212 @ ( push @ X215 @ id ) ) ) ) ) )
| ? [X210: term] :
( ( sub @ X210 @ id )
!= X210 )
| ? [X203: term,X204: term > $o] :
( ( $true
!= ( X204 @ X203 ) )
& ! [X207: term] :
( ? [X208: term] :
( ( $true
!= ( X204 @ ( sub @ X207 @ ( push @ X208 @ id ) ) ) )
& ( $true
= ( X204 @ X208 ) ) )
| ( $true
= ( X204 @ ( lam @ X207 ) ) ) )
& ! [X209: term] :
( ( ( X204 @ X209 )
= $true )
| ( ( var @ X209 )
!= $true ) )
& ! [X205: term,X206: term] :
( ( $true
!= ( X204 @ X206 ) )
| ( ( X204 @ X205 )
!= $true )
| ( ( X204 @ ( ap @ X206 @ X205 ) )
= $true ) ) )
| ? [X200: subst > term > subst > $o] :
! [X201: term > $o] :
? [X202: term] :
( ( X201 @ X202 )
!= ( X200 @ id @ X202 @ id ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( hoaslamnotvar = $true )
& ( ulamvar1 != $true )
& ( ! [X265: term,X266: subst > term > subst > $o,X267: term > $o] :
( ? [X269: term] :
( ( X267 @ X269 )
!= ( X266 @ id @ X269 @ id ) )
| ( $true
= ( X267 @ ( lam @ X265 ) ) )
| ? [X270: subst > term > term] :
( ! [X272: subst,X273: subst,X274: term] :
( ( sub @ ( X270 @ X272 @ X274 ) @ X273 )
= ( X270 @ ( comp @ X272 @ X273 ) @ ( sub @ X274 @ X273 ) ) )
& ! [X271: term] :
( ( ( X266 @ id @ X271 @ id )
!= $true )
| ( $true
= ( X266 @ id @ ( X270 @ id @ X271 ) @ id ) ) )
& ( ( X266 @ id @ ( lam @ ( X270 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X268: term] :
( ( ( X267 @ X268 )
= $true )
& ( $true
!= ( X267 @ ( sub @ X265 @ ( push @ X268 @ id ) ) ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( axapp = $true )
& ( hoasapnotvar_gthm = $true )
& ( axvarshift = $true )
& ( ( ! [X160: term > $o,X161: term] :
( ? [X162: term,X163: term] :
( ( ( X160 @ X163 )
= $true )
& ( $true
!= ( X160 @ ( ap @ X163 @ X162 ) ) )
& ( $true
= ( X160 @ X162 ) ) )
| ? [X164: term] :
( ( ( var @ X164 )
= $true )
& ( ( X160 @ X164 )
!= $true ) )
| ( ( X160 @ X161 )
= $true )
| ? [X165: term] :
( ( $true
!= ( X160 @ ( lam @ X165 ) ) )
& ! [X166: term] :
( ( $true
!= ( X160 @ X166 ) )
| ( ( X160 @ ( sub @ X165 @ ( push @ X166 @ id ) ) )
= $true ) ) ) )
| ? [X152: term > $o,X153: term,X154: subst] :
( ! [X158: term] :
( ? [X159: term] :
( ( $true
!= ( X152 @ ( sub @ X158 @ ( push @ X159 @ id ) ) ) )
& ( ( X152 @ X159 )
= $true ) )
| ( ( X152 @ ( lam @ X158 ) )
= $true ) )
& ! [X157: term] :
( ( ( X152 @ ( sub @ X157 @ X154 ) )
= $true )
| ( ( var @ X157 )
!= $true ) )
& ( ( X152 @ ( sub @ X153 @ X154 ) )
!= $true )
& ! [X155: term,X156: term] :
( ( $true
= ( X152 @ ( ap @ X155 @ X156 ) ) )
| ( ( X152 @ X156 )
!= $true )
| ( $true
!= ( X152 @ X155 ) ) ) )
| ? [X151: term] :
( ( sub @ X151 @ id )
!= X151 ) )
<=> ( induction2_lthm = $true ) )
& ( ( pushprop_lem0 = $true )
<=> ! [X37: term > $o,X38: term,X39: subst] :
? [X40: term > $o] :
! [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
= ( X40 @ X41 ) ) )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X336: term,X337: term,X338: subst > term > term] :
( ? [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
!= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
| ( ( ap @ ( sub @ X336 @ id ) @ X337 )
!= ( lam @ ( X338 @ sh @ one ) ) ) ) )
& ( induction2_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( ! [X0: subst,X1: term > $o,X2: term] :
( ? [X6: term] :
( ! [X7: term] :
( ( $true
!= ( X1 @ X7 ) )
| ( $true
= ( X1 @ ( sub @ X6 @ ( push @ X7 @ id ) ) ) ) )
& ( ( X1 @ ( lam @ X6 ) )
!= $true ) )
| ( ( X1 @ ( sub @ X2 @ X0 ) )
= $true )
| ? [X5: term] :
( ( ( var @ X5 )
= $true )
& ( ( X1 @ ( sub @ X5 @ X0 ) )
!= $true ) )
| ? [X3: term,X4: term] :
( ( ( X1 @ X3 )
= $true )
& ( ( X1 @ ( ap @ X3 @ X4 ) )
!= $true )
& ( ( X1 @ X4 )
= $true ) ) )
<=> ( induction2lem = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( axassoc = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
= $true )
| ( $true
!= ( X137 @ X134 ) ) ) )
& ! [X450: d_term] : ( d_one = X450 )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( lamnotvar = $true )
& ( pushprop = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ulamvarsh = $true )
& ( pushprop_gthm = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X98: term,X99: term,X100: term,X101: term] :
( ( X100 = X101 )
| ( ( ap @ ( sub @ X98 @ id ) @ X100 )
!= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) ) )
& ( ( ! [X440: term,X441: subst > term > term,X442: subst > term > term,X443: subst] :
( ? [X444: term,X445: subst,X446: subst] :
( ( X441 @ ( comp @ X446 @ X445 ) @ ( sub @ X444 @ X445 ) )
!= ( sub @ ( X441 @ X446 @ X444 ) @ X445 ) )
| ( ( X441 @ X443 @ X440 )
= ( X442 @ X443 @ X440 ) )
| ( ( lam @ ( X441 @ sh @ one ) )
!= ( lam @ ( X442 @ sh @ one ) ) )
| ? [X447: subst,X448: subst,X449: term] :
( ( X442 @ ( comp @ X448 @ X447 ) @ ( sub @ X449 @ X447 ) )
!= ( sub @ ( X442 @ X448 @ X449 ) @ X447 ) ) )
| ? [X438: term,X439: term] :
( ( X438 != X439 )
& ( ( lam @ X438 )
= ( lam @ X439 ) ) )
| ? [X436: term,X437: subst] :
( ( comp @ sh @ ( push @ X436 @ X437 ) )
!= X437 )
| ? [X434: term,X435: subst] :
( ( sub @ one @ ( push @ X434 @ X435 ) )
!= X434 ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X393: term,X394: subst > term > subst > $o,X395: term > $o] :
( ? [X402: term] :
( ( X395 @ X402 )
!= ( X394 @ id @ X402 @ id ) )
| ? [X397: subst > term > term] :
( ! [X399: term,X400: subst,X401: subst] :
( ( sub @ ( X397 @ X400 @ X399 ) @ X401 )
= ( X397 @ ( comp @ X400 @ X401 ) @ ( sub @ X399 @ X401 ) ) )
& ! [X398: term] :
( ( ( X394 @ id @ X398 @ id )
!= $true )
| ( ( X394 @ id @ ( X397 @ id @ X398 ) @ id )
= $true ) )
& ( ( X394 @ id @ ( lam @ ( X397 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X395 @ ( lam @ X393 ) )
= $true )
| ? [X396: term] :
( ( ( X395 @ ( sub @ X393 @ ( push @ X396 @ id ) ) )
!= $true )
& ( ( X395 @ X396 )
= $true ) ) )
| ? [X392: term] :
( ( sub @ X392 @ id )
!= X392 )
| ? [X388: term] :
! [X389: subst > term > term] :
? [X390: term,X391: subst] :
( ( sub @ X388 @ ( push @ X390 @ X391 ) )
!= ( X389 @ X391 @ X390 ) ) ) )
& ( axabs = $true )
& ( axmap = $true )
& ! [X464: subst] :
? [X465: d_subst] :
( ( d2subst @ X465 )
= X464 )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X342: term,X343: subst > term > subst > $o] :
( ? [X344: term] :
( ( ( X343 @ id @ X344 @ id )
= $true )
& ( ( X343 @ id @ ( sub @ X342 @ ( push @ X344 @ id ) ) @ id )
!= $true ) )
| ? [X345: subst > term > term] :
( ! [X346: term,X347: subst,X348: subst] :
( ( sub @ ( X345 @ X347 @ X346 ) @ X348 )
= ( X345 @ ( comp @ X347 @ X348 ) @ ( sub @ X346 @ X348 ) ) )
& ! [X349: term] :
( ( ( X343 @ id @ X349 @ id )
!= $true )
| ( $true
= ( X343 @ id @ ( X345 @ id @ X349 ) @ id ) ) )
& ( ( X343 @ id @ ( lam @ ( X345 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X343 @ id @ ( lam @ ( sub @ X342 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( ( substmonoid = $true )
<=> ( ! [X246: subst] :
( ( comp @ id @ X246 )
= X246 )
& ! [X245: subst] :
( ( comp @ X245 @ id )
= X245 ) ) )
& ( ( ! [X315: term,X316: subst > term > subst > $o] :
( ? [X328: subst,X329: subst,X330: subst,X331: term] :
( ( ( X316 @ X328 @ X331 @ ( comp @ X329 @ X330 ) )
!= $true )
& ( ( X316 @ ( comp @ X328 @ X329 ) @ ( sub @ X331 @ X329 ) @ X330 )
= $true ) )
| ? [X319: subst > term > term] :
( ! [X323: term] :
( ( ( X316 @ id @ ( X319 @ id @ X323 ) @ id )
= $true )
| ( ( X316 @ id @ X323 @ id )
!= $true ) )
& ! [X320: term,X321: subst,X322: subst] :
( ( sub @ ( X319 @ X321 @ X320 ) @ X322 )
= ( X319 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) ) )
& ( $true
!= ( X316 @ id @ ( lam @ ( X319 @ sh @ one ) ) @ id ) ) )
| ? [X324: subst,X325: term,X326: subst,X327: subst] :
( ( $true
= ( X316 @ X324 @ X325 @ ( comp @ X327 @ X326 ) ) )
& ( ( X316 @ ( comp @ X324 @ X327 ) @ ( sub @ X325 @ X327 ) @ X326 )
!= $true ) )
| ( ( X316 @ id @ X315 @ id )
= $true )
| ? [X332: term] :
( ( ( var @ ( sub @ X332 @ id ) )
= $true )
& ( ( X316 @ id @ X332 @ id )
!= $true ) )
| ? [X317: term,X318: term] :
( ( ( X316 @ id @ X318 @ id )
= $true )
& ( ( X316 @ id @ X317 @ id )
= $true )
& ( $true
!= ( X316 @ id @ ( ap @ ( sub @ X318 @ id ) @ X317 ) @ id ) ) ) )
| ? [X299: term,X300: subst > term > subst > $o] :
( ! [X302: subst,X303: subst,X304: term,X305: subst] :
( ( ( X300 @ ( comp @ X305 @ X302 ) @ ( sub @ X304 @ X302 ) @ X303 )
= $true )
| ( ( X300 @ X305 @ X304 @ ( comp @ X302 @ X303 ) )
!= $true ) )
& ( ( X300 @ id @ ( lam @ X299 ) @ id )
!= $true )
& ! [X301: term] :
( ( $true
!= ( X300 @ id @ X301 @ id ) )
| ( ( X300 @ id @ ( sub @ X299 @ ( push @ X301 @ id ) ) @ id )
= $true ) )
& ! [X310: subst > term > term] :
( ? [X312: term,X313: subst,X314: subst] :
( ( X310 @ ( comp @ X314 @ X313 ) @ ( sub @ X312 @ X313 ) )
!= ( sub @ ( X310 @ X314 @ X312 ) @ X313 ) )
| ( ( X300 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id )
= $true )
| ? [X311: term] :
( ( ( X300 @ id @ ( X310 @ id @ X311 ) @ id )
!= $true )
& ( $true
= ( X300 @ id @ X311 @ id ) ) ) )
& ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( ( X300 @ ( comp @ X307 @ X308 ) @ ( sub @ X306 @ X308 ) @ X309 )
!= $true )
| ( $true
= ( X300 @ X307 @ X306 @ ( comp @ X308 @ X309 ) ) ) ) )
| ? [X286: term,X287: subst > term > subst > $o,X288: term] :
( ! [X295: subst,X296: subst,X297: term,X298: subst] :
( ( $true
= ( X287 @ X295 @ X297 @ ( comp @ X296 @ X298 ) ) )
| ( ( X287 @ ( comp @ X295 @ X296 ) @ ( sub @ X297 @ X296 ) @ X298 )
!= $true ) )
& ( $true
= ( X287 @ id @ X286 @ id ) )
& ( ( X287 @ id @ X288 @ id )
= $true )
& ! [X289: subst,X290: subst,X291: term,X292: subst] :
( ( ( X287 @ ( comp @ X289 @ X290 ) @ ( sub @ X291 @ X290 ) @ X292 )
= $true )
| ( $true
!= ( X287 @ X289 @ X291 @ ( comp @ X290 @ X292 ) ) ) )
& ( $true
!= ( X287 @ id @ ( ap @ X286 @ X288 ) @ id ) )
& ! [X293: term,X294: term] :
( ( ( X287 @ id @ X294 @ id )
!= $true )
| ( ( X287 @ id @ ( ap @ ( sub @ X293 @ id ) @ X294 ) @ id )
= $true )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) ) )
| ? [X279: term,X280: term > $o] :
( ( ( X280 @ X279 )
!= $true )
& ! [X282: term] :
( ? [X283: term] :
( ( $true
!= ( X280 @ ( sub @ X282 @ ( push @ X283 @ id ) ) ) )
& ( ( X280 @ X283 )
= $true ) )
| ( ( X280 @ ( lam @ X282 ) )
= $true ) )
& ! [X284: term,X285: term] :
( ( $true
!= ( X280 @ X285 ) )
| ( ( X280 @ ( ap @ X285 @ X284 ) )
= $true )
| ( $true
!= ( X280 @ X284 ) ) )
& ! [X281: term] :
( ( ( X280 @ X281 )
= $true )
| ( $true
!= ( var @ X281 ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( ! [X10: subst > term > subst > $o] :
? [X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( hoasapinj1_gthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ( termmset = $true )
<=> ! [X247: term] :
( ( sub @ X247 @ id )
= X247 ) )
& ( ! [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
!= ( ap @ X276 @ X277 ) )
| ( X276 = X278 ) )
<=> ( apinj1 = $true ) )
& ( ( ! [X80: term,X81: subst > term > subst > $o,X82: term > $o] :
( ( ( X82 @ ( lam @ X80 ) )
= $true )
| ? [X89: subst,X90: subst,X91: subst,X92: term] :
( ( $true
= ( X81 @ X90 @ X92 @ ( comp @ X91 @ X89 ) ) )
& ( ( X81 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) @ X89 )
!= $true ) )
| ? [X97: term] :
( ( ( X82 @ ( sub @ X80 @ ( push @ X97 @ id ) ) )
!= $true )
& ( ( X82 @ X97 )
= $true ) )
| ? [X83: term] :
( ( X82 @ X83 )
!= ( X81 @ id @ X83 @ id ) )
| ? [X93: term,X94: subst,X95: subst,X96: subst] :
( ( $true
!= ( X81 @ X94 @ X93 @ ( comp @ X95 @ X96 ) ) )
& ( ( X81 @ ( comp @ X94 @ X95 ) @ ( sub @ X93 @ X95 ) @ X96 )
= $true ) )
| ? [X84: subst > term > term] :
( ! [X86: term,X87: subst,X88: subst] :
( ( sub @ ( X84 @ X88 @ X86 ) @ X87 )
= ( X84 @ ( comp @ X88 @ X87 ) @ ( sub @ X86 @ X87 ) ) )
& ! [X85: term] :
( ( ( X81 @ id @ X85 @ id )
!= $true )
| ( ( X81 @ id @ ( X84 @ id @ X85 ) @ id )
= $true ) )
& ( ( X81 @ id @ ( lam @ ( X84 @ sh @ one ) ) @ id )
!= $true ) ) )
| ? [X79: term] :
( ( sub @ X79 @ id )
!= X79 ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( pushprop_lthm_orig = $true )
& ( ( substmonoid_lthm = $true )
<=> ( ( ! [X538: subst] :
( ( comp @ X538 @ id )
= X538 )
& ! [X539: subst] :
( ( comp @ id @ X539 )
= X539 ) )
| ? [X537: subst] :
( ( comp @ X537 @ id )
!= X537 )
| ? [X536: subst] :
( ( comp @ id @ X536 )
!= X536 ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( substmonoid_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( laminj = $true )
<=> ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) ) )
& ( ! [X369: term > $o,X370: subst > term > subst > $o,X371: term,X372: term] :
( ? [X382: term,X383: term] :
( ( ( X370 @ id @ ( ap @ ( sub @ X383 @ id ) @ X382 ) @ id )
!= $true )
& ( ( X370 @ id @ X383 @ id )
= $true )
& ( ( X370 @ id @ X382 @ id )
= $true ) )
| ( $true
!= ( X369 @ X371 ) )
| ( ( X369 @ X372 )
!= $true )
| ( ( X369 @ ( ap @ X372 @ X371 ) )
= $true )
| ? [X373: subst,X374: subst,X375: term,X376: subst] :
( ( ( X370 @ ( comp @ X374 @ X373 ) @ ( sub @ X375 @ X373 ) @ X376 )
!= $true )
& ( ( X370 @ X374 @ X375 @ ( comp @ X373 @ X376 ) )
= $true ) )
| ? [X378: term,X379: subst,X380: subst,X381: subst] :
( ( $true
!= ( X370 @ X380 @ X378 @ ( comp @ X381 @ X379 ) ) )
& ( ( X370 @ ( comp @ X380 @ X381 ) @ ( sub @ X378 @ X381 ) @ X379 )
= $true ) )
| ? [X377: term] :
( ( X369 @ X377 )
!= ( X370 @ id @ X377 @ id ) ) )
<=> ( hoasinduction_lem2v2 = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X451: subst > term > subst > $o,X452: term,X453: term] :
( ? [X460: subst,X461: subst,X462: term,X463: subst] :
( ( ( X451 @ X460 @ X462 @ ( comp @ X463 @ X461 ) )
!= $true )
& ( ( X451 @ ( comp @ X460 @ X463 ) @ ( sub @ X462 @ X463 ) @ X461 )
= $true ) )
| ( ( X451 @ id @ X453 @ id )
!= $true )
| ? [X458: term,X459: term] :
( ( $true
!= ( X451 @ id @ ( ap @ ( sub @ X458 @ id ) @ X459 ) @ id ) )
& ( $true
= ( X451 @ id @ X459 @ id ) )
& ( ( X451 @ id @ X458 @ id )
= $true ) )
| ? [X454: subst,X455: subst,X456: term,X457: subst] :
( ( ( X451 @ X457 @ X456 @ ( comp @ X455 @ X454 ) )
= $true )
& ( $true
!= ( X451 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 ) ) )
| ( ( X451 @ id @ ( ap @ X453 @ X452 ) @ id )
= $true )
| ( ( X451 @ id @ X452 @ id )
!= $true ) ) )
& ( ! [X121: term,X122: subst > term > subst > $o] :
( ? [X123: term] :
( ( ( X122 @ id @ X123 @ id )
= $true )
& ( ( X122 @ id @ ( sub @ X121 @ ( push @ X123 @ id ) ) @ id )
!= $true ) )
| ( $true
= ( X122 @ id @ ( lam @ ( sub @ X121 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X124: term,X125: subst > term > term] :
( ! [X126: subst,X127: term,X128: subst] :
( ( sub @ ( sub @ X124 @ ( push @ X127 @ X128 ) ) @ X126 )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
& ( ( X122 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id )
!= $true )
& ! [X129: term] :
( ( ( X122 @ id @ X129 @ id )
!= $true )
| ( ( X122 @ id @ ( X125 @ id @ X129 ) @ id )
= $true ) )
& ! [X130: term,X131: subst,X132: subst] :
( ( sub @ X124 @ ( push @ ( sub @ X130 @ X132 ) @ ( comp @ X131 @ X132 ) ) )
= ( X125 @ ( comp @ X131 @ X132 ) @ ( sub @ X130 @ X132 ) ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ! [X467: term] :
? [X468: subst > term > term] :
( ( X468 @ sh @ one )
= ( sub @ X467 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem3v2 = $true )
& ( ( induction2lem_lthm = $true )
<=> ( ! [X361: term,X362: term > $o,X363: subst] :
( ? [X365: term] :
( ! [X366: term] :
( ( $true
= ( X362 @ ( sub @ X365 @ ( push @ X366 @ id ) ) ) )
| ( ( X362 @ X366 )
!= $true ) )
& ( $true
!= ( X362 @ ( lam @ X365 ) ) ) )
| ? [X364: term] :
( ( ( var @ X364 )
= $true )
& ( $true
!= ( X362 @ ( sub @ X364 @ X363 ) ) ) )
| ? [X367: term,X368: term] :
( ( ( X362 @ X367 )
= $true )
& ( ( X362 @ X368 )
= $true )
& ( $true
!= ( X362 @ ( ap @ X368 @ X367 ) ) ) )
| ( ( X362 @ ( sub @ X361 @ X363 ) )
= $true ) )
| ? [X355: term,X356: term > $o] :
( ( ( X356 @ X355 )
!= $true )
& ! [X358: term] :
( ( $true
= ( X356 @ ( lam @ X358 ) ) )
| ( ( X356 @ X358 )
!= $true ) )
& ! [X357: term] :
( ( ( var @ X357 )
!= $true )
| ( $true
= ( X356 @ X357 ) ) )
& ! [X359: term,X360: term] :
( ( $true
!= ( X356 @ X360 ) )
| ( $true
= ( X356 @ ( ap @ X360 @ X359 ) ) )
| ( $true
!= ( X356 @ X359 ) ) ) )
| ? [X354: subst] :
( ( comp @ X354 @ id )
!= X354 )
| ? [X352: term,X353: subst] :
( ( comp @ sh @ ( push @ X352 @ X353 ) )
!= X353 )
| ? [X350: subst,X351: term] :
( ( sub @ one @ ( push @ X351 @ X350 ) )
!= X351 ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X178: term] :
? [X179: subst > term > term] :
! [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
= ( X179 @ X180 @ X181 ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X116: term,X117: subst,X118: term > $o] :
? [X119: term > $o] :
! [X120: term] :
( ( X119 @ X120 )
= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X384: term] :
? [X385: subst > term > term] :
! [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
= ( X385 @ X386 @ X387 ) ) )
& ( ! [X42: term > $o,X43: term] :
( ? [X44: term] :
( ( ( var @ X44 )
= $true )
& ( ( X42 @ X44 )
!= $true ) )
| ? [X47: term,X48: term] :
( ( $true
!= ( X42 @ ( ap @ X47 @ X48 ) ) )
& ( $true
= ( X42 @ X47 ) )
& ( ( X42 @ X48 )
= $true ) )
| ? [X45: term] :
( ( ( X42 @ ( lam @ X45 ) )
!= $true )
& ! [X46: term] :
( ( $true
!= ( X42 @ X46 ) )
| ( $true
= ( X42 @ ( sub @ X45 @ ( push @ X46 @ id ) ) ) ) ) )
| ( ( X42 @ X43 )
= $true ) )
<=> ( induction2 = $true ) )
& ( ulamvarind = $true )
& ( termmset_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ? [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
!= X168 )
| ! [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ? [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
!= ( X170 @ X173 ) )
| ( ( X170 @ one )
= $true )
| ( ( X171 @ X172 )
!= $true ) ) ) )
& ( ( ! [X257: subst > term > subst > $o,X258: term] :
( ( ( X257 @ id @ ( lam @ X258 ) @ id )
= $true )
| ? [X259: term] :
( ( ( X257 @ id @ X259 @ id )
= $true )
& ( ( X257 @ id @ ( sub @ X258 @ ( push @ X259 @ id ) ) @ id )
!= $true ) )
| ? [X260: subst > term > term] :
( ! [X264: term] :
( ( ( X257 @ id @ ( X260 @ id @ X264 ) @ id )
= $true )
| ( ( X257 @ id @ X264 @ id )
!= $true ) )
& ! [X261: term,X262: subst,X263: subst] :
( ( X260 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) )
= ( sub @ ( X260 @ X262 @ X261 ) @ X263 ) )
& ( ( X257 @ id @ ( lam @ ( X260 @ sh @ one ) ) @ id )
!= $true ) ) )
| ? [X249: subst > term > subst > $o,X250: term] :
( ! [X256: term] :
( ( ( X249 @ id @ ( sub @ X250 @ ( push @ X256 @ id ) ) @ id )
= $true )
| ( ( X249 @ id @ X256 @ id )
!= $true ) )
& ( ( X249 @ id @ ( lam @ ( sub @ X250 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X251: subst > term > term] :
( ? [X255: term] :
( ( ( X249 @ id @ ( X251 @ id @ X255 ) @ id )
!= $true )
& ( ( X249 @ id @ X255 @ id )
= $true ) )
| ( $true
= ( X249 @ id @ ( lam @ ( X251 @ sh @ one ) ) @ id ) )
| ? [X252: subst,X253: term,X254: subst] :
( ( sub @ ( X251 @ X252 @ X253 ) @ X254 )
!= ( X251 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) ) ) ) )
| ? [X248: term] :
( ( sub @ X248 @ id )
!= X248 ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X182: subst > term > subst > $o,X183: term] :
( ? [X185: subst > term > term] :
( ! [X186: term] :
( ( $true
!= ( X182 @ id @ X186 @ id ) )
| ( ( X182 @ id @ ( X185 @ id @ X186 ) @ id )
= $true ) )
& ! [X187: term,X188: subst,X189: subst] :
( ( X185 @ ( comp @ X189 @ X188 ) @ ( sub @ X187 @ X188 ) )
= ( sub @ ( X185 @ X189 @ X187 ) @ X188 ) )
& ( ( X182 @ id @ ( lam @ ( X185 @ sh @ one ) ) @ id )
!= $true ) )
| ( ( X182 @ id @ ( lam @ ( sub @ X183 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ? [X184: term] :
( ( ( X182 @ id @ X184 @ id )
= $true )
& ( ( X182 @ id @ ( sub @ X183 @ ( push @ X184 @ id ) ) @ id )
!= $true ) ) ) )
& ( ! [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
= X240 )
<=> ( axshiftcons = $true ) )
& ( ! [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ( ( X474 @ X475 )
!= $true )
| ? [X476: term] :
( ( X472 @ X476 )
!= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
| ( ( X472 @ one )
= $true ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( induction2lem_gthm = $true )
& ! [X177: d_subst] : ( d_id = X177 )
& ( lamnotap != $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ! [X469: d_term,X470: d_term] :
( ( ( d2term @ X469 )
!= ( d2term @ X470 ) )
| ( X469 = X470 ) )
& ( ( induction = $true )
<=> ! [X403: term > $o,X404: term] :
( ? [X405: term,X406: term] :
( ( ( X403 @ X405 )
= $true )
& ( ( X403 @ ( ap @ X405 @ X406 ) )
!= $true )
& ( $true
= ( X403 @ X406 ) ) )
| ? [X407: term] :
( ( $true
= ( var @ X407 ) )
& ( $true
!= ( X403 @ X407 ) ) )
| ( ( X403 @ X404 )
= $true )
| ? [X408: term] :
( ( ( X403 @ ( lam @ X408 ) )
!= $true )
& ( ( X403 @ X408 )
= $true ) ) ) )
& ( pushprop_lem0_gthm = $true )
& ( pushprop_lthm = $true )
& ( ! [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
!= ( ap @ X243 @ X241 ) )
| ( X241 = X242 ) )
<=> ( apinj2 = $true ) )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasinduction = $true )
<=> ! [X493: term,X494: subst > term > subst > $o] :
( ( $true
= ( X494 @ id @ X493 @ id ) )
| ? [X499: term,X500: term] :
( ( ( X494 @ id @ X500 @ id )
= $true )
& ( ( X494 @ id @ ( ap @ ( sub @ X500 @ id ) @ X499 ) @ id )
!= $true )
& ( ( X494 @ id @ X499 @ id )
= $true ) )
| ? [X507: subst,X508: subst,X509: term,X510: subst] :
( ( ( X494 @ X510 @ X509 @ ( comp @ X508 @ X507 ) )
= $true )
& ( $true
!= ( X494 @ ( comp @ X510 @ X508 ) @ ( sub @ X509 @ X508 ) @ X507 ) ) )
| ? [X501: subst > term > term] :
( ( $true
!= ( X494 @ id @ ( lam @ ( X501 @ sh @ one ) ) @ id ) )
& ! [X505: term] :
( ( ( X494 @ id @ X505 @ id )
!= $true )
| ( ( X494 @ id @ ( X501 @ id @ X505 ) @ id )
= $true ) )
& ! [X502: subst,X503: term,X504: subst] :
( ( X501 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X501 @ X502 @ X503 ) @ X504 ) ) )
| ? [X495: term,X496: subst,X497: subst,X498: subst] :
( ( $true
= ( X494 @ ( comp @ X498 @ X497 ) @ ( sub @ X495 @ X497 ) @ X496 ) )
& ( ( X494 @ X498 @ X495 @ ( comp @ X497 @ X496 ) )
!= $true ) )
| ? [X506: term] :
( ( ( var @ ( sub @ X506 @ id ) )
= $true )
& ( $true
!= ( X494 @ id @ X506 @ id ) ) ) ) )
& ( ( axidl = $true )
<=> ! [X471: subst] :
( ( comp @ id @ X471 )
= X471 ) )
& ( ! [X17: term,X18: subst > term > subst > $o,X19: term > $o] :
( ? [X22: subst > term > term] :
( ! [X26: term] :
( ( $true
!= ( X18 @ id @ X26 @ id ) )
| ( ( X18 @ id @ ( X22 @ id @ X26 ) @ id )
= $true ) )
& ! [X23: subst,X24: subst,X25: term] :
( ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( X22 @ X24 @ X25 ) @ X23 ) )
& ( $true
!= ( X18 @ id @ ( lam @ ( X22 @ sh @ one ) ) @ id ) ) )
| ? [X21: term] :
( ( X19 @ X21 )
!= ( X18 @ id @ X21 @ id ) )
| ? [X20: term] :
( ( $true
= ( X19 @ X20 ) )
& ( ( X19 @ ( sub @ X17 @ ( push @ X20 @ id ) ) )
!= $true ) )
| ( ( X19 @ ( lam @ X17 ) )
= $true )
| ? [X27: subst,X28: term,X29: subst,X30: subst] :
( ( $true
= ( X18 @ X27 @ X28 @ ( comp @ X29 @ X30 ) ) )
& ( ( X18 @ ( comp @ X27 @ X29 ) @ ( sub @ X28 @ X29 ) @ X30 )
!= $true ) )
| ? [X31: subst,X32: subst,X33: subst,X34: term] :
( ( $true
= ( X18 @ ( comp @ X33 @ X32 ) @ ( sub @ X34 @ X32 ) @ X31 ) )
& ( ( X18 @ X33 @ X34 @ ( comp @ X32 @ X31 ) )
!= $true ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoaslamnotap_lthm = $true )
& ( ! [X174: term] :
( ( sub @ X174 @ id )
= X174 )
<=> ( axvarid = $true ) )
& ( hoasinduction_lem1 = $true )
& ( one
= ( d2term @ d_one ) )
& ( ( ! [X150: term] :
( ( sub @ X150 @ id )
= X150 )
| ? [X149: term] :
( ( sub @ X149 @ id )
!= X149 ) )
<=> ( termmset_lthm = $true ) )
& ( ( ! [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) )
| ( $true
!= ( X523 @ X527 ) ) )
| ? [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
!= X521 )
| ? [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
!= X520 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ! [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
= X176 )
<=> ( axvarcons = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X466: subst] :
( ( comp @ X466 @ id )
= X466 )
<=> ( axidr = $true ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X8: term] :
? [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
= ( X9 @ sh @ one ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X532: term,X533: term,X534: term,X535: term] :
( ( X532 = X534 )
| ( ( ap @ ( sub @ X533 @ id ) @ X532 )
!= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) )
| ? [X528: term,X529: term,X530: term,X531: term] :
( ( ( ap @ X531 @ X529 )
= ( ap @ X528 @ X530 ) )
& ( X529 != X530 ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( ( shinj = $true )
<=> ! [X35: term,X36: term] :
( ( X35 = X36 )
| ( ( sub @ X36 @ sh )
!= ( sub @ X35 @ sh ) ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X108: term,X109: term,X110: term,X111: term] :
( ( X108 = X109 )
| ( ( ap @ ( sub @ X109 @ id ) @ X111 )
!= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) )
| ? [X104: term,X105: term,X106: term,X107: term] :
( ( X106 != X107 )
& ( ( ap @ X106 @ X105 )
= ( ap @ X107 @ X104 ) ) )
| ? [X103: term] :
( ( sub @ X103 @ id )
!= X103 ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ! [X511: term,X512: subst > term > subst > $o] :
( ? [X518: term] :
( ( ( X512 @ id @ ( sub @ X511 @ ( push @ X518 @ id ) ) @ id )
!= $true )
& ( ( X512 @ id @ X518 @ id )
= $true ) )
| ( ( X512 @ id @ ( lam @ X511 ) @ id )
= $true )
| ? [X513: subst > term > term] :
( ( ( X512 @ id @ ( lam @ ( X513 @ sh @ one ) ) @ id )
!= $true )
& ! [X514: subst,X515: term,X516: subst] :
( ( X513 @ ( comp @ X514 @ X516 ) @ ( sub @ X515 @ X516 ) )
= ( sub @ ( X513 @ X514 @ X515 ) @ X516 ) )
& ! [X517: term] :
( ( $true
= ( X512 @ id @ ( X513 @ id @ X517 ) @ id ) )
| ( ( X512 @ id @ X517 @ id )
!= $true ) ) ) )
<=> ( hoasinduction_lem3a = $true ) ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( ( ( hoasapinj1 = $true )
<=> ! [X13: term,X14: term,X15: term,X16: term] :
( ( X13 = X16 )
| ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) ) )
& ( ( ! [X49: subst > term > subst > $o] :
~ ! [X50: term > $o] :
~ ! [X51: term] :
( ( X49 @ id @ X51 @ id )
= ( X50 @ X51 ) )
=> ( ! [X52: term > $o,X53: term] :
( ~ ! [X57: term] :
( ~ ! [X58: term] :
( ( $true
!= ( X52 @ X58 ) )
| ( ( X52 @ ( sub @ X57 @ ( push @ X58 @ id ) ) )
= $true ) )
| ( ( X52 @ ( lam @ X57 ) )
= $true ) )
| ~ ! [X54: term,X55: term] :
( ( $true
!= ( X52 @ X54 ) )
| ( ( X52 @ ( ap @ X55 @ X54 ) )
= $true )
| ( ( X52 @ X55 )
!= $true ) )
| ( ( X52 @ X53 )
= $true )
| ~ ! [X56: term] :
( ( ( var @ X56 )
!= $true )
| ( $true
= ( X52 @ X56 ) ) ) )
=> ( ! [X59: term] :
( ( sub @ X59 @ id )
= X59 )
=> ( ! [X60: subst > term > subst > $o,X61: term > $o,X62: term] :
( ~ ! [X69: term] :
( ( X60 @ id @ X69 @ id )
= ( X61 @ X69 ) )
| ~ ! [X63: subst > term > term] :
( ( $true
= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) )
| ~ ! [X64: term] :
( ( ( X60 @ id @ ( X63 @ id @ X64 ) @ id )
= $true )
| ( ( X60 @ id @ X64 @ id )
!= $true ) )
| ~ ! [X65: subst,X66: subst,X67: term] :
( ( sub @ ( X63 @ X66 @ X67 ) @ X65 )
= ( X63 @ ( comp @ X66 @ X65 ) @ ( sub @ X67 @ X65 ) ) ) )
| ( ( X61 @ ( lam @ X62 ) )
= $true )
| ~ ! [X68: term] :
( ( $true
= ( X61 @ ( sub @ X62 @ ( push @ X68 @ id ) ) ) )
| ( ( X61 @ X68 )
!= $true ) ) )
=> ! [X70: subst > term > subst > $o,X71: term] :
( ~ ! [X72: subst > term > term] :
( ( ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X73: subst,X74: term,X75: subst] :
( ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) )
= ( sub @ ( X72 @ X75 @ X74 ) @ X73 ) )
| ~ ! [X76: term] :
( ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true )
| ( $true
!= ( X70 @ id @ X76 @ id ) ) ) )
| ~ ! [X77: term,X78: term] :
( ( $true
!= ( X70 @ id @ X77 @ id ) )
| ( ( X70 @ id @ X78 @ id )
!= $true )
| ( ( X70 @ id @ ( ap @ ( sub @ X78 @ id ) @ X77 ) @ id )
= $true ) )
| ( $true
= ( X70 @ id @ X71 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X333: subst > term > subst > $o] :
~ ! [X334: term > $o] :
~ ! [X335: term] :
( ( X334 @ X335 )
= ( X333 @ id @ X335 @ id ) ) )
& ( axclos = $true )
& ( ! [X477: term,X478: subst > term > subst > $o] :
( ( ( X478 @ id @ ( lam @ X477 ) @ id )
= $true )
| ~ ! [X484: subst > term > term] :
( ( $true
= ( X478 @ id @ ( lam @ ( X484 @ sh @ one ) ) @ id ) )
| ~ ! [X485: subst,X486: subst,X487: term] :
( ( sub @ ( X484 @ X486 @ X487 ) @ X485 )
= ( X484 @ ( comp @ X486 @ X485 ) @ ( sub @ X487 @ X485 ) ) )
| ~ ! [X488: term] :
( ( ( X478 @ id @ ( X484 @ id @ X488 ) @ id )
= $true )
| ( $true
!= ( X478 @ id @ X488 @ id ) ) ) )
| ~ ! [X483: term] :
( ( ( X478 @ id @ X483 @ id )
!= $true )
| ( $true
= ( X478 @ id @ ( sub @ X477 @ ( push @ X483 @ id ) ) @ id ) ) )
| ~ ! [X489: subst,X490: subst,X491: term,X492: subst] :
( ( ( X478 @ X492 @ X491 @ ( comp @ X490 @ X489 ) )
= $true )
| ( ( X478 @ ( comp @ X492 @ X490 ) @ ( sub @ X491 @ X490 ) @ X489 )
!= $true ) )
| ~ ! [X479: subst,X480: subst,X481: subst,X482: term] :
( ( ( X478 @ X481 @ X482 @ ( comp @ X479 @ X480 ) )
!= $true )
| ( ( X478 @ ( comp @ X481 @ X479 ) @ ( sub @ X482 @ X479 ) @ X480 )
= $true ) ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X138: subst > term > subst > $o,X139: term] :
( ~ ! [X145: term,X146: term] :
( ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
= $true )
| ( ( X138 @ id @ X146 @ id )
!= $true )
| ( $true
!= ( X138 @ id @ X145 @ id ) ) )
| ( ( X138 @ id @ X139 @ id )
= $true )
| ~ ! [X140: subst > term > term] :
( ( ( X138 @ id @ ( lam @ ( X140 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X142: subst,X143: term,X144: subst] :
( ( X140 @ ( comp @ X142 @ X144 ) @ ( sub @ X143 @ X144 ) )
= ( sub @ ( X140 @ X142 @ X143 ) @ X144 ) )
| ~ ! [X141: term] :
( ( ( X138 @ id @ X141 @ id )
!= $true )
| ( $true
= ( X138 @ id @ ( X140 @ id @ X141 ) @ id ) ) ) ) ) )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X409: term] :
( ( sub @ X409 @ id )
= X409 )
=> ( ! [X410: term,X411: subst > term > subst > $o] :
( ~ ! [X412: subst > term > term] :
( ~ ! [X413: term] :
( ( ( X411 @ id @ ( X412 @ id @ X413 ) @ id )
= $true )
| ( $true
!= ( X411 @ id @ X413 @ id ) ) )
| ( ( X411 @ id @ ( lam @ ( X412 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X414: subst,X415: subst,X416: term] :
( ( X412 @ ( comp @ X414 @ X415 ) @ ( sub @ X416 @ X415 ) )
= ( sub @ ( X412 @ X414 @ X416 ) @ X415 ) ) )
| ~ ! [X417: term] :
( ( $true
= ( X411 @ id @ ( sub @ X410 @ ( push @ X417 @ id ) ) @ id ) )
| ( ( X411 @ id @ X417 @ id )
!= $true ) )
| ( ( X411 @ id @ ( lam @ ( sub @ X410 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X418: term,X419: subst > term > subst > $o] :
( ~ ! [X420: subst,X421: subst,X422: term,X423: subst] :
( ( ( X419 @ ( comp @ X423 @ X421 ) @ ( sub @ X422 @ X421 ) @ X420 )
!= $true )
| ( ( X419 @ X423 @ X422 @ ( comp @ X421 @ X420 ) )
= $true ) )
| ( $true
= ( X419 @ id @ ( lam @ X418 ) @ id ) )
| ~ ! [X433: term] :
( ( $true
!= ( X419 @ id @ X433 @ id ) )
| ( ( X419 @ id @ ( sub @ X418 @ ( push @ X433 @ id ) ) @ id )
= $true ) )
| ~ ! [X428: subst > term > term] :
( ~ ! [X432: term] :
( ( $true
= ( X419 @ id @ ( X428 @ id @ X432 ) @ id ) )
| ( ( X419 @ id @ X432 @ id )
!= $true ) )
| ( $true
= ( X419 @ id @ ( lam @ ( X428 @ sh @ one ) ) @ id ) )
| ~ ! [X429: term,X430: subst,X431: subst] :
( ( sub @ ( X428 @ X431 @ X429 ) @ X430 )
= ( X428 @ ( comp @ X431 @ X430 ) @ ( sub @ X429 @ X430 ) ) ) )
| ~ ! [X424: subst,X425: subst,X426: subst,X427: term] :
( ( ( X419 @ X426 @ X427 @ ( comp @ X425 @ X424 ) )
!= $true )
| ( $true
= ( X419 @ ( comp @ X426 @ X425 ) @ ( sub @ X427 @ X425 ) @ X424 ) ) ) ) ) ) )
& ( hoaslaminj_gthm = $true )
& ! [X112: d_subst,X113: d_subst] :
( ( ( d2subst @ X112 )
= ( d2subst @ X113 ) )
=> ( X112 = X113 ) )
& ! [X114: term] :
? [X115: d_term] :
( ( d2term @ X115 )
= X114 )
& ( hoasapnotvar = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoaslaminj = $true )
<=> ! [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ~ ! [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
| ( ( X191 @ X192 @ X193 )
= ( X190 @ X192 @ X193 ) )
| ~ ! [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) )
| ( ( lam @ ( X191 @ sh @ one ) )
!= ( lam @ ( X190 @ sh @ one ) ) ) ) )
& ( ( d2subst @ d_id )
= id )
& ( ( ! [X200: subst > term > subst > $o] :
~ ! [X201: term > $o] :
~ ! [X202: term] :
( ( X201 @ X202 )
= ( X200 @ id @ X202 @ id ) )
=> ( ! [X203: term,X204: term > $o] :
( ~ ! [X207: term] :
( ~ ! [X208: term] :
( ( $true
!= ( X204 @ X208 ) )
| ( $true
= ( X204 @ ( sub @ X207 @ ( push @ X208 @ id ) ) ) ) )
| ( $true
= ( X204 @ ( lam @ X207 ) ) ) )
| ~ ! [X209: term] :
( ( ( X204 @ X209 )
= $true )
| ( ( var @ X209 )
!= $true ) )
| ( $true
= ( X204 @ X203 ) )
| ~ ! [X205: term,X206: term] :
( ( $true
!= ( X204 @ X206 ) )
| ( ( X204 @ X205 )
!= $true )
| ( ( X204 @ ( ap @ X206 @ X205 ) )
= $true ) ) )
=> ( ! [X210: term] :
( ( sub @ X210 @ id )
= X210 )
=> ( ! [X211: term > $o,X212: term,X213: subst > term > subst > $o] :
( ( $true
= ( X211 @ ( lam @ X212 ) ) )
| ~ ! [X216: subst > term > term] :
( ( ( X213 @ id @ ( lam @ ( X216 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X217: term,X218: subst,X219: subst] :
( ( sub @ ( X216 @ X219 @ X217 ) @ X218 )
= ( X216 @ ( comp @ X219 @ X218 ) @ ( sub @ X217 @ X218 ) ) )
| ~ ! [X220: term] :
( ( ( X213 @ id @ X220 @ id )
!= $true )
| ( ( X213 @ id @ ( X216 @ id @ X220 ) @ id )
= $true ) ) )
| ~ ! [X214: term] :
( ( X213 @ id @ X214 @ id )
= ( X211 @ X214 ) )
| ~ ! [X215: term] :
( ( $true
!= ( X211 @ X215 ) )
| ( $true
= ( X211 @ ( sub @ X212 @ ( push @ X215 @ id ) ) ) ) ) )
=> ! [X221: term,X222: subst > term > subst > $o] :
( ~ ! [X229: term] :
( ( ( var @ ( sub @ X229 @ id ) )
!= $true )
| ( $true
= ( X222 @ id @ X229 @ id ) ) )
| ~ ! [X234: subst > term > term] :
( ~ ! [X236: term,X237: subst,X238: subst] :
( ( X234 @ ( comp @ X238 @ X237 ) @ ( sub @ X236 @ X237 ) )
= ( sub @ ( X234 @ X238 @ X236 ) @ X237 ) )
| ~ ! [X235: term] :
( ( $true
= ( X222 @ id @ ( X234 @ id @ X235 ) @ id ) )
| ( ( X222 @ id @ X235 @ id )
!= $true ) )
| ( ( X222 @ id @ ( lam @ ( X234 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X222 @ id @ X221 @ id )
= $true )
| ~ ! [X230: subst,X231: subst,X232: term,X233: subst] :
( ( ( X222 @ X233 @ X232 @ ( comp @ X230 @ X231 ) )
= $true )
| ( $true
!= ( X222 @ ( comp @ X233 @ X230 ) @ ( sub @ X232 @ X230 ) @ X231 ) ) )
| ~ ! [X223: subst,X224: term,X225: subst,X226: subst] :
( ( ( X222 @ ( comp @ X225 @ X226 ) @ ( sub @ X224 @ X226 ) @ X223 )
= $true )
| ( $true
!= ( X222 @ X225 @ X224 @ ( comp @ X226 @ X223 ) ) ) )
| ~ ! [X227: term,X228: term] :
( ( ( X222 @ id @ X227 @ id )
!= $true )
| ( ( X222 @ id @ X228 @ id )
!= $true )
| ( ( X222 @ id @ ( ap @ ( sub @ X228 @ id ) @ X227 ) @ id )
= $true ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( hoaslamnotvar = $true )
& ( ulamvar1 != $true )
& ( ! [X265: term,X266: subst > term > subst > $o,X267: term > $o] :
( ~ ! [X269: term] :
( ( X267 @ X269 )
= ( X266 @ id @ X269 @ id ) )
| ~ ! [X268: term] :
( ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ id ) ) ) )
| ( ( X267 @ X268 )
!= $true ) )
| ( $true
= ( X267 @ ( lam @ X265 ) ) )
| ~ ! [X270: subst > term > term] :
( ( ( X266 @ id @ ( lam @ ( X270 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X272: subst,X273: subst,X274: term] :
( ( sub @ ( X270 @ X272 @ X274 ) @ X273 )
= ( X270 @ ( comp @ X272 @ X273 ) @ ( sub @ X274 @ X273 ) ) )
| ~ ! [X271: term] :
( ( ( X266 @ id @ X271 @ id )
!= $true )
| ( $true
= ( X266 @ id @ ( X270 @ id @ X271 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( axapp = $true )
& ( hoasapnotvar_gthm = $true )
& ( axvarshift = $true )
& ( ( ! [X151: term] :
( ( sub @ X151 @ id )
= X151 )
=> ( ! [X152: term > $o,X153: term,X154: subst] :
( ~ ! [X158: term] :
( ~ ! [X159: term] :
( ( ( X152 @ X159 )
!= $true )
| ( $true
= ( X152 @ ( sub @ X158 @ ( push @ X159 @ id ) ) ) ) )
| ( ( X152 @ ( lam @ X158 ) )
= $true ) )
| ~ ! [X155: term,X156: term] :
( ( $true
= ( X152 @ ( ap @ X155 @ X156 ) ) )
| ( ( X152 @ X156 )
!= $true )
| ( $true
!= ( X152 @ X155 ) ) )
| ~ ! [X157: term] :
( ( ( X152 @ ( sub @ X157 @ X154 ) )
= $true )
| ( ( var @ X157 )
!= $true ) )
| ( ( X152 @ ( sub @ X153 @ X154 ) )
= $true ) )
=> ! [X160: term > $o,X161: term] :
( ~ ! [X162: term,X163: term] :
( ( ( X160 @ X163 )
!= $true )
| ( $true
= ( X160 @ ( ap @ X163 @ X162 ) ) )
| ( $true
!= ( X160 @ X162 ) ) )
| ~ ! [X165: term] :
( ( $true
= ( X160 @ ( lam @ X165 ) ) )
| ~ ! [X166: term] :
( ( $true
!= ( X160 @ X166 ) )
| ( ( X160 @ ( sub @ X165 @ ( push @ X166 @ id ) ) )
= $true ) ) )
| ( ( X160 @ X161 )
= $true )
| ~ ! [X164: term] :
( ( ( var @ X164 )
!= $true )
| ( ( X160 @ X164 )
= $true ) ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( ! [X37: term > $o,X38: term,X39: subst] :
~ ! [X40: term > $o] :
~ ! [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
= ( X40 @ X41 ) )
<=> ( pushprop_lem0 = $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X336: term,X337: term,X338: subst > term > term] :
( ~ ! [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
| ( ( ap @ ( sub @ X336 @ id ) @ X337 )
!= ( lam @ ( X338 @ sh @ one ) ) ) ) )
& ( induction2_gthm = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( ! [X0: subst,X1: term > $o,X2: term] :
( ~ ! [X6: term] :
( ~ ! [X7: term] :
( ( $true
!= ( X1 @ X7 ) )
| ( $true
= ( X1 @ ( sub @ X6 @ ( push @ X7 @ id ) ) ) ) )
| ( ( X1 @ ( lam @ X6 ) )
= $true ) )
| ~ ! [X5: term] :
( ( ( var @ X5 )
!= $true )
| ( ( X1 @ ( sub @ X5 @ X0 ) )
= $true ) )
| ~ ! [X3: term,X4: term] :
( ( ( X1 @ X3 )
!= $true )
| ( ( X1 @ X4 )
!= $true )
| ( ( X1 @ ( ap @ X3 @ X4 ) )
= $true ) )
| ( ( X1 @ ( sub @ X2 @ X0 ) )
= $true ) )
<=> ( induction2lem = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( axassoc = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
= $true )
| ( $true
!= ( X137 @ X134 ) ) ) )
& ! [X450: d_term] : ( d_one = X450 )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( lamnotvar = $true )
& ( pushprop = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ulamvarsh = $true )
& ( pushprop_gthm = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X98: term,X99: term,X100: term,X101: term] :
( ( X100 = X101 )
| ( ( ap @ ( sub @ X98 @ id ) @ X100 )
!= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) ) )
& ( ( ! [X434: term,X435: subst] :
( ( sub @ one @ ( push @ X434 @ X435 ) )
= X434 )
=> ( ! [X436: term,X437: subst] :
( ( comp @ sh @ ( push @ X436 @ X437 ) )
= X437 )
=> ( ! [X438: term,X439: term] :
( ( ( lam @ X438 )
!= ( lam @ X439 ) )
| ( X438 = X439 ) )
=> ! [X440: term,X441: subst > term > term,X442: subst > term > term,X443: subst] :
( ( ( X441 @ X443 @ X440 )
= ( X442 @ X443 @ X440 ) )
| ~ ! [X447: subst,X448: subst,X449: term] :
( ( X442 @ ( comp @ X448 @ X447 ) @ ( sub @ X449 @ X447 ) )
= ( sub @ ( X442 @ X448 @ X449 ) @ X447 ) )
| ~ ! [X444: term,X445: subst,X446: subst] :
( ( X441 @ ( comp @ X446 @ X445 ) @ ( sub @ X444 @ X445 ) )
= ( sub @ ( X441 @ X446 @ X444 ) @ X445 ) )
| ( ( lam @ ( X441 @ sh @ one ) )
!= ( lam @ ( X442 @ sh @ one ) ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X388: term] :
~ ! [X389: subst > term > term] :
~ ! [X390: term,X391: subst] :
( ( sub @ X388 @ ( push @ X390 @ X391 ) )
= ( X389 @ X391 @ X390 ) )
=> ( ! [X392: term] :
( ( sub @ X392 @ id )
= X392 )
=> ! [X393: term,X394: subst > term > subst > $o,X395: term > $o] :
( ~ ! [X397: subst > term > term] :
( ~ ! [X399: term,X400: subst,X401: subst] :
( ( sub @ ( X397 @ X400 @ X399 ) @ X401 )
= ( X397 @ ( comp @ X400 @ X401 ) @ ( sub @ X399 @ X401 ) ) )
| ~ ! [X398: term] :
( ( ( X394 @ id @ X398 @ id )
!= $true )
| ( ( X394 @ id @ ( X397 @ id @ X398 ) @ id )
= $true ) )
| ( ( X394 @ id @ ( lam @ ( X397 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X396: term] :
( ( ( X395 @ X396 )
!= $true )
| ( ( X395 @ ( sub @ X393 @ ( push @ X396 @ id ) ) )
= $true ) )
| ( ( X395 @ ( lam @ X393 ) )
= $true )
| ~ ! [X402: term] :
( ( X395 @ X402 )
= ( X394 @ id @ X402 @ id ) ) ) ) ) )
& ( axabs = $true )
& ( axmap = $true )
& ! [X464: subst] :
? [X465: d_subst] :
( ( d2subst @ X465 )
= X464 )
& ( ! [X342: term,X343: subst > term > subst > $o] :
( ~ ! [X345: subst > term > term] :
( ~ ! [X349: term] :
( ( ( X343 @ id @ X349 @ id )
!= $true )
| ( $true
= ( X343 @ id @ ( X345 @ id @ X349 ) @ id ) ) )
| ~ ! [X346: term,X347: subst,X348: subst] :
( ( sub @ ( X345 @ X347 @ X346 ) @ X348 )
= ( X345 @ ( comp @ X347 @ X348 ) @ ( sub @ X346 @ X348 ) ) )
| ( ( X343 @ id @ ( lam @ ( X345 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X343 @ id @ ( lam @ ( sub @ X342 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X344: term] :
( ( ( X343 @ id @ ( sub @ X342 @ ( push @ X344 @ id ) ) @ id )
= $true )
| ( ( X343 @ id @ X344 @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ( substmonoid = $true )
<=> ( ! [X246: subst] :
( ( comp @ id @ X246 )
= X246 )
& ! [X245: subst] :
( ( comp @ X245 @ id )
= X245 ) ) )
& ( ( ! [X279: term,X280: term > $o] :
( ~ ! [X284: term,X285: term] :
( ( $true
!= ( X280 @ X285 ) )
| ( ( X280 @ ( ap @ X285 @ X284 ) )
= $true )
| ( $true
!= ( X280 @ X284 ) ) )
| ~ ! [X282: term] :
( ( ( X280 @ ( lam @ X282 ) )
= $true )
| ~ ! [X283: term] :
( ( $true
= ( X280 @ ( sub @ X282 @ ( push @ X283 @ id ) ) ) )
| ( ( X280 @ X283 )
!= $true ) ) )
| ( ( X280 @ X279 )
= $true )
| ~ ! [X281: term] :
( ( ( X280 @ X281 )
= $true )
| ( $true
!= ( var @ X281 ) ) ) )
=> ( ! [X286: term,X287: subst > term > subst > $o,X288: term] :
( ( $true
!= ( X287 @ id @ X286 @ id ) )
| ( $true
= ( X287 @ id @ ( ap @ X286 @ X288 ) @ id ) )
| ~ ! [X295: subst,X296: subst,X297: term,X298: subst] :
( ( $true
= ( X287 @ X295 @ X297 @ ( comp @ X296 @ X298 ) ) )
| ( ( X287 @ ( comp @ X295 @ X296 ) @ ( sub @ X297 @ X296 ) @ X298 )
!= $true ) )
| ( ( X287 @ id @ X288 @ id )
!= $true )
| ~ ! [X289: subst,X290: subst,X291: term,X292: subst] :
( ( ( X287 @ ( comp @ X289 @ X290 ) @ ( sub @ X291 @ X290 ) @ X292 )
= $true )
| ( $true
!= ( X287 @ X289 @ X291 @ ( comp @ X290 @ X292 ) ) ) )
| ~ ! [X293: term,X294: term] :
( ( ( X287 @ id @ X294 @ id )
!= $true )
| ( ( X287 @ id @ ( ap @ ( sub @ X293 @ id ) @ X294 ) @ id )
= $true )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) ) )
=> ( ! [X299: term,X300: subst > term > subst > $o] :
( ( ( X300 @ id @ ( lam @ X299 ) @ id )
= $true )
| ~ ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( ( X300 @ ( comp @ X307 @ X308 ) @ ( sub @ X306 @ X308 ) @ X309 )
!= $true )
| ( $true
= ( X300 @ X307 @ X306 @ ( comp @ X308 @ X309 ) ) ) )
| ~ ! [X301: term] :
( ( $true
!= ( X300 @ id @ X301 @ id ) )
| ( ( X300 @ id @ ( sub @ X299 @ ( push @ X301 @ id ) ) @ id )
= $true ) )
| ~ ! [X310: subst > term > term] :
( ( ( X300 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X312: term,X313: subst,X314: subst] :
( ( X310 @ ( comp @ X314 @ X313 ) @ ( sub @ X312 @ X313 ) )
= ( sub @ ( X310 @ X314 @ X312 ) @ X313 ) )
| ~ ! [X311: term] :
( ( ( X300 @ id @ ( X310 @ id @ X311 ) @ id )
= $true )
| ( $true
!= ( X300 @ id @ X311 @ id ) ) ) )
| ~ ! [X302: subst,X303: subst,X304: term,X305: subst] :
( ( ( X300 @ ( comp @ X305 @ X302 ) @ ( sub @ X304 @ X302 ) @ X303 )
= $true )
| ( ( X300 @ X305 @ X304 @ ( comp @ X302 @ X303 ) )
!= $true ) ) )
=> ! [X315: term,X316: subst > term > subst > $o] :
( ~ ! [X324: subst,X325: term,X326: subst,X327: subst] :
( ( ( X316 @ ( comp @ X324 @ X327 ) @ ( sub @ X325 @ X327 ) @ X326 )
= $true )
| ( $true
!= ( X316 @ X324 @ X325 @ ( comp @ X327 @ X326 ) ) ) )
| ~ ! [X328: subst,X329: subst,X330: subst,X331: term] :
( ( ( X316 @ X328 @ X331 @ ( comp @ X329 @ X330 ) )
= $true )
| ( ( X316 @ ( comp @ X328 @ X329 ) @ ( sub @ X331 @ X329 ) @ X330 )
!= $true ) )
| ( ( X316 @ id @ X315 @ id )
= $true )
| ~ ! [X317: term,X318: term] :
( ( ( X316 @ id @ X318 @ id )
!= $true )
| ( $true
= ( X316 @ id @ ( ap @ ( sub @ X318 @ id ) @ X317 ) @ id ) )
| ( ( X316 @ id @ X317 @ id )
!= $true ) )
| ~ ! [X319: subst > term > term] :
( ( $true
= ( X316 @ id @ ( lam @ ( X319 @ sh @ one ) ) @ id ) )
| ~ ! [X323: term] :
( ( ( X316 @ id @ ( X319 @ id @ X323 ) @ id )
= $true )
| ( ( X316 @ id @ X323 @ id )
!= $true ) )
| ~ ! [X320: term,X321: subst,X322: subst] :
( ( sub @ ( X319 @ X321 @ X320 ) @ X322 )
= ( X319 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) ) ) )
| ~ ! [X332: term] :
( ( ( X316 @ id @ X332 @ id )
= $true )
| ( ( var @ ( sub @ X332 @ id ) )
!= $true ) ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( hoasapinj1_gthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ( termmset = $true )
<=> ! [X247: term] :
( ( sub @ X247 @ id )
= X247 ) )
& ( ! [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
!= ( ap @ X276 @ X277 ) )
| ( X276 = X278 ) )
<=> ( apinj1 = $true ) )
& ( ( ! [X79: term] :
( ( sub @ X79 @ id )
= X79 )
=> ! [X80: term,X81: subst > term > subst > $o,X82: term > $o] :
( ~ ! [X97: term] :
( ( ( X82 @ ( sub @ X80 @ ( push @ X97 @ id ) ) )
= $true )
| ( ( X82 @ X97 )
!= $true ) )
| ~ ! [X89: subst,X90: subst,X91: subst,X92: term] :
( ( ( X81 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) @ X89 )
= $true )
| ( $true
!= ( X81 @ X90 @ X92 @ ( comp @ X91 @ X89 ) ) ) )
| ~ ! [X93: term,X94: subst,X95: subst,X96: subst] :
( ( $true
= ( X81 @ X94 @ X93 @ ( comp @ X95 @ X96 ) ) )
| ( ( X81 @ ( comp @ X94 @ X95 ) @ ( sub @ X93 @ X95 ) @ X96 )
!= $true ) )
| ~ ! [X83: term] :
( ( X82 @ X83 )
= ( X81 @ id @ X83 @ id ) )
| ~ ! [X84: subst > term > term] :
( ~ ! [X86: term,X87: subst,X88: subst] :
( ( sub @ ( X84 @ X88 @ X86 ) @ X87 )
= ( X84 @ ( comp @ X88 @ X87 ) @ ( sub @ X86 @ X87 ) ) )
| ( ( X81 @ id @ ( lam @ ( X84 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X85: term] :
( ( ( X81 @ id @ X85 @ id )
!= $true )
| ( ( X81 @ id @ ( X84 @ id @ X85 ) @ id )
= $true ) ) )
| ( ( X82 @ ( lam @ X80 ) )
= $true ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( pushprop_lthm_orig = $true )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X536: subst] :
( ( comp @ id @ X536 )
= X536 )
=> ( ! [X537: subst] :
( ( comp @ X537 @ id )
= X537 )
=> ( ! [X538: subst] :
( ( comp @ X538 @ id )
= X538 )
& ! [X539: subst] :
( ( comp @ id @ X539 )
= X539 ) ) ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( substmonoid_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( laminj = $true )
<=> ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X369: term > $o,X370: subst > term > subst > $o,X371: term,X372: term] :
( ~ ! [X382: term,X383: term] :
( ( ( X370 @ id @ X382 @ id )
!= $true )
| ( ( X370 @ id @ ( ap @ ( sub @ X383 @ id ) @ X382 ) @ id )
= $true )
| ( ( X370 @ id @ X383 @ id )
!= $true ) )
| ( $true
!= ( X369 @ X371 ) )
| ( ( X369 @ X372 )
!= $true )
| ~ ! [X373: subst,X374: subst,X375: term,X376: subst] :
( ( ( X370 @ ( comp @ X374 @ X373 ) @ ( sub @ X375 @ X373 ) @ X376 )
= $true )
| ( ( X370 @ X374 @ X375 @ ( comp @ X373 @ X376 ) )
!= $true ) )
| ( ( X369 @ ( ap @ X372 @ X371 ) )
= $true )
| ~ ! [X378: term,X379: subst,X380: subst,X381: subst] :
( ( ( X370 @ ( comp @ X380 @ X381 ) @ ( sub @ X378 @ X381 ) @ X379 )
!= $true )
| ( $true
= ( X370 @ X380 @ X378 @ ( comp @ X381 @ X379 ) ) ) )
| ~ ! [X377: term] :
( ( X369 @ X377 )
= ( X370 @ id @ X377 @ id ) ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X451: subst > term > subst > $o,X452: term,X453: term] :
( ( ( X451 @ id @ X452 @ id )
!= $true )
| ~ ! [X458: term,X459: term] :
( ( ( X451 @ id @ X458 @ id )
!= $true )
| ( $true
= ( X451 @ id @ ( ap @ ( sub @ X458 @ id ) @ X459 ) @ id ) )
| ( $true
!= ( X451 @ id @ X459 @ id ) ) )
| ~ ! [X460: subst,X461: subst,X462: term,X463: subst] :
( ( ( X451 @ ( comp @ X460 @ X463 ) @ ( sub @ X462 @ X463 ) @ X461 )
!= $true )
| ( ( X451 @ X460 @ X462 @ ( comp @ X463 @ X461 ) )
= $true ) )
| ( ( X451 @ id @ ( ap @ X453 @ X452 ) @ id )
= $true )
| ~ ! [X454: subst,X455: subst,X456: term,X457: subst] :
( ( $true
= ( X451 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 ) )
| ( ( X451 @ X457 @ X456 @ ( comp @ X455 @ X454 ) )
!= $true ) )
| ( ( X451 @ id @ X453 @ id )
!= $true ) ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X121: term,X122: subst > term > subst > $o] :
( ~ ! [X124: term,X125: subst > term > term] :
( ~ ! [X126: subst,X127: term,X128: subst] :
( ( sub @ ( sub @ X124 @ ( push @ X127 @ X128 ) ) @ X126 )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
| ( ( X122 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X130: term,X131: subst,X132: subst] :
( ( sub @ X124 @ ( push @ ( sub @ X130 @ X132 ) @ ( comp @ X131 @ X132 ) ) )
= ( X125 @ ( comp @ X131 @ X132 ) @ ( sub @ X130 @ X132 ) ) )
| ~ ! [X129: term] :
( ( ( X122 @ id @ X129 @ id )
!= $true )
| ( ( X122 @ id @ ( X125 @ id @ X129 ) @ id )
= $true ) ) )
| ( $true
= ( X122 @ id @ ( lam @ ( sub @ X121 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X123: term] :
( ( ( X122 @ id @ X123 @ id )
!= $true )
| ( ( X122 @ id @ ( sub @ X121 @ ( push @ X123 @ id ) ) @ id )
= $true ) ) ) )
& ( ! [X467: term] :
~ ! [X468: subst > term > term] :
( ( X468 @ sh @ one )
!= ( sub @ X467 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( pushprop_lem3v2 = $true )
& ( ( induction2lem_lthm = $true )
<=> ( ! [X350: subst,X351: term] :
( ( sub @ one @ ( push @ X351 @ X350 ) )
= X351 )
=> ( ! [X352: term,X353: subst] :
( ( comp @ sh @ ( push @ X352 @ X353 ) )
= X353 )
=> ( ! [X354: subst] :
( ( comp @ X354 @ id )
= X354 )
=> ( ! [X355: term,X356: term > $o] :
( ( ( X356 @ X355 )
= $true )
| ~ ! [X359: term,X360: term] :
( ( $true
!= ( X356 @ X360 ) )
| ( $true
= ( X356 @ ( ap @ X360 @ X359 ) ) )
| ( $true
!= ( X356 @ X359 ) ) )
| ~ ! [X358: term] :
( ( $true
= ( X356 @ ( lam @ X358 ) ) )
| ( ( X356 @ X358 )
!= $true ) )
| ~ ! [X357: term] :
( ( ( var @ X357 )
!= $true )
| ( $true
= ( X356 @ X357 ) ) ) )
=> ! [X361: term,X362: term > $o,X363: subst] :
( ~ ! [X365: term] :
( ~ ! [X366: term] :
( ( $true
= ( X362 @ ( sub @ X365 @ ( push @ X366 @ id ) ) ) )
| ( ( X362 @ X366 )
!= $true ) )
| ( $true
= ( X362 @ ( lam @ X365 ) ) ) )
| ~ ! [X364: term] :
( ( ( var @ X364 )
!= $true )
| ( $true
= ( X362 @ ( sub @ X364 @ X363 ) ) ) )
| ~ ! [X367: term,X368: term] :
( ( $true
= ( X362 @ ( ap @ X368 @ X367 ) ) )
| ( ( X362 @ X368 )
!= $true )
| ( ( X362 @ X367 )
!= $true ) )
| ( ( X362 @ ( sub @ X361 @ X363 ) )
= $true ) ) ) ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X178: term] :
~ ! [X179: subst > term > term] :
~ ! [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
= ( X179 @ X180 @ X181 ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X116: term,X117: subst,X118: term > $o] :
~ ! [X119: term > $o] :
~ ! [X120: term] :
( ( X119 @ X120 )
= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X384: term] :
~ ! [X385: subst > term > term] :
~ ! [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
= ( X385 @ X386 @ X387 ) ) )
& ( ( induction2 = $true )
<=> ! [X42: term > $o,X43: term] :
( ~ ! [X44: term] :
( ( ( var @ X44 )
!= $true )
| ( ( X42 @ X44 )
= $true ) )
| ~ ! [X47: term,X48: term] :
( ( $true
!= ( X42 @ X47 ) )
| ( ( X42 @ X48 )
!= $true )
| ( $true
= ( X42 @ ( ap @ X47 @ X48 ) ) ) )
| ( ( X42 @ X43 )
= $true )
| ~ ! [X45: term] :
( ~ ! [X46: term] :
( ( $true
!= ( X42 @ X46 ) )
| ( $true
= ( X42 @ ( sub @ X45 @ ( push @ X46 @ id ) ) ) ) )
| ( ( X42 @ ( lam @ X45 ) )
= $true ) ) ) )
& ( ulamvarind = $true )
& ( termmset_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
= X168 )
=> ! [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ~ ! [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
= ( X170 @ X173 ) )
| ( ( X171 @ X172 )
!= $true )
| ( ( X170 @ one )
= $true ) ) ) )
& ( ( ! [X248: term] :
( ( sub @ X248 @ id )
= X248 )
=> ( ! [X249: subst > term > subst > $o,X250: term] :
( ( ( X249 @ id @ ( lam @ ( sub @ X250 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X256: term] :
( ( ( X249 @ id @ ( sub @ X250 @ ( push @ X256 @ id ) ) @ id )
= $true )
| ( ( X249 @ id @ X256 @ id )
!= $true ) )
| ~ ! [X251: subst > term > term] :
( ~ ! [X252: subst,X253: term,X254: subst] :
( ( sub @ ( X251 @ X252 @ X253 ) @ X254 )
= ( X251 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) ) )
| ( $true
= ( X249 @ id @ ( lam @ ( X251 @ sh @ one ) ) @ id ) )
| ~ ! [X255: term] :
( ( ( X249 @ id @ ( X251 @ id @ X255 ) @ id )
= $true )
| ( ( X249 @ id @ X255 @ id )
!= $true ) ) ) )
=> ! [X257: subst > term > subst > $o,X258: term] :
( ~ ! [X260: subst > term > term] :
( ~ ! [X261: term,X262: subst,X263: subst] :
( ( X260 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) )
= ( sub @ ( X260 @ X262 @ X261 ) @ X263 ) )
| ( ( X257 @ id @ ( lam @ ( X260 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X264: term] :
( ( ( X257 @ id @ ( X260 @ id @ X264 ) @ id )
= $true )
| ( ( X257 @ id @ X264 @ id )
!= $true ) ) )
| ~ ! [X259: term] :
( ( ( X257 @ id @ X259 @ id )
!= $true )
| ( ( X257 @ id @ ( sub @ X258 @ ( push @ X259 @ id ) ) @ id )
= $true ) )
| ( ( X257 @ id @ ( lam @ X258 ) @ id )
= $true ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X182: subst > term > subst > $o,X183: term] :
( ~ ! [X184: term] :
( ( ( X182 @ id @ ( sub @ X183 @ ( push @ X184 @ id ) ) @ id )
= $true )
| ( ( X182 @ id @ X184 @ id )
!= $true ) )
| ~ ! [X185: subst > term > term] :
( ( ( X182 @ id @ ( lam @ ( X185 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X186: term] :
( ( $true
!= ( X182 @ id @ X186 @ id ) )
| ( ( X182 @ id @ ( X185 @ id @ X186 ) @ id )
= $true ) )
| ~ ! [X187: term,X188: subst,X189: subst] :
( ( X185 @ ( comp @ X189 @ X188 ) @ ( sub @ X187 @ X188 ) )
= ( sub @ ( X185 @ X189 @ X187 ) @ X188 ) ) )
| ( ( X182 @ id @ ( lam @ ( sub @ X183 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( ! [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
= X240 )
<=> ( axshiftcons = $true ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ( ( X472 @ one )
= $true )
| ~ ! [X476: term] :
( ( X472 @ X476 )
= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
| ( ( X474 @ X475 )
!= $true ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( induction2lem_gthm = $true )
& ! [X177: d_subst] : ( d_id = X177 )
& ( lamnotap != $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ! [X469: d_term,X470: d_term] :
( ( ( d2term @ X469 )
= ( d2term @ X470 ) )
=> ( X469 = X470 ) )
& ( ! [X403: term > $o,X404: term] :
( ~ ! [X408: term] :
( ( ( X403 @ X408 )
!= $true )
| ( ( X403 @ ( lam @ X408 ) )
= $true ) )
| ~ ! [X405: term,X406: term] :
( ( ( X403 @ ( ap @ X405 @ X406 ) )
= $true )
| ( $true
!= ( X403 @ X406 ) )
| ( ( X403 @ X405 )
!= $true ) )
| ~ ! [X407: term] :
( ( $true
= ( X403 @ X407 ) )
| ( $true
!= ( var @ X407 ) ) )
| ( ( X403 @ X404 )
= $true ) )
<=> ( induction = $true ) )
& ( pushprop_lem0_gthm = $true )
& ( pushprop_lthm = $true )
& ( ! [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
!= ( ap @ X243 @ X241 ) )
| ( X241 = X242 ) )
<=> ( apinj2 = $true ) )
& ( pushprop_lem1_gthm = $true )
& ( ! [X493: term,X494: subst > term > subst > $o] :
( ( $true
= ( X494 @ id @ X493 @ id ) )
| ~ ! [X501: subst > term > term] :
( ~ ! [X505: term] :
( ( ( X494 @ id @ X505 @ id )
!= $true )
| ( ( X494 @ id @ ( X501 @ id @ X505 ) @ id )
= $true ) )
| ( $true
= ( X494 @ id @ ( lam @ ( X501 @ sh @ one ) ) @ id ) )
| ~ ! [X502: subst,X503: term,X504: subst] :
( ( X501 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X501 @ X502 @ X503 ) @ X504 ) ) )
| ~ ! [X507: subst,X508: subst,X509: term,X510: subst] :
( ( $true
= ( X494 @ ( comp @ X510 @ X508 ) @ ( sub @ X509 @ X508 ) @ X507 ) )
| ( ( X494 @ X510 @ X509 @ ( comp @ X508 @ X507 ) )
!= $true ) )
| ~ ! [X506: term] :
( ( ( var @ ( sub @ X506 @ id ) )
!= $true )
| ( $true
= ( X494 @ id @ X506 @ id ) ) )
| ~ ! [X495: term,X496: subst,X497: subst,X498: subst] :
( ( $true
!= ( X494 @ ( comp @ X498 @ X497 ) @ ( sub @ X495 @ X497 ) @ X496 ) )
| ( ( X494 @ X498 @ X495 @ ( comp @ X497 @ X496 ) )
= $true ) )
| ~ ! [X499: term,X500: term] :
( ( ( X494 @ id @ X499 @ id )
!= $true )
| ( ( X494 @ id @ ( ap @ ( sub @ X500 @ id ) @ X499 ) @ id )
= $true )
| ( ( X494 @ id @ X500 @ id )
!= $true ) ) )
<=> ( hoasinduction = $true ) )
& ( ( axidl = $true )
<=> ! [X471: subst] :
( ( comp @ id @ X471 )
= X471 ) )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X17: term,X18: subst > term > subst > $o,X19: term > $o] :
( ~ ! [X20: term] :
( ( ( X19 @ ( sub @ X17 @ ( push @ X20 @ id ) ) )
= $true )
| ( $true
!= ( X19 @ X20 ) ) )
| ~ ! [X22: subst > term > term] :
( ~ ! [X26: term] :
( ( $true
!= ( X18 @ id @ X26 @ id ) )
| ( ( X18 @ id @ ( X22 @ id @ X26 ) @ id )
= $true ) )
| ( $true
= ( X18 @ id @ ( lam @ ( X22 @ sh @ one ) ) @ id ) )
| ~ ! [X23: subst,X24: subst,X25: term] :
( ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( X22 @ X24 @ X25 ) @ X23 ) ) )
| ~ ! [X21: term] :
( ( X19 @ X21 )
= ( X18 @ id @ X21 @ id ) )
| ( ( X19 @ ( lam @ X17 ) )
= $true )
| ~ ! [X31: subst,X32: subst,X33: subst,X34: term] :
( ( $true
!= ( X18 @ ( comp @ X33 @ X32 ) @ ( sub @ X34 @ X32 ) @ X31 ) )
| ( ( X18 @ X33 @ X34 @ ( comp @ X32 @ X31 ) )
= $true ) )
| ~ ! [X27: subst,X28: term,X29: subst,X30: subst] :
( ( $true
!= ( X18 @ X27 @ X28 @ ( comp @ X29 @ X30 ) ) )
| ( ( X18 @ ( comp @ X27 @ X29 ) @ ( sub @ X28 @ X29 ) @ X30 )
= $true ) ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoaslamnotap_lthm = $true )
& ( ! [X174: term] :
( ( sub @ X174 @ id )
= X174 )
<=> ( axvarid = $true ) )
& ( hoasinduction_lem1 = $true )
& ( one
= ( d2term @ d_one ) )
& ( ( termmset_lthm = $true )
<=> ( ! [X149: term] :
( ( sub @ X149 @ id )
= X149 )
=> ! [X150: term] :
( ( sub @ X150 @ id )
= X150 ) ) )
& ( ( ! [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
= X520 )
=> ( ! [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
= X521 )
=> ! [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) )
| ( $true
!= ( X523 @ X527 ) ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ! [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
= X176 )
<=> ( axvarcons = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X466: subst] :
( ( comp @ X466 @ id )
= X466 )
<=> ( axidr = $true ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X8: term] :
~ ! [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
!= ( X9 @ sh @ one ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X528: term,X529: term,X530: term,X531: term] :
( ( X529 = X530 )
| ( ( ap @ X531 @ X529 )
!= ( ap @ X528 @ X530 ) ) )
=> ! [X532: term,X533: term,X534: term,X535: term] :
( ( X532 = X534 )
| ( ( ap @ ( sub @ X533 @ id ) @ X532 )
!= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( ( shinj = $true )
<=> ! [X35: term,X36: term] :
( ( X35 = X36 )
| ( ( sub @ X36 @ sh )
!= ( sub @ X35 @ sh ) ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X103: term] :
( ( sub @ X103 @ id )
= X103 )
=> ( ! [X104: term,X105: term,X106: term,X107: term] :
( ( ( ap @ X106 @ X105 )
!= ( ap @ X107 @ X104 ) )
| ( X106 = X107 ) )
=> ! [X108: term,X109: term,X110: term,X111: term] :
( ( X108 = X109 )
| ( ( ap @ ( sub @ X109 @ id ) @ X111 )
!= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) ) ) ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X511: term,X512: subst > term > subst > $o] :
( ~ ! [X518: term] :
( ( ( X512 @ id @ X518 @ id )
!= $true )
| ( ( X512 @ id @ ( sub @ X511 @ ( push @ X518 @ id ) ) @ id )
= $true ) )
| ( ( X512 @ id @ ( lam @ X511 ) @ id )
= $true )
| ~ ! [X513: subst > term > term] :
( ~ ! [X514: subst,X515: term,X516: subst] :
( ( X513 @ ( comp @ X514 @ X516 ) @ ( sub @ X515 @ X516 ) )
= ( sub @ ( X513 @ X514 @ X515 ) @ X516 ) )
| ( ( X512 @ id @ ( lam @ ( X513 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X517: term] :
( ( $true
= ( X512 @ id @ ( X513 @ id @ X517 ) @ id ) )
| ( ( X512 @ id @ X517 @ id )
!= $true ) ) ) ) ) ),
inference(flattening,[status(thm)],[f10]) ).
thf(f10,plain,
( ( ( induction2lem = $true )
<=> ! [X0: subst,X1: term > $o,X2: term] :
( ~ ! [X3: term,X4: term] :
( ( ( X1 @ X3 )
!= $true )
| ( ( X1 @ ( ap @ X3 @ X4 ) )
= $true )
| ( ( X1 @ X4 )
!= $true ) )
| ( ( X1 @ ( sub @ X2 @ X0 ) )
= $true )
| ~ ! [X5: term] :
( ( ( var @ X5 )
!= $true )
| ( ( X1 @ ( sub @ X5 @ X0 ) )
= $true ) )
| ~ ! [X6: term] :
( ( ( X1 @ ( lam @ X6 ) )
= $true )
| ~ ! [X7: term] :
( ( $true
!= ( X1 @ X7 ) )
| ( $true
= ( X1 @ ( sub @ X6 @ ( push @ X7 @ id ) ) ) ) ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X8: term] :
~ ! [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
!= ( X9 @ sh @ one ) ) )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( one
= ( d2term @ d_one ) )
& ( ( hoasapinj1 = $true )
<=> ! [X13: term,X14: term,X15: term,X16: term] :
( ( X13 = X16 )
| ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem1 = $true )
& ( axvarshift = $true )
& ( ! [X17: term,X18: subst > term > subst > $o,X19: term > $o] :
( ~ ! [X20: term] :
( ( ( X19 @ ( sub @ X17 @ ( push @ X20 @ id ) ) )
= $true )
| ( $true
!= ( X19 @ X20 ) ) )
| ~ ! [X21: term] :
( ( X19 @ X21 )
= ( X18 @ id @ X21 @ id ) )
| ~ ! [X22: subst > term > term] :
( ~ ! [X23: subst,X24: subst,X25: term] :
( ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( X22 @ X24 @ X25 ) @ X23 ) )
| ~ ! [X26: term] :
( ( $true
!= ( X18 @ id @ X26 @ id ) )
| ( ( X18 @ id @ ( X22 @ id @ X26 ) @ id )
= $true ) )
| ( $true
= ( X18 @ id @ ( lam @ ( X22 @ sh @ one ) ) @ id ) ) )
| ~ ! [X27: subst,X28: term,X29: subst,X30: subst] :
( ( $true
!= ( X18 @ X27 @ X28 @ ( comp @ X29 @ X30 ) ) )
| ( ( X18 @ ( comp @ X27 @ X29 ) @ ( sub @ X28 @ X29 ) @ X30 )
= $true ) )
| ~ ! [X31: subst,X32: subst,X33: subst,X34: term] :
( ( $true
!= ( X18 @ ( comp @ X33 @ X32 ) @ ( sub @ X34 @ X32 ) @ X31 ) )
| ( ( X18 @ X33 @ X34 @ ( comp @ X32 @ X31 ) )
= $true ) )
| ( ( X19 @ ( lam @ X17 ) )
= $true ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( pushprop = $true )
& ( ( shinj = $true )
<=> ! [X35: term,X36: term] :
( ( X35 = X36 )
| ( ( sub @ X36 @ sh )
!= ( sub @ X35 @ sh ) ) ) )
& ( ! [X37: term > $o,X38: term,X39: subst] :
~ ! [X40: term > $o] :
~ ! [X41: term] :
( ( X37 @ ( sub @ X41 @ ( push @ X38 @ X39 ) ) )
= ( X40 @ X41 ) )
<=> ( pushprop_lem0 = $true ) )
& ( ! [X42: term > $o,X43: term] :
( ~ ! [X44: term] :
( ( ( var @ X44 )
!= $true )
| ( ( X42 @ X44 )
= $true ) )
| ~ ! [X45: term] :
( ~ ! [X46: term] :
( ( $true
!= ( X42 @ X46 ) )
| ( $true
= ( X42 @ ( sub @ X45 @ ( push @ X46 @ id ) ) ) ) )
| ( ( X42 @ ( lam @ X45 ) )
= $true ) )
| ~ ! [X47: term,X48: term] :
( ( $true
!= ( X42 @ X47 ) )
| ( $true
= ( X42 @ ( ap @ X47 @ X48 ) ) )
| ( ( X42 @ X48 )
!= $true ) )
| ( ( X42 @ X43 )
= $true ) )
<=> ( induction2 = $true ) )
& ( ( ! [X49: subst > term > subst > $o] :
~ ! [X50: term > $o] :
~ ! [X51: term] :
( ( X49 @ id @ X51 @ id )
= ( X50 @ X51 ) )
=> ( ! [X52: term > $o,X53: term] :
( ~ ! [X54: term,X55: term] :
( ( $true
!= ( X52 @ X54 ) )
| ( ( X52 @ ( ap @ X55 @ X54 ) )
= $true )
| ( ( X52 @ X55 )
!= $true ) )
| ~ ! [X56: term] :
( ( ( var @ X56 )
!= $true )
| ( $true
= ( X52 @ X56 ) ) )
| ( ( X52 @ X53 )
= $true )
| ~ ! [X57: term] :
( ( ( X52 @ ( lam @ X57 ) )
= $true )
| ~ ! [X58: term] :
( ( $true
!= ( X52 @ X58 ) )
| ( ( X52 @ ( sub @ X57 @ ( push @ X58 @ id ) ) )
= $true ) ) ) )
=> ( ! [X59: term] :
( ( sub @ X59 @ id )
= X59 )
=> ( ! [X60: subst > term > subst > $o,X61: term > $o,X62: term] :
( ~ ! [X63: subst > term > term] :
( ~ ! [X64: term] :
( ( ( X60 @ id @ X64 @ id )
!= $true )
| ( ( X60 @ id @ ( X63 @ id @ X64 ) @ id )
= $true ) )
| ~ ! [X65: subst,X66: subst,X67: term] :
( ( sub @ ( X63 @ X66 @ X67 ) @ X65 )
= ( X63 @ ( comp @ X66 @ X65 ) @ ( sub @ X67 @ X65 ) ) )
| ( $true
= ( X60 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id ) ) )
| ~ ! [X68: term] :
( ( $true
= ( X61 @ ( sub @ X62 @ ( push @ X68 @ id ) ) ) )
| ( ( X61 @ X68 )
!= $true ) )
| ~ ! [X69: term] :
( ( X60 @ id @ X69 @ id )
= ( X61 @ X69 ) )
| ( ( X61 @ ( lam @ X62 ) )
= $true ) )
=> ! [X70: subst > term > subst > $o,X71: term] :
( ( $true
= ( X70 @ id @ X71 @ id ) )
| ~ ! [X72: subst > term > term] :
( ( ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X73: subst,X74: term,X75: subst] :
( ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) )
= ( sub @ ( X72 @ X75 @ X74 ) @ X73 ) )
| ~ ! [X76: term] :
( ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true )
| ( $true
!= ( X70 @ id @ X76 @ id ) ) ) )
| ~ ! [X77: term,X78: term] :
( ( ( X70 @ id @ ( ap @ ( sub @ X78 @ id ) @ X77 ) @ id )
= $true )
| ( ( X70 @ id @ X78 @ id )
!= $true )
| ( $true
!= ( X70 @ id @ X77 @ id ) ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( ( ! [X79: term] :
( ( sub @ X79 @ id )
= X79 )
=> ! [X80: term,X81: subst > term > subst > $o,X82: term > $o] :
( ~ ! [X83: term] :
( ( X82 @ X83 )
= ( X81 @ id @ X83 @ id ) )
| ~ ! [X84: subst > term > term] :
( ~ ! [X85: term] :
( ( ( X81 @ id @ ( X84 @ id @ X85 ) @ id )
= $true )
| ( ( X81 @ id @ X85 @ id )
!= $true ) )
| ( ( X81 @ id @ ( lam @ ( X84 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X86: term,X87: subst,X88: subst] :
( ( sub @ ( X84 @ X88 @ X86 ) @ X87 )
= ( X84 @ ( comp @ X88 @ X87 ) @ ( sub @ X86 @ X87 ) ) ) )
| ~ ! [X89: subst,X90: subst,X91: subst,X92: term] :
( ( ( X81 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) @ X89 )
= $true )
| ( $true
!= ( X81 @ X90 @ X92 @ ( comp @ X91 @ X89 ) ) ) )
| ~ ! [X93: term,X94: subst,X95: subst,X96: subst] :
( ( $true
= ( X81 @ X94 @ X93 @ ( comp @ X95 @ X96 ) ) )
| ( ( X81 @ ( comp @ X94 @ X95 ) @ ( sub @ X93 @ X95 ) @ X96 )
!= $true ) )
| ( ( X82 @ ( lam @ X80 ) )
= $true )
| ~ ! [X97: term] :
( ( ( X82 @ ( sub @ X80 @ ( push @ X97 @ id ) ) )
= $true )
| ( ( X82 @ X97 )
!= $true ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( ( hoasapinj2 = $true )
<=> ! [X98: term,X99: term,X100: term,X101: term] :
( ( X100 = X101 )
| ( ( ap @ ( sub @ X98 @ id ) @ X100 )
!= ( ap @ ( sub @ X99 @ id ) @ X101 ) ) ) )
& ( hoaslaminj_gthm = $true )
& ( ! [X102: subst] :
( ( push @ ( sub @ one @ X102 ) @ ( comp @ sh @ X102 ) )
= X102 )
<=> ( axscons = $true ) )
& ( apnotvar = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X103: term] :
( ( sub @ X103 @ id )
= X103 )
=> ( ! [X104: term,X105: term,X106: term,X107: term] :
( ( ( ap @ X106 @ X105 )
!= ( ap @ X107 @ X104 ) )
| ( X106 = X107 ) )
=> ! [X108: term,X109: term,X110: term,X111: term] :
( ( X108 = X109 )
| ( ( ap @ ( sub @ X109 @ id ) @ X111 )
!= ( ap @ ( sub @ X108 @ id ) @ X110 ) ) ) ) ) )
& ( termmset_gthm = $true )
& ! [X112: d_subst,X113: d_subst] :
( ( ( d2subst @ X112 )
= ( d2subst @ X113 ) )
=> ( X112 = X113 ) )
& ! [X114: term] :
? [X115: d_term] :
( ( d2term @ X115 )
= X114 )
& ( pushprop_lem3v2_lthm = $true )
& ( axapp = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X116: term,X117: subst,X118: term > $o] :
~ ! [X119: term > $o] :
~ ! [X120: term] :
( ( X119 @ X120 )
= ( X118 @ ( sub @ X120 @ ( push @ X116 @ X117 ) ) ) ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X121: term,X122: subst > term > subst > $o] :
( ~ ! [X123: term] :
( ( ( X122 @ id @ X123 @ id )
!= $true )
| ( ( X122 @ id @ ( sub @ X121 @ ( push @ X123 @ id ) ) @ id )
= $true ) )
| ~ ! [X124: term,X125: subst > term > term] :
( ~ ! [X126: subst,X127: term,X128: subst] :
( ( sub @ ( sub @ X124 @ ( push @ X127 @ X128 ) ) @ X126 )
= ( sub @ ( X125 @ X128 @ X127 ) @ X126 ) )
| ( ( X122 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X129: term] :
( ( ( X122 @ id @ ( X125 @ id @ X129 ) @ id )
= $true )
| ( ( X122 @ id @ X129 @ id )
!= $true ) )
| ~ ! [X130: term,X131: subst,X132: subst] :
( ( sub @ X124 @ ( push @ ( sub @ X130 @ X132 ) @ ( comp @ X131 @ X132 ) ) )
= ( X125 @ ( comp @ X131 @ X132 ) @ ( sub @ X130 @ X132 ) ) ) )
| ( $true
= ( X122 @ id @ ( lam @ ( sub @ X121 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( ! [X133: term > $o,X134: term,X135: term,X136: subst,X137: term > $o] :
( ( $true
!= ( X137 @ X134 ) )
| ( ( X133 @ ( sub @ X134 @ ( push @ X135 @ X136 ) ) )
= $true ) )
<=> ( pushprop_lem1 = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X138: subst > term > subst > $o,X139: term] :
( ~ ! [X140: subst > term > term] :
( ( ( X138 @ id @ ( lam @ ( X140 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X141: term] :
( ( ( X138 @ id @ X141 @ id )
!= $true )
| ( $true
= ( X138 @ id @ ( X140 @ id @ X141 ) @ id ) ) )
| ~ ! [X142: subst,X143: term,X144: subst] :
( ( X140 @ ( comp @ X142 @ X144 ) @ ( sub @ X143 @ X144 ) )
= ( sub @ ( X140 @ X142 @ X143 ) @ X144 ) ) )
| ~ ! [X145: term,X146: term] :
( ( ( X138 @ id @ X146 @ id )
!= $true )
| ( ( X138 @ id @ ( ap @ ( sub @ X145 @ id ) @ X146 ) @ id )
= $true )
| ( $true
!= ( X138 @ id @ X145 @ id ) ) )
| ( ( X138 @ id @ X139 @ id )
= $true ) ) )
& ( ( laminj = $true )
<=> ! [X147: term,X148: term] :
( ( X147 = X148 )
| ( ( lam @ X147 )
!= ( lam @ X148 ) ) ) )
& ( ( termmset_lthm = $true )
<=> ( ! [X149: term] :
( ( sub @ X149 @ id )
= X149 )
=> ! [X150: term] :
( ( sub @ X150 @ id )
= X150 ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( ! [X151: term] :
( ( sub @ X151 @ id )
= X151 )
=> ( ! [X152: term > $o,X153: term,X154: subst] :
( ~ ! [X155: term,X156: term] :
( ( $true
= ( X152 @ ( ap @ X155 @ X156 ) ) )
| ( $true
!= ( X152 @ X155 ) )
| ( ( X152 @ X156 )
!= $true ) )
| ~ ! [X157: term] :
( ( ( var @ X157 )
!= $true )
| ( ( X152 @ ( sub @ X157 @ X154 ) )
= $true ) )
| ~ ! [X158: term] :
( ~ ! [X159: term] :
( ( $true
= ( X152 @ ( sub @ X158 @ ( push @ X159 @ id ) ) ) )
| ( ( X152 @ X159 )
!= $true ) )
| ( ( X152 @ ( lam @ X158 ) )
= $true ) )
| ( ( X152 @ ( sub @ X153 @ X154 ) )
= $true ) )
=> ! [X160: term > $o,X161: term] :
( ~ ! [X162: term,X163: term] :
( ( ( X160 @ X163 )
!= $true )
| ( $true
= ( X160 @ ( ap @ X163 @ X162 ) ) )
| ( $true
!= ( X160 @ X162 ) ) )
| ( ( X160 @ X161 )
= $true )
| ~ ! [X164: term] :
( ( ( var @ X164 )
!= $true )
| ( ( X160 @ X164 )
= $true ) )
| ~ ! [X165: term] :
( ~ ! [X166: term] :
( ( ( X160 @ ( sub @ X165 @ ( push @ X166 @ id ) ) )
= $true )
| ( $true
!= ( X160 @ X166 ) ) )
| ( $true
= ( X160 @ ( lam @ X165 ) ) ) ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( ulamvarind = $true )
& ( induction2lem_gthm = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( ! [X167: subst,X168: term] :
( ( sub @ one @ ( push @ X168 @ X167 ) )
= X168 )
=> ! [X169: subst,X170: term > $o,X171: term > $o,X172: term] :
( ~ ! [X173: term] :
( ( X171 @ ( sub @ X173 @ ( push @ X172 @ X169 ) ) )
= ( X170 @ X173 ) )
| ( ( X171 @ X172 )
!= $true )
| ( ( X170 @ one )
= $true ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( ! [X174: term] :
( ( sub @ X174 @ id )
= X174 )
<=> ( axvarid = $true ) )
& ( ! [X175: subst,X176: term] :
( ( sub @ one @ ( push @ X176 @ X175 ) )
= X176 )
<=> ( axvarcons = $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 ) ) ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ! [X177: d_subst] : ( d_id = X177 )
& ( hoasinduction_lem1_lthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X178: term] :
~ ! [X179: subst > term > term] :
~ ! [X180: subst,X181: term] :
( ( sub @ X178 @ ( push @ X181 @ X180 ) )
= ( X179 @ X180 @ X181 ) ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( pushprop_lem1_gthm = $true )
& ( axmap = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X182: subst > term > subst > $o,X183: term] :
( ~ ! [X184: term] :
( ( ( X182 @ id @ X184 @ id )
!= $true )
| ( ( X182 @ id @ ( sub @ X183 @ ( push @ X184 @ id ) ) @ id )
= $true ) )
| ~ ! [X185: subst > term > term] :
( ~ ! [X186: term] :
( ( $true
!= ( X182 @ id @ X186 @ id ) )
| ( ( X182 @ id @ ( X185 @ id @ X186 ) @ id )
= $true ) )
| ( ( X182 @ id @ ( lam @ ( X185 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X187: term,X188: subst,X189: subst] :
( ( X185 @ ( comp @ X189 @ X188 ) @ ( sub @ X187 @ X188 ) )
= ( sub @ ( X185 @ X189 @ X187 ) @ X188 ) ) )
| ( ( X182 @ id @ ( lam @ ( sub @ X183 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( ( hoaslaminj = $true )
<=> ! [X190: subst > term > term,X191: subst > term > term,X192: subst,X193: term] :
( ~ ! [X194: term,X195: subst,X196: subst] :
( ( X190 @ ( comp @ X196 @ X195 ) @ ( sub @ X194 @ X195 ) )
= ( sub @ ( X190 @ X196 @ X194 ) @ X195 ) )
| ( ( X191 @ X192 @ X193 )
= ( X190 @ X192 @ X193 ) )
| ~ ! [X197: term,X198: subst,X199: subst] :
( ( sub @ ( X191 @ X198 @ X197 ) @ X199 )
= ( X191 @ ( comp @ X198 @ X199 ) @ ( sub @ X197 @ X199 ) ) )
| ( ( lam @ ( X191 @ sh @ one ) )
!= ( lam @ ( X190 @ sh @ one ) ) ) ) )
& ( ( ! [X200: subst > term > subst > $o] :
~ ! [X201: term > $o] :
~ ! [X202: term] :
( ( X201 @ X202 )
= ( X200 @ id @ X202 @ id ) )
=> ( ! [X203: term,X204: term > $o] :
( ~ ! [X205: term,X206: term] :
( ( ( X204 @ X205 )
!= $true )
| ( $true
!= ( X204 @ X206 ) )
| ( ( X204 @ ( ap @ X206 @ X205 ) )
= $true ) )
| ( $true
= ( X204 @ X203 ) )
| ~ ! [X207: term] :
( ~ ! [X208: term] :
( ( $true
!= ( X204 @ X208 ) )
| ( $true
= ( X204 @ ( sub @ X207 @ ( push @ X208 @ id ) ) ) ) )
| ( $true
= ( X204 @ ( lam @ X207 ) ) ) )
| ~ ! [X209: term] :
( ( ( var @ X209 )
!= $true )
| ( ( X204 @ X209 )
= $true ) ) )
=> ( ! [X210: term] :
( ( sub @ X210 @ id )
= X210 )
=> ( ! [X211: term > $o,X212: term,X213: subst > term > subst > $o] :
( ~ ! [X214: term] :
( ( X213 @ id @ X214 @ id )
= ( X211 @ X214 ) )
| ~ ! [X215: term] :
( ( $true
= ( X211 @ ( sub @ X212 @ ( push @ X215 @ id ) ) ) )
| ( $true
!= ( X211 @ X215 ) ) )
| ~ ! [X216: subst > term > term] :
( ( ( X213 @ id @ ( lam @ ( X216 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X217: term,X218: subst,X219: subst] :
( ( sub @ ( X216 @ X219 @ X217 ) @ X218 )
= ( X216 @ ( comp @ X219 @ X218 ) @ ( sub @ X217 @ X218 ) ) )
| ~ ! [X220: term] :
( ( ( X213 @ id @ ( X216 @ id @ X220 ) @ id )
= $true )
| ( ( X213 @ id @ X220 @ id )
!= $true ) ) )
| ( $true
= ( X211 @ ( lam @ X212 ) ) ) )
=> ! [X221: term,X222: subst > term > subst > $o] :
( ~ ! [X223: subst,X224: term,X225: subst,X226: subst] :
( ( $true
!= ( X222 @ X225 @ X224 @ ( comp @ X226 @ X223 ) ) )
| ( ( X222 @ ( comp @ X225 @ X226 ) @ ( sub @ X224 @ X226 ) @ X223 )
= $true ) )
| ~ ! [X227: term,X228: term] :
( ( ( X222 @ id @ X227 @ id )
!= $true )
| ( ( X222 @ id @ ( ap @ ( sub @ X228 @ id ) @ X227 ) @ id )
= $true )
| ( ( X222 @ id @ X228 @ id )
!= $true ) )
| ~ ! [X229: term] :
( ( ( var @ ( sub @ X229 @ id ) )
!= $true )
| ( $true
= ( X222 @ id @ X229 @ id ) ) )
| ~ ! [X230: subst,X231: subst,X232: term,X233: subst] :
( ( $true
!= ( X222 @ ( comp @ X233 @ X230 ) @ ( sub @ X232 @ X230 ) @ X231 ) )
| ( ( X222 @ X233 @ X232 @ ( comp @ X230 @ X231 ) )
= $true ) )
| ~ ! [X234: subst > term > term] :
( ~ ! [X235: term] :
( ( $true
= ( X222 @ id @ ( X234 @ id @ X235 ) @ id ) )
| ( ( X222 @ id @ X235 @ id )
!= $true ) )
| ~ ! [X236: term,X237: subst,X238: subst] :
( ( X234 @ ( comp @ X238 @ X237 ) @ ( sub @ X236 @ X237 ) )
= ( sub @ ( X234 @ X238 @ X236 ) @ X237 ) )
| ( ( X222 @ id @ ( lam @ ( X234 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X222 @ id @ X221 @ id )
= $true ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ! [X239: term,X240: subst] :
( ( comp @ sh @ ( push @ X239 @ X240 ) )
= X240 )
<=> ( axshiftcons = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X241: term,X242: term,X243: term,X244: term] :
( ( ( ap @ X244 @ X242 )
!= ( ap @ X243 @ X241 ) )
| ( X241 = X242 ) )
<=> ( apinj2 = $true ) )
& ( ( substmonoid = $true )
<=> ( ! [X246: subst] :
( ( comp @ id @ X246 )
= X246 )
& ! [X245: subst] :
( ( comp @ X245 @ id )
= X245 ) ) )
& ( ( termmset = $true )
<=> ! [X247: term] :
( ( sub @ X247 @ id )
= X247 ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X248: term] :
( ( sub @ X248 @ id )
= X248 )
=> ( ! [X249: subst > term > subst > $o,X250: term] :
( ~ ! [X251: subst > term > term] :
( ( $true
= ( X249 @ id @ ( lam @ ( X251 @ sh @ one ) ) @ id ) )
| ~ ! [X252: subst,X253: term,X254: subst] :
( ( sub @ ( X251 @ X252 @ X253 ) @ X254 )
= ( X251 @ ( comp @ X252 @ X254 ) @ ( sub @ X253 @ X254 ) ) )
| ~ ! [X255: term] :
( ( ( X249 @ id @ X255 @ id )
!= $true )
| ( ( X249 @ id @ ( X251 @ id @ X255 ) @ id )
= $true ) ) )
| ~ ! [X256: term] :
( ( ( X249 @ id @ ( sub @ X250 @ ( push @ X256 @ id ) ) @ id )
= $true )
| ( ( X249 @ id @ X256 @ id )
!= $true ) )
| ( ( X249 @ id @ ( lam @ ( sub @ X250 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X257: subst > term > subst > $o,X258: term] :
( ~ ! [X259: term] :
( ( ( X257 @ id @ X259 @ id )
!= $true )
| ( ( X257 @ id @ ( sub @ X258 @ ( push @ X259 @ id ) ) @ id )
= $true ) )
| ~ ! [X260: subst > term > term] :
( ~ ! [X261: term,X262: subst,X263: subst] :
( ( X260 @ ( comp @ X262 @ X263 ) @ ( sub @ X261 @ X263 ) )
= ( sub @ ( X260 @ X262 @ X261 ) @ X263 ) )
| ~ ! [X264: term] :
( ( ( X257 @ id @ ( X260 @ id @ X264 ) @ id )
= $true )
| ( ( X257 @ id @ X264 @ id )
!= $true ) )
| ( ( X257 @ id @ ( lam @ ( X260 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X257 @ id @ ( lam @ X258 ) @ id )
= $true ) ) ) ) )
& ( lamnotap != $true )
& ( axclos = $true )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X265: term,X266: subst > term > subst > $o,X267: term > $o] :
( ( $true
= ( X267 @ ( lam @ X265 ) ) )
| ~ ! [X268: term] :
( ( ( X267 @ X268 )
!= $true )
| ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ id ) ) ) ) )
| ~ ! [X269: term] :
( ( X267 @ X269 )
= ( X266 @ id @ X269 @ id ) )
| ~ ! [X270: subst > term > term] :
( ~ ! [X271: term] :
( ( $true
= ( X266 @ id @ ( X270 @ id @ X271 ) @ id ) )
| ( ( X266 @ id @ X271 @ id )
!= $true ) )
| ( ( X266 @ id @ ( lam @ ( X270 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X272: subst,X273: subst,X274: term] :
( ( sub @ ( X270 @ X272 @ X274 ) @ X273 )
= ( X270 @ ( comp @ X272 @ X273 ) @ ( sub @ X274 @ X273 ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ! [X275: term,X276: term,X277: term,X278: term] :
( ( ( ap @ X278 @ X275 )
!= ( ap @ X276 @ X277 ) )
| ( X276 = X278 ) )
<=> ( apinj1 = $true ) )
& ( ( ! [X279: term,X280: term > $o] :
( ( ( X280 @ X279 )
= $true )
| ~ ! [X281: term] :
( ( $true
!= ( var @ X281 ) )
| ( ( X280 @ X281 )
= $true ) )
| ~ ! [X282: term] :
( ~ ! [X283: term] :
( ( ( X280 @ X283 )
!= $true )
| ( $true
= ( X280 @ ( sub @ X282 @ ( push @ X283 @ id ) ) ) ) )
| ( ( X280 @ ( lam @ X282 ) )
= $true ) )
| ~ ! [X284: term,X285: term] :
( ( ( X280 @ ( ap @ X285 @ X284 ) )
= $true )
| ( $true
!= ( X280 @ X285 ) )
| ( $true
!= ( X280 @ X284 ) ) ) )
=> ( ! [X286: term,X287: subst > term > subst > $o,X288: term] :
( ~ ! [X289: subst,X290: subst,X291: term,X292: subst] :
( ( $true
!= ( X287 @ X289 @ X291 @ ( comp @ X290 @ X292 ) ) )
| ( ( X287 @ ( comp @ X289 @ X290 ) @ ( sub @ X291 @ X290 ) @ X292 )
= $true ) )
| ~ ! [X293: term,X294: term] :
( ( ( X287 @ id @ ( ap @ ( sub @ X293 @ id ) @ X294 ) @ id )
= $true )
| ( $true
!= ( X287 @ id @ X293 @ id ) )
| ( ( X287 @ id @ X294 @ id )
!= $true ) )
| ( $true
!= ( X287 @ id @ X286 @ id ) )
| ~ ! [X295: subst,X296: subst,X297: term,X298: subst] :
( ( ( X287 @ ( comp @ X295 @ X296 ) @ ( sub @ X297 @ X296 ) @ X298 )
!= $true )
| ( $true
= ( X287 @ X295 @ X297 @ ( comp @ X296 @ X298 ) ) ) )
| ( ( X287 @ id @ X288 @ id )
!= $true )
| ( $true
= ( X287 @ id @ ( ap @ X286 @ X288 ) @ id ) ) )
=> ( ! [X299: term,X300: subst > term > subst > $o] :
( ~ ! [X301: term] :
( ( ( X300 @ id @ ( sub @ X299 @ ( push @ X301 @ id ) ) @ id )
= $true )
| ( $true
!= ( X300 @ id @ X301 @ id ) ) )
| ~ ! [X302: subst,X303: subst,X304: term,X305: subst] :
( ( ( X300 @ X305 @ X304 @ ( comp @ X302 @ X303 ) )
!= $true )
| ( ( X300 @ ( comp @ X305 @ X302 ) @ ( sub @ X304 @ X302 ) @ X303 )
= $true ) )
| ~ ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( $true
= ( X300 @ X307 @ X306 @ ( comp @ X308 @ X309 ) ) )
| ( ( X300 @ ( comp @ X307 @ X308 ) @ ( sub @ X306 @ X308 ) @ X309 )
!= $true ) )
| ( ( X300 @ id @ ( lam @ X299 ) @ id )
= $true )
| ~ ! [X310: subst > term > term] :
( ( ( X300 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X311: term] :
( ( ( X300 @ id @ ( X310 @ id @ X311 ) @ id )
= $true )
| ( $true
!= ( X300 @ id @ X311 @ id ) ) )
| ~ ! [X312: term,X313: subst,X314: subst] :
( ( X310 @ ( comp @ X314 @ X313 ) @ ( sub @ X312 @ X313 ) )
= ( sub @ ( X310 @ X314 @ X312 ) @ X313 ) ) ) )
=> ! [X315: term,X316: subst > term > subst > $o] :
( ~ ! [X317: term,X318: term] :
( ( $true
= ( X316 @ id @ ( ap @ ( sub @ X318 @ id ) @ X317 ) @ id ) )
| ( ( X316 @ id @ X317 @ id )
!= $true )
| ( ( X316 @ id @ X318 @ id )
!= $true ) )
| ~ ! [X319: subst > term > term] :
( ~ ! [X320: term,X321: subst,X322: subst] :
( ( sub @ ( X319 @ X321 @ X320 ) @ X322 )
= ( X319 @ ( comp @ X321 @ X322 ) @ ( sub @ X320 @ X322 ) ) )
| ( $true
= ( X316 @ id @ ( lam @ ( X319 @ sh @ one ) ) @ id ) )
| ~ ! [X323: term] :
( ( ( X316 @ id @ X323 @ id )
!= $true )
| ( ( X316 @ id @ ( X319 @ id @ X323 ) @ id )
= $true ) ) )
| ~ ! [X324: subst,X325: term,X326: subst,X327: subst] :
( ( ( X316 @ ( comp @ X324 @ X327 ) @ ( sub @ X325 @ X327 ) @ X326 )
= $true )
| ( $true
!= ( X316 @ X324 @ X325 @ ( comp @ X327 @ X326 ) ) ) )
| ~ ! [X328: subst,X329: subst,X330: subst,X331: term] :
( ( ( X316 @ X328 @ X331 @ ( comp @ X329 @ X330 ) )
= $true )
| ( ( X316 @ ( comp @ X328 @ X329 ) @ ( sub @ X331 @ X329 ) @ X330 )
!= $true ) )
| ~ ! [X332: term] :
( ( ( var @ ( sub @ X332 @ id ) )
!= $true )
| ( ( X316 @ id @ X332 @ id )
= $true ) )
| ( ( X316 @ id @ X315 @ id )
= $true ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X333: subst > term > subst > $o] :
~ ! [X334: term > $o] :
~ ! [X335: term] :
( ( X334 @ X335 )
= ( X333 @ id @ X335 @ id ) ) )
& ( pushprop_lem0_gthm = $true )
& ( ( hoaslamnotap = $true )
<=> ! [X336: term,X337: term,X338: subst > term > term] :
( ~ ! [X339: subst,X340: subst,X341: term] :
( ( sub @ ( X338 @ X340 @ X341 ) @ X339 )
= ( X338 @ ( comp @ X340 @ X339 ) @ ( sub @ X341 @ X339 ) ) )
| ( ( ap @ ( sub @ X336 @ id ) @ X337 )
!= ( lam @ ( X338 @ sh @ one ) ) ) ) )
& ( axabs = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ! [X342: term,X343: subst > term > subst > $o] :
( ~ ! [X344: term] :
( ( ( X343 @ id @ ( sub @ X342 @ ( push @ X344 @ id ) ) @ id )
= $true )
| ( ( X343 @ id @ X344 @ id )
!= $true ) )
| ~ ! [X345: subst > term > term] :
( ~ ! [X346: term,X347: subst,X348: subst] :
( ( sub @ ( X345 @ X347 @ X346 ) @ X348 )
= ( X345 @ ( comp @ X347 @ X348 ) @ ( sub @ X346 @ X348 ) ) )
| ~ ! [X349: term] :
( ( $true
= ( X343 @ id @ ( X345 @ id @ X349 ) @ id ) )
| ( ( X343 @ id @ X349 @ id )
!= $true ) )
| ( ( X343 @ id @ ( lam @ ( X345 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X343 @ id @ ( lam @ ( sub @ X342 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ( ! [X350: subst,X351: term] :
( ( sub @ one @ ( push @ X351 @ X350 ) )
= X351 )
=> ( ! [X352: term,X353: subst] :
( ( comp @ sh @ ( push @ X352 @ X353 ) )
= X353 )
=> ( ! [X354: subst] :
( ( comp @ X354 @ id )
= X354 )
=> ( ! [X355: term,X356: term > $o] :
( ~ ! [X357: term] :
( ( $true
= ( X356 @ X357 ) )
| ( ( var @ X357 )
!= $true ) )
| ~ ! [X358: term] :
( ( $true
= ( X356 @ ( lam @ X358 ) ) )
| ( ( X356 @ X358 )
!= $true ) )
| ~ ! [X359: term,X360: term] :
( ( $true
= ( X356 @ ( ap @ X360 @ X359 ) ) )
| ( $true
!= ( X356 @ X359 ) )
| ( $true
!= ( X356 @ X360 ) ) )
| ( ( X356 @ X355 )
= $true ) )
=> ! [X361: term,X362: term > $o,X363: subst] :
( ( ( X362 @ ( sub @ X361 @ X363 ) )
= $true )
| ~ ! [X364: term] :
( ( $true
= ( X362 @ ( sub @ X364 @ X363 ) ) )
| ( ( var @ X364 )
!= $true ) )
| ~ ! [X365: term] :
( ~ ! [X366: term] :
( ( $true
= ( X362 @ ( sub @ X365 @ ( push @ X366 @ id ) ) ) )
| ( ( X362 @ X366 )
!= $true ) )
| ( $true
= ( X362 @ ( lam @ X365 ) ) ) )
| ~ ! [X367: term,X368: term] :
( ( ( X362 @ X367 )
!= $true )
| ( $true
= ( X362 @ ( ap @ X368 @ X367 ) ) )
| ( ( X362 @ X368 )
!= $true ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( axassoc = $true )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X369: term > $o,X370: subst > term > subst > $o,X371: term,X372: term] :
( ~ ! [X373: subst,X374: subst,X375: term,X376: subst] :
( ( ( X370 @ X374 @ X375 @ ( comp @ X373 @ X376 ) )
!= $true )
| ( ( X370 @ ( comp @ X374 @ X373 ) @ ( sub @ X375 @ X373 ) @ X376 )
= $true ) )
| ~ ! [X377: term] :
( ( X369 @ X377 )
= ( X370 @ id @ X377 @ id ) )
| ~ ! [X378: term,X379: subst,X380: subst,X381: subst] :
( ( $true
= ( X370 @ X380 @ X378 @ ( comp @ X381 @ X379 ) ) )
| ( ( X370 @ ( comp @ X380 @ X381 ) @ ( sub @ X378 @ X381 ) @ X379 )
!= $true ) )
| ( ( X369 @ ( ap @ X372 @ X371 ) )
= $true )
| ( ( X369 @ X372 )
!= $true )
| ( $true
!= ( X369 @ X371 ) )
| ~ ! [X382: term,X383: term] :
( ( ( X370 @ id @ X382 @ id )
!= $true )
| ( ( X370 @ id @ ( ap @ ( sub @ X383 @ id ) @ X382 ) @ id )
= $true )
| ( ( X370 @ id @ X383 @ id )
!= $true ) ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X384: term] :
~ ! [X385: subst > term > term] :
~ ! [X386: subst,X387: term] :
( ( sub @ X384 @ ( push @ X387 @ X386 ) )
= ( X385 @ X386 @ X387 ) ) )
& ( ( ! [X388: term] :
~ ! [X389: subst > term > term] :
~ ! [X390: term,X391: subst] :
( ( sub @ X388 @ ( push @ X390 @ X391 ) )
= ( X389 @ X391 @ X390 ) )
=> ( ! [X392: term] :
( ( sub @ X392 @ id )
= X392 )
=> ! [X393: term,X394: subst > term > subst > $o,X395: term > $o] :
( ( ( X395 @ ( lam @ X393 ) )
= $true )
| ~ ! [X396: term] :
( ( ( X395 @ ( sub @ X393 @ ( push @ X396 @ id ) ) )
= $true )
| ( ( X395 @ X396 )
!= $true ) )
| ~ ! [X397: subst > term > term] :
( ( ( X394 @ id @ ( lam @ ( X397 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X398: term] :
( ( ( X394 @ id @ X398 @ id )
!= $true )
| ( ( X394 @ id @ ( X397 @ id @ X398 ) @ id )
= $true ) )
| ~ ! [X399: term,X400: subst,X401: subst] :
( ( sub @ ( X397 @ X400 @ X399 ) @ X401 )
= ( X397 @ ( comp @ X400 @ X401 ) @ ( sub @ X399 @ X401 ) ) ) )
| ~ ! [X402: term] :
( ( X395 @ X402 )
= ( X394 @ id @ X402 @ id ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( lamnotvar = $true )
& ( pushprop_lthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ! [X403: term > $o,X404: term] :
( ~ ! [X405: term,X406: term] :
( ( ( X403 @ ( ap @ X405 @ X406 ) )
= $true )
| ( ( X403 @ X405 )
!= $true )
| ( $true
!= ( X403 @ X406 ) ) )
| ~ ! [X407: term] :
( ( $true
= ( X403 @ X407 ) )
| ( $true
!= ( var @ X407 ) ) )
| ( ( X403 @ X404 )
= $true )
| ~ ! [X408: term] :
( ( ( X403 @ X408 )
!= $true )
| ( ( X403 @ ( lam @ X408 ) )
= $true ) ) )
<=> ( induction = $true ) )
& ( ( ! [X409: term] :
( ( sub @ X409 @ id )
= X409 )
=> ( ! [X410: term,X411: subst > term > subst > $o] :
( ~ ! [X412: subst > term > term] :
( ~ ! [X413: term] :
( ( $true
!= ( X411 @ id @ X413 @ id ) )
| ( ( X411 @ id @ ( X412 @ id @ X413 ) @ id )
= $true ) )
| ( ( X411 @ id @ ( lam @ ( X412 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X414: subst,X415: subst,X416: term] :
( ( X412 @ ( comp @ X414 @ X415 ) @ ( sub @ X416 @ X415 ) )
= ( sub @ ( X412 @ X414 @ X416 ) @ X415 ) ) )
| ~ ! [X417: term] :
( ( ( X411 @ id @ X417 @ id )
!= $true )
| ( $true
= ( X411 @ id @ ( sub @ X410 @ ( push @ X417 @ id ) ) @ id ) ) )
| ( ( X411 @ id @ ( lam @ ( sub @ X410 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X418: term,X419: subst > term > subst > $o] :
( ( $true
= ( X419 @ id @ ( lam @ X418 ) @ id ) )
| ~ ! [X420: subst,X421: subst,X422: term,X423: subst] :
( ( ( X419 @ X423 @ X422 @ ( comp @ X421 @ X420 ) )
= $true )
| ( ( X419 @ ( comp @ X423 @ X421 ) @ ( sub @ X422 @ X421 ) @ X420 )
!= $true ) )
| ~ ! [X424: subst,X425: subst,X426: subst,X427: term] :
( ( $true
= ( X419 @ ( comp @ X426 @ X425 ) @ ( sub @ X427 @ X425 ) @ X424 ) )
| ( ( X419 @ X426 @ X427 @ ( comp @ X425 @ X424 ) )
!= $true ) )
| ~ ! [X428: subst > term > term] :
( ~ ! [X429: term,X430: subst,X431: subst] :
( ( sub @ ( X428 @ X431 @ X429 ) @ X430 )
= ( X428 @ ( comp @ X431 @ X430 ) @ ( sub @ X429 @ X430 ) ) )
| ( $true
= ( X419 @ id @ ( lam @ ( X428 @ sh @ one ) ) @ id ) )
| ~ ! [X432: term] :
( ( ( X419 @ id @ X432 @ id )
!= $true )
| ( $true
= ( X419 @ id @ ( X428 @ id @ X432 ) @ id ) ) ) )
| ~ ! [X433: term] :
( ( $true
!= ( X419 @ id @ X433 @ id ) )
| ( ( X419 @ id @ ( sub @ X418 @ ( push @ X433 @ id ) ) @ id )
= $true ) ) ) ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( ( ! [X434: term,X435: subst] :
( ( sub @ one @ ( push @ X434 @ X435 ) )
= X434 )
=> ( ! [X436: term,X437: subst] :
( ( comp @ sh @ ( push @ X436 @ X437 ) )
= X437 )
=> ( ! [X438: term,X439: term] :
( ( ( lam @ X438 )
!= ( lam @ X439 ) )
| ( X438 = X439 ) )
=> ! [X440: term,X441: subst > term > term,X442: subst > term > term,X443: subst] :
( ( ( X441 @ X443 @ X440 )
= ( X442 @ X443 @ X440 ) )
| ~ ! [X447: subst,X448: subst,X449: term] :
( ( X442 @ ( comp @ X448 @ X447 ) @ ( sub @ X449 @ X447 ) )
= ( sub @ ( X442 @ X448 @ X449 ) @ X447 ) )
| ~ ! [X444: term,X445: subst,X446: subst] :
( ( X441 @ ( comp @ X446 @ X445 ) @ ( sub @ X444 @ X445 ) )
= ( sub @ ( X441 @ X446 @ X444 ) @ X445 ) )
| ( ( lam @ ( X441 @ sh @ one ) )
!= ( lam @ ( X442 @ sh @ one ) ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ! [X450: d_term] : ( d_one = X450 )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X451: subst > term > subst > $o,X452: term,X453: term] :
( ( ( X451 @ id @ X452 @ id )
!= $true )
| ( ( X451 @ id @ ( ap @ X453 @ X452 ) @ id )
= $true )
| ( ( X451 @ id @ X453 @ id )
!= $true )
| ~ ! [X454: subst,X455: subst,X456: term,X457: subst] :
( ( $true
= ( X451 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 ) )
| ( ( X451 @ X457 @ X456 @ ( comp @ X455 @ X454 ) )
!= $true ) )
| ~ ! [X458: term,X459: term] :
( ( $true
= ( X451 @ id @ ( ap @ ( sub @ X458 @ id ) @ X459 ) @ id ) )
| ( $true
!= ( X451 @ id @ X459 @ id ) )
| ( ( X451 @ id @ X458 @ id )
!= $true ) )
| ~ ! [X460: subst,X461: subst,X462: term,X463: subst] :
( ( ( X451 @ ( comp @ X460 @ X463 ) @ ( sub @ X462 @ X463 ) @ X461 )
!= $true )
| ( ( X451 @ X460 @ X462 @ ( comp @ X463 @ X461 ) )
= $true ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1_gthm = $true )
& ! [X464: subst] :
? [X465: d_subst] :
( ( d2subst @ X465 )
= X464 )
& ( ! [X466: subst] :
( ( comp @ X466 @ id )
= X466 )
<=> ( axidr = $true ) )
& ( ! [X467: term] :
~ ! [X468: subst > term > term] :
( ( X468 @ sh @ one )
!= ( sub @ X467 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ! [X469: d_term,X470: d_term] :
( ( ( d2term @ X469 )
= ( d2term @ X470 ) )
=> ( X469 = X470 ) )
& ( ( axidl = $true )
<=> ! [X471: subst] :
( ( comp @ id @ X471 )
= X471 ) )
& ( ! [X472: term > $o,X473: subst,X474: term > $o,X475: term] :
( ~ ! [X476: term] :
( ( X472 @ X476 )
= ( X474 @ ( sub @ X476 @ ( push @ X475 @ X473 ) ) ) )
| ( ( X474 @ X475 )
!= $true )
| ( ( X472 @ one )
= $true ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( hoasapinj1_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X477: term,X478: subst > term > subst > $o] :
( ~ ! [X479: subst,X480: subst,X481: subst,X482: term] :
( ( ( X478 @ ( comp @ X481 @ X479 ) @ ( sub @ X482 @ X479 ) @ X480 )
= $true )
| ( ( X478 @ X481 @ X482 @ ( comp @ X479 @ X480 ) )
!= $true ) )
| ( ( X478 @ id @ ( lam @ X477 ) @ id )
= $true )
| ~ ! [X483: term] :
( ( $true
= ( X478 @ id @ ( sub @ X477 @ ( push @ X483 @ id ) ) @ id ) )
| ( ( X478 @ id @ X483 @ id )
!= $true ) )
| ~ ! [X484: subst > term > term] :
( ( $true
= ( X478 @ id @ ( lam @ ( X484 @ sh @ one ) ) @ id ) )
| ~ ! [X485: subst,X486: subst,X487: term] :
( ( sub @ ( X484 @ X486 @ X487 ) @ X485 )
= ( X484 @ ( comp @ X486 @ X485 ) @ ( sub @ X487 @ X485 ) ) )
| ~ ! [X488: term] :
( ( ( X478 @ id @ ( X484 @ id @ X488 ) @ id )
= $true )
| ( $true
!= ( X478 @ id @ X488 @ id ) ) ) )
| ~ ! [X489: subst,X490: subst,X491: term,X492: subst] :
( ( ( X478 @ ( comp @ X492 @ X490 ) @ ( sub @ X491 @ X490 ) @ X489 )
!= $true )
| ( ( X478 @ X492 @ X491 @ ( comp @ X490 @ X489 ) )
= $true ) ) ) )
& ( ( hoasinduction = $true )
<=> ! [X493: term,X494: subst > term > subst > $o] :
( ~ ! [X495: term,X496: subst,X497: subst,X498: subst] :
( ( ( X494 @ X498 @ X495 @ ( comp @ X497 @ X496 ) )
= $true )
| ( $true
!= ( X494 @ ( comp @ X498 @ X497 ) @ ( sub @ X495 @ X497 ) @ X496 ) ) )
| ~ ! [X499: term,X500: term] :
( ( ( X494 @ id @ X499 @ id )
!= $true )
| ( ( X494 @ id @ X500 @ id )
!= $true )
| ( ( X494 @ id @ ( ap @ ( sub @ X500 @ id ) @ X499 ) @ id )
= $true ) )
| ~ ! [X501: subst > term > term] :
( ~ ! [X502: subst,X503: term,X504: subst] :
( ( X501 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X501 @ X502 @ X503 ) @ X504 ) )
| ~ ! [X505: term] :
( ( ( X494 @ id @ X505 @ id )
!= $true )
| ( ( X494 @ id @ ( X501 @ id @ X505 ) @ id )
= $true ) )
| ( $true
= ( X494 @ id @ ( lam @ ( X501 @ sh @ one ) ) @ id ) ) )
| ~ ! [X506: term] :
( ( ( var @ ( sub @ X506 @ id ) )
!= $true )
| ( $true
= ( X494 @ id @ X506 @ id ) ) )
| ~ ! [X507: subst,X508: subst,X509: term,X510: subst] :
( ( $true
= ( X494 @ ( comp @ X510 @ X508 ) @ ( sub @ X509 @ X508 ) @ X507 ) )
| ( ( X494 @ X510 @ X509 @ ( comp @ X508 @ X507 ) )
!= $true ) )
| ( $true
= ( X494 @ id @ X493 @ id ) ) ) )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X511: term,X512: subst > term > subst > $o] :
( ( ( X512 @ id @ ( lam @ X511 ) @ id )
= $true )
| ~ ! [X513: subst > term > term] :
( ( ( X512 @ id @ ( lam @ ( X513 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X514: subst,X515: term,X516: subst] :
( ( X513 @ ( comp @ X514 @ X516 ) @ ( sub @ X515 @ X516 ) )
= ( sub @ ( X513 @ X514 @ X515 ) @ X516 ) )
| ~ ! [X517: term] :
( ( ( X512 @ id @ X517 @ id )
!= $true )
| ( $true
= ( X512 @ id @ ( X513 @ id @ X517 ) @ id ) ) ) )
| ~ ! [X518: term] :
( ( ( X512 @ id @ X518 @ id )
!= $true )
| ( ( X512 @ id @ ( sub @ X511 @ ( push @ X518 @ id ) ) @ id )
= $true ) ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ulamvar1 != $true )
& ( hoasapinj2_gthm = $true )
& ( ( ! [X519: subst,X520: term] :
( ( sub @ one @ ( push @ X520 @ X519 ) )
= X520 )
=> ( ! [X521: subst,X522: term] :
( ( comp @ sh @ ( push @ X522 @ X521 ) )
= X521 )
=> ! [X523: term > $o,X524: term > $o,X525: term,X526: subst,X527: term] :
( ( $true
!= ( X523 @ X527 ) )
| ( $true
= ( X524 @ ( sub @ X527 @ ( push @ X525 @ X526 ) ) ) ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( induction2_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X528: term,X529: term,X530: term,X531: term] :
( ( X529 = X530 )
| ( ( ap @ X531 @ X529 )
!= ( ap @ X528 @ X530 ) ) )
=> ! [X532: term,X533: term,X534: term,X535: term] :
( ( X532 = X534 )
| ( ( ap @ ( sub @ X533 @ id ) @ X532 )
!= ( ap @ ( sub @ X535 @ id ) @ X534 ) ) ) ) )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X536: subst] :
( ( comp @ id @ X536 )
= X536 )
=> ( ! [X537: subst] :
( ( comp @ X537 @ id )
= X537 )
=> ( ! [X538: subst] :
( ( comp @ X538 @ id )
= X538 )
& ! [X539: subst] :
( ( comp @ id @ X539 )
= X539 ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( pushprop_lem1v2_gthm = $true ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,plain,
( ( ( induction2lem = $true )
<=> ! [X0: subst,X1: term > $o,X2: term] :
( ~ ! [X3: term,X4: term] :
( ( ( X1 @ X3 )
!= $true )
| ( ( X1 @ ( ap @ X3 @ X4 ) )
= $true )
| ( ( X1 @ X4 )
!= $true ) )
| ( ( X1 @ ( sub @ X2 @ X0 ) )
= $true )
| ~ ! [X5: term] :
( ( ( var @ X5 )
!= $true )
| ( ( X1 @ ( sub @ X5 @ X0 ) )
= $true ) )
| ~ ! [X6: term] :
( ( ( X1 @ ( lam @ X6 ) )
= $true )
| ~ ! [X7: term] :
( ( $true
!= ( X1 @ X7 ) )
| ( $true
= ( X1 @ ( sub @ X6 @ ( push @ X7 @ id ) ) ) ) ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( pushprop_lthm_orig = $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X8: term] :
~ ! [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
!= ( X9 @ sh @ one ) ) )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( one
= ( d2term @ d_one ) )
& ( ( hoasapinj1 = $true )
<=> ! [X13: term,X14: term,X15: term,X16: term] :
( ( X13 = X16 )
| ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( hoasinduction_lem1 = $true )
& ( axvarshift = $true )
& ( ! [X17: term,X18: subst > term > subst > $o,X19: term > $o] :
( ~ ! [X20: term] :
( ( ( X19 @ ( sub @ X17 @ ( push @ X20 @ id ) ) )
= $true )
| ( $true
!= ( X19 @ X20 ) ) )
| ~ ! [X21: term] :
( ( X19 @ X21 )
= ( X18 @ id @ X21 @ id ) )
| ~ ! [X22: subst > term > term] :
( ~ ! [X23: subst,X24: subst,X25: term] :
( ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( X22 @ X24 @ X25 ) @ X23 ) )
| ~ ! [X26: term] :
( ( $true
!= ( X18 @ id @ X26 @ id ) )
| ( ( X18 @ id @ ( X22 @ id @ X26 ) @ id )
= $true ) )
| ( $true
= ( X18 @ id @ ( lam @ ( X22 @ sh @ one ) ) @ id ) ) )
| ~ ! [X27: subst,X28: term,X29: subst,X30: subst] :
( ( $true
!= ( X18 @ X27 @ X28 @ ( comp @ X29 @ X30 ) ) )
| ( ( X18 @ ( comp @ X27 @ X29 ) @ ( sub @ X28 @ X29 ) @ X30 )
= $true ) )
| ~ ! [X31: subst,X32: subst,X33: subst,X34: term] :
( ( $true
!= ( X18 @ ( comp @ X33 @ X32 ) @ ( sub @ X34 @ X32 ) @ X31 ) )
| ( ( X18 @ X33 @ X34 @ ( comp @ X32 @ X31 ) )
= $true ) )
| ( ( X19 @ ( lam @ X17 ) )
= $true ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) )
= ( Y3 @ Y4 ) ) ) ) )
& ( pushprop = $true )
& ( ! [X40: term,X41: term] :
( ( X40 = X41 )
| ( ( sub @ X40 @ sh )
!= ( sub @ X41 @ sh ) ) )
<=> ( shinj = $true ) )
& ( ( pushprop_lem0 = $true )
<=> ! [X42: term > $o,X43: term,X44: subst] :
~ ! [X45: term > $o] :
~ ! [X46: term] :
( ( X45 @ X46 )
= ( X42 @ ( sub @ X46 @ ( push @ X43 @ X44 ) ) ) ) )
& ( ! [X47: term > $o,X48: term] :
( ~ ! [X49: term] :
( ( ( var @ X49 )
!= $true )
| ( ( X47 @ X49 )
= $true ) )
| ~ ! [X50: term] :
( ~ ! [X51: term] :
( ( $true
!= ( X47 @ X51 ) )
| ( ( X47 @ ( sub @ X50 @ ( push @ X51 @ id ) ) )
= $true ) )
| ( ( X47 @ ( lam @ X50 ) )
= $true ) )
| ~ ! [X52: term,X53: term] :
( ( $true
!= ( X47 @ X52 ) )
| ( $true
= ( X47 @ ( ap @ X52 @ X53 ) ) )
| ( $true
!= ( X47 @ X53 ) ) )
| ( $true
= ( X47 @ X48 ) ) )
<=> ( induction2 = $true ) )
& ( ( ! [X54: subst > term > subst > $o] :
~ ! [X55: term > $o] :
~ ! [X56: term] :
( ( X55 @ X56 )
= ( X54 @ id @ X56 @ id ) )
=> ( ! [X57: term > $o,X58: term] :
( ~ ! [X59: term,X60: term] :
( ( $true
!= ( X57 @ X59 ) )
| ( $true
= ( X57 @ ( ap @ X60 @ X59 ) ) )
| ( ( X57 @ X60 )
!= $true ) )
| ~ ! [X61: term] :
( ( $true
!= ( var @ X61 ) )
| ( ( X57 @ X61 )
= $true ) )
| ( ( X57 @ X58 )
= $true )
| ~ ! [X62: term] :
( ( $true
= ( X57 @ ( lam @ X62 ) ) )
| ~ ! [X63: term] :
( ( $true
!= ( X57 @ X63 ) )
| ( $true
= ( X57 @ ( sub @ X62 @ ( push @ X63 @ id ) ) ) ) ) ) )
=> ( ! [X64: term] :
( ( sub @ X64 @ id )
= X64 )
=> ( ! [X65: subst > term > subst > $o,X66: term > $o,X67: term] :
( ~ ! [X68: subst > term > term] :
( ~ ! [X69: term] :
( ( ( X65 @ id @ X69 @ id )
!= $true )
| ( ( X65 @ id @ ( X68 @ id @ X69 ) @ id )
= $true ) )
| ~ ! [X70: subst,X71: subst,X72: term] :
( ( X68 @ ( comp @ X71 @ X70 ) @ ( sub @ X72 @ X70 ) )
= ( sub @ ( X68 @ X71 @ X72 ) @ X70 ) )
| ( $true
= ( X65 @ id @ ( lam @ ( X68 @ sh @ one ) ) @ id ) ) )
| ~ ! [X73: term] :
( ( ( X66 @ ( sub @ X67 @ ( push @ X73 @ id ) ) )
= $true )
| ( ( X66 @ X73 )
!= $true ) )
| ~ ! [X74: term] :
( ( X65 @ id @ X74 @ id )
= ( X66 @ X74 ) )
| ( $true
= ( X66 @ ( lam @ X67 ) ) ) )
=> ! [X75: subst > term > subst > $o,X76: term] :
( ( ( X75 @ id @ X76 @ id )
= $true )
| ~ ! [X77: subst > term > term] :
( ( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X78: subst,X79: term,X80: subst] :
( ( X77 @ ( comp @ X80 @ X78 ) @ ( sub @ X79 @ X78 ) )
= ( sub @ ( X77 @ X80 @ X79 ) @ X78 ) )
| ~ ! [X81: term] :
( ( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
= $true )
| ( $true
!= ( X75 @ id @ X81 @ id ) ) ) )
| ~ ! [X82: term,X83: term] :
( ( $true
= ( X75 @ id @ ( ap @ ( sub @ X83 @ id ) @ X82 ) @ id ) )
| ( ( X75 @ id @ X83 @ id )
!= $true )
| ( ( X75 @ id @ X82 @ id )
!= $true ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( ( ! [X84: term] :
( ( sub @ X84 @ id )
= X84 )
=> ! [X85: term,X86: subst > term > subst > $o,X87: term > $o] :
( ~ ! [X88: term] :
( ( X87 @ X88 )
= ( X86 @ id @ X88 @ id ) )
| ~ ! [X89: subst > term > term] :
( ~ ! [X90: term] :
( ( $true
= ( X86 @ id @ ( X89 @ id @ X90 ) @ id ) )
| ( ( X86 @ id @ X90 @ id )
!= $true ) )
| ( ( X86 @ id @ ( lam @ ( X89 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X91: term,X92: subst,X93: subst] :
( ( sub @ ( X89 @ X93 @ X91 ) @ X92 )
= ( X89 @ ( comp @ X93 @ X92 ) @ ( sub @ X91 @ X92 ) ) ) )
| ~ ! [X94: subst,X95: subst,X96: subst,X97: term] :
( ( ( X86 @ ( comp @ X95 @ X96 ) @ ( sub @ X97 @ X96 ) @ X94 )
= $true )
| ( ( X86 @ X95 @ X97 @ ( comp @ X96 @ X94 ) )
!= $true ) )
| ~ ! [X98: term,X99: subst,X100: subst,X101: subst] :
( ( ( X86 @ X99 @ X98 @ ( comp @ X100 @ X101 ) )
= $true )
| ( $true
!= ( X86 @ ( comp @ X99 @ X100 ) @ ( sub @ X98 @ X100 ) @ X101 ) ) )
| ( ( X87 @ ( lam @ X85 ) )
= $true )
| ~ ! [X102: term] :
( ( ( X87 @ ( sub @ X85 @ ( push @ X102 @ id ) ) )
= $true )
| ( ( X87 @ X102 )
!= $true ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( ( hoasapinj2 = $true )
<=> ! [X103: term,X104: term,X105: term,X106: term] :
( ( X105 = X106 )
| ( ( ap @ ( sub @ X104 @ id ) @ X106 )
!= ( ap @ ( sub @ X103 @ id ) @ X105 ) ) ) )
& ( hoaslaminj_gthm = $true )
& ( ! [X107: subst] :
( ( push @ ( sub @ one @ X107 ) @ ( comp @ sh @ X107 ) )
= X107 )
<=> ( axscons = $true ) )
& ( apnotvar = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X108: term] :
( ( sub @ X108 @ id )
= X108 )
=> ( ! [X109: term,X110: term,X111: term,X112: term] :
( ( ( ap @ X111 @ X110 )
!= ( ap @ X112 @ X109 ) )
| ( X111 = X112 ) )
=> ! [X113: term,X114: term,X115: term,X116: term] :
( ( ( ap @ ( sub @ X114 @ id ) @ X116 )
!= ( ap @ ( sub @ X113 @ id ) @ X115 ) )
| ( X113 = X114 ) ) ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( termmset_gthm = $true )
& ! [X117: d_subst,X118: d_subst] :
( ( ( d2subst @ X118 )
= ( d2subst @ X117 ) )
=> ( X117 = X118 ) )
& ! [X119: term] :
? [X120: d_term] :
( ( d2term @ X120 )
= X119 )
& ( pushprop_lem3v2_lthm = $true )
& ( axapp = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X121: term,X122: subst,X123: term > $o] :
~ ! [X124: term > $o] :
~ ! [X125: term] :
( ( X124 @ X125 )
= ( X123 @ ( sub @ X125 @ ( push @ X121 @ X122 ) ) ) ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X126: term,X127: subst > term > subst > $o] :
( ~ ! [X128: term] :
( ( ( X127 @ id @ X128 @ id )
!= $true )
| ( $true
= ( X127 @ id @ ( sub @ X126 @ ( push @ X128 @ id ) ) @ id ) ) )
| ~ ! [X129: term,X130: subst > term > term] :
( ~ ! [X131: subst,X132: term,X133: subst] :
( ( sub @ ( X130 @ X133 @ X132 ) @ X131 )
= ( sub @ ( sub @ X129 @ ( push @ X132 @ X133 ) ) @ X131 ) )
| ( ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X134: term] :
( ( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
= $true )
| ( ( X127 @ id @ X134 @ id )
!= $true ) )
| ~ ! [X135: term,X136: subst,X137: subst] :
( ( sub @ X129 @ ( push @ ( sub @ X135 @ X137 ) @ ( comp @ X136 @ X137 ) ) )
= ( X130 @ ( comp @ X136 @ X137 ) @ ( sub @ X135 @ X137 ) ) ) )
| ( ( X127 @ id @ ( lam @ ( sub @ X126 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( ! [X138: term > $o,X139: term,X140: term,X141: subst,X142: term > $o] :
( ( ( X142 @ X139 )
!= $true )
| ( ( X138 @ ( sub @ X139 @ ( push @ X140 @ X141 ) ) )
= $true ) )
<=> ( pushprop_lem1 = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X143: subst > term > subst > $o,X144: term] :
( ~ ! [X145: subst > term > term] :
( ( $true
= ( X143 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id ) )
| ~ ! [X146: term] :
( ( ( X143 @ id @ X146 @ id )
!= $true )
| ( ( X143 @ id @ ( X145 @ id @ X146 ) @ id )
= $true ) )
| ~ ! [X147: subst,X148: term,X149: subst] :
( ( sub @ ( X145 @ X147 @ X148 ) @ X149 )
= ( X145 @ ( comp @ X147 @ X149 ) @ ( sub @ X148 @ X149 ) ) ) )
| ~ ! [X150: term,X151: term] :
( ( ( X143 @ id @ X151 @ id )
!= $true )
| ( ( X143 @ id @ ( ap @ ( sub @ X150 @ id ) @ X151 ) @ id )
= $true )
| ( ( X143 @ id @ X150 @ id )
!= $true ) )
| ( ( X143 @ id @ X144 @ id )
= $true ) ) )
& ( ! [X152: term,X153: term] :
( ( X152 = X153 )
| ( ( lam @ X152 )
!= ( lam @ X153 ) ) )
<=> ( laminj = $true ) )
& ( ( ! [X154: term] :
( ( sub @ X154 @ id )
= X154 )
=> ! [X155: term] :
( ( sub @ X155 @ id )
= X155 ) )
<=> ( termmset_lthm = $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( ! [X156: term] :
( ( sub @ X156 @ id )
= X156 )
=> ( ! [X157: term > $o,X158: term,X159: subst] :
( ~ ! [X160: term,X161: term] :
( ( $true
= ( X157 @ ( ap @ X160 @ X161 ) ) )
| ( ( X157 @ X160 )
!= $true )
| ( ( X157 @ X161 )
!= $true ) )
| ~ ! [X162: term] :
( ( ( var @ X162 )
!= $true )
| ( ( X157 @ ( sub @ X162 @ X159 ) )
= $true ) )
| ~ ! [X163: term] :
( ~ ! [X164: term] :
( ( ( X157 @ ( sub @ X163 @ ( push @ X164 @ id ) ) )
= $true )
| ( ( X157 @ X164 )
!= $true ) )
| ( ( X157 @ ( lam @ X163 ) )
= $true ) )
| ( ( X157 @ ( sub @ X158 @ X159 ) )
= $true ) )
=> ! [X165: term > $o,X166: term] :
( ~ ! [X167: term,X168: term] :
( ( ( X165 @ X168 )
!= $true )
| ( $true
= ( X165 @ ( ap @ X168 @ X167 ) ) )
| ( ( X165 @ X167 )
!= $true ) )
| ( ( X165 @ X166 )
= $true )
| ~ ! [X169: term] :
( ( ( var @ X169 )
!= $true )
| ( ( X165 @ X169 )
= $true ) )
| ~ ! [X170: term] :
( ~ ! [X171: term] :
( ( ( X165 @ ( sub @ X170 @ ( push @ X171 @ id ) ) )
= $true )
| ( ( X165 @ X171 )
!= $true ) )
| ( ( X165 @ ( lam @ X170 ) )
= $true ) ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( ulamvarind = $true )
& ( induction2lem_gthm = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( ! [X172: subst,X173: term] :
( ( sub @ one @ ( push @ X173 @ X172 ) )
= X173 )
=> ! [X174: subst,X175: term > $o,X176: term > $o,X177: term] :
( ~ ! [X178: term] :
( ( X175 @ X178 )
= ( X176 @ ( sub @ X178 @ ( push @ X177 @ X174 ) ) ) )
| ( ( X176 @ X177 )
!= $true )
| ( $true
= ( X175 @ one ) ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( ! [X179: term] :
( ( sub @ X179 @ id )
= X179 )
<=> ( axvarid = $true ) )
& ( ! [X180: subst,X181: term] :
( ( sub @ one @ ( push @ X181 @ X180 ) )
= X181 )
<=> ( axvarcons = $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 ) ) ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ! [X185: d_subst] : ( d_id = X185 )
& ( hoasinduction_lem1_lthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X186: term] :
~ ! [X187: subst > term > term] :
~ ! [X188: subst,X189: term] :
( ( X187 @ X188 @ X189 )
= ( sub @ X186 @ ( push @ X189 @ X188 ) ) ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( pushprop_lem1_gthm = $true )
& ( axmap = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X190: subst > term > subst > $o,X191: term] :
( ~ ! [X192: term] :
( ( ( X190 @ id @ X192 @ id )
!= $true )
| ( ( X190 @ id @ ( sub @ X191 @ ( push @ X192 @ id ) ) @ id )
= $true ) )
| ~ ! [X193: subst > term > term] :
( ~ ! [X194: term] :
( ( ( X190 @ id @ X194 @ id )
!= $true )
| ( ( X190 @ id @ ( X193 @ id @ X194 ) @ id )
= $true ) )
| ( ( X190 @ id @ ( lam @ ( X193 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X195: term,X196: subst,X197: subst] :
( ( X193 @ ( comp @ X197 @ X196 ) @ ( sub @ X195 @ X196 ) )
= ( sub @ ( X193 @ X197 @ X195 ) @ X196 ) ) )
| ( ( X190 @ id @ ( lam @ ( sub @ X191 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( ( hoaslaminj = $true )
<=> ! [X198: subst > term > term,X199: subst > term > term,X200: subst,X201: term] :
( ~ ! [X202: term,X203: subst,X204: subst] :
( ( sub @ ( X198 @ X204 @ X202 ) @ X203 )
= ( X198 @ ( comp @ X204 @ X203 ) @ ( sub @ X202 @ X203 ) ) )
| ( ( lam @ ( X198 @ sh @ one ) )
!= ( lam @ ( X199 @ sh @ one ) ) )
| ~ ! [X205: term,X206: subst,X207: subst] :
( ( sub @ ( X199 @ X206 @ X205 ) @ X207 )
= ( X199 @ ( comp @ X206 @ X207 ) @ ( sub @ X205 @ X207 ) ) )
| ( ( X199 @ X200 @ X201 )
= ( X198 @ X200 @ X201 ) ) ) )
& ( ( ! [X208: subst > term > subst > $o] :
~ ! [X209: term > $o] :
~ ! [X210: term] :
( ( X208 @ id @ X210 @ id )
= ( X209 @ X210 ) )
=> ( ! [X211: term,X212: term > $o] :
( ~ ! [X213: term,X214: term] :
( ( $true
!= ( X212 @ X213 ) )
| ( ( X212 @ X214 )
!= $true )
| ( $true
= ( X212 @ ( ap @ X214 @ X213 ) ) ) )
| ( $true
= ( X212 @ X211 ) )
| ~ ! [X215: term] :
( ~ ! [X216: term] :
( ( $true
!= ( X212 @ X216 ) )
| ( ( X212 @ ( sub @ X215 @ ( push @ X216 @ id ) ) )
= $true ) )
| ( $true
= ( X212 @ ( lam @ X215 ) ) ) )
| ~ ! [X217: term] :
( ( $true
!= ( var @ X217 ) )
| ( ( X212 @ X217 )
= $true ) ) )
=> ( ! [X218: term] :
( ( sub @ X218 @ id )
= X218 )
=> ( ! [X219: term > $o,X220: term,X221: subst > term > subst > $o] :
( ~ ! [X222: term] :
( ( X221 @ id @ X222 @ id )
= ( X219 @ X222 ) )
| ~ ! [X223: term] :
( ( $true
= ( X219 @ ( sub @ X220 @ ( push @ X223 @ id ) ) ) )
| ( ( X219 @ X223 )
!= $true ) )
| ~ ! [X224: subst > term > term] :
( ( $true
= ( X221 @ id @ ( lam @ ( X224 @ sh @ one ) ) @ id ) )
| ~ ! [X225: term,X226: subst,X227: subst] :
( ( X224 @ ( comp @ X227 @ X226 ) @ ( sub @ X225 @ X226 ) )
= ( sub @ ( X224 @ X227 @ X225 ) @ X226 ) )
| ~ ! [X228: term] :
( ( ( X221 @ id @ ( X224 @ id @ X228 ) @ id )
= $true )
| ( $true
!= ( X221 @ id @ X228 @ id ) ) ) )
| ( ( X219 @ ( lam @ X220 ) )
= $true ) )
=> ! [X229: term,X230: subst > term > subst > $o] :
( ~ ! [X231: subst,X232: term,X233: subst,X234: subst] :
( ( ( X230 @ X233 @ X232 @ ( comp @ X234 @ X231 ) )
!= $true )
| ( $true
= ( X230 @ ( comp @ X233 @ X234 ) @ ( sub @ X232 @ X234 ) @ X231 ) ) )
| ~ ! [X235: term,X236: term] :
( ( ( X230 @ id @ X235 @ id )
!= $true )
| ( ( X230 @ id @ ( ap @ ( sub @ X236 @ id ) @ X235 ) @ id )
= $true )
| ( ( X230 @ id @ X236 @ id )
!= $true ) )
| ~ ! [X237: term] :
( ( ( var @ ( sub @ X237 @ id ) )
!= $true )
| ( ( X230 @ id @ X237 @ id )
= $true ) )
| ~ ! [X238: subst,X239: subst,X240: term,X241: subst] :
( ( ( X230 @ ( comp @ X241 @ X238 ) @ ( sub @ X240 @ X238 ) @ X239 )
!= $true )
| ( $true
= ( X230 @ X241 @ X240 @ ( comp @ X238 @ X239 ) ) ) )
| ~ ! [X242: subst > term > term] :
( ~ ! [X243: term] :
( ( $true
= ( X230 @ id @ ( X242 @ id @ X243 ) @ id ) )
| ( $true
!= ( X230 @ id @ X243 @ id ) ) )
| ~ ! [X244: term,X245: subst,X246: subst] :
( ( sub @ ( X242 @ X246 @ X244 ) @ X245 )
= ( X242 @ ( comp @ X246 @ X245 ) @ ( sub @ X244 @ X245 ) ) )
| ( $true
= ( X230 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id ) ) )
| ( ( X230 @ id @ X229 @ id )
= $true ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ! [X247: term,X248: subst] :
( ( comp @ sh @ ( push @ X247 @ X248 ) )
= X248 )
<=> ( axshiftcons = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X251: term,X252: term,X253: term,X254: term] :
( ( ( ap @ X253 @ X251 )
!= ( ap @ X254 @ X252 ) )
| ( X251 = X252 ) )
<=> ( apinj2 = $true ) )
& ( ( substmonoid = $true )
<=> ( ! [X255: subst] :
( ( comp @ X255 @ id )
= X255 )
& ! [X256: subst] :
( ( comp @ id @ X256 )
= X256 ) ) )
& ( ! [X257: term] :
( ( sub @ X257 @ id )
= X257 )
<=> ( termmset = $true ) )
& ( hoasinduction_gthm = $true )
& ( pushprop_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X258: term] :
( ( sub @ X258 @ id )
= X258 )
=> ( ! [X259: subst > term > subst > $o,X260: term] :
( ~ ! [X261: subst > term > term] :
( ( $true
= ( X259 @ id @ ( lam @ ( X261 @ sh @ one ) ) @ id ) )
| ~ ! [X262: subst,X263: term,X264: subst] :
( ( sub @ ( X261 @ X262 @ X263 ) @ X264 )
= ( X261 @ ( comp @ X262 @ X264 ) @ ( sub @ X263 @ X264 ) ) )
| ~ ! [X265: term] :
( ( ( X259 @ id @ X265 @ id )
!= $true )
| ( ( X259 @ id @ ( X261 @ id @ X265 ) @ id )
= $true ) ) )
| ~ ! [X266: term] :
( ( ( X259 @ id @ ( sub @ X260 @ ( push @ X266 @ id ) ) @ id )
= $true )
| ( ( X259 @ id @ X266 @ id )
!= $true ) )
| ( $true
= ( X259 @ id @ ( lam @ ( sub @ X260 @ ( push @ one @ sh ) ) ) @ id ) ) )
=> ! [X267: subst > term > subst > $o,X268: term] :
( ~ ! [X269: term] :
( ( $true
!= ( X267 @ id @ X269 @ id ) )
| ( $true
= ( X267 @ id @ ( sub @ X268 @ ( push @ X269 @ id ) ) @ id ) ) )
| ~ ! [X270: subst > term > term] :
( ~ ! [X271: term,X272: subst,X273: subst] :
( ( X270 @ ( comp @ X272 @ X273 ) @ ( sub @ X271 @ X273 ) )
= ( sub @ ( X270 @ X272 @ X271 ) @ X273 ) )
| ~ ! [X274: term] :
( ( ( X267 @ id @ ( X270 @ id @ X274 ) @ id )
= $true )
| ( ( X267 @ id @ X274 @ id )
!= $true ) )
| ( ( X267 @ id @ ( lam @ ( X270 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X267 @ id @ ( lam @ X268 ) @ id )
= $true ) ) ) ) )
& ( lamnotap != $true )
& ( axclos = $true )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X275: term,X276: subst > term > subst > $o,X277: term > $o] :
( ( ( X277 @ ( lam @ X275 ) )
= $true )
| ~ ! [X278: term] :
( ( $true
!= ( X277 @ X278 ) )
| ( ( X277 @ ( sub @ X275 @ ( push @ X278 @ id ) ) )
= $true ) )
| ~ ! [X279: term] :
( ( X276 @ id @ X279 @ id )
= ( X277 @ X279 ) )
| ~ ! [X280: subst > term > term] :
( ~ ! [X281: term] :
( ( ( X276 @ id @ ( X280 @ id @ X281 ) @ id )
= $true )
| ( ( X276 @ id @ X281 @ id )
!= $true ) )
| ( ( X276 @ id @ ( lam @ ( X280 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X282: subst,X283: subst,X284: term] :
( ( X280 @ ( comp @ X282 @ X283 ) @ ( sub @ X284 @ X283 ) )
= ( sub @ ( X280 @ X282 @ X284 ) @ X283 ) ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( apinj1 = $true )
<=> ! [X285: term,X286: term,X287: term,X288: term] :
( ( ( ap @ X286 @ X287 )
!= ( ap @ X288 @ X285 ) )
| ( X286 = X288 ) ) )
& ( ( ! [X289: term,X290: term > $o] :
( ( $true
= ( X290 @ X289 ) )
| ~ ! [X291: term] :
( ( ( var @ X291 )
!= $true )
| ( ( X290 @ X291 )
= $true ) )
| ~ ! [X292: term] :
( ~ ! [X293: term] :
( ( ( X290 @ X293 )
!= $true )
| ( $true
= ( X290 @ ( sub @ X292 @ ( push @ X293 @ id ) ) ) ) )
| ( ( X290 @ ( lam @ X292 ) )
= $true ) )
| ~ ! [X294: term,X295: term] :
( ( ( X290 @ ( ap @ X295 @ X294 ) )
= $true )
| ( $true
!= ( X290 @ X295 ) )
| ( ( X290 @ X294 )
!= $true ) ) )
=> ( ! [X296: term,X297: subst > term > subst > $o,X298: term] :
( ~ ! [X299: subst,X300: subst,X301: term,X302: subst] :
( ( ( X297 @ X299 @ X301 @ ( comp @ X300 @ X302 ) )
!= $true )
| ( $true
= ( X297 @ ( comp @ X299 @ X300 ) @ ( sub @ X301 @ X300 ) @ X302 ) ) )
| ~ ! [X303: term,X304: term] :
( ( ( X297 @ id @ ( ap @ ( sub @ X303 @ id ) @ X304 ) @ id )
= $true )
| ( $true
!= ( X297 @ id @ X303 @ id ) )
| ( ( X297 @ id @ X304 @ id )
!= $true ) )
| ( ( X297 @ id @ X296 @ id )
!= $true )
| ~ ! [X305: subst,X306: subst,X307: term,X308: subst] :
( ( ( X297 @ ( comp @ X305 @ X306 ) @ ( sub @ X307 @ X306 ) @ X308 )
!= $true )
| ( ( X297 @ X305 @ X307 @ ( comp @ X306 @ X308 ) )
= $true ) )
| ( ( X297 @ id @ X298 @ id )
!= $true )
| ( ( X297 @ id @ ( ap @ X296 @ X298 ) @ id )
= $true ) )
=> ( ! [X309: term,X310: subst > term > subst > $o] :
( ~ ! [X311: term] :
( ( ( X310 @ id @ ( sub @ X309 @ ( push @ X311 @ id ) ) @ id )
= $true )
| ( ( X310 @ id @ X311 @ id )
!= $true ) )
| ~ ! [X312: subst,X313: subst,X314: term,X315: subst] :
( ( ( X310 @ X315 @ X314 @ ( comp @ X312 @ X313 ) )
!= $true )
| ( ( X310 @ ( comp @ X315 @ X312 ) @ ( sub @ X314 @ X312 ) @ X313 )
= $true ) )
| ~ ! [X316: term,X317: subst,X318: subst,X319: subst] :
( ( ( X310 @ X317 @ X316 @ ( comp @ X318 @ X319 ) )
= $true )
| ( $true
!= ( X310 @ ( comp @ X317 @ X318 ) @ ( sub @ X316 @ X318 ) @ X319 ) ) )
| ( ( X310 @ id @ ( lam @ X309 ) @ id )
= $true )
| ~ ! [X320: subst > term > term] :
( ( $true
= ( X310 @ id @ ( lam @ ( X320 @ sh @ one ) ) @ id ) )
| ~ ! [X321: term] :
( ( $true
= ( X310 @ id @ ( X320 @ id @ X321 ) @ id ) )
| ( ( X310 @ id @ X321 @ id )
!= $true ) )
| ~ ! [X322: term,X323: subst,X324: subst] :
( ( X320 @ ( comp @ X324 @ X323 ) @ ( sub @ X322 @ X323 ) )
= ( sub @ ( X320 @ X324 @ X322 ) @ X323 ) ) ) )
=> ! [X325: term,X326: subst > term > subst > $o] :
( ~ ! [X327: term,X328: term] :
( ( ( X326 @ id @ ( ap @ ( sub @ X328 @ id ) @ X327 ) @ id )
= $true )
| ( ( X326 @ id @ X327 @ id )
!= $true )
| ( ( X326 @ id @ X328 @ id )
!= $true ) )
| ~ ! [X329: subst > term > term] :
( ~ ! [X330: term,X331: subst,X332: subst] :
( ( X329 @ ( comp @ X331 @ X332 ) @ ( sub @ X330 @ X332 ) )
= ( sub @ ( X329 @ X331 @ X330 ) @ X332 ) )
| ( $true
= ( X326 @ id @ ( lam @ ( X329 @ sh @ one ) ) @ id ) )
| ~ ! [X333: term] :
( ( ( X326 @ id @ X333 @ id )
!= $true )
| ( ( X326 @ id @ ( X329 @ id @ X333 ) @ id )
= $true ) ) )
| ~ ! [X334: subst,X335: term,X336: subst,X337: subst] :
( ( $true
= ( X326 @ ( comp @ X334 @ X337 ) @ ( sub @ X335 @ X337 ) @ X336 ) )
| ( ( X326 @ X334 @ X335 @ ( comp @ X337 @ X336 ) )
!= $true ) )
| ~ ! [X338: subst,X339: subst,X340: subst,X341: term] :
( ( ( X326 @ X338 @ X341 @ ( comp @ X339 @ X340 ) )
= $true )
| ( $true
!= ( X326 @ ( comp @ X338 @ X339 ) @ ( sub @ X341 @ X339 ) @ X340 ) ) )
| ~ ! [X342: term] :
( ( $true
!= ( var @ ( sub @ X342 @ id ) ) )
| ( ( X326 @ id @ X342 @ id )
= $true ) )
| ( ( X326 @ id @ X325 @ id )
= $true ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( ! [X343: subst > term > subst > $o] :
~ ! [X344: term > $o] :
~ ! [X345: term] :
( ( X344 @ X345 )
= ( X343 @ id @ X345 @ id ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( pushprop_lem0_gthm = $true )
& ( ! [X346: term,X347: term,X348: subst > term > term] :
( ~ ! [X349: subst,X350: subst,X351: term] :
( ( sub @ ( X348 @ X350 @ X351 ) @ X349 )
= ( X348 @ ( comp @ X350 @ X349 ) @ ( sub @ X351 @ X349 ) ) )
| ( ( ap @ ( sub @ X346 @ id ) @ X347 )
!= ( lam @ ( X348 @ sh @ one ) ) ) )
<=> ( hoaslamnotap = $true ) )
& ( axabs = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ! [X352: term,X353: subst > term > subst > $o] :
( ~ ! [X354: term] :
( ( ( X353 @ id @ ( sub @ X352 @ ( push @ X354 @ id ) ) @ id )
= $true )
| ( ( X353 @ id @ X354 @ id )
!= $true ) )
| ~ ! [X355: subst > term > term] :
( ~ ! [X356: term,X357: subst,X358: subst] :
( ( sub @ ( X355 @ X357 @ X356 ) @ X358 )
= ( X355 @ ( comp @ X357 @ X358 ) @ ( sub @ X356 @ X358 ) ) )
| ~ ! [X359: term] :
( ( ( X353 @ id @ ( X355 @ id @ X359 ) @ id )
= $true )
| ( ( X353 @ id @ X359 @ id )
!= $true ) )
| ( ( X353 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X353 @ id @ ( lam @ ( sub @ X352 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ( ! [X360: subst,X361: term] :
( ( sub @ one @ ( push @ X361 @ X360 ) )
= X361 )
=> ( ! [X362: term,X363: subst] :
( ( comp @ sh @ ( push @ X362 @ X363 ) )
= X363 )
=> ( ! [X364: subst] :
( ( comp @ X364 @ id )
= X364 )
=> ( ! [X365: term,X366: term > $o] :
( ~ ! [X367: term] :
( ( ( X366 @ X367 )
= $true )
| ( ( var @ X367 )
!= $true ) )
| ~ ! [X368: term] :
( ( $true
= ( X366 @ ( lam @ X368 ) ) )
| ( ( X366 @ X368 )
!= $true ) )
| ~ ! [X369: term,X370: term] :
( ( ( X366 @ ( ap @ X370 @ X369 ) )
= $true )
| ( ( X366 @ X369 )
!= $true )
| ( $true
!= ( X366 @ X370 ) ) )
| ( ( X366 @ X365 )
= $true ) )
=> ! [X371: term,X372: term > $o,X373: subst] :
( ( $true
= ( X372 @ ( sub @ X371 @ X373 ) ) )
| ~ ! [X374: term] :
( ( ( X372 @ ( sub @ X374 @ X373 ) )
= $true )
| ( ( var @ X374 )
!= $true ) )
| ~ ! [X375: term] :
( ~ ! [X376: term] :
( ( ( X372 @ ( sub @ X375 @ ( push @ X376 @ id ) ) )
= $true )
| ( $true
!= ( X372 @ X376 ) ) )
| ( ( X372 @ ( lam @ X375 ) )
= $true ) )
| ~ ! [X377: term,X378: term] :
( ( ( X372 @ X377 )
!= $true )
| ( ( X372 @ ( ap @ X378 @ X377 ) )
= $true )
| ( $true
!= ( X372 @ X378 ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( axassoc = $true )
& ( pushprop_lem2v2 = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X379: term > $o,X380: subst > term > subst > $o,X381: term,X382: term] :
( ~ ! [X383: subst,X384: subst,X385: term,X386: subst] :
( ( ( X380 @ X384 @ X385 @ ( comp @ X383 @ X386 ) )
!= $true )
| ( $true
= ( X380 @ ( comp @ X384 @ X383 ) @ ( sub @ X385 @ X383 ) @ X386 ) ) )
| ~ ! [X387: term] :
( ( X379 @ X387 )
= ( X380 @ id @ X387 @ id ) )
| ~ ! [X388: term,X389: subst,X390: subst,X391: subst] :
( ( ( X380 @ X390 @ X388 @ ( comp @ X391 @ X389 ) )
= $true )
| ( ( X380 @ ( comp @ X390 @ X391 ) @ ( sub @ X388 @ X391 ) @ X389 )
!= $true ) )
| ( ( X379 @ ( ap @ X382 @ X381 ) )
= $true )
| ( ( X379 @ X382 )
!= $true )
| ( ( X379 @ X381 )
!= $true )
| ~ ! [X392: term,X393: term] :
( ( $true
!= ( X380 @ id @ X392 @ id ) )
| ( $true
= ( X380 @ id @ ( ap @ ( sub @ X393 @ id ) @ X392 ) @ id ) )
| ( ( X380 @ id @ X393 @ id )
!= $true ) ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X394: term] :
~ ! [X395: subst > term > term] :
~ ! [X396: subst,X397: term] :
( ( sub @ X394 @ ( push @ X397 @ X396 ) )
= ( X395 @ X396 @ X397 ) ) )
& ( ( ! [X398: term] :
~ ! [X399: subst > term > term] :
~ ! [X400: term,X401: subst] :
( ( X399 @ X401 @ X400 )
= ( sub @ X398 @ ( push @ X400 @ X401 ) ) )
=> ( ! [X402: term] :
( ( sub @ X402 @ id )
= X402 )
=> ! [X403: term,X404: subst > term > subst > $o,X405: term > $o] :
( ( $true
= ( X405 @ ( lam @ X403 ) ) )
| ~ ! [X406: term] :
( ( ( X405 @ ( sub @ X403 @ ( push @ X406 @ id ) ) )
= $true )
| ( ( X405 @ X406 )
!= $true ) )
| ~ ! [X407: subst > term > term] :
( ( ( X404 @ id @ ( lam @ ( X407 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X408: term] :
( ( ( X404 @ id @ X408 @ id )
!= $true )
| ( $true
= ( X404 @ id @ ( X407 @ id @ X408 ) @ id ) ) )
| ~ ! [X409: term,X410: subst,X411: subst] :
( ( X407 @ ( comp @ X410 @ X411 ) @ ( sub @ X409 @ X411 ) )
= ( sub @ ( X407 @ X410 @ X409 ) @ X411 ) ) )
| ~ ! [X412: term] :
( ( X404 @ id @ X412 @ id )
= ( X405 @ X412 ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( lamnotvar = $true )
& ( pushprop_lthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( ! [X413: term > $o,X414: term] :
( ~ ! [X415: term,X416: term] :
( ( $true
= ( X413 @ ( ap @ X415 @ X416 ) ) )
| ( $true
!= ( X413 @ X415 ) )
| ( ( X413 @ X416 )
!= $true ) )
| ~ ! [X417: term] :
( ( ( X413 @ X417 )
= $true )
| ( $true
!= ( var @ X417 ) ) )
| ( ( X413 @ X414 )
= $true )
| ~ ! [X418: term] :
( ( ( X413 @ X418 )
!= $true )
| ( ( X413 @ ( lam @ X418 ) )
= $true ) ) )
<=> ( induction = $true ) )
& ( ( ! [X419: term] :
( ( sub @ X419 @ id )
= X419 )
=> ( ! [X420: term,X421: subst > term > subst > $o] :
( ~ ! [X422: subst > term > term] :
( ~ ! [X423: term] :
( ( ( X421 @ id @ X423 @ id )
!= $true )
| ( ( X421 @ id @ ( X422 @ id @ X423 ) @ id )
= $true ) )
| ( ( X421 @ id @ ( lam @ ( X422 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X424: subst,X425: subst,X426: term] :
( ( X422 @ ( comp @ X424 @ X425 ) @ ( sub @ X426 @ X425 ) )
= ( sub @ ( X422 @ X424 @ X426 ) @ X425 ) ) )
| ~ ! [X427: term] :
( ( $true
!= ( X421 @ id @ X427 @ id ) )
| ( $true
= ( X421 @ id @ ( sub @ X420 @ ( push @ X427 @ id ) ) @ id ) ) )
| ( ( X421 @ id @ ( lam @ ( sub @ X420 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X428: term,X429: subst > term > subst > $o] :
( ( ( X429 @ id @ ( lam @ X428 ) @ id )
= $true )
| ~ ! [X430: subst,X431: subst,X432: term,X433: subst] :
( ( ( X429 @ X433 @ X432 @ ( comp @ X431 @ X430 ) )
= $true )
| ( ( X429 @ ( comp @ X433 @ X431 ) @ ( sub @ X432 @ X431 ) @ X430 )
!= $true ) )
| ~ ! [X434: subst,X435: subst,X436: subst,X437: term] :
( ( ( X429 @ ( comp @ X436 @ X435 ) @ ( sub @ X437 @ X435 ) @ X434 )
= $true )
| ( ( X429 @ X436 @ X437 @ ( comp @ X435 @ X434 ) )
!= $true ) )
| ~ ! [X438: subst > term > term] :
( ~ ! [X439: term,X440: subst,X441: subst] :
( ( sub @ ( X438 @ X441 @ X439 ) @ X440 )
= ( X438 @ ( comp @ X441 @ X440 ) @ ( sub @ X439 @ X440 ) ) )
| ( ( X429 @ id @ ( lam @ ( X438 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X442: term] :
( ( ( X429 @ id @ X442 @ id )
!= $true )
| ( ( X429 @ id @ ( X438 @ id @ X442 ) @ id )
= $true ) ) )
| ~ ! [X443: term] :
( ( ( X429 @ id @ X443 @ id )
!= $true )
| ( ( X429 @ id @ ( sub @ X428 @ ( push @ X443 @ id ) ) @ id )
= $true ) ) ) ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X444: term,X445: subst] :
( ( sub @ one @ ( push @ X444 @ X445 ) )
= X444 )
=> ( ! [X446: term,X447: subst] :
( ( comp @ sh @ ( push @ X446 @ X447 ) )
= X447 )
=> ( ! [X448: term,X449: term] :
( ( ( lam @ X449 )
!= ( lam @ X448 ) )
| ( X448 = X449 ) )
=> ! [X450: term,X451: subst > term > term,X452: subst > term > term,X453: subst] :
( ( ( lam @ ( X452 @ sh @ one ) )
!= ( lam @ ( X451 @ sh @ one ) ) )
| ~ ! [X454: term,X455: subst,X456: subst] :
( ( sub @ ( X451 @ X456 @ X454 ) @ X455 )
= ( X451 @ ( comp @ X456 @ X455 ) @ ( sub @ X454 @ X455 ) ) )
| ~ ! [X457: subst,X458: subst,X459: term] :
( ( X452 @ ( comp @ X458 @ X457 ) @ ( sub @ X459 @ X457 ) )
= ( sub @ ( X452 @ X458 @ X459 ) @ X457 ) )
| ( ( X451 @ X453 @ X450 )
= ( X452 @ X453 @ X450 ) ) ) ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ! [X460: d_term] : ( d_one = X460 )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X461: subst > term > subst > $o,X462: term,X463: term] :
( ( ( X461 @ id @ X462 @ id )
!= $true )
| ( $true
= ( X461 @ id @ ( ap @ X463 @ X462 ) @ id ) )
| ( $true
!= ( X461 @ id @ X463 @ id ) )
| ~ ! [X464: subst,X465: subst,X466: term,X467: subst] :
( ( $true
= ( X461 @ ( comp @ X467 @ X465 ) @ ( sub @ X466 @ X465 ) @ X464 ) )
| ( ( X461 @ X467 @ X466 @ ( comp @ X465 @ X464 ) )
!= $true ) )
| ~ ! [X468: term,X469: term] :
( ( ( X461 @ id @ ( ap @ ( sub @ X468 @ id ) @ X469 ) @ id )
= $true )
| ( ( X461 @ id @ X469 @ id )
!= $true )
| ( $true
!= ( X461 @ id @ X468 @ id ) ) )
| ~ ! [X470: subst,X471: subst,X472: term,X473: subst] :
( ( $true
!= ( X461 @ ( comp @ X470 @ X473 ) @ ( sub @ X472 @ X473 ) @ X471 ) )
| ( $true
= ( X461 @ X470 @ X472 @ ( comp @ X473 @ X471 ) ) ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1_gthm = $true )
& ! [X474: subst] :
? [X475: d_subst] :
( ( d2subst @ X475 )
= X474 )
& ( ! [X476: subst] :
( ( comp @ X476 @ id )
= X476 )
<=> ( axidr = $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X477: term] :
~ ! [X478: subst > term > term] :
( ( X478 @ sh @ one )
!= ( sub @ X477 @ ( push @ one @ sh ) ) ) )
& ! [X479: d_term,X480: d_term] :
( ( ( d2term @ X479 )
= ( d2term @ X480 ) )
=> ( X479 = X480 ) )
& ( ! [X481: subst] :
( ( comp @ id @ X481 )
= X481 )
<=> ( axidl = $true ) )
& ( ! [X482: term > $o,X483: subst,X484: term > $o,X485: term] :
( ~ ! [X486: term] :
( ( X484 @ ( sub @ X486 @ ( push @ X485 @ X483 ) ) )
= ( X482 @ X486 ) )
| ( ( X484 @ X485 )
!= $true )
| ( ( X482 @ one )
= $true ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( hoasapinj1_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X487: term,X488: subst > term > subst > $o] :
( ~ ! [X489: subst,X490: subst,X491: subst,X492: term] :
( ( ( X488 @ ( comp @ X491 @ X489 ) @ ( sub @ X492 @ X489 ) @ X490 )
= $true )
| ( ( X488 @ X491 @ X492 @ ( comp @ X489 @ X490 ) )
!= $true ) )
| ( $true
= ( X488 @ id @ ( lam @ X487 ) @ id ) )
| ~ ! [X493: term] :
( ( ( X488 @ id @ ( sub @ X487 @ ( push @ X493 @ id ) ) @ id )
= $true )
| ( ( X488 @ id @ X493 @ id )
!= $true ) )
| ~ ! [X494: subst > term > term] :
( ( ( X488 @ id @ ( lam @ ( X494 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X495: subst,X496: subst,X497: term] :
( ( sub @ ( X494 @ X496 @ X497 ) @ X495 )
= ( X494 @ ( comp @ X496 @ X495 ) @ ( sub @ X497 @ X495 ) ) )
| ~ ! [X498: term] :
( ( $true
= ( X488 @ id @ ( X494 @ id @ X498 ) @ id ) )
| ( ( X488 @ id @ X498 @ id )
!= $true ) ) )
| ~ ! [X499: subst,X500: subst,X501: term,X502: subst] :
( ( $true
!= ( X488 @ ( comp @ X502 @ X500 ) @ ( sub @ X501 @ X500 ) @ X499 ) )
| ( ( X488 @ X502 @ X501 @ ( comp @ X500 @ X499 ) )
= $true ) ) ) )
& ( ( hoasinduction = $true )
<=> ! [X503: term,X504: subst > term > subst > $o] :
( ~ ! [X505: term,X506: subst,X507: subst,X508: subst] :
( ( $true
= ( X504 @ X508 @ X505 @ ( comp @ X507 @ X506 ) ) )
| ( $true
!= ( X504 @ ( comp @ X508 @ X507 ) @ ( sub @ X505 @ X507 ) @ X506 ) ) )
| ~ ! [X509: term,X510: term] :
( ( ( X504 @ id @ X509 @ id )
!= $true )
| ( ( X504 @ id @ X510 @ id )
!= $true )
| ( ( X504 @ id @ ( ap @ ( sub @ X510 @ id ) @ X509 ) @ id )
= $true ) )
| ~ ! [X511: subst > term > term] :
( ~ ! [X512: subst,X513: term,X514: subst] :
( ( X511 @ ( comp @ X512 @ X514 ) @ ( sub @ X513 @ X514 ) )
= ( sub @ ( X511 @ X512 @ X513 ) @ X514 ) )
| ~ ! [X515: term] :
( ( ( X504 @ id @ X515 @ id )
!= $true )
| ( $true
= ( X504 @ id @ ( X511 @ id @ X515 ) @ id ) ) )
| ( ( X504 @ id @ ( lam @ ( X511 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X516: term] :
( ( $true
!= ( var @ ( sub @ X516 @ id ) ) )
| ( ( X504 @ id @ X516 @ id )
= $true ) )
| ~ ! [X517: subst,X518: subst,X519: term,X520: subst] :
( ( ( X504 @ ( comp @ X520 @ X518 ) @ ( sub @ X519 @ X518 ) @ X517 )
= $true )
| ( ( X504 @ X520 @ X519 @ ( comp @ X518 @ X517 ) )
!= $true ) )
| ( ( X504 @ id @ X503 @ id )
= $true ) ) )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X521: term,X522: subst > term > subst > $o] :
( ( ( X522 @ id @ ( lam @ X521 ) @ id )
= $true )
| ~ ! [X523: subst > term > term] :
( ( $true
= ( X522 @ id @ ( lam @ ( X523 @ sh @ one ) ) @ id ) )
| ~ ! [X524: subst,X525: term,X526: subst] :
( ( X523 @ ( comp @ X524 @ X526 ) @ ( sub @ X525 @ X526 ) )
= ( sub @ ( X523 @ X524 @ X525 ) @ X526 ) )
| ~ ! [X527: term] :
( ( ( X522 @ id @ X527 @ id )
!= $true )
| ( ( X522 @ id @ ( X523 @ id @ X527 ) @ id )
= $true ) ) )
| ~ ! [X528: term] :
( ( ( X522 @ id @ X528 @ id )
!= $true )
| ( $true
= ( X522 @ id @ ( sub @ X521 @ ( push @ X528 @ id ) ) @ id ) ) ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ulamvar1 != $true )
& ( hoasapinj2_gthm = $true )
& ( ( ! [X529: subst,X530: term] :
( ( sub @ one @ ( push @ X530 @ X529 ) )
= X530 )
=> ( ! [X531: subst,X532: term] :
( ( comp @ sh @ ( push @ X532 @ X531 ) )
= X531 )
=> ! [X533: term > $o,X534: term > $o,X535: term,X536: subst,X537: term] :
( ( $true
!= ( X533 @ X537 ) )
| ( ( X534 @ ( sub @ X537 @ ( push @ X535 @ X536 ) ) )
= $true ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( induction2_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X538: term,X539: term,X540: term,X541: term] :
( ( ( ap @ X541 @ X539 )
!= ( ap @ X538 @ X540 ) )
| ( X539 = X540 ) )
=> ! [X542: term,X543: term,X544: term,X545: term] :
( ( ( ap @ ( sub @ X545 @ id ) @ X544 )
!= ( ap @ ( sub @ X543 @ id ) @ X542 ) )
| ( X542 = X544 ) ) ) )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X546: subst] :
( ( comp @ id @ X546 )
= X546 )
=> ( ! [X547: subst] :
( ( comp @ X547 @ id )
= X547 )
=> ( ! [X548: subst] :
( ( comp @ X548 @ id )
= X548 )
& ! [X549: subst] :
( ( comp @ id @ X549 )
= X549 ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( pushprop_lem1v2_gthm = $true ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
( ( induction2lem
<=> ! [X0: subst,X1: term > $o,X2: term] :
( ~ ! [X3: term,X4: term] :
( ~ ( X1 @ X3 )
| ( X1 @ ( ap @ X3 @ X4 ) )
| ~ ( X1 @ X4 ) )
| ( X1 @ ( sub @ X2 @ X0 ) )
| ~ ! [X5: term] :
( ~ ( var @ X5 )
| ( X1 @ ( sub @ X5 @ X0 ) ) )
| ~ ! [X6: term] :
( ( X1 @ ( lam @ X6 ) )
| ~ ! [X7: term] :
( ~ ( X1 @ X7 )
| ( X1 @ ( sub @ X6 @ ( push @ X7 @ id ) ) ) ) ) ) )
& hoaslamnotap_gthm
& hoaslamnotvar_gthm
& pushprop_lthm_orig
& ( hoasinduction_lem3b
<=> ! [X8: term] :
~ ! [X9: subst > term > term] :
( ( sub @ X8 @ ( push @ one @ sh ) )
!= ( X9 @ sh @ one ) ) )
& ( hoasinduction_lem0_lthm
<=> ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
& ( one
= ( d2term @ d_one ) )
& ( hoasapinj1
<=> ! [X13: term,X14: term,X15: term,X16: term] :
( ( ( ap @ ( sub @ X13 @ id ) @ X14 )
!= ( ap @ ( sub @ X16 @ id ) @ X15 ) )
| ( X13 = X16 ) ) )
& hoasapnotvar_lthm
& hoasinduction_lem1
& axvarshift
& ( ! [X17: term,X18: subst > term > subst > $o,X19: term > $o] :
( ~ ! [X20: term] :
( ( X19 @ ( sub @ X17 @ ( push @ X20 @ id ) ) )
| ~ ( X19 @ X20 ) )
| ~ ! [X21: term] :
( ( X19 @ X21 )
= ( X18 @ id @ X21 @ id ) )
| ~ ! [X22: subst > term > term] :
( ~ ! [X23: subst,X24: subst,X25: term] :
( ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( X22 @ X24 @ X25 ) @ X23 ) )
| ~ ! [X26: term] :
( ~ ( X18 @ id @ X26 @ id )
| ( X18 @ id @ ( X22 @ id @ X26 ) @ id ) )
| ( X18 @ id @ ( lam @ ( X22 @ sh @ one ) ) @ id ) )
| ~ ! [X27: subst,X28: term,X29: subst,X30: subst] :
( ~ ( X18 @ X27 @ X28 @ ( comp @ X29 @ X30 ) )
| ( X18 @ ( comp @ X27 @ X29 ) @ ( sub @ X28 @ X29 ) @ X30 ) )
| ~ ! [X31: subst,X32: subst,X33: subst,X34: term] :
( ~ ( X18 @ ( comp @ X33 @ X32 ) @ ( sub @ X34 @ X32 ) @ X31 )
| ( X18 @ X33 @ X34 @ ( comp @ X32 @ X31 ) ) )
| ( X19 @ ( lam @ X17 ) ) )
<=> hoasinduction_lem3v2 )
& ( ( ^ [X35: term,X36: subst,X37: term > $o,X38: term > $o] :
! [X39: term] :
( ( X37 @ ( sub @ X39 @ ( push @ X35 @ X36 ) ) )
= ( X38 @ X39 ) ) )
= pushprop_p_and_p_prime )
& pushprop
& ( ! [X40: term,X41: term] :
( ( X40 = X41 )
| ( ( sub @ X40 @ sh )
!= ( sub @ X41 @ sh ) ) )
<=> shinj )
& ( pushprop_lem0
<=> ! [X42: term > $o,X43: term,X44: subst] :
~ ! [X45: term > $o] :
~ ! [X46: term] :
( ( X45 @ X46 )
= ( X42 @ ( sub @ X46 @ ( push @ X43 @ X44 ) ) ) ) )
& ( ! [X47: term > $o,X48: term] :
( ~ ! [X49: term] :
( ~ ( var @ X49 )
| ( X47 @ X49 ) )
| ~ ! [X50: term] :
( ~ ! [X51: term] :
( ~ ( X47 @ X51 )
| ( X47 @ ( sub @ X50 @ ( push @ X51 @ id ) ) ) )
| ( X47 @ ( lam @ X50 ) ) )
| ~ ! [X52: term,X53: term] :
( ~ ( X47 @ X52 )
| ( X47 @ ( ap @ X52 @ X53 ) )
| ~ ( X47 @ X53 ) )
| ( X47 @ X48 ) )
<=> induction2 )
& ( ( ! [X54: subst > term > subst > $o] :
~ ! [X55: term > $o] :
~ ! [X56: term] :
( ( X55 @ X56 )
= ( X54 @ id @ X56 @ id ) )
=> ( ! [X57: term > $o,X58: term] :
( ~ ! [X59: term,X60: term] :
( ~ ( X57 @ X59 )
| ( X57 @ ( ap @ X60 @ X59 ) )
| ~ ( X57 @ X60 ) )
| ~ ! [X61: term] :
( ~ ( var @ X61 )
| ( X57 @ X61 ) )
| ( X57 @ X58 )
| ~ ! [X62: term] :
( ( X57 @ ( lam @ X62 ) )
| ~ ! [X63: term] :
( ~ ( X57 @ X63 )
| ( X57 @ ( sub @ X62 @ ( push @ X63 @ id ) ) ) ) ) )
=> ( ! [X64: term] :
( ( sub @ X64 @ id )
= X64 )
=> ( ! [X65: subst > term > subst > $o,X66: term > $o,X67: term] :
( ~ ! [X68: subst > term > term] :
( ~ ! [X69: term] :
( ~ ( X65 @ id @ X69 @ id )
| ( X65 @ id @ ( X68 @ id @ X69 ) @ id ) )
| ~ ! [X70: subst,X71: subst,X72: term] :
( ( X68 @ ( comp @ X71 @ X70 ) @ ( sub @ X72 @ X70 ) )
= ( sub @ ( X68 @ X71 @ X72 ) @ X70 ) )
| ( X65 @ id @ ( lam @ ( X68 @ sh @ one ) ) @ id ) )
| ~ ! [X73: term] :
( ( X66 @ ( sub @ X67 @ ( push @ X73 @ id ) ) )
| ~ ( X66 @ X73 ) )
| ~ ! [X74: term] :
( ( X65 @ id @ X74 @ id )
= ( X66 @ X74 ) )
| ( X66 @ ( lam @ X67 ) ) )
=> ! [X75: subst > term > subst > $o,X76: term] :
( ( X75 @ id @ X76 @ id )
| ~ ! [X77: subst > term > term] :
( ( X75 @ id @ ( lam @ ( X77 @ sh @ one ) ) @ id )
| ~ ! [X78: subst,X79: term,X80: subst] :
( ( X77 @ ( comp @ X80 @ X78 ) @ ( sub @ X79 @ X78 ) )
= ( sub @ ( X77 @ X80 @ X79 ) @ X78 ) )
| ~ ! [X81: term] :
( ( X75 @ id @ ( X77 @ id @ X81 ) @ id )
| ~ ( X75 @ id @ X81 @ id ) ) )
| ~ ! [X82: term,X83: term] :
( ( X75 @ id @ ( ap @ ( sub @ X83 @ id ) @ X82 ) @ id )
| ~ ( X75 @ id @ X83 @ id )
| ~ ( X75 @ id @ X82 @ id ) ) ) ) ) ) )
<=> hoasinduction_no_psi_cond_lthm )
& ( ( ! [X84: term] :
( ( sub @ X84 @ id )
= X84 )
=> ! [X85: term,X86: subst > term > subst > $o,X87: term > $o] :
( ~ ! [X88: term] :
( ( X87 @ X88 )
= ( X86 @ id @ X88 @ id ) )
| ~ ! [X89: subst > term > term] :
( ~ ! [X90: term] :
( ( X86 @ id @ ( X89 @ id @ X90 ) @ id )
| ~ ( X86 @ id @ X90 @ id ) )
| ( X86 @ id @ ( lam @ ( X89 @ sh @ one ) ) @ id )
| ~ ! [X91: term,X92: subst,X93: subst] :
( ( sub @ ( X89 @ X93 @ X91 ) @ X92 )
= ( X89 @ ( comp @ X93 @ X92 ) @ ( sub @ X91 @ X92 ) ) ) )
| ~ ! [X94: subst,X95: subst,X96: subst,X97: term] :
( ( X86 @ ( comp @ X95 @ X96 ) @ ( sub @ X97 @ X96 ) @ X94 )
| ~ ( X86 @ X95 @ X97 @ ( comp @ X96 @ X94 ) ) )
| ~ ! [X98: term,X99: subst,X100: subst,X101: subst] :
( ( X86 @ X99 @ X98 @ ( comp @ X100 @ X101 ) )
| ~ ( X86 @ ( comp @ X99 @ X100 ) @ ( sub @ X98 @ X100 ) @ X101 ) )
| ( X87 @ ( lam @ X85 ) )
| ~ ! [X102: term] :
( ( X87 @ ( sub @ X85 @ ( push @ X102 @ id ) ) )
| ~ ( X87 @ X102 ) ) ) )
<=> hoasinduction_lem3v2_lthm )
& ( hoasapinj2
<=> ! [X103: term,X104: term,X105: term,X106: term] :
( ( X105 = X106 )
| ( ( ap @ ( sub @ X104 @ id ) @ X106 )
!= ( ap @ ( sub @ X103 @ id ) @ X105 ) ) ) )
& hoaslaminj_gthm
& ( ! [X107: subst] :
( ( push @ ( sub @ one @ X107 ) @ ( comp @ sh @ X107 ) )
= X107 )
<=> axscons )
& apnotvar
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X108: term] :
( ( sub @ X108 @ id )
= X108 )
=> ( ! [X109: term,X110: term,X111: term,X112: term] :
( ( ( ap @ X111 @ X110 )
!= ( ap @ X112 @ X109 ) )
| ( X111 = X112 ) )
=> ! [X113: term,X114: term,X115: term,X116: term] :
( ( ( ap @ ( sub @ X114 @ id ) @ X116 )
!= ( ap @ ( sub @ X113 @ id ) @ X115 ) )
| ( X113 = X114 ) ) ) )
<=> hoasapinj1_lthm )
& termmset_gthm
& ! [X117: d_subst,X118: d_subst] :
( ( ( d2subst @ X118 )
= ( d2subst @ X117 ) )
=> ( X117 = X118 ) )
& ! [X119: term] :
? [X120: d_term] :
( ( d2term @ X120 )
= X119 )
& pushprop_lem3v2_lthm
& axapp
& ( pushprop_lem0_lthm
<=> ! [X121: term,X122: subst,X123: term > $o] :
~ ! [X124: term > $o] :
~ ! [X125: term] :
( ( X124 @ X125 )
= ( X123 @ ( sub @ X125 @ ( push @ X121 @ X122 ) ) ) ) )
& ( hoasinduction_lem3aaa
<=> ! [X126: term,X127: subst > term > subst > $o] :
( ~ ! [X128: term] :
( ~ ( X127 @ id @ X128 @ id )
| ( X127 @ id @ ( sub @ X126 @ ( push @ X128 @ id ) ) @ id ) )
| ~ ! [X129: term,X130: subst > term > term] :
( ~ ! [X131: subst,X132: term,X133: subst] :
( ( sub @ ( X130 @ X133 @ X132 ) @ X131 )
= ( sub @ ( sub @ X129 @ ( push @ X132 @ X133 ) ) @ X131 ) )
| ( X127 @ id @ ( lam @ ( X130 @ sh @ one ) ) @ id )
| ~ ! [X134: term] :
( ( X127 @ id @ ( X130 @ id @ X134 ) @ id )
| ~ ( X127 @ id @ X134 @ id ) )
| ~ ! [X135: term,X136: subst,X137: subst] :
( ( sub @ X129 @ ( push @ ( sub @ X135 @ X137 ) @ ( comp @ X136 @ X137 ) ) )
= ( X130 @ ( comp @ X136 @ X137 ) @ ( sub @ X135 @ X137 ) ) ) )
| ( X127 @ id @ ( lam @ ( sub @ X126 @ ( push @ one @ sh ) ) ) @ id ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( ! [X138: term > $o,X139: term,X140: term,X141: subst,X142: term > $o] :
( ~ ( X142 @ X139 )
| ( X138 @ ( sub @ X139 @ ( push @ X140 @ X141 ) ) ) )
<=> pushprop_lem1 )
& hoasinduction_lem2_lthm
& ( hoasinduction_no_psi_cond
<=> ! [X143: subst > term > subst > $o,X144: term] :
( ~ ! [X145: subst > term > term] :
( ( X143 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id )
| ~ ! [X146: term] :
( ~ ( X143 @ id @ X146 @ id )
| ( X143 @ id @ ( X145 @ id @ X146 ) @ id ) )
| ~ ! [X147: subst,X148: term,X149: subst] :
( ( sub @ ( X145 @ X147 @ X148 ) @ X149 )
= ( X145 @ ( comp @ X147 @ X149 ) @ ( sub @ X148 @ X149 ) ) ) )
| ~ ! [X150: term,X151: term] :
( ~ ( X143 @ id @ X151 @ id )
| ( X143 @ id @ ( ap @ ( sub @ X150 @ id ) @ X151 ) @ id )
| ~ ( X143 @ id @ X150 @ id ) )
| ( X143 @ id @ X144 @ id ) ) )
& ( ! [X152: term,X153: term] :
( ( X152 = X153 )
| ( ( lam @ X152 )
!= ( lam @ X153 ) ) )
<=> laminj )
& ( ( ! [X154: term] :
( ( sub @ X154 @ id )
= X154 )
=> ! [X155: term] :
( ( sub @ X155 @ id )
= X155 ) )
<=> termmset_lthm )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& hoasinduction_lem3_gthm
& pushprop_lem2v2_gthm
& hoasinduction_lem3v2_gthm
& ( ( ! [X156: term] :
( ( sub @ X156 @ id )
= X156 )
=> ( ! [X157: term > $o,X158: term,X159: subst] :
( ~ ! [X160: term,X161: term] :
( ( X157 @ ( ap @ X160 @ X161 ) )
| ~ ( X157 @ X160 )
| ~ ( X157 @ X161 ) )
| ~ ! [X162: term] :
( ~ ( var @ X162 )
| ( X157 @ ( sub @ X162 @ X159 ) ) )
| ~ ! [X163: term] :
( ~ ! [X164: term] :
( ( X157 @ ( sub @ X163 @ ( push @ X164 @ id ) ) )
| ~ ( X157 @ X164 ) )
| ( X157 @ ( lam @ X163 ) ) )
| ( X157 @ ( sub @ X158 @ X159 ) ) )
=> ! [X165: term > $o,X166: term] :
( ~ ! [X167: term,X168: term] :
( ~ ( X165 @ X168 )
| ( X165 @ ( ap @ X168 @ X167 ) )
| ~ ( X165 @ X167 ) )
| ( X165 @ X166 )
| ~ ! [X169: term] :
( ~ ( var @ X169 )
| ( X165 @ X169 ) )
| ~ ! [X170: term] :
( ~ ! [X171: term] :
( ( X165 @ ( sub @ X170 @ ( push @ X171 @ id ) ) )
| ~ ( X165 @ X171 ) )
| ( X165 @ ( lam @ X170 ) ) ) ) ) )
<=> induction2_lthm )
& ulamvarind
& induction2lem_gthm
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ( ( ! [X172: subst,X173: term] :
( ( sub @ one @ ( push @ X173 @ X172 ) )
= X173 )
=> ! [X174: subst,X175: term > $o,X176: term > $o,X177: term] :
( ~ ! [X178: term] :
( ( X175 @ X178 )
= ( X176 @ ( sub @ X178 @ ( push @ X177 @ X174 ) ) ) )
| ~ ( X176 @ X177 )
| ( X175 @ one ) ) )
<=> pushprop_lem1v2_lthm )
& ( ! [X179: term] :
( ( sub @ X179 @ id )
= X179 )
<=> axvarid )
& ( ! [X180: subst,X181: term] :
( ( sub @ one @ ( push @ X181 @ X180 ) )
= X181 )
<=> axvarcons )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [X182: subst > term > subst > $o,X183: term > $o] :
! [X184: term] :
( ( X183 @ X184 )
= ( X182 @ id @ X184 @ id ) ) ) )
& ~ ( var @ ( d2term @ d_one ) )
& ! [X185: d_subst] : ( d_id = X185 )
& hoasinduction_lem1_lthm
& hoasinduction_lem1v2
& hoasinduction_lem3b_gthm
& ( hoasinduction_lem3v2_f
<=> ! [X186: term] :
~ ! [X187: subst > term > term] :
~ ! [X188: subst,X189: term] :
( ( X187 @ X188 @ X189 )
= ( sub @ X186 @ ( push @ X189 @ X188 ) ) ) )
& substmonoid_gthm
& pushprop_lem3v2
& pushprop_lem1_gthm
& axmap
& hoasapnotvar
& ( hoasinduction_lem3aa
<=> ! [X190: subst > term > subst > $o,X191: term] :
( ~ ! [X192: term] :
( ~ ( X190 @ id @ X192 @ id )
| ( X190 @ id @ ( sub @ X191 @ ( push @ X192 @ id ) ) @ id ) )
| ~ ! [X193: subst > term > term] :
( ~ ! [X194: term] :
( ~ ( X190 @ id @ X194 @ id )
| ( X190 @ id @ ( X193 @ id @ X194 ) @ id ) )
| ( X190 @ id @ ( lam @ ( X193 @ sh @ one ) ) @ id )
| ~ ! [X195: term,X196: subst,X197: subst] :
( ( X193 @ ( comp @ X197 @ X196 ) @ ( sub @ X195 @ X196 ) )
= ( sub @ ( X193 @ X197 @ X195 ) @ X196 ) ) )
| ( X190 @ id @ ( lam @ ( sub @ X191 @ ( push @ one @ sh ) ) ) @ id ) ) )
& ( hoaslaminj
<=> ! [X198: subst > term > term,X199: subst > term > term,X200: subst,X201: term] :
( ~ ! [X202: term,X203: subst,X204: subst] :
( ( sub @ ( X198 @ X204 @ X202 ) @ X203 )
= ( X198 @ ( comp @ X204 @ X203 ) @ ( sub @ X202 @ X203 ) ) )
| ( ( lam @ ( X198 @ sh @ one ) )
!= ( lam @ ( X199 @ sh @ one ) ) )
| ~ ! [X205: term,X206: subst,X207: subst] :
( ( sub @ ( X199 @ X206 @ X205 ) @ X207 )
= ( X199 @ ( comp @ X206 @ X207 ) @ ( sub @ X205 @ X207 ) ) )
| ( ( X199 @ X200 @ X201 )
= ( X198 @ X200 @ X201 ) ) ) )
& ( ( ! [X208: subst > term > subst > $o] :
~ ! [X209: term > $o] :
~ ! [X210: term] :
( ( X208 @ id @ X210 @ id )
= ( X209 @ X210 ) )
=> ( ! [X211: term,X212: term > $o] :
( ~ ! [X213: term,X214: term] :
( ~ ( X212 @ X213 )
| ~ ( X212 @ X214 )
| ( X212 @ ( ap @ X214 @ X213 ) ) )
| ( X212 @ X211 )
| ~ ! [X215: term] :
( ~ ! [X216: term] :
( ~ ( X212 @ X216 )
| ( X212 @ ( sub @ X215 @ ( push @ X216 @ id ) ) ) )
| ( X212 @ ( lam @ X215 ) ) )
| ~ ! [X217: term] :
( ~ ( var @ X217 )
| ( X212 @ X217 ) ) )
=> ( ! [X218: term] :
( ( sub @ X218 @ id )
= X218 )
=> ( ! [X219: term > $o,X220: term,X221: subst > term > subst > $o] :
( ~ ! [X222: term] :
( ( X221 @ id @ X222 @ id )
= ( X219 @ X222 ) )
| ~ ! [X223: term] :
( ( X219 @ ( sub @ X220 @ ( push @ X223 @ id ) ) )
| ~ ( X219 @ X223 ) )
| ~ ! [X224: subst > term > term] :
( ( X221 @ id @ ( lam @ ( X224 @ sh @ one ) ) @ id )
| ~ ! [X225: term,X226: subst,X227: subst] :
( ( X224 @ ( comp @ X227 @ X226 ) @ ( sub @ X225 @ X226 ) )
= ( sub @ ( X224 @ X227 @ X225 ) @ X226 ) )
| ~ ! [X228: term] :
( ( X221 @ id @ ( X224 @ id @ X228 ) @ id )
| ~ ( X221 @ id @ X228 @ id ) ) )
| ( X219 @ ( lam @ X220 ) ) )
=> ! [X229: term,X230: subst > term > subst > $o] :
( ~ ! [X231: subst,X232: term,X233: subst,X234: subst] :
( ~ ( X230 @ X233 @ X232 @ ( comp @ X234 @ X231 ) )
| ( X230 @ ( comp @ X233 @ X234 ) @ ( sub @ X232 @ X234 ) @ X231 ) )
| ~ ! [X235: term,X236: term] :
( ~ ( X230 @ id @ X235 @ id )
| ( X230 @ id @ ( ap @ ( sub @ X236 @ id ) @ X235 ) @ id )
| ~ ( X230 @ id @ X236 @ id ) )
| ~ ! [X237: term] :
( ~ ( var @ ( sub @ X237 @ id ) )
| ( X230 @ id @ X237 @ id ) )
| ~ ! [X238: subst,X239: subst,X240: term,X241: subst] :
( ~ ( X230 @ ( comp @ X241 @ X238 ) @ ( sub @ X240 @ X238 ) @ X239 )
| ( X230 @ X241 @ X240 @ ( comp @ X238 @ X239 ) ) )
| ~ ! [X242: subst > term > term] :
( ~ ! [X243: term] :
( ( X230 @ id @ ( X242 @ id @ X243 ) @ id )
| ~ ( X230 @ id @ X243 @ id ) )
| ~ ! [X244: term,X245: subst,X246: subst] :
( ( sub @ ( X242 @ X246 @ X244 ) @ X245 )
= ( X242 @ ( comp @ X246 @ X245 ) @ ( sub @ X244 @ X245 ) ) )
| ( X230 @ id @ ( lam @ ( X242 @ sh @ one ) ) @ id ) )
| ( X230 @ id @ X229 @ id ) ) ) ) ) )
<=> hoasinduction_lthm_3 )
& ( ! [X247: term,X248: subst] :
( ( comp @ sh @ ( push @ X247 @ X248 ) )
= X248 )
<=> axshiftcons )
& ( hoaslam
= ( ^ [X249: subst,X250: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X251: term,X252: term,X253: term,X254: term] :
( ( ( ap @ X253 @ X251 )
!= ( ap @ X254 @ X252 ) )
| ( X251 = X252 ) )
<=> apinj2 )
& ( substmonoid
<=> ( ! [X255: subst] :
( ( comp @ X255 @ id )
= X255 )
& ! [X256: subst] :
( ( comp @ id @ X256 )
= X256 ) ) )
& ( ! [X257: term] :
( ( sub @ X257 @ id )
= X257 )
<=> termmset )
& hoasinduction_gthm
& pushprop_gthm
& hoaslamnotvar
& ( hoasinduction_lem3a_lthm
<=> ( ! [X258: term] :
( ( sub @ X258 @ id )
= X258 )
=> ( ! [X259: subst > term > subst > $o,X260: term] :
( ~ ! [X261: subst > term > term] :
( ( X259 @ id @ ( lam @ ( X261 @ sh @ one ) ) @ id )
| ~ ! [X262: subst,X263: term,X264: subst] :
( ( sub @ ( X261 @ X262 @ X263 ) @ X264 )
= ( X261 @ ( comp @ X262 @ X264 ) @ ( sub @ X263 @ X264 ) ) )
| ~ ! [X265: term] :
( ~ ( X259 @ id @ X265 @ id )
| ( X259 @ id @ ( X261 @ id @ X265 ) @ id ) ) )
| ~ ! [X266: term] :
( ( X259 @ id @ ( sub @ X260 @ ( push @ X266 @ id ) ) @ id )
| ~ ( X259 @ id @ X266 @ id ) )
| ( X259 @ id @ ( lam @ ( sub @ X260 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X267: subst > term > subst > $o,X268: term] :
( ~ ! [X269: term] :
( ~ ( X267 @ id @ X269 @ id )
| ( X267 @ id @ ( sub @ X268 @ ( push @ X269 @ id ) ) @ id ) )
| ~ ! [X270: subst > term > term] :
( ~ ! [X271: term,X272: subst,X273: subst] :
( ( X270 @ ( comp @ X272 @ X273 ) @ ( sub @ X271 @ X273 ) )
= ( sub @ ( X270 @ X272 @ X271 ) @ X273 ) )
| ~ ! [X274: term] :
( ( X267 @ id @ ( X270 @ id @ X274 ) @ id )
| ~ ( X267 @ id @ X274 @ id ) )
| ( X267 @ id @ ( lam @ ( X270 @ sh @ one ) ) @ id ) )
| ( X267 @ id @ ( lam @ X268 ) @ id ) ) ) ) )
& ~ lamnotap
& axclos
& ( hoasinduction_lem3v2a
<=> ! [X275: term,X276: subst > term > subst > $o,X277: term > $o] :
( ( X277 @ ( lam @ X275 ) )
| ~ ! [X278: term] :
( ~ ( X277 @ X278 )
| ( X277 @ ( sub @ X275 @ ( push @ X278 @ id ) ) ) )
| ~ ! [X279: term] :
( ( X276 @ id @ X279 @ id )
= ( X277 @ X279 ) )
| ~ ! [X280: subst > term > term] :
( ~ ! [X281: term] :
( ( X276 @ id @ ( X280 @ id @ X281 ) @ id )
| ~ ( X276 @ id @ X281 @ id ) )
| ( X276 @ id @ ( lam @ ( X280 @ sh @ one ) ) @ id )
| ~ ! [X282: subst,X283: subst,X284: term] :
( ( X280 @ ( comp @ X282 @ X283 ) @ ( sub @ X284 @ X283 ) )
= ( sub @ ( X280 @ X282 @ X284 ) @ X283 ) ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( apinj1
<=> ! [X285: term,X286: term,X287: term,X288: term] :
( ( ( ap @ X286 @ X287 )
!= ( ap @ X288 @ X285 ) )
| ( X286 = X288 ) ) )
& ( ( ! [X289: term,X290: term > $o] :
( ( X290 @ X289 )
| ~ ! [X291: term] :
( ~ ( var @ X291 )
| ( X290 @ X291 ) )
| ~ ! [X292: term] :
( ~ ! [X293: term] :
( ~ ( X290 @ X293 )
| ( X290 @ ( sub @ X292 @ ( push @ X293 @ id ) ) ) )
| ( X290 @ ( lam @ X292 ) ) )
| ~ ! [X294: term,X295: term] :
( ( X290 @ ( ap @ X295 @ X294 ) )
| ~ ( X290 @ X295 )
| ~ ( X290 @ X294 ) ) )
=> ( ! [X296: term,X297: subst > term > subst > $o,X298: term] :
( ~ ! [X299: subst,X300: subst,X301: term,X302: subst] :
( ~ ( X297 @ X299 @ X301 @ ( comp @ X300 @ X302 ) )
| ( X297 @ ( comp @ X299 @ X300 ) @ ( sub @ X301 @ X300 ) @ X302 ) )
| ~ ! [X303: term,X304: term] :
( ( X297 @ id @ ( ap @ ( sub @ X303 @ id ) @ X304 ) @ id )
| ~ ( X297 @ id @ X303 @ id )
| ~ ( X297 @ id @ X304 @ id ) )
| ~ ( X297 @ id @ X296 @ id )
| ~ ! [X305: subst,X306: subst,X307: term,X308: subst] :
( ~ ( X297 @ ( comp @ X305 @ X306 ) @ ( sub @ X307 @ X306 ) @ X308 )
| ( X297 @ X305 @ X307 @ ( comp @ X306 @ X308 ) ) )
| ~ ( X297 @ id @ X298 @ id )
| ( X297 @ id @ ( ap @ X296 @ X298 ) @ id ) )
=> ( ! [X309: term,X310: subst > term > subst > $o] :
( ~ ! [X311: term] :
( ( X310 @ id @ ( sub @ X309 @ ( push @ X311 @ id ) ) @ id )
| ~ ( X310 @ id @ X311 @ id ) )
| ~ ! [X312: subst,X313: subst,X314: term,X315: subst] :
( ~ ( X310 @ X315 @ X314 @ ( comp @ X312 @ X313 ) )
| ( X310 @ ( comp @ X315 @ X312 ) @ ( sub @ X314 @ X312 ) @ X313 ) )
| ~ ! [X316: term,X317: subst,X318: subst,X319: subst] :
( ( X310 @ X317 @ X316 @ ( comp @ X318 @ X319 ) )
| ~ ( X310 @ ( comp @ X317 @ X318 ) @ ( sub @ X316 @ X318 ) @ X319 ) )
| ( X310 @ id @ ( lam @ X309 ) @ id )
| ~ ! [X320: subst > term > term] :
( ( X310 @ id @ ( lam @ ( X320 @ sh @ one ) ) @ id )
| ~ ! [X321: term] :
( ( X310 @ id @ ( X320 @ id @ X321 ) @ id )
| ~ ( X310 @ id @ X321 @ id ) )
| ~ ! [X322: term,X323: subst,X324: subst] :
( ( X320 @ ( comp @ X324 @ X323 ) @ ( sub @ X322 @ X323 ) )
= ( sub @ ( X320 @ X324 @ X322 ) @ X323 ) ) ) )
=> ! [X325: term,X326: subst > term > subst > $o] :
( ~ ! [X327: term,X328: term] :
( ( X326 @ id @ ( ap @ ( sub @ X328 @ id ) @ X327 ) @ id )
| ~ ( X326 @ id @ X327 @ id )
| ~ ( X326 @ id @ X328 @ id ) )
| ~ ! [X329: subst > term > term] :
( ~ ! [X330: term,X331: subst,X332: subst] :
( ( X329 @ ( comp @ X331 @ X332 ) @ ( sub @ X330 @ X332 ) )
= ( sub @ ( X329 @ X331 @ X330 ) @ X332 ) )
| ( X326 @ id @ ( lam @ ( X329 @ sh @ one ) ) @ id )
| ~ ! [X333: term] :
( ~ ( X326 @ id @ X333 @ id )
| ( X326 @ id @ ( X329 @ id @ X333 ) @ id ) ) )
| ~ ! [X334: subst,X335: term,X336: subst,X337: subst] :
( ( X326 @ ( comp @ X334 @ X337 ) @ ( sub @ X335 @ X337 ) @ X336 )
| ~ ( X326 @ X334 @ X335 @ ( comp @ X337 @ X336 ) ) )
| ~ ! [X338: subst,X339: subst,X340: subst,X341: term] :
( ( X326 @ X338 @ X341 @ ( comp @ X339 @ X340 ) )
| ~ ( X326 @ ( comp @ X338 @ X339 ) @ ( sub @ X341 @ X339 ) @ X340 ) )
| ~ ! [X342: term] :
( ~ ( var @ ( sub @ X342 @ id ) )
| ( X326 @ id @ X342 @ id ) )
| ( X326 @ id @ X325 @ id ) ) ) ) )
<=> hoasinduction_lthm )
& ( ! [X343: subst > term > subst > $o] :
~ ! [X344: term > $o] :
~ ! [X345: term] :
( ( X344 @ X345 )
= ( X343 @ id @ X345 @ id ) )
<=> hoasinduction_lem0 )
& pushprop_lem0_gthm
& ( ! [X346: term,X347: term,X348: subst > term > term] :
( ~ ! [X349: subst,X350: subst,X351: term] :
( ( sub @ ( X348 @ X350 @ X351 ) @ X349 )
= ( X348 @ ( comp @ X350 @ X349 ) @ ( sub @ X351 @ X349 ) ) )
| ( ( ap @ ( sub @ X346 @ id ) @ X347 )
!= ( lam @ ( X348 @ sh @ one ) ) ) )
<=> hoaslamnotap )
& axabs
& hoasinduction_lem1v2_gthm
& ( ! [X352: term,X353: subst > term > subst > $o] :
( ~ ! [X354: term] :
( ( X353 @ id @ ( sub @ X352 @ ( push @ X354 @ id ) ) @ id )
| ~ ( X353 @ id @ X354 @ id ) )
| ~ ! [X355: subst > term > term] :
( ~ ! [X356: term,X357: subst,X358: subst] :
( ( sub @ ( X355 @ X357 @ X356 ) @ X358 )
= ( X355 @ ( comp @ X357 @ X358 ) @ ( sub @ X356 @ X358 ) ) )
| ~ ! [X359: term] :
( ( X353 @ id @ ( X355 @ id @ X359 ) @ id )
| ~ ( X353 @ id @ X359 @ id ) )
| ( X353 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id ) )
| ( X353 @ id @ ( lam @ ( sub @ X352 @ ( push @ one @ sh ) ) ) @ id ) )
<=> hoasinduction_lem3aa_lthm )
& ( ( ! [X360: subst,X361: term] :
( ( sub @ one @ ( push @ X361 @ X360 ) )
= X361 )
=> ( ! [X362: term,X363: subst] :
( ( comp @ sh @ ( push @ X362 @ X363 ) )
= X363 )
=> ( ! [X364: subst] :
( ( comp @ X364 @ id )
= X364 )
=> ( ! [X365: term,X366: term > $o] :
( ~ ! [X367: term] :
( ( X366 @ X367 )
| ~ ( var @ X367 ) )
| ~ ! [X368: term] :
( ( X366 @ ( lam @ X368 ) )
| ~ ( X366 @ X368 ) )
| ~ ! [X369: term,X370: term] :
( ( X366 @ ( ap @ X370 @ X369 ) )
| ~ ( X366 @ X369 )
| ~ ( X366 @ X370 ) )
| ( X366 @ X365 ) )
=> ! [X371: term,X372: term > $o,X373: subst] :
( ( X372 @ ( sub @ X371 @ X373 ) )
| ~ ! [X374: term] :
( ( X372 @ ( sub @ X374 @ X373 ) )
| ~ ( var @ X374 ) )
| ~ ! [X375: term] :
( ~ ! [X376: term] :
( ( X372 @ ( sub @ X375 @ ( push @ X376 @ id ) ) )
| ~ ( X372 @ X376 ) )
| ( X372 @ ( lam @ X375 ) ) )
| ~ ! [X377: term,X378: term] :
( ~ ( X372 @ X377 )
| ( X372 @ ( ap @ X378 @ X377 ) )
| ~ ( X372 @ X378 ) ) ) ) ) ) )
<=> induction2lem_lthm )
& axassoc
& pushprop_lem2v2
& pushprop_lem2v2_lthm
& ( hoasinduction_lem2v2
<=> ! [X379: term > $o,X380: subst > term > subst > $o,X381: term,X382: term] :
( ~ ! [X383: subst,X384: subst,X385: term,X386: subst] :
( ~ ( X380 @ X384 @ X385 @ ( comp @ X383 @ X386 ) )
| ( X380 @ ( comp @ X384 @ X383 ) @ ( sub @ X385 @ X383 ) @ X386 ) )
| ~ ! [X387: term] :
( ( X379 @ X387 )
= ( X380 @ id @ X387 @ id ) )
| ~ ! [X388: term,X389: subst,X390: subst,X391: subst] :
( ( X380 @ X390 @ X388 @ ( comp @ X391 @ X389 ) )
| ~ ( X380 @ ( comp @ X390 @ X391 ) @ ( sub @ X388 @ X391 ) @ X389 ) )
| ( X379 @ ( ap @ X382 @ X381 ) )
| ~ ( X379 @ X382 )
| ~ ( X379 @ X381 )
| ~ ! [X392: term,X393: term] :
( ~ ( X380 @ id @ X392 @ id )
| ( X380 @ id @ ( ap @ ( sub @ X393 @ id ) @ X392 ) @ id )
| ~ ( X380 @ id @ X393 @ id ) ) ) )
& ( hoasinduction_lem3v2_f_lthm
<=> ! [X394: term] :
~ ! [X395: subst > term > term] :
~ ! [X396: subst,X397: term] :
( ( sub @ X394 @ ( push @ X397 @ X396 ) )
= ( X395 @ X396 @ X397 ) ) )
& ( ( ! [X398: term] :
~ ! [X399: subst > term > term] :
~ ! [X400: term,X401: subst] :
( ( X399 @ X401 @ X400 )
= ( sub @ X398 @ ( push @ X400 @ X401 ) ) )
=> ( ! [X402: term] :
( ( sub @ X402 @ id )
= X402 )
=> ! [X403: term,X404: subst > term > subst > $o,X405: term > $o] :
( ( X405 @ ( lam @ X403 ) )
| ~ ! [X406: term] :
( ( X405 @ ( sub @ X403 @ ( push @ X406 @ id ) ) )
| ~ ( X405 @ X406 ) )
| ~ ! [X407: subst > term > term] :
( ( X404 @ id @ ( lam @ ( X407 @ sh @ one ) ) @ id )
| ~ ! [X408: term] :
( ~ ( X404 @ id @ X408 @ id )
| ( X404 @ id @ ( X407 @ id @ X408 ) @ id ) )
| ~ ! [X409: term,X410: subst,X411: subst] :
( ( X407 @ ( comp @ X410 @ X411 ) @ ( sub @ X409 @ X411 ) )
= ( sub @ ( X407 @ X410 @ X409 ) @ X411 ) ) )
| ~ ! [X412: term] :
( ( X404 @ id @ X412 @ id )
= ( X405 @ X412 ) ) ) ) )
<=> hoasinduction_lem3v2a_lthm )
& lamnotvar
& pushprop_lthm
& hoasapnotvar_gthm
& ( ! [X413: term > $o,X414: term] :
( ~ ! [X415: term,X416: term] :
( ( X413 @ ( ap @ X415 @ X416 ) )
| ~ ( X413 @ X415 )
| ~ ( X413 @ X416 ) )
| ~ ! [X417: term] :
( ( X413 @ X417 )
| ~ ( var @ X417 ) )
| ( X413 @ X414 )
| ~ ! [X418: term] :
( ~ ( X413 @ X418 )
| ( X413 @ ( lam @ X418 ) ) ) )
<=> induction )
& ( ( ! [X419: term] :
( ( sub @ X419 @ id )
= X419 )
=> ( ! [X420: term,X421: subst > term > subst > $o] :
( ~ ! [X422: subst > term > term] :
( ~ ! [X423: term] :
( ~ ( X421 @ id @ X423 @ id )
| ( X421 @ id @ ( X422 @ id @ X423 ) @ id ) )
| ( X421 @ id @ ( lam @ ( X422 @ sh @ one ) ) @ id )
| ~ ! [X424: subst,X425: subst,X426: term] :
( ( X422 @ ( comp @ X424 @ X425 ) @ ( sub @ X426 @ X425 ) )
= ( sub @ ( X422 @ X424 @ X426 ) @ X425 ) ) )
| ~ ! [X427: term] :
( ~ ( X421 @ id @ X427 @ id )
| ( X421 @ id @ ( sub @ X420 @ ( push @ X427 @ id ) ) @ id ) )
| ( X421 @ id @ ( lam @ ( sub @ X420 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X428: term,X429: subst > term > subst > $o] :
( ( X429 @ id @ ( lam @ X428 ) @ id )
| ~ ! [X430: subst,X431: subst,X432: term,X433: subst] :
( ( X429 @ X433 @ X432 @ ( comp @ X431 @ X430 ) )
| ~ ( X429 @ ( comp @ X433 @ X431 ) @ ( sub @ X432 @ X431 ) @ X430 ) )
| ~ ! [X434: subst,X435: subst,X436: subst,X437: term] :
( ( X429 @ ( comp @ X436 @ X435 ) @ ( sub @ X437 @ X435 ) @ X434 )
| ~ ( X429 @ X436 @ X437 @ ( comp @ X435 @ X434 ) ) )
| ~ ! [X438: subst > term > term] :
( ~ ! [X439: term,X440: subst,X441: subst] :
( ( sub @ ( X438 @ X441 @ X439 ) @ X440 )
= ( X438 @ ( comp @ X441 @ X440 ) @ ( sub @ X439 @ X440 ) ) )
| ( X429 @ id @ ( lam @ ( X438 @ sh @ one ) ) @ id )
| ~ ! [X442: term] :
( ~ ( X429 @ id @ X442 @ id )
| ( X429 @ id @ ( X438 @ id @ X442 ) @ id ) ) )
| ~ ! [X443: term] :
( ~ ( X429 @ id @ X443 @ id )
| ( X429 @ id @ ( sub @ X428 @ ( push @ X443 @ id ) ) @ id ) ) ) ) )
<=> hoasinduction_lem3_lthm )
& hoaslamnotap_lthm
& ( hoaslaminj_lthm
<=> ( ! [X444: term,X445: subst] :
( ( sub @ one @ ( push @ X444 @ X445 ) )
= X444 )
=> ( ! [X446: term,X447: subst] :
( ( comp @ sh @ ( push @ X446 @ X447 ) )
= X447 )
=> ( ! [X448: term,X449: term] :
( ( ( lam @ X449 )
!= ( lam @ X448 ) )
| ( X448 = X449 ) )
=> ! [X450: term,X451: subst > term > term,X452: subst > term > term,X453: subst] :
( ( ( lam @ ( X452 @ sh @ one ) )
!= ( lam @ ( X451 @ sh @ one ) ) )
| ~ ! [X454: term,X455: subst,X456: subst] :
( ( sub @ ( X451 @ X456 @ X454 ) @ X455 )
= ( X451 @ ( comp @ X456 @ X455 ) @ ( sub @ X454 @ X455 ) ) )
| ~ ! [X457: subst,X458: subst,X459: term] :
( ( X452 @ ( comp @ X458 @ X457 ) @ ( sub @ X459 @ X457 ) )
= ( sub @ ( X452 @ X458 @ X459 ) @ X457 ) )
| ( ( X451 @ X453 @ X450 )
= ( X452 @ X453 @ X450 ) ) ) ) ) ) )
& hoasinduction_lem3a_gthm
& ! [X460: d_term] : ( d_one = X460 )
& ( hoasinduction_lem2
<=> ! [X461: subst > term > subst > $o,X462: term,X463: term] :
( ~ ( X461 @ id @ X462 @ id )
| ( X461 @ id @ ( ap @ X463 @ X462 ) @ id )
| ~ ( X461 @ id @ X463 @ id )
| ~ ! [X464: subst,X465: subst,X466: term,X467: subst] :
( ( X461 @ ( comp @ X467 @ X465 ) @ ( sub @ X466 @ X465 ) @ X464 )
| ~ ( X461 @ X467 @ X466 @ ( comp @ X465 @ X464 ) ) )
| ~ ! [X468: term,X469: term] :
( ( X461 @ id @ ( ap @ ( sub @ X468 @ id ) @ X469 ) @ id )
| ~ ( X461 @ id @ X469 @ id )
| ~ ( X461 @ id @ X468 @ id ) )
| ~ ! [X470: subst,X471: subst,X472: term,X473: subst] :
( ~ ( X461 @ ( comp @ X470 @ X473 ) @ ( sub @ X472 @ X473 ) @ X471 )
| ( X461 @ X470 @ X472 @ ( comp @ X473 @ X471 ) ) ) ) )
& hoasinduction_lem2_gthm
& ulamvarsh
& hoasinduction_lem1_gthm
& ! [X474: subst] :
? [X475: d_subst] :
( ( d2subst @ X475 )
= X474 )
& ( ! [X476: subst] :
( ( comp @ X476 @ id )
= X476 )
<=> axidr )
& ( hoasinduction_lem3b_lthm
<=> ! [X477: term] :
~ ! [X478: subst > term > term] :
( ( X478 @ sh @ one )
!= ( sub @ X477 @ ( push @ one @ sh ) ) ) )
& ! [X479: d_term,X480: d_term] :
( ( ( d2term @ X479 )
= ( d2term @ X480 ) )
=> ( X479 = X480 ) )
& ( ! [X481: subst] :
( ( comp @ id @ X481 )
= X481 )
<=> axidl )
& ( ! [X482: term > $o,X483: subst,X484: term > $o,X485: term] :
( ~ ! [X486: term] :
( ( X484 @ ( sub @ X486 @ ( push @ X485 @ X483 ) ) )
= ( X482 @ X486 ) )
| ~ ( X484 @ X485 )
| ( X482 @ one ) )
<=> pushprop_lem1v2 )
& hoasapinj1_gthm
& ( hoasinduction_lem3
<=> ! [X487: term,X488: subst > term > subst > $o] :
( ~ ! [X489: subst,X490: subst,X491: subst,X492: term] :
( ( X488 @ ( comp @ X491 @ X489 ) @ ( sub @ X492 @ X489 ) @ X490 )
| ~ ( X488 @ X491 @ X492 @ ( comp @ X489 @ X490 ) ) )
| ( X488 @ id @ ( lam @ X487 ) @ id )
| ~ ! [X493: term] :
( ( X488 @ id @ ( sub @ X487 @ ( push @ X493 @ id ) ) @ id )
| ~ ( X488 @ id @ X493 @ id ) )
| ~ ! [X494: subst > term > term] :
( ( X488 @ id @ ( lam @ ( X494 @ sh @ one ) ) @ id )
| ~ ! [X495: subst,X496: subst,X497: term] :
( ( sub @ ( X494 @ X496 @ X497 ) @ X495 )
= ( X494 @ ( comp @ X496 @ X495 ) @ ( sub @ X497 @ X495 ) ) )
| ~ ! [X498: term] :
( ( X488 @ id @ ( X494 @ id @ X498 ) @ id )
| ~ ( X488 @ id @ X498 @ id ) ) )
| ~ ! [X499: subst,X500: subst,X501: term,X502: subst] :
( ~ ( X488 @ ( comp @ X502 @ X500 ) @ ( sub @ X501 @ X500 ) @ X499 )
| ( X488 @ X502 @ X501 @ ( comp @ X500 @ X499 ) ) ) ) )
& ( hoasinduction
<=> ! [X503: term,X504: subst > term > subst > $o] :
( ~ ! [X505: term,X506: subst,X507: subst,X508: subst] :
( ( X504 @ X508 @ X505 @ ( comp @ X507 @ X506 ) )
| ~ ( X504 @ ( comp @ X508 @ X507 ) @ ( sub @ X505 @ X507 ) @ X506 ) )
| ~ ! [X509: term,X510: term] :
( ~ ( X504 @ id @ X509 @ id )
| ~ ( X504 @ id @ X510 @ id )
| ( X504 @ id @ ( ap @ ( sub @ X510 @ id ) @ X509 ) @ id ) )
| ~ ! [X511: subst > term > term] :
( ~ ! [X512: subst,X513: term,X514: subst] :
( ( X511 @ ( comp @ X512 @ X514 ) @ ( sub @ X513 @ X514 ) )
= ( sub @ ( X511 @ X512 @ X513 ) @ X514 ) )
| ~ ! [X515: term] :
( ~ ( X504 @ id @ X515 @ id )
| ( X504 @ id @ ( X511 @ id @ X515 ) @ id ) )
| ( X504 @ id @ ( lam @ ( X511 @ sh @ one ) ) @ id ) )
| ~ ! [X516: term] :
( ~ ( var @ ( sub @ X516 @ id ) )
| ( X504 @ id @ X516 @ id ) )
| ~ ! [X517: subst,X518: subst,X519: term,X520: subst] :
( ( X504 @ ( comp @ X520 @ X518 ) @ ( sub @ X519 @ X518 ) @ X517 )
| ~ ( X504 @ X520 @ X519 @ ( comp @ X518 @ X517 ) ) )
| ( X504 @ id @ X503 @ id ) ) )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lem3a
<=> ! [X521: term,X522: subst > term > subst > $o] :
( ( X522 @ id @ ( lam @ X521 ) @ id )
| ~ ! [X523: subst > term > term] :
( ( X522 @ id @ ( lam @ ( X523 @ sh @ one ) ) @ id )
| ~ ! [X524: subst,X525: term,X526: subst] :
( ( X523 @ ( comp @ X524 @ X526 ) @ ( sub @ X525 @ X526 ) )
= ( sub @ ( X523 @ X524 @ X525 ) @ X526 ) )
| ~ ! [X527: term] :
( ~ ( X522 @ id @ X527 @ id )
| ( X522 @ id @ ( X523 @ id @ X527 ) @ id ) ) )
| ~ ! [X528: term] :
( ~ ( X522 @ id @ X528 @ id )
| ( X522 @ id @ ( sub @ X521 @ ( push @ X528 @ id ) ) @ id ) ) ) )
& hoaslamnotvar_lthm
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ~ ulamvar1
& hoasapinj2_gthm
& ( ( ! [X529: subst,X530: term] :
( ( sub @ one @ ( push @ X530 @ X529 ) )
= X530 )
=> ( ! [X531: subst,X532: term] :
( ( comp @ sh @ ( push @ X532 @ X531 ) )
= X531 )
=> ! [X533: term > $o,X534: term > $o,X535: term,X536: subst,X537: term] :
( ~ ( X533 @ X537 )
| ( X534 @ ( sub @ X537 @ ( push @ X535 @ X536 ) ) ) ) ) )
<=> pushprop_lem1_lthm )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& induction2_gthm
& ( hoasapinj2_lthm
<=> ( ! [X538: term,X539: term,X540: term,X541: term] :
( ( ( ap @ X541 @ X539 )
!= ( ap @ X538 @ X540 ) )
| ( X539 = X540 ) )
=> ! [X542: term,X543: term,X544: term,X545: term] :
( ( ( ap @ ( sub @ X545 @ id ) @ X544 )
!= ( ap @ ( sub @ X543 @ id ) @ X542 ) )
| ( X542 = X544 ) ) ) )
& ( substmonoid_lthm
<=> ( ! [X546: subst] :
( ( comp @ id @ X546 )
= X546 )
=> ( ! [X547: subst] :
( ( comp @ X547 @ id )
= X547 )
=> ( ! [X548: subst] :
( ( comp @ X548 @ id )
= X548 )
& ! [X549: subst] :
( ( comp @ id @ X549 )
= X549 ) ) ) ) )
& hoasinduction_lem2v2_gthm
& pushprop_lem1v2_gthm ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( induction2lem
<=> ! [X17: subst,X10: term > $o,X16: term] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) ) ) )
& hoaslamnotap_gthm
& hoaslamnotvar_gthm
& pushprop_lthm_orig
& ( hoasinduction_lem3b
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) ) )
& ( hoasinduction_lem0_lthm
<=> ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
& ( one
= ( d2term @ d_one ) )
& ( hoasapinj1
<=> ! [X15: term,X20: term,X19: term,X13: term] :
( ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) )
| ( X13 = X15 ) ) )
& hoasapnotvar_lthm
& hoasinduction_lem1
& axvarshift
& ( ! [X48: term,X10: subst > term > subst > $o,X11: term > $o] :
( ~ ! [X15: term] :
( ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X14: subst,X13: term,X26: subst,X25: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X25: subst,X26: subst,X14: subst,X13: term] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X11 @ ( lam @ X48 ) ) )
<=> hoasinduction_lem3v2 )
& ( ( ^ [X13: term,X14: subst,X10: term > $o,X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) )
= pushprop_p_and_p_prime )
& pushprop
& ( ! [X15: term,X13: term] :
( ( X13 = X15 )
| ( ( sub @ X13 @ sh )
!= ( sub @ X15 @ sh ) ) )
<=> shinj )
& ( pushprop_lem0
<=> ! [X10: term > $o,X13: term,X14: subst] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) )
& ( ! [X10: term > $o,X22: term] :
( ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ X22 ) )
<=> induction2 )
& ( ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X10: term > $o,X22: term] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: subst > term > subst > $o,X11: term > $o,X29: term] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ 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 ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X29 ) ) )
=> ! [X10: subst > term > subst > $o,X30: term] :
( ( X10 @ id @ X30 @ id )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) ) ) ) ) ) )
<=> hoasinduction_no_psi_cond_lthm )
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X48: term,X10: subst > term > subst > $o,X11: term > $o] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X25: subst,X14: subst,X26: subst,X13: term] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term,X14: subst,X26: subst,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) ) ) )
<=> hoasinduction_lem3v2_lthm )
& ( hoasapinj2
<=> ! [X15: term,X13: term,X20: term,X19: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) )
& hoaslaminj_gthm
& ( ! [X14: subst] :
( ( push @ ( sub @ one @ X14 ) @ ( comp @ sh @ X14 ) )
= X14 )
<=> axscons )
& apnotvar
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X20: term,X19: term,X13: term,X15: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X13 = X15 ) )
=> ! [X15: term,X13: term,X20: term,X19: term] :
( ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) )
| ( X13 = X15 ) ) ) )
<=> hoasapinj1_lthm )
& termmset_gthm
& ! [X7: d_subst,X6: d_subst] :
( ( ( d2subst @ X6 )
= ( d2subst @ X7 ) )
=> ( X6 = X7 ) )
& ! [X0: term] :
? [X1: d_term] :
( ( d2term @ X1 )
= X0 )
& pushprop_lem3v2_lthm
& axapp
& ( pushprop_lem0_lthm
<=> ! [X13: term,X14: subst,X10: term > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) )
& ( hoasinduction_lem3aaa
<=> ! [X34: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X34 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X35: term,X18: subst > term > term] :
( ~ ! [X38: subst,X37: term,X36: subst] :
( ( sub @ ( X18 @ X36 @ X37 ) @ X38 )
= ( sub @ ( sub @ X35 @ ( push @ X37 @ X36 ) ) @ X38 ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X40: term,X39: subst,X41: subst] :
( ( X18 @ ( comp @ X39 @ X41 ) @ ( sub @ X40 @ X41 ) )
= ( sub @ X35 @ ( push @ ( sub @ X40 @ X41 ) @ ( comp @ X39 @ X41 ) ) ) ) )
| ( X10 @ id @ ( lam @ ( sub @ X34 @ ( push @ one @ sh ) ) ) @ id ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( ! [X26: term > $o,X13: term,X15: term,X14: subst,X10: term > $o] :
( ~ ( X10 @ X13 )
| ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) ) )
<=> pushprop_lem1 )
& hoasinduction_lem2_lthm
& ( hoasinduction_no_psi_cond
<=> ! [X10: subst > term > subst > $o,X30: term] :
( ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ X30 @ id ) ) )
& ( ! [X15: term,X13: term] :
( ( X13 = X15 )
| ( ( lam @ X13 )
!= ( lam @ X15 ) ) )
<=> laminj )
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) )
<=> termmset_lthm )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& hoasinduction_lem3_gthm
& pushprop_lem2v2_gthm
& hoasinduction_lem3v2_gthm
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: term > $o,X16: term,X17: subst] :
( ~ ! [X13: term,X15: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) ) )
=> ! [X10: term > $o,X22: term] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) ) ) ) )
<=> induction2_lthm )
& ulamvarind
& induction2lem_gthm
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ( ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ! [X14: subst,X11: term > $o,X10: term > $o,X13: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
| ~ ( X10 @ X13 )
| ( X11 @ one ) ) )
<=> pushprop_lem1v2_lthm )
& ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
<=> axvarid )
& ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
<=> axvarcons )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [X10: subst > term > subst > $o,X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) ) )
& ~ ( var @ ( d2term @ d_one ) )
& ! [X5: d_subst] : ( d_id = X5 )
& hoasinduction_lem1_lthm
& hoasinduction_lem1v2
& hoasinduction_lem3b_gthm
& ( hoasinduction_lem3v2_f
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) ) )
& substmonoid_gthm
& pushprop_lem3v2
& pushprop_lem1_gthm
& axmap
& hoasapnotvar
& ( hoasinduction_lem3aa
<=> ! [X10: subst > term > subst > $o,X42: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) ) )
& ( hoaslaminj
<=> ! [X31: subst > term > term,X18: subst > term > term,X32: subst,X33: term] :
( ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) )
| ~ ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) ) ) )
& ( ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X22: term,X10: term > $o] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X11: term > $o,X29: term,X10: subst > term > subst > $o] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X18: subst > term > term] :
( ( 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 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ( X11 @ ( lam @ X29 ) ) )
=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X25: subst,X13: term,X14: subst,X26: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X10 @ id @ X28 @ id ) ) ) ) ) )
<=> hoasinduction_lthm_3 )
& ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
<=> axshiftcons )
& ( hoaslam
= ( ^ [X8: subst,X9: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X19: term,X20: term,X13: term,X15: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X19 = X20 ) )
<=> apinj2 )
& ( substmonoid
<=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) ) )
& ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
<=> termmset )
& hoasinduction_gthm
& pushprop_gthm
& hoaslamnotvar
& ( hoasinduction_lem3a_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: subst > term > subst > $o,X42: term] :
( ~ ! [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 ) ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [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] :
( ~ ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X10 @ id @ ( lam @ X43 ) @ id ) ) ) ) )
& ~ lamnotap
& axclos
& ( hoasinduction_lem3v2a
<=> ! [X29: term,X10: subst > term > subst > $o,X11: term > $o] :
( ( X11 @ ( lam @ X29 ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( apinj1
<=> ! [X20: term,X13: term,X19: term,X15: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X13 = X15 ) ) )
& ( ( ! [X22: term,X10: term > $o] :
( ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) ) )
=> ( ! [X23: term,X10: subst > term > subst > $o,X24: term] :
( ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ( X10 @ id @ X23 @ id )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ( X10 @ id @ X24 @ id )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id ) )
=> ( ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term,X14: subst,X26: subst,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) )
=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term,X13: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [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 ) ) )
| ~ ! [X14: subst,X13: term,X25: subst,X26: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X14: subst,X26: subst,X25: subst,X13: term] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ X28 @ id ) ) ) ) )
<=> hoasinduction_lthm )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0 )
& pushprop_lem0_gthm
& ( ! [X46: term,X47: term,X18: subst > term > term] :
( ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( ap @ ( sub @ X46 @ id ) @ X47 ) ) )
<=> hoaslamnotap )
& axabs
& hoasinduction_lem1v2_gthm
& ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
<=> hoasinduction_lem3aa_lthm )
& ( ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X21: term,X10: term > $o] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 )
| ~ ( X10 @ X13 ) )
| ( X10 @ X21 ) )
=> ! [X16: term,X10: term > $o,X17: subst] :
( ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X15 @ X17 ) )
| ~ ( var @ X15 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) ) ) ) ) ) )
<=> induction2lem_lthm )
& axassoc
& pushprop_lem2v2
& pushprop_lem2v2_lthm
& ( hoasinduction_lem2v2
<=> ! [X11: term > $o,X10: subst > term > subst > $o,X45: term,X44: term] :
( ~ ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst,X26: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X11 @ ( ap @ X44 @ X45 ) )
| ~ ( X11 @ X44 )
| ~ ( X11 @ X45 )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) ) )
& ( hoasinduction_lem3v2_f_lthm
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) ) )
& ( ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X29: term,X10: subst > term > subst > $o,X11: term > $o] :
( ( X11 @ ( lam @ X29 ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) ) ) )
<=> hoasinduction_lem3v2a_lthm )
& lamnotvar
& pushprop_lthm
& hoasapnotvar_gthm
& ( ! [X10: term > $o,X21: term] :
( ~ ! [X13: term,X15: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ( X10 @ X21 )
| ~ ! [X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( lam @ X13 ) ) ) )
<=> induction )
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [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 ) ) ) )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X27: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X25: subst,X26: subst,X14: subst,X13: term] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id ) ) ) ) )
<=> hoasinduction_lem3_lthm )
& hoaslamnotap_lthm
& ( hoaslaminj_lthm
<=> ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X13: term,X15: term] :
( ( ( lam @ X13 )
!= ( lam @ X15 ) )
| ( X13 = X15 ) )
=> ! [X33: term,X31: subst > term > term,X18: subst > term > term,X32: subst] :
( ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) ) ) ) ) ) )
& hoasinduction_lem3a_gthm
& ! [X1: d_term] : ( d_one = X1 )
& ( hoasinduction_lem2
<=> ! [X10: subst > term > subst > $o,X24: term,X23: term] :
( ~ ( X10 @ id @ X24 @ id )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ( X10 @ id @ X23 @ id )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ 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 ) ) ) ) )
& hoasinduction_lem2_gthm
& ulamvarsh
& hoasinduction_lem1_gthm
& ! [X4: subst] :
? [X5: d_subst] :
( ( d2subst @ X5 )
= X4 )
& ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
<=> axidr )
& ( hoasinduction_lem3b_lthm
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) ) )
& ! [X2: d_term,X3: d_term] :
( ( ( d2term @ X2 )
= ( d2term @ X3 ) )
=> ( X2 = X3 ) )
& ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
<=> axidl )
& ( ! [X11: term > $o,X14: subst,X10: term > $o,X13: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
| ~ ( X10 @ X13 )
| ( X11 @ one ) )
<=> pushprop_lem1v2 )
& hoasapinj1_gthm
& ( hoasinduction_lem3
<=> ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X26: subst,X25: subst,X14: subst,X13: term] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) ) )
& ( hoasinduction
<=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X13: term,X25: subst,X26: subst,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ X28 @ id ) ) )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lem3a
<=> ! [X43: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ X43 ) @ 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 ) ) )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id ) ) ) )
& hoaslamnotvar_lthm
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ~ ulamvar1
& hoasapinj2_gthm
& ( ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ! [X10: term > $o,X26: term > $o,X15: term,X14: subst,X13: term] :
( ~ ( X10 @ X13 )
| ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) ) ) ) )
<=> pushprop_lem1_lthm )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& induction2_gthm
& ( hoasapinj2_lthm
<=> ( ! [X15: term,X19: term,X20: term,X13: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X19 = X20 ) )
=> ! [X19: term,X13: term,X20: term,X15: term] :
( ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) )
| ( X19 = X20 ) ) ) )
& ( 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 ) ) ) ) )
& hoasinduction_lem2v2_gthm
& pushprop_lem1v2_gthm ),
file('/export/starexec/sandbox/tmp/tmp.6IcC7TwM2m/DTF2THF_2950.p',alg444_1) ).
thf(f1212,plain,
! [X62: term] :
( ( d2term @ ( sK487 @ X62 ) )
= X62 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f2674,plain,
( ( d2term @ d_one )
!= ( sub @ ( sub @ sK594 @ sK592 ) @ sK593 ) ),
inference(forward_demodulation,[status(thm)],[f1784,f2587]) ).
thf(f1784,plain,
( ( sub @ ( sub @ sK594 @ sK592 ) @ sK593 )
!= ( sub @ sK594 @ ( comp @ sK592 @ sK593 ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1783]) ).
thf(f1783,plain,
( ( ( sub @ sK594 @ ( comp @ sK592 @ sK593 ) )
= ( sub @ ( sub @ sK594 @ sK592 ) @ sK593 ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f1782]) ).
thf(f1782,plain,
( ( ^ [Y0: term] :
( ( sub @ Y0 @ ( comp @ sK592 @ sK593 ) )
= ( sub @ ( sub @ Y0 @ sK592 ) @ sK593 ) )
@ sK594 )
= $false ),
inference(sigma_clausification,[status(thm)],[f1781]) ).
thf(f1781,plain,
( ( !! @ term
@ ^ [Y0: term] :
( ( sub @ Y0 @ ( comp @ sK592 @ sK593 ) )
= ( sub @ ( sub @ Y0 @ sK592 ) @ sK593 ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f1780]) ).
thf(f1780,plain,
( ( ^ [Y0: subst] :
( !! @ term
@ ^ [Y1: term] :
( ( sub @ Y1 @ ( comp @ sK592 @ Y0 ) )
= ( sub @ ( sub @ Y1 @ sK592 ) @ Y0 ) ) )
@ sK593 )
= $false ),
inference(sigma_clausification,[status(thm)],[f1779]) ).
thf(f1779,plain,
( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ term
@ ^ [Y1: term] :
( ( sub @ Y1 @ ( comp @ sK592 @ Y0 ) )
= ( sub @ ( sub @ Y1 @ sK592 ) @ Y0 ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f1778]) ).
thf(f1778,plain,
( $false
= ( ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ Y2 @ ( comp @ Y0 @ Y1 ) )
= ( sub @ ( sub @ Y2 @ Y0 ) @ Y1 ) ) ) )
@ sK592 ) ),
inference(sigma_clausification,[status(thm)],[f1262]) ).
thf(f1262,plain,
( ( !! @ subst
@ ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ Y2 @ ( comp @ Y0 @ Y1 ) )
= ( sub @ ( sub @ Y2 @ Y0 ) @ Y1 ) ) ) ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f546,f1254]) ).
thf(f1254,plain,
axclos = $true,
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f546,plain,
( axclos
!= ( !! @ subst
@ ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ Y2 @ ( comp @ Y0 @ Y1 ) )
= ( sub @ ( sub @ Y2 @ Y0 ) @ Y1 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( axclos
!= ( !! @ subst
@ ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ Y2 @ ( comp @ Y0 @ Y1 ) )
= ( sub @ ( sub @ Y2 @ Y0 ) @ Y1 ) ) ) ) ) ),
inference(flattening,[status(thm)],[f6]) ).
thf(f6,plain,
( axclos
!= ( !! @ subst
@ ^ [Y0: subst] :
( !! @ subst
@ ^ [Y1: subst] :
( !! @ term
@ ^ [Y2: term] :
( ( sub @ Y2 @ ( comp @ Y0 @ Y1 ) )
= ( sub @ ( sub @ Y2 @ Y0 ) @ Y1 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
( axclos
!= ( ! [X0: term,X1: subst,X2: subst] :
( ( sub @ X0 @ ( comp @ X2 @ X1 ) )
= ( sub @ ( sub @ X0 @ X2 ) @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
( axclos
!= ( ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( sub @ X13 @ X14 ) @ X25 )
= ( sub @ X13 @ ( comp @ X14 @ X25 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
( axclos
= ( ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( sub @ X13 @ X14 ) @ X25 )
= ( sub @ X13 @ ( comp @ X14 @ X25 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.6IcC7TwM2m/DTF2THF_2950.p',axclos) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.33 % Computer : n021.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Tue May 5 08:57:25 EDT 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.44 ---- Original DTF file ---
% 0.18/0.44 thf(spec,logic,$$dhol).
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % File : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.18/0.44 % Domain : Software Verification
% 0.18/0.44 % Problem : Benchmark: ALG444^1, Active target: axclos
% 0.18/0.44 % Version : Especial.
% 0.18/0.44 % English :
% 0.18/0.44
% 0.18/0.44 % Refs : [Kon26] Kondylidou (2026), Email to Geoff Sutcliffe
% 0.18/0.44 % Source : [Kon26]
% 0.18/0.44 % Names : ALG444^1__005__axclos.verify [Kon26]
% 0.18/0.44
% 0.18/0.44 % Status : Theorem
% 0.18/0.44 % Rating : ? v9.3.0
% 0.18/0.44 % Syntax : Number of formulae : 132 ( 1 unt; 130 typ; 0 def)
% 0.18/0.44 % Number of atoms : 255 ( 129 equ; 0 cnn)
% 0.18/0.44 % Maximal formula atoms : 248 ( 127 avg)
% 0.18/0.44 % Number of connectives : 2577 ( 333 ~; 306 |; 129 &;1715 @)
% 0.18/0.44 % ( 56 <=>; 38 =>; 0 <=; 0 <~>)
% 0.18/0.44 % Maximal formula depth : 151 ( 76 avg)
% 0.18/0.44 % Number of types : 5 ( 4 usr)
% 0.18/0.44 % Number of type conns : 222 ( 222 >; 0 *; 0 +; 0 <<)
% 0.18/0.44 % Number of symbols : 127 ( 126 usr; 113 con; 0-4 aty)
% 0.18/0.44 % Number of variables : 553 ( 8 ^; 543 !; 2 ?; 553 :)
% 0.18/0.44 % SPC : TH0_THM_EQU_NAR
% 0.18/0.44
% 0.18/0.44 % Comments :
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 thf(term_type,type,
% 0.18/0.44 term: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(subst_type,type,
% 0.18/0.44 subst: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(one_decl,type,
% 0.18/0.44 one: term ).
% 0.18/0.44
% 0.18/0.44 thf(ap_decl,type,
% 0.18/0.44 ap: term > term > term ).
% 0.18/0.44
% 0.18/0.44 thf(lam_decl,type,
% 0.18/0.44 lam: term > term ).
% 0.18/0.44
% 0.18/0.44 thf(sub_decl,type,
% 0.18/0.44 sub: term > subst > term ).
% 0.18/0.44
% 0.18/0.44 thf(id_decl,type,
% 0.18/0.44 id: subst ).
% 0.18/0.44
% 0.18/0.44 thf(sh_decl,type,
% 0.18/0.44 sh: subst ).
% 0.18/0.44
% 0.18/0.44 thf(push_decl,type,
% 0.18/0.44 push: term > subst > subst ).
% 0.18/0.44
% 0.18/0.44 thf(comp_decl,type,
% 0.18/0.44 comp: subst > subst > subst ).
% 0.18/0.44
% 0.18/0.44 thf(var_decl,type,
% 0.18/0.44 var: term > $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem1v2_decl,type,
% 0.18/0.44 pushprop_lem1v2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem1_gthm_decl,type,
% 0.18/0.44 pushprop_lem1_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axmap_decl,type,
% 0.18/0.44 axmap: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem0_gthm_decl,type,
% 0.18/0.44 pushprop_lem0_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(shinj_decl,type,
% 0.18/0.44 shinj: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem1v2_decl,type,
% 0.18/0.44 hoasinduction_lem1v2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem1v2_gthm_decl,type,
% 0.18/0.44 hoasinduction_lem1v2_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasap_decl,type,
% 0.18/0.44 hoasap: subst > term > subst > term > term ).
% 0.18/0.44
% 0.18/0.44 thf(induction2lem_decl,type,
% 0.18/0.44 induction2lem: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3v2_f_decl,type,
% 0.18/0.44 hoasinduction_lem3v2_f: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axvarshift_decl,type,
% 0.18/0.44 axvarshift: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapinj2_decl,type,
% 0.18/0.44 hoasapinj2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapnotvar_gthm_decl,type,
% 0.18/0.44 hoasapnotvar_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapinj1_decl,type,
% 0.18/0.44 hoasapinj1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ulamvar1_decl,type,
% 0.18/0.44 ulamvar1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(induction2lem_lthm_decl,type,
% 0.18/0.44 induction2lem_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3v2_gthm_decl,type,
% 0.18/0.44 hoasinduction_lem3v2_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(apnotvar_decl,type,
% 0.18/0.44 apnotvar: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lthm_orig_decl,type,
% 0.18/0.44 pushprop_lthm_orig: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem3v2_f_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lthm_decl,type,
% 0.18/0.44 hoasinduction_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 0.18/0.44 hoasinduction_no_psi_cond_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslaminj_decl,type,
% 0.18/0.44 hoaslaminj: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3aaa_decl,type,
% 0.18/0.44 hoasinduction_lem3aaa: $o ).
% 0.18/0.44
% 0.18/0.44 thf(induction2lem_gthm_decl,type,
% 0.18/0.44 induction2lem_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3aa_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem3aa_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3_decl,type,
% 0.18/0.44 hoasinduction_lem3: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem2_decl,type,
% 0.18/0.44 hoasinduction_lem2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(termmset_lthm_decl,type,
% 0.18/0.44 termmset_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem1_decl,type,
% 0.18/0.44 hoasinduction_lem1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslamnotap_lthm_decl,type,
% 0.18/0.44 hoaslamnotap_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem1v2_lthm_decl,type,
% 0.18/0.44 pushprop_lem1v2_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapnotvar_decl,type,
% 0.18/0.44 hoasapnotvar: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem0_decl,type,
% 0.18/0.44 hoasinduction_lem0: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_decl,type,
% 0.18/0.44 hoasinduction: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_gthm_decl,type,
% 0.18/0.44 hoasinduction_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axapp_decl,type,
% 0.18/0.44 axapp: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslamnotvar_lthm_decl,type,
% 0.18/0.44 hoaslamnotvar_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem3v2_lthm_decl,type,
% 0.18/0.44 pushprop_lem3v2_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3b_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem3b_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ulamvarind_decl,type,
% 0.18/0.44 ulamvarind: $o ).
% 0.18/0.44
% 0.18/0.44 thf(induction_decl,type,
% 0.18/0.44 induction: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3a_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem3a_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(termmset_gthm_decl,type,
% 0.18/0.44 termmset_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3aa_decl,type,
% 0.18/0.44 hoasinduction_lem3aa: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem1v2_gthm_decl,type,
% 0.18/0.44 pushprop_lem1v2_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslamnotap_gthm_decl,type,
% 0.18/0.44 hoaslamnotap_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslamnotvar_gthm_decl,type,
% 0.18/0.44 hoaslamnotvar_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3b_gthm_decl,type,
% 0.18/0.44 hoasinduction_lem3b_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem2v2_decl,type,
% 0.18/0.44 pushprop_lem2v2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3a_gthm_decl,type,
% 0.18/0.44 hoasinduction_lem3a_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axclos_decl,type,
% 0.18/0.44 axclos: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axassoc_decl,type,
% 0.18/0.44 axassoc: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem2v2_decl,type,
% 0.18/0.44 hoasinduction_lem2v2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lthm_decl,type,
% 0.18/0.44 pushprop_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(apinj2_decl,type,
% 0.18/0.44 apinj2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(apinj1_decl,type,
% 0.18/0.44 apinj1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapinj2_lthm_decl,type,
% 0.18/0.44 hoasapinj2_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3v2a_decl,type,
% 0.18/0.44 hoasinduction_lem3v2a: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapinj1_lthm_decl,type,
% 0.18/0.44 hoasapinj1_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslaminj_lthm_decl,type,
% 0.18/0.44 hoaslaminj_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axvarcons_decl,type,
% 0.18/0.44 axvarcons: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslam_decl,type,
% 0.18/0.44 hoaslam: subst > ( subst > term > term ) > term ).
% 0.18/0.44
% 0.18/0.44 thf(axscons_decl,type,
% 0.18/0.44 axscons: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem2v2_gthm_decl,type,
% 0.18/0.44 hoasinduction_lem2v2_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axidr_decl,type,
% 0.18/0.44 axidr: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem1_decl,type,
% 0.18/0.44 pushprop_lem1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(laminj_decl,type,
% 0.18/0.44 laminj: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem3_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem0_decl,type,
% 0.18/0.44 pushprop_lem0: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_gthm_decl,type,
% 0.18/0.44 pushprop_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axabs_decl,type,
% 0.18/0.44 axabs: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem3v2a_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem2_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem2_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapinj2_gthm_decl,type,
% 0.18/0.44 hoasapinj2_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_p_and_p_prime_decl,type,
% 0.18/0.44 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem1_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem1_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(lamnotap_decl,type,
% 0.18/0.44 lamnotap: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapinj1_gthm_decl,type,
% 0.18/0.44 hoasapinj1_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslamnotvar_decl,type,
% 0.18/0.44 hoaslamnotvar: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axidl_decl,type,
% 0.18/0.44 axidl: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslaminj_gthm_decl,type,
% 0.18/0.44 hoaslaminj_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(induction2_lthm_decl,type,
% 0.18/0.44 induction2_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem0_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem0_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(substmonoid_lthm_decl,type,
% 0.18/0.44 substmonoid_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_decl,type,
% 0.18/0.44 pushprop: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3_gthm_decl,type,
% 0.18/0.44 hoasinduction_lem3_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem2_gthm_decl,type,
% 0.18/0.44 hoasinduction_lem2_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3b_decl,type,
% 0.18/0.44 hoasinduction_lem3b: $o ).
% 0.18/0.44
% 0.18/0.44 thf(substmonoid_decl,type,
% 0.18/0.44 substmonoid: $o ).
% 0.18/0.44
% 0.18/0.44 thf(lamnotvar_decl,type,
% 0.18/0.44 lamnotvar: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3a_decl,type,
% 0.18/0.44 hoasinduction_lem3a: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem1_gthm_decl,type,
% 0.18/0.44 hoasinduction_lem1_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_no_psi_cond_decl,type,
% 0.18/0.44 hoasinduction_no_psi_cond: $o ).
% 0.18/0.44
% 0.18/0.44 thf(induction2_gthm_decl,type,
% 0.18/0.44 induction2_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem2v2_lthm_decl,type,
% 0.18/0.44 pushprop_lem2v2_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasvar_decl,type,
% 0.18/0.44 hoasvar: subst > term > subst > $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoaslamnotap_decl,type,
% 0.18/0.44 hoaslamnotap: $o ).
% 0.18/0.44
% 0.18/0.44 thf(substmonoid_gthm_decl,type,
% 0.18/0.44 substmonoid_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ulamvarsh_decl,type,
% 0.18/0.44 ulamvarsh: $o ).
% 0.18/0.44
% 0.18/0.44 thf(induction2_decl,type,
% 0.18/0.44 induction2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem3v2_decl,type,
% 0.18/0.44 pushprop_lem3v2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem2v2_gthm_decl,type,
% 0.18/0.44 pushprop_lem2v2_gthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem1_lthm_decl,type,
% 0.18/0.44 pushprop_lem1_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3v2_decl,type,
% 0.18/0.44 hoasinduction_lem3v2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(axshiftcons_decl,type,
% 0.18/0.44 axshiftcons: $o ).
% 0.18/0.44
% 0.18/0.44 thf(termmset_decl,type,
% 0.18/0.44 termmset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_lem0_lthm_decl,type,
% 0.18/0.44 pushprop_lem0_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasapnotvar_lthm_decl,type,
% 0.18/0.44 hoasapnotvar_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lem3v2_lthm_decl,type,
% 0.18/0.44 hoasinduction_lem3v2_lthm: $o ).
% 0.18/0.44
% 0.18/0.44 thf(pushprop_p_and_p_prime_decl,type,
% 0.18/0.44 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 0.18/0.44
% 0.18/0.44 thf(axvarid_decl,type,
% 0.18/0.44 axvarid: $o ).
% 0.18/0.44
% 0.18/0.44 thf(hoasinduction_lthm_3_decl,type,
% 0.18/0.44 hoasinduction_lthm_3: $o ).
% 0.18/0.44
% 0.18/0.44 %----Types of the domains
% 0.18/0.44 thf(d_term_type,type,
% 0.18/0.44 d_term: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(d_subst_type,type,
% 0.18/0.44 d_subst: $tType ).
% 0.18/0.44
% 0.18/0.44 %----Types of the promotion functions
% 0.18/0.44 thf(d2term_decl,type,
% 0.18/0.44 d2term: d_term > term ).
% 0.18/0.44
% 0.18/0.44 thf(d2subst_decl,type,
% 0.18/0.44 d2subst: d_subst > subst ).
% 0.18/0.44
% 0.18/0.44 %----Types of the domain elements
% 0.18/0.44 thf(d_one_decl,type,
% 0.18/0.44 d_one: d_term ).
% 0.18/0.44
% 0.18/0.44 thf(d_id_decl,type,
% 0.18/0.44 d_id: d_subst ).
% 0.18/0.44
% 0.18/0.44 thf(alg444_1,axiom,
% 0.18/0.44 ( ! [T: term] :
% 0.18/0.44 ? [DT: d_term] :
% 0.18/0.44 ( T
% 0.18/0.44 = ( d2term @ DT ) )
% 0.18/0.44 & ! [DT: d_term] : ( DT = d_one )
% 0.18/0.44 & ! [DT1: d_term,DT2: d_term] :
% 0.18/0.44 ( ( ( d2term @ DT1 )
% 0.18/0.44 = ( d2term @ DT2 ) )
% 0.18/0.44 => ( DT1 = DT2 ) )
% 0.18/0.44 & ! [S: subst] :
% 0.18/0.44 ? [DS: d_subst] :
% 0.18/0.44 ( S
% 0.18/0.44 = ( d2subst @ DS ) )
% 0.18/0.44 & ! [DS: d_subst] : ( DS = d_id )
% 0.18/0.44 & ! [DS1: d_subst,DS2: d_subst] :
% 0.18/0.44 ( ( ( d2subst @ DS1 )
% 0.18/0.44 = ( d2subst @ DS2 ) )
% 0.18/0.44 => ( DS1 = DS2 ) )
% 0.18/0.44 & ( one
% 0.18/0.44 = ( d2term @ d_one ) )
% 0.18/0.44 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 0.18/0.44 = ( d2term @ d_one ) )
% 0.18/0.44 & ( ( lam @ ( d2term @ d_one ) )
% 0.18/0.44 = ( d2term @ d_one ) )
% 0.18/0.44 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.18/0.44 = ( d2term @ d_one ) )
% 0.18/0.44 & ( id
% 0.18/0.44 = ( d2subst @ d_id ) )
% 0.18/0.44 & ( sh
% 0.18/0.44 = ( d2subst @ d_id ) )
% 0.18/0.44 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.18/0.44 = ( d2subst @ d_id ) )
% 0.18/0.44 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 0.18/0.44 = ( d2subst @ d_id ) )
% 0.18/0.44 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 0.18/0.44 = ( d2term @ d_one ) )
% 0.18/0.44 & ( hoaslam
% 0.18/0.44 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 0.18/0.44 & ( hoasinduction_p_and_p_prime
% 0.18/0.44 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 0.18/0.44 ! [X: term] :
% 0.18/0.44 ( ( Q @ X )
% 0.18/0.44 = ( P @ id @ X @ id ) ) ) )
% 0.18/0.44 & ( pushprop_p_and_p_prime
% 0.18/0.44 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 0.18/0.44 ! [X: term] :
% 0.18/0.44 ( ( Q @ X )
% 0.18/0.44 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 0.18/0.44 & ~ ( var @ ( d2term @ d_one ) )
% 0.18/0.44 & ( pushprop_lem1v2
% 0.18/0.44 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ~ ! [X: term] :
% 0.18/0.44 ( ( Q @ X )
% 0.18/0.44 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.18/0.44 | ( Q @ one ) ) )
% 0.18/0.44 & pushprop_lem1_gthm
% 0.18/0.44 & axmap
% 0.18/0.44 & pushprop_lem0_gthm
% 0.18/0.44 & ( shinj
% 0.18/0.44 <=> ! [A: term,B: term] :
% 0.18/0.44 ( ( ( sub @ A @ sh )
% 0.18/0.44 != ( sub @ B @ sh ) )
% 0.18/0.44 | ( A = B ) ) )
% 0.18/0.44 & hoasinduction_lem1v2
% 0.18/0.44 & hoasinduction_lem1v2_gthm
% 0.18/0.44 & ( induction2lem
% 0.18/0.44 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.18/0.44 ( ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ~ ( P @ B )
% 0.18/0.44 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ! [B: term] :
% 0.18/0.44 ( ~ ( P @ B )
% 0.18/0.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.18/0.44 | ( P @ ( lam @ A ) ) )
% 0.18/0.44 | ~ ! [B: term] :
% 0.18/0.44 ( ~ ( var @ B )
% 0.18/0.44 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.18/0.44 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 0.18/0.44 & ( hoasinduction_lem3v2_f
% 0.18/0.44 <=> ! [B: term] :
% 0.18/0.44 ~ ! [F: subst > term > term] :
% 0.18/0.44 ~ ! [A: term,M: subst] :
% 0.18/0.44 ( ( sub @ B @ ( push @ A @ M ) )
% 0.18/0.44 = ( F @ M @ A ) ) )
% 0.18/0.44 & axvarshift
% 0.18/0.44 & ( hoasapinj2
% 0.18/0.44 <=> ! [A: term,B: term,C: term,D: term] :
% 0.18/0.44 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.18/0.44 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.18/0.44 | ( C = D ) ) )
% 0.18/0.44 & hoasapnotvar_gthm
% 0.18/0.44 & ( hoasapinj1
% 0.18/0.44 <=> ! [A: term,B: term,C: term,D: term] :
% 0.18/0.44 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.18/0.44 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.18/0.44 | ( A = B ) ) )
% 0.18/0.44 & ~ ulamvar1
% 0.18/0.44 & ( induction2lem_lthm
% 0.18/0.44 <=> ( ! [A: term,M: subst] :
% 0.18/0.44 ( A
% 0.18/0.44 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.18/0.44 => ( ! [A: term,M: subst] :
% 0.18/0.44 ( M
% 0.18/0.44 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.18/0.44 => ( ! [M: subst] :
% 0.18/0.44 ( M
% 0.18/0.44 = ( comp @ M @ id ) )
% 0.18/0.44 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 0.18/0.44 ( ~ ! [A: term] :
% 0.18/0.44 ( ~ ( var @ A )
% 0.18/0.44 | ( P @ A ) )
% 0.18/0.44 | ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ~ ( P @ B )
% 0.18/0.44 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ( P @ ( lam @ A ) ) )
% 0.18/0.44 | ( P @ Bound_variable_2140 ) )
% 0.18/0.44 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.18/0.44 ( ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ~ ( P @ B )
% 0.18/0.44 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ! [B: term] :
% 0.18/0.44 ( ~ ( P @ B )
% 0.18/0.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.18/0.44 | ( P @ ( lam @ A ) ) )
% 0.18/0.44 | ~ ! [B: term] :
% 0.18/0.44 ( ~ ( var @ B )
% 0.18/0.44 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.18/0.44 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 0.18/0.44 & hoasinduction_lem3v2_gthm
% 0.18/0.44 & apnotvar
% 0.18/0.44 & pushprop_lthm_orig
% 0.18/0.44 & ( hoasinduction_lem3v2_f_lthm
% 0.18/0.44 <=> ! [B: term] :
% 0.18/0.44 ~ ! [F: subst > term > term] :
% 0.18/0.44 ~ ! [A: term,M: subst] :
% 0.18/0.44 ( ( sub @ B @ ( push @ A @ M ) )
% 0.18/0.44 = ( F @ M @ A ) ) )
% 0.18/0.44 & ( hoasinduction_lthm
% 0.18/0.44 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.18/0.44 ( ~ ! [A: term] :
% 0.18/0.44 ( ~ ( var @ A )
% 0.18/0.44 | ( P @ A ) )
% 0.18/0.44 | ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ~ ( P @ B )
% 0.18/0.44 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ! [B: term] :
% 0.18/0.44 ( ~ ( P @ B )
% 0.18/0.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.18/0.44 | ( P @ ( lam @ A ) ) )
% 0.18/0.44 | ( P @ Bound_variable_2367 ) )
% 0.18/0.44 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.44 | ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.18/0.44 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.18/0.44 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.18/0.44 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 0.18/0.44 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.44 | ~ ! [F: subst > term > term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.44 | ~ ! [B: term] :
% 0.18/0.44 ( ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 0.18/0.44 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( var @ ( sub @ A @ id ) )
% 0.18/0.44 | ( P @ id @ A @ id ) )
% 0.18/0.44 | ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.18/0.44 | ~ ! [F: subst > term > term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.44 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 0.18/0.44 & ( hoasinduction_no_psi_cond_lthm
% 0.18/0.44 <=> ( ! [P: subst > term > subst > $o] :
% 0.18/0.44 ~ ! [Q: term > $o] :
% 0.18/0.44 ~ ! [X: term] :
% 0.18/0.44 ( ( Q @ X )
% 0.18/0.44 = ( P @ id @ X @ id ) )
% 0.18/0.44 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.18/0.44 ( ~ ! [A: term] :
% 0.18/0.44 ( ~ ( var @ A )
% 0.18/0.44 | ( P @ A ) )
% 0.18/0.44 | ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ~ ( P @ B )
% 0.18/0.44 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ! [B: term] :
% 0.18/0.44 ( ~ ( P @ B )
% 0.18/0.44 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.18/0.44 | ( P @ ( lam @ A ) ) )
% 0.18/0.44 | ( P @ Bound_variable_2367 ) )
% 0.18/0.44 => ( ! [A: term] :
% 0.18/0.44 ( A
% 0.18/0.44 = ( sub @ A @ id ) )
% 0.18/0.44 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.18/0.44 ( ~ ! [F: subst > term > term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.44 | ~ ! [X: term] :
% 0.18/0.44 ( ( Q @ X )
% 0.18/0.44 = ( P @ id @ X @ id ) )
% 0.18/0.44 | ~ ! [B: term] :
% 0.18/0.44 ( ~ ( Q @ B )
% 0.18/0.44 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.18/0.44 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.18/0.44 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.18/0.44 ( ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.18/0.44 | ~ ! [F: subst > term > term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.44 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 0.18/0.44 & ( hoaslaminj
% 0.18/0.44 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.18/0.44 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ( ( lam @ ( F @ sh @ one ) )
% 0.18/0.44 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.18/0.44 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.18/0.44 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 0.18/0.44 & ( hoasinduction_lem3aaa
% 0.18/0.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 0.18/0.44 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 0.18/0.44 ( ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 0.18/0.44 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 0.18/0.44 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 0.18/0.44 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 0.18/0.44 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 0.18/0.44 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 0.18/0.44 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 0.18/0.44 | ~ ! [B: term] :
% 0.18/0.44 ( ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.18/0.44 & induction2lem_gthm
% 0.18/0.44 & ( hoasinduction_lem3aa_lthm
% 0.18/0.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.18/0.44 ( ~ ! [F: subst > term > term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.44 | ~ ! [B: term] :
% 0.18/0.44 ( ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.18/0.44 & ( hoasinduction_lem3
% 0.18/0.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.44 | ~ ! [F: subst > term > term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.44 | ~ ! [B: term] :
% 0.18/0.44 ( ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 0.18/0.44 & ( hoasinduction_lem2
% 0.18/0.44 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.44 | ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.18/0.44 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.18/0.44 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.18/0.44 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 0.18/0.44 & ( termmset_lthm
% 0.18/0.44 <=> ( ! [A: term] :
% 0.18/0.44 ( A
% 0.18/0.44 = ( sub @ A @ id ) )
% 0.18/0.44 => ! [A: term] :
% 0.18/0.44 ( A
% 0.18/0.44 = ( sub @ A @ id ) ) ) )
% 0.18/0.44 & hoasinduction_lem1
% 0.18/0.44 & hoaslamnotap_lthm
% 0.18/0.44 & ( pushprop_lem1v2_lthm
% 0.18/0.44 <=> ( ! [A: term,M: subst] :
% 0.18/0.44 ( A
% 0.18/0.44 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.18/0.44 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ~ ! [X: term] :
% 0.18/0.44 ( ( Q @ X )
% 0.18/0.44 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.18/0.44 | ( Q @ one ) ) ) )
% 0.18/0.44 & hoasapnotvar
% 0.18/0.44 & ( hoasinduction_lem0
% 0.18/0.44 <=> ! [P: subst > term > subst > $o] :
% 0.18/0.44 ~ ! [Q: term > $o] :
% 0.18/0.44 ~ ! [X: term] :
% 0.18/0.44 ( ( Q @ X )
% 0.18/0.44 = ( P @ id @ X @ id ) ) )
% 0.18/0.44 & ( hoasinduction
% 0.18/0.44 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.44 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.44 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.44 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.44 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( var @ ( sub @ A @ id ) )
% 0.18/0.44 | ( P @ id @ A @ id ) )
% 0.18/0.44 | ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ~ ( P @ id @ B @ id )
% 0.18/0.44 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.18/0.44 | ~ ! [F: subst > term > term] :
% 0.18/0.44 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.44 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.44 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ id @ A @ id )
% 0.18/0.44 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.44 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.44 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 0.18/0.44 & hoasinduction_gthm
% 0.18/0.44 & axapp
% 0.18/0.44 & hoaslamnotvar_lthm
% 0.18/0.44 & pushprop_lem3v2_lthm
% 0.18/0.44 & ( hoasinduction_lem3b_lthm
% 0.18/0.44 <=> ! [B: term] :
% 0.18/0.44 ~ ! [F: subst > term > term] :
% 0.18/0.44 ( ( F @ sh @ one )
% 0.18/0.44 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.18/0.44 & ulamvarind
% 0.18/0.44 & ( induction
% 0.18/0.44 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 0.18/0.44 ( ~ ! [A: term] :
% 0.18/0.44 ( ~ ( var @ A )
% 0.18/0.44 | ( P @ A ) )
% 0.18/0.44 | ~ ! [A: term,B: term] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.44 | ~ ( P @ B )
% 0.18/0.44 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.44 | ~ ! [A: term] :
% 0.18/0.44 ( ~ ( P @ A )
% 0.18/0.45 | ( P @ ( lam @ A ) ) )
% 0.18/0.45 | ( P @ Bound_variable_2140 ) ) )
% 0.18/0.45 & ( hoasinduction_lem3a_lthm
% 0.18/0.45 <=> ( ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) )
% 0.18/0.45 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.18/0.45 ( ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.18/0.45 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.18/0.45 ( ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 0.18/0.45 & termmset_gthm
% 0.18/0.45 & ( hoasinduction_lem3aa
% 0.18/0.45 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.18/0.45 ( ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.18/0.45 & pushprop_lem1v2_gthm
% 0.18/0.45 & hoaslamnotap_gthm
% 0.18/0.45 & hoaslamnotvar_gthm
% 0.18/0.45 & hoasinduction_lem3b_gthm
% 0.18/0.45 & pushprop_lem2v2
% 0.18/0.45 & hoasinduction_lem3a_gthm
% 0.18/0.45 & axclos
% 0.18/0.45 & axassoc
% 0.18/0.45 & ( hoasinduction_lem2v2
% 0.18/0.45 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.45 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.45 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.45 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.45 | ~ ! [A: term,B: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.18/0.45 | ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ id @ X @ id ) )
% 0.18/0.45 | ~ ( Q @ Bound_variable_2820 )
% 0.18/0.45 | ~ ( Q @ Bound_variable_2822 )
% 0.18/0.45 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 0.18/0.45 & pushprop_lthm
% 0.18/0.45 & ( apinj2
% 0.18/0.45 <=> ! [A: term,B: term,C: term,D: term] :
% 0.18/0.45 ( ( ( ap @ A @ C )
% 0.18/0.45 != ( ap @ B @ D ) )
% 0.18/0.45 | ( C = D ) ) )
% 0.18/0.45 & ( apinj1
% 0.18/0.45 <=> ! [A: term,B: term,C: term,D: term] :
% 0.18/0.45 ( ( ( ap @ A @ C )
% 0.18/0.45 != ( ap @ B @ D ) )
% 0.18/0.45 | ( A = B ) ) )
% 0.18/0.45 & ( hoasapinj2_lthm
% 0.18/0.45 <=> ( ! [A: term,B: term,C: term,D: term] :
% 0.18/0.45 ( ( ( ap @ A @ C )
% 0.18/0.45 != ( ap @ B @ D ) )
% 0.18/0.45 | ( C = D ) )
% 0.18/0.45 => ! [A: term,B: term,C: term,D: term] :
% 0.18/0.45 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.18/0.45 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.18/0.45 | ( C = D ) ) ) )
% 0.18/0.45 & ( hoasinduction_lem3v2a
% 0.18/0.45 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.18/0.45 ( ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ id @ X @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( Q @ B )
% 0.18/0.45 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 0.18/0.45 & ( hoasapinj1_lthm
% 0.18/0.45 <=> ( ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) )
% 0.18/0.45 => ( ! [A: term,B: term,C: term,D: term] :
% 0.18/0.45 ( ( ( ap @ A @ C )
% 0.18/0.45 != ( ap @ B @ D ) )
% 0.18/0.45 | ( A = B ) )
% 0.18/0.45 => ! [A: term,B: term,C: term,D: term] :
% 0.18/0.45 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.18/0.45 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.18/0.45 | ( A = B ) ) ) ) )
% 0.18/0.45 & ( hoaslaminj_lthm
% 0.18/0.45 <=> ( ! [A: term,M: subst] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.18/0.45 => ( ! [A: term,M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.18/0.45 => ( ! [A: term,B: term] :
% 0.18/0.45 ( ( ( lam @ A )
% 0.18/0.45 != ( lam @ B ) )
% 0.18/0.45 | ( A = B ) )
% 0.18/0.45 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.18/0.45 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ( ( lam @ ( F @ sh @ one ) )
% 0.18/0.45 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.18/0.45 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.18/0.45 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 0.18/0.45 & ( axvarcons
% 0.18/0.45 <=> ! [A: term,M: subst] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 0.18/0.45 & ( axscons
% 0.18/0.45 <=> ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 0.18/0.45 & hoasinduction_lem2v2_gthm
% 0.18/0.45 & ( axidr
% 0.18/0.45 <=> ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ M @ id ) ) )
% 0.18/0.45 & ( pushprop_lem1
% 0.18/0.45 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.18/0.45 ( ~ ( P @ A )
% 0.18/0.45 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 0.18/0.45 & ( laminj
% 0.18/0.45 <=> ! [A: term,B: term] :
% 0.18/0.45 ( ( ( lam @ A )
% 0.18/0.45 != ( lam @ B ) )
% 0.18/0.45 | ( A = B ) ) )
% 0.18/0.45 & ( hoasinduction_lem3_lthm
% 0.18/0.45 <=> ( ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) )
% 0.18/0.45 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.18/0.45 ( ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.18/0.45 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.45 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.45 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.45 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.45 | ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 0.18/0.45 & ( pushprop_lem0
% 0.18/0.45 <=> ! [P: term > $o,A: term,M: subst] :
% 0.18/0.45 ~ ! [Q: term > $o] :
% 0.18/0.45 ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.18/0.45 & pushprop_gthm
% 0.18/0.45 & axabs
% 0.18/0.45 & ( hoasinduction_lem3v2a_lthm
% 0.18/0.45 <=> ( ! [B: term] :
% 0.18/0.45 ~ ! [F: subst > term > term] :
% 0.18/0.45 ~ ! [A: term,M: subst] :
% 0.18/0.45 ( ( sub @ B @ ( push @ A @ M ) )
% 0.18/0.45 = ( F @ M @ A ) )
% 0.18/0.45 => ( ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) )
% 0.18/0.45 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.18/0.45 ( ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ id @ X @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( Q @ B )
% 0.18/0.45 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 0.18/0.45 & hoasinduction_lem2_lthm
% 0.18/0.45 & hoasapinj2_gthm
% 0.18/0.45 & hoasinduction_lem1_lthm
% 0.18/0.45 & ~ lamnotap
% 0.18/0.45 & hoasapinj1_gthm
% 0.18/0.45 & hoaslamnotvar
% 0.18/0.45 & ( axidl
% 0.18/0.45 <=> ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ id @ M ) ) )
% 0.18/0.45 & hoaslaminj_gthm
% 0.18/0.45 & ( induction2_lthm
% 0.18/0.45 <=> ( ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) )
% 0.18/0.45 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.18/0.45 ( ~ ! [A: term,B: term] :
% 0.18/0.45 ( ~ ( P @ A )
% 0.18/0.45 | ~ ( P @ B )
% 0.18/0.45 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ B )
% 0.18/0.45 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( P @ ( lam @ A ) ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( var @ B )
% 0.18/0.45 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.18/0.45 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 0.18/0.45 => ! [P: term > $o,Bound_variable_2367: term] :
% 0.18/0.45 ( ~ ! [A: term] :
% 0.18/0.45 ( ~ ( var @ A )
% 0.18/0.45 | ( P @ A ) )
% 0.18/0.45 | ~ ! [A: term,B: term] :
% 0.18/0.45 ( ~ ( P @ A )
% 0.18/0.45 | ~ ( P @ B )
% 0.18/0.45 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ B )
% 0.18/0.45 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( P @ ( lam @ A ) ) )
% 0.18/0.45 | ( P @ Bound_variable_2367 ) ) ) ) )
% 0.18/0.45 & ( hoasinduction_lem0_lthm
% 0.18/0.45 <=> ! [P: subst > term > subst > $o] :
% 0.18/0.45 ~ ! [Q: term > $o] :
% 0.18/0.45 ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ id @ X @ id ) ) )
% 0.18/0.45 & ( substmonoid_lthm
% 0.18/0.45 <=> ( ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ id @ M ) )
% 0.18/0.45 => ( ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ M @ id ) )
% 0.18/0.45 => ( ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ id @ M ) )
% 0.18/0.45 & ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ M @ id ) ) ) ) ) )
% 0.18/0.45 & pushprop
% 0.18/0.45 & hoasinduction_lem3_gthm
% 0.18/0.45 & hoasinduction_lem2_gthm
% 0.18/0.45 & ( hoasinduction_lem3b
% 0.18/0.45 <=> ! [B: term] :
% 0.18/0.45 ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ( F @ sh @ one )
% 0.18/0.45 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.18/0.45 & ( substmonoid
% 0.18/0.45 <=> ( ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ id @ M ) )
% 0.18/0.45 & ! [M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ M @ id ) ) ) )
% 0.18/0.45 & lamnotvar
% 0.18/0.45 & ( hoasinduction_lem3a
% 0.18/0.45 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.18/0.45 ( ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 0.18/0.45 & hoasinduction_lem1_gthm
% 0.18/0.45 & ( hoasinduction_no_psi_cond
% 0.18/0.45 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.18/0.45 ( ~ ! [A: term,B: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.18/0.45 | ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 0.18/0.45 & induction2_gthm
% 0.18/0.45 & pushprop_lem2v2_lthm
% 0.18/0.45 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.18/0.45 & ( hoaslamnotap
% 0.18/0.45 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ( ( lam @ ( F @ sh @ one ) )
% 0.18/0.45 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 0.18/0.45 & substmonoid_gthm
% 0.18/0.45 & ulamvarsh
% 0.18/0.45 & ( induction2
% 0.18/0.45 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 0.18/0.45 ( ~ ! [A: term] :
% 0.18/0.45 ( ~ ( var @ A )
% 0.18/0.45 | ( P @ A ) )
% 0.18/0.45 | ~ ! [A: term,B: term] :
% 0.18/0.45 ( ~ ( P @ A )
% 0.18/0.45 | ~ ( P @ B )
% 0.18/0.45 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ B )
% 0.18/0.45 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( P @ ( lam @ A ) ) )
% 0.18/0.45 | ( P @ Bound_variable_2367 ) ) )
% 0.18/0.45 & pushprop_lem3v2
% 0.18/0.45 & pushprop_lem2v2_gthm
% 0.18/0.45 & ( pushprop_lem1_lthm
% 0.18/0.45 <=> ( ! [A: term,M: subst] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.18/0.45 => ( ! [A: term,M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.18/0.45 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.18/0.45 ( ~ ( P @ A )
% 0.18/0.45 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 0.18/0.45 & ( hoasinduction_lem3v2
% 0.18/0.45 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.45 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.45 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.45 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.45 | ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ id @ X @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( Q @ B )
% 0.18/0.45 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 0.18/0.45 & ( axshiftcons
% 0.18/0.45 <=> ! [A: term,M: subst] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 0.18/0.45 & ( termmset
% 0.18/0.45 <=> ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) ) )
% 0.18/0.45 & ( pushprop_lem0_lthm
% 0.18/0.45 <=> ! [P: term > $o,A: term,M: subst] :
% 0.18/0.45 ~ ! [Q: term > $o] :
% 0.18/0.45 ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.18/0.45 & hoasapnotvar_lthm
% 0.18/0.45 & ( hoasinduction_lem3v2_lthm
% 0.18/0.45 <=> ( ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) )
% 0.18/0.45 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.45 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.45 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.45 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.45 | ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ id @ X @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( Q @ B )
% 0.18/0.45 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 0.18/0.45 & ( axvarid
% 0.18/0.45 <=> ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) ) )
% 0.18/0.45 & ( hoasinduction_lthm_3
% 0.18/0.45 <=> ( ! [P: subst > term > subst > $o] :
% 0.18/0.45 ~ ! [Q: term > $o] :
% 0.18/0.45 ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ id @ X @ id ) )
% 0.18/0.45 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.18/0.45 ( ~ ! [A: term] :
% 0.18/0.45 ( ~ ( var @ A )
% 0.18/0.45 | ( P @ A ) )
% 0.18/0.45 | ~ ! [A: term,B: term] :
% 0.18/0.45 ( ~ ( P @ A )
% 0.18/0.45 | ~ ( P @ B )
% 0.18/0.45 | ( P @ ( ap @ A @ B ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ! [B: term] :
% 0.18/0.45 ( ~ ( P @ B )
% 0.18/0.45 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( P @ ( lam @ A ) ) )
% 0.18/0.45 | ( P @ Bound_variable_2367 ) )
% 0.18/0.45 => ( ! [A: term] :
% 0.18/0.45 ( A
% 0.18/0.45 = ( sub @ A @ id ) )
% 0.18/0.45 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.18/0.45 ( ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ~ ! [X: term] :
% 0.18/0.45 ( ( Q @ X )
% 0.18/0.45 = ( P @ id @ X @ id ) )
% 0.18/0.45 | ~ ! [B: term] :
% 0.18/0.45 ( ~ ( Q @ B )
% 0.18/0.45 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.18/0.45 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.18/0.45 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.18/0.45 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.18/0.45 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.18/0.45 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.18/0.45 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( var @ ( sub @ A @ id ) )
% 0.18/0.45 | ( P @ id @ A @ id ) )
% 0.18/0.45 | ~ ! [A: term,B: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ~ ( P @ id @ B @ id )
% 0.18/0.45 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.18/0.45 | ~ ! [F: subst > term > term] :
% 0.18/0.45 ( ~ ! [M: subst,A: term,N: subst] :
% 0.18/0.45 ( ( sub @ ( F @ M @ A ) @ N )
% 0.18/0.45 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.18/0.45 | ~ ! [A: term] :
% 0.18/0.45 ( ~ ( P @ id @ A @ id )
% 0.18/0.45 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.18/0.45 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.18/0.45 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(axclos,conjecture,
% 0.18/0.45 ( axclos
% 0.18/0.45 = ( ! [A: term,M: subst,N: subst] :
% 0.18/0.45 ( ( sub @ ( sub @ A @ M ) @ N )
% 0.18/0.45 = ( sub @ A @ ( comp @ M @ N ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 ------------------------
% 2.39/1.48 ---- Embedded in THF ---
% 2.39/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.39/1.48 %%% Generated on Tue May 05 08:57:26 EDT 2026
% 2.39/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.39/1.48 %%% Logic specification used:
% 2.39/1.48 %%% thf(spec, logic, $$dhol).
% 2.39/1.48
% 2.39/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.6IcC7TwM2m/DTF2DTF2950.p
% See solution above
% 2.39/1.48 ------------------------
% 2.61/1.52 ---- Cleaned THF ---
% 2.61/1.52 thf(term_type,type,
% 2.61/1.52 term: $tType ).
% 2.61/1.52
% 2.61/1.52 thf(subst_type,type,
% 2.61/1.52 subst: $tType ).
% 2.61/1.52
% 2.61/1.52 thf(one_decl,type,
% 2.61/1.52 one: term ).
% 2.61/1.52
% 2.61/1.52 thf(ap_decl,type,
% 2.61/1.52 ap: term > term > term ).
% 2.61/1.52
% 2.61/1.52 thf(lam_decl,type,
% 2.61/1.52 lam: term > term ).
% 2.61/1.52
% 2.61/1.52 thf(sub_decl,type,
% 2.61/1.52 sub: term > subst > term ).
% 2.61/1.52
% 2.61/1.52 thf(id_decl,type,
% 2.61/1.52 id: subst ).
% 2.61/1.52
% 2.61/1.52 thf(sh_decl,type,
% 2.61/1.52 sh: subst ).
% 2.61/1.52
% 2.61/1.52 thf(push_decl,type,
% 2.61/1.52 push: term > subst > subst ).
% 2.61/1.52
% 2.61/1.52 thf(comp_decl,type,
% 2.61/1.52 comp: subst > subst > subst ).
% 2.61/1.52
% 2.61/1.52 thf(var_decl,type,
% 2.61/1.52 var: term > $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem1v2_decl,type,
% 2.61/1.52 pushprop_lem1v2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem1_gthm_decl,type,
% 2.61/1.52 pushprop_lem1_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axmap_decl,type,
% 2.61/1.52 axmap: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem0_gthm_decl,type,
% 2.61/1.52 pushprop_lem0_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(shinj_decl,type,
% 2.61/1.52 shinj: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem1v2_decl,type,
% 2.61/1.52 hoasinduction_lem1v2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem1v2_gthm_decl,type,
% 2.61/1.52 hoasinduction_lem1v2_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasap_decl,type,
% 2.61/1.52 hoasap: subst > term > subst > term > term ).
% 2.61/1.52
% 2.61/1.52 thf(induction2lem_decl,type,
% 2.61/1.52 induction2lem: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3v2_f_decl,type,
% 2.61/1.52 hoasinduction_lem3v2_f: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axvarshift_decl,type,
% 2.61/1.52 axvarshift: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapinj2_decl,type,
% 2.61/1.52 hoasapinj2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapnotvar_gthm_decl,type,
% 2.61/1.52 hoasapnotvar_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapinj1_decl,type,
% 2.61/1.52 hoasapinj1: $o ).
% 2.61/1.52
% 2.61/1.52 thf(ulamvar1_decl,type,
% 2.61/1.52 ulamvar1: $o ).
% 2.61/1.52
% 2.61/1.52 thf(induction2lem_lthm_decl,type,
% 2.61/1.52 induction2lem_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3v2_gthm_decl,type,
% 2.61/1.52 hoasinduction_lem3v2_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(apnotvar_decl,type,
% 2.61/1.52 apnotvar: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lthm_orig_decl,type,
% 2.61/1.52 pushprop_lthm_orig: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem3v2_f_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lthm_decl,type,
% 2.61/1.52 hoasinduction_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 2.61/1.52 hoasinduction_no_psi_cond_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslaminj_decl,type,
% 2.61/1.52 hoaslaminj: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3aaa_decl,type,
% 2.61/1.52 hoasinduction_lem3aaa: $o ).
% 2.61/1.52
% 2.61/1.52 thf(induction2lem_gthm_decl,type,
% 2.61/1.52 induction2lem_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3aa_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem3aa_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3_decl,type,
% 2.61/1.52 hoasinduction_lem3: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem2_decl,type,
% 2.61/1.52 hoasinduction_lem2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(termmset_lthm_decl,type,
% 2.61/1.52 termmset_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem1_decl,type,
% 2.61/1.52 hoasinduction_lem1: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslamnotap_lthm_decl,type,
% 2.61/1.52 hoaslamnotap_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem1v2_lthm_decl,type,
% 2.61/1.52 pushprop_lem1v2_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapnotvar_decl,type,
% 2.61/1.52 hoasapnotvar: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem0_decl,type,
% 2.61/1.52 hoasinduction_lem0: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_decl,type,
% 2.61/1.52 hoasinduction: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_gthm_decl,type,
% 2.61/1.52 hoasinduction_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axapp_decl,type,
% 2.61/1.52 axapp: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslamnotvar_lthm_decl,type,
% 2.61/1.52 hoaslamnotvar_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem3v2_lthm_decl,type,
% 2.61/1.52 pushprop_lem3v2_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3b_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem3b_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(ulamvarind_decl,type,
% 2.61/1.52 ulamvarind: $o ).
% 2.61/1.52
% 2.61/1.52 thf(induction_decl,type,
% 2.61/1.52 induction: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3a_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem3a_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(termmset_gthm_decl,type,
% 2.61/1.52 termmset_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3aa_decl,type,
% 2.61/1.52 hoasinduction_lem3aa: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem1v2_gthm_decl,type,
% 2.61/1.52 pushprop_lem1v2_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslamnotap_gthm_decl,type,
% 2.61/1.52 hoaslamnotap_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslamnotvar_gthm_decl,type,
% 2.61/1.52 hoaslamnotvar_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3b_gthm_decl,type,
% 2.61/1.52 hoasinduction_lem3b_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem2v2_decl,type,
% 2.61/1.52 pushprop_lem2v2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3a_gthm_decl,type,
% 2.61/1.52 hoasinduction_lem3a_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axclos_decl,type,
% 2.61/1.52 axclos: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axassoc_decl,type,
% 2.61/1.52 axassoc: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem2v2_decl,type,
% 2.61/1.52 hoasinduction_lem2v2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lthm_decl,type,
% 2.61/1.52 pushprop_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(apinj2_decl,type,
% 2.61/1.52 apinj2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(apinj1_decl,type,
% 2.61/1.52 apinj1: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapinj2_lthm_decl,type,
% 2.61/1.52 hoasapinj2_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3v2a_decl,type,
% 2.61/1.52 hoasinduction_lem3v2a: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapinj1_lthm_decl,type,
% 2.61/1.52 hoasapinj1_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslaminj_lthm_decl,type,
% 2.61/1.52 hoaslaminj_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axvarcons_decl,type,
% 2.61/1.52 axvarcons: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslam_decl,type,
% 2.61/1.52 hoaslam: subst > ( subst > term > term ) > term ).
% 2.61/1.52
% 2.61/1.52 thf(axscons_decl,type,
% 2.61/1.52 axscons: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem2v2_gthm_decl,type,
% 2.61/1.52 hoasinduction_lem2v2_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axidr_decl,type,
% 2.61/1.52 axidr: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem1_decl,type,
% 2.61/1.52 pushprop_lem1: $o ).
% 2.61/1.52
% 2.61/1.52 thf(laminj_decl,type,
% 2.61/1.52 laminj: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem3_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem0_decl,type,
% 2.61/1.52 pushprop_lem0: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_gthm_decl,type,
% 2.61/1.52 pushprop_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axabs_decl,type,
% 2.61/1.52 axabs: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem3v2a_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem2_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem2_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapinj2_gthm_decl,type,
% 2.61/1.52 hoasapinj2_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_p_and_p_prime_decl,type,
% 2.61/1.52 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem1_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem1_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(lamnotap_decl,type,
% 2.61/1.52 lamnotap: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapinj1_gthm_decl,type,
% 2.61/1.52 hoasapinj1_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslamnotvar_decl,type,
% 2.61/1.52 hoaslamnotvar: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axidl_decl,type,
% 2.61/1.52 axidl: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslaminj_gthm_decl,type,
% 2.61/1.52 hoaslaminj_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(induction2_lthm_decl,type,
% 2.61/1.52 induction2_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem0_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem0_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(substmonoid_lthm_decl,type,
% 2.61/1.52 substmonoid_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_decl,type,
% 2.61/1.52 pushprop: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3_gthm_decl,type,
% 2.61/1.52 hoasinduction_lem3_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem2_gthm_decl,type,
% 2.61/1.52 hoasinduction_lem2_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3b_decl,type,
% 2.61/1.52 hoasinduction_lem3b: $o ).
% 2.61/1.52
% 2.61/1.52 thf(substmonoid_decl,type,
% 2.61/1.52 substmonoid: $o ).
% 2.61/1.52
% 2.61/1.52 thf(lamnotvar_decl,type,
% 2.61/1.52 lamnotvar: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3a_decl,type,
% 2.61/1.52 hoasinduction_lem3a: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem1_gthm_decl,type,
% 2.61/1.52 hoasinduction_lem1_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_no_psi_cond_decl,type,
% 2.61/1.52 hoasinduction_no_psi_cond: $o ).
% 2.61/1.52
% 2.61/1.52 thf(induction2_gthm_decl,type,
% 2.61/1.52 induction2_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem2v2_lthm_decl,type,
% 2.61/1.52 pushprop_lem2v2_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasvar_decl,type,
% 2.61/1.52 hoasvar: subst > term > subst > $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoaslamnotap_decl,type,
% 2.61/1.52 hoaslamnotap: $o ).
% 2.61/1.52
% 2.61/1.52 thf(substmonoid_gthm_decl,type,
% 2.61/1.52 substmonoid_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(ulamvarsh_decl,type,
% 2.61/1.52 ulamvarsh: $o ).
% 2.61/1.52
% 2.61/1.52 thf(induction2_decl,type,
% 2.61/1.52 induction2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem3v2_decl,type,
% 2.61/1.52 pushprop_lem3v2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem2v2_gthm_decl,type,
% 2.61/1.52 pushprop_lem2v2_gthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem1_lthm_decl,type,
% 2.61/1.52 pushprop_lem1_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3v2_decl,type,
% 2.61/1.52 hoasinduction_lem3v2: $o ).
% 2.61/1.52
% 2.61/1.52 thf(axshiftcons_decl,type,
% 2.61/1.52 axshiftcons: $o ).
% 2.61/1.52
% 2.61/1.52 thf(termmset_decl,type,
% 2.61/1.52 termmset: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_lem0_lthm_decl,type,
% 2.61/1.52 pushprop_lem0_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasapnotvar_lthm_decl,type,
% 2.61/1.52 hoasapnotvar_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lem3v2_lthm_decl,type,
% 2.61/1.52 hoasinduction_lem3v2_lthm: $o ).
% 2.61/1.52
% 2.61/1.52 thf(pushprop_p_and_p_prime_decl,type,
% 2.61/1.52 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 2.61/1.52
% 2.61/1.52 thf(axvarid_decl,type,
% 2.61/1.52 axvarid: $o ).
% 2.61/1.52
% 2.61/1.52 thf(hoasinduction_lthm_3_decl,type,
% 2.61/1.52 hoasinduction_lthm_3: $o ).
% 2.61/1.52
% 2.61/1.52 thf(d_term_type,type,
% 2.61/1.52 d_term: $tType ).
% 2.61/1.52
% 2.61/1.52 thf(d_subst_type,type,
% 2.61/1.52 d_subst: $tType ).
% 2.61/1.52
% 2.61/1.52 thf(d2term_decl,type,
% 2.61/1.52 d2term: d_term > term ).
% 2.61/1.52
% 2.61/1.52 thf(d2subst_decl,type,
% 2.61/1.52 d2subst: d_subst > subst ).
% 2.61/1.52
% 2.61/1.52 thf(d_one_decl,type,
% 2.61/1.52 d_one: d_term ).
% 2.61/1.52
% 2.61/1.52 thf(d_id_decl,type,
% 2.61/1.52 d_id: d_subst ).
% 2.61/1.52
% 2.61/1.52 thf(alg444_1,axiom,
% 2.61/1.52 ( ! [T: term] :
% 2.61/1.52 ? [DT: d_term] :
% 2.61/1.52 ( T
% 2.61/1.52 = ( d2term @ DT ) )
% 2.61/1.52 & ! [DT: d_term] : ( DT = d_one )
% 2.61/1.52 & ! [DT1: d_term,DT2: d_term] :
% 2.61/1.52 ( ( ( d2term @ DT1 )
% 2.61/1.52 = ( d2term @ DT2 ) )
% 2.61/1.52 => ( DT1 = DT2 ) )
% 2.61/1.52 & ! [S: subst] :
% 2.61/1.52 ? [DS: d_subst] :
% 2.61/1.52 ( S
% 2.61/1.52 = ( d2subst @ DS ) )
% 2.61/1.52 & ! [DS: d_subst] : ( DS = d_id )
% 2.61/1.52 & ! [DS1: d_subst,DS2: d_subst] :
% 2.61/1.52 ( ( ( d2subst @ DS1 )
% 2.61/1.52 = ( d2subst @ DS2 ) )
% 2.61/1.52 => ( DS1 = DS2 ) )
% 2.61/1.52 & ( one
% 2.61/1.52 = ( d2term @ d_one ) )
% 2.61/1.52 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 2.61/1.52 = ( d2term @ d_one ) )
% 2.61/1.52 & ( ( lam @ ( d2term @ d_one ) )
% 2.61/1.52 = ( d2term @ d_one ) )
% 2.61/1.52 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.61/1.52 = ( d2term @ d_one ) )
% 2.61/1.52 & ( id
% 2.61/1.52 = ( d2subst @ d_id ) )
% 2.61/1.52 & ( sh
% 2.61/1.52 = ( d2subst @ d_id ) )
% 2.61/1.52 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.61/1.52 = ( d2subst @ d_id ) )
% 2.61/1.52 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 2.61/1.52 = ( d2subst @ d_id ) )
% 2.61/1.52 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 2.61/1.52 = ( d2term @ d_one ) )
% 2.61/1.52 & ( hoaslam
% 2.61/1.52 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 2.61/1.52 & ( hoasinduction_p_and_p_prime
% 2.61/1.52 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 2.61/1.52 ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) ) ) )
% 2.61/1.52 & ( pushprop_p_and_p_prime
% 2.61/1.52 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 2.61/1.52 ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 2.61/1.52 & ~ ( var @ ( d2term @ d_one ) )
% 2.61/1.52 & ( pushprop_lem1v2
% 2.61/1.52 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.61/1.52 | ( Q @ one ) ) )
% 2.61/1.52 & pushprop_lem1_gthm
% 2.61/1.52 & axmap
% 2.61/1.52 & pushprop_lem0_gthm
% 2.61/1.52 & ( shinj
% 2.61/1.52 <=> ! [A: term,B: term] :
% 2.61/1.52 ( ( ( sub @ A @ sh )
% 2.61/1.52 != ( sub @ B @ sh ) )
% 2.61/1.52 | ( A = B ) ) )
% 2.61/1.52 & hoasinduction_lem1v2
% 2.61/1.52 & hoasinduction_lem1v2_gthm
% 2.61/1.52 & ( induction2lem
% 2.61/1.52 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.61/1.52 ( ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ B )
% 2.61/1.52 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( var @ B )
% 2.61/1.52 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.61/1.52 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 2.61/1.52 & ( hoasinduction_lem3v2_f
% 2.61/1.52 <=> ! [B: term] :
% 2.61/1.52 ~ ! [F: subst > term > term] :
% 2.61/1.52 ~ ! [A: term,M: subst] :
% 2.61/1.52 ( ( sub @ B @ ( push @ A @ M ) )
% 2.61/1.52 = ( F @ M @ A ) ) )
% 2.61/1.52 & axvarshift
% 2.61/1.52 & ( hoasapinj2
% 2.61/1.52 <=> ! [A: term,B: term,C: term,D: term] :
% 2.61/1.52 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.61/1.52 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.61/1.52 | ( C = D ) ) )
% 2.61/1.52 & hoasapnotvar_gthm
% 2.61/1.52 & ( hoasapinj1
% 2.61/1.52 <=> ! [A: term,B: term,C: term,D: term] :
% 2.61/1.52 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.61/1.52 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.61/1.52 | ( A = B ) ) )
% 2.61/1.52 & ~ ulamvar1
% 2.61/1.52 & ( induction2lem_lthm
% 2.61/1.52 <=> ( ! [A: term,M: subst] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.61/1.52 => ( ! [A: term,M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.61/1.52 => ( ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ M @ id ) )
% 2.61/1.52 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 2.61/1.52 ( ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ A )
% 2.61/1.52 | ( P @ A ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ( P @ Bound_variable_2140 ) )
% 2.61/1.52 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.61/1.52 ( ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ B )
% 2.61/1.52 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( var @ B )
% 2.61/1.52 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.61/1.52 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 2.61/1.52 & hoasinduction_lem3v2_gthm
% 2.61/1.52 & apnotvar
% 2.61/1.52 & pushprop_lthm_orig
% 2.61/1.52 & ( hoasinduction_lem3v2_f_lthm
% 2.61/1.52 <=> ! [B: term] :
% 2.61/1.52 ~ ! [F: subst > term > term] :
% 2.61/1.52 ~ ! [A: term,M: subst] :
% 2.61/1.52 ( ( sub @ B @ ( push @ A @ M ) )
% 2.61/1.52 = ( F @ M @ A ) ) )
% 2.61/1.52 & ( hoasinduction_lthm
% 2.61/1.52 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.61/1.52 ( ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ A )
% 2.61/1.52 | ( P @ A ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ B )
% 2.61/1.52 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ( P @ Bound_variable_2367 ) )
% 2.61/1.52 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.61/1.52 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.61/1.52 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 2.61/1.52 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 2.61/1.52 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ ( sub @ A @ id ) )
% 2.61/1.52 | ( P @ id @ A @ id ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 2.61/1.52 & ( hoasinduction_no_psi_cond_lthm
% 2.61/1.52 <=> ( ! [P: subst > term > subst > $o] :
% 2.61/1.52 ~ ! [Q: term > $o] :
% 2.61/1.52 ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.61/1.52 ( ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ A )
% 2.61/1.52 | ( P @ A ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ B )
% 2.61/1.52 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ( P @ Bound_variable_2367 ) )
% 2.61/1.52 => ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( Q @ B )
% 2.61/1.52 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.61/1.52 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.61/1.52 ( ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 2.61/1.52 & ( hoaslaminj
% 2.61/1.52 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.61/1.52 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ( ( lam @ ( F @ sh @ one ) )
% 2.61/1.52 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.61/1.52 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.61/1.52 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 2.61/1.52 & ( hoasinduction_lem3aaa
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 2.61/1.52 ( ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 2.61/1.52 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 2.61/1.52 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 2.61/1.52 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 2.61/1.52 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 2.61/1.52 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 2.61/1.52 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.61/1.52 & induction2lem_gthm
% 2.61/1.52 & ( hoasinduction_lem3aa_lthm
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.61/1.52 & ( hoasinduction_lem3
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 2.61/1.52 & ( hoasinduction_lem2
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.61/1.52 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.61/1.52 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 2.61/1.52 & ( termmset_lthm
% 2.61/1.52 <=> ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) ) ) )
% 2.61/1.52 & hoasinduction_lem1
% 2.61/1.52 & hoaslamnotap_lthm
% 2.61/1.52 & ( pushprop_lem1v2_lthm
% 2.61/1.52 <=> ( ! [A: term,M: subst] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.61/1.52 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.61/1.52 | ( Q @ one ) ) ) )
% 2.61/1.52 & hoasapnotvar
% 2.61/1.52 & ( hoasinduction_lem0
% 2.61/1.52 <=> ! [P: subst > term > subst > $o] :
% 2.61/1.52 ~ ! [Q: term > $o] :
% 2.61/1.52 ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) ) )
% 2.61/1.52 & ( hoasinduction
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ ( sub @ A @ id ) )
% 2.61/1.52 | ( P @ id @ A @ id ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 2.61/1.52 & hoasinduction_gthm
% 2.61/1.52 & axapp
% 2.61/1.52 & hoaslamnotvar_lthm
% 2.61/1.52 & pushprop_lem3v2_lthm
% 2.61/1.52 & ( hoasinduction_lem3b_lthm
% 2.61/1.52 <=> ! [B: term] :
% 2.61/1.52 ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ( F @ sh @ one )
% 2.61/1.52 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.61/1.52 & ulamvarind
% 2.61/1.52 & ( induction
% 2.61/1.52 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 2.61/1.52 ( ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ A )
% 2.61/1.52 | ( P @ A ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ( P @ Bound_variable_2140 ) ) )
% 2.61/1.52 & ( hoasinduction_lem3a_lthm
% 2.61/1.52 <=> ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.61/1.52 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 2.61/1.52 & termmset_gthm
% 2.61/1.52 & ( hoasinduction_lem3aa
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.61/1.52 & pushprop_lem1v2_gthm
% 2.61/1.52 & hoaslamnotap_gthm
% 2.61/1.52 & hoaslamnotvar_gthm
% 2.61/1.52 & hoasinduction_lem3b_gthm
% 2.61/1.52 & pushprop_lem2v2
% 2.61/1.52 & hoasinduction_lem3a_gthm
% 2.61/1.52 & axclos
% 2.61/1.52 & axassoc
% 2.61/1.52 & ( hoasinduction_lem2v2
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 | ~ ( Q @ Bound_variable_2820 )
% 2.61/1.52 | ~ ( Q @ Bound_variable_2822 )
% 2.61/1.52 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 2.61/1.52 & pushprop_lthm
% 2.61/1.52 & ( apinj2
% 2.61/1.52 <=> ! [A: term,B: term,C: term,D: term] :
% 2.61/1.52 ( ( ( ap @ A @ C )
% 2.61/1.52 != ( ap @ B @ D ) )
% 2.61/1.52 | ( C = D ) ) )
% 2.61/1.52 & ( apinj1
% 2.61/1.52 <=> ! [A: term,B: term,C: term,D: term] :
% 2.61/1.52 ( ( ( ap @ A @ C )
% 2.61/1.52 != ( ap @ B @ D ) )
% 2.61/1.52 | ( A = B ) ) )
% 2.61/1.52 & ( hoasapinj2_lthm
% 2.61/1.52 <=> ( ! [A: term,B: term,C: term,D: term] :
% 2.61/1.52 ( ( ( ap @ A @ C )
% 2.61/1.52 != ( ap @ B @ D ) )
% 2.61/1.52 | ( C = D ) )
% 2.61/1.52 => ! [A: term,B: term,C: term,D: term] :
% 2.61/1.52 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.61/1.52 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.61/1.52 | ( C = D ) ) ) )
% 2.61/1.52 & ( hoasinduction_lem3v2a
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( Q @ B )
% 2.61/1.52 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 2.61/1.52 & ( hoasapinj1_lthm
% 2.61/1.52 <=> ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ( ! [A: term,B: term,C: term,D: term] :
% 2.61/1.52 ( ( ( ap @ A @ C )
% 2.61/1.52 != ( ap @ B @ D ) )
% 2.61/1.52 | ( A = B ) )
% 2.61/1.52 => ! [A: term,B: term,C: term,D: term] :
% 2.61/1.52 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.61/1.52 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.61/1.52 | ( A = B ) ) ) ) )
% 2.61/1.52 & ( hoaslaminj_lthm
% 2.61/1.52 <=> ( ! [A: term,M: subst] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.61/1.52 => ( ! [A: term,M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.61/1.52 => ( ! [A: term,B: term] :
% 2.61/1.52 ( ( ( lam @ A )
% 2.61/1.52 != ( lam @ B ) )
% 2.61/1.52 | ( A = B ) )
% 2.61/1.52 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.61/1.52 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ( ( lam @ ( F @ sh @ one ) )
% 2.61/1.52 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.61/1.52 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.61/1.52 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 2.61/1.52 & ( axvarcons
% 2.61/1.52 <=> ! [A: term,M: subst] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 2.61/1.52 & ( axscons
% 2.61/1.52 <=> ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 2.61/1.52 & hoasinduction_lem2v2_gthm
% 2.61/1.52 & ( axidr
% 2.61/1.52 <=> ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ M @ id ) ) )
% 2.61/1.52 & ( pushprop_lem1
% 2.61/1.52 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 2.61/1.52 & ( laminj
% 2.61/1.52 <=> ! [A: term,B: term] :
% 2.61/1.52 ( ( ( lam @ A )
% 2.61/1.52 != ( lam @ B ) )
% 2.61/1.52 | ( A = B ) ) )
% 2.61/1.52 & ( hoasinduction_lem3_lthm
% 2.61/1.52 <=> ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.61/1.52 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 2.61/1.52 & ( pushprop_lem0
% 2.61/1.52 <=> ! [P: term > $o,A: term,M: subst] :
% 2.61/1.52 ~ ! [Q: term > $o] :
% 2.61/1.52 ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.61/1.52 & pushprop_gthm
% 2.61/1.52 & axabs
% 2.61/1.52 & ( hoasinduction_lem3v2a_lthm
% 2.61/1.52 <=> ( ! [B: term] :
% 2.61/1.52 ~ ! [F: subst > term > term] :
% 2.61/1.52 ~ ! [A: term,M: subst] :
% 2.61/1.52 ( ( sub @ B @ ( push @ A @ M ) )
% 2.61/1.52 = ( F @ M @ A ) )
% 2.61/1.52 => ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( Q @ B )
% 2.61/1.52 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 2.61/1.52 & hoasinduction_lem2_lthm
% 2.61/1.52 & hoasapinj2_gthm
% 2.61/1.52 & hoasinduction_lem1_lthm
% 2.61/1.52 & ~ lamnotap
% 2.61/1.52 & hoasapinj1_gthm
% 2.61/1.52 & hoaslamnotvar
% 2.61/1.52 & ( axidl
% 2.61/1.52 <=> ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ id @ M ) ) )
% 2.61/1.52 & hoaslaminj_gthm
% 2.61/1.52 & ( induction2_lthm
% 2.61/1.52 <=> ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.61/1.52 ( ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ B )
% 2.61/1.52 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( var @ B )
% 2.61/1.52 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.61/1.52 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 2.61/1.52 => ! [P: term > $o,Bound_variable_2367: term] :
% 2.61/1.52 ( ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ A )
% 2.61/1.52 | ( P @ A ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ B )
% 2.61/1.52 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ( P @ Bound_variable_2367 ) ) ) ) )
% 2.61/1.52 & ( hoasinduction_lem0_lthm
% 2.61/1.52 <=> ! [P: subst > term > subst > $o] :
% 2.61/1.52 ~ ! [Q: term > $o] :
% 2.61/1.52 ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) ) )
% 2.61/1.52 & ( substmonoid_lthm
% 2.61/1.52 <=> ( ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ id @ M ) )
% 2.61/1.52 => ( ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ M @ id ) )
% 2.61/1.52 => ( ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ id @ M ) )
% 2.61/1.52 & ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ M @ id ) ) ) ) ) )
% 2.61/1.52 & pushprop
% 2.61/1.52 & hoasinduction_lem3_gthm
% 2.61/1.52 & hoasinduction_lem2_gthm
% 2.61/1.52 & ( hoasinduction_lem3b
% 2.61/1.52 <=> ! [B: term] :
% 2.61/1.52 ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ( F @ sh @ one )
% 2.61/1.52 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.61/1.52 & ( substmonoid
% 2.61/1.52 <=> ( ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ id @ M ) )
% 2.61/1.52 & ! [M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ M @ id ) ) ) )
% 2.61/1.52 & lamnotvar
% 2.61/1.52 & ( hoasinduction_lem3a
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 2.61/1.52 & hoasinduction_lem1_gthm
% 2.61/1.52 & ( hoasinduction_no_psi_cond
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.61/1.52 ( ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 2.61/1.52 & induction2_gthm
% 2.61/1.52 & pushprop_lem2v2_lthm
% 2.61/1.52 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.61/1.52 & ( hoaslamnotap
% 2.61/1.52 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ( ( lam @ ( F @ sh @ one ) )
% 2.61/1.52 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 2.61/1.52 & substmonoid_gthm
% 2.61/1.52 & ulamvarsh
% 2.61/1.52 & ( induction2
% 2.61/1.52 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 2.61/1.52 ( ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ A )
% 2.61/1.52 | ( P @ A ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ B )
% 2.61/1.52 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ( P @ Bound_variable_2367 ) ) )
% 2.61/1.52 & pushprop_lem3v2
% 2.61/1.52 & pushprop_lem2v2_gthm
% 2.61/1.52 & ( pushprop_lem1_lthm
% 2.61/1.52 <=> ( ! [A: term,M: subst] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.61/1.52 => ( ! [A: term,M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.61/1.52 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 2.61/1.52 & ( hoasinduction_lem3v2
% 2.61/1.52 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( Q @ B )
% 2.61/1.52 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 2.61/1.52 & ( axshiftcons
% 2.61/1.52 <=> ! [A: term,M: subst] :
% 2.61/1.52 ( M
% 2.61/1.52 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 2.61/1.52 & ( termmset
% 2.61/1.52 <=> ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) ) )
% 2.61/1.52 & ( pushprop_lem0_lthm
% 2.61/1.52 <=> ! [P: term > $o,A: term,M: subst] :
% 2.61/1.52 ~ ! [Q: term > $o] :
% 2.61/1.52 ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.61/1.52 & hoasapnotvar_lthm
% 2.61/1.52 & ( hoasinduction_lem3v2_lthm
% 2.61/1.52 <=> ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( Q @ B )
% 2.61/1.52 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 2.61/1.52 & ( axvarid
% 2.61/1.52 <=> ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) ) )
% 2.61/1.52 & ( hoasinduction_lthm_3
% 2.61/1.52 <=> ( ! [P: subst > term > subst > $o] :
% 2.61/1.52 ~ ! [Q: term > $o] :
% 2.61/1.52 ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.61/1.52 ( ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ A )
% 2.61/1.52 | ( P @ A ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ A )
% 2.61/1.52 | ~ ( P @ B )
% 2.61/1.52 | ( P @ ( ap @ A @ B ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ! [B: term] :
% 2.61/1.52 ( ~ ( P @ B )
% 2.61/1.52 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( P @ ( lam @ A ) ) )
% 2.61/1.52 | ( P @ Bound_variable_2367 ) )
% 2.61/1.52 => ( ! [A: term] :
% 2.61/1.52 ( A
% 2.61/1.52 = ( sub @ A @ id ) )
% 2.61/1.52 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.61/1.52 ( ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ~ ! [X: term] :
% 2.61/1.52 ( ( Q @ X )
% 2.61/1.52 = ( P @ id @ X @ id ) )
% 2.61/1.52 | ~ ! [B: term] :
% 2.61/1.52 ( ~ ( Q @ B )
% 2.61/1.52 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.61/1.52 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.61/1.52 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.61/1.52 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.61/1.52 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.61/1.52 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.61/1.52 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( var @ ( sub @ A @ id ) )
% 2.61/1.52 | ( P @ id @ A @ id ) )
% 2.61/1.52 | ~ ! [A: term,B: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ~ ( P @ id @ B @ id )
% 2.61/1.52 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.61/1.52 | ~ ! [F: subst > term > term] :
% 2.61/1.52 ( ~ ! [M: subst,A: term,N: subst] :
% 2.61/1.52 ( ( sub @ ( F @ M @ A ) @ N )
% 2.61/1.52 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.61/1.52 | ~ ! [A: term] :
% 2.61/1.52 ( ~ ( P @ id @ A @ id )
% 2.61/1.52 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.61/1.52 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.61/1.52 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 2.61/1.52
% 2.61/1.52 thf(axclos,conjecture,
% 2.61/1.52 ( axclos
% 2.61/1.52 = ( ! [A: term,M: subst,N: subst] :
% 2.61/1.52 ( ( sub @ ( sub @ A @ M ) @ N )
% 2.61/1.52 = ( sub @ A @ ( comp @ M @ N ) ) ) ) ) ).
% 2.61/1.52 ------------------------
% 2.61/1.54 % (3422)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_2950 for (2999ds/2Mi)
% 2.61/1.54 % (3422)Instruction limit reached!
% 2.61/1.54 % (3422)------------------------------
% 2.61/1.54 % (3422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.54 % (3422)Termination reason: Unknown
% 2.61/1.54 % (3422)Termination phase: shuffling
% 2.61/1.54
% 2.61/1.54 % (3422)Memory used [KB]: 1151
% 2.61/1.54 % (3422)Time elapsed: 0.002 s
% 2.61/1.54 % (3422)Instructions burned: 2 (million)
% 2.61/1.54 % (3422)------------------------------
% 2.61/1.54 % (3422)------------------------------
% 2.61/1.54 % (3418)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/183Mi)
% 2.61/1.54 % (3419)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_2950 for (2999ds/4Mi)
% 2.61/1.54 % (3420)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_2950 for (2999ds/27Mi)
% 2.61/1.54 % (3421)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/2Mi)
% 2.61/1.54 % (3423)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_2950 for (2999ds/275Mi)
% 2.61/1.54 % (3424)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/18Mi)
% 2.61/1.55 % (3421)Instruction limit reached!
% 2.61/1.55 % (3421)------------------------------
% 2.61/1.55 % (3421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.55 % (3421)Termination reason: Unknown
% 2.61/1.55 % (3421)Termination phase: shuffling
% 2.61/1.55
% 2.61/1.55 % (3421)Memory used [KB]: 1279
% 2.61/1.55 % (3421)Time elapsed: 0.004 s
% 2.61/1.55 % (3421)Instructions burned: 3 (million)
% 2.61/1.55 % (3421)------------------------------
% 2.61/1.55 % (3421)------------------------------
% 2.61/1.55 % (3419)Instruction limit reached!
% 2.61/1.55 % (3419)------------------------------
% 2.61/1.55 % (3419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.55 % (3419)Termination reason: Unknown
% 2.61/1.55 % (3419)Termination phase: shuffling
% 2.61/1.55
% 2.61/1.55 % (3419)Memory used [KB]: 1279
% 2.61/1.55 % (3419)Time elapsed: 0.005 s
% 2.61/1.55 % (3419)Instructions burned: 5 (million)
% 2.61/1.55 % (3419)------------------------------
% 2.61/1.55 % (3419)------------------------------
% 2.61/1.55 % (3425)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/3Mi)
% 2.61/1.55 % (3425)Instruction limit reached!
% 2.61/1.55 % (3425)------------------------------
% 2.61/1.55 % (3425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.55 % (3425)Termination reason: Unknown
% 2.61/1.55 % (3425)Termination phase: shuffling
% 2.61/1.55
% 2.61/1.55 % (3425)Memory used [KB]: 1279
% 2.61/1.55 % (3425)Time elapsed: 0.003 s
% 2.61/1.55 % (3425)Instructions burned: 4 (million)
% 2.61/1.55 % (3425)------------------------------
% 2.61/1.55 % (3425)------------------------------
% 2.61/1.55 % (3424)Instruction limit reached!
% 2.61/1.55 % (3424)------------------------------
% 2.61/1.55 % (3424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.55 % (3424)Termination reason: Unknown
% 2.61/1.55 % (3424)Termination phase: shuffling
% 2.61/1.55
% 2.61/1.55 % (3424)Memory used [KB]: 2046
% 2.61/1.55 % (3424)Time elapsed: 0.013 s
% 2.61/1.55 % (3424)Instructions burned: 18 (million)
% 2.61/1.55 % (3424)------------------------------
% 2.61/1.55 % (3424)------------------------------
% 2.61/1.56 % (3428)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_2950 for (2999ds/3Mi)
% 2.61/1.56 % (3420)Instruction limit reached!
% 2.61/1.56 % (3420)------------------------------
% 2.61/1.56 % (3420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.56 % (3420)Termination reason: Unknown
% 2.61/1.56 % (3420)Termination phase: Property scanning
% 2.61/1.56
% 2.61/1.56 % (3420)Memory used [KB]: 2174
% 2.61/1.56 % (3420)Time elapsed: 0.018 s
% 2.61/1.56 % (3420)Instructions burned: 27 (million)
% 2.61/1.56 % (3420)------------------------------
% 2.61/1.56 % (3420)------------------------------
% 2.61/1.56 % (3428)Instruction limit reached!
% 2.61/1.56 % (3428)------------------------------
% 2.61/1.56 % (3428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.56 % (3428)Termination reason: Unknown
% 2.61/1.56 % (3428)Termination phase: shuffling
% 2.61/1.56
% 2.61/1.56 % (3428)Memory used [KB]: 1279
% 2.61/1.56 % (3428)Time elapsed: 0.003 s
% 2.61/1.56 % (3428)Instructions burned: 4 (million)
% 2.61/1.56 % (3428)------------------------------
% 2.61/1.56 % (3428)------------------------------
% 2.61/1.56 % (3426)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_2950 for (2999ds/37Mi)
% 2.61/1.56 % (3427)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_2950 for (2999ds/15Mi)
% 2.61/1.57 % (3423)Refutation not found, incomplete strategy
% 2.61/1.57 % (3423)------------------------------
% 2.61/1.57 % (3423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.57 % (3423)Termination reason: Refutation not found, incomplete strategy
% 2.61/1.57
% 2.61/1.57
% 2.61/1.57 % (3423)Memory used [KB]: 6652
% 2.61/1.57 % (3423)Time elapsed: 0.023 s
% 2.61/1.57 % (3423)Instructions burned: 37 (million)
% 2.61/1.57 % (3423)------------------------------
% 2.61/1.57 % (3423)------------------------------
% 2.61/1.57 % (3431)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/16Mi)
% 2.61/1.57 % (3427)Instruction limit reached!
% 2.61/1.57 % (3427)------------------------------
% 2.61/1.57 % (3427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.57 % (3427)Termination reason: Unknown
% 2.61/1.57 % (3427)Termination phase: shuffling
% 2.61/1.57
% 2.61/1.57 % (3427)Memory used [KB]: 1791
% 2.61/1.57 % (3427)Time elapsed: 0.010 s
% 2.61/1.57 % (3427)Instructions burned: 15 (million)
% 2.61/1.57 % (3427)------------------------------
% 2.61/1.57 % (3427)------------------------------
% 2.61/1.57 % (3429)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_2950 for (2999ds/1041Mi)
% 2.61/1.57 % (3430)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/7Mi)
% 2.61/1.58 % (3431)Instruction limit reached!
% 2.61/1.58 % (3431)------------------------------
% 2.61/1.58 % (3431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.58 % (3431)Termination reason: Unknown
% 2.61/1.58 % (3431)Termination phase: shuffling
% 2.61/1.58
% 2.61/1.58 % (3431)Memory used [KB]: 1918
% 2.61/1.58 % (3431)Time elapsed: 0.007 s
% 2.61/1.58 % (3431)Instructions burned: 16 (million)
% 2.61/1.58 % (3431)------------------------------
% 2.61/1.58 % (3431)------------------------------
% 2.61/1.58 % (3430)Instruction limit reached!
% 2.61/1.58 % (3430)------------------------------
% 2.61/1.58 % (3430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.58 % (3430)Termination reason: Unknown
% 2.61/1.58 % (3430)Termination phase: shuffling
% 2.61/1.58
% 2.61/1.58 % (3430)Memory used [KB]: 1407
% 2.61/1.58 % (3430)Time elapsed: 0.004 s
% 2.61/1.58 % (3430)Instructions burned: 7 (million)
% 2.61/1.58 % (3430)------------------------------
% 2.61/1.58 % (3430)------------------------------
% 2.61/1.58 % (3426)Instruction limit reached!
% 2.61/1.58 % (3426)------------------------------
% 2.61/1.58 % (3426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.58 % (3426)Termination reason: Unknown
% 2.61/1.58 % (3426)Termination phase: shuffling
% 2.61/1.58
% 2.61/1.58 % (3426)Memory used [KB]: 1918
% 2.61/1.58 % (3426)Time elapsed: 0.020 s
% 2.61/1.58 % (3426)Instructions burned: 37 (million)
% 2.61/1.58 % (3426)------------------------------
% 2.61/1.58 % (3426)------------------------------
% 2.61/1.58 % (3433)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_2950 for (2999ds/3Mi)
% 2.61/1.58 % (3433)Instruction limit reached!
% 2.61/1.58 % (3433)------------------------------
% 2.61/1.58 % (3433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.58 % (3433)Termination reason: Unknown
% 2.61/1.58 % (3433)Termination phase: shuffling
% 2.61/1.58
% 2.61/1.58 % (3433)Memory used [KB]: 1279
% 2.61/1.58 % (3433)Time elapsed: 0.003 s
% 2.61/1.58 % (3433)Instructions burned: 5 (million)
% 2.61/1.58 % (3433)------------------------------
% 2.61/1.58 % (3433)------------------------------
% 2.61/1.58 % (3432)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_2950 for (2999ds/3Mi)
% 2.61/1.59 % (3434)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/7Mi)
% 2.61/1.59 % (3432)Instruction limit reached!
% 2.61/1.59 % (3432)------------------------------
% 2.61/1.59 % (3432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.59 % (3432)Termination reason: Unknown
% 2.61/1.59 % (3432)Termination phase: shuffling
% 2.61/1.59
% 2.61/1.59 % (3432)Memory used [KB]: 1279
% 2.61/1.59 % (3432)Time elapsed: 0.004 s
% 2.61/1.59 % (3432)Instructions burned: 4 (million)
% 2.61/1.59 % (3432)------------------------------
% 2.61/1.59 % (3432)------------------------------
% 2.61/1.59 % (3435)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/3Mi)
% 2.61/1.59 % (3435)Instruction limit reached!
% 2.61/1.59 % (3435)------------------------------
% 2.61/1.59 % (3435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.59 % (3435)Termination reason: Unknown
% 2.61/1.59 % (3435)Termination phase: shuffling
% 2.61/1.59
% 2.61/1.59 % (3435)Memory used [KB]: 1279
% 2.61/1.59 % (3435)Time elapsed: 0.002 s
% 2.61/1.59 % (3435)Instructions burned: 3 (million)
% 2.61/1.59 % (3435)------------------------------
% 2.61/1.59 % (3435)------------------------------
% 2.61/1.59 % (3434)Instruction limit reached!
% 2.61/1.59 % (3434)------------------------------
% 2.61/1.59 % (3434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.59 % (3434)Termination reason: Unknown
% 2.61/1.59 % (3434)Termination phase: shuffling
% 2.61/1.59
% 2.61/1.59 % (3434)Memory used [KB]: 1407
% 2.61/1.59 % (3434)Time elapsed: 0.004 s
% 2.61/1.59 % (3434)Instructions burned: 8 (million)
% 2.61/1.59 % (3434)------------------------------
% 2.61/1.59 % (3434)------------------------------
% 2.61/1.59 % (3437)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/18Mi)
% 2.61/1.59 % (3436)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/4Mi)
% 2.61/1.60 % (3436)Instruction limit reached!
% 2.61/1.60 % (3436)------------------------------
% 2.61/1.60 % (3436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.60 % (3436)Termination reason: Unknown
% 2.61/1.60 % (3436)Termination phase: shuffling
% 2.61/1.60
% 2.61/1.60 % (3436)Memory used [KB]: 1279
% 2.61/1.60 % (3436)Time elapsed: 0.003 s
% 2.61/1.60 % (3436)Instructions burned: 4 (million)
% 2.61/1.60 % (3436)------------------------------
% 2.61/1.60 % (3436)------------------------------
% 2.65/1.60 % (3439)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/6Mi)
% 2.65/1.60 % (3440)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_2950 for (2999ds/902Mi)
% 2.65/1.60 % (3437)Instruction limit reached!
% 2.65/1.60 % (3437)------------------------------
% 2.65/1.60 % (3437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (3437)Termination reason: Unknown
% 2.65/1.60 % (3437)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (3437)Memory used [KB]: 2046
% 2.65/1.60 % (3437)Time elapsed: 0.007 s
% 2.65/1.60 % (3437)Instructions burned: 18 (million)
% 2.65/1.60 % (3437)------------------------------
% 2.65/1.60 % (3437)------------------------------
% 2.65/1.60 % (3438)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/710Mi)
% 2.65/1.60 % (3439)Instruction limit reached!
% 2.65/1.60 % (3439)------------------------------
% 2.65/1.60 % (3439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (3439)Termination reason: Unknown
% 2.65/1.60 % (3439)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (3439)Memory used [KB]: 1407
% 2.65/1.60 % (3439)Time elapsed: 0.004 s
% 2.65/1.60 % (3439)Instructions burned: 7 (million)
% 2.65/1.60 % (3439)------------------------------
% 2.65/1.60 % (3439)------------------------------
% 2.65/1.61 % (3441)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_2950 for (2999ds/21Mi)
% 2.65/1.61 % (3442)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/5Mi)
% 2.65/1.61 % (3443)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_2950 for (2999ds/6Mi)
% 2.65/1.61 % (3442)Instruction limit reached!
% 2.65/1.61 % (3442)------------------------------
% 2.65/1.61 % (3442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (3442)Termination reason: Unknown
% 2.65/1.61 % (3442)Termination phase: shuffling
% 2.65/1.61
% 2.65/1.61 % (3442)Memory used [KB]: 1407
% 2.65/1.61 % (3442)Time elapsed: 0.003 s
% 2.65/1.61 % (3442)Instructions burned: 7 (million)
% 2.65/1.61 % (3442)------------------------------
% 2.65/1.61 % (3442)------------------------------
% 2.65/1.61 % (3418)Instruction limit reached!
% 2.65/1.61 % (3418)------------------------------
% 2.65/1.61 % (3418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (3418)Termination reason: Unknown
% 2.65/1.61 % (3418)Termination phase: Property scanning
% 2.65/1.61
% 2.65/1.61 % (3418)Memory used [KB]: 4605
% 2.65/1.61 % (3418)Time elapsed: 0.070 s
% 2.65/1.61 % (3418)Instructions burned: 183 (million)
% 2.65/1.61 % (3418)------------------------------
% 2.65/1.61 % (3418)------------------------------
% 2.65/1.61 % (3443)Instruction limit reached!
% 2.65/1.61 % (3443)------------------------------
% 2.65/1.61 % (3443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (3443)Termination reason: Unknown
% 2.65/1.61 % (3443)Termination phase: shuffling
% 2.65/1.61
% 2.65/1.61 % (3443)Memory used [KB]: 1407
% 2.65/1.61 % (3443)Time elapsed: 0.004 s
% 2.65/1.61 % (3443)Instructions burned: 7 (million)
% 2.65/1.61 % (3443)------------------------------
% 2.65/1.61 % (3443)------------------------------
% 2.65/1.61 % (3441)Instruction limit reached!
% 2.65/1.61 % (3441)------------------------------
% 2.65/1.61 % (3441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (3441)Termination reason: Unknown
% 2.65/1.61 % (3441)Termination phase: shuffling
% 2.65/1.61
% 2.65/1.61 % (3441)Memory used [KB]: 2046
% 2.65/1.61 % (3441)Time elapsed: 0.009 s
% 2.65/1.61 % (3441)Instructions burned: 21 (million)
% 2.65/1.61 % (3441)------------------------------
% 2.65/1.61 % (3441)------------------------------
% 2.65/1.62 % (3444)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/377Mi)
% 2.65/1.62 % (3445)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_2950 for (2999ds/779Mi)
% 2.65/1.62 % (3446)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_2950 for (2999ds/19Mi)
% 2.65/1.62 % (3447)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/879Mi)
% 2.65/1.63 % (3446)Instruction limit reached!
% 2.65/1.63 % (3446)------------------------------
% 2.65/1.63 % (3446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.63 % (3446)Termination reason: Unknown
% 2.65/1.63 % (3446)Termination phase: shuffling
% 2.65/1.63
% 2.65/1.63 % (3446)Memory used [KB]: 1663
% 2.65/1.63 % (3446)Time elapsed: 0.008 s
% 2.65/1.63 % (3446)Instructions burned: 21 (million)
% 2.65/1.63 % (3446)------------------------------
% 2.65/1.63 % (3446)------------------------------
% 2.65/1.64 % (3448)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_2950 for (2999ds/17Mi)
% 2.65/1.64 % (3448)Instruction limit reached!
% 2.65/1.64 % (3448)------------------------------
% 2.65/1.64 % (3448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.64 % (3448)Termination reason: Unknown
% 2.65/1.64 % (3448)Termination phase: shuffling
% 2.65/1.64
% 2.65/1.64 % (3448)Memory used [KB]: 2046
% 2.65/1.64 % (3448)Time elapsed: 0.008 s
% 2.65/1.64 % (3448)Instructions burned: 18 (million)
% 2.65/1.64 % (3448)------------------------------
% 2.65/1.64 % (3448)------------------------------
% 2.65/1.65 % (3440)Refutation not found, incomplete strategy
% 2.65/1.65 % (3440)------------------------------
% 2.65/1.65 % (3440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.65 % (3440)Termination reason: Refutation not found, incomplete strategy
% 2.65/1.65
% 2.65/1.65
% 2.65/1.65 % (3440)Memory used [KB]: 9338
% 2.65/1.65 % (3440)Time elapsed: 0.057 s
% 2.65/1.65 % (3440)Instructions burned: 193 (million)
% 2.65/1.65 % (3440)------------------------------
% 2.65/1.65 % (3440)------------------------------
% 2.65/1.65 % (3449)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_2950 for (2998ds/3Mi)
% 2.65/1.66 % (3449)Instruction limit reached!
% 2.65/1.66 % (3449)------------------------------
% 2.65/1.66 % (3449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.66 % (3449)Termination reason: Unknown
% 2.65/1.66 % (3449)Termination phase: shuffling
% 2.65/1.66
% 2.65/1.66 % (3449)Memory used [KB]: 1279
% 2.65/1.66 % (3449)Time elapsed: 0.003 s
% 2.65/1.66 % (3449)Instructions burned: 4 (million)
% 2.65/1.66 % (3449)------------------------------
% 2.65/1.66 % (3449)------------------------------
% 2.65/1.66 % (3450)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_2950 for (2998ds/30Mi)
% 2.65/1.67 % (3451)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_2950 for (2998ds/127Mi)
% 2.65/1.67 % (3450)Instruction limit reached!
% 2.65/1.67 % (3450)------------------------------
% 2.65/1.67 % (3450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.67 % (3450)Termination reason: Unknown
% 2.65/1.67 % (3450)Termination phase: Property scanning
% 2.65/1.67
% 2.65/1.67 % (3450)Memory used [KB]: 2174
% 2.65/1.67 % (3450)Time elapsed: 0.010 s
% 2.65/1.67 % (3450)Instructions burned: 30 (million)
% 2.65/1.67 % (3450)------------------------------
% 2.65/1.67 % (3450)------------------------------
% 2.65/1.68 % (3452)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_2950 for (2998ds/100Mi)
% 2.65/1.70 % (3451)Instruction limit reached!
% 2.65/1.70 % (3451)------------------------------
% 2.65/1.70 % (3451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.70 % (3451)Termination reason: Unknown
% 2.65/1.70 % (3451)Termination phase: Property scanning
% 2.65/1.70
% 2.65/1.70 % (3451)Memory used [KB]: 4221
% 2.65/1.70 % (3451)Time elapsed: 0.040 s
% 2.65/1.70 % (3451)Instructions burned: 128 (million)
% 2.65/1.70 % (3451)------------------------------
% 2.65/1.70 % (3451)------------------------------
% 2.65/1.71 % (3452)Instruction limit reached!
% 2.65/1.71 % (3452)------------------------------
% 2.65/1.71 % (3452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.71 % (3452)Termination reason: Unknown
% 2.65/1.71 % (3452)Termination phase: Saturation
% 2.65/1.71
% 2.65/1.71 % (3452)Memory used [KB]: 6524
% 2.65/1.71 % (3452)Time elapsed: 0.027 s
% 2.65/1.71 % (3452)Instructions burned: 103 (million)
% 2.65/1.71 % (3452)------------------------------
% 2.65/1.71 % (3452)------------------------------
% 2.65/1.72 % (3453)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_2950 for (2998ds/3Mi)
% 2.65/1.72 % (3453)Instruction limit reached!
% 2.65/1.72 % (3453)------------------------------
% 2.65/1.72 % (3453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.72 % (3453)Termination reason: Unknown
% 2.65/1.72 % (3453)Termination phase: shuffling
% 2.65/1.72
% 2.65/1.72 % (3453)Memory used [KB]: 1279
% 2.65/1.72 % (3453)Time elapsed: 0.002 s
% 2.65/1.72 % (3453)Instructions burned: 4 (million)
% 2.65/1.72 % (3453)------------------------------
% 2.65/1.72 % (3453)------------------------------
% 2.65/1.72 % (3454)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_2950 for (2998ds/20Mi)
% 2.65/1.72 % (3447)First to succeed.
% 2.65/1.73 % (3454)Instruction limit reached!
% 2.65/1.73 % (3454)------------------------------
% 2.65/1.73 % (3454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.73 % (3454)Termination reason: Unknown
% 2.65/1.73 % (3454)Termination phase: shuffling
% 2.65/1.73
% 2.65/1.73 % (3454)Memory used [KB]: 2174
% 2.65/1.73 % (3454)Time elapsed: 0.008 s
% 2.65/1.73 % (3454)Instructions burned: 22 (million)
% 2.65/1.73 % (3454)------------------------------
% 2.65/1.73 % (3454)------------------------------
% 2.65/1.73 % (3455)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_2950 for (2998ds/86Mi)
% 2.65/1.73 % (3444)Instruction limit reached!
% 2.65/1.73 % (3444)------------------------------
% 2.65/1.73 % (3444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.73 % (3444)Termination reason: Unknown
% 2.65/1.73 % (3444)Termination phase: Saturation
% 2.65/1.73
% 2.65/1.73 % (3444)Memory used [KB]: 10618
% 2.65/1.73 % (3444)Time elapsed: 0.110 s
% 2.65/1.73 % (3444)Instructions burned: 379 (million)
% 2.65/1.73 % (3444)------------------------------
% 2.65/1.73 % (3444)------------------------------
% 2.65/1.74 % (3456)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_2950 for (2997ds/107Mi)
% 2.65/1.74 % (3457)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_2950 for (2997ds/39Mi)
% 2.65/1.75 % (3457)Instruction limit reached!
% 2.65/1.75 % (3457)------------------------------
% 2.65/1.75 % (3457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.75 % (3457)Termination reason: Unknown
% 2.65/1.75 % (3457)Termination phase: Property scanning
% 2.65/1.75
% 2.65/1.75 % (3457)Memory used [KB]: 1918
% 2.65/1.75 % (3457)Time elapsed: 0.012 s
% 2.65/1.75 % (3457)Instructions burned: 40 (million)
% 2.65/1.75 % (3457)------------------------------
% 2.65/1.75 % (3457)------------------------------
% 2.65/1.75 % (3455)Instruction limit reached!
% 2.65/1.75 % (3455)------------------------------
% 2.65/1.75 % (3455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.75 % (3455)Termination reason: Unknown
% 2.65/1.75 % (3455)Termination phase: Preprocessing 3
% 2.65/1.75
% 2.65/1.75 % (3455)Memory used [KB]: 3965
% 2.65/1.75 % (3455)Time elapsed: 0.029 s
% 2.65/1.75 % (3455)Instructions burned: 86 (million)
% 2.65/1.75 % (3455)------------------------------
% 2.65/1.75 % (3455)------------------------------
% 2.65/1.76 % (3458)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_2950 for (2997ds/448Mi)
% 2.65/1.76 % (3447)Refutation found. Thanks to Tanya!
% 2.65/1.76 % SZS status Theorem for DTF2THF_2950
% 2.65/1.76 % SZS output start Proof for DTF2THF_2950
% 2.65/1.76 thf(type_def_5, type, term: $tType).
% 2.65/1.76 thf(type_def_7, type, subst: $tType).
% 2.65/1.76 thf(type_def_8, type, d_term: $tType).
% 2.65/1.76 thf(type_def_9, type, d_subst: $tType).
% 2.65/1.76 thf(func_def_0, type, term: $tType).
% 2.65/1.76 thf(func_def_1, type, subst: $tType).
% 2.65/1.76 thf(func_def_2, type, one: term).
% 2.65/1.76 thf(func_def_3, type, ap: term > term > term).
% 2.65/1.76 thf(func_def_4, type, lam: term > term).
% 2.65/1.76 thf(func_def_5, type, sub: term > subst > term).
% 2.65/1.76 thf(func_def_6, type, id: subst).
% 2.65/1.76 thf(func_def_7, type, sh: subst).
% 2.65/1.76 thf(func_def_8, type, push: term > subst > subst).
% 2.65/1.76 thf(func_def_9, type, comp: subst > subst > subst).
% 2.65/1.76 thf(func_def_10, type, var: term > $o).
% 2.65/1.76 thf(func_def_18, type, hoasap: subst > term > subst > term > term).
% 2.65/1.76 thf(func_def_73, type, hoaslam: subst > (subst > term > term) > term).
% 2.65/1.76 thf(func_def_86, type, hoasinduction_p_and_p_prime: (subst > term > subst > $o) > (term > $o) > $o).
% 2.65/1.76 thf(func_def_107, type, hoasvar: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_121, type, pushprop_p_and_p_prime: term > subst > (term > $o) > (term > $o) > $o).
% 2.65/1.76 thf(func_def_124, type, d_term: $tType).
% 2.65/1.76 thf(func_def_125, type, d_subst: $tType).
% 2.65/1.76 thf(func_def_126, type, d2term: d_term > term).
% 2.65/1.76 thf(func_def_127, type, d2subst: d_subst > subst).
% 2.65/1.76 thf(func_def_128, type, d_one: d_term).
% 2.65/1.76 thf(func_def_129, type, d_id: d_subst).
% 2.65/1.76 thf(func_def_141, type, sP0: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_143, type, sP2: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_144, type, sP3: (term > $o) > $o).
% 2.65/1.76 thf(func_def_148, type, sP7: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_149, type, sP8: (term > $o) > $o).
% 2.65/1.76 thf(func_def_150, type, sP9: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_151, type, sP10: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_152, type, sP11: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_157, type, sP16: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_161, type, sP20: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_163, type, sP22: (term > $o) > $o).
% 2.65/1.76 thf(func_def_166, type, sP25: (term > $o) > $o).
% 2.65/1.76 thf(func_def_167, type, sP26: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_168, type, sP27: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_172, type, sP31: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_173, type, sP32: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_175, type, sP34: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_176, type, sP35: (term > $o) > $o).
% 2.65/1.76 thf(func_def_177, type, sP36: (term > $o) > $o).
% 2.65/1.76 thf(func_def_181, type, sP40: (term > $o) > $o).
% 2.65/1.76 thf(func_def_182, type, sP41: (term > $o) > $o).
% 2.65/1.76 thf(func_def_186, type, sP45: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_188, type, sP47: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_191, type, sP50: (subst > term > subst > $o) > $o).
% 2.65/1.76 thf(func_def_197, type, sP56: (term > $o) > $o).
% 2.65/1.76 thf(func_def_198, type, sP57: (term > $o) > $o).
% 2.65/1.76 thf(func_def_200, type, sK59: subst).
% 2.65/1.76 thf(func_def_201, type, sK60: term > $o).
% 2.65/1.76 thf(func_def_202, type, sK61: term).
% 2.65/1.76 thf(func_def_203, type, sK62: term > term).
% 2.65/1.76 thf(func_def_204, type, sK63: (term > $o) > term).
% 2.65/1.76 thf(func_def_205, type, sK64: (term > $o) > subst > term).
% 2.65/1.76 thf(func_def_206, type, sK65: (term > $o) > term).
% 2.65/1.76 thf(func_def_207, type, sK66: (term > $o) > term).
% 2.65/1.76 thf(func_def_208, type, sK67: (term > $o) > term).
% 2.65/1.76 thf(func_def_209, type, sK68: (term > $o) > term).
% 2.65/1.76 thf(func_def_210, type, sK69: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_211, type, sK70: term).
% 2.65/1.76 thf(func_def_212, type, sK71: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_213, type, sK72: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_214, type, sK73: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_215, type, sK74: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_216, type, sK75: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_217, type, sK76: term > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_218, type, sK77: term).
% 2.65/1.76 thf(func_def_219, type, sK78: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_220, type, sK79: term > $o).
% 2.65/1.76 thf(func_def_221, type, sK80: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_222, type, sK81: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_223, type, sK82: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_224, type, sK83: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_225, type, sK84: (term > $o) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_226, type, sK85: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_227, type, sK86: (term > $o) > term > term).
% 2.65/1.76 thf(func_def_228, type, sK87: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_229, type, sK88: term).
% 2.65/1.76 thf(func_def_230, type, sK89: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_231, type, sK90: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_232, type, sK91: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_233, type, sK92: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_234, type, sK93: term).
% 2.65/1.76 thf(func_def_235, type, sK94: term > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_236, type, sK95: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_237, type, sK96: term > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_238, type, sK97: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_239, type, sK98: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_240, type, sK99: term).
% 2.65/1.76 thf(func_def_241, type, sK100: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_242, type, sK101: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_243, type, sK102: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_244, type, sK103: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_245, type, sK104: term).
% 2.65/1.76 thf(func_def_246, type, sK105: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_247, type, sK106: term > $o).
% 2.65/1.76 thf(func_def_248, type, sK107: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_249, type, sK108: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_250, type, sK109: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_251, type, sK110: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_252, type, sK111: (term > $o) > term > term).
% 2.65/1.76 thf(func_def_253, type, sK112: (term > $o) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_254, type, sK113: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_255, type, sK114: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_256, type, sK115: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_257, type, sK116: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_258, type, sK117: term).
% 2.65/1.76 thf(func_def_259, type, sK118: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_260, type, sK119: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_261, type, sK120: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_262, type, sK121: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_263, type, sK122: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_264, type, sK123: term).
% 2.65/1.76 thf(func_def_265, type, sK124: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_266, type, sK125: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_267, type, sK126: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_268, type, sK127: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_269, type, sK128: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_270, type, sK129: (subst > term > subst > $o) > term > term).
% 2.65/1.76 thf(func_def_271, type, sK130: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_272, type, sK131: term).
% 2.65/1.76 thf(func_def_273, type, sK132: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_274, type, sK133: term > $o).
% 2.65/1.76 thf(func_def_275, type, sK134: (term > $o) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_276, type, sK135: (term > $o) > term > term).
% 2.65/1.76 thf(func_def_277, type, sK136: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_278, type, sK137: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_279, type, sK138: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_280, type, sK139: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_281, type, sK140: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_282, type, sK141: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_283, type, sK142: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_284, type, sK143: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_285, type, sK144: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_286, type, sK145: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_287, type, sK146: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_288, type, sK147: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_289, type, sK148: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_290, type, sK149: term).
% 2.65/1.76 thf(func_def_291, type, sK150: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_292, type, sK151: (subst > term > subst > $o) > term > term).
% 2.65/1.76 thf(func_def_293, type, sK152: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_294, type, sK153: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_295, type, sK154: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_296, type, sK155: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_297, type, sK156: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_298, type, sK157: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_299, type, sK158: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_300, type, sK159: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_301, type, sK160: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_302, type, sK161: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_303, type, sK162: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_304, type, sK163: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_305, type, sK164: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_306, type, sK165: term).
% 2.65/1.76 thf(func_def_307, type, sK166: subst > term > term).
% 2.65/1.76 thf(func_def_308, type, sK167: subst > term > term).
% 2.65/1.76 thf(func_def_309, type, sK168: subst).
% 2.65/1.76 thf(func_def_310, type, sK169: term).
% 2.65/1.76 thf(func_def_311, type, sK170: term).
% 2.65/1.76 thf(func_def_312, type, sK171: term).
% 2.65/1.76 thf(func_def_313, type, sK172: subst).
% 2.65/1.76 thf(func_def_314, type, sK173: term).
% 2.65/1.76 thf(func_def_315, type, sK174: subst).
% 2.65/1.76 thf(func_def_316, type, sK175: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_317, type, sK176: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_318, type, sK177: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_319, type, sK178: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_320, type, sK179: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_321, type, sK180: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_322, type, sK181: (term > $o) > term).
% 2.65/1.76 thf(func_def_323, type, sK182: (term > $o) > term).
% 2.65/1.76 thf(func_def_324, type, sK183: term).
% 2.65/1.76 thf(func_def_325, type, sK184: term > $o).
% 2.65/1.76 thf(func_def_326, type, sK185: subst).
% 2.65/1.76 thf(func_def_327, type, sK186: term).
% 2.65/1.76 thf(func_def_328, type, sK187: subst).
% 2.65/1.76 thf(func_def_329, type, sK188: subst).
% 2.65/1.76 thf(func_def_330, type, sK189: term).
% 2.65/1.76 thf(func_def_331, type, sK190: term).
% 2.65/1.76 thf(func_def_332, type, sK191: term > $o).
% 2.65/1.76 thf(func_def_333, type, sK192: subst).
% 2.65/1.76 thf(func_def_334, type, sK193: term > term).
% 2.65/1.76 thf(func_def_335, type, sK194: (term > $o) > term).
% 2.65/1.76 thf(func_def_336, type, sK195: subst > (term > $o) > term).
% 2.65/1.76 thf(func_def_337, type, sK196: (term > $o) > term).
% 2.65/1.76 thf(func_def_338, type, sK197: (term > $o) > term).
% 2.65/1.76 thf(func_def_339, type, sK198: (term > $o) > term).
% 2.65/1.76 thf(func_def_340, type, sK199: (term > $o) > term).
% 2.65/1.76 thf(func_def_341, type, sK200: term).
% 2.65/1.76 thf(func_def_342, type, sK201: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_343, type, sK202: (subst > term > term) > term > subst).
% 2.65/1.76 thf(func_def_344, type, sK203: (subst > term > term) > term > term).
% 2.65/1.76 thf(func_def_345, type, sK204: (subst > term > term) > term > subst).
% 2.65/1.76 thf(func_def_346, type, sK205: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_347, type, sK206: (subst > term > term) > term > term).
% 2.65/1.76 thf(func_def_348, type, sK207: (subst > term > term) > term > subst).
% 2.65/1.76 thf(func_def_349, type, sK208: (subst > term > term) > term > subst).
% 2.65/1.76 thf(func_def_350, type, sK209: (subst > term > subst > $o) > term > term).
% 2.65/1.76 thf(func_def_351, type, sK210: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_352, type, sK211: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_353, type, sK212: (term > $o) > term).
% 2.65/1.76 thf(func_def_354, type, sK213: subst > (term > $o) > term).
% 2.65/1.76 thf(func_def_355, type, sK214: term > $o).
% 2.65/1.76 thf(func_def_356, type, sK215: term).
% 2.65/1.76 thf(func_def_357, type, sK216: subst).
% 2.65/1.76 thf(func_def_358, type, sK217: term > term).
% 2.65/1.76 thf(func_def_359, type, sK218: term).
% 2.65/1.76 thf(func_def_360, type, sK219: term > $o).
% 2.65/1.76 thf(func_def_361, type, sK220: term).
% 2.65/1.76 thf(func_def_362, type, sK221: term > term).
% 2.65/1.76 thf(func_def_363, type, sK222: (term > $o) > term).
% 2.65/1.76 thf(func_def_364, type, sK223: (term > $o) > term).
% 2.65/1.76 thf(func_def_365, type, sK224: (term > $o) > term).
% 2.65/1.76 thf(func_def_366, type, sK225: (term > $o) > term).
% 2.65/1.76 thf(func_def_367, type, sK226: (term > $o) > term).
% 2.65/1.76 thf(func_def_368, type, sK227: (term > $o) > term).
% 2.65/1.76 thf(func_def_369, type, sK228: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_370, type, sK229: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_371, type, sK230: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_372, type, sK231: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_373, type, sK232: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_374, type, sK233: term).
% 2.65/1.76 thf(func_def_375, type, sK234: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_376, type, sK235: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_377, type, sK236: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_378, type, sK237: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_379, type, sK238: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_380, type, sK239: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_381, type, sK240: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_382, type, sK241: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_383, type, sK242: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_384, type, sK243: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_385, type, sK244: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_386, type, sK245: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_387, type, sK246: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_388, type, sK247: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_389, type, sK248: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_390, type, sK249: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_391, type, sK250: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_392, type, sK251: (term > $o) > term).
% 2.65/1.76 thf(func_def_393, type, sK252: (term > $o) > term).
% 2.65/1.76 thf(func_def_394, type, sK253: (subst > term > subst > $o) > term > $o).
% 2.65/1.76 thf(func_def_395, type, sK254: term).
% 2.65/1.76 thf(func_def_396, type, sK255: term).
% 2.65/1.76 thf(func_def_397, type, sK256: term > $o).
% 2.65/1.76 thf(func_def_398, type, sK257: term > term).
% 2.65/1.76 thf(func_def_399, type, sK258: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_400, type, sK259: (term > $o) > term).
% 2.65/1.76 thf(func_def_401, type, sK260: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_402, type, sK261: (subst > term > subst > $o) > (term > $o) > term).
% 2.65/1.76 thf(func_def_403, type, sK262: term > (term > $o) > term).
% 2.65/1.76 thf(func_def_404, type, sK263: term > $o).
% 2.65/1.76 thf(func_def_405, type, sK264: term).
% 2.65/1.76 thf(func_def_406, type, sK265: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_407, type, sK266: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_408, type, sK267: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_409, type, sK268: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_410, type, sK269: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_411, type, sK270: term).
% 2.65/1.76 thf(func_def_412, type, sK271: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_413, type, sK272: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_414, type, sK273: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_415, type, sK274: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_416, type, sK275: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_417, type, sK276: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_418, type, sK277: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_419, type, sK278: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_420, type, sK279: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_421, type, sK280: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_422, type, sK281: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_423, type, sK282: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_424, type, sK283: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_425, type, sK284: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_426, type, sK285: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_427, type, sK286: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_428, type, sK287: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_429, type, sK288: (term > $o) > term).
% 2.65/1.76 thf(func_def_430, type, sK289: (term > $o) > term).
% 2.65/1.76 thf(func_def_431, type, sK290: term).
% 2.65/1.76 thf(func_def_432, type, sK291: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_433, type, sK292: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_434, type, sK293: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_435, type, sK294: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_436, type, sK295: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_437, type, sK296: (subst > term > subst > $o) > term > term).
% 2.65/1.76 thf(func_def_438, type, sK297: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_439, type, sK298: term > $o).
% 2.65/1.76 thf(func_def_440, type, sK299: term).
% 2.65/1.76 thf(func_def_441, type, sK300: term > term).
% 2.65/1.76 thf(func_def_442, type, sK301: (term > $o) > term).
% 2.65/1.76 thf(func_def_443, type, sK302: (term > $o) > term).
% 2.65/1.76 thf(func_def_444, type, sK303: (term > $o) > term).
% 2.65/1.76 thf(func_def_445, type, sK304: (term > $o) > term).
% 2.65/1.76 thf(func_def_446, type, sK305: term > $o).
% 2.65/1.76 thf(func_def_447, type, sK306: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_448, type, sK307: term).
% 2.65/1.76 thf(func_def_449, type, sK308: term).
% 2.65/1.76 thf(func_def_450, type, sK309: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_451, type, sK310: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_452, type, sK311: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_453, type, sK312: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_454, type, sK313: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_455, type, sK314: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_456, type, sK315: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_457, type, sK316: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_458, type, sK317: (subst > term > subst > $o) > (term > $o) > term).
% 2.65/1.76 thf(func_def_459, type, sK318: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_460, type, sK319: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_461, type, sK320: term).
% 2.65/1.76 thf(func_def_462, type, sK321: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_463, type, sK322: term > $o).
% 2.65/1.76 thf(func_def_464, type, sK323: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_465, type, sK324: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_466, type, sK325: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_467, type, sK326: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_468, type, sK327: term > subst > term > term).
% 2.65/1.76 thf(func_def_469, type, sK328: (term > $o) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_470, type, sK329: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_471, type, sK330: (term > $o) > term > term).
% 2.65/1.76 thf(func_def_472, type, sK331: term).
% 2.65/1.76 thf(func_def_473, type, sK332: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_474, type, sK333: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_475, type, sK334: term).
% 2.65/1.76 thf(func_def_476, type, sK335: term).
% 2.65/1.76 thf(func_def_477, type, sK336: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_478, type, sK337: term).
% 2.65/1.76 thf(func_def_479, type, sK338: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_480, type, sK339: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_481, type, sK340: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_482, type, sK341: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_483, type, sK342: (subst > term > subst > $o) > term > term).
% 2.65/1.76 thf(func_def_484, type, sK343: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_485, type, sK344: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_486, type, sK345: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_487, type, sK346: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_488, type, sK347: (subst > term > subst > $o) > term > term).
% 2.65/1.76 thf(func_def_489, type, sK348: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_490, type, sK349: term).
% 2.65/1.76 thf(func_def_491, type, sK350: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_492, type, sK351: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_493, type, sK352: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_494, type, sK353: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_495, type, sK354: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_496, type, sK355: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_497, type, sK356: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_498, type, sK357: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_499, type, sK358: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_500, type, sK359: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_501, type, sK360: (term > $o) > term).
% 2.65/1.76 thf(func_def_502, type, sK361: (term > $o) > term).
% 2.65/1.76 thf(func_def_503, type, sK362: term).
% 2.65/1.76 thf(func_def_504, type, sK363: term > $o).
% 2.65/1.76 thf(func_def_505, type, sK364: term > term).
% 2.65/1.76 thf(func_def_506, type, sK365: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_507, type, sK366: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_508, type, sK367: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_509, type, sK368: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_510, type, sK369: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_511, type, sK370: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_512, type, sK371: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_513, type, sK372: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_514, type, sK373: term).
% 2.65/1.76 thf(func_def_515, type, sK374: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_516, type, sK375: term).
% 2.65/1.76 thf(func_def_517, type, sK376: term).
% 2.65/1.76 thf(func_def_518, type, sK377: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_519, type, sK378: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_520, type, sK379: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_521, type, sK380: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_522, type, sK381: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_523, type, sK382: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_524, type, sK383: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_525, type, sK384: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_526, type, sK385: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_527, type, sK386: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_528, type, sK387: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_529, type, sK388: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_530, type, sK389: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_531, type, sK390: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_532, type, sK391: (subst > term > subst > $o) > term > term).
% 2.65/1.76 thf(func_def_533, type, sK392: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_534, type, sK393: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_535, type, sK394: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_536, type, sK395: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_537, type, sK396: term).
% 2.65/1.76 thf(func_def_538, type, sK397: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_539, type, sK398: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_540, type, sK399: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_541, type, sK400: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_542, type, sK401: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_543, type, sK402: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_544, type, sK403: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_545, type, sK404: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_546, type, sK405: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_547, type, sK406: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_548, type, sK407: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_549, type, sK408: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_550, type, sK409: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_551, type, sK410: (term > $o) > term).
% 2.65/1.76 thf(func_def_552, type, sK411: (term > $o) > term).
% 2.65/1.76 thf(func_def_553, type, sK412: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_554, type, sK413: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_555, type, sK414: (term > $o) > term).
% 2.65/1.76 thf(func_def_556, type, sK415: (term > $o) > term).
% 2.65/1.76 thf(func_def_557, type, sK416: (subst > term > subst > $o) > term > $o).
% 2.65/1.76 thf(func_def_558, type, sK417: term > $o).
% 2.65/1.76 thf(func_def_559, type, sK418: term).
% 2.65/1.76 thf(func_def_560, type, sK419: term > term).
% 2.65/1.76 thf(func_def_561, type, sK420: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_562, type, sK421: (term > $o) > term).
% 2.65/1.76 thf(func_def_563, type, sK422: term).
% 2.65/1.76 thf(func_def_564, type, sK423: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_565, type, sK424: term).
% 2.65/1.76 thf(func_def_566, type, sK425: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_567, type, sK426: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_568, type, sK427: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_569, type, sK428: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_570, type, sK429: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_571, type, sK430: term > (term > $o) > term).
% 2.65/1.76 thf(func_def_572, type, sK431: (term > $o) > (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_573, type, sK432: (subst > term > subst > $o) > subst > term > term).
% 2.65/1.76 thf(func_def_574, type, sK433: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_575, type, sK434: term > $o).
% 2.65/1.76 thf(func_def_576, type, sK435: term).
% 2.65/1.76 thf(func_def_577, type, sK436: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_578, type, sK437: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_579, type, sK438: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_580, type, sK439: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_581, type, sK440: (term > $o) > term).
% 2.65/1.76 thf(func_def_582, type, sK441: (term > $o) > term).
% 2.65/1.76 thf(func_def_583, type, sK442: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_584, type, sK443: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_585, type, sK444: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_586, type, sK445: term).
% 2.65/1.76 thf(func_def_587, type, sK446: term).
% 2.65/1.76 thf(func_def_588, type, sK447: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_589, type, sK448: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_590, type, sK449: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_591, type, sK450: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_592, type, sK451: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_593, type, sK452: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_594, type, sK453: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_595, type, sK454: (subst > term > subst > $o) > subst).
% 2.65/1.76 thf(func_def_596, type, sK455: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_597, type, sK456: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_598, type, sK457: term).
% 2.65/1.76 thf(func_def_599, type, sK458: subst).
% 2.65/1.76 thf(func_def_600, type, sK459: term > $o).
% 2.65/1.76 thf(func_def_601, type, sK460: (term > $o) > term).
% 2.65/1.76 thf(func_def_602, type, sK461: (term > $o) > subst > term > term > $o).
% 2.65/1.76 thf(func_def_603, type, sK462: subst).
% 2.65/1.76 thf(func_def_604, type, sK463: term).
% 2.65/1.76 thf(func_def_605, type, sK464: term > $o).
% 2.65/1.76 thf(func_def_606, type, sK465: term).
% 2.65/1.76 thf(func_def_607, type, sK466: (term > $o) > term).
% 2.65/1.76 thf(func_def_608, type, sK467: (term > $o) > term).
% 2.65/1.76 thf(func_def_609, type, sK468: term > $o).
% 2.65/1.76 thf(func_def_610, type, sK469: term).
% 2.65/1.76 thf(func_def_611, type, sK470: term).
% 2.65/1.76 thf(func_def_612, type, sK471: subst).
% 2.65/1.76 thf(func_def_613, type, sK472: term > $o).
% 2.65/1.76 thf(func_def_614, type, sK473: term).
% 2.65/1.76 thf(func_def_615, type, sK474: term).
% 2.65/1.76 thf(func_def_616, type, sK475: term).
% 2.65/1.76 thf(func_def_617, type, sK476: term).
% 2.65/1.76 thf(func_def_618, type, sK477: subst > d_subst).
% 2.65/1.76 thf(func_def_619, type, sK478: term).
% 2.65/1.76 thf(func_def_620, type, sK479: term).
% 2.65/1.76 thf(func_def_621, type, sK480: term).
% 2.65/1.76 thf(func_def_622, type, sK481: term).
% 2.65/1.76 thf(func_def_623, type, sK482: term).
% 2.65/1.76 thf(func_def_624, type, sK483: term).
% 2.65/1.76 thf(func_def_625, type, sK484: term).
% 2.65/1.76 thf(func_def_626, type, sK485: term).
% 2.65/1.76 thf(func_def_627, type, sK486: term).
% 2.65/1.76 thf(func_def_628, type, sK487: term > d_term).
% 2.65/1.76 thf(func_def_629, type, sK488: term).
% 2.65/1.76 thf(func_def_630, type, sK489: term).
% 2.65/1.76 thf(func_def_631, type, sK490: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_632, type, sK491: (term > $o) > term).
% 2.65/1.76 thf(func_def_633, type, sK492: (subst > term > subst > $o) > term > $o).
% 2.65/1.76 thf(func_def_634, type, sK493: term > subst > term > term).
% 2.65/1.76 thf(func_def_635, type, sK494: term).
% 2.65/1.76 thf(func_def_636, type, sK495: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_637, type, sK496: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_638, type, sK497: term).
% 2.65/1.76 thf(func_def_639, type, sK498: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_640, type, sK499: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_641, type, sK500: term > subst > term > term).
% 2.65/1.76 thf(func_def_642, type, sK501: subst).
% 2.65/1.76 thf(func_def_643, type, sK502: term).
% 2.65/1.76 thf(func_def_644, type, sK503: term).
% 2.65/1.76 thf(func_def_645, type, sK504: term).
% 2.65/1.76 thf(func_def_646, type, sK505: term).
% 2.65/1.76 thf(func_def_647, type, sK506: term).
% 2.65/1.76 thf(func_def_648, type, sK507: term > subst > term > term).
% 2.65/1.76 thf(func_def_649, type, sK508: subst).
% 2.65/1.76 thf(func_def_650, type, sK509: subst).
% 2.65/1.76 thf(func_def_651, type, sK510: subst).
% 2.65/1.76 thf(func_def_652, type, sK511: subst).
% 2.65/1.76 thf(func_def_653, type, sK512: term).
% 2.65/1.76 thf(func_def_654, type, sK513: subst).
% 2.65/1.76 thf(func_def_655, type, sK514: subst).
% 2.65/1.76 thf(func_def_656, type, sK515: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_657, type, sK516: term).
% 2.65/1.76 thf(func_def_658, type, sK517: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_659, type, sK518: (subst > term > subst > $o) > term).
% 2.65/1.76 thf(func_def_660, type, sK519: term).
% 2.65/1.76 thf(func_def_661, type, sK520: term).
% 2.65/1.76 thf(func_def_662, type, sK521: subst).
% 2.65/1.76 thf(func_def_663, type, sK522: term).
% 2.65/1.76 thf(func_def_664, type, sK523: term).
% 2.65/1.76 thf(func_def_665, type, sK524: term).
% 2.65/1.76 thf(func_def_666, type, sK525: term).
% 2.65/1.76 thf(func_def_667, type, sK526: term).
% 2.65/1.76 thf(func_def_668, type, sK527: term).
% 2.65/1.76 thf(func_def_669, type, sK528: subst > term > term).
% 2.65/1.76 thf(func_def_670, type, sK529: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_671, type, sK530: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_672, type, sK531: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_673, type, sK532: term).
% 2.65/1.76 thf(func_def_674, type, sK533: term).
% 2.65/1.76 thf(func_def_675, type, sK534: term).
% 2.65/1.76 thf(func_def_676, type, sK535: term).
% 2.65/1.76 thf(func_def_677, type, sK536: term).
% 2.65/1.76 thf(func_def_678, type, sK537: term).
% 2.65/1.76 thf(func_def_679, type, sK538: term).
% 2.65/1.76 thf(func_def_680, type, sK539: term).
% 2.65/1.76 thf(func_def_681, type, sK540: term).
% 2.65/1.76 thf(func_def_682, type, sK541: term).
% 2.65/1.76 thf(func_def_683, type, sK542: term).
% 2.65/1.76 thf(func_def_684, type, sK543: term).
% 2.65/1.76 thf(func_def_685, type, sK544: term > subst > term > term).
% 2.65/1.76 thf(func_def_686, type, sK545: term).
% 2.65/1.76 thf(func_def_687, type, sK546: subst).
% 2.65/1.76 thf(func_def_688, type, sK547: term > $o).
% 2.65/1.76 thf(func_def_689, type, sK548: term > $o).
% 2.65/1.76 thf(func_def_690, type, sK549: term).
% 2.65/1.76 thf(func_def_691, type, sK550: subst).
% 2.65/1.76 thf(func_def_692, type, sK551: term).
% 2.65/1.76 thf(func_def_693, type, sK552: term > (term > $o) > (term > $o) > subst > term).
% 2.65/1.76 thf(func_def_694, type, sK553: term > (term > $o) > subst > (term > $o) > term).
% 2.65/1.76 thf(func_def_695, type, sK554: term > $o).
% 2.65/1.76 thf(func_def_696, type, sK555: subst).
% 2.65/1.76 thf(func_def_697, type, sK556: term > $o).
% 2.65/1.76 thf(func_def_698, type, sK557: term).
% 2.65/1.76 thf(func_def_699, type, sK558: (subst > term > subst > $o) > term > $o).
% 2.65/1.76 thf(func_def_700, type, sK559: subst > term > subst > $o).
% 2.65/1.76 thf(func_def_701, type, sK560: (term > $o) > term).
% 2.65/1.76 thf(func_def_702, type, sK561: term).
% 2.65/1.76 thf(func_def_703, type, sK562: term).
% 2.65/1.76 thf(func_def_704, type, sK563: term).
% 2.65/1.76 thf(func_def_705, type, sK564: term).
% 2.65/1.76 thf(func_def_706, type, sK565: subst).
% 2.65/1.76 thf(func_def_707, type, sK566: subst).
% 2.65/1.76 thf(func_def_708, type, sK567: subst > term > term).
% 2.65/1.76 thf(func_def_709, type, sK568: subst > term > term).
% 2.65/1.76 thf(func_def_710, type, sK569: subst).
% 2.65/1.76 thf(func_def_711, type, sK570: term).
% 2.65/1.76 thf(func_def_712, type, sK571: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_713, type, sK572: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_714, type, sK573: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_715, type, sK574: (subst > term > term) > term).
% 2.65/1.76 thf(func_def_716, type, sK575: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_717, type, sK576: (subst > term > term) > subst).
% 2.65/1.76 thf(func_def_718, type, sK577: subst).
% 2.65/1.76 thf(func_def_719, type, sK578: term).
% 2.65/1.76 thf(func_def_720, type, sK579: subst).
% 2.65/1.76 thf(func_def_721, type, sK580: term).
% 2.65/1.76 thf(func_def_722, type, sK581: term > $o).
% 2.65/1.76 thf(func_def_723, type, sK582: term > $o).
% 2.65/1.76 thf(func_def_724, type, sK583: term).
% 2.65/1.76 thf(func_def_725, type, sK584: subst).
% 2.65/1.76 thf(func_def_726, type, sK585: term).
% 2.65/1.76 thf(func_def_727, type, sK586: term > $o).
% 2.65/1.76 thf(func_def_728, type, sK587: term).
% 2.65/1.76 thf(func_def_729, type, sK588: subst).
% 2.65/1.76 thf(func_def_730, type, sK589: (term > $o) > term).
% 2.65/1.76 thf(func_def_731, type, sK590: subst > term > (term > $o) > term > $o).
% 2.65/1.76 thf(func_def_733, type, sK592: subst).
% 2.65/1.76 thf(func_def_734, type, sK593: subst).
% 2.65/1.76 thf(func_def_735, type, sK594: term).
% 2.65/1.76 thf(f2675,plain,(
% 2.65/1.76 $false),
% 2.65/1.76 inference(subsumption_resolution,[status(thm)],[f2674,f2587])).
% 2.65/1.76 thf(f2587,plain,(
% 2.65/1.76 ( ! [X62 : term] : (((d2term @ d_one) = X62)) )),
% 2.65/1.76 inference(forward_demodulation,[status(thm)],[f1212,f1038])).
% 2.65/1.76 thf(f1038,plain,(
% 2.65/1.76 ( ! [X263 : d_term] : ((d_one = X263)) )),
% 2.65/1.76 inference(cnf_transformation,[status(thm)],[f545])).
% 2.65/1.76 thf(f545,plain,(
% 2.65/1.76 ((pushprop_lem0_lthm = $true) | ! [X3 : term > $o] : ((sK459 @ (sub @ (sK460 @ X3) @ (push @ sK457 @ sK458))) != (X3 @ (sK460 @ X3)))) & (! [X5 : term,X6 : subst,X7 : term > $o] : ! [X9 : term] : ((X7 @ (sub @ X9 @ (push @ X5 @ X6))) = (sK461 @ X7 @ X6 @ X5 @ X9)) | (pushprop_lem0_lthm != $true)) & (! [X10 : subst] : ((comp @ X10 @ id) = X10) | (axidr != $true)) & ((axidr = $true) | (sK462 != (comp @ sK462 @ id))) & ! [X12 : d_subst,X13 : d_subst] : (((d2subst @ X13) != (d2subst @ X12)) | (X12 = X13)) & (hoasinduction_lem2_lthm = $true) & (axclos = $true) & (($true = sP58) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP58)) & (! [X14 : term] : ((sub @ X14 @ id) = X14) | (axvarid != $true)) & ((axvarid = $true) | (sK463 != (sub @ sK463 @ id))) & (substmonoid_gthm = $true) & ((induction = $true) | (((sP56 @ sK464) != $true) & ! [X18 : term] : (((var @ X18) != $true) | ($true = (sK464 @ X18))) & ((sK464 @ sK465) != $true) & ! [X19 : term] : (($true = (sK464 @ (lam @ X19))) | ((sK464 @ X19) != $true)))) & (! [X20 : term > $o,X21 : term] : (((sP56 @ X20) = $true) | (($true = (var @ (sK466 @ X20))) & ((X20 @ (sK466 @ X20)) != $true)) | ((X20 @ X21) = $true) | (((X20 @ (lam @ (sK467 @ X20))) != $true) & ((X20 @ (sK467 @ X20)) = $true))) | (induction != $true)) & ((pushprop_lem1 = $true) | (($true != (sK468 @ (sub @ sK469 @ (push @ sK470 @ sK471)))) & ($true = (sK472 @ sK469)))) & (! [X29 : term > $o,X30 : term,X31 : term,X32 : subst,X33 : term > $o] : (((X29 @ (sub @ X30 @ (push @ X31 @ X32))) = $true) | ((X33 @ X30) != $true)) | (pushprop_lem1 != $true)) & (hoaslamnotvar_gthm = $true) & (axassoc = $true) & (! [X34 : term,X35 : term,X36 : term,X37 : term] : (((ap @ X37 @ X34) != (ap @ X35 @ X36)) | (X35 = X37)) | (apinj1 != $true)) & ((apinj1 = $true) | (((ap @ sK474 @ sK475) = (ap @ sK476 @ sK473)) & (sK476 != sK474))) & (hoasapinj2_gthm = $true) & ! [X42 : subst] : ((d2subst @ (sK477 @ X42)) = X42) & (((sK481 != sK480) & ((ap @ sK481 @ sK478) = (ap @ sK480 @ sK479))) | ((sub @ sK482 @ id) != sK482) | ! [X49 : term,X50 : term,X51 : term,X52 : term] : ((X49 = X50) | ((ap @ (sub @ X49 @ id) @ X51) != (ap @ (sub @ X50 @ id) @ X52))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X53 : term,X54 : term,X55 : term,X56 : term] : ((X55 = X56) | ((ap @ X55 @ X54) != (ap @ X56 @ X53))) & ! [X57 : term] : ((sub @ X57 @ id) = X57) & ((sK484 != sK483) & ((ap @ (sub @ sK484 @ id) @ sK486) = (ap @ (sub @ sK483 @ id) @ sK485))))) & (hoasapnotvar = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem3aa = $true) | ($true != sP55)) & (($true = sP55) | (hoasinduction_lem3aa != $true)) & ((var @ (d2term @ d_one)) != $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar_lthm = $true) & (pushprop_lem3v2 = $true) & ((sP54 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP54 != $true)) & (termmset_gthm = $true) & (one = (d2term @ d_one)) & ! [X62 : term] : ((d2term @ (sK487 @ X62)) = X62) & ((shinj = $true) | ((sK488 != sK489) & ((sub @ sK488 @ sh) = (sub @ sK489 @ sh)))) & (! [X66 : term,X67 : term] : ((X66 = X67) | ((sub @ X66 @ sh) != (sub @ X67 @ sh))) | (shinj != $true)) & ((hoasinduction_lem0 = $true) | ! [X69 : term > $o] : ((sK490 @ id @ (sK491 @ X69) @ id) != (X69 @ (sK491 @ X69)))) & (! [X71 : subst > term > subst > $o] : ! [X73 : term] : ((sK492 @ X71 @ X73) = (X71 @ id @ X73 @ id)) | (hoasinduction_lem0 != $true)) & (hoasinduction_lem1_lthm = $true) & (induction2_gthm = $true) & (! [X74 : term] : ! [X76 : subst,X77 : term] : ((sub @ X74 @ (push @ X77 @ X76)) = (sK493 @ X74 @ X76 @ X77)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X79 : subst > term > term] : ((X79 @ (sK495 @ X79) @ (sK496 @ X79)) != (sub @ sK494 @ (push @ (sK496 @ X79) @ (sK495 @ X79))))) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X83 : subst > term > term] : ((sub @ sK497 @ (push @ (sK499 @ X83) @ (sK498 @ X83))) != (X83 @ (sK498 @ X83) @ (sK499 @ X83)))) & (! [X86 : term] : ! [X88 : subst,X89 : term] : ((sub @ X86 @ (push @ X89 @ X88)) = (sK500 @ X86 @ X88 @ X89)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((sP53 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP53 != $true)) & ! [X90 : d_subst] : (d_id = X90) & (hoasinduction_lem1v2 = $true) & (! [X91 : subst,X92 : term] : ((sub @ one @ (push @ X92 @ X91)) = X92) | (axvarcons != $true)) & ((axvarcons = $true) | ((sub @ one @ (push @ sK502 @ sK501)) != sK502)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (($true = sP51) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP51)) & (axapp = $true) & (induction2lem_gthm = $true) & (ulamvarsh = $true) & (axvarshift = $true) & (ulamvarind = $true) & ((d2subst @ d_id) = id) & ((termmset = $true) | ((sub @ sK503 @ id) != sK503)) & (! [X96 : term] : ((sub @ X96 @ id) = X96) | (termmset != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP49 != $true)) & ((sP49 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & (apnotvar = $true) & (! [X97 : term] : ((sub @ X97 @ id) = X97) | (sK504 != (sub @ sK504 @ id)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (((sub @ sK505 @ id) != sK505) & ! [X100 : term] : ((sub @ X100 @ id) = X100))) & ((sP48 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP48 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP46)) & (($true = sP46) | (hoasinduction_lem3 != $true)) & (hoasinduction_lem3_gthm = $true) & ((sP44 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP44 != $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3b = $true) | ! [X102 : subst > term > term] : ((X102 @ sh @ one) != (sub @ sK506 @ (push @ one @ sh)))) & (! [X103 : term] : ((sK507 @ X103 @ sh @ one) = (sub @ X103 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & (hoasinduction_lem3v2_gthm = $true) & (axmap = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop = $true) & ((substmonoid_lthm = $true) | (! [X105 : subst] : ((comp @ X105 @ id) = X105) & (((comp @ sK508 @ id) != sK508) | (sK509 != (comp @ id @ sK509))) & ! [X108 : subst] : ((comp @ id @ X108) = X108))) & (((comp @ sK510 @ id) != sK510) | (! [X110 : subst] : ((comp @ X110 @ id) = X110) & ! [X111 : subst] : ((comp @ id @ X111) = X111)) | (sK511 != (comp @ id @ sK511)) | (substmonoid_lthm != $true)) & (axabs = $true) & (lamnotvar = $true) & (! [X113 : term,X114 : subst] : ((comp @ sh @ (push @ X113 @ X114)) = X114) | (axshiftcons != $true)) & ((axshiftcons = $true) | (sK513 != (comp @ sh @ (push @ sK512 @ sK513)))) & ((sP43 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP43 != $true)) & (($true = sP39) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP39)) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X117 : subst] : ((push @ (sub @ one @ X117) @ (comp @ sh @ X117)) = X117) | (axscons != $true)) & ((axscons = $true) | (sK514 != (push @ (sub @ one @ sK514) @ (comp @ sh @ sK514)))) & (hoasinduction_lem1_gthm = $true) & (pushprop_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) | (((sK515 @ id @ sK516 @ id) != $true) & ! [X121 : term,X122 : term] : (((sK515 @ id @ (ap @ (sub @ X121 @ id) @ X122) @ id) = $true) | ((sK515 @ id @ X121 @ id) != $true) | ((sK515 @ id @ X122 @ id) != $true)) & ((sP34 @ sK515) != $true))) & (! [X123 : subst > term > subst > $o,X124 : term] : (($true = (X123 @ id @ X124 @ id)) | (((X123 @ id @ (ap @ (sub @ (sK517 @ X123) @ id) @ (sK518 @ X123)) @ id) != $true) & ((X123 @ id @ (sK517 @ X123) @ id) = $true) & ((X123 @ id @ (sK518 @ X123) @ id) = $true)) | ($true = (sP34 @ X123))) | (hoasinduction_no_psi_cond != $true)) & (lamnotap != $true) & ((hoasinduction = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction != $true)) & ((laminj = $true) | ((sK520 != sK519) & ((lam @ sK520) = (lam @ sK519)))) & (! [X129 : term,X130 : term] : ((X129 = X130) | ((lam @ X129) != (lam @ X130))) | (laminj != $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((axidl = $true) | ((comp @ id @ sK521) != sK521)) & (! [X132 : subst] : ((comp @ id @ X132) = X132) | (axidl != $true)) & ((hoasapinj2 = $true) | ((sK525 != sK524) & ((ap @ (sub @ sK523 @ id) @ sK525) = (ap @ (sub @ sK522 @ id) @ sK524)))) & (! [X137 : term,X138 : term,X139 : term,X140 : term] : ((X139 = X140) | ((ap @ (sub @ X138 @ id) @ X140) != (ap @ (sub @ X137 @ id) @ X139))) | (hoasapinj2 != $true)) & ((hoaslamnotap = $true) | (! [X144 : subst,X145 : subst,X146 : term] : ((sK528 @ (comp @ X145 @ X144) @ (sub @ X146 @ X144)) = (sub @ (sK528 @ X145 @ X146) @ X144)) & ((lam @ (sK528 @ sh @ one)) = (ap @ (sub @ sK526 @ id) @ sK527)))) & (! [X147 : term,X148 : term,X149 : subst > term > term] : (((X149 @ (comp @ (sK530 @ X149) @ (sK529 @ X149)) @ (sub @ (sK531 @ X149) @ (sK529 @ X149))) != (sub @ (X149 @ (sK530 @ X149) @ (sK531 @ X149)) @ (sK529 @ X149))) | ((ap @ (sub @ X147 @ id) @ X148) != (lam @ (X149 @ sh @ one)))) | (hoaslamnotap != $true)) & ((hoasinduction_lthm_3 = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lthm_3 != $true)) & (pushprop_lem3v2_lthm = $true) & ((sP24 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP24 != $true)) & ((hoasapinj2_lthm = $true) | (((sK532 != sK534) & ((ap @ (sub @ sK533 @ id) @ sK532) = (ap @ (sub @ sK535 @ id) @ sK534))) & ! [X157 : term,X158 : term,X159 : term,X160 : term] : (((ap @ X157 @ X159) != (ap @ X160 @ X158)) | (X158 = X159)))) & (! [X161 : term,X162 : term,X163 : term,X164 : term] : ((X161 = X163) | ((ap @ (sub @ X162 @ id) @ X161) != (ap @ (sub @ X164 @ id) @ X163))) | (((ap @ sK539 @ sK537) = (ap @ sK536 @ sK538)) & (sK537 != sK538)) | (hoasapinj2_lthm != $true)) & ((hoasapinj1 = $true) | ((sK543 != sK540) & ((ap @ (sub @ sK540 @ id) @ sK541) = (ap @ (sub @ sK543 @ id) @ sK542)))) & (! [X173 : term,X174 : term,X175 : term,X176 : term] : ((X173 = X176) | ((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X176 @ id) @ X175))) | (hoasapinj1 != $true)) & (hoaslamnotap_gthm = $true) & (hoasinduction_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X177 : term] : ((sub @ X177 @ (push @ one @ sh)) = (sK544 @ X177 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X180 : subst > term > term] : ((X180 @ sh @ one) != (sub @ sK545 @ (push @ one @ sh)))) & ((sP23 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP23 != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X181 : subst,X182 : term] : ((sub @ one @ (push @ X182 @ X181)) = X182) & (! [X187 : term] : ((sK548 @ (sub @ X187 @ (push @ sK549 @ sK546))) = (sK547 @ X187)) & ((sK547 @ one) != $true) & ($true = (sK548 @ sK549))))) & (((sub @ one @ (push @ sK551 @ sK550)) != sK551) | ! [X190 : subst,X191 : term > $o,X192 : term > $o,X193 : term] : (((X191 @ (sK552 @ X193 @ X192 @ X191 @ X190)) != (X192 @ (sub @ (sK552 @ X193 @ X192 @ X191 @ X190) @ (push @ X193 @ X190)))) | ((X191 @ one) = $true) | ($true != (X192 @ X193))) | (pushprop_lem1v2_lthm != $true)) & (! [X195 : term > $o,X196 : subst,X197 : term > $o,X198 : term] : (((X197 @ X198) != $true) | ((X197 @ (sub @ (sK553 @ X198 @ X197 @ X196 @ X195) @ (push @ X198 @ X196))) != (X195 @ (sK553 @ X198 @ X197 @ X196 @ X195))) | ($true = (X195 @ one))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | (($true = (sK556 @ sK557)) & ! [X204 : term] : ((sK556 @ (sub @ X204 @ (push @ sK557 @ sK555))) = (sK554 @ X204)) & ((sK554 @ one) != $true))) & (hoaslamnotap_lthm = $true) & ((sP21 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP21 != $true)) & (pushprop_gthm = $true) & (! [X205 : subst > term > subst > $o] : ! [X207 : term] : ((sK558 @ X205 @ X207) = (X205 @ id @ X207 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X209 : term > $o] : ((sK559 @ id @ (sK560 @ X209) @ id) != (X209 @ (sK560 @ X209)))) & (($true = sP19) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP19)) & (pushprop_lem2v2_lthm = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((d2subst @ d_id) = sh) & (hoasapinj1_gthm = $true) & (! [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X214 @ X212) != (ap @ X213 @ X211)) | (X211 = X212)) | (apinj2 != $true)) & ((apinj2 = $true) | (((ap @ sK564 @ sK562) = (ap @ sK563 @ sK561)) & (sK561 != sK562))) & ((substmonoid = $true) | (sK565 != (comp @ id @ sK565)) | (sK566 != (comp @ sK566 @ id))) & ((! [X221 : subst] : ((comp @ id @ X221) = X221) & ! [X222 : subst] : ((comp @ X222 @ id) = X222)) | (substmonoid != $true)) & (($true = sP18) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP18)) & ((hoasinduction_lthm = $true) | ($true != sP15)) & (($true = sP15) | (hoasinduction_lthm != $true)) & (hoasinduction_lem1 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP6 != $true)) & ((sP6 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoaslaminj = $true) | (((lam @ (sK568 @ sh @ one)) = (lam @ (sK567 @ sh @ one))) & ! [X227 : term,X228 : subst,X229 : subst] : ((sub @ (sK567 @ X229 @ X227) @ X228) = (sK567 @ (comp @ X229 @ X228) @ (sub @ X227 @ X228))) & ((sK567 @ sK569 @ sK570) != (sK568 @ sK569 @ sK570)) & ! [X230 : term,X231 : subst,X232 : subst] : ((sK568 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232)) = (sub @ (sK568 @ X231 @ X230) @ X232)))) & (! [X233 : subst > term > term,X234 : subst > term > term,X235 : subst,X236 : term] : (((lam @ (X233 @ sh @ one)) != (lam @ (X234 @ sh @ one))) | ((sub @ (X233 @ (sK573 @ X233) @ (sK571 @ X233)) @ (sK572 @ X233)) != (X233 @ (comp @ (sK573 @ X233) @ (sK572 @ X233)) @ (sub @ (sK571 @ X233) @ (sK572 @ X233)))) | ((X233 @ X235 @ X236) = (X234 @ X235 @ X236)) | ((X234 @ (comp @ (sK575 @ X234) @ (sK576 @ X234)) @ (sub @ (sK574 @ X234) @ (sK576 @ X234))) != (sub @ (X234 @ (sK575 @ X234) @ (sK574 @ X234)) @ (sK576 @ X234)))) | (hoaslaminj != $true)) & (! [X243 : term > $o,X244 : term > $o,X245 : term,X246 : subst,X247 : term] : (((X244 @ (sub @ X247 @ (push @ X245 @ X246))) = $true) | ((X243 @ X247) != $true)) | (sK577 != (comp @ sh @ (push @ sK578 @ sK577))) | (sK580 != (sub @ one @ (push @ sK580 @ sK579))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | ((($true != (sK582 @ (sub @ sK585 @ (push @ sK583 @ sK584)))) & ($true = (sK581 @ sK585))) & ! [X257 : subst,X258 : term] : ((comp @ sh @ (push @ X258 @ X257)) = X257) & ! [X259 : subst,X260 : term] : ((sub @ one @ (push @ X260 @ X259)) = X260))) & (hoasinduction_lem3a_gthm = $true) & (ulamvar1 != $true) & (hoasinduction_lem2_gthm = $true) & ! [X261 : d_term,X262 : d_term] : (((d2term @ X262) != (d2term @ X261)) | (X261 = X262)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem2 != $true)) & (pushprop_lthm_orig = $true) & ! [X263 : d_term] : (d_one = X263) & ((pushprop_lem0 = $true) | ! [X267 : term > $o] : ((sK586 @ (sub @ (sK589 @ X267) @ (push @ sK587 @ sK588))) != (X267 @ (sK589 @ X267)))) & (! [X269 : term > $o,X270 : term,X271 : subst] : ! [X273 : term] : ((X269 @ (sub @ X273 @ (push @ X270 @ X271))) = (sK590 @ X271 @ X270 @ X269 @ X273)) | (pushprop_lem0 != $true))),
% 2.65/1.76 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])).
% 2.65/1.76 thf(f476,plain,(
% 2.65/1.76 ? [X0 : term,X1 : subst,X2 : term > $o] : ! [X3 : term > $o] : ? [X4 : term] : ((X3 @ X4) != (X2 @ (sub @ X4 @ (push @ X0 @ X1)))) => ! [X3 : term > $o] : ? [X4 : term] : ((sK459 @ (sub @ X4 @ (push @ sK457 @ sK458))) != (X3 @ X4))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f477,plain,(
% 2.65/1.76 ! [X3 : term > $o] : (? [X4 : term] : ((sK459 @ (sub @ X4 @ (push @ sK457 @ sK458))) != (X3 @ X4)) => ((sK459 @ (sub @ (sK460 @ X3) @ (push @ sK457 @ sK458))) != (X3 @ (sK460 @ X3))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f478,plain,(
% 2.65/1.76 ! [X5 : term,X6 : subst,X7 : term > $o] : (? [X8 : term > $o] : ! [X9 : term] : ((X7 @ (sub @ X9 @ (push @ X5 @ X6))) = (X8 @ X9)) => ! [X9 : term] : ((X7 @ (sub @ X9 @ (push @ X5 @ X6))) = (sK461 @ X7 @ X6 @ X5 @ X9)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f479,plain,(
% 2.65/1.76 ? [X11 : subst] : ((comp @ X11 @ id) != X11) => (sK462 != (comp @ sK462 @ id))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f480,plain,(
% 2.65/1.76 ? [X15 : term] : ((sub @ X15 @ id) != X15) => (sK463 != (sub @ sK463 @ id))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f481,plain,(
% 2.65/1.76 ? [X16 : term > $o,X17 : term] : (((sP56 @ X16) != $true) & ! [X18 : term] : (((var @ X18) != $true) | ($true = (X16 @ X18))) & ($true != (X16 @ X17)) & ! [X19 : term] : (((X16 @ (lam @ X19)) = $true) | ($true != (X16 @ X19)))) => (((sP56 @ sK464) != $true) & ! [X18 : term] : (((var @ X18) != $true) | ($true = (sK464 @ X18))) & ((sK464 @ sK465) != $true) & ! [X19 : term] : (($true = (sK464 @ (lam @ X19))) | ((sK464 @ X19) != $true)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f482,plain,(
% 2.65/1.76 ! [X20 : term > $o] : (? [X22 : term] : (($true = (var @ X22)) & ((X20 @ X22) != $true)) => (($true = (var @ (sK466 @ X20))) & ((X20 @ (sK466 @ X20)) != $true)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f483,plain,(
% 2.65/1.76 ! [X20 : term > $o] : (? [X23 : term] : (($true != (X20 @ (lam @ X23))) & ($true = (X20 @ X23))) => (((X20 @ (lam @ (sK467 @ X20))) != $true) & ((X20 @ (sK467 @ X20)) = $true)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f484,plain,(
% 2.65/1.76 ? [X24 : term > $o,X25 : term,X26 : term,X27 : subst,X28 : term > $o] : (((X24 @ (sub @ X25 @ (push @ X26 @ X27))) != $true) & ($true = (X28 @ X25))) => (($true != (sK468 @ (sub @ sK469 @ (push @ sK470 @ sK471)))) & ($true = (sK472 @ sK469)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f485,plain,(
% 2.65/1.76 ? [X38 : term,X39 : term,X40 : term,X41 : term] : (((ap @ X41 @ X38) = (ap @ X39 @ X40)) & (X39 != X41)) => (((ap @ sK474 @ sK475) = (ap @ sK476 @ sK473)) & (sK476 != sK474))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f486,plain,(
% 2.65/1.76 ! [X42 : subst] : (? [X43 : d_subst] : ((d2subst @ X43) = X42) => ((d2subst @ (sK477 @ X42)) = X42))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f487,plain,(
% 2.65/1.76 ? [X44 : term,X45 : term,X46 : term,X47 : term] : ((X46 != X47) & ((ap @ X46 @ X45) = (ap @ X47 @ X44))) => ((sK481 != sK480) & ((ap @ sK481 @ sK478) = (ap @ sK480 @ sK479)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f488,plain,(
% 2.65/1.76 ? [X48 : term] : ((sub @ X48 @ id) != X48) => ((sub @ sK482 @ id) != sK482)),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f489,plain,(
% 2.65/1.76 ? [X58 : term,X59 : term,X60 : term,X61 : term] : ((X58 != X59) & ((ap @ (sub @ X59 @ id) @ X61) = (ap @ (sub @ X58 @ id) @ X60))) => ((sK484 != sK483) & ((ap @ (sub @ sK484 @ id) @ sK486) = (ap @ (sub @ sK483 @ id) @ sK485)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f490,plain,(
% 2.65/1.76 ! [X62 : term] : (? [X63 : d_term] : ((d2term @ X63) = X62) => ((d2term @ (sK487 @ X62)) = X62))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f491,plain,(
% 2.65/1.76 ? [X64 : term,X65 : term] : ((X64 != X65) & ((sub @ X65 @ sh) = (sub @ X64 @ sh))) => ((sK488 != sK489) & ((sub @ sK488 @ sh) = (sub @ sK489 @ sh)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f492,plain,(
% 2.65/1.76 ? [X68 : subst > term > subst > $o] : ! [X69 : term > $o] : ? [X70 : term] : ((X68 @ id @ X70 @ id) != (X69 @ X70)) => ! [X69 : term > $o] : ? [X70 : term] : ((sK490 @ id @ X70 @ id) != (X69 @ X70))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f493,plain,(
% 2.65/1.76 ! [X69 : term > $o] : (? [X70 : term] : ((sK490 @ id @ X70 @ id) != (X69 @ X70)) => ((sK490 @ id @ (sK491 @ X69) @ id) != (X69 @ (sK491 @ X69))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f494,plain,(
% 2.65/1.76 ! [X71 : subst > term > subst > $o] : (? [X72 : term > $o] : ! [X73 : term] : ((X72 @ X73) = (X71 @ id @ X73 @ id)) => ! [X73 : term] : ((sK492 @ X71 @ X73) = (X71 @ id @ X73 @ id)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f495,plain,(
% 2.65/1.76 ! [X74 : term] : (? [X75 : subst > term > term] : ! [X76 : subst,X77 : term] : ((X75 @ X76 @ X77) = (sub @ X74 @ (push @ X77 @ X76))) => ! [X77 : term,X76 : subst] : ((sub @ X74 @ (push @ X77 @ X76)) = (sK493 @ X74 @ X76 @ X77)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f496,plain,(
% 2.65/1.76 ? [X78 : term] : ! [X79 : subst > term > term] : ? [X80 : subst,X81 : term] : ((sub @ X78 @ (push @ X81 @ X80)) != (X79 @ X80 @ X81)) => ! [X79 : subst > term > term] : ? [X81 : term,X80 : subst] : ((sub @ sK494 @ (push @ X81 @ X80)) != (X79 @ X80 @ X81))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f497,plain,(
% 2.65/1.76 ! [X79 : subst > term > term] : (? [X81 : term,X80 : subst] : ((sub @ sK494 @ (push @ X81 @ X80)) != (X79 @ X80 @ X81)) => ((X79 @ (sK495 @ X79) @ (sK496 @ X79)) != (sub @ sK494 @ (push @ (sK496 @ X79) @ (sK495 @ X79)))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f498,plain,(
% 2.65/1.76 ? [X82 : term] : ! [X83 : subst > term > term] : ? [X84 : subst,X85 : term] : ((X83 @ X84 @ X85) != (sub @ X82 @ (push @ X85 @ X84))) => ! [X83 : subst > term > term] : ? [X85 : term,X84 : subst] : ((X83 @ X84 @ X85) != (sub @ sK497 @ (push @ X85 @ X84)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f499,plain,(
% 2.65/1.76 ! [X83 : subst > term > term] : (? [X85 : term,X84 : subst] : ((X83 @ X84 @ X85) != (sub @ sK497 @ (push @ X85 @ X84))) => ((sub @ sK497 @ (push @ (sK499 @ X83) @ (sK498 @ X83))) != (X83 @ (sK498 @ X83) @ (sK499 @ X83))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f500,plain,(
% 2.65/1.76 ! [X86 : term] : (? [X87 : subst > term > term] : ! [X88 : subst,X89 : term] : ((sub @ X86 @ (push @ X89 @ X88)) = (X87 @ X88 @ X89)) => ! [X89 : term,X88 : subst] : ((sub @ X86 @ (push @ X89 @ X88)) = (sK500 @ X86 @ X88 @ X89)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f501,plain,(
% 2.65/1.76 ? [X93 : subst,X94 : term] : ((sub @ one @ (push @ X94 @ X93)) != X94) => ((sub @ one @ (push @ sK502 @ sK501)) != sK502)),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f502,plain,(
% 2.65/1.76 ? [X95 : term] : ((sub @ X95 @ id) != X95) => ((sub @ sK503 @ id) != sK503)),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f503,plain,(
% 2.65/1.76 ? [X98 : term] : ((sub @ X98 @ id) != X98) => (sK504 != (sub @ sK504 @ id))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f504,plain,(
% 2.65/1.76 ? [X99 : term] : ((sub @ X99 @ id) != X99) => ((sub @ sK505 @ id) != sK505)),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f505,plain,(
% 2.65/1.76 ? [X101 : term] : ! [X102 : subst > term > term] : ((sub @ X101 @ (push @ one @ sh)) != (X102 @ sh @ one)) => ! [X102 : subst > term > term] : ((X102 @ sh @ one) != (sub @ sK506 @ (push @ one @ sh)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f506,plain,(
% 2.65/1.76 ! [X103 : term] : (? [X104 : subst > term > term] : ((X104 @ sh @ one) = (sub @ X103 @ (push @ one @ sh))) => ((sK507 @ X103 @ sh @ one) = (sub @ X103 @ (push @ one @ sh))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f507,plain,(
% 2.65/1.76 ? [X106 : subst] : ((comp @ X106 @ id) != X106) => ((comp @ sK508 @ id) != sK508)),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f508,plain,(
% 2.65/1.76 ? [X107 : subst] : ((comp @ id @ X107) != X107) => (sK509 != (comp @ id @ sK509))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f509,plain,(
% 2.65/1.76 ? [X109 : subst] : ((comp @ X109 @ id) != X109) => ((comp @ sK510 @ id) != sK510)),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f510,plain,(
% 2.65/1.76 ? [X112 : subst] : ((comp @ id @ X112) != X112) => (sK511 != (comp @ id @ sK511))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f511,plain,(
% 2.65/1.76 ? [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) != X116) => (sK513 != (comp @ sh @ (push @ sK512 @ sK513)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f512,plain,(
% 2.65/1.76 ? [X118 : subst] : ((push @ (sub @ one @ X118) @ (comp @ sh @ X118)) != X118) => (sK514 != (push @ (sub @ one @ sK514) @ (comp @ sh @ sK514)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f513,plain,(
% 2.65/1.76 ? [X119 : subst > term > subst > $o,X120 : term] : (($true != (X119 @ id @ X120 @ id)) & ! [X121 : term,X122 : term] : (((X119 @ id @ (ap @ (sub @ X121 @ id) @ X122) @ id) = $true) | ((X119 @ id @ X121 @ id) != $true) | ((X119 @ id @ X122 @ id) != $true)) & ($true != (sP34 @ X119))) => (((sK515 @ id @ sK516 @ id) != $true) & ! [X122 : term,X121 : term] : (((sK515 @ id @ (ap @ (sub @ X121 @ id) @ X122) @ id) = $true) | ((sK515 @ id @ X121 @ id) != $true) | ((sK515 @ id @ X122 @ id) != $true)) & ((sP34 @ sK515) != $true))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f514,plain,(
% 2.65/1.76 ! [X123 : subst > term > subst > $o] : (? [X125 : term,X126 : term] : (((X123 @ id @ (ap @ (sub @ X125 @ id) @ X126) @ id) != $true) & ($true = (X123 @ id @ X125 @ id)) & ((X123 @ id @ X126 @ id) = $true)) => (((X123 @ id @ (ap @ (sub @ (sK517 @ X123) @ id) @ (sK518 @ X123)) @ id) != $true) & ((X123 @ id @ (sK517 @ X123) @ id) = $true) & ((X123 @ id @ (sK518 @ X123) @ id) = $true)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f515,plain,(
% 2.65/1.76 ? [X127 : term,X128 : term] : ((X127 != X128) & ((lam @ X127) = (lam @ X128))) => ((sK520 != sK519) & ((lam @ sK520) = (lam @ sK519)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f516,plain,(
% 2.65/1.76 ? [X131 : subst] : ((comp @ id @ X131) != X131) => ((comp @ id @ sK521) != sK521)),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f517,plain,(
% 2.65/1.76 ? [X133 : term,X134 : term,X135 : term,X136 : term] : ((X135 != X136) & ((ap @ (sub @ X134 @ id) @ X136) = (ap @ (sub @ X133 @ id) @ X135))) => ((sK525 != sK524) & ((ap @ (sub @ sK523 @ id) @ sK525) = (ap @ (sub @ sK522 @ id) @ sK524)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f518,plain,(
% 2.65/1.76 ? [X141 : term,X142 : term,X143 : subst > term > term] : (! [X144 : subst,X145 : subst,X146 : term] : ((sub @ (X143 @ X145 @ X146) @ X144) = (X143 @ (comp @ X145 @ X144) @ (sub @ X146 @ X144))) & ((lam @ (X143 @ sh @ one)) = (ap @ (sub @ X141 @ id) @ X142))) => (! [X146 : term,X145 : subst,X144 : subst] : ((sK528 @ (comp @ X145 @ X144) @ (sub @ X146 @ X144)) = (sub @ (sK528 @ X145 @ X146) @ X144)) & ((lam @ (sK528 @ sh @ one)) = (ap @ (sub @ sK526 @ id) @ sK527)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f519,plain,(
% 2.65/1.76 ! [X149 : subst > term > term] : (? [X150 : subst,X151 : subst,X152 : term] : ((sub @ (X149 @ X151 @ X152) @ X150) != (X149 @ (comp @ X151 @ X150) @ (sub @ X152 @ X150))) => ((X149 @ (comp @ (sK530 @ X149) @ (sK529 @ X149)) @ (sub @ (sK531 @ X149) @ (sK529 @ X149))) != (sub @ (X149 @ (sK530 @ X149) @ (sK531 @ X149)) @ (sK529 @ X149))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f520,plain,(
% 2.65/1.76 ? [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 != X155) & ((ap @ (sub @ X156 @ id) @ X155) = (ap @ (sub @ X154 @ id) @ X153))) => ((sK532 != sK534) & ((ap @ (sub @ sK533 @ id) @ sK532) = (ap @ (sub @ sK535 @ id) @ sK534)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f521,plain,(
% 2.65/1.76 ? [X165 : term,X166 : term,X167 : term,X168 : term] : (((ap @ X168 @ X166) = (ap @ X165 @ X167)) & (X166 != X167)) => (((ap @ sK539 @ sK537) = (ap @ sK536 @ sK538)) & (sK537 != sK538))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f522,plain,(
% 2.65/1.76 ? [X169 : term,X170 : term,X171 : term,X172 : term] : ((X169 != X172) & ((ap @ (sub @ X172 @ id) @ X171) = (ap @ (sub @ X169 @ id) @ X170))) => ((sK543 != sK540) & ((ap @ (sub @ sK540 @ id) @ sK541) = (ap @ (sub @ sK543 @ id) @ sK542)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f523,plain,(
% 2.65/1.76 ! [X177 : term] : (? [X178 : subst > term > term] : ((sub @ X177 @ (push @ one @ sh)) = (X178 @ sh @ one)) => ((sub @ X177 @ (push @ one @ sh)) = (sK544 @ X177 @ sh @ one)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f524,plain,(
% 2.65/1.76 ? [X179 : term] : ! [X180 : subst > term > term] : ((sub @ X179 @ (push @ one @ sh)) != (X180 @ sh @ one)) => ! [X180 : subst > term > term] : ((X180 @ sh @ one) != (sub @ sK545 @ (push @ one @ sh)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f525,plain,(
% 2.65/1.76 ? [X183 : subst,X184 : term > $o,X185 : term > $o,X186 : term] : (! [X187 : term] : ((X184 @ X187) = (X185 @ (sub @ X187 @ (push @ X186 @ X183)))) & ((X184 @ one) != $true) & ($true = (X185 @ X186))) => (! [X187 : term] : ((sK548 @ (sub @ X187 @ (push @ sK549 @ sK546))) = (sK547 @ X187)) & ((sK547 @ one) != $true) & ($true = (sK548 @ sK549)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f526,plain,(
% 2.65/1.76 ? [X188 : subst,X189 : term] : ((sub @ one @ (push @ X189 @ X188)) != X189) => ((sub @ one @ (push @ sK551 @ sK550)) != sK551)),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f527,plain,(
% 2.65/1.76 ! [X190 : subst,X191 : term > $o,X192 : term > $o,X193 : term] : (? [X194 : term] : ((X191 @ X194) != (X192 @ (sub @ X194 @ (push @ X193 @ X190)))) => ((X191 @ (sK552 @ X193 @ X192 @ X191 @ X190)) != (X192 @ (sub @ (sK552 @ X193 @ X192 @ X191 @ X190) @ (push @ X193 @ X190)))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f528,plain,(
% 2.65/1.76 ! [X195 : term > $o,X196 : subst,X197 : term > $o,X198 : term] : (? [X199 : term] : ((X197 @ (sub @ X199 @ (push @ X198 @ X196))) != (X195 @ X199)) => ((X197 @ (sub @ (sK553 @ X198 @ X197 @ X196 @ X195) @ (push @ X198 @ X196))) != (X195 @ (sK553 @ X198 @ X197 @ X196 @ X195))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f529,plain,(
% 2.65/1.76 ? [X200 : term > $o,X201 : subst,X202 : term > $o,X203 : term] : (((X202 @ X203) = $true) & ! [X204 : term] : ((X200 @ X204) = (X202 @ (sub @ X204 @ (push @ X203 @ X201)))) & ((X200 @ one) != $true)) => (($true = (sK556 @ sK557)) & ! [X204 : term] : ((sK556 @ (sub @ X204 @ (push @ sK557 @ sK555))) = (sK554 @ X204)) & ((sK554 @ one) != $true))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f530,plain,(
% 2.65/1.76 ! [X205 : subst > term > subst > $o] : (? [X206 : term > $o] : ! [X207 : term] : ((X206 @ X207) = (X205 @ id @ X207 @ id)) => ! [X207 : term] : ((sK558 @ X205 @ X207) = (X205 @ id @ X207 @ id)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f531,plain,(
% 2.65/1.76 ? [X208 : subst > term > subst > $o] : ! [X209 : term > $o] : ? [X210 : term] : ((X208 @ id @ X210 @ id) != (X209 @ X210)) => ! [X209 : term > $o] : ? [X210 : term] : ((sK559 @ id @ X210 @ id) != (X209 @ X210))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f532,plain,(
% 2.65/1.76 ! [X209 : term > $o] : (? [X210 : term] : ((sK559 @ id @ X210 @ id) != (X209 @ X210)) => ((sK559 @ id @ (sK560 @ X209) @ id) != (X209 @ (sK560 @ X209))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f533,plain,(
% 2.65/1.76 ? [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ X217 @ X215) = (ap @ X218 @ X216)) & (X215 != X216)) => (((ap @ sK564 @ sK562) = (ap @ sK563 @ sK561)) & (sK561 != sK562))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f534,plain,(
% 2.65/1.76 ? [X219 : subst] : ((comp @ id @ X219) != X219) => (sK565 != (comp @ id @ sK565))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f535,plain,(
% 2.65/1.76 ? [X220 : subst] : ((comp @ X220 @ id) != X220) => (sK566 != (comp @ sK566 @ id))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f536,plain,(
% 2.65/1.76 ? [X223 : subst > term > term,X224 : subst > term > term,X225 : subst,X226 : term] : (((lam @ (X224 @ sh @ one)) = (lam @ (X223 @ sh @ one))) & ! [X227 : term,X228 : subst,X229 : subst] : ((X223 @ (comp @ X229 @ X228) @ (sub @ X227 @ X228)) = (sub @ (X223 @ X229 @ X227) @ X228)) & ((X224 @ X225 @ X226) != (X223 @ X225 @ X226)) & ! [X230 : term,X231 : subst,X232 : subst] : ((sub @ (X224 @ X231 @ X230) @ X232) = (X224 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232)))) => (((lam @ (sK568 @ sh @ one)) = (lam @ (sK567 @ sh @ one))) & ! [X229 : subst,X228 : subst,X227 : term] : ((sub @ (sK567 @ X229 @ X227) @ X228) = (sK567 @ (comp @ X229 @ X228) @ (sub @ X227 @ X228))) & ((sK567 @ sK569 @ sK570) != (sK568 @ sK569 @ sK570)) & ! [X232 : subst,X231 : subst,X230 : term] : ((sK568 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232)) = (sub @ (sK568 @ X231 @ X230) @ X232)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f537,plain,(
% 2.65/1.76 ! [X233 : subst > term > term] : (? [X237 : term,X238 : subst,X239 : subst] : ((sub @ (X233 @ X239 @ X237) @ X238) != (X233 @ (comp @ X239 @ X238) @ (sub @ X237 @ X238))) => ((sub @ (X233 @ (sK573 @ X233) @ (sK571 @ X233)) @ (sK572 @ X233)) != (X233 @ (comp @ (sK573 @ X233) @ (sK572 @ X233)) @ (sub @ (sK571 @ X233) @ (sK572 @ X233)))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f538,plain,(
% 2.65/1.76 ! [X234 : subst > term > term] : (? [X240 : term,X241 : subst,X242 : subst] : ((sub @ (X234 @ X241 @ X240) @ X242) != (X234 @ (comp @ X241 @ X242) @ (sub @ X240 @ X242))) => ((X234 @ (comp @ (sK575 @ X234) @ (sK576 @ X234)) @ (sub @ (sK574 @ X234) @ (sK576 @ X234))) != (sub @ (X234 @ (sK575 @ X234) @ (sK574 @ X234)) @ (sK576 @ X234))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f539,plain,(
% 2.65/1.76 ? [X248 : subst,X249 : term] : ((comp @ sh @ (push @ X249 @ X248)) != X248) => (sK577 != (comp @ sh @ (push @ sK578 @ sK577)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f540,plain,(
% 2.65/1.76 ? [X250 : subst,X251 : term] : ((sub @ one @ (push @ X251 @ X250)) != X251) => (sK580 != (sub @ one @ (push @ sK580 @ sK579)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f541,plain,(
% 2.65/1.76 ? [X252 : term > $o,X253 : term > $o,X254 : term,X255 : subst,X256 : term] : (((X253 @ (sub @ X256 @ (push @ X254 @ X255))) != $true) & ((X252 @ X256) = $true)) => (($true != (sK582 @ (sub @ sK585 @ (push @ sK583 @ sK584)))) & ($true = (sK581 @ sK585)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f542,plain,(
% 2.65/1.76 ? [X264 : term > $o,X265 : term,X266 : subst] : ! [X267 : term > $o] : ? [X268 : term] : ((X267 @ X268) != (X264 @ (sub @ X268 @ (push @ X265 @ X266)))) => ! [X267 : term > $o] : ? [X268 : term] : ((X267 @ X268) != (sK586 @ (sub @ X268 @ (push @ sK587 @ sK588))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f543,plain,(
% 2.65/1.76 ! [X267 : term > $o] : (? [X268 : term] : ((X267 @ X268) != (sK586 @ (sub @ X268 @ (push @ sK587 @ sK588)))) => ((sK586 @ (sub @ (sK589 @ X267) @ (push @ sK587 @ sK588))) != (X267 @ (sK589 @ X267))))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f544,plain,(
% 2.65/1.76 ! [X269 : term > $o,X270 : term,X271 : subst] : (? [X272 : term > $o] : ! [X273 : term] : ((X269 @ (sub @ X273 @ (push @ X270 @ X271))) = (X272 @ X273)) => ! [X273 : term] : ((X269 @ (sub @ X273 @ (push @ X270 @ X271))) = (sK590 @ X271 @ X270 @ X269 @ X273)))),
% 2.65/1.76 introduced(definition,[],[choice_axiom])).
% 2.65/1.76 thf(f475,plain,(
% 2.65/1.76 ((pushprop_lem0_lthm = $true) | ? [X0 : term,X1 : subst,X2 : term > $o] : ! [X3 : term > $o] : ? [X4 : term] : ((X3 @ X4) != (X2 @ (sub @ X4 @ (push @ X0 @ X1))))) & (! [X5 : term,X6 : subst,X7 : term > $o] : ? [X8 : term > $o] : ! [X9 : term] : ((X7 @ (sub @ X9 @ (push @ X5 @ X6))) = (X8 @ X9)) | (pushprop_lem0_lthm != $true)) & (! [X10 : subst] : ((comp @ X10 @ id) = X10) | (axidr != $true)) & ((axidr = $true) | ? [X11 : subst] : ((comp @ X11 @ id) != X11)) & ! [X12 : d_subst,X13 : d_subst] : (((d2subst @ X13) != (d2subst @ X12)) | (X12 = X13)) & (hoasinduction_lem2_lthm = $true) & (axclos = $true) & (($true = sP58) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP58)) & (! [X14 : term] : ((sub @ X14 @ id) = X14) | (axvarid != $true)) & ((axvarid = $true) | ? [X15 : term] : ((sub @ X15 @ id) != X15)) & (substmonoid_gthm = $true) & ((induction = $true) | ? [X16 : term > $o,X17 : term] : (((sP56 @ X16) != $true) & ! [X18 : term] : (((var @ X18) != $true) | ($true = (X16 @ X18))) & ($true != (X16 @ X17)) & ! [X19 : term] : (((X16 @ (lam @ X19)) = $true) | ($true != (X16 @ X19))))) & (! [X20 : term > $o,X21 : term] : (((sP56 @ X20) = $true) | ? [X22 : term] : (($true = (var @ X22)) & ((X20 @ X22) != $true)) | ((X20 @ X21) = $true) | ? [X23 : term] : (($true != (X20 @ (lam @ X23))) & ($true = (X20 @ X23)))) | (induction != $true)) & ((pushprop_lem1 = $true) | ? [X24 : term > $o,X25 : term,X26 : term,X27 : subst,X28 : term > $o] : (((X24 @ (sub @ X25 @ (push @ X26 @ X27))) != $true) & ($true = (X28 @ X25)))) & (! [X29 : term > $o,X30 : term,X31 : term,X32 : subst,X33 : term > $o] : (((X29 @ (sub @ X30 @ (push @ X31 @ X32))) = $true) | ((X33 @ X30) != $true)) | (pushprop_lem1 != $true)) & (hoaslamnotvar_gthm = $true) & (axassoc = $true) & (! [X34 : term,X35 : term,X36 : term,X37 : term] : (((ap @ X37 @ X34) != (ap @ X35 @ X36)) | (X35 = X37)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X38 : term,X39 : term,X40 : term,X41 : term] : (((ap @ X41 @ X38) = (ap @ X39 @ X40)) & (X39 != X41))) & (hoasapinj2_gthm = $true) & ! [X42 : subst] : ? [X43 : d_subst] : ((d2subst @ X43) = X42) & (? [X44 : term,X45 : term,X46 : term,X47 : term] : ((X46 != X47) & ((ap @ X46 @ X45) = (ap @ X47 @ X44))) | ? [X48 : term] : ((sub @ X48 @ id) != X48) | ! [X49 : term,X50 : term,X51 : term,X52 : term] : ((X49 = X50) | ((ap @ (sub @ X49 @ id) @ X51) != (ap @ (sub @ X50 @ id) @ X52))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X53 : term,X54 : term,X55 : term,X56 : term] : ((X55 = X56) | ((ap @ X55 @ X54) != (ap @ X56 @ X53))) & ! [X57 : term] : ((sub @ X57 @ id) = X57) & ? [X58 : term,X59 : term,X60 : term,X61 : term] : ((X58 != X59) & ((ap @ (sub @ X59 @ id) @ X61) = (ap @ (sub @ X58 @ id) @ X60))))) & (hoasapnotvar = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem3aa = $true) | ($true != sP55)) & (($true = sP55) | (hoasinduction_lem3aa != $true)) & ((var @ (d2term @ d_one)) != $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar_lthm = $true) & (pushprop_lem3v2 = $true) & ((sP54 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP54 != $true)) & (termmset_gthm = $true) & (one = (d2term @ d_one)) & ! [X62 : term] : ? [X63 : d_term] : ((d2term @ X63) = X62) & ((shinj = $true) | ? [X64 : term,X65 : term] : ((X64 != X65) & ((sub @ X65 @ sh) = (sub @ X64 @ sh)))) & (! [X66 : term,X67 : term] : ((X66 = X67) | ((sub @ X66 @ sh) != (sub @ X67 @ sh))) | (shinj != $true)) & ((hoasinduction_lem0 = $true) | ? [X68 : subst > term > subst > $o] : ! [X69 : term > $o] : ? [X70 : term] : ((X68 @ id @ X70 @ id) != (X69 @ X70))) & (! [X71 : subst > term > subst > $o] : ? [X72 : term > $o] : ! [X73 : term] : ((X72 @ X73) = (X71 @ id @ X73 @ id)) | (hoasinduction_lem0 != $true)) & (hoasinduction_lem1_lthm = $true) & (induction2_gthm = $true) & (! [X74 : term] : ? [X75 : subst > term > term] : ! [X76 : subst,X77 : term] : ((X75 @ X76 @ X77) = (sub @ X74 @ (push @ X77 @ X76))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X78 : term] : ! [X79 : subst > term > term] : ? [X80 : subst,X81 : term] : ((sub @ X78 @ (push @ X81 @ X80)) != (X79 @ X80 @ X81))) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X82 : term] : ! [X83 : subst > term > term] : ? [X84 : subst,X85 : term] : ((X83 @ X84 @ X85) != (sub @ X82 @ (push @ X85 @ X84)))) & (! [X86 : term] : ? [X87 : subst > term > term] : ! [X88 : subst,X89 : term] : ((sub @ X86 @ (push @ X89 @ X88)) = (X87 @ X88 @ X89)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((sP53 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP53 != $true)) & ! [X90 : d_subst] : (d_id = X90) & (hoasinduction_lem1v2 = $true) & (! [X91 : subst,X92 : term] : ((sub @ one @ (push @ X92 @ X91)) = X92) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X93 : subst,X94 : term] : ((sub @ one @ (push @ X94 @ X93)) != X94)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (($true = sP51) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP51)) & (axapp = $true) & (induction2lem_gthm = $true) & (ulamvarsh = $true) & (axvarshift = $true) & (ulamvarind = $true) & ((d2subst @ d_id) = id) & ((termmset = $true) | ? [X95 : term] : ((sub @ X95 @ id) != X95)) & (! [X96 : term] : ((sub @ X96 @ id) = X96) | (termmset != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP49 != $true)) & ((sP49 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & (apnotvar = $true) & (! [X97 : term] : ((sub @ X97 @ id) = X97) | ? [X98 : term] : ((sub @ X98 @ id) != X98) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X99 : term] : ((sub @ X99 @ id) != X99) & ! [X100 : term] : ((sub @ X100 @ id) = X100))) & ((sP48 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP48 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP46)) & (($true = sP46) | (hoasinduction_lem3 != $true)) & (hoasinduction_lem3_gthm = $true) & ((sP44 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP44 != $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3b = $true) | ? [X101 : term] : ! [X102 : subst > term > term] : ((sub @ X101 @ (push @ one @ sh)) != (X102 @ sh @ one))) & (! [X103 : term] : ? [X104 : subst > term > term] : ((X104 @ sh @ one) = (sub @ X103 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & (hoasinduction_lem3v2_gthm = $true) & (axmap = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop = $true) & ((substmonoid_lthm = $true) | (! [X105 : subst] : ((comp @ X105 @ id) = X105) & (? [X106 : subst] : ((comp @ X106 @ id) != X106) | ? [X107 : subst] : ((comp @ id @ X107) != X107)) & ! [X108 : subst] : ((comp @ id @ X108) = X108))) & (? [X109 : subst] : ((comp @ X109 @ id) != X109) | (! [X110 : subst] : ((comp @ X110 @ id) = X110) & ! [X111 : subst] : ((comp @ id @ X111) = X111)) | ? [X112 : subst] : ((comp @ id @ X112) != X112) | (substmonoid_lthm != $true)) & (axabs = $true) & (lamnotvar = $true) & (! [X113 : term,X114 : subst] : ((comp @ sh @ (push @ X113 @ X114)) = X114) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X115 : term,X116 : subst] : ((comp @ sh @ (push @ X115 @ X116)) != X116)) & ((sP43 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP43 != $true)) & (($true = sP39) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP39)) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X117 : subst] : ((push @ (sub @ one @ X117) @ (comp @ sh @ X117)) = X117) | (axscons != $true)) & ((axscons = $true) | ? [X118 : subst] : ((push @ (sub @ one @ X118) @ (comp @ sh @ X118)) != X118)) & (hoasinduction_lem1_gthm = $true) & (pushprop_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X119 : subst > term > subst > $o,X120 : term] : (($true != (X119 @ id @ X120 @ id)) & ! [X121 : term,X122 : term] : (((X119 @ id @ (ap @ (sub @ X121 @ id) @ X122) @ id) = $true) | ((X119 @ id @ X121 @ id) != $true) | ((X119 @ id @ X122 @ id) != $true)) & ($true != (sP34 @ X119)))) & (! [X123 : subst > term > subst > $o,X124 : term] : (($true = (X123 @ id @ X124 @ id)) | ? [X125 : term,X126 : term] : (((X123 @ id @ (ap @ (sub @ X125 @ id) @ X126) @ id) != $true) & ($true = (X123 @ id @ X125 @ id)) & ((X123 @ id @ X126 @ id) = $true)) | ($true = (sP34 @ X123))) | (hoasinduction_no_psi_cond != $true)) & (lamnotap != $true) & ((hoasinduction = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction != $true)) & ((laminj = $true) | ? [X127 : term,X128 : term] : ((X127 != X128) & ((lam @ X127) = (lam @ X128)))) & (! [X129 : term,X130 : term] : ((X129 = X130) | ((lam @ X129) != (lam @ X130))) | (laminj != $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((axidl = $true) | ? [X131 : subst] : ((comp @ id @ X131) != X131)) & (! [X132 : subst] : ((comp @ id @ X132) = X132) | (axidl != $true)) & ((hoasapinj2 = $true) | ? [X133 : term,X134 : term,X135 : term,X136 : term] : ((X135 != X136) & ((ap @ (sub @ X134 @ id) @ X136) = (ap @ (sub @ X133 @ id) @ X135)))) & (! [X137 : term,X138 : term,X139 : term,X140 : term] : ((X139 = X140) | ((ap @ (sub @ X138 @ id) @ X140) != (ap @ (sub @ X137 @ id) @ X139))) | (hoasapinj2 != $true)) & ((hoaslamnotap = $true) | ? [X141 : term,X142 : term,X143 : subst > term > term] : (! [X144 : subst,X145 : subst,X146 : term] : ((sub @ (X143 @ X145 @ X146) @ X144) = (X143 @ (comp @ X145 @ X144) @ (sub @ X146 @ X144))) & ((lam @ (X143 @ sh @ one)) = (ap @ (sub @ X141 @ id) @ X142)))) & (! [X147 : term,X148 : term,X149 : subst > term > term] : (? [X150 : subst,X151 : subst,X152 : term] : ((sub @ (X149 @ X151 @ X152) @ X150) != (X149 @ (comp @ X151 @ X150) @ (sub @ X152 @ X150))) | ((ap @ (sub @ X147 @ id) @ X148) != (lam @ (X149 @ sh @ one)))) | (hoaslamnotap != $true)) & ((hoasinduction_lthm_3 = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lthm_3 != $true)) & (pushprop_lem3v2_lthm = $true) & ((sP24 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP24 != $true)) & ((hoasapinj2_lthm = $true) | (? [X153 : term,X154 : term,X155 : term,X156 : term] : ((X153 != X155) & ((ap @ (sub @ X156 @ id) @ X155) = (ap @ (sub @ X154 @ id) @ X153))) & ! [X157 : term,X158 : term,X159 : term,X160 : term] : (((ap @ X157 @ X159) != (ap @ X160 @ X158)) | (X158 = X159)))) & (! [X161 : term,X162 : term,X163 : term,X164 : term] : ((X161 = X163) | ((ap @ (sub @ X162 @ id) @ X161) != (ap @ (sub @ X164 @ id) @ X163))) | ? [X165 : term,X166 : term,X167 : term,X168 : term] : (((ap @ X168 @ X166) = (ap @ X165 @ X167)) & (X166 != X167)) | (hoasapinj2_lthm != $true)) & ((hoasapinj1 = $true) | ? [X169 : term,X170 : term,X171 : term,X172 : term] : ((X169 != X172) & ((ap @ (sub @ X172 @ id) @ X171) = (ap @ (sub @ X169 @ id) @ X170)))) & (! [X173 : term,X174 : term,X175 : term,X176 : term] : ((X173 = X176) | ((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X176 @ id) @ X175))) | (hoasapinj1 != $true)) & (hoaslamnotap_gthm = $true) & (hoasinduction_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X177 : term] : ? [X178 : subst > term > term] : ((sub @ X177 @ (push @ one @ sh)) = (X178 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X179 : term] : ! [X180 : subst > term > term] : ((sub @ X179 @ (push @ one @ sh)) != (X180 @ sh @ one))) & ((sP23 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP23 != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X181 : subst,X182 : term] : ((sub @ one @ (push @ X182 @ X181)) = X182) & ? [X183 : subst,X184 : term > $o,X185 : term > $o,X186 : term] : (! [X187 : term] : ((X184 @ X187) = (X185 @ (sub @ X187 @ (push @ X186 @ X183)))) & ((X184 @ one) != $true) & ($true = (X185 @ X186))))) & (? [X188 : subst,X189 : term] : ((sub @ one @ (push @ X189 @ X188)) != X189) | ! [X190 : subst,X191 : term > $o,X192 : term > $o,X193 : term] : (? [X194 : term] : ((X191 @ X194) != (X192 @ (sub @ X194 @ (push @ X193 @ X190)))) | ((X191 @ one) = $true) | ($true != (X192 @ X193))) | (pushprop_lem1v2_lthm != $true)) & (! [X195 : term > $o,X196 : subst,X197 : term > $o,X198 : term] : (((X197 @ X198) != $true) | ? [X199 : term] : ((X197 @ (sub @ X199 @ (push @ X198 @ X196))) != (X195 @ X199)) | ($true = (X195 @ one))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X200 : term > $o,X201 : subst,X202 : term > $o,X203 : term] : (((X202 @ X203) = $true) & ! [X204 : term] : ((X200 @ X204) = (X202 @ (sub @ X204 @ (push @ X203 @ X201)))) & ((X200 @ one) != $true))) & (hoaslamnotap_lthm = $true) & ((sP21 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP21 != $true)) & (pushprop_gthm = $true) & (! [X205 : subst > term > subst > $o] : ? [X206 : term > $o] : ! [X207 : term] : ((X206 @ X207) = (X205 @ id @ X207 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X208 : subst > term > subst > $o] : ! [X209 : term > $o] : ? [X210 : term] : ((X208 @ id @ X210 @ id) != (X209 @ X210))) & (($true = sP19) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP19)) & (pushprop_lem2v2_lthm = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((d2subst @ d_id) = sh) & (hoasapinj1_gthm = $true) & (! [X211 : term,X212 : term,X213 : term,X214 : term] : (((ap @ X214 @ X212) != (ap @ X213 @ X211)) | (X211 = X212)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X215 : term,X216 : term,X217 : term,X218 : term] : (((ap @ X217 @ X215) = (ap @ X218 @ X216)) & (X215 != X216))) & ((substmonoid = $true) | ? [X219 : subst] : ((comp @ id @ X219) != X219) | ? [X220 : subst] : ((comp @ X220 @ id) != X220)) & ((! [X221 : subst] : ((comp @ id @ X221) = X221) & ! [X222 : subst] : ((comp @ X222 @ id) = X222)) | (substmonoid != $true)) & (($true = sP18) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP18)) & ((hoasinduction_lthm = $true) | ($true != sP15)) & (($true = sP15) | (hoasinduction_lthm != $true)) & (hoasinduction_lem1 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP6 != $true)) & ((sP6 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoaslaminj = $true) | ? [X223 : subst > term > term,X224 : subst > term > term,X225 : subst,X226 : term] : (((lam @ (X224 @ sh @ one)) = (lam @ (X223 @ sh @ one))) & ! [X227 : term,X228 : subst,X229 : subst] : ((X223 @ (comp @ X229 @ X228) @ (sub @ X227 @ X228)) = (sub @ (X223 @ X229 @ X227) @ X228)) & ((X224 @ X225 @ X226) != (X223 @ X225 @ X226)) & ! [X230 : term,X231 : subst,X232 : subst] : ((sub @ (X224 @ X231 @ X230) @ X232) = (X224 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232))))) & (! [X233 : subst > term > term,X234 : subst > term > term,X235 : subst,X236 : term] : (((lam @ (X233 @ sh @ one)) != (lam @ (X234 @ sh @ one))) | ? [X237 : term,X238 : subst,X239 : subst] : ((sub @ (X233 @ X239 @ X237) @ X238) != (X233 @ (comp @ X239 @ X238) @ (sub @ X237 @ X238))) | ((X233 @ X235 @ X236) = (X234 @ X235 @ X236)) | ? [X240 : term,X241 : subst,X242 : subst] : ((sub @ (X234 @ X241 @ X240) @ X242) != (X234 @ (comp @ X241 @ X242) @ (sub @ X240 @ X242)))) | (hoaslaminj != $true)) & (! [X243 : term > $o,X244 : term > $o,X245 : term,X246 : subst,X247 : term] : (((X244 @ (sub @ X247 @ (push @ X245 @ X246))) = $true) | ((X243 @ X247) != $true)) | ? [X248 : subst,X249 : term] : ((comp @ sh @ (push @ X249 @ X248)) != X248) | ? [X250 : subst,X251 : term] : ((sub @ one @ (push @ X251 @ X250)) != X251) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X252 : term > $o,X253 : term > $o,X254 : term,X255 : subst,X256 : term] : (((X253 @ (sub @ X256 @ (push @ X254 @ X255))) != $true) & ((X252 @ X256) = $true)) & ! [X257 : subst,X258 : term] : ((comp @ sh @ (push @ X258 @ X257)) = X257) & ! [X259 : subst,X260 : term] : ((sub @ one @ (push @ X260 @ X259)) = X260))) & (hoasinduction_lem3a_gthm = $true) & (ulamvar1 != $true) & (hoasinduction_lem2_gthm = $true) & ! [X261 : d_term,X262 : d_term] : (((d2term @ X262) != (d2term @ X261)) | (X261 = X262)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem2 != $true)) & (pushprop_lthm_orig = $true) & ! [X263 : d_term] : (d_one = X263) & ((pushprop_lem0 = $true) | ? [X264 : term > $o,X265 : term,X266 : subst] : ! [X267 : term > $o] : ? [X268 : term] : ((X267 @ X268) != (X264 @ (sub @ X268 @ (push @ X265 @ X266))))) & (! [X269 : term > $o,X270 : term,X271 : subst] : ? [X272 : term > $o] : ! [X273 : term] : ((X269 @ (sub @ X273 @ (push @ X270 @ X271))) = (X272 @ X273)) | (pushprop_lem0 != $true))),
% 2.65/1.76 inference(rectify,[status(thm)],[f474])).
% 2.65/1.76 thf(f474,plain,(
% 2.65/1.76 ((pushprop_lem0_lthm = $true) | ? [X116 : term,X117 : subst,X118 : term > $o] : ! [X119 : term > $o] : ? [X120 : term] : ((X119 @ X120) != (X118 @ (sub @ X120 @ (push @ X116 @ X117))))) & (! [X116 : term,X117 : subst,X118 : term > $o] : ? [X119 : term > $o] : ! [X120 : term] : ((X119 @ X120) = (X118 @ (sub @ X120 @ (push @ X116 @ X117)))) | (pushprop_lem0_lthm != $true)) & (! [X466 : subst] : ((comp @ X466 @ id) = X466) | (axidr != $true)) & ((axidr = $true) | ? [X466 : subst] : ((comp @ X466 @ id) != X466)) & ! [X112 : d_subst,X113 : d_subst] : (((d2subst @ X112) != (d2subst @ X113)) | (X112 = X113)) & (hoasinduction_lem2_lthm = $true) & (axclos = $true) & (($true = sP58) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP58)) & (! [X174 : term] : ((sub @ X174 @ id) = X174) | (axvarid != $true)) & ((axvarid = $true) | ? [X174 : term] : ((sub @ X174 @ id) != X174)) & (substmonoid_gthm = $true) & ((induction = $true) | ? [X403 : term > $o,X404 : term] : (((sP56 @ X403) != $true) & ! [X407 : term] : (($true != (var @ X407)) | ($true = (X403 @ X407))) & ((X403 @ X404) != $true) & ! [X408 : term] : (((X403 @ (lam @ X408)) = $true) | ((X403 @ X408) != $true)))) & (! [X403 : term > $o,X404 : term] : (((sP56 @ X403) = $true) | ? [X407 : term] : (($true = (var @ X407)) & ($true != (X403 @ X407))) | ((X403 @ X404) = $true) | ? [X408 : term] : (((X403 @ (lam @ X408)) != $true) & ((X403 @ X408) = $true))) | (induction != $true)) & ((pushprop_lem1 = $true) | ? [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (((X133 @ (sub @ X134 @ (push @ X135 @ X136))) != $true) & ($true = (X137 @ X134)))) & (! [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (((X133 @ (sub @ X134 @ (push @ X135 @ X136))) = $true) | ($true != (X137 @ X134))) | (pushprop_lem1 != $true)) & (hoaslamnotvar_gthm = $true) & (axassoc = $true) & (! [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) != (ap @ X276 @ X277)) | (X276 = X278)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) = (ap @ X276 @ X277)) & (X276 != X278))) & (hoasapinj2_gthm = $true) & ! [X464 : subst] : ? [X465 : d_subst] : ((d2subst @ X465) = X464) & (? [X104 : term,X105 : term,X106 : term,X107 : term] : ((X106 != X107) & ((ap @ X106 @ X105) = (ap @ X107 @ X104))) | ? [X103 : term] : ((sub @ X103 @ id) != X103) | ! [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 = X109) | ((ap @ (sub @ X109 @ id) @ X111) != (ap @ (sub @ X108 @ id) @ X110))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X104 : term,X105 : term,X106 : term,X107 : term] : ((X106 = X107) | ((ap @ X106 @ X105) != (ap @ X107 @ X104))) & ! [X103 : term] : ((sub @ X103 @ id) = X103) & ? [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 != X109) & ((ap @ (sub @ X109 @ id) @ X111) = (ap @ (sub @ X108 @ id) @ X110))))) & (hoasapnotvar = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem3aa = $true) | ($true != sP55)) & (($true = sP55) | (hoasinduction_lem3aa != $true)) & ((var @ (d2term @ d_one)) != $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar_lthm = $true) & (pushprop_lem3v2 = $true) & ((sP54 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP54 != $true)) & (termmset_gthm = $true) & (one = (d2term @ d_one)) & ! [X114 : term] : ? [X115 : d_term] : ((d2term @ X115) = X114) & ((shinj = $true) | ? [X35 : term,X36 : term] : ((X35 != X36) & ((sub @ X36 @ sh) = (sub @ X35 @ sh)))) & (! [X35 : term,X36 : term] : ((X35 = X36) | ((sub @ X36 @ sh) != (sub @ X35 @ sh))) | (shinj != $true)) & ((hoasinduction_lem0 = $true) | ? [X333 : subst > term > subst > $o] : ! [X334 : term > $o] : ? [X335 : term] : ((X334 @ X335) != (X333 @ id @ X335 @ id))) & (! [X333 : subst > term > subst > $o] : ? [X334 : term > $o] : ! [X335 : term] : ((X334 @ X335) = (X333 @ id @ X335 @ id)) | (hoasinduction_lem0 != $true)) & (hoasinduction_lem1_lthm = $true) & (induction2_gthm = $true) & (! [X178 : term] : ? [X179 : subst > term > term] : ! [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) = (X179 @ X180 @ X181)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X178 : term] : ! [X179 : subst > term > term] : ? [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) != (X179 @ X180 @ X181))) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X384 : term] : ! [X385 : subst > term > term] : ? [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) != (X385 @ X386 @ X387))) & (! [X384 : term] : ? [X385 : subst > term > term] : ! [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) = (X385 @ X386 @ X387)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((sP53 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP53 != $true)) & ! [X177 : d_subst] : (d_id = X177) & (hoasinduction_lem1v2 = $true) & (! [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) = X176) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) != X176)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (($true = sP51) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP51)) & (axapp = $true) & (induction2lem_gthm = $true) & (ulamvarsh = $true) & (axvarshift = $true) & (ulamvarind = $true) & ((d2subst @ d_id) = id) & ((termmset = $true) | ? [X247 : term] : ((sub @ X247 @ id) != X247)) & (! [X247 : term] : ((sub @ X247 @ id) = X247) | (termmset != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP49 != $true)) & ((sP49 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & (apnotvar = $true) & (! [X150 : term] : ((sub @ X150 @ id) = X150) | ? [X149 : term] : ((sub @ X149 @ id) != X149) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X150 : term] : ((sub @ X150 @ id) != X150) & ! [X149 : term] : ((sub @ X149 @ id) = X149))) & ((sP48 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP48 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP46)) & (($true = sP46) | (hoasinduction_lem3 != $true)) & (hoasinduction_lem3_gthm = $true) & ((sP44 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP44 != $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3b = $true) | ? [X8 : term] : ! [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) != (X9 @ sh @ one))) & (! [X8 : term] : ? [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) = (X9 @ sh @ one)) | (hoasinduction_lem3b != $true)) & (hoasinduction_lem3v2_gthm = $true) & (axmap = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop = $true) & ((substmonoid_lthm = $true) | (! [X537 : subst] : ((comp @ X537 @ id) = X537) & (? [X538 : subst] : ((comp @ X538 @ id) != X538) | ? [X539 : subst] : ((comp @ id @ X539) != X539)) & ! [X536 : subst] : ((comp @ id @ X536) = X536))) & (? [X537 : subst] : ((comp @ X537 @ id) != X537) | (! [X538 : subst] : ((comp @ X538 @ id) = X538) & ! [X539 : subst] : ((comp @ id @ X539) = X539)) | ? [X536 : subst] : ((comp @ id @ X536) != X536) | (substmonoid_lthm != $true)) & (axabs = $true) & (lamnotvar = $true) & (! [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) = X240) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) != X240)) & ((sP43 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP43 != $true)) & (($true = sP39) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP39)) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) | (axscons != $true)) & ((axscons = $true) | ? [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) != X102)) & (hoasinduction_lem1_gthm = $true) & (pushprop_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X138 : subst > term > subst > $o,X139 : term] : (((X138 @ id @ X139 @ id) != $true) & ! [X145 : term,X146 : term] : (((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) = $true) | ($true != (X138 @ id @ X145 @ id)) | ((X138 @ id @ X146 @ id) != $true)) & ($true != (sP34 @ X138)))) & (! [X138 : subst > term > subst > $o,X139 : term] : (((X138 @ id @ X139 @ id) = $true) | ? [X145 : term,X146 : term] : (((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) != $true) & ($true = (X138 @ id @ X145 @ id)) & ((X138 @ id @ X146 @ id) = $true)) | ($true = (sP34 @ X138))) | (hoasinduction_no_psi_cond != $true)) & (lamnotap != $true) & ((hoasinduction = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction != $true)) & ((laminj = $true) | ? [X147 : term,X148 : term] : ((X147 != X148) & ((lam @ X147) = (lam @ X148)))) & (! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148))) | (laminj != $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((axidl = $true) | ? [X471 : subst] : ((comp @ id @ X471) != X471)) & (! [X471 : subst] : ((comp @ id @ X471) = X471) | (axidl != $true)) & ((hoasapinj2 = $true) | ? [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 != X101) & ((ap @ (sub @ X98 @ id) @ X100) = (ap @ (sub @ X99 @ id) @ X101)))) & (! [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 = X101) | ((ap @ (sub @ X98 @ id) @ X100) != (ap @ (sub @ X99 @ id) @ X101))) | (hoasapinj2 != $true)) & ((hoaslamnotap = $true) | ? [X336 : term,X337 : term,X338 : subst > term > term] : (! [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) = (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) & ((ap @ (sub @ X336 @ id) @ X337) = (lam @ (X338 @ sh @ one))))) & (! [X336 : term,X337 : term,X338 : subst > term > term] : (? [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) != (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) | ((ap @ (sub @ X336 @ id) @ X337) != (lam @ (X338 @ sh @ one)))) | (hoaslamnotap != $true)) & ((hoasinduction_lthm_3 = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lthm_3 != $true)) & (pushprop_lem3v2_lthm = $true) & ((sP24 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP24 != $true)) & ((hoasapinj2_lthm = $true) | (? [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 != X534) & ((ap @ (sub @ X533 @ id) @ X532) = (ap @ (sub @ X535 @ id) @ X534))) & ! [X528 : term,X529 : term,X530 : term,X531 : term] : (((ap @ X531 @ X529) != (ap @ X528 @ X530)) | (X529 = X530)))) & (! [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 = X534) | ((ap @ (sub @ X533 @ id) @ X532) != (ap @ (sub @ X535 @ id) @ X534))) | ? [X528 : term,X529 : term,X530 : term,X531 : term] : (((ap @ X531 @ X529) = (ap @ X528 @ X530)) & (X529 != X530)) | (hoasapinj2_lthm != $true)) & ((hoasapinj1 = $true) | ? [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 != X16) & ((ap @ (sub @ X13 @ id) @ X14) = (ap @ (sub @ X16 @ id) @ X15)))) & (! [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 = X16) | ((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15))) | (hoasapinj1 != $true)) & (hoaslamnotap_gthm = $true) & (hoasinduction_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X467 : term] : ? [X468 : subst > term > term] : ((X468 @ sh @ one) = (sub @ X467 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X467 : term] : ! [X468 : subst > term > term] : ((X468 @ sh @ one) != (sub @ X467 @ (push @ one @ sh)))) & ((sP23 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP23 != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) = X168) & ? [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (! [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) = (X170 @ X173)) & ((X170 @ one) != $true) & ((X171 @ X172) = $true)))) & (? [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) != X168) | ! [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (? [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) != (X170 @ X173)) | ((X170 @ one) = $true) | ((X171 @ X172) != $true)) | (pushprop_lem1v2_lthm != $true)) & (! [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (((X474 @ X475) != $true) | ? [X476 : term] : ((X472 @ X476) != (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) | ((X472 @ one) = $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (((X474 @ X475) = $true) & ! [X476 : term] : ((X472 @ X476) = (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) & ((X472 @ one) != $true))) & (hoaslamnotap_lthm = $true) & ((sP21 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP21 != $true)) & (pushprop_gthm = $true) & (! [X10 : subst > term > subst > $o] : ? [X11 : term > $o] : ! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X10 : subst > term > subst > $o] : ! [X11 : term > $o] : ? [X12 : term] : ((X11 @ X12) != (X10 @ id @ X12 @ id))) & (($true = sP19) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP19)) & (pushprop_lem2v2_lthm = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((d2subst @ d_id) = sh) & (hoasapinj1_gthm = $true) & (! [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) != (ap @ X243 @ X241)) | (X241 = X242)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) = (ap @ X243 @ X241)) & (X241 != X242))) & ((substmonoid = $true) | ? [X246 : subst] : ((comp @ id @ X246) != X246) | ? [X245 : subst] : ((comp @ X245 @ id) != X245)) & ((! [X246 : subst] : ((comp @ id @ X246) = X246) & ! [X245 : subst] : ((comp @ X245 @ id) = X245)) | (substmonoid != $true)) & (($true = sP18) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP18)) & ((hoasinduction_lthm = $true) | ($true != sP15)) & (($true = sP15) | (hoasinduction_lthm != $true)) & (hoasinduction_lem1 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP6 != $true)) & ((sP6 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoaslaminj = $true) | ? [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (((lam @ (X191 @ sh @ one)) = (lam @ (X190 @ sh @ one))) & ! [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) = (sub @ (X190 @ X196 @ X194) @ X195)) & ((X191 @ X192 @ X193) != (X190 @ X192 @ X193)) & ! [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) = (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199))))) & (! [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (((lam @ (X191 @ sh @ one)) != (lam @ (X190 @ sh @ one))) | ? [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) != (sub @ (X190 @ X196 @ X194) @ X195)) | ((X191 @ X192 @ X193) = (X190 @ X192 @ X193)) | ? [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) != (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199)))) | (hoaslaminj != $true)) & (! [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (($true = (X524 @ (sub @ X527 @ (push @ X525 @ X526)))) | ($true != (X523 @ X527))) | ? [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) != X521) | ? [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) != X520) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (($true != (X524 @ (sub @ X527 @ (push @ X525 @ X526)))) & ($true = (X523 @ X527))) & ! [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) = X521) & ! [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) = X520))) & (hoasinduction_lem3a_gthm = $true) & (ulamvar1 != $true) & (hoasinduction_lem2_gthm = $true) & ! [X469 : d_term,X470 : d_term] : (((d2term @ X469) != (d2term @ X470)) | (X469 = X470)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem2 != $true)) & (pushprop_lthm_orig = $true) & ! [X450 : d_term] : (d_one = X450) & ((pushprop_lem0 = $true) | ? [X37 : term > $o,X38 : term,X39 : subst] : ! [X40 : term > $o] : ? [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) != (X40 @ X41))) & (! [X37 : term > $o,X38 : term,X39 : subst] : ? [X40 : term > $o] : ! [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) = (X40 @ X41)) | (pushprop_lem0 != $true))),
% 2.65/1.76 inference(flattening,[status(thm)],[f473])).
% 2.65/1.76 thf(f473,plain,(
% 2.65/1.76 (((pushprop_lem0_lthm = $true) | ? [X116 : term,X117 : subst,X118 : term > $o] : ! [X119 : term > $o] : ? [X120 : term] : ((X119 @ X120) != (X118 @ (sub @ X120 @ (push @ X116 @ X117))))) & (! [X116 : term,X117 : subst,X118 : term > $o] : ? [X119 : term > $o] : ! [X120 : term] : ((X119 @ X120) = (X118 @ (sub @ X120 @ (push @ X116 @ X117)))) | (pushprop_lem0_lthm != $true))) & ((! [X466 : subst] : ((comp @ X466 @ id) = X466) | (axidr != $true)) & ((axidr = $true) | ? [X466 : subst] : ((comp @ X466 @ id) != X466))) & ! [X112 : d_subst,X113 : d_subst] : (((d2subst @ X112) != (d2subst @ X113)) | (X112 = X113)) & (hoasinduction_lem2_lthm = $true) & (axclos = $true) & ((($true = sP58) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP58))) & ((! [X174 : term] : ((sub @ X174 @ id) = X174) | (axvarid != $true)) & ((axvarid = $true) | ? [X174 : term] : ((sub @ X174 @ id) != X174))) & (substmonoid_gthm = $true) & (((induction = $true) | ? [X403 : term > $o,X404 : term] : (((sP56 @ X403) != $true) & ! [X407 : term] : (($true != (var @ X407)) | ($true = (X403 @ X407))) & ((X403 @ X404) != $true) & ! [X408 : term] : (((X403 @ (lam @ X408)) = $true) | ((X403 @ X408) != $true)))) & (! [X403 : term > $o,X404 : term] : (((sP56 @ X403) = $true) | ? [X407 : term] : (($true = (var @ X407)) & ($true != (X403 @ X407))) | ((X403 @ X404) = $true) | ? [X408 : term] : (((X403 @ (lam @ X408)) != $true) & ((X403 @ X408) = $true))) | (induction != $true))) & (((pushprop_lem1 = $true) | ? [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (((X133 @ (sub @ X134 @ (push @ X135 @ X136))) != $true) & ($true = (X137 @ X134)))) & (! [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (((X133 @ (sub @ X134 @ (push @ X135 @ X136))) = $true) | ($true != (X137 @ X134))) | (pushprop_lem1 != $true))) & (hoaslamnotvar_gthm = $true) & (axassoc = $true) & ((! [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) != (ap @ X276 @ X277)) | (X276 = X278)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) = (ap @ X276 @ X277)) & (X276 != X278)))) & (hoasapinj2_gthm = $true) & ! [X464 : subst] : ? [X465 : d_subst] : ((d2subst @ X465) = X464) & (((? [X104 : term,X105 : term,X106 : term,X107 : term] : ((X106 != X107) & ((ap @ X106 @ X105) = (ap @ X107 @ X104))) | ? [X103 : term] : ((sub @ X103 @ id) != X103) | ! [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 = X109) | ((ap @ (sub @ X109 @ id) @ X111) != (ap @ (sub @ X108 @ id) @ X110)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X104 : term,X105 : term,X106 : term,X107 : term] : ((X106 = X107) | ((ap @ X106 @ X105) != (ap @ X107 @ X104))) & ! [X103 : term] : ((sub @ X103 @ id) = X103) & ? [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 != X109) & ((ap @ (sub @ X109 @ id) @ X111) = (ap @ (sub @ X108 @ id) @ X110)))))) & (hoasapnotvar = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (((hoasinduction_lem3aa = $true) | ($true != sP55)) & (($true = sP55) | (hoasinduction_lem3aa != $true))) & ((var @ (d2term @ d_one)) != $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar_lthm = $true) & (pushprop_lem3v2 = $true) & (((sP54 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP54 != $true))) & (termmset_gthm = $true) & (one = (d2term @ d_one)) & ! [X114 : term] : ? [X115 : d_term] : ((d2term @ X115) = X114) & (((shinj = $true) | ? [X35 : term,X36 : term] : ((X35 != X36) & ((sub @ X36 @ sh) = (sub @ X35 @ sh)))) & (! [X35 : term,X36 : term] : ((X35 = X36) | ((sub @ X36 @ sh) != (sub @ X35 @ sh))) | (shinj != $true))) & (((hoasinduction_lem0 = $true) | ? [X333 : subst > term > subst > $o] : ! [X334 : term > $o] : ? [X335 : term] : ((X334 @ X335) != (X333 @ id @ X335 @ id))) & (! [X333 : subst > term > subst > $o] : ? [X334 : term > $o] : ! [X335 : term] : ((X334 @ X335) = (X333 @ id @ X335 @ id)) | (hoasinduction_lem0 != $true))) & (hoasinduction_lem1_lthm = $true) & (induction2_gthm = $true) & ((! [X178 : term] : ? [X179 : subst > term > term] : ! [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) = (X179 @ X180 @ X181)) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X178 : term] : ! [X179 : subst > term > term] : ? [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) != (X179 @ X180 @ X181)))) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X384 : term] : ! [X385 : subst > term > term] : ? [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) != (X385 @ X386 @ X387))) & (! [X384 : term] : ? [X385 : subst > term > term] : ! [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) = (X385 @ X386 @ X387)) | (hoasinduction_lem3v2_f_lthm != $true))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (((sP53 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP53 != $true))) & ! [X177 : d_subst] : (d_id = X177) & (hoasinduction_lem1v2 = $true) & ((! [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) = X176) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) != X176))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((($true = sP51) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP51))) & (axapp = $true) & (induction2lem_gthm = $true) & (ulamvarsh = $true) & (axvarshift = $true) & (ulamvarind = $true) & ((d2subst @ d_id) = id) & (((termmset = $true) | ? [X247 : term] : ((sub @ X247 @ id) != X247)) & (! [X247 : term] : ((sub @ X247 @ id) = X247) | (termmset != $true))) & (((hoasinduction_lem3aa_lthm = $true) | (sP49 != $true)) & ((sP49 = $true) | (hoasinduction_lem3aa_lthm != $true))) & (hoasinduction_lem3b_gthm = $true) & (apnotvar = $true) & (((! [X150 : term] : ((sub @ X150 @ id) = X150) | ? [X149 : term] : ((sub @ X149 @ id) != X149)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X150 : term] : ((sub @ X150 @ id) != X150) & ! [X149 : term] : ((sub @ X149 @ id) = X149)))) & (((sP48 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP48 != $true))) & (((hoasinduction_lem3 = $true) | ($true != sP46)) & (($true = sP46) | (hoasinduction_lem3 != $true))) & (hoasinduction_lem3_gthm = $true) & (((sP44 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP44 != $true))) & (pushprop_lem0_gthm = $true) & (((hoasinduction_lem3b = $true) | ? [X8 : term] : ! [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) != (X9 @ sh @ one))) & (! [X8 : term] : ? [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) = (X9 @ sh @ one)) | (hoasinduction_lem3b != $true))) & (hoasinduction_lem3v2_gthm = $true) & (axmap = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop = $true) & (((substmonoid_lthm = $true) | (! [X537 : subst] : ((comp @ X537 @ id) = X537) & (? [X538 : subst] : ((comp @ X538 @ id) != X538) | ? [X539 : subst] : ((comp @ id @ X539) != X539)) & ! [X536 : subst] : ((comp @ id @ X536) = X536))) & ((? [X537 : subst] : ((comp @ X537 @ id) != X537) | (! [X538 : subst] : ((comp @ X538 @ id) = X538) & ! [X539 : subst] : ((comp @ id @ X539) = X539)) | ? [X536 : subst] : ((comp @ id @ X536) != X536)) | (substmonoid_lthm != $true))) & (axabs = $true) & (lamnotvar = $true) & ((! [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) = X240) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) != X240))) & (((sP43 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP43 != $true))) & ((($true = sP39) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP39))) & (((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) | (axscons != $true)) & ((axscons = $true) | ? [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) != X102))) & (hoasinduction_lem1_gthm = $true) & (pushprop_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotvar = $true) & (((hoasinduction_no_psi_cond = $true) | ? [X138 : subst > term > subst > $o,X139 : term] : (((X138 @ id @ X139 @ id) != $true) & ! [X145 : term,X146 : term] : (((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) = $true) | ($true != (X138 @ id @ X145 @ id)) | ((X138 @ id @ X146 @ id) != $true)) & ($true != (sP34 @ X138)))) & (! [X138 : subst > term > subst > $o,X139 : term] : (((X138 @ id @ X139 @ id) = $true) | ? [X145 : term,X146 : term] : (((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) != $true) & ($true = (X138 @ id @ X145 @ id)) & ((X138 @ id @ X146 @ id) = $true)) | ($true = (sP34 @ X138))) | (hoasinduction_no_psi_cond != $true))) & (lamnotap != $true) & (((hoasinduction = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction != $true))) & (((laminj = $true) | ? [X147 : term,X148 : term] : ((X147 != X148) & ((lam @ X147) = (lam @ X148)))) & (! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148))) | (laminj != $true))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (((axidl = $true) | ? [X471 : subst] : ((comp @ id @ X471) != X471)) & (! [X471 : subst] : ((comp @ id @ X471) = X471) | (axidl != $true))) & (((hoasapinj2 = $true) | ? [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 != X101) & ((ap @ (sub @ X98 @ id) @ X100) = (ap @ (sub @ X99 @ id) @ X101)))) & (! [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 = X101) | ((ap @ (sub @ X98 @ id) @ X100) != (ap @ (sub @ X99 @ id) @ X101))) | (hoasapinj2 != $true))) & (((hoaslamnotap = $true) | ? [X336 : term,X337 : term,X338 : subst > term > term] : (! [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) = (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) & ((ap @ (sub @ X336 @ id) @ X337) = (lam @ (X338 @ sh @ one))))) & (! [X336 : term,X337 : term,X338 : subst > term > term] : (? [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) != (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) | ((ap @ (sub @ X336 @ id) @ X337) != (lam @ (X338 @ sh @ one)))) | (hoaslamnotap != $true))) & (((hoasinduction_lthm_3 = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lthm_3 != $true))) & (pushprop_lem3v2_lthm = $true) & (((sP24 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP24 != $true))) & (((hoasapinj2_lthm = $true) | (? [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 != X534) & ((ap @ (sub @ X533 @ id) @ X532) = (ap @ (sub @ X535 @ id) @ X534))) & ! [X528 : term,X529 : term,X530 : term,X531 : term] : (((ap @ X531 @ X529) != (ap @ X528 @ X530)) | (X529 = X530)))) & ((! [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 = X534) | ((ap @ (sub @ X533 @ id) @ X532) != (ap @ (sub @ X535 @ id) @ X534))) | ? [X528 : term,X529 : term,X530 : term,X531 : term] : (((ap @ X531 @ X529) = (ap @ X528 @ X530)) & (X529 != X530))) | (hoasapinj2_lthm != $true))) & (((hoasapinj1 = $true) | ? [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 != X16) & ((ap @ (sub @ X13 @ id) @ X14) = (ap @ (sub @ X16 @ id) @ X15)))) & (! [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 = X16) | ((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15))) | (hoasapinj1 != $true))) & (hoaslamnotap_gthm = $true) & (hoasinduction_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((! [X467 : term] : ? [X468 : subst > term > term] : ((X468 @ sh @ one) = (sub @ X467 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X467 : term] : ! [X468 : subst > term > term] : ((X468 @ sh @ one) != (sub @ X467 @ (push @ one @ sh))))) & (((sP23 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP23 != $true))) & (((pushprop_lem1v2_lthm = $true) | (! [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) = X168) & ? [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (! [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) = (X170 @ X173)) & ((X170 @ one) != $true) & ((X171 @ X172) = $true)))) & ((? [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) != X168) | ! [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (? [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) != (X170 @ X173)) | ((X170 @ one) = $true) | ((X171 @ X172) != $true))) | (pushprop_lem1v2_lthm != $true))) & ((! [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (((X474 @ X475) != $true) | ? [X476 : term] : ((X472 @ X476) != (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) | ((X472 @ one) = $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (((X474 @ X475) = $true) & ! [X476 : term] : ((X472 @ X476) = (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) & ((X472 @ one) != $true)))) & (hoaslamnotap_lthm = $true) & (((sP21 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP21 != $true))) & (pushprop_gthm = $true) & ((! [X10 : subst > term > subst > $o] : ? [X11 : term > $o] : ! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X10 : subst > term > subst > $o] : ! [X11 : term > $o] : ? [X12 : term] : ((X11 @ X12) != (X10 @ id @ X12 @ id)))) & ((($true = sP19) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP19))) & (pushprop_lem2v2_lthm = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((d2subst @ d_id) = sh) & (hoasapinj1_gthm = $true) & ((! [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) != (ap @ X243 @ X241)) | (X241 = X242)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) = (ap @ X243 @ X241)) & (X241 != X242)))) & (((substmonoid = $true) | (? [X246 : subst] : ((comp @ id @ X246) != X246) | ? [X245 : subst] : ((comp @ X245 @ id) != X245))) & ((! [X246 : subst] : ((comp @ id @ X246) = X246) & ! [X245 : subst] : ((comp @ X245 @ id) = X245)) | (substmonoid != $true))) & ((($true = sP18) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP18))) & (((hoasinduction_lthm = $true) | ($true != sP15)) & (($true = sP15) | (hoasinduction_lthm != $true))) & (hoasinduction_lem1 = $true) & (((hoasinduction_no_psi_cond_lthm = $true) | (sP6 != $true)) & ((sP6 = $true) | (hoasinduction_no_psi_cond_lthm != $true))) & (((hoaslaminj = $true) | ? [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (((lam @ (X191 @ sh @ one)) = (lam @ (X190 @ sh @ one))) & ! [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) = (sub @ (X190 @ X196 @ X194) @ X195)) & ((X191 @ X192 @ X193) != (X190 @ X192 @ X193)) & ! [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) = (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199))))) & (! [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (((lam @ (X191 @ sh @ one)) != (lam @ (X190 @ sh @ one))) | ? [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) != (sub @ (X190 @ X196 @ X194) @ X195)) | ((X191 @ X192 @ X193) = (X190 @ X192 @ X193)) | ? [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) != (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199)))) | (hoaslaminj != $true))) & (((! [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (($true = (X524 @ (sub @ X527 @ (push @ X525 @ X526)))) | ($true != (X523 @ X527))) | ? [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) != X521) | ? [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) != X520)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (($true != (X524 @ (sub @ X527 @ (push @ X525 @ X526)))) & ($true = (X523 @ X527))) & ! [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) = X521) & ! [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) = X520)))) & (hoasinduction_lem3a_gthm = $true) & (ulamvar1 != $true) & (hoasinduction_lem2_gthm = $true) & ! [X469 : d_term,X470 : d_term] : (((d2term @ X469) != (d2term @ X470)) | (X469 = X470)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & (((hoasinduction_lem2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem2 != $true))) & (pushprop_lthm_orig = $true) & ! [X450 : d_term] : (d_one = X450) & (((pushprop_lem0 = $true) | ? [X37 : term > $o,X38 : term,X39 : subst] : ! [X40 : term > $o] : ? [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) != (X40 @ X41))) & (! [X37 : term > $o,X38 : term,X39 : subst] : ? [X40 : term > $o] : ! [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) = (X40 @ X41)) | (pushprop_lem0 != $true)))),
% 2.65/1.76 inference(nnf_transformation,[status(thm)],[f73])).
% 2.65/1.76 thf(f73,plain,(
% 2.65/1.76 ((pushprop_lem0_lthm = $true) <=> ! [X116 : term,X117 : subst,X118 : term > $o] : ? [X119 : term > $o] : ! [X120 : term] : ((X119 @ X120) = (X118 @ (sub @ X120 @ (push @ X116 @ X117))))) & (! [X466 : subst] : ((comp @ X466 @ id) = X466) <=> (axidr = $true)) & ! [X112 : d_subst,X113 : d_subst] : (((d2subst @ X112) != (d2subst @ X113)) | (X112 = X113)) & (hoasinduction_lem2_lthm = $true) & (axclos = $true) & (($true = sP58) <=> (induction2lem = $true)) & (! [X174 : term] : ((sub @ X174 @ id) = X174) <=> (axvarid = $true)) & (substmonoid_gthm = $true) & ((induction = $true) <=> ! [X403 : term > $o,X404 : term] : (((sP56 @ X403) = $true) | ? [X407 : term] : (($true = (var @ X407)) & ($true != (X403 @ X407))) | ((X403 @ X404) = $true) | ? [X408 : term] : (((X403 @ (lam @ X408)) != $true) & ((X403 @ X408) = $true)))) & ((pushprop_lem1 = $true) <=> ! [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (((X133 @ (sub @ X134 @ (push @ X135 @ X136))) = $true) | ($true != (X137 @ X134)))) & (hoaslamnotvar_gthm = $true) & (axassoc = $true) & (! [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) != (ap @ X276 @ X277)) | (X276 = X278)) <=> (apinj1 = $true)) & (hoasapinj2_gthm = $true) & ! [X464 : subst] : ? [X465 : d_subst] : ((d2subst @ X465) = X464) & ((? [X104 : term,X105 : term,X106 : term,X107 : term] : ((X106 != X107) & ((ap @ X106 @ X105) = (ap @ X107 @ X104))) | ? [X103 : term] : ((sub @ X103 @ id) != X103) | ! [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 = X109) | ((ap @ (sub @ X109 @ id) @ X111) != (ap @ (sub @ X108 @ id) @ X110)))) <=> (hoasapinj1_lthm = $true)) & (hoasapnotvar = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem3aa = $true) <=> ($true = sP55)) & ((var @ (d2term @ d_one)) != $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar_lthm = $true) & (pushprop_lem3v2 = $true) & ((sP54 = $true) <=> (hoasinduction_lem3v2a = $true)) & (termmset_gthm = $true) & (one = (d2term @ d_one)) & ! [X114 : term] : ? [X115 : d_term] : ((d2term @ X115) = X114) & ((shinj = $true) <=> ! [X35 : term,X36 : term] : ((X35 = X36) | ((sub @ X36 @ sh) != (sub @ X35 @ sh)))) & ((hoasinduction_lem0 = $true) <=> ! [X333 : subst > term > subst > $o] : ? [X334 : term > $o] : ! [X335 : term] : ((X334 @ X335) = (X333 @ id @ X335 @ id))) & (hoasinduction_lem1_lthm = $true) & (induction2_gthm = $true) & (! [X178 : term] : ? [X179 : subst > term > term] : ! [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) = (X179 @ X180 @ X181)) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X384 : term] : ? [X385 : subst > term > term] : ! [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) = (X385 @ X386 @ X387))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((sP53 = $true) <=> (hoasinduction_lem3a_lthm = $true)) & ! [X177 : d_subst] : (d_id = X177) & (hoasinduction_lem1v2 = $true) & (! [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) = X176) <=> (axvarcons = $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (($true = sP51) <=> (hoasinduction_lem3v2_lthm = $true)) & (axapp = $true) & (induction2lem_gthm = $true) & (ulamvarsh = $true) & (axvarshift = $true) & (ulamvarind = $true) & ((d2subst @ d_id) = id) & ((termmset = $true) <=> ! [X247 : term] : ((sub @ X247 @ id) = X247)) & ((hoasinduction_lem3aa_lthm = $true) <=> (sP49 = $true)) & (hoasinduction_lem3b_gthm = $true) & (apnotvar = $true) & ((! [X150 : term] : ((sub @ X150 @ id) = X150) | ? [X149 : term] : ((sub @ X149 @ id) != X149)) <=> (termmset_lthm = $true)) & ((sP48 = $true) <=> (hoasinduction_lem3v2 = $true)) & ((hoasinduction_lem3 = $true) <=> ($true = sP46)) & (hoasinduction_lem3_gthm = $true) & ((sP44 = $true) <=> (hoaslaminj_lthm = $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3b = $true) <=> ! [X8 : term] : ? [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) = (X9 @ sh @ one))) & (hoasinduction_lem3v2_gthm = $true) & (axmap = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop = $true) & ((substmonoid_lthm = $true) <=> (? [X537 : subst] : ((comp @ X537 @ id) != X537) | (! [X538 : subst] : ((comp @ X538 @ id) = X538) & ! [X539 : subst] : ((comp @ id @ X539) = X539)) | ? [X536 : subst] : ((comp @ id @ X536) != X536))) & (axabs = $true) & (lamnotvar = $true) & (! [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) = X240) <=> (axshiftcons = $true)) & ((sP43 = $true) <=> (induction2lem_lthm = $true)) & (($true = sP39) <=> (hoasinduction_lem3aaa = $true)) & ((sP38 = $true) <=> (induction2_lthm = $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & (hoasinduction_lem1_gthm = $true) & (pushprop_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X138 : subst > term > subst > $o,X139 : term] : (((X138 @ id @ X139 @ id) = $true) | ? [X145 : term,X146 : term] : (((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) != $true) & ($true = (X138 @ id @ X145 @ id)) & ((X138 @ id @ X146 @ id) = $true)) | ($true = (sP34 @ X138)))) & (lamnotap != $true) & ((hoasinduction = $true) <=> (sP33 = $true)) & ((laminj = $true) <=> ! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148)))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((axidl = $true) <=> ! [X471 : subst] : ((comp @ id @ X471) = X471)) & ((hoasapinj2 = $true) <=> ! [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 = X101) | ((ap @ (sub @ X98 @ id) @ X100) != (ap @ (sub @ X99 @ id) @ X101)))) & ((hoaslamnotap = $true) <=> ! [X336 : term,X337 : term,X338 : subst > term > term] : (? [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) != (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) | ((ap @ (sub @ X336 @ id) @ X337) != (lam @ (X338 @ sh @ one))))) & ((hoasinduction_lthm_3 = $true) <=> (sP30 = $true)) & (pushprop_lem3v2_lthm = $true) & ((sP24 = $true) <=> (hoasinduction_lem3a = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 = X534) | ((ap @ (sub @ X533 @ id) @ X532) != (ap @ (sub @ X535 @ id) @ X534))) | ? [X528 : term,X529 : term,X530 : term,X531 : term] : (((ap @ X531 @ X529) = (ap @ X528 @ X530)) & (X529 != X530)))) & ((hoasapinj1 = $true) <=> ! [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 = X16) | ((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15)))) & (hoaslamnotap_gthm = $true) & (hoasinduction_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X467 : term] : ? [X468 : subst > term > term] : ((X468 @ sh @ one) = (sub @ X467 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & ((sP23 = $true) <=> (induction2 = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (? [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) != X168) | ! [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (? [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) != (X170 @ X173)) | ((X170 @ one) = $true) | ((X171 @ X172) != $true)))) & (! [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (((X474 @ X475) != $true) | ? [X476 : term] : ((X472 @ X476) != (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) | ((X472 @ one) = $true)) <=> (pushprop_lem1v2 = $true)) & (hoaslamnotap_lthm = $true) & ((sP21 = $true) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_gthm = $true) & (! [X10 : subst > term > subst > $o] : ? [X11 : term > $o] : ! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (($true = sP19) <=> (hoasinduction_lem3v2a_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((d2subst @ d_id) = sh) & (hoasapinj1_gthm = $true) & (! [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) != (ap @ X243 @ X241)) | (X241 = X242)) <=> (apinj2 = $true)) & ((substmonoid = $true) <=> (! [X246 : subst] : ((comp @ id @ X246) = X246) & ! [X245 : subst] : ((comp @ X245 @ id) = X245))) & (($true = sP18) <=> (hoasinduction_lem3_lthm = $true)) & ((hoasinduction_lthm = $true) <=> ($true = sP15)) & (hoasinduction_lem1 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (sP6 = $true)) & ((hoaslaminj = $true) <=> ! [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (((lam @ (X191 @ sh @ one)) != (lam @ (X190 @ sh @ one))) | ? [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) != (sub @ (X190 @ X196 @ X194) @ X195)) | ((X191 @ X192 @ X193) = (X190 @ X192 @ X193)) | ? [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) != (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199))))) & ((! [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (($true = (X524 @ (sub @ X527 @ (push @ X525 @ X526)))) | ($true != (X523 @ X527))) | ? [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) != X521) | ? [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) != X520)) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (ulamvar1 != $true) & (hoasinduction_lem2_gthm = $true) & ! [X469 : d_term,X470 : d_term] : (((d2term @ X469) != (d2term @ X470)) | (X469 = X470)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem2 = $true) <=> (sP1 = $true)) & (pushprop_lthm_orig = $true) & ! [X450 : d_term] : (d_one = X450) & ((pushprop_lem0 = $true) <=> ! [X37 : term > $o,X38 : term,X39 : subst] : ? [X40 : term > $o] : ! [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) = (X40 @ X41)))),
% 2.65/1.76 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])).
% 2.65/1.76 thf(f14,plain,(
% 2.65/1.76 ! [X451 : subst > term > subst > $o] : (((sP0 @ X451) = $true) <=> ? [X458 : term,X459 : term] : (($true != (X451 @ id @ (ap @ (sub @ X458 @ id) @ X459) @ id)) & ($true = (X451 @ id @ X459 @ id)) & ((X451 @ id @ X458 @ id) = $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f15,plain,(
% 2.65/1.76 (sP1 = $true) <=> ! [X451 : subst > term > subst > $o,X452 : term,X453 : term] : (? [X460 : subst,X461 : subst,X462 : term,X463 : subst] : (((X451 @ X460 @ X462 @ (comp @ X463 @ X461)) != $true) & ((X451 @ (comp @ X460 @ X463) @ (sub @ X462 @ X463) @ X461) = $true)) | ((X451 @ id @ X453 @ id) != $true) | ((sP0 @ X451) = $true) | ? [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (((X451 @ X457 @ X456 @ (comp @ X455 @ X454)) = $true) & ($true != (X451 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454))) | ((X451 @ id @ (ap @ X453 @ X452) @ id) = $true) | ((X451 @ id @ X452 @ id) != $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f16,plain,(
% 2.65/1.76 ! [X70 : subst > term > subst > $o] : (((sP2 @ X70) = $true) <=> ? [X77 : term,X78 : term] : (($true = (X70 @ id @ X77 @ id)) & ((X70 @ id @ (ap @ (sub @ X78 @ id) @ X77) @ id) != $true) & ((X70 @ id @ X78 @ id) = $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f17,plain,(
% 2.65/1.76 ! [X52 : term > $o] : (((sP3 @ X52) = $true) <=> ! [X54 : term,X55 : term] : (($true != (X52 @ X54)) | ((X52 @ (ap @ X55 @ X54)) = $true) | ((X52 @ X55) != $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f18,plain,(
% 2.65/1.76 (sP4 = $true) <=> ? [X60 : subst > term > subst > $o,X61 : term > $o,X62 : term] : (! [X68 : term] : (($true = (X61 @ (sub @ X62 @ (push @ X68 @ id)))) | ((X61 @ X68) != $true)) & ! [X69 : term] : ((X60 @ id @ X69 @ id) = (X61 @ X69)) & ! [X63 : subst > term > term] : (($true = (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) | ? [X65 : subst,X66 : subst,X67 : term] : ((sub @ (X63 @ X66 @ X67) @ X65) != (X63 @ (comp @ X66 @ X65) @ (sub @ X67 @ X65))) | ? [X64 : term] : (((X60 @ id @ X64 @ id) = $true) & ((X60 @ id @ (X63 @ id @ X64) @ id) != $true))) & ((X61 @ (lam @ X62)) != $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f19,plain,(
% 2.65/1.76 (sP5 = $true) <=> ! [X70 : subst > term > subst > $o,X71 : term] : (? [X72 : subst > term > term] : (! [X76 : term] : (((X70 @ id @ (X72 @ id @ X76) @ id) = $true) | ($true != (X70 @ id @ X76 @ id))) & ! [X73 : subst,X74 : term,X75 : subst] : ((X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73)) = (sub @ (X72 @ X75 @ X74) @ X73)) & ((X70 @ id @ (lam @ (X72 @ sh @ one)) @ id) != $true)) | ((sP2 @ X70) = $true) | ($true = (X70 @ id @ X71 @ id)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f20,plain,(
% 2.65/1.76 (sP6 = $true) <=> (? [X52 : term > $o,X53 : term] : (((sP3 @ X52) = $true) & ! [X57 : term] : (? [X58 : term] : (($true = (X52 @ X58)) & ((X52 @ (sub @ X57 @ (push @ X58 @ id))) != $true)) | ((X52 @ (lam @ X57)) = $true)) & ! [X56 : term] : (((var @ X56) != $true) | ($true = (X52 @ X56))) & ((X52 @ X53) != $true)) | (sP5 = $true) | ? [X49 : subst > term > subst > $o] : ! [X50 : term > $o] : ? [X51 : term] : ((X49 @ id @ X51 @ id) != (X50 @ X51)) | ? [X59 : term] : ((sub @ X59 @ id) != X59) | (sP4 = $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f21,plain,(
% 2.65/1.76 ! [X287 : subst > term > subst > $o] : (((sP7 @ X287) = $true) <=> ! [X293 : term,X294 : term] : (((X287 @ id @ X294 @ id) != $true) | ((X287 @ id @ (ap @ (sub @ X293 @ id) @ X294) @ id) = $true) | ($true != (X287 @ id @ X293 @ id))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f22,plain,(
% 2.65/1.76 ! [X280 : term > $o] : (($true = (sP8 @ X280)) <=> ! [X284 : term,X285 : term] : (($true != (X280 @ X285)) | ((X280 @ (ap @ X285 @ X284)) = $true) | ($true != (X280 @ X284))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f23,plain,(
% 2.65/1.76 ! [X300 : subst > term > subst > $o] : (($true = (sP9 @ X300)) <=> ! [X310 : subst > term > term] : (? [X312 : term,X313 : subst,X314 : subst] : ((X310 @ (comp @ X314 @ X313) @ (sub @ X312 @ X313)) != (sub @ (X310 @ X314 @ X312) @ X313)) | ((X300 @ id @ (lam @ (X310 @ sh @ one)) @ id) = $true) | ? [X311 : term] : (((X300 @ id @ (X310 @ id @ X311) @ id) != $true) & ($true = (X300 @ id @ X311 @ id)))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f24,plain,(
% 2.65/1.76 ! [X316 : subst > term > subst > $o] : (($true = (sP10 @ X316)) <=> ? [X317 : term,X318 : term] : (((X316 @ id @ X318 @ id) = $true) & ((X316 @ id @ X317 @ id) = $true) & ($true != (X316 @ id @ (ap @ (sub @ X318 @ id) @ X317) @ id))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f25,plain,(
% 2.65/1.76 ! [X316 : subst > term > subst > $o] : (((sP11 @ X316) = $true) <=> ? [X319 : subst > term > term] : (! [X323 : term] : (((X316 @ id @ (X319 @ id @ X323) @ id) = $true) | ((X316 @ id @ X323 @ id) != $true)) & ! [X320 : term,X321 : subst,X322 : subst] : ((sub @ (X319 @ X321 @ X320) @ X322) = (X319 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322))) & ($true != (X316 @ id @ (lam @ (X319 @ sh @ one)) @ id))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f26,plain,(
% 2.65/1.76 (sP12 = $true) <=> ? [X299 : term,X300 : subst > term > subst > $o] : (! [X302 : subst,X303 : subst,X304 : term,X305 : subst] : (((X300 @ (comp @ X305 @ X302) @ (sub @ X304 @ X302) @ X303) = $true) | ((X300 @ X305 @ X304 @ (comp @ X302 @ X303)) != $true)) & ((X300 @ id @ (lam @ X299) @ id) != $true) & ! [X301 : term] : (($true != (X300 @ id @ X301 @ id)) | ((X300 @ id @ (sub @ X299 @ (push @ X301 @ id)) @ id) = $true)) & ($true = (sP9 @ X300)) & ! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (((X300 @ (comp @ X307 @ X308) @ (sub @ X306 @ X308) @ X309) != $true) | ($true = (X300 @ X307 @ X306 @ (comp @ X308 @ X309)))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f27,plain,(
% 2.65/1.76 (sP13 = $true) <=> ! [X315 : term,X316 : subst > term > subst > $o] : (? [X328 : subst,X329 : subst,X330 : subst,X331 : term] : (((X316 @ X328 @ X331 @ (comp @ X329 @ X330)) != $true) & ((X316 @ (comp @ X328 @ X329) @ (sub @ X331 @ X329) @ X330) = $true)) | ((sP11 @ X316) = $true) | ? [X324 : subst,X325 : term,X326 : subst,X327 : subst] : (($true = (X316 @ X324 @ X325 @ (comp @ X327 @ X326))) & ((X316 @ (comp @ X324 @ X327) @ (sub @ X325 @ X327) @ X326) != $true)) | ((X316 @ id @ X315 @ id) = $true) | ? [X332 : term] : (((var @ (sub @ X332 @ id)) = $true) & ((X316 @ id @ X332 @ id) != $true)) | ($true = (sP10 @ X316)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f28,plain,(
% 2.65/1.76 (sP14 = $true) <=> ? [X286 : term,X287 : subst > term > subst > $o,X288 : term] : (! [X295 : subst,X296 : subst,X297 : term,X298 : subst] : (($true = (X287 @ X295 @ X297 @ (comp @ X296 @ X298))) | ((X287 @ (comp @ X295 @ X296) @ (sub @ X297 @ X296) @ X298) != $true)) & ($true = (X287 @ id @ X286 @ id)) & ((X287 @ id @ X288 @ id) = $true) & ! [X289 : subst,X290 : subst,X291 : term,X292 : subst] : (((X287 @ (comp @ X289 @ X290) @ (sub @ X291 @ X290) @ X292) = $true) | ($true != (X287 @ X289 @ X291 @ (comp @ X290 @ X292)))) & ($true != (X287 @ id @ (ap @ X286 @ X288) @ id)) & ((sP7 @ X287) = $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f29,plain,(
% 2.65/1.76 ($true = sP15) <=> ((sP13 = $true) | (sP12 = $true) | ? [X279 : term,X280 : term > $o] : (((X280 @ X279) != $true) & ! [X282 : term] : (? [X283 : term] : (($true != (X280 @ (sub @ X282 @ (push @ X283 @ id)))) & ((X280 @ X283) = $true)) | ((X280 @ (lam @ X282)) = $true)) & ($true = (sP8 @ X280)) & ! [X281 : term] : (((X280 @ X281) = $true) | ($true != (var @ X281)))) | (sP14 = $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f30,plain,(
% 2.65/1.76 ! [X419 : subst > term > subst > $o] : (($true = (sP16 @ X419)) <=> ? [X428 : subst > term > term] : (($true != (X419 @ id @ (lam @ (X428 @ sh @ one)) @ id)) & ! [X429 : term,X430 : subst,X431 : subst] : ((sub @ (X428 @ X431 @ X429) @ X430) = (X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430))) & ! [X432 : term] : (($true = (X419 @ id @ (X428 @ id @ X432) @ id)) | ((X419 @ id @ X432 @ id) != $true))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f31,plain,(
% 2.65/1.76 (sP17 = $true) <=> ? [X410 : term,X411 : subst > term > subst > $o] : (! [X417 : term] : (($true = (X411 @ id @ (sub @ X410 @ (push @ X417 @ id)) @ id)) | ((X411 @ id @ X417 @ id) != $true)) & ((X411 @ id @ (lam @ (sub @ X410 @ (push @ one @ sh))) @ id) != $true) & ! [X412 : subst > term > term] : (? [X414 : subst,X415 : subst,X416 : term] : ((X412 @ (comp @ X414 @ X415) @ (sub @ X416 @ X415)) != (sub @ (X412 @ X414 @ X416) @ X415)) | ? [X413 : term] : (((X411 @ id @ (X412 @ id @ X413) @ id) != $true) & ($true = (X411 @ id @ X413 @ id))) | ((X411 @ id @ (lam @ (X412 @ sh @ one)) @ id) = $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f32,plain,(
% 2.65/1.76 ($true = sP18) <=> (? [X409 : term] : ((sub @ X409 @ id) != X409) | ! [X418 : term,X419 : subst > term > subst > $o] : (? [X420 : subst,X421 : subst,X422 : term,X423 : subst] : (((X419 @ X423 @ X422 @ (comp @ X421 @ X420)) != $true) & ((X419 @ (comp @ X423 @ X421) @ (sub @ X422 @ X421) @ X420) = $true)) | ? [X433 : term] : (((X419 @ id @ (sub @ X418 @ (push @ X433 @ id)) @ id) != $true) & ($true = (X419 @ id @ X433 @ id))) | ($true = (X419 @ id @ (lam @ X418) @ id)) | ($true = (sP16 @ X419)) | ? [X424 : subst,X425 : subst,X426 : subst,X427 : term] : (($true != (X419 @ (comp @ X426 @ X425) @ (sub @ X427 @ X425) @ X424)) & ((X419 @ X426 @ X427 @ (comp @ X425 @ X424)) = $true))) | (sP17 = $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f33,plain,(
% 2.65/1.76 ($true = sP19) <=> (! [X393 : term,X394 : subst > term > subst > $o,X395 : term > $o] : (? [X402 : term] : ((X395 @ X402) != (X394 @ id @ X402 @ id)) | ? [X397 : subst > term > term] : (! [X399 : term,X400 : subst,X401 : subst] : ((sub @ (X397 @ X400 @ X399) @ X401) = (X397 @ (comp @ X400 @ X401) @ (sub @ X399 @ X401))) & ! [X398 : term] : (((X394 @ id @ X398 @ id) != $true) | ((X394 @ id @ (X397 @ id @ X398) @ id) = $true)) & ((X394 @ id @ (lam @ (X397 @ sh @ one)) @ id) != $true)) | ((X395 @ (lam @ X393)) = $true) | ? [X396 : term] : (((X395 @ (sub @ X393 @ (push @ X396 @ id))) != $true) & ((X395 @ X396) = $true))) | ? [X388 : term] : ! [X389 : subst > term > term] : ? [X390 : term,X391 : subst] : ((sub @ X388 @ (push @ X390 @ X391)) != (X389 @ X391 @ X390)) | ? [X392 : term] : ((sub @ X392 @ id) != X392))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f34,plain,(
% 2.65/1.76 ! [X370 : subst > term > subst > $o] : (((sP20 @ X370) = $true) <=> ? [X382 : term,X383 : term] : (((X370 @ id @ (ap @ (sub @ X383 @ id) @ X382) @ id) != $true) & ((X370 @ id @ X383 @ id) = $true) & ((X370 @ id @ X382 @ id) = $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f35,plain,(
% 2.65/1.76 (sP21 = $true) <=> ! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term,X372 : term] : (((sP20 @ X370) = $true) | ($true != (X369 @ X371)) | ((X369 @ X372) != $true) | ((X369 @ (ap @ X372 @ X371)) = $true) | ? [X373 : subst,X374 : subst,X375 : term,X376 : subst] : (((X370 @ (comp @ X374 @ X373) @ (sub @ X375 @ X373) @ X376) != $true) & ((X370 @ X374 @ X375 @ (comp @ X373 @ X376)) = $true)) | ? [X378 : term,X379 : subst,X380 : subst,X381 : subst] : (($true != (X370 @ X380 @ X378 @ (comp @ X381 @ X379))) & ((X370 @ (comp @ X380 @ X381) @ (sub @ X378 @ X381) @ X379) = $true)) | ? [X377 : term] : ((X369 @ X377) != (X370 @ id @ X377 @ id)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f36,plain,(
% 2.65/1.76 ! [X42 : term > $o] : (((sP22 @ X42) = $true) <=> ? [X47 : term,X48 : term] : (($true != (X42 @ (ap @ X47 @ X48))) & ($true = (X42 @ X47)) & ((X42 @ X48) = $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f37,plain,(
% 2.65/1.76 (sP23 = $true) <=> ! [X42 : term > $o,X43 : term] : (? [X44 : term] : (((var @ X44) = $true) & ((X42 @ X44) != $true)) | ((sP22 @ X42) = $true) | ? [X45 : term] : (((X42 @ (lam @ X45)) != $true) & ! [X46 : term] : (($true != (X42 @ X46)) | ($true = (X42 @ (sub @ X45 @ (push @ X46 @ id)))))) | ((X42 @ X43) = $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f38,plain,(
% 2.65/1.76 (sP24 = $true) <=> ! [X511 : term,X512 : subst > term > subst > $o] : (? [X518 : term] : (((X512 @ id @ (sub @ X511 @ (push @ X518 @ id)) @ id) != $true) & ((X512 @ id @ X518 @ id) = $true)) | ((X512 @ id @ (lam @ X511) @ id) = $true) | ? [X513 : subst > term > term] : (((X512 @ id @ (lam @ (X513 @ sh @ one)) @ id) != $true) & ! [X514 : subst,X515 : term,X516 : subst] : ((X513 @ (comp @ X514 @ X516) @ (sub @ X515 @ X516)) = (sub @ (X513 @ X514 @ X515) @ X516)) & ! [X517 : term] : (($true = (X512 @ id @ (X513 @ id @ X517) @ id)) | ((X512 @ id @ X517 @ id) != $true))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f39,plain,(
% 2.65/1.76 ! [X204 : term > $o] : (($true = (sP25 @ X204)) <=> ! [X205 : term,X206 : term] : (($true != (X204 @ X206)) | ((X204 @ X205) != $true) | ((X204 @ (ap @ X206 @ X205)) = $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f40,plain,(
% 2.65/1.76 ! [X222 : subst > term > subst > $o] : (((sP26 @ X222) = $true) <=> ? [X234 : subst > term > term] : (((X222 @ id @ (lam @ (X234 @ sh @ one)) @ id) != $true) & ! [X236 : term,X237 : subst,X238 : subst] : ((X234 @ (comp @ X238 @ X237) @ (sub @ X236 @ X237)) = (sub @ (X234 @ X238 @ X236) @ X237)) & ! [X235 : term] : (($true = (X222 @ id @ (X234 @ id @ X235) @ id)) | ((X222 @ id @ X235 @ id) != $true))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f41,plain,(
% 2.65/1.76 ! [X222 : subst > term > subst > $o] : (((sP27 @ X222) = $true) <=> ? [X227 : term,X228 : term] : (((X222 @ id @ X227 @ id) = $true) & ((X222 @ id @ X228 @ id) = $true) & ((X222 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id) != $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f42,plain,(
% 2.65/1.76 (sP28 = $true) <=> ! [X221 : term,X222 : subst > term > subst > $o] : (? [X223 : subst,X224 : term,X225 : subst,X226 : subst] : (($true = (X222 @ X225 @ X224 @ (comp @ X226 @ X223))) & ((X222 @ (comp @ X225 @ X226) @ (sub @ X224 @ X226) @ X223) != $true)) | ? [X229 : term] : (($true != (X222 @ id @ X229 @ id)) & ((var @ (sub @ X229 @ id)) = $true)) | ((X222 @ id @ X221 @ id) = $true) | ((sP27 @ X222) = $true) | ((sP26 @ X222) = $true) | ? [X230 : subst,X231 : subst,X232 : term,X233 : subst] : (((X222 @ X233 @ X232 @ (comp @ X230 @ X231)) != $true) & ($true = (X222 @ (comp @ X233 @ X230) @ (sub @ X232 @ X230) @ X231))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f43,plain,(
% 2.65/1.76 ($true = sP29) <=> ? [X211 : term > $o,X212 : term,X213 : subst > term > subst > $o] : (! [X216 : subst > term > term] : (((X213 @ id @ (lam @ (X216 @ sh @ one)) @ id) = $true) | ? [X220 : term] : (((X213 @ id @ X220 @ id) = $true) & ((X213 @ id @ (X216 @ id @ X220) @ id) != $true)) | ? [X217 : term,X218 : subst,X219 : subst] : ((sub @ (X216 @ X219 @ X217) @ X218) != (X216 @ (comp @ X219 @ X218) @ (sub @ X217 @ X218)))) & ! [X214 : term] : ((X213 @ id @ X214 @ id) = (X211 @ X214)) & ($true != (X211 @ (lam @ X212))) & ! [X215 : term] : (($true != (X211 @ X215)) | ($true = (X211 @ (sub @ X212 @ (push @ X215 @ id))))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f44,plain,(
% 2.65/1.76 (sP30 = $true) <=> (($true = sP29) | (sP28 = $true) | ? [X210 : term] : ((sub @ X210 @ id) != X210) | ? [X203 : term,X204 : term > $o] : (($true != (X204 @ X203)) & ! [X207 : term] : (? [X208 : term] : (($true != (X204 @ (sub @ X207 @ (push @ X208 @ id)))) & ($true = (X204 @ X208))) | ($true = (X204 @ (lam @ X207)))) & ! [X209 : term] : (((X204 @ X209) = $true) | ((var @ X209) != $true)) & ($true = (sP25 @ X204))) | ? [X200 : subst > term > subst > $o] : ! [X201 : term > $o] : ? [X202 : term] : ((X201 @ X202) != (X200 @ id @ X202 @ id)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f45,plain,(
% 2.65/1.76 ! [X494 : subst > term > subst > $o] : (($true = (sP31 @ X494)) <=> ? [X501 : subst > term > term] : (($true != (X494 @ id @ (lam @ (X501 @ sh @ one)) @ id)) & ! [X505 : term] : (((X494 @ id @ X505 @ id) != $true) | ((X494 @ id @ (X501 @ id @ X505) @ id) = $true)) & ! [X502 : subst,X503 : term,X504 : subst] : ((X501 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X501 @ X502 @ X503) @ X504))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f46,plain,(
% 2.65/1.76 ! [X494 : subst > term > subst > $o] : (((sP32 @ X494) = $true) <=> ? [X499 : term,X500 : term] : (((X494 @ id @ X500 @ id) = $true) & ((X494 @ id @ (ap @ (sub @ X500 @ id) @ X499) @ id) != $true) & ((X494 @ id @ X499 @ id) = $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f47,plain,(
% 2.65/1.76 (sP33 = $true) <=> ! [X493 : term,X494 : subst > term > subst > $o] : (($true = (X494 @ id @ X493 @ id)) | ((sP32 @ X494) = $true) | ? [X507 : subst,X508 : subst,X509 : term,X510 : subst] : (((X494 @ X510 @ X509 @ (comp @ X508 @ X507)) = $true) & ($true != (X494 @ (comp @ X510 @ X508) @ (sub @ X509 @ X508) @ X507))) | ($true = (sP31 @ X494)) | ? [X495 : term,X496 : subst,X497 : subst,X498 : subst] : (($true = (X494 @ (comp @ X498 @ X497) @ (sub @ X495 @ X497) @ X496)) & ((X494 @ X498 @ X495 @ (comp @ X497 @ X496)) != $true)) | ? [X506 : term] : (((var @ (sub @ X506 @ id)) = $true) & ($true != (X494 @ id @ X506 @ id))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f48,plain,(
% 2.65/1.76 ! [X138 : subst > term > subst > $o] : (($true = (sP34 @ X138)) <=> ? [X140 : subst > term > term] : (((X138 @ id @ (lam @ (X140 @ sh @ one)) @ id) != $true) & ! [X142 : subst,X143 : term,X144 : subst] : ((X140 @ (comp @ X142 @ X144) @ (sub @ X143 @ X144)) = (sub @ (X140 @ X142 @ X143) @ X144)) & ! [X141 : term] : (((X138 @ id @ X141 @ id) != $true) | ($true = (X138 @ id @ (X140 @ id @ X141) @ id)))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f49,plain,(
% 2.65/1.76 ! [X152 : term > $o] : (((sP35 @ X152) = $true) <=> ! [X155 : term,X156 : term] : (($true = (X152 @ (ap @ X155 @ X156))) | ((X152 @ X156) != $true) | ($true != (X152 @ X155))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f50,plain,(
% 2.65/1.76 ! [X160 : term > $o] : (((sP36 @ X160) = $true) <=> ? [X162 : term,X163 : term] : (((X160 @ X163) = $true) & ($true != (X160 @ (ap @ X163 @ X162))) & ($true = (X160 @ X162))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f51,plain,(
% 2.65/1.76 (sP37 = $true) <=> ! [X160 : term > $o,X161 : term] : (((sP36 @ X160) = $true) | ? [X164 : term] : (((var @ X164) = $true) & ((X160 @ X164) != $true)) | ((X160 @ X161) = $true) | ? [X165 : term] : (($true != (X160 @ (lam @ X165))) & ! [X166 : term] : (($true != (X160 @ X166)) | ((X160 @ (sub @ X165 @ (push @ X166 @ id))) = $true))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f52,plain,(
% 2.65/1.76 (sP38 = $true) <=> ((sP37 = $true) | ? [X152 : term > $o,X153 : term,X154 : subst] : (! [X158 : term] : (? [X159 : term] : (($true != (X152 @ (sub @ X158 @ (push @ X159 @ id)))) & ((X152 @ X159) = $true)) | ((X152 @ (lam @ X158)) = $true)) & ! [X157 : term] : (((X152 @ (sub @ X157 @ X154)) = $true) | ((var @ X157) != $true)) & ((X152 @ (sub @ X153 @ X154)) != $true) & ((sP35 @ X152) = $true)) | ? [X151 : term] : ((sub @ X151 @ id) != X151))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f53,plain,(
% 2.65/1.76 ($true = sP39) <=> ! [X121 : term,X122 : subst > term > subst > $o] : (? [X123 : term] : (((X122 @ id @ X123 @ id) = $true) & ((X122 @ id @ (sub @ X121 @ (push @ X123 @ id)) @ id) != $true)) | ($true = (X122 @ id @ (lam @ (sub @ X121 @ (push @ one @ sh))) @ id)) | ? [X124 : term,X125 : subst > term > term] : (! [X126 : subst,X127 : term,X128 : subst] : ((sub @ (sub @ X124 @ (push @ X127 @ X128)) @ X126) = (sub @ (X125 @ X128 @ X127) @ X126)) & ((X122 @ id @ (lam @ (X125 @ sh @ one)) @ id) != $true) & ! [X129 : term] : (((X122 @ id @ X129 @ id) != $true) | ((X122 @ id @ (X125 @ id @ X129) @ id) = $true)) & ! [X130 : term,X131 : subst,X132 : subst] : ((sub @ X124 @ (push @ (sub @ X130 @ X132) @ (comp @ X131 @ X132))) = (X125 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132)))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f54,plain,(
% 2.65/1.76 ! [X362 : term > $o] : (($true = (sP40 @ X362)) <=> ? [X367 : term,X368 : term] : (((X362 @ X367) = $true) & ((X362 @ X368) = $true) & ($true != (X362 @ (ap @ X368 @ X367)))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f55,plain,(
% 2.65/1.76 ! [X356 : term > $o] : (($true = (sP41 @ X356)) <=> ! [X359 : term,X360 : term] : (($true != (X356 @ X360)) | ($true = (X356 @ (ap @ X360 @ X359))) | ($true != (X356 @ X359))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f56,plain,(
% 2.65/1.76 ($true = sP42) <=> ! [X361 : term,X362 : term > $o,X363 : subst] : (? [X365 : term] : (! [X366 : term] : (($true = (X362 @ (sub @ X365 @ (push @ X366 @ id)))) | ((X362 @ X366) != $true)) & ($true != (X362 @ (lam @ X365)))) | ? [X364 : term] : (((var @ X364) = $true) & ($true != (X362 @ (sub @ X364 @ X363)))) | ($true = (sP40 @ X362)) | ((X362 @ (sub @ X361 @ X363)) = $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f57,plain,(
% 2.65/1.76 (sP43 = $true) <=> (? [X355 : term,X356 : term > $o] : (((X356 @ X355) != $true) & ! [X358 : term] : (($true = (X356 @ (lam @ X358))) | ((X356 @ X358) != $true)) & ! [X357 : term] : (((var @ X357) != $true) | ($true = (X356 @ X357))) & ($true = (sP41 @ X356))) | ? [X354 : subst] : ((comp @ X354 @ id) != X354) | ? [X352 : term,X353 : subst] : ((comp @ sh @ (push @ X352 @ X353)) != X353) | ? [X350 : subst,X351 : term] : ((sub @ one @ (push @ X351 @ X350)) != X351) | ($true = sP42))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f58,plain,(
% 2.65/1.76 (sP44 = $true) <=> (? [X438 : term,X439 : term] : ((X438 != X439) & ((lam @ X438) = (lam @ X439))) | ? [X436 : term,X437 : subst] : ((comp @ sh @ (push @ X436 @ X437)) != X437) | ? [X434 : term,X435 : subst] : ((sub @ one @ (push @ X434 @ X435)) != X434) | ! [X440 : term,X441 : subst > term > term,X442 : subst > term > term,X443 : subst] : (? [X444 : term,X445 : subst,X446 : subst] : ((X441 @ (comp @ X446 @ X445) @ (sub @ X444 @ X445)) != (sub @ (X441 @ X446 @ X444) @ X445)) | ((X441 @ X443 @ X440) = (X442 @ X443 @ X440)) | ((lam @ (X441 @ sh @ one)) != (lam @ (X442 @ sh @ one))) | ? [X447 : subst,X448 : subst,X449 : term] : ((X442 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447)) != (sub @ (X442 @ X448 @ X449) @ X447))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f59,plain,(
% 2.65/1.76 ! [X478 : subst > term > subst > $o] : (((sP45 @ X478) = $true) <=> ? [X484 : subst > term > term] : (! [X488 : term] : (((X478 @ id @ (X484 @ id @ X488) @ id) = $true) | ($true != (X478 @ id @ X488 @ id))) & ! [X485 : subst,X486 : subst,X487 : term] : ((sub @ (X484 @ X486 @ X487) @ X485) = (X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485))) & ($true != (X478 @ id @ (lam @ (X484 @ sh @ one)) @ id))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f60,plain,(
% 2.65/1.76 ($true = sP46) <=> ! [X477 : term,X478 : subst > term > subst > $o] : (((X478 @ id @ (lam @ X477) @ id) = $true) | ? [X483 : term] : (($true != (X478 @ id @ (sub @ X477 @ (push @ X483 @ id)) @ id)) & ((X478 @ id @ X483 @ id) = $true)) | ? [X489 : subst,X490 : subst,X491 : term,X492 : subst] : (((X478 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490) @ X489) = $true) & ((X478 @ X492 @ X491 @ (comp @ X490 @ X489)) != $true)) | ? [X479 : subst,X480 : subst,X481 : subst,X482 : term] : (((X478 @ (comp @ X481 @ X479) @ (sub @ X482 @ X479) @ X480) != $true) & ((X478 @ X481 @ X482 @ (comp @ X479 @ X480)) = $true)) | ((sP45 @ X478) = $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f61,plain,(
% 2.65/1.76 ! [X18 : subst > term > subst > $o] : (((sP47 @ X18) = $true) <=> ? [X22 : subst > term > term] : (! [X26 : term] : (($true != (X18 @ id @ X26 @ id)) | ((X18 @ id @ (X22 @ id @ X26) @ id) = $true)) & ! [X23 : subst,X24 : subst,X25 : term] : ((X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (X22 @ X24 @ X25) @ X23)) & ($true != (X18 @ id @ (lam @ (X22 @ sh @ one)) @ id))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f62,plain,(
% 2.65/1.76 (sP48 = $true) <=> ! [X17 : term,X18 : subst > term > subst > $o,X19 : term > $o] : (((sP47 @ X18) = $true) | ? [X21 : term] : ((X19 @ X21) != (X18 @ id @ X21 @ id)) | ? [X20 : term] : (($true = (X19 @ X20)) & ((X19 @ (sub @ X17 @ (push @ X20 @ id))) != $true)) | ((X19 @ (lam @ X17)) = $true) | ? [X27 : subst,X28 : term,X29 : subst,X30 : subst] : (($true = (X18 @ X27 @ X28 @ (comp @ X29 @ X30))) & ((X18 @ (comp @ X27 @ X29) @ (sub @ X28 @ X29) @ X30) != $true)) | ? [X31 : subst,X32 : subst,X33 : subst,X34 : term] : (($true = (X18 @ (comp @ X33 @ X32) @ (sub @ X34 @ X32) @ X31)) & ((X18 @ X33 @ X34 @ (comp @ X32 @ X31)) != $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f63,plain,(
% 2.65/1.76 (sP49 = $true) <=> ! [X342 : term,X343 : subst > term > subst > $o] : (? [X344 : term] : (((X343 @ id @ X344 @ id) = $true) & ((X343 @ id @ (sub @ X342 @ (push @ X344 @ id)) @ id) != $true)) | ? [X345 : subst > term > term] : (! [X346 : term,X347 : subst,X348 : subst] : ((sub @ (X345 @ X347 @ X346) @ X348) = (X345 @ (comp @ X347 @ X348) @ (sub @ X346 @ X348))) & ! [X349 : term] : (((X343 @ id @ X349 @ id) != $true) | ($true = (X343 @ id @ (X345 @ id @ X349) @ id))) & ((X343 @ id @ (lam @ (X345 @ sh @ one)) @ id) != $true)) | ((X343 @ id @ (lam @ (sub @ X342 @ (push @ one @ sh))) @ id) = $true))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f64,plain,(
% 2.65/1.76 ! [X81 : subst > term > subst > $o] : (((sP50 @ X81) = $true) <=> ? [X84 : subst > term > term] : (! [X86 : term,X87 : subst,X88 : subst] : ((sub @ (X84 @ X88 @ X86) @ X87) = (X84 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) & ! [X85 : term] : (((X81 @ id @ X85 @ id) != $true) | ((X81 @ id @ (X84 @ id @ X85) @ id) = $true)) & ((X81 @ id @ (lam @ (X84 @ sh @ one)) @ id) != $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f65,plain,(
% 2.65/1.76 ($true = sP51) <=> (! [X80 : term,X81 : subst > term > subst > $o,X82 : term > $o] : (((X82 @ (lam @ X80)) = $true) | ? [X89 : subst,X90 : subst,X91 : subst,X92 : term] : (($true = (X81 @ X90 @ X92 @ (comp @ X91 @ X89))) & ((X81 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91) @ X89) != $true)) | ? [X97 : term] : (((X82 @ (sub @ X80 @ (push @ X97 @ id))) != $true) & ((X82 @ X97) = $true)) | ? [X83 : term] : ((X82 @ X83) != (X81 @ id @ X83 @ id)) | ? [X93 : term,X94 : subst,X95 : subst,X96 : subst] : (($true != (X81 @ X94 @ X93 @ (comp @ X95 @ X96))) & ((X81 @ (comp @ X94 @ X95) @ (sub @ X93 @ X95) @ X96) = $true)) | ((sP50 @ X81) = $true)) | ? [X79 : term] : ((sub @ X79 @ id) != X79))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f66,plain,(
% 2.65/1.76 ($true = sP52) <=> ? [X249 : subst > term > subst > $o,X250 : term] : (! [X256 : term] : (((X249 @ id @ (sub @ X250 @ (push @ X256 @ id)) @ id) = $true) | ((X249 @ id @ X256 @ id) != $true)) & ((X249 @ id @ (lam @ (sub @ X250 @ (push @ one @ sh))) @ id) != $true) & ! [X251 : subst > term > term] : (? [X255 : term] : (((X249 @ id @ (X251 @ id @ X255) @ id) != $true) & ((X249 @ id @ X255 @ id) = $true)) | ($true = (X249 @ id @ (lam @ (X251 @ sh @ one)) @ id)) | ? [X252 : subst,X253 : term,X254 : subst] : ((sub @ (X251 @ X252 @ X253) @ X254) != (X251 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254)))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f67,plain,(
% 2.65/1.76 (sP53 = $true) <=> (? [X248 : term] : ((sub @ X248 @ id) != X248) | ! [X257 : subst > term > subst > $o,X258 : term] : (((X257 @ id @ (lam @ X258) @ id) = $true) | ? [X259 : term] : (((X257 @ id @ X259 @ id) = $true) & ((X257 @ id @ (sub @ X258 @ (push @ X259 @ id)) @ id) != $true)) | ? [X260 : subst > term > term] : (! [X264 : term] : (((X257 @ id @ (X260 @ id @ X264) @ id) = $true) | ((X257 @ id @ X264 @ id) != $true)) & ! [X261 : term,X262 : subst,X263 : subst] : ((X260 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263)) = (sub @ (X260 @ X262 @ X261) @ X263)) & ((X257 @ id @ (lam @ (X260 @ sh @ one)) @ id) != $true))) | ($true = sP52))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f68,plain,(
% 2.65/1.76 (sP54 = $true) <=> ! [X265 : term,X266 : subst > term > subst > $o,X267 : term > $o] : (? [X269 : term] : ((X267 @ X269) != (X266 @ id @ X269 @ id)) | ($true = (X267 @ (lam @ X265))) | ? [X270 : subst > term > term] : (! [X272 : subst,X273 : subst,X274 : term] : ((sub @ (X270 @ X272 @ X274) @ X273) = (X270 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273))) & ! [X271 : term] : (((X266 @ id @ X271 @ id) != $true) | ($true = (X266 @ id @ (X270 @ id @ X271) @ id))) & ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id) != $true)) | ? [X268 : term] : (((X267 @ X268) = $true) & ($true != (X267 @ (sub @ X265 @ (push @ X268 @ id))))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f69,plain,(
% 2.65/1.76 ($true = sP55) <=> ! [X182 : subst > term > subst > $o,X183 : term] : (? [X185 : subst > term > term] : (! [X186 : term] : (($true != (X182 @ id @ X186 @ id)) | ((X182 @ id @ (X185 @ id @ X186) @ id) = $true)) & ! [X187 : term,X188 : subst,X189 : subst] : ((X185 @ (comp @ X189 @ X188) @ (sub @ X187 @ X188)) = (sub @ (X185 @ X189 @ X187) @ X188)) & ((X182 @ id @ (lam @ (X185 @ sh @ one)) @ id) != $true)) | ((X182 @ id @ (lam @ (sub @ X183 @ (push @ one @ sh))) @ id) = $true) | ? [X184 : term] : (((X182 @ id @ X184 @ id) = $true) & ((X182 @ id @ (sub @ X183 @ (push @ X184 @ id)) @ id) != $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f70,plain,(
% 2.65/1.76 ! [X403 : term > $o] : (((sP56 @ X403) = $true) <=> ? [X405 : term,X406 : term] : (((X403 @ X405) = $true) & ((X403 @ (ap @ X405 @ X406)) != $true) & ($true = (X403 @ X406))))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f71,plain,(
% 2.65/1.76 ! [X1 : term > $o] : (($true = (sP57 @ X1)) <=> ? [X3 : term,X4 : term] : (((X1 @ X3) = $true) & ((X1 @ (ap @ X3 @ X4)) != $true) & ((X1 @ X4) = $true)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f72,plain,(
% 2.65/1.76 ($true = sP58) <=> ! [X0 : subst,X1 : term > $o,X2 : term] : (? [X6 : term] : (! [X7 : term] : (($true != (X1 @ X7)) | ($true = (X1 @ (sub @ X6 @ (push @ X7 @ id))))) & ((X1 @ (lam @ X6)) != $true)) | ((X1 @ (sub @ X2 @ X0)) = $true) | ? [X5 : term] : (((var @ X5) = $true) & ((X1 @ (sub @ X5 @ X0)) != $true)) | ($true = (sP57 @ X1)))),
% 2.65/1.76 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.65/1.76 thf(f13,plain,(
% 2.65/1.76 ((pushprop_lem0_lthm = $true) <=> ! [X116 : term,X117 : subst,X118 : term > $o] : ? [X119 : term > $o] : ! [X120 : term] : ((X119 @ X120) = (X118 @ (sub @ X120 @ (push @ X116 @ X117))))) & (! [X466 : subst] : ((comp @ X466 @ id) = X466) <=> (axidr = $true)) & ! [X112 : d_subst,X113 : d_subst] : (((d2subst @ X112) != (d2subst @ X113)) | (X112 = X113)) & (hoasinduction_lem2_lthm = $true) & (axclos = $true) & (! [X0 : subst,X1 : term > $o,X2 : term] : (? [X6 : term] : (! [X7 : term] : (($true != (X1 @ X7)) | ($true = (X1 @ (sub @ X6 @ (push @ X7 @ id))))) & ((X1 @ (lam @ X6)) != $true)) | ((X1 @ (sub @ X2 @ X0)) = $true) | ? [X5 : term] : (((var @ X5) = $true) & ((X1 @ (sub @ X5 @ X0)) != $true)) | ? [X3 : term,X4 : term] : (((X1 @ X3) = $true) & ((X1 @ (ap @ X3 @ X4)) != $true) & ((X1 @ X4) = $true))) <=> (induction2lem = $true)) & (! [X174 : term] : ((sub @ X174 @ id) = X174) <=> (axvarid = $true)) & (substmonoid_gthm = $true) & ((induction = $true) <=> ! [X403 : term > $o,X404 : term] : (? [X405 : term,X406 : term] : (((X403 @ X405) = $true) & ((X403 @ (ap @ X405 @ X406)) != $true) & ($true = (X403 @ X406))) | ? [X407 : term] : (($true = (var @ X407)) & ($true != (X403 @ X407))) | ((X403 @ X404) = $true) | ? [X408 : term] : (((X403 @ (lam @ X408)) != $true) & ((X403 @ X408) = $true)))) & ((pushprop_lem1 = $true) <=> ! [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (((X133 @ (sub @ X134 @ (push @ X135 @ X136))) = $true) | ($true != (X137 @ X134)))) & (hoaslamnotvar_gthm = $true) & (axassoc = $true) & (! [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) != (ap @ X276 @ X277)) | (X276 = X278)) <=> (apinj1 = $true)) & (hoasapinj2_gthm = $true) & ! [X464 : subst] : ? [X465 : d_subst] : ((d2subst @ X465) = X464) & ((? [X104 : term,X105 : term,X106 : term,X107 : term] : ((X106 != X107) & ((ap @ X106 @ X105) = (ap @ X107 @ X104))) | ? [X103 : term] : ((sub @ X103 @ id) != X103) | ! [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 = X109) | ((ap @ (sub @ X109 @ id) @ X111) != (ap @ (sub @ X108 @ id) @ X110)))) <=> (hoasapinj1_lthm = $true)) & (hoasapnotvar = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasinduction_lem3aa = $true) <=> ! [X182 : subst > term > subst > $o,X183 : term] : (? [X185 : subst > term > term] : (! [X186 : term] : (($true != (X182 @ id @ X186 @ id)) | ((X182 @ id @ (X185 @ id @ X186) @ id) = $true)) & ! [X187 : term,X188 : subst,X189 : subst] : ((X185 @ (comp @ X189 @ X188) @ (sub @ X187 @ X188)) = (sub @ (X185 @ X189 @ X187) @ X188)) & ((X182 @ id @ (lam @ (X185 @ sh @ one)) @ id) != $true)) | ((X182 @ id @ (lam @ (sub @ X183 @ (push @ one @ sh))) @ id) = $true) | ? [X184 : term] : (((X182 @ id @ X184 @ id) = $true) & ((X182 @ id @ (sub @ X183 @ (push @ X184 @ id)) @ id) != $true)))) & ((var @ (d2term @ d_one)) != $true) & (hoaslaminj_gthm = $true) & (hoasapnotvar_lthm = $true) & (pushprop_lem3v2 = $true) & (! [X265 : term,X266 : subst > term > subst > $o,X267 : term > $o] : (? [X269 : term] : ((X267 @ X269) != (X266 @ id @ X269 @ id)) | ($true = (X267 @ (lam @ X265))) | ? [X270 : subst > term > term] : (! [X272 : subst,X273 : subst,X274 : term] : ((sub @ (X270 @ X272 @ X274) @ X273) = (X270 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273))) & ! [X271 : term] : (((X266 @ id @ X271 @ id) != $true) | ($true = (X266 @ id @ (X270 @ id @ X271) @ id))) & ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id) != $true)) | ? [X268 : term] : (((X267 @ X268) = $true) & ($true != (X267 @ (sub @ X265 @ (push @ X268 @ id)))))) <=> (hoasinduction_lem3v2a = $true)) & (termmset_gthm = $true) & (one = (d2term @ d_one)) & ! [X114 : term] : ? [X115 : d_term] : ((d2term @ X115) = X114) & ((shinj = $true) <=> ! [X35 : term,X36 : term] : ((X35 = X36) | ((sub @ X36 @ sh) != (sub @ X35 @ sh)))) & ((hoasinduction_lem0 = $true) <=> ! [X333 : subst > term > subst > $o] : ? [X334 : term > $o] : ! [X335 : term] : ((X334 @ X335) = (X333 @ id @ X335 @ id))) & (hoasinduction_lem1_lthm = $true) & (induction2_gthm = $true) & (! [X178 : term] : ? [X179 : subst > term > term] : ! [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) = (X179 @ X180 @ X181)) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X384 : term] : ? [X385 : subst > term > term] : ! [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) = (X385 @ X386 @ X387))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((? [X248 : term] : ((sub @ X248 @ id) != X248) | ! [X257 : subst > term > subst > $o,X258 : term] : (((X257 @ id @ (lam @ X258) @ id) = $true) | ? [X259 : term] : (((X257 @ id @ X259 @ id) = $true) & ((X257 @ id @ (sub @ X258 @ (push @ X259 @ id)) @ id) != $true)) | ? [X260 : subst > term > term] : (! [X264 : term] : (((X257 @ id @ (X260 @ id @ X264) @ id) = $true) | ((X257 @ id @ X264 @ id) != $true)) & ! [X261 : term,X262 : subst,X263 : subst] : ((X260 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263)) = (sub @ (X260 @ X262 @ X261) @ X263)) & ((X257 @ id @ (lam @ (X260 @ sh @ one)) @ id) != $true))) | ? [X249 : subst > term > subst > $o,X250 : term] : (! [X256 : term] : (((X249 @ id @ (sub @ X250 @ (push @ X256 @ id)) @ id) = $true) | ((X249 @ id @ X256 @ id) != $true)) & ((X249 @ id @ (lam @ (sub @ X250 @ (push @ one @ sh))) @ id) != $true) & ! [X251 : subst > term > term] : (? [X255 : term] : (((X249 @ id @ (X251 @ id @ X255) @ id) != $true) & ((X249 @ id @ X255 @ id) = $true)) | ($true = (X249 @ id @ (lam @ (X251 @ sh @ one)) @ id)) | ? [X252 : subst,X253 : term,X254 : subst] : ((sub @ (X251 @ X252 @ X253) @ X254) != (X251 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254)))))) <=> (hoasinduction_lem3a_lthm = $true)) & ! [X177 : d_subst] : (d_id = X177) & (hoasinduction_lem1v2 = $true) & (! [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) = X176) <=> (axvarcons = $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((! [X80 : term,X81 : subst > term > subst > $o,X82 : term > $o] : (((X82 @ (lam @ X80)) = $true) | ? [X89 : subst,X90 : subst,X91 : subst,X92 : term] : (($true = (X81 @ X90 @ X92 @ (comp @ X91 @ X89))) & ((X81 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91) @ X89) != $true)) | ? [X97 : term] : (((X82 @ (sub @ X80 @ (push @ X97 @ id))) != $true) & ((X82 @ X97) = $true)) | ? [X83 : term] : ((X82 @ X83) != (X81 @ id @ X83 @ id)) | ? [X93 : term,X94 : subst,X95 : subst,X96 : subst] : (($true != (X81 @ X94 @ X93 @ (comp @ X95 @ X96))) & ((X81 @ (comp @ X94 @ X95) @ (sub @ X93 @ X95) @ X96) = $true)) | ? [X84 : subst > term > term] : (! [X86 : term,X87 : subst,X88 : subst] : ((sub @ (X84 @ X88 @ X86) @ X87) = (X84 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) & ! [X85 : term] : (((X81 @ id @ X85 @ id) != $true) | ((X81 @ id @ (X84 @ id @ X85) @ id) = $true)) & ((X81 @ id @ (lam @ (X84 @ sh @ one)) @ id) != $true))) | ? [X79 : term] : ((sub @ X79 @ id) != X79)) <=> (hoasinduction_lem3v2_lthm = $true)) & (axapp = $true) & (induction2lem_gthm = $true) & (ulamvarsh = $true) & (axvarshift = $true) & (ulamvarind = $true) & ((d2subst @ d_id) = id) & ((termmset = $true) <=> ! [X247 : term] : ((sub @ X247 @ id) = X247)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X342 : term,X343 : subst > term > subst > $o] : (? [X344 : term] : (((X343 @ id @ X344 @ id) = $true) & ((X343 @ id @ (sub @ X342 @ (push @ X344 @ id)) @ id) != $true)) | ? [X345 : subst > term > term] : (! [X346 : term,X347 : subst,X348 : subst] : ((sub @ (X345 @ X347 @ X346) @ X348) = (X345 @ (comp @ X347 @ X348) @ (sub @ X346 @ X348))) & ! [X349 : term] : (((X343 @ id @ X349 @ id) != $true) | ($true = (X343 @ id @ (X345 @ id @ X349) @ id))) & ((X343 @ id @ (lam @ (X345 @ sh @ one)) @ id) != $true)) | ((X343 @ id @ (lam @ (sub @ X342 @ (push @ one @ sh))) @ id) = $true))) & (hoasinduction_lem3b_gthm = $true) & (apnotvar = $true) & ((! [X150 : term] : ((sub @ X150 @ id) = X150) | ? [X149 : term] : ((sub @ X149 @ id) != X149)) <=> (termmset_lthm = $true)) & (! [X17 : term,X18 : subst > term > subst > $o,X19 : term > $o] : (? [X22 : subst > term > term] : (! [X26 : term] : (($true != (X18 @ id @ X26 @ id)) | ((X18 @ id @ (X22 @ id @ X26) @ id) = $true)) & ! [X23 : subst,X24 : subst,X25 : term] : ((X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (X22 @ X24 @ X25) @ X23)) & ($true != (X18 @ id @ (lam @ (X22 @ sh @ one)) @ id))) | ? [X21 : term] : ((X19 @ X21) != (X18 @ id @ X21 @ id)) | ? [X20 : term] : (($true = (X19 @ X20)) & ((X19 @ (sub @ X17 @ (push @ X20 @ id))) != $true)) | ((X19 @ (lam @ X17)) = $true) | ? [X27 : subst,X28 : term,X29 : subst,X30 : subst] : (($true = (X18 @ X27 @ X28 @ (comp @ X29 @ X30))) & ((X18 @ (comp @ X27 @ X29) @ (sub @ X28 @ X29) @ X30) != $true)) | ? [X31 : subst,X32 : subst,X33 : subst,X34 : term] : (($true = (X18 @ (comp @ X33 @ X32) @ (sub @ X34 @ X32) @ X31)) & ((X18 @ X33 @ X34 @ (comp @ X32 @ X31)) != $true))) <=> (hoasinduction_lem3v2 = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X477 : term,X478 : subst > term > subst > $o] : (((X478 @ id @ (lam @ X477) @ id) = $true) | ? [X483 : term] : (($true != (X478 @ id @ (sub @ X477 @ (push @ X483 @ id)) @ id)) & ((X478 @ id @ X483 @ id) = $true)) | ? [X489 : subst,X490 : subst,X491 : term,X492 : subst] : (((X478 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490) @ X489) = $true) & ((X478 @ X492 @ X491 @ (comp @ X490 @ X489)) != $true)) | ? [X479 : subst,X480 : subst,X481 : subst,X482 : term] : (((X478 @ (comp @ X481 @ X479) @ (sub @ X482 @ X479) @ X480) != $true) & ((X478 @ X481 @ X482 @ (comp @ X479 @ X480)) = $true)) | ? [X484 : subst > term > term] : (! [X488 : term] : (((X478 @ id @ (X484 @ id @ X488) @ id) = $true) | ($true != (X478 @ id @ X488 @ id))) & ! [X485 : subst,X486 : subst,X487 : term] : ((sub @ (X484 @ X486 @ X487) @ X485) = (X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485))) & ($true != (X478 @ id @ (lam @ (X484 @ sh @ one)) @ id))))) & (hoasinduction_lem3_gthm = $true) & ((? [X438 : term,X439 : term] : ((X438 != X439) & ((lam @ X438) = (lam @ X439))) | ? [X436 : term,X437 : subst] : ((comp @ sh @ (push @ X436 @ X437)) != X437) | ? [X434 : term,X435 : subst] : ((sub @ one @ (push @ X434 @ X435)) != X434) | ! [X440 : term,X441 : subst > term > term,X442 : subst > term > term,X443 : subst] : (? [X444 : term,X445 : subst,X446 : subst] : ((X441 @ (comp @ X446 @ X445) @ (sub @ X444 @ X445)) != (sub @ (X441 @ X446 @ X444) @ X445)) | ((X441 @ X443 @ X440) = (X442 @ X443 @ X440)) | ((lam @ (X441 @ sh @ one)) != (lam @ (X442 @ sh @ one))) | ? [X447 : subst,X448 : subst,X449 : term] : ((X442 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447)) != (sub @ (X442 @ X448 @ X449) @ X447)))) <=> (hoaslaminj_lthm = $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3b = $true) <=> ! [X8 : term] : ? [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) = (X9 @ sh @ one))) & (hoasinduction_lem3v2_gthm = $true) & (axmap = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop = $true) & ((substmonoid_lthm = $true) <=> (? [X537 : subst] : ((comp @ X537 @ id) != X537) | (! [X538 : subst] : ((comp @ X538 @ id) = X538) & ! [X539 : subst] : ((comp @ id @ X539) = X539)) | ? [X536 : subst] : ((comp @ id @ X536) != X536))) & (axabs = $true) & (lamnotvar = $true) & (! [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) = X240) <=> (axshiftcons = $true)) & ((? [X355 : term,X356 : term > $o] : (((X356 @ X355) != $true) & ! [X358 : term] : (($true = (X356 @ (lam @ X358))) | ((X356 @ X358) != $true)) & ! [X357 : term] : (((var @ X357) != $true) | ($true = (X356 @ X357))) & ! [X359 : term,X360 : term] : (($true != (X356 @ X360)) | ($true = (X356 @ (ap @ X360 @ X359))) | ($true != (X356 @ X359)))) | ? [X354 : subst] : ((comp @ X354 @ id) != X354) | ? [X352 : term,X353 : subst] : ((comp @ sh @ (push @ X352 @ X353)) != X353) | ? [X350 : subst,X351 : term] : ((sub @ one @ (push @ X351 @ X350)) != X351) | ! [X361 : term,X362 : term > $o,X363 : subst] : (? [X365 : term] : (! [X366 : term] : (($true = (X362 @ (sub @ X365 @ (push @ X366 @ id)))) | ((X362 @ X366) != $true)) & ($true != (X362 @ (lam @ X365)))) | ? [X364 : term] : (((var @ X364) = $true) & ($true != (X362 @ (sub @ X364 @ X363)))) | ? [X367 : term,X368 : term] : (((X362 @ X367) = $true) & ((X362 @ X368) = $true) & ($true != (X362 @ (ap @ X368 @ X367)))) | ((X362 @ (sub @ X361 @ X363)) = $true))) <=> (induction2lem_lthm = $true)) & (! [X121 : term,X122 : subst > term > subst > $o] : (? [X123 : term] : (((X122 @ id @ X123 @ id) = $true) & ((X122 @ id @ (sub @ X121 @ (push @ X123 @ id)) @ id) != $true)) | ($true = (X122 @ id @ (lam @ (sub @ X121 @ (push @ one @ sh))) @ id)) | ? [X124 : term,X125 : subst > term > term] : (! [X126 : subst,X127 : term,X128 : subst] : ((sub @ (sub @ X124 @ (push @ X127 @ X128)) @ X126) = (sub @ (X125 @ X128 @ X127) @ X126)) & ((X122 @ id @ (lam @ (X125 @ sh @ one)) @ id) != $true) & ! [X129 : term] : (((X122 @ id @ X129 @ id) != $true) | ((X122 @ id @ (X125 @ id @ X129) @ id) = $true)) & ! [X130 : term,X131 : subst,X132 : subst] : ((sub @ X124 @ (push @ (sub @ X130 @ X132) @ (comp @ X131 @ X132))) = (X125 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))))) <=> (hoasinduction_lem3aaa = $true)) & ((! [X160 : term > $o,X161 : term] : (? [X162 : term,X163 : term] : (((X160 @ X163) = $true) & ($true != (X160 @ (ap @ X163 @ X162))) & ($true = (X160 @ X162))) | ? [X164 : term] : (((var @ X164) = $true) & ((X160 @ X164) != $true)) | ((X160 @ X161) = $true) | ? [X165 : term] : (($true != (X160 @ (lam @ X165))) & ! [X166 : term] : (($true != (X160 @ X166)) | ((X160 @ (sub @ X165 @ (push @ X166 @ id))) = $true)))) | ? [X152 : term > $o,X153 : term,X154 : subst] : (! [X158 : term] : (? [X159 : term] : (($true != (X152 @ (sub @ X158 @ (push @ X159 @ id)))) & ((X152 @ X159) = $true)) | ((X152 @ (lam @ X158)) = $true)) & ! [X157 : term] : (((X152 @ (sub @ X157 @ X154)) = $true) | ((var @ X157) != $true)) & ((X152 @ (sub @ X153 @ X154)) != $true) & ! [X155 : term,X156 : term] : (($true = (X152 @ (ap @ X155 @ X156))) | ((X152 @ X156) != $true) | ($true != (X152 @ X155)))) | ? [X151 : term] : ((sub @ X151 @ id) != X151)) <=> (induction2_lthm = $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & (hoasinduction_lem1_gthm = $true) & (pushprop_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X138 : subst > term > subst > $o,X139 : term] : (((X138 @ id @ X139 @ id) = $true) | ? [X145 : term,X146 : term] : (((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) != $true) & ($true = (X138 @ id @ X145 @ id)) & ((X138 @ id @ X146 @ id) = $true)) | ? [X140 : subst > term > term] : (((X138 @ id @ (lam @ (X140 @ sh @ one)) @ id) != $true) & ! [X142 : subst,X143 : term,X144 : subst] : ((X140 @ (comp @ X142 @ X144) @ (sub @ X143 @ X144)) = (sub @ (X140 @ X142 @ X143) @ X144)) & ! [X141 : term] : (((X138 @ id @ X141 @ id) != $true) | ($true = (X138 @ id @ (X140 @ id @ X141) @ id)))))) & (lamnotap != $true) & ((hoasinduction = $true) <=> ! [X493 : term,X494 : subst > term > subst > $o] : (($true = (X494 @ id @ X493 @ id)) | ? [X499 : term,X500 : term] : (((X494 @ id @ X500 @ id) = $true) & ((X494 @ id @ (ap @ (sub @ X500 @ id) @ X499) @ id) != $true) & ((X494 @ id @ X499 @ id) = $true)) | ? [X507 : subst,X508 : subst,X509 : term,X510 : subst] : (((X494 @ X510 @ X509 @ (comp @ X508 @ X507)) = $true) & ($true != (X494 @ (comp @ X510 @ X508) @ (sub @ X509 @ X508) @ X507))) | ? [X501 : subst > term > term] : (($true != (X494 @ id @ (lam @ (X501 @ sh @ one)) @ id)) & ! [X505 : term] : (((X494 @ id @ X505 @ id) != $true) | ((X494 @ id @ (X501 @ id @ X505) @ id) = $true)) & ! [X502 : subst,X503 : term,X504 : subst] : ((X501 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X501 @ X502 @ X503) @ X504))) | ? [X495 : term,X496 : subst,X497 : subst,X498 : subst] : (($true = (X494 @ (comp @ X498 @ X497) @ (sub @ X495 @ X497) @ X496)) & ((X494 @ X498 @ X495 @ (comp @ X497 @ X496)) != $true)) | ? [X506 : term] : (((var @ (sub @ X506 @ id)) = $true) & ($true != (X494 @ id @ X506 @ id))))) & ((laminj = $true) <=> ! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148)))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((axidl = $true) <=> ! [X471 : subst] : ((comp @ id @ X471) = X471)) & ((hoasapinj2 = $true) <=> ! [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 = X101) | ((ap @ (sub @ X98 @ id) @ X100) != (ap @ (sub @ X99 @ id) @ X101)))) & ((hoaslamnotap = $true) <=> ! [X336 : term,X337 : term,X338 : subst > term > term] : (? [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) != (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) | ((ap @ (sub @ X336 @ id) @ X337) != (lam @ (X338 @ sh @ one))))) & ((hoasinduction_lthm_3 = $true) <=> (? [X211 : term > $o,X212 : term,X213 : subst > term > subst > $o] : (! [X216 : subst > term > term] : (((X213 @ id @ (lam @ (X216 @ sh @ one)) @ id) = $true) | ? [X220 : term] : (((X213 @ id @ X220 @ id) = $true) & ((X213 @ id @ (X216 @ id @ X220) @ id) != $true)) | ? [X217 : term,X218 : subst,X219 : subst] : ((sub @ (X216 @ X219 @ X217) @ X218) != (X216 @ (comp @ X219 @ X218) @ (sub @ X217 @ X218)))) & ! [X214 : term] : ((X213 @ id @ X214 @ id) = (X211 @ X214)) & ($true != (X211 @ (lam @ X212))) & ! [X215 : term] : (($true != (X211 @ X215)) | ($true = (X211 @ (sub @ X212 @ (push @ X215 @ id)))))) | ! [X221 : term,X222 : subst > term > subst > $o] : (? [X223 : subst,X224 : term,X225 : subst,X226 : subst] : (($true = (X222 @ X225 @ X224 @ (comp @ X226 @ X223))) & ((X222 @ (comp @ X225 @ X226) @ (sub @ X224 @ X226) @ X223) != $true)) | ? [X229 : term] : (($true != (X222 @ id @ X229 @ id)) & ((var @ (sub @ X229 @ id)) = $true)) | ((X222 @ id @ X221 @ id) = $true) | ? [X227 : term,X228 : term] : (((X222 @ id @ X227 @ id) = $true) & ((X222 @ id @ X228 @ id) = $true) & ((X222 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id) != $true)) | ? [X234 : subst > term > term] : (((X222 @ id @ (lam @ (X234 @ sh @ one)) @ id) != $true) & ! [X236 : term,X237 : subst,X238 : subst] : ((X234 @ (comp @ X238 @ X237) @ (sub @ X236 @ X237)) = (sub @ (X234 @ X238 @ X236) @ X237)) & ! [X235 : term] : (($true = (X222 @ id @ (X234 @ id @ X235) @ id)) | ((X222 @ id @ X235 @ id) != $true))) | ? [X230 : subst,X231 : subst,X232 : term,X233 : subst] : (((X222 @ X233 @ X232 @ (comp @ X230 @ X231)) != $true) & ($true = (X222 @ (comp @ X233 @ X230) @ (sub @ X232 @ X230) @ X231)))) | ? [X210 : term] : ((sub @ X210 @ id) != X210) | ? [X203 : term,X204 : term > $o] : (($true != (X204 @ X203)) & ! [X207 : term] : (? [X208 : term] : (($true != (X204 @ (sub @ X207 @ (push @ X208 @ id)))) & ($true = (X204 @ X208))) | ($true = (X204 @ (lam @ X207)))) & ! [X209 : term] : (((X204 @ X209) = $true) | ((var @ X209) != $true)) & ! [X205 : term,X206 : term] : (($true != (X204 @ X206)) | ((X204 @ X205) != $true) | ((X204 @ (ap @ X206 @ X205)) = $true))) | ? [X200 : subst > term > subst > $o] : ! [X201 : term > $o] : ? [X202 : term] : ((X201 @ X202) != (X200 @ id @ X202 @ id)))) & (pushprop_lem3v2_lthm = $true) & (! [X511 : term,X512 : subst > term > subst > $o] : (? [X518 : term] : (((X512 @ id @ (sub @ X511 @ (push @ X518 @ id)) @ id) != $true) & ((X512 @ id @ X518 @ id) = $true)) | ((X512 @ id @ (lam @ X511) @ id) = $true) | ? [X513 : subst > term > term] : (((X512 @ id @ (lam @ (X513 @ sh @ one)) @ id) != $true) & ! [X514 : subst,X515 : term,X516 : subst] : ((X513 @ (comp @ X514 @ X516) @ (sub @ X515 @ X516)) = (sub @ (X513 @ X514 @ X515) @ X516)) & ! [X517 : term] : (($true = (X512 @ id @ (X513 @ id @ X517) @ id)) | ((X512 @ id @ X517 @ id) != $true)))) <=> (hoasinduction_lem3a = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 = X534) | ((ap @ (sub @ X533 @ id) @ X532) != (ap @ (sub @ X535 @ id) @ X534))) | ? [X528 : term,X529 : term,X530 : term,X531 : term] : (((ap @ X531 @ X529) = (ap @ X528 @ X530)) & (X529 != X530)))) & ((hoasapinj1 = $true) <=> ! [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 = X16) | ((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15)))) & (hoaslamnotap_gthm = $true) & (hoasinduction_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X467 : term] : ? [X468 : subst > term > term] : ((X468 @ sh @ one) = (sub @ X467 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X42 : term > $o,X43 : term] : (? [X44 : term] : (((var @ X44) = $true) & ((X42 @ X44) != $true)) | ? [X47 : term,X48 : term] : (($true != (X42 @ (ap @ X47 @ X48))) & ($true = (X42 @ X47)) & ((X42 @ X48) = $true)) | ? [X45 : term] : (((X42 @ (lam @ X45)) != $true) & ! [X46 : term] : (($true != (X42 @ X46)) | ($true = (X42 @ (sub @ X45 @ (push @ X46 @ id)))))) | ((X42 @ X43) = $true)) <=> (induction2 = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (? [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) != X168) | ! [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (? [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) != (X170 @ X173)) | ((X170 @ one) = $true) | ((X171 @ X172) != $true)))) & (! [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (((X474 @ X475) != $true) | ? [X476 : term] : ((X472 @ X476) != (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) | ((X472 @ one) = $true)) <=> (pushprop_lem1v2 = $true)) & (hoaslamnotap_lthm = $true) & (! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term,X372 : term] : (? [X382 : term,X383 : term] : (((X370 @ id @ (ap @ (sub @ X383 @ id) @ X382) @ id) != $true) & ((X370 @ id @ X383 @ id) = $true) & ((X370 @ id @ X382 @ id) = $true)) | ($true != (X369 @ X371)) | ((X369 @ X372) != $true) | ((X369 @ (ap @ X372 @ X371)) = $true) | ? [X373 : subst,X374 : subst,X375 : term,X376 : subst] : (((X370 @ (comp @ X374 @ X373) @ (sub @ X375 @ X373) @ X376) != $true) & ((X370 @ X374 @ X375 @ (comp @ X373 @ X376)) = $true)) | ? [X378 : term,X379 : subst,X380 : subst,X381 : subst] : (($true != (X370 @ X380 @ X378 @ (comp @ X381 @ X379))) & ((X370 @ (comp @ X380 @ X381) @ (sub @ X378 @ X381) @ X379) = $true)) | ? [X377 : term] : ((X369 @ X377) != (X370 @ id @ X377 @ id))) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_gthm = $true) & (! [X10 : subst > term > subst > $o] : ? [X11 : term > $o] : ! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & ((! [X393 : term,X394 : subst > term > subst > $o,X395 : term > $o] : (? [X402 : term] : ((X395 @ X402) != (X394 @ id @ X402 @ id)) | ? [X397 : subst > term > term] : (! [X399 : term,X400 : subst,X401 : subst] : ((sub @ (X397 @ X400 @ X399) @ X401) = (X397 @ (comp @ X400 @ X401) @ (sub @ X399 @ X401))) & ! [X398 : term] : (((X394 @ id @ X398 @ id) != $true) | ((X394 @ id @ (X397 @ id @ X398) @ id) = $true)) & ((X394 @ id @ (lam @ (X397 @ sh @ one)) @ id) != $true)) | ((X395 @ (lam @ X393)) = $true) | ? [X396 : term] : (((X395 @ (sub @ X393 @ (push @ X396 @ id))) != $true) & ((X395 @ X396) = $true))) | ? [X388 : term] : ! [X389 : subst > term > term] : ? [X390 : term,X391 : subst] : ((sub @ X388 @ (push @ X390 @ X391)) != (X389 @ X391 @ X390)) | ? [X392 : term] : ((sub @ X392 @ id) != X392)) <=> (hoasinduction_lem3v2a_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((d2subst @ d_id) = sh) & (hoasapinj1_gthm = $true) & (! [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) != (ap @ X243 @ X241)) | (X241 = X242)) <=> (apinj2 = $true)) & ((substmonoid = $true) <=> (! [X246 : subst] : ((comp @ id @ X246) = X246) & ! [X245 : subst] : ((comp @ X245 @ id) = X245))) & ((? [X409 : term] : ((sub @ X409 @ id) != X409) | ! [X418 : term,X419 : subst > term > subst > $o] : (? [X420 : subst,X421 : subst,X422 : term,X423 : subst] : (((X419 @ X423 @ X422 @ (comp @ X421 @ X420)) != $true) & ((X419 @ (comp @ X423 @ X421) @ (sub @ X422 @ X421) @ X420) = $true)) | ? [X433 : term] : (((X419 @ id @ (sub @ X418 @ (push @ X433 @ id)) @ id) != $true) & ($true = (X419 @ id @ X433 @ id))) | ($true = (X419 @ id @ (lam @ X418) @ id)) | ? [X428 : subst > term > term] : (($true != (X419 @ id @ (lam @ (X428 @ sh @ one)) @ id)) & ! [X429 : term,X430 : subst,X431 : subst] : ((sub @ (X428 @ X431 @ X429) @ X430) = (X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430))) & ! [X432 : term] : (($true = (X419 @ id @ (X428 @ id @ X432) @ id)) | ((X419 @ id @ X432 @ id) != $true))) | ? [X424 : subst,X425 : subst,X426 : subst,X427 : term] : (($true != (X419 @ (comp @ X426 @ X425) @ (sub @ X427 @ X425) @ X424)) & ((X419 @ X426 @ X427 @ (comp @ X425 @ X424)) = $true))) | ? [X410 : term,X411 : subst > term > subst > $o] : (! [X417 : term] : (($true = (X411 @ id @ (sub @ X410 @ (push @ X417 @ id)) @ id)) | ((X411 @ id @ X417 @ id) != $true)) & ((X411 @ id @ (lam @ (sub @ X410 @ (push @ one @ sh))) @ id) != $true) & ! [X412 : subst > term > term] : (? [X414 : subst,X415 : subst,X416 : term] : ((X412 @ (comp @ X414 @ X415) @ (sub @ X416 @ X415)) != (sub @ (X412 @ X414 @ X416) @ X415)) | ? [X413 : term] : (((X411 @ id @ (X412 @ id @ X413) @ id) != $true) & ($true = (X411 @ id @ X413 @ id))) | ((X411 @ id @ (lam @ (X412 @ sh @ one)) @ id) = $true)))) <=> (hoasinduction_lem3_lthm = $true)) & ((hoasinduction_lthm = $true) <=> (! [X315 : term,X316 : subst > term > subst > $o] : (? [X328 : subst,X329 : subst,X330 : subst,X331 : term] : (((X316 @ X328 @ X331 @ (comp @ X329 @ X330)) != $true) & ((X316 @ (comp @ X328 @ X329) @ (sub @ X331 @ X329) @ X330) = $true)) | ? [X319 : subst > term > term] : (! [X323 : term] : (((X316 @ id @ (X319 @ id @ X323) @ id) = $true) | ((X316 @ id @ X323 @ id) != $true)) & ! [X320 : term,X321 : subst,X322 : subst] : ((sub @ (X319 @ X321 @ X320) @ X322) = (X319 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322))) & ($true != (X316 @ id @ (lam @ (X319 @ sh @ one)) @ id))) | ? [X324 : subst,X325 : term,X326 : subst,X327 : subst] : (($true = (X316 @ X324 @ X325 @ (comp @ X327 @ X326))) & ((X316 @ (comp @ X324 @ X327) @ (sub @ X325 @ X327) @ X326) != $true)) | ((X316 @ id @ X315 @ id) = $true) | ? [X332 : term] : (((var @ (sub @ X332 @ id)) = $true) & ((X316 @ id @ X332 @ id) != $true)) | ? [X317 : term,X318 : term] : (((X316 @ id @ X318 @ id) = $true) & ((X316 @ id @ X317 @ id) = $true) & ($true != (X316 @ id @ (ap @ (sub @ X318 @ id) @ X317) @ id)))) | ? [X299 : term,X300 : subst > term > subst > $o] : (! [X302 : subst,X303 : subst,X304 : term,X305 : subst] : (((X300 @ (comp @ X305 @ X302) @ (sub @ X304 @ X302) @ X303) = $true) | ((X300 @ X305 @ X304 @ (comp @ X302 @ X303)) != $true)) & ((X300 @ id @ (lam @ X299) @ id) != $true) & ! [X301 : term] : (($true != (X300 @ id @ X301 @ id)) | ((X300 @ id @ (sub @ X299 @ (push @ X301 @ id)) @ id) = $true)) & ! [X310 : subst > term > term] : (? [X312 : term,X313 : subst,X314 : subst] : ((X310 @ (comp @ X314 @ X313) @ (sub @ X312 @ X313)) != (sub @ (X310 @ X314 @ X312) @ X313)) | ((X300 @ id @ (lam @ (X310 @ sh @ one)) @ id) = $true) | ? [X311 : term] : (((X300 @ id @ (X310 @ id @ X311) @ id) != $true) & ($true = (X300 @ id @ X311 @ id)))) & ! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (((X300 @ (comp @ X307 @ X308) @ (sub @ X306 @ X308) @ X309) != $true) | ($true = (X300 @ X307 @ X306 @ (comp @ X308 @ X309))))) | ? [X279 : term,X280 : term > $o] : (((X280 @ X279) != $true) & ! [X282 : term] : (? [X283 : term] : (($true != (X280 @ (sub @ X282 @ (push @ X283 @ id)))) & ((X280 @ X283) = $true)) | ((X280 @ (lam @ X282)) = $true)) & ! [X284 : term,X285 : term] : (($true != (X280 @ X285)) | ((X280 @ (ap @ X285 @ X284)) = $true) | ($true != (X280 @ X284))) & ! [X281 : term] : (((X280 @ X281) = $true) | ($true != (var @ X281)))) | ? [X286 : term,X287 : subst > term > subst > $o,X288 : term] : (! [X295 : subst,X296 : subst,X297 : term,X298 : subst] : (($true = (X287 @ X295 @ X297 @ (comp @ X296 @ X298))) | ((X287 @ (comp @ X295 @ X296) @ (sub @ X297 @ X296) @ X298) != $true)) & ($true = (X287 @ id @ X286 @ id)) & ((X287 @ id @ X288 @ id) = $true) & ! [X289 : subst,X290 : subst,X291 : term,X292 : subst] : (((X287 @ (comp @ X289 @ X290) @ (sub @ X291 @ X290) @ X292) = $true) | ($true != (X287 @ X289 @ X291 @ (comp @ X290 @ X292)))) & ($true != (X287 @ id @ (ap @ X286 @ X288) @ id)) & ! [X293 : term,X294 : term] : (((X287 @ id @ X294 @ id) != $true) | ((X287 @ id @ (ap @ (sub @ X293 @ id) @ X294) @ id) = $true) | ($true != (X287 @ id @ X293 @ id)))))) & (hoasinduction_lem1 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X52 : term > $o,X53 : term] : (! [X54 : term,X55 : term] : (($true != (X52 @ X54)) | ((X52 @ (ap @ X55 @ X54)) = $true) | ((X52 @ X55) != $true)) & ! [X57 : term] : (? [X58 : term] : (($true = (X52 @ X58)) & ((X52 @ (sub @ X57 @ (push @ X58 @ id))) != $true)) | ((X52 @ (lam @ X57)) = $true)) & ! [X56 : term] : (((var @ X56) != $true) | ($true = (X52 @ X56))) & ((X52 @ X53) != $true)) | ! [X70 : subst > term > subst > $o,X71 : term] : (? [X72 : subst > term > term] : (! [X76 : term] : (((X70 @ id @ (X72 @ id @ X76) @ id) = $true) | ($true != (X70 @ id @ X76 @ id))) & ! [X73 : subst,X74 : term,X75 : subst] : ((X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73)) = (sub @ (X72 @ X75 @ X74) @ X73)) & ((X70 @ id @ (lam @ (X72 @ sh @ one)) @ id) != $true)) | ? [X77 : term,X78 : term] : (($true = (X70 @ id @ X77 @ id)) & ((X70 @ id @ (ap @ (sub @ X78 @ id) @ X77) @ id) != $true) & ((X70 @ id @ X78 @ id) = $true)) | ($true = (X70 @ id @ X71 @ id))) | ? [X49 : subst > term > subst > $o] : ! [X50 : term > $o] : ? [X51 : term] : ((X49 @ id @ X51 @ id) != (X50 @ X51)) | ? [X59 : term] : ((sub @ X59 @ id) != X59) | ? [X60 : subst > term > subst > $o,X61 : term > $o,X62 : term] : (! [X68 : term] : (($true = (X61 @ (sub @ X62 @ (push @ X68 @ id)))) | ((X61 @ X68) != $true)) & ! [X69 : term] : ((X60 @ id @ X69 @ id) = (X61 @ X69)) & ! [X63 : subst > term > term] : (($true = (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) | ? [X65 : subst,X66 : subst,X67 : term] : ((sub @ (X63 @ X66 @ X67) @ X65) != (X63 @ (comp @ X66 @ X65) @ (sub @ X67 @ X65))) | ? [X64 : term] : (((X60 @ id @ X64 @ id) = $true) & ((X60 @ id @ (X63 @ id @ X64) @ id) != $true))) & ((X61 @ (lam @ X62)) != $true)))) & ((hoaslaminj = $true) <=> ! [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (((lam @ (X191 @ sh @ one)) != (lam @ (X190 @ sh @ one))) | ? [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) != (sub @ (X190 @ X196 @ X194) @ X195)) | ((X191 @ X192 @ X193) = (X190 @ X192 @ X193)) | ? [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) != (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199))))) & ((! [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (($true = (X524 @ (sub @ X527 @ (push @ X525 @ X526)))) | ($true != (X523 @ X527))) | ? [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) != X521) | ? [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) != X520)) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (ulamvar1 != $true) & (hoasinduction_lem2_gthm = $true) & ! [X469 : d_term,X470 : d_term] : (((d2term @ X469) != (d2term @ X470)) | (X469 = X470)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X451 : subst > term > subst > $o,X452 : term,X453 : term] : (? [X460 : subst,X461 : subst,X462 : term,X463 : subst] : (((X451 @ X460 @ X462 @ (comp @ X463 @ X461)) != $true) & ((X451 @ (comp @ X460 @ X463) @ (sub @ X462 @ X463) @ X461) = $true)) | ((X451 @ id @ X453 @ id) != $true) | ? [X458 : term,X459 : term] : (($true != (X451 @ id @ (ap @ (sub @ X458 @ id) @ X459) @ id)) & ($true = (X451 @ id @ X459 @ id)) & ((X451 @ id @ X458 @ id) = $true)) | ? [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (((X451 @ X457 @ X456 @ (comp @ X455 @ X454)) = $true) & ($true != (X451 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454))) | ((X451 @ id @ (ap @ X453 @ X452) @ id) = $true) | ((X451 @ id @ X452 @ id) != $true))) & (pushprop_lthm_orig = $true) & ! [X450 : d_term] : (d_one = X450) & ((pushprop_lem0 = $true) <=> ! [X37 : term > $o,X38 : term,X39 : subst] : ? [X40 : term > $o] : ! [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) = (X40 @ X41)))),
% 2.65/1.76 inference(flattening,[status(thm)],[f12])).
% 2.65/1.76 thf(f12,plain,(
% 2.65/1.76 ((hoasapinj1 = $true) <=> ! [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 = X16) | ((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15)))) & (((((! [X70 : subst > term > subst > $o,X71 : term] : (? [X72 : subst > term > term] : (! [X76 : term] : (((X70 @ id @ (X72 @ id @ X76) @ id) = $true) | ($true != (X70 @ id @ X76 @ id))) & ! [X73 : subst,X74 : term,X75 : subst] : ((X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73)) = (sub @ (X72 @ X75 @ X74) @ X73)) & ((X70 @ id @ (lam @ (X72 @ sh @ one)) @ id) != $true)) | ? [X77 : term,X78 : term] : (($true = (X70 @ id @ X77 @ id)) & ((X70 @ id @ (ap @ (sub @ X78 @ id) @ X77) @ id) != $true) & ((X70 @ id @ X78 @ id) = $true)) | ($true = (X70 @ id @ X71 @ id))) | ? [X60 : subst > term > subst > $o,X61 : term > $o,X62 : term] : (! [X68 : term] : (($true = (X61 @ (sub @ X62 @ (push @ X68 @ id)))) | ((X61 @ X68) != $true)) & ! [X69 : term] : ((X60 @ id @ X69 @ id) = (X61 @ X69)) & ! [X63 : subst > term > term] : (($true = (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) | ? [X65 : subst,X66 : subst,X67 : term] : ((sub @ (X63 @ X66 @ X67) @ X65) != (X63 @ (comp @ X66 @ X65) @ (sub @ X67 @ X65))) | ? [X64 : term] : (((X60 @ id @ X64 @ id) = $true) & ((X60 @ id @ (X63 @ id @ X64) @ id) != $true))) & ((X61 @ (lam @ X62)) != $true))) | ? [X59 : term] : ((sub @ X59 @ id) != X59)) | ? [X52 : term > $o,X53 : term] : (! [X54 : term,X55 : term] : (($true != (X52 @ X54)) | ((X52 @ (ap @ X55 @ X54)) = $true) | ((X52 @ X55) != $true)) & ! [X57 : term] : (? [X58 : term] : (($true = (X52 @ X58)) & ((X52 @ (sub @ X57 @ (push @ X58 @ id))) != $true)) | ((X52 @ (lam @ X57)) = $true)) & ! [X56 : term] : (((var @ X56) != $true) | ($true = (X52 @ X56))) & ((X52 @ X53) != $true))) | ? [X49 : subst > term > subst > $o] : ! [X50 : term > $o] : ? [X51 : term] : ((X49 @ id @ X51 @ id) != (X50 @ X51))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lem0 = $true) <=> ! [X333 : subst > term > subst > $o] : ? [X334 : term > $o] : ! [X335 : term] : ((X334 @ X335) = (X333 @ id @ X335 @ id))) & (axclos = $true) & ((hoasinduction_lem3 = $true) <=> ! [X477 : term,X478 : subst > term > subst > $o] : (((X478 @ id @ (lam @ X477) @ id) = $true) | ? [X483 : term] : (($true != (X478 @ id @ (sub @ X477 @ (push @ X483 @ id)) @ id)) & ((X478 @ id @ X483 @ id) = $true)) | ? [X489 : subst,X490 : subst,X491 : term,X492 : subst] : (((X478 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490) @ X489) = $true) & ((X478 @ X492 @ X491 @ (comp @ X490 @ X489)) != $true)) | ? [X479 : subst,X480 : subst,X481 : subst,X482 : term] : (((X478 @ (comp @ X481 @ X479) @ (sub @ X482 @ X479) @ X480) != $true) & ((X478 @ X481 @ X482 @ (comp @ X479 @ X480)) = $true)) | ? [X484 : subst > term > term] : (! [X488 : term] : (((X478 @ id @ (X484 @ id @ X488) @ id) = $true) | ($true != (X478 @ id @ X488 @ id))) & ! [X485 : subst,X486 : subst,X487 : term] : ((sub @ (X484 @ X486 @ X487) @ X485) = (X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485))) & ($true != (X478 @ id @ (lam @ (X484 @ sh @ one)) @ id))))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X138 : subst > term > subst > $o,X139 : term] : (((X138 @ id @ X139 @ id) = $true) | ? [X145 : term,X146 : term] : (((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) != $true) & ($true = (X138 @ id @ X145 @ id)) & ((X138 @ id @ X146 @ id) = $true)) | ? [X140 : subst > term > term] : (((X138 @ id @ (lam @ (X140 @ sh @ one)) @ id) != $true) & ! [X142 : subst,X143 : term,X144 : subst] : ((X140 @ (comp @ X142 @ X144) @ (sub @ X143 @ X144)) = (sub @ (X140 @ X142 @ X143) @ X144)) & ! [X141 : term] : (((X138 @ id @ X141 @ id) != $true) | ($true = (X138 @ id @ (X140 @ id @ X141) @ id)))))) & (apnotvar = $true) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X418 : term,X419 : subst > term > subst > $o] : (? [X420 : subst,X421 : subst,X422 : term,X423 : subst] : (((X419 @ X423 @ X422 @ (comp @ X421 @ X420)) != $true) & ((X419 @ (comp @ X423 @ X421) @ (sub @ X422 @ X421) @ X420) = $true)) | ? [X433 : term] : (((X419 @ id @ (sub @ X418 @ (push @ X433 @ id)) @ id) != $true) & ($true = (X419 @ id @ X433 @ id))) | ($true = (X419 @ id @ (lam @ X418) @ id)) | ? [X428 : subst > term > term] : (($true != (X419 @ id @ (lam @ (X428 @ sh @ one)) @ id)) & ! [X429 : term,X430 : subst,X431 : subst] : ((sub @ (X428 @ X431 @ X429) @ X430) = (X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430))) & ! [X432 : term] : (($true = (X419 @ id @ (X428 @ id @ X432) @ id)) | ((X419 @ id @ X432 @ id) != $true))) | ? [X424 : subst,X425 : subst,X426 : subst,X427 : term] : (($true != (X419 @ (comp @ X426 @ X425) @ (sub @ X427 @ X425) @ X424)) & ((X419 @ X426 @ X427 @ (comp @ X425 @ X424)) = $true))) | ? [X410 : term,X411 : subst > term > subst > $o] : (! [X417 : term] : (($true = (X411 @ id @ (sub @ X410 @ (push @ X417 @ id)) @ id)) | ((X411 @ id @ X417 @ id) != $true)) & ((X411 @ id @ (lam @ (sub @ X410 @ (push @ one @ sh))) @ id) != $true) & ! [X412 : subst > term > term] : (? [X414 : subst,X415 : subst,X416 : term] : ((X412 @ (comp @ X414 @ X415) @ (sub @ X416 @ X415)) != (sub @ (X412 @ X414 @ X416) @ X415)) | ? [X413 : term] : (((X411 @ id @ (X412 @ id @ X413) @ id) != $true) & ($true = (X411 @ id @ X413 @ id))) | ((X411 @ id @ (lam @ (X412 @ sh @ one)) @ id) = $true)))) | ? [X409 : term] : ((sub @ X409 @ id) != X409))) & (hoaslaminj_gthm = $true) & ! [X112 : d_subst,X113 : d_subst] : (((d2subst @ X112) != (d2subst @ X113)) | (X112 = X113)) & ! [X114 : term] : ? [X115 : d_term] : ((d2term @ X115) = X114) & (hoasapnotvar = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoaslaminj = $true) <=> ! [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (((lam @ (X191 @ sh @ one)) != (lam @ (X190 @ sh @ one))) | ? [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) != (sub @ (X190 @ X196 @ X194) @ X195)) | ((X191 @ X192 @ X193) = (X190 @ X192 @ X193)) | ? [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) != (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199))))) & ((d2subst @ d_id) = id) & (((((! [X221 : term,X222 : subst > term > subst > $o] : (? [X223 : subst,X224 : term,X225 : subst,X226 : subst] : (($true = (X222 @ X225 @ X224 @ (comp @ X226 @ X223))) & ((X222 @ (comp @ X225 @ X226) @ (sub @ X224 @ X226) @ X223) != $true)) | ? [X229 : term] : (($true != (X222 @ id @ X229 @ id)) & ((var @ (sub @ X229 @ id)) = $true)) | ((X222 @ id @ X221 @ id) = $true) | ? [X227 : term,X228 : term] : (((X222 @ id @ X227 @ id) = $true) & ((X222 @ id @ X228 @ id) = $true) & ((X222 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id) != $true)) | ? [X234 : subst > term > term] : (((X222 @ id @ (lam @ (X234 @ sh @ one)) @ id) != $true) & ! [X236 : term,X237 : subst,X238 : subst] : ((X234 @ (comp @ X238 @ X237) @ (sub @ X236 @ X237)) = (sub @ (X234 @ X238 @ X236) @ X237)) & ! [X235 : term] : (($true = (X222 @ id @ (X234 @ id @ X235) @ id)) | ((X222 @ id @ X235 @ id) != $true))) | ? [X230 : subst,X231 : subst,X232 : term,X233 : subst] : (((X222 @ X233 @ X232 @ (comp @ X230 @ X231)) != $true) & ($true = (X222 @ (comp @ X233 @ X230) @ (sub @ X232 @ X230) @ X231)))) | ? [X211 : term > $o,X212 : term,X213 : subst > term > subst > $o] : (! [X216 : subst > term > term] : (((X213 @ id @ (lam @ (X216 @ sh @ one)) @ id) = $true) | ? [X220 : term] : (((X213 @ id @ X220 @ id) = $true) & ((X213 @ id @ (X216 @ id @ X220) @ id) != $true)) | ? [X217 : term,X218 : subst,X219 : subst] : ((sub @ (X216 @ X219 @ X217) @ X218) != (X216 @ (comp @ X219 @ X218) @ (sub @ X217 @ X218)))) & ! [X214 : term] : ((X213 @ id @ X214 @ id) = (X211 @ X214)) & ($true != (X211 @ (lam @ X212))) & ! [X215 : term] : (($true != (X211 @ X215)) | ($true = (X211 @ (sub @ X212 @ (push @ X215 @ id))))))) | ? [X210 : term] : ((sub @ X210 @ id) != X210)) | ? [X203 : term,X204 : term > $o] : (($true != (X204 @ X203)) & ! [X207 : term] : (? [X208 : term] : (($true != (X204 @ (sub @ X207 @ (push @ X208 @ id)))) & ($true = (X204 @ X208))) | ($true = (X204 @ (lam @ X207)))) & ! [X209 : term] : (((X204 @ X209) = $true) | ((var @ X209) != $true)) & ! [X205 : term,X206 : term] : (($true != (X204 @ X206)) | ((X204 @ X205) != $true) | ((X204 @ (ap @ X206 @ X205)) = $true)))) | ? [X200 : subst > term > subst > $o] : ! [X201 : term > $o] : ? [X202 : term] : ((X201 @ X202) != (X200 @ id @ X202 @ id))) <=> (hoasinduction_lthm_3 = $true)) & (hoaslamnotvar = $true) & (ulamvar1 != $true) & (! [X265 : term,X266 : subst > term > subst > $o,X267 : term > $o] : (? [X269 : term] : ((X267 @ X269) != (X266 @ id @ X269 @ id)) | ($true = (X267 @ (lam @ X265))) | ? [X270 : subst > term > term] : (! [X272 : subst,X273 : subst,X274 : term] : ((sub @ (X270 @ X272 @ X274) @ X273) = (X270 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273))) & ! [X271 : term] : (((X266 @ id @ X271 @ id) != $true) | ($true = (X266 @ id @ (X270 @ id @ X271) @ id))) & ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id) != $true)) | ? [X268 : term] : (((X267 @ X268) = $true) & ($true != (X267 @ (sub @ X265 @ (push @ X268 @ id)))))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (axapp = $true) & (hoasapnotvar_gthm = $true) & (axvarshift = $true) & (((! [X160 : term > $o,X161 : term] : (? [X162 : term,X163 : term] : (((X160 @ X163) = $true) & ($true != (X160 @ (ap @ X163 @ X162))) & ($true = (X160 @ X162))) | ? [X164 : term] : (((var @ X164) = $true) & ((X160 @ X164) != $true)) | ((X160 @ X161) = $true) | ? [X165 : term] : (($true != (X160 @ (lam @ X165))) & ! [X166 : term] : (($true != (X160 @ X166)) | ((X160 @ (sub @ X165 @ (push @ X166 @ id))) = $true)))) | ? [X152 : term > $o,X153 : term,X154 : subst] : (! [X158 : term] : (? [X159 : term] : (($true != (X152 @ (sub @ X158 @ (push @ X159 @ id)))) & ((X152 @ X159) = $true)) | ((X152 @ (lam @ X158)) = $true)) & ! [X157 : term] : (((X152 @ (sub @ X157 @ X154)) = $true) | ((var @ X157) != $true)) & ((X152 @ (sub @ X153 @ X154)) != $true) & ! [X155 : term,X156 : term] : (($true = (X152 @ (ap @ X155 @ X156))) | ((X152 @ X156) != $true) | ($true != (X152 @ X155))))) | ? [X151 : term] : ((sub @ X151 @ id) != X151)) <=> (induction2_lthm = $true)) & ((pushprop_lem0 = $true) <=> ! [X37 : term > $o,X38 : term,X39 : subst] : ? [X40 : term > $o] : ! [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) = (X40 @ X41))) & (pushprop_lem2v2_lthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((hoaslamnotap = $true) <=> ! [X336 : term,X337 : term,X338 : subst > term > term] : (? [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) != (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) | ((ap @ (sub @ X336 @ id) @ X337) != (lam @ (X338 @ sh @ one))))) & (induction2_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (hoasinduction_lem1v2 = $true) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & (! [X0 : subst,X1 : term > $o,X2 : term] : (? [X6 : term] : (! [X7 : term] : (($true != (X1 @ X7)) | ($true = (X1 @ (sub @ X6 @ (push @ X7 @ id))))) & ((X1 @ (lam @ X6)) != $true)) | ((X1 @ (sub @ X2 @ X0)) = $true) | ? [X5 : term] : (((var @ X5) = $true) & ((X1 @ (sub @ X5 @ X0)) != $true)) | ? [X3 : term,X4 : term] : (((X1 @ X3) = $true) & ((X1 @ (ap @ X3 @ X4)) != $true) & ((X1 @ X4) = $true))) <=> (induction2lem = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (axassoc = $true) & ((pushprop_lem1 = $true) <=> ! [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (((X133 @ (sub @ X134 @ (push @ X135 @ X136))) = $true) | ($true != (X137 @ X134)))) & ! [X450 : d_term] : (d_one = X450) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (lamnotvar = $true) & (pushprop = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvarsh = $true) & (pushprop_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 = X101) | ((ap @ (sub @ X98 @ id) @ X100) != (ap @ (sub @ X99 @ id) @ X101)))) & ((((! [X440 : term,X441 : subst > term > term,X442 : subst > term > term,X443 : subst] : (? [X444 : term,X445 : subst,X446 : subst] : ((X441 @ (comp @ X446 @ X445) @ (sub @ X444 @ X445)) != (sub @ (X441 @ X446 @ X444) @ X445)) | ((X441 @ X443 @ X440) = (X442 @ X443 @ X440)) | ((lam @ (X441 @ sh @ one)) != (lam @ (X442 @ sh @ one))) | ? [X447 : subst,X448 : subst,X449 : term] : ((X442 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447)) != (sub @ (X442 @ X448 @ X449) @ X447))) | ? [X438 : term,X439 : term] : ((X438 != X439) & ((lam @ X438) = (lam @ X439)))) | ? [X436 : term,X437 : subst] : ((comp @ sh @ (push @ X436 @ X437)) != X437)) | ? [X434 : term,X435 : subst] : ((sub @ one @ (push @ X434 @ X435)) != X434)) <=> (hoaslaminj_lthm = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X393 : term,X394 : subst > term > subst > $o,X395 : term > $o] : (? [X402 : term] : ((X395 @ X402) != (X394 @ id @ X402 @ id)) | ? [X397 : subst > term > term] : (! [X399 : term,X400 : subst,X401 : subst] : ((sub @ (X397 @ X400 @ X399) @ X401) = (X397 @ (comp @ X400 @ X401) @ (sub @ X399 @ X401))) & ! [X398 : term] : (((X394 @ id @ X398 @ id) != $true) | ((X394 @ id @ (X397 @ id @ X398) @ id) = $true)) & ((X394 @ id @ (lam @ (X397 @ sh @ one)) @ id) != $true)) | ((X395 @ (lam @ X393)) = $true) | ? [X396 : term] : (((X395 @ (sub @ X393 @ (push @ X396 @ id))) != $true) & ((X395 @ X396) = $true))) | ? [X392 : term] : ((sub @ X392 @ id) != X392)) | ? [X388 : term] : ! [X389 : subst > term > term] : ? [X390 : term,X391 : subst] : ((sub @ X388 @ (push @ X390 @ X391)) != (X389 @ X391 @ X390)))) & (axabs = $true) & (axmap = $true) & ! [X464 : subst] : ? [X465 : d_subst] : ((d2subst @ X465) = X464) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X342 : term,X343 : subst > term > subst > $o] : (? [X344 : term] : (((X343 @ id @ X344 @ id) = $true) & ((X343 @ id @ (sub @ X342 @ (push @ X344 @ id)) @ id) != $true)) | ? [X345 : subst > term > term] : (! [X346 : term,X347 : subst,X348 : subst] : ((sub @ (X345 @ X347 @ X346) @ X348) = (X345 @ (comp @ X347 @ X348) @ (sub @ X346 @ X348))) & ! [X349 : term] : (((X343 @ id @ X349 @ id) != $true) | ($true = (X343 @ id @ (X345 @ id @ X349) @ id))) & ((X343 @ id @ (lam @ (X345 @ sh @ one)) @ id) != $true)) | ((X343 @ id @ (lam @ (sub @ X342 @ (push @ one @ sh))) @ id) = $true))) & ((substmonoid = $true) <=> (! [X246 : subst] : ((comp @ id @ X246) = X246) & ! [X245 : subst] : ((comp @ X245 @ id) = X245))) & ((((! [X315 : term,X316 : subst > term > subst > $o] : (? [X328 : subst,X329 : subst,X330 : subst,X331 : term] : (((X316 @ X328 @ X331 @ (comp @ X329 @ X330)) != $true) & ((X316 @ (comp @ X328 @ X329) @ (sub @ X331 @ X329) @ X330) = $true)) | ? [X319 : subst > term > term] : (! [X323 : term] : (((X316 @ id @ (X319 @ id @ X323) @ id) = $true) | ((X316 @ id @ X323 @ id) != $true)) & ! [X320 : term,X321 : subst,X322 : subst] : ((sub @ (X319 @ X321 @ X320) @ X322) = (X319 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322))) & ($true != (X316 @ id @ (lam @ (X319 @ sh @ one)) @ id))) | ? [X324 : subst,X325 : term,X326 : subst,X327 : subst] : (($true = (X316 @ X324 @ X325 @ (comp @ X327 @ X326))) & ((X316 @ (comp @ X324 @ X327) @ (sub @ X325 @ X327) @ X326) != $true)) | ((X316 @ id @ X315 @ id) = $true) | ? [X332 : term] : (((var @ (sub @ X332 @ id)) = $true) & ((X316 @ id @ X332 @ id) != $true)) | ? [X317 : term,X318 : term] : (((X316 @ id @ X318 @ id) = $true) & ((X316 @ id @ X317 @ id) = $true) & ($true != (X316 @ id @ (ap @ (sub @ X318 @ id) @ X317) @ id)))) | ? [X299 : term,X300 : subst > term > subst > $o] : (! [X302 : subst,X303 : subst,X304 : term,X305 : subst] : (((X300 @ (comp @ X305 @ X302) @ (sub @ X304 @ X302) @ X303) = $true) | ((X300 @ X305 @ X304 @ (comp @ X302 @ X303)) != $true)) & ((X300 @ id @ (lam @ X299) @ id) != $true) & ! [X301 : term] : (($true != (X300 @ id @ X301 @ id)) | ((X300 @ id @ (sub @ X299 @ (push @ X301 @ id)) @ id) = $true)) & ! [X310 : subst > term > term] : (? [X312 : term,X313 : subst,X314 : subst] : ((X310 @ (comp @ X314 @ X313) @ (sub @ X312 @ X313)) != (sub @ (X310 @ X314 @ X312) @ X313)) | ((X300 @ id @ (lam @ (X310 @ sh @ one)) @ id) = $true) | ? [X311 : term] : (((X300 @ id @ (X310 @ id @ X311) @ id) != $true) & ($true = (X300 @ id @ X311 @ id)))) & ! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (((X300 @ (comp @ X307 @ X308) @ (sub @ X306 @ X308) @ X309) != $true) | ($true = (X300 @ X307 @ X306 @ (comp @ X308 @ X309)))))) | ? [X286 : term,X287 : subst > term > subst > $o,X288 : term] : (! [X295 : subst,X296 : subst,X297 : term,X298 : subst] : (($true = (X287 @ X295 @ X297 @ (comp @ X296 @ X298))) | ((X287 @ (comp @ X295 @ X296) @ (sub @ X297 @ X296) @ X298) != $true)) & ($true = (X287 @ id @ X286 @ id)) & ((X287 @ id @ X288 @ id) = $true) & ! [X289 : subst,X290 : subst,X291 : term,X292 : subst] : (((X287 @ (comp @ X289 @ X290) @ (sub @ X291 @ X290) @ X292) = $true) | ($true != (X287 @ X289 @ X291 @ (comp @ X290 @ X292)))) & ($true != (X287 @ id @ (ap @ X286 @ X288) @ id)) & ! [X293 : term,X294 : term] : (((X287 @ id @ X294 @ id) != $true) | ((X287 @ id @ (ap @ (sub @ X293 @ id) @ X294) @ id) = $true) | ($true != (X287 @ id @ X293 @ id))))) | ? [X279 : term,X280 : term > $o] : (((X280 @ X279) != $true) & ! [X282 : term] : (? [X283 : term] : (($true != (X280 @ (sub @ X282 @ (push @ X283 @ id)))) & ((X280 @ X283) = $true)) | ((X280 @ (lam @ X282)) = $true)) & ! [X284 : term,X285 : term] : (($true != (X280 @ X285)) | ((X280 @ (ap @ X285 @ X284)) = $true) | ($true != (X280 @ X284))) & ! [X281 : term] : (((X280 @ X281) = $true) | ($true != (var @ X281))))) <=> (hoasinduction_lthm = $true)) & (! [X10 : subst > term > subst > $o] : ? [X11 : term > $o] : ! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (hoasinduction_lem1v2_gthm = $true) & (hoasapinj1_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((termmset = $true) <=> ! [X247 : term] : ((sub @ X247 @ id) = X247)) & (! [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) != (ap @ X276 @ X277)) | (X276 = X278)) <=> (apinj1 = $true)) & ((! [X80 : term,X81 : subst > term > subst > $o,X82 : term > $o] : (((X82 @ (lam @ X80)) = $true) | ? [X89 : subst,X90 : subst,X91 : subst,X92 : term] : (($true = (X81 @ X90 @ X92 @ (comp @ X91 @ X89))) & ((X81 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91) @ X89) != $true)) | ? [X97 : term] : (((X82 @ (sub @ X80 @ (push @ X97 @ id))) != $true) & ((X82 @ X97) = $true)) | ? [X83 : term] : ((X82 @ X83) != (X81 @ id @ X83 @ id)) | ? [X93 : term,X94 : subst,X95 : subst,X96 : subst] : (($true != (X81 @ X94 @ X93 @ (comp @ X95 @ X96))) & ((X81 @ (comp @ X94 @ X95) @ (sub @ X93 @ X95) @ X96) = $true)) | ? [X84 : subst > term > term] : (! [X86 : term,X87 : subst,X88 : subst] : ((sub @ (X84 @ X88 @ X86) @ X87) = (X84 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) & ! [X85 : term] : (((X81 @ id @ X85 @ id) != $true) | ((X81 @ id @ (X84 @ id @ X85) @ id) = $true)) & ((X81 @ id @ (lam @ (X84 @ sh @ one)) @ id) != $true))) | ? [X79 : term] : ((sub @ X79 @ id) != X79)) <=> (hoasinduction_lem3v2_lthm = $true)) & (pushprop_lthm_orig = $true) & ((substmonoid_lthm = $true) <=> (((! [X538 : subst] : ((comp @ X538 @ id) = X538) & ! [X539 : subst] : ((comp @ id @ X539) = X539)) | ? [X537 : subst] : ((comp @ X537 @ id) != X537)) | ? [X536 : subst] : ((comp @ id @ X536) != X536))) & (hoasinduction_lem3b_gthm = $true) & (substmonoid_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop_lem2v2 = $true) & ((d2subst @ d_id) = sh) & ((laminj = $true) <=> ! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148)))) & (! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term,X372 : term] : (? [X382 : term,X383 : term] : (((X370 @ id @ (ap @ (sub @ X383 @ id) @ X382) @ id) != $true) & ((X370 @ id @ X383 @ id) = $true) & ((X370 @ id @ X382 @ id) = $true)) | ($true != (X369 @ X371)) | ((X369 @ X372) != $true) | ((X369 @ (ap @ X372 @ X371)) = $true) | ? [X373 : subst,X374 : subst,X375 : term,X376 : subst] : (((X370 @ (comp @ X374 @ X373) @ (sub @ X375 @ X373) @ X376) != $true) & ((X370 @ X374 @ X375 @ (comp @ X373 @ X376)) = $true)) | ? [X378 : term,X379 : subst,X380 : subst,X381 : subst] : (($true != (X370 @ X380 @ X378 @ (comp @ X381 @ X379))) & ((X370 @ (comp @ X380 @ X381) @ (sub @ X378 @ X381) @ X379) = $true)) | ? [X377 : term] : ((X369 @ X377) != (X370 @ id @ X377 @ id))) <=> (hoasinduction_lem2v2 = $true)) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X451 : subst > term > subst > $o,X452 : term,X453 : term] : (? [X460 : subst,X461 : subst,X462 : term,X463 : subst] : (((X451 @ X460 @ X462 @ (comp @ X463 @ X461)) != $true) & ((X451 @ (comp @ X460 @ X463) @ (sub @ X462 @ X463) @ X461) = $true)) | ((X451 @ id @ X453 @ id) != $true) | ? [X458 : term,X459 : term] : (($true != (X451 @ id @ (ap @ (sub @ X458 @ id) @ X459) @ id)) & ($true = (X451 @ id @ X459 @ id)) & ((X451 @ id @ X458 @ id) = $true)) | ? [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (((X451 @ X457 @ X456 @ (comp @ X455 @ X454)) = $true) & ($true != (X451 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454))) | ((X451 @ id @ (ap @ X453 @ X452) @ id) = $true) | ((X451 @ id @ X452 @ id) != $true))) & (! [X121 : term,X122 : subst > term > subst > $o] : (? [X123 : term] : (((X122 @ id @ X123 @ id) = $true) & ((X122 @ id @ (sub @ X121 @ (push @ X123 @ id)) @ id) != $true)) | ($true = (X122 @ id @ (lam @ (sub @ X121 @ (push @ one @ sh))) @ id)) | ? [X124 : term,X125 : subst > term > term] : (! [X126 : subst,X127 : term,X128 : subst] : ((sub @ (sub @ X124 @ (push @ X127 @ X128)) @ X126) = (sub @ (X125 @ X128 @ X127) @ X126)) & ((X122 @ id @ (lam @ (X125 @ sh @ one)) @ id) != $true) & ! [X129 : term] : (((X122 @ id @ X129 @ id) != $true) | ((X122 @ id @ (X125 @ id @ X129) @ id) = $true)) & ! [X130 : term,X131 : subst,X132 : subst] : ((sub @ X124 @ (push @ (sub @ X130 @ X132) @ (comp @ X131 @ X132))) = (X125 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))))) <=> (hoasinduction_lem3aaa = $true)) & (! [X467 : term] : ? [X468 : subst > term > term] : ((X468 @ sh @ one) = (sub @ X467 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & ((var @ (d2term @ d_one)) != $true) & (pushprop_lem3v2 = $true) & ((induction2lem_lthm = $true) <=> ((((! [X361 : term,X362 : term > $o,X363 : subst] : (? [X365 : term] : (! [X366 : term] : (($true = (X362 @ (sub @ X365 @ (push @ X366 @ id)))) | ((X362 @ X366) != $true)) & ($true != (X362 @ (lam @ X365)))) | ? [X364 : term] : (((var @ X364) = $true) & ($true != (X362 @ (sub @ X364 @ X363)))) | ? [X367 : term,X368 : term] : (((X362 @ X367) = $true) & ((X362 @ X368) = $true) & ($true != (X362 @ (ap @ X368 @ X367)))) | ((X362 @ (sub @ X361 @ X363)) = $true)) | ? [X355 : term,X356 : term > $o] : (((X356 @ X355) != $true) & ! [X358 : term] : (($true = (X356 @ (lam @ X358))) | ((X356 @ X358) != $true)) & ! [X357 : term] : (((var @ X357) != $true) | ($true = (X356 @ X357))) & ! [X359 : term,X360 : term] : (($true != (X356 @ X360)) | ($true = (X356 @ (ap @ X360 @ X359))) | ($true != (X356 @ X359))))) | ? [X354 : subst] : ((comp @ X354 @ id) != X354)) | ? [X352 : term,X353 : subst] : ((comp @ sh @ (push @ X352 @ X353)) != X353)) | ? [X350 : subst,X351 : term] : ((sub @ one @ (push @ X351 @ X350)) != X351))) & (hoasapnotvar_lthm = $true) & (! [X178 : term] : ? [X179 : subst > term > term] : ! [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) = (X179 @ X180 @ X181)) <=> (hoasinduction_lem3v2_f = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X116 : term,X117 : subst,X118 : term > $o] : ? [X119 : term > $o] : ! [X120 : term] : ((X119 @ X120) = (X118 @ (sub @ X120 @ (push @ X116 @ X117))))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X384 : term] : ? [X385 : subst > term > term] : ! [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) = (X385 @ X386 @ X387))) & (! [X42 : term > $o,X43 : term] : (? [X44 : term] : (((var @ X44) = $true) & ((X42 @ X44) != $true)) | ? [X47 : term,X48 : term] : (($true != (X42 @ (ap @ X47 @ X48))) & ($true = (X42 @ X47)) & ((X42 @ X48) = $true)) | ? [X45 : term] : (((X42 @ (lam @ X45)) != $true) & ! [X46 : term] : (($true != (X42 @ X46)) | ($true = (X42 @ (sub @ X45 @ (push @ X46 @ id)))))) | ((X42 @ X43) = $true)) <=> (induction2 = $true)) & (ulamvarind = $true) & (termmset_gthm = $true) & (hoasinduction_gthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (? [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) != X168) | ! [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (? [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) != (X170 @ X173)) | ((X170 @ one) = $true) | ((X171 @ X172) != $true)))) & (((! [X257 : subst > term > subst > $o,X258 : term] : (((X257 @ id @ (lam @ X258) @ id) = $true) | ? [X259 : term] : (((X257 @ id @ X259 @ id) = $true) & ((X257 @ id @ (sub @ X258 @ (push @ X259 @ id)) @ id) != $true)) | ? [X260 : subst > term > term] : (! [X264 : term] : (((X257 @ id @ (X260 @ id @ X264) @ id) = $true) | ((X257 @ id @ X264 @ id) != $true)) & ! [X261 : term,X262 : subst,X263 : subst] : ((X260 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263)) = (sub @ (X260 @ X262 @ X261) @ X263)) & ((X257 @ id @ (lam @ (X260 @ sh @ one)) @ id) != $true))) | ? [X249 : subst > term > subst > $o,X250 : term] : (! [X256 : term] : (((X249 @ id @ (sub @ X250 @ (push @ X256 @ id)) @ id) = $true) | ((X249 @ id @ X256 @ id) != $true)) & ((X249 @ id @ (lam @ (sub @ X250 @ (push @ one @ sh))) @ id) != $true) & ! [X251 : subst > term > term] : (? [X255 : term] : (((X249 @ id @ (X251 @ id @ X255) @ id) != $true) & ((X249 @ id @ X255 @ id) = $true)) | ($true = (X249 @ id @ (lam @ (X251 @ sh @ one)) @ id)) | ? [X252 : subst,X253 : term,X254 : subst] : ((sub @ (X251 @ X252 @ X253) @ X254) != (X251 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254)))))) | ? [X248 : term] : ((sub @ X248 @ id) != X248)) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X182 : subst > term > subst > $o,X183 : term] : (? [X185 : subst > term > term] : (! [X186 : term] : (($true != (X182 @ id @ X186 @ id)) | ((X182 @ id @ (X185 @ id @ X186) @ id) = $true)) & ! [X187 : term,X188 : subst,X189 : subst] : ((X185 @ (comp @ X189 @ X188) @ (sub @ X187 @ X188)) = (sub @ (X185 @ X189 @ X187) @ X188)) & ((X182 @ id @ (lam @ (X185 @ sh @ one)) @ id) != $true)) | ((X182 @ id @ (lam @ (sub @ X183 @ (push @ one @ sh))) @ id) = $true) | ? [X184 : term] : (((X182 @ id @ X184 @ id) = $true) & ((X182 @ id @ (sub @ X183 @ (push @ X184 @ id)) @ id) != $true)))) & (! [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) = X240) <=> (axshiftcons = $true)) & (! [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (((X474 @ X475) != $true) | ? [X476 : term] : ((X472 @ X476) != (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) | ((X472 @ one) = $true)) <=> (pushprop_lem1v2 = $true)) & (hoaslamnotvar_lthm = $true) & (induction2lem_gthm = $true) & ! [X177 : d_subst] : (d_id = X177) & (lamnotap != $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ! [X469 : d_term,X470 : d_term] : (((d2term @ X469) != (d2term @ X470)) | (X469 = X470)) & ((induction = $true) <=> ! [X403 : term > $o,X404 : term] : (? [X405 : term,X406 : term] : (((X403 @ X405) = $true) & ((X403 @ (ap @ X405 @ X406)) != $true) & ($true = (X403 @ X406))) | ? [X407 : term] : (($true = (var @ X407)) & ($true != (X403 @ X407))) | ((X403 @ X404) = $true) | ? [X408 : term] : (((X403 @ (lam @ X408)) != $true) & ((X403 @ X408) = $true)))) & (pushprop_lem0_gthm = $true) & (pushprop_lthm = $true) & (! [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) != (ap @ X243 @ X241)) | (X241 = X242)) <=> (apinj2 = $true)) & (pushprop_lem1_gthm = $true) & ((hoasinduction = $true) <=> ! [X493 : term,X494 : subst > term > subst > $o] : (($true = (X494 @ id @ X493 @ id)) | ? [X499 : term,X500 : term] : (((X494 @ id @ X500 @ id) = $true) & ((X494 @ id @ (ap @ (sub @ X500 @ id) @ X499) @ id) != $true) & ((X494 @ id @ X499 @ id) = $true)) | ? [X507 : subst,X508 : subst,X509 : term,X510 : subst] : (((X494 @ X510 @ X509 @ (comp @ X508 @ X507)) = $true) & ($true != (X494 @ (comp @ X510 @ X508) @ (sub @ X509 @ X508) @ X507))) | ? [X501 : subst > term > term] : (($true != (X494 @ id @ (lam @ (X501 @ sh @ one)) @ id)) & ! [X505 : term] : (((X494 @ id @ X505 @ id) != $true) | ((X494 @ id @ (X501 @ id @ X505) @ id) = $true)) & ! [X502 : subst,X503 : term,X504 : subst] : ((X501 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X501 @ X502 @ X503) @ X504))) | ? [X495 : term,X496 : subst,X497 : subst,X498 : subst] : (($true = (X494 @ (comp @ X498 @ X497) @ (sub @ X495 @ X497) @ X496)) & ((X494 @ X498 @ X495 @ (comp @ X497 @ X496)) != $true)) | ? [X506 : term] : (((var @ (sub @ X506 @ id)) = $true) & ($true != (X494 @ id @ X506 @ id))))) & ((axidl = $true) <=> ! [X471 : subst] : ((comp @ id @ X471) = X471)) & (! [X17 : term,X18 : subst > term > subst > $o,X19 : term > $o] : (? [X22 : subst > term > term] : (! [X26 : term] : (($true != (X18 @ id @ X26 @ id)) | ((X18 @ id @ (X22 @ id @ X26) @ id) = $true)) & ! [X23 : subst,X24 : subst,X25 : term] : ((X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (X22 @ X24 @ X25) @ X23)) & ($true != (X18 @ id @ (lam @ (X22 @ sh @ one)) @ id))) | ? [X21 : term] : ((X19 @ X21) != (X18 @ id @ X21 @ id)) | ? [X20 : term] : (($true = (X19 @ X20)) & ((X19 @ (sub @ X17 @ (push @ X20 @ id))) != $true)) | ((X19 @ (lam @ X17)) = $true) | ? [X27 : subst,X28 : term,X29 : subst,X30 : subst] : (($true = (X18 @ X27 @ X28 @ (comp @ X29 @ X30))) & ((X18 @ (comp @ X27 @ X29) @ (sub @ X28 @ X29) @ X30) != $true)) | ? [X31 : subst,X32 : subst,X33 : subst,X34 : term] : (($true = (X18 @ (comp @ X33 @ X32) @ (sub @ X34 @ X32) @ X31)) & ((X18 @ X33 @ X34 @ (comp @ X32 @ X31)) != $true))) <=> (hoasinduction_lem3v2 = $true)) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoaslamnotap_lthm = $true) & (! [X174 : term] : ((sub @ X174 @ id) = X174) <=> (axvarid = $true)) & (hoasinduction_lem1 = $true) & (one = (d2term @ d_one)) & ((! [X150 : term] : ((sub @ X150 @ id) = X150) | ? [X149 : term] : ((sub @ X149 @ id) != X149)) <=> (termmset_lthm = $true)) & (((! [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (($true = (X524 @ (sub @ X527 @ (push @ X525 @ X526)))) | ($true != (X523 @ X527))) | ? [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) != X521)) | ? [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) != X520)) <=> (pushprop_lem1_lthm = $true)) & (! [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) = X176) <=> (axvarcons = $true)) & (hoasinduction_lem2_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X466 : subst] : ((comp @ X466 @ id) = X466) <=> (axidr = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X8 : term] : ? [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) = (X9 @ sh @ one))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasapinj2_lthm = $true) <=> (! [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 = X534) | ((ap @ (sub @ X533 @ id) @ X532) != (ap @ (sub @ X535 @ id) @ X534))) | ? [X528 : term,X529 : term,X530 : term,X531 : term] : (((ap @ X531 @ X529) = (ap @ X528 @ X530)) & (X529 != X530)))) & (hoaslamnotvar_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((shinj = $true) <=> ! [X35 : term,X36 : term] : ((X35 = X36) | ((sub @ X36 @ sh) != (sub @ X35 @ sh)))) & ((hoasapinj1_lthm = $true) <=> ((! [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 = X109) | ((ap @ (sub @ X109 @ id) @ X111) != (ap @ (sub @ X108 @ id) @ X110))) | ? [X104 : term,X105 : term,X106 : term,X107 : term] : ((X106 != X107) & ((ap @ X106 @ X105) = (ap @ X107 @ X104)))) | ? [X103 : term] : ((sub @ X103 @ id) != X103))) & (hoasinduction_lem1_gthm = $true) & (! [X511 : term,X512 : subst > term > subst > $o] : (? [X518 : term] : (((X512 @ id @ (sub @ X511 @ (push @ X518 @ id)) @ id) != $true) & ((X512 @ id @ X518 @ id) = $true)) | ((X512 @ id @ (lam @ X511) @ id) = $true) | ? [X513 : subst > term > term] : (((X512 @ id @ (lam @ (X513 @ sh @ one)) @ id) != $true) & ! [X514 : subst,X515 : term,X516 : subst] : ((X513 @ (comp @ X514 @ X516) @ (sub @ X515 @ X516)) = (sub @ (X513 @ X514 @ X515) @ X516)) & ! [X517 : term] : (($true = (X512 @ id @ (X513 @ id @ X517) @ id)) | ((X512 @ id @ X517 @ id) != $true)))) <=> (hoasinduction_lem3a = $true))),
% 2.65/1.76 inference(ennf_transformation,[status(thm)],[f11])).
% 2.65/1.76 thf(f11,plain,(
% 2.65/1.76 ((hoasapinj1 = $true) <=> ! [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 = X16) | ((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15)))) & ((! [X49 : subst > term > subst > $o] : ~! [X50 : term > $o] : ~! [X51 : term] : ((X49 @ id @ X51 @ id) = (X50 @ X51)) => (! [X52 : term > $o,X53 : term] : (~! [X57 : term] : (~! [X58 : term] : (($true != (X52 @ X58)) | ((X52 @ (sub @ X57 @ (push @ X58 @ id))) = $true)) | ((X52 @ (lam @ X57)) = $true)) | ~! [X54 : term,X55 : term] : (($true != (X52 @ X54)) | ((X52 @ (ap @ X55 @ X54)) = $true) | ((X52 @ X55) != $true)) | ((X52 @ X53) = $true) | ~! [X56 : term] : (((var @ X56) != $true) | ($true = (X52 @ X56)))) => (! [X59 : term] : ((sub @ X59 @ id) = X59) => (! [X60 : subst > term > subst > $o,X61 : term > $o,X62 : term] : (~! [X69 : term] : ((X60 @ id @ X69 @ id) = (X61 @ X69)) | ~! [X63 : subst > term > term] : (($true = (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id)) | ~! [X64 : term] : (((X60 @ id @ (X63 @ id @ X64) @ id) = $true) | ((X60 @ id @ X64 @ id) != $true)) | ~! [X65 : subst,X66 : subst,X67 : term] : ((sub @ (X63 @ X66 @ X67) @ X65) = (X63 @ (comp @ X66 @ X65) @ (sub @ X67 @ X65)))) | ((X61 @ (lam @ X62)) = $true) | ~! [X68 : term] : (($true = (X61 @ (sub @ X62 @ (push @ X68 @ id)))) | ((X61 @ X68) != $true))) => ! [X70 : subst > term > subst > $o,X71 : term] : (~! [X72 : subst > term > term] : (((X70 @ id @ (lam @ (X72 @ sh @ one)) @ id) = $true) | ~! [X73 : subst,X74 : term,X75 : subst] : ((X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73)) = (sub @ (X72 @ X75 @ X74) @ X73)) | ~! [X76 : term] : (((X70 @ id @ (X72 @ id @ X76) @ id) = $true) | ($true != (X70 @ id @ X76 @ id)))) | ~! [X77 : term,X78 : term] : (($true != (X70 @ id @ X77 @ id)) | ((X70 @ id @ X78 @ id) != $true) | ((X70 @ id @ (ap @ (sub @ X78 @ id) @ X77) @ id) = $true)) | ($true = (X70 @ id @ X71 @ id))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lem0 = $true) <=> ! [X333 : subst > term > subst > $o] : ~! [X334 : term > $o] : ~! [X335 : term] : ((X334 @ X335) = (X333 @ id @ X335 @ id))) & (axclos = $true) & (! [X477 : term,X478 : subst > term > subst > $o] : (((X478 @ id @ (lam @ X477) @ id) = $true) | ~! [X484 : subst > term > term] : (($true = (X478 @ id @ (lam @ (X484 @ sh @ one)) @ id)) | ~! [X485 : subst,X486 : subst,X487 : term] : ((sub @ (X484 @ X486 @ X487) @ X485) = (X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485))) | ~! [X488 : term] : (((X478 @ id @ (X484 @ id @ X488) @ id) = $true) | ($true != (X478 @ id @ X488 @ id)))) | ~! [X483 : term] : (((X478 @ id @ X483 @ id) != $true) | ($true = (X478 @ id @ (sub @ X477 @ (push @ X483 @ id)) @ id))) | ~! [X489 : subst,X490 : subst,X491 : term,X492 : subst] : (((X478 @ X492 @ X491 @ (comp @ X490 @ X489)) = $true) | ((X478 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490) @ X489) != $true)) | ~! [X479 : subst,X480 : subst,X481 : subst,X482 : term] : (((X478 @ X481 @ X482 @ (comp @ X479 @ X480)) != $true) | ((X478 @ (comp @ X481 @ X479) @ (sub @ X482 @ X479) @ X480) = $true))) <=> (hoasinduction_lem3 = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X138 : subst > term > subst > $o,X139 : term] : (~! [X145 : term,X146 : term] : (((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) = $true) | ((X138 @ id @ X146 @ id) != $true) | ($true != (X138 @ id @ X145 @ id))) | ((X138 @ id @ X139 @ id) = $true) | ~! [X140 : subst > term > term] : (((X138 @ id @ (lam @ (X140 @ sh @ one)) @ id) = $true) | ~! [X142 : subst,X143 : term,X144 : subst] : ((X140 @ (comp @ X142 @ X144) @ (sub @ X143 @ X144)) = (sub @ (X140 @ X142 @ X143) @ X144)) | ~! [X141 : term] : (((X138 @ id @ X141 @ id) != $true) | ($true = (X138 @ id @ (X140 @ id @ X141) @ id)))))) & (apnotvar = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X409 : term] : ((sub @ X409 @ id) = X409) => (! [X410 : term,X411 : subst > term > subst > $o] : (~! [X412 : subst > term > term] : (~! [X413 : term] : (((X411 @ id @ (X412 @ id @ X413) @ id) = $true) | ($true != (X411 @ id @ X413 @ id))) | ((X411 @ id @ (lam @ (X412 @ sh @ one)) @ id) = $true) | ~! [X414 : subst,X415 : subst,X416 : term] : ((X412 @ (comp @ X414 @ X415) @ (sub @ X416 @ X415)) = (sub @ (X412 @ X414 @ X416) @ X415))) | ~! [X417 : term] : (($true = (X411 @ id @ (sub @ X410 @ (push @ X417 @ id)) @ id)) | ((X411 @ id @ X417 @ id) != $true)) | ((X411 @ id @ (lam @ (sub @ X410 @ (push @ one @ sh))) @ id) = $true)) => ! [X418 : term,X419 : subst > term > subst > $o] : (~! [X420 : subst,X421 : subst,X422 : term,X423 : subst] : (((X419 @ (comp @ X423 @ X421) @ (sub @ X422 @ X421) @ X420) != $true) | ((X419 @ X423 @ X422 @ (comp @ X421 @ X420)) = $true)) | ($true = (X419 @ id @ (lam @ X418) @ id)) | ~! [X433 : term] : (($true != (X419 @ id @ X433 @ id)) | ((X419 @ id @ (sub @ X418 @ (push @ X433 @ id)) @ id) = $true)) | ~! [X428 : subst > term > term] : (~! [X432 : term] : (($true = (X419 @ id @ (X428 @ id @ X432) @ id)) | ((X419 @ id @ X432 @ id) != $true)) | ($true = (X419 @ id @ (lam @ (X428 @ sh @ one)) @ id)) | ~! [X429 : term,X430 : subst,X431 : subst] : ((sub @ (X428 @ X431 @ X429) @ X430) = (X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430)))) | ~! [X424 : subst,X425 : subst,X426 : subst,X427 : term] : (((X419 @ X426 @ X427 @ (comp @ X425 @ X424)) != $true) | ($true = (X419 @ (comp @ X426 @ X425) @ (sub @ X427 @ X425) @ X424))))))) & (hoaslaminj_gthm = $true) & ! [X112 : d_subst,X113 : d_subst] : (((d2subst @ X112) = (d2subst @ X113)) => (X112 = X113)) & ! [X114 : term] : ? [X115 : d_term] : ((d2term @ X115) = X114) & (hoasapnotvar = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoaslaminj = $true) <=> ! [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (~! [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) = (sub @ (X190 @ X196 @ X194) @ X195)) | ((X191 @ X192 @ X193) = (X190 @ X192 @ X193)) | ~! [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) = (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199))) | ((lam @ (X191 @ sh @ one)) != (lam @ (X190 @ sh @ one))))) & ((d2subst @ d_id) = id) & ((! [X200 : subst > term > subst > $o] : ~! [X201 : term > $o] : ~! [X202 : term] : ((X201 @ X202) = (X200 @ id @ X202 @ id)) => (! [X203 : term,X204 : term > $o] : (~! [X207 : term] : (~! [X208 : term] : (($true != (X204 @ X208)) | ($true = (X204 @ (sub @ X207 @ (push @ X208 @ id))))) | ($true = (X204 @ (lam @ X207)))) | ~! [X209 : term] : (((X204 @ X209) = $true) | ((var @ X209) != $true)) | ($true = (X204 @ X203)) | ~! [X205 : term,X206 : term] : (($true != (X204 @ X206)) | ((X204 @ X205) != $true) | ((X204 @ (ap @ X206 @ X205)) = $true))) => (! [X210 : term] : ((sub @ X210 @ id) = X210) => (! [X211 : term > $o,X212 : term,X213 : subst > term > subst > $o] : (($true = (X211 @ (lam @ X212))) | ~! [X216 : subst > term > term] : (((X213 @ id @ (lam @ (X216 @ sh @ one)) @ id) = $true) | ~! [X217 : term,X218 : subst,X219 : subst] : ((sub @ (X216 @ X219 @ X217) @ X218) = (X216 @ (comp @ X219 @ X218) @ (sub @ X217 @ X218))) | ~! [X220 : term] : (((X213 @ id @ X220 @ id) != $true) | ((X213 @ id @ (X216 @ id @ X220) @ id) = $true))) | ~! [X214 : term] : ((X213 @ id @ X214 @ id) = (X211 @ X214)) | ~! [X215 : term] : (($true != (X211 @ X215)) | ($true = (X211 @ (sub @ X212 @ (push @ X215 @ id)))))) => ! [X221 : term,X222 : subst > term > subst > $o] : (~! [X229 : term] : (((var @ (sub @ X229 @ id)) != $true) | ($true = (X222 @ id @ X229 @ id))) | ~! [X234 : subst > term > term] : (~! [X236 : term,X237 : subst,X238 : subst] : ((X234 @ (comp @ X238 @ X237) @ (sub @ X236 @ X237)) = (sub @ (X234 @ X238 @ X236) @ X237)) | ~! [X235 : term] : (($true = (X222 @ id @ (X234 @ id @ X235) @ id)) | ((X222 @ id @ X235 @ id) != $true)) | ((X222 @ id @ (lam @ (X234 @ sh @ one)) @ id) = $true)) | ((X222 @ id @ X221 @ id) = $true) | ~! [X230 : subst,X231 : subst,X232 : term,X233 : subst] : (((X222 @ X233 @ X232 @ (comp @ X230 @ X231)) = $true) | ($true != (X222 @ (comp @ X233 @ X230) @ (sub @ X232 @ X230) @ X231))) | ~! [X223 : subst,X224 : term,X225 : subst,X226 : subst] : (((X222 @ (comp @ X225 @ X226) @ (sub @ X224 @ X226) @ X223) = $true) | ($true != (X222 @ X225 @ X224 @ (comp @ X226 @ X223)))) | ~! [X227 : term,X228 : term] : (((X222 @ id @ X227 @ id) != $true) | ((X222 @ id @ X228 @ id) != $true) | ((X222 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (hoaslamnotvar = $true) & (ulamvar1 != $true) & (! [X265 : term,X266 : subst > term > subst > $o,X267 : term > $o] : (~! [X269 : term] : ((X267 @ X269) = (X266 @ id @ X269 @ id)) | ~! [X268 : term] : (($true = (X267 @ (sub @ X265 @ (push @ X268 @ id)))) | ((X267 @ X268) != $true)) | ($true = (X267 @ (lam @ X265))) | ~! [X270 : subst > term > term] : (((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id) = $true) | ~! [X272 : subst,X273 : subst,X274 : term] : ((sub @ (X270 @ X272 @ X274) @ X273) = (X270 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273))) | ~! [X271 : term] : (((X266 @ id @ X271 @ id) != $true) | ($true = (X266 @ id @ (X270 @ id @ X271) @ id))))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (axapp = $true) & (hoasapnotvar_gthm = $true) & (axvarshift = $true) & ((! [X151 : term] : ((sub @ X151 @ id) = X151) => (! [X152 : term > $o,X153 : term,X154 : subst] : (~! [X158 : term] : (~! [X159 : term] : (((X152 @ X159) != $true) | ($true = (X152 @ (sub @ X158 @ (push @ X159 @ id))))) | ((X152 @ (lam @ X158)) = $true)) | ~! [X155 : term,X156 : term] : (($true = (X152 @ (ap @ X155 @ X156))) | ((X152 @ X156) != $true) | ($true != (X152 @ X155))) | ~! [X157 : term] : (((X152 @ (sub @ X157 @ X154)) = $true) | ((var @ X157) != $true)) | ((X152 @ (sub @ X153 @ X154)) = $true)) => ! [X160 : term > $o,X161 : term] : (~! [X162 : term,X163 : term] : (((X160 @ X163) != $true) | ($true = (X160 @ (ap @ X163 @ X162))) | ($true != (X160 @ X162))) | ~! [X165 : term] : (($true = (X160 @ (lam @ X165))) | ~! [X166 : term] : (($true != (X160 @ X166)) | ((X160 @ (sub @ X165 @ (push @ X166 @ id))) = $true))) | ((X160 @ X161) = $true) | ~! [X164 : term] : (((var @ X164) != $true) | ((X160 @ X164) = $true))))) <=> (induction2_lthm = $true)) & (! [X37 : term > $o,X38 : term,X39 : subst] : ~! [X40 : term > $o] : ~! [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) = (X40 @ X41)) <=> (pushprop_lem0 = $true)) & (pushprop_lem2v2_lthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((hoaslamnotap = $true) <=> ! [X336 : term,X337 : term,X338 : subst > term > term] : (~! [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) = (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) | ((ap @ (sub @ X336 @ id) @ X337) != (lam @ (X338 @ sh @ one))))) & (induction2_gthm = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (hoasinduction_lem1v2 = $true) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & (! [X0 : subst,X1 : term > $o,X2 : term] : (~! [X6 : term] : (~! [X7 : term] : (($true != (X1 @ X7)) | ($true = (X1 @ (sub @ X6 @ (push @ X7 @ id))))) | ((X1 @ (lam @ X6)) = $true)) | ~! [X5 : term] : (((var @ X5) != $true) | ((X1 @ (sub @ X5 @ X0)) = $true)) | ~! [X3 : term,X4 : term] : (((X1 @ X3) != $true) | ((X1 @ X4) != $true) | ((X1 @ (ap @ X3 @ X4)) = $true)) | ((X1 @ (sub @ X2 @ X0)) = $true)) <=> (induction2lem = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (axassoc = $true) & ((pushprop_lem1 = $true) <=> ! [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (((X133 @ (sub @ X134 @ (push @ X135 @ X136))) = $true) | ($true != (X137 @ X134)))) & ! [X450 : d_term] : (d_one = X450) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (lamnotvar = $true) & (pushprop = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvarsh = $true) & (pushprop_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 = X101) | ((ap @ (sub @ X98 @ id) @ X100) != (ap @ (sub @ X99 @ id) @ X101)))) & ((! [X434 : term,X435 : subst] : ((sub @ one @ (push @ X434 @ X435)) = X434) => (! [X436 : term,X437 : subst] : ((comp @ sh @ (push @ X436 @ X437)) = X437) => (! [X438 : term,X439 : term] : (((lam @ X438) != (lam @ X439)) | (X438 = X439)) => ! [X440 : term,X441 : subst > term > term,X442 : subst > term > term,X443 : subst] : (((X441 @ X443 @ X440) = (X442 @ X443 @ X440)) | ~! [X447 : subst,X448 : subst,X449 : term] : ((X442 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447)) = (sub @ (X442 @ X448 @ X449) @ X447)) | ~! [X444 : term,X445 : subst,X446 : subst] : ((X441 @ (comp @ X446 @ X445) @ (sub @ X444 @ X445)) = (sub @ (X441 @ X446 @ X444) @ X445)) | ((lam @ (X441 @ sh @ one)) != (lam @ (X442 @ sh @ one))))))) <=> (hoaslaminj_lthm = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X388 : term] : ~! [X389 : subst > term > term] : ~! [X390 : term,X391 : subst] : ((sub @ X388 @ (push @ X390 @ X391)) = (X389 @ X391 @ X390)) => (! [X392 : term] : ((sub @ X392 @ id) = X392) => ! [X393 : term,X394 : subst > term > subst > $o,X395 : term > $o] : (~! [X397 : subst > term > term] : (~! [X399 : term,X400 : subst,X401 : subst] : ((sub @ (X397 @ X400 @ X399) @ X401) = (X397 @ (comp @ X400 @ X401) @ (sub @ X399 @ X401))) | ~! [X398 : term] : (((X394 @ id @ X398 @ id) != $true) | ((X394 @ id @ (X397 @ id @ X398) @ id) = $true)) | ((X394 @ id @ (lam @ (X397 @ sh @ one)) @ id) = $true)) | ~! [X396 : term] : (((X395 @ X396) != $true) | ((X395 @ (sub @ X393 @ (push @ X396 @ id))) = $true)) | ((X395 @ (lam @ X393)) = $true) | ~! [X402 : term] : ((X395 @ X402) = (X394 @ id @ X402 @ id)))))) & (axabs = $true) & (axmap = $true) & ! [X464 : subst] : ? [X465 : d_subst] : ((d2subst @ X465) = X464) & (! [X342 : term,X343 : subst > term > subst > $o] : (~! [X345 : subst > term > term] : (~! [X349 : term] : (((X343 @ id @ X349 @ id) != $true) | ($true = (X343 @ id @ (X345 @ id @ X349) @ id))) | ~! [X346 : term,X347 : subst,X348 : subst] : ((sub @ (X345 @ X347 @ X346) @ X348) = (X345 @ (comp @ X347 @ X348) @ (sub @ X346 @ X348))) | ((X343 @ id @ (lam @ (X345 @ sh @ one)) @ id) = $true)) | ((X343 @ id @ (lam @ (sub @ X342 @ (push @ one @ sh))) @ id) = $true) | ~! [X344 : term] : (((X343 @ id @ (sub @ X342 @ (push @ X344 @ id)) @ id) = $true) | ((X343 @ id @ X344 @ id) != $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((substmonoid = $true) <=> (! [X246 : subst] : ((comp @ id @ X246) = X246) & ! [X245 : subst] : ((comp @ X245 @ id) = X245))) & ((! [X279 : term,X280 : term > $o] : (~! [X284 : term,X285 : term] : (($true != (X280 @ X285)) | ((X280 @ (ap @ X285 @ X284)) = $true) | ($true != (X280 @ X284))) | ~! [X282 : term] : (((X280 @ (lam @ X282)) = $true) | ~! [X283 : term] : (($true = (X280 @ (sub @ X282 @ (push @ X283 @ id)))) | ((X280 @ X283) != $true))) | ((X280 @ X279) = $true) | ~! [X281 : term] : (((X280 @ X281) = $true) | ($true != (var @ X281)))) => (! [X286 : term,X287 : subst > term > subst > $o,X288 : term] : (($true != (X287 @ id @ X286 @ id)) | ($true = (X287 @ id @ (ap @ X286 @ X288) @ id)) | ~! [X295 : subst,X296 : subst,X297 : term,X298 : subst] : (($true = (X287 @ X295 @ X297 @ (comp @ X296 @ X298))) | ((X287 @ (comp @ X295 @ X296) @ (sub @ X297 @ X296) @ X298) != $true)) | ((X287 @ id @ X288 @ id) != $true) | ~! [X289 : subst,X290 : subst,X291 : term,X292 : subst] : (((X287 @ (comp @ X289 @ X290) @ (sub @ X291 @ X290) @ X292) = $true) | ($true != (X287 @ X289 @ X291 @ (comp @ X290 @ X292)))) | ~! [X293 : term,X294 : term] : (((X287 @ id @ X294 @ id) != $true) | ((X287 @ id @ (ap @ (sub @ X293 @ id) @ X294) @ id) = $true) | ($true != (X287 @ id @ X293 @ id)))) => (! [X299 : term,X300 : subst > term > subst > $o] : (((X300 @ id @ (lam @ X299) @ id) = $true) | ~! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (((X300 @ (comp @ X307 @ X308) @ (sub @ X306 @ X308) @ X309) != $true) | ($true = (X300 @ X307 @ X306 @ (comp @ X308 @ X309)))) | ~! [X301 : term] : (($true != (X300 @ id @ X301 @ id)) | ((X300 @ id @ (sub @ X299 @ (push @ X301 @ id)) @ id) = $true)) | ~! [X310 : subst > term > term] : (((X300 @ id @ (lam @ (X310 @ sh @ one)) @ id) = $true) | ~! [X312 : term,X313 : subst,X314 : subst] : ((X310 @ (comp @ X314 @ X313) @ (sub @ X312 @ X313)) = (sub @ (X310 @ X314 @ X312) @ X313)) | ~! [X311 : term] : (((X300 @ id @ (X310 @ id @ X311) @ id) = $true) | ($true != (X300 @ id @ X311 @ id)))) | ~! [X302 : subst,X303 : subst,X304 : term,X305 : subst] : (((X300 @ (comp @ X305 @ X302) @ (sub @ X304 @ X302) @ X303) = $true) | ((X300 @ X305 @ X304 @ (comp @ X302 @ X303)) != $true))) => ! [X315 : term,X316 : subst > term > subst > $o] : (~! [X324 : subst,X325 : term,X326 : subst,X327 : subst] : (((X316 @ (comp @ X324 @ X327) @ (sub @ X325 @ X327) @ X326) = $true) | ($true != (X316 @ X324 @ X325 @ (comp @ X327 @ X326)))) | ~! [X328 : subst,X329 : subst,X330 : subst,X331 : term] : (((X316 @ X328 @ X331 @ (comp @ X329 @ X330)) = $true) | ((X316 @ (comp @ X328 @ X329) @ (sub @ X331 @ X329) @ X330) != $true)) | ((X316 @ id @ X315 @ id) = $true) | ~! [X317 : term,X318 : term] : (((X316 @ id @ X318 @ id) != $true) | ($true = (X316 @ id @ (ap @ (sub @ X318 @ id) @ X317) @ id)) | ((X316 @ id @ X317 @ id) != $true)) | ~! [X319 : subst > term > term] : (($true = (X316 @ id @ (lam @ (X319 @ sh @ one)) @ id)) | ~! [X323 : term] : (((X316 @ id @ (X319 @ id @ X323) @ id) = $true) | ((X316 @ id @ X323 @ id) != $true)) | ~! [X320 : term,X321 : subst,X322 : subst] : ((sub @ (X319 @ X321 @ X320) @ X322) = (X319 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322)))) | ~! [X332 : term] : (((X316 @ id @ X332 @ id) = $true) | ((var @ (sub @ X332 @ id)) != $true)))))) <=> (hoasinduction_lthm = $true)) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (hoasinduction_lem1v2_gthm = $true) & (hoasapinj1_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((termmset = $true) <=> ! [X247 : term] : ((sub @ X247 @ id) = X247)) & (! [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) != (ap @ X276 @ X277)) | (X276 = X278)) <=> (apinj1 = $true)) & ((! [X79 : term] : ((sub @ X79 @ id) = X79) => ! [X80 : term,X81 : subst > term > subst > $o,X82 : term > $o] : (~! [X97 : term] : (((X82 @ (sub @ X80 @ (push @ X97 @ id))) = $true) | ((X82 @ X97) != $true)) | ~! [X89 : subst,X90 : subst,X91 : subst,X92 : term] : (((X81 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91) @ X89) = $true) | ($true != (X81 @ X90 @ X92 @ (comp @ X91 @ X89)))) | ~! [X93 : term,X94 : subst,X95 : subst,X96 : subst] : (($true = (X81 @ X94 @ X93 @ (comp @ X95 @ X96))) | ((X81 @ (comp @ X94 @ X95) @ (sub @ X93 @ X95) @ X96) != $true)) | ~! [X83 : term] : ((X82 @ X83) = (X81 @ id @ X83 @ id)) | ~! [X84 : subst > term > term] : (~! [X86 : term,X87 : subst,X88 : subst] : ((sub @ (X84 @ X88 @ X86) @ X87) = (X84 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) | ((X81 @ id @ (lam @ (X84 @ sh @ one)) @ id) = $true) | ~! [X85 : term] : (((X81 @ id @ X85 @ id) != $true) | ((X81 @ id @ (X84 @ id @ X85) @ id) = $true))) | ((X82 @ (lam @ X80)) = $true))) <=> (hoasinduction_lem3v2_lthm = $true)) & (pushprop_lthm_orig = $true) & ((substmonoid_lthm = $true) <=> (! [X536 : subst] : ((comp @ id @ X536) = X536) => (! [X537 : subst] : ((comp @ X537 @ id) = X537) => (! [X538 : subst] : ((comp @ X538 @ id) = X538) & ! [X539 : subst] : ((comp @ id @ X539) = X539))))) & (hoasinduction_lem3b_gthm = $true) & (substmonoid_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop_lem2v2 = $true) & ((d2subst @ d_id) = sh) & ((laminj = $true) <=> ! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148)))) & ((hoasinduction_lem2v2 = $true) <=> ! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term,X372 : term] : (~! [X382 : term,X383 : term] : (((X370 @ id @ X382 @ id) != $true) | ((X370 @ id @ (ap @ (sub @ X383 @ id) @ X382) @ id) = $true) | ((X370 @ id @ X383 @ id) != $true)) | ($true != (X369 @ X371)) | ((X369 @ X372) != $true) | ~! [X373 : subst,X374 : subst,X375 : term,X376 : subst] : (((X370 @ (comp @ X374 @ X373) @ (sub @ X375 @ X373) @ X376) = $true) | ((X370 @ X374 @ X375 @ (comp @ X373 @ X376)) != $true)) | ((X369 @ (ap @ X372 @ X371)) = $true) | ~! [X378 : term,X379 : subst,X380 : subst,X381 : subst] : (((X370 @ (comp @ X380 @ X381) @ (sub @ X378 @ X381) @ X379) != $true) | ($true = (X370 @ X380 @ X378 @ (comp @ X381 @ X379)))) | ~! [X377 : term] : ((X369 @ X377) = (X370 @ id @ X377 @ id)))) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X451 : subst > term > subst > $o,X452 : term,X453 : term] : (((X451 @ id @ X452 @ id) != $true) | ~! [X458 : term,X459 : term] : (((X451 @ id @ X458 @ id) != $true) | ($true = (X451 @ id @ (ap @ (sub @ X458 @ id) @ X459) @ id)) | ($true != (X451 @ id @ X459 @ id))) | ~! [X460 : subst,X461 : subst,X462 : term,X463 : subst] : (((X451 @ (comp @ X460 @ X463) @ (sub @ X462 @ X463) @ X461) != $true) | ((X451 @ X460 @ X462 @ (comp @ X463 @ X461)) = $true)) | ((X451 @ id @ (ap @ X453 @ X452) @ id) = $true) | ~! [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (($true = (X451 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454)) | ((X451 @ X457 @ X456 @ (comp @ X455 @ X454)) != $true)) | ((X451 @ id @ X453 @ id) != $true))) & ((hoasinduction_lem3aaa = $true) <=> ! [X121 : term,X122 : subst > term > subst > $o] : (~! [X124 : term,X125 : subst > term > term] : (~! [X126 : subst,X127 : term,X128 : subst] : ((sub @ (sub @ X124 @ (push @ X127 @ X128)) @ X126) = (sub @ (X125 @ X128 @ X127) @ X126)) | ((X122 @ id @ (lam @ (X125 @ sh @ one)) @ id) = $true) | ~! [X130 : term,X131 : subst,X132 : subst] : ((sub @ X124 @ (push @ (sub @ X130 @ X132) @ (comp @ X131 @ X132))) = (X125 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))) | ~! [X129 : term] : (((X122 @ id @ X129 @ id) != $true) | ((X122 @ id @ (X125 @ id @ X129) @ id) = $true))) | ($true = (X122 @ id @ (lam @ (sub @ X121 @ (push @ one @ sh))) @ id)) | ~! [X123 : term] : (((X122 @ id @ X123 @ id) != $true) | ((X122 @ id @ (sub @ X121 @ (push @ X123 @ id)) @ id) = $true)))) & (! [X467 : term] : ~! [X468 : subst > term > term] : ((X468 @ sh @ one) != (sub @ X467 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & ((var @ (d2term @ d_one)) != $true) & (pushprop_lem3v2 = $true) & ((induction2lem_lthm = $true) <=> (! [X350 : subst,X351 : term] : ((sub @ one @ (push @ X351 @ X350)) = X351) => (! [X352 : term,X353 : subst] : ((comp @ sh @ (push @ X352 @ X353)) = X353) => (! [X354 : subst] : ((comp @ X354 @ id) = X354) => (! [X355 : term,X356 : term > $o] : (((X356 @ X355) = $true) | ~! [X359 : term,X360 : term] : (($true != (X356 @ X360)) | ($true = (X356 @ (ap @ X360 @ X359))) | ($true != (X356 @ X359))) | ~! [X358 : term] : (($true = (X356 @ (lam @ X358))) | ((X356 @ X358) != $true)) | ~! [X357 : term] : (((var @ X357) != $true) | ($true = (X356 @ X357)))) => ! [X361 : term,X362 : term > $o,X363 : subst] : (~! [X365 : term] : (~! [X366 : term] : (($true = (X362 @ (sub @ X365 @ (push @ X366 @ id)))) | ((X362 @ X366) != $true)) | ($true = (X362 @ (lam @ X365)))) | ~! [X364 : term] : (((var @ X364) != $true) | ($true = (X362 @ (sub @ X364 @ X363)))) | ~! [X367 : term,X368 : term] : (($true = (X362 @ (ap @ X368 @ X367))) | ((X362 @ X368) != $true) | ((X362 @ X367) != $true)) | ((X362 @ (sub @ X361 @ X363)) = $true))))))) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X178 : term] : ~! [X179 : subst > term > term] : ~! [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) = (X179 @ X180 @ X181))) & ((pushprop_lem0_lthm = $true) <=> ! [X116 : term,X117 : subst,X118 : term > $o] : ~! [X119 : term > $o] : ~! [X120 : term] : ((X119 @ X120) = (X118 @ (sub @ X120 @ (push @ X116 @ X117))))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X384 : term] : ~! [X385 : subst > term > term] : ~! [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) = (X385 @ X386 @ X387))) & ((induction2 = $true) <=> ! [X42 : term > $o,X43 : term] : (~! [X44 : term] : (((var @ X44) != $true) | ((X42 @ X44) = $true)) | ~! [X47 : term,X48 : term] : (($true != (X42 @ X47)) | ((X42 @ X48) != $true) | ($true = (X42 @ (ap @ X47 @ X48)))) | ((X42 @ X43) = $true) | ~! [X45 : term] : (~! [X46 : term] : (($true != (X42 @ X46)) | ($true = (X42 @ (sub @ X45 @ (push @ X46 @ id))))) | ((X42 @ (lam @ X45)) = $true)))) & (ulamvarind = $true) & (termmset_gthm = $true) & (hoasinduction_gthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) = X168) => ! [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (~! [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) = (X170 @ X173)) | ((X171 @ X172) != $true) | ((X170 @ one) = $true)))) & ((! [X248 : term] : ((sub @ X248 @ id) = X248) => (! [X249 : subst > term > subst > $o,X250 : term] : (((X249 @ id @ (lam @ (sub @ X250 @ (push @ one @ sh))) @ id) = $true) | ~! [X256 : term] : (((X249 @ id @ (sub @ X250 @ (push @ X256 @ id)) @ id) = $true) | ((X249 @ id @ X256 @ id) != $true)) | ~! [X251 : subst > term > term] : (~! [X252 : subst,X253 : term,X254 : subst] : ((sub @ (X251 @ X252 @ X253) @ X254) = (X251 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254))) | ($true = (X249 @ id @ (lam @ (X251 @ sh @ one)) @ id)) | ~! [X255 : term] : (((X249 @ id @ (X251 @ id @ X255) @ id) = $true) | ((X249 @ id @ X255 @ id) != $true)))) => ! [X257 : subst > term > subst > $o,X258 : term] : (~! [X260 : subst > term > term] : (~! [X261 : term,X262 : subst,X263 : subst] : ((X260 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263)) = (sub @ (X260 @ X262 @ X261) @ X263)) | ((X257 @ id @ (lam @ (X260 @ sh @ one)) @ id) = $true) | ~! [X264 : term] : (((X257 @ id @ (X260 @ id @ X264) @ id) = $true) | ((X257 @ id @ X264 @ id) != $true))) | ~! [X259 : term] : (((X257 @ id @ X259 @ id) != $true) | ((X257 @ id @ (sub @ X258 @ (push @ X259 @ id)) @ id) = $true)) | ((X257 @ id @ (lam @ X258) @ id) = $true)))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X182 : subst > term > subst > $o,X183 : term] : (~! [X184 : term] : (((X182 @ id @ (sub @ X183 @ (push @ X184 @ id)) @ id) = $true) | ((X182 @ id @ X184 @ id) != $true)) | ~! [X185 : subst > term > term] : (((X182 @ id @ (lam @ (X185 @ sh @ one)) @ id) = $true) | ~! [X186 : term] : (($true != (X182 @ id @ X186 @ id)) | ((X182 @ id @ (X185 @ id @ X186) @ id) = $true)) | ~! [X187 : term,X188 : subst,X189 : subst] : ((X185 @ (comp @ X189 @ X188) @ (sub @ X187 @ X188)) = (sub @ (X185 @ X189 @ X187) @ X188))) | ((X182 @ id @ (lam @ (sub @ X183 @ (push @ one @ sh))) @ id) = $true))) & (! [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) = X240) <=> (axshiftcons = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (((X472 @ one) = $true) | ~! [X476 : term] : ((X472 @ X476) = (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) | ((X474 @ X475) != $true))) & (hoaslamnotvar_lthm = $true) & (induction2lem_gthm = $true) & ! [X177 : d_subst] : (d_id = X177) & (lamnotap != $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ! [X469 : d_term,X470 : d_term] : (((d2term @ X469) = (d2term @ X470)) => (X469 = X470)) & (! [X403 : term > $o,X404 : term] : (~! [X408 : term] : (((X403 @ X408) != $true) | ((X403 @ (lam @ X408)) = $true)) | ~! [X405 : term,X406 : term] : (((X403 @ (ap @ X405 @ X406)) = $true) | ($true != (X403 @ X406)) | ((X403 @ X405) != $true)) | ~! [X407 : term] : (($true = (X403 @ X407)) | ($true != (var @ X407))) | ((X403 @ X404) = $true)) <=> (induction = $true)) & (pushprop_lem0_gthm = $true) & (pushprop_lthm = $true) & (! [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) != (ap @ X243 @ X241)) | (X241 = X242)) <=> (apinj2 = $true)) & (pushprop_lem1_gthm = $true) & (! [X493 : term,X494 : subst > term > subst > $o] : (($true = (X494 @ id @ X493 @ id)) | ~! [X501 : subst > term > term] : (~! [X505 : term] : (((X494 @ id @ X505 @ id) != $true) | ((X494 @ id @ (X501 @ id @ X505) @ id) = $true)) | ($true = (X494 @ id @ (lam @ (X501 @ sh @ one)) @ id)) | ~! [X502 : subst,X503 : term,X504 : subst] : ((X501 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X501 @ X502 @ X503) @ X504))) | ~! [X507 : subst,X508 : subst,X509 : term,X510 : subst] : (($true = (X494 @ (comp @ X510 @ X508) @ (sub @ X509 @ X508) @ X507)) | ((X494 @ X510 @ X509 @ (comp @ X508 @ X507)) != $true)) | ~! [X506 : term] : (((var @ (sub @ X506 @ id)) != $true) | ($true = (X494 @ id @ X506 @ id))) | ~! [X495 : term,X496 : subst,X497 : subst,X498 : subst] : (($true != (X494 @ (comp @ X498 @ X497) @ (sub @ X495 @ X497) @ X496)) | ((X494 @ X498 @ X495 @ (comp @ X497 @ X496)) = $true)) | ~! [X499 : term,X500 : term] : (((X494 @ id @ X499 @ id) != $true) | ((X494 @ id @ (ap @ (sub @ X500 @ id) @ X499) @ id) = $true) | ((X494 @ id @ X500 @ id) != $true))) <=> (hoasinduction = $true)) & ((axidl = $true) <=> ! [X471 : subst] : ((comp @ id @ X471) = X471)) & ((hoasinduction_lem3v2 = $true) <=> ! [X17 : term,X18 : subst > term > subst > $o,X19 : term > $o] : (~! [X20 : term] : (((X19 @ (sub @ X17 @ (push @ X20 @ id))) = $true) | ($true != (X19 @ X20))) | ~! [X22 : subst > term > term] : (~! [X26 : term] : (($true != (X18 @ id @ X26 @ id)) | ((X18 @ id @ (X22 @ id @ X26) @ id) = $true)) | ($true = (X18 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | ~! [X23 : subst,X24 : subst,X25 : term] : ((X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (X22 @ X24 @ X25) @ X23))) | ~! [X21 : term] : ((X19 @ X21) = (X18 @ id @ X21 @ id)) | ((X19 @ (lam @ X17)) = $true) | ~! [X31 : subst,X32 : subst,X33 : subst,X34 : term] : (($true != (X18 @ (comp @ X33 @ X32) @ (sub @ X34 @ X32) @ X31)) | ((X18 @ X33 @ X34 @ (comp @ X32 @ X31)) = $true)) | ~! [X27 : subst,X28 : term,X29 : subst,X30 : subst] : (($true != (X18 @ X27 @ X28 @ (comp @ X29 @ X30))) | ((X18 @ (comp @ X27 @ X29) @ (sub @ X28 @ X29) @ X30) = $true)))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoaslamnotap_lthm = $true) & (! [X174 : term] : ((sub @ X174 @ id) = X174) <=> (axvarid = $true)) & (hoasinduction_lem1 = $true) & (one = (d2term @ d_one)) & ((termmset_lthm = $true) <=> (! [X149 : term] : ((sub @ X149 @ id) = X149) => ! [X150 : term] : ((sub @ X150 @ id) = X150))) & ((! [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) = X520) => (! [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) = X521) => ! [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (($true = (X524 @ (sub @ X527 @ (push @ X525 @ X526)))) | ($true != (X523 @ X527))))) <=> (pushprop_lem1_lthm = $true)) & (! [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) = X176) <=> (axvarcons = $true)) & (hoasinduction_lem2_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X466 : subst] : ((comp @ X466 @ id) = X466) <=> (axidr = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X8 : term] : ~! [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) != (X9 @ sh @ one))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((hoasapinj2_lthm = $true) <=> (! [X528 : term,X529 : term,X530 : term,X531 : term] : ((X529 = X530) | ((ap @ X531 @ X529) != (ap @ X528 @ X530))) => ! [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 = X534) | ((ap @ (sub @ X533 @ id) @ X532) != (ap @ (sub @ X535 @ id) @ X534))))) & (hoaslamnotvar_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((shinj = $true) <=> ! [X35 : term,X36 : term] : ((X35 = X36) | ((sub @ X36 @ sh) != (sub @ X35 @ sh)))) & ((hoasapinj1_lthm = $true) <=> (! [X103 : term] : ((sub @ X103 @ id) = X103) => (! [X104 : term,X105 : term,X106 : term,X107 : term] : (((ap @ X106 @ X105) != (ap @ X107 @ X104)) | (X106 = X107)) => ! [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 = X109) | ((ap @ (sub @ X109 @ id) @ X111) != (ap @ (sub @ X108 @ id) @ X110)))))) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X511 : term,X512 : subst > term > subst > $o] : (~! [X518 : term] : (((X512 @ id @ X518 @ id) != $true) | ((X512 @ id @ (sub @ X511 @ (push @ X518 @ id)) @ id) = $true)) | ((X512 @ id @ (lam @ X511) @ id) = $true) | ~! [X513 : subst > term > term] : (~! [X514 : subst,X515 : term,X516 : subst] : ((X513 @ (comp @ X514 @ X516) @ (sub @ X515 @ X516)) = (sub @ (X513 @ X514 @ X515) @ X516)) | ((X512 @ id @ (lam @ (X513 @ sh @ one)) @ id) = $true) | ~! [X517 : term] : (($true = (X512 @ id @ (X513 @ id @ X517) @ id)) | ((X512 @ id @ X517 @ id) != $true)))))),
% 2.65/1.76 inference(flattening,[status(thm)],[f10])).
% 2.65/1.76 thf(f10,plain,(
% 2.65/1.76 ((induction2lem = $true) <=> ! [X0 : subst,X1 : term > $o,X2 : term] : (~! [X3 : term,X4 : term] : (~((X1 @ X3) = $true) | ((X1 @ (ap @ X3 @ X4)) = $true) | ~((X1 @ X4) = $true)) | ((X1 @ (sub @ X2 @ X0)) = $true) | ~! [X5 : term] : (~((var @ X5) = $true) | ((X1 @ (sub @ X5 @ X0)) = $true)) | ~! [X6 : term] : (((X1 @ (lam @ X6)) = $true) | ~! [X7 : term] : (~($true = (X1 @ X7)) | ($true = (X1 @ (sub @ X6 @ (push @ X7 @ id)))))))) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar_gthm = $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem3b = $true) <=> ! [X8 : term] : ~! [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) != (X9 @ sh @ one))) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (one = (d2term @ d_one)) & ((hoasapinj1 = $true) <=> ! [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 = X16) | ((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15)))) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1 = $true) & (axvarshift = $true) & (! [X17 : term,X18 : subst > term > subst > $o,X19 : term > $o] : (~! [X20 : term] : (((X19 @ (sub @ X17 @ (push @ X20 @ id))) = $true) | ~($true = (X19 @ X20))) | ~! [X21 : term] : ((X19 @ X21) = (X18 @ id @ X21 @ id)) | ~! [X22 : subst > term > term] : (~! [X23 : subst,X24 : subst,X25 : term] : ((X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (X22 @ X24 @ X25) @ X23)) | ~! [X26 : term] : (~($true = (X18 @ id @ X26 @ id)) | ((X18 @ id @ (X22 @ id @ X26) @ id) = $true)) | ($true = (X18 @ id @ (lam @ (X22 @ sh @ one)) @ id))) | ~! [X27 : subst,X28 : term,X29 : subst,X30 : subst] : (~($true = (X18 @ X27 @ X28 @ (comp @ X29 @ X30))) | ((X18 @ (comp @ X27 @ X29) @ (sub @ X28 @ X29) @ X30) = $true)) | ~! [X31 : subst,X32 : subst,X33 : subst,X34 : term] : (~($true = (X18 @ (comp @ X33 @ X32) @ (sub @ X34 @ X32) @ X31)) | ((X18 @ X33 @ X34 @ (comp @ X32 @ X31)) = $true)) | ((X19 @ (lam @ X17)) = $true)) <=> (hoasinduction_lem3v2 = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop = $true) & ((shinj = $true) <=> ! [X35 : term,X36 : term] : ((X35 = X36) | ((sub @ X36 @ sh) != (sub @ X35 @ sh)))) & (! [X37 : term > $o,X38 : term,X39 : subst] : ~! [X40 : term > $o] : ~! [X41 : term] : ((X37 @ (sub @ X41 @ (push @ X38 @ X39))) = (X40 @ X41)) <=> (pushprop_lem0 = $true)) & (! [X42 : term > $o,X43 : term] : (~! [X44 : term] : (~((var @ X44) = $true) | ((X42 @ X44) = $true)) | ~! [X45 : term] : (~! [X46 : term] : (~($true = (X42 @ X46)) | ($true = (X42 @ (sub @ X45 @ (push @ X46 @ id))))) | ((X42 @ (lam @ X45)) = $true)) | ~! [X47 : term,X48 : term] : (~($true = (X42 @ X47)) | ($true = (X42 @ (ap @ X47 @ X48))) | ~((X42 @ X48) = $true)) | ((X42 @ X43) = $true)) <=> (induction2 = $true)) & ((! [X49 : subst > term > subst > $o] : ~! [X50 : term > $o] : ~! [X51 : term] : ((X49 @ id @ X51 @ id) = (X50 @ X51)) => (! [X52 : term > $o,X53 : term] : (~! [X54 : term,X55 : term] : (~($true = (X52 @ X54)) | ((X52 @ (ap @ X55 @ X54)) = $true) | ~((X52 @ X55) = $true)) | ~! [X56 : term] : (~((var @ X56) = $true) | ($true = (X52 @ X56))) | ((X52 @ X53) = $true) | ~! [X57 : term] : (((X52 @ (lam @ X57)) = $true) | ~! [X58 : term] : (~($true = (X52 @ X58)) | ((X52 @ (sub @ X57 @ (push @ X58 @ id))) = $true)))) => (! [X59 : term] : ((sub @ X59 @ id) = X59) => (! [X60 : subst > term > subst > $o,X61 : term > $o,X62 : term] : (~! [X63 : subst > term > term] : (~! [X64 : term] : (~((X60 @ id @ X64 @ id) = $true) | ((X60 @ id @ (X63 @ id @ X64) @ id) = $true)) | ~! [X65 : subst,X66 : subst,X67 : term] : ((sub @ (X63 @ X66 @ X67) @ X65) = (X63 @ (comp @ X66 @ X65) @ (sub @ X67 @ X65))) | ($true = (X60 @ id @ (lam @ (X63 @ sh @ one)) @ id))) | ~! [X68 : term] : (($true = (X61 @ (sub @ X62 @ (push @ X68 @ id)))) | ~((X61 @ X68) = $true)) | ~! [X69 : term] : ((X60 @ id @ X69 @ id) = (X61 @ X69)) | ((X61 @ (lam @ X62)) = $true)) => ! [X70 : subst > term > subst > $o,X71 : term] : (($true = (X70 @ id @ X71 @ id)) | ~! [X72 : subst > term > term] : (((X70 @ id @ (lam @ (X72 @ sh @ one)) @ id) = $true) | ~! [X73 : subst,X74 : term,X75 : subst] : ((X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73)) = (sub @ (X72 @ X75 @ X74) @ X73)) | ~! [X76 : term] : (((X70 @ id @ (X72 @ id @ X76) @ id) = $true) | ~($true = (X70 @ id @ X76 @ id)))) | ~! [X77 : term,X78 : term] : (((X70 @ id @ (ap @ (sub @ X78 @ id) @ X77) @ id) = $true) | ~((X70 @ id @ X78 @ id) = $true) | ~($true = (X70 @ id @ X77 @ id)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((! [X79 : term] : ((sub @ X79 @ id) = X79) => ! [X80 : term,X81 : subst > term > subst > $o,X82 : term > $o] : (~! [X83 : term] : ((X82 @ X83) = (X81 @ id @ X83 @ id)) | ~! [X84 : subst > term > term] : (~! [X85 : term] : (((X81 @ id @ (X84 @ id @ X85) @ id) = $true) | ~((X81 @ id @ X85 @ id) = $true)) | ((X81 @ id @ (lam @ (X84 @ sh @ one)) @ id) = $true) | ~! [X86 : term,X87 : subst,X88 : subst] : ((sub @ (X84 @ X88 @ X86) @ X87) = (X84 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87)))) | ~! [X89 : subst,X90 : subst,X91 : subst,X92 : term] : (((X81 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91) @ X89) = $true) | ~($true = (X81 @ X90 @ X92 @ (comp @ X91 @ X89)))) | ~! [X93 : term,X94 : subst,X95 : subst,X96 : subst] : (($true = (X81 @ X94 @ X93 @ (comp @ X95 @ X96))) | ~((X81 @ (comp @ X94 @ X95) @ (sub @ X93 @ X95) @ X96) = $true)) | ((X82 @ (lam @ X80)) = $true) | ~! [X97 : term] : (((X82 @ (sub @ X80 @ (push @ X97 @ id))) = $true) | ~((X82 @ X97) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((hoasapinj2 = $true) <=> ! [X98 : term,X99 : term,X100 : term,X101 : term] : ((X100 = X101) | ((ap @ (sub @ X98 @ id) @ X100) != (ap @ (sub @ X99 @ id) @ X101)))) & (hoaslaminj_gthm = $true) & (! [X102 : subst] : ((push @ (sub @ one @ X102) @ (comp @ sh @ X102)) = X102) <=> (axscons = $true)) & (apnotvar = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasapinj1_lthm = $true) <=> (! [X103 : term] : ((sub @ X103 @ id) = X103) => (! [X104 : term,X105 : term,X106 : term,X107 : term] : (((ap @ X106 @ X105) != (ap @ X107 @ X104)) | (X106 = X107)) => ! [X108 : term,X109 : term,X110 : term,X111 : term] : ((X108 = X109) | ((ap @ (sub @ X109 @ id) @ X111) != (ap @ (sub @ X108 @ id) @ X110)))))) & (termmset_gthm = $true) & ! [X112 : d_subst,X113 : d_subst] : (((d2subst @ X112) = (d2subst @ X113)) => (X112 = X113)) & ! [X114 : term] : ? [X115 : d_term] : ((d2term @ X115) = X114) & (pushprop_lem3v2_lthm = $true) & (axapp = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X116 : term,X117 : subst,X118 : term > $o] : ~! [X119 : term > $o] : ~! [X120 : term] : ((X119 @ X120) = (X118 @ (sub @ X120 @ (push @ X116 @ X117))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X121 : term,X122 : subst > term > subst > $o] : (~! [X123 : term] : (~((X122 @ id @ X123 @ id) = $true) | ((X122 @ id @ (sub @ X121 @ (push @ X123 @ id)) @ id) = $true)) | ~! [X124 : term,X125 : subst > term > term] : (~! [X126 : subst,X127 : term,X128 : subst] : ((sub @ (sub @ X124 @ (push @ X127 @ X128)) @ X126) = (sub @ (X125 @ X128 @ X127) @ X126)) | ((X122 @ id @ (lam @ (X125 @ sh @ one)) @ id) = $true) | ~! [X129 : term] : (((X122 @ id @ (X125 @ id @ X129) @ id) = $true) | ~((X122 @ id @ X129 @ id) = $true)) | ~! [X130 : term,X131 : subst,X132 : subst] : ((sub @ X124 @ (push @ (sub @ X130 @ X132) @ (comp @ X131 @ X132))) = (X125 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132)))) | ($true = (X122 @ id @ (lam @ (sub @ X121 @ (push @ one @ sh))) @ id)))) & ((d2subst @ d_id) = sh) & (! [X133 : term > $o,X134 : term,X135 : term,X136 : subst,X137 : term > $o] : (~($true = (X137 @ X134)) | ((X133 @ (sub @ X134 @ (push @ X135 @ X136))) = $true)) <=> (pushprop_lem1 = $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X138 : subst > term > subst > $o,X139 : term] : (~! [X140 : subst > term > term] : (((X138 @ id @ (lam @ (X140 @ sh @ one)) @ id) = $true) | ~! [X141 : term] : (~((X138 @ id @ X141 @ id) = $true) | ($true = (X138 @ id @ (X140 @ id @ X141) @ id))) | ~! [X142 : subst,X143 : term,X144 : subst] : ((X140 @ (comp @ X142 @ X144) @ (sub @ X143 @ X144)) = (sub @ (X140 @ X142 @ X143) @ X144))) | ~! [X145 : term,X146 : term] : (~((X138 @ id @ X146 @ id) = $true) | ((X138 @ id @ (ap @ (sub @ X145 @ id) @ X146) @ id) = $true) | ~($true = (X138 @ id @ X145 @ id))) | ((X138 @ id @ X139 @ id) = $true))) & ((laminj = $true) <=> ! [X147 : term,X148 : term] : ((X147 = X148) | ((lam @ X147) != (lam @ X148)))) & ((termmset_lthm = $true) <=> (! [X149 : term] : ((sub @ X149 @ id) = X149) => ! [X150 : term] : ((sub @ X150 @ id) = X150))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((! [X151 : term] : ((sub @ X151 @ id) = X151) => (! [X152 : term > $o,X153 : term,X154 : subst] : (~! [X155 : term,X156 : term] : (($true = (X152 @ (ap @ X155 @ X156))) | ~($true = (X152 @ X155)) | ~((X152 @ X156) = $true)) | ~! [X157 : term] : (~((var @ X157) = $true) | ((X152 @ (sub @ X157 @ X154)) = $true)) | ~! [X158 : term] : (~! [X159 : term] : (($true = (X152 @ (sub @ X158 @ (push @ X159 @ id)))) | ~((X152 @ X159) = $true)) | ((X152 @ (lam @ X158)) = $true)) | ((X152 @ (sub @ X153 @ X154)) = $true)) => ! [X160 : term > $o,X161 : term] : (~! [X162 : term,X163 : term] : (~((X160 @ X163) = $true) | ($true = (X160 @ (ap @ X163 @ X162))) | ~($true = (X160 @ X162))) | ((X160 @ X161) = $true) | ~! [X164 : term] : (~((var @ X164) = $true) | ((X160 @ X164) = $true)) | ~! [X165 : term] : (~! [X166 : term] : (((X160 @ (sub @ X165 @ (push @ X166 @ id))) = $true) | ~($true = (X160 @ X166))) | ($true = (X160 @ (lam @ X165))))))) <=> (induction2_lthm = $true)) & (ulamvarind = $true) & (induction2lem_gthm = $true) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ((! [X167 : subst,X168 : term] : ((sub @ one @ (push @ X168 @ X167)) = X168) => ! [X169 : subst,X170 : term > $o,X171 : term > $o,X172 : term] : (~! [X173 : term] : ((X171 @ (sub @ X173 @ (push @ X172 @ X169))) = (X170 @ X173)) | ~((X171 @ X172) = $true) | ((X170 @ one) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & (! [X174 : term] : ((sub @ X174 @ id) = X174) <=> (axvarid = $true)) & (! [X175 : subst,X176 : term] : ((sub @ one @ (push @ X176 @ X175)) = X176) <=> (axvarcons = $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)))))))) & ~((var @ (d2term @ d_one)) = $true) & ! [X177 : d_subst] : (d_id = X177) & (hoasinduction_lem1_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X178 : term] : ~! [X179 : subst > term > term] : ~! [X180 : subst,X181 : term] : ((sub @ X178 @ (push @ X181 @ X180)) = (X179 @ X180 @ X181))) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & (pushprop_lem1_gthm = $true) & (axmap = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X182 : subst > term > subst > $o,X183 : term] : (~! [X184 : term] : (~((X182 @ id @ X184 @ id) = $true) | ((X182 @ id @ (sub @ X183 @ (push @ X184 @ id)) @ id) = $true)) | ~! [X185 : subst > term > term] : (~! [X186 : term] : (~($true = (X182 @ id @ X186 @ id)) | ((X182 @ id @ (X185 @ id @ X186) @ id) = $true)) | ((X182 @ id @ (lam @ (X185 @ sh @ one)) @ id) = $true) | ~! [X187 : term,X188 : subst,X189 : subst] : ((X185 @ (comp @ X189 @ X188) @ (sub @ X187 @ X188)) = (sub @ (X185 @ X189 @ X187) @ X188))) | ((X182 @ id @ (lam @ (sub @ X183 @ (push @ one @ sh))) @ id) = $true))) & ((hoaslaminj = $true) <=> ! [X190 : subst > term > term,X191 : subst > term > term,X192 : subst,X193 : term] : (~! [X194 : term,X195 : subst,X196 : subst] : ((X190 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195)) = (sub @ (X190 @ X196 @ X194) @ X195)) | ((X191 @ X192 @ X193) = (X190 @ X192 @ X193)) | ~! [X197 : term,X198 : subst,X199 : subst] : ((sub @ (X191 @ X198 @ X197) @ X199) = (X191 @ (comp @ X198 @ X199) @ (sub @ X197 @ X199))) | ((lam @ (X191 @ sh @ one)) != (lam @ (X190 @ sh @ one))))) & ((! [X200 : subst > term > subst > $o] : ~! [X201 : term > $o] : ~! [X202 : term] : ((X201 @ X202) = (X200 @ id @ X202 @ id)) => (! [X203 : term,X204 : term > $o] : (~! [X205 : term,X206 : term] : (~((X204 @ X205) = $true) | ~($true = (X204 @ X206)) | ((X204 @ (ap @ X206 @ X205)) = $true)) | ($true = (X204 @ X203)) | ~! [X207 : term] : (~! [X208 : term] : (~($true = (X204 @ X208)) | ($true = (X204 @ (sub @ X207 @ (push @ X208 @ id))))) | ($true = (X204 @ (lam @ X207)))) | ~! [X209 : term] : (~((var @ X209) = $true) | ((X204 @ X209) = $true))) => (! [X210 : term] : ((sub @ X210 @ id) = X210) => (! [X211 : term > $o,X212 : term,X213 : subst > term > subst > $o] : (~! [X214 : term] : ((X213 @ id @ X214 @ id) = (X211 @ X214)) | ~! [X215 : term] : (($true = (X211 @ (sub @ X212 @ (push @ X215 @ id)))) | ~($true = (X211 @ X215))) | ~! [X216 : subst > term > term] : (((X213 @ id @ (lam @ (X216 @ sh @ one)) @ id) = $true) | ~! [X217 : term,X218 : subst,X219 : subst] : ((sub @ (X216 @ X219 @ X217) @ X218) = (X216 @ (comp @ X219 @ X218) @ (sub @ X217 @ X218))) | ~! [X220 : term] : (((X213 @ id @ (X216 @ id @ X220) @ id) = $true) | ~((X213 @ id @ X220 @ id) = $true))) | ($true = (X211 @ (lam @ X212)))) => ! [X221 : term,X222 : subst > term > subst > $o] : (~! [X223 : subst,X224 : term,X225 : subst,X226 : subst] : (~($true = (X222 @ X225 @ X224 @ (comp @ X226 @ X223))) | ((X222 @ (comp @ X225 @ X226) @ (sub @ X224 @ X226) @ X223) = $true)) | ~! [X227 : term,X228 : term] : (~((X222 @ id @ X227 @ id) = $true) | ((X222 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id) = $true) | ~((X222 @ id @ X228 @ id) = $true)) | ~! [X229 : term] : (~((var @ (sub @ X229 @ id)) = $true) | ($true = (X222 @ id @ X229 @ id))) | ~! [X230 : subst,X231 : subst,X232 : term,X233 : subst] : (~($true = (X222 @ (comp @ X233 @ X230) @ (sub @ X232 @ X230) @ X231)) | ((X222 @ X233 @ X232 @ (comp @ X230 @ X231)) = $true)) | ~! [X234 : subst > term > term] : (~! [X235 : term] : (($true = (X222 @ id @ (X234 @ id @ X235) @ id)) | ~((X222 @ id @ X235 @ id) = $true)) | ~! [X236 : term,X237 : subst,X238 : subst] : ((X234 @ (comp @ X238 @ X237) @ (sub @ X236 @ X237)) = (sub @ (X234 @ X238 @ X236) @ X237)) | ((X222 @ id @ (lam @ (X234 @ sh @ one)) @ id) = $true)) | ((X222 @ id @ X221 @ id) = $true)))))) <=> (hoasinduction_lthm_3 = $true)) & (! [X239 : term,X240 : subst] : ((comp @ sh @ (push @ X239 @ X240)) = X240) <=> (axshiftcons = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X241 : term,X242 : term,X243 : term,X244 : term] : (((ap @ X244 @ X242) != (ap @ X243 @ X241)) | (X241 = X242)) <=> (apinj2 = $true)) & ((substmonoid = $true) <=> (! [X246 : subst] : ((comp @ id @ X246) = X246) & ! [X245 : subst] : ((comp @ X245 @ id) = X245))) & ((termmset = $true) <=> ! [X247 : term] : ((sub @ X247 @ id) = X247)) & (hoasinduction_gthm = $true) & (pushprop_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X248 : term] : ((sub @ X248 @ id) = X248) => (! [X249 : subst > term > subst > $o,X250 : term] : (~! [X251 : subst > term > term] : (($true = (X249 @ id @ (lam @ (X251 @ sh @ one)) @ id)) | ~! [X252 : subst,X253 : term,X254 : subst] : ((sub @ (X251 @ X252 @ X253) @ X254) = (X251 @ (comp @ X252 @ X254) @ (sub @ X253 @ X254))) | ~! [X255 : term] : (~((X249 @ id @ X255 @ id) = $true) | ((X249 @ id @ (X251 @ id @ X255) @ id) = $true))) | ~! [X256 : term] : (((X249 @ id @ (sub @ X250 @ (push @ X256 @ id)) @ id) = $true) | ~((X249 @ id @ X256 @ id) = $true)) | ((X249 @ id @ (lam @ (sub @ X250 @ (push @ one @ sh))) @ id) = $true)) => ! [X257 : subst > term > subst > $o,X258 : term] : (~! [X259 : term] : (~((X257 @ id @ X259 @ id) = $true) | ((X257 @ id @ (sub @ X258 @ (push @ X259 @ id)) @ id) = $true)) | ~! [X260 : subst > term > term] : (~! [X261 : term,X262 : subst,X263 : subst] : ((X260 @ (comp @ X262 @ X263) @ (sub @ X261 @ X263)) = (sub @ (X260 @ X262 @ X261) @ X263)) | ~! [X264 : term] : (((X257 @ id @ (X260 @ id @ X264) @ id) = $true) | ~((X257 @ id @ X264 @ id) = $true)) | ((X257 @ id @ (lam @ (X260 @ sh @ one)) @ id) = $true)) | ((X257 @ id @ (lam @ X258) @ id) = $true))))) & ~(lamnotap = $true) & (axclos = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X265 : term,X266 : subst > term > subst > $o,X267 : term > $o] : (($true = (X267 @ (lam @ X265))) | ~! [X268 : term] : (~((X267 @ X268) = $true) | ($true = (X267 @ (sub @ X265 @ (push @ X268 @ id))))) | ~! [X269 : term] : ((X267 @ X269) = (X266 @ id @ X269 @ id)) | ~! [X270 : subst > term > term] : (~! [X271 : term] : (($true = (X266 @ id @ (X270 @ id @ X271) @ id)) | ~((X266 @ id @ X271 @ id) = $true)) | ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id) = $true) | ~! [X272 : subst,X273 : subst,X274 : term] : ((sub @ (X270 @ X272 @ X274) @ X273) = (X270 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273)))))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (! [X275 : term,X276 : term,X277 : term,X278 : term] : (((ap @ X278 @ X275) != (ap @ X276 @ X277)) | (X276 = X278)) <=> (apinj1 = $true)) & ((! [X279 : term,X280 : term > $o] : (((X280 @ X279) = $true) | ~! [X281 : term] : (~($true = (var @ X281)) | ((X280 @ X281) = $true)) | ~! [X282 : term] : (~! [X283 : term] : (~((X280 @ X283) = $true) | ($true = (X280 @ (sub @ X282 @ (push @ X283 @ id))))) | ((X280 @ (lam @ X282)) = $true)) | ~! [X284 : term,X285 : term] : (((X280 @ (ap @ X285 @ X284)) = $true) | ~($true = (X280 @ X285)) | ~($true = (X280 @ X284)))) => (! [X286 : term,X287 : subst > term > subst > $o,X288 : term] : (~! [X289 : subst,X290 : subst,X291 : term,X292 : subst] : (~($true = (X287 @ X289 @ X291 @ (comp @ X290 @ X292))) | ((X287 @ (comp @ X289 @ X290) @ (sub @ X291 @ X290) @ X292) = $true)) | ~! [X293 : term,X294 : term] : (((X287 @ id @ (ap @ (sub @ X293 @ id) @ X294) @ id) = $true) | ~($true = (X287 @ id @ X293 @ id)) | ~((X287 @ id @ X294 @ id) = $true)) | ~($true = (X287 @ id @ X286 @ id)) | ~! [X295 : subst,X296 : subst,X297 : term,X298 : subst] : (~((X287 @ (comp @ X295 @ X296) @ (sub @ X297 @ X296) @ X298) = $true) | ($true = (X287 @ X295 @ X297 @ (comp @ X296 @ X298)))) | ~((X287 @ id @ X288 @ id) = $true) | ($true = (X287 @ id @ (ap @ X286 @ X288) @ id))) => (! [X299 : term,X300 : subst > term > subst > $o] : (~! [X301 : term] : (((X300 @ id @ (sub @ X299 @ (push @ X301 @ id)) @ id) = $true) | ~($true = (X300 @ id @ X301 @ id))) | ~! [X302 : subst,X303 : subst,X304 : term,X305 : subst] : (~((X300 @ X305 @ X304 @ (comp @ X302 @ X303)) = $true) | ((X300 @ (comp @ X305 @ X302) @ (sub @ X304 @ X302) @ X303) = $true)) | ~! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (($true = (X300 @ X307 @ X306 @ (comp @ X308 @ X309))) | ~((X300 @ (comp @ X307 @ X308) @ (sub @ X306 @ X308) @ X309) = $true)) | ((X300 @ id @ (lam @ X299) @ id) = $true) | ~! [X310 : subst > term > term] : (((X300 @ id @ (lam @ (X310 @ sh @ one)) @ id) = $true) | ~! [X311 : term] : (((X300 @ id @ (X310 @ id @ X311) @ id) = $true) | ~($true = (X300 @ id @ X311 @ id))) | ~! [X312 : term,X313 : subst,X314 : subst] : ((X310 @ (comp @ X314 @ X313) @ (sub @ X312 @ X313)) = (sub @ (X310 @ X314 @ X312) @ X313)))) => ! [X315 : term,X316 : subst > term > subst > $o] : (~! [X317 : term,X318 : term] : (($true = (X316 @ id @ (ap @ (sub @ X318 @ id) @ X317) @ id)) | ~((X316 @ id @ X317 @ id) = $true) | ~((X316 @ id @ X318 @ id) = $true)) | ~! [X319 : subst > term > term] : (~! [X320 : term,X321 : subst,X322 : subst] : ((sub @ (X319 @ X321 @ X320) @ X322) = (X319 @ (comp @ X321 @ X322) @ (sub @ X320 @ X322))) | ($true = (X316 @ id @ (lam @ (X319 @ sh @ one)) @ id)) | ~! [X323 : term] : (~((X316 @ id @ X323 @ id) = $true) | ((X316 @ id @ (X319 @ id @ X323) @ id) = $true))) | ~! [X324 : subst,X325 : term,X326 : subst,X327 : subst] : (((X316 @ (comp @ X324 @ X327) @ (sub @ X325 @ X327) @ X326) = $true) | ~($true = (X316 @ X324 @ X325 @ (comp @ X327 @ X326)))) | ~! [X328 : subst,X329 : subst,X330 : subst,X331 : term] : (((X316 @ X328 @ X331 @ (comp @ X329 @ X330)) = $true) | ~((X316 @ (comp @ X328 @ X329) @ (sub @ X331 @ X329) @ X330) = $true)) | ~! [X332 : term] : (~((var @ (sub @ X332 @ id)) = $true) | ((X316 @ id @ X332 @ id) = $true)) | ((X316 @ id @ X315 @ id) = $true))))) <=> (hoasinduction_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X333 : subst > term > subst > $o] : ~! [X334 : term > $o] : ~! [X335 : term] : ((X334 @ X335) = (X333 @ id @ X335 @ id))) & (pushprop_lem0_gthm = $true) & ((hoaslamnotap = $true) <=> ! [X336 : term,X337 : term,X338 : subst > term > term] : (~! [X339 : subst,X340 : subst,X341 : term] : ((sub @ (X338 @ X340 @ X341) @ X339) = (X338 @ (comp @ X340 @ X339) @ (sub @ X341 @ X339))) | ((ap @ (sub @ X336 @ id) @ X337) != (lam @ (X338 @ sh @ one))))) & (axabs = $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X342 : term,X343 : subst > term > subst > $o] : (~! [X344 : term] : (((X343 @ id @ (sub @ X342 @ (push @ X344 @ id)) @ id) = $true) | ~((X343 @ id @ X344 @ id) = $true)) | ~! [X345 : subst > term > term] : (~! [X346 : term,X347 : subst,X348 : subst] : ((sub @ (X345 @ X347 @ X346) @ X348) = (X345 @ (comp @ X347 @ X348) @ (sub @ X346 @ X348))) | ~! [X349 : term] : (($true = (X343 @ id @ (X345 @ id @ X349) @ id)) | ~((X343 @ id @ X349 @ id) = $true)) | ((X343 @ id @ (lam @ (X345 @ sh @ one)) @ id) = $true)) | ((X343 @ id @ (lam @ (sub @ X342 @ (push @ one @ sh))) @ id) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & ((! [X350 : subst,X351 : term] : ((sub @ one @ (push @ X351 @ X350)) = X351) => (! [X352 : term,X353 : subst] : ((comp @ sh @ (push @ X352 @ X353)) = X353) => (! [X354 : subst] : ((comp @ X354 @ id) = X354) => (! [X355 : term,X356 : term > $o] : (~! [X357 : term] : (($true = (X356 @ X357)) | ~((var @ X357) = $true)) | ~! [X358 : term] : (($true = (X356 @ (lam @ X358))) | ~((X356 @ X358) = $true)) | ~! [X359 : term,X360 : term] : (($true = (X356 @ (ap @ X360 @ X359))) | ~($true = (X356 @ X359)) | ~($true = (X356 @ X360))) | ((X356 @ X355) = $true)) => ! [X361 : term,X362 : term > $o,X363 : subst] : (((X362 @ (sub @ X361 @ X363)) = $true) | ~! [X364 : term] : (($true = (X362 @ (sub @ X364 @ X363))) | ~((var @ X364) = $true)) | ~! [X365 : term] : (~! [X366 : term] : (($true = (X362 @ (sub @ X365 @ (push @ X366 @ id)))) | ~((X362 @ X366) = $true)) | ($true = (X362 @ (lam @ X365)))) | ~! [X367 : term,X368 : term] : (~((X362 @ X367) = $true) | ($true = (X362 @ (ap @ X368 @ X367))) | ~((X362 @ X368) = $true))))))) <=> (induction2lem_lthm = $true)) & (axassoc = $true) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X369 : term > $o,X370 : subst > term > subst > $o,X371 : term,X372 : term] : (~! [X373 : subst,X374 : subst,X375 : term,X376 : subst] : (~((X370 @ X374 @ X375 @ (comp @ X373 @ X376)) = $true) | ((X370 @ (comp @ X374 @ X373) @ (sub @ X375 @ X373) @ X376) = $true)) | ~! [X377 : term] : ((X369 @ X377) = (X370 @ id @ X377 @ id)) | ~! [X378 : term,X379 : subst,X380 : subst,X381 : subst] : (($true = (X370 @ X380 @ X378 @ (comp @ X381 @ X379))) | ~((X370 @ (comp @ X380 @ X381) @ (sub @ X378 @ X381) @ X379) = $true)) | ((X369 @ (ap @ X372 @ X371)) = $true) | ~((X369 @ X372) = $true) | ~($true = (X369 @ X371)) | ~! [X382 : term,X383 : term] : (~((X370 @ id @ X382 @ id) = $true) | ((X370 @ id @ (ap @ (sub @ X383 @ id) @ X382) @ id) = $true) | ~((X370 @ id @ X383 @ id) = $true)))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X384 : term] : ~! [X385 : subst > term > term] : ~! [X386 : subst,X387 : term] : ((sub @ X384 @ (push @ X387 @ X386)) = (X385 @ X386 @ X387))) & ((! [X388 : term] : ~! [X389 : subst > term > term] : ~! [X390 : term,X391 : subst] : ((sub @ X388 @ (push @ X390 @ X391)) = (X389 @ X391 @ X390)) => (! [X392 : term] : ((sub @ X392 @ id) = X392) => ! [X393 : term,X394 : subst > term > subst > $o,X395 : term > $o] : (((X395 @ (lam @ X393)) = $true) | ~! [X396 : term] : (((X395 @ (sub @ X393 @ (push @ X396 @ id))) = $true) | ~((X395 @ X396) = $true)) | ~! [X397 : subst > term > term] : (((X394 @ id @ (lam @ (X397 @ sh @ one)) @ id) = $true) | ~! [X398 : term] : (~((X394 @ id @ X398 @ id) = $true) | ((X394 @ id @ (X397 @ id @ X398) @ id) = $true)) | ~! [X399 : term,X400 : subst,X401 : subst] : ((sub @ (X397 @ X400 @ X399) @ X401) = (X397 @ (comp @ X400 @ X401) @ (sub @ X399 @ X401)))) | ~! [X402 : term] : ((X395 @ X402) = (X394 @ id @ X402 @ id))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (lamnotvar = $true) & (pushprop_lthm = $true) & (hoasapnotvar_gthm = $true) & (! [X403 : term > $o,X404 : term] : (~! [X405 : term,X406 : term] : (((X403 @ (ap @ X405 @ X406)) = $true) | ~((X403 @ X405) = $true) | ~($true = (X403 @ X406))) | ~! [X407 : term] : (($true = (X403 @ X407)) | ~($true = (var @ X407))) | ((X403 @ X404) = $true) | ~! [X408 : term] : (~((X403 @ X408) = $true) | ((X403 @ (lam @ X408)) = $true))) <=> (induction = $true)) & ((! [X409 : term] : ((sub @ X409 @ id) = X409) => (! [X410 : term,X411 : subst > term > subst > $o] : (~! [X412 : subst > term > term] : (~! [X413 : term] : (~($true = (X411 @ id @ X413 @ id)) | ((X411 @ id @ (X412 @ id @ X413) @ id) = $true)) | ((X411 @ id @ (lam @ (X412 @ sh @ one)) @ id) = $true) | ~! [X414 : subst,X415 : subst,X416 : term] : ((X412 @ (comp @ X414 @ X415) @ (sub @ X416 @ X415)) = (sub @ (X412 @ X414 @ X416) @ X415))) | ~! [X417 : term] : (~((X411 @ id @ X417 @ id) = $true) | ($true = (X411 @ id @ (sub @ X410 @ (push @ X417 @ id)) @ id))) | ((X411 @ id @ (lam @ (sub @ X410 @ (push @ one @ sh))) @ id) = $true)) => ! [X418 : term,X419 : subst > term > subst > $o] : (($true = (X419 @ id @ (lam @ X418) @ id)) | ~! [X420 : subst,X421 : subst,X422 : term,X423 : subst] : (((X419 @ X423 @ X422 @ (comp @ X421 @ X420)) = $true) | ~((X419 @ (comp @ X423 @ X421) @ (sub @ X422 @ X421) @ X420) = $true)) | ~! [X424 : subst,X425 : subst,X426 : subst,X427 : term] : (($true = (X419 @ (comp @ X426 @ X425) @ (sub @ X427 @ X425) @ X424)) | ~((X419 @ X426 @ X427 @ (comp @ X425 @ X424)) = $true)) | ~! [X428 : subst > term > term] : (~! [X429 : term,X430 : subst,X431 : subst] : ((sub @ (X428 @ X431 @ X429) @ X430) = (X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430))) | ($true = (X419 @ id @ (lam @ (X428 @ sh @ one)) @ id)) | ~! [X432 : term] : (~((X419 @ id @ X432 @ id) = $true) | ($true = (X419 @ id @ (X428 @ id @ X432) @ id)))) | ~! [X433 : term] : (~($true = (X419 @ id @ X433 @ id)) | ((X419 @ id @ (sub @ X418 @ (push @ X433 @ id)) @ id) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & (hoaslamnotap_lthm = $true) & ((! [X434 : term,X435 : subst] : ((sub @ one @ (push @ X434 @ X435)) = X434) => (! [X436 : term,X437 : subst] : ((comp @ sh @ (push @ X436 @ X437)) = X437) => (! [X438 : term,X439 : term] : (((lam @ X438) != (lam @ X439)) | (X438 = X439)) => ! [X440 : term,X441 : subst > term > term,X442 : subst > term > term,X443 : subst] : (((X441 @ X443 @ X440) = (X442 @ X443 @ X440)) | ~! [X447 : subst,X448 : subst,X449 : term] : ((X442 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447)) = (sub @ (X442 @ X448 @ X449) @ X447)) | ~! [X444 : term,X445 : subst,X446 : subst] : ((X441 @ (comp @ X446 @ X445) @ (sub @ X444 @ X445)) = (sub @ (X441 @ X446 @ X444) @ X445)) | ((lam @ (X441 @ sh @ one)) != (lam @ (X442 @ sh @ one))))))) <=> (hoaslaminj_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & ! [X450 : d_term] : (d_one = X450) & ((hoasinduction_lem2 = $true) <=> ! [X451 : subst > term > subst > $o,X452 : term,X453 : term] : (~((X451 @ id @ X452 @ id) = $true) | ((X451 @ id @ (ap @ X453 @ X452) @ id) = $true) | ~((X451 @ id @ X453 @ id) = $true) | ~! [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (($true = (X451 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454)) | ~((X451 @ X457 @ X456 @ (comp @ X455 @ X454)) = $true)) | ~! [X458 : term,X459 : term] : (($true = (X451 @ id @ (ap @ (sub @ X458 @ id) @ X459) @ id)) | ~($true = (X451 @ id @ X459 @ id)) | ~((X451 @ id @ X458 @ id) = $true)) | ~! [X460 : subst,X461 : subst,X462 : term,X463 : subst] : (~((X451 @ (comp @ X460 @ X463) @ (sub @ X462 @ X463) @ X461) = $true) | ((X451 @ X460 @ X462 @ (comp @ X463 @ X461)) = $true)))) & (hoasinduction_lem2_gthm = $true) & (ulamvarsh = $true) & (hoasinduction_lem1_gthm = $true) & ! [X464 : subst] : ? [X465 : d_subst] : ((d2subst @ X465) = X464) & (! [X466 : subst] : ((comp @ X466 @ id) = X466) <=> (axidr = $true)) & (! [X467 : term] : ~! [X468 : subst > term > term] : ((X468 @ sh @ one) != (sub @ X467 @ (push @ one @ sh))) <=> (hoasinduction_lem3b_lthm = $true)) & ! [X469 : d_term,X470 : d_term] : (((d2term @ X469) = (d2term @ X470)) => (X469 = X470)) & ((axidl = $true) <=> ! [X471 : subst] : ((comp @ id @ X471) = X471)) & (! [X472 : term > $o,X473 : subst,X474 : term > $o,X475 : term] : (~! [X476 : term] : ((X472 @ X476) = (X474 @ (sub @ X476 @ (push @ X475 @ X473)))) | ~((X474 @ X475) = $true) | ((X472 @ one) = $true)) <=> (pushprop_lem1v2 = $true)) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X477 : term,X478 : subst > term > subst > $o] : (~! [X479 : subst,X480 : subst,X481 : subst,X482 : term] : (((X478 @ (comp @ X481 @ X479) @ (sub @ X482 @ X479) @ X480) = $true) | ~((X478 @ X481 @ X482 @ (comp @ X479 @ X480)) = $true)) | ((X478 @ id @ (lam @ X477) @ id) = $true) | ~! [X483 : term] : (($true = (X478 @ id @ (sub @ X477 @ (push @ X483 @ id)) @ id)) | ~((X478 @ id @ X483 @ id) = $true)) | ~! [X484 : subst > term > term] : (($true = (X478 @ id @ (lam @ (X484 @ sh @ one)) @ id)) | ~! [X485 : subst,X486 : subst,X487 : term] : ((sub @ (X484 @ X486 @ X487) @ X485) = (X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485))) | ~! [X488 : term] : (((X478 @ id @ (X484 @ id @ X488) @ id) = $true) | ~($true = (X478 @ id @ X488 @ id)))) | ~! [X489 : subst,X490 : subst,X491 : term,X492 : subst] : (~((X478 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490) @ X489) = $true) | ((X478 @ X492 @ X491 @ (comp @ X490 @ X489)) = $true)))) & ((hoasinduction = $true) <=> ! [X493 : term,X494 : subst > term > subst > $o] : (~! [X495 : term,X496 : subst,X497 : subst,X498 : subst] : (((X494 @ X498 @ X495 @ (comp @ X497 @ X496)) = $true) | ~($true = (X494 @ (comp @ X498 @ X497) @ (sub @ X495 @ X497) @ X496))) | ~! [X499 : term,X500 : term] : (~((X494 @ id @ X499 @ id) = $true) | ~((X494 @ id @ X500 @ id) = $true) | ((X494 @ id @ (ap @ (sub @ X500 @ id) @ X499) @ id) = $true)) | ~! [X501 : subst > term > term] : (~! [X502 : subst,X503 : term,X504 : subst] : ((X501 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X501 @ X502 @ X503) @ X504)) | ~! [X505 : term] : (~((X494 @ id @ X505 @ id) = $true) | ((X494 @ id @ (X501 @ id @ X505) @ id) = $true)) | ($true = (X494 @ id @ (lam @ (X501 @ sh @ one)) @ id))) | ~! [X506 : term] : (~((var @ (sub @ X506 @ id)) = $true) | ($true = (X494 @ id @ X506 @ id))) | ~! [X507 : subst,X508 : subst,X509 : term,X510 : subst] : (($true = (X494 @ (comp @ X510 @ X508) @ (sub @ X509 @ X508) @ X507)) | ~((X494 @ X510 @ X509 @ (comp @ X508 @ X507)) = $true)) | ($true = (X494 @ id @ X493 @ id)))) & ((d2subst @ d_id) = id) & ((hoasinduction_lem3a = $true) <=> ! [X511 : term,X512 : subst > term > subst > $o] : (((X512 @ id @ (lam @ X511) @ id) = $true) | ~! [X513 : subst > term > term] : (((X512 @ id @ (lam @ (X513 @ sh @ one)) @ id) = $true) | ~! [X514 : subst,X515 : term,X516 : subst] : ((X513 @ (comp @ X514 @ X516) @ (sub @ X515 @ X516)) = (sub @ (X513 @ X514 @ X515) @ X516)) | ~! [X517 : term] : (~((X512 @ id @ X517 @ id) = $true) | ($true = (X512 @ id @ (X513 @ id @ X517) @ id)))) | ~! [X518 : term] : (~((X512 @ id @ X518 @ id) = $true) | ((X512 @ id @ (sub @ X511 @ (push @ X518 @ id)) @ id) = $true)))) & (hoaslamnotvar_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ~(ulamvar1 = $true) & (hoasapinj2_gthm = $true) & ((! [X519 : subst,X520 : term] : ((sub @ one @ (push @ X520 @ X519)) = X520) => (! [X521 : subst,X522 : term] : ((comp @ sh @ (push @ X522 @ X521)) = X521) => ! [X523 : term > $o,X524 : term > $o,X525 : term,X526 : subst,X527 : term] : (~($true = (X523 @ X527)) | ($true = (X524 @ (sub @ X527 @ (push @ X525 @ X526))))))) <=> (pushprop_lem1_lthm = $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (induction2_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X528 : term,X529 : term,X530 : term,X531 : term] : ((X529 = X530) | ((ap @ X531 @ X529) != (ap @ X528 @ X530))) => ! [X532 : term,X533 : term,X534 : term,X535 : term] : ((X532 = X534) | ((ap @ (sub @ X533 @ id) @ X532) != (ap @ (sub @ X535 @ id) @ X534))))) & ((substmonoid_lthm = $true) <=> (! [X536 : subst] : ((comp @ id @ X536) = X536) => (! [X537 : subst] : ((comp @ X537 @ id) = X537) => (! [X538 : subst] : ((comp @ X538 @ id) = X538) & ! [X539 : subst] : ((comp @ id @ X539) = X539))))) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem1v2_gthm = $true)),
% 2.65/1.76 inference(rectify,[status(thm)],[f8])).
% 2.65/1.76 thf(f8,plain,(
% 2.65/1.76 ((induction2lem = $true) <=> ! [X0 : subst,X1 : term > $o,X2 : term] : (~! [X3 : term,X4 : term] : (~((X1 @ X3) = $true) | ((X1 @ (ap @ X3 @ X4)) = $true) | ~((X1 @ X4) = $true)) | ((X1 @ (sub @ X2 @ X0)) = $true) | ~! [X5 : term] : (~((var @ X5) = $true) | ((X1 @ (sub @ X5 @ X0)) = $true)) | ~! [X6 : term] : (((X1 @ (lam @ X6)) = $true) | ~! [X7 : term] : (~($true = (X1 @ X7)) | ($true = (X1 @ (sub @ X6 @ (push @ X7 @ id)))))))) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar_gthm = $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem3b = $true) <=> ! [X8 : term] : ~! [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) != (X9 @ sh @ one))) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (one = (d2term @ d_one)) & ((hoasapinj1 = $true) <=> ! [X13 : term,X14 : term,X15 : term,X16 : term] : ((X13 = X16) | ((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15)))) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1 = $true) & (axvarshift = $true) & (! [X17 : term,X18 : subst > term > subst > $o,X19 : term > $o] : (~! [X20 : term] : (((X19 @ (sub @ X17 @ (push @ X20 @ id))) = $true) | ~($true = (X19 @ X20))) | ~! [X21 : term] : ((X19 @ X21) = (X18 @ id @ X21 @ id)) | ~! [X22 : subst > term > term] : (~! [X23 : subst,X24 : subst,X25 : term] : ((X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (X22 @ X24 @ X25) @ X23)) | ~! [X26 : term] : (~($true = (X18 @ id @ X26 @ id)) | ((X18 @ id @ (X22 @ id @ X26) @ id) = $true)) | ($true = (X18 @ id @ (lam @ (X22 @ sh @ one)) @ id))) | ~! [X27 : subst,X28 : term,X29 : subst,X30 : subst] : (~($true = (X18 @ X27 @ X28 @ (comp @ X29 @ X30))) | ((X18 @ (comp @ X27 @ X29) @ (sub @ X28 @ X29) @ X30) = $true)) | ~! [X31 : subst,X32 : subst,X33 : subst,X34 : term] : (~($true = (X18 @ (comp @ X33 @ X32) @ (sub @ X34 @ X32) @ X31)) | ((X18 @ X33 @ X34 @ (comp @ X32 @ X31)) = $true)) | ((X19 @ (lam @ X17)) = $true)) <=> (hoasinduction_lem3v2 = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop = $true) & (! [X40 : term,X41 : term] : ((X40 = X41) | ((sub @ X40 @ sh) != (sub @ X41 @ sh))) <=> (shinj = $true)) & ((pushprop_lem0 = $true) <=> ! [X42 : term > $o,X43 : term,X44 : subst] : ~! [X45 : term > $o] : ~! [X46 : term] : ((X45 @ X46) = (X42 @ (sub @ X46 @ (push @ X43 @ X44))))) & (! [X47 : term > $o,X48 : term] : (~! [X49 : term] : (~((var @ X49) = $true) | ((X47 @ X49) = $true)) | ~! [X50 : term] : (~! [X51 : term] : (~($true = (X47 @ X51)) | ((X47 @ (sub @ X50 @ (push @ X51 @ id))) = $true)) | ((X47 @ (lam @ X50)) = $true)) | ~! [X52 : term,X53 : term] : (~($true = (X47 @ X52)) | ($true = (X47 @ (ap @ X52 @ X53))) | ~($true = (X47 @ X53))) | ($true = (X47 @ X48))) <=> (induction2 = $true)) & ((! [X54 : subst > term > subst > $o] : ~! [X55 : term > $o] : ~! [X56 : term] : ((X55 @ X56) = (X54 @ id @ X56 @ id)) => (! [X57 : term > $o,X58 : term] : (~! [X59 : term,X60 : term] : (~($true = (X57 @ X59)) | ($true = (X57 @ (ap @ X60 @ X59))) | ~((X57 @ X60) = $true)) | ~! [X61 : term] : (~($true = (var @ X61)) | ((X57 @ X61) = $true)) | ((X57 @ X58) = $true) | ~! [X62 : term] : (($true = (X57 @ (lam @ X62))) | ~! [X63 : term] : (~($true = (X57 @ X63)) | ($true = (X57 @ (sub @ X62 @ (push @ X63 @ id))))))) => (! [X64 : term] : ((sub @ X64 @ id) = X64) => (! [X65 : subst > term > subst > $o,X66 : term > $o,X67 : term] : (~! [X68 : subst > term > term] : (~! [X69 : term] : (~((X65 @ id @ X69 @ id) = $true) | ((X65 @ id @ (X68 @ id @ X69) @ id) = $true)) | ~! [X70 : subst,X71 : subst,X72 : term] : ((X68 @ (comp @ X71 @ X70) @ (sub @ X72 @ X70)) = (sub @ (X68 @ X71 @ X72) @ X70)) | ($true = (X65 @ id @ (lam @ (X68 @ sh @ one)) @ id))) | ~! [X73 : term] : (((X66 @ (sub @ X67 @ (push @ X73 @ id))) = $true) | ~((X66 @ X73) = $true)) | ~! [X74 : term] : ((X65 @ id @ X74 @ id) = (X66 @ X74)) | ($true = (X66 @ (lam @ X67)))) => ! [X75 : subst > term > subst > $o,X76 : term] : (((X75 @ id @ X76 @ id) = $true) | ~! [X77 : subst > term > term] : (((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) = $true) | ~! [X78 : subst,X79 : term,X80 : subst] : ((X77 @ (comp @ X80 @ X78) @ (sub @ X79 @ X78)) = (sub @ (X77 @ X80 @ X79) @ X78)) | ~! [X81 : term] : (((X75 @ id @ (X77 @ id @ X81) @ id) = $true) | ~($true = (X75 @ id @ X81 @ id)))) | ~! [X82 : term,X83 : term] : (($true = (X75 @ id @ (ap @ (sub @ X83 @ id) @ X82) @ id)) | ~((X75 @ id @ X83 @ id) = $true) | ~((X75 @ id @ X82 @ id) = $true))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((! [X84 : term] : ((sub @ X84 @ id) = X84) => ! [X85 : term,X86 : subst > term > subst > $o,X87 : term > $o] : (~! [X88 : term] : ((X87 @ X88) = (X86 @ id @ X88 @ id)) | ~! [X89 : subst > term > term] : (~! [X90 : term] : (($true = (X86 @ id @ (X89 @ id @ X90) @ id)) | ~((X86 @ id @ X90 @ id) = $true)) | ((X86 @ id @ (lam @ (X89 @ sh @ one)) @ id) = $true) | ~! [X91 : term,X92 : subst,X93 : subst] : ((sub @ (X89 @ X93 @ X91) @ X92) = (X89 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92)))) | ~! [X94 : subst,X95 : subst,X96 : subst,X97 : term] : (((X86 @ (comp @ X95 @ X96) @ (sub @ X97 @ X96) @ X94) = $true) | ~((X86 @ X95 @ X97 @ (comp @ X96 @ X94)) = $true)) | ~! [X98 : term,X99 : subst,X100 : subst,X101 : subst] : (((X86 @ X99 @ X98 @ (comp @ X100 @ X101)) = $true) | ~($true = (X86 @ (comp @ X99 @ X100) @ (sub @ X98 @ X100) @ X101))) | ((X87 @ (lam @ X85)) = $true) | ~! [X102 : term] : (((X87 @ (sub @ X85 @ (push @ X102 @ id))) = $true) | ~((X87 @ X102) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((hoasapinj2 = $true) <=> ! [X103 : term,X104 : term,X105 : term,X106 : term] : ((X105 = X106) | ((ap @ (sub @ X104 @ id) @ X106) != (ap @ (sub @ X103 @ id) @ X105)))) & (hoaslaminj_gthm = $true) & (! [X107 : subst] : ((push @ (sub @ one @ X107) @ (comp @ sh @ X107)) = X107) <=> (axscons = $true)) & (apnotvar = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X108 : term] : ((sub @ X108 @ id) = X108) => (! [X109 : term,X110 : term,X111 : term,X112 : term] : (((ap @ X111 @ X110) != (ap @ X112 @ X109)) | (X111 = X112)) => ! [X113 : term,X114 : term,X115 : term,X116 : term] : (((ap @ (sub @ X114 @ id) @ X116) != (ap @ (sub @ X113 @ id) @ X115)) | (X113 = X114)))) <=> (hoasapinj1_lthm = $true)) & (termmset_gthm = $true) & ! [X117 : d_subst,X118 : d_subst] : (((d2subst @ X118) = (d2subst @ X117)) => (X117 = X118)) & ! [X119 : term] : ? [X120 : d_term] : ((d2term @ X120) = X119) & (pushprop_lem3v2_lthm = $true) & (axapp = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X121 : term,X122 : subst,X123 : term > $o] : ~! [X124 : term > $o] : ~! [X125 : term] : ((X124 @ X125) = (X123 @ (sub @ X125 @ (push @ X121 @ X122))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X126 : term,X127 : subst > term > subst > $o] : (~! [X128 : term] : (~((X127 @ id @ X128 @ id) = $true) | ($true = (X127 @ id @ (sub @ X126 @ (push @ X128 @ id)) @ id))) | ~! [X129 : term,X130 : subst > term > term] : (~! [X131 : subst,X132 : term,X133 : subst] : ((sub @ (X130 @ X133 @ X132) @ X131) = (sub @ (sub @ X129 @ (push @ X132 @ X133)) @ X131)) | ((X127 @ id @ (lam @ (X130 @ sh @ one)) @ id) = $true) | ~! [X134 : term] : (((X127 @ id @ (X130 @ id @ X134) @ id) = $true) | ~((X127 @ id @ X134 @ id) = $true)) | ~! [X135 : term,X136 : subst,X137 : subst] : ((sub @ X129 @ (push @ (sub @ X135 @ X137) @ (comp @ X136 @ X137))) = (X130 @ (comp @ X136 @ X137) @ (sub @ X135 @ X137)))) | ((X127 @ id @ (lam @ (sub @ X126 @ (push @ one @ sh))) @ id) = $true))) & ((d2subst @ d_id) = sh) & (! [X138 : term > $o,X139 : term,X140 : term,X141 : subst,X142 : term > $o] : (~((X142 @ X139) = $true) | ((X138 @ (sub @ X139 @ (push @ X140 @ X141))) = $true)) <=> (pushprop_lem1 = $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X143 : subst > term > subst > $o,X144 : term] : (~! [X145 : subst > term > term] : (($true = (X143 @ id @ (lam @ (X145 @ sh @ one)) @ id)) | ~! [X146 : term] : (~((X143 @ id @ X146 @ id) = $true) | ((X143 @ id @ (X145 @ id @ X146) @ id) = $true)) | ~! [X147 : subst,X148 : term,X149 : subst] : ((sub @ (X145 @ X147 @ X148) @ X149) = (X145 @ (comp @ X147 @ X149) @ (sub @ X148 @ X149)))) | ~! [X150 : term,X151 : term] : (~((X143 @ id @ X151 @ id) = $true) | ((X143 @ id @ (ap @ (sub @ X150 @ id) @ X151) @ id) = $true) | ~((X143 @ id @ X150 @ id) = $true)) | ((X143 @ id @ X144 @ id) = $true))) & (! [X152 : term,X153 : term] : ((X152 = X153) | ((lam @ X152) != (lam @ X153))) <=> (laminj = $true)) & ((! [X154 : term] : ((sub @ X154 @ id) = X154) => ! [X155 : term] : ((sub @ X155 @ id) = X155)) <=> (termmset_lthm = $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((! [X156 : term] : ((sub @ X156 @ id) = X156) => (! [X157 : term > $o,X158 : term,X159 : subst] : (~! [X160 : term,X161 : term] : (($true = (X157 @ (ap @ X160 @ X161))) | ~((X157 @ X160) = $true) | ~((X157 @ X161) = $true)) | ~! [X162 : term] : (~((var @ X162) = $true) | ((X157 @ (sub @ X162 @ X159)) = $true)) | ~! [X163 : term] : (~! [X164 : term] : (((X157 @ (sub @ X163 @ (push @ X164 @ id))) = $true) | ~((X157 @ X164) = $true)) | ((X157 @ (lam @ X163)) = $true)) | ((X157 @ (sub @ X158 @ X159)) = $true)) => ! [X165 : term > $o,X166 : term] : (~! [X167 : term,X168 : term] : (~((X165 @ X168) = $true) | ($true = (X165 @ (ap @ X168 @ X167))) | ~((X165 @ X167) = $true)) | ((X165 @ X166) = $true) | ~! [X169 : term] : (~((var @ X169) = $true) | ((X165 @ X169) = $true)) | ~! [X170 : term] : (~! [X171 : term] : (((X165 @ (sub @ X170 @ (push @ X171 @ id))) = $true) | ~((X165 @ X171) = $true)) | ((X165 @ (lam @ X170)) = $true))))) <=> (induction2_lthm = $true)) & (ulamvarind = $true) & (induction2lem_gthm = $true) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ((! [X172 : subst,X173 : term] : ((sub @ one @ (push @ X173 @ X172)) = X173) => ! [X174 : subst,X175 : term > $o,X176 : term > $o,X177 : term] : (~! [X178 : term] : ((X175 @ X178) = (X176 @ (sub @ X178 @ (push @ X177 @ X174)))) | ~((X176 @ X177) = $true) | ($true = (X175 @ one)))) <=> (pushprop_lem1v2_lthm = $true)) & (! [X179 : term] : ((sub @ X179 @ id) = X179) <=> (axvarid = $true)) & (! [X180 : subst,X181 : term] : ((sub @ one @ (push @ X181 @ X180)) = X181) <=> (axvarcons = $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)))))))) & ~((var @ (d2term @ d_one)) = $true) & ! [X185 : d_subst] : (d_id = X185) & (hoasinduction_lem1_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X186 : term] : ~! [X187 : subst > term > term] : ~! [X188 : subst,X189 : term] : ((X187 @ X188 @ X189) = (sub @ X186 @ (push @ X189 @ X188)))) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & (pushprop_lem1_gthm = $true) & (axmap = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X190 : subst > term > subst > $o,X191 : term] : (~! [X192 : term] : (~((X190 @ id @ X192 @ id) = $true) | ((X190 @ id @ (sub @ X191 @ (push @ X192 @ id)) @ id) = $true)) | ~! [X193 : subst > term > term] : (~! [X194 : term] : (~((X190 @ id @ X194 @ id) = $true) | ((X190 @ id @ (X193 @ id @ X194) @ id) = $true)) | ((X190 @ id @ (lam @ (X193 @ sh @ one)) @ id) = $true) | ~! [X195 : term,X196 : subst,X197 : subst] : ((X193 @ (comp @ X197 @ X196) @ (sub @ X195 @ X196)) = (sub @ (X193 @ X197 @ X195) @ X196))) | ((X190 @ id @ (lam @ (sub @ X191 @ (push @ one @ sh))) @ id) = $true))) & ((hoaslaminj = $true) <=> ! [X198 : subst > term > term,X199 : subst > term > term,X200 : subst,X201 : term] : (~! [X202 : term,X203 : subst,X204 : subst] : ((sub @ (X198 @ X204 @ X202) @ X203) = (X198 @ (comp @ X204 @ X203) @ (sub @ X202 @ X203))) | ((lam @ (X198 @ sh @ one)) != (lam @ (X199 @ sh @ one))) | ~! [X205 : term,X206 : subst,X207 : subst] : ((sub @ (X199 @ X206 @ X205) @ X207) = (X199 @ (comp @ X206 @ X207) @ (sub @ X205 @ X207))) | ((X199 @ X200 @ X201) = (X198 @ X200 @ X201)))) & ((! [X208 : subst > term > subst > $o] : ~! [X209 : term > $o] : ~! [X210 : term] : ((X208 @ id @ X210 @ id) = (X209 @ X210)) => (! [X211 : term,X212 : term > $o] : (~! [X213 : term,X214 : term] : (~($true = (X212 @ X213)) | ~((X212 @ X214) = $true) | ($true = (X212 @ (ap @ X214 @ X213)))) | ($true = (X212 @ X211)) | ~! [X215 : term] : (~! [X216 : term] : (~($true = (X212 @ X216)) | ((X212 @ (sub @ X215 @ (push @ X216 @ id))) = $true)) | ($true = (X212 @ (lam @ X215)))) | ~! [X217 : term] : (~($true = (var @ X217)) | ((X212 @ X217) = $true))) => (! [X218 : term] : ((sub @ X218 @ id) = X218) => (! [X219 : term > $o,X220 : term,X221 : subst > term > subst > $o] : (~! [X222 : term] : ((X221 @ id @ X222 @ id) = (X219 @ X222)) | ~! [X223 : term] : (($true = (X219 @ (sub @ X220 @ (push @ X223 @ id)))) | ~((X219 @ X223) = $true)) | ~! [X224 : subst > term > term] : (($true = (X221 @ id @ (lam @ (X224 @ sh @ one)) @ id)) | ~! [X225 : term,X226 : subst,X227 : subst] : ((X224 @ (comp @ X227 @ X226) @ (sub @ X225 @ X226)) = (sub @ (X224 @ X227 @ X225) @ X226)) | ~! [X228 : term] : (((X221 @ id @ (X224 @ id @ X228) @ id) = $true) | ~($true = (X221 @ id @ X228 @ id)))) | ((X219 @ (lam @ X220)) = $true)) => ! [X229 : term,X230 : subst > term > subst > $o] : (~! [X231 : subst,X232 : term,X233 : subst,X234 : subst] : (~((X230 @ X233 @ X232 @ (comp @ X234 @ X231)) = $true) | ($true = (X230 @ (comp @ X233 @ X234) @ (sub @ X232 @ X234) @ X231))) | ~! [X235 : term,X236 : term] : (~((X230 @ id @ X235 @ id) = $true) | ((X230 @ id @ (ap @ (sub @ X236 @ id) @ X235) @ id) = $true) | ~((X230 @ id @ X236 @ id) = $true)) | ~! [X237 : term] : (~((var @ (sub @ X237 @ id)) = $true) | ((X230 @ id @ X237 @ id) = $true)) | ~! [X238 : subst,X239 : subst,X240 : term,X241 : subst] : (~((X230 @ (comp @ X241 @ X238) @ (sub @ X240 @ X238) @ X239) = $true) | ($true = (X230 @ X241 @ X240 @ (comp @ X238 @ X239)))) | ~! [X242 : subst > term > term] : (~! [X243 : term] : (($true = (X230 @ id @ (X242 @ id @ X243) @ id)) | ~($true = (X230 @ id @ X243 @ id))) | ~! [X244 : term,X245 : subst,X246 : subst] : ((sub @ (X242 @ X246 @ X244) @ X245) = (X242 @ (comp @ X246 @ X245) @ (sub @ X244 @ X245))) | ($true = (X230 @ id @ (lam @ (X242 @ sh @ one)) @ id))) | ((X230 @ id @ X229 @ id) = $true)))))) <=> (hoasinduction_lthm_3 = $true)) & (! [X247 : term,X248 : subst] : ((comp @ sh @ (push @ X247 @ X248)) = X248) <=> (axshiftcons = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X251 : term,X252 : term,X253 : term,X254 : term] : (((ap @ X253 @ X251) != (ap @ X254 @ X252)) | (X251 = X252)) <=> (apinj2 = $true)) & ((substmonoid = $true) <=> (! [X255 : subst] : ((comp @ X255 @ id) = X255) & ! [X256 : subst] : ((comp @ id @ X256) = X256))) & (! [X257 : term] : ((sub @ X257 @ id) = X257) <=> (termmset = $true)) & (hoasinduction_gthm = $true) & (pushprop_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X258 : term] : ((sub @ X258 @ id) = X258) => (! [X259 : subst > term > subst > $o,X260 : term] : (~! [X261 : subst > term > term] : (($true = (X259 @ id @ (lam @ (X261 @ sh @ one)) @ id)) | ~! [X262 : subst,X263 : term,X264 : subst] : ((sub @ (X261 @ X262 @ X263) @ X264) = (X261 @ (comp @ X262 @ X264) @ (sub @ X263 @ X264))) | ~! [X265 : term] : (~((X259 @ id @ X265 @ id) = $true) | ((X259 @ id @ (X261 @ id @ X265) @ id) = $true))) | ~! [X266 : term] : (((X259 @ id @ (sub @ X260 @ (push @ X266 @ id)) @ id) = $true) | ~((X259 @ id @ X266 @ id) = $true)) | ($true = (X259 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id))) => ! [X267 : subst > term > subst > $o,X268 : term] : (~! [X269 : term] : (~($true = (X267 @ id @ X269 @ id)) | ($true = (X267 @ id @ (sub @ X268 @ (push @ X269 @ id)) @ id))) | ~! [X270 : subst > term > term] : (~! [X271 : term,X272 : subst,X273 : subst] : ((X270 @ (comp @ X272 @ X273) @ (sub @ X271 @ X273)) = (sub @ (X270 @ X272 @ X271) @ X273)) | ~! [X274 : term] : (((X267 @ id @ (X270 @ id @ X274) @ id) = $true) | ~((X267 @ id @ X274 @ id) = $true)) | ((X267 @ id @ (lam @ (X270 @ sh @ one)) @ id) = $true)) | ((X267 @ id @ (lam @ X268) @ id) = $true))))) & ~(lamnotap = $true) & (axclos = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X275 : term,X276 : subst > term > subst > $o,X277 : term > $o] : (((X277 @ (lam @ X275)) = $true) | ~! [X278 : term] : (~($true = (X277 @ X278)) | ((X277 @ (sub @ X275 @ (push @ X278 @ id))) = $true)) | ~! [X279 : term] : ((X276 @ id @ X279 @ id) = (X277 @ X279)) | ~! [X280 : subst > term > term] : (~! [X281 : term] : (((X276 @ id @ (X280 @ id @ X281) @ id) = $true) | ~((X276 @ id @ X281 @ id) = $true)) | ((X276 @ id @ (lam @ (X280 @ sh @ one)) @ id) = $true) | ~! [X282 : subst,X283 : subst,X284 : term] : ((X280 @ (comp @ X282 @ X283) @ (sub @ X284 @ X283)) = (sub @ (X280 @ X282 @ X284) @ X283))))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((apinj1 = $true) <=> ! [X285 : term,X286 : term,X287 : term,X288 : term] : (((ap @ X286 @ X287) != (ap @ X288 @ X285)) | (X286 = X288))) & ((! [X289 : term,X290 : term > $o] : (($true = (X290 @ X289)) | ~! [X291 : term] : (~((var @ X291) = $true) | ((X290 @ X291) = $true)) | ~! [X292 : term] : (~! [X293 : term] : (~((X290 @ X293) = $true) | ($true = (X290 @ (sub @ X292 @ (push @ X293 @ id))))) | ((X290 @ (lam @ X292)) = $true)) | ~! [X294 : term,X295 : term] : (((X290 @ (ap @ X295 @ X294)) = $true) | ~($true = (X290 @ X295)) | ~((X290 @ X294) = $true))) => (! [X296 : term,X297 : subst > term > subst > $o,X298 : term] : (~! [X299 : subst,X300 : subst,X301 : term,X302 : subst] : (~((X297 @ X299 @ X301 @ (comp @ X300 @ X302)) = $true) | ($true = (X297 @ (comp @ X299 @ X300) @ (sub @ X301 @ X300) @ X302))) | ~! [X303 : term,X304 : term] : (((X297 @ id @ (ap @ (sub @ X303 @ id) @ X304) @ id) = $true) | ~($true = (X297 @ id @ X303 @ id)) | ~((X297 @ id @ X304 @ id) = $true)) | ~((X297 @ id @ X296 @ id) = $true) | ~! [X305 : subst,X306 : subst,X307 : term,X308 : subst] : (~((X297 @ (comp @ X305 @ X306) @ (sub @ X307 @ X306) @ X308) = $true) | ((X297 @ X305 @ X307 @ (comp @ X306 @ X308)) = $true)) | ~((X297 @ id @ X298 @ id) = $true) | ((X297 @ id @ (ap @ X296 @ X298) @ id) = $true)) => (! [X309 : term,X310 : subst > term > subst > $o] : (~! [X311 : term] : (((X310 @ id @ (sub @ X309 @ (push @ X311 @ id)) @ id) = $true) | ~((X310 @ id @ X311 @ id) = $true)) | ~! [X312 : subst,X313 : subst,X314 : term,X315 : subst] : (~((X310 @ X315 @ X314 @ (comp @ X312 @ X313)) = $true) | ((X310 @ (comp @ X315 @ X312) @ (sub @ X314 @ X312) @ X313) = $true)) | ~! [X316 : term,X317 : subst,X318 : subst,X319 : subst] : (((X310 @ X317 @ X316 @ (comp @ X318 @ X319)) = $true) | ~($true = (X310 @ (comp @ X317 @ X318) @ (sub @ X316 @ X318) @ X319))) | ((X310 @ id @ (lam @ X309) @ id) = $true) | ~! [X320 : subst > term > term] : (($true = (X310 @ id @ (lam @ (X320 @ sh @ one)) @ id)) | ~! [X321 : term] : (($true = (X310 @ id @ (X320 @ id @ X321) @ id)) | ~((X310 @ id @ X321 @ id) = $true)) | ~! [X322 : term,X323 : subst,X324 : subst] : ((X320 @ (comp @ X324 @ X323) @ (sub @ X322 @ X323)) = (sub @ (X320 @ X324 @ X322) @ X323)))) => ! [X325 : term,X326 : subst > term > subst > $o] : (~! [X327 : term,X328 : term] : (((X326 @ id @ (ap @ (sub @ X328 @ id) @ X327) @ id) = $true) | ~((X326 @ id @ X327 @ id) = $true) | ~((X326 @ id @ X328 @ id) = $true)) | ~! [X329 : subst > term > term] : (~! [X330 : term,X331 : subst,X332 : subst] : ((X329 @ (comp @ X331 @ X332) @ (sub @ X330 @ X332)) = (sub @ (X329 @ X331 @ X330) @ X332)) | ($true = (X326 @ id @ (lam @ (X329 @ sh @ one)) @ id)) | ~! [X333 : term] : (~((X326 @ id @ X333 @ id) = $true) | ((X326 @ id @ (X329 @ id @ X333) @ id) = $true))) | ~! [X334 : subst,X335 : term,X336 : subst,X337 : subst] : (($true = (X326 @ (comp @ X334 @ X337) @ (sub @ X335 @ X337) @ X336)) | ~((X326 @ X334 @ X335 @ (comp @ X337 @ X336)) = $true)) | ~! [X338 : subst,X339 : subst,X340 : subst,X341 : term] : (((X326 @ X338 @ X341 @ (comp @ X339 @ X340)) = $true) | ~($true = (X326 @ (comp @ X338 @ X339) @ (sub @ X341 @ X339) @ X340))) | ~! [X342 : term] : (~($true = (var @ (sub @ X342 @ id))) | ((X326 @ id @ X342 @ id) = $true)) | ((X326 @ id @ X325 @ id) = $true))))) <=> (hoasinduction_lthm = $true)) & (! [X343 : subst > term > subst > $o] : ~! [X344 : term > $o] : ~! [X345 : term] : ((X344 @ X345) = (X343 @ id @ X345 @ id)) <=> (hoasinduction_lem0 = $true)) & (pushprop_lem0_gthm = $true) & (! [X346 : term,X347 : term,X348 : subst > term > term] : (~! [X349 : subst,X350 : subst,X351 : term] : ((sub @ (X348 @ X350 @ X351) @ X349) = (X348 @ (comp @ X350 @ X349) @ (sub @ X351 @ X349))) | ((ap @ (sub @ X346 @ id) @ X347) != (lam @ (X348 @ sh @ one)))) <=> (hoaslamnotap = $true)) & (axabs = $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X352 : term,X353 : subst > term > subst > $o] : (~! [X354 : term] : (((X353 @ id @ (sub @ X352 @ (push @ X354 @ id)) @ id) = $true) | ~((X353 @ id @ X354 @ id) = $true)) | ~! [X355 : subst > term > term] : (~! [X356 : term,X357 : subst,X358 : subst] : ((sub @ (X355 @ X357 @ X356) @ X358) = (X355 @ (comp @ X357 @ X358) @ (sub @ X356 @ X358))) | ~! [X359 : term] : (((X353 @ id @ (X355 @ id @ X359) @ id) = $true) | ~((X353 @ id @ X359 @ id) = $true)) | ((X353 @ id @ (lam @ (X355 @ sh @ one)) @ id) = $true)) | ((X353 @ id @ (lam @ (sub @ X352 @ (push @ one @ sh))) @ id) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & ((! [X360 : subst,X361 : term] : ((sub @ one @ (push @ X361 @ X360)) = X361) => (! [X362 : term,X363 : subst] : ((comp @ sh @ (push @ X362 @ X363)) = X363) => (! [X364 : subst] : ((comp @ X364 @ id) = X364) => (! [X365 : term,X366 : term > $o] : (~! [X367 : term] : (((X366 @ X367) = $true) | ~((var @ X367) = $true)) | ~! [X368 : term] : (($true = (X366 @ (lam @ X368))) | ~((X366 @ X368) = $true)) | ~! [X369 : term,X370 : term] : (((X366 @ (ap @ X370 @ X369)) = $true) | ~((X366 @ X369) = $true) | ~($true = (X366 @ X370))) | ((X366 @ X365) = $true)) => ! [X371 : term,X372 : term > $o,X373 : subst] : (($true = (X372 @ (sub @ X371 @ X373))) | ~! [X374 : term] : (((X372 @ (sub @ X374 @ X373)) = $true) | ~((var @ X374) = $true)) | ~! [X375 : term] : (~! [X376 : term] : (((X372 @ (sub @ X375 @ (push @ X376 @ id))) = $true) | ~($true = (X372 @ X376))) | ((X372 @ (lam @ X375)) = $true)) | ~! [X377 : term,X378 : term] : (~((X372 @ X377) = $true) | ((X372 @ (ap @ X378 @ X377)) = $true) | ~($true = (X372 @ X378)))))))) <=> (induction2lem_lthm = $true)) & (axassoc = $true) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X379 : term > $o,X380 : subst > term > subst > $o,X381 : term,X382 : term] : (~! [X383 : subst,X384 : subst,X385 : term,X386 : subst] : (~((X380 @ X384 @ X385 @ (comp @ X383 @ X386)) = $true) | ($true = (X380 @ (comp @ X384 @ X383) @ (sub @ X385 @ X383) @ X386))) | ~! [X387 : term] : ((X379 @ X387) = (X380 @ id @ X387 @ id)) | ~! [X388 : term,X389 : subst,X390 : subst,X391 : subst] : (((X380 @ X390 @ X388 @ (comp @ X391 @ X389)) = $true) | ~((X380 @ (comp @ X390 @ X391) @ (sub @ X388 @ X391) @ X389) = $true)) | ((X379 @ (ap @ X382 @ X381)) = $true) | ~((X379 @ X382) = $true) | ~((X379 @ X381) = $true) | ~! [X392 : term,X393 : term] : (~($true = (X380 @ id @ X392 @ id)) | ($true = (X380 @ id @ (ap @ (sub @ X393 @ id) @ X392) @ id)) | ~((X380 @ id @ X393 @ id) = $true)))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X394 : term] : ~! [X395 : subst > term > term] : ~! [X396 : subst,X397 : term] : ((sub @ X394 @ (push @ X397 @ X396)) = (X395 @ X396 @ X397))) & ((! [X398 : term] : ~! [X399 : subst > term > term] : ~! [X400 : term,X401 : subst] : ((X399 @ X401 @ X400) = (sub @ X398 @ (push @ X400 @ X401))) => (! [X402 : term] : ((sub @ X402 @ id) = X402) => ! [X403 : term,X404 : subst > term > subst > $o,X405 : term > $o] : (($true = (X405 @ (lam @ X403))) | ~! [X406 : term] : (((X405 @ (sub @ X403 @ (push @ X406 @ id))) = $true) | ~((X405 @ X406) = $true)) | ~! [X407 : subst > term > term] : (((X404 @ id @ (lam @ (X407 @ sh @ one)) @ id) = $true) | ~! [X408 : term] : (~((X404 @ id @ X408 @ id) = $true) | ($true = (X404 @ id @ (X407 @ id @ X408) @ id))) | ~! [X409 : term,X410 : subst,X411 : subst] : ((X407 @ (comp @ X410 @ X411) @ (sub @ X409 @ X411)) = (sub @ (X407 @ X410 @ X409) @ X411))) | ~! [X412 : term] : ((X404 @ id @ X412 @ id) = (X405 @ X412))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (lamnotvar = $true) & (pushprop_lthm = $true) & (hoasapnotvar_gthm = $true) & (! [X413 : term > $o,X414 : term] : (~! [X415 : term,X416 : term] : (($true = (X413 @ (ap @ X415 @ X416))) | ~($true = (X413 @ X415)) | ~((X413 @ X416) = $true)) | ~! [X417 : term] : (((X413 @ X417) = $true) | ~($true = (var @ X417))) | ((X413 @ X414) = $true) | ~! [X418 : term] : (~((X413 @ X418) = $true) | ((X413 @ (lam @ X418)) = $true))) <=> (induction = $true)) & ((! [X419 : term] : ((sub @ X419 @ id) = X419) => (! [X420 : term,X421 : subst > term > subst > $o] : (~! [X422 : subst > term > term] : (~! [X423 : term] : (~((X421 @ id @ X423 @ id) = $true) | ((X421 @ id @ (X422 @ id @ X423) @ id) = $true)) | ((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id) = $true) | ~! [X424 : subst,X425 : subst,X426 : term] : ((X422 @ (comp @ X424 @ X425) @ (sub @ X426 @ X425)) = (sub @ (X422 @ X424 @ X426) @ X425))) | ~! [X427 : term] : (~($true = (X421 @ id @ X427 @ id)) | ($true = (X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id))) | ((X421 @ id @ (lam @ (sub @ X420 @ (push @ one @ sh))) @ id) = $true)) => ! [X428 : term,X429 : subst > term > subst > $o] : (((X429 @ id @ (lam @ X428) @ id) = $true) | ~! [X430 : subst,X431 : subst,X432 : term,X433 : subst] : (((X429 @ X433 @ X432 @ (comp @ X431 @ X430)) = $true) | ~((X429 @ (comp @ X433 @ X431) @ (sub @ X432 @ X431) @ X430) = $true)) | ~! [X434 : subst,X435 : subst,X436 : subst,X437 : term] : (((X429 @ (comp @ X436 @ X435) @ (sub @ X437 @ X435) @ X434) = $true) | ~((X429 @ X436 @ X437 @ (comp @ X435 @ X434)) = $true)) | ~! [X438 : subst > term > term] : (~! [X439 : term,X440 : subst,X441 : subst] : ((sub @ (X438 @ X441 @ X439) @ X440) = (X438 @ (comp @ X441 @ X440) @ (sub @ X439 @ X440))) | ((X429 @ id @ (lam @ (X438 @ sh @ one)) @ id) = $true) | ~! [X442 : term] : (~((X429 @ id @ X442 @ id) = $true) | ((X429 @ id @ (X438 @ id @ X442) @ id) = $true))) | ~! [X443 : term] : (~((X429 @ id @ X443 @ id) = $true) | ((X429 @ id @ (sub @ X428 @ (push @ X443 @ id)) @ id) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & (hoaslamnotap_lthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X444 : term,X445 : subst] : ((sub @ one @ (push @ X444 @ X445)) = X444) => (! [X446 : term,X447 : subst] : ((comp @ sh @ (push @ X446 @ X447)) = X447) => (! [X448 : term,X449 : term] : (((lam @ X449) != (lam @ X448)) | (X448 = X449)) => ! [X450 : term,X451 : subst > term > term,X452 : subst > term > term,X453 : subst] : (((lam @ (X452 @ sh @ one)) != (lam @ (X451 @ sh @ one))) | ~! [X454 : term,X455 : subst,X456 : subst] : ((sub @ (X451 @ X456 @ X454) @ X455) = (X451 @ (comp @ X456 @ X455) @ (sub @ X454 @ X455))) | ~! [X457 : subst,X458 : subst,X459 : term] : ((X452 @ (comp @ X458 @ X457) @ (sub @ X459 @ X457)) = (sub @ (X452 @ X458 @ X459) @ X457)) | ((X451 @ X453 @ X450) = (X452 @ X453 @ X450))))))) & (hoasinduction_lem3a_gthm = $true) & ! [X460 : d_term] : (d_one = X460) & ((hoasinduction_lem2 = $true) <=> ! [X461 : subst > term > subst > $o,X462 : term,X463 : term] : (~((X461 @ id @ X462 @ id) = $true) | ($true = (X461 @ id @ (ap @ X463 @ X462) @ id)) | ~($true = (X461 @ id @ X463 @ id)) | ~! [X464 : subst,X465 : subst,X466 : term,X467 : subst] : (($true = (X461 @ (comp @ X467 @ X465) @ (sub @ X466 @ X465) @ X464)) | ~((X461 @ X467 @ X466 @ (comp @ X465 @ X464)) = $true)) | ~! [X468 : term,X469 : term] : (((X461 @ id @ (ap @ (sub @ X468 @ id) @ X469) @ id) = $true) | ~((X461 @ id @ X469 @ id) = $true) | ~($true = (X461 @ id @ X468 @ id))) | ~! [X470 : subst,X471 : subst,X472 : term,X473 : subst] : (~($true = (X461 @ (comp @ X470 @ X473) @ (sub @ X472 @ X473) @ X471)) | ($true = (X461 @ X470 @ X472 @ (comp @ X473 @ X471)))))) & (hoasinduction_lem2_gthm = $true) & (ulamvarsh = $true) & (hoasinduction_lem1_gthm = $true) & ! [X474 : subst] : ? [X475 : d_subst] : ((d2subst @ X475) = X474) & (! [X476 : subst] : ((comp @ X476 @ id) = X476) <=> (axidr = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X477 : term] : ~! [X478 : subst > term > term] : ((X478 @ sh @ one) != (sub @ X477 @ (push @ one @ sh)))) & ! [X479 : d_term,X480 : d_term] : (((d2term @ X479) = (d2term @ X480)) => (X479 = X480)) & (! [X481 : subst] : ((comp @ id @ X481) = X481) <=> (axidl = $true)) & (! [X482 : term > $o,X483 : subst,X484 : term > $o,X485 : term] : (~! [X486 : term] : ((X484 @ (sub @ X486 @ (push @ X485 @ X483))) = (X482 @ X486)) | ~((X484 @ X485) = $true) | ((X482 @ one) = $true)) <=> (pushprop_lem1v2 = $true)) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X487 : term,X488 : subst > term > subst > $o] : (~! [X489 : subst,X490 : subst,X491 : subst,X492 : term] : (((X488 @ (comp @ X491 @ X489) @ (sub @ X492 @ X489) @ X490) = $true) | ~((X488 @ X491 @ X492 @ (comp @ X489 @ X490)) = $true)) | ($true = (X488 @ id @ (lam @ X487) @ id)) | ~! [X493 : term] : (((X488 @ id @ (sub @ X487 @ (push @ X493 @ id)) @ id) = $true) | ~((X488 @ id @ X493 @ id) = $true)) | ~! [X494 : subst > term > term] : (((X488 @ id @ (lam @ (X494 @ sh @ one)) @ id) = $true) | ~! [X495 : subst,X496 : subst,X497 : term] : ((sub @ (X494 @ X496 @ X497) @ X495) = (X494 @ (comp @ X496 @ X495) @ (sub @ X497 @ X495))) | ~! [X498 : term] : (($true = (X488 @ id @ (X494 @ id @ X498) @ id)) | ~((X488 @ id @ X498 @ id) = $true))) | ~! [X499 : subst,X500 : subst,X501 : term,X502 : subst] : (~($true = (X488 @ (comp @ X502 @ X500) @ (sub @ X501 @ X500) @ X499)) | ((X488 @ X502 @ X501 @ (comp @ X500 @ X499)) = $true)))) & ((hoasinduction = $true) <=> ! [X503 : term,X504 : subst > term > subst > $o] : (~! [X505 : term,X506 : subst,X507 : subst,X508 : subst] : (($true = (X504 @ X508 @ X505 @ (comp @ X507 @ X506))) | ~($true = (X504 @ (comp @ X508 @ X507) @ (sub @ X505 @ X507) @ X506))) | ~! [X509 : term,X510 : term] : (~((X504 @ id @ X509 @ id) = $true) | ~((X504 @ id @ X510 @ id) = $true) | ((X504 @ id @ (ap @ (sub @ X510 @ id) @ X509) @ id) = $true)) | ~! [X511 : subst > term > term] : (~! [X512 : subst,X513 : term,X514 : subst] : ((X511 @ (comp @ X512 @ X514) @ (sub @ X513 @ X514)) = (sub @ (X511 @ X512 @ X513) @ X514)) | ~! [X515 : term] : (~((X504 @ id @ X515 @ id) = $true) | ($true = (X504 @ id @ (X511 @ id @ X515) @ id))) | ((X504 @ id @ (lam @ (X511 @ sh @ one)) @ id) = $true)) | ~! [X516 : term] : (~($true = (var @ (sub @ X516 @ id))) | ((X504 @ id @ X516 @ id) = $true)) | ~! [X517 : subst,X518 : subst,X519 : term,X520 : subst] : (((X504 @ (comp @ X520 @ X518) @ (sub @ X519 @ X518) @ X517) = $true) | ~((X504 @ X520 @ X519 @ (comp @ X518 @ X517)) = $true)) | ((X504 @ id @ X503 @ id) = $true))) & ((d2subst @ d_id) = id) & ((hoasinduction_lem3a = $true) <=> ! [X521 : term,X522 : subst > term > subst > $o] : (((X522 @ id @ (lam @ X521) @ id) = $true) | ~! [X523 : subst > term > term] : (($true = (X522 @ id @ (lam @ (X523 @ sh @ one)) @ id)) | ~! [X524 : subst,X525 : term,X526 : subst] : ((X523 @ (comp @ X524 @ X526) @ (sub @ X525 @ X526)) = (sub @ (X523 @ X524 @ X525) @ X526)) | ~! [X527 : term] : (~((X522 @ id @ X527 @ id) = $true) | ((X522 @ id @ (X523 @ id @ X527) @ id) = $true))) | ~! [X528 : term] : (~((X522 @ id @ X528 @ id) = $true) | ($true = (X522 @ id @ (sub @ X521 @ (push @ X528 @ id)) @ id))))) & (hoaslamnotvar_lthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ~(ulamvar1 = $true) & (hoasapinj2_gthm = $true) & ((! [X529 : subst,X530 : term] : ((sub @ one @ (push @ X530 @ X529)) = X530) => (! [X531 : subst,X532 : term] : ((comp @ sh @ (push @ X532 @ X531)) = X531) => ! [X533 : term > $o,X534 : term > $o,X535 : term,X536 : subst,X537 : term] : (~($true = (X533 @ X537)) | ((X534 @ (sub @ X537 @ (push @ X535 @ X536))) = $true)))) <=> (pushprop_lem1_lthm = $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (induction2_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X538 : term,X539 : term,X540 : term,X541 : term] : (((ap @ X541 @ X539) != (ap @ X538 @ X540)) | (X539 = X540)) => ! [X542 : term,X543 : term,X544 : term,X545 : term] : (((ap @ (sub @ X545 @ id) @ X544) != (ap @ (sub @ X543 @ id) @ X542)) | (X542 = X544)))) & ((substmonoid_lthm = $true) <=> (! [X546 : subst] : ((comp @ id @ X546) = X546) => (! [X547 : subst] : ((comp @ X547 @ id) = X547) => (! [X548 : subst] : ((comp @ X548 @ id) = X548) & ! [X549 : subst] : ((comp @ id @ X549) = X549))))) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem1v2_gthm = $true)),
% 2.65/1.76 inference(fool_elimination,[status(thm)],[f7])).
% 2.65/1.76 thf(f7,plain,(
% 2.65/1.76 (induction2lem <=> ! [X0 : subst,X1 : term > $o,X2 : term] : (~! [X3 : term,X4 : term] : (~(X1 @ X3) | (X1 @ (ap @ X3 @ X4)) | ~(X1 @ X4)) | (X1 @ (sub @ X2 @ X0)) | ~! [X5 : term] : (~(var @ X5) | (X1 @ (sub @ X5 @ X0))) | ~! [X6 : term] : ((X1 @ (lam @ X6)) | ~! [X7 : term] : (~(X1 @ X7) | (X1 @ (sub @ X6 @ (push @ X7 @ id))))))) & hoaslamnotap_gthm & hoaslamnotvar_gthm & pushprop_lthm_orig & (hoasinduction_lem3b <=> ! [X8 : term] : ~! [X9 : subst > term > term] : ((sub @ X8 @ (push @ one @ sh)) != (X9 @ sh @ one))) & (hoasinduction_lem0_lthm <=> ! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))) & (one = (d2term @ d_one)) & (hoasapinj1 <=> ! [X13 : term,X14 : term,X15 : term,X16 : term] : (((ap @ (sub @ X13 @ id) @ X14) != (ap @ (sub @ X16 @ id) @ X15)) | (X13 = X16))) & hoasapnotvar_lthm & hoasinduction_lem1 & axvarshift & (! [X17 : term,X18 : subst > term > subst > $o,X19 : term > $o] : (~! [X20 : term] : ((X19 @ (sub @ X17 @ (push @ X20 @ id))) | ~(X19 @ X20)) | ~! [X21 : term] : ((X19 @ X21) = (X18 @ id @ X21 @ id)) | ~! [X22 : subst > term > term] : (~! [X23 : subst,X24 : subst,X25 : term] : ((X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (X22 @ X24 @ X25) @ X23)) | ~! [X26 : term] : (~(X18 @ id @ X26 @ id) | (X18 @ id @ (X22 @ id @ X26) @ id)) | (X18 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | ~! [X27 : subst,X28 : term,X29 : subst,X30 : subst] : (~(X18 @ X27 @ X28 @ (comp @ X29 @ X30)) | (X18 @ (comp @ X27 @ X29) @ (sub @ X28 @ X29) @ X30)) | ~! [X31 : subst,X32 : subst,X33 : subst,X34 : term] : (~(X18 @ (comp @ X33 @ X32) @ (sub @ X34 @ X32) @ X31) | (X18 @ X33 @ X34 @ (comp @ X32 @ X31))) | (X19 @ (lam @ X17))) <=> hoasinduction_lem3v2) & ((^[X35 : term, X36 : subst, X37 : term > $o, X38 : term > $o] : (! [X39 : term] : ((X37 @ (sub @ X39 @ (push @ X35 @ X36))) = (X38 @ X39)))) = pushprop_p_and_p_prime) & pushprop & (! [X40 : term,X41 : term] : ((X40 = X41) | ((sub @ X40 @ sh) != (sub @ X41 @ sh))) <=> shinj) & (pushprop_lem0 <=> ! [X42 : term > $o,X43 : term,X44 : subst] : ~! [X45 : term > $o] : ~! [X46 : term] : ((X45 @ X46) = (X42 @ (sub @ X46 @ (push @ X43 @ X44))))) & (! [X47 : term > $o,X48 : term] : (~! [X49 : term] : (~(var @ X49) | (X47 @ X49)) | ~! [X50 : term] : (~! [X51 : term] : (~(X47 @ X51) | (X47 @ (sub @ X50 @ (push @ X51 @ id)))) | (X47 @ (lam @ X50))) | ~! [X52 : term,X53 : term] : (~(X47 @ X52) | (X47 @ (ap @ X52 @ X53)) | ~(X47 @ X53)) | (X47 @ X48)) <=> induction2) & ((! [X54 : subst > term > subst > $o] : ~! [X55 : term > $o] : ~! [X56 : term] : ((X55 @ X56) = (X54 @ id @ X56 @ id)) => (! [X57 : term > $o,X58 : term] : (~! [X59 : term,X60 : term] : (~(X57 @ X59) | (X57 @ (ap @ X60 @ X59)) | ~(X57 @ X60)) | ~! [X61 : term] : (~(var @ X61) | (X57 @ X61)) | (X57 @ X58) | ~! [X62 : term] : ((X57 @ (lam @ X62)) | ~! [X63 : term] : (~(X57 @ X63) | (X57 @ (sub @ X62 @ (push @ X63 @ id)))))) => (! [X64 : term] : ((sub @ X64 @ id) = X64) => (! [X65 : subst > term > subst > $o,X66 : term > $o,X67 : term] : (~! [X68 : subst > term > term] : (~! [X69 : term] : (~(X65 @ id @ X69 @ id) | (X65 @ id @ (X68 @ id @ X69) @ id)) | ~! [X70 : subst,X71 : subst,X72 : term] : ((X68 @ (comp @ X71 @ X70) @ (sub @ X72 @ X70)) = (sub @ (X68 @ X71 @ X72) @ X70)) | (X65 @ id @ (lam @ (X68 @ sh @ one)) @ id)) | ~! [X73 : term] : ((X66 @ (sub @ X67 @ (push @ X73 @ id))) | ~(X66 @ X73)) | ~! [X74 : term] : ((X65 @ id @ X74 @ id) = (X66 @ X74)) | (X66 @ (lam @ X67))) => ! [X75 : subst > term > subst > $o,X76 : term] : ((X75 @ id @ X76 @ id) | ~! [X77 : subst > term > term] : ((X75 @ id @ (lam @ (X77 @ sh @ one)) @ id) | ~! [X78 : subst,X79 : term,X80 : subst] : ((X77 @ (comp @ X80 @ X78) @ (sub @ X79 @ X78)) = (sub @ (X77 @ X80 @ X79) @ X78)) | ~! [X81 : term] : ((X75 @ id @ (X77 @ id @ X81) @ id) | ~(X75 @ id @ X81 @ id))) | ~! [X82 : term,X83 : term] : ((X75 @ id @ (ap @ (sub @ X83 @ id) @ X82) @ id) | ~(X75 @ id @ X83 @ id) | ~(X75 @ id @ X82 @ id))))))) <=> hoasinduction_no_psi_cond_lthm) & ((! [X84 : term] : ((sub @ X84 @ id) = X84) => ! [X85 : term,X86 : subst > term > subst > $o,X87 : term > $o] : (~! [X88 : term] : ((X87 @ X88) = (X86 @ id @ X88 @ id)) | ~! [X89 : subst > term > term] : (~! [X90 : term] : ((X86 @ id @ (X89 @ id @ X90) @ id) | ~(X86 @ id @ X90 @ id)) | (X86 @ id @ (lam @ (X89 @ sh @ one)) @ id) | ~! [X91 : term,X92 : subst,X93 : subst] : ((sub @ (X89 @ X93 @ X91) @ X92) = (X89 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92)))) | ~! [X94 : subst,X95 : subst,X96 : subst,X97 : term] : ((X86 @ (comp @ X95 @ X96) @ (sub @ X97 @ X96) @ X94) | ~(X86 @ X95 @ X97 @ (comp @ X96 @ X94))) | ~! [X98 : term,X99 : subst,X100 : subst,X101 : subst] : ((X86 @ X99 @ X98 @ (comp @ X100 @ X101)) | ~(X86 @ (comp @ X99 @ X100) @ (sub @ X98 @ X100) @ X101)) | (X87 @ (lam @ X85)) | ~! [X102 : term] : ((X87 @ (sub @ X85 @ (push @ X102 @ id))) | ~(X87 @ X102)))) <=> hoasinduction_lem3v2_lthm) & (hoasapinj2 <=> ! [X103 : term,X104 : term,X105 : term,X106 : term] : ((X105 = X106) | ((ap @ (sub @ X104 @ id) @ X106) != (ap @ (sub @ X103 @ id) @ X105)))) & hoaslaminj_gthm & (! [X107 : subst] : ((push @ (sub @ one @ X107) @ (comp @ sh @ X107)) = X107) <=> axscons) & apnotvar & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X108 : term] : ((sub @ X108 @ id) = X108) => (! [X109 : term,X110 : term,X111 : term,X112 : term] : (((ap @ X111 @ X110) != (ap @ X112 @ X109)) | (X111 = X112)) => ! [X113 : term,X114 : term,X115 : term,X116 : term] : (((ap @ (sub @ X114 @ id) @ X116) != (ap @ (sub @ X113 @ id) @ X115)) | (X113 = X114)))) <=> hoasapinj1_lthm) & termmset_gthm & ! [X117 : d_subst,X118 : d_subst] : (((d2subst @ X118) = (d2subst @ X117)) => (X117 = X118)) & ! [X119 : term] : ? [X120 : d_term] : ((d2term @ X120) = X119) & pushprop_lem3v2_lthm & axapp & (pushprop_lem0_lthm <=> ! [X121 : term,X122 : subst,X123 : term > $o] : ~! [X124 : term > $o] : ~! [X125 : term] : ((X124 @ X125) = (X123 @ (sub @ X125 @ (push @ X121 @ X122))))) & (hoasinduction_lem3aaa <=> ! [X126 : term,X127 : subst > term > subst > $o] : (~! [X128 : term] : (~(X127 @ id @ X128 @ id) | (X127 @ id @ (sub @ X126 @ (push @ X128 @ id)) @ id)) | ~! [X129 : term,X130 : subst > term > term] : (~! [X131 : subst,X132 : term,X133 : subst] : ((sub @ (X130 @ X133 @ X132) @ X131) = (sub @ (sub @ X129 @ (push @ X132 @ X133)) @ X131)) | (X127 @ id @ (lam @ (X130 @ sh @ one)) @ id) | ~! [X134 : term] : ((X127 @ id @ (X130 @ id @ X134) @ id) | ~(X127 @ id @ X134 @ id)) | ~! [X135 : term,X136 : subst,X137 : subst] : ((sub @ X129 @ (push @ (sub @ X135 @ X137) @ (comp @ X136 @ X137))) = (X130 @ (comp @ X136 @ X137) @ (sub @ X135 @ X137)))) | (X127 @ id @ (lam @ (sub @ X126 @ (push @ one @ sh))) @ id))) & ((d2subst @ d_id) = sh) & (! [X138 : term > $o,X139 : term,X140 : term,X141 : subst,X142 : term > $o] : (~(X142 @ X139) | (X138 @ (sub @ X139 @ (push @ X140 @ X141)))) <=> pushprop_lem1) & hoasinduction_lem2_lthm & (hoasinduction_no_psi_cond <=> ! [X143 : subst > term > subst > $o,X144 : term] : (~! [X145 : subst > term > term] : ((X143 @ id @ (lam @ (X145 @ sh @ one)) @ id) | ~! [X146 : term] : (~(X143 @ id @ X146 @ id) | (X143 @ id @ (X145 @ id @ X146) @ id)) | ~! [X147 : subst,X148 : term,X149 : subst] : ((sub @ (X145 @ X147 @ X148) @ X149) = (X145 @ (comp @ X147 @ X149) @ (sub @ X148 @ X149)))) | ~! [X150 : term,X151 : term] : (~(X143 @ id @ X151 @ id) | (X143 @ id @ (ap @ (sub @ X150 @ id) @ X151) @ id) | ~(X143 @ id @ X150 @ id)) | (X143 @ id @ X144 @ id))) & (! [X152 : term,X153 : term] : ((X152 = X153) | ((lam @ X152) != (lam @ X153))) <=> laminj) & ((! [X154 : term] : ((sub @ X154 @ id) = X154) => ! [X155 : term] : ((sub @ X155 @ id) = X155)) <=> termmset_lthm) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & hoasinduction_lem3_gthm & pushprop_lem2v2_gthm & hoasinduction_lem3v2_gthm & ((! [X156 : term] : ((sub @ X156 @ id) = X156) => (! [X157 : term > $o,X158 : term,X159 : subst] : (~! [X160 : term,X161 : term] : ((X157 @ (ap @ X160 @ X161)) | ~(X157 @ X160) | ~(X157 @ X161)) | ~! [X162 : term] : (~(var @ X162) | (X157 @ (sub @ X162 @ X159))) | ~! [X163 : term] : (~! [X164 : term] : ((X157 @ (sub @ X163 @ (push @ X164 @ id))) | ~(X157 @ X164)) | (X157 @ (lam @ X163))) | (X157 @ (sub @ X158 @ X159))) => ! [X165 : term > $o,X166 : term] : (~! [X167 : term,X168 : term] : (~(X165 @ X168) | (X165 @ (ap @ X168 @ X167)) | ~(X165 @ X167)) | (X165 @ X166) | ~! [X169 : term] : (~(var @ X169) | (X165 @ X169)) | ~! [X170 : term] : (~! [X171 : term] : ((X165 @ (sub @ X170 @ (push @ X171 @ id))) | ~(X165 @ X171)) | (X165 @ (lam @ X170)))))) <=> induction2_lthm) & ulamvarind & induction2lem_gthm & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ((! [X172 : subst,X173 : term] : ((sub @ one @ (push @ X173 @ X172)) = X173) => ! [X174 : subst,X175 : term > $o,X176 : term > $o,X177 : term] : (~! [X178 : term] : ((X175 @ X178) = (X176 @ (sub @ X178 @ (push @ X177 @ X174)))) | ~(X176 @ X177) | (X175 @ one))) <=> pushprop_lem1v2_lthm) & (! [X179 : term] : ((sub @ X179 @ id) = X179) <=> axvarid) & (! [X180 : subst,X181 : term] : ((sub @ one @ (push @ X181 @ X180)) = X181) <=> axvarcons) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_p_and_p_prime = (^[X182 : subst > term > subst > $o, X183 : term > $o] : (! [X184 : term] : ((X183 @ X184) = (X182 @ id @ X184 @ id))))) & ~(var @ (d2term @ d_one)) & ! [X185 : d_subst] : (d_id = X185) & hoasinduction_lem1_lthm & hoasinduction_lem1v2 & hoasinduction_lem3b_gthm & (hoasinduction_lem3v2_f <=> ! [X186 : term] : ~! [X187 : subst > term > term] : ~! [X188 : subst,X189 : term] : ((X187 @ X188 @ X189) = (sub @ X186 @ (push @ X189 @ X188)))) & substmonoid_gthm & pushprop_lem3v2 & pushprop_lem1_gthm & axmap & hoasapnotvar & (hoasinduction_lem3aa <=> ! [X190 : subst > term > subst > $o,X191 : term] : (~! [X192 : term] : (~(X190 @ id @ X192 @ id) | (X190 @ id @ (sub @ X191 @ (push @ X192 @ id)) @ id)) | ~! [X193 : subst > term > term] : (~! [X194 : term] : (~(X190 @ id @ X194 @ id) | (X190 @ id @ (X193 @ id @ X194) @ id)) | (X190 @ id @ (lam @ (X193 @ sh @ one)) @ id) | ~! [X195 : term,X196 : subst,X197 : subst] : ((X193 @ (comp @ X197 @ X196) @ (sub @ X195 @ X196)) = (sub @ (X193 @ X197 @ X195) @ X196))) | (X190 @ id @ (lam @ (sub @ X191 @ (push @ one @ sh))) @ id))) & (hoaslaminj <=> ! [X198 : subst > term > term,X199 : subst > term > term,X200 : subst,X201 : term] : (~! [X202 : term,X203 : subst,X204 : subst] : ((sub @ (X198 @ X204 @ X202) @ X203) = (X198 @ (comp @ X204 @ X203) @ (sub @ X202 @ X203))) | ((lam @ (X198 @ sh @ one)) != (lam @ (X199 @ sh @ one))) | ~! [X205 : term,X206 : subst,X207 : subst] : ((sub @ (X199 @ X206 @ X205) @ X207) = (X199 @ (comp @ X206 @ X207) @ (sub @ X205 @ X207))) | ((X199 @ X200 @ X201) = (X198 @ X200 @ X201)))) & ((! [X208 : subst > term > subst > $o] : ~! [X209 : term > $o] : ~! [X210 : term] : ((X208 @ id @ X210 @ id) = (X209 @ X210)) => (! [X211 : term,X212 : term > $o] : (~! [X213 : term,X214 : term] : (~(X212 @ X213) | ~(X212 @ X214) | (X212 @ (ap @ X214 @ X213))) | (X212 @ X211) | ~! [X215 : term] : (~! [X216 : term] : (~(X212 @ X216) | (X212 @ (sub @ X215 @ (push @ X216 @ id)))) | (X212 @ (lam @ X215))) | ~! [X217 : term] : (~(var @ X217) | (X212 @ X217))) => (! [X218 : term] : ((sub @ X218 @ id) = X218) => (! [X219 : term > $o,X220 : term,X221 : subst > term > subst > $o] : (~! [X222 : term] : ((X221 @ id @ X222 @ id) = (X219 @ X222)) | ~! [X223 : term] : ((X219 @ (sub @ X220 @ (push @ X223 @ id))) | ~(X219 @ X223)) | ~! [X224 : subst > term > term] : ((X221 @ id @ (lam @ (X224 @ sh @ one)) @ id) | ~! [X225 : term,X226 : subst,X227 : subst] : ((X224 @ (comp @ X227 @ X226) @ (sub @ X225 @ X226)) = (sub @ (X224 @ X227 @ X225) @ X226)) | ~! [X228 : term] : ((X221 @ id @ (X224 @ id @ X228) @ id) | ~(X221 @ id @ X228 @ id))) | (X219 @ (lam @ X220))) => ! [X229 : term,X230 : subst > term > subst > $o] : (~! [X231 : subst,X232 : term,X233 : subst,X234 : subst] : (~(X230 @ X233 @ X232 @ (comp @ X234 @ X231)) | (X230 @ (comp @ X233 @ X234) @ (sub @ X232 @ X234) @ X231)) | ~! [X235 : term,X236 : term] : (~(X230 @ id @ X235 @ id) | (X230 @ id @ (ap @ (sub @ X236 @ id) @ X235) @ id) | ~(X230 @ id @ X236 @ id)) | ~! [X237 : term] : (~(var @ (sub @ X237 @ id)) | (X230 @ id @ X237 @ id)) | ~! [X238 : subst,X239 : subst,X240 : term,X241 : subst] : (~(X230 @ (comp @ X241 @ X238) @ (sub @ X240 @ X238) @ X239) | (X230 @ X241 @ X240 @ (comp @ X238 @ X239))) | ~! [X242 : subst > term > term] : (~! [X243 : term] : ((X230 @ id @ (X242 @ id @ X243) @ id) | ~(X230 @ id @ X243 @ id)) | ~! [X244 : term,X245 : subst,X246 : subst] : ((sub @ (X242 @ X246 @ X244) @ X245) = (X242 @ (comp @ X246 @ X245) @ (sub @ X244 @ X245))) | (X230 @ id @ (lam @ (X242 @ sh @ one)) @ id)) | (X230 @ id @ X229 @ id)))))) <=> hoasinduction_lthm_3) & (! [X247 : term,X248 : subst] : ((comp @ sh @ (push @ X247 @ X248)) = X248) <=> axshiftcons) & (hoaslam = (^[X249 : subst, X250 : subst > term > term] : (d2term @ d_one))) & (! [X251 : term,X252 : term,X253 : term,X254 : term] : (((ap @ X253 @ X251) != (ap @ X254 @ X252)) | (X251 = X252)) <=> apinj2) & (substmonoid <=> (! [X255 : subst] : ((comp @ X255 @ id) = X255) & ! [X256 : subst] : ((comp @ id @ X256) = X256))) & (! [X257 : term] : ((sub @ X257 @ id) = X257) <=> termmset) & hoasinduction_gthm & pushprop_gthm & hoaslamnotvar & (hoasinduction_lem3a_lthm <=> (! [X258 : term] : ((sub @ X258 @ id) = X258) => (! [X259 : subst > term > subst > $o,X260 : term] : (~! [X261 : subst > term > term] : ((X259 @ id @ (lam @ (X261 @ sh @ one)) @ id) | ~! [X262 : subst,X263 : term,X264 : subst] : ((sub @ (X261 @ X262 @ X263) @ X264) = (X261 @ (comp @ X262 @ X264) @ (sub @ X263 @ X264))) | ~! [X265 : term] : (~(X259 @ id @ X265 @ id) | (X259 @ id @ (X261 @ id @ X265) @ id))) | ~! [X266 : term] : ((X259 @ id @ (sub @ X260 @ (push @ X266 @ id)) @ id) | ~(X259 @ id @ X266 @ id)) | (X259 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)) => ! [X267 : subst > term > subst > $o,X268 : term] : (~! [X269 : term] : (~(X267 @ id @ X269 @ id) | (X267 @ id @ (sub @ X268 @ (push @ X269 @ id)) @ id)) | ~! [X270 : subst > term > term] : (~! [X271 : term,X272 : subst,X273 : subst] : ((X270 @ (comp @ X272 @ X273) @ (sub @ X271 @ X273)) = (sub @ (X270 @ X272 @ X271) @ X273)) | ~! [X274 : term] : ((X267 @ id @ (X270 @ id @ X274) @ id) | ~(X267 @ id @ X274 @ id)) | (X267 @ id @ (lam @ (X270 @ sh @ one)) @ id)) | (X267 @ id @ (lam @ X268) @ id))))) & ~lamnotap & axclos & (hoasinduction_lem3v2a <=> ! [X275 : term,X276 : subst > term > subst > $o,X277 : term > $o] : ((X277 @ (lam @ X275)) | ~! [X278 : term] : (~(X277 @ X278) | (X277 @ (sub @ X275 @ (push @ X278 @ id)))) | ~! [X279 : term] : ((X276 @ id @ X279 @ id) = (X277 @ X279)) | ~! [X280 : subst > term > term] : (~! [X281 : term] : ((X276 @ id @ (X280 @ id @ X281) @ id) | ~(X276 @ id @ X281 @ id)) | (X276 @ id @ (lam @ (X280 @ sh @ one)) @ id) | ~! [X282 : subst,X283 : subst,X284 : term] : ((X280 @ (comp @ X282 @ X283) @ (sub @ X284 @ X283)) = (sub @ (X280 @ X282 @ X284) @ X283))))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (apinj1 <=> ! [X285 : term,X286 : term,X287 : term,X288 : term] : (((ap @ X286 @ X287) != (ap @ X288 @ X285)) | (X286 = X288))) & ((! [X289 : term,X290 : term > $o] : ((X290 @ X289) | ~! [X291 : term] : (~(var @ X291) | (X290 @ X291)) | ~! [X292 : term] : (~! [X293 : term] : (~(X290 @ X293) | (X290 @ (sub @ X292 @ (push @ X293 @ id)))) | (X290 @ (lam @ X292))) | ~! [X294 : term,X295 : term] : ((X290 @ (ap @ X295 @ X294)) | ~(X290 @ X295) | ~(X290 @ X294))) => (! [X296 : term,X297 : subst > term > subst > $o,X298 : term] : (~! [X299 : subst,X300 : subst,X301 : term,X302 : subst] : (~(X297 @ X299 @ X301 @ (comp @ X300 @ X302)) | (X297 @ (comp @ X299 @ X300) @ (sub @ X301 @ X300) @ X302)) | ~! [X303 : term,X304 : term] : ((X297 @ id @ (ap @ (sub @ X303 @ id) @ X304) @ id) | ~(X297 @ id @ X303 @ id) | ~(X297 @ id @ X304 @ id)) | ~(X297 @ id @ X296 @ id) | ~! [X305 : subst,X306 : subst,X307 : term,X308 : subst] : (~(X297 @ (comp @ X305 @ X306) @ (sub @ X307 @ X306) @ X308) | (X297 @ X305 @ X307 @ (comp @ X306 @ X308))) | ~(X297 @ id @ X298 @ id) | (X297 @ id @ (ap @ X296 @ X298) @ id)) => (! [X309 : term,X310 : subst > term > subst > $o] : (~! [X311 : term] : ((X310 @ id @ (sub @ X309 @ (push @ X311 @ id)) @ id) | ~(X310 @ id @ X311 @ id)) | ~! [X312 : subst,X313 : subst,X314 : term,X315 : subst] : (~(X310 @ X315 @ X314 @ (comp @ X312 @ X313)) | (X310 @ (comp @ X315 @ X312) @ (sub @ X314 @ X312) @ X313)) | ~! [X316 : term,X317 : subst,X318 : subst,X319 : subst] : ((X310 @ X317 @ X316 @ (comp @ X318 @ X319)) | ~(X310 @ (comp @ X317 @ X318) @ (sub @ X316 @ X318) @ X319)) | (X310 @ id @ (lam @ X309) @ id) | ~! [X320 : subst > term > term] : ((X310 @ id @ (lam @ (X320 @ sh @ one)) @ id) | ~! [X321 : term] : ((X310 @ id @ (X320 @ id @ X321) @ id) | ~(X310 @ id @ X321 @ id)) | ~! [X322 : term,X323 : subst,X324 : subst] : ((X320 @ (comp @ X324 @ X323) @ (sub @ X322 @ X323)) = (sub @ (X320 @ X324 @ X322) @ X323)))) => ! [X325 : term,X326 : subst > term > subst > $o] : (~! [X327 : term,X328 : term] : ((X326 @ id @ (ap @ (sub @ X328 @ id) @ X327) @ id) | ~(X326 @ id @ X327 @ id) | ~(X326 @ id @ X328 @ id)) | ~! [X329 : subst > term > term] : (~! [X330 : term,X331 : subst,X332 : subst] : ((X329 @ (comp @ X331 @ X332) @ (sub @ X330 @ X332)) = (sub @ (X329 @ X331 @ X330) @ X332)) | (X326 @ id @ (lam @ (X329 @ sh @ one)) @ id) | ~! [X333 : term] : (~(X326 @ id @ X333 @ id) | (X326 @ id @ (X329 @ id @ X333) @ id))) | ~! [X334 : subst,X335 : term,X336 : subst,X337 : subst] : ((X326 @ (comp @ X334 @ X337) @ (sub @ X335 @ X337) @ X336) | ~(X326 @ X334 @ X335 @ (comp @ X337 @ X336))) | ~! [X338 : subst,X339 : subst,X340 : subst,X341 : term] : ((X326 @ X338 @ X341 @ (comp @ X339 @ X340)) | ~(X326 @ (comp @ X338 @ X339) @ (sub @ X341 @ X339) @ X340)) | ~! [X342 : term] : (~(var @ (sub @ X342 @ id)) | (X326 @ id @ X342 @ id)) | (X326 @ id @ X325 @ id))))) <=> hoasinduction_lthm) & (! [X343 : subst > term > subst > $o] : ~! [X344 : term > $o] : ~! [X345 : term] : ((X344 @ X345) = (X343 @ id @ X345 @ id)) <=> hoasinduction_lem0) & pushprop_lem0_gthm & (! [X346 : term,X347 : term,X348 : subst > term > term] : (~! [X349 : subst,X350 : subst,X351 : term] : ((sub @ (X348 @ X350 @ X351) @ X349) = (X348 @ (comp @ X350 @ X349) @ (sub @ X351 @ X349))) | ((ap @ (sub @ X346 @ id) @ X347) != (lam @ (X348 @ sh @ one)))) <=> hoaslamnotap) & axabs & hoasinduction_lem1v2_gthm & (! [X352 : term,X353 : subst > term > subst > $o] : (~! [X354 : term] : ((X353 @ id @ (sub @ X352 @ (push @ X354 @ id)) @ id) | ~(X353 @ id @ X354 @ id)) | ~! [X355 : subst > term > term] : (~! [X356 : term,X357 : subst,X358 : subst] : ((sub @ (X355 @ X357 @ X356) @ X358) = (X355 @ (comp @ X357 @ X358) @ (sub @ X356 @ X358))) | ~! [X359 : term] : ((X353 @ id @ (X355 @ id @ X359) @ id) | ~(X353 @ id @ X359 @ id)) | (X353 @ id @ (lam @ (X355 @ sh @ one)) @ id)) | (X353 @ id @ (lam @ (sub @ X352 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aa_lthm) & ((! [X360 : subst,X361 : term] : ((sub @ one @ (push @ X361 @ X360)) = X361) => (! [X362 : term,X363 : subst] : ((comp @ sh @ (push @ X362 @ X363)) = X363) => (! [X364 : subst] : ((comp @ X364 @ id) = X364) => (! [X365 : term,X366 : term > $o] : (~! [X367 : term] : ((X366 @ X367) | ~(var @ X367)) | ~! [X368 : term] : ((X366 @ (lam @ X368)) | ~(X366 @ X368)) | ~! [X369 : term,X370 : term] : ((X366 @ (ap @ X370 @ X369)) | ~(X366 @ X369) | ~(X366 @ X370)) | (X366 @ X365)) => ! [X371 : term,X372 : term > $o,X373 : subst] : ((X372 @ (sub @ X371 @ X373)) | ~! [X374 : term] : ((X372 @ (sub @ X374 @ X373)) | ~(var @ X374)) | ~! [X375 : term] : (~! [X376 : term] : ((X372 @ (sub @ X375 @ (push @ X376 @ id))) | ~(X372 @ X376)) | (X372 @ (lam @ X375))) | ~! [X377 : term,X378 : term] : (~(X372 @ X377) | (X372 @ (ap @ X378 @ X377)) | ~(X372 @ X378))))))) <=> induction2lem_lthm) & axassoc & pushprop_lem2v2 & pushprop_lem2v2_lthm & (hoasinduction_lem2v2 <=> ! [X379 : term > $o,X380 : subst > term > subst > $o,X381 : term,X382 : term] : (~! [X383 : subst,X384 : subst,X385 : term,X386 : subst] : (~(X380 @ X384 @ X385 @ (comp @ X383 @ X386)) | (X380 @ (comp @ X384 @ X383) @ (sub @ X385 @ X383) @ X386)) | ~! [X387 : term] : ((X379 @ X387) = (X380 @ id @ X387 @ id)) | ~! [X388 : term,X389 : subst,X390 : subst,X391 : subst] : ((X380 @ X390 @ X388 @ (comp @ X391 @ X389)) | ~(X380 @ (comp @ X390 @ X391) @ (sub @ X388 @ X391) @ X389)) | (X379 @ (ap @ X382 @ X381)) | ~(X379 @ X382) | ~(X379 @ X381) | ~! [X392 : term,X393 : term] : (~(X380 @ id @ X392 @ id) | (X380 @ id @ (ap @ (sub @ X393 @ id) @ X392) @ id) | ~(X380 @ id @ X393 @ id)))) & (hoasinduction_lem3v2_f_lthm <=> ! [X394 : term] : ~! [X395 : subst > term > term] : ~! [X396 : subst,X397 : term] : ((sub @ X394 @ (push @ X397 @ X396)) = (X395 @ X396 @ X397))) & ((! [X398 : term] : ~! [X399 : subst > term > term] : ~! [X400 : term,X401 : subst] : ((X399 @ X401 @ X400) = (sub @ X398 @ (push @ X400 @ X401))) => (! [X402 : term] : ((sub @ X402 @ id) = X402) => ! [X403 : term,X404 : subst > term > subst > $o,X405 : term > $o] : ((X405 @ (lam @ X403)) | ~! [X406 : term] : ((X405 @ (sub @ X403 @ (push @ X406 @ id))) | ~(X405 @ X406)) | ~! [X407 : subst > term > term] : ((X404 @ id @ (lam @ (X407 @ sh @ one)) @ id) | ~! [X408 : term] : (~(X404 @ id @ X408 @ id) | (X404 @ id @ (X407 @ id @ X408) @ id)) | ~! [X409 : term,X410 : subst,X411 : subst] : ((X407 @ (comp @ X410 @ X411) @ (sub @ X409 @ X411)) = (sub @ (X407 @ X410 @ X409) @ X411))) | ~! [X412 : term] : ((X404 @ id @ X412 @ id) = (X405 @ X412))))) <=> hoasinduction_lem3v2a_lthm) & lamnotvar & pushprop_lthm & hoasapnotvar_gthm & (! [X413 : term > $o,X414 : term] : (~! [X415 : term,X416 : term] : ((X413 @ (ap @ X415 @ X416)) | ~(X413 @ X415) | ~(X413 @ X416)) | ~! [X417 : term] : ((X413 @ X417) | ~(var @ X417)) | (X413 @ X414) | ~! [X418 : term] : (~(X413 @ X418) | (X413 @ (lam @ X418)))) <=> induction) & ((! [X419 : term] : ((sub @ X419 @ id) = X419) => (! [X420 : term,X421 : subst > term > subst > $o] : (~! [X422 : subst > term > term] : (~! [X423 : term] : (~(X421 @ id @ X423 @ id) | (X421 @ id @ (X422 @ id @ X423) @ id)) | (X421 @ id @ (lam @ (X422 @ sh @ one)) @ id) | ~! [X424 : subst,X425 : subst,X426 : term] : ((X422 @ (comp @ X424 @ X425) @ (sub @ X426 @ X425)) = (sub @ (X422 @ X424 @ X426) @ X425))) | ~! [X427 : term] : (~(X421 @ id @ X427 @ id) | (X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) | (X421 @ id @ (lam @ (sub @ X420 @ (push @ one @ sh))) @ id)) => ! [X428 : term,X429 : subst > term > subst > $o] : ((X429 @ id @ (lam @ X428) @ id) | ~! [X430 : subst,X431 : subst,X432 : term,X433 : subst] : ((X429 @ X433 @ X432 @ (comp @ X431 @ X430)) | ~(X429 @ (comp @ X433 @ X431) @ (sub @ X432 @ X431) @ X430)) | ~! [X434 : subst,X435 : subst,X436 : subst,X437 : term] : ((X429 @ (comp @ X436 @ X435) @ (sub @ X437 @ X435) @ X434) | ~(X429 @ X436 @ X437 @ (comp @ X435 @ X434))) | ~! [X438 : subst > term > term] : (~! [X439 : term,X440 : subst,X441 : subst] : ((sub @ (X438 @ X441 @ X439) @ X440) = (X438 @ (comp @ X441 @ X440) @ (sub @ X439 @ X440))) | (X429 @ id @ (lam @ (X438 @ sh @ one)) @ id) | ~! [X442 : term] : (~(X429 @ id @ X442 @ id) | (X429 @ id @ (X438 @ id @ X442) @ id))) | ~! [X443 : term] : (~(X429 @ id @ X443 @ id) | (X429 @ id @ (sub @ X428 @ (push @ X443 @ id)) @ id))))) <=> hoasinduction_lem3_lthm) & hoaslamnotap_lthm & (hoaslaminj_lthm <=> (! [X444 : term,X445 : subst] : ((sub @ one @ (push @ X444 @ X445)) = X444) => (! [X446 : term,X447 : subst] : ((comp @ sh @ (push @ X446 @ X447)) = X447) => (! [X448 : term,X449 : term] : (((lam @ X449) != (lam @ X448)) | (X448 = X449)) => ! [X450 : term,X451 : subst > term > term,X452 : subst > term > term,X453 : subst] : (((lam @ (X452 @ sh @ one)) != (lam @ (X451 @ sh @ one))) | ~! [X454 : term,X455 : subst,X456 : subst] : ((sub @ (X451 @ X456 @ X454) @ X455) = (X451 @ (comp @ X456 @ X455) @ (sub @ X454 @ X455))) | ~! [X457 : subst,X458 : subst,X459 : term] : ((X452 @ (comp @ X458 @ X457) @ (sub @ X459 @ X457)) = (sub @ (X452 @ X458 @ X459) @ X457)) | ((X451 @ X453 @ X450) = (X452 @ X453 @ X450))))))) & hoasinduction_lem3a_gthm & ! [X460 : d_term] : (d_one = X460) & (hoasinduction_lem2 <=> ! [X461 : subst > term > subst > $o,X462 : term,X463 : term] : (~(X461 @ id @ X462 @ id) | (X461 @ id @ (ap @ X463 @ X462) @ id) | ~(X461 @ id @ X463 @ id) | ~! [X464 : subst,X465 : subst,X466 : term,X467 : subst] : ((X461 @ (comp @ X467 @ X465) @ (sub @ X466 @ X465) @ X464) | ~(X461 @ X467 @ X466 @ (comp @ X465 @ X464))) | ~! [X468 : term,X469 : term] : ((X461 @ id @ (ap @ (sub @ X468 @ id) @ X469) @ id) | ~(X461 @ id @ X469 @ id) | ~(X461 @ id @ X468 @ id)) | ~! [X470 : subst,X471 : subst,X472 : term,X473 : subst] : (~(X461 @ (comp @ X470 @ X473) @ (sub @ X472 @ X473) @ X471) | (X461 @ X470 @ X472 @ (comp @ X473 @ X471))))) & hoasinduction_lem2_gthm & ulamvarsh & hoasinduction_lem1_gthm & ! [X474 : subst] : ? [X475 : d_subst] : ((d2subst @ X475) = X474) & (! [X476 : subst] : ((comp @ X476 @ id) = X476) <=> axidr) & (hoasinduction_lem3b_lthm <=> ! [X477 : term] : ~! [X478 : subst > term > term] : ((X478 @ sh @ one) != (sub @ X477 @ (push @ one @ sh)))) & ! [X479 : d_term,X480 : d_term] : (((d2term @ X479) = (d2term @ X480)) => (X479 = X480)) & (! [X481 : subst] : ((comp @ id @ X481) = X481) <=> axidl) & (! [X482 : term > $o,X483 : subst,X484 : term > $o,X485 : term] : (~! [X486 : term] : ((X484 @ (sub @ X486 @ (push @ X485 @ X483))) = (X482 @ X486)) | ~(X484 @ X485) | (X482 @ one)) <=> pushprop_lem1v2) & hoasapinj1_gthm & (hoasinduction_lem3 <=> ! [X487 : term,X488 : subst > term > subst > $o] : (~! [X489 : subst,X490 : subst,X491 : subst,X492 : term] : ((X488 @ (comp @ X491 @ X489) @ (sub @ X492 @ X489) @ X490) | ~(X488 @ X491 @ X492 @ (comp @ X489 @ X490))) | (X488 @ id @ (lam @ X487) @ id) | ~! [X493 : term] : ((X488 @ id @ (sub @ X487 @ (push @ X493 @ id)) @ id) | ~(X488 @ id @ X493 @ id)) | ~! [X494 : subst > term > term] : ((X488 @ id @ (lam @ (X494 @ sh @ one)) @ id) | ~! [X495 : subst,X496 : subst,X497 : term] : ((sub @ (X494 @ X496 @ X497) @ X495) = (X494 @ (comp @ X496 @ X495) @ (sub @ X497 @ X495))) | ~! [X498 : term] : ((X488 @ id @ (X494 @ id @ X498) @ id) | ~(X488 @ id @ X498 @ id))) | ~! [X499 : subst,X500 : subst,X501 : term,X502 : subst] : (~(X488 @ (comp @ X502 @ X500) @ (sub @ X501 @ X500) @ X499) | (X488 @ X502 @ X501 @ (comp @ X500 @ X499))))) & (hoasinduction <=> ! [X503 : term,X504 : subst > term > subst > $o] : (~! [X505 : term,X506 : subst,X507 : subst,X508 : subst] : ((X504 @ X508 @ X505 @ (comp @ X507 @ X506)) | ~(X504 @ (comp @ X508 @ X507) @ (sub @ X505 @ X507) @ X506)) | ~! [X509 : term,X510 : term] : (~(X504 @ id @ X509 @ id) | ~(X504 @ id @ X510 @ id) | (X504 @ id @ (ap @ (sub @ X510 @ id) @ X509) @ id)) | ~! [X511 : subst > term > term] : (~! [X512 : subst,X513 : term,X514 : subst] : ((X511 @ (comp @ X512 @ X514) @ (sub @ X513 @ X514)) = (sub @ (X511 @ X512 @ X513) @ X514)) | ~! [X515 : term] : (~(X504 @ id @ X515 @ id) | (X504 @ id @ (X511 @ id @ X515) @ id)) | (X504 @ id @ (lam @ (X511 @ sh @ one)) @ id)) | ~! [X516 : term] : (~(var @ (sub @ X516 @ id)) | (X504 @ id @ X516 @ id)) | ~! [X517 : subst,X518 : subst,X519 : term,X520 : subst] : ((X504 @ (comp @ X520 @ X518) @ (sub @ X519 @ X518) @ X517) | ~(X504 @ X520 @ X519 @ (comp @ X518 @ X517))) | (X504 @ id @ X503 @ id))) & ((d2subst @ d_id) = id) & (hoasinduction_lem3a <=> ! [X521 : term,X522 : subst > term > subst > $o] : ((X522 @ id @ (lam @ X521) @ id) | ~! [X523 : subst > term > term] : ((X522 @ id @ (lam @ (X523 @ sh @ one)) @ id) | ~! [X524 : subst,X525 : term,X526 : subst] : ((X523 @ (comp @ X524 @ X526) @ (sub @ X525 @ X526)) = (sub @ (X523 @ X524 @ X525) @ X526)) | ~! [X527 : term] : (~(X522 @ id @ X527 @ id) | (X522 @ id @ (X523 @ id @ X527) @ id))) | ~! [X528 : term] : (~(X522 @ id @ X528 @ id) | (X522 @ id @ (sub @ X521 @ (push @ X528 @ id)) @ id)))) & hoaslamnotvar_lthm & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ~ulamvar1 & hoasapinj2_gthm & ((! [X529 : subst,X530 : term] : ((sub @ one @ (push @ X530 @ X529)) = X530) => (! [X531 : subst,X532 : term] : ((comp @ sh @ (push @ X532 @ X531)) = X531) => ! [X533 : term > $o,X534 : term > $o,X535 : term,X536 : subst,X537 : term] : (~(X533 @ X537) | (X534 @ (sub @ X537 @ (push @ X535 @ X536)))))) <=> pushprop_lem1_lthm) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & induction2_gthm & (hoasapinj2_lthm <=> (! [X538 : term,X539 : term,X540 : term,X541 : term] : (((ap @ X541 @ X539) != (ap @ X538 @ X540)) | (X539 = X540)) => ! [X542 : term,X543 : term,X544 : term,X545 : term] : (((ap @ (sub @ X545 @ id) @ X544) != (ap @ (sub @ X543 @ id) @ X542)) | (X542 = X544)))) & (substmonoid_lthm <=> (! [X546 : subst] : ((comp @ id @ X546) = X546) => (! [X547 : subst] : ((comp @ X547 @ id) = X547) => (! [X548 : subst] : ((comp @ X548 @ id) = X548) & ! [X549 : subst] : ((comp @ id @ X549) = X549))))) & hoasinduction_lem2v2_gthm & pushprop_lem1v2_gthm),
% 2.65/1.76 inference(rectify,[status(thm)],[f1])).
% 2.65/1.76 thf(f1,axiom,(
% 2.65/1.76 (induction2lem <=> ! [X17 : subst,X10 : term > $o,X16 : term] : (~! [X13 : term,X15 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | (X10 @ (sub @ X16 @ X17)) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id))))))) & hoaslamnotap_gthm & hoaslamnotvar_gthm & pushprop_lthm_orig & (hoasinduction_lem3b <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh)))) & (hoasinduction_lem0_lthm <=> ! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))) & (one = (d2term @ d_one)) & (hoasapinj1 <=> ! [X15 : term,X20 : term,X19 : term,X13 : term] : (((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)) | (X13 = X15))) & hoasapnotvar_lthm & hoasinduction_lem1 & axvarshift & (! [X48 : term,X10 : subst > term > subst > $o,X11 : term > $o] : (~! [X15 : term] : ((X11 @ (sub @ X48 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X18 : subst > term > term] : (~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X14 : subst,X13 : term,X26 : subst,X25 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X25 : subst,X26 : subst,X14 : subst,X13 : term] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X11 @ (lam @ X48))) <=> hoasinduction_lem3v2) & ((^[X13 : term, X14 : subst, X10 : term > $o, X11 : term > $o] : (! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))))) = pushprop_p_and_p_prime) & pushprop & (! [X15 : term,X13 : term] : ((X13 = X15) | ((sub @ X13 @ sh) != (sub @ X15 @ sh))) <=> shinj) & (pushprop_lem0 <=> ! [X10 : term > $o,X13 : term,X14 : subst] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14))))) & (! [X10 : term > $o,X22 : term] : (~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | ~! [X13 : term,X15 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | (X10 @ X22)) <=> induction2) & ((! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X10 : term > $o,X22 : term] : (~! [X15 : term,X13 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13)) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | (X10 @ X22) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : subst > term > subst > $o,X11 : term > $o,X29 : term] : (~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ 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)) | ~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X29))) => ! [X10 : subst > term > subst > $o,X30 : term] : ((X10 @ id @ X30 @ id) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X15 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id))))))) <=> hoasinduction_no_psi_cond_lthm) & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X48 : term,X10 : subst > term > subst > $o,X11 : term > $o] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X25 : subst,X14 : subst,X26 : subst,X13 : term] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term,X14 : subst,X26 : subst,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X11 @ (lam @ X48)) | ~! [X15 : term] : ((X11 @ (sub @ X48 @ (push @ X15 @ id))) | ~(X11 @ X15)))) <=> hoasinduction_lem3v2_lthm) & (hoasapinj2 <=> ! [X15 : term,X13 : term,X20 : term,X19 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)))) & hoaslaminj_gthm & (! [X14 : subst] : ((push @ (sub @ one @ X14) @ (comp @ sh @ X14)) = X14) <=> axscons) & apnotvar & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X20 : term,X19 : term,X13 : term,X15 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X13 = X15)) => ! [X15 : term,X13 : term,X20 : term,X19 : term] : (((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)) | (X13 = X15)))) <=> hoasapinj1_lthm) & termmset_gthm & ! [X7 : d_subst,X6 : d_subst] : (((d2subst @ X6) = (d2subst @ X7)) => (X6 = X7)) & ! [X0 : term] : ? [X1 : d_term] : ((d2term @ X1) = X0) & pushprop_lem3v2_lthm & axapp & (pushprop_lem0_lthm <=> ! [X13 : term,X14 : subst,X10 : term > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14))))) & (hoasinduction_lem3aaa <=> ! [X34 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X34 @ (push @ X15 @ id)) @ id)) | ~! [X35 : term,X18 : subst > term > term] : (~! [X38 : subst,X37 : term,X36 : subst] : ((sub @ (X18 @ X36 @ X37) @ X38) = (sub @ (sub @ X35 @ (push @ X37 @ X36)) @ X38)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X40 : term,X39 : subst,X41 : subst] : ((X18 @ (comp @ X39 @ X41) @ (sub @ X40 @ X41)) = (sub @ X35 @ (push @ (sub @ X40 @ X41) @ (comp @ X39 @ X41))))) | (X10 @ id @ (lam @ (sub @ X34 @ (push @ one @ sh))) @ id))) & ((d2subst @ d_id) = sh) & (! [X26 : term > $o,X13 : term,X15 : term,X14 : subst,X10 : term > $o] : (~(X10 @ X13) | (X26 @ (sub @ X13 @ (push @ X15 @ X14)))) <=> pushprop_lem1) & hoasinduction_lem2_lthm & (hoasinduction_no_psi_cond <=> ! [X10 : subst > term > subst > $o,X30 : term] : (~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ X30 @ id))) & (! [X15 : term,X13 : term] : ((X13 = X15) | ((lam @ X13) != (lam @ X15))) <=> laminj) & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X13 : term] : ((sub @ X13 @ id) = X13)) <=> termmset_lthm) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & hoasinduction_lem3_gthm & pushprop_lem2v2_gthm & hoasinduction_lem3v2_gthm & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : term > $o,X16 : term,X17 : subst] : (~! [X13 : term,X15 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | (X10 @ (sub @ X16 @ X17))) => ! [X10 : term > $o,X22 : term] : (~! [X15 : term,X13 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | (X10 @ X22) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13)))))) <=> induction2_lthm) & ulamvarind & induction2lem_gthm & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ((! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => ! [X14 : subst,X11 : term > $o,X10 : term > $o,X13 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) | ~(X10 @ X13) | (X11 @ one))) <=> pushprop_lem1v2_lthm) & (! [X13 : term] : ((sub @ X13 @ id) = X13) <=> axvarid) & (! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) <=> axvarcons) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_p_and_p_prime = (^[X10 : subst > term > subst > $o, X11 : term > $o] : (! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))))) & ~(var @ (d2term @ d_one)) & ! [X5 : d_subst] : (d_id = X5) & hoasinduction_lem1_lthm & hoasinduction_lem1v2 & hoasinduction_lem3b_gthm & (hoasinduction_lem3v2_f <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13))) & substmonoid_gthm & pushprop_lem3v2 & pushprop_lem1_gthm & axmap & hoasapnotvar & (hoasinduction_lem3aa <=> ! [X10 : subst > term > subst > $o,X42 : term] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id))) & (hoaslaminj <=> ! [X31 : subst > term > term,X18 : subst > term > term,X32 : subst,X33 : term] : (~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))) | ~! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33)))) & ((! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X22 : term,X10 : term > $o] : (~! [X15 : term,X13 : term] : (~(X10 @ X15) | ~(X10 @ X13) | (X10 @ (ap @ X13 @ X15))) | (X10 @ X22) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X11 : term > $o,X29 : term,X10 : subst > term > subst > $o] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X18 : subst > term > term] : ((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))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | (X11 @ (lam @ X29))) => ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X25 : subst,X13 : term,X14 : subst,X26 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | ~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X10 @ id @ X28 @ id)))))) <=> hoasinduction_lthm_3) & (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) <=> axshiftcons) & (hoaslam = (^[X8 : subst, X9 : subst > term > term] : (d2term @ d_one))) & (! [X19 : term,X20 : term,X13 : term,X15 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X19 = X20)) <=> apinj2) & (substmonoid <=> (! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X14 : subst] : ((comp @ id @ X14) = X14))) & (! [X13 : term] : ((sub @ X13 @ id) = X13) <=> termmset) & hoasinduction_gthm & pushprop_gthm & hoaslamnotvar & (hoasinduction_lem3a_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : subst > term > subst > $o,X42 : term] : (~! [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))) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) => ! [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] : (~! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X10 @ id @ (lam @ X43) @ id))))) & ~lamnotap & axclos & (hoasinduction_lem3v2a <=> ! [X29 : term,X10 : subst > term > subst > $o,X11 : term > $o] : ((X11 @ (lam @ X29)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (apinj1 <=> ! [X20 : term,X13 : term,X19 : term,X15 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X13 = X15))) & ((! [X22 : term,X10 : term > $o] : ((X10 @ X22) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | ~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15))) => (! [X23 : term,X10 : subst > term > subst > $o,X24 : term] : (~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id)) | ~(X10 @ id @ X23 @ id) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~(X10 @ id @ X24 @ id) | (X10 @ id @ (ap @ X23 @ X24) @ id)) => (! [X27 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term,X14 : subst,X26 : subst,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X10 @ id @ (lam @ X27) @ id) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))))) => ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X15 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id)) | ~! [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))) | ~! [X14 : subst,X13 : term,X25 : subst,X26 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X14 : subst,X26 : subst,X25 : subst,X13 : term] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | (X10 @ id @ X28 @ id))))) <=> hoasinduction_lthm) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0) & pushprop_lem0_gthm & (! [X46 : term,X47 : term,X18 : subst > term > term] : (~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (ap @ (sub @ X46 @ id) @ X47))) <=> hoaslamnotap) & axabs & hoasinduction_lem1v2_gthm & (! [X42 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aa_lthm) & ((! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X21 : term,X10 : term > $o] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~(X10 @ X13)) | ~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15) | ~(X10 @ X13)) | (X10 @ X21)) => ! [X16 : term,X10 : term > $o,X17 : subst] : ((X10 @ (sub @ X16 @ X17)) | ~! [X15 : term] : ((X10 @ (sub @ X15 @ X17)) | ~(var @ X15)) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | ~! [X15 : term,X13 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13))))))) <=> induction2lem_lthm) & axassoc & pushprop_lem2v2 & pushprop_lem2v2_lthm & (hoasinduction_lem2v2 <=> ! [X11 : term > $o,X10 : subst > term > subst > $o,X45 : term,X44 : term] : (~! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X13 : term,X25 : subst,X14 : subst,X26 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X11 @ (ap @ X44 @ X45)) | ~(X11 @ X44) | ~(X11 @ X45) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id)))) & (hoasinduction_lem3v2_f_lthm <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13))) & ((! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X29 : term,X10 : subst > term > subst > $o,X11 : term > $o] : ((X11 @ (lam @ X29)) | ~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))))) <=> hoasinduction_lem3v2a_lthm) & lamnotvar & pushprop_lthm & hoasapnotvar_gthm & (! [X10 : term > $o,X21 : term] : (~! [X13 : term,X15 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | (X10 @ X21) | ~! [X13 : term] : (~(X10 @ X13) | (X10 @ (lam @ X13)))) <=> induction) & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : (~! [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)))) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) => ! [X27 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ X27) @ id) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X25 : subst,X26 : subst,X14 : subst,X13 : term] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : (~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id))))) <=> hoasinduction_lem3_lthm) & hoaslamnotap_lthm & (hoaslaminj_lthm <=> (! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X13 : term,X15 : term] : (((lam @ X13) != (lam @ X15)) | (X13 = X15)) => ! [X33 : term,X31 : subst > term > term,X18 : subst > term > term,X32 : subst] : (((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33))))))) & hoasinduction_lem3a_gthm & ! [X1 : d_term] : (d_one = X1) & (hoasinduction_lem2 <=> ! [X10 : subst > term > subst > $o,X24 : term,X23 : term] : (~(X10 @ id @ X24 @ id) | (X10 @ id @ (ap @ X23 @ X24) @ id) | ~(X10 @ id @ X23 @ id) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ 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))))) & hoasinduction_lem2_gthm & ulamvarsh & hoasinduction_lem1_gthm & ! [X4 : subst] : ? [X5 : d_subst] : ((d2subst @ X5) = X4) & (! [X14 : subst] : ((comp @ X14 @ id) = X14) <=> axidr) & (hoasinduction_lem3b_lthm <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh)))) & ! [X2 : d_term,X3 : d_term] : (((d2term @ X2) = (d2term @ X3)) => (X2 = X3)) & (! [X14 : subst] : ((comp @ id @ X14) = X14) <=> axidl) & (! [X11 : term > $o,X14 : subst,X10 : term > $o,X13 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) | ~(X10 @ X13) | (X11 @ one)) <=> pushprop_lem1v2) & hoasapinj1_gthm & (hoasinduction_lem3 <=> ! [X27 : term,X10 : subst > term > subst > $o] : (~! [X26 : subst,X25 : subst,X14 : subst,X13 : term] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ (lam @ X27) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))))) & (hoasinduction <=> ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X13 : term,X25 : subst,X26 : subst,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~! [X18 : subst > term > term] : (~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ X28 @ id))) & ((d2subst @ d_id) = id) & (hoasinduction_lem3a <=> ! [X43 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ X43) @ 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))) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id)))) & hoaslamnotvar_lthm & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ~ulamvar1 & hoasapinj2_gthm & ((! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => ! [X10 : term > $o,X26 : term > $o,X15 : term,X14 : subst,X13 : term] : (~(X10 @ X13) | (X26 @ (sub @ X13 @ (push @ X15 @ X14)))))) <=> pushprop_lem1_lthm) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & induction2_gthm & (hoasapinj2_lthm <=> (! [X15 : term,X19 : term,X20 : term,X13 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X19 = X20)) => ! [X19 : term,X13 : term,X20 : term,X15 : term] : (((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)) | (X19 = X20)))) & (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))))) & hoasinduction_lem2v2_gthm & pushprop_lem1v2_gthm),
% 2.65/1.76 file('/export/starexec/sandbox/tmp/tmp.6IcC7TwM2m/DTF2THF_2950.p',alg444_1)).
% 2.65/1.76 thf(f1212,plain,(
% 2.65/1.76 ( ! [X62 : term] : (((d2term @ (sK487 @ X62)) = X62)) )),
% 2.65/1.76 inference(cnf_transformation,[status(thm)],[f545])).
% 2.65/1.76 thf(f2674,plain,(
% 2.65/1.76 ((d2term @ d_one) != (sub @ (sub @ sK594 @ sK592) @ sK593))),
% 2.65/1.76 inference(forward_demodulation,[status(thm)],[f1784,f2587])).
% 2.65/1.76 thf(f1784,plain,(
% 2.65/1.76 ((sub @ (sub @ sK594 @ sK592) @ sK593) != (sub @ sK594 @ (comp @ sK592 @ sK593)))),
% 2.65/1.76 inference(equality_proxy_clausification,[status(thm)],[f1783])).
% 2.65/1.76 thf(f1783,plain,(
% 2.65/1.76 (((sub @ sK594 @ (comp @ sK592 @ sK593)) = (sub @ (sub @ sK594 @ sK592) @ sK593)) = $false)),
% 2.65/1.76 inference(beta_eta_normalization,[status(thm)],[f1782])).
% 2.65/1.76 thf(f1782,plain,(
% 2.65/1.76 (((^[Y0 : term]: ((sub @ Y0 @ (comp @ sK592 @ sK593)) = (sub @ (sub @ Y0 @ sK592) @ sK593))) @ sK594) = $false)),
% 2.65/1.76 inference(sigma_clausification,[status(thm)],[f1781])).
% 2.65/1.76 thf(f1781,plain,(
% 2.65/1.76 ((!! @ term @ (^[Y0 : term]: ((sub @ Y0 @ (comp @ sK592 @ sK593)) = (sub @ (sub @ Y0 @ sK592) @ sK593)))) = $false)),
% 2.65/1.76 inference(beta_eta_normalization,[status(thm)],[f1780])).
% 2.65/1.76 thf(f1780,plain,(
% 2.65/1.76 (((^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sub @ Y1 @ (comp @ sK592 @ Y0)) = (sub @ (sub @ Y1 @ sK592) @ Y0))))) @ sK593) = $false)),
% 2.65/1.76 inference(sigma_clausification,[status(thm)],[f1779])).
% 2.65/1.76 thf(f1779,plain,(
% 2.65/1.76 ((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sub @ Y1 @ (comp @ sK592 @ Y0)) = (sub @ (sub @ Y1 @ sK592) @ Y0)))))) = $false)),
% 2.65/1.76 inference(beta_eta_normalization,[status(thm)],[f1778])).
% 2.65/1.76 thf(f1778,plain,(
% 2.65/1.76 ($false = ((^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ Y2 @ (comp @ Y0 @ Y1)) = (sub @ (sub @ Y2 @ Y0) @ Y1))))))) @ sK592))),
% 2.65/1.76 inference(sigma_clausification,[status(thm)],[f1262])).
% 2.65/1.76 thf(f1262,plain,(
% 2.65/1.76 ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ Y2 @ (comp @ Y0 @ Y1)) = (sub @ (sub @ Y2 @ Y0) @ Y1)))))))) != $true)),
% 2.65/1.76 inference(definition_unfolding,[status(thm)],[f546,f1254])).
% 2.65/1.76 thf(f1254,plain,(
% 2.65/1.76 (axclos = $true)),
% 2.65/1.76 inference(cnf_transformation,[status(thm)],[f545])).
% 2.65/1.76 thf(f546,plain,(
% 2.65/1.76 (axclos != (!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ Y2 @ (comp @ Y0 @ Y1)) = (sub @ (sub @ Y2 @ Y0) @ Y1)))))))))),
% 2.65/1.76 inference(cnf_transformation,[status(thm)],[f9])).
% 2.65/1.76 thf(f9,plain,(
% 2.65/1.76 (axclos != (!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ Y2 @ (comp @ Y0 @ Y1)) = (sub @ (sub @ Y2 @ Y0) @ Y1)))))))))),
% 2.65/1.76 inference(flattening,[status(thm)],[f6])).
% 2.65/1.76 thf(f6,plain,(
% 2.65/1.76 ~(axclos = (!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ Y2 @ (comp @ Y0 @ Y1)) = (sub @ (sub @ Y2 @ Y0) @ Y1)))))))))),
% 2.65/1.76 inference(fool_elimination,[status(thm)],[f5])).
% 2.65/1.76 thf(f5,plain,(
% 2.65/1.76 ~(axclos = ! [X0 : term,X1 : subst,X2 : subst] : ((sub @ X0 @ (comp @ X2 @ X1)) = (sub @ (sub @ X0 @ X2) @ X1)))),
% 2.65/1.76 inference(rectify,[status(thm)],[f3])).
% 2.65/1.76 thf(f3,negated_conjecture,(
% 2.65/1.76 ~(axclos = ! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (sub @ X13 @ X14) @ X25) = (sub @ X13 @ (comp @ X14 @ X25))))),
% 2.65/1.76 inference(negated_conjecture,[status(cth)],[f2])).
% 2.65/1.76 thf(f2,conjecture,(
% 2.65/1.76 (axclos = ! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (sub @ X13 @ X14) @ X25) = (sub @ X13 @ (comp @ X14 @ X25))))),
% 2.65/1.76 file('/export/starexec/sandbox/tmp/tmp.6IcC7TwM2m/DTF2THF_2950.p',axclos)).
% 2.65/1.76 % SZS output end Proof for DTF2THF_2950
% 2.65/1.76 % (3447)------------------------------
% 2.65/1.76 % (3447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.76 % (3447)Termination reason: Refutation
% 2.65/1.76
% 2.65/1.76 % (3447)Memory used [KB]: 11257
% 2.65/1.76 % (3447)Time elapsed: 0.137 s
% 2.65/1.76 % (3447)Instructions burned: 330 (million)
% 2.65/1.76 % (3447)------------------------------
% 2.65/1.76 % (3447)------------------------------
% 2.65/1.76 % (3417)Success in time 0.245 s
%------------------------------------------------------------------------------