%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n029.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 06:58:49 PM UTC 2026
% Result : Theorem 2.43s 1.93s
% Output : Refutation 2.43s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 196
% Syntax : Number of formulae : 239 ( 14 unt; 0 typ; 2 def)
% Number of atoms : 11840 (5571 equ; 0 cnn)
% Maximal formula atoms : 530 ( 49 avg)
% Number of connectives : 26640 (3131 ~;2822 |;2464 &;17339 @)
% ( 565 <=>; 296 =>; 0 <=; 3 <~>)
% Maximal formula depth : 186 ( 15 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 2205 (2205 >; 0 *; 0 +; 0 <<)
% Number of symbols : 790 ( 786 usr; 302 con; 0-4 aty)
% ( 20 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 5835 ( 116 ^;4338 !;1381 ?;5835 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
term: $tType ).
thf(type_def_7,type,
subst: $tType ).
thf(type_def_8,type,
d_term: $tType ).
thf(type_def_9,type,
d_subst: $tType ).
thf(func_def_0,type,
term: $tType ).
thf(func_def_1,type,
subst: $tType ).
thf(func_def_2,type,
one: term ).
thf(func_def_3,type,
ap: term > term > term ).
thf(func_def_4,type,
lam: term > term ).
thf(func_def_5,type,
sub: term > subst > term ).
thf(func_def_6,type,
id: subst ).
thf(func_def_7,type,
sh: subst ).
thf(func_def_8,type,
push: term > subst > subst ).
thf(func_def_9,type,
comp: subst > subst > subst ).
thf(func_def_10,type,
var: term > $o ).
thf(func_def_18,type,
hoasap: subst > term > subst > term > term ).
thf(func_def_73,type,
hoaslam: subst > ( subst > term > term ) > term ).
thf(func_def_86,type,
hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
thf(func_def_107,type,
hoasvar: subst > term > subst > $o ).
thf(func_def_121,type,
pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
thf(func_def_124,type,
d_term: $tType ).
thf(func_def_125,type,
d_subst: $tType ).
thf(func_def_126,type,
d2term: d_term > term ).
thf(func_def_127,type,
d2subst: d_subst > subst ).
thf(func_def_128,type,
d_one: d_term ).
thf(func_def_129,type,
d_id: d_subst ).
thf(func_def_142,type,
sP1: ( subst > term > subst > $o ) > $o ).
thf(func_def_144,type,
sP3: ( term > $o ) > $o ).
thf(func_def_145,type,
sP4: ( term > $o ) > $o ).
thf(func_def_146,type,
sP5: ( term > $o ) > $o ).
thf(func_def_147,type,
sP6: ( term > $o ) > subst > $o ).
thf(func_def_148,type,
sP7: ( term > $o ) > $o ).
thf(func_def_151,type,
sP10: ( subst > term > subst > $o ) > $o ).
thf(func_def_153,type,
sP12: ( term > $o ) > $o ).
thf(func_def_154,type,
sP13: ( term > $o ) > $o ).
thf(func_def_156,type,
sP15: ( subst > term > subst > $o ) > $o ).
thf(func_def_157,type,
sP16: ( subst > term > subst > $o ) > $o ).
thf(func_def_158,type,
sP17: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_160,type,
sP19: ( term > $o ) > $o ).
thf(func_def_161,type,
sP20: ( term > $o ) > $o ).
thf(func_def_162,type,
sP21: ( subst > term > subst > $o ) > $o ).
thf(func_def_163,type,
sP22: ( subst > term > subst > $o ) > $o ).
thf(func_def_164,type,
sP23: ( subst > term > subst > $o ) > $o ).
thf(func_def_165,type,
sP24: ( subst > term > subst > $o ) > $o ).
thf(func_def_166,type,
sP25: ( subst > term > subst > $o ) > $o ).
thf(func_def_170,type,
sP29: ( term > $o ) > $o ).
thf(func_def_171,type,
sP30: ( term > $o ) > $o ).
thf(func_def_175,type,
sP34: ( term > $o ) > $o ).
thf(func_def_176,type,
sP35: ( term > $o ) > $o ).
thf(func_def_177,type,
sP36: ( term > $o ) > $o ).
thf(func_def_178,type,
sP37: ( term > $o ) > $o ).
thf(func_def_181,type,
sP40: ( subst > term > subst > $o ) > $o ).
thf(func_def_182,type,
sP41: ( subst > term > subst > $o ) > $o ).
thf(func_def_183,type,
sP42: ( subst > term > subst > $o ) > $o ).
thf(func_def_184,type,
sP43: ( subst > term > subst > $o ) > $o ).
thf(func_def_185,type,
sP44: ( term > $o ) > $o ).
thf(func_def_186,type,
sP45: ( term > $o ) > $o ).
thf(func_def_187,type,
sP46: ( subst > term > subst > $o ) > $o ).
thf(func_def_188,type,
sP47: ( subst > term > subst > $o ) > $o ).
thf(func_def_189,type,
sP48: ( subst > term > subst > $o ) > $o ).
thf(func_def_190,type,
sP49: ( subst > term > subst > $o ) > $o ).
thf(func_def_191,type,
sP50: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_196,type,
sP55: ( subst > term > subst > $o ) > $o ).
thf(func_def_197,type,
sP56: ( subst > term > subst > $o ) > $o ).
thf(func_def_198,type,
sP57: ( subst > term > subst > $o ) > $o ).
thf(func_def_200,type,
sP59: ( subst > term > subst > $o ) > $o ).
thf(func_def_201,type,
sP60: ( subst > term > subst > $o ) > $o ).
thf(func_def_202,type,
sP61: term > ( term > $o ) > $o ).
thf(func_def_204,type,
sP63: ( subst > term > subst > $o ) > $o ).
thf(func_def_205,type,
sP64: ( subst > term > subst > $o ) > $o ).
thf(func_def_208,type,
sP67: ( subst > term > subst > $o ) > $o ).
thf(func_def_209,type,
sP68: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_210,type,
sP69: ( subst > term > subst > $o ) > $o ).
thf(func_def_211,type,
sP70: ( subst > term > subst > $o ) > $o ).
thf(func_def_216,type,
sP75: ( subst > term > subst > $o ) > $o ).
thf(func_def_218,type,
sP77: ( term > $o ) > $o ).
thf(func_def_219,type,
sP78: ( term > $o ) > $o ).
thf(func_def_220,type,
sP79: ( term > $o ) > $o ).
thf(func_def_221,type,
sP80: ( subst > term > subst > $o ) > $o ).
thf(func_def_222,type,
sP81: ( subst > term > subst > $o ) > $o ).
thf(func_def_223,type,
sP82: ( subst > term > subst > $o ) > $o ).
thf(func_def_229,type,
sP88: ( subst > term > subst > $o ) > $o ).
thf(func_def_230,type,
sP89: ( subst > term > subst > $o ) > $o ).
thf(func_def_231,type,
sP90: ( subst > term > subst > $o ) > $o ).
thf(func_def_232,type,
sP91: ( subst > term > subst > $o ) > $o ).
thf(func_def_234,type,
sP93: ( subst > term > subst > $o ) > $o ).
thf(func_def_236,type,
sP95: ( term > $o ) > $o ).
thf(func_def_237,type,
sP96: ( term > $o ) > $o ).
thf(func_def_239,type,
sP98: ( subst > term > subst > $o ) > $o ).
thf(func_def_241,type,
sP100: ( subst > term > subst > $o ) > $o ).
thf(func_def_242,type,
sP101: ( subst > term > subst > $o ) > $o ).
thf(func_def_244,type,
sP103: ( subst > term > subst > $o ) > $o ).
thf(func_def_245,type,
sP104: ( subst > term > subst > $o ) > $o ).
thf(func_def_246,type,
sP105: ( subst > term > subst > $o ) > $o ).
thf(func_def_248,type,
sP107: ( subst > term > subst > $o ) > $o ).
thf(func_def_249,type,
sP108: ( subst > term > subst > $o ) > $o ).
thf(func_def_251,type,
sP110: ( subst > term > subst > $o ) > $o ).
thf(func_def_252,type,
sP111: ( subst > term > subst > $o ) > $o ).
thf(func_def_253,type,
sP112: ( subst > term > subst > $o ) > $o ).
thf(func_def_255,type,
sK114: term > $o ).
thf(func_def_256,type,
sK115: term ).
thf(func_def_257,type,
sK116: subst > term > subst > $o ).
thf(func_def_258,type,
sK117: term ).
thf(func_def_259,type,
sK118: ( subst > term > subst > $o ) > subst ).
thf(func_def_260,type,
sK119: ( subst > term > subst > $o ) > subst ).
thf(func_def_261,type,
sK120: ( subst > term > subst > $o ) > subst ).
thf(func_def_262,type,
sK121: ( subst > term > subst > $o ) > term ).
thf(func_def_263,type,
sK122: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_264,type,
sK123: ( subst > term > subst > $o ) > subst ).
thf(func_def_265,type,
sK124: ( subst > term > subst > $o ) > term ).
thf(func_def_266,type,
sK125: ( subst > term > subst > $o ) > subst ).
thf(func_def_267,type,
sK126: ( subst > term > subst > $o ) > subst ).
thf(func_def_268,type,
sK127: ( subst > term > subst > $o ) > term ).
thf(func_def_269,type,
sK128: ( subst > term > subst > $o ) > term ).
thf(func_def_270,type,
sK129: term > $o ).
thf(func_def_271,type,
sK130: subst > term > subst > $o ).
thf(func_def_272,type,
sK131: term ).
thf(func_def_273,type,
sK132: term > ( term > $o ) > term ).
thf(func_def_274,type,
sK133: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_275,type,
sK134: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_276,type,
sK135: ( subst > term > term ) > term ).
thf(func_def_277,type,
sK136: ( subst > term > term ) > subst ).
thf(func_def_278,type,
sK137: ( subst > term > term ) > subst ).
thf(func_def_279,type,
sK138: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_280,type,
sK139: ( subst > term > subst > $o ) > term ).
thf(func_def_281,type,
sK140: ( subst > term > subst > $o ) > term ).
thf(func_def_282,type,
sK141: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_283,type,
sK142: ( subst > term > term ) > subst ).
thf(func_def_284,type,
sK143: ( subst > term > term ) > term ).
thf(func_def_285,type,
sK144: ( subst > term > term ) > subst ).
thf(func_def_286,type,
sK145: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_287,type,
sK146: subst > term > subst > $o ).
thf(func_def_288,type,
sK147: term ).
thf(func_def_289,type,
sK148: term ).
thf(func_def_290,type,
sK149: ( subst > term > subst > $o ) > subst ).
thf(func_def_291,type,
sK150: ( subst > term > subst > $o ) > subst ).
thf(func_def_292,type,
sK151: ( subst > term > subst > $o ) > 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 ) > subst ).
thf(func_def_296,type,
sK155: ( subst > term > subst > $o ) > term ).
thf(func_def_297,type,
sK156: ( subst > term > subst > $o ) > subst ).
thf(func_def_298,type,
sK157: ( subst > term > subst > $o ) > term ).
thf(func_def_299,type,
sK158: ( subst > term > subst > $o ) > term ).
thf(func_def_300,type,
sK159: subst > term > subst > $o ).
thf(func_def_301,type,
sK160: term ).
thf(func_def_302,type,
sK161: term > ( subst > term > subst > $o ) > term ).
thf(func_def_303,type,
sK162: ( subst > term > term ) > subst ).
thf(func_def_304,type,
sK163: ( subst > term > term ) > subst ).
thf(func_def_305,type,
sK164: ( subst > term > term ) > term ).
thf(func_def_306,type,
sK165: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_307,type,
sK166: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_308,type,
sK167: term > $o ).
thf(func_def_309,type,
sK168: term ).
thf(func_def_310,type,
sK169: ( term > $o ) > term ).
thf(func_def_311,type,
sK170: ( term > $o ) > term ).
thf(func_def_312,type,
sK171: ( term > $o ) > term ).
thf(func_def_313,type,
sK172: ( term > $o ) > term ).
thf(func_def_314,type,
sK173: term ).
thf(func_def_315,type,
sK174: subst > term > subst > $o ).
thf(func_def_316,type,
sK175: ( subst > term > subst > $o ) > term > term ).
thf(func_def_317,type,
sK176: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_318,type,
sK177: ( subst > term > term ) > term ).
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 > subst > $o ) > subst > term > term ).
thf(func_def_322,type,
sK181: subst > term > subst > $o ).
thf(func_def_323,type,
sK182: term ).
thf(func_def_324,type,
sK183: ( subst > term > subst > $o ) > subst ).
thf(func_def_325,type,
sK184: ( subst > term > subst > $o ) > subst ).
thf(func_def_326,type,
sK185: ( subst > term > subst > $o ) > subst ).
thf(func_def_327,type,
sK186: ( subst > term > subst > $o ) > term ).
thf(func_def_328,type,
sK187: ( subst > term > subst > $o ) > term ).
thf(func_def_329,type,
sK188: ( subst > term > subst > $o ) > subst ).
thf(func_def_330,type,
sK189: ( subst > term > subst > $o ) > subst ).
thf(func_def_331,type,
sK190: ( subst > term > subst > $o ) > term ).
thf(func_def_332,type,
sK191: ( subst > term > subst > $o ) > subst ).
thf(func_def_333,type,
sK192: ( subst > term > subst > $o ) > term ).
thf(func_def_334,type,
sK193: ( subst > term > subst > $o ) > term ).
thf(func_def_335,type,
sK194: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_336,type,
sK195: ( subst > term > term ) > term ).
thf(func_def_337,type,
sK196: ( subst > term > term ) > subst ).
thf(func_def_338,type,
sK197: ( subst > term > term ) > subst ).
thf(func_def_339,type,
sK198: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_340,type,
sK199: term ).
thf(func_def_341,type,
sK200: term ).
thf(func_def_342,type,
sK201: term ).
thf(func_def_343,type,
sK202: term ).
thf(func_def_344,type,
sK203: term ).
thf(func_def_345,type,
sK204: term ).
thf(func_def_346,type,
sK205: term ).
thf(func_def_347,type,
sK206: term ).
thf(func_def_348,type,
sK207: term ).
thf(func_def_349,type,
sK208: term ).
thf(func_def_350,type,
sK209: term > $o ).
thf(func_def_351,type,
sK210: term > ( term > $o ) > term ).
thf(func_def_352,type,
sK211: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_353,type,
sK212: term > $o ).
thf(func_def_354,type,
sK213: term ).
thf(func_def_355,type,
sK214: subst > term > subst > $o ).
thf(func_def_356,type,
sK215: term ).
thf(func_def_357,type,
sK216: subst > term > subst > $o ).
thf(func_def_358,type,
sK217: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_359,type,
sK218: ( subst > term > term ) > subst ).
thf(func_def_360,type,
sK219: ( subst > term > term ) > subst ).
thf(func_def_361,type,
sK220: ( subst > term > term ) > term ).
thf(func_def_362,type,
sK221: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_363,type,
sK222: ( subst > term > subst > $o ) > term ).
thf(func_def_364,type,
sK223: ( subst > term > subst > $o ) > term ).
thf(func_def_365,type,
sK224: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_366,type,
sK225: ( subst > term > term ) > subst ).
thf(func_def_367,type,
sK226: ( subst > term > term ) > subst ).
thf(func_def_368,type,
sK227: ( subst > term > term ) > term ).
thf(func_def_369,type,
sK228: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_370,type,
sK229: ( term > $o ) > term ).
thf(func_def_371,type,
sK230: term > ( term > $o ) > term ).
thf(func_def_372,type,
sK231: ( term > $o ) > term ).
thf(func_def_373,type,
sK232: ( term > $o ) > term ).
thf(func_def_374,type,
sK233: ( term > $o ) > term ).
thf(func_def_375,type,
sK234: term ).
thf(func_def_376,type,
sK235: term > $o ).
thf(func_def_377,type,
sK236: subst > term > subst > $o ).
thf(func_def_378,type,
sK237: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_379,type,
sK238: ( term > $o ) > term > term ).
thf(func_def_380,type,
sK239: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_381,type,
sK240: ( subst > term > term ) > term ).
thf(func_def_382,type,
sK241: ( subst > term > term ) > subst ).
thf(func_def_383,type,
sK242: ( subst > term > term ) > subst ).
thf(func_def_384,type,
sK243: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_385,type,
sK244: term ).
thf(func_def_386,type,
sK245: term ).
thf(func_def_387,type,
sK246: term ).
thf(func_def_388,type,
sK247: term ).
thf(func_def_389,type,
sK248: term ).
thf(func_def_390,type,
sK249: term ).
thf(func_def_391,type,
sK250: term ).
thf(func_def_392,type,
sK251: term ).
thf(func_def_393,type,
sK252: ( subst > term > subst > $o ) > term > term ).
thf(func_def_394,type,
sK253: term ).
thf(func_def_395,type,
sK254: subst > term > subst > $o ).
thf(func_def_396,type,
sK255: term ).
thf(func_def_397,type,
sK256: subst > term > subst > $o ).
thf(func_def_398,type,
sK257: ( subst > term > subst > $o ) > term ).
thf(func_def_399,type,
sK258: ( subst > term > subst > $o ) > subst ).
thf(func_def_400,type,
sK259: ( subst > term > subst > $o ) > subst ).
thf(func_def_401,type,
sK260: ( subst > term > subst > $o ) > subst ).
thf(func_def_402,type,
sK261: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_403,type,
sK262: ( subst > term > term ) > term ).
thf(func_def_404,type,
sK263: ( subst > term > term ) > subst ).
thf(func_def_405,type,
sK264: ( subst > term > term ) > subst ).
thf(func_def_406,type,
sK265: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_407,type,
sK266: ( subst > term > subst > $o ) > term ).
thf(func_def_408,type,
sK267: ( subst > term > subst > $o ) > subst ).
thf(func_def_409,type,
sK268: ( subst > term > subst > $o ) > subst ).
thf(func_def_410,type,
sK269: ( subst > term > subst > $o ) > subst ).
thf(func_def_411,type,
sK270: ( subst > term > subst > $o ) > term > term ).
thf(func_def_412,type,
sK271: ( subst > term > term ) > term ).
thf(func_def_413,type,
sK272: ( subst > term > term ) > subst ).
thf(func_def_414,type,
sK273: ( subst > term > term ) > subst ).
thf(func_def_415,type,
sK274: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_416,type,
sK275: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_417,type,
sK276: ( subst > term > subst > $o ) > term > term ).
thf(func_def_418,type,
sK277: term ).
thf(func_def_419,type,
sK278: subst > term > subst > $o ).
thf(func_def_420,type,
sK279: term ).
thf(func_def_421,type,
sK280: subst > term > subst > $o ).
thf(func_def_422,type,
sK281: ( subst > term > subst > $o ) > term > term ).
thf(func_def_423,type,
sK282: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_424,type,
sK283: ( subst > term > term ) > subst ).
thf(func_def_425,type,
sK284: ( subst > term > term ) > term ).
thf(func_def_426,type,
sK285: ( subst > term > term ) > subst ).
thf(func_def_427,type,
sK286: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_428,type,
sK287: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_429,type,
sK288: ( subst > term > term ) > subst ).
thf(func_def_430,type,
sK289: ( subst > term > term ) > subst ).
thf(func_def_431,type,
sK290: ( subst > term > term ) > term ).
thf(func_def_432,type,
sK291: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_433,type,
sK292: subst > term > subst > $o ).
thf(func_def_434,type,
sK293: term ).
thf(func_def_435,type,
sK294: term > $o ).
thf(func_def_436,type,
sK295: ( subst > term > subst > $o ) > subst ).
thf(func_def_437,type,
sK296: ( subst > term > subst > $o ) > subst ).
thf(func_def_438,type,
sK297: ( subst > term > subst > $o ) > subst ).
thf(func_def_439,type,
sK298: ( subst > term > subst > $o ) > term ).
thf(func_def_440,type,
sK299: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_441,type,
sK300: term > ( term > $o ) > term ).
thf(func_def_442,type,
sK301: ( subst > term > subst > $o ) > subst ).
thf(func_def_443,type,
sK302: ( subst > term > subst > $o ) > subst ).
thf(func_def_444,type,
sK303: ( subst > term > subst > $o ) > subst ).
thf(func_def_445,type,
sK304: ( subst > term > subst > $o ) > term ).
thf(func_def_446,type,
sK305: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_447,type,
sK306: ( subst > term > term ) > subst ).
thf(func_def_448,type,
sK307: ( subst > term > term ) > subst ).
thf(func_def_449,type,
sK308: ( subst > term > term ) > term ).
thf(func_def_450,type,
sK309: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_451,type,
sK310: term ).
thf(func_def_452,type,
sK311: subst > term > subst > $o ).
thf(func_def_453,type,
sK312: term > $o ).
thf(func_def_454,type,
sK313: ( term > $o ) > term > term ).
thf(func_def_455,type,
sK314: ( term > $o ) > ( subst > term > subst > $o ) > term ).
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 ) > subst ).
thf(func_def_459,type,
sK318: ( subst > term > subst > $o ) > term ).
thf(func_def_460,type,
sK319: ( subst > term > subst > $o ) > subst ).
thf(func_def_461,type,
sK320: ( subst > term > subst > $o ) > subst ).
thf(func_def_462,type,
sK321: ( subst > term > subst > $o ) > term ).
thf(func_def_463,type,
sK322: ( subst > term > subst > $o ) > subst ).
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 ) > ( subst > term > subst > $o ) > term ).
thf(func_def_468,type,
sK327: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_469,type,
sK328: ( subst > term > subst > $o ) > subst ).
thf(func_def_470,type,
sK329: ( subst > term > subst > $o ) > term ).
thf(func_def_471,type,
sK330: ( subst > term > subst > $o ) > subst ).
thf(func_def_472,type,
sK331: ( subst > term > subst > $o ) > subst ).
thf(func_def_473,type,
sK332: term ).
thf(func_def_474,type,
sK333: subst > term > subst > $o ).
thf(func_def_475,type,
sK334: ( subst > term > subst > $o ) > subst ).
thf(func_def_476,type,
sK335: ( subst > term > subst > $o ) > subst ).
thf(func_def_477,type,
sK336: ( subst > term > subst > $o ) > term ).
thf(func_def_478,type,
sK337: ( subst > term > subst > $o ) > subst ).
thf(func_def_479,type,
sK338: subst > term > subst > $o ).
thf(func_def_480,type,
sK339: term ).
thf(func_def_481,type,
sK340: term ).
thf(func_def_482,type,
sK341: ( term > $o ) > term ).
thf(func_def_483,type,
sK342: term > $o ).
thf(func_def_484,type,
sK343: term ).
thf(func_def_485,type,
sK344: term ).
thf(func_def_486,type,
sK345: subst > term > subst > $o ).
thf(func_def_487,type,
sK346: ( subst > term > subst > $o ) > subst ).
thf(func_def_488,type,
sK347: ( subst > term > subst > $o ) > subst ).
thf(func_def_489,type,
sK348: ( subst > term > subst > $o ) > subst ).
thf(func_def_490,type,
sK349: ( subst > term > subst > $o ) > term ).
thf(func_def_491,type,
sK350: ( subst > term > subst > $o ) > term > term ).
thf(func_def_492,type,
sK351: ( subst > term > subst > $o ) > subst ).
thf(func_def_493,type,
sK352: ( subst > term > subst > $o ) > subst ).
thf(func_def_494,type,
sK353: ( subst > term > subst > $o ) > term ).
thf(func_def_495,type,
sK354: ( subst > term > subst > $o ) > subst ).
thf(func_def_496,type,
sK355: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_497,type,
sK356: ( subst > term > term ) > subst ).
thf(func_def_498,type,
sK357: ( subst > term > term ) > term ).
thf(func_def_499,type,
sK358: ( subst > term > term ) > subst ).
thf(func_def_500,type,
sK359: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_501,type,
sK360: ( subst > term > subst > $o ) > term ).
thf(func_def_502,type,
sK361: ( subst > term > subst > $o ) > subst ).
thf(func_def_503,type,
sK362: ( subst > term > subst > $o ) > subst ).
thf(func_def_504,type,
sK363: ( subst > term > subst > $o ) > subst ).
thf(func_def_505,type,
sK364: ( subst > term > subst > $o ) > term ).
thf(func_def_506,type,
sK365: ( subst > term > subst > $o ) > term ).
thf(func_def_507,type,
sK366: ( term > $o ) > term ).
thf(func_def_508,type,
sK367: term > ( term > $o ) > term ).
thf(func_def_509,type,
sK368: ( term > $o ) > term ).
thf(func_def_510,type,
sK369: ( term > $o ) > term ).
thf(func_def_511,type,
sK370: ( subst > term > subst > $o ) > term ).
thf(func_def_512,type,
sK371: ( subst > term > subst > $o ) > subst ).
thf(func_def_513,type,
sK372: ( subst > term > subst > $o ) > subst ).
thf(func_def_514,type,
sK373: ( subst > term > subst > $o ) > subst ).
thf(func_def_515,type,
sK374: ( subst > term > subst > $o ) > term ).
thf(func_def_516,type,
sK375: ( subst > term > subst > $o ) > term ).
thf(func_def_517,type,
sK376: ( subst > term > subst > $o ) > term ).
thf(func_def_518,type,
sK377: ( subst > term > term ) > subst ).
thf(func_def_519,type,
sK378: ( subst > term > term ) > term ).
thf(func_def_520,type,
sK379: ( subst > term > term ) > subst ).
thf(func_def_521,type,
sK380: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_522,type,
sK381: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_523,type,
sK382: ( term > $o ) > term ).
thf(func_def_524,type,
sK383: term ).
thf(func_def_525,type,
sK384: term > $o ).
thf(func_def_526,type,
sK385: term ).
thf(func_def_527,type,
sK386: term > $o ).
thf(func_def_528,type,
sK387: subst ).
thf(func_def_529,type,
sK388: subst > ( term > $o ) > term ).
thf(func_def_530,type,
sK389: ( term > $o ) > term ).
thf(func_def_531,type,
sK390: ( term > $o ) > term ).
thf(func_def_532,type,
sK391: ( term > $o ) > term ).
thf(func_def_533,type,
sK392: ( term > $o ) > term ).
thf(func_def_534,type,
sK393: ( term > $o ) > term ).
thf(func_def_535,type,
sK394: term > ( term > $o ) > term ).
thf(func_def_536,type,
sK395: ( term > $o ) > term ).
thf(func_def_537,type,
sK396: subst ).
thf(func_def_538,type,
sK397: term ).
thf(func_def_539,type,
sK398: term ).
thf(func_def_540,type,
sK399: term ).
thf(func_def_541,type,
sK400: subst ).
thf(func_def_542,type,
sK401: term ).
thf(func_def_543,type,
sK402: term ).
thf(func_def_544,type,
sK403: subst > term > term ).
thf(func_def_545,type,
sK404: subst ).
thf(func_def_546,type,
sK405: subst > term > term ).
thf(func_def_547,type,
sK406: ( subst > term > term ) > subst ).
thf(func_def_548,type,
sK407: ( subst > term > term ) > subst ).
thf(func_def_549,type,
sK408: ( subst > term > term ) > term ).
thf(func_def_550,type,
sK409: ( subst > term > term ) > subst ).
thf(func_def_551,type,
sK410: ( subst > term > term ) > subst ).
thf(func_def_552,type,
sK411: ( subst > term > term ) > term ).
thf(func_def_553,type,
sK412: term ).
thf(func_def_554,type,
sK413: term > $o ).
thf(func_def_555,type,
sK414: ( term > $o ) > term ).
thf(func_def_556,type,
sK415: ( term > $o ) > term ).
thf(func_def_557,type,
sK416: ( term > $o ) > term ).
thf(func_def_558,type,
sK417: term > ( term > $o ) > term ).
thf(func_def_559,type,
sK418: ( term > $o ) > term ).
thf(func_def_560,type,
sK419: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_561,type,
sK420: ( term > $o ) > term > term ).
thf(func_def_562,type,
sK421: subst > term > subst > $o ).
thf(func_def_563,type,
sK422: term ).
thf(func_def_564,type,
sK423: term > $o ).
thf(func_def_565,type,
sK424: ( term > $o ) > term ).
thf(func_def_566,type,
sK425: term > $o ).
thf(func_def_567,type,
sK426: term ).
thf(func_def_568,type,
sK427: term ).
thf(func_def_569,type,
sK428: subst > term > subst > $o ).
thf(func_def_570,type,
sK429: ( subst > term > subst > $o ) > term ).
thf(func_def_571,type,
sK430: ( subst > term > subst > $o ) > subst ).
thf(func_def_572,type,
sK431: ( subst > term > subst > $o ) > subst ).
thf(func_def_573,type,
sK432: ( subst > term > subst > $o ) > term ).
thf(func_def_574,type,
sK433: ( subst > term > subst > $o ) > subst ).
thf(func_def_575,type,
sK434: ( subst > term > subst > $o ) > subst ).
thf(func_def_576,type,
sK435: ( subst > term > subst > $o ) > subst ).
thf(func_def_577,type,
sK436: ( subst > term > subst > $o ) > term ).
thf(func_def_578,type,
sK437: ( subst > term > subst > $o ) > subst ).
thf(func_def_579,type,
sK438: ( subst > term > subst > $o ) > term ).
thf(func_def_580,type,
sK439: ( subst > term > subst > $o ) > term ).
thf(func_def_581,type,
sK440: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_582,type,
sK441: ( subst > term > term ) > subst ).
thf(func_def_583,type,
sK442: ( subst > term > term ) > term ).
thf(func_def_584,type,
sK443: ( subst > term > term ) > subst ).
thf(func_def_585,type,
sK444: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_586,type,
sK445: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_587,type,
sK446: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_588,type,
sK447: ( subst > term > term ) > term ).
thf(func_def_589,type,
sK448: ( subst > term > term ) > subst ).
thf(func_def_590,type,
sK449: ( subst > term > term ) > subst ).
thf(func_def_591,type,
sK450: ( term > $o ) > term ).
thf(func_def_592,type,
sK451: term > ( term > $o ) > term ).
thf(func_def_593,type,
sK452: ( term > $o ) > term ).
thf(func_def_594,type,
sK453: ( term > $o ) > term ).
thf(func_def_595,type,
sK454: subst > term > subst > $o ).
thf(func_def_596,type,
sK455: term ).
thf(func_def_597,type,
sK456: ( subst > term > subst > $o ) > subst ).
thf(func_def_598,type,
sK457: ( subst > term > subst > $o ) > subst ).
thf(func_def_599,type,
sK458: ( subst > term > subst > $o ) > subst ).
thf(func_def_600,type,
sK459: ( subst > term > subst > $o ) > term ).
thf(func_def_601,type,
sK460: ( subst > term > subst > $o ) > term > term ).
thf(func_def_602,type,
sK461: ( subst > term > subst > $o ) > subst ).
thf(func_def_603,type,
sK462: ( subst > term > subst > $o ) > subst ).
thf(func_def_604,type,
sK463: ( subst > term > subst > $o ) > subst ).
thf(func_def_605,type,
sK464: ( subst > term > subst > $o ) > term ).
thf(func_def_606,type,
sK465: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_607,type,
sK466: ( subst > term > term ) > term ).
thf(func_def_608,type,
sK467: ( subst > term > term ) > subst ).
thf(func_def_609,type,
sK468: ( subst > term > term ) > subst ).
thf(func_def_610,type,
sK469: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_611,type,
sK470: term ).
thf(func_def_612,type,
sK471: term > $o ).
thf(func_def_613,type,
sK472: subst ).
thf(func_def_614,type,
sK473: subst > ( term > $o ) > term ).
thf(func_def_615,type,
sK474: ( term > $o ) > term ).
thf(func_def_616,type,
sK475: ( term > $o ) > term ).
thf(func_def_617,type,
sK476: term > ( term > $o ) > term ).
thf(func_def_618,type,
sK477: ( term > $o ) > term ).
thf(func_def_619,type,
sK478: subst > term > subst > $o ).
thf(func_def_620,type,
sK479: term ).
thf(func_def_621,type,
sK480: term > ( subst > term > subst > $o ) > term ).
thf(func_def_622,type,
sK481: ( subst > term > term ) > term > subst ).
thf(func_def_623,type,
sK482: ( subst > term > term ) > term > subst ).
thf(func_def_624,type,
sK483: ( subst > term > term ) > term > term ).
thf(func_def_625,type,
sK484: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_626,type,
sK485: ( subst > term > term ) > term > subst ).
thf(func_def_627,type,
sK486: ( subst > term > term ) > term > term ).
thf(func_def_628,type,
sK487: ( subst > term > term ) > term > subst ).
thf(func_def_629,type,
sK488: ( subst > term > subst > $o ) > term ).
thf(func_def_630,type,
sK489: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_631,type,
sK490: term ).
thf(func_def_632,type,
sK491: subst ).
thf(func_def_633,type,
sK492: term > $o ).
thf(func_def_634,type,
sK493: term ).
thf(func_def_635,type,
sK494: term > $o ).
thf(func_def_636,type,
sK495: ( term > $o ) > term ).
thf(func_def_637,type,
sK496: ( term > $o ) > term ).
thf(func_def_638,type,
sK497: term > ( term > $o ) > term ).
thf(func_def_639,type,
sK498: ( term > $o ) > subst > term ).
thf(func_def_640,type,
sK499: ( term > $o ) > term ).
thf(func_def_641,type,
sK500: ( term > $o ) > term ).
thf(func_def_642,type,
sK501: ( term > $o ) > term ).
thf(func_def_643,type,
sK502: ( term > $o ) > term ).
thf(func_def_644,type,
sK503: term > ( term > $o ) > term ).
thf(func_def_645,type,
sK504: ( term > $o ) > term ).
thf(func_def_646,type,
sK505: term ).
thf(func_def_647,type,
sK506: subst > term > subst > $o ).
thf(func_def_648,type,
sK507: ( subst > term > subst > $o ) > term > term ).
thf(func_def_649,type,
sK508: ( subst > term > term ) > subst ).
thf(func_def_650,type,
sK509: ( subst > term > term ) > subst ).
thf(func_def_651,type,
sK510: ( subst > term > term ) > term ).
thf(func_def_652,type,
sK511: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_653,type,
sK512: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_654,type,
sK513: subst ).
thf(func_def_655,type,
sK514: subst ).
thf(func_def_656,type,
sK515: subst ).
thf(func_def_657,type,
sK516: subst ).
thf(func_def_658,type,
sK517: term > $o ).
thf(func_def_659,type,
sK518: term ).
thf(func_def_660,type,
sK519: subst ).
thf(func_def_661,type,
sK520: term ).
thf(func_def_662,type,
sK521: term > $o ).
thf(func_def_663,type,
sK522: subst ).
thf(func_def_664,type,
sK523: term ).
thf(func_def_665,type,
sK524: subst ).
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 ) > term ).
thf(func_def_670,type,
sK529: ( subst > term > term ) > subst ).
thf(func_def_671,type,
sK530: term > subst > term > term ).
thf(func_def_672,type,
sK531: subst > term > subst > $o ).
thf(func_def_673,type,
sK532: ( term > $o ) > term ).
thf(func_def_674,type,
sK533: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_675,type,
sK534: term ).
thf(func_def_676,type,
sK535: term > subst > term > term ).
thf(func_def_677,type,
sK536: subst ).
thf(func_def_678,type,
sK537: term ).
thf(func_def_679,type,
sK538: term ).
thf(func_def_680,type,
sK539: subst > term > subst > $o ).
thf(func_def_681,type,
sK540: term ).
thf(func_def_682,type,
sK541: term > $o ).
thf(func_def_683,type,
sK542: subst ).
thf(func_def_684,type,
sK543: ( term > $o ) > term ).
thf(func_def_685,type,
sK544: subst > ( term > $o ) > term > term > $o ).
thf(func_def_686,type,
sK545: subst ).
thf(func_def_687,type,
sK546: subst ).
thf(func_def_688,type,
sK547: term ).
thf(func_def_689,type,
sK548: term ).
thf(func_def_690,type,
sK549: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_691,type,
sK550: subst > term > subst > $o ).
thf(func_def_692,type,
sK551: ( term > $o ) > term ).
thf(func_def_693,type,
sK552: term ).
thf(func_def_694,type,
sK553: term ).
thf(func_def_695,type,
sK554: term ).
thf(func_def_696,type,
sK555: term ).
thf(func_def_697,type,
sK556: term ).
thf(func_def_698,type,
sK557: subst > d_subst ).
thf(func_def_699,type,
sK558: term ).
thf(func_def_700,type,
sK559: term > $o ).
thf(func_def_701,type,
sK560: term > $o ).
thf(func_def_702,type,
sK561: term ).
thf(func_def_703,type,
sK562: subst ).
thf(func_def_704,type,
sK563: term ).
thf(func_def_705,type,
sK564: subst > term > subst > $o ).
thf(func_def_706,type,
sK565: ( term > $o ) > term ).
thf(func_def_707,type,
sK566: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_708,type,
sK567: term ).
thf(func_def_709,type,
sK568: term ).
thf(func_def_710,type,
sK569: term ).
thf(func_def_711,type,
sK570: term ).
thf(func_def_712,type,
sK571: term > subst > term > term ).
thf(func_def_713,type,
sK572: term ).
thf(func_def_714,type,
sK573: ( subst > term > term ) > term ).
thf(func_def_715,type,
sK574: ( subst > term > term ) > subst ).
thf(func_def_716,type,
sK575: term ).
thf(func_def_717,type,
sK576: term > d_term ).
thf(func_def_718,type,
sK577: subst ).
thf(func_def_719,type,
sK578: term ).
thf(func_def_720,type,
sK579: term ).
thf(func_def_721,type,
sK580: subst ).
thf(func_def_722,type,
sK581: subst ).
thf(func_def_723,type,
sK582: term ).
thf(func_def_724,type,
sK583: term ).
thf(func_def_725,type,
sK584: term ).
thf(func_def_726,type,
sK585: term ).
thf(func_def_727,type,
sK586: subst ).
thf(func_def_728,type,
sK587: subst ).
thf(func_def_729,type,
sK588: term ).
thf(func_def_730,type,
sK589: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_731,type,
sK590: subst > term > subst > $o ).
thf(func_def_732,type,
sK591: ( term > $o ) > term ).
thf(func_def_733,type,
sK592: term ).
thf(func_def_734,type,
sK593: term ).
thf(func_def_735,type,
sK594: term ).
thf(func_def_736,type,
sK595: term ).
thf(func_def_737,type,
sK596: ( subst > term > term ) > term ).
thf(func_def_738,type,
sK597: ( subst > term > term ) > subst ).
thf(func_def_739,type,
sK598: ( subst > term > term ) > subst ).
thf(func_def_740,type,
sK599: term ).
thf(func_def_741,type,
sK600: subst > term > term ).
thf(func_def_742,type,
sK601: term ).
thf(func_def_743,type,
sK602: term ).
thf(func_def_744,type,
sK603: term > $o ).
thf(func_def_745,type,
sK604: term > $o ).
thf(func_def_746,type,
sK605: subst ).
thf(func_def_747,type,
sK606: subst > ( term > $o ) > ( term > $o ) > term > term ).
thf(func_def_748,type,
sK607: term > subst > term > term ).
thf(func_def_749,type,
sK608: term ).
thf(func_def_750,type,
sK609: subst ).
thf(func_def_751,type,
sK610: ( subst > term > term ) > subst ).
thf(func_def_752,type,
sK611: ( subst > term > term ) > term ).
thf(func_def_753,type,
sK612: ( subst > term > term ) > subst ).
thf(func_def_754,type,
sK613: ( subst > term > term ) > subst ).
thf(func_def_755,type,
sK614: ( subst > term > term ) > term ).
thf(func_def_756,type,
sK615: ( subst > term > term ) > subst ).
thf(func_def_757,type,
sK616: term ).
thf(func_def_758,type,
sK617: subst > term > term ).
thf(func_def_759,type,
sK618: subst ).
thf(func_def_760,type,
sK619: subst > term > term ).
thf(func_def_761,type,
sK620: subst > term > ( term > $o ) > term > $o ).
thf(func_def_762,type,
sK621: term > $o ).
thf(func_def_763,type,
sK622: term ).
thf(func_def_764,type,
sK623: subst ).
thf(func_def_765,type,
sK624: ( term > $o ) > term ).
thf(func_def_766,type,
sK625: term ).
thf(func_def_767,type,
sK626: term ).
thf(func_def_768,type,
sK627: term ).
thf(func_def_769,type,
sK628: term ).
thf(func_def_770,type,
sK629: term ).
thf(func_def_771,type,
sK630: term > subst > ( term > $o ) > ( term > $o ) > term ).
thf(func_def_772,type,
sK631: term ).
thf(func_def_773,type,
sK632: subst ).
thf(func_def_774,type,
sK633: term > $o ).
thf(func_def_775,type,
sK634: term > $o ).
thf(func_def_776,type,
sK635: subst ).
thf(func_def_777,type,
sK636: term ).
thf(func_def_778,type,
sK637: subst ).
thf(func_def_779,type,
sK638: term > subst > term > term ).
thf(func_def_780,type,
sK639: term ).
thf(func_def_781,type,
sK640: ( subst > term > term ) > term ).
thf(func_def_782,type,
sK641: ( subst > term > term ) > subst ).
thf(func_def_783,type,
sK642: term ).
thf(func_def_784,type,
sK643: term ).
thf(func_def_785,type,
sK644: term ).
thf(func_def_786,type,
sK645: term ).
thf(func_def_787,type,
sK646: subst > term > subst > $o ).
thf(func_def_788,type,
sK647: ( subst > term > subst > $o ) > subst ).
thf(func_def_789,type,
sK648: ( subst > term > subst > $o ) > subst ).
thf(func_def_790,type,
sK649: ( subst > term > subst > $o ) > term ).
thf(func_def_791,type,
sK650: ( subst > term > subst > $o ) > subst ).
thf(func_def_792,type,
sK651: ( subst > term > subst > $o ) > term ).
thf(func_def_793,type,
sK652: ( subst > term > subst > $o ) > subst ).
thf(func_def_794,type,
sK653: ( subst > term > subst > $o ) > subst ).
thf(func_def_795,type,
sK654: ( subst > term > subst > $o ) > term ).
thf(func_def_796,type,
sK655: ( subst > term > subst > $o ) > subst ).
thf(func_def_797,type,
sK656: ( subst > term > subst > $o ) > term ).
thf(func_def_800,type,
ph658:
!>[X0: $tType] : X0 ).
thf(f4958,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f2942,f3332,f4957]) ).
thf(f4957,plain,
spl657_345,
inference(avatar_contradiction_clause,[status(thm)],[f4956]) ).
thf(f4956,plain,
( $false
| spl657_345 ),
inference(trivial_inequality_removal,[status(thm)],[f4955]) ).
thf(f4955,plain,
( ( $true != $true )
| spl657_345 ),
inference(superposition,[status(thm)],[f3331,f4954]) ).
thf(f4954,plain,
! [X0: subst > term > subst > $o] :
( $true
= ( sP110 @ X0 ) ),
inference(subsumption_resolution,[status(thm)],[f1530,f3880]) ).
thf(f3880,plain,
! [X0: term] :
( ( var @ X0 )
!= $true ),
inference(superposition,[status(thm)],[f1395,f3872]) ).
thf(f3872,plain,
! [X124: term] :
( ( d2term @ d_one )
= X124 ),
inference(forward_demodulation,[status(thm)],[f1393,f1474]) ).
thf(f1474,plain,
! [X62: d_term] : ( d_one = X62 ),
inference(cnf_transformation,[status(thm)],[f749]) ).
thf(f749,plain,
( ( ( pushprop_lem1_lthm = $true )
| ( ( ( sK517 @ ( sub @ sK518 @ ( push @ sK520 @ sK519 ) ) )
!= $true )
& ( ( sK521 @ sK518 )
= $true )
& ! [X5: subst,X6: term] :
( ( sub @ one @ ( push @ X6 @ X5 ) )
= X6 )
& ! [X7: subst,X8: term] :
( ( comp @ sh @ ( push @ X8 @ X7 ) )
= X7 ) ) )
& ( ! [X9: term > $o,X10: term,X11: subst,X12: term,X13: term > $o] :
( ( ( X9 @ ( sub @ X10 @ ( push @ X12 @ X11 ) ) )
= $true )
| ( ( X13 @ X10 )
!= $true ) )
| ( sK523
!= ( sub @ one @ ( push @ sK523 @ sK522 ) ) )
| ( ( comp @ sh @ ( push @ sK525 @ sK524 ) )
!= sK524 )
| ( pushprop_lem1_lthm != $true ) )
& ! [X18: d_subst,X19: d_subst] :
( ( ( d2subst @ X18 )
!= ( d2subst @ X19 ) )
| ( X18 = X19 ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP109 ) )
& ( ( $true = sP109 )
| ( hoasinduction_lem2v2 != $true ) )
& ( ( ( sub @ sK526 @ id )
!= sK526 )
| ! [X22: subst > term > term] :
( ( sub @ sK527 @ ( push @ ( sK528 @ X22 ) @ ( sK529 @ X22 ) ) )
!= ( X22 @ ( sK529 @ X22 ) @ ( sK528 @ X22 ) ) )
| ( $true = sP106 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( ! [X25: term] :
( ( sub @ X25 @ id )
= X25 )
& ! [X26: term,X28: term,X29: subst] :
( ( sub @ X26 @ ( push @ X28 @ X29 ) )
= ( sK530 @ X26 @ X29 @ X28 ) )
& ( $true != sP106 ) ) )
& ( ( hoasinduction_lem0 = $true )
| ! [X31: term > $o] :
( ( X31 @ ( sK532 @ X31 ) )
!= ( sK531 @ id @ ( sK532 @ X31 ) @ id ) ) )
& ( ! [X33: subst > term > subst > $o,X35: term] :
( ( sK533 @ X33 @ X35 )
= ( X33 @ id @ X35 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ! [X37: subst > term > term] :
( ( X37 @ sh @ one )
!= ( sub @ sK534 @ ( push @ one @ sh ) ) ) )
& ( ! [X38: term] :
( ( sK535 @ X38 @ sh @ one )
= ( sub @ X38 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( induction2lem_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X40: subst,X41: term] :
( ( comp @ sh @ ( push @ X41 @ X40 ) )
= X40 )
| ( axshiftcons != $true ) )
& ( ( axshiftcons = $true )
| ( ( comp @ sh @ ( push @ sK537 @ sK536 ) )
!= sK536 ) )
& ( ! [X44: term,X45: subst > term > subst > $o] :
( ( ( sP103 @ X45 )
= $true )
| ( ( X45 @ id @ X44 @ id )
= $true )
| ( ( sP104 @ X45 )
= $true ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
| ( ( $true
!= ( sP103 @ sK539 ) )
& ( ( sK539 @ id @ sK538 @ id )
!= $true )
& ( $true
!= ( sP104 @ sK539 ) ) ) )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP102 ) )
& ( ( $true = sP102 )
| ( hoasinduction_lem2 != $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( $true = sP99 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP99 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
| ! [X51: term > $o] :
( ( X51 @ ( sK543 @ X51 ) )
!= ( sK541 @ ( sub @ ( sK543 @ X51 ) @ ( push @ sK540 @ sK542 ) ) ) ) )
& ( ! [X53: term,X54: term > $o,X55: subst,X57: term] :
( ( X54 @ ( sub @ X57 @ ( push @ X53 @ X55 ) ) )
= ( sK544 @ X55 @ X54 @ X53 @ X57 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X58: subst] :
( ( comp @ id @ X58 )
= X58 )
& ! [X59: subst] :
( ( comp @ X59 @ id )
= X59 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ( sK545
!= ( comp @ id @ sK545 ) )
| ( ( comp @ sK546 @ id )
!= sK546 ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lthm = $true )
& ! [X62: d_term] : ( d_one = X62 )
& ( lamnotvar = $true )
& ( ( induction = $true )
| ( sP97 != $true ) )
& ( ( sP97 = $true )
| ( induction != $true ) )
& ( ! [X63: term,X64: term] :
( ( ( sub @ X63 @ sh )
!= ( sub @ X64 @ sh ) )
| ( X63 = X64 ) )
| ( shinj != $true ) )
& ( ( shinj = $true )
| ( ( ( sub @ sK547 @ sh )
= ( sub @ sK548 @ sh ) )
& ( sK548 != sK547 ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X67: subst > term > subst > $o,X69: term] :
( ( X67 @ id @ X69 @ id )
= ( sK549 @ X67 @ X69 ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ! [X71: term > $o] :
( ( sK550 @ id @ ( sK551 @ X71 ) @ id )
!= ( X71 @ ( sK551 @ X71 ) ) ) )
& ( ( sP94 = $true )
| ( hoasinduction_lem3aa != $true ) )
& ( ( hoasinduction_lem3aa = $true )
| ( sP94 != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoasinduction = $true )
| ( sP92 != $true ) )
& ( ( sP92 = $true )
| ( hoasinduction != $true ) )
& ( ( sP87 = $true )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( sP87 != $true ) )
& ( lamnotap != $true )
& ( ( apinj2 = $true )
| ( ( ( ap @ sK554 @ sK552 )
= ( ap @ sK555 @ sK553 ) )
& ( sK553 != sK552 ) ) )
& ( ! [X77: term,X78: term,X79: term,X80: term] :
( ( ( ap @ X80 @ X78 )
!= ( ap @ X79 @ X77 ) )
| ( X77 = X78 ) )
| ( apinj2 != $true ) )
& ( ( termmset = $true )
| ( ( sub @ sK556 @ id )
!= sK556 ) )
& ( ! [X82: term] :
( ( sub @ X82 @ id )
= X82 )
| ( termmset != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ulamvar1 != $true )
& ( axapp = $true )
& ! [X83: subst] :
( ( d2subst @ ( sK557 @ X83 ) )
= X83 )
& ( ! [X85: term,X86: term > $o,X87: term > $o,X88: term,X89: subst] :
( ( ( X87 @ ( sub @ X85 @ ( push @ X88 @ X89 ) ) )
= $true )
| ( ( X86 @ X85 )
!= $true ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ( ( ( sK560 @ ( sub @ sK558 @ ( push @ sK561 @ sK562 ) ) )
!= $true )
& ( ( sK559 @ sK558 )
= $true ) ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ! [X95: d_subst] : ( d_id = X95 )
& ! [X96: d_term,X97: d_term] :
( ( X96 = X97 )
| ( ( d2term @ X97 )
!= ( d2term @ X96 ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( sK563
!= ( sub @ sK563 @ id ) )
| ( sP84 = $true )
| ! [X100: term > $o] :
( ( sK564 @ id @ ( sK565 @ X100 ) @ id )
!= ( X100 @ ( sK565 @ X100 ) ) )
| ( sP83 = $true )
| ( sP85 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( ! [X102: term] :
( ( sub @ X102 @ id )
= X102 )
& ( sP84 != $true )
& ! [X103: subst > term > subst > $o,X105: term] :
( ( sK566 @ X103 @ X105 )
= ( X103 @ id @ X105 @ id ) )
& ( sP83 != $true )
& ( sP85 != $true ) ) )
& ( axclos = $true )
& ( ( laminj = $true )
| ( ( sK567 != sK568 )
& ( ( lam @ sK567 )
= ( lam @ sK568 ) ) ) )
& ( ! [X108: term,X109: term] :
( ( X108 = X109 )
| ( ( lam @ X109 )
!= ( lam @ X108 ) ) )
| ( laminj != $true ) )
& ( ( sP76 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP76 != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( sP74 != $true ) )
& ( ( sP74 = $true )
| ( hoasapinj2_lthm != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( hoasinduction_lem3_lthm = $true )
| ( ( sP72 != $true )
& ! [X110: term] :
( ( sub @ X110 @ id )
= X110 )
& ( sP71 != $true ) ) )
& ( ( sP72 = $true )
| ( sK569
!= ( sub @ sK569 @ id ) )
| ( sP71 = $true )
| ( hoasinduction_lem3_lthm != $true ) )
& ( hoaslamnotap_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( pushprop_lthm_orig = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( ! [X112: term] :
( ( sub @ X112 @ id )
= X112 )
& ( $true != sP65 )
& ( $true != sP66 ) ) )
& ( ( ( sub @ sK570 @ id )
!= sK570 )
| ( $true = sP65 )
| ( $true = sP66 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ! [X114: term,X116: term,X117: subst] :
( ( sub @ X114 @ ( push @ X116 @ X117 ) )
= ( sK571 @ X114 @ X117 @ X116 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ! [X119: subst > term > term] :
( ( X119 @ ( sK574 @ X119 ) @ ( sK573 @ X119 ) )
!= ( sub @ sK572 @ ( push @ ( sK573 @ X119 ) @ ( sK574 @ X119 ) ) ) ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP62 != $true ) )
& ( ( sP62 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( ! [X122: term] :
( ( sub @ X122 @ id )
= X122 )
& ( sP58 != $true ) ) )
& ( ( ( sub @ sK575 @ id )
!= sK575 )
| ( sP58 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lthm = $true )
| ( ( sP54 != $true )
& ( $true != sP53 )
& ( sP52 != $true )
& ( sP51 != $true ) ) )
& ( ( sP54 = $true )
| ( $true = sP53 )
| ( sP52 = $true )
| ( sP51 = $true )
| ( hoasinduction_lthm != $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasapinj1_gthm = $true )
& ! [X124: term] :
( ( d2term @ ( sK576 @ X124 ) )
= X124 )
& ( ( induction2lem_lthm = $true )
| ( ! [X126: subst,X127: term] :
( ( comp @ sh @ ( push @ X127 @ X126 ) )
= X126 )
& ! [X128: term,X129: subst] :
( ( sub @ one @ ( push @ X128 @ X129 ) )
= X128 )
& ( sP39 != $true )
& ( sP38 != $true )
& ! [X130: subst] :
( ( comp @ X130 @ id )
= X130 ) ) )
& ( ( sK577
!= ( comp @ sh @ ( push @ sK578 @ sK577 ) ) )
| ( sK579
!= ( sub @ one @ ( push @ sK579 @ sK580 ) ) )
| ( sP39 = $true )
| ( sP38 = $true )
| ( sK581
!= ( comp @ sK581 @ id ) )
| ( induction2lem_lthm != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj2 = $true )
| ( ( sK583 != sK585 )
& ( ( ap @ ( sub @ sK582 @ id ) @ sK583 )
= ( ap @ ( sub @ sK584 @ id ) @ sK585 ) ) ) )
& ( ! [X140: term,X141: term,X142: term,X143: term] :
( ( X141 = X143 )
| ( ( ap @ ( sub @ X142 @ id ) @ X143 )
!= ( ap @ ( sub @ X140 @ id ) @ X141 ) ) )
| ( hoasapinj2 != $true ) )
& ( hoasinduction_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X144: subst] :
( ( comp @ id @ X144 )
= X144 )
| ( axidl != $true ) )
& ( ( axidl = $true )
| ( sK586
!= ( comp @ id @ sK586 ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
| ( sP33 != $true ) )
& ( ( sP33 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( ( $true = sP31 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP31 ) )
& ( ! [X146: subst,X147: term] :
( ( sub @ one @ ( push @ X147 @ X146 ) )
= X147 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ( sK588
!= ( sub @ one @ ( push @ sK588 @ sK587 ) ) ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_lthm_3 = $true )
| ( ( sP26 != $true )
& ( $true != sP28 )
& ! [X150: subst > term > subst > $o,X152: term] :
( ( sK589 @ X150 @ X152 )
= ( X150 @ id @ X152 @ id ) )
& ! [X153: term] :
( ( sub @ X153 @ id )
= X153 )
& ( sP27 != $true ) ) )
& ( ( sP26 = $true )
| ( $true = sP28 )
| ! [X155: term > $o] :
( ( X155 @ ( sK591 @ X155 ) )
!= ( sK590 @ id @ ( sK591 @ X155 ) @ id ) )
| ( sK592
!= ( sub @ sK592 @ id ) )
| ( sP27 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3 = $true )
| ( sP18 != $true ) )
& ( ( sP18 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ! [X158: term] :
( ( sub @ X158 @ id )
= X158 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ( ( sub @ sK593 @ id )
!= sK593 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ( termmset_lthm = $true )
| ( ! [X160: term] :
( ( sub @ X160 @ id )
= X160 )
& ( sK594
!= ( sub @ sK594 @ id ) ) ) )
& ( ( sK595
!= ( sub @ sK595 @ id ) )
| ! [X163: term] :
( ( sub @ X163 @ id )
= X163 )
| ( termmset_lthm != $true ) )
& ( ! [X164: term,X165: subst > term > term,X166: term] :
( ( ( ap @ ( sub @ X166 @ id ) @ X164 )
!= ( lam @ ( X165 @ sh @ one ) ) )
| ( ( sub @ ( X165 @ ( sK598 @ X165 ) @ ( sK596 @ X165 ) ) @ ( sK597 @ X165 ) )
!= ( X165 @ ( comp @ ( sK598 @ X165 ) @ ( sK597 @ X165 ) ) @ ( sub @ ( sK596 @ X165 ) @ ( sK597 @ X165 ) ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoaslamnotap = $true )
| ( ( ( ap @ ( sub @ sK601 @ id ) @ sK599 )
= ( lam @ ( sK600 @ sh @ one ) ) )
& ! [X173: term,X174: subst,X175: subst] :
( ( sub @ ( sK600 @ X175 @ X173 ) @ X174 )
= ( sK600 @ ( comp @ X175 @ X174 ) @ ( sub @ X173 @ X174 ) ) ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop = $true )
& ( axmap = $true )
& ( ( pushprop_lem1v2 = $true )
| ( ( ( sK604 @ sK602 )
= $true )
& ! [X180: term] :
( ( sK603 @ X180 )
= ( sK604 @ ( sub @ X180 @ ( push @ sK602 @ sK605 ) ) ) )
& ( $true
!= ( sK603 @ one ) ) ) )
& ( ! [X181: term,X182: term > $o,X183: term > $o,X184: subst] :
( ( ( X183 @ X181 )
!= $true )
| ( ( X182 @ ( sK606 @ X184 @ X183 @ X182 @ X181 ) )
!= ( X183 @ ( sub @ ( sK606 @ X184 @ X183 @ X182 @ X181 ) @ ( push @ X181 @ X184 ) ) ) )
| ( $true
= ( X182 @ one ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ! [X186: term] :
( ( sK607 @ X186 @ sh @ one )
= ( sub @ X186 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ! [X189: subst > term > term] :
( ( X189 @ sh @ one )
!= ( sub @ sK608 @ ( push @ one @ sh ) ) ) )
& ( ! [X190: subst] :
( ( push @ ( sub @ one @ X190 ) @ ( comp @ sh @ X190 ) )
= X190 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ( sK609
!= ( push @ ( sub @ one @ sK609 ) @ ( comp @ sh @ sK609 ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( induction2lem = $true )
| ( $true != sP14 ) )
& ( ( $true = sP14 )
| ( induction2lem != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( axassoc = $true )
& ( hoaslamnotvar = $true )
& ( ! [X192: term,X193: subst > term > term,X194: subst,X195: subst > term > term] :
( ( ( sub @ ( X193 @ ( sK610 @ X193 ) @ ( sK611 @ X193 ) ) @ ( sK612 @ X193 ) )
!= ( X193 @ ( comp @ ( sK610 @ X193 ) @ ( sK612 @ X193 ) ) @ ( sub @ ( sK611 @ X193 ) @ ( sK612 @ X193 ) ) ) )
| ( ( sub @ ( X195 @ ( sK613 @ X195 ) @ ( sK614 @ X195 ) ) @ ( sK615 @ X195 ) )
!= ( X195 @ ( comp @ ( sK613 @ X195 ) @ ( sK615 @ X195 ) ) @ ( sub @ ( sK614 @ X195 ) @ ( sK615 @ X195 ) ) ) )
| ( ( X193 @ X194 @ X192 )
= ( X195 @ X194 @ X192 ) )
| ( ( lam @ ( X195 @ sh @ one ) )
!= ( lam @ ( X193 @ sh @ one ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ( ! [X206: subst,X207: term,X208: subst] :
( ( sK617 @ ( comp @ X206 @ X208 ) @ ( sub @ X207 @ X208 ) )
= ( sub @ ( sK617 @ X206 @ X207 ) @ X208 ) )
& ! [X209: subst,X210: term,X211: subst] :
( ( sK619 @ ( comp @ X209 @ X211 ) @ ( sub @ X210 @ X211 ) )
= ( sub @ ( sK619 @ X209 @ X210 ) @ X211 ) )
& ( ( sK619 @ sK618 @ sK616 )
!= ( sK617 @ sK618 @ sK616 ) )
& ( ( lam @ ( sK617 @ sh @ one ) )
= ( lam @ ( sK619 @ sh @ one ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( axvarshift = $true )
& ( ( hoasinduction_lem3aaa = $true )
| ( sP11 != $true ) )
& ( ( sP11 = $true )
| ( hoasinduction_lem3aaa != $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( induction2_gthm = $true )
& ( ! [X212: term > $o,X213: term,X214: subst,X216: term] :
( ( sK620 @ X214 @ X213 @ X212 @ X216 )
= ( X212 @ ( sub @ X216 @ ( push @ X213 @ X214 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ! [X220: term > $o] :
( ( sK621 @ ( sub @ ( sK624 @ X220 ) @ ( push @ sK622 @ sK623 ) ) )
!= ( X220 @ ( sK624 @ X220 ) ) ) )
& ( ulamvarind = $true )
& ( ( induction2_lthm = $true )
| ( ( sP9 != $true )
& ! [X222: term] :
( ( sub @ X222 @ id )
= X222 )
& ( sP8 != $true ) ) )
& ( ( sP9 = $true )
| ( sK625
!= ( sub @ sK625 @ id ) )
| ( sP8 = $true )
| ( induction2_lthm != $true ) )
& ( axabs = $true )
& ( ( apinj1 = $true )
| ( ( sK627 != sK629 )
& ( ( ap @ sK629 @ sK628 )
= ( ap @ sK627 @ sK626 ) ) ) )
& ( ! [X228: term,X229: term,X230: term,X231: term] :
( ( X229 = X231 )
| ( ( ap @ X229 @ X228 )
!= ( ap @ X231 @ X230 ) ) )
| ( apinj1 != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X232: term > $o,X233: term > $o,X234: subst,X235: term] :
( ( ( X233 @ X235 )
!= $true )
| ( ( X233 @ ( sub @ ( sK630 @ X235 @ X234 @ X233 @ X232 ) @ ( push @ X235 @ X234 ) ) )
!= ( X232 @ ( sK630 @ X235 @ X234 @ X233 @ X232 ) ) )
| ( ( X232 @ one )
= $true ) )
| ( ( sub @ one @ ( push @ sK631 @ sK632 ) )
!= sK631 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ( ( sK634 @ sK636 )
= $true )
& ! [X243: term] :
( ( sK633 @ X243 )
= ( sK634 @ ( sub @ X243 @ ( push @ sK636 @ sK635 ) ) ) )
& ( $true
!= ( sK633 @ one ) )
& ! [X244: term,X245: subst] :
( ( sub @ one @ ( push @ X244 @ X245 ) )
= X244 ) ) )
& ( ( d2subst @ d_id )
= id )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X246: subst] :
( ( comp @ X246 @ id )
= X246 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ( sK637
!= ( comp @ sK637 @ id ) ) )
& ( ! [X248: term,X250: term,X251: subst] :
( ( sK638 @ X248 @ X251 @ X250 )
= ( sub @ X248 @ ( push @ X250 @ X251 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ! [X253: subst > term > term] :
( ( X253 @ ( sK641 @ X253 ) @ ( sK640 @ X253 ) )
!= ( sub @ sK639 @ ( push @ ( sK640 @ X253 ) @ ( sK641 @ X253 ) ) ) ) )
& ( ( hoasapinj1 = $true )
| ( ( sK643 != sK644 )
& ( ( ap @ ( sub @ sK643 @ id ) @ sK642 )
= ( ap @ ( sub @ sK644 @ id ) @ sK645 ) ) ) )
& ( ! [X260: term,X261: term,X262: term,X263: term] :
( ( X261 = X262 )
| ( ( ap @ ( sub @ X261 @ id ) @ X260 )
!= ( ap @ ( sub @ X262 @ id ) @ X263 ) ) )
| ( hoasapinj1 != $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( $true = sP2 )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( $true != sP2 ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP0 )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( $true != sP0 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[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,sK591,sK592,sK593,sK594,sK595,sK596,sK597,sK598,sK599,sK600,sK601,sK602,sK603,sK604,sK605,sK606,sK607,sK608,sK609,sK610,sK611,sK612,sK613,sK614,sK615,sK616,sK617,sK618,sK619,sK620,sK621,sK622,sK623,sK624,sK625,sK626,sK627,sK628,sK629,sK630,sK631,sK632,sK633,sK634,sK635,sK636,sK637,sK638,sK639,sK640,sK641,sK642,sK643,sK644,sK645])],[f673,f748,f747,f746,f745,f744,f743,f742,f741,f740,f739,f738,f737,f736,f735,f734,f733,f732,f731,f730,f729,f728,f727,f726,f725,f724,f723,f722,f721,f720,f719,f718,f717,f716,f715,f714,f713,f712,f711,f710,f709,f708,f707,f706,f705,f704,f703,f702,f701,f700,f699,f698,f697,f696,f695,f694,f693,f692,f691,f690,f689,f688,f687,f686,f685,f684,f683,f682,f681,f680,f679,f678,f677,f676,f675,f674]) ).
thf(f674,plain,
( ? [X0: term > $o,X1: term,X2: subst,X3: term,X4: term > $o] :
( ( $true
!= ( X0 @ ( sub @ X1 @ ( push @ X3 @ X2 ) ) ) )
& ( $true
= ( X4 @ X1 ) ) )
=> ( ( ( sK517 @ ( sub @ sK518 @ ( push @ sK520 @ sK519 ) ) )
!= $true )
& ( ( sK521 @ sK518 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f675,plain,
( ? [X14: subst,X15: term] :
( ( sub @ one @ ( push @ X15 @ X14 ) )
!= X15 )
=> ( sK523
!= ( sub @ one @ ( push @ sK523 @ sK522 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f676,plain,
( ? [X16: subst,X17: term] :
( ( comp @ sh @ ( push @ X17 @ X16 ) )
!= X16 )
=> ( ( comp @ sh @ ( push @ sK525 @ sK524 ) )
!= sK524 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f677,plain,
( ? [X20: term] :
( ( sub @ X20 @ id )
!= X20 )
=> ( ( sub @ sK526 @ id )
!= sK526 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f678,plain,
( ? [X21: term] :
! [X22: subst > term > term] :
? [X23: term,X24: subst] :
( ( sub @ X21 @ ( push @ X23 @ X24 ) )
!= ( X22 @ X24 @ X23 ) )
=> ! [X22: subst > term > term] :
? [X24: subst,X23: term] :
( ( sub @ sK527 @ ( push @ X23 @ X24 ) )
!= ( X22 @ X24 @ X23 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f679,plain,
! [X22: subst > term > term] :
( ? [X24: subst,X23: term] :
( ( sub @ sK527 @ ( push @ X23 @ X24 ) )
!= ( X22 @ X24 @ X23 ) )
=> ( ( sub @ sK527 @ ( push @ ( sK528 @ X22 ) @ ( sK529 @ X22 ) ) )
!= ( X22 @ ( sK529 @ X22 ) @ ( sK528 @ X22 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f680,plain,
! [X26: term] :
( ? [X27: subst > term > term] :
! [X28: term,X29: subst] :
( ( sub @ X26 @ ( push @ X28 @ X29 ) )
= ( X27 @ X29 @ X28 ) )
=> ! [X29: subst,X28: term] :
( ( sub @ X26 @ ( push @ X28 @ X29 ) )
= ( sK530 @ X26 @ X29 @ X28 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f681,plain,
( ? [X30: subst > term > subst > $o] :
! [X31: term > $o] :
? [X32: term] :
( ( X31 @ X32 )
!= ( X30 @ id @ X32 @ id ) )
=> ! [X31: term > $o] :
? [X32: term] :
( ( X31 @ X32 )
!= ( sK531 @ id @ X32 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f682,plain,
! [X31: term > $o] :
( ? [X32: term] :
( ( X31 @ X32 )
!= ( sK531 @ id @ X32 @ id ) )
=> ( ( X31 @ ( sK532 @ X31 ) )
!= ( sK531 @ id @ ( sK532 @ X31 ) @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f683,plain,
! [X33: subst > term > subst > $o] :
( ? [X34: term > $o] :
! [X35: term] :
( ( X33 @ id @ X35 @ id )
= ( X34 @ X35 ) )
=> ! [X35: term] :
( ( sK533 @ X33 @ X35 )
= ( X33 @ id @ X35 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f684,plain,
( ? [X36: term] :
! [X37: subst > term > term] :
( ( sub @ X36 @ ( push @ one @ sh ) )
!= ( X37 @ sh @ one ) )
=> ! [X37: subst > term > term] :
( ( X37 @ sh @ one )
!= ( sub @ sK534 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f685,plain,
! [X38: term] :
( ? [X39: subst > term > term] :
( ( X39 @ sh @ one )
= ( sub @ X38 @ ( push @ one @ sh ) ) )
=> ( ( sK535 @ X38 @ sh @ one )
= ( sub @ X38 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f686,plain,
( ? [X42: subst,X43: term] :
( ( comp @ sh @ ( push @ X43 @ X42 ) )
!= X42 )
=> ( ( comp @ sh @ ( push @ sK537 @ sK536 ) )
!= sK536 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f687,plain,
( ? [X46: term,X47: subst > term > subst > $o] :
( ( ( sP103 @ X47 )
!= $true )
& ( $true
!= ( X47 @ id @ X46 @ id ) )
& ( $true
!= ( sP104 @ X47 ) ) )
=> ( ( $true
!= ( sP103 @ sK539 ) )
& ( ( sK539 @ id @ sK538 @ id )
!= $true )
& ( $true
!= ( sP104 @ sK539 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f688,plain,
( ? [X48: term,X49: term > $o,X50: subst] :
! [X51: term > $o] :
? [X52: term] :
( ( X51 @ X52 )
!= ( X49 @ ( sub @ X52 @ ( push @ X48 @ X50 ) ) ) )
=> ! [X51: term > $o] :
? [X52: term] :
( ( sK541 @ ( sub @ X52 @ ( push @ sK540 @ sK542 ) ) )
!= ( X51 @ X52 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f689,plain,
! [X51: term > $o] :
( ? [X52: term] :
( ( sK541 @ ( sub @ X52 @ ( push @ sK540 @ sK542 ) ) )
!= ( X51 @ X52 ) )
=> ( ( X51 @ ( sK543 @ X51 ) )
!= ( sK541 @ ( sub @ ( sK543 @ X51 ) @ ( push @ sK540 @ sK542 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f690,plain,
! [X53: term,X54: term > $o,X55: subst] :
( ? [X56: term > $o] :
! [X57: term] :
( ( X54 @ ( sub @ X57 @ ( push @ X53 @ X55 ) ) )
= ( X56 @ X57 ) )
=> ! [X57: term] :
( ( X54 @ ( sub @ X57 @ ( push @ X53 @ X55 ) ) )
= ( sK544 @ X55 @ X54 @ X53 @ X57 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f691,plain,
( ? [X60: subst] :
( ( comp @ id @ X60 )
!= X60 )
=> ( sK545
!= ( comp @ id @ sK545 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f692,plain,
( ? [X61: subst] :
( ( comp @ X61 @ id )
!= X61 )
=> ( ( comp @ sK546 @ id )
!= sK546 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f693,plain,
( ? [X65: term,X66: term] :
( ( ( sub @ X65 @ sh )
= ( sub @ X66 @ sh ) )
& ( X65 != X66 ) )
=> ( ( ( sub @ sK547 @ sh )
= ( sub @ sK548 @ sh ) )
& ( sK548 != sK547 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f694,plain,
! [X67: subst > term > subst > $o] :
( ? [X68: term > $o] :
! [X69: term] :
( ( X67 @ id @ X69 @ id )
= ( X68 @ X69 ) )
=> ! [X69: term] :
( ( X67 @ id @ X69 @ id )
= ( sK549 @ X67 @ X69 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f695,plain,
( ? [X70: subst > term > subst > $o] :
! [X71: term > $o] :
? [X72: term] :
( ( X70 @ id @ X72 @ id )
!= ( X71 @ X72 ) )
=> ! [X71: term > $o] :
? [X72: term] :
( ( X71 @ X72 )
!= ( sK550 @ id @ X72 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f696,plain,
! [X71: term > $o] :
( ? [X72: term] :
( ( X71 @ X72 )
!= ( sK550 @ id @ X72 @ id ) )
=> ( ( sK550 @ id @ ( sK551 @ X71 ) @ id )
!= ( X71 @ ( sK551 @ X71 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f697,plain,
( ? [X73: term,X74: term,X75: term,X76: term] :
( ( ( ap @ X75 @ X73 )
= ( ap @ X76 @ X74 ) )
& ( X73 != X74 ) )
=> ( ( ( ap @ sK554 @ sK552 )
= ( ap @ sK555 @ sK553 ) )
& ( sK553 != sK552 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f698,plain,
( ? [X81: term] :
( ( sub @ X81 @ id )
!= X81 )
=> ( ( sub @ sK556 @ id )
!= sK556 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f699,plain,
! [X83: subst] :
( ? [X84: d_subst] :
( ( d2subst @ X84 )
= X83 )
=> ( ( d2subst @ ( sK557 @ X83 ) )
= X83 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f700,plain,
( ? [X90: term,X91: term > $o,X92: term > $o,X93: term,X94: subst] :
( ( $true
!= ( X92 @ ( sub @ X90 @ ( push @ X93 @ X94 ) ) ) )
& ( ( X91 @ X90 )
= $true ) )
=> ( ( ( sK560 @ ( sub @ sK558 @ ( push @ sK561 @ sK562 ) ) )
!= $true )
& ( ( sK559 @ sK558 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f701,plain,
( ? [X98: term] :
( ( sub @ X98 @ id )
!= X98 )
=> ( sK563
!= ( sub @ sK563 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f702,plain,
( ? [X99: subst > term > subst > $o] :
! [X100: term > $o] :
? [X101: term] :
( ( X100 @ X101 )
!= ( X99 @ id @ X101 @ id ) )
=> ! [X100: term > $o] :
? [X101: term] :
( ( X100 @ X101 )
!= ( sK564 @ id @ X101 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f703,plain,
! [X100: term > $o] :
( ? [X101: term] :
( ( X100 @ X101 )
!= ( sK564 @ id @ X101 @ id ) )
=> ( ( sK564 @ id @ ( sK565 @ X100 ) @ id )
!= ( X100 @ ( sK565 @ X100 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f704,plain,
! [X103: subst > term > subst > $o] :
( ? [X104: term > $o] :
! [X105: term] :
( ( X104 @ X105 )
= ( X103 @ id @ X105 @ id ) )
=> ! [X105: term] :
( ( sK566 @ X103 @ X105 )
= ( X103 @ id @ X105 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f705,plain,
( ? [X106: term,X107: term] :
( ( X106 != X107 )
& ( ( lam @ X106 )
= ( lam @ X107 ) ) )
=> ( ( sK567 != sK568 )
& ( ( lam @ sK567 )
= ( lam @ sK568 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f706,plain,
( ? [X111: term] :
( ( sub @ X111 @ id )
!= X111 )
=> ( sK569
!= ( sub @ sK569 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f707,plain,
( ? [X113: term] :
( ( sub @ X113 @ id )
!= X113 )
=> ( ( sub @ sK570 @ id )
!= sK570 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f708,plain,
! [X114: term] :
( ? [X115: subst > term > term] :
! [X116: term,X117: subst] :
( ( sub @ X114 @ ( push @ X116 @ X117 ) )
= ( X115 @ X117 @ X116 ) )
=> ! [X117: subst,X116: term] :
( ( sub @ X114 @ ( push @ X116 @ X117 ) )
= ( sK571 @ X114 @ X117 @ X116 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f709,plain,
( ? [X118: term] :
! [X119: subst > term > term] :
? [X120: term,X121: subst] :
( ( X119 @ X121 @ X120 )
!= ( sub @ X118 @ ( push @ X120 @ X121 ) ) )
=> ! [X119: subst > term > term] :
? [X121: subst,X120: term] :
( ( sub @ sK572 @ ( push @ X120 @ X121 ) )
!= ( X119 @ X121 @ X120 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f710,plain,
! [X119: subst > term > term] :
( ? [X121: subst,X120: term] :
( ( sub @ sK572 @ ( push @ X120 @ X121 ) )
!= ( X119 @ X121 @ X120 ) )
=> ( ( X119 @ ( sK574 @ X119 ) @ ( sK573 @ X119 ) )
!= ( sub @ sK572 @ ( push @ ( sK573 @ X119 ) @ ( sK574 @ X119 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f711,plain,
( ? [X123: term] :
( ( sub @ X123 @ id )
!= X123 )
=> ( ( sub @ sK575 @ id )
!= sK575 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f712,plain,
! [X124: term] :
( ? [X125: d_term] :
( ( d2term @ X125 )
= X124 )
=> ( ( d2term @ ( sK576 @ X124 ) )
= X124 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f713,plain,
( ? [X131: subst,X132: term] :
( ( comp @ sh @ ( push @ X132 @ X131 ) )
!= X131 )
=> ( sK577
!= ( comp @ sh @ ( push @ sK578 @ sK577 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f714,plain,
( ? [X133: term,X134: subst] :
( ( sub @ one @ ( push @ X133 @ X134 ) )
!= X133 )
=> ( sK579
!= ( sub @ one @ ( push @ sK579 @ sK580 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f715,plain,
( ? [X135: subst] :
( ( comp @ X135 @ id )
!= X135 )
=> ( sK581
!= ( comp @ sK581 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f716,plain,
( ? [X136: term,X137: term,X138: term,X139: term] :
( ( X137 != X139 )
& ( ( ap @ ( sub @ X136 @ id ) @ X137 )
= ( ap @ ( sub @ X138 @ id ) @ X139 ) ) )
=> ( ( sK583 != sK585 )
& ( ( ap @ ( sub @ sK582 @ id ) @ sK583 )
= ( ap @ ( sub @ sK584 @ id ) @ sK585 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f717,plain,
( ? [X145: subst] :
( ( comp @ id @ X145 )
!= X145 )
=> ( sK586
!= ( comp @ id @ sK586 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f718,plain,
( ? [X148: subst,X149: term] :
( ( sub @ one @ ( push @ X149 @ X148 ) )
!= X149 )
=> ( sK588
!= ( sub @ one @ ( push @ sK588 @ sK587 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f719,plain,
! [X150: subst > term > subst > $o] :
( ? [X151: term > $o] :
! [X152: term] :
( ( X151 @ X152 )
= ( X150 @ id @ X152 @ id ) )
=> ! [X152: term] :
( ( sK589 @ X150 @ X152 )
= ( X150 @ id @ X152 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f720,plain,
( ? [X154: subst > term > subst > $o] :
! [X155: term > $o] :
? [X156: term] :
( ( X154 @ id @ X156 @ id )
!= ( X155 @ X156 ) )
=> ! [X155: term > $o] :
? [X156: term] :
( ( X155 @ X156 )
!= ( sK590 @ id @ X156 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f721,plain,
! [X155: term > $o] :
( ? [X156: term] :
( ( X155 @ X156 )
!= ( sK590 @ id @ X156 @ id ) )
=> ( ( X155 @ ( sK591 @ X155 ) )
!= ( sK590 @ id @ ( sK591 @ X155 ) @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f722,plain,
( ? [X157: term] :
( ( sub @ X157 @ id )
!= X157 )
=> ( sK592
!= ( sub @ sK592 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f723,plain,
( ? [X159: term] :
( ( sub @ X159 @ id )
!= X159 )
=> ( ( sub @ sK593 @ id )
!= sK593 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f724,plain,
( ? [X161: term] :
( ( sub @ X161 @ id )
!= X161 )
=> ( sK594
!= ( sub @ sK594 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f725,plain,
( ? [X162: term] :
( ( sub @ X162 @ id )
!= X162 )
=> ( sK595
!= ( sub @ sK595 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f726,plain,
! [X165: subst > term > term] :
( ? [X167: term,X168: subst,X169: subst] :
( ( X165 @ ( comp @ X169 @ X168 ) @ ( sub @ X167 @ X168 ) )
!= ( sub @ ( X165 @ X169 @ X167 ) @ X168 ) )
=> ( ( sub @ ( X165 @ ( sK598 @ X165 ) @ ( sK596 @ X165 ) ) @ ( sK597 @ X165 ) )
!= ( X165 @ ( comp @ ( sK598 @ X165 ) @ ( sK597 @ X165 ) ) @ ( sub @ ( sK596 @ X165 ) @ ( sK597 @ X165 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f727,plain,
( ? [X170: term,X171: subst > term > term,X172: term] :
( ( ( lam @ ( X171 @ sh @ one ) )
= ( ap @ ( sub @ X172 @ id ) @ X170 ) )
& ! [X173: term,X174: subst,X175: subst] :
( ( sub @ ( X171 @ X175 @ X173 ) @ X174 )
= ( X171 @ ( comp @ X175 @ X174 ) @ ( sub @ X173 @ X174 ) ) ) )
=> ( ( ( ap @ ( sub @ sK601 @ id ) @ sK599 )
= ( lam @ ( sK600 @ sh @ one ) ) )
& ! [X175: subst,X174: subst,X173: term] :
( ( sub @ ( sK600 @ X175 @ X173 ) @ X174 )
= ( sK600 @ ( comp @ X175 @ X174 ) @ ( sub @ X173 @ X174 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f728,plain,
( ? [X176: term,X177: term > $o,X178: term > $o,X179: subst] :
( ( $true
= ( X178 @ X176 ) )
& ! [X180: term] :
( ( X178 @ ( sub @ X180 @ ( push @ X176 @ X179 ) ) )
= ( X177 @ X180 ) )
& ( ( X177 @ one )
!= $true ) )
=> ( ( ( sK604 @ sK602 )
= $true )
& ! [X180: term] :
( ( sK603 @ X180 )
= ( sK604 @ ( sub @ X180 @ ( push @ sK602 @ sK605 ) ) ) )
& ( $true
!= ( sK603 @ one ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f729,plain,
! [X181: term,X182: term > $o,X183: term > $o,X184: subst] :
( ? [X185: term] :
( ( X182 @ X185 )
!= ( X183 @ ( sub @ X185 @ ( push @ X181 @ X184 ) ) ) )
=> ( ( X182 @ ( sK606 @ X184 @ X183 @ X182 @ X181 ) )
!= ( X183 @ ( sub @ ( sK606 @ X184 @ X183 @ X182 @ X181 ) @ ( push @ X181 @ X184 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f730,plain,
! [X186: term] :
( ? [X187: subst > term > term] :
( ( X187 @ sh @ one )
= ( sub @ X186 @ ( push @ one @ sh ) ) )
=> ( ( sK607 @ X186 @ sh @ one )
= ( sub @ X186 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f731,plain,
( ? [X188: term] :
! [X189: subst > term > term] :
( ( sub @ X188 @ ( push @ one @ sh ) )
!= ( X189 @ sh @ one ) )
=> ! [X189: subst > term > term] :
( ( X189 @ sh @ one )
!= ( sub @ sK608 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f732,plain,
( ? [X191: subst] :
( ( push @ ( sub @ one @ X191 ) @ ( comp @ sh @ X191 ) )
!= X191 )
=> ( sK609
!= ( push @ ( sub @ one @ sK609 ) @ ( comp @ sh @ sK609 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f733,plain,
! [X193: subst > term > term] :
( ? [X196: subst,X197: term,X198: subst] :
( ( sub @ ( X193 @ X196 @ X197 ) @ X198 )
!= ( X193 @ ( comp @ X196 @ X198 ) @ ( sub @ X197 @ X198 ) ) )
=> ( ( sub @ ( X193 @ ( sK610 @ X193 ) @ ( sK611 @ X193 ) ) @ ( sK612 @ X193 ) )
!= ( X193 @ ( comp @ ( sK610 @ X193 ) @ ( sK612 @ X193 ) ) @ ( sub @ ( sK611 @ X193 ) @ ( sK612 @ X193 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f734,plain,
! [X195: subst > term > term] :
( ? [X199: subst,X200: term,X201: subst] :
( ( X195 @ ( comp @ X199 @ X201 ) @ ( sub @ X200 @ X201 ) )
!= ( sub @ ( X195 @ X199 @ X200 ) @ X201 ) )
=> ( ( sub @ ( X195 @ ( sK613 @ X195 ) @ ( sK614 @ X195 ) ) @ ( sK615 @ X195 ) )
!= ( X195 @ ( comp @ ( sK613 @ X195 ) @ ( sK615 @ X195 ) ) @ ( sub @ ( sK614 @ X195 ) @ ( sK615 @ X195 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f735,plain,
( ? [X202: term,X203: subst > term > term,X204: subst,X205: subst > term > term] :
( ! [X206: subst,X207: term,X208: subst] :
( ( sub @ ( X203 @ X206 @ X207 ) @ X208 )
= ( X203 @ ( comp @ X206 @ X208 ) @ ( sub @ X207 @ X208 ) ) )
& ! [X209: subst,X210: term,X211: subst] :
( ( sub @ ( X205 @ X209 @ X210 ) @ X211 )
= ( X205 @ ( comp @ X209 @ X211 ) @ ( sub @ X210 @ X211 ) ) )
& ( ( X203 @ X204 @ X202 )
!= ( X205 @ X204 @ X202 ) )
& ( ( lam @ ( X205 @ sh @ one ) )
= ( lam @ ( X203 @ sh @ one ) ) ) )
=> ( ! [X208: subst,X207: term,X206: subst] :
( ( sK617 @ ( comp @ X206 @ X208 ) @ ( sub @ X207 @ X208 ) )
= ( sub @ ( sK617 @ X206 @ X207 ) @ X208 ) )
& ! [X211: subst,X210: term,X209: subst] :
( ( sK619 @ ( comp @ X209 @ X211 ) @ ( sub @ X210 @ X211 ) )
= ( sub @ ( sK619 @ X209 @ X210 ) @ X211 ) )
& ( ( sK619 @ sK618 @ sK616 )
!= ( sK617 @ sK618 @ sK616 ) )
& ( ( lam @ ( sK617 @ sh @ one ) )
= ( lam @ ( sK619 @ sh @ one ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f736,plain,
! [X212: term > $o,X213: term,X214: subst] :
( ? [X215: term > $o] :
! [X216: term] :
( ( X212 @ ( sub @ X216 @ ( push @ X213 @ X214 ) ) )
= ( X215 @ X216 ) )
=> ! [X216: term] :
( ( sK620 @ X214 @ X213 @ X212 @ X216 )
= ( X212 @ ( sub @ X216 @ ( push @ X213 @ X214 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f737,plain,
( ? [X217: term > $o,X218: term,X219: subst] :
! [X220: term > $o] :
? [X221: term] :
( ( X220 @ X221 )
!= ( X217 @ ( sub @ X221 @ ( push @ X218 @ X219 ) ) ) )
=> ! [X220: term > $o] :
? [X221: term] :
( ( sK621 @ ( sub @ X221 @ ( push @ sK622 @ sK623 ) ) )
!= ( X220 @ X221 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f738,plain,
! [X220: term > $o] :
( ? [X221: term] :
( ( sK621 @ ( sub @ X221 @ ( push @ sK622 @ sK623 ) ) )
!= ( X220 @ X221 ) )
=> ( ( sK621 @ ( sub @ ( sK624 @ X220 ) @ ( push @ sK622 @ sK623 ) ) )
!= ( X220 @ ( sK624 @ X220 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f739,plain,
( ? [X223: term] :
( ( sub @ X223 @ id )
!= X223 )
=> ( sK625
!= ( sub @ sK625 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f740,plain,
( ? [X224: term,X225: term,X226: term,X227: term] :
( ( X225 != X227 )
& ( ( ap @ X225 @ X224 )
= ( ap @ X227 @ X226 ) ) )
=> ( ( sK627 != sK629 )
& ( ( ap @ sK629 @ sK628 )
= ( ap @ sK627 @ sK626 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f741,plain,
! [X232: term > $o,X233: term > $o,X234: subst,X235: term] :
( ? [X236: term] :
( ( X233 @ ( sub @ X236 @ ( push @ X235 @ X234 ) ) )
!= ( X232 @ X236 ) )
=> ( ( X233 @ ( sub @ ( sK630 @ X235 @ X234 @ X233 @ X232 ) @ ( push @ X235 @ X234 ) ) )
!= ( X232 @ ( sK630 @ X235 @ X234 @ X233 @ X232 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f742,plain,
( ? [X237: term,X238: subst] :
( ( sub @ one @ ( push @ X237 @ X238 ) )
!= X237 )
=> ( ( sub @ one @ ( push @ sK631 @ sK632 ) )
!= sK631 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f743,plain,
( ? [X239: term > $o,X240: term > $o,X241: subst,X242: term] :
( ( ( X240 @ X242 )
= $true )
& ! [X243: term] :
( ( X240 @ ( sub @ X243 @ ( push @ X242 @ X241 ) ) )
= ( X239 @ X243 ) )
& ( $true
!= ( X239 @ one ) ) )
=> ( ( ( sK634 @ sK636 )
= $true )
& ! [X243: term] :
( ( sK633 @ X243 )
= ( sK634 @ ( sub @ X243 @ ( push @ sK636 @ sK635 ) ) ) )
& ( $true
!= ( sK633 @ one ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f744,plain,
( ? [X247: subst] :
( ( comp @ X247 @ id )
!= X247 )
=> ( sK637
!= ( comp @ sK637 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f745,plain,
! [X248: term] :
( ? [X249: subst > term > term] :
! [X250: term,X251: subst] :
( ( X249 @ X251 @ X250 )
= ( sub @ X248 @ ( push @ X250 @ X251 ) ) )
=> ! [X251: subst,X250: term] :
( ( sK638 @ X248 @ X251 @ X250 )
= ( sub @ X248 @ ( push @ X250 @ X251 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f746,plain,
( ? [X252: term] :
! [X253: subst > term > term] :
? [X254: term,X255: subst] :
( ( X253 @ X255 @ X254 )
!= ( sub @ X252 @ ( push @ X254 @ X255 ) ) )
=> ! [X253: subst > term > term] :
? [X255: subst,X254: term] :
( ( X253 @ X255 @ X254 )
!= ( sub @ sK639 @ ( push @ X254 @ X255 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f747,plain,
! [X253: subst > term > term] :
( ? [X255: subst,X254: term] :
( ( X253 @ X255 @ X254 )
!= ( sub @ sK639 @ ( push @ X254 @ X255 ) ) )
=> ( ( X253 @ ( sK641 @ X253 ) @ ( sK640 @ X253 ) )
!= ( sub @ sK639 @ ( push @ ( sK640 @ X253 ) @ ( sK641 @ X253 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f748,plain,
( ? [X256: term,X257: term,X258: term,X259: term] :
( ( X257 != X258 )
& ( ( ap @ ( sub @ X257 @ id ) @ X256 )
= ( ap @ ( sub @ X258 @ id ) @ X259 ) ) )
=> ( ( sK643 != sK644 )
& ( ( ap @ ( sub @ sK643 @ id ) @ sK642 )
= ( ap @ ( sub @ sK644 @ id ) @ sK645 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f673,plain,
( ( ( pushprop_lem1_lthm = $true )
| ( ? [X0: term > $o,X1: term,X2: subst,X3: term,X4: term > $o] :
( ( $true
!= ( X0 @ ( sub @ X1 @ ( push @ X3 @ X2 ) ) ) )
& ( $true
= ( X4 @ X1 ) ) )
& ! [X5: subst,X6: term] :
( ( sub @ one @ ( push @ X6 @ X5 ) )
= X6 )
& ! [X7: subst,X8: term] :
( ( comp @ sh @ ( push @ X8 @ X7 ) )
= X7 ) ) )
& ( ! [X9: term > $o,X10: term,X11: subst,X12: term,X13: term > $o] :
( ( ( X9 @ ( sub @ X10 @ ( push @ X12 @ X11 ) ) )
= $true )
| ( ( X13 @ X10 )
!= $true ) )
| ? [X14: subst,X15: term] :
( ( sub @ one @ ( push @ X15 @ X14 ) )
!= X15 )
| ? [X16: subst,X17: term] :
( ( comp @ sh @ ( push @ X17 @ X16 ) )
!= X16 )
| ( pushprop_lem1_lthm != $true ) )
& ! [X18: d_subst,X19: d_subst] :
( ( ( d2subst @ X18 )
!= ( d2subst @ X19 ) )
| ( X18 = X19 ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP109 ) )
& ( ( $true = sP109 )
| ( hoasinduction_lem2v2 != $true ) )
& ( ? [X20: term] :
( ( sub @ X20 @ id )
!= X20 )
| ? [X21: term] :
! [X22: subst > term > term] :
? [X23: term,X24: subst] :
( ( sub @ X21 @ ( push @ X23 @ X24 ) )
!= ( X22 @ X24 @ X23 ) )
| ( $true = sP106 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( ! [X25: term] :
( ( sub @ X25 @ id )
= X25 )
& ! [X26: term] :
? [X27: subst > term > term] :
! [X28: term,X29: subst] :
( ( sub @ X26 @ ( push @ X28 @ X29 ) )
= ( X27 @ X29 @ X28 ) )
& ( $true != sP106 ) ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X30: subst > term > subst > $o] :
! [X31: term > $o] :
? [X32: term] :
( ( X31 @ X32 )
!= ( X30 @ id @ X32 @ id ) ) )
& ( ! [X33: subst > term > subst > $o] :
? [X34: term > $o] :
! [X35: term] :
( ( X33 @ id @ X35 @ id )
= ( X34 @ X35 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X36: term] :
! [X37: subst > term > term] :
( ( sub @ X36 @ ( push @ one @ sh ) )
!= ( X37 @ sh @ one ) ) )
& ( ! [X38: term] :
? [X39: subst > term > term] :
( ( X39 @ sh @ one )
= ( sub @ X38 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( induction2lem_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X40: subst,X41: term] :
( ( comp @ sh @ ( push @ X41 @ X40 ) )
= X40 )
| ( axshiftcons != $true ) )
& ( ( axshiftcons = $true )
| ? [X42: subst,X43: term] :
( ( comp @ sh @ ( push @ X43 @ X42 ) )
!= X42 ) )
& ( ! [X44: term,X45: subst > term > subst > $o] :
( ( ( sP103 @ X45 )
= $true )
| ( ( X45 @ id @ X44 @ id )
= $true )
| ( ( sP104 @ X45 )
= $true ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X46: term,X47: subst > term > subst > $o] :
( ( ( sP103 @ X47 )
!= $true )
& ( $true
!= ( X47 @ id @ X46 @ id ) )
& ( $true
!= ( sP104 @ X47 ) ) ) )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP102 ) )
& ( ( $true = sP102 )
| ( hoasinduction_lem2 != $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( $true = sP99 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP99 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
| ? [X48: term,X49: term > $o,X50: subst] :
! [X51: term > $o] :
? [X52: term] :
( ( X51 @ X52 )
!= ( X49 @ ( sub @ X52 @ ( push @ X48 @ X50 ) ) ) ) )
& ( ! [X53: term,X54: term > $o,X55: subst] :
? [X56: term > $o] :
! [X57: term] :
( ( X54 @ ( sub @ X57 @ ( push @ X53 @ X55 ) ) )
= ( X56 @ X57 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X58: subst] :
( ( comp @ id @ X58 )
= X58 )
& ! [X59: subst] :
( ( comp @ X59 @ id )
= X59 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X60: subst] :
( ( comp @ id @ X60 )
!= X60 )
| ? [X61: subst] :
( ( comp @ X61 @ id )
!= X61 ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lthm = $true )
& ! [X62: d_term] : ( d_one = X62 )
& ( lamnotvar = $true )
& ( ( induction = $true )
| ( sP97 != $true ) )
& ( ( sP97 = $true )
| ( induction != $true ) )
& ( ! [X63: term,X64: term] :
( ( ( sub @ X63 @ sh )
!= ( sub @ X64 @ sh ) )
| ( X63 = X64 ) )
| ( shinj != $true ) )
& ( ( shinj = $true )
| ? [X65: term,X66: term] :
( ( ( sub @ X65 @ sh )
= ( sub @ X66 @ sh ) )
& ( X65 != X66 ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X67: subst > term > subst > $o] :
? [X68: term > $o] :
! [X69: term] :
( ( X67 @ id @ X69 @ id )
= ( X68 @ X69 ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X70: subst > term > subst > $o] :
! [X71: term > $o] :
? [X72: term] :
( ( X70 @ id @ X72 @ id )
!= ( X71 @ X72 ) ) )
& ( ( sP94 = $true )
| ( hoasinduction_lem3aa != $true ) )
& ( ( hoasinduction_lem3aa = $true )
| ( sP94 != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoasinduction = $true )
| ( sP92 != $true ) )
& ( ( sP92 = $true )
| ( hoasinduction != $true ) )
& ( ( sP87 = $true )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( sP87 != $true ) )
& ( lamnotap != $true )
& ( ( apinj2 = $true )
| ? [X73: term,X74: term,X75: term,X76: term] :
( ( ( ap @ X75 @ X73 )
= ( ap @ X76 @ X74 ) )
& ( X73 != X74 ) ) )
& ( ! [X77: term,X78: term,X79: term,X80: term] :
( ( ( ap @ X80 @ X78 )
!= ( ap @ X79 @ X77 ) )
| ( X77 = X78 ) )
| ( apinj2 != $true ) )
& ( ( termmset = $true )
| ? [X81: term] :
( ( sub @ X81 @ id )
!= X81 ) )
& ( ! [X82: term] :
( ( sub @ X82 @ id )
= X82 )
| ( termmset != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ulamvar1 != $true )
& ( axapp = $true )
& ! [X83: subst] :
? [X84: d_subst] :
( ( d2subst @ X84 )
= X83 )
& ( ! [X85: term,X86: term > $o,X87: term > $o,X88: term,X89: subst] :
( ( ( X87 @ ( sub @ X85 @ ( push @ X88 @ X89 ) ) )
= $true )
| ( ( X86 @ X85 )
!= $true ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X90: term,X91: term > $o,X92: term > $o,X93: term,X94: subst] :
( ( $true
!= ( X92 @ ( sub @ X90 @ ( push @ X93 @ X94 ) ) ) )
& ( ( X91 @ X90 )
= $true ) ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ! [X95: d_subst] : ( d_id = X95 )
& ! [X96: d_term,X97: d_term] :
( ( X96 = X97 )
| ( ( d2term @ X97 )
!= ( d2term @ X96 ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ? [X98: term] :
( ( sub @ X98 @ id )
!= X98 )
| ( sP84 = $true )
| ? [X99: subst > term > subst > $o] :
! [X100: term > $o] :
? [X101: term] :
( ( X100 @ X101 )
!= ( X99 @ id @ X101 @ id ) )
| ( sP83 = $true )
| ( sP85 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( ! [X102: term] :
( ( sub @ X102 @ id )
= X102 )
& ( sP84 != $true )
& ! [X103: subst > term > subst > $o] :
? [X104: term > $o] :
! [X105: term] :
( ( X104 @ X105 )
= ( X103 @ id @ X105 @ id ) )
& ( sP83 != $true )
& ( sP85 != $true ) ) )
& ( axclos = $true )
& ( ( laminj = $true )
| ? [X106: term,X107: term] :
( ( X106 != X107 )
& ( ( lam @ X106 )
= ( lam @ X107 ) ) ) )
& ( ! [X108: term,X109: term] :
( ( X108 = X109 )
| ( ( lam @ X109 )
!= ( lam @ X108 ) ) )
| ( laminj != $true ) )
& ( ( sP76 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP76 != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( sP74 != $true ) )
& ( ( sP74 = $true )
| ( hoasapinj2_lthm != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( hoasinduction_lem3_lthm = $true )
| ( ( sP72 != $true )
& ! [X110: term] :
( ( sub @ X110 @ id )
= X110 )
& ( sP71 != $true ) ) )
& ( ( sP72 = $true )
| ? [X111: term] :
( ( sub @ X111 @ id )
!= X111 )
| ( sP71 = $true )
| ( hoasinduction_lem3_lthm != $true ) )
& ( hoaslamnotap_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( pushprop_lthm_orig = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( ! [X112: term] :
( ( sub @ X112 @ id )
= X112 )
& ( $true != sP65 )
& ( $true != sP66 ) ) )
& ( ? [X113: term] :
( ( sub @ X113 @ id )
!= X113 )
| ( $true = sP65 )
| ( $true = sP66 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ! [X114: term] :
? [X115: subst > term > term] :
! [X116: term,X117: subst] :
( ( sub @ X114 @ ( push @ X116 @ X117 ) )
= ( X115 @ X117 @ X116 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X118: term] :
! [X119: subst > term > term] :
? [X120: term,X121: subst] :
( ( X119 @ X121 @ X120 )
!= ( sub @ X118 @ ( push @ X120 @ X121 ) ) ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP62 != $true ) )
& ( ( sP62 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( ! [X122: term] :
( ( sub @ X122 @ id )
= X122 )
& ( sP58 != $true ) ) )
& ( ? [X123: term] :
( ( sub @ X123 @ id )
!= X123 )
| ( sP58 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lthm = $true )
| ( ( sP54 != $true )
& ( $true != sP53 )
& ( sP52 != $true )
& ( sP51 != $true ) ) )
& ( ( sP54 = $true )
| ( $true = sP53 )
| ( sP52 = $true )
| ( sP51 = $true )
| ( hoasinduction_lthm != $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasapinj1_gthm = $true )
& ! [X124: term] :
? [X125: d_term] :
( ( d2term @ X125 )
= X124 )
& ( ( induction2lem_lthm = $true )
| ( ! [X126: subst,X127: term] :
( ( comp @ sh @ ( push @ X127 @ X126 ) )
= X126 )
& ! [X128: term,X129: subst] :
( ( sub @ one @ ( push @ X128 @ X129 ) )
= X128 )
& ( sP39 != $true )
& ( sP38 != $true )
& ! [X130: subst] :
( ( comp @ X130 @ id )
= X130 ) ) )
& ( ? [X131: subst,X132: term] :
( ( comp @ sh @ ( push @ X132 @ X131 ) )
!= X131 )
| ? [X133: term,X134: subst] :
( ( sub @ one @ ( push @ X133 @ X134 ) )
!= X133 )
| ( sP39 = $true )
| ( sP38 = $true )
| ? [X135: subst] :
( ( comp @ X135 @ id )
!= X135 )
| ( induction2lem_lthm != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj2 = $true )
| ? [X136: term,X137: term,X138: term,X139: term] :
( ( X137 != X139 )
& ( ( ap @ ( sub @ X136 @ id ) @ X137 )
= ( ap @ ( sub @ X138 @ id ) @ X139 ) ) ) )
& ( ! [X140: term,X141: term,X142: term,X143: term] :
( ( X141 = X143 )
| ( ( ap @ ( sub @ X142 @ id ) @ X143 )
!= ( ap @ ( sub @ X140 @ id ) @ X141 ) ) )
| ( hoasapinj2 != $true ) )
& ( hoasinduction_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X144: subst] :
( ( comp @ id @ X144 )
= X144 )
| ( axidl != $true ) )
& ( ( axidl = $true )
| ? [X145: subst] :
( ( comp @ id @ X145 )
!= X145 ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
| ( sP33 != $true ) )
& ( ( sP33 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( ( $true = sP31 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP31 ) )
& ( ! [X146: subst,X147: term] :
( ( sub @ one @ ( push @ X147 @ X146 ) )
= X147 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X148: subst,X149: term] :
( ( sub @ one @ ( push @ X149 @ X148 ) )
!= X149 ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_lthm_3 = $true )
| ( ( sP26 != $true )
& ( $true != sP28 )
& ! [X150: subst > term > subst > $o] :
? [X151: term > $o] :
! [X152: term] :
( ( X151 @ X152 )
= ( X150 @ id @ X152 @ id ) )
& ! [X153: term] :
( ( sub @ X153 @ id )
= X153 )
& ( sP27 != $true ) ) )
& ( ( sP26 = $true )
| ( $true = sP28 )
| ? [X154: subst > term > subst > $o] :
! [X155: term > $o] :
? [X156: term] :
( ( X154 @ id @ X156 @ id )
!= ( X155 @ X156 ) )
| ? [X157: term] :
( ( sub @ X157 @ id )
!= X157 )
| ( sP27 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3 = $true )
| ( sP18 != $true ) )
& ( ( sP18 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ! [X158: term] :
( ( sub @ X158 @ id )
= X158 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X159: term] :
( ( sub @ X159 @ id )
!= X159 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ( termmset_lthm = $true )
| ( ! [X160: term] :
( ( sub @ X160 @ id )
= X160 )
& ? [X161: term] :
( ( sub @ X161 @ id )
!= X161 ) ) )
& ( ? [X162: term] :
( ( sub @ X162 @ id )
!= X162 )
| ! [X163: term] :
( ( sub @ X163 @ id )
= X163 )
| ( termmset_lthm != $true ) )
& ( ! [X164: term,X165: subst > term > term,X166: term] :
( ( ( ap @ ( sub @ X166 @ id ) @ X164 )
!= ( lam @ ( X165 @ sh @ one ) ) )
| ? [X167: term,X168: subst,X169: subst] :
( ( X165 @ ( comp @ X169 @ X168 ) @ ( sub @ X167 @ X168 ) )
!= ( sub @ ( X165 @ X169 @ X167 ) @ X168 ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X170: term,X171: subst > term > term,X172: term] :
( ( ( lam @ ( X171 @ sh @ one ) )
= ( ap @ ( sub @ X172 @ id ) @ X170 ) )
& ! [X173: term,X174: subst,X175: subst] :
( ( sub @ ( X171 @ X175 @ X173 ) @ X174 )
= ( X171 @ ( comp @ X175 @ X174 ) @ ( sub @ X173 @ X174 ) ) ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop = $true )
& ( axmap = $true )
& ( ( pushprop_lem1v2 = $true )
| ? [X176: term,X177: term > $o,X178: term > $o,X179: subst] :
( ( $true
= ( X178 @ X176 ) )
& ! [X180: term] :
( ( X178 @ ( sub @ X180 @ ( push @ X176 @ X179 ) ) )
= ( X177 @ X180 ) )
& ( ( X177 @ one )
!= $true ) ) )
& ( ! [X181: term,X182: term > $o,X183: term > $o,X184: subst] :
( ( ( X183 @ X181 )
!= $true )
| ? [X185: term] :
( ( X182 @ X185 )
!= ( X183 @ ( sub @ X185 @ ( push @ X181 @ X184 ) ) ) )
| ( $true
= ( X182 @ one ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( ! [X186: term] :
? [X187: subst > term > term] :
( ( X187 @ sh @ one )
= ( sub @ X186 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X188: term] :
! [X189: subst > term > term] :
( ( sub @ X188 @ ( push @ one @ sh ) )
!= ( X189 @ sh @ one ) ) )
& ( ! [X190: subst] :
( ( push @ ( sub @ one @ X190 ) @ ( comp @ sh @ X190 ) )
= X190 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X191: subst] :
( ( push @ ( sub @ one @ X191 ) @ ( comp @ sh @ X191 ) )
!= X191 ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( induction2lem = $true )
| ( $true != sP14 ) )
& ( ( $true = sP14 )
| ( induction2lem != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( axassoc = $true )
& ( hoaslamnotvar = $true )
& ( ! [X192: term,X193: subst > term > term,X194: subst,X195: subst > term > term] :
( ? [X196: subst,X197: term,X198: subst] :
( ( sub @ ( X193 @ X196 @ X197 ) @ X198 )
!= ( X193 @ ( comp @ X196 @ X198 ) @ ( sub @ X197 @ X198 ) ) )
| ? [X199: subst,X200: term,X201: subst] :
( ( X195 @ ( comp @ X199 @ X201 ) @ ( sub @ X200 @ X201 ) )
!= ( sub @ ( X195 @ X199 @ X200 ) @ X201 ) )
| ( ( X193 @ X194 @ X192 )
= ( X195 @ X194 @ X192 ) )
| ( ( lam @ ( X195 @ sh @ one ) )
!= ( lam @ ( X193 @ sh @ one ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X202: term,X203: subst > term > term,X204: subst,X205: subst > term > term] :
( ! [X206: subst,X207: term,X208: subst] :
( ( sub @ ( X203 @ X206 @ X207 ) @ X208 )
= ( X203 @ ( comp @ X206 @ X208 ) @ ( sub @ X207 @ X208 ) ) )
& ! [X209: subst,X210: term,X211: subst] :
( ( sub @ ( X205 @ X209 @ X210 ) @ X211 )
= ( X205 @ ( comp @ X209 @ X211 ) @ ( sub @ X210 @ X211 ) ) )
& ( ( X203 @ X204 @ X202 )
!= ( X205 @ X204 @ X202 ) )
& ( ( lam @ ( X205 @ sh @ one ) )
= ( lam @ ( X203 @ sh @ one ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( axvarshift = $true )
& ( ( hoasinduction_lem3aaa = $true )
| ( sP11 != $true ) )
& ( ( sP11 = $true )
| ( hoasinduction_lem3aaa != $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( induction2_gthm = $true )
& ( ! [X212: term > $o,X213: term,X214: subst] :
? [X215: term > $o] :
! [X216: term] :
( ( X212 @ ( sub @ X216 @ ( push @ X213 @ X214 ) ) )
= ( X215 @ X216 ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X217: term > $o,X218: term,X219: subst] :
! [X220: term > $o] :
? [X221: term] :
( ( X220 @ X221 )
!= ( X217 @ ( sub @ X221 @ ( push @ X218 @ X219 ) ) ) ) )
& ( ulamvarind = $true )
& ( ( induction2_lthm = $true )
| ( ( sP9 != $true )
& ! [X222: term] :
( ( sub @ X222 @ id )
= X222 )
& ( sP8 != $true ) ) )
& ( ( sP9 = $true )
| ? [X223: term] :
( ( sub @ X223 @ id )
!= X223 )
| ( sP8 = $true )
| ( induction2_lthm != $true ) )
& ( axabs = $true )
& ( ( apinj1 = $true )
| ? [X224: term,X225: term,X226: term,X227: term] :
( ( X225 != X227 )
& ( ( ap @ X225 @ X224 )
= ( ap @ X227 @ X226 ) ) ) )
& ( ! [X228: term,X229: term,X230: term,X231: term] :
( ( X229 = X231 )
| ( ( ap @ X229 @ X228 )
!= ( ap @ X231 @ X230 ) ) )
| ( apinj1 != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X232: term > $o,X233: term > $o,X234: subst,X235: term] :
( ( ( X233 @ X235 )
!= $true )
| ? [X236: term] :
( ( X233 @ ( sub @ X236 @ ( push @ X235 @ X234 ) ) )
!= ( X232 @ X236 ) )
| ( ( X232 @ one )
= $true ) )
| ? [X237: term,X238: subst] :
( ( sub @ one @ ( push @ X237 @ X238 ) )
!= X237 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X239: term > $o,X240: term > $o,X241: subst,X242: term] :
( ( ( X240 @ X242 )
= $true )
& ! [X243: term] :
( ( X240 @ ( sub @ X243 @ ( push @ X242 @ X241 ) ) )
= ( X239 @ X243 ) )
& ( $true
!= ( X239 @ one ) ) )
& ! [X244: term,X245: subst] :
( ( sub @ one @ ( push @ X244 @ X245 ) )
= X244 ) ) )
& ( ( d2subst @ d_id )
= id )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X246: subst] :
( ( comp @ X246 @ id )
= X246 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X247: subst] :
( ( comp @ X247 @ id )
!= X247 ) )
& ( ! [X248: term] :
? [X249: subst > term > term] :
! [X250: term,X251: subst] :
( ( X249 @ X251 @ X250 )
= ( sub @ X248 @ ( push @ X250 @ X251 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X252: term] :
! [X253: subst > term > term] :
? [X254: term,X255: subst] :
( ( X253 @ X255 @ X254 )
!= ( sub @ X252 @ ( push @ X254 @ X255 ) ) ) )
& ( ( hoasapinj1 = $true )
| ? [X256: term,X257: term,X258: term,X259: term] :
( ( X257 != X258 )
& ( ( ap @ ( sub @ X257 @ id ) @ X256 )
= ( ap @ ( sub @ X258 @ id ) @ X259 ) ) ) )
& ( ! [X260: term,X261: term,X262: term,X263: term] :
( ( X261 = X262 )
| ( ( ap @ ( sub @ X261 @ id ) @ X260 )
!= ( ap @ ( sub @ X262 @ id ) @ X263 ) ) )
| ( hoasapinj1 != $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( $true = sP2 )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( $true != sP2 ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP0 )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( $true != sP0 ) ) ),
inference(rectify,[status(thm)],[f672]) ).
thf(f672,plain,
( ( ( pushprop_lem1_lthm = $true )
| ( ? [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
!= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
& ( ( X18 @ X15 )
= $true ) )
& ! [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
= X11 )
& ! [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
= X12 ) ) )
& ( ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
| ( ( X18 @ X15 )
!= $true ) )
| ? [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
!= X11 )
| ? [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
!= X12 )
| ( pushprop_lem1_lthm != $true ) )
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
!= ( d2subst @ X55 ) )
| ( X55 = X56 ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP109 ) )
& ( ( $true = sP109 )
| ( hoasinduction_lem2v2 != $true ) )
& ( ? [X281: term] :
( ( sub @ X281 @ id )
!= X281 )
| ? [X277: term] :
! [X278: subst > term > term] :
? [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
!= ( X278 @ X280 @ X279 ) )
| ( $true = sP106 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( ! [X281: term] :
( ( sub @ X281 @ id )
= X281 )
& ! [X277: term] :
? [X278: subst > term > term] :
! [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
= ( X278 @ X280 @ X279 ) )
& ( $true != sP106 ) ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X7: subst > term > subst > $o] :
! [X8: term > $o] :
? [X9: term] :
( ( X8 @ X9 )
!= ( X7 @ id @ X9 @ id ) ) )
& ( ! [X7: subst > term > subst > $o] :
? [X8: term > $o] :
! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X338: term] :
! [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
!= ( X339 @ sh @ one ) ) )
& ( ! [X338: term] :
? [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
= ( X339 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( induction2lem_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
= X273 )
| ( axshiftcons != $true ) )
& ( ( axshiftcons = $true )
| ? [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
!= X273 ) )
& ( ! [X105: term,X106: subst > term > subst > $o] :
( ( $true
= ( sP103 @ X106 ) )
| ( $true
= ( X106 @ id @ X105 @ id ) )
| ( $true
= ( sP104 @ X106 ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X105: term,X106: subst > term > subst > $o] :
( ( $true
!= ( sP103 @ X106 ) )
& ( $true
!= ( X106 @ id @ X105 @ id ) )
& ( $true
!= ( sP104 @ X106 ) ) ) )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP102 ) )
& ( ( $true = sP102 )
| ( hoasinduction_lem2 != $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( $true = sP99 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP99 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
| ? [X513: term,X514: term > $o,X515: subst] :
! [X516: term > $o] :
? [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
!= ( X516 @ X517 ) ) )
& ( ! [X513: term,X514: term > $o,X515: subst] :
? [X516: term > $o] :
! [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
= ( X516 @ X517 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 )
& ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X59: subst] :
( ( comp @ id @ X59 )
!= X59 )
| ? [X58: subst] :
( ( comp @ X58 @ id )
!= X58 ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lthm = $true )
& ! [X96: d_term] : ( d_one = X96 )
& ( lamnotvar = $true )
& ( ( induction = $true )
| ( sP97 != $true ) )
& ( ( sP97 = $true )
| ( induction != $true ) )
& ( ! [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
!= ( sub @ X275 @ sh ) )
| ( X275 = X276 ) )
| ( shinj != $true ) )
& ( ( shinj = $true )
| ? [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
= ( sub @ X275 @ sh ) )
& ( X275 != X276 ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X93: subst > term > subst > $o] :
? [X94: term > $o] :
! [X95: term] :
( ( X94 @ X95 )
= ( X93 @ id @ X95 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X93: subst > term > subst > $o] :
! [X94: term > $o] :
? [X95: term] :
( ( X94 @ X95 )
!= ( X93 @ id @ X95 @ id ) ) )
& ( ( sP94 = $true )
| ( hoasinduction_lem3aa != $true ) )
& ( ( hoasinduction_lem3aa = $true )
| ( sP94 != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoasinduction = $true )
| ( sP92 != $true ) )
& ( ( sP92 = $true )
| ( hoasinduction != $true ) )
& ( ( sP87 = $true )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( sP87 != $true ) )
& ( lamnotap != $true )
& ( ( apinj2 = $true )
| ? [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
= ( ap @ X100 @ X98 ) )
& ( X97 != X98 ) ) )
& ( ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X98 ) )
| ( apinj2 != $true ) )
& ( ( termmset = $true )
| ? [X57: term] :
( ( sub @ X57 @ id )
!= X57 ) )
& ( ! [X57: term] :
( ( sub @ X57 @ id )
= X57 )
| ( termmset != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ulamvar1 != $true )
& ( axapp = $true )
& ! [X382: subst] :
? [X383: d_subst] :
( ( d2subst @ X383 )
= X382 )
& ( ! [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) )
| ( ( X266 @ X265 )
!= $true ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( $true
!= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) )
& ( ( X266 @ X265 )
= $true ) ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ! [X270: d_subst] : ( d_id = X270 )
& ! [X23: d_term,X24: d_term] :
( ( X23 = X24 )
| ( ( d2term @ X23 )
!= ( d2term @ X24 ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ? [X394: term] :
( ( sub @ X394 @ id )
!= X394 )
| ( sP84 = $true )
| ? [X384: subst > term > subst > $o] :
! [X385: term > $o] :
? [X386: term] :
( ( X384 @ id @ X386 @ id )
!= ( X385 @ X386 ) )
| ( sP83 = $true )
| ( sP85 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( ! [X394: term] :
( ( sub @ X394 @ id )
= X394 )
& ( sP84 != $true )
& ! [X384: subst > term > subst > $o] :
? [X385: term > $o] :
! [X386: term] :
( ( X384 @ id @ X386 @ id )
= ( X385 @ X386 ) )
& ( sP83 != $true )
& ( sP85 != $true ) ) )
& ( axclos = $true )
& ( ( laminj = $true )
| ? [X167: term,X168: term] :
( ( X167 != X168 )
& ( ( lam @ X168 )
= ( lam @ X167 ) ) ) )
& ( ! [X167: term,X168: term] :
( ( X167 = X168 )
| ( ( lam @ X168 )
!= ( lam @ X167 ) ) )
| ( laminj != $true ) )
& ( ( sP76 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP76 != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( sP74 != $true ) )
& ( ( sP74 = $true )
| ( hoasapinj2_lthm != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( hoasinduction_lem3_lthm = $true )
| ( ( sP72 != $true )
& ! [X114: term] :
( ( sub @ X114 @ id )
= X114 )
& ( sP71 != $true ) ) )
& ( ( sP72 = $true )
| ? [X114: term] :
( ( sub @ X114 @ id )
!= X114 )
| ( sP71 = $true )
| ( hoasinduction_lem3_lthm != $true ) )
& ( hoaslamnotap_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( pushprop_lthm_orig = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( ! [X60: term] :
( ( sub @ X60 @ id )
= X60 )
& ( $true != sP65 )
& ( $true != sP66 ) ) )
& ( ? [X60: term] :
( ( sub @ X60 @ id )
!= X60 )
| ( $true = sP65 )
| ( $true = sP66 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ! [X101: term] :
? [X102: subst > term > term] :
! [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
= ( X102 @ X104 @ X103 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X101: term] :
! [X102: subst > term > term] :
? [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
!= ( X102 @ X104 @ X103 ) ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP62 != $true ) )
& ( ( sP62 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( ! [X349: term] :
( ( sub @ X349 @ id )
= X349 )
& ( sP58 != $true ) ) )
& ( ? [X349: term] :
( ( sub @ X349 @ id )
!= X349 )
| ( sP58 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lthm = $true )
| ( ( sP54 != $true )
& ( $true != sP53 )
& ( sP52 != $true )
& ( sP51 != $true ) ) )
& ( ( sP54 = $true )
| ( $true = sP53 )
| ( sP52 = $true )
| ( sP51 = $true )
| ( hoasinduction_lthm != $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasapinj1_gthm = $true )
& ! [X271: term] :
? [X272: d_term] :
( ( d2term @ X272 )
= X271 )
& ( ( induction2lem_lthm = $true )
| ( ! [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
= X171 )
& ! [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
= X169 )
& ( sP39 != $true )
& ( sP38 != $true )
& ! [X173: subst] :
( ( comp @ X173 @ id )
= X173 ) ) )
& ( ? [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
!= X171 )
| ? [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
!= X169 )
| ( sP39 = $true )
| ( sP38 = $true )
| ? [X173: subst] :
( ( comp @ X173 @ id )
!= X173 )
| ( induction2lem_lthm != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj2 = $true )
| ? [X19: term,X20: term,X21: term,X22: term] :
( ( X20 != X22 )
& ( ( ap @ ( sub @ X19 @ id ) @ X20 )
= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) ) )
& ( ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) )
| ( hoasapinj2 != $true ) )
& ( hoasinduction_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
| ( axidl != $true ) )
& ( ( axidl = $true )
| ? [X531: subst] :
( ( comp @ id @ X531 )
!= X531 ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
| ( sP33 != $true ) )
& ( ( sP33 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( ( $true = sP31 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP31 ) )
& ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
!= X337 ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_lthm_3 = $true )
| ( ( sP26 != $true )
& ( $true != sP28 )
& ! [X292: subst > term > subst > $o] :
? [X293: term > $o] :
! [X294: term] :
( ( X292 @ id @ X294 @ id )
= ( X293 @ X294 ) )
& ! [X302: term] :
( ( sub @ X302 @ id )
= X302 )
& ( sP27 != $true ) ) )
& ( ( sP26 = $true )
| ( $true = sP28 )
| ? [X292: subst > term > subst > $o] :
! [X293: term > $o] :
? [X294: term] :
( ( X292 @ id @ X294 @ id )
!= ( X293 @ X294 ) )
| ? [X302: term] :
( ( sub @ X302 @ id )
!= X302 )
| ( sP27 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3 = $true )
| ( sP18 != $true ) )
& ( ( sP18 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ! [X264: term] :
( ( sub @ X264 @ id )
= X264 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X264: term] :
( ( sub @ X264 @ id )
!= X264 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ( termmset_lthm = $true )
| ( ! [X488: term] :
( ( sub @ X488 @ id )
= X488 )
& ? [X489: term] :
( ( sub @ X489 @ id )
!= X489 ) ) )
& ( ? [X488: term] :
( ( sub @ X488 @ id )
!= X488 )
| ! [X489: term] :
( ( sub @ X489 @ id )
= X489 )
| ( termmset_lthm != $true ) )
& ( ! [X258: term,X259: subst > term > term,X260: term] :
( ( ( lam @ ( X259 @ sh @ one ) )
!= ( ap @ ( sub @ X260 @ id ) @ X258 ) )
| ? [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
!= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X258: term,X259: subst > term > term,X260: term] :
( ( ( lam @ ( X259 @ sh @ one ) )
= ( ap @ ( sub @ X260 @ id ) @ X258 ) )
& ! [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop = $true )
& ( axmap = $true )
& ( ( pushprop_lem1v2 = $true )
| ? [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( $true
= ( X217 @ X215 ) )
& ! [X219: term] :
( ( X216 @ X219 )
= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) )
& ( ( X216 @ one )
!= $true ) ) )
& ( ! [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( $true
!= ( X217 @ X215 ) )
| ? [X219: term] :
( ( X216 @ X219 )
!= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) )
| ( ( X216 @ one )
= $true ) )
| ( pushprop_lem1v2 != $true ) )
& ( ! [X212: term] :
? [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
= ( X213 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X212: term] :
! [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
!= ( X213 @ sh @ one ) ) )
& ( ! [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
= X335 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
!= X335 ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( induction2lem = $true )
| ( $true != sP14 ) )
& ( ( $true = sP14 )
| ( induction2lem != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( axassoc = $true )
& ( hoaslamnotvar = $true )
& ( ! [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ? [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
!= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) )
| ? [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
!= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
| ( ( X498 @ X497 @ X495 )
= ( X496 @ X497 @ X495 ) )
| ( ( lam @ ( X498 @ sh @ one ) )
!= ( lam @ ( X496 @ sh @ one ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ! [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) )
& ! [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
& ( ( X498 @ X497 @ X495 )
!= ( X496 @ X497 @ X495 ) )
& ( ( lam @ ( X498 @ sh @ one ) )
= ( lam @ ( X496 @ sh @ one ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( axvarshift = $true )
& ( ( hoasinduction_lem3aaa = $true )
| ( sP11 != $true ) )
& ( ( sP11 = $true )
| ( hoasinduction_lem3aaa != $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( induction2_gthm = $true )
& ( ! [X490: term > $o,X491: term,X492: subst] :
? [X493: term > $o] :
! [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
= ( X493 @ X494 ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X490: term > $o,X491: term,X492: subst] :
! [X493: term > $o] :
? [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
!= ( X493 @ X494 ) ) )
& ( ulamvarind = $true )
& ( ( induction2_lthm = $true )
| ( ( sP9 != $true )
& ! [X151: term] :
( ( sub @ X151 @ id )
= X151 )
& ( sP8 != $true ) ) )
& ( ( sP9 = $true )
| ? [X151: term] :
( ( sub @ X151 @ id )
!= X151 )
| ( sP8 = $true )
| ( induction2_lthm != $true ) )
& ( axabs = $true )
& ( ( apinj1 = $true )
| ? [X484: term,X485: term,X486: term,X487: term] :
( ( X485 != X487 )
& ( ( ap @ X485 @ X484 )
= ( ap @ X487 @ X486 ) ) ) )
& ( ! [X484: term,X485: term,X486: term,X487: term] :
( ( X485 = X487 )
| ( ( ap @ X485 @ X484 )
!= ( ap @ X487 @ X486 ) ) )
| ( apinj1 != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
!= $true )
| ? [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
!= ( X2 @ X6 ) )
| ( ( X2 @ one )
= $true ) )
| ? [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
!= X0 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
= $true )
& ! [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
= ( X2 @ X6 ) )
& ( ( X2 @ one )
!= $true ) )
& ! [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
= X0 ) ) )
& ( ( d2subst @ d_id )
= id )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X214: subst] :
( ( comp @ X214 @ id )
= X214 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X214: subst] :
( ( comp @ X214 @ id )
!= X214 ) )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
= ( sub @ X147 @ ( push @ X149 @ X150 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X147: term] :
! [X148: subst > term > term] :
? [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
!= ( sub @ X147 @ ( push @ X149 @ X150 ) ) ) )
& ( ( hoasapinj1 = $true )
| ? [X378: term,X379: term,X380: term,X381: term] :
( ( X379 != X380 )
& ( ( ap @ ( sub @ X379 @ id ) @ X378 )
= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) ) )
& ( ! [X378: term,X379: term,X380: term,X381: term] :
( ( X379 = X380 )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) )
| ( hoasapinj1 != $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( $true = sP2 )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( $true != sP2 ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP0 )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( $true != sP0 ) ) ),
inference(flattening,[status(thm)],[f671]) ).
thf(f671,plain,
( ( ( pushprop_lem1_lthm = $true )
| ( ? [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
!= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
& ( ( X18 @ X15 )
= $true ) )
& ! [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
= X11 )
& ! [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
= X12 ) ) )
& ( ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
| ( ( X18 @ X15 )
!= $true ) )
| ? [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
!= X11 )
| ? [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
!= X12 )
| ( pushprop_lem1_lthm != $true ) )
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
!= ( d2subst @ X55 ) )
| ( X55 = X56 ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( $true != sP109 ) )
& ( ( $true = sP109 )
| ( hoasinduction_lem2v2 != $true ) )
& ( ? [X281: term] :
( ( sub @ X281 @ id )
!= X281 )
| ? [X277: term] :
! [X278: subst > term > term] :
? [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
!= ( X278 @ X280 @ X279 ) )
| ( $true = sP106 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( ! [X281: term] :
( ( sub @ X281 @ id )
= X281 )
& ! [X277: term] :
? [X278: subst > term > term] :
! [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
= ( X278 @ X280 @ X279 ) )
& ( $true != sP106 ) ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X7: subst > term > subst > $o] :
! [X8: term > $o] :
? [X9: term] :
( ( X8 @ X9 )
!= ( X7 @ id @ X9 @ id ) ) )
& ( ! [X7: subst > term > subst > $o] :
? [X8: term > $o] :
! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X338: term] :
! [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
!= ( X339 @ sh @ one ) ) )
& ( ! [X338: term] :
? [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
= ( X339 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( induction2lem_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
= X273 )
| ( axshiftcons != $true ) )
& ( ( axshiftcons = $true )
| ? [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
!= X273 ) )
& ( ! [X105: term,X106: subst > term > subst > $o] :
( ( $true
= ( sP103 @ X106 ) )
| ( $true
= ( X106 @ id @ X105 @ id ) )
| ( $true
= ( sP104 @ X106 ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X105: term,X106: subst > term > subst > $o] :
( ( $true
!= ( sP103 @ X106 ) )
& ( $true
!= ( X106 @ id @ X105 @ id ) )
& ( $true
!= ( sP104 @ X106 ) ) ) )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP102 ) )
& ( ( $true = sP102 )
| ( hoasinduction_lem2 != $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( $true = sP99 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP99 ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
| ? [X513: term,X514: term > $o,X515: subst] :
! [X516: term > $o] :
? [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
!= ( X516 @ X517 ) ) )
& ( ! [X513: term,X514: term > $o,X515: subst] :
? [X516: term > $o] :
! [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
= ( X516 @ X517 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 )
& ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X59: subst] :
( ( comp @ id @ X59 )
!= X59 )
| ? [X58: subst] :
( ( comp @ X58 @ id )
!= X58 ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lthm = $true )
& ! [X96: d_term] : ( d_one = X96 )
& ( lamnotvar = $true )
& ( ( induction = $true )
| ( sP97 != $true ) )
& ( ( sP97 = $true )
| ( induction != $true ) )
& ( ! [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
!= ( sub @ X275 @ sh ) )
| ( X275 = X276 ) )
| ( shinj != $true ) )
& ( ( shinj = $true )
| ? [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
= ( sub @ X275 @ sh ) )
& ( X275 != X276 ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X93: subst > term > subst > $o] :
? [X94: term > $o] :
! [X95: term] :
( ( X94 @ X95 )
= ( X93 @ id @ X95 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X93: subst > term > subst > $o] :
! [X94: term > $o] :
? [X95: term] :
( ( X94 @ X95 )
!= ( X93 @ id @ X95 @ id ) ) )
& ( ( sP94 = $true )
| ( hoasinduction_lem3aa != $true ) )
& ( ( hoasinduction_lem3aa = $true )
| ( sP94 != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoasinduction = $true )
| ( sP92 != $true ) )
& ( ( sP92 = $true )
| ( hoasinduction != $true ) )
& ( ( sP87 = $true )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( sP87 != $true ) )
& ( lamnotap != $true )
& ( ( apinj2 = $true )
| ? [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
= ( ap @ X100 @ X98 ) )
& ( X97 != X98 ) ) )
& ( ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X98 ) )
| ( apinj2 != $true ) )
& ( ( termmset = $true )
| ? [X57: term] :
( ( sub @ X57 @ id )
!= X57 ) )
& ( ! [X57: term] :
( ( sub @ X57 @ id )
= X57 )
| ( termmset != $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ulamvar1 != $true )
& ( axapp = $true )
& ! [X382: subst] :
? [X383: d_subst] :
( ( d2subst @ X383 )
= X382 )
& ( ! [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) )
| ( ( X266 @ X265 )
!= $true ) )
| ( pushprop_lem1 != $true ) )
& ( ( pushprop_lem1 = $true )
| ? [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( $true
!= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) )
& ( ( X266 @ X265 )
= $true ) ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ! [X270: d_subst] : ( d_id = X270 )
& ! [X23: d_term,X24: d_term] :
( ( X23 = X24 )
| ( ( d2term @ X23 )
!= ( d2term @ X24 ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ? [X394: term] :
( ( sub @ X394 @ id )
!= X394 )
| ( sP84 = $true )
| ? [X384: subst > term > subst > $o] :
! [X385: term > $o] :
? [X386: term] :
( ( X384 @ id @ X386 @ id )
!= ( X385 @ X386 ) )
| ( sP83 = $true )
| ( sP85 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( ! [X394: term] :
( ( sub @ X394 @ id )
= X394 )
& ( sP84 != $true )
& ! [X384: subst > term > subst > $o] :
? [X385: term > $o] :
! [X386: term] :
( ( X384 @ id @ X386 @ id )
= ( X385 @ X386 ) )
& ( sP83 != $true )
& ( sP85 != $true ) ) )
& ( axclos = $true )
& ( ( laminj = $true )
| ? [X167: term,X168: term] :
( ( X167 != X168 )
& ( ( lam @ X168 )
= ( lam @ X167 ) ) ) )
& ( ! [X167: term,X168: term] :
( ( X167 = X168 )
| ( ( lam @ X168 )
!= ( lam @ X167 ) ) )
| ( laminj != $true ) )
& ( ( sP76 = $true )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( sP76 != $true ) )
& ( ( hoasapinj2_lthm = $true )
| ( sP74 != $true ) )
& ( ( sP74 = $true )
| ( hoasapinj2_lthm != $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( hoasinduction_lem3_lthm = $true )
| ( ( sP72 != $true )
& ! [X114: term] :
( ( sub @ X114 @ id )
= X114 )
& ( sP71 != $true ) ) )
& ( ( sP72 = $true )
| ? [X114: term] :
( ( sub @ X114 @ id )
!= X114 )
| ( sP71 = $true )
| ( hoasinduction_lem3_lthm != $true ) )
& ( hoaslamnotap_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( pushprop_lthm_orig = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( ! [X60: term] :
( ( sub @ X60 @ id )
= X60 )
& ( $true != sP65 )
& ( $true != sP66 ) ) )
& ( ? [X60: term] :
( ( sub @ X60 @ id )
!= X60 )
| ( $true = sP65 )
| ( $true = sP66 )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ! [X101: term] :
? [X102: subst > term > term] :
! [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
= ( X102 @ X104 @ X103 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X101: term] :
! [X102: subst > term > term] :
? [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
!= ( X102 @ X104 @ X103 ) ) )
& ( ( hoasinduction_lem3v2 = $true )
| ( sP62 != $true ) )
& ( ( sP62 = $true )
| ( hoasinduction_lem3v2 != $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( ! [X349: term] :
( ( sub @ X349 @ id )
= X349 )
& ( sP58 != $true ) ) )
& ( ? [X349: term] :
( ( sub @ X349 @ id )
!= X349 )
| ( sP58 = $true )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lthm = $true )
| ( ( sP54 != $true )
& ( $true != sP53 )
& ( sP52 != $true )
& ( sP51 != $true ) ) )
& ( ( sP54 = $true )
| ( $true = sP53 )
| ( sP52 = $true )
| ( sP51 = $true )
| ( hoasinduction_lthm != $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasapinj1_gthm = $true )
& ! [X271: term] :
? [X272: d_term] :
( ( d2term @ X272 )
= X271 )
& ( ( induction2lem_lthm = $true )
| ( ! [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
= X171 )
& ! [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
= X169 )
& ( sP39 != $true )
& ( sP38 != $true )
& ! [X173: subst] :
( ( comp @ X173 @ id )
= X173 ) ) )
& ( ? [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
!= X171 )
| ? [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
!= X169 )
| ( sP39 = $true )
| ( sP38 = $true )
| ? [X173: subst] :
( ( comp @ X173 @ id )
!= X173 )
| ( induction2lem_lthm != $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj2 = $true )
| ? [X19: term,X20: term,X21: term,X22: term] :
( ( X20 != X22 )
& ( ( ap @ ( sub @ X19 @ id ) @ X20 )
= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) ) )
& ( ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) )
| ( hoasapinj2 != $true ) )
& ( hoasinduction_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
| ( axidl != $true ) )
& ( ( axidl = $true )
| ? [X531: subst] :
( ( comp @ id @ X531 )
!= X531 ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
| ( sP33 != $true ) )
& ( ( sP33 = $true )
| ( hoaslaminj_lthm != $true ) )
& ( ( $true = sP31 )
| ( induction2 != $true ) )
& ( ( induction2 = $true )
| ( $true != sP31 ) )
& ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
| ( axvarcons != $true ) )
& ( ( axvarcons = $true )
| ? [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
!= X337 ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_lthm_3 = $true )
| ( ( sP26 != $true )
& ( $true != sP28 )
& ! [X292: subst > term > subst > $o] :
? [X293: term > $o] :
! [X294: term] :
( ( X292 @ id @ X294 @ id )
= ( X293 @ X294 ) )
& ! [X302: term] :
( ( sub @ X302 @ id )
= X302 )
& ( sP27 != $true ) ) )
& ( ( sP26 = $true )
| ( $true = sP28 )
| ? [X292: subst > term > subst > $o] :
! [X293: term > $o] :
? [X294: term] :
( ( X292 @ id @ X294 @ id )
!= ( X293 @ X294 ) )
| ? [X302: term] :
( ( sub @ X302 @ id )
!= X302 )
| ( sP27 = $true )
| ( hoasinduction_lthm_3 != $true ) )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3 = $true )
| ( sP18 != $true ) )
& ( ( sP18 = $true )
| ( hoasinduction_lem3 != $true ) )
& ( ! [X264: term] :
( ( sub @ X264 @ id )
= X264 )
| ( axvarid != $true ) )
& ( ( axvarid = $true )
| ? [X264: term] :
( ( sub @ X264 @ id )
!= X264 ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ( termmset_lthm = $true )
| ( ! [X488: term] :
( ( sub @ X488 @ id )
= X488 )
& ? [X489: term] :
( ( sub @ X489 @ id )
!= X489 ) ) )
& ( ? [X488: term] :
( ( sub @ X488 @ id )
!= X488 )
| ! [X489: term] :
( ( sub @ X489 @ id )
= X489 )
| ( termmset_lthm != $true ) )
& ( ! [X258: term,X259: subst > term > term,X260: term] :
( ( ( lam @ ( X259 @ sh @ one ) )
!= ( ap @ ( sub @ X260 @ id ) @ X258 ) )
| ? [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
!= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X258: term,X259: subst > term > term,X260: term] :
( ( ( lam @ ( X259 @ sh @ one ) )
= ( ap @ ( sub @ X260 @ id ) @ X258 ) )
& ! [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop = $true )
& ( axmap = $true )
& ( ( pushprop_lem1v2 = $true )
| ? [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( $true
= ( X217 @ X215 ) )
& ! [X219: term] :
( ( X216 @ X219 )
= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) )
& ( ( X216 @ one )
!= $true ) ) )
& ( ! [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( $true
!= ( X217 @ X215 ) )
| ? [X219: term] :
( ( X216 @ X219 )
!= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) )
| ( ( X216 @ one )
= $true ) )
| ( pushprop_lem1v2 != $true ) )
& ( ! [X212: term] :
? [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
= ( X213 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X212: term] :
! [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
!= ( X213 @ sh @ one ) ) )
& ( ! [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
= X335 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
!= X335 ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( induction2lem = $true )
| ( $true != sP14 ) )
& ( ( $true = sP14 )
| ( induction2lem != $true ) )
& ( pushprop_lem0_gthm = $true )
& ( axassoc = $true )
& ( hoaslamnotvar = $true )
& ( ! [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ? [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
!= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) )
| ? [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
!= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
| ( ( X498 @ X497 @ X495 )
= ( X496 @ X497 @ X495 ) )
| ( ( lam @ ( X498 @ sh @ one ) )
!= ( lam @ ( X496 @ sh @ one ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ! [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) )
& ! [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
& ( ( X498 @ X497 @ X495 )
!= ( X496 @ X497 @ X495 ) )
& ( ( lam @ ( X498 @ sh @ one ) )
= ( lam @ ( X496 @ sh @ one ) ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( axvarshift = $true )
& ( ( hoasinduction_lem3aaa = $true )
| ( sP11 != $true ) )
& ( ( sP11 = $true )
| ( hoasinduction_lem3aaa != $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( induction2_gthm = $true )
& ( ! [X490: term > $o,X491: term,X492: subst] :
? [X493: term > $o] :
! [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
= ( X493 @ X494 ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X490: term > $o,X491: term,X492: subst] :
! [X493: term > $o] :
? [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
!= ( X493 @ X494 ) ) )
& ( ulamvarind = $true )
& ( ( induction2_lthm = $true )
| ( ( sP9 != $true )
& ! [X151: term] :
( ( sub @ X151 @ id )
= X151 )
& ( sP8 != $true ) ) )
& ( ( sP9 = $true )
| ? [X151: term] :
( ( sub @ X151 @ id )
!= X151 )
| ( sP8 = $true )
| ( induction2_lthm != $true ) )
& ( axabs = $true )
& ( ( apinj1 = $true )
| ? [X484: term,X485: term,X486: term,X487: term] :
( ( X485 != X487 )
& ( ( ap @ X485 @ X484 )
= ( ap @ X487 @ X486 ) ) ) )
& ( ! [X484: term,X485: term,X486: term,X487: term] :
( ( X485 = X487 )
| ( ( ap @ X485 @ X484 )
!= ( ap @ X487 @ X486 ) ) )
| ( apinj1 != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
!= $true )
| ? [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
!= ( X2 @ X6 ) )
| ( ( X2 @ one )
= $true ) )
| ? [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
!= X0 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
= $true )
& ! [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
= ( X2 @ X6 ) )
& ( ( X2 @ one )
!= $true ) )
& ! [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
= X0 ) ) )
& ( ( d2subst @ d_id )
= id )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X214: subst] :
( ( comp @ X214 @ id )
= X214 )
| ( axidr != $true ) )
& ( ( axidr = $true )
| ? [X214: subst] :
( ( comp @ X214 @ id )
!= X214 ) )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
= ( sub @ X147 @ ( push @ X149 @ X150 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X147: term] :
! [X148: subst > term > term] :
? [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
!= ( sub @ X147 @ ( push @ X149 @ X150 ) ) ) )
& ( ( hoasapinj1 = $true )
| ? [X378: term,X379: term,X380: term,X381: term] :
( ( X379 != X380 )
& ( ( ap @ ( sub @ X379 @ id ) @ X378 )
= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) ) )
& ( ! [X378: term,X379: term,X380: term,X381: term] :
( ( X379 = X380 )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) )
| ( hoasapinj1 != $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( $true = sP2 )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( $true != sP2 ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP0 )
| ( substmonoid_lthm != $true ) )
& ( ( substmonoid_lthm = $true )
| ( $true != sP0 ) ) ),
inference(nnf_transformation,[status(thm)],[f125]) ).
thf(f125,plain,
( ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
| ( ( X18 @ X15 )
!= $true ) )
| ? [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
!= X11 )
| ? [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
!= X12 ) ) )
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
!= ( d2subst @ X55 ) )
| ( X55 = X56 ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ( $true = sP109 ) )
& ( ( ? [X281: term] :
( ( sub @ X281 @ id )
!= X281 )
| ? [X277: term] :
! [X278: subst > term > term] :
? [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
!= ( X278 @ X280 @ X279 ) )
| ( $true = sP106 ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X7: subst > term > subst > $o] :
? [X8: term > $o] :
! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X338: term] :
? [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
= ( X339 @ sh @ one ) ) )
& ( induction2lem_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
= X273 )
<=> ( axshiftcons = $true ) )
& ( ! [X105: term,X106: subst > term > subst > $o] :
( ( $true
= ( sP103 @ X106 ) )
| ( $true
= ( X106 @ id @ X105 @ id ) )
| ( $true
= ( sP104 @ X106 ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ( $true = sP102 ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( $true = sP99 )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X513: term,X514: term > $o,X515: subst] :
? [X516: term > $o] :
! [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
= ( X516 @ X517 ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 )
& ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 ) )
<=> ( substmonoid = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lthm = $true )
& ! [X96: d_term] : ( d_one = X96 )
& ( lamnotvar = $true )
& ( ( induction = $true )
<=> ( sP97 = $true ) )
& ( ! [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
!= ( sub @ X275 @ sh ) )
| ( X275 = X276 ) )
<=> ( shinj = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X93: subst > term > subst > $o] :
? [X94: term > $o] :
! [X95: term] :
( ( X94 @ X95 )
= ( X93 @ id @ X95 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( sP94 = $true )
<=> ( hoasinduction_lem3aa = $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoasinduction = $true )
<=> ( sP92 = $true ) )
& ( ( sP87 = $true )
<=> ( hoasapinj1_lthm = $true ) )
& ( lamnotap != $true )
& ( ( apinj2 = $true )
<=> ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X98 ) ) )
& ( ( termmset = $true )
<=> ! [X57: term] :
( ( sub @ X57 @ id )
= X57 ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ulamvar1 != $true )
& ( axapp = $true )
& ! [X382: subst] :
? [X383: d_subst] :
( ( d2subst @ X383 )
= X382 )
& ( ! [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) )
| ( ( X266 @ X265 )
!= $true ) )
<=> ( pushprop_lem1 = $true ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ! [X270: d_subst] : ( d_id = X270 )
& ! [X23: d_term,X24: d_term] :
( ( X23 = X24 )
| ( ( d2term @ X23 )
!= ( d2term @ X24 ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( ? [X394: term] :
( ( sub @ X394 @ id )
!= X394 )
| ( sP84 = $true )
| ? [X384: subst > term > subst > $o] :
! [X385: term > $o] :
? [X386: term] :
( ( X384 @ id @ X386 @ id )
!= ( X385 @ X386 ) )
| ( sP83 = $true )
| ( sP85 = $true ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( axclos = $true )
& ( ( laminj = $true )
<=> ! [X167: term,X168: term] :
( ( X167 = X168 )
| ( ( lam @ X168 )
!= ( lam @ X167 ) ) ) )
& ( ( sP76 = $true )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( sP74 = $true ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ( sP72 = $true )
| ? [X114: term] :
( ( sub @ X114 @ id )
!= X114 )
| ( sP71 = $true ) ) )
& ( hoaslamnotap_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( pushprop_lthm_orig = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ? [X60: term] :
( ( sub @ X60 @ id )
!= X60 )
| ( $true = sP65 )
| ( $true = sP66 ) ) )
& ( ! [X101: term] :
? [X102: subst > term > term] :
! [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
= ( X102 @ X104 @ X103 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( hoasinduction_lem3v2 = $true )
<=> ( sP62 = $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ? [X349: term] :
( ( sub @ X349 @ id )
!= X349 )
| ( sP58 = $true ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ( sP54 = $true )
| ( $true = sP53 )
| ( sP52 = $true )
| ( sP51 = $true ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasapinj1_gthm = $true )
& ! [X271: term] :
? [X272: d_term] :
( ( d2term @ X272 )
= X271 )
& ( ( induction2lem_lthm = $true )
<=> ( ? [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
!= X171 )
| ? [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
!= X169 )
| ( sP39 = $true )
| ( sP38 = $true )
| ? [X173: subst] :
( ( comp @ X173 @ id )
!= X173 ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) ) )
& ( hoasinduction_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
<=> ( axidl = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( sP33 = $true ) )
& ( ( $true = sP31 )
<=> ( induction2 = $true ) )
& ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
<=> ( axvarcons = $true ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ( sP26 = $true )
| ( $true = sP28 )
| ? [X292: subst > term > subst > $o] :
! [X293: term > $o] :
? [X294: term] :
( ( X292 @ id @ X294 @ id )
!= ( X293 @ X294 ) )
| ? [X302: term] :
( ( sub @ X302 @ id )
!= X302 )
| ( sP27 = $true ) ) )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ( sP18 = $true ) )
& ( ! [X264: term] :
( ( sub @ X264 @ id )
= X264 )
<=> ( axvarid = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ( termmset_lthm = $true )
<=> ( ? [X488: term] :
( ( sub @ X488 @ id )
!= X488 )
| ! [X489: term] :
( ( sub @ X489 @ id )
= X489 ) ) )
& ( ! [X258: term,X259: subst > term > term,X260: term] :
( ( ( lam @ ( X259 @ sh @ one ) )
!= ( ap @ ( sub @ X260 @ id ) @ X258 ) )
| ? [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
!= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) ) )
<=> ( hoaslamnotap = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop = $true )
& ( axmap = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( $true
!= ( X217 @ X215 ) )
| ? [X219: term] :
( ( X216 @ X219 )
!= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) )
| ( ( X216 @ one )
= $true ) ) )
& ( ! [X212: term] :
? [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
= ( X213 @ sh @ one ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( ! [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
= X335 )
<=> ( axscons = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( induction2lem = $true )
<=> ( $true = sP14 ) )
& ( pushprop_lem0_gthm = $true )
& ( axassoc = $true )
& ( hoaslamnotvar = $true )
& ( ! [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ? [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
!= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) )
| ? [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
!= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
| ( ( X498 @ X497 @ X495 )
= ( X496 @ X497 @ X495 ) )
| ( ( lam @ ( X498 @ sh @ one ) )
!= ( lam @ ( X496 @ sh @ one ) ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( axvarshift = $true )
& ( ( hoasinduction_lem3aaa = $true )
<=> ( sP11 = $true ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( induction2_gthm = $true )
& ( ! [X490: term > $o,X491: term,X492: subst] :
? [X493: term > $o] :
! [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
= ( X493 @ X494 ) )
<=> ( pushprop_lem0 = $true ) )
& ( ulamvarind = $true )
& ( ( induction2_lthm = $true )
<=> ( ( sP9 = $true )
| ? [X151: term] :
( ( sub @ X151 @ id )
!= X151 )
| ( sP8 = $true ) ) )
& ( axabs = $true )
& ( ( apinj1 = $true )
<=> ! [X484: term,X485: term,X486: term,X487: term] :
( ( X485 = X487 )
| ( ( ap @ X485 @ X484 )
!= ( ap @ X487 @ X486 ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
!= $true )
| ? [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
!= ( X2 @ X6 ) )
| ( ( X2 @ one )
= $true ) )
| ? [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
!= X0 ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( ( d2subst @ d_id )
= id )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X214: subst] :
( ( comp @ X214 @ id )
= X214 )
<=> ( axidr = $true ) )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
= ( sub @ X147 @ ( push @ X149 @ X150 ) ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( hoasapinj1 = $true )
<=> ! [X378: term,X379: term,X380: term,X381: term] :
( ( X379 = X380 )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ( $true = sP2 )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( $true = sP0 )
<=> ( substmonoid_lthm = $true ) ) ),
inference(definition_folding,[status(thm)],[f14,f124,f123,f122,f121,f120,f119,f118,f117,f116,f115,f114,f113,f112,f111,f110,f109,f108,f107,f106,f105,f104,f103,f102,f101,f100,f99,f98,f97,f96,f95,f94,f93,f92,f91,f90,f89,f88,f87,f86,f85,f84,f83,f82,f81,f80,f79,f78,f77,f76,f75,f74,f73,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]) ).
thf(f15,plain,
( ( $true = sP0 )
<=> ( ? [X332: subst] :
( ( comp @ X332 @ id )
!= X332 )
| ( ! [X333: subst] :
( ( comp @ X333 @ id )
= X333 )
& ! [X334: subst] :
( ( comp @ id @ X334 )
= X334 ) )
| ? [X331: subst] :
( ( comp @ id @ X331 )
!= X331 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f16,plain,
! [X533: subst > term > subst > $o] :
( ( $true
= ( sP1 @ X533 ) )
<=> ? [X534: subst > term > term] :
( ( ( X533 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id )
!= $true )
& ! [X535: subst,X536: subst,X537: term] :
( ( sub @ ( X534 @ X535 @ X537 ) @ X536 )
= ( X534 @ ( comp @ X535 @ X536 ) @ ( sub @ X537 @ X536 ) ) )
& ! [X538: term] :
( ( ( X533 @ id @ X538 @ id )
!= $true )
| ( ( X533 @ id @ ( X534 @ id @ X538 ) @ id )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f17,plain,
( ( $true = sP2 )
<=> ! [X532: term,X533: subst > term > subst > $o] :
( ( $true
= ( X533 @ id @ ( lam @ X532 ) @ id ) )
| ( $true
= ( sP1 @ X533 ) )
| ? [X539: term] :
( ( ( X533 @ id @ X539 @ id )
= $true )
& ( ( X533 @ id @ ( sub @ X532 @ ( push @ X539 @ id ) ) @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f18,plain,
! [X161: term > $o] :
( ( ( sP3 @ X161 )
= $true )
<=> ? [X165: term] :
( ( $true
!= ( X161 @ ( lam @ X165 ) ) )
& ! [X166: term] :
( ( ( X161 @ X166 )
!= $true )
| ( $true
= ( X161 @ ( sub @ X165 @ ( push @ X166 @ id ) ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f19,plain,
! [X161: term > $o] :
( ( $true
= ( sP4 @ X161 ) )
<=> ? [X163: term,X164: term] :
( ( ( X161 @ ( ap @ X164 @ X163 ) )
!= $true )
& ( $true
= ( X161 @ X163 ) )
& ( ( X161 @ X164 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f20,plain,
! [X154: term > $o] :
( ( ( sP5 @ X154 )
= $true )
<=> ! [X157: term,X158: term] :
( ( $true
!= ( X154 @ X158 ) )
| ( ( X154 @ X157 )
!= $true )
| ( ( X154 @ ( ap @ X158 @ X157 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f21,plain,
! [X153: subst,X154: term > $o] :
( ( ( sP6 @ X154 @ X153 )
= $true )
<=> ! [X159: term] :
( ( ( var @ X159 )
!= $true )
| ( ( X154 @ ( sub @ X159 @ X153 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f22,plain,
! [X154: term > $o] :
( ( $true
= ( sP7 @ X154 ) )
<=> ! [X155: term] :
( ( $true
= ( X154 @ ( lam @ X155 ) ) )
| ? [X156: term] :
( ( ( X154 @ X156 )
= $true )
& ( ( X154 @ ( sub @ X155 @ ( push @ X156 @ id ) ) )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f23,plain,
( ( sP8 = $true )
<=> ! [X160: term,X161: term > $o] :
( ( ( sP3 @ X161 )
= $true )
| ? [X162: term] :
( ( $true
!= ( X161 @ X162 ) )
& ( $true
= ( var @ X162 ) ) )
| ( $true
= ( sP4 @ X161 ) )
| ( ( X161 @ X160 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f24,plain,
( ( sP9 = $true )
<=> ? [X152: term,X153: subst,X154: term > $o] :
( ( $true
= ( sP7 @ X154 ) )
& ( ( sP6 @ X154 @ X153 )
= $true )
& ( ( sP5 @ X154 )
= $true )
& ( ( X154 @ ( sub @ X152 @ X153 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f25,plain,
! [X25: subst > term > subst > $o] :
( ( ( sP10 @ X25 )
= $true )
<=> ? [X28: term,X29: subst > term > term] :
( ( ( X25 @ id @ ( lam @ ( X29 @ sh @ one ) ) @ id )
!= $true )
& ! [X34: subst,X35: subst,X36: term] :
( ( sub @ X28 @ ( push @ ( sub @ X36 @ X34 ) @ ( comp @ X35 @ X34 ) ) )
= ( X29 @ ( comp @ X35 @ X34 ) @ ( sub @ X36 @ X34 ) ) )
& ! [X30: term] :
( ( ( X25 @ id @ ( X29 @ id @ X30 ) @ id )
= $true )
| ( ( X25 @ id @ X30 @ id )
!= $true ) )
& ! [X31: subst,X32: term,X33: subst] :
( ( sub @ ( sub @ X28 @ ( push @ X32 @ X33 ) ) @ X31 )
= ( sub @ ( X29 @ X33 @ X32 ) @ X31 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f26,plain,
( ( sP11 = $true )
<=> ! [X25: subst > term > subst > $o,X26: term] :
( ? [X27: term] :
( ( $true
= ( X25 @ id @ X27 @ id ) )
& ( $true
!= ( X25 @ id @ ( sub @ X26 @ ( push @ X27 @ id ) ) @ id ) ) )
| ( ( sP10 @ X25 )
= $true )
| ( $true
= ( X25 @ id @ ( lam @ ( sub @ X26 @ ( push @ one @ sh ) ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f27,plain,
! [X140: term > $o] :
( ( ( sP12 @ X140 )
= $true )
<=> ? [X143: term] :
( ! [X144: term] :
( ( ( X140 @ X144 )
!= $true )
| ( ( X140 @ ( sub @ X143 @ ( push @ X144 @ id ) ) )
= $true ) )
& ( $true
!= ( X140 @ ( lam @ X143 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f28,plain,
! [X140: term > $o] :
( ( ( sP13 @ X140 )
= $true )
<=> ? [X145: term,X146: term] :
( ( ( X140 @ X146 )
= $true )
& ( $true
!= ( X140 @ ( ap @ X145 @ X146 ) ) )
& ( ( X140 @ X145 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f29,plain,
( ( $true = sP14 )
<=> ! [X139: term,X140: term > $o,X141: subst] :
( ( ( sP13 @ X140 )
= $true )
| ? [X142: term] :
( ( ( var @ X142 )
= $true )
& ( ( X140 @ ( sub @ X142 @ X141 ) )
!= $true ) )
| ( ( sP12 @ X140 )
= $true )
| ( $true
= ( X140 @ ( sub @ X139 @ X141 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f30,plain,
! [X235: subst > term > subst > $o] :
( ( $true
= ( sP15 @ X235 ) )
<=> ? [X241: subst > term > term] :
( ( ( X235 @ id @ ( lam @ ( X241 @ sh @ one ) ) @ id )
!= $true )
& ! [X245: term] :
( ( $true
= ( X235 @ id @ ( X241 @ id @ X245 ) @ id ) )
| ( ( X235 @ id @ X245 @ id )
!= $true ) )
& ! [X242: term,X243: subst,X244: subst] :
( ( X241 @ ( comp @ X244 @ X243 ) @ ( sub @ X242 @ X243 ) )
= ( sub @ ( X241 @ X244 @ X242 ) @ X243 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f31,plain,
! [X235: subst > term > subst > $o] :
( ( $true
= ( sP16 @ X235 ) )
<=> ? [X237: subst,X238: subst,X239: subst,X240: term] :
( ( ( X235 @ X238 @ X240 @ ( comp @ X239 @ X237 ) )
!= $true )
& ( $true
= ( X235 @ ( comp @ X238 @ X239 ) @ ( sub @ X240 @ X239 ) @ X237 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f32,plain,
! [X236: term,X235: subst > term > subst > $o] :
( ( ( sP17 @ X235 @ X236 )
= $true )
<=> ? [X246: term] :
( ( $true
!= ( X235 @ id @ ( sub @ X236 @ ( push @ X246 @ id ) ) @ id ) )
& ( ( X235 @ id @ X246 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f33,plain,
( ( sP18 = $true )
<=> ! [X235: subst > term > subst > $o,X236: term] :
( ( ( X235 @ id @ ( lam @ X236 ) @ id )
= $true )
| ? [X247: subst,X248: subst,X249: subst,X250: term] :
( ( ( X235 @ X248 @ X250 @ ( comp @ X249 @ X247 ) )
= $true )
& ( $true
!= ( X235 @ ( comp @ X248 @ X249 ) @ ( sub @ X250 @ X249 ) @ X247 ) ) )
| ( ( sP17 @ X235 @ X236 )
= $true )
| ( $true
= ( sP16 @ X235 ) )
| ( $true
= ( sP15 @ X235 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f34,plain,
! [X295: term > $o] :
( ( ( sP19 @ X295 )
= $true )
<=> ! [X297: term,X298: term] :
( ( ( X295 @ X297 )
!= $true )
| ( ( X295 @ X298 )
!= $true )
| ( $true
= ( X295 @ ( ap @ X297 @ X298 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f35,plain,
! [X295: term > $o] :
( ( $true
= ( sP20 @ X295 ) )
<=> ! [X300: term] :
( ? [X301: term] :
( ( ( X295 @ X301 )
= $true )
& ( $true
!= ( X295 @ ( sub @ X300 @ ( push @ X301 @ id ) ) ) ) )
| ( ( X295 @ ( lam @ X300 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f36,plain,
! [X303: subst > term > subst > $o] :
( ( $true
= ( sP21 @ X303 ) )
<=> ! [X306: subst > term > term] :
( ? [X307: term] :
( ( $true
= ( X303 @ id @ X307 @ id ) )
& ( $true
!= ( X303 @ id @ ( X306 @ id @ X307 ) @ id ) ) )
| ? [X308: term,X309: subst,X310: subst] :
( ( X306 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) )
!= ( sub @ ( X306 @ X310 @ X308 ) @ X309 ) )
| ( $true
= ( X303 @ id @ ( lam @ ( X306 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f37,plain,
! [X314: subst > term > subst > $o] :
( ( ( sP22 @ X314 )
= $true )
<=> ? [X326: subst > term > term] :
( ( ( X314 @ id @ ( lam @ ( X326 @ sh @ one ) ) @ id )
!= $true )
& ! [X330: term] :
( ( ( X314 @ id @ X330 @ id )
!= $true )
| ( ( X314 @ id @ ( X326 @ id @ X330 ) @ id )
= $true ) )
& ! [X327: subst,X328: term,X329: subst] :
( ( X326 @ ( comp @ X327 @ X329 ) @ ( sub @ X328 @ X329 ) )
= ( sub @ ( X326 @ X327 @ X328 ) @ X329 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f38,plain,
! [X314: subst > term > subst > $o] :
( ( $true
= ( sP23 @ X314 ) )
<=> ? [X320: term,X321: term] :
( ( ( X314 @ id @ ( ap @ ( sub @ X320 @ id ) @ X321 ) @ id )
!= $true )
& ( $true
= ( X314 @ id @ X321 @ id ) )
& ( ( X314 @ id @ X320 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f39,plain,
! [X314: subst > term > subst > $o] :
( ( $true
= ( sP24 @ X314 ) )
<=> ? [X315: subst,X316: subst,X317: term,X318: subst] :
( ( $true
!= ( X314 @ X318 @ X317 @ ( comp @ X315 @ X316 ) ) )
& ( $true
= ( X314 @ ( comp @ X318 @ X315 ) @ ( sub @ X317 @ X315 ) @ X316 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f40,plain,
! [X314: subst > term > subst > $o] :
( ( $true
= ( sP25 @ X314 ) )
<=> ? [X322: subst,X323: subst,X324: term,X325: subst] :
( ( ( X314 @ X322 @ X324 @ ( comp @ X323 @ X325 ) )
= $true )
& ( ( X314 @ ( comp @ X322 @ X323 ) @ ( sub @ X324 @ X323 ) @ X325 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f41,plain,
( ( sP26 = $true )
<=> ! [X313: term,X314: subst > term > subst > $o] :
( ? [X319: term] :
( ( ( X314 @ id @ X319 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X319 @ id ) ) ) )
| ( ( sP22 @ X314 )
= $true )
| ( ( X314 @ id @ X313 @ id )
= $true )
| ( $true
= ( sP25 @ X314 ) )
| ( $true
= ( sP24 @ X314 ) )
| ( $true
= ( sP23 @ X314 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f42,plain,
( ( sP27 = $true )
<=> ? [X295: term > $o,X296: term] :
( ( ( sP19 @ X295 )
= $true )
& ! [X299: term] :
( ( $true
!= ( var @ X299 ) )
| ( ( X295 @ X299 )
= $true ) )
& ( $true
= ( sP20 @ X295 ) )
& ( ( X295 @ X296 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f43,plain,
( ( $true = sP28 )
<=> ? [X303: subst > term > subst > $o,X304: term,X305: term > $o] :
( ( ( X305 @ ( lam @ X304 ) )
!= $true )
& ! [X312: term] :
( ( X303 @ id @ X312 @ id )
= ( X305 @ X312 ) )
& ( $true
= ( sP21 @ X303 ) )
& ! [X311: term] :
( ( ( X305 @ X311 )
!= $true )
| ( ( X305 @ ( sub @ X304 @ ( push @ X311 @ id ) ) )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f44,plain,
! [X252: term > $o] :
( ( $true
= ( sP29 @ X252 ) )
<=> ? [X255: term] :
( ( ( X252 @ ( lam @ X255 ) )
!= $true )
& ! [X256: term] :
( ( $true
!= ( X252 @ X256 ) )
| ( ( X252 @ ( sub @ X255 @ ( push @ X256 @ id ) ) )
= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f45,plain,
! [X252: term > $o] :
( ( ( sP30 @ X252 )
= $true )
<=> ? [X253: term,X254: term] :
( ( $true
!= ( X252 @ ( ap @ X254 @ X253 ) ) )
& ( ( X252 @ X254 )
= $true )
& ( $true
= ( X252 @ X253 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f46,plain,
( ( $true = sP31 )
<=> ! [X251: term,X252: term > $o] :
( ? [X257: term] :
( ( $true
!= ( X252 @ X257 ) )
& ( $true
= ( var @ X257 ) ) )
| ( ( sP30 @ X252 )
= $true )
| ( $true
= ( sP29 @ X252 ) )
| ( $true
= ( X252 @ X251 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f47,plain,
( ( sP32 = $true )
<=> ! [X83: term,X84: subst > term > term,X85: subst,X86: subst > term > term] :
( ( ( lam @ ( X84 @ sh @ one ) )
!= ( lam @ ( X86 @ sh @ one ) ) )
| ? [X90: subst,X91: subst,X92: term] :
( ( X84 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) )
!= ( sub @ ( X84 @ X90 @ X92 ) @ X91 ) )
| ( ( X84 @ X85 @ X83 )
= ( X86 @ X85 @ X83 ) )
| ? [X87: subst,X88: subst,X89: term] :
( ( X86 @ ( comp @ X87 @ X88 ) @ ( sub @ X89 @ X88 ) )
!= ( sub @ ( X86 @ X87 @ X89 ) @ X88 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f48,plain,
( ( sP33 = $true )
<=> ( ? [X79: subst,X80: term] :
( ( comp @ sh @ ( push @ X80 @ X79 ) )
!= X79 )
| ? [X81: term,X82: term] :
( ( ( lam @ X81 )
= ( lam @ X82 ) )
& ( X81 != X82 ) )
| ( sP32 = $true )
| ? [X77: subst,X78: term] :
( ( sub @ one @ ( push @ X78 @ X77 ) )
!= X78 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f49,plain,
! [X181: term > $o] :
( ( ( sP34 @ X181 )
= $true )
<=> ? [X186: term] :
( ! [X187: term] :
( ( ( X181 @ ( sub @ X186 @ ( push @ X187 @ id ) ) )
= $true )
| ( ( X181 @ X187 )
!= $true ) )
& ( $true
!= ( X181 @ ( lam @ X186 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f50,plain,
! [X181: term > $o] :
( ( ( sP35 @ X181 )
= $true )
<=> ? [X183: term,X184: term] :
( ( $true
= ( X181 @ X183 ) )
& ( $true
!= ( X181 @ ( ap @ X184 @ X183 ) ) )
& ( ( X181 @ X184 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f51,plain,
! [X175: term > $o] :
( ( $true
= ( sP36 @ X175 ) )
<=> ! [X176: term,X177: term] :
( ( $true
= ( X175 @ ( ap @ X177 @ X176 ) ) )
| ( ( X175 @ X177 )
!= $true )
| ( ( X175 @ X176 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f52,plain,
! [X175: term > $o] :
( ( ( sP37 @ X175 )
= $true )
<=> ! [X178: term] :
( ( ( X175 @ X178 )
!= $true )
| ( $true
= ( X175 @ ( lam @ X178 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f53,plain,
( ( sP38 = $true )
<=> ! [X180: term,X181: term > $o,X182: subst] :
( ( ( sP35 @ X181 )
= $true )
| ? [X185: term] :
( ( ( var @ X185 )
= $true )
& ( ( X181 @ ( sub @ X185 @ X182 ) )
!= $true ) )
| ( ( sP34 @ X181 )
= $true )
| ( ( X181 @ ( sub @ X180 @ X182 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f54,plain,
( ( sP39 = $true )
<=> ? [X174: term,X175: term > $o] :
( ( ( X175 @ X174 )
!= $true )
& ! [X179: term] :
( ( $true
= ( X175 @ X179 ) )
| ( $true
!= ( var @ X179 ) ) )
& ( $true
= ( sP36 @ X175 ) )
& ( ( sP37 @ X175 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f55,plain,
! [X459: subst > term > subst > $o] :
( ( $true
= ( sP40 @ X459 ) )
<=> ? [X462: subst > term > term] :
( ! [X464: subst,X465: term,X466: subst] :
( ( X462 @ ( comp @ X466 @ X464 ) @ ( sub @ X465 @ X464 ) )
= ( sub @ ( X462 @ X466 @ X465 ) @ X464 ) )
& ! [X463: term] :
( ( ( X459 @ id @ ( X462 @ id @ X463 ) @ id )
= $true )
| ( ( X459 @ id @ X463 @ id )
!= $true ) )
& ( $true
!= ( X459 @ id @ ( lam @ ( X462 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f56,plain,
! [X459: subst > term > subst > $o] :
( ( $true
= ( sP41 @ X459 ) )
<=> ? [X460: term,X461: term] :
( ( $true
= ( X459 @ id @ X460 @ id ) )
& ( $true
= ( X459 @ id @ X461 @ id ) )
& ( $true
!= ( X459 @ id @ ( ap @ ( sub @ X460 @ id ) @ X461 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f57,plain,
! [X459: subst > term > subst > $o] :
( ( ( sP42 @ X459 )
= $true )
<=> ? [X475: term] :
( ( $true
= ( var @ ( sub @ X475 @ id ) ) )
& ( $true
!= ( X459 @ id @ X475 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f58,plain,
! [X459: subst > term > subst > $o] :
( ( ( sP43 @ X459 )
= $true )
<=> ? [X467: term,X468: subst,X469: subst,X470: subst] :
( ( ( X459 @ ( comp @ X469 @ X470 ) @ ( sub @ X467 @ X470 ) @ X468 )
= $true )
& ( ( X459 @ X469 @ X467 @ ( comp @ X470 @ X468 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f59,plain,
! [X422: term > $o] :
( ( ( sP44 @ X422 )
= $true )
<=> ! [X425: term,X426: term] :
( ( $true
!= ( X422 @ X425 ) )
| ( ( X422 @ X426 )
!= $true )
| ( ( X422 @ ( ap @ X426 @ X425 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f60,plain,
! [X422: term > $o] :
( ( $true
= ( sP45 @ X422 ) )
<=> ! [X427: term] :
( ? [X428: term] :
( ( ( X422 @ ( sub @ X427 @ ( push @ X428 @ id ) ) )
!= $true )
& ( ( X422 @ X428 )
= $true ) )
| ( $true
= ( X422 @ ( lam @ X427 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f61,plain,
! [X429: subst > term > subst > $o] :
( ( $true
= ( sP46 @ X429 ) )
<=> ! [X440: term,X441: term] :
( ( ( X429 @ id @ X440 @ id )
!= $true )
| ( $true
!= ( X429 @ id @ X441 @ id ) )
| ( $true
= ( X429 @ id @ ( ap @ ( sub @ X440 @ id ) @ X441 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f62,plain,
! [X429: subst > term > subst > $o] :
( ( ( sP47 @ X429 )
= $true )
<=> ! [X432: term,X433: subst,X434: subst,X435: subst] :
( ( ( X429 @ X435 @ X432 @ ( comp @ X434 @ X433 ) )
= $true )
| ( ( X429 @ ( comp @ X435 @ X434 ) @ ( sub @ X432 @ X434 ) @ X433 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f63,plain,
! [X443: subst > term > subst > $o] :
( ( ( sP48 @ X443 )
= $true )
<=> ! [X444: subst > term > term] :
( ? [X445: subst,X446: term,X447: subst] :
( ( X444 @ ( comp @ X447 @ X445 ) @ ( sub @ X446 @ X445 ) )
!= ( sub @ ( X444 @ X447 @ X446 ) @ X445 ) )
| ( $true
= ( X443 @ id @ ( lam @ ( X444 @ sh @ one ) ) @ id ) )
| ? [X448: term] :
( ( ( X443 @ id @ X448 @ id )
= $true )
& ( $true
!= ( X443 @ id @ ( X444 @ id @ X448 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f64,plain,
! [X443: subst > term > subst > $o] :
( ( ( sP49 @ X443 )
= $true )
<=> ! [X454: subst,X455: subst,X456: term,X457: subst] :
( ( $true
!= ( X443 @ X457 @ X456 @ ( comp @ X455 @ X454 ) ) )
| ( ( X443 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f65,plain,
! [X442: term,X443: subst > term > subst > $o] :
( ( ( sP50 @ X443 @ X442 )
= $true )
<=> ! [X449: term] :
( ( ( X443 @ id @ ( sub @ X442 @ ( push @ X449 @ id ) ) @ id )
= $true )
| ( $true
!= ( X443 @ id @ X449 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f66,plain,
( ( sP51 = $true )
<=> ! [X458: term,X459: subst > term > subst > $o] :
( ? [X471: subst,X472: subst,X473: subst,X474: term] :
( ( ( X459 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 )
!= $true )
& ( ( X459 @ X473 @ X474 @ ( comp @ X472 @ X471 ) )
= $true ) )
| ( $true
= ( sP40 @ X459 ) )
| ( ( X459 @ id @ X458 @ id )
= $true )
| ( ( sP43 @ X459 )
= $true )
| ( $true
= ( sP41 @ X459 ) )
| ( ( sP42 @ X459 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f67,plain,
( ( sP52 = $true )
<=> ? [X422: term > $o,X423: term] :
( ! [X424: term] :
( ( $true
= ( X422 @ X424 ) )
| ( ( var @ X424 )
!= $true ) )
& ( ( sP44 @ X422 )
= $true )
& ( $true
= ( sP45 @ X422 ) )
& ( ( X422 @ X423 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f68,plain,
( ( $true = sP53 )
<=> ? [X429: subst > term > subst > $o,X430: term,X431: term] :
( ( $true
= ( sP46 @ X429 ) )
& ( ( X429 @ id @ X430 @ id )
= $true )
& ! [X436: subst,X437: subst,X438: term,X439: subst] :
( ( $true
!= ( X429 @ X437 @ X438 @ ( comp @ X436 @ X439 ) ) )
| ( ( X429 @ ( comp @ X437 @ X436 ) @ ( sub @ X438 @ X436 ) @ X439 )
= $true ) )
& ( ( X429 @ id @ X431 @ id )
= $true )
& ( ( X429 @ id @ ( ap @ X431 @ X430 ) @ id )
!= $true )
& ( ( sP47 @ X429 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f69,plain,
( ( sP54 = $true )
<=> ? [X442: term,X443: subst > term > subst > $o] :
( ! [X450: subst,X451: term,X452: subst,X453: subst] :
( ( ( X443 @ ( comp @ X452 @ X453 ) @ ( sub @ X451 @ X453 ) @ X450 )
!= $true )
| ( $true
= ( X443 @ X452 @ X451 @ ( comp @ X453 @ X450 ) ) ) )
& ( ( sP50 @ X443 @ X442 )
= $true )
& ( ( X443 @ id @ ( lam @ X442 ) @ id )
!= $true )
& ( ( sP49 @ X443 )
= $true )
& ( ( sP48 @ X443 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f70,plain,
! [X351: subst > term > subst > $o] :
( ( ( sP55 @ X351 )
= $true )
<=> ? [X362: subst > term > term] :
( ! [X363: term,X364: subst,X365: subst] :
( ( X362 @ ( comp @ X365 @ X364 ) @ ( sub @ X363 @ X364 ) )
= ( sub @ ( X362 @ X365 @ X363 ) @ X364 ) )
& ! [X366: term] :
( ( ( X351 @ id @ ( X362 @ id @ X366 ) @ id )
= $true )
| ( $true
!= ( X351 @ id @ X366 @ id ) ) )
& ( ( X351 @ id @ ( lam @ ( X362 @ sh @ one ) ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f71,plain,
! [X351: subst > term > subst > $o] :
( ( ( sP56 @ X351 )
= $true )
<=> ? [X358: subst,X359: subst,X360: term,X361: subst] :
( ( $true
!= ( X351 @ X358 @ X360 @ ( comp @ X359 @ X361 ) ) )
& ( $true
= ( X351 @ ( comp @ X358 @ X359 ) @ ( sub @ X360 @ X359 ) @ X361 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f72,plain,
! [X351: subst > term > subst > $o] :
( ( ( sP57 @ X351 )
= $true )
<=> ? [X354: subst,X355: subst,X356: subst,X357: term] :
( ( ( X351 @ ( comp @ X356 @ X354 ) @ ( sub @ X357 @ X354 ) @ X355 )
!= $true )
& ( ( X351 @ X356 @ X357 @ ( comp @ X354 @ X355 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f73,plain,
( ( sP58 = $true )
<=> ! [X350: term,X351: subst > term > subst > $o,X352: term > $o] :
( ? [X367: term] :
( ( ( X352 @ ( sub @ X350 @ ( push @ X367 @ id ) ) )
!= $true )
& ( ( X352 @ X367 )
= $true ) )
| ( ( sP55 @ X351 )
= $true )
| ( ( sP57 @ X351 )
= $true )
| ( ( sP56 @ X351 )
= $true )
| ? [X353: term] :
( ( X351 @ id @ X353 @ id )
!= ( X352 @ X353 ) )
| ( ( X352 @ ( lam @ X350 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f74,plain,
! [X188: subst > term > subst > $o] :
( ( ( sP59 @ X188 )
= $true )
<=> ? [X200: subst > term > term] :
( ! [X204: term] :
( ( ( X188 @ id @ X204 @ id )
!= $true )
| ( ( X188 @ id @ ( X200 @ id @ X204 ) @ id )
= $true ) )
& ( ( X188 @ id @ ( lam @ ( X200 @ sh @ one ) ) @ id )
!= $true )
& ! [X201: subst,X202: subst,X203: term] :
( ( X200 @ ( comp @ X202 @ X201 ) @ ( sub @ X203 @ X201 ) )
= ( sub @ ( X200 @ X202 @ X203 ) @ X201 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f75,plain,
! [X188: subst > term > subst > $o] :
( ( $true
= ( sP60 @ X188 ) )
<=> ? [X195: subst,X196: subst,X197: subst,X198: term] :
( ( ( X188 @ ( comp @ X197 @ X196 ) @ ( sub @ X198 @ X196 ) @ X195 )
= $true )
& ( ( X188 @ X197 @ X198 @ ( comp @ X196 @ X195 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f76,plain,
! [X190: term > $o,X189: term] :
( ( $true
= ( sP61 @ X189 @ X190 ) )
<=> ? [X199: term] :
( ( ( X190 @ X199 )
= $true )
& ( ( X190 @ ( sub @ X189 @ ( push @ X199 @ id ) ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f77,plain,
( ( sP62 = $true )
<=> ! [X188: subst > term > subst > $o,X189: term,X190: term > $o] :
( ( ( X190 @ ( lam @ X189 ) )
= $true )
| ? [X191: subst,X192: subst,X193: subst,X194: term] :
( ( ( X188 @ ( comp @ X193 @ X191 ) @ ( sub @ X194 @ X191 ) @ X192 )
!= $true )
& ( ( X188 @ X193 @ X194 @ ( comp @ X191 @ X192 ) )
= $true ) )
| ( ( sP59 @ X188 )
= $true )
| ? [X205: term] :
( ( X188 @ id @ X205 @ id )
!= ( X190 @ X205 ) )
| ( $true
= ( sP61 @ X189 @ X190 ) )
| ( $true
= ( sP60 @ X188 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f78,plain,
! [X62: subst > term > subst > $o] :
( ( $true
= ( sP63 @ X62 ) )
<=> ! [X63: subst > term > term] :
( ( ( X62 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true )
| ? [X64: subst,X65: subst,X66: term] :
( ( X63 @ ( comp @ X65 @ X64 ) @ ( sub @ X66 @ X64 ) )
!= ( sub @ ( X63 @ X65 @ X66 ) @ X64 ) )
| ? [X67: term] :
( ( $true
!= ( X62 @ id @ ( X63 @ id @ X67 ) @ id ) )
& ( $true
= ( X62 @ id @ X67 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f79,plain,
! [X70: subst > term > subst > $o] :
( ( $true
= ( sP64 @ X70 ) )
<=> ? [X72: subst > term > term] :
( ! [X76: term] :
( ( $true
!= ( X70 @ id @ X76 @ id ) )
| ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true ) )
& ( $true
!= ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id ) )
& ! [X73: subst,X74: term,X75: subst] :
( ( sub @ ( X72 @ X75 @ X74 ) @ X73 )
= ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f80,plain,
( ( $true = sP65 )
<=> ! [X69: term,X70: subst > term > subst > $o] :
( ( $true
= ( sP64 @ X70 ) )
| ? [X71: term] :
( ( $true
= ( X70 @ id @ X71 @ id ) )
& ( $true
!= ( X70 @ id @ ( sub @ X69 @ ( push @ X71 @ id ) ) @ id ) ) )
| ( ( X70 @ id @ ( lam @ X69 ) @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f81,plain,
( ( $true = sP66 )
<=> ? [X61: term,X62: subst > term > subst > $o] :
( ! [X68: term] :
( ( ( X62 @ id @ ( sub @ X61 @ ( push @ X68 @ id ) ) @ id )
= $true )
| ( ( X62 @ id @ X68 @ id )
!= $true ) )
& ( ( X62 @ id @ ( lam @ ( sub @ X61 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ( $true
= ( sP63 @ X62 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f82,plain,
! [X124: subst > term > subst > $o] :
( ( ( sP67 @ X124 )
= $true )
<=> ? [X125: subst > term > term] :
( ! [X127: term,X128: subst,X129: subst] :
( ( X125 @ ( comp @ X129 @ X128 ) @ ( sub @ X127 @ X128 ) )
= ( sub @ ( X125 @ X129 @ X127 ) @ X128 ) )
& ! [X126: term] :
( ( ( X124 @ id @ X126 @ id )
!= $true )
| ( ( X124 @ id @ ( X125 @ id @ X126 ) @ id )
= $true ) )
& ( $true
!= ( X124 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f83,plain,
! [X123: term,X124: subst > term > subst > $o] :
( ( $true
= ( sP68 @ X124 @ X123 ) )
<=> ? [X138: term] :
( ( ( X124 @ id @ ( sub @ X123 @ ( push @ X138 @ id ) ) @ id )
!= $true )
& ( ( X124 @ id @ X138 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f84,plain,
! [X124: subst > term > subst > $o] :
( ( $true
= ( sP69 @ X124 ) )
<=> ? [X134: term,X135: subst,X136: subst,X137: subst] :
( ( ( X124 @ X137 @ X134 @ ( comp @ X136 @ X135 ) )
= $true )
& ( $true
!= ( X124 @ ( comp @ X137 @ X136 ) @ ( sub @ X134 @ X136 ) @ X135 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f85,plain,
! [X116: subst > term > subst > $o] :
( ( $true
= ( sP70 @ X116 ) )
<=> ! [X117: subst > term > term] :
( ( ( X116 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id )
= $true )
| ? [X119: term,X120: subst,X121: subst] :
( ( X117 @ ( comp @ X121 @ X120 ) @ ( sub @ X119 @ X120 ) )
!= ( sub @ ( X117 @ X121 @ X119 ) @ X120 ) )
| ? [X118: term] :
( ( $true
= ( X116 @ id @ X118 @ id ) )
& ( $true
!= ( X116 @ id @ ( X117 @ id @ X118 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f86,plain,
( ( sP71 = $true )
<=> ! [X123: term,X124: subst > term > subst > $o] :
( ( $true
= ( X124 @ id @ ( lam @ X123 ) @ id ) )
| ? [X130: term,X131: subst,X132: subst,X133: subst] :
( ( ( X124 @ X132 @ X130 @ ( comp @ X131 @ X133 ) )
!= $true )
& ( $true
= ( X124 @ ( comp @ X132 @ X131 ) @ ( sub @ X130 @ X131 ) @ X133 ) ) )
| ( $true
= ( sP69 @ X124 ) )
| ( $true
= ( sP68 @ X124 @ X123 ) )
| ( ( sP67 @ X124 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f87,plain,
( ( sP72 = $true )
<=> ? [X115: term,X116: subst > term > subst > $o] :
( ! [X122: term] :
( ( ( X116 @ id @ X122 @ id )
!= $true )
| ( ( X116 @ id @ ( sub @ X115 @ ( push @ X122 @ id ) ) @ id )
= $true ) )
& ( ( X116 @ id @ ( lam @ ( sub @ X115 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ( $true
= ( sP70 @ X116 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f88,plain,
( ( sP73 = $true )
<=> ! [X480: term,X481: term,X482: term,X483: term] :
( ( X482 = X483 )
| ( ( ap @ ( sub @ X481 @ id ) @ X483 )
!= ( ap @ ( sub @ X480 @ id ) @ X482 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f89,plain,
( ( sP74 = $true )
<=> ( ? [X476: term,X477: term,X478: term,X479: term] :
( ( ( ap @ X477 @ X476 )
= ( ap @ X478 @ X479 ) )
& ( X476 != X479 ) )
| ( sP73 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f90,plain,
! [X370: subst > term > subst > $o] :
( ( ( sP75 @ X370 )
= $true )
<=> ? [X371: subst > term > term] :
( ( $true
!= ( X370 @ id @ ( lam @ ( X371 @ sh @ one ) ) @ id ) )
& ! [X372: term] :
( ( $true
!= ( X370 @ id @ X372 @ id ) )
| ( $true
= ( X370 @ id @ ( X371 @ id @ X372 ) @ id ) ) )
& ! [X373: term,X374: subst,X375: subst] :
( ( sub @ ( X371 @ X375 @ X373 ) @ X374 )
= ( X371 @ ( comp @ X375 @ X374 ) @ ( sub @ X373 @ X374 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f91,plain,
( ( sP76 = $true )
<=> ! [X368: term,X369: term > $o,X370: subst > term > subst > $o] :
( ? [X376: term] :
( ( X370 @ id @ X376 @ id )
!= ( X369 @ X376 ) )
| ? [X377: term] :
( ( ( X369 @ ( sub @ X368 @ ( push @ X377 @ id ) ) )
!= $true )
& ( ( X369 @ X377 )
= $true ) )
| ( ( sP75 @ X370 )
= $true )
| ( $true
= ( X369 @ ( lam @ X368 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f92,plain,
! [X388: term > $o] :
( ( ( sP77 @ X388 )
= $true )
<=> ! [X392: term,X393: term] :
( ( ( X388 @ X393 )
!= $true )
| ( $true
= ( X388 @ ( ap @ X393 @ X392 ) ) )
| ( $true
!= ( X388 @ X392 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f93,plain,
! [X388: term > $o] :
( ( $true
= ( sP78 @ X388 ) )
<=> ! [X389: term] :
( ( $true
!= ( var @ X389 ) )
| ( $true
= ( X388 @ X389 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f94,plain,
! [X388: term > $o] :
( ( ( sP79 @ X388 )
= $true )
<=> ! [X390: term] :
( ? [X391: term] :
( ( ( X388 @ X391 )
= $true )
& ( ( X388 @ ( sub @ X390 @ ( push @ X391 @ id ) ) )
!= $true ) )
| ( ( X388 @ ( lam @ X390 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f95,plain,
! [X406: subst > term > subst > $o] :
( ( ( sP80 @ X406 )
= $true )
<=> ? [X407: subst > term > term] :
( ! [X408: term] :
( ( ( X406 @ id @ ( X407 @ id @ X408 ) @ id )
= $true )
| ( ( X406 @ id @ X408 @ id )
!= $true ) )
& ( $true
!= ( X406 @ id @ ( lam @ ( X407 @ sh @ one ) ) @ id ) )
& ! [X409: subst,X410: subst,X411: term] :
( ( sub @ ( X407 @ X410 @ X411 ) @ X409 )
= ( X407 @ ( comp @ X410 @ X409 ) @ ( sub @ X411 @ X409 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f96,plain,
! [X406: subst > term > subst > $o] :
( ( ( sP81 @ X406 )
= $true )
<=> ? [X412: term,X413: term] :
( ( ( X406 @ id @ ( ap @ ( sub @ X412 @ id ) @ X413 ) @ id )
!= $true )
& ( $true
= ( X406 @ id @ X413 @ id ) )
& ( ( X406 @ id @ X412 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f97,plain,
! [X397: subst > term > subst > $o] :
( ( ( sP82 @ X397 )
= $true )
<=> ! [X399: subst > term > term] :
( ? [X400: subst,X401: subst,X402: term] :
( ( sub @ ( X399 @ X400 @ X402 ) @ X401 )
!= ( X399 @ ( comp @ X400 @ X401 ) @ ( sub @ X402 @ X401 ) ) )
| ? [X403: term] :
( ( $true
!= ( X397 @ id @ ( X399 @ id @ X403 ) @ id ) )
& ( ( X397 @ id @ X403 @ id )
= $true ) )
| ( $true
= ( X397 @ id @ ( lam @ ( X399 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f98,plain,
( ( sP83 = $true )
<=> ! [X405: term,X406: subst > term > subst > $o] :
( ( $true
= ( X406 @ id @ X405 @ id ) )
| ( ( sP80 @ X406 )
= $true )
| ( ( sP81 @ X406 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f99,plain,
( ( sP84 = $true )
<=> ? [X395: term > $o,X396: term,X397: subst > term > subst > $o] :
( ! [X404: term] :
( ( ( X395 @ ( sub @ X396 @ ( push @ X404 @ id ) ) )
= $true )
| ( ( X395 @ X404 )
!= $true ) )
& ! [X398: term] :
( ( X397 @ id @ X398 @ id )
= ( X395 @ X398 ) )
& ( ( sP82 @ X397 )
= $true )
& ( $true
!= ( X395 @ ( lam @ X396 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f100,plain,
( ( sP85 = $true )
<=> ? [X387: term,X388: term > $o] :
( ( ( X388 @ X387 )
!= $true )
& ( ( sP79 @ X388 )
= $true )
& ( $true
= ( sP78 @ X388 ) )
& ( ( sP77 @ X388 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f101,plain,
( ( sP86 = $true )
<=> ! [X345: term,X346: term,X347: term,X348: term] :
( ( ( ap @ ( sub @ X345 @ id ) @ X347 )
!= ( ap @ ( sub @ X348 @ id ) @ X346 ) )
| ( X345 = X348 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f102,plain,
( ( sP87 = $true )
<=> ( ? [X341: term,X342: term,X343: term,X344: term] :
( ( X341 != X342 )
& ( ( ap @ X342 @ X343 )
= ( ap @ X341 @ X344 ) ) )
| ( sP86 = $true )
| ? [X340: term] :
( ( sub @ X340 @ id )
!= X340 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f103,plain,
! [X37: subst > term > subst > $o] :
( ( $true
= ( sP88 @ X37 ) )
<=> ? [X41: subst > term > term] :
( ( ( X37 @ id @ ( lam @ ( X41 @ sh @ one ) ) @ id )
!= $true )
& ! [X45: term] :
( ( $true
= ( X37 @ id @ ( X41 @ id @ X45 ) @ id ) )
| ( ( X37 @ id @ X45 @ id )
!= $true ) )
& ! [X42: term,X43: subst,X44: subst] :
( ( X41 @ ( comp @ X43 @ X44 ) @ ( sub @ X42 @ X44 ) )
= ( sub @ ( X41 @ X43 @ X42 ) @ X44 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f104,plain,
! [X37: subst > term > subst > $o] :
( ( ( sP89 @ X37 )
= $true )
<=> ? [X39: term,X40: term] :
( ( $true
= ( X37 @ id @ X40 @ id ) )
& ( $true
= ( X37 @ id @ X39 @ id ) )
& ( $true
!= ( X37 @ id @ ( ap @ ( sub @ X40 @ id ) @ X39 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f105,plain,
! [X37: subst > term > subst > $o] :
( ( ( sP90 @ X37 )
= $true )
<=> ? [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
= ( X37 @ ( comp @ X51 @ X50 ) @ ( sub @ X52 @ X50 ) @ X53 ) )
& ( ( X37 @ X51 @ X52 @ ( comp @ X50 @ X53 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f106,plain,
! [X37: subst > term > subst > $o] :
( ( ( sP91 @ X37 )
= $true )
<=> ? [X54: term] :
( ( $true
= ( var @ ( sub @ X54 @ id ) ) )
& ( $true
!= ( X37 @ id @ X54 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f107,plain,
( ( sP92 = $true )
<=> ! [X37: subst > term > subst > $o,X38: term] :
( ? [X46: subst,X47: subst,X48: subst,X49: term] :
( ( ( X37 @ X46 @ X49 @ ( comp @ X47 @ X48 ) )
= $true )
& ( ( X37 @ ( comp @ X46 @ X47 ) @ ( sub @ X49 @ X47 ) @ X48 )
!= $true ) )
| ( $true
= ( X37 @ id @ X38 @ id ) )
| ( ( sP89 @ X37 )
= $true )
| ( $true
= ( sP88 @ X37 ) )
| ( ( sP91 @ X37 )
= $true )
| ( ( sP90 @ X37 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f108,plain,
! [X415: subst > term > subst > $o] :
( ( ( sP93 @ X415 )
= $true )
<=> ? [X416: subst > term > term] :
( ! [X417: term] :
( ( ( X415 @ id @ ( X416 @ id @ X417 ) @ id )
= $true )
| ( ( X415 @ id @ X417 @ id )
!= $true ) )
& ( ( X415 @ id @ ( lam @ ( X416 @ sh @ one ) ) @ id )
!= $true )
& ! [X418: term,X419: subst,X420: subst] :
( ( sub @ ( X416 @ X419 @ X418 ) @ X420 )
= ( X416 @ ( comp @ X419 @ X420 ) @ ( sub @ X418 @ X420 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f109,plain,
( ( sP94 = $true )
<=> ! [X414: term,X415: subst > term > subst > $o] :
( ( ( sP93 @ X415 )
= $true )
| ( $true
= ( X415 @ id @ ( lam @ ( sub @ X414 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X421: term] :
( ( $true
= ( X415 @ id @ X421 @ id ) )
& ( ( X415 @ id @ ( sub @ X414 @ ( push @ X421 @ id ) ) @ id )
!= $true ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f110,plain,
! [X206: term > $o] :
( ( ( sP95 @ X206 )
= $true )
<=> ? [X209: term,X210: term] :
( ( $true
= ( X206 @ X210 ) )
& ( ( X206 @ ( ap @ X209 @ X210 ) )
!= $true )
& ( $true
= ( X206 @ X209 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f111,plain,
! [X206: term > $o] :
( ( ( sP96 @ X206 )
= $true )
<=> ? [X208: term] :
( ( $true
!= ( X206 @ ( lam @ X208 ) ) )
& ( $true
= ( X206 @ X208 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f112,plain,
( ( sP97 = $true )
<=> ! [X206: term > $o,X207: term] :
( ( ( sP95 @ X206 )
= $true )
| ? [X211: term] :
( ( $true
= ( var @ X211 ) )
& ( $true
!= ( X206 @ X211 ) ) )
| ( ( sP96 @ X206 )
= $true )
| ( ( X206 @ X207 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f113,plain,
! [X505: subst > term > subst > $o] :
( ( $true
= ( sP98 @ X505 ) )
<=> ? [X508: subst > term > term] :
( ( ( X505 @ id @ ( lam @ ( X508 @ sh @ one ) ) @ id )
!= $true )
& ! [X510: subst,X511: subst,X512: term] :
( ( X508 @ ( comp @ X511 @ X510 ) @ ( sub @ X512 @ X510 ) )
= ( sub @ ( X508 @ X511 @ X512 ) @ X510 ) )
& ! [X509: term] :
( ( $true
!= ( X505 @ id @ X509 @ id ) )
| ( $true
= ( X505 @ id @ ( X508 @ id @ X509 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f114,plain,
( ( $true = sP99 )
<=> ! [X505: subst > term > subst > $o,X506: term] :
( ( $true
= ( sP98 @ X505 ) )
| ? [X507: term] :
( ( $true
= ( X505 @ id @ X507 @ id ) )
& ( ( X505 @ id @ ( sub @ X506 @ ( push @ X507 @ id ) ) @ id )
!= $true ) )
| ( ( X505 @ id @ ( lam @ ( sub @ X506 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f115,plain,
! [X518: subst > term > subst > $o] :
( ( $true
= ( sP100 @ X518 ) )
<=> ? [X525: term,X526: term] :
( ( $true
= ( X518 @ id @ X525 @ id ) )
& ( $true
= ( X518 @ id @ X526 @ id ) )
& ( ( X518 @ id @ ( ap @ ( sub @ X525 @ id ) @ X526 ) @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f116,plain,
! [X518: subst > term > subst > $o] :
( ( ( sP101 @ X518 )
= $true )
<=> ? [X527: subst,X528: subst,X529: term,X530: subst] :
( ( $true
= ( X518 @ X528 @ X529 @ ( comp @ X530 @ X527 ) ) )
& ( $true
!= ( X518 @ ( comp @ X528 @ X530 ) @ ( sub @ X529 @ X530 ) @ X527 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f117,plain,
( ( $true = sP102 )
<=> ! [X518: subst > term > subst > $o,X519: term,X520: term] :
( ( ( X518 @ id @ X519 @ id )
!= $true )
| ( ( X518 @ id @ ( ap @ X519 @ X520 ) @ id )
= $true )
| ( $true
!= ( X518 @ id @ X520 @ id ) )
| ? [X521: subst,X522: subst,X523: term,X524: subst] :
( ( $true
!= ( X518 @ X522 @ X523 @ ( comp @ X524 @ X521 ) ) )
& ( $true
= ( X518 @ ( comp @ X522 @ X524 ) @ ( sub @ X523 @ X524 ) @ X521 ) ) )
| ( $true
= ( sP100 @ X518 ) )
| ( ( sP101 @ X518 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f118,plain,
! [X106: subst > term > subst > $o] :
( ( $true
= ( sP103 @ X106 ) )
<=> ? [X107: subst > term > term] :
( ! [X108: term] :
( ( $true
= ( X106 @ id @ ( X107 @ id @ X108 ) @ id ) )
| ( $true
!= ( X106 @ id @ X108 @ id ) ) )
& ( ( X106 @ id @ ( lam @ ( X107 @ sh @ one ) ) @ id )
!= $true )
& ! [X109: subst,X110: term,X111: subst] :
( ( X107 @ ( comp @ X111 @ X109 ) @ ( sub @ X110 @ X109 ) )
= ( sub @ ( X107 @ X111 @ X110 ) @ X109 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f119,plain,
! [X106: subst > term > subst > $o] :
( ( $true
= ( sP104 @ X106 ) )
<=> ? [X112: term,X113: term] :
( ( ( X106 @ id @ X113 @ id )
= $true )
& ( ( X106 @ id @ X112 @ id )
= $true )
& ( $true
!= ( X106 @ id @ ( ap @ ( sub @ X113 @ id ) @ X112 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f120,plain,
! [X283: subst > term > subst > $o] :
( ( ( sP105 @ X283 )
= $true )
<=> ? [X285: subst > term > term] :
( ! [X286: term] :
( ( $true
= ( X283 @ id @ ( X285 @ id @ X286 ) @ id ) )
| ( $true
!= ( X283 @ id @ X286 @ id ) ) )
& ( $true
!= ( X283 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id ) )
& ! [X287: term,X288: subst,X289: subst] :
( ( X285 @ ( comp @ X289 @ X288 ) @ ( sub @ X287 @ X288 ) )
= ( sub @ ( X285 @ X289 @ X287 ) @ X288 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f121,plain,
( ( $true = sP106 )
<=> ! [X282: term > $o,X283: subst > term > subst > $o,X284: term] :
( ( ( sP105 @ X283 )
= $true )
| ( ( X282 @ ( lam @ X284 ) )
= $true )
| ? [X291: term] :
( ( ( X282 @ X291 )
= $true )
& ( $true
!= ( X282 @ ( sub @ X284 @ ( push @ X291 @ id ) ) ) ) )
| ? [X290: term] :
( ( X282 @ X290 )
!= ( X283 @ id @ X290 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f122,plain,
! [X222: subst > term > subst > $o] :
( ( ( sP107 @ X222 )
= $true )
<=> ? [X233: term,X234: term] :
( ( $true
!= ( X222 @ id @ ( ap @ ( sub @ X234 @ id ) @ X233 ) @ id ) )
& ( ( X222 @ id @ X234 @ id )
= $true )
& ( $true
= ( X222 @ id @ X233 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f123,plain,
! [X222: subst > term > subst > $o] :
( ( ( sP108 @ X222 )
= $true )
<=> ? [X224: subst,X225: term,X226: subst,X227: subst] :
( ( ( X222 @ ( comp @ X224 @ X227 ) @ ( sub @ X225 @ X227 ) @ X226 )
!= $true )
& ( ( X222 @ X224 @ X225 @ ( comp @ X227 @ X226 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f124,plain,
( ( $true = sP109 )
<=> ! [X220: term > $o,X221: term,X222: subst > term > subst > $o,X223: term] :
( ( ( X220 @ X223 )
!= $true )
| ( ( sP107 @ X222 )
= $true )
| ? [X228: subst,X229: subst,X230: subst,X231: term] :
( ( $true
!= ( X222 @ X230 @ X231 @ ( comp @ X228 @ X229 ) ) )
& ( ( X222 @ ( comp @ X230 @ X228 ) @ ( sub @ X231 @ X228 ) @ X229 )
= $true ) )
| ( ( X220 @ X221 )
!= $true )
| ( ( sP108 @ X222 )
= $true )
| ( ( X220 @ ( ap @ X221 @ X223 ) )
= $true )
| ? [X232: term] :
( ( X220 @ X232 )
!= ( X222 @ id @ X232 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f14,plain,
( ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
| ( ( X18 @ X15 )
!= $true ) )
| ? [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
!= X11 )
| ? [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
!= X12 ) ) )
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
!= ( d2subst @ X55 ) )
| ( X55 = X56 ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X220: term > $o,X221: term,X222: subst > term > subst > $o,X223: term] :
( ( ( X220 @ X223 )
!= $true )
| ? [X233: term,X234: term] :
( ( $true
!= ( X222 @ id @ ( ap @ ( sub @ X234 @ id ) @ X233 ) @ id ) )
& ( ( X222 @ id @ X234 @ id )
= $true )
& ( $true
= ( X222 @ id @ X233 @ id ) ) )
| ? [X228: subst,X229: subst,X230: subst,X231: term] :
( ( $true
!= ( X222 @ X230 @ X231 @ ( comp @ X228 @ X229 ) ) )
& ( ( X222 @ ( comp @ X230 @ X228 ) @ ( sub @ X231 @ X228 ) @ X229 )
= $true ) )
| ( ( X220 @ X221 )
!= $true )
| ? [X224: subst,X225: term,X226: subst,X227: subst] :
( ( ( X222 @ ( comp @ X224 @ X227 ) @ ( sub @ X225 @ X227 ) @ X226 )
!= $true )
& ( ( X222 @ X224 @ X225 @ ( comp @ X227 @ X226 ) )
= $true ) )
| ( ( X220 @ ( ap @ X221 @ X223 ) )
= $true )
| ? [X232: term] :
( ( X220 @ X232 )
!= ( X222 @ id @ X232 @ id ) ) ) )
& ( ( ? [X281: term] :
( ( sub @ X281 @ id )
!= X281 )
| ? [X277: term] :
! [X278: subst > term > term] :
? [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
!= ( X278 @ X280 @ X279 ) )
| ! [X282: term > $o,X283: subst > term > subst > $o,X284: term] :
( ? [X285: subst > term > term] :
( ! [X286: term] :
( ( $true
= ( X283 @ id @ ( X285 @ id @ X286 ) @ id ) )
| ( $true
!= ( X283 @ id @ X286 @ id ) ) )
& ( $true
!= ( X283 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id ) )
& ! [X287: term,X288: subst,X289: subst] :
( ( X285 @ ( comp @ X289 @ X288 ) @ ( sub @ X287 @ X288 ) )
= ( sub @ ( X285 @ X289 @ X287 ) @ X288 ) ) )
| ( ( X282 @ ( lam @ X284 ) )
= $true )
| ? [X291: term] :
( ( ( X282 @ X291 )
= $true )
& ( $true
!= ( X282 @ ( sub @ X284 @ ( push @ X291 @ id ) ) ) ) )
| ? [X290: term] :
( ( X282 @ X290 )
!= ( X283 @ id @ X290 @ id ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X7: subst > term > subst > $o] :
? [X8: term > $o] :
! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X338: term] :
? [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
= ( X339 @ sh @ one ) ) )
& ( induction2lem_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
= X273 )
<=> ( axshiftcons = $true ) )
& ( ! [X105: term,X106: subst > term > subst > $o] :
( ? [X107: subst > term > term] :
( ! [X108: term] :
( ( $true
= ( X106 @ id @ ( X107 @ id @ X108 ) @ id ) )
| ( $true
!= ( X106 @ id @ X108 @ id ) ) )
& ( ( X106 @ id @ ( lam @ ( X107 @ sh @ one ) ) @ id )
!= $true )
& ! [X109: subst,X110: term,X111: subst] :
( ( X107 @ ( comp @ X111 @ X109 ) @ ( sub @ X110 @ X109 ) )
= ( sub @ ( X107 @ X111 @ X110 ) @ X109 ) ) )
| ( $true
= ( X106 @ id @ X105 @ id ) )
| ? [X112: term,X113: term] :
( ( ( X106 @ id @ X113 @ id )
= $true )
& ( ( X106 @ id @ X112 @ id )
= $true )
& ( $true
!= ( X106 @ id @ ( ap @ ( sub @ X113 @ id ) @ X112 ) @ id ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( termmset_gthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X518: subst > term > subst > $o,X519: term,X520: term] :
( ( ( X518 @ id @ X519 @ id )
!= $true )
| ( ( X518 @ id @ ( ap @ X519 @ X520 ) @ id )
= $true )
| ( $true
!= ( X518 @ id @ X520 @ id ) )
| ? [X521: subst,X522: subst,X523: term,X524: subst] :
( ( $true
!= ( X518 @ X522 @ X523 @ ( comp @ X524 @ X521 ) ) )
& ( $true
= ( X518 @ ( comp @ X522 @ X524 ) @ ( sub @ X523 @ X524 ) @ X521 ) ) )
| ? [X525: term,X526: term] :
( ( $true
= ( X518 @ id @ X525 @ id ) )
& ( $true
= ( X518 @ id @ X526 @ id ) )
& ( ( X518 @ id @ ( ap @ ( sub @ X525 @ id ) @ X526 ) @ id )
!= $true ) )
| ? [X527: subst,X528: subst,X529: term,X530: subst] :
( ( $true
= ( X518 @ X528 @ X529 @ ( comp @ X530 @ X527 ) ) )
& ( $true
!= ( X518 @ ( comp @ X528 @ X530 ) @ ( sub @ X529 @ X530 ) @ X527 ) ) ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X505: subst > term > subst > $o,X506: term] :
( ? [X508: subst > term > term] :
( ( ( X505 @ id @ ( lam @ ( X508 @ sh @ one ) ) @ id )
!= $true )
& ! [X510: subst,X511: subst,X512: term] :
( ( X508 @ ( comp @ X511 @ X510 ) @ ( sub @ X512 @ X510 ) )
= ( sub @ ( X508 @ X511 @ X512 ) @ X510 ) )
& ! [X509: term] :
( ( $true
!= ( X505 @ id @ X509 @ id ) )
| ( $true
= ( X505 @ id @ ( X508 @ id @ X509 ) @ id ) ) ) )
| ? [X507: term] :
( ( $true
= ( X505 @ id @ X507 @ id ) )
& ( ( X505 @ id @ ( sub @ X506 @ ( push @ X507 @ id ) ) @ id )
!= $true ) )
| ( ( X505 @ id @ ( lam @ ( sub @ X506 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X513: term,X514: term > $o,X515: subst] :
? [X516: term > $o] :
! [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
= ( X516 @ X517 ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 )
& ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 ) )
<=> ( substmonoid = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lthm = $true )
& ! [X96: d_term] : ( d_one = X96 )
& ( lamnotvar = $true )
& ( ( induction = $true )
<=> ! [X206: term > $o,X207: term] :
( ? [X209: term,X210: term] :
( ( $true
= ( X206 @ X210 ) )
& ( ( X206 @ ( ap @ X209 @ X210 ) )
!= $true )
& ( $true
= ( X206 @ X209 ) ) )
| ? [X211: term] :
( ( $true
= ( var @ X211 ) )
& ( $true
!= ( X206 @ X211 ) ) )
| ? [X208: term] :
( ( $true
!= ( X206 @ ( lam @ X208 ) ) )
& ( $true
= ( X206 @ X208 ) ) )
| ( ( X206 @ X207 )
= $true ) ) )
& ( ! [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
!= ( sub @ X275 @ sh ) )
| ( X275 = X276 ) )
<=> ( shinj = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ! [X93: subst > term > subst > $o] :
? [X94: term > $o] :
! [X95: term] :
( ( X94 @ X95 )
= ( X93 @ id @ X95 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ! [X414: term,X415: subst > term > subst > $o] :
( ? [X416: subst > term > term] :
( ! [X417: term] :
( ( ( X415 @ id @ ( X416 @ id @ X417 ) @ id )
= $true )
| ( ( X415 @ id @ X417 @ id )
!= $true ) )
& ( ( X415 @ id @ ( lam @ ( X416 @ sh @ one ) ) @ id )
!= $true )
& ! [X418: term,X419: subst,X420: subst] :
( ( sub @ ( X416 @ X419 @ X418 ) @ X420 )
= ( X416 @ ( comp @ X419 @ X420 ) @ ( sub @ X418 @ X420 ) ) ) )
| ( $true
= ( X415 @ id @ ( lam @ ( sub @ X414 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X421: term] :
( ( $true
= ( X415 @ id @ X421 @ id ) )
& ( ( X415 @ id @ ( sub @ X414 @ ( push @ X421 @ id ) ) @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ( hoasinduction = $true )
<=> ! [X37: subst > term > subst > $o,X38: term] :
( ? [X46: subst,X47: subst,X48: subst,X49: term] :
( ( ( X37 @ X46 @ X49 @ ( comp @ X47 @ X48 ) )
= $true )
& ( ( X37 @ ( comp @ X46 @ X47 ) @ ( sub @ X49 @ X47 ) @ X48 )
!= $true ) )
| ( $true
= ( X37 @ id @ X38 @ id ) )
| ? [X39: term,X40: term] :
( ( $true
= ( X37 @ id @ X40 @ id ) )
& ( $true
= ( X37 @ id @ X39 @ id ) )
& ( $true
!= ( X37 @ id @ ( ap @ ( sub @ X40 @ id ) @ X39 ) @ id ) ) )
| ? [X41: subst > term > term] :
( ( ( X37 @ id @ ( lam @ ( X41 @ sh @ one ) ) @ id )
!= $true )
& ! [X45: term] :
( ( $true
= ( X37 @ id @ ( X41 @ id @ X45 ) @ id ) )
| ( ( X37 @ id @ X45 @ id )
!= $true ) )
& ! [X42: term,X43: subst,X44: subst] :
( ( X41 @ ( comp @ X43 @ X44 ) @ ( sub @ X42 @ X44 ) )
= ( sub @ ( X41 @ X43 @ X42 ) @ X44 ) ) )
| ? [X54: term] :
( ( $true
= ( var @ ( sub @ X54 @ id ) ) )
& ( $true
!= ( X37 @ id @ X54 @ id ) ) )
| ? [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
= ( X37 @ ( comp @ X51 @ X50 ) @ ( sub @ X52 @ X50 ) @ X53 ) )
& ( ( X37 @ X51 @ X52 @ ( comp @ X50 @ X53 ) )
!= $true ) ) ) )
& ( ( ? [X341: term,X342: term,X343: term,X344: term] :
( ( X341 != X342 )
& ( ( ap @ X342 @ X343 )
= ( ap @ X341 @ X344 ) ) )
| ! [X345: term,X346: term,X347: term,X348: term] :
( ( ( ap @ ( sub @ X345 @ id ) @ X347 )
!= ( ap @ ( sub @ X348 @ id ) @ X346 ) )
| ( X345 = X348 ) )
| ? [X340: term] :
( ( sub @ X340 @ id )
!= X340 ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( lamnotap != $true )
& ( ( apinj2 = $true )
<=> ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X98 ) ) )
& ( ( termmset = $true )
<=> ! [X57: term] :
( ( sub @ X57 @ id )
= X57 ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ulamvar1 != $true )
& ( axapp = $true )
& ! [X382: subst] :
? [X383: d_subst] :
( ( d2subst @ X383 )
= X382 )
& ( ! [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) )
| ( ( X266 @ X265 )
!= $true ) )
<=> ( pushprop_lem1 = $true ) )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ! [X270: d_subst] : ( d_id = X270 )
& ! [X23: d_term,X24: d_term] :
( ( X23 = X24 )
| ( ( d2term @ X23 )
!= ( d2term @ X24 ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( ? [X394: term] :
( ( sub @ X394 @ id )
!= X394 )
| ? [X395: term > $o,X396: term,X397: subst > term > subst > $o] :
( ! [X404: term] :
( ( ( X395 @ ( sub @ X396 @ ( push @ X404 @ id ) ) )
= $true )
| ( ( X395 @ X404 )
!= $true ) )
& ! [X398: term] :
( ( X397 @ id @ X398 @ id )
= ( X395 @ X398 ) )
& ! [X399: subst > term > term] :
( ? [X400: subst,X401: subst,X402: term] :
( ( sub @ ( X399 @ X400 @ X402 ) @ X401 )
!= ( X399 @ ( comp @ X400 @ X401 ) @ ( sub @ X402 @ X401 ) ) )
| ? [X403: term] :
( ( $true
!= ( X397 @ id @ ( X399 @ id @ X403 ) @ id ) )
& ( ( X397 @ id @ X403 @ id )
= $true ) )
| ( $true
= ( X397 @ id @ ( lam @ ( X399 @ sh @ one ) ) @ id ) ) )
& ( $true
!= ( X395 @ ( lam @ X396 ) ) ) )
| ? [X384: subst > term > subst > $o] :
! [X385: term > $o] :
? [X386: term] :
( ( X384 @ id @ X386 @ id )
!= ( X385 @ X386 ) )
| ! [X405: term,X406: subst > term > subst > $o] :
( ( $true
= ( X406 @ id @ X405 @ id ) )
| ? [X407: subst > term > term] :
( ! [X408: term] :
( ( ( X406 @ id @ ( X407 @ id @ X408 ) @ id )
= $true )
| ( ( X406 @ id @ X408 @ id )
!= $true ) )
& ( $true
!= ( X406 @ id @ ( lam @ ( X407 @ sh @ one ) ) @ id ) )
& ! [X409: subst,X410: subst,X411: term] :
( ( sub @ ( X407 @ X410 @ X411 ) @ X409 )
= ( X407 @ ( comp @ X410 @ X409 ) @ ( sub @ X411 @ X409 ) ) ) )
| ? [X412: term,X413: term] :
( ( ( X406 @ id @ ( ap @ ( sub @ X412 @ id ) @ X413 ) @ id )
!= $true )
& ( $true
= ( X406 @ id @ X413 @ id ) )
& ( ( X406 @ id @ X412 @ id )
= $true ) ) )
| ? [X387: term,X388: term > $o] :
( ( ( X388 @ X387 )
!= $true )
& ! [X390: term] :
( ? [X391: term] :
( ( ( X388 @ X391 )
= $true )
& ( ( X388 @ ( sub @ X390 @ ( push @ X391 @ id ) ) )
!= $true ) )
| ( ( X388 @ ( lam @ X390 ) )
= $true ) )
& ! [X389: term] :
( ( $true
!= ( var @ X389 ) )
| ( $true
= ( X388 @ X389 ) ) )
& ! [X392: term,X393: term] :
( ( ( X388 @ X393 )
!= $true )
| ( $true
= ( X388 @ ( ap @ X393 @ X392 ) ) )
| ( $true
!= ( X388 @ X392 ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( axclos = $true )
& ( ( laminj = $true )
<=> ! [X167: term,X168: term] :
( ( X167 = X168 )
| ( ( lam @ X168 )
!= ( lam @ X167 ) ) ) )
& ( ! [X368: term,X369: term > $o,X370: subst > term > subst > $o] :
( ? [X376: term] :
( ( X370 @ id @ X376 @ id )
!= ( X369 @ X376 ) )
| ? [X377: term] :
( ( ( X369 @ ( sub @ X368 @ ( push @ X377 @ id ) ) )
!= $true )
& ( ( X369 @ X377 )
= $true ) )
| ? [X371: subst > term > term] :
( ( $true
!= ( X370 @ id @ ( lam @ ( X371 @ sh @ one ) ) @ id ) )
& ! [X372: term] :
( ( $true
!= ( X370 @ id @ X372 @ id ) )
| ( $true
= ( X370 @ id @ ( X371 @ id @ X372 ) @ id ) ) )
& ! [X373: term,X374: subst,X375: subst] :
( ( sub @ ( X371 @ X375 @ X373 ) @ X374 )
= ( X371 @ ( comp @ X375 @ X374 ) @ ( sub @ X373 @ X374 ) ) ) )
| ( $true
= ( X369 @ ( lam @ X368 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ? [X476: term,X477: term,X478: term,X479: term] :
( ( ( ap @ X477 @ X476 )
= ( ap @ X478 @ X479 ) )
& ( X476 != X479 ) )
| ! [X480: term,X481: term,X482: term,X483: term] :
( ( X482 = X483 )
| ( ( ap @ ( sub @ X481 @ id ) @ X483 )
!= ( ap @ ( sub @ X480 @ id ) @ X482 ) ) ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ? [X115: term,X116: subst > term > subst > $o] :
( ! [X122: term] :
( ( ( X116 @ id @ X122 @ id )
!= $true )
| ( ( X116 @ id @ ( sub @ X115 @ ( push @ X122 @ id ) ) @ id )
= $true ) )
& ( ( X116 @ id @ ( lam @ ( sub @ X115 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X117: subst > term > term] :
( ( ( X116 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id )
= $true )
| ? [X119: term,X120: subst,X121: subst] :
( ( X117 @ ( comp @ X121 @ X120 ) @ ( sub @ X119 @ X120 ) )
!= ( sub @ ( X117 @ X121 @ X119 ) @ X120 ) )
| ? [X118: term] :
( ( $true
= ( X116 @ id @ X118 @ id ) )
& ( $true
!= ( X116 @ id @ ( X117 @ id @ X118 ) @ id ) ) ) ) )
| ? [X114: term] :
( ( sub @ X114 @ id )
!= X114 )
| ! [X123: term,X124: subst > term > subst > $o] :
( ( $true
= ( X124 @ id @ ( lam @ X123 ) @ id ) )
| ? [X130: term,X131: subst,X132: subst,X133: subst] :
( ( ( X124 @ X132 @ X130 @ ( comp @ X131 @ X133 ) )
!= $true )
& ( $true
= ( X124 @ ( comp @ X132 @ X131 ) @ ( sub @ X130 @ X131 ) @ X133 ) ) )
| ? [X134: term,X135: subst,X136: subst,X137: subst] :
( ( ( X124 @ X137 @ X134 @ ( comp @ X136 @ X135 ) )
= $true )
& ( $true
!= ( X124 @ ( comp @ X137 @ X136 ) @ ( sub @ X134 @ X136 ) @ X135 ) ) )
| ? [X138: term] :
( ( ( X124 @ id @ ( sub @ X123 @ ( push @ X138 @ id ) ) @ id )
!= $true )
& ( ( X124 @ id @ X138 @ id )
= $true ) )
| ? [X125: subst > term > term] :
( ! [X127: term,X128: subst,X129: subst] :
( ( X125 @ ( comp @ X129 @ X128 ) @ ( sub @ X127 @ X128 ) )
= ( sub @ ( X125 @ X129 @ X127 ) @ X128 ) )
& ! [X126: term] :
( ( ( X124 @ id @ X126 @ id )
!= $true )
| ( ( X124 @ id @ ( X125 @ id @ X126 ) @ id )
= $true ) )
& ( $true
!= ( X124 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) ) ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( pushprop_lthm_orig = $true )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ? [X60: term] :
( ( sub @ X60 @ id )
!= X60 )
| ! [X69: term,X70: subst > term > subst > $o] :
( ? [X72: subst > term > term] :
( ! [X76: term] :
( ( $true
!= ( X70 @ id @ X76 @ id ) )
| ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true ) )
& ( $true
!= ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id ) )
& ! [X73: subst,X74: term,X75: subst] :
( ( sub @ ( X72 @ X75 @ X74 ) @ X73 )
= ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) ) ) )
| ? [X71: term] :
( ( $true
= ( X70 @ id @ X71 @ id ) )
& ( $true
!= ( X70 @ id @ ( sub @ X69 @ ( push @ X71 @ id ) ) @ id ) ) )
| ( ( X70 @ id @ ( lam @ X69 ) @ id )
= $true ) )
| ? [X61: term,X62: subst > term > subst > $o] :
( ! [X68: term] :
( ( ( X62 @ id @ ( sub @ X61 @ ( push @ X68 @ id ) ) @ id )
= $true )
| ( ( X62 @ id @ X68 @ id )
!= $true ) )
& ( ( X62 @ id @ ( lam @ ( sub @ X61 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X63: subst > term > term] :
( ( ( X62 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true )
| ? [X64: subst,X65: subst,X66: term] :
( ( X63 @ ( comp @ X65 @ X64 ) @ ( sub @ X66 @ X64 ) )
!= ( sub @ ( X63 @ X65 @ X66 ) @ X64 ) )
| ? [X67: term] :
( ( $true
!= ( X62 @ id @ ( X63 @ id @ X67 ) @ id ) )
& ( $true
= ( X62 @ id @ X67 @ id ) ) ) ) ) ) )
& ( ! [X101: term] :
? [X102: subst > term > term] :
! [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
= ( X102 @ X104 @ X103 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X188: subst > term > subst > $o,X189: term,X190: term > $o] :
( ( ( X190 @ ( lam @ X189 ) )
= $true )
| ? [X191: subst,X192: subst,X193: subst,X194: term] :
( ( ( X188 @ ( comp @ X193 @ X191 ) @ ( sub @ X194 @ X191 ) @ X192 )
!= $true )
& ( ( X188 @ X193 @ X194 @ ( comp @ X191 @ X192 ) )
= $true ) )
| ? [X200: subst > term > term] :
( ! [X204: term] :
( ( ( X188 @ id @ X204 @ id )
!= $true )
| ( ( X188 @ id @ ( X200 @ id @ X204 ) @ id )
= $true ) )
& ( ( X188 @ id @ ( lam @ ( X200 @ sh @ one ) ) @ id )
!= $true )
& ! [X201: subst,X202: subst,X203: term] :
( ( X200 @ ( comp @ X202 @ X201 ) @ ( sub @ X203 @ X201 ) )
= ( sub @ ( X200 @ X202 @ X203 ) @ X201 ) ) )
| ? [X205: term] :
( ( X188 @ id @ X205 @ id )
!= ( X190 @ X205 ) )
| ? [X199: term] :
( ( ( X190 @ X199 )
= $true )
& ( ( X190 @ ( sub @ X189 @ ( push @ X199 @ id ) ) )
!= $true ) )
| ? [X195: subst,X196: subst,X197: subst,X198: term] :
( ( ( X188 @ ( comp @ X197 @ X196 ) @ ( sub @ X198 @ X196 ) @ X195 )
= $true )
& ( ( X188 @ X197 @ X198 @ ( comp @ X196 @ X195 ) )
!= $true ) ) ) )
& ( hoaslaminj_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ? [X349: term] :
( ( sub @ X349 @ id )
!= X349 )
| ! [X350: term,X351: subst > term > subst > $o,X352: term > $o] :
( ? [X367: term] :
( ( ( X352 @ ( sub @ X350 @ ( push @ X367 @ id ) ) )
!= $true )
& ( ( X352 @ X367 )
= $true ) )
| ? [X362: subst > term > term] :
( ! [X363: term,X364: subst,X365: subst] :
( ( X362 @ ( comp @ X365 @ X364 ) @ ( sub @ X363 @ X364 ) )
= ( sub @ ( X362 @ X365 @ X363 ) @ X364 ) )
& ! [X366: term] :
( ( ( X351 @ id @ ( X362 @ id @ X366 ) @ id )
= $true )
| ( $true
!= ( X351 @ id @ X366 @ id ) ) )
& ( ( X351 @ id @ ( lam @ ( X362 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X354: subst,X355: subst,X356: subst,X357: term] :
( ( ( X351 @ ( comp @ X356 @ X354 ) @ ( sub @ X357 @ X354 ) @ X355 )
!= $true )
& ( ( X351 @ X356 @ X357 @ ( comp @ X354 @ X355 ) )
= $true ) )
| ? [X358: subst,X359: subst,X360: term,X361: subst] :
( ( $true
!= ( X351 @ X358 @ X360 @ ( comp @ X359 @ X361 ) ) )
& ( $true
= ( X351 @ ( comp @ X358 @ X359 ) @ ( sub @ X360 @ X359 ) @ X361 ) ) )
| ? [X353: term] :
( ( X351 @ id @ X353 @ id )
!= ( X352 @ X353 ) )
| ( ( X352 @ ( lam @ X350 ) )
= $true ) ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ? [X442: term,X443: subst > term > subst > $o] :
( ! [X450: subst,X451: term,X452: subst,X453: subst] :
( ( ( X443 @ ( comp @ X452 @ X453 ) @ ( sub @ X451 @ X453 ) @ X450 )
!= $true )
| ( $true
= ( X443 @ X452 @ X451 @ ( comp @ X453 @ X450 ) ) ) )
& ! [X449: term] :
( ( ( X443 @ id @ ( sub @ X442 @ ( push @ X449 @ id ) ) @ id )
= $true )
| ( $true
!= ( X443 @ id @ X449 @ id ) ) )
& ( ( X443 @ id @ ( lam @ X442 ) @ id )
!= $true )
& ! [X454: subst,X455: subst,X456: term,X457: subst] :
( ( $true
!= ( X443 @ X457 @ X456 @ ( comp @ X455 @ X454 ) ) )
| ( ( X443 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 )
= $true ) )
& ! [X444: subst > term > term] :
( ? [X445: subst,X446: term,X447: subst] :
( ( X444 @ ( comp @ X447 @ X445 ) @ ( sub @ X446 @ X445 ) )
!= ( sub @ ( X444 @ X447 @ X446 ) @ X445 ) )
| ( $true
= ( X443 @ id @ ( lam @ ( X444 @ sh @ one ) ) @ id ) )
| ? [X448: term] :
( ( ( X443 @ id @ X448 @ id )
= $true )
& ( $true
!= ( X443 @ id @ ( X444 @ id @ X448 ) @ id ) ) ) ) )
| ? [X429: subst > term > subst > $o,X430: term,X431: term] :
( ! [X440: term,X441: term] :
( ( ( X429 @ id @ X440 @ id )
!= $true )
| ( $true
!= ( X429 @ id @ X441 @ id ) )
| ( $true
= ( X429 @ id @ ( ap @ ( sub @ X440 @ id ) @ X441 ) @ id ) ) )
& ( ( X429 @ id @ X430 @ id )
= $true )
& ! [X436: subst,X437: subst,X438: term,X439: subst] :
( ( $true
!= ( X429 @ X437 @ X438 @ ( comp @ X436 @ X439 ) ) )
| ( ( X429 @ ( comp @ X437 @ X436 ) @ ( sub @ X438 @ X436 ) @ X439 )
= $true ) )
& ( ( X429 @ id @ X431 @ id )
= $true )
& ( ( X429 @ id @ ( ap @ X431 @ X430 ) @ id )
!= $true )
& ! [X432: term,X433: subst,X434: subst,X435: subst] :
( ( ( X429 @ X435 @ X432 @ ( comp @ X434 @ X433 ) )
= $true )
| ( ( X429 @ ( comp @ X435 @ X434 ) @ ( sub @ X432 @ X434 ) @ X433 )
!= $true ) ) )
| ? [X422: term > $o,X423: term] :
( ! [X424: term] :
( ( $true
= ( X422 @ X424 ) )
| ( ( var @ X424 )
!= $true ) )
& ! [X425: term,X426: term] :
( ( $true
!= ( X422 @ X425 ) )
| ( ( X422 @ X426 )
!= $true )
| ( ( X422 @ ( ap @ X426 @ X425 ) )
= $true ) )
& ! [X427: term] :
( ? [X428: term] :
( ( ( X422 @ ( sub @ X427 @ ( push @ X428 @ id ) ) )
!= $true )
& ( ( X422 @ X428 )
= $true ) )
| ( $true
= ( X422 @ ( lam @ X427 ) ) ) )
& ( ( X422 @ X423 )
!= $true ) )
| ! [X458: term,X459: subst > term > subst > $o] :
( ? [X471: subst,X472: subst,X473: subst,X474: term] :
( ( ( X459 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 )
!= $true )
& ( ( X459 @ X473 @ X474 @ ( comp @ X472 @ X471 ) )
= $true ) )
| ? [X462: subst > term > term] :
( ! [X464: subst,X465: term,X466: subst] :
( ( X462 @ ( comp @ X466 @ X464 ) @ ( sub @ X465 @ X464 ) )
= ( sub @ ( X462 @ X466 @ X465 ) @ X464 ) )
& ! [X463: term] :
( ( ( X459 @ id @ ( X462 @ id @ X463 ) @ id )
= $true )
| ( ( X459 @ id @ X463 @ id )
!= $true ) )
& ( $true
!= ( X459 @ id @ ( lam @ ( X462 @ sh @ one ) ) @ id ) ) )
| ( ( X459 @ id @ X458 @ id )
= $true )
| ? [X467: term,X468: subst,X469: subst,X470: subst] :
( ( ( X459 @ ( comp @ X469 @ X470 ) @ ( sub @ X467 @ X470 ) @ X468 )
= $true )
& ( ( X459 @ X469 @ X467 @ ( comp @ X470 @ X468 ) )
!= $true ) )
| ? [X460: term,X461: term] :
( ( $true
= ( X459 @ id @ X460 @ id ) )
& ( $true
= ( X459 @ id @ X461 @ id ) )
& ( $true
!= ( X459 @ id @ ( ap @ ( sub @ X460 @ id ) @ X461 ) @ id ) ) )
| ? [X475: term] :
( ( $true
= ( var @ ( sub @ X475 @ id ) ) )
& ( $true
!= ( X459 @ id @ X475 @ id ) ) ) ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoasapinj1_gthm = $true )
& ! [X271: term] :
? [X272: d_term] :
( ( d2term @ X272 )
= X271 )
& ( ( induction2lem_lthm = $true )
<=> ( ? [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
!= X171 )
| ? [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
!= X169 )
| ? [X174: term,X175: term > $o] :
( ( ( X175 @ X174 )
!= $true )
& ! [X179: term] :
( ( $true
= ( X175 @ X179 ) )
| ( $true
!= ( var @ X179 ) ) )
& ! [X176: term,X177: term] :
( ( $true
= ( X175 @ ( ap @ X177 @ X176 ) ) )
| ( ( X175 @ X177 )
!= $true )
| ( ( X175 @ X176 )
!= $true ) )
& ! [X178: term] :
( ( ( X175 @ X178 )
!= $true )
| ( $true
= ( X175 @ ( lam @ X178 ) ) ) ) )
| ! [X180: term,X181: term > $o,X182: subst] :
( ? [X183: term,X184: term] :
( ( $true
= ( X181 @ X183 ) )
& ( $true
!= ( X181 @ ( ap @ X184 @ X183 ) ) )
& ( ( X181 @ X184 )
= $true ) )
| ? [X185: term] :
( ( ( var @ X185 )
= $true )
& ( ( X181 @ ( sub @ X185 @ X182 ) )
!= $true ) )
| ? [X186: term] :
( ! [X187: term] :
( ( ( X181 @ ( sub @ X186 @ ( push @ X187 @ id ) ) )
= $true )
| ( ( X181 @ X187 )
!= $true ) )
& ( $true
!= ( X181 @ ( lam @ X186 ) ) ) )
| ( ( X181 @ ( sub @ X180 @ X182 ) )
= $true ) )
| ? [X173: subst] :
( ( comp @ X173 @ id )
!= X173 ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) ) )
& ( hoasinduction_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem2_lthm = $true )
& ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
<=> ( axidl = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ? [X79: subst,X80: term] :
( ( comp @ sh @ ( push @ X80 @ X79 ) )
!= X79 )
| ? [X81: term,X82: term] :
( ( ( lam @ X81 )
= ( lam @ X82 ) )
& ( X81 != X82 ) )
| ! [X83: term,X84: subst > term > term,X85: subst,X86: subst > term > term] :
( ( ( lam @ ( X84 @ sh @ one ) )
!= ( lam @ ( X86 @ sh @ one ) ) )
| ? [X90: subst,X91: subst,X92: term] :
( ( X84 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) )
!= ( sub @ ( X84 @ X90 @ X92 ) @ X91 ) )
| ( ( X84 @ X85 @ X83 )
= ( X86 @ X85 @ X83 ) )
| ? [X87: subst,X88: subst,X89: term] :
( ( X86 @ ( comp @ X87 @ X88 ) @ ( sub @ X89 @ X88 ) )
!= ( sub @ ( X86 @ X87 @ X89 ) @ X88 ) ) )
| ? [X77: subst,X78: term] :
( ( sub @ one @ ( push @ X78 @ X77 ) )
!= X78 ) ) )
& ( ! [X251: term,X252: term > $o] :
( ? [X257: term] :
( ( $true
!= ( X252 @ X257 ) )
& ( $true
= ( var @ X257 ) ) )
| ? [X253: term,X254: term] :
( ( $true
!= ( X252 @ ( ap @ X254 @ X253 ) ) )
& ( ( X252 @ X254 )
= $true )
& ( $true
= ( X252 @ X253 ) ) )
| ? [X255: term] :
( ( ( X252 @ ( lam @ X255 ) )
!= $true )
& ! [X256: term] :
( ( $true
!= ( X252 @ X256 ) )
| ( ( X252 @ ( sub @ X255 @ ( push @ X256 @ id ) ) )
= $true ) ) )
| ( $true
= ( X252 @ X251 ) ) )
<=> ( induction2 = $true ) )
& ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
<=> ( axvarcons = $true ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1 = $true )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ! [X313: term,X314: subst > term > subst > $o] :
( ? [X319: term] :
( ( ( X314 @ id @ X319 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X319 @ id ) ) ) )
| ? [X326: subst > term > term] :
( ( ( X314 @ id @ ( lam @ ( X326 @ sh @ one ) ) @ id )
!= $true )
& ! [X330: term] :
( ( ( X314 @ id @ X330 @ id )
!= $true )
| ( ( X314 @ id @ ( X326 @ id @ X330 ) @ id )
= $true ) )
& ! [X327: subst,X328: term,X329: subst] :
( ( X326 @ ( comp @ X327 @ X329 ) @ ( sub @ X328 @ X329 ) )
= ( sub @ ( X326 @ X327 @ X328 ) @ X329 ) ) )
| ( ( X314 @ id @ X313 @ id )
= $true )
| ? [X322: subst,X323: subst,X324: term,X325: subst] :
( ( ( X314 @ X322 @ X324 @ ( comp @ X323 @ X325 ) )
= $true )
& ( ( X314 @ ( comp @ X322 @ X323 ) @ ( sub @ X324 @ X323 ) @ X325 )
!= $true ) )
| ? [X315: subst,X316: subst,X317: term,X318: subst] :
( ( $true
!= ( X314 @ X318 @ X317 @ ( comp @ X315 @ X316 ) ) )
& ( $true
= ( X314 @ ( comp @ X318 @ X315 ) @ ( sub @ X317 @ X315 ) @ X316 ) ) )
| ? [X320: term,X321: term] :
( ( ( X314 @ id @ ( ap @ ( sub @ X320 @ id ) @ X321 ) @ id )
!= $true )
& ( $true
= ( X314 @ id @ X321 @ id ) )
& ( ( X314 @ id @ X320 @ id )
= $true ) ) )
| ? [X303: subst > term > subst > $o,X304: term,X305: term > $o] :
( ( ( X305 @ ( lam @ X304 ) )
!= $true )
& ! [X312: term] :
( ( X303 @ id @ X312 @ id )
= ( X305 @ X312 ) )
& ! [X306: subst > term > term] :
( ? [X307: term] :
( ( $true
= ( X303 @ id @ X307 @ id ) )
& ( $true
!= ( X303 @ id @ ( X306 @ id @ X307 ) @ id ) ) )
| ? [X308: term,X309: subst,X310: subst] :
( ( X306 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) )
!= ( sub @ ( X306 @ X310 @ X308 ) @ X309 ) )
| ( $true
= ( X303 @ id @ ( lam @ ( X306 @ sh @ one ) ) @ id ) ) )
& ! [X311: term] :
( ( ( X305 @ X311 )
!= $true )
| ( ( X305 @ ( sub @ X304 @ ( push @ X311 @ id ) ) )
= $true ) ) )
| ? [X292: subst > term > subst > $o] :
! [X293: term > $o] :
? [X294: term] :
( ( X292 @ id @ X294 @ id )
!= ( X293 @ X294 ) )
| ? [X302: term] :
( ( sub @ X302 @ id )
!= X302 )
| ? [X295: term > $o,X296: term] :
( ! [X297: term,X298: term] :
( ( ( X295 @ X297 )
!= $true )
| ( ( X295 @ X298 )
!= $true )
| ( $true
= ( X295 @ ( ap @ X297 @ X298 ) ) ) )
& ! [X299: term] :
( ( $true
!= ( var @ X299 ) )
| ( ( X295 @ X299 )
= $true ) )
& ! [X300: term] :
( ? [X301: term] :
( ( ( X295 @ X301 )
= $true )
& ( $true
!= ( X295 @ ( sub @ X300 @ ( push @ X301 @ id ) ) ) ) )
| ( ( X295 @ ( lam @ X300 ) )
= $true ) )
& ( ( X295 @ X296 )
!= $true ) ) ) )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X235: subst > term > subst > $o,X236: term] :
( ( ( X235 @ id @ ( lam @ X236 ) @ id )
= $true )
| ? [X247: subst,X248: subst,X249: subst,X250: term] :
( ( ( X235 @ X248 @ X250 @ ( comp @ X249 @ X247 ) )
= $true )
& ( $true
!= ( X235 @ ( comp @ X248 @ X249 ) @ ( sub @ X250 @ X249 ) @ X247 ) ) )
| ? [X246: term] :
( ( $true
!= ( X235 @ id @ ( sub @ X236 @ ( push @ X246 @ id ) ) @ id ) )
& ( ( X235 @ id @ X246 @ id )
= $true ) )
| ? [X237: subst,X238: subst,X239: subst,X240: term] :
( ( ( X235 @ X238 @ X240 @ ( comp @ X239 @ X237 ) )
!= $true )
& ( $true
= ( X235 @ ( comp @ X238 @ X239 ) @ ( sub @ X240 @ X239 ) @ X237 ) ) )
| ? [X241: subst > term > term] :
( ( ( X235 @ id @ ( lam @ ( X241 @ sh @ one ) ) @ id )
!= $true )
& ! [X245: term] :
( ( $true
= ( X235 @ id @ ( X241 @ id @ X245 ) @ id ) )
| ( ( X235 @ id @ X245 @ id )
!= $true ) )
& ! [X242: term,X243: subst,X244: subst] :
( ( X241 @ ( comp @ X244 @ X243 ) @ ( sub @ X242 @ X243 ) )
= ( sub @ ( X241 @ X244 @ X242 ) @ X243 ) ) ) ) )
& ( ! [X264: term] :
( ( sub @ X264 @ id )
= X264 )
<=> ( axvarid = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ( termmset_lthm = $true )
<=> ( ? [X488: term] :
( ( sub @ X488 @ id )
!= X488 )
| ! [X489: term] :
( ( sub @ X489 @ id )
= X489 ) ) )
& ( ! [X258: term,X259: subst > term > term,X260: term] :
( ( ( lam @ ( X259 @ sh @ one ) )
!= ( ap @ ( sub @ X260 @ id ) @ X258 ) )
| ? [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
!= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) ) )
<=> ( hoaslamnotap = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_gthm = $true )
& ( hoasapnotvar_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( pushprop = $true )
& ( axmap = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( $true
!= ( X217 @ X215 ) )
| ? [X219: term] :
( ( X216 @ X219 )
!= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) )
| ( ( X216 @ one )
= $true ) ) )
& ( ! [X212: term] :
? [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
= ( X213 @ sh @ one ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( ! [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
= X335 )
<=> ( axscons = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( induction2lem = $true )
<=> ! [X139: term,X140: term > $o,X141: subst] :
( ? [X145: term,X146: term] :
( ( ( X140 @ X146 )
= $true )
& ( $true
!= ( X140 @ ( ap @ X145 @ X146 ) ) )
& ( ( X140 @ X145 )
= $true ) )
| ? [X142: term] :
( ( ( var @ X142 )
= $true )
& ( ( X140 @ ( sub @ X142 @ X141 ) )
!= $true ) )
| ? [X143: term] :
( ! [X144: term] :
( ( ( X140 @ X144 )
!= $true )
| ( ( X140 @ ( sub @ X143 @ ( push @ X144 @ id ) ) )
= $true ) )
& ( $true
!= ( X140 @ ( lam @ X143 ) ) ) )
| ( $true
= ( X140 @ ( sub @ X139 @ X141 ) ) ) ) )
& ( pushprop_lem0_gthm = $true )
& ( axassoc = $true )
& ( hoaslamnotvar = $true )
& ( ! [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ? [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
!= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) )
| ? [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
!= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
| ( ( X498 @ X497 @ X495 )
= ( X496 @ X497 @ X495 ) )
| ( ( lam @ ( X498 @ sh @ one ) )
!= ( lam @ ( X496 @ sh @ one ) ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( axvarshift = $true )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X25: subst > term > subst > $o,X26: term] :
( ? [X27: term] :
( ( $true
= ( X25 @ id @ X27 @ id ) )
& ( $true
!= ( X25 @ id @ ( sub @ X26 @ ( push @ X27 @ id ) ) @ id ) ) )
| ? [X28: term,X29: subst > term > term] :
( ( ( X25 @ id @ ( lam @ ( X29 @ sh @ one ) ) @ id )
!= $true )
& ! [X34: subst,X35: subst,X36: term] :
( ( sub @ X28 @ ( push @ ( sub @ X36 @ X34 ) @ ( comp @ X35 @ X34 ) ) )
= ( X29 @ ( comp @ X35 @ X34 ) @ ( sub @ X36 @ X34 ) ) )
& ! [X30: term] :
( ( ( X25 @ id @ ( X29 @ id @ X30 ) @ id )
= $true )
| ( ( X25 @ id @ X30 @ id )
!= $true ) )
& ! [X31: subst,X32: term,X33: subst] :
( ( sub @ ( sub @ X28 @ ( push @ X32 @ X33 ) ) @ X31 )
= ( sub @ ( X29 @ X33 @ X32 ) @ X31 ) ) )
| ( $true
= ( X25 @ id @ ( lam @ ( sub @ X26 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( induction2_gthm = $true )
& ( ! [X490: term > $o,X491: term,X492: subst] :
? [X493: term > $o] :
! [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
= ( X493 @ X494 ) )
<=> ( pushprop_lem0 = $true ) )
& ( ulamvarind = $true )
& ( ( induction2_lthm = $true )
<=> ( ? [X152: term,X153: subst,X154: term > $o] :
( ! [X155: term] :
( ( $true
= ( X154 @ ( lam @ X155 ) ) )
| ? [X156: term] :
( ( ( X154 @ X156 )
= $true )
& ( ( X154 @ ( sub @ X155 @ ( push @ X156 @ id ) ) )
!= $true ) ) )
& ! [X159: term] :
( ( ( var @ X159 )
!= $true )
| ( ( X154 @ ( sub @ X159 @ X153 ) )
= $true ) )
& ! [X157: term,X158: term] :
( ( $true
!= ( X154 @ X158 ) )
| ( ( X154 @ X157 )
!= $true )
| ( ( X154 @ ( ap @ X158 @ X157 ) )
= $true ) )
& ( ( X154 @ ( sub @ X152 @ X153 ) )
!= $true ) )
| ? [X151: term] :
( ( sub @ X151 @ id )
!= X151 )
| ! [X160: term,X161: term > $o] :
( ? [X165: term] :
( ( $true
!= ( X161 @ ( lam @ X165 ) ) )
& ! [X166: term] :
( ( ( X161 @ X166 )
!= $true )
| ( $true
= ( X161 @ ( sub @ X165 @ ( push @ X166 @ id ) ) ) ) ) )
| ? [X162: term] :
( ( $true
!= ( X161 @ X162 ) )
& ( $true
= ( var @ X162 ) ) )
| ? [X163: term,X164: term] :
( ( ( X161 @ ( ap @ X164 @ X163 ) )
!= $true )
& ( $true
= ( X161 @ X163 ) )
& ( ( X161 @ X164 )
= $true ) )
| ( ( X161 @ X160 )
= $true ) ) ) )
& ( axabs = $true )
& ( ( apinj1 = $true )
<=> ! [X484: term,X485: term,X486: term,X487: term] :
( ( X485 = X487 )
| ( ( ap @ X485 @ X484 )
!= ( ap @ X487 @ X486 ) ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
!= $true )
| ? [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
!= ( X2 @ X6 ) )
| ( ( X2 @ one )
= $true ) )
| ? [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
!= X0 ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( ( d2subst @ d_id )
= id )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X214: subst] :
( ( comp @ X214 @ id )
= X214 )
<=> ( axidr = $true ) )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
= ( sub @ X147 @ ( push @ X149 @ X150 ) ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( hoasapinj1 = $true )
<=> ! [X378: term,X379: term,X380: term,X381: term] :
( ( X379 = X380 )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ! [X532: term,X533: subst > term > subst > $o] :
( ( $true
= ( X533 @ id @ ( lam @ X532 ) @ id ) )
| ? [X534: subst > term > term] :
( ( ( X533 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id )
!= $true )
& ! [X535: subst,X536: subst,X537: term] :
( ( sub @ ( X534 @ X535 @ X537 ) @ X536 )
= ( X534 @ ( comp @ X535 @ X536 ) @ ( sub @ X537 @ X536 ) ) )
& ! [X538: term] :
( ( ( X533 @ id @ X538 @ id )
!= $true )
| ( ( X533 @ id @ ( X534 @ id @ X538 ) @ id )
= $true ) ) )
| ? [X539: term] :
( ( ( X533 @ id @ X539 @ id )
= $true )
& ( ( X533 @ id @ ( sub @ X532 @ ( push @ X539 @ id ) ) @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ? [X332: subst] :
( ( comp @ X332 @ id )
!= X332 )
| ( ! [X333: subst] :
( ( comp @ X333 @ id )
= X333 )
& ! [X334: subst] :
( ( comp @ id @ X334 )
= X334 ) )
| ? [X331: subst] :
( ( comp @ id @ X331 )
!= X331 ) )
<=> ( substmonoid_lthm = $true ) ) ),
inference(flattening,[status(thm)],[f13]) ).
thf(f13,plain,
( ( hoasinduction_lem1 = $true )
& ( lamnotap != $true )
& ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
<=> ( axidl = $true ) )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ! [X313: term,X314: subst > term > subst > $o] :
( ? [X319: term] :
( ( ( X314 @ id @ X319 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X319 @ id ) ) ) )
| ? [X326: subst > term > term] :
( ( ( X314 @ id @ ( lam @ ( X326 @ sh @ one ) ) @ id )
!= $true )
& ! [X330: term] :
( ( ( X314 @ id @ X330 @ id )
!= $true )
| ( ( X314 @ id @ ( X326 @ id @ X330 ) @ id )
= $true ) )
& ! [X327: subst,X328: term,X329: subst] :
( ( X326 @ ( comp @ X327 @ X329 ) @ ( sub @ X328 @ X329 ) )
= ( sub @ ( X326 @ X327 @ X328 ) @ X329 ) ) )
| ( ( X314 @ id @ X313 @ id )
= $true )
| ? [X322: subst,X323: subst,X324: term,X325: subst] :
( ( ( X314 @ X322 @ X324 @ ( comp @ X323 @ X325 ) )
= $true )
& ( ( X314 @ ( comp @ X322 @ X323 ) @ ( sub @ X324 @ X323 ) @ X325 )
!= $true ) )
| ? [X315: subst,X316: subst,X317: term,X318: subst] :
( ( $true
!= ( X314 @ X318 @ X317 @ ( comp @ X315 @ X316 ) ) )
& ( $true
= ( X314 @ ( comp @ X318 @ X315 ) @ ( sub @ X317 @ X315 ) @ X316 ) ) )
| ? [X320: term,X321: term] :
( ( ( X314 @ id @ ( ap @ ( sub @ X320 @ id ) @ X321 ) @ id )
!= $true )
& ( $true
= ( X314 @ id @ X321 @ id ) )
& ( ( X314 @ id @ X320 @ id )
= $true ) ) )
| ? [X303: subst > term > subst > $o,X304: term,X305: term > $o] :
( ( ( X305 @ ( lam @ X304 ) )
!= $true )
& ! [X312: term] :
( ( X303 @ id @ X312 @ id )
= ( X305 @ X312 ) )
& ! [X306: subst > term > term] :
( ? [X307: term] :
( ( $true
= ( X303 @ id @ X307 @ id ) )
& ( $true
!= ( X303 @ id @ ( X306 @ id @ X307 ) @ id ) ) )
| ? [X308: term,X309: subst,X310: subst] :
( ( X306 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) )
!= ( sub @ ( X306 @ X310 @ X308 ) @ X309 ) )
| ( $true
= ( X303 @ id @ ( lam @ ( X306 @ sh @ one ) ) @ id ) ) )
& ! [X311: term] :
( ( ( X305 @ X311 )
!= $true )
| ( ( X305 @ ( sub @ X304 @ ( push @ X311 @ id ) ) )
= $true ) ) )
| ? [X302: term] :
( ( sub @ X302 @ id )
!= X302 )
| ? [X295: term > $o,X296: term] :
( ! [X297: term,X298: term] :
( ( ( X295 @ X297 )
!= $true )
| ( ( X295 @ X298 )
!= $true )
| ( $true
= ( X295 @ ( ap @ X297 @ X298 ) ) ) )
& ! [X299: term] :
( ( $true
!= ( var @ X299 ) )
| ( ( X295 @ X299 )
= $true ) )
& ! [X300: term] :
( ? [X301: term] :
( ( ( X295 @ X301 )
= $true )
& ( $true
!= ( X295 @ ( sub @ X300 @ ( push @ X301 @ id ) ) ) ) )
| ( ( X295 @ ( lam @ X300 ) )
= $true ) )
& ( ( X295 @ X296 )
!= $true ) )
| ? [X292: subst > term > subst > $o] :
! [X293: term > $o] :
? [X294: term] :
( ( X292 @ id @ X294 @ id )
!= ( X293 @ X294 ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_gthm = $true )
& ( ! [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) )
| ( ( X266 @ X265 )
!= $true ) )
<=> ( pushprop_lem1 = $true ) )
& ! [X96: d_term] : ( d_one = X96 )
& ( ! [X368: term,X369: term > $o,X370: subst > term > subst > $o] :
( ? [X376: term] :
( ( X370 @ id @ X376 @ id )
!= ( X369 @ X376 ) )
| ? [X377: term] :
( ( ( X369 @ ( sub @ X368 @ ( push @ X377 @ id ) ) )
!= $true )
& ( ( X369 @ X377 )
= $true ) )
| ? [X371: subst > term > term] :
( ( $true
!= ( X370 @ id @ ( lam @ ( X371 @ sh @ one ) ) @ id ) )
& ! [X372: term] :
( ( $true
!= ( X370 @ id @ X372 @ id ) )
| ( $true
= ( X370 @ id @ ( X371 @ id @ X372 ) @ id ) ) )
& ! [X373: term,X374: subst,X375: subst] :
( ( sub @ ( X371 @ X375 @ X373 ) @ X374 )
= ( X371 @ ( comp @ X375 @ X374 ) @ ( sub @ X373 @ X374 ) ) ) )
| ( $true
= ( X369 @ ( lam @ X368 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( ulamvar1 != $true )
& ( ! [X147: term] :
? [X148: subst > term > term] :
! [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
= ( sub @ X147 @ ( push @ X149 @ X150 ) ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
| ( ( X18 @ X15 )
!= $true ) )
| ? [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
!= X12 )
| ? [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
!= X11 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X220: term > $o,X221: term,X222: subst > term > subst > $o,X223: term] :
( ( ( X220 @ X223 )
!= $true )
| ? [X233: term,X234: term] :
( ( $true
!= ( X222 @ id @ ( ap @ ( sub @ X234 @ id ) @ X233 ) @ id ) )
& ( ( X222 @ id @ X234 @ id )
= $true )
& ( $true
= ( X222 @ id @ X233 @ id ) ) )
| ? [X228: subst,X229: subst,X230: subst,X231: term] :
( ( $true
!= ( X222 @ X230 @ X231 @ ( comp @ X228 @ X229 ) ) )
& ( ( X222 @ ( comp @ X230 @ X228 ) @ ( sub @ X231 @ X228 ) @ X229 )
= $true ) )
| ( ( X220 @ X221 )
!= $true )
| ? [X224: subst,X225: term,X226: subst,X227: subst] :
( ( ( X222 @ ( comp @ X224 @ X227 ) @ ( sub @ X225 @ X227 ) @ X226 )
!= $true )
& ( ( X222 @ X224 @ X225 @ ( comp @ X227 @ X226 ) )
= $true ) )
| ( ( X220 @ ( ap @ X221 @ X223 ) )
= $true )
| ? [X232: term] :
( ( X220 @ X232 )
!= ( X222 @ id @ X232 @ id ) ) ) )
& ( ( apinj1 = $true )
<=> ! [X484: term,X485: term,X486: term,X487: term] :
( ( X485 = X487 )
| ( ( ap @ X485 @ X484 )
!= ( ap @ X487 @ X486 ) ) ) )
& ( ! [X251: term,X252: term > $o] :
( ? [X257: term] :
( ( $true
!= ( X252 @ X257 ) )
& ( $true
= ( var @ X257 ) ) )
| ? [X253: term,X254: term] :
( ( $true
!= ( X252 @ ( ap @ X254 @ X253 ) ) )
& ( ( X252 @ X254 )
= $true )
& ( $true
= ( X252 @ X253 ) ) )
| ? [X255: term] :
( ( ( X252 @ ( lam @ X255 ) )
!= $true )
& ! [X256: term] :
( ( $true
!= ( X252 @ X256 ) )
| ( ( X252 @ ( sub @ X255 @ ( push @ X256 @ id ) ) )
= $true ) ) )
| ( $true
= ( X252 @ X251 ) ) )
<=> ( induction2 = $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X93: subst > term > subst > $o] :
? [X94: term > $o] :
! [X95: term] :
( ( X94 @ X95 )
= ( X93 @ id @ X95 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( apnotvar = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ! [X258: term,X259: subst > term > term,X260: term] :
( ( ( lam @ ( X259 @ sh @ one ) )
!= ( ap @ ( sub @ X260 @ id ) @ X258 ) )
| ? [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
!= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) ) )
<=> ( hoaslamnotap = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( axassoc = $true )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X338: term] :
? [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
= ( X339 @ sh @ one ) ) )
& ( ( termmset = $true )
<=> ! [X57: term] :
( ( sub @ X57 @ id )
= X57 ) )
& ( ! [X505: subst > term > subst > $o,X506: term] :
( ? [X508: subst > term > term] :
( ( ( X505 @ id @ ( lam @ ( X508 @ sh @ one ) ) @ id )
!= $true )
& ! [X510: subst,X511: subst,X512: term] :
( ( X508 @ ( comp @ X511 @ X510 ) @ ( sub @ X512 @ X510 ) )
= ( sub @ ( X508 @ X511 @ X512 ) @ X510 ) )
& ! [X509: term] :
( ( $true
!= ( X505 @ id @ X509 @ id ) )
| ( $true
= ( X505 @ id @ ( X508 @ id @ X509 ) @ id ) ) ) )
| ? [X507: term] :
( ( $true
= ( X505 @ id @ X507 @ id ) )
& ( ( X505 @ id @ ( sub @ X506 @ ( push @ X507 @ id ) ) @ id )
!= $true ) )
| ( ( X505 @ id @ ( lam @ ( sub @ X506 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( axclos = $true )
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
!= ( d2subst @ X55 ) )
| ( X55 = X56 ) )
& ( ( termmset_lthm = $true )
<=> ( ? [X488: term] :
( ( sub @ X488 @ id )
!= X488 )
| ! [X489: term] :
( ( sub @ X489 @ id )
= X489 ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X532: term,X533: subst > term > subst > $o] :
( ( $true
= ( X533 @ id @ ( lam @ X532 ) @ id ) )
| ? [X534: subst > term > term] :
( ( ( X533 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id )
!= $true )
& ! [X535: subst,X536: subst,X537: term] :
( ( sub @ ( X534 @ X535 @ X537 ) @ X536 )
= ( X534 @ ( comp @ X535 @ X536 ) @ ( sub @ X537 @ X536 ) ) )
& ! [X538: term] :
( ( ( X533 @ id @ X538 @ id )
!= $true )
| ( ( X533 @ id @ ( X534 @ id @ X538 ) @ id )
= $true ) ) )
| ? [X539: term] :
( ( ( X533 @ id @ X539 @ id )
= $true )
& ( ( X533 @ id @ ( sub @ X532 @ ( push @ X539 @ id ) ) @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
= X273 )
<=> ( axshiftcons = $true ) )
& ( ! [X214: subst] :
( ( comp @ X214 @ id )
= X214 )
<=> ( axidr = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X25: subst > term > subst > $o,X26: term] :
( ? [X27: term] :
( ( $true
= ( X25 @ id @ X27 @ id ) )
& ( $true
!= ( X25 @ id @ ( sub @ X26 @ ( push @ X27 @ id ) ) @ id ) ) )
| ? [X28: term,X29: subst > term > term] :
( ( ( X25 @ id @ ( lam @ ( X29 @ sh @ one ) ) @ id )
!= $true )
& ! [X34: subst,X35: subst,X36: term] :
( ( sub @ X28 @ ( push @ ( sub @ X36 @ X34 ) @ ( comp @ X35 @ X34 ) ) )
= ( X29 @ ( comp @ X35 @ X34 ) @ ( sub @ X36 @ X34 ) ) )
& ! [X30: term] :
( ( ( X25 @ id @ ( X29 @ id @ X30 ) @ id )
= $true )
| ( ( X25 @ id @ X30 @ id )
!= $true ) )
& ! [X31: subst,X32: term,X33: subst] :
( ( sub @ ( sub @ X28 @ ( push @ X32 @ X33 ) ) @ X31 )
= ( sub @ ( X29 @ X33 @ X32 ) @ X31 ) ) )
| ( $true
= ( X25 @ id @ ( lam @ ( sub @ X26 @ ( push @ one @ sh ) ) ) @ id ) ) ) )
& ( ( ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 )
& ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 ) )
<=> ( substmonoid = $true ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X458: term,X459: subst > term > subst > $o] :
( ? [X471: subst,X472: subst,X473: subst,X474: term] :
( ( ( X459 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 )
!= $true )
& ( ( X459 @ X473 @ X474 @ ( comp @ X472 @ X471 ) )
= $true ) )
| ? [X462: subst > term > term] :
( ! [X464: subst,X465: term,X466: subst] :
( ( X462 @ ( comp @ X466 @ X464 ) @ ( sub @ X465 @ X464 ) )
= ( sub @ ( X462 @ X466 @ X465 ) @ X464 ) )
& ! [X463: term] :
( ( ( X459 @ id @ ( X462 @ id @ X463 ) @ id )
= $true )
| ( ( X459 @ id @ X463 @ id )
!= $true ) )
& ( $true
!= ( X459 @ id @ ( lam @ ( X462 @ sh @ one ) ) @ id ) ) )
| ( ( X459 @ id @ X458 @ id )
= $true )
| ? [X467: term,X468: subst,X469: subst,X470: subst] :
( ( ( X459 @ ( comp @ X469 @ X470 ) @ ( sub @ X467 @ X470 ) @ X468 )
= $true )
& ( ( X459 @ X469 @ X467 @ ( comp @ X470 @ X468 ) )
!= $true ) )
| ? [X460: term,X461: term] :
( ( $true
= ( X459 @ id @ X460 @ id ) )
& ( $true
= ( X459 @ id @ X461 @ id ) )
& ( $true
!= ( X459 @ id @ ( ap @ ( sub @ X460 @ id ) @ X461 ) @ id ) ) )
| ? [X475: term] :
( ( $true
= ( var @ ( sub @ X475 @ id ) ) )
& ( $true
!= ( X459 @ id @ X475 @ id ) ) ) )
| ? [X442: term,X443: subst > term > subst > $o] :
( ! [X450: subst,X451: term,X452: subst,X453: subst] :
( ( ( X443 @ ( comp @ X452 @ X453 ) @ ( sub @ X451 @ X453 ) @ X450 )
!= $true )
| ( $true
= ( X443 @ X452 @ X451 @ ( comp @ X453 @ X450 ) ) ) )
& ! [X449: term] :
( ( ( X443 @ id @ ( sub @ X442 @ ( push @ X449 @ id ) ) @ id )
= $true )
| ( $true
!= ( X443 @ id @ X449 @ id ) ) )
& ( ( X443 @ id @ ( lam @ X442 ) @ id )
!= $true )
& ! [X454: subst,X455: subst,X456: term,X457: subst] :
( ( $true
!= ( X443 @ X457 @ X456 @ ( comp @ X455 @ X454 ) ) )
| ( ( X443 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 )
= $true ) )
& ! [X444: subst > term > term] :
( ? [X445: subst,X446: term,X447: subst] :
( ( X444 @ ( comp @ X447 @ X445 ) @ ( sub @ X446 @ X445 ) )
!= ( sub @ ( X444 @ X447 @ X446 ) @ X445 ) )
| ( $true
= ( X443 @ id @ ( lam @ ( X444 @ sh @ one ) ) @ id ) )
| ? [X448: term] :
( ( ( X443 @ id @ X448 @ id )
= $true )
& ( $true
!= ( X443 @ id @ ( X444 @ id @ X448 ) @ id ) ) ) ) )
| ? [X429: subst > term > subst > $o,X430: term,X431: term] :
( ! [X440: term,X441: term] :
( ( ( X429 @ id @ X440 @ id )
!= $true )
| ( $true
!= ( X429 @ id @ X441 @ id ) )
| ( $true
= ( X429 @ id @ ( ap @ ( sub @ X440 @ id ) @ X441 ) @ id ) ) )
& ( ( X429 @ id @ X430 @ id )
= $true )
& ! [X436: subst,X437: subst,X438: term,X439: subst] :
( ( $true
!= ( X429 @ X437 @ X438 @ ( comp @ X436 @ X439 ) ) )
| ( ( X429 @ ( comp @ X437 @ X436 ) @ ( sub @ X438 @ X436 ) @ X439 )
= $true ) )
& ( ( X429 @ id @ X431 @ id )
= $true )
& ( ( X429 @ id @ ( ap @ X431 @ X430 ) @ id )
!= $true )
& ! [X432: term,X433: subst,X434: subst,X435: subst] :
( ( ( X429 @ X435 @ X432 @ ( comp @ X434 @ X433 ) )
= $true )
| ( ( X429 @ ( comp @ X435 @ X434 ) @ ( sub @ X432 @ X434 ) @ X433 )
!= $true ) ) )
| ? [X422: term > $o,X423: term] :
( ! [X424: term] :
( ( $true
= ( X422 @ X424 ) )
| ( ( var @ X424 )
!= $true ) )
& ! [X425: term,X426: term] :
( ( $true
!= ( X422 @ X425 ) )
| ( ( X422 @ X426 )
!= $true )
| ( ( X422 @ ( ap @ X426 @ X425 ) )
= $true ) )
& ! [X427: term] :
( ? [X428: term] :
( ( ( X422 @ ( sub @ X427 @ ( push @ X428 @ id ) ) )
!= $true )
& ( ( X422 @ X428 )
= $true ) )
| ( $true
= ( X422 @ ( lam @ X427 ) ) ) )
& ( ( X422 @ X423 )
!= $true ) ) ) )
& ( ! [X212: term] :
? [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
= ( X213 @ sh @ one ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( axmap = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( $true
!= ( X217 @ X215 ) )
| ? [X219: term] :
( ( X216 @ X219 )
!= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) )
| ( ( X216 @ one )
= $true ) ) )
& ( axvarshift = $true )
& ! [X270: d_subst] : ( d_id = X270 )
& ( ! [X490: term > $o,X491: term,X492: subst] :
? [X493: term > $o] :
! [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
= ( X493 @ X494 ) )
<=> ( pushprop_lem0 = $true ) )
& ( pushprop = $true )
& ( hoasinduction_lem1_gthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( ( induction2_lthm = $true )
<=> ( ! [X160: term,X161: term > $o] :
( ? [X165: term] :
( ( $true
!= ( X161 @ ( lam @ X165 ) ) )
& ! [X166: term] :
( ( ( X161 @ X166 )
!= $true )
| ( $true
= ( X161 @ ( sub @ X165 @ ( push @ X166 @ id ) ) ) ) ) )
| ? [X162: term] :
( ( $true
!= ( X161 @ X162 ) )
& ( $true
= ( var @ X162 ) ) )
| ? [X163: term,X164: term] :
( ( ( X161 @ ( ap @ X164 @ X163 ) )
!= $true )
& ( $true
= ( X161 @ X163 ) )
& ( ( X161 @ X164 )
= $true ) )
| ( ( X161 @ X160 )
= $true ) )
| ? [X152: term,X153: subst,X154: term > $o] :
( ! [X155: term] :
( ( $true
= ( X154 @ ( lam @ X155 ) ) )
| ? [X156: term] :
( ( ( X154 @ X156 )
= $true )
& ( ( X154 @ ( sub @ X155 @ ( push @ X156 @ id ) ) )
!= $true ) ) )
& ! [X159: term] :
( ( ( var @ X159 )
!= $true )
| ( ( X154 @ ( sub @ X159 @ X153 ) )
= $true ) )
& ! [X157: term,X158: term] :
( ( $true
!= ( X154 @ X158 ) )
| ( ( X154 @ X157 )
!= $true )
| ( ( X154 @ ( ap @ X158 @ X157 ) )
= $true ) )
& ( ( X154 @ ( sub @ X152 @ X153 ) )
!= $true ) )
| ? [X151: term] :
( ( sub @ X151 @ id )
!= X151 ) ) )
& ( hoasapnotvar = $true )
& ( ! [X105: term,X106: subst > term > subst > $o] :
( ? [X107: subst > term > term] :
( ! [X108: term] :
( ( $true
= ( X106 @ id @ ( X107 @ id @ X108 ) @ id ) )
| ( $true
!= ( X106 @ id @ X108 @ id ) ) )
& ( ( X106 @ id @ ( lam @ ( X107 @ sh @ one ) ) @ id )
!= $true )
& ! [X109: subst,X110: term,X111: subst] :
( ( X107 @ ( comp @ X111 @ X109 ) @ ( sub @ X110 @ X109 ) )
= ( sub @ ( X107 @ X111 @ X110 ) @ X109 ) ) )
| ( $true
= ( X106 @ id @ X105 @ id ) )
| ? [X112: term,X113: term] :
( ( ( X106 @ id @ X113 @ id )
= $true )
& ( ( X106 @ id @ X112 @ id )
= $true )
& ( $true
!= ( X106 @ id @ ( ap @ ( sub @ X113 @ id ) @ X112 ) @ id ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( ! [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
!= ( sub @ X275 @ sh ) )
| ( X275 = X276 ) )
<=> ( shinj = $true ) )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X235: subst > term > subst > $o,X236: term] :
( ( ( X235 @ id @ ( lam @ X236 ) @ id )
= $true )
| ? [X247: subst,X248: subst,X249: subst,X250: term] :
( ( ( X235 @ X248 @ X250 @ ( comp @ X249 @ X247 ) )
= $true )
& ( $true
!= ( X235 @ ( comp @ X248 @ X249 ) @ ( sub @ X250 @ X249 ) @ X247 ) ) )
| ? [X246: term] :
( ( $true
!= ( X235 @ id @ ( sub @ X236 @ ( push @ X246 @ id ) ) @ id ) )
& ( ( X235 @ id @ X246 @ id )
= $true ) )
| ? [X237: subst,X238: subst,X239: subst,X240: term] :
( ( ( X235 @ X238 @ X240 @ ( comp @ X239 @ X237 ) )
!= $true )
& ( $true
= ( X235 @ ( comp @ X238 @ X239 ) @ ( sub @ X240 @ X239 ) @ X237 ) ) )
| ? [X241: subst > term > term] :
( ( ( X235 @ id @ ( lam @ ( X241 @ sh @ one ) ) @ id )
!= $true )
& ! [X245: term] :
( ( $true
= ( X235 @ id @ ( X241 @ id @ X245 ) @ id ) )
| ( ( X235 @ id @ X245 @ id )
!= $true ) )
& ! [X242: term,X243: subst,X244: subst] :
( ( X241 @ ( comp @ X244 @ X243 ) @ ( sub @ X242 @ X243 ) )
= ( sub @ ( X241 @ X244 @ X242 ) @ X243 ) ) ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X188: subst > term > subst > $o,X189: term,X190: term > $o] :
( ( ( X190 @ ( lam @ X189 ) )
= $true )
| ? [X191: subst,X192: subst,X193: subst,X194: term] :
( ( ( X188 @ ( comp @ X193 @ X191 ) @ ( sub @ X194 @ X191 ) @ X192 )
!= $true )
& ( ( X188 @ X193 @ X194 @ ( comp @ X191 @ X192 ) )
= $true ) )
| ? [X200: subst > term > term] :
( ! [X204: term] :
( ( ( X188 @ id @ X204 @ id )
!= $true )
| ( ( X188 @ id @ ( X200 @ id @ X204 ) @ id )
= $true ) )
& ( ( X188 @ id @ ( lam @ ( X200 @ sh @ one ) ) @ id )
!= $true )
& ! [X201: subst,X202: subst,X203: term] :
( ( X200 @ ( comp @ X202 @ X201 ) @ ( sub @ X203 @ X201 ) )
= ( sub @ ( X200 @ X202 @ X203 ) @ X201 ) ) )
| ? [X205: term] :
( ( X188 @ id @ X205 @ id )
!= ( X190 @ X205 ) )
| ? [X199: term] :
( ( ( X190 @ X199 )
= $true )
& ( ( X190 @ ( sub @ X189 @ ( push @ X199 @ id ) ) )
!= $true ) )
| ? [X195: subst,X196: subst,X197: subst,X198: term] :
( ( ( X188 @ ( comp @ X197 @ X196 ) @ ( sub @ X198 @ X196 ) @ X195 )
= $true )
& ( ( X188 @ X197 @ X198 @ ( comp @ X196 @ X195 ) )
!= $true ) ) ) )
& ( ! [X264: term] :
( ( sub @ X264 @ id )
= X264 )
<=> ( axvarid = $true ) )
& ( pushprop_gthm = $true )
& ( 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 )
& ( hoasinduction_lem2v2_gthm = $true )
& ( lamnotvar = $true )
& ( ! [X101: term] :
? [X102: subst > term > term] :
! [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
= ( X102 @ X104 @ X103 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X518: subst > term > subst > $o,X519: term,X520: term] :
( ( ( X518 @ id @ X519 @ id )
!= $true )
| ( ( X518 @ id @ ( ap @ X519 @ X520 ) @ id )
= $true )
| ( $true
!= ( X518 @ id @ X520 @ id ) )
| ? [X521: subst,X522: subst,X523: term,X524: subst] :
( ( $true
!= ( X518 @ X522 @ X523 @ ( comp @ X524 @ X521 ) ) )
& ( $true
= ( X518 @ ( comp @ X522 @ X524 ) @ ( sub @ X523 @ X524 ) @ X521 ) ) )
| ? [X525: term,X526: term] :
( ( $true
= ( X518 @ id @ X525 @ id ) )
& ( $true
= ( X518 @ id @ X526 @ id ) )
& ( ( X518 @ id @ ( ap @ ( sub @ X525 @ id ) @ X526 ) @ id )
!= $true ) )
| ? [X527: subst,X528: subst,X529: term,X530: subst] :
( ( $true
= ( X518 @ X528 @ X529 @ ( comp @ X530 @ X527 ) ) )
& ( $true
!= ( X518 @ ( comp @ X528 @ X530 ) @ ( sub @ X529 @ X530 ) @ X527 ) ) ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X282: term > $o,X283: subst > term > subst > $o,X284: term] :
( ? [X285: subst > term > term] :
( ! [X286: term] :
( ( $true
= ( X283 @ id @ ( X285 @ id @ X286 ) @ id ) )
| ( $true
!= ( X283 @ id @ X286 @ id ) ) )
& ( $true
!= ( X283 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id ) )
& ! [X287: term,X288: subst,X289: subst] :
( ( X285 @ ( comp @ X289 @ X288 ) @ ( sub @ X287 @ X288 ) )
= ( sub @ ( X285 @ X289 @ X287 ) @ X288 ) ) )
| ( ( X282 @ ( lam @ X284 ) )
= $true )
| ? [X291: term] :
( ( ( X282 @ X291 )
= $true )
& ( $true
!= ( X282 @ ( sub @ X284 @ ( push @ X291 @ id ) ) ) ) )
| ? [X290: term] :
( ( X282 @ X290 )
!= ( X283 @ id @ X290 @ id ) ) )
| ? [X281: term] :
( ( sub @ X281 @ id )
!= X281 )
| ? [X277: term] :
! [X278: subst > term > term] :
? [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
!= ( X278 @ X280 @ X279 ) ) ) )
& ( ulamvarind = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ! [X405: term,X406: subst > term > subst > $o] :
( ( $true
= ( X406 @ id @ X405 @ id ) )
| ? [X407: subst > term > term] :
( ! [X408: term] :
( ( ( X406 @ id @ ( X407 @ id @ X408 ) @ id )
= $true )
| ( ( X406 @ id @ X408 @ id )
!= $true ) )
& ( $true
!= ( X406 @ id @ ( lam @ ( X407 @ sh @ one ) ) @ id ) )
& ! [X409: subst,X410: subst,X411: term] :
( ( sub @ ( X407 @ X410 @ X411 ) @ X409 )
= ( X407 @ ( comp @ X410 @ X409 ) @ ( sub @ X411 @ X409 ) ) ) )
| ? [X412: term,X413: term] :
( ( ( X406 @ id @ ( ap @ ( sub @ X412 @ id ) @ X413 ) @ id )
!= $true )
& ( $true
= ( X406 @ id @ X413 @ id ) )
& ( ( X406 @ id @ X412 @ id )
= $true ) ) )
| ? [X395: term > $o,X396: term,X397: subst > term > subst > $o] :
( ! [X404: term] :
( ( ( X395 @ ( sub @ X396 @ ( push @ X404 @ id ) ) )
= $true )
| ( ( X395 @ X404 )
!= $true ) )
& ! [X398: term] :
( ( X397 @ id @ X398 @ id )
= ( X395 @ X398 ) )
& ! [X399: subst > term > term] :
( ? [X400: subst,X401: subst,X402: term] :
( ( sub @ ( X399 @ X400 @ X402 ) @ X401 )
!= ( X399 @ ( comp @ X400 @ X401 ) @ ( sub @ X402 @ X401 ) ) )
| ? [X403: term] :
( ( $true
!= ( X397 @ id @ ( X399 @ id @ X403 ) @ id ) )
& ( ( X397 @ id @ X403 @ id )
= $true ) )
| ( $true
= ( X397 @ id @ ( lam @ ( X399 @ sh @ one ) ) @ id ) ) )
& ( $true
!= ( X395 @ ( lam @ X396 ) ) ) )
| ? [X394: term] :
( ( sub @ X394 @ id )
!= X394 )
| ? [X387: term,X388: term > $o] :
( ( ( X388 @ X387 )
!= $true )
& ! [X390: term] :
( ? [X391: term] :
( ( ( X388 @ X391 )
= $true )
& ( ( X388 @ ( sub @ X390 @ ( push @ X391 @ id ) ) )
!= $true ) )
| ( ( X388 @ ( lam @ X390 ) )
= $true ) )
& ! [X389: term] :
( ( $true
!= ( var @ X389 ) )
| ( $true
= ( X388 @ X389 ) ) )
& ! [X392: term,X393: term] :
( ( ( X388 @ X393 )
!= $true )
| ( $true
= ( X388 @ ( ap @ X393 @ X392 ) ) )
| ( $true
!= ( X388 @ X392 ) ) ) )
| ? [X384: subst > term > subst > $o] :
! [X385: term > $o] :
? [X386: term] :
( ( X384 @ id @ X386 @ id )
!= ( X385 @ X386 ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( axapp = $true )
& ! [X23: d_term,X24: d_term] :
( ( X23 = X24 )
| ( ( d2term @ X23 )
!= ( d2term @ X24 ) ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X513: term,X514: term > $o,X515: subst] :
? [X516: term > $o] :
! [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
= ( X516 @ X517 ) ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ? [X476: term,X477: term,X478: term,X479: term] :
( ( ( ap @ X477 @ X476 )
= ( ap @ X478 @ X479 ) )
& ( X476 != X479 ) )
| ! [X480: term,X481: term,X482: term,X483: term] :
( ( X482 = X483 )
| ( ( ap @ ( sub @ X481 @ id ) @ X483 )
!= ( ap @ ( sub @ X480 @ id ) @ X482 ) ) ) ) )
& ( pushprop_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ? [X349: term] :
( ( sub @ X349 @ id )
!= X349 )
| ! [X350: term,X351: subst > term > subst > $o,X352: term > $o] :
( ? [X367: term] :
( ( ( X352 @ ( sub @ X350 @ ( push @ X367 @ id ) ) )
!= $true )
& ( ( X352 @ X367 )
= $true ) )
| ? [X362: subst > term > term] :
( ! [X363: term,X364: subst,X365: subst] :
( ( X362 @ ( comp @ X365 @ X364 ) @ ( sub @ X363 @ X364 ) )
= ( sub @ ( X362 @ X365 @ X363 ) @ X364 ) )
& ! [X366: term] :
( ( ( X351 @ id @ ( X362 @ id @ X366 ) @ id )
= $true )
| ( $true
!= ( X351 @ id @ X366 @ id ) ) )
& ( ( X351 @ id @ ( lam @ ( X362 @ sh @ one ) ) @ id )
!= $true ) )
| ? [X354: subst,X355: subst,X356: subst,X357: term] :
( ( ( X351 @ ( comp @ X356 @ X354 ) @ ( sub @ X357 @ X354 ) @ X355 )
!= $true )
& ( ( X351 @ X356 @ X357 @ ( comp @ X354 @ X355 ) )
= $true ) )
| ? [X358: subst,X359: subst,X360: term,X361: subst] :
( ( $true
!= ( X351 @ X358 @ X360 @ ( comp @ X359 @ X361 ) ) )
& ( $true
= ( X351 @ ( comp @ X358 @ X359 ) @ ( sub @ X360 @ X359 ) @ X361 ) ) )
| ? [X353: term] :
( ( X351 @ id @ X353 @ id )
!= ( X352 @ X353 ) )
| ( ( X352 @ ( lam @ X350 ) )
= $true ) ) ) )
& ( pushprop_lem0_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X180: term,X181: term > $o,X182: subst] :
( ? [X183: term,X184: term] :
( ( $true
= ( X181 @ X183 ) )
& ( $true
!= ( X181 @ ( ap @ X184 @ X183 ) ) )
& ( ( X181 @ X184 )
= $true ) )
| ? [X185: term] :
( ( ( var @ X185 )
= $true )
& ( ( X181 @ ( sub @ X185 @ X182 ) )
!= $true ) )
| ? [X186: term] :
( ! [X187: term] :
( ( ( X181 @ ( sub @ X186 @ ( push @ X187 @ id ) ) )
= $true )
| ( ( X181 @ X187 )
!= $true ) )
& ( $true
!= ( X181 @ ( lam @ X186 ) ) ) )
| ( ( X181 @ ( sub @ X180 @ X182 ) )
= $true ) )
| ? [X174: term,X175: term > $o] :
( ( ( X175 @ X174 )
!= $true )
& ! [X179: term] :
( ( $true
= ( X175 @ X179 ) )
| ( $true
!= ( var @ X179 ) ) )
& ! [X176: term,X177: term] :
( ( $true
= ( X175 @ ( ap @ X177 @ X176 ) ) )
| ( ( X175 @ X177 )
!= $true )
| ( ( X175 @ X176 )
!= $true ) )
& ! [X178: term] :
( ( ( X175 @ X178 )
!= $true )
| ( $true
= ( X175 @ ( lam @ X178 ) ) ) ) )
| ? [X173: subst] :
( ( comp @ X173 @ id )
!= X173 )
| ? [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
!= X171 )
| ? [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
!= X169 ) )
<=> ( induction2lem_lthm = $true ) )
& ( substmonoid_gthm = $true )
& ! [X271: term] :
? [X272: d_term] :
( ( d2term @ X272 )
= X271 )
& ( ( laminj = $true )
<=> ! [X167: term,X168: term] :
( ( X167 = X168 )
| ( ( lam @ X168 )
!= ( lam @ X167 ) ) ) )
& ( axabs = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X123: term,X124: subst > term > subst > $o] :
( ( $true
= ( X124 @ id @ ( lam @ X123 ) @ id ) )
| ? [X130: term,X131: subst,X132: subst,X133: subst] :
( ( ( X124 @ X132 @ X130 @ ( comp @ X131 @ X133 ) )
!= $true )
& ( $true
= ( X124 @ ( comp @ X132 @ X131 ) @ ( sub @ X130 @ X131 ) @ X133 ) ) )
| ? [X134: term,X135: subst,X136: subst,X137: subst] :
( ( ( X124 @ X137 @ X134 @ ( comp @ X136 @ X135 ) )
= $true )
& ( $true
!= ( X124 @ ( comp @ X137 @ X136 ) @ ( sub @ X134 @ X136 ) @ X135 ) ) )
| ? [X138: term] :
( ( ( X124 @ id @ ( sub @ X123 @ ( push @ X138 @ id ) ) @ id )
!= $true )
& ( ( X124 @ id @ X138 @ id )
= $true ) )
| ? [X125: subst > term > term] :
( ! [X127: term,X128: subst,X129: subst] :
( ( X125 @ ( comp @ X129 @ X128 ) @ ( sub @ X127 @ X128 ) )
= ( sub @ ( X125 @ X129 @ X127 ) @ X128 ) )
& ! [X126: term] :
( ( ( X124 @ id @ X126 @ id )
!= $true )
| ( ( X124 @ id @ ( X125 @ id @ X126 ) @ id )
= $true ) )
& ( $true
!= ( X124 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) ) ) )
| ? [X115: term,X116: subst > term > subst > $o] :
( ! [X122: term] :
( ( ( X116 @ id @ X122 @ id )
!= $true )
| ( ( X116 @ id @ ( sub @ X115 @ ( push @ X122 @ id ) ) @ id )
= $true ) )
& ( ( X116 @ id @ ( lam @ ( sub @ X115 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X117: subst > term > term] :
( ( ( X116 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id )
= $true )
| ? [X119: term,X120: subst,X121: subst] :
( ( X117 @ ( comp @ X121 @ X120 ) @ ( sub @ X119 @ X120 ) )
!= ( sub @ ( X117 @ X121 @ X119 ) @ X120 ) )
| ? [X118: term] :
( ( $true
= ( X116 @ id @ X118 @ id ) )
& ( $true
!= ( X116 @ id @ ( X117 @ id @ X118 ) @ id ) ) ) ) )
| ? [X114: term] :
( ( sub @ X114 @ id )
!= X114 ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj1_gthm = $true )
& ( ! [X414: term,X415: subst > term > subst > $o] :
( ? [X416: subst > term > term] :
( ! [X417: term] :
( ( ( X415 @ id @ ( X416 @ id @ X417 ) @ id )
= $true )
| ( ( X415 @ id @ X417 @ id )
!= $true ) )
& ( ( X415 @ id @ ( lam @ ( X416 @ sh @ one ) ) @ id )
!= $true )
& ! [X418: term,X419: subst,X420: subst] :
( ( sub @ ( X416 @ X419 @ X418 ) @ X420 )
= ( X416 @ ( comp @ X419 @ X420 ) @ ( sub @ X418 @ X420 ) ) ) )
| ( $true
= ( X415 @ id @ ( lam @ ( sub @ X414 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X421: term] :
( ( $true
= ( X415 @ id @ X421 @ id ) )
& ( ( X415 @ id @ ( sub @ X414 @ ( push @ X421 @ id ) ) @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( ( apinj2 = $true )
<=> ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X98 ) ) )
& ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
<=> ( axvarcons = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X7: subst > term > subst > $o] :
? [X8: term > $o] :
! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X345: term,X346: term,X347: term,X348: term] :
( ( ( ap @ ( sub @ X345 @ id ) @ X347 )
!= ( ap @ ( sub @ X348 @ id ) @ X346 ) )
| ( X345 = X348 ) )
| ? [X341: term,X342: term,X343: term,X344: term] :
( ( X341 != X342 )
& ( ( ap @ X342 @ X343 )
= ( ap @ X341 @ X344 ) ) )
| ? [X340: term] :
( ( sub @ X340 @ id )
!= X340 ) ) )
& ( ( substmonoid_lthm = $true )
<=> ( ( ! [X333: subst] :
( ( comp @ X333 @ id )
= X333 )
& ! [X334: subst] :
( ( comp @ id @ X334 )
= X334 ) )
| ? [X332: subst] :
( ( comp @ X332 @ id )
!= X332 )
| ? [X331: subst] :
( ( comp @ id @ X331 )
!= X331 ) ) )
& ( ( hoasapinj1 = $true )
<=> ! [X378: term,X379: term,X380: term,X381: term] :
( ( X379 = X380 )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) ) )
& ! [X382: subst] :
? [X383: d_subst] :
( ( d2subst @ X383 )
= X382 )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
!= $true )
| ? [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
!= ( X2 @ X6 ) )
| ( ( X2 @ one )
= $true ) )
| ? [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
!= X0 ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lem2_lthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X69: term,X70: subst > term > subst > $o] :
( ? [X72: subst > term > term] :
( ! [X76: term] :
( ( $true
!= ( X70 @ id @ X76 @ id ) )
| ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true ) )
& ( $true
!= ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id ) )
& ! [X73: subst,X74: term,X75: subst] :
( ( sub @ ( X72 @ X75 @ X74 ) @ X73 )
= ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) ) ) )
| ? [X71: term] :
( ( $true
= ( X70 @ id @ X71 @ id ) )
& ( $true
!= ( X70 @ id @ ( sub @ X69 @ ( push @ X71 @ id ) ) @ id ) ) )
| ( ( X70 @ id @ ( lam @ X69 ) @ id )
= $true ) )
| ? [X61: term,X62: subst > term > subst > $o] :
( ! [X68: term] :
( ( ( X62 @ id @ ( sub @ X61 @ ( push @ X68 @ id ) ) @ id )
= $true )
| ( ( X62 @ id @ X68 @ id )
!= $true ) )
& ( ( X62 @ id @ ( lam @ ( sub @ X61 @ ( push @ one @ sh ) ) ) @ id )
!= $true )
& ! [X63: subst > term > term] :
( ( ( X62 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true )
| ? [X64: subst,X65: subst,X66: term] :
( ( X63 @ ( comp @ X65 @ X64 ) @ ( sub @ X66 @ X64 ) )
!= ( sub @ ( X63 @ X65 @ X66 ) @ X64 ) )
| ? [X67: term] :
( ( $true
!= ( X62 @ id @ ( X63 @ id @ X67 ) @ id ) )
& ( $true
= ( X62 @ id @ X67 @ id ) ) ) ) )
| ? [X60: term] :
( ( sub @ X60 @ id )
!= X60 ) ) )
& ( ( induction2lem = $true )
<=> ! [X139: term,X140: term > $o,X141: subst] :
( ? [X145: term,X146: term] :
( ( ( X140 @ X146 )
= $true )
& ( $true
!= ( X140 @ ( ap @ X145 @ X146 ) ) )
& ( ( X140 @ X145 )
= $true ) )
| ? [X142: term] :
( ( ( var @ X142 )
= $true )
& ( ( X140 @ ( sub @ X142 @ X141 ) )
!= $true ) )
| ? [X143: term] :
( ! [X144: term] :
( ( ( X140 @ X144 )
!= $true )
| ( ( X140 @ ( sub @ X143 @ ( push @ X144 @ id ) ) )
= $true ) )
& ( $true
!= ( X140 @ ( lam @ X143 ) ) ) )
| ( $true
= ( X140 @ ( sub @ X139 @ X141 ) ) ) ) )
& ( ( hoasinduction = $true )
<=> ! [X37: subst > term > subst > $o,X38: term] :
( ? [X46: subst,X47: subst,X48: subst,X49: term] :
( ( ( X37 @ X46 @ X49 @ ( comp @ X47 @ X48 ) )
= $true )
& ( ( X37 @ ( comp @ X46 @ X47 ) @ ( sub @ X49 @ X47 ) @ X48 )
!= $true ) )
| ( $true
= ( X37 @ id @ X38 @ id ) )
| ? [X39: term,X40: term] :
( ( $true
= ( X37 @ id @ X40 @ id ) )
& ( $true
= ( X37 @ id @ X39 @ id ) )
& ( $true
!= ( X37 @ id @ ( ap @ ( sub @ X40 @ id ) @ X39 ) @ id ) ) )
| ? [X41: subst > term > term] :
( ( ( X37 @ id @ ( lam @ ( X41 @ sh @ one ) ) @ id )
!= $true )
& ! [X45: term] :
( ( $true
= ( X37 @ id @ ( X41 @ id @ X45 ) @ id ) )
| ( ( X37 @ id @ X45 @ id )
!= $true ) )
& ! [X42: term,X43: subst,X44: subst] :
( ( X41 @ ( comp @ X43 @ X44 ) @ ( sub @ X42 @ X44 ) )
= ( sub @ ( X41 @ X43 @ X42 ) @ X44 ) ) )
| ? [X54: term] :
( ( $true
= ( var @ ( sub @ X54 @ id ) ) )
& ( $true
!= ( X37 @ id @ X54 @ id ) ) )
| ? [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
= ( X37 @ ( comp @ X51 @ X50 ) @ ( sub @ X52 @ X50 ) @ X53 ) )
& ( ( X37 @ X51 @ X52 @ ( comp @ X50 @ X53 ) )
!= $true ) ) ) )
& ( ( hoasapinj2 = $true )
<=> ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) ) )
& ( hoasapinj2_gthm = $true )
& ( termmset_gthm = $true )
& ( pushprop_lem2v2 = $true )
& ( induction2_gthm = $true )
& ( ! [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
= X335 )
<=> ( axscons = $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ! [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ? [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
!= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) )
| ? [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
!= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
| ( ( X498 @ X497 @ X495 )
= ( X496 @ X497 @ X495 ) )
| ( ( lam @ ( X498 @ sh @ one ) )
!= ( lam @ ( X496 @ sh @ one ) ) ) )
<=> ( hoaslaminj = $true ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ulamvarsh = $true )
& ( ( ! [X83: term,X84: subst > term > term,X85: subst,X86: subst > term > term] :
( ( ( lam @ ( X84 @ sh @ one ) )
!= ( lam @ ( X86 @ sh @ one ) ) )
| ? [X90: subst,X91: subst,X92: term] :
( ( X84 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) )
!= ( sub @ ( X84 @ X90 @ X92 ) @ X91 ) )
| ( ( X84 @ X85 @ X83 )
= ( X86 @ X85 @ X83 ) )
| ? [X87: subst,X88: subst,X89: term] :
( ( X86 @ ( comp @ X87 @ X88 ) @ ( sub @ X89 @ X88 ) )
!= ( sub @ ( X86 @ X87 @ X89 ) @ X88 ) ) )
| ? [X81: term,X82: term] :
( ( ( lam @ X81 )
= ( lam @ X82 ) )
& ( X81 != X82 ) )
| ? [X79: subst,X80: term] :
( ( comp @ sh @ ( push @ X80 @ X79 ) )
!= X79 )
| ? [X77: subst,X78: term] :
( ( sub @ one @ ( push @ X78 @ X77 ) )
!= X78 ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( ( induction = $true )
<=> ! [X206: term > $o,X207: term] :
( ? [X209: term,X210: term] :
( ( $true
= ( X206 @ X210 ) )
& ( ( X206 @ ( ap @ X209 @ X210 ) )
!= $true )
& ( $true
= ( X206 @ X209 ) ) )
| ? [X211: term] :
( ( $true
= ( var @ X211 ) )
& ( $true
!= ( X206 @ X211 ) ) )
| ? [X208: term] :
( ( $true
!= ( X206 @ ( lam @ X208 ) ) )
& ( $true
= ( X206 @ X208 ) ) )
| ( ( X206 @ X207 )
= $true ) ) )
& ( pushprop_lem1v2_gthm = $true ) ),
inference(ennf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
( ( hoasinduction_lem1 = $true )
& ( lamnotap != $true )
& ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
<=> ( axidl = $true ) )
& ( ( hoasinduction_lthm_3 = $true )
<=> ( ! [X292: subst > term > subst > $o] :
~ ! [X293: term > $o] :
~ ! [X294: term] :
( ( X292 @ id @ X294 @ id )
= ( X293 @ X294 ) )
=> ( ! [X295: term > $o,X296: term] :
( ~ ! [X299: term] :
( ( $true
!= ( var @ X299 ) )
| ( ( X295 @ X299 )
= $true ) )
| ~ ! [X297: term,X298: term] :
( ( ( X295 @ X297 )
!= $true )
| ( ( X295 @ X298 )
!= $true )
| ( $true
= ( X295 @ ( ap @ X297 @ X298 ) ) ) )
| ( ( X295 @ X296 )
= $true )
| ~ ! [X300: term] :
( ( ( X295 @ ( lam @ X300 ) )
= $true )
| ~ ! [X301: term] :
( ( ( X295 @ X301 )
!= $true )
| ( $true
= ( X295 @ ( sub @ X300 @ ( push @ X301 @ id ) ) ) ) ) ) )
=> ( ! [X302: term] :
( ( sub @ X302 @ id )
= X302 )
=> ( ! [X303: subst > term > subst > $o,X304: term,X305: term > $o] :
( ( ( X305 @ ( lam @ X304 ) )
= $true )
| ~ ! [X311: term] :
( ( ( X305 @ X311 )
!= $true )
| ( ( X305 @ ( sub @ X304 @ ( push @ X311 @ id ) ) )
= $true ) )
| ~ ! [X306: subst > term > term] :
( ( $true
= ( X303 @ id @ ( lam @ ( X306 @ sh @ one ) ) @ id ) )
| ~ ! [X308: term,X309: subst,X310: subst] :
( ( X306 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) )
= ( sub @ ( X306 @ X310 @ X308 ) @ X309 ) )
| ~ ! [X307: term] :
( ( $true
= ( X303 @ id @ ( X306 @ id @ X307 ) @ id ) )
| ( $true
!= ( X303 @ id @ X307 @ id ) ) ) )
| ~ ! [X312: term] :
( ( X303 @ id @ X312 @ id )
= ( X305 @ X312 ) ) )
=> ! [X313: term,X314: subst > term > subst > $o] :
( ~ ! [X326: subst > term > term] :
( ( ( X314 @ id @ ( lam @ ( X326 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X327: subst,X328: term,X329: subst] :
( ( X326 @ ( comp @ X327 @ X329 ) @ ( sub @ X328 @ X329 ) )
= ( sub @ ( X326 @ X327 @ X328 ) @ X329 ) )
| ~ ! [X330: term] :
( ( ( X314 @ id @ X330 @ id )
!= $true )
| ( ( X314 @ id @ ( X326 @ id @ X330 ) @ id )
= $true ) ) )
| ~ ! [X319: term] :
( ( $true
!= ( var @ ( sub @ X319 @ id ) ) )
| ( ( X314 @ id @ X319 @ id )
= $true ) )
| ~ ! [X322: subst,X323: subst,X324: term,X325: subst] :
( ( ( X314 @ ( comp @ X322 @ X323 ) @ ( sub @ X324 @ X323 ) @ X325 )
= $true )
| ( ( X314 @ X322 @ X324 @ ( comp @ X323 @ X325 ) )
!= $true ) )
| ~ ! [X320: term,X321: term] :
( ( ( X314 @ id @ ( ap @ ( sub @ X320 @ id ) @ X321 ) @ id )
= $true )
| ( ( X314 @ id @ X320 @ id )
!= $true )
| ( $true
!= ( X314 @ id @ X321 @ id ) ) )
| ( ( X314 @ id @ X313 @ id )
= $true )
| ~ ! [X315: subst,X316: subst,X317: term,X318: subst] :
( ( $true
= ( X314 @ X318 @ X317 @ ( comp @ X315 @ X316 ) ) )
| ( $true
!= ( X314 @ ( comp @ X318 @ X315 ) @ ( sub @ X317 @ X315 ) @ X316 ) ) ) ) ) ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_gthm = $true )
& ( ! [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) )
| ( ( X266 @ X265 )
!= $true ) )
<=> ( pushprop_lem1 = $true ) )
& ! [X96: d_term] : ( d_one = X96 )
& ( ! [X368: term,X369: term > $o,X370: subst > term > subst > $o] :
( ~ ! [X371: subst > term > term] :
( ( $true
= ( X370 @ id @ ( lam @ ( X371 @ sh @ one ) ) @ id ) )
| ~ ! [X373: term,X374: subst,X375: subst] :
( ( sub @ ( X371 @ X375 @ X373 ) @ X374 )
= ( X371 @ ( comp @ X375 @ X374 ) @ ( sub @ X373 @ X374 ) ) )
| ~ ! [X372: term] :
( ( $true
!= ( X370 @ id @ X372 @ id ) )
| ( $true
= ( X370 @ id @ ( X371 @ id @ X372 ) @ id ) ) ) )
| ~ ! [X377: term] :
( ( ( X369 @ X377 )
!= $true )
| ( ( X369 @ ( sub @ X368 @ ( push @ X377 @ id ) ) )
= $true ) )
| ~ ! [X376: term] :
( ( X370 @ id @ X376 @ id )
= ( X369 @ X376 ) )
| ( $true
= ( X369 @ ( lam @ X368 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( ulamvar1 != $true )
& ( ! [X147: term] :
~ ! [X148: subst > term > term] :
~ ! [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
= ( sub @ X147 @ ( push @ X149 @ X150 ) ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( ! [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
= X11 )
=> ( ! [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
= X12 )
=> ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
| ( ( X18 @ X15 )
!= $true ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoaslamnotap_lthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X220: term > $o,X221: term,X222: subst > term > subst > $o,X223: term] :
( ~ ! [X224: subst,X225: term,X226: subst,X227: subst] :
( ( ( X222 @ ( comp @ X224 @ X227 ) @ ( sub @ X225 @ X227 ) @ X226 )
= $true )
| ( ( X222 @ X224 @ X225 @ ( comp @ X227 @ X226 ) )
!= $true ) )
| ~ ! [X232: term] :
( ( X220 @ X232 )
= ( X222 @ id @ X232 @ id ) )
| ( ( X220 @ X221 )
!= $true )
| ( ( X220 @ X223 )
!= $true )
| ~ ! [X228: subst,X229: subst,X230: subst,X231: term] :
( ( ( X222 @ ( comp @ X230 @ X228 ) @ ( sub @ X231 @ X228 ) @ X229 )
!= $true )
| ( $true
= ( X222 @ X230 @ X231 @ ( comp @ X228 @ X229 ) ) ) )
| ~ ! [X233: term,X234: term] :
( ( ( X222 @ id @ X234 @ id )
!= $true )
| ( $true
= ( X222 @ id @ ( ap @ ( sub @ X234 @ id ) @ X233 ) @ id ) )
| ( $true
!= ( X222 @ id @ X233 @ id ) ) )
| ( ( X220 @ ( ap @ X221 @ X223 ) )
= $true ) ) )
& ( ( apinj1 = $true )
<=> ! [X484: term,X485: term,X486: term,X487: term] :
( ( X485 = X487 )
| ( ( ap @ X485 @ X484 )
!= ( ap @ X487 @ X486 ) ) ) )
& ( ( induction2 = $true )
<=> ! [X251: term,X252: term > $o] :
( ~ ! [X257: term] :
( ( $true
!= ( var @ X257 ) )
| ( $true
= ( X252 @ X257 ) ) )
| ~ ! [X253: term,X254: term] :
( ( $true
!= ( X252 @ X253 ) )
| ( ( X252 @ X254 )
!= $true )
| ( $true
= ( X252 @ ( ap @ X254 @ X253 ) ) ) )
| ~ ! [X255: term] :
( ( ( X252 @ ( lam @ X255 ) )
= $true )
| ~ ! [X256: term] :
( ( $true
!= ( X252 @ X256 ) )
| ( ( X252 @ ( sub @ X255 @ ( push @ X256 @ id ) ) )
= $true ) ) )
| ( $true
= ( X252 @ X251 ) ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lthm_orig = $true )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X93: subst > term > subst > $o] :
~ ! [X94: term > $o] :
~ ! [X95: term] :
( ( X94 @ X95 )
= ( X93 @ id @ X95 @ id ) ) )
& ( apnotvar = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ! [X258: term,X259: subst > term > term,X260: term] :
( ~ ! [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) )
| ( ( lam @ ( X259 @ sh @ one ) )
!= ( ap @ ( sub @ X260 @ id ) @ X258 ) ) )
<=> ( hoaslamnotap = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( axassoc = $true )
& ( ! [X338: term] :
~ ! [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
!= ( X339 @ sh @ one ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( ( termmset = $true )
<=> ! [X57: term] :
( ( sub @ X57 @ id )
= X57 ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X505: subst > term > subst > $o,X506: term] :
( ~ ! [X507: term] :
( ( $true
!= ( X505 @ id @ X507 @ id ) )
| ( ( X505 @ id @ ( sub @ X506 @ ( push @ X507 @ id ) ) @ id )
= $true ) )
| ~ ! [X508: subst > term > term] :
( ( ( X505 @ id @ ( lam @ ( X508 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X509: term] :
( ( $true
!= ( X505 @ id @ X509 @ id ) )
| ( $true
= ( X505 @ id @ ( X508 @ id @ X509 ) @ id ) ) )
| ~ ! [X510: subst,X511: subst,X512: term] :
( ( X508 @ ( comp @ X511 @ X510 ) @ ( sub @ X512 @ X510 ) )
= ( sub @ ( X508 @ X511 @ X512 ) @ X510 ) ) )
| ( ( X505 @ id @ ( lam @ ( sub @ X506 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( axclos = $true )
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
= ( d2subst @ X55 ) )
=> ( X55 = X56 ) )
& ( ( ! [X488: term] :
( ( sub @ X488 @ id )
= X488 )
=> ! [X489: term] :
( ( sub @ X489 @ id )
= X489 ) )
<=> ( termmset_lthm = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3a = $true )
<=> ! [X532: term,X533: subst > term > subst > $o] :
( ~ ! [X539: term] :
( ( ( X533 @ id @ X539 @ id )
!= $true )
| ( ( X533 @ id @ ( sub @ X532 @ ( push @ X539 @ id ) ) @ id )
= $true ) )
| ( $true
= ( X533 @ id @ ( lam @ X532 ) @ id ) )
| ~ ! [X534: subst > term > term] :
( ( ( X533 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X535: subst,X536: subst,X537: term] :
( ( sub @ ( X534 @ X535 @ X537 ) @ X536 )
= ( X534 @ ( comp @ X535 @ X536 ) @ ( sub @ X537 @ X536 ) ) )
| ~ ! [X538: term] :
( ( ( X533 @ id @ X538 @ id )
!= $true )
| ( ( X533 @ id @ ( X534 @ id @ X538 ) @ id )
= $true ) ) ) ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
= X273 )
<=> ( axshiftcons = $true ) )
& ( ! [X214: subst] :
( ( comp @ X214 @ id )
= X214 )
<=> ( axidr = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ! [X25: subst > term > subst > $o,X26: term] :
( ~ ! [X27: term] :
( ( $true
!= ( X25 @ id @ X27 @ id ) )
| ( $true
= ( X25 @ id @ ( sub @ X26 @ ( push @ X27 @ id ) ) @ id ) ) )
| ~ ! [X28: term,X29: subst > term > term] :
( ~ ! [X30: term] :
( ( ( X25 @ id @ ( X29 @ id @ X30 ) @ id )
= $true )
| ( ( X25 @ id @ X30 @ id )
!= $true ) )
| ~ ! [X34: subst,X35: subst,X36: term] :
( ( sub @ X28 @ ( push @ ( sub @ X36 @ X34 ) @ ( comp @ X35 @ X34 ) ) )
= ( X29 @ ( comp @ X35 @ X34 ) @ ( sub @ X36 @ X34 ) ) )
| ( ( X25 @ id @ ( lam @ ( X29 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X31: subst,X32: term,X33: subst] :
( ( sub @ ( sub @ X28 @ ( push @ X32 @ X33 ) ) @ X31 )
= ( sub @ ( X29 @ X33 @ X32 ) @ X31 ) ) )
| ( $true
= ( X25 @ id @ ( lam @ ( sub @ X26 @ ( push @ one @ sh ) ) ) @ id ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( ( ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 )
& ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 ) )
<=> ( substmonoid = $true ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X422: term > $o,X423: term] :
( ( ( X422 @ X423 )
= $true )
| ~ ! [X425: term,X426: term] :
( ( $true
!= ( X422 @ X425 ) )
| ( ( X422 @ X426 )
!= $true )
| ( ( X422 @ ( ap @ X426 @ X425 ) )
= $true ) )
| ~ ! [X424: term] :
( ( $true
= ( X422 @ X424 ) )
| ( ( var @ X424 )
!= $true ) )
| ~ ! [X427: term] :
( ( $true
= ( X422 @ ( lam @ X427 ) ) )
| ~ ! [X428: term] :
( ( ( X422 @ X428 )
!= $true )
| ( ( X422 @ ( sub @ X427 @ ( push @ X428 @ id ) ) )
= $true ) ) ) )
=> ( ! [X429: subst > term > subst > $o,X430: term,X431: term] :
( ~ ! [X436: subst,X437: subst,X438: term,X439: subst] :
( ( $true
!= ( X429 @ X437 @ X438 @ ( comp @ X436 @ X439 ) ) )
| ( ( X429 @ ( comp @ X437 @ X436 ) @ ( sub @ X438 @ X436 ) @ X439 )
= $true ) )
| ~ ! [X432: term,X433: subst,X434: subst,X435: subst] :
( ( ( X429 @ X435 @ X432 @ ( comp @ X434 @ X433 ) )
= $true )
| ( ( X429 @ ( comp @ X435 @ X434 ) @ ( sub @ X432 @ X434 ) @ X433 )
!= $true ) )
| ( ( X429 @ id @ X430 @ id )
!= $true )
| ( ( X429 @ id @ ( ap @ X431 @ X430 ) @ id )
= $true )
| ( ( X429 @ id @ X431 @ id )
!= $true )
| ~ ! [X440: term,X441: term] :
( ( ( X429 @ id @ X440 @ id )
!= $true )
| ( $true
!= ( X429 @ id @ X441 @ id ) )
| ( $true
= ( X429 @ id @ ( ap @ ( sub @ X440 @ id ) @ X441 ) @ id ) ) ) )
=> ( ! [X442: term,X443: subst > term > subst > $o] :
( ~ ! [X449: term] :
( ( ( X443 @ id @ ( sub @ X442 @ ( push @ X449 @ id ) ) @ id )
= $true )
| ( $true
!= ( X443 @ id @ X449 @ id ) ) )
| ~ ! [X454: subst,X455: subst,X456: term,X457: subst] :
( ( $true
!= ( X443 @ X457 @ X456 @ ( comp @ X455 @ X454 ) ) )
| ( ( X443 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 )
= $true ) )
| ~ ! [X450: subst,X451: term,X452: subst,X453: subst] :
( ( ( X443 @ ( comp @ X452 @ X453 ) @ ( sub @ X451 @ X453 ) @ X450 )
!= $true )
| ( $true
= ( X443 @ X452 @ X451 @ ( comp @ X453 @ X450 ) ) ) )
| ( ( X443 @ id @ ( lam @ X442 ) @ id )
= $true )
| ~ ! [X444: subst > term > term] :
( ~ ! [X445: subst,X446: term,X447: subst] :
( ( X444 @ ( comp @ X447 @ X445 ) @ ( sub @ X446 @ X445 ) )
= ( sub @ ( X444 @ X447 @ X446 ) @ X445 ) )
| ~ ! [X448: term] :
( ( ( X443 @ id @ X448 @ id )
!= $true )
| ( $true
= ( X443 @ id @ ( X444 @ id @ X448 ) @ id ) ) )
| ( $true
= ( X443 @ id @ ( lam @ ( X444 @ sh @ one ) ) @ id ) ) ) )
=> ! [X458: term,X459: subst > term > subst > $o] :
( ~ ! [X462: subst > term > term] :
( ~ ! [X464: subst,X465: term,X466: subst] :
( ( X462 @ ( comp @ X466 @ X464 ) @ ( sub @ X465 @ X464 ) )
= ( sub @ ( X462 @ X466 @ X465 ) @ X464 ) )
| ( $true
= ( X459 @ id @ ( lam @ ( X462 @ sh @ one ) ) @ id ) )
| ~ ! [X463: term] :
( ( ( X459 @ id @ ( X462 @ id @ X463 ) @ id )
= $true )
| ( ( X459 @ id @ X463 @ id )
!= $true ) ) )
| ~ ! [X460: term,X461: term] :
( ( $true
= ( X459 @ id @ ( ap @ ( sub @ X460 @ id ) @ X461 ) @ id ) )
| ( $true
!= ( X459 @ id @ X461 @ id ) )
| ( $true
!= ( X459 @ id @ X460 @ id ) ) )
| ( ( X459 @ id @ X458 @ id )
= $true )
| ~ ! [X475: term] :
( ( $true
!= ( var @ ( sub @ X475 @ id ) ) )
| ( $true
= ( X459 @ id @ X475 @ id ) ) )
| ~ ! [X467: term,X468: subst,X469: subst,X470: subst] :
( ( ( X459 @ X469 @ X467 @ ( comp @ X470 @ X468 ) )
= $true )
| ( ( X459 @ ( comp @ X469 @ X470 ) @ ( sub @ X467 @ X470 ) @ X468 )
!= $true ) )
| ~ ! [X471: subst,X472: subst,X473: subst,X474: term] :
( ( ( X459 @ X473 @ X474 @ ( comp @ X472 @ X471 ) )
!= $true )
| ( ( X459 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 )
= $true ) ) ) ) ) ) )
& ( ! [X212: term] :
~ ! [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
!= ( X213 @ sh @ one ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( axmap = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( ( X216 @ one )
= $true )
| ( $true
!= ( X217 @ X215 ) )
| ~ ! [X219: term] :
( ( X216 @ X219 )
= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) ) ) )
& ( axvarshift = $true )
& ! [X270: d_subst] : ( d_id = X270 )
& ( ! [X490: term > $o,X491: term,X492: subst] :
~ ! [X493: term > $o] :
~ ! [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
= ( X493 @ X494 ) )
<=> ( pushprop_lem0 = $true ) )
& ( pushprop = $true )
& ( hoasinduction_lem1_gthm = $true )
& ( hoaslamnotvar_lthm = $true )
& ( ( induction2_lthm = $true )
<=> ( ! [X151: term] :
( ( sub @ X151 @ id )
= X151 )
=> ( ! [X152: term,X153: subst,X154: term > $o] :
( ~ ! [X159: term] :
( ( ( var @ X159 )
!= $true )
| ( ( X154 @ ( sub @ X159 @ X153 ) )
= $true ) )
| ( ( X154 @ ( sub @ X152 @ X153 ) )
= $true )
| ~ ! [X155: term] :
( ~ ! [X156: term] :
( ( ( X154 @ ( sub @ X155 @ ( push @ X156 @ id ) ) )
= $true )
| ( ( X154 @ X156 )
!= $true ) )
| ( $true
= ( X154 @ ( lam @ X155 ) ) ) )
| ~ ! [X157: term,X158: term] :
( ( $true
!= ( X154 @ X158 ) )
| ( ( X154 @ X157 )
!= $true )
| ( ( X154 @ ( ap @ X158 @ X157 ) )
= $true ) ) )
=> ! [X160: term,X161: term > $o] :
( ( ( X161 @ X160 )
= $true )
| ~ ! [X162: term] :
( ( $true
!= ( var @ X162 ) )
| ( $true
= ( X161 @ X162 ) ) )
| ~ ! [X163: term,X164: term] :
( ( $true
!= ( X161 @ X163 ) )
| ( ( X161 @ X164 )
!= $true )
| ( ( X161 @ ( ap @ X164 @ X163 ) )
= $true ) )
| ~ ! [X165: term] :
( ~ ! [X166: term] :
( ( ( X161 @ X166 )
!= $true )
| ( $true
= ( X161 @ ( sub @ X165 @ ( push @ X166 @ id ) ) ) ) )
| ( $true
= ( X161 @ ( lam @ X165 ) ) ) ) ) ) ) )
& ( hoasapnotvar = $true )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X105: term,X106: subst > term > subst > $o] :
( ~ ! [X112: term,X113: term] :
( ( ( X106 @ id @ X113 @ id )
!= $true )
| ( ( X106 @ id @ X112 @ id )
!= $true )
| ( $true
= ( X106 @ id @ ( ap @ ( sub @ X113 @ id ) @ X112 ) @ id ) ) )
| ( $true
= ( X106 @ id @ X105 @ id ) )
| ~ ! [X107: subst > term > term] :
( ~ ! [X109: subst,X110: term,X111: subst] :
( ( X107 @ ( comp @ X111 @ X109 ) @ ( sub @ X110 @ X109 ) )
= ( sub @ ( X107 @ X111 @ X110 ) @ X109 ) )
| ~ ! [X108: term] :
( ( $true
= ( X106 @ id @ ( X107 @ id @ X108 ) @ id ) )
| ( $true
!= ( X106 @ id @ X108 @ id ) ) )
| ( ( X106 @ id @ ( lam @ ( X107 @ sh @ one ) ) @ id )
= $true ) ) ) )
& ( ! [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
!= ( sub @ X275 @ sh ) )
| ( X275 = X276 ) )
<=> ( shinj = $true ) )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X235: subst > term > subst > $o,X236: term] :
( ~ ! [X237: subst,X238: subst,X239: subst,X240: term] :
( ( $true
!= ( X235 @ ( comp @ X238 @ X239 ) @ ( sub @ X240 @ X239 ) @ X237 ) )
| ( ( X235 @ X238 @ X240 @ ( comp @ X239 @ X237 ) )
= $true ) )
| ~ ! [X246: term] :
( ( $true
= ( X235 @ id @ ( sub @ X236 @ ( push @ X246 @ id ) ) @ id ) )
| ( ( X235 @ id @ X246 @ id )
!= $true ) )
| ~ ! [X241: subst > term > term] :
( ( ( X235 @ id @ ( lam @ ( X241 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X245: term] :
( ( $true
= ( X235 @ id @ ( X241 @ id @ X245 ) @ id ) )
| ( ( X235 @ id @ X245 @ id )
!= $true ) )
| ~ ! [X242: term,X243: subst,X244: subst] :
( ( X241 @ ( comp @ X244 @ X243 ) @ ( sub @ X242 @ X243 ) )
= ( sub @ ( X241 @ X244 @ X242 ) @ X243 ) ) )
| ~ ! [X247: subst,X248: subst,X249: subst,X250: term] :
( ( ( X235 @ X248 @ X250 @ ( comp @ X249 @ X247 ) )
!= $true )
| ( $true
= ( X235 @ ( comp @ X248 @ X249 ) @ ( sub @ X250 @ X249 ) @ X247 ) ) )
| ( ( X235 @ id @ ( lam @ X236 ) @ id )
= $true ) ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X188: subst > term > subst > $o,X189: term,X190: term > $o] :
( ( ( X190 @ ( lam @ X189 ) )
= $true )
| ~ ! [X200: subst > term > term] :
( ~ ! [X201: subst,X202: subst,X203: term] :
( ( X200 @ ( comp @ X202 @ X201 ) @ ( sub @ X203 @ X201 ) )
= ( sub @ ( X200 @ X202 @ X203 ) @ X201 ) )
| ( ( X188 @ id @ ( lam @ ( X200 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X204: term] :
( ( ( X188 @ id @ X204 @ id )
!= $true )
| ( ( X188 @ id @ ( X200 @ id @ X204 ) @ id )
= $true ) ) )
| ~ ! [X191: subst,X192: subst,X193: subst,X194: term] :
( ( ( X188 @ ( comp @ X193 @ X191 ) @ ( sub @ X194 @ X191 ) @ X192 )
= $true )
| ( ( X188 @ X193 @ X194 @ ( comp @ X191 @ X192 ) )
!= $true ) )
| ~ ! [X195: subst,X196: subst,X197: subst,X198: term] :
( ( ( X188 @ X197 @ X198 @ ( comp @ X196 @ X195 ) )
= $true )
| ( ( X188 @ ( comp @ X197 @ X196 ) @ ( sub @ X198 @ X196 ) @ X195 )
!= $true ) )
| ~ ! [X199: term] :
( ( ( X190 @ ( sub @ X189 @ ( push @ X199 @ id ) ) )
= $true )
| ( ( X190 @ X199 )
!= $true ) )
| ~ ! [X205: term] :
( ( X188 @ id @ X205 @ id )
= ( X190 @ X205 ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ! [X264: term] :
( ( sub @ X264 @ id )
= X264 )
<=> ( axvarid = $true ) )
& ( pushprop_gthm = $true )
& ( 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 )
& ( hoasinduction_lem2v2_gthm = $true )
& ( lamnotvar = $true )
& ( ! [X101: term] :
~ ! [X102: subst > term > term] :
~ ! [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
= ( X102 @ X104 @ X103 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X518: subst > term > subst > $o,X519: term,X520: term] :
( ( ( X518 @ id @ ( ap @ X519 @ X520 ) @ id )
= $true )
| ~ ! [X527: subst,X528: subst,X529: term,X530: subst] :
( ( $true
!= ( X518 @ X528 @ X529 @ ( comp @ X530 @ X527 ) ) )
| ( $true
= ( X518 @ ( comp @ X528 @ X530 ) @ ( sub @ X529 @ X530 ) @ X527 ) ) )
| ( $true
!= ( X518 @ id @ X520 @ id ) )
| ( ( X518 @ id @ X519 @ id )
!= $true )
| ~ ! [X525: term,X526: term] :
( ( $true
!= ( X518 @ id @ X525 @ id ) )
| ( ( X518 @ id @ ( ap @ ( sub @ X525 @ id ) @ X526 ) @ id )
= $true )
| ( $true
!= ( X518 @ id @ X526 @ id ) ) )
| ~ ! [X521: subst,X522: subst,X523: term,X524: subst] :
( ( $true
!= ( X518 @ ( comp @ X522 @ X524 ) @ ( sub @ X523 @ X524 ) @ X521 ) )
| ( $true
= ( X518 @ X522 @ X523 @ ( comp @ X524 @ X521 ) ) ) ) ) )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X277: term] :
~ ! [X278: subst > term > term] :
~ ! [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
= ( X278 @ X280 @ X279 ) )
=> ( ! [X281: term] :
( ( sub @ X281 @ id )
= X281 )
=> ! [X282: term > $o,X283: subst > term > subst > $o,X284: term] :
( ~ ! [X291: term] :
( ( $true
= ( X282 @ ( sub @ X284 @ ( push @ X291 @ id ) ) ) )
| ( ( X282 @ X291 )
!= $true ) )
| ( ( X282 @ ( lam @ X284 ) )
= $true )
| ~ ! [X285: subst > term > term] :
( ~ ! [X287: term,X288: subst,X289: subst] :
( ( X285 @ ( comp @ X289 @ X288 ) @ ( sub @ X287 @ X288 ) )
= ( sub @ ( X285 @ X289 @ X287 ) @ X288 ) )
| ~ ! [X286: term] :
( ( $true
= ( X283 @ id @ ( X285 @ id @ X286 ) @ id ) )
| ( $true
!= ( X283 @ id @ X286 @ id ) ) )
| ( $true
= ( X283 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id ) ) )
| ~ ! [X290: term] :
( ( X282 @ X290 )
= ( X283 @ id @ X290 @ id ) ) ) ) ) )
& ( ulamvarind = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ! [X384: subst > term > subst > $o] :
~ ! [X385: term > $o] :
~ ! [X386: term] :
( ( X384 @ id @ X386 @ id )
= ( X385 @ X386 ) )
=> ( ! [X387: term,X388: term > $o] :
( ~ ! [X390: term] :
( ~ ! [X391: term] :
( ( ( X388 @ X391 )
!= $true )
| ( ( X388 @ ( sub @ X390 @ ( push @ X391 @ id ) ) )
= $true ) )
| ( ( X388 @ ( lam @ X390 ) )
= $true ) )
| ( ( X388 @ X387 )
= $true )
| ~ ! [X392: term,X393: term] :
( ( ( X388 @ X393 )
!= $true )
| ( $true
= ( X388 @ ( ap @ X393 @ X392 ) ) )
| ( $true
!= ( X388 @ X392 ) ) )
| ~ ! [X389: term] :
( ( $true
!= ( var @ X389 ) )
| ( $true
= ( X388 @ X389 ) ) ) )
=> ( ! [X394: term] :
( ( sub @ X394 @ id )
= X394 )
=> ( ! [X395: term > $o,X396: term,X397: subst > term > subst > $o] :
( ~ ! [X398: term] :
( ( X397 @ id @ X398 @ id )
= ( X395 @ X398 ) )
| ( $true
= ( X395 @ ( lam @ X396 ) ) )
| ~ ! [X399: subst > term > term] :
( ~ ! [X400: subst,X401: subst,X402: term] :
( ( sub @ ( X399 @ X400 @ X402 ) @ X401 )
= ( X399 @ ( comp @ X400 @ X401 ) @ ( sub @ X402 @ X401 ) ) )
| ( $true
= ( X397 @ id @ ( lam @ ( X399 @ sh @ one ) ) @ id ) )
| ~ ! [X403: term] :
( ( $true
= ( X397 @ id @ ( X399 @ id @ X403 ) @ id ) )
| ( ( X397 @ id @ X403 @ id )
!= $true ) ) )
| ~ ! [X404: term] :
( ( ( X395 @ ( sub @ X396 @ ( push @ X404 @ id ) ) )
= $true )
| ( ( X395 @ X404 )
!= $true ) ) )
=> ! [X405: term,X406: subst > term > subst > $o] :
( ( $true
= ( X406 @ id @ X405 @ id ) )
| ~ ! [X407: subst > term > term] :
( ~ ! [X408: term] :
( ( ( X406 @ id @ ( X407 @ id @ X408 ) @ id )
= $true )
| ( ( X406 @ id @ X408 @ id )
!= $true ) )
| ( $true
= ( X406 @ id @ ( lam @ ( X407 @ sh @ one ) ) @ id ) )
| ~ ! [X409: subst,X410: subst,X411: term] :
( ( sub @ ( X407 @ X410 @ X411 ) @ X409 )
= ( X407 @ ( comp @ X410 @ X409 ) @ ( sub @ X411 @ X409 ) ) ) )
| ~ ! [X412: term,X413: term] :
( ( $true
!= ( X406 @ id @ X413 @ id ) )
| ( ( X406 @ id @ X412 @ id )
!= $true )
| ( ( X406 @ id @ ( ap @ ( sub @ X412 @ id ) @ X413 ) @ id )
= $true ) ) ) ) ) ) ) )
& ( hoasinduction_lem2_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( axapp = $true )
& ! [X23: d_term,X24: d_term] :
( ( ( d2term @ X23 )
= ( d2term @ X24 ) )
=> ( X23 = X24 ) )
& ( ! [X513: term,X514: term > $o,X515: subst] :
~ ! [X516: term > $o] :
~ ! [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
= ( X516 @ X517 ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X476: term,X477: term,X478: term,X479: term] :
( ( X476 = X479 )
| ( ( ap @ X477 @ X476 )
!= ( ap @ X478 @ X479 ) ) )
=> ! [X480: term,X481: term,X482: term,X483: term] :
( ( X482 = X483 )
| ( ( ap @ ( sub @ X481 @ id ) @ X483 )
!= ( ap @ ( sub @ X480 @ id ) @ X482 ) ) ) ) )
& ( pushprop_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X349: term] :
( ( sub @ X349 @ id )
= X349 )
=> ! [X350: term,X351: subst > term > subst > $o,X352: term > $o] :
( ~ ! [X367: term] :
( ( ( X352 @ X367 )
!= $true )
| ( ( X352 @ ( sub @ X350 @ ( push @ X367 @ id ) ) )
= $true ) )
| ~ ! [X354: subst,X355: subst,X356: subst,X357: term] :
( ( ( X351 @ X356 @ X357 @ ( comp @ X354 @ X355 ) )
!= $true )
| ( ( X351 @ ( comp @ X356 @ X354 ) @ ( sub @ X357 @ X354 ) @ X355 )
= $true ) )
| ( ( X352 @ ( lam @ X350 ) )
= $true )
| ~ ! [X353: term] :
( ( X351 @ id @ X353 @ id )
= ( X352 @ X353 ) )
| ~ ! [X358: subst,X359: subst,X360: term,X361: subst] :
( ( $true
!= ( X351 @ ( comp @ X358 @ X359 ) @ ( sub @ X360 @ X359 ) @ X361 ) )
| ( $true
= ( X351 @ X358 @ X360 @ ( comp @ X359 @ X361 ) ) ) )
| ~ ! [X362: subst > term > term] :
( ~ ! [X366: term] :
( ( ( X351 @ id @ ( X362 @ id @ X366 ) @ id )
= $true )
| ( $true
!= ( X351 @ id @ X366 @ id ) ) )
| ~ ! [X363: term,X364: subst,X365: subst] :
( ( X362 @ ( comp @ X365 @ X364 ) @ ( sub @ X363 @ X364 ) )
= ( sub @ ( X362 @ X365 @ X363 ) @ X364 ) )
| ( ( X351 @ id @ ( lam @ ( X362 @ sh @ one ) ) @ id )
= $true ) ) ) ) )
& ( pushprop_lem0_gthm = $true )
& ( hoaslamnotvar = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
= X169 )
=> ( ! [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
= X171 )
=> ( ! [X173: subst] :
( ( comp @ X173 @ id )
= X173 )
=> ( ! [X174: term,X175: term > $o] :
( ~ ! [X178: term] :
( ( ( X175 @ X178 )
!= $true )
| ( $true
= ( X175 @ ( lam @ X178 ) ) ) )
| ~ ! [X176: term,X177: term] :
( ( $true
= ( X175 @ ( ap @ X177 @ X176 ) ) )
| ( ( X175 @ X177 )
!= $true )
| ( ( X175 @ X176 )
!= $true ) )
| ~ ! [X179: term] :
( ( $true
= ( X175 @ X179 ) )
| ( $true
!= ( var @ X179 ) ) )
| ( ( X175 @ X174 )
= $true ) )
=> ! [X180: term,X181: term > $o,X182: subst] :
( ~ ! [X185: term] :
( ( ( var @ X185 )
!= $true )
| ( ( X181 @ ( sub @ X185 @ X182 ) )
= $true ) )
| ~ ! [X186: term] :
( ( $true
= ( X181 @ ( lam @ X186 ) ) )
| ~ ! [X187: term] :
( ( ( X181 @ ( sub @ X186 @ ( push @ X187 @ id ) ) )
= $true )
| ( ( X181 @ X187 )
!= $true ) ) )
| ~ ! [X183: term,X184: term] :
( ( $true
= ( X181 @ ( ap @ X184 @ X183 ) ) )
| ( $true
!= ( X181 @ X183 ) )
| ( ( X181 @ X184 )
!= $true ) )
| ( ( X181 @ ( sub @ X180 @ X182 ) )
= $true ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( substmonoid_gthm = $true )
& ! [X271: term] :
? [X272: d_term] :
( ( d2term @ X272 )
= X271 )
& ( ( laminj = $true )
<=> ! [X167: term,X168: term] :
( ( X167 = X168 )
| ( ( lam @ X168 )
!= ( lam @ X167 ) ) ) )
& ( axabs = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X114: term] :
( ( sub @ X114 @ id )
= X114 )
=> ( ! [X115: term,X116: subst > term > subst > $o] :
( ( ( X116 @ id @ ( lam @ ( sub @ X115 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X122: term] :
( ( ( X116 @ id @ X122 @ id )
!= $true )
| ( ( X116 @ id @ ( sub @ X115 @ ( push @ X122 @ id ) ) @ id )
= $true ) )
| ~ ! [X117: subst > term > term] :
( ~ ! [X118: term] :
( ( $true
= ( X116 @ id @ ( X117 @ id @ X118 ) @ id ) )
| ( $true
!= ( X116 @ id @ X118 @ id ) ) )
| ~ ! [X119: term,X120: subst,X121: subst] :
( ( X117 @ ( comp @ X121 @ X120 ) @ ( sub @ X119 @ X120 ) )
= ( sub @ ( X117 @ X121 @ X119 ) @ X120 ) )
| ( ( X116 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id )
= $true ) ) )
=> ! [X123: term,X124: subst > term > subst > $o] :
( ~ ! [X125: subst > term > term] :
( ~ ! [X127: term,X128: subst,X129: subst] :
( ( X125 @ ( comp @ X129 @ X128 ) @ ( sub @ X127 @ X128 ) )
= ( sub @ ( X125 @ X129 @ X127 ) @ X128 ) )
| ( $true
= ( X124 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) )
| ~ ! [X126: term] :
( ( ( X124 @ id @ X126 @ id )
!= $true )
| ( ( X124 @ id @ ( X125 @ id @ X126 ) @ id )
= $true ) ) )
| ( $true
= ( X124 @ id @ ( lam @ X123 ) @ id ) )
| ~ ! [X130: term,X131: subst,X132: subst,X133: subst] :
( ( $true
!= ( X124 @ ( comp @ X132 @ X131 ) @ ( sub @ X130 @ X131 ) @ X133 ) )
| ( ( X124 @ X132 @ X130 @ ( comp @ X131 @ X133 ) )
= $true ) )
| ~ ! [X134: term,X135: subst,X136: subst,X137: subst] :
( ( $true
= ( X124 @ ( comp @ X137 @ X136 ) @ ( sub @ X134 @ X136 ) @ X135 ) )
| ( ( X124 @ X137 @ X134 @ ( comp @ X136 @ X135 ) )
!= $true ) )
| ~ ! [X138: term] :
( ( ( X124 @ id @ X138 @ id )
!= $true )
| ( ( X124 @ id @ ( sub @ X123 @ ( push @ X138 @ id ) ) @ id )
= $true ) ) ) ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( hoasapinj1_gthm = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X414: term,X415: subst > term > subst > $o] :
( ~ ! [X416: subst > term > term] :
( ( ( X415 @ id @ ( lam @ ( X416 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X418: term,X419: subst,X420: subst] :
( ( sub @ ( X416 @ X419 @ X418 ) @ X420 )
= ( X416 @ ( comp @ X419 @ X420 ) @ ( sub @ X418 @ X420 ) ) )
| ~ ! [X417: term] :
( ( ( X415 @ id @ ( X416 @ id @ X417 ) @ id )
= $true )
| ( ( X415 @ id @ X417 @ id )
!= $true ) ) )
| ( $true
= ( X415 @ id @ ( lam @ ( sub @ X414 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X421: term] :
( ( ( X415 @ id @ ( sub @ X414 @ ( push @ X421 @ id ) ) @ id )
= $true )
| ( $true
!= ( X415 @ id @ X421 @ id ) ) ) ) )
& ( ( apinj2 = $true )
<=> ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X98 ) ) )
& ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
<=> ( axvarcons = $true ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X7: subst > term > subst > $o] :
~ ! [X8: term > $o] :
~ ! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X340: term] :
( ( sub @ X340 @ id )
= X340 )
=> ( ! [X341: term,X342: term,X343: term,X344: term] :
( ( X341 = X342 )
| ( ( ap @ X342 @ X343 )
!= ( ap @ X341 @ X344 ) ) )
=> ! [X345: term,X346: term,X347: term,X348: term] :
( ( ( ap @ ( sub @ X345 @ id ) @ X347 )
!= ( ap @ ( sub @ X348 @ id ) @ X346 ) )
| ( X345 = X348 ) ) ) ) )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X331: subst] :
( ( comp @ id @ X331 )
= X331 )
=> ( ! [X332: subst] :
( ( comp @ X332 @ id )
= X332 )
=> ( ! [X333: subst] :
( ( comp @ X333 @ id )
= X333 )
& ! [X334: subst] :
( ( comp @ id @ X334 )
= X334 ) ) ) ) )
& ( ( hoasapinj1 = $true )
<=> ! [X378: term,X379: term,X380: term,X381: term] :
( ( X379 = X380 )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) ) )
& ! [X382: subst] :
? [X383: d_subst] :
( ( d2subst @ X383 )
= X382 )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( ! [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
= X0 )
=> ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ~ ! [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
= ( X2 @ X6 ) )
| ( ( X3 @ X5 )
!= $true )
| ( ( X2 @ one )
= $true ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lem2_lthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( pushprop_lem1_gthm = $true )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X60: term] :
( ( sub @ X60 @ id )
= X60 )
=> ( ! [X61: term,X62: subst > term > subst > $o] :
( ( ( X62 @ id @ ( lam @ ( sub @ X61 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X68: term] :
( ( ( X62 @ id @ ( sub @ X61 @ ( push @ X68 @ id ) ) @ id )
= $true )
| ( ( X62 @ id @ X68 @ id )
!= $true ) )
| ~ ! [X63: subst > term > term] :
( ~ ! [X67: term] :
( ( $true
= ( X62 @ id @ ( X63 @ id @ X67 ) @ id ) )
| ( $true
!= ( X62 @ id @ X67 @ id ) ) )
| ~ ! [X64: subst,X65: subst,X66: term] :
( ( X63 @ ( comp @ X65 @ X64 ) @ ( sub @ X66 @ X64 ) )
= ( sub @ ( X63 @ X65 @ X66 ) @ X64 ) )
| ( ( X62 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true ) ) )
=> ! [X69: term,X70: subst > term > subst > $o] :
( ~ ! [X71: term] :
( ( $true
= ( X70 @ id @ ( sub @ X69 @ ( push @ X71 @ id ) ) @ id ) )
| ( $true
!= ( X70 @ id @ X71 @ id ) ) )
| ~ ! [X72: subst > term > term] :
( ~ ! [X76: term] :
( ( $true
!= ( X70 @ id @ X76 @ id ) )
| ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true ) )
| ~ ! [X73: subst,X74: term,X75: subst] :
( ( sub @ ( X72 @ X75 @ X74 ) @ X73 )
= ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) ) )
| ( $true
= ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id ) ) )
| ( ( X70 @ id @ ( lam @ X69 ) @ id )
= $true ) ) ) ) )
& ( ! [X139: term,X140: term > $o,X141: subst] :
( ~ ! [X142: term] :
( ( ( var @ X142 )
!= $true )
| ( ( X140 @ ( sub @ X142 @ X141 ) )
= $true ) )
| ~ ! [X145: term,X146: term] :
( ( ( X140 @ X146 )
!= $true )
| ( $true
= ( X140 @ ( ap @ X145 @ X146 ) ) )
| ( ( X140 @ X145 )
!= $true ) )
| ( $true
= ( X140 @ ( sub @ X139 @ X141 ) ) )
| ~ ! [X143: term] :
( ( $true
= ( X140 @ ( lam @ X143 ) ) )
| ~ ! [X144: term] :
( ( ( X140 @ X144 )
!= $true )
| ( ( X140 @ ( sub @ X143 @ ( push @ X144 @ id ) ) )
= $true ) ) ) )
<=> ( induction2lem = $true ) )
& ( ! [X37: subst > term > subst > $o,X38: term] :
( ~ ! [X41: subst > term > term] :
( ~ ! [X42: term,X43: subst,X44: subst] :
( ( X41 @ ( comp @ X43 @ X44 ) @ ( sub @ X42 @ X44 ) )
= ( sub @ ( X41 @ X43 @ X42 ) @ X44 ) )
| ~ ! [X45: term] :
( ( $true
= ( X37 @ id @ ( X41 @ id @ X45 ) @ id ) )
| ( ( X37 @ id @ X45 @ id )
!= $true ) )
| ( ( X37 @ id @ ( lam @ ( X41 @ sh @ one ) ) @ id )
= $true ) )
| ( $true
= ( X37 @ id @ X38 @ id ) )
| ~ ! [X50: subst,X51: subst,X52: term,X53: subst] :
( ( ( X37 @ X51 @ X52 @ ( comp @ X50 @ X53 ) )
= $true )
| ( $true
!= ( X37 @ ( comp @ X51 @ X50 ) @ ( sub @ X52 @ X50 ) @ X53 ) ) )
| ~ ! [X54: term] :
( ( $true
!= ( var @ ( sub @ X54 @ id ) ) )
| ( $true
= ( X37 @ id @ X54 @ id ) ) )
| ~ ! [X39: term,X40: term] :
( ( $true
= ( X37 @ id @ ( ap @ ( sub @ X40 @ id ) @ X39 ) @ id ) )
| ( $true
!= ( X37 @ id @ X40 @ id ) )
| ( $true
!= ( X37 @ id @ X39 @ id ) ) )
| ~ ! [X46: subst,X47: subst,X48: subst,X49: term] :
( ( ( X37 @ X46 @ X49 @ ( comp @ X47 @ X48 ) )
!= $true )
| ( ( X37 @ ( comp @ X46 @ X47 ) @ ( sub @ X49 @ X47 ) @ X48 )
= $true ) ) )
<=> ( hoasinduction = $true ) )
& ( ( hoasapinj2 = $true )
<=> ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) ) )
& ( hoasapinj2_gthm = $true )
& ( termmset_gthm = $true )
& ( pushprop_lem2v2 = $true )
& ( induction2_gthm = $true )
& ( ! [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
= X335 )
<=> ( axscons = $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( hoaslaminj = $true )
<=> ! [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ~ ! [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
| ( ( X498 @ X497 @ X495 )
= ( X496 @ X497 @ X495 ) )
| ( ( lam @ ( X498 @ sh @ one ) )
!= ( lam @ ( X496 @ sh @ one ) ) )
| ~ ! [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ulamvarsh = $true )
& ( ( ! [X77: subst,X78: term] :
( ( sub @ one @ ( push @ X78 @ X77 ) )
= X78 )
=> ( ! [X79: subst,X80: term] :
( ( comp @ sh @ ( push @ X80 @ X79 ) )
= X79 )
=> ( ! [X81: term,X82: term] :
( ( X81 = X82 )
| ( ( lam @ X81 )
!= ( lam @ X82 ) ) )
=> ! [X83: term,X84: subst > term > term,X85: subst,X86: subst > term > term] :
( ( ( X84 @ X85 @ X83 )
= ( X86 @ X85 @ X83 ) )
| ( ( lam @ ( X84 @ sh @ one ) )
!= ( lam @ ( X86 @ sh @ one ) ) )
| ~ ! [X90: subst,X91: subst,X92: term] :
( ( X84 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) )
= ( sub @ ( X84 @ X90 @ X92 ) @ X91 ) )
| ~ ! [X87: subst,X88: subst,X89: term] :
( ( X86 @ ( comp @ X87 @ X88 ) @ ( sub @ X89 @ X88 ) )
= ( sub @ ( X86 @ X87 @ X89 ) @ X88 ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( ( induction = $true )
<=> ! [X206: term > $o,X207: term] :
( ~ ! [X209: term,X210: term] :
( ( ( X206 @ ( ap @ X209 @ X210 ) )
= $true )
| ( $true
!= ( X206 @ X209 ) )
| ( $true
!= ( X206 @ X210 ) ) )
| ( ( X206 @ X207 )
= $true )
| ~ ! [X211: term] :
( ( $true
!= ( var @ X211 ) )
| ( $true
= ( X206 @ X211 ) ) )
| ~ ! [X208: term] :
( ( $true
= ( X206 @ ( lam @ X208 ) ) )
| ( $true
!= ( X206 @ X208 ) ) ) ) )
& ( pushprop_lem1v2_gthm = $true ) ),
inference(flattening,[status(thm)],[f9]) ).
thf(f9,plain,
( ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
= X0 )
=> ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
!= $true )
| ~ ! [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
= ( X2 @ X6 ) )
| ( ( X2 @ one )
= $true ) ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X7: subst > term > subst > $o] :
~ ! [X8: term > $o] :
~ ! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) ) )
& ( ( ! [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
= X11 )
=> ( ! [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
= X12 )
=> ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
| ( ( X18 @ X15 )
!= $true ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) ) )
& ( axclos = $true )
& ! [X23: d_term,X24: d_term] :
( ( ( d2term @ X23 )
= ( d2term @ X24 ) )
=> ( X23 = X24 ) )
& ( ! [X25: subst > term > subst > $o,X26: term] :
( ~ ! [X27: term] :
( ( $true
!= ( X25 @ id @ X27 @ id ) )
| ( $true
= ( X25 @ id @ ( sub @ X26 @ ( push @ X27 @ id ) ) @ id ) ) )
| ( $true
= ( X25 @ id @ ( lam @ ( sub @ X26 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X28: term,X29: subst > term > term] :
( ( ( X25 @ id @ ( lam @ ( X29 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X30: term] :
( ( ( X25 @ id @ ( X29 @ id @ X30 ) @ id )
= $true )
| ( ( X25 @ id @ X30 @ id )
!= $true ) )
| ~ ! [X31: subst,X32: term,X33: subst] :
( ( sub @ ( sub @ X28 @ ( push @ X32 @ X33 ) ) @ X31 )
= ( sub @ ( X29 @ X33 @ X32 ) @ X31 ) )
| ~ ! [X34: subst,X35: subst,X36: term] :
( ( sub @ X28 @ ( push @ ( sub @ X36 @ X34 ) @ ( comp @ X35 @ X34 ) ) )
= ( X29 @ ( comp @ X35 @ X34 ) @ ( sub @ X36 @ X34 ) ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( pushprop_gthm = $true )
& ( ( hoasinduction = $true )
<=> ! [X37: subst > term > subst > $o,X38: term] :
( ~ ! [X39: term,X40: term] :
( ( $true
!= ( X37 @ id @ X40 @ id ) )
| ( $true
!= ( X37 @ id @ X39 @ id ) )
| ( $true
= ( X37 @ id @ ( ap @ ( sub @ X40 @ id ) @ X39 ) @ id ) ) )
| ~ ! [X41: subst > term > term] :
( ~ ! [X42: term,X43: subst,X44: subst] :
( ( X41 @ ( comp @ X43 @ X44 ) @ ( sub @ X42 @ X44 ) )
= ( sub @ ( X41 @ X43 @ X42 ) @ X44 ) )
| ~ ! [X45: term] :
( ( $true
= ( X37 @ id @ ( X41 @ id @ X45 ) @ id ) )
| ( ( X37 @ id @ X45 @ id )
!= $true ) )
| ( ( X37 @ id @ ( lam @ ( X41 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X46: subst,X47: subst,X48: subst,X49: term] :
( ( ( X37 @ ( comp @ X46 @ X47 ) @ ( sub @ X49 @ X47 ) @ X48 )
= $true )
| ( ( X37 @ X46 @ X49 @ ( comp @ X47 @ X48 ) )
!= $true ) )
| ~ ! [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
!= ( X37 @ ( comp @ X51 @ X50 ) @ ( sub @ X52 @ X50 ) @ X53 ) )
| ( ( X37 @ X51 @ X52 @ ( comp @ X50 @ X53 ) )
= $true ) )
| ~ ! [X54: term] :
( ( $true
!= ( var @ ( sub @ X54 @ id ) ) )
| ( $true
= ( X37 @ id @ X54 @ id ) ) )
| ( $true
= ( X37 @ id @ X38 @ id ) ) ) )
& ( ulamvarsh = $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
= ( d2subst @ X55 ) )
=> ( X55 = X56 ) )
& ( pushprop_lem2v2_lthm = $true )
& ( ( termmset = $true )
<=> ! [X57: term] :
( ( sub @ X57 @ id )
= X57 ) )
& ( axvarshift = $true )
& ( ( ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 )
& ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 ) )
<=> ( substmonoid = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( ! [X60: term] :
( ( sub @ X60 @ id )
= X60 )
=> ( ! [X61: term,X62: subst > term > subst > $o] :
( ~ ! [X63: subst > term > term] :
( ( ( X62 @ id @ ( lam @ ( X63 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X64: subst,X65: subst,X66: term] :
( ( X63 @ ( comp @ X65 @ X64 ) @ ( sub @ X66 @ X64 ) )
= ( sub @ ( X63 @ X65 @ X66 ) @ X64 ) )
| ~ ! [X67: term] :
( ( $true
!= ( X62 @ id @ X67 @ id ) )
| ( $true
= ( X62 @ id @ ( X63 @ id @ X67 ) @ id ) ) ) )
| ~ ! [X68: term] :
( ( ( X62 @ id @ X68 @ id )
!= $true )
| ( ( X62 @ id @ ( sub @ X61 @ ( push @ X68 @ id ) ) @ id )
= $true ) )
| ( ( X62 @ id @ ( lam @ ( sub @ X61 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X69: term,X70: subst > term > subst > $o] :
( ~ ! [X71: term] :
( ( $true
!= ( X70 @ id @ X71 @ id ) )
| ( $true
= ( X70 @ id @ ( sub @ X69 @ ( push @ X71 @ id ) ) @ id ) ) )
| ~ ! [X72: subst > term > term] :
( ~ ! [X73: subst,X74: term,X75: subst] :
( ( sub @ ( X72 @ X75 @ X74 ) @ X73 )
= ( X72 @ ( comp @ X75 @ X73 ) @ ( sub @ X74 @ X73 ) ) )
| ( $true
= ( X70 @ id @ ( lam @ ( X72 @ sh @ one ) ) @ id ) )
| ~ ! [X76: term] :
( ( $true
!= ( X70 @ id @ X76 @ id ) )
| ( ( X70 @ id @ ( X72 @ id @ X76 ) @ id )
= $true ) ) )
| ( ( X70 @ id @ ( lam @ X69 ) @ id )
= $true ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( ( ! [X77: subst,X78: term] :
( ( sub @ one @ ( push @ X78 @ X77 ) )
= X78 )
=> ( ! [X79: subst,X80: term] :
( ( comp @ sh @ ( push @ X80 @ X79 ) )
= X79 )
=> ( ! [X81: term,X82: term] :
( ( X81 = X82 )
| ( ( lam @ X81 )
!= ( lam @ X82 ) ) )
=> ! [X83: term,X84: subst > term > term,X85: subst,X86: subst > term > term] :
( ( ( X84 @ X85 @ X83 )
= ( X86 @ X85 @ X83 ) )
| ( ( lam @ ( X84 @ sh @ one ) )
!= ( lam @ ( X86 @ sh @ one ) ) )
| ~ ! [X90: subst,X91: subst,X92: term] :
( ( X84 @ ( comp @ X90 @ X91 ) @ ( sub @ X92 @ X91 ) )
= ( sub @ ( X84 @ X90 @ X92 ) @ X91 ) )
| ~ ! [X87: subst,X88: subst,X89: term] :
( ( X86 @ ( comp @ X87 @ X88 ) @ ( sub @ X89 @ X88 ) )
= ( sub @ ( X86 @ X87 @ X89 ) @ X88 ) ) ) ) ) )
<=> ( hoaslaminj_lthm = $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
<=> ! [X93: subst > term > subst > $o] :
~ ! [X94: term > $o] :
~ ! [X95: term] :
( ( X94 @ X95 )
= ( X93 @ id @ X95 @ id ) ) )
& ( apnotvar = $true )
& ( pushprop = $true )
& ( ( d2subst @ d_id )
= sh )
& ! [X96: d_term] : ( d_one = X96 )
& ( ( apinj2 = $true )
<=> ! [X97: term,X98: term,X99: term,X100: term] :
( ( ( ap @ X99 @ X97 )
!= ( ap @ X100 @ X98 ) )
| ( X97 = X98 ) ) )
& ( ulamvarind = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( ! [X101: term] :
~ ! [X102: subst > term > term] :
~ ! [X103: term,X104: subst] :
( ( sub @ X101 @ ( push @ X103 @ X104 ) )
= ( X102 @ X104 @ X103 ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X105: term,X106: subst > term > subst > $o] :
( ~ ! [X107: subst > term > term] :
( ~ ! [X108: term] :
( ( $true
= ( X106 @ id @ ( X107 @ id @ X108 ) @ id ) )
| ( $true
!= ( X106 @ id @ X108 @ id ) ) )
| ~ ! [X109: subst,X110: term,X111: subst] :
( ( X107 @ ( comp @ X111 @ X109 ) @ ( sub @ X110 @ X109 ) )
= ( sub @ ( X107 @ X111 @ X110 ) @ X109 ) )
| ( ( X106 @ id @ ( lam @ ( X107 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X112: term,X113: term] :
( ( ( X106 @ id @ X113 @ id )
!= $true )
| ( ( X106 @ id @ X112 @ id )
!= $true )
| ( $true
= ( X106 @ id @ ( ap @ ( sub @ X113 @ id ) @ X112 ) @ id ) ) )
| ( $true
= ( X106 @ id @ X105 @ id ) ) ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X114: term] :
( ( sub @ X114 @ id )
= X114 )
=> ( ! [X115: term,X116: subst > term > subst > $o] :
( ~ ! [X117: subst > term > term] :
( ~ ! [X118: term] :
( ( $true
= ( X116 @ id @ ( X117 @ id @ X118 ) @ id ) )
| ( $true
!= ( X116 @ id @ X118 @ id ) ) )
| ~ ! [X119: term,X120: subst,X121: subst] :
( ( X117 @ ( comp @ X121 @ X120 ) @ ( sub @ X119 @ X120 ) )
= ( sub @ ( X117 @ X121 @ X119 ) @ X120 ) )
| ( ( X116 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X116 @ id @ ( lam @ ( sub @ X115 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X122: term] :
( ( ( X116 @ id @ X122 @ id )
!= $true )
| ( ( X116 @ id @ ( sub @ X115 @ ( push @ X122 @ id ) ) @ id )
= $true ) ) )
=> ! [X123: term,X124: subst > term > subst > $o] :
( ( $true
= ( X124 @ id @ ( lam @ X123 ) @ id ) )
| ~ ! [X125: subst > term > term] :
( ( $true
= ( X124 @ id @ ( lam @ ( X125 @ sh @ one ) ) @ id ) )
| ~ ! [X126: term] :
( ( ( X124 @ id @ X126 @ id )
!= $true )
| ( ( X124 @ id @ ( X125 @ id @ X126 ) @ id )
= $true ) )
| ~ ! [X127: term,X128: subst,X129: subst] :
( ( X125 @ ( comp @ X129 @ X128 ) @ ( sub @ X127 @ X128 ) )
= ( sub @ ( X125 @ X129 @ X127 ) @ X128 ) ) )
| ~ ! [X130: term,X131: subst,X132: subst,X133: subst] :
( ( ( X124 @ X132 @ X130 @ ( comp @ X131 @ X133 ) )
= $true )
| ( $true
!= ( X124 @ ( comp @ X132 @ X131 ) @ ( sub @ X130 @ X131 ) @ X133 ) ) )
| ~ ! [X134: term,X135: subst,X136: subst,X137: subst] :
( ( $true
= ( X124 @ ( comp @ X137 @ X136 ) @ ( sub @ X134 @ X136 ) @ X135 ) )
| ( ( X124 @ X137 @ X134 @ ( comp @ X136 @ X135 ) )
!= $true ) )
| ~ ! [X138: term] :
( ( ( X124 @ id @ ( sub @ X123 @ ( push @ X138 @ id ) ) @ id )
= $true )
| ( ( X124 @ id @ X138 @ id )
!= $true ) ) ) ) ) )
& ( ( induction2lem = $true )
<=> ! [X139: term,X140: term > $o,X141: subst] :
( ~ ! [X142: term] :
( ( ( X140 @ ( sub @ X142 @ X141 ) )
= $true )
| ( ( var @ X142 )
!= $true ) )
| ~ ! [X143: term] :
( ~ ! [X144: term] :
( ( ( X140 @ ( sub @ X143 @ ( push @ X144 @ id ) ) )
= $true )
| ( ( X140 @ X144 )
!= $true ) )
| ( $true
= ( X140 @ ( lam @ X143 ) ) ) )
| ( $true
= ( X140 @ ( sub @ X139 @ X141 ) ) )
| ~ ! [X145: term,X146: term] :
( ( ( X140 @ X145 )
!= $true )
| ( ( X140 @ X146 )
!= $true )
| ( $true
= ( X140 @ ( ap @ X145 @ X146 ) ) ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ! [X147: term] :
~ ! [X148: subst > term > term] :
~ ! [X149: term,X150: subst] :
( ( X148 @ X150 @ X149 )
= ( sub @ X147 @ ( push @ X149 @ X150 ) ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( ! [X151: term] :
( ( sub @ X151 @ id )
= X151 )
=> ( ! [X152: term,X153: subst,X154: term > $o] :
( ~ ! [X155: term] :
( ( $true
= ( X154 @ ( lam @ X155 ) ) )
| ~ ! [X156: term] :
( ( ( X154 @ ( sub @ X155 @ ( push @ X156 @ id ) ) )
= $true )
| ( ( X154 @ X156 )
!= $true ) ) )
| ~ ! [X157: term,X158: term] :
( ( $true
!= ( X154 @ X158 ) )
| ( ( X154 @ ( ap @ X158 @ X157 ) )
= $true )
| ( ( X154 @ X157 )
!= $true ) )
| ( ( X154 @ ( sub @ X152 @ X153 ) )
= $true )
| ~ ! [X159: term] :
( ( ( var @ X159 )
!= $true )
| ( ( X154 @ ( sub @ X159 @ X153 ) )
= $true ) ) )
=> ! [X160: term,X161: term > $o] :
( ~ ! [X162: term] :
( ( $true
= ( X161 @ X162 ) )
| ( $true
!= ( var @ X162 ) ) )
| ~ ! [X163: term,X164: term] :
( ( ( X161 @ ( ap @ X164 @ X163 ) )
= $true )
| ( $true
!= ( X161 @ X163 ) )
| ( ( X161 @ X164 )
!= $true ) )
| ~ ! [X165: term] :
( ~ ! [X166: term] :
( ( $true
= ( X161 @ ( sub @ X165 @ ( push @ X166 @ id ) ) ) )
| ( ( X161 @ X166 )
!= $true ) )
| ( $true
= ( X161 @ ( lam @ X165 ) ) ) )
| ( ( X161 @ X160 )
= $true ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( ( laminj = $true )
<=> ! [X167: term,X168: term] :
( ( X167 = X168 )
| ( ( lam @ X168 )
!= ( lam @ X167 ) ) ) )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ulamvar1 != $true )
& ( ( ! [X169: term,X170: subst] :
( ( sub @ one @ ( push @ X169 @ X170 ) )
= X169 )
=> ( ! [X171: subst,X172: term] :
( ( comp @ sh @ ( push @ X172 @ X171 ) )
= X171 )
=> ( ! [X173: subst] :
( ( comp @ X173 @ id )
= X173 )
=> ( ! [X174: term,X175: term > $o] :
( ~ ! [X176: term,X177: term] :
( ( ( X175 @ X176 )
!= $true )
| ( ( X175 @ X177 )
!= $true )
| ( $true
= ( X175 @ ( ap @ X177 @ X176 ) ) ) )
| ( ( X175 @ X174 )
= $true )
| ~ ! [X178: term] :
( ( $true
= ( X175 @ ( lam @ X178 ) ) )
| ( ( X175 @ X178 )
!= $true ) )
| ~ ! [X179: term] :
( ( $true
= ( X175 @ X179 ) )
| ( $true
!= ( var @ X179 ) ) ) )
=> ! [X180: term,X181: term > $o,X182: subst] :
( ( ( X181 @ ( sub @ X180 @ X182 ) )
= $true )
| ~ ! [X183: term,X184: term] :
( ( $true
= ( X181 @ ( ap @ X184 @ X183 ) ) )
| ( ( X181 @ X184 )
!= $true )
| ( $true
!= ( X181 @ X183 ) ) )
| ~ ! [X185: term] :
( ( ( X181 @ ( sub @ X185 @ X182 ) )
= $true )
| ( ( var @ X185 )
!= $true ) )
| ~ ! [X186: term] :
( ( $true
= ( X181 @ ( lam @ X186 ) ) )
| ~ ! [X187: term] :
( ( ( X181 @ X187 )
!= $true )
| ( ( X181 @ ( sub @ X186 @ ( push @ X187 @ id ) ) )
= $true ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( ! [X188: subst > term > subst > $o,X189: term,X190: term > $o] :
( ~ ! [X191: subst,X192: subst,X193: subst,X194: term] :
( ( ( X188 @ X193 @ X194 @ ( comp @ X191 @ X192 ) )
!= $true )
| ( ( X188 @ ( comp @ X193 @ X191 ) @ ( sub @ X194 @ X191 ) @ X192 )
= $true ) )
| ~ ! [X195: subst,X196: subst,X197: subst,X198: term] :
( ( ( X188 @ ( comp @ X197 @ X196 ) @ ( sub @ X198 @ X196 ) @ X195 )
!= $true )
| ( ( X188 @ X197 @ X198 @ ( comp @ X196 @ X195 ) )
= $true ) )
| ~ ! [X199: term] :
( ( ( X190 @ ( sub @ X189 @ ( push @ X199 @ id ) ) )
= $true )
| ( ( X190 @ X199 )
!= $true ) )
| ( ( X190 @ ( lam @ X189 ) )
= $true )
| ~ ! [X200: subst > term > term] :
( ( ( X188 @ id @ ( lam @ ( X200 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X201: subst,X202: subst,X203: term] :
( ( X200 @ ( comp @ X202 @ X201 ) @ ( sub @ X203 @ X201 ) )
= ( sub @ ( X200 @ X202 @ X203 ) @ X201 ) )
| ~ ! [X204: term] :
( ( ( X188 @ id @ X204 @ id )
!= $true )
| ( ( X188 @ id @ ( X200 @ id @ X204 ) @ id )
= $true ) ) )
| ~ ! [X205: term] :
( ( X188 @ id @ X205 @ id )
= ( X190 @ X205 ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ! [X206: term > $o,X207: term] :
( ( ( X206 @ X207 )
= $true )
| ~ ! [X208: term] :
( ( $true
= ( X206 @ ( lam @ X208 ) ) )
| ( $true
!= ( X206 @ X208 ) ) )
| ~ ! [X209: term,X210: term] :
( ( $true
!= ( X206 @ X209 ) )
| ( ( X206 @ ( ap @ X209 @ X210 ) )
= $true )
| ( $true
!= ( X206 @ X210 ) ) )
| ~ ! [X211: term] :
( ( $true
!= ( var @ X211 ) )
| ( $true
= ( X206 @ X211 ) ) ) )
<=> ( induction = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X212: term] :
~ ! [X213: subst > term > term] :
( ( sub @ X212 @ ( push @ one @ sh ) )
!= ( X213 @ sh @ one ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( ! [X214: subst] :
( ( comp @ X214 @ id )
= X214 )
<=> ( axidr = $true ) )
& ( termmset_gthm = $true )
& ( ! [X215: term,X216: term > $o,X217: term > $o,X218: subst] :
( ( ( X216 @ one )
= $true )
| ( $true
!= ( X217 @ X215 ) )
| ~ ! [X219: term] :
( ( X216 @ X219 )
= ( X217 @ ( sub @ X219 @ ( push @ X215 @ X218 ) ) ) ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X220: term > $o,X221: term,X222: subst > term > subst > $o,X223: term] :
( ( ( X220 @ X221 )
!= $true )
| ~ ! [X224: subst,X225: term,X226: subst,X227: subst] :
( ( ( X222 @ ( comp @ X224 @ X227 ) @ ( sub @ X225 @ X227 ) @ X226 )
= $true )
| ( ( X222 @ X224 @ X225 @ ( comp @ X227 @ X226 ) )
!= $true ) )
| ( ( X220 @ X223 )
!= $true )
| ~ ! [X228: subst,X229: subst,X230: subst,X231: term] :
( ( $true
= ( X222 @ X230 @ X231 @ ( comp @ X228 @ X229 ) ) )
| ( ( X222 @ ( comp @ X230 @ X228 ) @ ( sub @ X231 @ X228 ) @ X229 )
!= $true ) )
| ( ( X220 @ ( ap @ X221 @ X223 ) )
= $true )
| ~ ! [X232: term] :
( ( X220 @ X232 )
= ( X222 @ id @ X232 @ id ) )
| ~ ! [X233: term,X234: term] :
( ( $true
= ( X222 @ id @ ( ap @ ( sub @ X234 @ id ) @ X233 ) @ id ) )
| ( $true
!= ( X222 @ id @ X233 @ id ) )
| ( ( X222 @ id @ X234 @ id )
!= $true ) ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X235: subst > term > subst > $o,X236: term] :
( ~ ! [X237: subst,X238: subst,X239: subst,X240: term] :
( ( ( X235 @ X238 @ X240 @ ( comp @ X239 @ X237 ) )
= $true )
| ( $true
!= ( X235 @ ( comp @ X238 @ X239 ) @ ( sub @ X240 @ X239 ) @ X237 ) ) )
| ( ( X235 @ id @ ( lam @ X236 ) @ id )
= $true )
| ~ ! [X241: subst > term > term] :
( ~ ! [X242: term,X243: subst,X244: subst] :
( ( X241 @ ( comp @ X244 @ X243 ) @ ( sub @ X242 @ X243 ) )
= ( sub @ ( X241 @ X244 @ X242 ) @ X243 ) )
| ( ( X235 @ id @ ( lam @ ( X241 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X245: term] :
( ( $true
= ( X235 @ id @ ( X241 @ id @ X245 ) @ id ) )
| ( ( X235 @ id @ X245 @ id )
!= $true ) ) )
| ~ ! [X246: term] :
( ( ( X235 @ id @ X246 @ id )
!= $true )
| ( $true
= ( X235 @ id @ ( sub @ X236 @ ( push @ X246 @ id ) ) @ id ) ) )
| ~ ! [X247: subst,X248: subst,X249: subst,X250: term] :
( ( $true
= ( X235 @ ( comp @ X248 @ X249 ) @ ( sub @ X250 @ X249 ) @ X247 ) )
| ( ( X235 @ X248 @ X250 @ ( comp @ X249 @ X247 ) )
!= $true ) ) ) )
& ( hoasapnotvar_gthm = $true )
& ( lamnotap != $true )
& ( ! [X251: term,X252: term > $o] :
( ( $true
= ( X252 @ X251 ) )
| ~ ! [X253: term,X254: term] :
( ( ( X252 @ X254 )
!= $true )
| ( $true
!= ( X252 @ X253 ) )
| ( $true
= ( X252 @ ( ap @ X254 @ X253 ) ) ) )
| ~ ! [X255: term] :
( ~ ! [X256: term] :
( ( $true
!= ( X252 @ X256 ) )
| ( ( X252 @ ( sub @ X255 @ ( push @ X256 @ id ) ) )
= $true ) )
| ( ( X252 @ ( lam @ X255 ) )
= $true ) )
| ~ ! [X257: term] :
( ( $true
= ( X252 @ X257 ) )
| ( $true
!= ( var @ X257 ) ) ) )
<=> ( induction2 = $true ) )
& ( ! [X258: term,X259: subst > term > term,X260: term] :
( ~ ! [X261: term,X262: subst,X263: subst] :
( ( sub @ ( X259 @ X263 @ X261 ) @ X262 )
= ( X259 @ ( comp @ X263 @ X262 ) @ ( sub @ X261 @ X262 ) ) )
| ( ( lam @ ( X259 @ sh @ one ) )
!= ( ap @ ( sub @ X260 @ id ) @ X258 ) ) )
<=> ( hoaslamnotap = $true ) )
& ( axmap = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X264: term] :
( ( sub @ X264 @ id )
= X264 )
<=> ( axvarid = $true ) )
& ( ( pushprop_lem1 = $true )
<=> ! [X265: term,X266: term > $o,X267: term > $o,X268: term,X269: subst] :
( ( ( X266 @ X265 )
!= $true )
| ( $true
= ( X267 @ ( sub @ X265 @ ( push @ X268 @ X269 ) ) ) ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( lamnotvar = $true )
& ! [X270: d_subst] : ( d_id = X270 )
& ! [X271: term] :
? [X272: d_term] :
( ( d2term @ X272 )
= X271 )
& ( ! [X273: subst,X274: term] :
( ( comp @ sh @ ( push @ X274 @ X273 ) )
= X273 )
<=> ( axshiftcons = $true ) )
& ( ! [X275: term,X276: term] :
( ( ( sub @ X276 @ sh )
!= ( sub @ X275 @ sh ) )
| ( X275 = X276 ) )
<=> ( shinj = $true ) )
& ( pushprop_lthm_orig = $true )
& ( ( ! [X277: term] :
~ ! [X278: subst > term > term] :
~ ! [X279: term,X280: subst] :
( ( sub @ X277 @ ( push @ X279 @ X280 ) )
= ( X278 @ X280 @ X279 ) )
=> ( ! [X281: term] :
( ( sub @ X281 @ id )
= X281 )
=> ! [X282: term > $o,X283: subst > term > subst > $o,X284: term] :
( ~ ! [X285: subst > term > term] :
( ( $true
= ( X283 @ id @ ( lam @ ( X285 @ sh @ one ) ) @ id ) )
| ~ ! [X286: term] :
( ( $true
!= ( X283 @ id @ X286 @ id ) )
| ( $true
= ( X283 @ id @ ( X285 @ id @ X286 ) @ id ) ) )
| ~ ! [X287: term,X288: subst,X289: subst] :
( ( X285 @ ( comp @ X289 @ X288 ) @ ( sub @ X287 @ X288 ) )
= ( sub @ ( X285 @ X289 @ X287 ) @ X288 ) ) )
| ( ( X282 @ ( lam @ X284 ) )
= $true )
| ~ ! [X290: term] :
( ( X282 @ X290 )
= ( X283 @ id @ X290 @ id ) )
| ~ ! [X291: term] :
( ( ( X282 @ X291 )
!= $true )
| ( $true
= ( X282 @ ( sub @ X284 @ ( push @ X291 @ id ) ) ) ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X292: subst > term > subst > $o] :
~ ! [X293: term > $o] :
~ ! [X294: term] :
( ( X292 @ id @ X294 @ id )
= ( X293 @ X294 ) )
=> ( ! [X295: term > $o,X296: term] :
( ~ ! [X297: term,X298: term] :
( ( ( X295 @ X297 )
!= $true )
| ( $true
= ( X295 @ ( ap @ X297 @ X298 ) ) )
| ( ( X295 @ X298 )
!= $true ) )
| ~ ! [X299: term] :
( ( ( X295 @ X299 )
= $true )
| ( $true
!= ( var @ X299 ) ) )
| ~ ! [X300: term] :
( ( ( X295 @ ( lam @ X300 ) )
= $true )
| ~ ! [X301: term] :
( ( ( X295 @ X301 )
!= $true )
| ( $true
= ( X295 @ ( sub @ X300 @ ( push @ X301 @ id ) ) ) ) ) )
| ( ( X295 @ X296 )
= $true ) )
=> ( ! [X302: term] :
( ( sub @ X302 @ id )
= X302 )
=> ( ! [X303: subst > term > subst > $o,X304: term,X305: term > $o] :
( ~ ! [X306: subst > term > term] :
( ( $true
= ( X303 @ id @ ( lam @ ( X306 @ sh @ one ) ) @ id ) )
| ~ ! [X307: term] :
( ( $true
= ( X303 @ id @ ( X306 @ id @ X307 ) @ id ) )
| ( $true
!= ( X303 @ id @ X307 @ id ) ) )
| ~ ! [X308: term,X309: subst,X310: subst] :
( ( X306 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) )
= ( sub @ ( X306 @ X310 @ X308 ) @ X309 ) ) )
| ( ( X305 @ ( lam @ X304 ) )
= $true )
| ~ ! [X311: term] :
( ( ( X305 @ X311 )
!= $true )
| ( ( X305 @ ( sub @ X304 @ ( push @ X311 @ id ) ) )
= $true ) )
| ~ ! [X312: term] :
( ( X303 @ id @ X312 @ id )
= ( X305 @ X312 ) ) )
=> ! [X313: term,X314: subst > term > subst > $o] :
( ~ ! [X315: subst,X316: subst,X317: term,X318: subst] :
( ( $true
= ( X314 @ X318 @ X317 @ ( comp @ X315 @ X316 ) ) )
| ( $true
!= ( X314 @ ( comp @ X318 @ X315 ) @ ( sub @ X317 @ X315 ) @ X316 ) ) )
| ~ ! [X319: term] :
( ( $true
!= ( var @ ( sub @ X319 @ id ) ) )
| ( ( X314 @ id @ X319 @ id )
= $true ) )
| ~ ! [X320: term,X321: term] :
( ( ( X314 @ id @ X320 @ id )
!= $true )
| ( ( X314 @ id @ ( ap @ ( sub @ X320 @ id ) @ X321 ) @ id )
= $true )
| ( $true
!= ( X314 @ id @ X321 @ id ) ) )
| ~ ! [X322: subst,X323: subst,X324: term,X325: subst] :
( ( ( X314 @ ( comp @ X322 @ X323 ) @ ( sub @ X324 @ X323 ) @ X325 )
= $true )
| ( ( X314 @ X322 @ X324 @ ( comp @ X323 @ X325 ) )
!= $true ) )
| ~ ! [X326: subst > term > term] :
( ~ ! [X327: subst,X328: term,X329: subst] :
( ( X326 @ ( comp @ X327 @ X329 ) @ ( sub @ X328 @ X329 ) )
= ( sub @ ( X326 @ X327 @ X328 ) @ X329 ) )
| ( ( X314 @ id @ ( lam @ ( X326 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X330: term] :
( ( ( X314 @ id @ X330 @ id )
!= $true )
| ( ( X314 @ id @ ( X326 @ id @ X330 ) @ id )
= $true ) ) )
| ( ( X314 @ id @ X313 @ id )
= $true ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X331: subst] :
( ( comp @ id @ X331 )
= X331 )
=> ( ! [X332: subst] :
( ( comp @ X332 @ id )
= X332 )
=> ( ! [X333: subst] :
( ( comp @ X333 @ id )
= X333 )
& ! [X334: subst] :
( ( comp @ id @ X334 )
= X334 ) ) ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( axassoc = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( induction2_gthm = $true )
& ( ! [X335: subst] :
( ( push @ ( sub @ one @ X335 ) @ ( comp @ sh @ X335 ) )
= X335 )
<=> ( axscons = $true ) )
& ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
<=> ( axvarcons = $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ! [X338: term] :
~ ! [X339: subst > term > term] :
( ( sub @ X338 @ ( push @ one @ sh ) )
!= ( X339 @ sh @ one ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( ( hoasapinj1_lthm = $true )
<=> ( ! [X340: term] :
( ( sub @ X340 @ id )
= X340 )
=> ( ! [X341: term,X342: term,X343: term,X344: term] :
( ( X341 = X342 )
| ( ( ap @ X342 @ X343 )
!= ( ap @ X341 @ X344 ) ) )
=> ! [X345: term,X346: term,X347: term,X348: term] :
( ( ( ap @ ( sub @ X345 @ id ) @ X347 )
!= ( ap @ ( sub @ X348 @ id ) @ X346 ) )
| ( X345 = X348 ) ) ) ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X349: term] :
( ( sub @ X349 @ id )
= X349 )
=> ! [X350: term,X351: subst > term > subst > $o,X352: term > $o] :
( ~ ! [X353: term] :
( ( X351 @ id @ X353 @ id )
= ( X352 @ X353 ) )
| ( ( X352 @ ( lam @ X350 ) )
= $true )
| ~ ! [X354: subst,X355: subst,X356: subst,X357: term] :
( ( ( X351 @ X356 @ X357 @ ( comp @ X354 @ X355 ) )
!= $true )
| ( ( X351 @ ( comp @ X356 @ X354 ) @ ( sub @ X357 @ X354 ) @ X355 )
= $true ) )
| ~ ! [X358: subst,X359: subst,X360: term,X361: subst] :
( ( $true
= ( X351 @ X358 @ X360 @ ( comp @ X359 @ X361 ) ) )
| ( $true
!= ( X351 @ ( comp @ X358 @ X359 ) @ ( sub @ X360 @ X359 ) @ X361 ) ) )
| ~ ! [X362: subst > term > term] :
( ~ ! [X363: term,X364: subst,X365: subst] :
( ( X362 @ ( comp @ X365 @ X364 ) @ ( sub @ X363 @ X364 ) )
= ( sub @ ( X362 @ X365 @ X363 ) @ X364 ) )
| ( ( X351 @ id @ ( lam @ ( X362 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X366: term] :
( ( ( X351 @ id @ ( X362 @ id @ X366 ) @ id )
= $true )
| ( $true
!= ( X351 @ id @ X366 @ id ) ) ) )
| ~ ! [X367: term] :
( ( ( X352 @ X367 )
!= $true )
| ( ( X352 @ ( sub @ X350 @ ( push @ X367 @ id ) ) )
= $true ) ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X368: term,X369: term > $o,X370: subst > term > subst > $o] :
( ~ ! [X371: subst > term > term] :
( ( $true
= ( X370 @ id @ ( lam @ ( X371 @ sh @ one ) ) @ id ) )
| ~ ! [X372: term] :
( ( $true
!= ( X370 @ id @ X372 @ id ) )
| ( $true
= ( X370 @ id @ ( X371 @ id @ X372 ) @ id ) ) )
| ~ ! [X373: term,X374: subst,X375: subst] :
( ( sub @ ( X371 @ X375 @ X373 ) @ X374 )
= ( X371 @ ( comp @ X375 @ X374 ) @ ( sub @ X373 @ X374 ) ) ) )
| ~ ! [X376: term] :
( ( X370 @ id @ X376 @ id )
= ( X369 @ X376 ) )
| ~ ! [X377: term] :
( ( ( X369 @ ( sub @ X368 @ ( push @ X377 @ id ) ) )
= $true )
| ( ( X369 @ X377 )
!= $true ) )
| ( $true
= ( X369 @ ( lam @ X368 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( substmonoid_gthm = $true )
& ( ( hoasapinj1 = $true )
<=> ! [X378: term,X379: term,X380: term,X381: term] :
( ( X379 = X380 )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( ap @ ( sub @ X380 @ id ) @ X381 ) ) ) )
& ! [X382: subst] :
? [X383: d_subst] :
( ( d2subst @ X383 )
= X382 )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( hoasinduction_lem1 = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ! [X384: subst > term > subst > $o] :
~ ! [X385: term > $o] :
~ ! [X386: term] :
( ( X384 @ id @ X386 @ id )
= ( X385 @ X386 ) )
=> ( ! [X387: term,X388: term > $o] :
( ~ ! [X389: term] :
( ( $true
= ( X388 @ X389 ) )
| ( $true
!= ( var @ X389 ) ) )
| ( ( X388 @ X387 )
= $true )
| ~ ! [X390: term] :
( ( ( X388 @ ( lam @ X390 ) )
= $true )
| ~ ! [X391: term] :
( ( ( X388 @ ( sub @ X390 @ ( push @ X391 @ id ) ) )
= $true )
| ( ( X388 @ X391 )
!= $true ) ) )
| ~ ! [X392: term,X393: term] :
( ( $true
!= ( X388 @ X392 ) )
| ( ( X388 @ X393 )
!= $true )
| ( $true
= ( X388 @ ( ap @ X393 @ X392 ) ) ) ) )
=> ( ! [X394: term] :
( ( sub @ X394 @ id )
= X394 )
=> ( ! [X395: term > $o,X396: term,X397: subst > term > subst > $o] :
( ~ ! [X398: term] :
( ( X397 @ id @ X398 @ id )
= ( X395 @ X398 ) )
| ~ ! [X399: subst > term > term] :
( ~ ! [X400: subst,X401: subst,X402: term] :
( ( sub @ ( X399 @ X400 @ X402 ) @ X401 )
= ( X399 @ ( comp @ X400 @ X401 ) @ ( sub @ X402 @ X401 ) ) )
| ~ ! [X403: term] :
( ( ( X397 @ id @ X403 @ id )
!= $true )
| ( $true
= ( X397 @ id @ ( X399 @ id @ X403 ) @ id ) ) )
| ( $true
= ( X397 @ id @ ( lam @ ( X399 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X395 @ ( lam @ X396 ) ) )
| ~ ! [X404: term] :
( ( ( X395 @ X404 )
!= $true )
| ( ( X395 @ ( sub @ X396 @ ( push @ X404 @ id ) ) )
= $true ) ) )
=> ! [X405: term,X406: subst > term > subst > $o] :
( ~ ! [X407: subst > term > term] :
( ~ ! [X408: term] :
( ( ( X406 @ id @ ( X407 @ id @ X408 ) @ id )
= $true )
| ( ( X406 @ id @ X408 @ id )
!= $true ) )
| ~ ! [X409: subst,X410: subst,X411: term] :
( ( sub @ ( X407 @ X410 @ X411 ) @ X409 )
= ( X407 @ ( comp @ X410 @ X409 ) @ ( sub @ X411 @ X409 ) ) )
| ( $true
= ( X406 @ id @ ( lam @ ( X407 @ sh @ one ) ) @ id ) ) )
| ~ ! [X412: term,X413: term] :
( ( ( X406 @ id @ ( ap @ ( sub @ X412 @ id ) @ X413 ) @ id )
= $true )
| ( ( X406 @ id @ X412 @ id )
!= $true )
| ( $true
!= ( X406 @ id @ X413 @ id ) ) )
| ( $true
= ( X406 @ id @ X405 @ id ) ) ) ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoasapnotvar_lthm = $true )
& ( axabs = $true )
& ( ! [X414: term,X415: subst > term > subst > $o] :
( ( $true
= ( X415 @ id @ ( lam @ ( sub @ X414 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X416: subst > term > term] :
( ~ ! [X417: term] :
( ( ( X415 @ id @ X417 @ id )
!= $true )
| ( ( X415 @ id @ ( X416 @ id @ X417 ) @ id )
= $true ) )
| ( ( X415 @ id @ ( lam @ ( X416 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X418: term,X419: subst,X420: subst] :
( ( sub @ ( X416 @ X419 @ X418 ) @ X420 )
= ( X416 @ ( comp @ X419 @ X420 ) @ ( sub @ X418 @ X420 ) ) ) )
| ~ ! [X421: term] :
( ( $true
!= ( X415 @ id @ X421 @ id ) )
| ( ( X415 @ id @ ( sub @ X414 @ ( push @ X421 @ id ) ) @ id )
= $true ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( hoasapnotvar = $true )
& ( ( ! [X422: term > $o,X423: term] :
( ~ ! [X424: term] :
( ( $true
= ( X422 @ X424 ) )
| ( ( var @ X424 )
!= $true ) )
| ~ ! [X425: term,X426: term] :
( ( ( X422 @ X426 )
!= $true )
| ( ( X422 @ ( ap @ X426 @ X425 ) )
= $true )
| ( $true
!= ( X422 @ X425 ) ) )
| ~ ! [X427: term] :
( ( $true
= ( X422 @ ( lam @ X427 ) ) )
| ~ ! [X428: term] :
( ( ( X422 @ ( sub @ X427 @ ( push @ X428 @ id ) ) )
= $true )
| ( ( X422 @ X428 )
!= $true ) ) )
| ( ( X422 @ X423 )
= $true ) )
=> ( ! [X429: subst > term > subst > $o,X430: term,X431: term] :
( ~ ! [X432: term,X433: subst,X434: subst,X435: subst] :
( ( ( X429 @ X435 @ X432 @ ( comp @ X434 @ X433 ) )
= $true )
| ( ( X429 @ ( comp @ X435 @ X434 ) @ ( sub @ X432 @ X434 ) @ X433 )
!= $true ) )
| ( ( X429 @ id @ X430 @ id )
!= $true )
| ( ( X429 @ id @ X431 @ id )
!= $true )
| ~ ! [X436: subst,X437: subst,X438: term,X439: subst] :
( ( ( X429 @ ( comp @ X437 @ X436 ) @ ( sub @ X438 @ X436 ) @ X439 )
= $true )
| ( $true
!= ( X429 @ X437 @ X438 @ ( comp @ X436 @ X439 ) ) ) )
| ~ ! [X440: term,X441: term] :
( ( $true
= ( X429 @ id @ ( ap @ ( sub @ X440 @ id ) @ X441 ) @ id ) )
| ( ( X429 @ id @ X440 @ id )
!= $true )
| ( $true
!= ( X429 @ id @ X441 @ id ) ) )
| ( ( X429 @ id @ ( ap @ X431 @ X430 ) @ id )
= $true ) )
=> ( ! [X442: term,X443: subst > term > subst > $o] :
( ~ ! [X444: subst > term > term] :
( ( $true
= ( X443 @ id @ ( lam @ ( X444 @ sh @ one ) ) @ id ) )
| ~ ! [X445: subst,X446: term,X447: subst] :
( ( X444 @ ( comp @ X447 @ X445 ) @ ( sub @ X446 @ X445 ) )
= ( sub @ ( X444 @ X447 @ X446 ) @ X445 ) )
| ~ ! [X448: term] :
( ( ( X443 @ id @ X448 @ id )
!= $true )
| ( $true
= ( X443 @ id @ ( X444 @ id @ X448 ) @ id ) ) ) )
| ~ ! [X449: term] :
( ( $true
!= ( X443 @ id @ X449 @ id ) )
| ( ( X443 @ id @ ( sub @ X442 @ ( push @ X449 @ id ) ) @ id )
= $true ) )
| ~ ! [X450: subst,X451: term,X452: subst,X453: subst] :
( ( $true
= ( X443 @ X452 @ X451 @ ( comp @ X453 @ X450 ) ) )
| ( ( X443 @ ( comp @ X452 @ X453 ) @ ( sub @ X451 @ X453 ) @ X450 )
!= $true ) )
| ~ ! [X454: subst,X455: subst,X456: term,X457: subst] :
( ( ( X443 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) @ X454 )
= $true )
| ( $true
!= ( X443 @ X457 @ X456 @ ( comp @ X455 @ X454 ) ) ) )
| ( ( X443 @ id @ ( lam @ X442 ) @ id )
= $true ) )
=> ! [X458: term,X459: subst > term > subst > $o] :
( ~ ! [X460: term,X461: term] :
( ( $true
= ( X459 @ id @ ( ap @ ( sub @ X460 @ id ) @ X461 ) @ id ) )
| ( $true
!= ( X459 @ id @ X460 @ id ) )
| ( $true
!= ( X459 @ id @ X461 @ id ) ) )
| ~ ! [X462: subst > term > term] :
( ( $true
= ( X459 @ id @ ( lam @ ( X462 @ sh @ one ) ) @ id ) )
| ~ ! [X463: term] :
( ( ( X459 @ id @ ( X462 @ id @ X463 ) @ id )
= $true )
| ( ( X459 @ id @ X463 @ id )
!= $true ) )
| ~ ! [X464: subst,X465: term,X466: subst] :
( ( X462 @ ( comp @ X466 @ X464 ) @ ( sub @ X465 @ X464 ) )
= ( sub @ ( X462 @ X466 @ X465 ) @ X464 ) ) )
| ~ ! [X467: term,X468: subst,X469: subst,X470: subst] :
( ( ( X459 @ ( comp @ X469 @ X470 ) @ ( sub @ X467 @ X470 ) @ X468 )
!= $true )
| ( ( X459 @ X469 @ X467 @ ( comp @ X470 @ X468 ) )
= $true ) )
| ~ ! [X471: subst,X472: subst,X473: subst,X474: term] :
( ( ( X459 @ X473 @ X474 @ ( comp @ X472 @ X471 ) )
!= $true )
| ( ( X459 @ ( comp @ X473 @ X472 ) @ ( sub @ X474 @ X472 ) @ X471 )
= $true ) )
| ( ( X459 @ id @ X458 @ id )
= $true )
| ~ ! [X475: term] :
( ( $true
!= ( var @ ( sub @ X475 @ id ) ) )
| ( $true
= ( X459 @ id @ X475 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X476: term,X477: term,X478: term,X479: term] :
( ( X476 = X479 )
| ( ( ap @ X477 @ X476 )
!= ( ap @ X478 @ X479 ) ) )
=> ! [X480: term,X481: term,X482: term,X483: term] :
( ( X482 = X483 )
| ( ( ap @ ( sub @ X481 @ id ) @ X483 )
!= ( ap @ ( sub @ X480 @ id ) @ X482 ) ) ) ) )
& ( ( apinj1 = $true )
<=> ! [X484: term,X485: term,X486: term,X487: term] :
( ( X485 = X487 )
| ( ( ap @ X485 @ X484 )
!= ( ap @ X487 @ X486 ) ) ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( ! [X488: term] :
( ( sub @ X488 @ id )
= X488 )
=> ! [X489: term] :
( ( sub @ X489 @ id )
= X489 ) )
<=> ( termmset_lthm = $true ) )
& ( axapp = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ! [X490: term > $o,X491: term,X492: subst] :
~ ! [X493: term > $o] :
~ ! [X494: term] :
( ( X490 @ ( sub @ X494 @ ( push @ X491 @ X492 ) ) )
= ( X493 @ X494 ) )
<=> ( pushprop_lem0 = $true ) )
& ( ( hoaslaminj = $true )
<=> ! [X495: term,X496: subst > term > term,X497: subst,X498: subst > term > term] :
( ~ ! [X502: subst,X503: term,X504: subst] :
( ( X498 @ ( comp @ X502 @ X504 ) @ ( sub @ X503 @ X504 ) )
= ( sub @ ( X498 @ X502 @ X503 ) @ X504 ) )
| ( ( X498 @ X497 @ X495 )
= ( X496 @ X497 @ X495 ) )
| ( ( lam @ ( X498 @ sh @ one ) )
!= ( lam @ ( X496 @ sh @ one ) ) )
| ~ ! [X499: subst,X500: term,X501: subst] :
( ( X496 @ ( comp @ X499 @ X501 ) @ ( sub @ X500 @ X501 ) )
= ( sub @ ( X496 @ X499 @ X500 ) @ X501 ) ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X505: subst > term > subst > $o,X506: term] :
( ~ ! [X507: term] :
( ( ( X505 @ id @ ( sub @ X506 @ ( push @ X507 @ id ) ) @ id )
= $true )
| ( $true
!= ( X505 @ id @ X507 @ id ) ) )
| ( ( X505 @ id @ ( lam @ ( sub @ X506 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X508: subst > term > term] :
( ~ ! [X509: term] :
( ( $true
= ( X505 @ id @ ( X508 @ id @ X509 ) @ id ) )
| ( $true
!= ( X505 @ id @ X509 @ id ) ) )
| ( ( X505 @ id @ ( lam @ ( X508 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X510: subst,X511: subst,X512: term] :
( ( X508 @ ( comp @ X511 @ X510 ) @ ( sub @ X512 @ X510 ) )
= ( sub @ ( X508 @ X511 @ X512 ) @ X510 ) ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ! [X513: term,X514: term > $o,X515: subst] :
~ ! [X516: term > $o] :
~ ! [X517: term] :
( ( X514 @ ( sub @ X517 @ ( push @ X513 @ X515 ) ) )
= ( X516 @ X517 ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( ! [X518: subst > term > subst > $o,X519: term,X520: term] :
( ~ ! [X521: subst,X522: subst,X523: term,X524: subst] :
( ( $true
!= ( X518 @ ( comp @ X522 @ X524 ) @ ( sub @ X523 @ X524 ) @ X521 ) )
| ( $true
= ( X518 @ X522 @ X523 @ ( comp @ X524 @ X521 ) ) ) )
| ( $true
!= ( X518 @ id @ X520 @ id ) )
| ( ( X518 @ id @ ( ap @ X519 @ X520 ) @ id )
= $true )
| ~ ! [X525: term,X526: term] :
( ( $true
!= ( X518 @ id @ X525 @ id ) )
| ( $true
!= ( X518 @ id @ X526 @ id ) )
| ( ( X518 @ id @ ( ap @ ( sub @ X525 @ id ) @ X526 ) @ id )
= $true ) )
| ~ ! [X527: subst,X528: subst,X529: term,X530: subst] :
( ( $true
= ( X518 @ ( comp @ X528 @ X530 ) @ ( sub @ X529 @ X530 ) @ X527 ) )
| ( $true
!= ( X518 @ X528 @ X529 @ ( comp @ X530 @ X527 ) ) ) )
| ( ( X518 @ id @ X519 @ id )
!= $true ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
<=> ( axidl = $true ) )
& ( ! [X532: term,X533: subst > term > subst > $o] :
( ( $true
= ( X533 @ id @ ( lam @ X532 ) @ id ) )
| ~ ! [X534: subst > term > term] :
( ( ( X533 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X535: subst,X536: subst,X537: term] :
( ( sub @ ( X534 @ X535 @ X537 ) @ X536 )
= ( X534 @ ( comp @ X535 @ X536 ) @ ( sub @ X537 @ X536 ) ) )
| ~ ! [X538: term] :
( ( ( X533 @ id @ ( X534 @ id @ X538 ) @ id )
= $true )
| ( ( X533 @ id @ X538 @ id )
!= $true ) ) )
| ~ ! [X539: term] :
( ( ( X533 @ id @ ( sub @ X532 @ ( push @ X539 @ id ) ) @ id )
= $true )
| ( ( X533 @ id @ X539 @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( hoasapinj1_gthm = $true ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,plain,
( ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
= X0 )
=> ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ( ( X3 @ X5 )
!= $true )
| ~ ! [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
= ( X2 @ X6 ) )
| ( ( X2 @ one )
= $true ) ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X7: subst > term > subst > $o] :
~ ! [X8: term > $o] :
~ ! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) ) )
& ( ( ! [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
= X11 )
=> ( ! [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
= X12 )
=> ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( $true
= ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) ) )
| ( ( X18 @ X15 )
!= $true ) ) ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) ) )
& ( axclos = $true )
& ! [X23: d_term,X24: d_term] :
( ( ( d2term @ X23 )
= ( d2term @ X24 ) )
=> ( X23 = X24 ) )
& ( ! [X25: subst > term > subst > $o,X26: term] :
( ~ ! [X27: term] :
( ( $true
!= ( X25 @ id @ X27 @ id ) )
| ( $true
= ( X25 @ id @ ( sub @ X26 @ ( push @ X27 @ id ) ) @ id ) ) )
| ( $true
= ( X25 @ id @ ( lam @ ( sub @ X26 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X28: term,X29: subst > term > term] :
( ( ( X25 @ id @ ( lam @ ( X29 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X30: term] :
( ( ( X25 @ id @ ( X29 @ id @ X30 ) @ id )
= $true )
| ( ( X25 @ id @ X30 @ id )
!= $true ) )
| ~ ! [X31: subst,X32: term,X33: subst] :
( ( sub @ ( sub @ X28 @ ( push @ X32 @ X33 ) ) @ X31 )
= ( sub @ ( X29 @ X33 @ X32 ) @ X31 ) )
| ~ ! [X34: subst,X35: subst,X36: term] :
( ( sub @ X28 @ ( push @ ( sub @ X36 @ X34 ) @ ( comp @ X35 @ X34 ) ) )
= ( X29 @ ( comp @ X35 @ X34 ) @ ( sub @ X36 @ X34 ) ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ( pushprop_gthm = $true )
& ( ( hoasinduction = $true )
<=> ! [X37: subst > term > subst > $o,X38: term] :
( ~ ! [X39: term,X40: term] :
( ( $true
!= ( X37 @ id @ X40 @ id ) )
| ( $true
!= ( X37 @ id @ X39 @ id ) )
| ( $true
= ( X37 @ id @ ( ap @ ( sub @ X40 @ id ) @ X39 ) @ id ) ) )
| ~ ! [X41: subst > term > term] :
( ~ ! [X42: term,X43: subst,X44: subst] :
( ( X41 @ ( comp @ X43 @ X44 ) @ ( sub @ X42 @ X44 ) )
= ( sub @ ( X41 @ X43 @ X42 ) @ X44 ) )
| ~ ! [X45: term] :
( ( $true
= ( X37 @ id @ ( X41 @ id @ X45 ) @ id ) )
| ( ( X37 @ id @ X45 @ id )
!= $true ) )
| ( ( X37 @ id @ ( lam @ ( X41 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X46: subst,X47: subst,X48: subst,X49: term] :
( ( ( X37 @ ( comp @ X46 @ X47 ) @ ( sub @ X49 @ X47 ) @ X48 )
= $true )
| ( ( X37 @ X46 @ X49 @ ( comp @ X47 @ X48 ) )
!= $true ) )
| ~ ! [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
!= ( X37 @ ( comp @ X51 @ X50 ) @ ( sub @ X52 @ X50 ) @ X53 ) )
| ( ( X37 @ X51 @ X52 @ ( comp @ X50 @ X53 ) )
= $true ) )
| ~ ! [X54: term] :
( ( $true
!= ( var @ ( sub @ X54 @ id ) ) )
| ( $true
= ( X37 @ id @ X54 @ id ) ) )
| ( $true
= ( X37 @ id @ X38 @ id ) ) ) )
& ( ulamvarsh = $true )
& ( hoasapinj2_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
= ( d2subst @ X55 ) )
=> ( X55 = X56 ) )
& ( pushprop_lem2v2_lthm = $true )
& ( ( termmset = $true )
<=> ! [X57: term] :
( ( sub @ X57 @ id )
= X57 ) )
& ( axvarshift = $true )
& ( ( ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 )
& ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 ) )
<=> ( substmonoid = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( pushprop_lthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( ( ! [X62: term] :
( ( sub @ X62 @ id )
= X62 )
=> ( ! [X63: term,X64: subst > term > subst > $o] :
( ~ ! [X65: subst > term > term] :
( ( $true
= ( X64 @ id @ ( lam @ ( X65 @ sh @ one ) ) @ id ) )
| ~ ! [X66: subst,X67: subst,X68: term] :
( ( X65 @ ( comp @ X67 @ X66 ) @ ( sub @ X68 @ X66 ) )
= ( sub @ ( X65 @ X67 @ X68 ) @ X66 ) )
| ~ ! [X69: term] :
( ( $true
!= ( X64 @ id @ X69 @ id ) )
| ( ( X64 @ id @ ( X65 @ id @ X69 ) @ id )
= $true ) ) )
| ~ ! [X70: term] :
( ( $true
!= ( X64 @ id @ X70 @ id ) )
| ( ( X64 @ id @ ( sub @ X63 @ ( push @ X70 @ id ) ) @ id )
= $true ) )
| ( ( X64 @ id @ ( lam @ ( sub @ X63 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
=> ! [X71: term,X72: subst > term > subst > $o] :
( ~ ! [X73: term] :
( ( ( X72 @ id @ X73 @ id )
!= $true )
| ( ( X72 @ id @ ( sub @ X71 @ ( push @ X73 @ id ) ) @ id )
= $true ) )
| ~ ! [X74: subst > term > term] :
( ~ ! [X75: subst,X76: term,X77: subst] :
( ( sub @ ( X74 @ X77 @ X76 ) @ X75 )
= ( X74 @ ( comp @ X77 @ X75 ) @ ( sub @ X76 @ X75 ) ) )
| ( ( X72 @ id @ ( lam @ ( X74 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X78: term] :
( ( ( X72 @ id @ X78 @ id )
!= $true )
| ( ( X72 @ id @ ( X74 @ id @ X78 ) @ id )
= $true ) ) )
| ( $true
= ( X72 @ id @ ( lam @ X71 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X79: subst,X80: term] :
( ( sub @ one @ ( push @ X80 @ X79 ) )
= X80 )
=> ( ! [X81: subst,X82: term] :
( ( comp @ sh @ ( push @ X82 @ X81 ) )
= X81 )
=> ( ! [X83: term,X84: term] :
( ( ( lam @ X83 )
!= ( lam @ X84 ) )
| ( X83 = X84 ) )
=> ! [X85: term,X86: subst > term > term,X87: subst,X88: subst > term > term] :
( ( ( lam @ ( X86 @ sh @ one ) )
!= ( lam @ ( X88 @ sh @ one ) ) )
| ( ( X88 @ X87 @ X85 )
= ( X86 @ X87 @ X85 ) )
| ~ ! [X89: subst,X90: subst,X91: term] :
( ( sub @ ( X88 @ X89 @ X91 ) @ X90 )
= ( X88 @ ( comp @ X89 @ X90 ) @ ( sub @ X91 @ X90 ) ) )
| ~ ! [X92: subst,X93: subst,X94: term] :
( ( X86 @ ( comp @ X92 @ X93 ) @ ( sub @ X94 @ X93 ) )
= ( sub @ ( X86 @ X92 @ X94 ) @ X93 ) ) ) ) ) ) )
& ( ! [X95: subst > term > subst > $o] :
~ ! [X96: term > $o] :
~ ! [X97: term] :
( ( X95 @ id @ X97 @ id )
= ( X96 @ X97 ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( apnotvar = $true )
& ( pushprop = $true )
& ( ( d2subst @ d_id )
= sh )
& ! [X98: d_term] : ( d_one = X98 )
& ( ! [X99: term,X100: term,X101: term,X102: term] :
( ( X99 = X100 )
| ( ( ap @ X102 @ X100 )
!= ( ap @ X101 @ X99 ) ) )
<=> ( apinj2 = $true ) )
& ( ulamvarind = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X108: term] :
~ ! [X109: subst > term > term] :
~ ! [X110: term,X111: subst] :
( ( sub @ X108 @ ( push @ X110 @ X111 ) )
= ( X109 @ X111 @ X110 ) ) )
& ( ( hoasinduction_no_psi_cond = $true )
<=> ! [X112: term,X113: subst > term > subst > $o] :
( ~ ! [X114: subst > term > term] :
( ~ ! [X115: term] :
( ( ( X113 @ id @ ( X114 @ id @ X115 ) @ id )
= $true )
| ( ( X113 @ id @ X115 @ id )
!= $true ) )
| ~ ! [X116: subst,X117: term,X118: subst] :
( ( sub @ ( X114 @ X118 @ X117 ) @ X116 )
= ( X114 @ ( comp @ X118 @ X116 ) @ ( sub @ X117 @ X116 ) ) )
| ( ( X113 @ id @ ( lam @ ( X114 @ sh @ one ) ) @ id )
= $true ) )
| ~ ! [X119: term,X120: term] :
( ( $true
!= ( X113 @ id @ X120 @ id ) )
| ( $true
!= ( X113 @ id @ X119 @ id ) )
| ( ( X113 @ id @ ( ap @ ( sub @ X120 @ id ) @ X119 ) @ id )
= $true ) )
| ( $true
= ( X113 @ id @ X112 @ id ) ) ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X121: term] :
( ( sub @ X121 @ id )
= X121 )
=> ( ! [X122: term,X123: subst > term > subst > $o] :
( ~ ! [X124: subst > term > term] :
( ~ ! [X125: term] :
( ( ( X123 @ id @ ( X124 @ id @ X125 ) @ id )
= $true )
| ( $true
!= ( X123 @ id @ X125 @ id ) ) )
| ~ ! [X126: term,X127: subst,X128: subst] :
( ( X124 @ ( comp @ X128 @ X127 ) @ ( sub @ X126 @ X127 ) )
= ( sub @ ( X124 @ X128 @ X126 ) @ X127 ) )
| ( ( X123 @ id @ ( lam @ ( X124 @ sh @ one ) ) @ id )
= $true ) )
| ( ( X123 @ id @ ( lam @ ( sub @ X122 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X129: term] :
( ( ( X123 @ id @ X129 @ id )
!= $true )
| ( ( X123 @ id @ ( sub @ X122 @ ( push @ X129 @ id ) ) @ id )
= $true ) ) )
=> ! [X130: term,X131: subst > term > subst > $o] :
( ( $true
= ( X131 @ id @ ( lam @ X130 ) @ id ) )
| ~ ! [X132: subst > term > term] :
( ( $true
= ( X131 @ id @ ( lam @ ( X132 @ sh @ one ) ) @ id ) )
| ~ ! [X133: term] :
( ( $true
!= ( X131 @ id @ X133 @ id ) )
| ( ( X131 @ id @ ( X132 @ id @ X133 ) @ id )
= $true ) )
| ~ ! [X134: term,X135: subst,X136: subst] :
( ( sub @ ( X132 @ X136 @ X134 ) @ X135 )
= ( X132 @ ( comp @ X136 @ X135 ) @ ( sub @ X134 @ X135 ) ) ) )
| ~ ! [X137: term,X138: subst,X139: subst,X140: subst] :
( ( ( X131 @ X139 @ X137 @ ( comp @ X138 @ X140 ) )
= $true )
| ( ( X131 @ ( comp @ X139 @ X138 ) @ ( sub @ X137 @ X138 ) @ X140 )
!= $true ) )
| ~ ! [X141: term,X142: subst,X143: subst,X144: subst] :
( ( $true
= ( X131 @ ( comp @ X144 @ X143 ) @ ( sub @ X141 @ X143 ) @ X142 ) )
| ( $true
!= ( X131 @ X144 @ X141 @ ( comp @ X143 @ X142 ) ) ) )
| ~ ! [X145: term] :
( ( ( X131 @ id @ ( sub @ X130 @ ( push @ X145 @ id ) ) @ id )
= $true )
| ( ( X131 @ id @ X145 @ id )
!= $true ) ) ) ) ) )
& ( ( induction2lem = $true )
<=> ! [X146: term,X147: term > $o,X148: subst] :
( ~ ! [X149: term] :
( ( $true
= ( X147 @ ( sub @ X149 @ X148 ) ) )
| ( $true
!= ( var @ X149 ) ) )
| ~ ! [X150: term] :
( ~ ! [X151: term] :
( ( ( X147 @ ( sub @ X150 @ ( push @ X151 @ id ) ) )
= $true )
| ( $true
!= ( X147 @ X151 ) ) )
| ( ( X147 @ ( lam @ X150 ) )
= $true ) )
| ( ( X147 @ ( sub @ X146 @ X148 ) )
= $true )
| ~ ! [X152: term,X153: term] :
( ( ( X147 @ X152 )
!= $true )
| ( $true
!= ( X147 @ X153 ) )
| ( ( X147 @ ( ap @ X152 @ X153 ) )
= $true ) ) ) )
& ( pushprop_lem3v2 = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X154: term] :
~ ! [X155: subst > term > term] :
~ ! [X156: term,X157: subst] :
( ( sub @ X154 @ ( push @ X156 @ X157 ) )
= ( X155 @ X157 @ X156 ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( ( ! [X158: term] :
( ( sub @ X158 @ id )
= X158 )
=> ( ! [X159: term,X160: subst,X161: term > $o] :
( ~ ! [X162: term] :
( ( $true
= ( X161 @ ( lam @ X162 ) ) )
| ~ ! [X163: term] :
( ( $true
= ( X161 @ ( sub @ X162 @ ( push @ X163 @ id ) ) ) )
| ( $true
!= ( X161 @ X163 ) ) ) )
| ~ ! [X164: term,X165: term] :
( ( ( X161 @ X165 )
!= $true )
| ( ( X161 @ ( ap @ X165 @ X164 ) )
= $true )
| ( ( X161 @ X164 )
!= $true ) )
| ( $true
= ( X161 @ ( sub @ X159 @ X160 ) ) )
| ~ ! [X166: term] :
( ( ( var @ X166 )
!= $true )
| ( ( X161 @ ( sub @ X166 @ X160 ) )
= $true ) ) )
=> ! [X167: term,X168: term > $o] :
( ~ ! [X169: term] :
( ( $true
= ( X168 @ X169 ) )
| ( ( var @ X169 )
!= $true ) )
| ~ ! [X170: term,X171: term] :
( ( ( X168 @ ( ap @ X171 @ X170 ) )
= $true )
| ( ( X168 @ X170 )
!= $true )
| ( $true
!= ( X168 @ X171 ) ) )
| ~ ! [X172: term] :
( ~ ! [X173: term] :
( ( ( X168 @ ( sub @ X172 @ ( push @ X173 @ id ) ) )
= $true )
| ( ( X168 @ X173 )
!= $true ) )
| ( ( X168 @ ( lam @ X172 ) )
= $true ) )
| ( ( X168 @ X167 )
= $true ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( ! [X174: term,X175: term] :
( ( X174 = X175 )
| ( ( lam @ X175 )
!= ( lam @ X174 ) ) )
<=> ( laminj = $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( pushprop_lem3v2_lthm = $true )
& ( ulamvar1 != $true )
& ( ( ! [X176: term,X177: subst] :
( ( sub @ one @ ( push @ X176 @ X177 ) )
= X176 )
=> ( ! [X178: subst,X179: term] :
( ( comp @ sh @ ( push @ X179 @ X178 ) )
= X178 )
=> ( ! [X180: subst] :
( ( comp @ X180 @ id )
= X180 )
=> ( ! [X181: term,X182: term > $o] :
( ~ ! [X183: term,X184: term] :
( ( $true
!= ( X182 @ X183 ) )
| ( ( X182 @ X184 )
!= $true )
| ( $true
= ( X182 @ ( ap @ X184 @ X183 ) ) ) )
| ( ( X182 @ X181 )
= $true )
| ~ ! [X185: term] :
( ( ( X182 @ ( lam @ X185 ) )
= $true )
| ( $true
!= ( X182 @ X185 ) ) )
| ~ ! [X186: term] :
( ( $true
= ( X182 @ X186 ) )
| ( $true
!= ( var @ X186 ) ) ) )
=> ! [X187: term,X188: term > $o,X189: subst] :
( ( $true
= ( X188 @ ( sub @ X187 @ X189 ) ) )
| ~ ! [X190: term,X191: term] :
( ( $true
= ( X188 @ ( ap @ X191 @ X190 ) ) )
| ( $true
!= ( X188 @ X191 ) )
| ( $true
!= ( X188 @ X190 ) ) )
| ~ ! [X192: term] :
( ( $true
= ( X188 @ ( sub @ X192 @ X189 ) ) )
| ( ( var @ X192 )
!= $true ) )
| ~ ! [X193: term] :
( ( ( X188 @ ( lam @ X193 ) )
= $true )
| ~ ! [X194: term] :
( ( ( X188 @ X194 )
!= $true )
| ( $true
= ( X188 @ ( sub @ X193 @ ( push @ X194 @ id ) ) ) ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( ! [X195: subst > term > subst > $o,X196: term,X197: term > $o] :
( ~ ! [X198: subst,X199: subst,X200: subst,X201: term] :
( ( $true
!= ( X195 @ X200 @ X201 @ ( comp @ X198 @ X199 ) ) )
| ( $true
= ( X195 @ ( comp @ X200 @ X198 ) @ ( sub @ X201 @ X198 ) @ X199 ) ) )
| ~ ! [X202: subst,X203: subst,X204: subst,X205: term] :
( ( $true
!= ( X195 @ ( comp @ X204 @ X203 ) @ ( sub @ X205 @ X203 ) @ X202 ) )
| ( $true
= ( X195 @ X204 @ X205 @ ( comp @ X203 @ X202 ) ) ) )
| ~ ! [X206: term] :
( ( ( X197 @ ( sub @ X196 @ ( push @ X206 @ id ) ) )
= $true )
| ( $true
!= ( X197 @ X206 ) ) )
| ( ( X197 @ ( lam @ X196 ) )
= $true )
| ~ ! [X207: subst > term > term] :
( ( ( X195 @ id @ ( lam @ ( X207 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X208: subst,X209: subst,X210: term] :
( ( X207 @ ( comp @ X209 @ X208 ) @ ( sub @ X210 @ X208 ) )
= ( sub @ ( X207 @ X209 @ X210 ) @ X208 ) )
| ~ ! [X211: term] :
( ( $true
!= ( X195 @ id @ X211 @ id ) )
| ( ( X195 @ id @ ( X207 @ id @ X211 ) @ id )
= $true ) ) )
| ~ ! [X212: term] :
( ( X195 @ id @ X212 @ id )
= ( X197 @ X212 ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ! [X213: term > $o,X214: term] :
( ( ( X213 @ X214 )
= $true )
| ~ ! [X215: term] :
( ( $true
= ( X213 @ ( lam @ X215 ) ) )
| ( $true
!= ( X213 @ X215 ) ) )
| ~ ! [X216: term,X217: term] :
( ( $true
!= ( X213 @ X216 ) )
| ( $true
= ( X213 @ ( ap @ X216 @ X217 ) ) )
| ( $true
!= ( X213 @ X217 ) ) )
| ~ ! [X218: term] :
( ( $true
!= ( var @ X218 ) )
| ( $true
= ( X213 @ X218 ) ) ) )
<=> ( induction = $true ) )
& ( hoasinduction_lem1_gthm = $true )
& ( induction2lem_gthm = $true )
& ( ! [X219: term] :
~ ! [X220: subst > term > term] :
( ( X220 @ sh @ one )
!= ( sub @ X219 @ ( push @ one @ sh ) ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( ! [X221: subst] :
( ( comp @ X221 @ id )
= X221 )
<=> ( axidr = $true ) )
& ( termmset_gthm = $true )
& ( ! [X222: term,X223: term > $o,X224: term > $o,X225: subst] :
( ( ( X223 @ one )
= $true )
| ( $true
!= ( X224 @ X222 ) )
| ~ ! [X226: term] :
( ( X223 @ X226 )
= ( X224 @ ( sub @ X226 @ ( push @ X222 @ X225 ) ) ) ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( ( d2subst @ d_id )
= id )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X227: term > $o,X228: term,X229: subst > term > subst > $o,X230: term] :
( ( ( X227 @ X228 )
!= $true )
| ~ ! [X231: subst,X232: term,X233: subst,X234: subst] :
( ( ( X229 @ ( comp @ X231 @ X234 ) @ ( sub @ X232 @ X234 ) @ X233 )
= $true )
| ( ( X229 @ X231 @ X232 @ ( comp @ X234 @ X233 ) )
!= $true ) )
| ( ( X227 @ X230 )
!= $true )
| ~ ! [X235: subst,X236: subst,X237: subst,X238: term] :
( ( ( X229 @ X237 @ X238 @ ( comp @ X235 @ X236 ) )
= $true )
| ( ( X229 @ ( comp @ X237 @ X235 ) @ ( sub @ X238 @ X235 ) @ X236 )
!= $true ) )
| ( ( X227 @ ( ap @ X228 @ X230 ) )
= $true )
| ~ ! [X239: term] :
( ( X229 @ id @ X239 @ id )
= ( X227 @ X239 ) )
| ~ ! [X240: term,X241: term] :
( ( $true
= ( X229 @ id @ ( ap @ ( sub @ X241 @ id ) @ X240 ) @ id ) )
| ( ( X229 @ id @ X240 @ id )
!= $true )
| ( ( X229 @ id @ X241 @ id )
!= $true ) ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X242: subst > term > subst > $o,X243: term] :
( ~ ! [X244: subst,X245: subst,X246: subst,X247: term] :
( ( $true
= ( X242 @ X245 @ X247 @ ( comp @ X246 @ X244 ) ) )
| ( ( X242 @ ( comp @ X245 @ X246 ) @ ( sub @ X247 @ X246 ) @ X244 )
!= $true ) )
| ( ( X242 @ id @ ( lam @ X243 ) @ id )
= $true )
| ~ ! [X248: subst > term > term] :
( ~ ! [X249: term,X250: subst,X251: subst] :
( ( X248 @ ( comp @ X251 @ X250 ) @ ( sub @ X249 @ X250 ) )
= ( sub @ ( X248 @ X251 @ X249 ) @ X250 ) )
| ( ( X242 @ id @ ( lam @ ( X248 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X252: term] :
( ( ( X242 @ id @ ( X248 @ id @ X252 ) @ id )
= $true )
| ( ( X242 @ id @ X252 @ id )
!= $true ) ) )
| ~ ! [X253: term] :
( ( ( X242 @ id @ X253 @ id )
!= $true )
| ( ( X242 @ id @ ( sub @ X243 @ ( push @ X253 @ id ) ) @ id )
= $true ) )
| ~ ! [X254: subst,X255: subst,X256: subst,X257: term] :
( ( ( X242 @ ( comp @ X255 @ X256 ) @ ( sub @ X257 @ X256 ) @ X254 )
= $true )
| ( ( X242 @ X255 @ X257 @ ( comp @ X256 @ X254 ) )
!= $true ) ) ) )
& ( hoasapnotvar_gthm = $true )
& ( lamnotap != $true )
& ( ! [X258: term,X259: term > $o] :
( ( ( X259 @ X258 )
= $true )
| ~ ! [X260: term,X261: term] :
( ( ( X259 @ X261 )
!= $true )
| ( $true
!= ( X259 @ X260 ) )
| ( ( X259 @ ( ap @ X261 @ X260 ) )
= $true ) )
| ~ ! [X262: term] :
( ~ ! [X263: term] :
( ( ( X259 @ X263 )
!= $true )
| ( $true
= ( X259 @ ( sub @ X262 @ ( push @ X263 @ id ) ) ) ) )
| ( ( X259 @ ( lam @ X262 ) )
= $true ) )
| ~ ! [X264: term] :
( ( $true
= ( X259 @ X264 ) )
| ( $true
!= ( var @ X264 ) ) ) )
<=> ( induction2 = $true ) )
& ( ! [X265: term,X266: subst > term > term,X267: term] :
( ~ ! [X268: term,X269: subst,X270: subst] :
( ( sub @ ( X266 @ X270 @ X268 ) @ X269 )
= ( X266 @ ( comp @ X270 @ X269 ) @ ( sub @ X268 @ X269 ) ) )
| ( ( ap @ ( sub @ X267 @ id ) @ X265 )
!= ( lam @ ( X266 @ sh @ one ) ) ) )
<=> ( hoaslamnotap = $true ) )
& ( axmap = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X271: term] :
( ( sub @ X271 @ id )
= X271 )
<=> ( axvarid = $true ) )
& ( ( pushprop_lem1 = $true )
<=> ! [X272: term,X273: term > $o,X274: term > $o,X275: term,X276: subst] :
( ( ( X273 @ X272 )
!= $true )
| ( ( X274 @ ( sub @ X272 @ ( push @ X275 @ X276 ) ) )
= $true ) ) )
& ( hoaslamnotvar = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( hoaslaminj_gthm = $true )
& ( lamnotvar = $true )
& ! [X277: d_subst] : ( d_id = X277 )
& ! [X278: term] :
? [X279: d_term] :
( ( d2term @ X279 )
= X278 )
& ( ( axshiftcons = $true )
<=> ! [X280: subst,X281: term] :
( ( comp @ sh @ ( push @ X281 @ X280 ) )
= X280 ) )
& ( ( shinj = $true )
<=> ! [X282: term,X283: term] :
( ( X282 = X283 )
| ( ( sub @ X282 @ sh )
!= ( sub @ X283 @ sh ) ) ) )
& ( pushprop_lthm_orig = $true )
& ( ( ! [X284: term] :
~ ! [X285: subst > term > term] :
~ ! [X286: term,X287: subst] :
( ( sub @ X284 @ ( push @ X286 @ X287 ) )
= ( X285 @ X287 @ X286 ) )
=> ( ! [X288: term] :
( ( sub @ X288 @ id )
= X288 )
=> ! [X289: term > $o,X290: subst > term > subst > $o,X291: term] :
( ~ ! [X292: subst > term > term] :
( ( $true
= ( X290 @ id @ ( lam @ ( X292 @ sh @ one ) ) @ id ) )
| ~ ! [X293: term] :
( ( $true
!= ( X290 @ id @ X293 @ id ) )
| ( ( X290 @ id @ ( X292 @ id @ X293 ) @ id )
= $true ) )
| ~ ! [X294: term,X295: subst,X296: subst] :
( ( X292 @ ( comp @ X296 @ X295 ) @ ( sub @ X294 @ X295 ) )
= ( sub @ ( X292 @ X296 @ X294 ) @ X295 ) ) )
| ( ( X289 @ ( lam @ X291 ) )
= $true )
| ~ ! [X297: term] :
( ( X289 @ X297 )
= ( X290 @ id @ X297 @ id ) )
| ~ ! [X298: term] :
( ( ( X289 @ X298 )
!= $true )
| ( ( X289 @ ( sub @ X291 @ ( push @ X298 @ id ) ) )
= $true ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X299: subst > term > subst > $o] :
~ ! [X300: term > $o] :
~ ! [X301: term] :
( ( X299 @ id @ X301 @ id )
= ( X300 @ X301 ) )
=> ( ! [X302: term > $o,X303: term] :
( ~ ! [X304: term,X305: term] :
( ( ( X302 @ X304 )
!= $true )
| ( $true
= ( X302 @ ( ap @ X304 @ X305 ) ) )
| ( ( X302 @ X305 )
!= $true ) )
| ~ ! [X306: term] :
( ( ( X302 @ X306 )
= $true )
| ( ( var @ X306 )
!= $true ) )
| ~ ! [X307: term] :
( ( $true
= ( X302 @ ( lam @ X307 ) ) )
| ~ ! [X308: term] :
( ( ( X302 @ X308 )
!= $true )
| ( ( X302 @ ( sub @ X307 @ ( push @ X308 @ id ) ) )
= $true ) ) )
| ( $true
= ( X302 @ X303 ) ) )
=> ( ! [X309: term] :
( ( sub @ X309 @ id )
= X309 )
=> ( ! [X310: subst > term > subst > $o,X311: term,X312: term > $o] :
( ~ ! [X313: subst > term > term] :
( ( $true
= ( X310 @ id @ ( lam @ ( X313 @ sh @ one ) ) @ id ) )
| ~ ! [X314: term] :
( ( $true
= ( X310 @ id @ ( X313 @ id @ X314 ) @ id ) )
| ( $true
!= ( X310 @ id @ X314 @ id ) ) )
| ~ ! [X315: term,X316: subst,X317: subst] :
( ( sub @ ( X313 @ X317 @ X315 ) @ X316 )
= ( X313 @ ( comp @ X317 @ X316 ) @ ( sub @ X315 @ X316 ) ) ) )
| ( ( X312 @ ( lam @ X311 ) )
= $true )
| ~ ! [X318: term] :
( ( $true
!= ( X312 @ X318 ) )
| ( $true
= ( X312 @ ( sub @ X311 @ ( push @ X318 @ id ) ) ) ) )
| ~ ! [X319: term] :
( ( X310 @ id @ X319 @ id )
= ( X312 @ X319 ) ) )
=> ! [X320: term,X321: subst > term > subst > $o] :
( ~ ! [X322: subst,X323: subst,X324: term,X325: subst] :
( ( $true
= ( X321 @ X325 @ X324 @ ( comp @ X322 @ X323 ) ) )
| ( ( X321 @ ( comp @ X325 @ X322 ) @ ( sub @ X324 @ X322 ) @ X323 )
!= $true ) )
| ~ ! [X326: term] :
( ( $true
!= ( var @ ( sub @ X326 @ id ) ) )
| ( ( X321 @ id @ X326 @ id )
= $true ) )
| ~ ! [X327: term,X328: term] :
( ( ( X321 @ id @ X327 @ id )
!= $true )
| ( ( X321 @ id @ ( ap @ ( sub @ X327 @ id ) @ X328 ) @ id )
= $true )
| ( ( X321 @ id @ X328 @ id )
!= $true ) )
| ~ ! [X329: subst,X330: subst,X331: term,X332: subst] :
( ( ( X321 @ ( comp @ X329 @ X330 ) @ ( sub @ X331 @ X330 ) @ X332 )
= $true )
| ( ( X321 @ X329 @ X331 @ ( comp @ X330 @ X332 ) )
!= $true ) )
| ~ ! [X333: subst > term > term] :
( ~ ! [X334: subst,X335: term,X336: subst] :
( ( sub @ ( X333 @ X334 @ X335 ) @ X336 )
= ( X333 @ ( comp @ X334 @ X336 ) @ ( sub @ X335 @ X336 ) ) )
| ( ( X321 @ id @ ( lam @ ( X333 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X337: term] :
( ( ( X321 @ id @ X337 @ id )
!= $true )
| ( ( X321 @ id @ ( X333 @ id @ X337 ) @ id )
= $true ) ) )
| ( ( X321 @ id @ X320 @ id )
= $true ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( ! [X338: subst] :
( ( comp @ id @ X338 )
= X338 )
=> ( ! [X339: subst] :
( ( comp @ X339 @ id )
= X339 )
=> ( ! [X340: subst] :
( ( comp @ X340 @ id )
= X340 )
& ! [X341: subst] :
( ( comp @ id @ X341 )
= X341 ) ) ) )
<=> ( substmonoid_lthm = $true ) )
& ( hoasinduction_lem1_lthm = $true )
& ( axassoc = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( induction2_gthm = $true )
& ( ( axscons = $true )
<=> ! [X342: subst] :
( ( push @ ( sub @ one @ X342 ) @ ( comp @ sh @ X342 ) )
= X342 ) )
& ( ! [X343: subst,X344: term] :
( ( sub @ one @ ( push @ X344 @ X343 ) )
= X344 )
<=> ( axvarcons = $true ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X345: term] :
~ ! [X346: subst > term > term] :
( ( X346 @ sh @ one )
!= ( sub @ X345 @ ( push @ one @ sh ) ) ) )
& ( ( ! [X347: term] :
( ( sub @ X347 @ id )
= X347 )
=> ( ! [X348: term,X349: term,X350: term,X351: term] :
( ( X348 = X349 )
| ( ( ap @ X349 @ X350 )
!= ( ap @ X348 @ X351 ) ) )
=> ! [X352: term,X353: term,X354: term,X355: term] :
( ( X352 = X355 )
| ( ( ap @ ( sub @ X352 @ id ) @ X354 )
!= ( ap @ ( sub @ X355 @ id ) @ X353 ) ) ) ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X356: term] :
( ( sub @ X356 @ id )
= X356 )
=> ! [X357: term,X358: subst > term > subst > $o,X359: term > $o] :
( ~ ! [X360: term] :
( ( X358 @ id @ X360 @ id )
= ( X359 @ X360 ) )
| ( ( X359 @ ( lam @ X357 ) )
= $true )
| ~ ! [X361: subst,X362: subst,X363: subst,X364: term] :
( ( ( X358 @ X363 @ X364 @ ( comp @ X361 @ X362 ) )
!= $true )
| ( $true
= ( X358 @ ( comp @ X363 @ X361 ) @ ( sub @ X364 @ X361 ) @ X362 ) ) )
| ~ ! [X365: subst,X366: subst,X367: term,X368: subst] :
( ( $true
= ( X358 @ X365 @ X367 @ ( comp @ X366 @ X368 ) ) )
| ( ( X358 @ ( comp @ X365 @ X366 ) @ ( sub @ X367 @ X366 ) @ X368 )
!= $true ) )
| ~ ! [X369: subst > term > term] :
( ~ ! [X370: term,X371: subst,X372: subst] :
( ( sub @ ( X369 @ X372 @ X370 ) @ X371 )
= ( X369 @ ( comp @ X372 @ X371 ) @ ( sub @ X370 @ X371 ) ) )
| ( $true
= ( X358 @ id @ ( lam @ ( X369 @ sh @ one ) ) @ id ) )
| ~ ! [X373: term] :
( ( ( X358 @ id @ ( X369 @ id @ X373 ) @ id )
= $true )
| ( ( X358 @ id @ X373 @ id )
!= $true ) ) )
| ~ ! [X374: term] :
( ( $true
!= ( X359 @ X374 ) )
| ( ( X359 @ ( sub @ X357 @ ( push @ X374 @ id ) ) )
= $true ) ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ! [X375: term,X376: term > $o,X377: subst > term > subst > $o] :
( ~ ! [X378: subst > term > term] :
( ( ( X377 @ id @ ( lam @ ( X378 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X379: term] :
( ( ( X377 @ id @ X379 @ id )
!= $true )
| ( ( X377 @ id @ ( X378 @ id @ X379 ) @ id )
= $true ) )
| ~ ! [X380: term,X381: subst,X382: subst] :
( ( sub @ ( X378 @ X382 @ X380 ) @ X381 )
= ( X378 @ ( comp @ X382 @ X381 ) @ ( sub @ X380 @ X381 ) ) ) )
| ~ ! [X383: term] :
( ( X377 @ id @ X383 @ id )
= ( X376 @ X383 ) )
| ~ ! [X384: term] :
( ( ( X376 @ ( sub @ X375 @ ( push @ X384 @ id ) ) )
= $true )
| ( ( X376 @ X384 )
!= $true ) )
| ( $true
= ( X376 @ ( lam @ X375 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( substmonoid_gthm = $true )
& ( ( hoasapinj1 = $true )
<=> ! [X385: term,X386: term,X387: term,X388: term] :
( ( ( ap @ ( sub @ X386 @ id ) @ X385 )
!= ( ap @ ( sub @ X387 @ id ) @ X388 ) )
| ( X386 = X387 ) ) )
& ! [X389: subst] :
? [X390: d_subst] :
( ( d2subst @ X390 )
= X389 )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y1 @ Y2 )
= ( Y0 @ id @ Y2 @ id ) ) ) ) )
& ( hoasinduction_lem1 = $true )
& ( pushprop_lem0_gthm = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ! [X394: subst > term > subst > $o] :
~ ! [X395: term > $o] :
~ ! [X396: term] :
( ( X394 @ id @ X396 @ id )
= ( X395 @ X396 ) )
=> ( ! [X397: term,X398: term > $o] :
( ~ ! [X399: term] :
( ( $true
= ( X398 @ X399 ) )
| ( ( var @ X399 )
!= $true ) )
| ( $true
= ( X398 @ X397 ) )
| ~ ! [X400: term] :
( ( ( X398 @ ( lam @ X400 ) )
= $true )
| ~ ! [X401: term] :
( ( $true
= ( X398 @ ( sub @ X400 @ ( push @ X401 @ id ) ) ) )
| ( $true
!= ( X398 @ X401 ) ) ) )
| ~ ! [X402: term,X403: term] :
( ( ( X398 @ X402 )
!= $true )
| ( $true
!= ( X398 @ X403 ) )
| ( $true
= ( X398 @ ( ap @ X403 @ X402 ) ) ) ) )
=> ( ! [X404: term] :
( ( sub @ X404 @ id )
= X404 )
=> ( ! [X405: term > $o,X406: term,X407: subst > term > subst > $o] :
( ~ ! [X408: term] :
( ( X405 @ X408 )
= ( X407 @ id @ X408 @ id ) )
| ~ ! [X409: subst > term > term] :
( ~ ! [X410: subst,X411: subst,X412: term] :
( ( sub @ ( X409 @ X410 @ X412 ) @ X411 )
= ( X409 @ ( comp @ X410 @ X411 ) @ ( sub @ X412 @ X411 ) ) )
| ~ ! [X413: term] :
( ( $true
!= ( X407 @ id @ X413 @ id ) )
| ( $true
= ( X407 @ id @ ( X409 @ id @ X413 ) @ id ) ) )
| ( $true
= ( X407 @ id @ ( lam @ ( X409 @ sh @ one ) ) @ id ) ) )
| ( ( X405 @ ( lam @ X406 ) )
= $true )
| ~ ! [X414: term] :
( ( ( X405 @ X414 )
!= $true )
| ( ( X405 @ ( sub @ X406 @ ( push @ X414 @ id ) ) )
= $true ) ) )
=> ! [X415: term,X416: subst > term > subst > $o] :
( ~ ! [X417: subst > term > term] :
( ~ ! [X418: term] :
( ( ( X416 @ id @ ( X417 @ id @ X418 ) @ id )
= $true )
| ( ( X416 @ id @ X418 @ id )
!= $true ) )
| ~ ! [X419: subst,X420: subst,X421: term] :
( ( X417 @ ( comp @ X420 @ X419 ) @ ( sub @ X421 @ X419 ) )
= ( sub @ ( X417 @ X420 @ X421 ) @ X419 ) )
| ( $true
= ( X416 @ id @ ( lam @ ( X417 @ sh @ one ) ) @ id ) ) )
| ~ ! [X422: term,X423: term] :
( ( $true
= ( X416 @ id @ ( ap @ ( sub @ X422 @ id ) @ X423 ) @ id ) )
| ( ( X416 @ id @ X422 @ id )
!= $true )
| ( ( X416 @ id @ X423 @ id )
!= $true ) )
| ( $true
= ( X416 @ id @ X415 @ id ) ) ) ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoasapnotvar_lthm = $true )
& ( axabs = $true )
& ( ! [X424: term,X425: subst > term > subst > $o] :
( ( ( X425 @ id @ ( lam @ ( sub @ X424 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X426: subst > term > term] :
( ~ ! [X427: term] :
( ( ( X425 @ id @ X427 @ id )
!= $true )
| ( ( X425 @ id @ ( X426 @ id @ X427 ) @ id )
= $true ) )
| ( $true
= ( X425 @ id @ ( lam @ ( X426 @ sh @ one ) ) @ id ) )
| ~ ! [X428: term,X429: subst,X430: subst] :
( ( X426 @ ( comp @ X429 @ X430 ) @ ( sub @ X428 @ X430 ) )
= ( sub @ ( X426 @ X429 @ X428 ) @ X430 ) ) )
| ~ ! [X431: term] :
( ( ( X425 @ id @ X431 @ id )
!= $true )
| ( $true
= ( X425 @ id @ ( sub @ X424 @ ( push @ X431 @ id ) ) @ id ) ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( hoasapnotvar = $true )
& ( ( ! [X432: term > $o,X433: term] :
( ~ ! [X434: term] :
( ( ( X432 @ X434 )
= $true )
| ( ( var @ X434 )
!= $true ) )
| ~ ! [X435: term,X436: term] :
( ( ( X432 @ X436 )
!= $true )
| ( ( X432 @ ( ap @ X436 @ X435 ) )
= $true )
| ( $true
!= ( X432 @ X435 ) ) )
| ~ ! [X437: term] :
( ( ( X432 @ ( lam @ X437 ) )
= $true )
| ~ ! [X438: term] :
( ( ( X432 @ ( sub @ X437 @ ( push @ X438 @ id ) ) )
= $true )
| ( ( X432 @ X438 )
!= $true ) ) )
| ( ( X432 @ X433 )
= $true ) )
=> ( ! [X439: subst > term > subst > $o,X440: term,X441: term] :
( ~ ! [X442: term,X443: subst,X444: subst,X445: subst] :
( ( ( X439 @ X445 @ X442 @ ( comp @ X444 @ X443 ) )
= $true )
| ( ( X439 @ ( comp @ X445 @ X444 ) @ ( sub @ X442 @ X444 ) @ X443 )
!= $true ) )
| ( ( X439 @ id @ X440 @ id )
!= $true )
| ( ( X439 @ id @ X441 @ id )
!= $true )
| ~ ! [X446: subst,X447: subst,X448: term,X449: subst] :
( ( ( X439 @ ( comp @ X447 @ X446 ) @ ( sub @ X448 @ X446 ) @ X449 )
= $true )
| ( ( X439 @ X447 @ X448 @ ( comp @ X446 @ X449 ) )
!= $true ) )
| ~ ! [X450: term,X451: term] :
( ( ( X439 @ id @ ( ap @ ( sub @ X450 @ id ) @ X451 ) @ id )
= $true )
| ( ( X439 @ id @ X450 @ id )
!= $true )
| ( $true
!= ( X439 @ id @ X451 @ id ) ) )
| ( ( X439 @ id @ ( ap @ X441 @ X440 ) @ id )
= $true ) )
=> ( ! [X452: term,X453: subst > term > subst > $o] :
( ~ ! [X454: subst > term > term] :
( ( ( X453 @ id @ ( lam @ ( X454 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X455: subst,X456: term,X457: subst] :
( ( X454 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) )
= ( sub @ ( X454 @ X457 @ X456 ) @ X455 ) )
| ~ ! [X458: term] :
( ( $true
!= ( X453 @ id @ X458 @ id ) )
| ( $true
= ( X453 @ id @ ( X454 @ id @ X458 ) @ id ) ) ) )
| ~ ! [X459: term] :
( ( ( X453 @ id @ X459 @ id )
!= $true )
| ( ( X453 @ id @ ( sub @ X452 @ ( push @ X459 @ id ) ) @ id )
= $true ) )
| ~ ! [X460: subst,X461: term,X462: subst,X463: subst] :
( ( $true
= ( X453 @ X462 @ X461 @ ( comp @ X463 @ X460 ) ) )
| ( ( X453 @ ( comp @ X462 @ X463 ) @ ( sub @ X461 @ X463 ) @ X460 )
!= $true ) )
| ~ ! [X464: subst,X465: subst,X466: term,X467: subst] :
( ( ( X453 @ ( comp @ X467 @ X465 ) @ ( sub @ X466 @ X465 ) @ X464 )
= $true )
| ( $true
!= ( X453 @ X467 @ X466 @ ( comp @ X465 @ X464 ) ) ) )
| ( ( X453 @ id @ ( lam @ X452 ) @ id )
= $true ) )
=> ! [X468: term,X469: subst > term > subst > $o] :
( ~ ! [X470: term,X471: term] :
( ( ( X469 @ id @ ( ap @ ( sub @ X470 @ id ) @ X471 ) @ id )
= $true )
| ( ( X469 @ id @ X470 @ id )
!= $true )
| ( ( X469 @ id @ X471 @ id )
!= $true ) )
| ~ ! [X472: subst > term > term] :
( ( $true
= ( X469 @ id @ ( lam @ ( X472 @ sh @ one ) ) @ id ) )
| ~ ! [X473: term] :
( ( ( X469 @ id @ ( X472 @ id @ X473 ) @ id )
= $true )
| ( ( X469 @ id @ X473 @ id )
!= $true ) )
| ~ ! [X474: subst,X475: term,X476: subst] :
( ( sub @ ( X472 @ X476 @ X475 ) @ X474 )
= ( X472 @ ( comp @ X476 @ X474 ) @ ( sub @ X475 @ X474 ) ) ) )
| ~ ! [X477: term,X478: subst,X479: subst,X480: subst] :
( ( ( X469 @ ( comp @ X479 @ X480 ) @ ( sub @ X477 @ X480 ) @ X478 )
!= $true )
| ( ( X469 @ X479 @ X477 @ ( comp @ X480 @ X478 ) )
= $true ) )
| ~ ! [X481: subst,X482: subst,X483: subst,X484: term] :
( ( $true
!= ( X469 @ X483 @ X484 @ ( comp @ X482 @ X481 ) ) )
| ( ( X469 @ ( comp @ X483 @ X482 ) @ ( sub @ X484 @ X482 ) @ X481 )
= $true ) )
| ( ( X469 @ id @ X468 @ id )
= $true )
| ~ ! [X485: term] :
( ( ( var @ ( sub @ X485 @ id ) )
!= $true )
| ( $true
= ( X469 @ id @ X485 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_lthm = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X486: term,X487: term,X488: term,X489: term] :
( ( X486 = X489 )
| ( ( ap @ X488 @ X489 )
!= ( ap @ X487 @ X486 ) ) )
=> ! [X490: term,X491: term,X492: term,X493: term] :
( ( X492 = X493 )
| ( ( ap @ ( sub @ X491 @ id ) @ X493 )
!= ( ap @ ( sub @ X490 @ id ) @ X492 ) ) ) ) )
& ( ! [X494: term,X495: term,X496: term,X497: term] :
( ( ( ap @ X497 @ X496 )
!= ( ap @ X495 @ X494 ) )
| ( X495 = X497 ) )
<=> ( apinj1 = $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( ( termmset_lthm = $true )
<=> ( ! [X498: term] :
( ( sub @ X498 @ id )
= X498 )
=> ! [X499: term] :
( ( sub @ X499 @ id )
= X499 ) ) )
& ( axapp = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ! [X500: term > $o,X501: term,X502: subst] :
~ ! [X503: term > $o] :
~ ! [X504: term] :
( ( X500 @ ( sub @ X504 @ ( push @ X501 @ X502 ) ) )
= ( X503 @ X504 ) )
<=> ( pushprop_lem0 = $true ) )
& ( ! [X505: term,X506: subst > term > term,X507: subst,X508: subst > term > term] :
( ~ ! [X509: subst,X510: term,X511: subst] :
( ( X506 @ ( comp @ X509 @ X511 ) @ ( sub @ X510 @ X511 ) )
= ( sub @ ( X506 @ X509 @ X510 ) @ X511 ) )
| ~ ! [X512: subst,X513: term,X514: subst] :
( ( sub @ ( X508 @ X512 @ X513 ) @ X514 )
= ( X508 @ ( comp @ X512 @ X514 ) @ ( sub @ X513 @ X514 ) ) )
| ( ( X508 @ X507 @ X505 )
= ( X506 @ X507 @ X505 ) )
| ( ( lam @ ( X506 @ sh @ one ) )
!= ( lam @ ( X508 @ sh @ one ) ) ) )
<=> ( hoaslaminj = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X515: subst > term > subst > $o,X516: term] :
( ~ ! [X517: term] :
( ( ( X515 @ id @ ( sub @ X516 @ ( push @ X517 @ id ) ) @ id )
= $true )
| ( $true
!= ( X515 @ id @ X517 @ id ) ) )
| ( ( X515 @ id @ ( lam @ ( sub @ X516 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X518: subst > term > term] :
( ~ ! [X519: term] :
( ( ( X515 @ id @ ( X518 @ id @ X519 ) @ id )
= $true )
| ( ( X515 @ id @ X519 @ id )
!= $true ) )
| ( ( X515 @ id @ ( lam @ ( X518 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X520: subst,X521: subst,X522: term] :
( ( X518 @ ( comp @ X521 @ X520 ) @ ( sub @ X522 @ X520 ) )
= ( sub @ ( X518 @ X521 @ X522 ) @ X520 ) ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ! [X523: term,X524: term > $o,X525: subst] :
~ ! [X526: term > $o] :
~ ! [X527: term] :
( ( X524 @ ( sub @ X527 @ ( push @ X523 @ X525 ) ) )
= ( X526 @ X527 ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( ! [X528: subst > term > subst > $o,X529: term,X530: term] :
( ~ ! [X531: subst,X532: subst,X533: term,X534: subst] :
( ( $true
!= ( X528 @ ( comp @ X532 @ X534 ) @ ( sub @ X533 @ X534 ) @ X531 ) )
| ( ( X528 @ X532 @ X533 @ ( comp @ X534 @ X531 ) )
= $true ) )
| ( ( X528 @ id @ X530 @ id )
!= $true )
| ( ( X528 @ id @ ( ap @ X529 @ X530 ) @ id )
= $true )
| ~ ! [X535: term,X536: term] :
( ( ( X528 @ id @ X535 @ id )
!= $true )
| ( ( X528 @ id @ X536 @ id )
!= $true )
| ( $true
= ( X528 @ id @ ( ap @ ( sub @ X535 @ id ) @ X536 ) @ id ) ) )
| ~ ! [X537: subst,X538: subst,X539: term,X540: subst] :
( ( $true
= ( X528 @ ( comp @ X538 @ X540 ) @ ( sub @ X539 @ X540 ) @ X537 ) )
| ( ( X528 @ X538 @ X539 @ ( comp @ X540 @ X537 ) )
!= $true ) )
| ( ( X528 @ id @ X529 @ id )
!= $true ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( ( axidl = $true )
<=> ! [X541: subst] :
( ( comp @ id @ X541 )
= X541 ) )
& ( ! [X542: term,X543: subst > term > subst > $o] :
( ( $true
= ( X543 @ id @ ( lam @ X542 ) @ id ) )
| ~ ! [X544: subst > term > term] :
( ( $true
= ( X543 @ id @ ( lam @ ( X544 @ sh @ one ) ) @ id ) )
| ~ ! [X545: subst,X546: subst,X547: term] :
( ( X544 @ ( comp @ X545 @ X546 ) @ ( sub @ X547 @ X546 ) )
= ( sub @ ( X544 @ X545 @ X547 ) @ X546 ) )
| ~ ! [X548: term] :
( ( $true
= ( X543 @ id @ ( X544 @ id @ X548 ) @ id ) )
| ( $true
!= ( X543 @ id @ X548 @ id ) ) ) )
| ~ ! [X549: term] :
( ( ( X543 @ id @ ( sub @ X542 @ ( push @ X549 @ id ) ) @ id )
= $true )
| ( ( X543 @ id @ X549 @ id )
!= $true ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( hoasapinj1_gthm = $true ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
( ( pushprop_lem1v2_lthm
<=> ( ! [X0: term,X1: subst] :
( ( sub @ one @ ( push @ X0 @ X1 ) )
= X0 )
=> ! [X2: term > $o,X3: term > $o,X4: subst,X5: term] :
( ~ ( X3 @ X5 )
| ~ ! [X6: term] :
( ( X3 @ ( sub @ X6 @ ( push @ X5 @ X4 ) ) )
= ( X2 @ X6 ) )
| ( X2 @ one ) ) ) )
& ( ! [X7: subst > term > subst > $o] :
~ ! [X8: term > $o] :
~ ! [X9: term] :
( ( X8 @ X9 )
= ( X7 @ id @ X9 @ id ) )
<=> hoasinduction_lem0 )
& ( ( ! [X10: subst,X11: term] :
( ( sub @ one @ ( push @ X11 @ X10 ) )
= X11 )
=> ( ! [X12: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X12 ) )
= X12 )
=> ! [X14: term > $o,X15: term,X16: subst,X17: term,X18: term > $o] :
( ( X14 @ ( sub @ X15 @ ( push @ X17 @ X16 ) ) )
| ~ ( X18 @ X15 ) ) ) )
<=> pushprop_lem1_lthm )
& hoasinduction_lem3_gthm
& ( ! [X19: term,X20: term,X21: term,X22: term] :
( ( X20 = X22 )
| ( ( ap @ ( sub @ X19 @ id ) @ X20 )
!= ( ap @ ( sub @ X21 @ id ) @ X22 ) ) )
<=> hoasapinj2 )
& axclos
& ! [X23: d_term,X24: d_term] :
( ( ( d2term @ X23 )
= ( d2term @ X24 ) )
=> ( X23 = X24 ) )
& ( ! [X25: subst > term > subst > $o,X26: term] :
( ~ ! [X27: term] :
( ~ ( X25 @ id @ X27 @ id )
| ( X25 @ id @ ( sub @ X26 @ ( push @ X27 @ id ) ) @ id ) )
| ( X25 @ id @ ( lam @ ( sub @ X26 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X28: term,X29: subst > term > term] :
( ( X25 @ id @ ( lam @ ( X29 @ sh @ one ) ) @ id )
| ~ ! [X30: term] :
( ( X25 @ id @ ( X29 @ id @ X30 ) @ id )
| ~ ( X25 @ id @ X30 @ id ) )
| ~ ! [X31: subst,X32: term,X33: subst] :
( ( sub @ ( sub @ X28 @ ( push @ X32 @ X33 ) ) @ X31 )
= ( sub @ ( X29 @ X33 @ X32 ) @ X31 ) )
| ~ ! [X34: subst,X35: subst,X36: term] :
( ( sub @ X28 @ ( push @ ( sub @ X36 @ X34 ) @ ( comp @ X35 @ X34 ) ) )
= ( X29 @ ( comp @ X35 @ X34 ) @ ( sub @ X36 @ X34 ) ) ) ) )
<=> hoasinduction_lem3aaa )
& pushprop_gthm
& ( hoasinduction
<=> ! [X37: subst > term > subst > $o,X38: term] :
( ~ ! [X39: term,X40: term] :
( ~ ( X37 @ id @ X40 @ id )
| ~ ( X37 @ id @ X39 @ id )
| ( X37 @ id @ ( ap @ ( sub @ X40 @ id ) @ X39 ) @ id ) )
| ~ ! [X41: subst > term > term] :
( ~ ! [X42: term,X43: subst,X44: subst] :
( ( X41 @ ( comp @ X43 @ X44 ) @ ( sub @ X42 @ X44 ) )
= ( sub @ ( X41 @ X43 @ X42 ) @ X44 ) )
| ~ ! [X45: term] :
( ( X37 @ id @ ( X41 @ id @ X45 ) @ id )
| ~ ( X37 @ id @ X45 @ id ) )
| ( X37 @ id @ ( lam @ ( X41 @ sh @ one ) ) @ id ) )
| ~ ! [X46: subst,X47: subst,X48: subst,X49: term] :
( ( X37 @ ( comp @ X46 @ X47 ) @ ( sub @ X49 @ X47 ) @ X48 )
| ~ ( X37 @ X46 @ X49 @ ( comp @ X47 @ X48 ) ) )
| ~ ! [X50: subst,X51: subst,X52: term,X53: subst] :
( ~ ( X37 @ ( comp @ X51 @ X50 ) @ ( sub @ X52 @ X50 ) @ X53 )
| ( X37 @ X51 @ X52 @ ( comp @ X50 @ X53 ) ) )
| ~ ! [X54: term] :
( ~ ( var @ ( sub @ X54 @ id ) )
| ( X37 @ id @ X54 @ id ) )
| ( X37 @ id @ X38 @ id ) ) )
& ulamvarsh
& hoasapinj2_gthm
& hoasinduction_lem3v2_gthm
& ! [X55: d_subst,X56: d_subst] :
( ( ( d2subst @ X56 )
= ( d2subst @ X55 ) )
=> ( X55 = X56 ) )
& pushprop_lem2v2_lthm
& ( termmset
<=> ! [X57: term] :
( ( sub @ X57 @ id )
= X57 ) )
& axvarshift
& ( substmonoid
<=> ( ! [X58: subst] :
( ( comp @ X58 @ id )
= X58 )
& ! [X59: subst] :
( ( comp @ id @ X59 )
= X59 ) ) )
& ( ( ^ [X60: subst,X61: subst > term > term] : ( d2term @ d_one ) )
= hoaslam )
& pushprop_lthm
& pushprop_lem1v2_gthm
& ( ( ! [X62: term] :
( ( sub @ X62 @ id )
= X62 )
=> ( ! [X63: term,X64: subst > term > subst > $o] :
( ~ ! [X65: subst > term > term] :
( ( X64 @ id @ ( lam @ ( X65 @ sh @ one ) ) @ id )
| ~ ! [X66: subst,X67: subst,X68: term] :
( ( X65 @ ( comp @ X67 @ X66 ) @ ( sub @ X68 @ X66 ) )
= ( sub @ ( X65 @ X67 @ X68 ) @ X66 ) )
| ~ ! [X69: term] :
( ~ ( X64 @ id @ X69 @ id )
| ( X64 @ id @ ( X65 @ id @ X69 ) @ id ) ) )
| ~ ! [X70: term] :
( ~ ( X64 @ id @ X70 @ id )
| ( X64 @ id @ ( sub @ X63 @ ( push @ X70 @ id ) ) @ id ) )
| ( X64 @ id @ ( lam @ ( sub @ X63 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X71: term,X72: subst > term > subst > $o] :
( ~ ! [X73: term] :
( ~ ( X72 @ id @ X73 @ id )
| ( X72 @ id @ ( sub @ X71 @ ( push @ X73 @ id ) ) @ id ) )
| ~ ! [X74: subst > term > term] :
( ~ ! [X75: subst,X76: term,X77: subst] :
( ( sub @ ( X74 @ X77 @ X76 ) @ X75 )
= ( X74 @ ( comp @ X77 @ X75 ) @ ( sub @ X76 @ X75 ) ) )
| ( X72 @ id @ ( lam @ ( X74 @ sh @ one ) ) @ id )
| ~ ! [X78: term] :
( ~ ( X72 @ id @ X78 @ id )
| ( X72 @ id @ ( X74 @ id @ X78 ) @ id ) ) )
| ( X72 @ id @ ( lam @ X71 ) @ id ) ) ) )
<=> hoasinduction_lem3a_lthm )
& ( hoaslaminj_lthm
<=> ( ! [X79: subst,X80: term] :
( ( sub @ one @ ( push @ X80 @ X79 ) )
= X80 )
=> ( ! [X81: subst,X82: term] :
( ( comp @ sh @ ( push @ X82 @ X81 ) )
= X81 )
=> ( ! [X83: term,X84: term] :
( ( ( lam @ X83 )
!= ( lam @ X84 ) )
| ( X83 = X84 ) )
=> ! [X85: term,X86: subst > term > term,X87: subst,X88: subst > term > term] :
( ( ( lam @ ( X86 @ sh @ one ) )
!= ( lam @ ( X88 @ sh @ one ) ) )
| ( ( X88 @ X87 @ X85 )
= ( X86 @ X87 @ X85 ) )
| ~ ! [X89: subst,X90: subst,X91: term] :
( ( sub @ ( X88 @ X89 @ X91 ) @ X90 )
= ( X88 @ ( comp @ X89 @ X90 ) @ ( sub @ X91 @ X90 ) ) )
| ~ ! [X92: subst,X93: subst,X94: term] :
( ( X86 @ ( comp @ X92 @ X93 ) @ ( sub @ X94 @ X93 ) )
= ( sub @ ( X86 @ X92 @ X94 ) @ X93 ) ) ) ) ) ) )
& ( ! [X95: subst > term > subst > $o] :
~ ! [X96: term > $o] :
~ ! [X97: term] :
( ( X95 @ id @ X97 @ id )
= ( X96 @ X97 ) )
<=> hoasinduction_lem0_lthm )
& apnotvar
& pushprop
& ( ( d2subst @ d_id )
= sh )
& ! [X98: d_term] : ( d_one = X98 )
& ( ! [X99: term,X100: term,X101: term,X102: term] :
( ( X99 = X100 )
| ( ( ap @ X102 @ X100 )
!= ( ap @ X101 @ X99 ) ) )
<=> apinj2 )
& ulamvarind
& ( ( ^ [X103: term,X104: subst,X105: term > $o,X106: term > $o] :
! [X107: term] :
( ( X106 @ X107 )
= ( X105 @ ( sub @ X107 @ ( push @ X103 @ X104 ) ) ) ) )
= pushprop_p_and_p_prime )
& ( hoasinduction_lem3v2_f_lthm
<=> ! [X108: term] :
~ ! [X109: subst > term > term] :
~ ! [X110: term,X111: subst] :
( ( sub @ X108 @ ( push @ X110 @ X111 ) )
= ( X109 @ X111 @ X110 ) ) )
& ( hoasinduction_no_psi_cond
<=> ! [X112: term,X113: subst > term > subst > $o] :
( ~ ! [X114: subst > term > term] :
( ~ ! [X115: term] :
( ( X113 @ id @ ( X114 @ id @ X115 ) @ id )
| ~ ( X113 @ id @ X115 @ id ) )
| ~ ! [X116: subst,X117: term,X118: subst] :
( ( sub @ ( X114 @ X118 @ X117 ) @ X116 )
= ( X114 @ ( comp @ X118 @ X116 ) @ ( sub @ X117 @ X116 ) ) )
| ( X113 @ id @ ( lam @ ( X114 @ sh @ one ) ) @ id ) )
| ~ ! [X119: term,X120: term] :
( ~ ( X113 @ id @ X120 @ id )
| ~ ( X113 @ id @ X119 @ id )
| ( X113 @ id @ ( ap @ ( sub @ X120 @ id ) @ X119 ) @ id ) )
| ( X113 @ id @ X112 @ id ) ) )
& ( hoasinduction_lem3_lthm
<=> ( ! [X121: term] :
( ( sub @ X121 @ id )
= X121 )
=> ( ! [X122: term,X123: subst > term > subst > $o] :
( ~ ! [X124: subst > term > term] :
( ~ ! [X125: term] :
( ( X123 @ id @ ( X124 @ id @ X125 ) @ id )
| ~ ( X123 @ id @ X125 @ id ) )
| ~ ! [X126: term,X127: subst,X128: subst] :
( ( X124 @ ( comp @ X128 @ X127 ) @ ( sub @ X126 @ X127 ) )
= ( sub @ ( X124 @ X128 @ X126 ) @ X127 ) )
| ( X123 @ id @ ( lam @ ( X124 @ sh @ one ) ) @ id ) )
| ( X123 @ id @ ( lam @ ( sub @ X122 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X129: term] :
( ~ ( X123 @ id @ X129 @ id )
| ( X123 @ id @ ( sub @ X122 @ ( push @ X129 @ id ) ) @ id ) ) )
=> ! [X130: term,X131: subst > term > subst > $o] :
( ( X131 @ id @ ( lam @ X130 ) @ id )
| ~ ! [X132: subst > term > term] :
( ( X131 @ id @ ( lam @ ( X132 @ sh @ one ) ) @ id )
| ~ ! [X133: term] :
( ~ ( X131 @ id @ X133 @ id )
| ( X131 @ id @ ( X132 @ id @ X133 ) @ id ) )
| ~ ! [X134: term,X135: subst,X136: subst] :
( ( sub @ ( X132 @ X136 @ X134 ) @ X135 )
= ( X132 @ ( comp @ X136 @ X135 ) @ ( sub @ X134 @ X135 ) ) ) )
| ~ ! [X137: term,X138: subst,X139: subst,X140: subst] :
( ( X131 @ X139 @ X137 @ ( comp @ X138 @ X140 ) )
| ~ ( X131 @ ( comp @ X139 @ X138 ) @ ( sub @ X137 @ X138 ) @ X140 ) )
| ~ ! [X141: term,X142: subst,X143: subst,X144: subst] :
( ( X131 @ ( comp @ X144 @ X143 ) @ ( sub @ X141 @ X143 ) @ X142 )
| ~ ( X131 @ X144 @ X141 @ ( comp @ X143 @ X142 ) ) )
| ~ ! [X145: term] :
( ( X131 @ id @ ( sub @ X130 @ ( push @ X145 @ id ) ) @ id )
| ~ ( X131 @ id @ X145 @ id ) ) ) ) ) )
& ( induction2lem
<=> ! [X146: term,X147: term > $o,X148: subst] :
( ~ ! [X149: term] :
( ( X147 @ ( sub @ X149 @ X148 ) )
| ~ ( var @ X149 ) )
| ~ ! [X150: term] :
( ~ ! [X151: term] :
( ( X147 @ ( sub @ X150 @ ( push @ X151 @ id ) ) )
| ~ ( X147 @ X151 ) )
| ( X147 @ ( lam @ X150 ) ) )
| ( X147 @ ( sub @ X146 @ X148 ) )
| ~ ! [X152: term,X153: term] :
( ~ ( X147 @ X152 )
| ~ ( X147 @ X153 )
| ( X147 @ ( ap @ X152 @ X153 ) ) ) ) )
& pushprop_lem3v2
& hoasinduction_lem1v2
& ( hoasinduction_lem3v2_f
<=> ! [X154: term] :
~ ! [X155: subst > term > term] :
~ ! [X156: term,X157: subst] :
( ( sub @ X154 @ ( push @ X156 @ X157 ) )
= ( X155 @ X157 @ X156 ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ( ( ! [X158: term] :
( ( sub @ X158 @ id )
= X158 )
=> ( ! [X159: term,X160: subst,X161: term > $o] :
( ~ ! [X162: term] :
( ( X161 @ ( lam @ X162 ) )
| ~ ! [X163: term] :
( ( X161 @ ( sub @ X162 @ ( push @ X163 @ id ) ) )
| ~ ( X161 @ X163 ) ) )
| ~ ! [X164: term,X165: term] :
( ~ ( X161 @ X165 )
| ( X161 @ ( ap @ X165 @ X164 ) )
| ~ ( X161 @ X164 ) )
| ( X161 @ ( sub @ X159 @ X160 ) )
| ~ ! [X166: term] :
( ~ ( var @ X166 )
| ( X161 @ ( sub @ X166 @ X160 ) ) ) )
=> ! [X167: term,X168: term > $o] :
( ~ ! [X169: term] :
( ( X168 @ X169 )
| ~ ( var @ X169 ) )
| ~ ! [X170: term,X171: term] :
( ( X168 @ ( ap @ X171 @ X170 ) )
| ~ ( X168 @ X170 )
| ~ ( X168 @ X171 ) )
| ~ ! [X172: term] :
( ~ ! [X173: term] :
( ( X168 @ ( sub @ X172 @ ( push @ X173 @ id ) ) )
| ~ ( X168 @ X173 ) )
| ( X168 @ ( lam @ X172 ) ) )
| ( X168 @ X167 ) ) ) )
<=> induction2_lthm )
& ( ! [X174: term,X175: term] :
( ( X174 = X175 )
| ( ( lam @ X175 )
!= ( lam @ X174 ) ) )
<=> laminj )
& hoaslamnotvar_lthm
& pushprop_lem3v2_lthm
& ~ ulamvar1
& ( ( ! [X176: term,X177: subst] :
( ( sub @ one @ ( push @ X176 @ X177 ) )
= X176 )
=> ( ! [X178: subst,X179: term] :
( ( comp @ sh @ ( push @ X179 @ X178 ) )
= X178 )
=> ( ! [X180: subst] :
( ( comp @ X180 @ id )
= X180 )
=> ( ! [X181: term,X182: term > $o] :
( ~ ! [X183: term,X184: term] :
( ~ ( X182 @ X183 )
| ~ ( X182 @ X184 )
| ( X182 @ ( ap @ X184 @ X183 ) ) )
| ( X182 @ X181 )
| ~ ! [X185: term] :
( ( X182 @ ( lam @ X185 ) )
| ~ ( X182 @ X185 ) )
| ~ ! [X186: term] :
( ( X182 @ X186 )
| ~ ( var @ X186 ) ) )
=> ! [X187: term,X188: term > $o,X189: subst] :
( ( X188 @ ( sub @ X187 @ X189 ) )
| ~ ! [X190: term,X191: term] :
( ( X188 @ ( ap @ X191 @ X190 ) )
| ~ ( X188 @ X191 )
| ~ ( X188 @ X190 ) )
| ~ ! [X192: term] :
( ( X188 @ ( sub @ X192 @ X189 ) )
| ~ ( var @ X192 ) )
| ~ ! [X193: term] :
( ( X188 @ ( lam @ X193 ) )
| ~ ! [X194: term] :
( ~ ( X188 @ X194 )
| ( X188 @ ( sub @ X193 @ ( push @ X194 @ id ) ) ) ) ) ) ) ) ) )
<=> induction2lem_lthm )
& ( ! [X195: subst > term > subst > $o,X196: term,X197: term > $o] :
( ~ ! [X198: subst,X199: subst,X200: subst,X201: term] :
( ~ ( X195 @ X200 @ X201 @ ( comp @ X198 @ X199 ) )
| ( X195 @ ( comp @ X200 @ X198 ) @ ( sub @ X201 @ X198 ) @ X199 ) )
| ~ ! [X202: subst,X203: subst,X204: subst,X205: term] :
( ~ ( X195 @ ( comp @ X204 @ X203 ) @ ( sub @ X205 @ X203 ) @ X202 )
| ( X195 @ X204 @ X205 @ ( comp @ X203 @ X202 ) ) )
| ~ ! [X206: term] :
( ( X197 @ ( sub @ X196 @ ( push @ X206 @ id ) ) )
| ~ ( X197 @ X206 ) )
| ( X197 @ ( lam @ X196 ) )
| ~ ! [X207: subst > term > term] :
( ( X195 @ id @ ( lam @ ( X207 @ sh @ one ) ) @ id )
| ~ ! [X208: subst,X209: subst,X210: term] :
( ( X207 @ ( comp @ X209 @ X208 ) @ ( sub @ X210 @ X208 ) )
= ( sub @ ( X207 @ X209 @ X210 ) @ X208 ) )
| ~ ! [X211: term] :
( ~ ( X195 @ id @ X211 @ id )
| ( X195 @ id @ ( X207 @ id @ X211 ) @ id ) ) )
| ~ ! [X212: term] :
( ( X195 @ id @ X212 @ id )
= ( X197 @ X212 ) ) )
<=> hoasinduction_lem3v2 )
& ( ! [X213: term > $o,X214: term] :
( ( X213 @ X214 )
| ~ ! [X215: term] :
( ( X213 @ ( lam @ X215 ) )
| ~ ( X213 @ X215 ) )
| ~ ! [X216: term,X217: term] :
( ~ ( X213 @ X216 )
| ( X213 @ ( ap @ X216 @ X217 ) )
| ~ ( X213 @ X217 ) )
| ~ ! [X218: term] :
( ~ ( var @ X218 )
| ( X213 @ X218 ) ) )
<=> induction )
& hoasinduction_lem1_gthm
& induction2lem_gthm
& ( ! [X219: term] :
~ ! [X220: subst > term > term] :
( ( X220 @ sh @ one )
!= ( sub @ X219 @ ( push @ one @ sh ) ) )
<=> hoasinduction_lem3b )
& ( ! [X221: subst] :
( ( comp @ X221 @ id )
= X221 )
<=> axidr )
& termmset_gthm
& ( ! [X222: term,X223: term > $o,X224: term > $o,X225: subst] :
( ( X223 @ one )
| ~ ( X224 @ X222 )
| ~ ! [X226: term] :
( ( X223 @ X226 )
= ( X224 @ ( sub @ X226 @ ( push @ X222 @ X225 ) ) ) ) )
<=> pushprop_lem1v2 )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lem2v2
<=> ! [X227: term > $o,X228: term,X229: subst > term > subst > $o,X230: term] :
( ~ ( X227 @ X228 )
| ~ ! [X231: subst,X232: term,X233: subst,X234: subst] :
( ( X229 @ ( comp @ X231 @ X234 ) @ ( sub @ X232 @ X234 ) @ X233 )
| ~ ( X229 @ X231 @ X232 @ ( comp @ X234 @ X233 ) ) )
| ~ ( X227 @ X230 )
| ~ ! [X235: subst,X236: subst,X237: subst,X238: term] :
( ( X229 @ X237 @ X238 @ ( comp @ X235 @ X236 ) )
| ~ ( X229 @ ( comp @ X237 @ X235 ) @ ( sub @ X238 @ X235 ) @ X236 ) )
| ( X227 @ ( ap @ X228 @ X230 ) )
| ~ ! [X239: term] :
( ( X229 @ id @ X239 @ id )
= ( X227 @ X239 ) )
| ~ ! [X240: term,X241: term] :
( ( X229 @ id @ ( ap @ ( sub @ X241 @ id ) @ X240 ) @ id )
| ~ ( X229 @ id @ X240 @ id )
| ~ ( X229 @ id @ X241 @ id ) ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& hoaslamnotap_lthm
& ( hoasinduction_lem3
<=> ! [X242: subst > term > subst > $o,X243: term] :
( ~ ! [X244: subst,X245: subst,X246: subst,X247: term] :
( ( X242 @ X245 @ X247 @ ( comp @ X246 @ X244 ) )
| ~ ( X242 @ ( comp @ X245 @ X246 ) @ ( sub @ X247 @ X246 ) @ X244 ) )
| ( X242 @ id @ ( lam @ X243 ) @ id )
| ~ ! [X248: subst > term > term] :
( ~ ! [X249: term,X250: subst,X251: subst] :
( ( X248 @ ( comp @ X251 @ X250 ) @ ( sub @ X249 @ X250 ) )
= ( sub @ ( X248 @ X251 @ X249 ) @ X250 ) )
| ( X242 @ id @ ( lam @ ( X248 @ sh @ one ) ) @ id )
| ~ ! [X252: term] :
( ( X242 @ id @ ( X248 @ id @ X252 ) @ id )
| ~ ( X242 @ id @ X252 @ id ) ) )
| ~ ! [X253: term] :
( ~ ( X242 @ id @ X253 @ id )
| ( X242 @ id @ ( sub @ X243 @ ( push @ X253 @ id ) ) @ id ) )
| ~ ! [X254: subst,X255: subst,X256: subst,X257: term] :
( ( X242 @ ( comp @ X255 @ X256 ) @ ( sub @ X257 @ X256 ) @ X254 )
| ~ ( X242 @ X255 @ X257 @ ( comp @ X256 @ X254 ) ) ) ) )
& hoasapnotvar_gthm
& ~ lamnotap
& ( ! [X258: term,X259: term > $o] :
( ( X259 @ X258 )
| ~ ! [X260: term,X261: term] :
( ~ ( X259 @ X261 )
| ~ ( X259 @ X260 )
| ( X259 @ ( ap @ X261 @ X260 ) ) )
| ~ ! [X262: term] :
( ~ ! [X263: term] :
( ~ ( X259 @ X263 )
| ( X259 @ ( sub @ X262 @ ( push @ X263 @ id ) ) ) )
| ( X259 @ ( lam @ X262 ) ) )
| ~ ! [X264: term] :
( ( X259 @ X264 )
| ~ ( var @ X264 ) ) )
<=> induction2 )
& ( ! [X265: term,X266: subst > term > term,X267: term] :
( ~ ! [X268: term,X269: subst,X270: subst] :
( ( sub @ ( X266 @ X270 @ X268 ) @ X269 )
= ( X266 @ ( comp @ X270 @ X269 ) @ ( sub @ X268 @ X269 ) ) )
| ( ( ap @ ( sub @ X267 @ id ) @ X265 )
!= ( lam @ ( X266 @ sh @ one ) ) ) )
<=> hoaslamnotap )
& axmap
& ( one
= ( d2term @ d_one ) )
& ( ! [X271: term] :
( ( sub @ X271 @ id )
= X271 )
<=> axvarid )
& ( pushprop_lem1
<=> ! [X272: term,X273: term > $o,X274: term > $o,X275: term,X276: subst] :
( ~ ( X273 @ X272 )
| ( X274 @ ( sub @ X272 @ ( push @ X275 @ X276 ) ) ) ) )
& hoaslamnotvar
& pushprop_lem2v2_gthm
& hoaslaminj_gthm
& lamnotvar
& ! [X277: d_subst] : ( d_id = X277 )
& ! [X278: term] :
? [X279: d_term] :
( ( d2term @ X279 )
= X278 )
& ( axshiftcons
<=> ! [X280: subst,X281: term] :
( ( comp @ sh @ ( push @ X281 @ X280 ) )
= X280 ) )
& ( shinj
<=> ! [X282: term,X283: term] :
( ( X282 = X283 )
| ( ( sub @ X282 @ sh )
!= ( sub @ X283 @ sh ) ) ) )
& pushprop_lthm_orig
& ( ( ! [X284: term] :
~ ! [X285: subst > term > term] :
~ ! [X286: term,X287: subst] :
( ( sub @ X284 @ ( push @ X286 @ X287 ) )
= ( X285 @ X287 @ X286 ) )
=> ( ! [X288: term] :
( ( sub @ X288 @ id )
= X288 )
=> ! [X289: term > $o,X290: subst > term > subst > $o,X291: term] :
( ~ ! [X292: subst > term > term] :
( ( X290 @ id @ ( lam @ ( X292 @ sh @ one ) ) @ id )
| ~ ! [X293: term] :
( ~ ( X290 @ id @ X293 @ id )
| ( X290 @ id @ ( X292 @ id @ X293 ) @ id ) )
| ~ ! [X294: term,X295: subst,X296: subst] :
( ( X292 @ ( comp @ X296 @ X295 ) @ ( sub @ X294 @ X295 ) )
= ( sub @ ( X292 @ X296 @ X294 ) @ X295 ) ) )
| ( X289 @ ( lam @ X291 ) )
| ~ ! [X297: term] :
( ( X289 @ X297 )
= ( X290 @ id @ X297 @ id ) )
| ~ ! [X298: term] :
( ~ ( X289 @ X298 )
| ( X289 @ ( sub @ X291 @ ( push @ X298 @ id ) ) ) ) ) ) )
<=> hoasinduction_lem3v2a_lthm )
& hoasinduction_lem2_gthm
& pushprop_lem1_gthm
& hoasinduction_lem2v2_gthm
& hoasinduction_lem3b_gthm
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X299: subst > term > subst > $o] :
~ ! [X300: term > $o] :
~ ! [X301: term] :
( ( X299 @ id @ X301 @ id )
= ( X300 @ X301 ) )
=> ( ! [X302: term > $o,X303: term] :
( ~ ! [X304: term,X305: term] :
( ~ ( X302 @ X304 )
| ( X302 @ ( ap @ X304 @ X305 ) )
| ~ ( X302 @ X305 ) )
| ~ ! [X306: term] :
( ( X302 @ X306 )
| ~ ( var @ X306 ) )
| ~ ! [X307: term] :
( ( X302 @ ( lam @ X307 ) )
| ~ ! [X308: term] :
( ~ ( X302 @ X308 )
| ( X302 @ ( sub @ X307 @ ( push @ X308 @ id ) ) ) ) )
| ( X302 @ X303 ) )
=> ( ! [X309: term] :
( ( sub @ X309 @ id )
= X309 )
=> ( ! [X310: subst > term > subst > $o,X311: term,X312: term > $o] :
( ~ ! [X313: subst > term > term] :
( ( X310 @ id @ ( lam @ ( X313 @ sh @ one ) ) @ id )
| ~ ! [X314: term] :
( ( X310 @ id @ ( X313 @ id @ X314 ) @ id )
| ~ ( X310 @ id @ X314 @ id ) )
| ~ ! [X315: term,X316: subst,X317: subst] :
( ( sub @ ( X313 @ X317 @ X315 ) @ X316 )
= ( X313 @ ( comp @ X317 @ X316 ) @ ( sub @ X315 @ X316 ) ) ) )
| ( X312 @ ( lam @ X311 ) )
| ~ ! [X318: term] :
( ~ ( X312 @ X318 )
| ( X312 @ ( sub @ X311 @ ( push @ X318 @ id ) ) ) )
| ~ ! [X319: term] :
( ( X310 @ id @ X319 @ id )
= ( X312 @ X319 ) ) )
=> ! [X320: term,X321: subst > term > subst > $o] :
( ~ ! [X322: subst,X323: subst,X324: term,X325: subst] :
( ( X321 @ X325 @ X324 @ ( comp @ X322 @ X323 ) )
| ~ ( X321 @ ( comp @ X325 @ X322 ) @ ( sub @ X324 @ X322 ) @ X323 ) )
| ~ ! [X326: term] :
( ~ ( var @ ( sub @ X326 @ id ) )
| ( X321 @ id @ X326 @ id ) )
| ~ ! [X327: term,X328: term] :
( ~ ( X321 @ id @ X327 @ id )
| ( X321 @ id @ ( ap @ ( sub @ X327 @ id ) @ X328 ) @ id )
| ~ ( X321 @ id @ X328 @ id ) )
| ~ ! [X329: subst,X330: subst,X331: term,X332: subst] :
( ( X321 @ ( comp @ X329 @ X330 ) @ ( sub @ X331 @ X330 ) @ X332 )
| ~ ( X321 @ X329 @ X331 @ ( comp @ X330 @ X332 ) ) )
| ~ ! [X333: subst > term > term] :
( ~ ! [X334: subst,X335: term,X336: subst] :
( ( sub @ ( X333 @ X334 @ X335 ) @ X336 )
= ( X333 @ ( comp @ X334 @ X336 ) @ ( sub @ X335 @ X336 ) ) )
| ( X321 @ id @ ( lam @ ( X333 @ sh @ one ) ) @ id )
| ~ ! [X337: term] :
( ~ ( X321 @ id @ X337 @ id )
| ( X321 @ id @ ( X333 @ id @ X337 ) @ id ) ) )
| ( X321 @ id @ X320 @ id ) ) ) ) ) )
<=> hoasinduction_lthm_3 )
& ( ( ! [X338: subst] :
( ( comp @ id @ X338 )
= X338 )
=> ( ! [X339: subst] :
( ( comp @ X339 @ id )
= X339 )
=> ( ! [X340: subst] :
( ( comp @ X340 @ id )
= X340 )
& ! [X341: subst] :
( ( comp @ id @ X341 )
= X341 ) ) ) )
<=> substmonoid_lthm )
& hoasinduction_lem1_lthm
& axassoc
& hoasinduction_lem1v2_gthm
& hoasinduction_gthm
& induction2_gthm
& ( axscons
<=> ! [X342: subst] :
( ( push @ ( sub @ one @ X342 ) @ ( comp @ sh @ X342 ) )
= X342 ) )
& ( ! [X343: subst,X344: term] :
( ( sub @ one @ ( push @ X344 @ X343 ) )
= X344 )
<=> axvarcons )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem3b_lthm
<=> ! [X345: term] :
~ ! [X346: subst > term > term] :
( ( X346 @ sh @ one )
!= ( sub @ X345 @ ( push @ one @ sh ) ) ) )
& ( ( ! [X347: term] :
( ( sub @ X347 @ id )
= X347 )
=> ( ! [X348: term,X349: term,X350: term,X351: term] :
( ( X348 = X349 )
| ( ( ap @ X349 @ X350 )
!= ( ap @ X348 @ X351 ) ) )
=> ! [X352: term,X353: term,X354: term,X355: term] :
( ( X352 = X355 )
| ( ( ap @ ( sub @ X352 @ id ) @ X354 )
!= ( ap @ ( sub @ X355 @ id ) @ X353 ) ) ) ) )
<=> hoasapinj1_lthm )
& ( hoasinduction_lem3v2_lthm
<=> ( ! [X356: term] :
( ( sub @ X356 @ id )
= X356 )
=> ! [X357: term,X358: subst > term > subst > $o,X359: term > $o] :
( ~ ! [X360: term] :
( ( X358 @ id @ X360 @ id )
= ( X359 @ X360 ) )
| ( X359 @ ( lam @ X357 ) )
| ~ ! [X361: subst,X362: subst,X363: subst,X364: term] :
( ~ ( X358 @ X363 @ X364 @ ( comp @ X361 @ X362 ) )
| ( X358 @ ( comp @ X363 @ X361 ) @ ( sub @ X364 @ X361 ) @ X362 ) )
| ~ ! [X365: subst,X366: subst,X367: term,X368: subst] :
( ( X358 @ X365 @ X367 @ ( comp @ X366 @ X368 ) )
| ~ ( X358 @ ( comp @ X365 @ X366 ) @ ( sub @ X367 @ X366 ) @ X368 ) )
| ~ ! [X369: subst > term > term] :
( ~ ! [X370: term,X371: subst,X372: subst] :
( ( sub @ ( X369 @ X372 @ X370 ) @ X371 )
= ( X369 @ ( comp @ X372 @ X371 ) @ ( sub @ X370 @ X371 ) ) )
| ( X358 @ id @ ( lam @ ( X369 @ sh @ one ) ) @ id )
| ~ ! [X373: term] :
( ( X358 @ id @ ( X369 @ id @ X373 ) @ id )
| ~ ( X358 @ id @ X373 @ id ) ) )
| ~ ! [X374: term] :
( ~ ( X359 @ X374 )
| ( X359 @ ( sub @ X357 @ ( push @ X374 @ id ) ) ) ) ) ) )
& hoaslamnotap_gthm
& ( ! [X375: term,X376: term > $o,X377: subst > term > subst > $o] :
( ~ ! [X378: subst > term > term] :
( ( X377 @ id @ ( lam @ ( X378 @ sh @ one ) ) @ id )
| ~ ! [X379: term] :
( ~ ( X377 @ id @ X379 @ id )
| ( X377 @ id @ ( X378 @ id @ X379 ) @ id ) )
| ~ ! [X380: term,X381: subst,X382: subst] :
( ( sub @ ( X378 @ X382 @ X380 ) @ X381 )
= ( X378 @ ( comp @ X382 @ X381 ) @ ( sub @ X380 @ X381 ) ) ) )
| ~ ! [X383: term] :
( ( X377 @ id @ X383 @ id )
= ( X376 @ X383 ) )
| ~ ! [X384: term] :
( ( X376 @ ( sub @ X375 @ ( push @ X384 @ id ) ) )
| ~ ( X376 @ X384 ) )
| ( X376 @ ( lam @ X375 ) ) )
<=> hoasinduction_lem3v2a )
& substmonoid_gthm
& ( hoasapinj1
<=> ! [X385: term,X386: term,X387: term,X388: term] :
( ( ( ap @ ( sub @ X386 @ id ) @ X385 )
!= ( ap @ ( sub @ X387 @ id ) @ X388 ) )
| ( X386 = X387 ) ) )
& ! [X389: subst] :
? [X390: d_subst] :
( ( d2subst @ X390 )
= X389 )
& ( hoasinduction_p_and_p_prime
= ( ^ [X391: subst > term > subst > $o,X392: term > $o] :
! [X393: term] :
( ( X392 @ X393 )
= ( X391 @ id @ X393 @ id ) ) ) )
& hoasinduction_lem1
& pushprop_lem0_gthm
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_no_psi_cond_lthm
<=> ( ! [X394: subst > term > subst > $o] :
~ ! [X395: term > $o] :
~ ! [X396: term] :
( ( X394 @ id @ X396 @ id )
= ( X395 @ X396 ) )
=> ( ! [X397: term,X398: term > $o] :
( ~ ! [X399: term] :
( ( X398 @ X399 )
| ~ ( var @ X399 ) )
| ( X398 @ X397 )
| ~ ! [X400: term] :
( ( X398 @ ( lam @ X400 ) )
| ~ ! [X401: term] :
( ( X398 @ ( sub @ X400 @ ( push @ X401 @ id ) ) )
| ~ ( X398 @ X401 ) ) )
| ~ ! [X402: term,X403: term] :
( ~ ( X398 @ X402 )
| ~ ( X398 @ X403 )
| ( X398 @ ( ap @ X403 @ X402 ) ) ) )
=> ( ! [X404: term] :
( ( sub @ X404 @ id )
= X404 )
=> ( ! [X405: term > $o,X406: term,X407: subst > term > subst > $o] :
( ~ ! [X408: term] :
( ( X405 @ X408 )
= ( X407 @ id @ X408 @ id ) )
| ~ ! [X409: subst > term > term] :
( ~ ! [X410: subst,X411: subst,X412: term] :
( ( sub @ ( X409 @ X410 @ X412 ) @ X411 )
= ( X409 @ ( comp @ X410 @ X411 ) @ ( sub @ X412 @ X411 ) ) )
| ~ ! [X413: term] :
( ~ ( X407 @ id @ X413 @ id )
| ( X407 @ id @ ( X409 @ id @ X413 ) @ id ) )
| ( X407 @ id @ ( lam @ ( X409 @ sh @ one ) ) @ id ) )
| ( X405 @ ( lam @ X406 ) )
| ~ ! [X414: term] :
( ~ ( X405 @ X414 )
| ( X405 @ ( sub @ X406 @ ( push @ X414 @ id ) ) ) ) )
=> ! [X415: term,X416: subst > term > subst > $o] :
( ~ ! [X417: subst > term > term] :
( ~ ! [X418: term] :
( ( X416 @ id @ ( X417 @ id @ X418 ) @ id )
| ~ ( X416 @ id @ X418 @ id ) )
| ~ ! [X419: subst,X420: subst,X421: term] :
( ( X417 @ ( comp @ X420 @ X419 ) @ ( sub @ X421 @ X419 ) )
= ( sub @ ( X417 @ X420 @ X421 ) @ X419 ) )
| ( X416 @ id @ ( lam @ ( X417 @ sh @ one ) ) @ id ) )
| ~ ! [X422: term,X423: term] :
( ( X416 @ id @ ( ap @ ( sub @ X422 @ id ) @ X423 ) @ id )
| ~ ( X416 @ id @ X422 @ id )
| ~ ( X416 @ id @ X423 @ id ) )
| ( X416 @ id @ X415 @ id ) ) ) ) ) ) )
& pushprop_lem2v2
& hoasapnotvar_lthm
& axabs
& ( ! [X424: term,X425: subst > term > subst > $o] :
( ( X425 @ id @ ( lam @ ( sub @ X424 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X426: subst > term > term] :
( ~ ! [X427: term] :
( ~ ( X425 @ id @ X427 @ id )
| ( X425 @ id @ ( X426 @ id @ X427 ) @ id ) )
| ( X425 @ id @ ( lam @ ( X426 @ sh @ one ) ) @ id )
| ~ ! [X428: term,X429: subst,X430: subst] :
( ( X426 @ ( comp @ X429 @ X430 ) @ ( sub @ X428 @ X430 ) )
= ( sub @ ( X426 @ X429 @ X428 ) @ X430 ) ) )
| ~ ! [X431: term] :
( ~ ( X425 @ id @ X431 @ id )
| ( X425 @ id @ ( sub @ X424 @ ( push @ X431 @ id ) ) @ id ) ) )
<=> hoasinduction_lem3aa )
& hoasapnotvar
& ( ( ! [X432: term > $o,X433: term] :
( ~ ! [X434: term] :
( ( X432 @ X434 )
| ~ ( var @ X434 ) )
| ~ ! [X435: term,X436: term] :
( ~ ( X432 @ X436 )
| ( X432 @ ( ap @ X436 @ X435 ) )
| ~ ( X432 @ X435 ) )
| ~ ! [X437: term] :
( ( X432 @ ( lam @ X437 ) )
| ~ ! [X438: term] :
( ( X432 @ ( sub @ X437 @ ( push @ X438 @ id ) ) )
| ~ ( X432 @ X438 ) ) )
| ( X432 @ X433 ) )
=> ( ! [X439: subst > term > subst > $o,X440: term,X441: term] :
( ~ ! [X442: term,X443: subst,X444: subst,X445: subst] :
( ( X439 @ X445 @ X442 @ ( comp @ X444 @ X443 ) )
| ~ ( X439 @ ( comp @ X445 @ X444 ) @ ( sub @ X442 @ X444 ) @ X443 ) )
| ~ ( X439 @ id @ X440 @ id )
| ~ ( X439 @ id @ X441 @ id )
| ~ ! [X446: subst,X447: subst,X448: term,X449: subst] :
( ( X439 @ ( comp @ X447 @ X446 ) @ ( sub @ X448 @ X446 ) @ X449 )
| ~ ( X439 @ X447 @ X448 @ ( comp @ X446 @ X449 ) ) )
| ~ ! [X450: term,X451: term] :
( ( X439 @ id @ ( ap @ ( sub @ X450 @ id ) @ X451 ) @ id )
| ~ ( X439 @ id @ X450 @ id )
| ~ ( X439 @ id @ X451 @ id ) )
| ( X439 @ id @ ( ap @ X441 @ X440 ) @ id ) )
=> ( ! [X452: term,X453: subst > term > subst > $o] :
( ~ ! [X454: subst > term > term] :
( ( X453 @ id @ ( lam @ ( X454 @ sh @ one ) ) @ id )
| ~ ! [X455: subst,X456: term,X457: subst] :
( ( X454 @ ( comp @ X457 @ X455 ) @ ( sub @ X456 @ X455 ) )
= ( sub @ ( X454 @ X457 @ X456 ) @ X455 ) )
| ~ ! [X458: term] :
( ~ ( X453 @ id @ X458 @ id )
| ( X453 @ id @ ( X454 @ id @ X458 ) @ id ) ) )
| ~ ! [X459: term] :
( ~ ( X453 @ id @ X459 @ id )
| ( X453 @ id @ ( sub @ X452 @ ( push @ X459 @ id ) ) @ id ) )
| ~ ! [X460: subst,X461: term,X462: subst,X463: subst] :
( ( X453 @ X462 @ X461 @ ( comp @ X463 @ X460 ) )
| ~ ( X453 @ ( comp @ X462 @ X463 ) @ ( sub @ X461 @ X463 ) @ X460 ) )
| ~ ! [X464: subst,X465: subst,X466: term,X467: subst] :
( ( X453 @ ( comp @ X467 @ X465 ) @ ( sub @ X466 @ X465 ) @ X464 )
| ~ ( X453 @ X467 @ X466 @ ( comp @ X465 @ X464 ) ) )
| ( X453 @ id @ ( lam @ X452 ) @ id ) )
=> ! [X468: term,X469: subst > term > subst > $o] :
( ~ ! [X470: term,X471: term] :
( ( X469 @ id @ ( ap @ ( sub @ X470 @ id ) @ X471 ) @ id )
| ~ ( X469 @ id @ X470 @ id )
| ~ ( X469 @ id @ X471 @ id ) )
| ~ ! [X472: subst > term > term] :
( ( X469 @ id @ ( lam @ ( X472 @ sh @ one ) ) @ id )
| ~ ! [X473: term] :
( ( X469 @ id @ ( X472 @ id @ X473 ) @ id )
| ~ ( X469 @ id @ X473 @ id ) )
| ~ ! [X474: subst,X475: term,X476: subst] :
( ( sub @ ( X472 @ X476 @ X475 ) @ X474 )
= ( X472 @ ( comp @ X476 @ X474 ) @ ( sub @ X475 @ X474 ) ) ) )
| ~ ! [X477: term,X478: subst,X479: subst,X480: subst] :
( ~ ( X469 @ ( comp @ X479 @ X480 ) @ ( sub @ X477 @ X480 ) @ X478 )
| ( X469 @ X479 @ X477 @ ( comp @ X480 @ X478 ) ) )
| ~ ! [X481: subst,X482: subst,X483: subst,X484: term] :
( ~ ( X469 @ X483 @ X484 @ ( comp @ X482 @ X481 ) )
| ( X469 @ ( comp @ X483 @ X482 ) @ ( sub @ X484 @ X482 ) @ X481 ) )
| ( X469 @ id @ X468 @ id )
| ~ ! [X485: term] :
( ~ ( var @ ( sub @ X485 @ id ) )
| ( X469 @ id @ X485 @ id ) ) ) ) ) )
<=> hoasinduction_lthm )
& ( hoasapinj2_lthm
<=> ( ! [X486: term,X487: term,X488: term,X489: term] :
( ( X486 = X489 )
| ( ( ap @ X488 @ X489 )
!= ( ap @ X487 @ X486 ) ) )
=> ! [X490: term,X491: term,X492: term,X493: term] :
( ( X492 = X493 )
| ( ( ap @ ( sub @ X491 @ id ) @ X493 )
!= ( ap @ ( sub @ X490 @ id ) @ X492 ) ) ) ) )
& ( ! [X494: term,X495: term,X496: term,X497: term] :
( ( ( ap @ X497 @ X496 )
!= ( ap @ X495 @ X494 ) )
| ( X495 = X497 ) )
<=> apinj1 )
& hoaslamnotvar_gthm
& ( termmset_lthm
<=> ( ! [X498: term] :
( ( sub @ X498 @ id )
= X498 )
=> ! [X499: term] :
( ( sub @ X499 @ id )
= X499 ) ) )
& axapp
& ~ ( var @ ( d2term @ d_one ) )
& ( ! [X500: term > $o,X501: term,X502: subst] :
~ ! [X503: term > $o] :
~ ! [X504: term] :
( ( X500 @ ( sub @ X504 @ ( push @ X501 @ X502 ) ) )
= ( X503 @ X504 ) )
<=> pushprop_lem0 )
& ( ! [X505: term,X506: subst > term > term,X507: subst,X508: subst > term > term] :
( ~ ! [X509: subst,X510: term,X511: subst] :
( ( X506 @ ( comp @ X509 @ X511 ) @ ( sub @ X510 @ X511 ) )
= ( sub @ ( X506 @ X509 @ X510 ) @ X511 ) )
| ~ ! [X512: subst,X513: term,X514: subst] :
( ( sub @ ( X508 @ X512 @ X513 ) @ X514 )
= ( X508 @ ( comp @ X512 @ X514 ) @ ( sub @ X513 @ X514 ) ) )
| ( ( X508 @ X507 @ X505 )
= ( X506 @ X507 @ X505 ) )
| ( ( lam @ ( X506 @ sh @ one ) )
!= ( lam @ ( X508 @ sh @ one ) ) ) )
<=> hoaslaminj )
& hoasinduction_lem2_lthm
& ( hoasinduction_lem3aa_lthm
<=> ! [X515: subst > term > subst > $o,X516: term] :
( ~ ! [X517: term] :
( ( X515 @ id @ ( sub @ X516 @ ( push @ X517 @ id ) ) @ id )
| ~ ( X515 @ id @ X517 @ id ) )
| ( X515 @ id @ ( lam @ ( sub @ X516 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X518: subst > term > term] :
( ~ ! [X519: term] :
( ( X515 @ id @ ( X518 @ id @ X519 ) @ id )
| ~ ( X515 @ id @ X519 @ id ) )
| ( X515 @ id @ ( lam @ ( X518 @ sh @ one ) ) @ id )
| ~ ! [X520: subst,X521: subst,X522: term] :
( ( X518 @ ( comp @ X521 @ X520 ) @ ( sub @ X522 @ X520 ) )
= ( sub @ ( X518 @ X521 @ X522 ) @ X520 ) ) ) ) )
& hoasinduction_lem3a_gthm
& ( ! [X523: term,X524: term > $o,X525: subst] :
~ ! [X526: term > $o] :
~ ! [X527: term] :
( ( X524 @ ( sub @ X527 @ ( push @ X523 @ X525 ) ) )
= ( X526 @ X527 ) )
<=> pushprop_lem0_lthm )
& ( ! [X528: subst > term > subst > $o,X529: term,X530: term] :
( ~ ! [X531: subst,X532: subst,X533: term,X534: subst] :
( ~ ( X528 @ ( comp @ X532 @ X534 ) @ ( sub @ X533 @ X534 ) @ X531 )
| ( X528 @ X532 @ X533 @ ( comp @ X534 @ X531 ) ) )
| ~ ( X528 @ id @ X530 @ id )
| ( X528 @ id @ ( ap @ X529 @ X530 ) @ id )
| ~ ! [X535: term,X536: term] :
( ~ ( X528 @ id @ X535 @ id )
| ~ ( X528 @ id @ X536 @ id )
| ( X528 @ id @ ( ap @ ( sub @ X535 @ id ) @ X536 ) @ id ) )
| ~ ! [X537: subst,X538: subst,X539: term,X540: subst] :
( ( X528 @ ( comp @ X538 @ X540 ) @ ( sub @ X539 @ X540 ) @ X537 )
| ~ ( X528 @ X538 @ X539 @ ( comp @ X540 @ X537 ) ) )
| ~ ( X528 @ id @ X529 @ id ) )
<=> hoasinduction_lem2 )
& ( axidl
<=> ! [X541: subst] :
( ( comp @ id @ X541 )
= X541 ) )
& ( ! [X542: term,X543: subst > term > subst > $o] :
( ( X543 @ id @ ( lam @ X542 ) @ id )
| ~ ! [X544: subst > term > term] :
( ( X543 @ id @ ( lam @ ( X544 @ sh @ one ) ) @ id )
| ~ ! [X545: subst,X546: subst,X547: term] :
( ( X544 @ ( comp @ X545 @ X546 ) @ ( sub @ X547 @ X546 ) )
= ( sub @ ( X544 @ X545 @ X547 ) @ X546 ) )
| ~ ! [X548: term] :
( ( X543 @ id @ ( X544 @ id @ X548 ) @ id )
| ~ ( X543 @ id @ X548 @ id ) ) )
| ~ ! [X549: term] :
( ( X543 @ id @ ( sub @ X542 @ ( push @ X549 @ id ) ) @ id )
| ~ ( X543 @ id @ X549 @ id ) ) )
<=> hoasinduction_lem3a )
& hoasapinj1_gthm ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( pushprop_lem1v2_lthm
<=> ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ! [X11: term > $o,X10: term > $o,X14: subst,X13: term] :
( ~ ( X10 @ X13 )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
| ( X11 @ one ) ) ) )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0 )
& ( ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ! [X26: term > $o,X13: term,X14: subst,X15: term,X10: term > $o] :
( ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) )
| ~ ( X10 @ X13 ) ) ) )
<=> pushprop_lem1_lthm )
& hoasinduction_lem3_gthm
& ( ! [X15: term,X20: term,X13: term,X19: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) )
<=> hoasapinj2 )
& axclos
& ! [X3: d_term,X2: d_term] :
( ( ( d2term @ X2 )
= ( d2term @ X3 ) )
=> ( X2 = X3 ) )
& ( ! [X10: subst > term > subst > $o,X34: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X34 @ ( push @ X15 @ id ) ) @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X34 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X35: term,X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X38: subst,X37: term,X36: subst] :
( ( sub @ ( X18 @ X36 @ X37 ) @ X38 )
= ( sub @ ( sub @ X35 @ ( push @ X37 @ X36 ) ) @ X38 ) )
| ~ ! [X41: subst,X39: subst,X40: term] :
( ( X18 @ ( comp @ X39 @ X41 ) @ ( sub @ X40 @ X41 ) )
= ( sub @ X35 @ ( push @ ( sub @ X40 @ X41 ) @ ( comp @ X39 @ X41 ) ) ) ) ) )
<=> hoasinduction_lem3aaa )
& pushprop_gthm
& ( hoasinduction
<=> ! [X10: subst > term > subst > $o,X28: term] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ 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 ) )
| ~ ! [X14: subst,X26: subst,X25: subst,X13: term] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ X28 @ id ) ) )
& ulamvarsh
& hoasapinj2_gthm
& hoasinduction_lem3v2_gthm
& ! [X6: d_subst,X7: d_subst] :
( ( ( d2subst @ X6 )
= ( d2subst @ X7 ) )
=> ( X6 = X7 ) )
& pushprop_lem2v2_lthm
& ( termmset
<=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) )
& axvarshift
& ( substmonoid
<=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) ) )
& ( ( ^ [X8: subst,X9: subst > term > term] : ( d2term @ d_one ) )
= hoaslam )
& pushprop_lthm
& pushprop_lem1v2_gthm
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id ) )
=> ! [X43: term,X10: subst > term > subst > $o] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ( X10 @ id @ ( lam @ X43 ) @ id ) ) ) )
<=> hoasinduction_lem3a_lthm )
& ( hoaslaminj_lthm
<=> ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X13: term,X15: term] :
( ( ( lam @ X13 )
!= ( lam @ X15 ) )
| ( X13 = X15 ) )
=> ! [X33: term,X31: subst > term > term,X32: subst,X18: subst > term > term] :
( ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) ) ) )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0_lthm )
& apnotvar
& pushprop
& ( ( d2subst @ d_id )
= sh )
& ! [X1: d_term] : ( d_one = X1 )
& ( ! [X19: term,X20: term,X13: term,X15: term] :
( ( X19 = X20 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
<=> apinj2 )
& ulamvarind
& ( ( ^ [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 )
& ( hoasinduction_lem3v2_f_lthm
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) ) )
& ( hoasinduction_no_psi_cond
<=> ! [X30: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ( X10 @ id @ X30 @ id ) ) )
& ( hoasinduction_lem3_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [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 @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id ) ) )
=> ! [X27: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X13: term,X26: subst,X14: subst,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term,X25: subst,X26: subst,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) ) ) ) )
& ( induction2lem
<=> ! [X16: term,X10: term > $o,X17: subst] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X15 @ X17 ) )
| ~ ( var @ X15 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) ) ) ) )
& pushprop_lem3v2
& hoasinduction_lem1v2
& ( hoasinduction_lem3v2_f
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X16: term,X17: subst,X10: term > $o] :
( ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) ) )
=> ! [X22: term,X10: term > $o] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ X22 ) ) ) )
<=> induction2_lthm )
& ( ! [X15: term,X13: term] :
( ( X13 = X15 )
| ( ( lam @ X13 )
!= ( lam @ X15 ) ) )
<=> laminj )
& hoaslamnotvar_lthm
& pushprop_lem3v2_lthm
& ~ ulamvar1
& ( ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X21: term,X10: term > $o] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ( X10 @ X21 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) ) )
=> ! [X16: term,X10: term > $o,X17: subst] :
( ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X15 @ X17 ) )
| ~ ( var @ X15 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) ) ) ) ) ) )
<=> induction2lem_lthm )
& ( ! [X10: subst > term > subst > $o,X48: term,X11: term > $o] :
( ~ ! [X26: subst,X25: subst,X14: subst,X13: term] :
( ~ ( 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 ) ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
<=> hoasinduction_lem3v2 )
& ( ! [X10: term > $o,X21: term] :
( ( X10 @ X21 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) ) )
<=> induction )
& hoasinduction_lem1_gthm
& induction2lem_gthm
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) )
<=> hoasinduction_lem3b )
& ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
<=> axidr )
& termmset_gthm
& ( ! [X13: term,X11: term > $o,X10: term > $o,X14: subst] :
( ( X11 @ one )
| ~ ( X10 @ X13 )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) )
<=> pushprop_lem1v2 )
& ( ( d2subst @ d_id )
= id )
& ( hoasinduction_lem2v2
<=> ! [X11: term > $o,X44: term,X10: subst > term > subst > $o,X45: term] :
( ~ ( X11 @ X44 )
| ~ ! [X14: subst,X13: term,X25: subst,X26: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ( X11 @ X45 )
| ~ ! [X26: subst,X25: subst,X14: subst,X13: term] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X11 @ ( ap @ X44 @ X45 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id ) ) ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& hoaslamnotap_lthm
& ( hoasinduction_lem3
<=> ! [X10: subst > term > subst > $o,X27: term] :
( ~ ! [X25: subst,X14: subst,X26: subst,X13: term] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X25: subst,X14: subst,X26: subst,X13: term] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) ) ) )
& hoasapnotvar_gthm
& ~ lamnotap
& ( ! [X22: term,X10: term > $o] :
( ( X10 @ X22 )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) ) )
<=> induction2 )
& ( ! [X47: term,X18: subst > term > term,X46: term] :
( ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( ap @ ( sub @ X46 @ id ) @ X47 ) ) )
<=> hoaslamnotap )
& axmap
& ( one
= ( d2term @ d_one ) )
& ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
<=> axvarid )
& ( pushprop_lem1
<=> ! [X13: term,X10: term > $o,X26: term > $o,X15: term,X14: subst] :
( ~ ( X10 @ X13 )
| ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) ) ) )
& hoaslamnotvar
& pushprop_lem2v2_gthm
& hoaslaminj_gthm
& lamnotvar
& ! [X5: d_subst] : ( d_id = X5 )
& ! [X0: term] :
? [X1: d_term] :
( ( d2term @ X1 )
= X0 )
& ( axshiftcons
<=> ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 ) )
& ( shinj
<=> ! [X13: term,X15: term] :
( ( X13 = X15 )
| ( ( sub @ X13 @ sh )
!= ( sub @ X15 @ sh ) ) ) )
& pushprop_lthm_orig
& ( ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X11: term > $o,X10: subst > term > subst > $o,X29: term] :
( ~ ! [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,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) ) ) ) )
<=> hoasinduction_lem3v2a_lthm )
& hoasinduction_lem2_gthm
& pushprop_lem1_gthm
& hoasinduction_lem2v2_gthm
& hoasinduction_lem3b_gthm
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X10: term > $o,X22: term] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) )
| ( X10 @ X22 ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: subst > term > subst > $o,X29: term,X11: term > $o] :
( ~ ! [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 ) ) ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) )
=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ( X10 @ id @ X28 @ id ) ) ) ) ) )
<=> hoasinduction_lthm_3 )
& ( ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) ) ) )
<=> substmonoid_lthm )
& hoasinduction_lem1_lthm
& axassoc
& hoasinduction_lem1v2_gthm
& hoasinduction_gthm
& induction2_gthm
& ( axscons
<=> ! [X14: subst] :
( ( push @ ( sub @ one @ X14 ) @ ( comp @ sh @ X14 ) )
= X14 ) )
& ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
<=> axvarcons )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_lem3b_lthm
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) ) )
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X15: term,X13: term,X19: term,X20: term] :
( ( X13 = X15 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
=> ! [X15: term,X19: term,X20: term,X13: term] :
( ( X13 = X15 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) ) )
<=> hoasapinj1_lthm )
& ( hoasinduction_lem3v2_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 ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X26: subst,X25: subst,X14: subst,X13: term] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X14: subst,X26: subst,X13: term,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ 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 @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) ) ) ) ) )
& hoaslamnotap_gthm
& ( ! [X29: term,X11: term > $o,X10: subst > term > subst > $o] :
( ~ ! [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,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ( X11 @ ( lam @ X29 ) ) )
<=> hoasinduction_lem3v2a )
& substmonoid_gthm
& ( hoasapinj1
<=> ! [X19: term,X13: term,X15: term,X20: term] :
( ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) )
| ( X13 = X15 ) ) )
& ! [X4: subst] :
? [X5: d_subst] :
( ( d2subst @ X5 )
= X4 )
& ( hoasinduction_p_and_p_prime
= ( ^ [X10: subst > term > subst > $o,X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) ) )
& hoasinduction_lem1
& pushprop_lem0_gthm
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( hoasinduction_no_psi_cond_lthm
<=> ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X22: term,X10: term > $o] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X11: term > $o,X29: term,X10: subst > term > subst > $o] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) ) )
=> ! [X30: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ X30 @ id ) ) ) ) ) ) )
& pushprop_lem2v2
& hoasapnotvar_lthm
& axabs
& ( ! [X42: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ 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,X14: subst,X25: subst] :
( ( 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 ) ) )
<=> hoasinduction_lem3aa )
& hoasapnotvar
& ( ( ! [X10: term > $o,X22: term] :
( ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) ) )
| ( X10 @ X22 ) )
=> ( ! [X10: subst > term > subst > $o,X24: term,X23: term] :
( ~ ! [X13: term,X25: subst,X26: subst,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ( X10 @ id @ X24 @ id )
| ~ ( X10 @ id @ X23 @ id )
| ~ ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id ) )
=> ( ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id ) )
| ~ ! [X25: subst,X13: term,X14: subst,X26: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ ( lam @ X27 ) @ id ) )
=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X13: term,X25: subst,X14: subst,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X25: subst,X26: subst,X14: subst,X13: term] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X10 @ id @ X28 @ id )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) ) ) ) ) )
<=> hoasinduction_lthm )
& ( hoasapinj2_lthm
<=> ( ! [X20: term,X15: term,X13: term,X19: term] :
( ( X19 = X20 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
=> ! [X15: term,X13: term,X20: term,X19: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) ) )
& ( ! [X20: term,X15: term,X19: term,X13: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X13 = X15 ) )
<=> apinj1 )
& hoaslamnotvar_gthm
& ( termmset_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) ) )
& axapp
& ~ ( var @ ( d2term @ d_one ) )
& ( ! [X10: term > $o,X13: term,X14: subst] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
<=> pushprop_lem0 )
& ( ! [X33: term,X31: subst > term > term,X32: subst,X18: subst > term > term] :
( ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) ) )
<=> hoaslaminj )
& hoasinduction_lem2_lthm
& ( hoasinduction_lem3aa_lthm
<=> ! [X10: subst > term > subst > $o,X42: term] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) )
& hoasinduction_lem3a_gthm
& ( ! [X13: term,X10: term > $o,X14: subst] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
<=> pushprop_lem0_lthm )
& ( ! [X10: subst > term > subst > $o,X23: term,X24: term] :
( ~ ! [X25: subst,X14: subst,X13: term,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ( X10 @ id @ X24 @ id )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ! [X25: subst,X14: subst,X13: term,X26: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ( X10 @ id @ X23 @ id ) )
<=> hoasinduction_lem2 )
& ( axidl
<=> ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) )
& ( ! [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,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) )
<=> hoasinduction_lem3a )
& hoasapinj1_gthm ),
file('/export/starexec/sandbox2/tmp/tmp.sO7hdZQLTS/DTF2THF_1662.p',alg444_1) ).
thf(f1393,plain,
! [X124: term] :
( ( d2term @ ( sK576 @ X124 ) )
= X124 ),
inference(cnf_transformation,[status(thm)],[f749]) ).
thf(f1395,plain,
( ( var @ ( d2term @ d_one ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f749]) ).
thf(f1530,plain,
! [X0: subst > term > subst > $o] :
( ( ( var @ ( sK656 @ X0 ) )
= $true )
| ( $true
= ( sP110 @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f766]) ).
thf(f766,plain,
! [X0: subst > term > subst > $o] :
( ( ( $true
= ( sP110 @ X0 ) )
| ( ( ( var @ ( sK656 @ X0 ) )
= $true )
& ( ( X0 @ id @ ( sK656 @ X0 ) @ id )
!= $true ) ) )
& ( ! [X2: term] :
( ( ( var @ X2 )
!= $true )
| ( ( X0 @ id @ X2 @ id )
= $true ) )
| ( $true
!= ( sP110 @ X0 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK656])],[f764,f765]) ).
thf(f765,plain,
! [X0: subst > term > subst > $o] :
( ? [X1: term] :
( ( ( var @ X1 )
= $true )
& ( $true
!= ( X0 @ id @ X1 @ id ) ) )
=> ( ( ( var @ ( sK656 @ X0 ) )
= $true )
& ( ( X0 @ id @ ( sK656 @ X0 ) @ id )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f764,plain,
! [X0: subst > term > subst > $o] :
( ( ( $true
= ( sP110 @ X0 ) )
| ? [X1: term] :
( ( ( var @ X1 )
= $true )
& ( $true
!= ( X0 @ id @ X1 @ id ) ) ) )
& ( ! [X2: term] :
( ( ( var @ X2 )
!= $true )
| ( ( X0 @ id @ X2 @ id )
= $true ) )
| ( $true
!= ( sP110 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f763]) ).
thf(f763,plain,
! [X0: subst > term > subst > $o] :
( ( ( $true
= ( sP110 @ X0 ) )
| ? [X10: term] :
( ( ( var @ X10 )
= $true )
& ( $true
!= ( X0 @ id @ X10 @ id ) ) ) )
& ( ! [X10: term] :
( ( ( var @ X10 )
!= $true )
| ( $true
= ( X0 @ id @ X10 @ id ) ) )
| ( $true
!= ( sP110 @ X0 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f126]) ).
thf(f126,plain,
! [X0: subst > term > subst > $o] :
( ( $true
= ( sP110 @ X0 ) )
<=> ! [X10: term] :
( ( ( var @ X10 )
!= $true )
| ( $true
= ( X0 @ id @ X10 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f3331,plain,
( ( $true
!= ( sP110 @ sK646 ) )
| spl657_345 ),
inference(avatar_component_clause,[status(thm)],[f3330]) ).
thf(f3330,definition,
( spl657_345
<=> ( $true
= ( sP110 @ sK646 ) ) ),
introduced(definition,[new_symbols(naming,[spl657_345])],[avatar_definition]) ).
thf(f3332,plain,
( ~ spl657_345
| spl657_173 ),
inference(avatar_split_clause,[status(thm)],[f1521,f2586,f3330]) ).
thf(f2586,definition,
( spl657_173
<=> ( sP113 = $true ) ),
introduced(definition,[new_symbols(naming,[spl657_173])],[avatar_definition]) ).
thf(f1521,plain,
( ( $true
!= ( sP110 @ sK646 ) )
| ( sP113 = $true ) ),
inference(cnf_transformation,[status(thm)],[f754]) ).
thf(f754,plain,
( ( ( sP113 = $true )
| ( ( $true
!= ( sP110 @ sK646 ) )
& ! [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( sK646 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
!= $true )
| ( ( sK646 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 )
= $true ) )
& ( ( sP112 @ sK646 )
!= $true )
& ( $true
!= ( sP111 @ sK646 ) ) ) )
& ( ! [X5: subst > term > subst > $o] :
( ( $true
= ( sP110 @ X5 ) )
| ( ( ( X5 @ ( sK648 @ X5 ) @ ( sK649 @ X5 ) @ ( comp @ ( sK647 @ X5 ) @ ( sK650 @ X5 ) ) )
= $true )
& ( $true
!= ( X5 @ ( comp @ ( sK648 @ X5 ) @ ( sK647 @ X5 ) ) @ ( sub @ ( sK649 @ X5 ) @ ( sK647 @ X5 ) ) @ ( sK650 @ X5 ) ) ) )
| ( $true
= ( sP112 @ X5 ) )
| ( $true
= ( sP111 @ X5 ) ) )
| ( sP113 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK646,sK647,sK648,sK649,sK650])],[f751,f753,f752]) ).
thf(f752,plain,
( ? [X0: subst > term > subst > $o] :
( ( $true
!= ( sP110 @ X0 ) )
& ! [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
!= $true )
| ( $true
= ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) ) )
& ( $true
!= ( sP112 @ X0 ) )
& ( $true
!= ( sP111 @ X0 ) ) )
=> ( ( $true
!= ( sP110 @ sK646 ) )
& ! [X4: subst,X3: term,X2: subst,X1: subst] :
( ( ( sK646 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
!= $true )
| ( ( sK646 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 )
= $true ) )
& ( ( sP112 @ sK646 )
!= $true )
& ( $true
!= ( sP111 @ sK646 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f753,plain,
! [X5: subst > term > subst > $o] :
( ? [X6: subst,X7: subst,X8: term,X9: subst] :
( ( $true
= ( X5 @ X7 @ X8 @ ( comp @ X6 @ X9 ) ) )
& ( $true
!= ( X5 @ ( comp @ X7 @ X6 ) @ ( sub @ X8 @ X6 ) @ X9 ) ) )
=> ( ( ( X5 @ ( sK648 @ X5 ) @ ( sK649 @ X5 ) @ ( comp @ ( sK647 @ X5 ) @ ( sK650 @ X5 ) ) )
= $true )
& ( $true
!= ( X5 @ ( comp @ ( sK648 @ X5 ) @ ( sK647 @ X5 ) ) @ ( sub @ ( sK649 @ X5 ) @ ( sK647 @ X5 ) ) @ ( sK650 @ X5 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f751,plain,
( ( ( sP113 = $true )
| ? [X0: subst > term > subst > $o] :
( ( $true
!= ( sP110 @ X0 ) )
& ! [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
!= $true )
| ( $true
= ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) ) )
& ( $true
!= ( sP112 @ X0 ) )
& ( $true
!= ( sP111 @ X0 ) ) ) )
& ( ! [X5: subst > term > subst > $o] :
( ( $true
= ( sP110 @ X5 ) )
| ? [X6: subst,X7: subst,X8: term,X9: subst] :
( ( $true
= ( X5 @ X7 @ X8 @ ( comp @ X6 @ X9 ) ) )
& ( $true
!= ( X5 @ ( comp @ X7 @ X6 ) @ ( sub @ X8 @ X6 ) @ X9 ) ) )
| ( $true
= ( sP112 @ X5 ) )
| ( $true
= ( sP111 @ X5 ) ) )
| ( sP113 != $true ) ) ),
inference(rectify,[status(thm)],[f750]) ).
thf(f750,plain,
( ( ( sP113 = $true )
| ? [X0: subst > term > subst > $o] :
( ( $true
!= ( sP110 @ X0 ) )
& ! [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
!= $true )
| ( $true
= ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) ) )
& ( $true
!= ( sP112 @ X0 ) )
& ( $true
!= ( sP111 @ X0 ) ) ) )
& ( ! [X0: subst > term > subst > $o] :
( ( $true
= ( sP110 @ X0 ) )
| ? [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
= $true )
& ( $true
!= ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) ) )
| ( $true
= ( sP112 @ X0 ) )
| ( $true
= ( sP111 @ X0 ) ) )
| ( sP113 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f129]) ).
thf(f129,plain,
( ( sP113 = $true )
<=> ! [X0: subst > term > subst > $o] :
( ( $true
= ( sP110 @ X0 ) )
| ? [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
= $true )
& ( $true
!= ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) ) )
| ( $true
= ( sP112 @ X0 ) )
| ( $true
= ( sP111 @ X0 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f2942,plain,
~ spl657_173,
inference(avatar_split_clause,[status(thm)],[f1899,f2586]) ).
thf(f1899,plain,
sP113 != $true,
inference(trivial_inequality_removal,[status(thm)],[f1859]) ).
thf(f1859,plain,
( ( $true != $true )
| ( sP113 != $true ) ),
inference(definition_unfolding,[status(thm)],[f1532,f1367]) ).
thf(f1367,plain,
hoasinduction_lem1 = $true,
inference(cnf_transformation,[status(thm)],[f749]) ).
thf(f1532,plain,
( ( hoasinduction_lem1 != $true )
| ( sP113 != $true ) ),
inference(cnf_transformation,[status(thm)],[f767]) ).
thf(f767,plain,
( ( ( hoasinduction_lem1 != $true )
| ( sP113 != $true ) )
& ( ( hoasinduction_lem1 = $true )
| ( sP113 = $true ) ) ),
inference(nnf_transformation,[status(thm)],[f130]) ).
thf(f130,plain,
( ( sP113 = $true )
<~> ( hoasinduction_lem1 = $true ) ),
inference(definition_folding,[status(thm)],[f12,f129,f128,f127,f126]) ).
thf(f127,plain,
! [X0: subst > term > subst > $o] :
( ( $true
= ( sP111 @ X0 ) )
<=> ? [X5: subst,X6: subst,X7: term,X8: subst] :
( ( ( X0 @ X8 @ X7 @ ( comp @ X6 @ X5 ) )
!= $true )
& ( $true
= ( X0 @ ( comp @ X8 @ X6 ) @ ( sub @ X7 @ X6 ) @ X5 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f128,plain,
! [X0: subst > term > subst > $o] :
( ( $true
= ( sP112 @ X0 ) )
<=> ? [X9: term] :
( ( ( hoasvar @ id @ X9 @ id )
= $true )
& ( ( X0 @ id @ X9 @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f12,plain,
( ! [X0: subst > term > subst > $o] :
( ! [X10: term] :
( ( ( var @ X10 )
!= $true )
| ( $true
= ( X0 @ id @ X10 @ id ) ) )
| ? [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
= $true )
& ( $true
!= ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) ) )
| ? [X9: term] :
( ( ( hoasvar @ id @ X9 @ id )
= $true )
& ( ( X0 @ id @ X9 @ id )
!= $true ) )
| ? [X5: subst,X6: subst,X7: term,X8: subst] :
( ( ( X0 @ X8 @ X7 @ ( comp @ X6 @ X5 ) )
!= $true )
& ( $true
= ( X0 @ ( comp @ X8 @ X6 ) @ ( sub @ X7 @ X6 ) @ X5 ) ) ) )
<~> ( hoasinduction_lem1 = $true ) ),
inference(flattening,[status(thm)],[f11]) ).
thf(f11,plain,
( ( hoasinduction_lem1 = $true )
<~> ! [X0: subst > term > subst > $o] :
( ! [X10: term] :
( ( ( var @ X10 )
!= $true )
| ( $true
= ( X0 @ id @ X10 @ id ) ) )
| ? [X9: term] :
( ( ( hoasvar @ id @ X9 @ id )
= $true )
& ( ( X0 @ id @ X9 @ id )
!= $true ) )
| ? [X5: subst,X6: subst,X7: term,X8: subst] :
( ( ( X0 @ X8 @ X7 @ ( comp @ X6 @ X5 ) )
!= $true )
& ( $true
= ( X0 @ ( comp @ X8 @ X6 ) @ ( sub @ X7 @ X6 ) @ X5 ) ) )
| ? [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
= $true )
& ( $true
!= ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f6]) ).
thf(f6,plain,
~ ( ( hoasinduction_lem1 = $true )
<=> ! [X0: subst > term > subst > $o] :
( ! [X1: subst,X2: subst,X3: term,X4: subst] :
( ( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
= $true )
=> ( $true
= ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) ) )
=> ( ! [X5: subst,X6: subst,X7: term,X8: subst] :
( ( $true
= ( X0 @ ( comp @ X8 @ X6 ) @ ( sub @ X7 @ X6 ) @ X5 ) )
=> ( ( X0 @ X8 @ X7 @ ( comp @ X6 @ X5 ) )
= $true ) )
=> ( ! [X9: term] :
( ( ( hoasvar @ id @ X9 @ id )
= $true )
=> ( ( X0 @ id @ X9 @ id )
= $true ) )
=> ! [X10: term] :
( ( ( var @ X10 )
= $true )
=> ( $true
= ( X0 @ id @ X10 @ id ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
( hoasinduction_lem1
!= ( ! [X0: subst > term > subst > $o] :
( ! [X1: subst,X2: subst,X3: term,X4: subst] :
( ( X0 @ X2 @ X3 @ ( comp @ X1 @ X4 ) )
=> ( X0 @ ( comp @ X2 @ X1 ) @ ( sub @ X3 @ X1 ) @ X4 ) )
=> ( ! [X5: subst,X6: subst,X7: term,X8: subst] :
( ( X0 @ ( comp @ X8 @ X6 ) @ ( sub @ X7 @ X6 ) @ X5 )
=> ( X0 @ X8 @ X7 @ ( comp @ X6 @ X5 ) ) )
=> ( ! [X9: term] :
( ( hoasvar @ id @ X9 @ id )
=> ( X0 @ id @ X9 @ id ) )
=> ! [X10: term] :
( ( var @ X10 )
=> ( X0 @ id @ X10 @ id ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
( hoasinduction_lem1
!= ( ! [X10: subst > term > subst > $o] :
( ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
=> ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
=> ( ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
=> ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
=> ( ! [X13: term] :
( ( hoasvar @ id @ X13 @ id )
=> ( X10 @ id @ X13 @ id ) )
=> ! [X13: term] :
( ( var @ X13 )
=> ( X10 @ id @ X13 @ id ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
( hoasinduction_lem1
= ( ! [X10: subst > term > subst > $o] :
( ! [X26: subst,X14: subst,X13: term,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
=> ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
=> ( ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
=> ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
=> ( ! [X13: term] :
( ( hoasvar @ id @ X13 @ id )
=> ( X10 @ id @ X13 @ id ) )
=> ! [X13: term] :
( ( var @ X13 )
=> ( X10 @ id @ X13 @ id ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.sO7hdZQLTS/DTF2THF_1662.p',hoasinduction_lem1) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n029.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue May 5 09:07:14 EDT 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.20/0.46 % Domain : Software Verification
% 0.20/0.46 % Problem : Benchmark: ALG444^1, Active target: hoasinduction_lem1
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [Kon26] Kondylidou (2026), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Kon26]
% 0.20/0.46 % Names : ALG444^1__005__axclos.verify [Kon26]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : ? v9.3.0
% 0.20/0.46 % Syntax : Number of formulae : 132 ( 1 unt; 130 typ; 0 def)
% 0.20/0.46 % Number of atoms : 256 ( 128 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 248 ( 128 avg)
% 0.20/0.46 % Number of connectives : 2610 ( 333 ~; 306 |; 129 &;1741 @)
% 0.20/0.46 % ( 56 <=>; 45 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 151 ( 76 avg)
% 0.20/0.46 % Number of types : 5 ( 4 usr)
% 0.20/0.46 % Number of type conns : 225 ( 225 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 127 ( 126 usr; 113 con; 0-4 aty)
% 0.20/0.46 % Number of variables : 561 ( 8 ^; 551 !; 2 ?; 561 :)
% 0.20/0.46 % SPC : TH0_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments :
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 thf(term_type,type,
% 0.20/0.46 term: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(subst_type,type,
% 0.20/0.46 subst: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(one_decl,type,
% 0.20/0.46 one: term ).
% 0.20/0.46
% 0.20/0.46 thf(ap_decl,type,
% 0.20/0.46 ap: term > term > term ).
% 0.20/0.46
% 0.20/0.46 thf(lam_decl,type,
% 0.20/0.46 lam: term > term ).
% 0.20/0.46
% 0.20/0.46 thf(sub_decl,type,
% 0.20/0.46 sub: term > subst > term ).
% 0.20/0.46
% 0.20/0.46 thf(id_decl,type,
% 0.20/0.46 id: subst ).
% 0.20/0.46
% 0.20/0.46 thf(sh_decl,type,
% 0.20/0.46 sh: subst ).
% 0.20/0.46
% 0.20/0.46 thf(push_decl,type,
% 0.20/0.46 push: term > subst > subst ).
% 0.20/0.46
% 0.20/0.46 thf(comp_decl,type,
% 0.20/0.46 comp: subst > subst > subst ).
% 0.20/0.46
% 0.20/0.46 thf(var_decl,type,
% 0.20/0.46 var: term > $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem1v2_decl,type,
% 0.20/0.46 pushprop_lem1v2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem1_gthm_decl,type,
% 0.20/0.46 pushprop_lem1_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axmap_decl,type,
% 0.20/0.46 axmap: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem0_gthm_decl,type,
% 0.20/0.46 pushprop_lem0_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(shinj_decl,type,
% 0.20/0.46 shinj: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem1v2_decl,type,
% 0.20/0.46 hoasinduction_lem1v2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem1v2_gthm_decl,type,
% 0.20/0.46 hoasinduction_lem1v2_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasap_decl,type,
% 0.20/0.46 hoasap: subst > term > subst > term > term ).
% 0.20/0.46
% 0.20/0.46 thf(induction2lem_decl,type,
% 0.20/0.46 induction2lem: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3v2_f_decl,type,
% 0.20/0.46 hoasinduction_lem3v2_f: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axvarshift_decl,type,
% 0.20/0.46 axvarshift: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapinj2_decl,type,
% 0.20/0.46 hoasapinj2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapnotvar_gthm_decl,type,
% 0.20/0.46 hoasapnotvar_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapinj1_decl,type,
% 0.20/0.46 hoasapinj1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ulamvar1_decl,type,
% 0.20/0.46 ulamvar1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(induction2lem_lthm_decl,type,
% 0.20/0.46 induction2lem_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3v2_gthm_decl,type,
% 0.20/0.46 hoasinduction_lem3v2_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(apnotvar_decl,type,
% 0.20/0.46 apnotvar: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lthm_orig_decl,type,
% 0.20/0.46 pushprop_lthm_orig: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem3v2_f_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lthm_decl,type,
% 0.20/0.46 hoasinduction_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 0.20/0.46 hoasinduction_no_psi_cond_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslaminj_decl,type,
% 0.20/0.46 hoaslaminj: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3aaa_decl,type,
% 0.20/0.46 hoasinduction_lem3aaa: $o ).
% 0.20/0.46
% 0.20/0.46 thf(induction2lem_gthm_decl,type,
% 0.20/0.46 induction2lem_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3aa_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem3aa_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3_decl,type,
% 0.20/0.46 hoasinduction_lem3: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem2_decl,type,
% 0.20/0.46 hoasinduction_lem2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(termmset_lthm_decl,type,
% 0.20/0.46 termmset_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem1_decl,type,
% 0.20/0.46 hoasinduction_lem1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslamnotap_lthm_decl,type,
% 0.20/0.46 hoaslamnotap_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem1v2_lthm_decl,type,
% 0.20/0.46 pushprop_lem1v2_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapnotvar_decl,type,
% 0.20/0.46 hoasapnotvar: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem0_decl,type,
% 0.20/0.46 hoasinduction_lem0: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_decl,type,
% 0.20/0.46 hoasinduction: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_gthm_decl,type,
% 0.20/0.46 hoasinduction_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axapp_decl,type,
% 0.20/0.46 axapp: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslamnotvar_lthm_decl,type,
% 0.20/0.46 hoaslamnotvar_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem3v2_lthm_decl,type,
% 0.20/0.46 pushprop_lem3v2_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3b_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem3b_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ulamvarind_decl,type,
% 0.20/0.46 ulamvarind: $o ).
% 0.20/0.46
% 0.20/0.46 thf(induction_decl,type,
% 0.20/0.46 induction: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3a_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem3a_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(termmset_gthm_decl,type,
% 0.20/0.46 termmset_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3aa_decl,type,
% 0.20/0.46 hoasinduction_lem3aa: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem1v2_gthm_decl,type,
% 0.20/0.46 pushprop_lem1v2_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslamnotap_gthm_decl,type,
% 0.20/0.46 hoaslamnotap_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslamnotvar_gthm_decl,type,
% 0.20/0.46 hoaslamnotvar_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3b_gthm_decl,type,
% 0.20/0.46 hoasinduction_lem3b_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem2v2_decl,type,
% 0.20/0.46 pushprop_lem2v2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3a_gthm_decl,type,
% 0.20/0.46 hoasinduction_lem3a_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axclos_decl,type,
% 0.20/0.46 axclos: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axassoc_decl,type,
% 0.20/0.46 axassoc: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem2v2_decl,type,
% 0.20/0.46 hoasinduction_lem2v2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lthm_decl,type,
% 0.20/0.46 pushprop_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(apinj2_decl,type,
% 0.20/0.46 apinj2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(apinj1_decl,type,
% 0.20/0.46 apinj1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapinj2_lthm_decl,type,
% 0.20/0.46 hoasapinj2_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3v2a_decl,type,
% 0.20/0.46 hoasinduction_lem3v2a: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapinj1_lthm_decl,type,
% 0.20/0.46 hoasapinj1_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslaminj_lthm_decl,type,
% 0.20/0.46 hoaslaminj_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axvarcons_decl,type,
% 0.20/0.46 axvarcons: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslam_decl,type,
% 0.20/0.46 hoaslam: subst > ( subst > term > term ) > term ).
% 0.20/0.46
% 0.20/0.46 thf(axscons_decl,type,
% 0.20/0.46 axscons: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem2v2_gthm_decl,type,
% 0.20/0.46 hoasinduction_lem2v2_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axidr_decl,type,
% 0.20/0.46 axidr: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem1_decl,type,
% 0.20/0.46 pushprop_lem1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(laminj_decl,type,
% 0.20/0.46 laminj: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem3_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem0_decl,type,
% 0.20/0.46 pushprop_lem0: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_gthm_decl,type,
% 0.20/0.46 pushprop_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axabs_decl,type,
% 0.20/0.46 axabs: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem3v2a_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem2_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem2_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapinj2_gthm_decl,type,
% 0.20/0.46 hoasapinj2_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_p_and_p_prime_decl,type,
% 0.20/0.46 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem1_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem1_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(lamnotap_decl,type,
% 0.20/0.46 lamnotap: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapinj1_gthm_decl,type,
% 0.20/0.46 hoasapinj1_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslamnotvar_decl,type,
% 0.20/0.46 hoaslamnotvar: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axidl_decl,type,
% 0.20/0.46 axidl: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslaminj_gthm_decl,type,
% 0.20/0.46 hoaslaminj_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(induction2_lthm_decl,type,
% 0.20/0.46 induction2_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem0_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem0_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(substmonoid_lthm_decl,type,
% 0.20/0.46 substmonoid_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_decl,type,
% 0.20/0.46 pushprop: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3_gthm_decl,type,
% 0.20/0.46 hoasinduction_lem3_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem2_gthm_decl,type,
% 0.20/0.46 hoasinduction_lem2_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3b_decl,type,
% 0.20/0.46 hoasinduction_lem3b: $o ).
% 0.20/0.46
% 0.20/0.46 thf(substmonoid_decl,type,
% 0.20/0.46 substmonoid: $o ).
% 0.20/0.46
% 0.20/0.46 thf(lamnotvar_decl,type,
% 0.20/0.46 lamnotvar: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3a_decl,type,
% 0.20/0.46 hoasinduction_lem3a: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem1_gthm_decl,type,
% 0.20/0.46 hoasinduction_lem1_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_no_psi_cond_decl,type,
% 0.20/0.46 hoasinduction_no_psi_cond: $o ).
% 0.20/0.46
% 0.20/0.46 thf(induction2_gthm_decl,type,
% 0.20/0.46 induction2_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem2v2_lthm_decl,type,
% 0.20/0.46 pushprop_lem2v2_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasvar_decl,type,
% 0.20/0.46 hoasvar: subst > term > subst > $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoaslamnotap_decl,type,
% 0.20/0.46 hoaslamnotap: $o ).
% 0.20/0.46
% 0.20/0.46 thf(substmonoid_gthm_decl,type,
% 0.20/0.46 substmonoid_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ulamvarsh_decl,type,
% 0.20/0.46 ulamvarsh: $o ).
% 0.20/0.46
% 0.20/0.46 thf(induction2_decl,type,
% 0.20/0.46 induction2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem3v2_decl,type,
% 0.20/0.46 pushprop_lem3v2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem2v2_gthm_decl,type,
% 0.20/0.46 pushprop_lem2v2_gthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem1_lthm_decl,type,
% 0.20/0.46 pushprop_lem1_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3v2_decl,type,
% 0.20/0.46 hoasinduction_lem3v2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(axshiftcons_decl,type,
% 0.20/0.46 axshiftcons: $o ).
% 0.20/0.46
% 0.20/0.46 thf(termmset_decl,type,
% 0.20/0.46 termmset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_lem0_lthm_decl,type,
% 0.20/0.46 pushprop_lem0_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasapnotvar_lthm_decl,type,
% 0.20/0.46 hoasapnotvar_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem3v2_lthm_decl,type,
% 0.20/0.46 hoasinduction_lem3v2_lthm: $o ).
% 0.20/0.46
% 0.20/0.46 thf(pushprop_p_and_p_prime_decl,type,
% 0.20/0.46 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 0.20/0.46
% 0.20/0.46 thf(axvarid_decl,type,
% 0.20/0.46 axvarid: $o ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lthm_3_decl,type,
% 0.20/0.46 hoasinduction_lthm_3: $o ).
% 0.20/0.46
% 0.20/0.46 %----Types of the domains
% 0.20/0.46 thf(d_term_type,type,
% 0.20/0.46 d_term: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(d_subst_type,type,
% 0.20/0.46 d_subst: $tType ).
% 0.20/0.46
% 0.20/0.46 %----Types of the promotion functions
% 0.20/0.46 thf(d2term_decl,type,
% 0.20/0.46 d2term: d_term > term ).
% 0.20/0.46
% 0.20/0.46 thf(d2subst_decl,type,
% 0.20/0.46 d2subst: d_subst > subst ).
% 0.20/0.46
% 0.20/0.46 %----Types of the domain elements
% 0.20/0.46 thf(d_one_decl,type,
% 0.20/0.46 d_one: d_term ).
% 0.20/0.46
% 0.20/0.46 thf(d_id_decl,type,
% 0.20/0.46 d_id: d_subst ).
% 0.20/0.46
% 0.20/0.46 thf(alg444_1,axiom,
% 0.20/0.46 ( ! [T: term] :
% 0.20/0.46 ? [DT: d_term] :
% 0.20/0.46 ( T
% 0.20/0.46 = ( d2term @ DT ) )
% 0.20/0.46 & ! [DT: d_term] : ( DT = d_one )
% 0.20/0.46 & ! [DT1: d_term,DT2: d_term] :
% 0.20/0.46 ( ( ( d2term @ DT1 )
% 0.20/0.46 = ( d2term @ DT2 ) )
% 0.20/0.46 => ( DT1 = DT2 ) )
% 0.20/0.46 & ! [S: subst] :
% 0.20/0.46 ? [DS: d_subst] :
% 0.20/0.46 ( S
% 0.20/0.46 = ( d2subst @ DS ) )
% 0.20/0.46 & ! [DS: d_subst] : ( DS = d_id )
% 0.20/0.46 & ! [DS1: d_subst,DS2: d_subst] :
% 0.20/0.46 ( ( ( d2subst @ DS1 )
% 0.20/0.46 = ( d2subst @ DS2 ) )
% 0.20/0.46 => ( DS1 = DS2 ) )
% 0.20/0.46 & ( one
% 0.20/0.46 = ( d2term @ d_one ) )
% 0.20/0.46 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 0.20/0.46 = ( d2term @ d_one ) )
% 0.20/0.46 & ( ( lam @ ( d2term @ d_one ) )
% 0.20/0.46 = ( d2term @ d_one ) )
% 0.20/0.46 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.20/0.46 = ( d2term @ d_one ) )
% 0.20/0.46 & ( id
% 0.20/0.46 = ( d2subst @ d_id ) )
% 0.20/0.46 & ( sh
% 0.20/0.46 = ( d2subst @ d_id ) )
% 0.20/0.46 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.20/0.46 = ( d2subst @ d_id ) )
% 0.20/0.46 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 0.20/0.46 = ( d2subst @ d_id ) )
% 0.20/0.46 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 0.20/0.46 = ( d2term @ d_one ) )
% 0.20/0.46 & ( hoaslam
% 0.20/0.46 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 0.20/0.46 & ( hoasinduction_p_and_p_prime
% 0.20/0.46 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 0.20/0.46 ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) ) ) )
% 0.20/0.46 & ( pushprop_p_and_p_prime
% 0.20/0.46 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 0.20/0.46 ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 0.20/0.46 & ~ ( var @ ( d2term @ d_one ) )
% 0.20/0.46 & ( pushprop_lem1v2
% 0.20/0.46 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.20/0.46 | ( Q @ one ) ) )
% 0.20/0.46 & pushprop_lem1_gthm
% 0.20/0.46 & axmap
% 0.20/0.46 & pushprop_lem0_gthm
% 0.20/0.46 & ( shinj
% 0.20/0.46 <=> ! [A: term,B: term] :
% 0.20/0.46 ( ( ( sub @ A @ sh )
% 0.20/0.46 != ( sub @ B @ sh ) )
% 0.20/0.46 | ( A = B ) ) )
% 0.20/0.46 & hoasinduction_lem1v2
% 0.20/0.46 & hoasinduction_lem1v2_gthm
% 0.20/0.46 & ( induction2lem
% 0.20/0.46 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.20/0.46 ( ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ B )
% 0.20/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( var @ B )
% 0.20/0.46 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.20/0.46 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 0.20/0.46 & ( hoasinduction_lem3v2_f
% 0.20/0.46 <=> ! [B: term] :
% 0.20/0.46 ~ ! [F: subst > term > term] :
% 0.20/0.46 ~ ! [A: term,M: subst] :
% 0.20/0.46 ( ( sub @ B @ ( push @ A @ M ) )
% 0.20/0.46 = ( F @ M @ A ) ) )
% 0.20/0.46 & axvarshift
% 0.20/0.46 & ( hoasapinj2
% 0.20/0.46 <=> ! [A: term,B: term,C: term,D: term] :
% 0.20/0.46 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.20/0.46 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.20/0.46 | ( C = D ) ) )
% 0.20/0.46 & hoasapnotvar_gthm
% 0.20/0.46 & ( hoasapinj1
% 0.20/0.46 <=> ! [A: term,B: term,C: term,D: term] :
% 0.20/0.46 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.20/0.46 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.20/0.46 | ( A = B ) ) )
% 0.20/0.46 & ~ ulamvar1
% 0.20/0.46 & ( induction2lem_lthm
% 0.20/0.46 <=> ( ! [A: term,M: subst] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.20/0.46 => ( ! [A: term,M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.20/0.46 => ( ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ M @ id ) )
% 0.20/0.46 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 0.20/0.46 ( ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ A )
% 0.20/0.46 | ( P @ A ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ( P @ Bound_variable_2140 ) )
% 0.20/0.46 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.20/0.46 ( ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ B )
% 0.20/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( var @ B )
% 0.20/0.46 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.20/0.46 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 0.20/0.46 & hoasinduction_lem3v2_gthm
% 0.20/0.46 & apnotvar
% 0.20/0.46 & pushprop_lthm_orig
% 0.20/0.46 & ( hoasinduction_lem3v2_f_lthm
% 0.20/0.46 <=> ! [B: term] :
% 0.20/0.46 ~ ! [F: subst > term > term] :
% 0.20/0.46 ~ ! [A: term,M: subst] :
% 0.20/0.46 ( ( sub @ B @ ( push @ A @ M ) )
% 0.20/0.46 = ( F @ M @ A ) ) )
% 0.20/0.46 & ( hoasinduction_lthm
% 0.20/0.46 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.20/0.46 ( ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ A )
% 0.20/0.46 | ( P @ A ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ B )
% 0.20/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ( P @ Bound_variable_2367 ) )
% 0.20/0.46 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.20/0.46 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.20/0.46 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 0.20/0.46 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 0.20/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ ( sub @ A @ id ) )
% 0.20/0.46 | ( P @ id @ A @ id ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 0.20/0.46 & ( hoasinduction_no_psi_cond_lthm
% 0.20/0.46 <=> ( ! [P: subst > term > subst > $o] :
% 0.20/0.46 ~ ! [Q: term > $o] :
% 0.20/0.46 ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.20/0.46 ( ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ A )
% 0.20/0.46 | ( P @ A ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ B )
% 0.20/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ( P @ Bound_variable_2367 ) )
% 0.20/0.46 => ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( Q @ B )
% 0.20/0.46 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.20/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.20/0.46 ( ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 0.20/0.46 & ( hoaslaminj
% 0.20/0.46 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.20/0.46 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ( ( lam @ ( F @ sh @ one ) )
% 0.20/0.46 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.20/0.46 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.20/0.46 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 0.20/0.46 & ( hoasinduction_lem3aaa
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 0.20/0.46 ( ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 0.20/0.46 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 0.20/0.46 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 0.20/0.46 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 0.20/0.46 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 0.20/0.46 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 0.20/0.46 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.20/0.46 & induction2lem_gthm
% 0.20/0.46 & ( hoasinduction_lem3aa_lthm
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.20/0.46 & ( hoasinduction_lem3
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 0.20/0.46 & ( hoasinduction_lem2
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.20/0.46 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.20/0.46 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 0.20/0.46 & ( termmset_lthm
% 0.20/0.46 <=> ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) ) ) )
% 0.20/0.46 & hoasinduction_lem1
% 0.20/0.46 & hoaslamnotap_lthm
% 0.20/0.46 & ( pushprop_lem1v2_lthm
% 0.20/0.46 <=> ( ! [A: term,M: subst] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.20/0.46 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.20/0.46 | ( Q @ one ) ) ) )
% 0.20/0.46 & hoasapnotvar
% 0.20/0.46 & ( hoasinduction_lem0
% 0.20/0.46 <=> ! [P: subst > term > subst > $o] :
% 0.20/0.46 ~ ! [Q: term > $o] :
% 0.20/0.46 ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) ) )
% 0.20/0.46 & ( hoasinduction
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ ( sub @ A @ id ) )
% 0.20/0.46 | ( P @ id @ A @ id ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 0.20/0.46 & hoasinduction_gthm
% 0.20/0.46 & axapp
% 0.20/0.46 & hoaslamnotvar_lthm
% 0.20/0.46 & pushprop_lem3v2_lthm
% 0.20/0.46 & ( hoasinduction_lem3b_lthm
% 0.20/0.46 <=> ! [B: term] :
% 0.20/0.46 ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ( F @ sh @ one )
% 0.20/0.46 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.20/0.46 & ulamvarind
% 0.20/0.46 & ( induction
% 0.20/0.46 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 0.20/0.46 ( ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ A )
% 0.20/0.46 | ( P @ A ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ( P @ Bound_variable_2140 ) ) )
% 0.20/0.46 & ( hoasinduction_lem3a_lthm
% 0.20/0.46 <=> ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.20/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 0.20/0.46 & termmset_gthm
% 0.20/0.46 & ( hoasinduction_lem3aa
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.20/0.46 & pushprop_lem1v2_gthm
% 0.20/0.46 & hoaslamnotap_gthm
% 0.20/0.46 & hoaslamnotvar_gthm
% 0.20/0.46 & hoasinduction_lem3b_gthm
% 0.20/0.46 & pushprop_lem2v2
% 0.20/0.46 & hoasinduction_lem3a_gthm
% 0.20/0.46 & axclos
% 0.20/0.46 & axassoc
% 0.20/0.46 & ( hoasinduction_lem2v2
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 | ~ ( Q @ Bound_variable_2820 )
% 0.20/0.46 | ~ ( Q @ Bound_variable_2822 )
% 0.20/0.46 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 0.20/0.46 & pushprop_lthm
% 0.20/0.46 & ( apinj2
% 0.20/0.46 <=> ! [A: term,B: term,C: term,D: term] :
% 0.20/0.46 ( ( ( ap @ A @ C )
% 0.20/0.46 != ( ap @ B @ D ) )
% 0.20/0.46 | ( C = D ) ) )
% 0.20/0.46 & ( apinj1
% 0.20/0.46 <=> ! [A: term,B: term,C: term,D: term] :
% 0.20/0.46 ( ( ( ap @ A @ C )
% 0.20/0.46 != ( ap @ B @ D ) )
% 0.20/0.46 | ( A = B ) ) )
% 0.20/0.46 & ( hoasapinj2_lthm
% 0.20/0.46 <=> ( ! [A: term,B: term,C: term,D: term] :
% 0.20/0.46 ( ( ( ap @ A @ C )
% 0.20/0.46 != ( ap @ B @ D ) )
% 0.20/0.46 | ( C = D ) )
% 0.20/0.46 => ! [A: term,B: term,C: term,D: term] :
% 0.20/0.46 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.20/0.46 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.20/0.46 | ( C = D ) ) ) )
% 0.20/0.46 & ( hoasinduction_lem3v2a
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( Q @ B )
% 0.20/0.46 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 0.20/0.46 & ( hoasapinj1_lthm
% 0.20/0.46 <=> ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ( ! [A: term,B: term,C: term,D: term] :
% 0.20/0.46 ( ( ( ap @ A @ C )
% 0.20/0.46 != ( ap @ B @ D ) )
% 0.20/0.46 | ( A = B ) )
% 0.20/0.46 => ! [A: term,B: term,C: term,D: term] :
% 0.20/0.46 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.20/0.46 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.20/0.46 | ( A = B ) ) ) ) )
% 0.20/0.46 & ( hoaslaminj_lthm
% 0.20/0.46 <=> ( ! [A: term,M: subst] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.20/0.46 => ( ! [A: term,M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.20/0.46 => ( ! [A: term,B: term] :
% 0.20/0.46 ( ( ( lam @ A )
% 0.20/0.46 != ( lam @ B ) )
% 0.20/0.46 | ( A = B ) )
% 0.20/0.46 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.20/0.46 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ( ( lam @ ( F @ sh @ one ) )
% 0.20/0.46 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.20/0.46 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.20/0.46 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 0.20/0.46 & ( axvarcons
% 0.20/0.46 <=> ! [A: term,M: subst] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 0.20/0.46 & ( axscons
% 0.20/0.46 <=> ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 0.20/0.46 & hoasinduction_lem2v2_gthm
% 0.20/0.46 & ( axidr
% 0.20/0.46 <=> ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ M @ id ) ) )
% 0.20/0.46 & ( pushprop_lem1
% 0.20/0.46 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 0.20/0.46 & ( laminj
% 0.20/0.46 <=> ! [A: term,B: term] :
% 0.20/0.46 ( ( ( lam @ A )
% 0.20/0.46 != ( lam @ B ) )
% 0.20/0.46 | ( A = B ) ) )
% 0.20/0.46 & ( hoasinduction_lem3_lthm
% 0.20/0.46 <=> ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.20/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 0.20/0.46 & ( pushprop_lem0
% 0.20/0.46 <=> ! [P: term > $o,A: term,M: subst] :
% 0.20/0.46 ~ ! [Q: term > $o] :
% 0.20/0.46 ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.20/0.46 & pushprop_gthm
% 0.20/0.46 & axabs
% 0.20/0.46 & ( hoasinduction_lem3v2a_lthm
% 0.20/0.46 <=> ( ! [B: term] :
% 0.20/0.46 ~ ! [F: subst > term > term] :
% 0.20/0.46 ~ ! [A: term,M: subst] :
% 0.20/0.46 ( ( sub @ B @ ( push @ A @ M ) )
% 0.20/0.46 = ( F @ M @ A ) )
% 0.20/0.46 => ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( Q @ B )
% 0.20/0.46 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 0.20/0.46 & hoasinduction_lem2_lthm
% 0.20/0.46 & hoasapinj2_gthm
% 0.20/0.46 & hoasinduction_lem1_lthm
% 0.20/0.46 & ~ lamnotap
% 0.20/0.46 & hoasapinj1_gthm
% 0.20/0.46 & hoaslamnotvar
% 0.20/0.46 & ( axidl
% 0.20/0.46 <=> ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ id @ M ) ) )
% 0.20/0.46 & hoaslaminj_gthm
% 0.20/0.46 & ( induction2_lthm
% 0.20/0.46 <=> ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.20/0.46 ( ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ B )
% 0.20/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( var @ B )
% 0.20/0.46 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.20/0.46 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 0.20/0.46 => ! [P: term > $o,Bound_variable_2367: term] :
% 0.20/0.46 ( ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ A )
% 0.20/0.46 | ( P @ A ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ B )
% 0.20/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ( P @ Bound_variable_2367 ) ) ) ) )
% 0.20/0.46 & ( hoasinduction_lem0_lthm
% 0.20/0.46 <=> ! [P: subst > term > subst > $o] :
% 0.20/0.46 ~ ! [Q: term > $o] :
% 0.20/0.46 ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) ) )
% 0.20/0.46 & ( substmonoid_lthm
% 0.20/0.46 <=> ( ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ id @ M ) )
% 0.20/0.46 => ( ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ M @ id ) )
% 0.20/0.46 => ( ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ id @ M ) )
% 0.20/0.46 & ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ M @ id ) ) ) ) ) )
% 0.20/0.46 & pushprop
% 0.20/0.46 & hoasinduction_lem3_gthm
% 0.20/0.46 & hoasinduction_lem2_gthm
% 0.20/0.46 & ( hoasinduction_lem3b
% 0.20/0.46 <=> ! [B: term] :
% 0.20/0.46 ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ( F @ sh @ one )
% 0.20/0.46 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.20/0.46 & ( substmonoid
% 0.20/0.46 <=> ( ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ id @ M ) )
% 0.20/0.46 & ! [M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ M @ id ) ) ) )
% 0.20/0.46 & lamnotvar
% 0.20/0.46 & ( hoasinduction_lem3a
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 0.20/0.46 & hoasinduction_lem1_gthm
% 0.20/0.46 & ( hoasinduction_no_psi_cond
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.20/0.46 ( ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 0.20/0.46 & induction2_gthm
% 0.20/0.46 & pushprop_lem2v2_lthm
% 0.20/0.46 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.20/0.46 & ( hoaslamnotap
% 0.20/0.46 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ( ( lam @ ( F @ sh @ one ) )
% 0.20/0.46 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 0.20/0.46 & substmonoid_gthm
% 0.20/0.46 & ulamvarsh
% 0.20/0.46 & ( induction2
% 0.20/0.46 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 0.20/0.46 ( ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ A )
% 0.20/0.46 | ( P @ A ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ B )
% 0.20/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ( P @ Bound_variable_2367 ) ) )
% 0.20/0.46 & pushprop_lem3v2
% 0.20/0.46 & pushprop_lem2v2_gthm
% 0.20/0.46 & ( pushprop_lem1_lthm
% 0.20/0.46 <=> ( ! [A: term,M: subst] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.20/0.46 => ( ! [A: term,M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.20/0.46 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 0.20/0.46 & ( hoasinduction_lem3v2
% 0.20/0.46 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( Q @ B )
% 0.20/0.46 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 0.20/0.46 & ( axshiftcons
% 0.20/0.46 <=> ! [A: term,M: subst] :
% 0.20/0.46 ( M
% 0.20/0.46 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 0.20/0.46 & ( termmset
% 0.20/0.46 <=> ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) ) )
% 0.20/0.46 & ( pushprop_lem0_lthm
% 0.20/0.46 <=> ! [P: term > $o,A: term,M: subst] :
% 0.20/0.46 ~ ! [Q: term > $o] :
% 0.20/0.46 ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.20/0.46 & hoasapnotvar_lthm
% 0.20/0.46 & ( hoasinduction_lem3v2_lthm
% 0.20/0.46 <=> ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( Q @ B )
% 0.20/0.46 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 0.20/0.46 & ( axvarid
% 0.20/0.46 <=> ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) ) )
% 0.20/0.46 & ( hoasinduction_lthm_3
% 0.20/0.46 <=> ( ! [P: subst > term > subst > $o] :
% 0.20/0.46 ~ ! [Q: term > $o] :
% 0.20/0.46 ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.20/0.46 ( ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ A )
% 0.20/0.46 | ( P @ A ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ A )
% 0.20/0.46 | ~ ( P @ B )
% 0.20/0.46 | ( P @ ( ap @ A @ B ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ! [B: term] :
% 0.20/0.46 ( ~ ( P @ B )
% 0.20/0.46 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( P @ ( lam @ A ) ) )
% 0.20/0.46 | ( P @ Bound_variable_2367 ) )
% 0.20/0.46 => ( ! [A: term] :
% 0.20/0.46 ( A
% 0.20/0.46 = ( sub @ A @ id ) )
% 0.20/0.46 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.20/0.46 ( ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ~ ! [X: term] :
% 0.20/0.46 ( ( Q @ X )
% 0.20/0.46 = ( P @ id @ X @ id ) )
% 0.20/0.46 | ~ ! [B: term] :
% 0.20/0.46 ( ~ ( Q @ B )
% 0.20/0.46 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.20/0.46 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.20/0.46 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( var @ ( sub @ A @ id ) )
% 0.20/0.46 | ( P @ id @ A @ id ) )
% 0.20/0.46 | ~ ! [A: term,B: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ~ ( P @ id @ B @ id )
% 0.20/0.46 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.20/0.46 | ~ ! [F: subst > term > term] :
% 0.20/0.46 ( ~ ! [M: subst,A: term,N: subst] :
% 0.20/0.46 ( ( sub @ ( F @ M @ A ) @ N )
% 0.20/0.46 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.20/0.46 | ~ ! [A: term] :
% 0.20/0.46 ( ~ ( P @ id @ A @ id )
% 0.20/0.46 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.20/0.46 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.20/0.46 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(hoasinduction_lem1,conjecture,
% 0.20/0.46 ( hoasinduction_lem1
% 0.20/0.46 = ( ! [P: subst > term > subst > $o] :
% 0.20/0.46 ( ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ( P @ M @ A @ ( comp @ K @ N ) )
% 0.20/0.46 => ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.20/0.46 => ( ! [M: subst,A: term,N: subst,K: subst] :
% 0.20/0.46 ( ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.20/0.46 => ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.20/0.46 => ( ! [A: term] :
% 0.20/0.46 ( ( hoasvar @ id @ A @ id )
% 0.20/0.46 => ( P @ id @ A @ id ) )
% 0.20/0.46 => ! [A: term] :
% 0.20/0.46 ( ( var @ A )
% 0.20/0.46 => ( P @ id @ A @ id ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 ------------------------
% 2.14/1.39 ---- Embedded in THF ---
% 2.14/1.39 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.14/1.39 %%% Generated on Tue May 05 09:07:15 EDT 2026
% 2.14/1.39 %%% using '$$dhol' embedding, version 1.3.0.
% 2.14/1.39 %%% Logic specification used:
% 2.14/1.39 %%% thf(spec, logic, $$dhol).
% 2.14/1.39
% 2.14/1.39 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.sO7hdZQLTS/DTF2DTF1662.p
% See solution above
% 2.14/1.39 ------------------------
% 2.41/1.41 ---- Cleaned THF ---
% 2.41/1.41 thf(term_type,type,
% 2.41/1.41 term: $tType ).
% 2.41/1.41
% 2.41/1.41 thf(subst_type,type,
% 2.41/1.41 subst: $tType ).
% 2.41/1.41
% 2.41/1.41 thf(one_decl,type,
% 2.41/1.41 one: term ).
% 2.41/1.41
% 2.41/1.41 thf(ap_decl,type,
% 2.41/1.41 ap: term > term > term ).
% 2.41/1.41
% 2.41/1.41 thf(lam_decl,type,
% 2.41/1.41 lam: term > term ).
% 2.41/1.41
% 2.41/1.41 thf(sub_decl,type,
% 2.41/1.41 sub: term > subst > term ).
% 2.41/1.41
% 2.41/1.41 thf(id_decl,type,
% 2.41/1.41 id: subst ).
% 2.41/1.41
% 2.41/1.41 thf(sh_decl,type,
% 2.41/1.41 sh: subst ).
% 2.41/1.41
% 2.41/1.41 thf(push_decl,type,
% 2.41/1.41 push: term > subst > subst ).
% 2.41/1.41
% 2.41/1.41 thf(comp_decl,type,
% 2.41/1.41 comp: subst > subst > subst ).
% 2.41/1.41
% 2.41/1.41 thf(var_decl,type,
% 2.41/1.41 var: term > $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem1v2_decl,type,
% 2.41/1.41 pushprop_lem1v2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem1_gthm_decl,type,
% 2.41/1.41 pushprop_lem1_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axmap_decl,type,
% 2.41/1.41 axmap: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem0_gthm_decl,type,
% 2.41/1.41 pushprop_lem0_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(shinj_decl,type,
% 2.41/1.41 shinj: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem1v2_decl,type,
% 2.41/1.41 hoasinduction_lem1v2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem1v2_gthm_decl,type,
% 2.41/1.41 hoasinduction_lem1v2_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasap_decl,type,
% 2.41/1.41 hoasap: subst > term > subst > term > term ).
% 2.41/1.41
% 2.41/1.41 thf(induction2lem_decl,type,
% 2.41/1.41 induction2lem: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3v2_f_decl,type,
% 2.41/1.41 hoasinduction_lem3v2_f: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axvarshift_decl,type,
% 2.41/1.41 axvarshift: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapinj2_decl,type,
% 2.41/1.41 hoasapinj2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapnotvar_gthm_decl,type,
% 2.41/1.41 hoasapnotvar_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapinj1_decl,type,
% 2.41/1.41 hoasapinj1: $o ).
% 2.41/1.41
% 2.41/1.41 thf(ulamvar1_decl,type,
% 2.41/1.41 ulamvar1: $o ).
% 2.41/1.41
% 2.41/1.41 thf(induction2lem_lthm_decl,type,
% 2.41/1.41 induction2lem_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3v2_gthm_decl,type,
% 2.41/1.41 hoasinduction_lem3v2_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(apnotvar_decl,type,
% 2.41/1.41 apnotvar: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lthm_orig_decl,type,
% 2.41/1.41 pushprop_lthm_orig: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem3v2_f_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lthm_decl,type,
% 2.41/1.41 hoasinduction_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 2.41/1.41 hoasinduction_no_psi_cond_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslaminj_decl,type,
% 2.41/1.41 hoaslaminj: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3aaa_decl,type,
% 2.41/1.41 hoasinduction_lem3aaa: $o ).
% 2.41/1.41
% 2.41/1.41 thf(induction2lem_gthm_decl,type,
% 2.41/1.41 induction2lem_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3aa_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem3aa_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3_decl,type,
% 2.41/1.41 hoasinduction_lem3: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem2_decl,type,
% 2.41/1.41 hoasinduction_lem2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(termmset_lthm_decl,type,
% 2.41/1.41 termmset_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem1_decl,type,
% 2.41/1.41 hoasinduction_lem1: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslamnotap_lthm_decl,type,
% 2.41/1.41 hoaslamnotap_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem1v2_lthm_decl,type,
% 2.41/1.41 pushprop_lem1v2_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapnotvar_decl,type,
% 2.41/1.41 hoasapnotvar: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem0_decl,type,
% 2.41/1.41 hoasinduction_lem0: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_decl,type,
% 2.41/1.41 hoasinduction: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_gthm_decl,type,
% 2.41/1.41 hoasinduction_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axapp_decl,type,
% 2.41/1.41 axapp: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslamnotvar_lthm_decl,type,
% 2.41/1.41 hoaslamnotvar_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem3v2_lthm_decl,type,
% 2.41/1.41 pushprop_lem3v2_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3b_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem3b_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(ulamvarind_decl,type,
% 2.41/1.41 ulamvarind: $o ).
% 2.41/1.41
% 2.41/1.41 thf(induction_decl,type,
% 2.41/1.41 induction: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3a_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem3a_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(termmset_gthm_decl,type,
% 2.41/1.41 termmset_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3aa_decl,type,
% 2.41/1.41 hoasinduction_lem3aa: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem1v2_gthm_decl,type,
% 2.41/1.41 pushprop_lem1v2_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslamnotap_gthm_decl,type,
% 2.41/1.41 hoaslamnotap_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslamnotvar_gthm_decl,type,
% 2.41/1.41 hoaslamnotvar_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3b_gthm_decl,type,
% 2.41/1.41 hoasinduction_lem3b_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem2v2_decl,type,
% 2.41/1.41 pushprop_lem2v2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3a_gthm_decl,type,
% 2.41/1.41 hoasinduction_lem3a_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axclos_decl,type,
% 2.41/1.41 axclos: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axassoc_decl,type,
% 2.41/1.41 axassoc: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem2v2_decl,type,
% 2.41/1.41 hoasinduction_lem2v2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lthm_decl,type,
% 2.41/1.41 pushprop_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(apinj2_decl,type,
% 2.41/1.41 apinj2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(apinj1_decl,type,
% 2.41/1.41 apinj1: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapinj2_lthm_decl,type,
% 2.41/1.41 hoasapinj2_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3v2a_decl,type,
% 2.41/1.41 hoasinduction_lem3v2a: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapinj1_lthm_decl,type,
% 2.41/1.41 hoasapinj1_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslaminj_lthm_decl,type,
% 2.41/1.41 hoaslaminj_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axvarcons_decl,type,
% 2.41/1.41 axvarcons: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslam_decl,type,
% 2.41/1.41 hoaslam: subst > ( subst > term > term ) > term ).
% 2.41/1.41
% 2.41/1.41 thf(axscons_decl,type,
% 2.41/1.41 axscons: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem2v2_gthm_decl,type,
% 2.41/1.41 hoasinduction_lem2v2_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axidr_decl,type,
% 2.41/1.41 axidr: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem1_decl,type,
% 2.41/1.41 pushprop_lem1: $o ).
% 2.41/1.41
% 2.41/1.41 thf(laminj_decl,type,
% 2.41/1.41 laminj: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem3_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem0_decl,type,
% 2.41/1.41 pushprop_lem0: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_gthm_decl,type,
% 2.41/1.41 pushprop_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axabs_decl,type,
% 2.41/1.41 axabs: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem3v2a_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem2_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem2_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapinj2_gthm_decl,type,
% 2.41/1.41 hoasapinj2_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_p_and_p_prime_decl,type,
% 2.41/1.41 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem1_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem1_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(lamnotap_decl,type,
% 2.41/1.41 lamnotap: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapinj1_gthm_decl,type,
% 2.41/1.41 hoasapinj1_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslamnotvar_decl,type,
% 2.41/1.41 hoaslamnotvar: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axidl_decl,type,
% 2.41/1.41 axidl: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslaminj_gthm_decl,type,
% 2.41/1.41 hoaslaminj_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(induction2_lthm_decl,type,
% 2.41/1.41 induction2_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem0_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem0_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(substmonoid_lthm_decl,type,
% 2.41/1.41 substmonoid_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_decl,type,
% 2.41/1.41 pushprop: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3_gthm_decl,type,
% 2.41/1.41 hoasinduction_lem3_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem2_gthm_decl,type,
% 2.41/1.41 hoasinduction_lem2_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3b_decl,type,
% 2.41/1.41 hoasinduction_lem3b: $o ).
% 2.41/1.41
% 2.41/1.41 thf(substmonoid_decl,type,
% 2.41/1.41 substmonoid: $o ).
% 2.41/1.41
% 2.41/1.41 thf(lamnotvar_decl,type,
% 2.41/1.41 lamnotvar: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3a_decl,type,
% 2.41/1.41 hoasinduction_lem3a: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem1_gthm_decl,type,
% 2.41/1.41 hoasinduction_lem1_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_no_psi_cond_decl,type,
% 2.41/1.41 hoasinduction_no_psi_cond: $o ).
% 2.41/1.41
% 2.41/1.41 thf(induction2_gthm_decl,type,
% 2.41/1.41 induction2_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem2v2_lthm_decl,type,
% 2.41/1.41 pushprop_lem2v2_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasvar_decl,type,
% 2.41/1.41 hoasvar: subst > term > subst > $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoaslamnotap_decl,type,
% 2.41/1.41 hoaslamnotap: $o ).
% 2.41/1.41
% 2.41/1.41 thf(substmonoid_gthm_decl,type,
% 2.41/1.41 substmonoid_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(ulamvarsh_decl,type,
% 2.41/1.41 ulamvarsh: $o ).
% 2.41/1.41
% 2.41/1.41 thf(induction2_decl,type,
% 2.41/1.41 induction2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem3v2_decl,type,
% 2.41/1.41 pushprop_lem3v2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem2v2_gthm_decl,type,
% 2.41/1.41 pushprop_lem2v2_gthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem1_lthm_decl,type,
% 2.41/1.41 pushprop_lem1_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3v2_decl,type,
% 2.41/1.41 hoasinduction_lem3v2: $o ).
% 2.41/1.41
% 2.41/1.41 thf(axshiftcons_decl,type,
% 2.41/1.41 axshiftcons: $o ).
% 2.41/1.41
% 2.41/1.41 thf(termmset_decl,type,
% 2.41/1.41 termmset: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_lem0_lthm_decl,type,
% 2.41/1.41 pushprop_lem0_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasapnotvar_lthm_decl,type,
% 2.41/1.41 hoasapnotvar_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem3v2_lthm_decl,type,
% 2.41/1.41 hoasinduction_lem3v2_lthm: $o ).
% 2.41/1.41
% 2.41/1.41 thf(pushprop_p_and_p_prime_decl,type,
% 2.41/1.41 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 2.41/1.41
% 2.41/1.41 thf(axvarid_decl,type,
% 2.41/1.41 axvarid: $o ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lthm_3_decl,type,
% 2.41/1.41 hoasinduction_lthm_3: $o ).
% 2.41/1.41
% 2.41/1.41 thf(d_term_type,type,
% 2.41/1.41 d_term: $tType ).
% 2.41/1.41
% 2.41/1.41 thf(d_subst_type,type,
% 2.41/1.41 d_subst: $tType ).
% 2.41/1.41
% 2.41/1.41 thf(d2term_decl,type,
% 2.41/1.41 d2term: d_term > term ).
% 2.41/1.41
% 2.41/1.41 thf(d2subst_decl,type,
% 2.41/1.41 d2subst: d_subst > subst ).
% 2.41/1.41
% 2.41/1.41 thf(d_one_decl,type,
% 2.41/1.41 d_one: d_term ).
% 2.41/1.41
% 2.41/1.41 thf(d_id_decl,type,
% 2.41/1.41 d_id: d_subst ).
% 2.41/1.41
% 2.41/1.41 thf(alg444_1,axiom,
% 2.41/1.41 ( ! [T: term] :
% 2.41/1.41 ? [DT: d_term] :
% 2.41/1.41 ( T
% 2.41/1.41 = ( d2term @ DT ) )
% 2.41/1.41 & ! [DT: d_term] : ( DT = d_one )
% 2.41/1.41 & ! [DT1: d_term,DT2: d_term] :
% 2.41/1.41 ( ( ( d2term @ DT1 )
% 2.41/1.41 = ( d2term @ DT2 ) )
% 2.41/1.41 => ( DT1 = DT2 ) )
% 2.41/1.41 & ! [S: subst] :
% 2.41/1.41 ? [DS: d_subst] :
% 2.41/1.41 ( S
% 2.41/1.41 = ( d2subst @ DS ) )
% 2.41/1.41 & ! [DS: d_subst] : ( DS = d_id )
% 2.41/1.41 & ! [DS1: d_subst,DS2: d_subst] :
% 2.41/1.41 ( ( ( d2subst @ DS1 )
% 2.41/1.41 = ( d2subst @ DS2 ) )
% 2.41/1.41 => ( DS1 = DS2 ) )
% 2.41/1.41 & ( one
% 2.41/1.41 = ( d2term @ d_one ) )
% 2.41/1.41 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 2.41/1.41 = ( d2term @ d_one ) )
% 2.41/1.41 & ( ( lam @ ( d2term @ d_one ) )
% 2.41/1.41 = ( d2term @ d_one ) )
% 2.41/1.41 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.41/1.41 = ( d2term @ d_one ) )
% 2.41/1.41 & ( id
% 2.41/1.41 = ( d2subst @ d_id ) )
% 2.41/1.41 & ( sh
% 2.41/1.41 = ( d2subst @ d_id ) )
% 2.41/1.41 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.41/1.41 = ( d2subst @ d_id ) )
% 2.41/1.41 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 2.41/1.41 = ( d2subst @ d_id ) )
% 2.41/1.41 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 2.41/1.41 = ( d2term @ d_one ) )
% 2.41/1.41 & ( hoaslam
% 2.41/1.41 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 2.41/1.41 & ( hoasinduction_p_and_p_prime
% 2.41/1.41 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 2.41/1.41 ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) ) ) )
% 2.41/1.41 & ( pushprop_p_and_p_prime
% 2.41/1.41 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 2.41/1.41 ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 2.41/1.41 & ~ ( var @ ( d2term @ d_one ) )
% 2.41/1.41 & ( pushprop_lem1v2
% 2.41/1.41 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.41/1.41 | ( Q @ one ) ) )
% 2.41/1.41 & pushprop_lem1_gthm
% 2.41/1.41 & axmap
% 2.41/1.41 & pushprop_lem0_gthm
% 2.41/1.41 & ( shinj
% 2.41/1.41 <=> ! [A: term,B: term] :
% 2.41/1.41 ( ( ( sub @ A @ sh )
% 2.41/1.41 != ( sub @ B @ sh ) )
% 2.41/1.41 | ( A = B ) ) )
% 2.41/1.41 & hoasinduction_lem1v2
% 2.41/1.41 & hoasinduction_lem1v2_gthm
% 2.41/1.41 & ( induction2lem
% 2.41/1.41 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.41/1.41 ( ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ B )
% 2.41/1.41 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( var @ B )
% 2.41/1.41 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.41/1.41 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 2.41/1.41 & ( hoasinduction_lem3v2_f
% 2.41/1.41 <=> ! [B: term] :
% 2.41/1.41 ~ ! [F: subst > term > term] :
% 2.41/1.41 ~ ! [A: term,M: subst] :
% 2.41/1.41 ( ( sub @ B @ ( push @ A @ M ) )
% 2.41/1.41 = ( F @ M @ A ) ) )
% 2.41/1.41 & axvarshift
% 2.41/1.41 & ( hoasapinj2
% 2.41/1.41 <=> ! [A: term,B: term,C: term,D: term] :
% 2.41/1.41 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.41/1.41 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.41/1.41 | ( C = D ) ) )
% 2.41/1.41 & hoasapnotvar_gthm
% 2.41/1.41 & ( hoasapinj1
% 2.41/1.41 <=> ! [A: term,B: term,C: term,D: term] :
% 2.41/1.41 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.41/1.41 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.41/1.41 | ( A = B ) ) )
% 2.41/1.41 & ~ ulamvar1
% 2.41/1.41 & ( induction2lem_lthm
% 2.41/1.41 <=> ( ! [A: term,M: subst] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.41/1.41 => ( ! [A: term,M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.41/1.41 => ( ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ M @ id ) )
% 2.41/1.41 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 2.41/1.41 ( ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ A )
% 2.41/1.41 | ( P @ A ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ( P @ Bound_variable_2140 ) )
% 2.41/1.41 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.41/1.41 ( ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ B )
% 2.41/1.41 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( var @ B )
% 2.41/1.41 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.41/1.41 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 2.41/1.41 & hoasinduction_lem3v2_gthm
% 2.41/1.41 & apnotvar
% 2.41/1.41 & pushprop_lthm_orig
% 2.41/1.41 & ( hoasinduction_lem3v2_f_lthm
% 2.41/1.41 <=> ! [B: term] :
% 2.41/1.41 ~ ! [F: subst > term > term] :
% 2.41/1.41 ~ ! [A: term,M: subst] :
% 2.41/1.41 ( ( sub @ B @ ( push @ A @ M ) )
% 2.41/1.41 = ( F @ M @ A ) ) )
% 2.41/1.41 & ( hoasinduction_lthm
% 2.41/1.41 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.41/1.41 ( ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ A )
% 2.41/1.41 | ( P @ A ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ B )
% 2.41/1.41 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ( P @ Bound_variable_2367 ) )
% 2.41/1.41 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.41/1.41 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.41/1.41 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 2.41/1.41 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 2.41/1.41 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ ( sub @ A @ id ) )
% 2.41/1.41 | ( P @ id @ A @ id ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 2.41/1.41 & ( hoasinduction_no_psi_cond_lthm
% 2.41/1.41 <=> ( ! [P: subst > term > subst > $o] :
% 2.41/1.41 ~ ! [Q: term > $o] :
% 2.41/1.41 ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.41/1.41 ( ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ A )
% 2.41/1.41 | ( P @ A ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ B )
% 2.41/1.41 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ( P @ Bound_variable_2367 ) )
% 2.41/1.41 => ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( Q @ B )
% 2.41/1.41 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.41/1.41 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.41/1.41 ( ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 2.41/1.41 & ( hoaslaminj
% 2.41/1.41 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.41/1.41 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ( ( lam @ ( F @ sh @ one ) )
% 2.41/1.41 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.41/1.41 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.41/1.41 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 2.41/1.41 & ( hoasinduction_lem3aaa
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 2.41/1.41 ( ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 2.41/1.41 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 2.41/1.41 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 2.41/1.41 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 2.41/1.41 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 2.41/1.41 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 2.41/1.41 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.41/1.41 & induction2lem_gthm
% 2.41/1.41 & ( hoasinduction_lem3aa_lthm
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.41/1.41 & ( hoasinduction_lem3
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 2.41/1.41 & ( hoasinduction_lem2
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.41/1.41 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 2.41/1.41 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 2.41/1.41 & ( termmset_lthm
% 2.41/1.41 <=> ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) ) ) )
% 2.41/1.41 & hoasinduction_lem1
% 2.41/1.41 & hoaslamnotap_lthm
% 2.41/1.41 & ( pushprop_lem1v2_lthm
% 2.41/1.41 <=> ( ! [A: term,M: subst] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.41/1.41 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 2.41/1.41 | ( Q @ one ) ) ) )
% 2.41/1.41 & hoasapnotvar
% 2.41/1.41 & ( hoasinduction_lem0
% 2.41/1.41 <=> ! [P: subst > term > subst > $o] :
% 2.41/1.41 ~ ! [Q: term > $o] :
% 2.41/1.41 ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) ) )
% 2.41/1.41 & ( hoasinduction
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ ( sub @ A @ id ) )
% 2.41/1.41 | ( P @ id @ A @ id ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 2.41/1.41 & hoasinduction_gthm
% 2.41/1.41 & axapp
% 2.41/1.41 & hoaslamnotvar_lthm
% 2.41/1.41 & pushprop_lem3v2_lthm
% 2.41/1.41 & ( hoasinduction_lem3b_lthm
% 2.41/1.41 <=> ! [B: term] :
% 2.41/1.41 ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ( F @ sh @ one )
% 2.41/1.41 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.41/1.41 & ulamvarind
% 2.41/1.41 & ( induction
% 2.41/1.41 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 2.41/1.41 ( ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ A )
% 2.41/1.41 | ( P @ A ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ( P @ Bound_variable_2140 ) ) )
% 2.41/1.41 & ( hoasinduction_lem3a_lthm
% 2.41/1.41 <=> ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.41/1.41 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 2.41/1.41 & termmset_gthm
% 2.41/1.41 & ( hoasinduction_lem3aa
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 2.41/1.41 & pushprop_lem1v2_gthm
% 2.41/1.41 & hoaslamnotap_gthm
% 2.41/1.41 & hoaslamnotvar_gthm
% 2.41/1.41 & hoasinduction_lem3b_gthm
% 2.41/1.41 & pushprop_lem2v2
% 2.41/1.41 & hoasinduction_lem3a_gthm
% 2.41/1.41 & axclos
% 2.41/1.41 & axassoc
% 2.41/1.41 & ( hoasinduction_lem2v2
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 | ~ ( Q @ Bound_variable_2820 )
% 2.41/1.41 | ~ ( Q @ Bound_variable_2822 )
% 2.41/1.41 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 2.41/1.41 & pushprop_lthm
% 2.41/1.41 & ( apinj2
% 2.41/1.41 <=> ! [A: term,B: term,C: term,D: term] :
% 2.41/1.41 ( ( ( ap @ A @ C )
% 2.41/1.41 != ( ap @ B @ D ) )
% 2.41/1.41 | ( C = D ) ) )
% 2.41/1.41 & ( apinj1
% 2.41/1.41 <=> ! [A: term,B: term,C: term,D: term] :
% 2.41/1.41 ( ( ( ap @ A @ C )
% 2.41/1.41 != ( ap @ B @ D ) )
% 2.41/1.41 | ( A = B ) ) )
% 2.41/1.41 & ( hoasapinj2_lthm
% 2.41/1.41 <=> ( ! [A: term,B: term,C: term,D: term] :
% 2.41/1.41 ( ( ( ap @ A @ C )
% 2.41/1.41 != ( ap @ B @ D ) )
% 2.41/1.41 | ( C = D ) )
% 2.41/1.41 => ! [A: term,B: term,C: term,D: term] :
% 2.41/1.41 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.41/1.41 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.41/1.41 | ( C = D ) ) ) )
% 2.41/1.41 & ( hoasinduction_lem3v2a
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( Q @ B )
% 2.41/1.41 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 2.41/1.41 & ( hoasapinj1_lthm
% 2.41/1.41 <=> ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ( ! [A: term,B: term,C: term,D: term] :
% 2.41/1.41 ( ( ( ap @ A @ C )
% 2.41/1.41 != ( ap @ B @ D ) )
% 2.41/1.41 | ( A = B ) )
% 2.41/1.41 => ! [A: term,B: term,C: term,D: term] :
% 2.41/1.41 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 2.41/1.41 != ( ap @ ( sub @ B @ id ) @ D ) )
% 2.41/1.41 | ( A = B ) ) ) ) )
% 2.41/1.41 & ( hoaslaminj_lthm
% 2.41/1.41 <=> ( ! [A: term,M: subst] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.41/1.41 => ( ! [A: term,M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.41/1.41 => ( ! [A: term,B: term] :
% 2.41/1.41 ( ( ( lam @ A )
% 2.41/1.41 != ( lam @ B ) )
% 2.41/1.41 | ( A = B ) )
% 2.41/1.41 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 2.41/1.41 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ( ( lam @ ( F @ sh @ one ) )
% 2.41/1.41 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 2.41/1.41 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 2.41/1.41 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 2.41/1.41 & ( axvarcons
% 2.41/1.41 <=> ! [A: term,M: subst] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 2.41/1.41 & ( axscons
% 2.41/1.41 <=> ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 2.41/1.41 & hoasinduction_lem2v2_gthm
% 2.41/1.41 & ( axidr
% 2.41/1.41 <=> ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ M @ id ) ) )
% 2.41/1.41 & ( pushprop_lem1
% 2.41/1.41 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 2.41/1.41 & ( laminj
% 2.41/1.41 <=> ! [A: term,B: term] :
% 2.41/1.41 ( ( ( lam @ A )
% 2.41/1.41 != ( lam @ B ) )
% 2.41/1.41 | ( A = B ) ) )
% 2.41/1.41 & ( hoasinduction_lem3_lthm
% 2.41/1.41 <=> ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 2.41/1.41 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 2.41/1.41 & ( pushprop_lem0
% 2.41/1.41 <=> ! [P: term > $o,A: term,M: subst] :
% 2.41/1.41 ~ ! [Q: term > $o] :
% 2.41/1.41 ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.41/1.41 & pushprop_gthm
% 2.41/1.41 & axabs
% 2.41/1.41 & ( hoasinduction_lem3v2a_lthm
% 2.41/1.41 <=> ( ! [B: term] :
% 2.41/1.41 ~ ! [F: subst > term > term] :
% 2.41/1.41 ~ ! [A: term,M: subst] :
% 2.41/1.41 ( ( sub @ B @ ( push @ A @ M ) )
% 2.41/1.41 = ( F @ M @ A ) )
% 2.41/1.41 => ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( Q @ B )
% 2.41/1.41 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 2.41/1.41 & hoasinduction_lem2_lthm
% 2.41/1.41 & hoasapinj2_gthm
% 2.41/1.41 & hoasinduction_lem1_lthm
% 2.41/1.41 & ~ lamnotap
% 2.41/1.41 & hoasapinj1_gthm
% 2.41/1.41 & hoaslamnotvar
% 2.41/1.41 & ( axidl
% 2.41/1.41 <=> ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ id @ M ) ) )
% 2.41/1.41 & hoaslaminj_gthm
% 2.41/1.41 & ( induction2_lthm
% 2.41/1.41 <=> ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 2.41/1.41 ( ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ B )
% 2.41/1.41 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( var @ B )
% 2.41/1.41 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 2.41/1.41 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 2.41/1.41 => ! [P: term > $o,Bound_variable_2367: term] :
% 2.41/1.41 ( ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ A )
% 2.41/1.41 | ( P @ A ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ B )
% 2.41/1.41 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ( P @ Bound_variable_2367 ) ) ) ) )
% 2.41/1.41 & ( hoasinduction_lem0_lthm
% 2.41/1.41 <=> ! [P: subst > term > subst > $o] :
% 2.41/1.41 ~ ! [Q: term > $o] :
% 2.41/1.41 ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) ) )
% 2.41/1.41 & ( substmonoid_lthm
% 2.41/1.41 <=> ( ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ id @ M ) )
% 2.41/1.41 => ( ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ M @ id ) )
% 2.41/1.41 => ( ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ id @ M ) )
% 2.41/1.41 & ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ M @ id ) ) ) ) ) )
% 2.41/1.41 & pushprop
% 2.41/1.41 & hoasinduction_lem3_gthm
% 2.41/1.41 & hoasinduction_lem2_gthm
% 2.41/1.41 & ( hoasinduction_lem3b
% 2.41/1.41 <=> ! [B: term] :
% 2.41/1.41 ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ( F @ sh @ one )
% 2.41/1.41 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 2.41/1.41 & ( substmonoid
% 2.41/1.41 <=> ( ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ id @ M ) )
% 2.41/1.41 & ! [M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ M @ id ) ) ) )
% 2.41/1.41 & lamnotvar
% 2.41/1.41 & ( hoasinduction_lem3a
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 2.41/1.41 & hoasinduction_lem1_gthm
% 2.41/1.41 & ( hoasinduction_no_psi_cond
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 2.41/1.41 ( ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 2.41/1.41 & induction2_gthm
% 2.41/1.41 & pushprop_lem2v2_lthm
% 2.41/1.41 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 2.41/1.41 & ( hoaslamnotap
% 2.41/1.41 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ( ( lam @ ( F @ sh @ one ) )
% 2.41/1.41 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 2.41/1.41 & substmonoid_gthm
% 2.41/1.41 & ulamvarsh
% 2.41/1.41 & ( induction2
% 2.41/1.41 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 2.41/1.41 ( ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ A )
% 2.41/1.41 | ( P @ A ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ B )
% 2.41/1.41 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ( P @ Bound_variable_2367 ) ) )
% 2.41/1.41 & pushprop_lem3v2
% 2.41/1.41 & pushprop_lem2v2_gthm
% 2.41/1.41 & ( pushprop_lem1_lthm
% 2.41/1.41 <=> ( ! [A: term,M: subst] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ one @ ( push @ A @ M ) ) )
% 2.41/1.41 => ( ! [A: term,M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ sh @ ( push @ A @ M ) ) )
% 2.41/1.41 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 2.41/1.41 & ( hoasinduction_lem3v2
% 2.41/1.41 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( Q @ B )
% 2.41/1.41 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 2.41/1.41 & ( axshiftcons
% 2.41/1.41 <=> ! [A: term,M: subst] :
% 2.41/1.41 ( M
% 2.41/1.41 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 2.41/1.41 & ( termmset
% 2.41/1.41 <=> ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) ) )
% 2.41/1.41 & ( pushprop_lem0_lthm
% 2.41/1.41 <=> ! [P: term > $o,A: term,M: subst] :
% 2.41/1.41 ~ ! [Q: term > $o] :
% 2.41/1.41 ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 2.41/1.41 & hoasapnotvar_lthm
% 2.41/1.41 & ( hoasinduction_lem3v2_lthm
% 2.41/1.41 <=> ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( Q @ B )
% 2.41/1.41 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 2.41/1.41 & ( axvarid
% 2.41/1.41 <=> ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) ) )
% 2.41/1.41 & ( hoasinduction_lthm_3
% 2.41/1.41 <=> ( ! [P: subst > term > subst > $o] :
% 2.41/1.41 ~ ! [Q: term > $o] :
% 2.41/1.41 ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 2.41/1.41 ( ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ A )
% 2.41/1.41 | ( P @ A ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ A )
% 2.41/1.41 | ~ ( P @ B )
% 2.41/1.41 | ( P @ ( ap @ A @ B ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ! [B: term] :
% 2.41/1.41 ( ~ ( P @ B )
% 2.41/1.41 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( P @ ( lam @ A ) ) )
% 2.41/1.41 | ( P @ Bound_variable_2367 ) )
% 2.41/1.41 => ( ! [A: term] :
% 2.41/1.41 ( A
% 2.41/1.41 = ( sub @ A @ id ) )
% 2.41/1.41 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 2.41/1.41 ( ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ~ ! [X: term] :
% 2.41/1.41 ( ( Q @ X )
% 2.41/1.41 = ( P @ id @ X @ id ) )
% 2.41/1.41 | ~ ! [B: term] :
% 2.41/1.41 ( ~ ( Q @ B )
% 2.41/1.41 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 2.41/1.41 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 2.41/1.41 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( var @ ( sub @ A @ id ) )
% 2.41/1.41 | ( P @ id @ A @ id ) )
% 2.41/1.41 | ~ ! [A: term,B: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ~ ( P @ id @ B @ id )
% 2.41/1.41 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 2.41/1.41 | ~ ! [F: subst > term > term] :
% 2.41/1.41 ( ~ ! [M: subst,A: term,N: subst] :
% 2.41/1.41 ( ( sub @ ( F @ M @ A ) @ N )
% 2.41/1.41 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 2.41/1.41 | ~ ! [A: term] :
% 2.41/1.41 ( ~ ( P @ id @ A @ id )
% 2.41/1.41 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 2.41/1.41 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 2.41/1.41 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 2.41/1.41
% 2.41/1.41 thf(hoasinduction_lem1,conjecture,
% 2.41/1.41 ( hoasinduction_lem1
% 2.41/1.41 = ( ! [P: subst > term > subst > $o] :
% 2.41/1.41 ( ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ( P @ M @ A @ ( comp @ K @ N ) )
% 2.41/1.41 => ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 2.41/1.41 => ( ! [M: subst,A: term,N: subst,K: subst] :
% 2.41/1.41 ( ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 2.41/1.41 => ( P @ M @ A @ ( comp @ K @ N ) ) )
% 2.41/1.41 => ( ! [A: term] :
% 2.41/1.41 ( ( hoasvar @ id @ A @ id )
% 2.41/1.41 => ( P @ id @ A @ id ) )
% 2.41/1.41 => ! [A: term] :
% 2.41/1.41 ( ( var @ A )
% 2.41/1.41 => ( P @ id @ A @ id ) ) ) ) ) ) ) ).
% 2.41/1.41 ------------------------
% 2.41/1.43 % (2030)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_1662 for (2999ds/4Mi)
% 2.41/1.43 % (2029)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/183Mi)
% 2.41/1.43 % (2031)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_1662 for (2999ds/27Mi)
% 2.41/1.43 % (2030)Instruction limit reached!
% 2.41/1.43 % (2030)------------------------------
% 2.41/1.43 % (2030)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.43 % (2030)Termination reason: Unknown
% 2.41/1.43 % (2030)Termination phase: shuffling
% 2.41/1.43
% 2.41/1.43 % (2030)Memory used [KB]: 1407
% 2.41/1.43 % (2030)Time elapsed: 0.003 s
% 2.41/1.43 % (2030)Instructions burned: 6 (million)
% 2.41/1.43 % (2030)------------------------------
% 2.41/1.43 % (2030)------------------------------
% 2.41/1.44 % (2032)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/2Mi)
% 2.41/1.44 % (2034)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_1662 for (2999ds/275Mi)
% 2.41/1.44 % (2035)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/18Mi)
% 2.41/1.44 % (2033)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_1662 for (2999ds/2Mi)
% 2.41/1.44 % (2032)Instruction limit reached!
% 2.41/1.44 % (2032)------------------------------
% 2.41/1.44 % (2032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.44 % (2032)Termination reason: Unknown
% 2.41/1.44 % (2032)Termination phase: shuffling
% 2.41/1.44
% 2.41/1.44 % (2032)Memory used [KB]: 1279
% 2.41/1.44 % (2032)Time elapsed: 0.004 s
% 2.41/1.44 % (2032)Instructions burned: 3 (million)
% 2.41/1.44 % (2032)------------------------------
% 2.41/1.44 % (2032)------------------------------
% 2.41/1.44 % (2033)Instruction limit reached!
% 2.41/1.44 % (2033)------------------------------
% 2.41/1.44 % (2033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.44 % (2033)Termination reason: Unknown
% 2.41/1.44 % (2033)Termination phase: shuffling
% 2.41/1.44
% 2.41/1.44 % (2033)Memory used [KB]: 1151
% 2.41/1.44 % (2033)Time elapsed: 0.003 s
% 2.41/1.44 % (2033)Instructions burned: 3 (million)
% 2.41/1.44 % (2033)------------------------------
% 2.41/1.44 % (2033)------------------------------
% 2.41/1.44 % (2031)Instruction limit reached!
% 2.41/1.44 % (2031)------------------------------
% 2.41/1.44 % (2031)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.44 % (2031)Termination reason: Unknown
% 2.41/1.44 % (2031)Termination phase: Property scanning
% 2.41/1.44
% 2.41/1.44 % (2031)Memory used [KB]: 2174
% 2.41/1.44 % (2031)Time elapsed: 0.011 s
% 2.41/1.44 % (2031)Instructions burned: 30 (million)
% 2.41/1.44 % (2031)------------------------------
% 2.41/1.44 % (2031)------------------------------
% 2.41/1.44 % (2036)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/3Mi)
% 2.41/1.44 % (2036)Instruction limit reached!
% 2.41/1.44 % (2036)------------------------------
% 2.41/1.44 % (2036)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.44 % (2036)Termination reason: Unknown
% 2.41/1.44 % (2036)Termination phase: shuffling
% 2.41/1.44
% 2.41/1.44 % (2036)Memory used [KB]: 1279
% 2.41/1.44 % (2036)Time elapsed: 0.002 s
% 2.41/1.44 % (2036)Instructions burned: 4 (million)
% 2.41/1.44 % (2036)------------------------------
% 2.41/1.44 % (2036)------------------------------
% 2.41/1.45 % (2035)Instruction limit reached!
% 2.41/1.45 % (2035)------------------------------
% 2.41/1.45 % (2035)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.45 % (2035)Termination reason: Unknown
% 2.41/1.45 % (2035)Termination phase: shuffling
% 2.41/1.45
% 2.41/1.45 % (2035)Memory used [KB]: 2046
% 2.41/1.45 % (2035)Time elapsed: 0.013 s
% 2.41/1.45 % (2035)Instructions burned: 18 (million)
% 2.41/1.45 % (2035)------------------------------
% 2.41/1.45 % (2035)------------------------------
% 2.41/1.45 % (2039)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_1662 for (2999ds/3Mi)
% 2.41/1.45 % (2040)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_1662 for (2999ds/1041Mi)
% 2.41/1.45 % (2039)Instruction limit reached!
% 2.41/1.45 % (2039)------------------------------
% 2.41/1.45 % (2039)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.45 % (2039)Termination reason: Unknown
% 2.41/1.45 % (2039)Termination phase: shuffling
% 2.41/1.45
% 2.41/1.45 % (2039)Memory used [KB]: 1279
% 2.41/1.45 % (2039)Time elapsed: 0.002 s
% 2.41/1.45 % (2039)Instructions burned: 4 (million)
% 2.41/1.45 % (2039)------------------------------
% 2.41/1.45 % (2039)------------------------------
% 2.41/1.46 % (2037)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_1662 for (2999ds/37Mi)
% 2.41/1.46 % (2038)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_1662 for (2999ds/15Mi)
% 2.41/1.46 % (2034)Refutation not found, incomplete strategy
% 2.41/1.46 % (2034)------------------------------
% 2.41/1.46 % (2034)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.46 % (2034)Termination reason: Refutation not found, incomplete strategy
% 2.41/1.46
% 2.41/1.46
% 2.41/1.46 % (2034)Memory used [KB]: 6652
% 2.41/1.46 % (2034)Time elapsed: 0.026 s
% 2.41/1.46 % (2034)Instructions burned: 41 (million)
% 2.41/1.46 % (2034)------------------------------
% 2.41/1.46 % (2034)------------------------------
% 2.41/1.46 % (2042)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/16Mi)
% 2.41/1.47 % (2038)Instruction limit reached!
% 2.41/1.47 % (2038)------------------------------
% 2.41/1.47 % (2038)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (2038)Termination reason: Unknown
% 2.41/1.47 % (2038)Termination phase: shuffling
% 2.41/1.47
% 2.41/1.47 % (2038)Memory used [KB]: 1918
% 2.41/1.47 % (2038)Time elapsed: 0.009 s
% 2.41/1.47 % (2038)Instructions burned: 15 (million)
% 2.41/1.47 % (2038)------------------------------
% 2.41/1.47 % (2038)------------------------------
% 2.41/1.47 % (2041)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/7Mi)
% 2.41/1.47 % (2042)Instruction limit reached!
% 2.41/1.47 % (2042)------------------------------
% 2.41/1.47 % (2042)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (2042)Termination reason: Unknown
% 2.41/1.47 % (2042)Termination phase: shuffling
% 2.41/1.47
% 2.41/1.47 % (2042)Memory used [KB]: 1791
% 2.41/1.47 % (2042)Time elapsed: 0.006 s
% 2.41/1.47 % (2042)Instructions burned: 16 (million)
% 2.41/1.47 % (2042)------------------------------
% 2.41/1.47 % (2042)------------------------------
% 2.41/1.47 % (2041)Instruction limit reached!
% 2.41/1.47 % (2041)------------------------------
% 2.41/1.47 % (2041)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (2041)Termination reason: Unknown
% 2.41/1.47 % (2041)Termination phase: shuffling
% 2.41/1.47
% 2.41/1.47 % (2041)Memory used [KB]: 1407
% 2.41/1.47 % (2041)Time elapsed: 0.006 s
% 2.41/1.47 % (2041)Instructions burned: 7 (million)
% 2.41/1.47 % (2041)------------------------------
% 2.41/1.47 % (2041)------------------------------
% 2.41/1.48 % (2044)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_1662 for (2999ds/3Mi)
% 2.41/1.48 % (2037)Instruction limit reached!
% 2.41/1.48 % (2037)------------------------------
% 2.41/1.48 % (2037)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (2037)Termination reason: Unknown
% 2.41/1.48 % (2037)Termination phase: shuffling
% 2.41/1.48
% 2.41/1.48 % (2037)Memory used [KB]: 1918
% 2.41/1.48 % (2037)Time elapsed: 0.021 s
% 2.41/1.48 % (2037)Instructions burned: 38 (million)
% 2.41/1.48 % (2037)------------------------------
% 2.41/1.48 % (2037)------------------------------
% 2.41/1.48 % (2044)Instruction limit reached!
% 2.41/1.48 % (2044)------------------------------
% 2.41/1.48 % (2044)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (2044)Termination reason: Unknown
% 2.41/1.48 % (2044)Termination phase: shuffling
% 2.41/1.48
% 2.41/1.48 % (2044)Memory used [KB]: 1151
% 2.41/1.48 % (2044)Time elapsed: 0.002 s
% 2.41/1.48 % (2044)Instructions burned: 3 (million)
% 2.41/1.48 % (2044)------------------------------
% 2.41/1.48 % (2044)------------------------------
% 2.41/1.48 % (2045)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/7Mi)
% 2.41/1.48 % (2045)Instruction limit reached!
% 2.41/1.48 % (2045)------------------------------
% 2.41/1.48 % (2045)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (2045)Termination reason: Unknown
% 2.41/1.48 % (2045)Termination phase: shuffling
% 2.41/1.48
% 2.41/1.48 % (2045)Memory used [KB]: 1407
% 2.41/1.48 % (2045)Time elapsed: 0.004 s
% 2.41/1.48 % (2045)Instructions burned: 7 (million)
% 2.41/1.48 % (2045)------------------------------
% 2.41/1.48 % (2045)------------------------------
% 2.41/1.48 % (2043)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_1662 for (2999ds/3Mi)
% 2.41/1.48 % (2043)Instruction limit reached!
% 2.41/1.48 % (2043)------------------------------
% 2.41/1.48 % (2043)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (2043)Termination reason: Unknown
% 2.41/1.48 % (2043)Termination phase: shuffling
% 2.41/1.48
% 2.41/1.48 % (2043)Memory used [KB]: 1279
% 2.41/1.48 % (2043)Time elapsed: 0.004 s
% 2.41/1.48 % (2043)Instructions burned: 4 (million)
% 2.41/1.48 % (2043)------------------------------
% 2.41/1.48 % (2043)------------------------------
% 2.41/1.48 % (2029)Instruction limit reached!
% 2.41/1.48 % (2029)------------------------------
% 2.41/1.48 % (2029)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (2029)Termination reason: Unknown
% 2.41/1.48 % (2029)Termination phase: Property scanning
% 2.41/1.48
% 2.41/1.48 % (2029)Memory used [KB]: 4605
% 2.41/1.48 % (2029)Time elapsed: 0.054 s
% 2.41/1.48 % (2029)Instructions burned: 186 (million)
% 2.41/1.48 % (2029)------------------------------
% 2.41/1.48 % (2029)------------------------------
% 2.41/1.49 % (2049)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/18Mi)
% 2.41/1.49 % (2046)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/3Mi)
% 2.41/1.49 % (2050)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/710Mi)
% 2.41/1.49 % (2046)Instruction limit reached!
% 2.41/1.49 % (2046)------------------------------
% 2.41/1.49 % (2046)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.49 % (2046)Termination reason: Unknown
% 2.41/1.49 % (2046)Termination phase: shuffling
% 2.41/1.49
% 2.41/1.49 % (2046)Memory used [KB]: 1279
% 2.41/1.49 % (2046)Time elapsed: 0.004 s
% 2.41/1.49 % (2046)Instructions burned: 4 (million)
% 2.41/1.49 % (2046)------------------------------
% 2.41/1.49 % (2046)------------------------------
% 2.41/1.49 % (2049)Instruction limit reached!
% 2.41/1.49 % (2049)------------------------------
% 2.41/1.49 % (2049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.49 % (2049)Termination reason: Unknown
% 2.41/1.49 % (2049)Termination phase: shuffling
% 2.41/1.49
% 2.41/1.49 % (2049)Memory used [KB]: 2046
% 2.41/1.49 % (2049)Time elapsed: 0.008 s
% 2.41/1.49 % (2049)Instructions burned: 19 (million)
% 2.41/1.49 % (2049)------------------------------
% 2.41/1.49 % (2049)------------------------------
% 2.41/1.49 % (2052)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_1662 for (2999ds/902Mi)
% 2.41/1.50 % (2048)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/4Mi)
% 2.43/1.50 % (2048)Instruction limit reached!
% 2.43/1.50 % (2048)------------------------------
% 2.43/1.50 % (2048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.50 % (2048)Termination reason: Unknown
% 2.43/1.50 % (2048)Termination phase: shuffling
% 2.43/1.50
% 2.43/1.50 % (2048)Memory used [KB]: 1279
% 2.43/1.50 % (2048)Time elapsed: 0.005 s
% 2.43/1.50 % (2048)Instructions burned: 5 (million)
% 2.43/1.50 % (2048)------------------------------
% 2.43/1.50 % (2048)------------------------------
% 2.43/1.50 % (2051)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/6Mi)
% 2.43/1.50 % (2051)Instruction limit reached!
% 2.43/1.50 % (2051)------------------------------
% 2.43/1.50 % (2051)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.50 % (2051)Termination reason: Unknown
% 2.43/1.50 % (2051)Termination phase: shuffling
% 2.43/1.50
% 2.43/1.50 % (2051)Memory used [KB]: 1407
% 2.43/1.50 % (2051)Time elapsed: 0.005 s
% 2.43/1.50 % (2051)Instructions burned: 6 (million)
% 2.43/1.50 % (2051)------------------------------
% 2.43/1.50 % (2051)------------------------------
% 2.43/1.50 % (2054)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/5Mi)
% 2.43/1.51 % (2054)Instruction limit reached!
% 2.43/1.51 % (2054)------------------------------
% 2.43/1.51 % (2054)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.51 % (2054)Termination reason: Unknown
% 2.43/1.51 % (2054)Termination phase: shuffling
% 2.43/1.51
% 2.43/1.51 % (2054)Memory used [KB]: 1407
% 2.43/1.51 % (2054)Time elapsed: 0.004 s
% 2.43/1.51 % (2054)Instructions burned: 7 (million)
% 2.43/1.51 % (2054)------------------------------
% 2.43/1.51 % (2054)------------------------------
% 2.43/1.51 % (2053)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_1662 for (2999ds/21Mi)
% 2.43/1.52 % (2055)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_1662 for (2999ds/6Mi)
% 2.43/1.52 % (2057)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_1662 for (2999ds/779Mi)
% 2.43/1.52 % (2055)Instruction limit reached!
% 2.43/1.52 % (2055)------------------------------
% 2.43/1.52 % (2055)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.52 % (2055)Termination reason: Unknown
% 2.43/1.52 % (2055)Termination phase: shuffling
% 2.43/1.52
% 2.43/1.52 % (2055)Memory used [KB]: 1407
% 2.43/1.52 % (2055)Time elapsed: 0.005 s
% 2.43/1.52 % (2055)Instructions burned: 6 (million)
% 2.43/1.52 % (2055)------------------------------
% 2.43/1.52 % (2055)------------------------------
% 2.43/1.52 % (2053)Instruction limit reached!
% 2.43/1.52 % (2053)------------------------------
% 2.43/1.52 % (2053)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.52 % (2053)Termination reason: Unknown
% 2.43/1.52 % (2053)Termination phase: shuffling
% 2.43/1.52
% 2.43/1.52 % (2053)Memory used [KB]: 2174
% 2.43/1.52 % (2053)Time elapsed: 0.014 s
% 2.43/1.52 % (2053)Instructions burned: 22 (million)
% 2.43/1.52 % (2053)------------------------------
% 2.43/1.52 % (2053)------------------------------
% 2.43/1.52 % (2056)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/377Mi)
% 2.43/1.53 % (2059)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_1662 for (2999ds/879Mi)
% 2.43/1.54 % (2058)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_1662 for (2999ds/19Mi)
% 2.43/1.54 % (2058)Instruction limit reached!
% 2.43/1.54 % (2058)------------------------------
% 2.43/1.54 % (2058)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.54 % (2058)Termination reason: Unknown
% 2.43/1.54 % (2058)Termination phase: shuffling
% 2.43/1.54
% 2.43/1.54 % (2058)Memory used [KB]: 1663
% 2.43/1.54 % (2058)Time elapsed: 0.010 s
% 2.43/1.54 % (2058)Instructions burned: 19 (million)
% 2.43/1.54 % (2058)------------------------------
% 2.43/1.54 % (2058)------------------------------
% 2.43/1.55 % (2052)Refutation not found, incomplete strategy
% 2.43/1.55 % (2052)------------------------------
% 2.43/1.55 % (2052)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.55 % (2052)Termination reason: Refutation not found, incomplete strategy
% 2.43/1.55
% 2.43/1.55
% 2.43/1.55 % (2052)Memory used [KB]: 9210
% 2.43/1.55 % (2052)Time elapsed: 0.056 s
% 2.43/1.55 % (2052)Instructions burned: 195 (million)
% 2.43/1.55 % (2052)------------------------------
% 2.43/1.55 % (2052)------------------------------
% 2.43/1.56 % (2060)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_1662 for (2998ds/17Mi)
% 2.43/1.56 % (2061)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_1662 for (2998ds/3Mi)
% 2.43/1.56 % (2061)Instruction limit reached!
% 2.43/1.56 % (2061)------------------------------
% 2.43/1.56 % (2061)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.56 % (2061)Termination reason: Unknown
% 2.43/1.56 % (2061)Termination phase: shuffling
% 2.43/1.56
% 2.43/1.56 % (2061)Memory used [KB]: 1279
% 2.43/1.56 % (2061)Time elapsed: 0.002 s
% 2.43/1.56 % (2061)Instructions burned: 4 (million)
% 2.43/1.56 % (2061)------------------------------
% 2.43/1.56 % (2061)------------------------------
% 2.43/1.56 % (2060)Instruction limit reached!
% 2.43/1.56 % (2060)------------------------------
% 2.43/1.56 % (2060)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.56 % (2060)Termination reason: Unknown
% 2.43/1.56 % (2060)Termination phase: shuffling
% 2.43/1.56
% 2.43/1.56 % (2060)Memory used [KB]: 2046
% 2.43/1.56 % (2060)Time elapsed: 0.008 s
% 2.43/1.56 % (2060)Instructions burned: 19 (million)
% 2.43/1.56 % (2060)------------------------------
% 2.43/1.56 % (2060)------------------------------
% 2.43/1.57 % (2062)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_1662 for (2998ds/30Mi)
% 2.43/1.57 % (2063)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_1662 for (2998ds/127Mi)
% 2.43/1.58 % (2062)Instruction limit reached!
% 2.43/1.58 % (2062)------------------------------
% 2.43/1.58 % (2062)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.58 % (2062)Termination reason: Unknown
% 2.43/1.58 % (2062)Termination phase: Property scanning
% 2.43/1.58
% 2.43/1.58 % (2062)Memory used [KB]: 2174
% 2.43/1.58 % (2062)Time elapsed: 0.010 s
% 2.43/1.58 % (2062)Instructions burned: 30 (million)
% 2.43/1.58 % (2062)------------------------------
% 2.43/1.58 % (2062)------------------------------
% 2.43/1.59 % (2064)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_1662 for (2998ds/100Mi)
% 2.43/1.61 % (2063)Instruction limit reached!
% 2.43/1.61 % (2063)------------------------------
% 2.43/1.61 % (2063)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.61 % (2063)Termination reason: Unknown
% 2.43/1.61 % (2063)Termination phase: Property scanning
% 2.43/1.61
% 2.43/1.61 % (2063)Memory used [KB]: 4221
% 2.43/1.61 % (2063)Time elapsed: 0.040 s
% 2.43/1.61 % (2063)Instructions burned: 127 (million)
% 2.43/1.61 % (2063)------------------------------
% 2.43/1.61 % (2063)------------------------------
% 2.43/1.61 % (2064)Instruction limit reached!
% 2.43/1.61 % (2064)------------------------------
% 2.43/1.61 % (2064)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.61 % (2064)Termination reason: Unknown
% 2.43/1.61 % (2064)Termination phase: Saturation
% 2.43/1.61
% 2.43/1.61 % (2064)Memory used [KB]: 6524
% 2.43/1.61 % (2064)Time elapsed: 0.026 s
% 2.43/1.61 % (2064)Instructions burned: 101 (million)
% 2.43/1.61 % (2064)------------------------------
% 2.43/1.61 % (2064)------------------------------
% 2.43/1.62 % (2065)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_1662 for (2998ds/3Mi)
% 2.43/1.62 % (2065)Instruction limit reached!
% 2.43/1.62 % (2065)------------------------------
% 2.43/1.62 % (2065)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.62 % (2065)Termination reason: Unknown
% 2.43/1.62 % (2065)Termination phase: shuffling
% 2.43/1.62
% 2.43/1.62 % (2065)Memory used [KB]: 1279
% 2.43/1.62 % (2065)Time elapsed: 0.002 s
% 2.43/1.62 % (2065)Instructions burned: 4 (million)
% 2.43/1.62 % (2065)------------------------------
% 2.43/1.62 % (2065)------------------------------
% 2.43/1.62 % (2066)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_1662 for (2998ds/20Mi)
% 2.43/1.63 % (2066)Instruction limit reached!
% 2.43/1.63 % (2066)------------------------------
% 2.43/1.63 % (2066)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.63 % (2066)Termination reason: Unknown
% 2.43/1.63 % (2066)Termination phase: shuffling
% 2.43/1.63
% 2.43/1.63 % (2066)Memory used [KB]: 2046
% 2.43/1.63 % (2066)Time elapsed: 0.007 s
% 2.43/1.63 % (2066)Instructions burned: 20 (million)
% 2.43/1.63 % (2066)------------------------------
% 2.43/1.63 % (2066)------------------------------
% 2.43/1.63 % (2067)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_1662 for (2998ds/86Mi)
% 2.43/1.64 % (2068)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_1662 for (2997ds/107Mi)
% 2.43/1.65 % (2056)Instruction limit reached!
% 2.43/1.65 % (2056)------------------------------
% 2.43/1.65 % (2056)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.65 % (2056)Termination reason: Unknown
% 2.43/1.65 % (2056)Termination phase: Saturation
% 2.43/1.65
% 2.43/1.65 % (2056)Memory used [KB]: 11001
% 2.43/1.65 % (2056)Time elapsed: 0.131 s
% 2.43/1.65 % (2056)Instructions burned: 377 (million)
% 2.43/1.65 % (2056)------------------------------
% 2.43/1.65 % (2056)------------------------------
% 2.43/1.66 % (2067)Instruction limit reached!
% 2.43/1.66 % (2067)------------------------------
% 2.43/1.66 % (2067)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.66 % (2067)Termination reason: Unknown
% 2.43/1.66 % (2067)Termination phase: Preprocessing 3
% 2.43/1.66
% 2.43/1.66 % (2067)Memory used [KB]: 4093
% 2.43/1.66 % (2067)Time elapsed: 0.029 s
% 2.43/1.66 % (2067)Instructions burned: 87 (million)
% 2.43/1.66 % (2067)------------------------------
% 2.43/1.66 % (2067)------------------------------
% 2.43/1.66 % (2069)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_1662 for (2997ds/39Mi)
% 2.43/1.67 % (2068)Instruction limit reached!
% 2.43/1.67 % (2068)------------------------------
% 2.43/1.67 % (2068)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.67 % (2068)Termination reason: Unknown
% 2.43/1.67 % (2068)Termination phase: Property scanning
% 2.43/1.67
% 2.43/1.67 % (2068)Memory used [KB]: 4477
% 2.43/1.67 % (2068)Time elapsed: 0.032 s
% 2.43/1.67 % (2068)Instructions burned: 108 (million)
% 2.43/1.67 % (2068)------------------------------
% 2.43/1.67 % (2068)------------------------------
% 2.43/1.67 % (2070)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_1662 for (2997ds/448Mi)
% 2.43/1.67 % (2069)Instruction limit reached!
% 2.43/1.67 % (2069)------------------------------
% 2.43/1.67 % (2069)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.67 % (2069)Termination reason: Unknown
% 2.43/1.67 % (2069)Termination phase: Property scanning
% 2.43/1.67
% 2.43/1.67 % (2069)Memory used [KB]: 1918
% 2.43/1.67 % (2069)Time elapsed: 0.012 s
% 2.43/1.67 % (2069)Instructions burned: 39 (million)
% 2.43/1.67 % (2069)------------------------------
% 2.43/1.67 % (2069)------------------------------
% 2.43/1.68 % (2071)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_1662 for (2997ds/46Mi)
% 2.43/1.68 % (2072)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_1662 for (2997ds/98Mi)
% 2.43/1.69 % (2071)Instruction limit reached!
% 2.43/1.69 % (2071)------------------------------
% 2.43/1.69 % (2071)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.69 % (2071)Termination reason: Unknown
% 2.43/1.69 % (2071)Termination phase: Preprocessing 1
% 2.43/1.69
% 2.43/1.69 % (2071)Memory used [KB]: 2302
% 2.43/1.69 % (2071)Time elapsed: 0.014 s
% 2.43/1.69 % (2071)Instructions burned: 46 (million)
% 2.43/1.69 % (2071)------------------------------
% 2.43/1.69 % (2071)------------------------------
% 2.43/1.70 % (2073)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_1662 for (2997ds/507Mi)
% 2.43/1.71 % (2072)Instruction limit reached!
% 2.43/1.71 % (2072)------------------------------
% 2.43/1.71 % (2072)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.71 % (2072)Termination reason: Unknown
% 2.43/1.71 % (2072)Termination phase: Saturation
% 2.43/1.71
% 2.43/1.71 % (2072)Memory used [KB]: 2046
% 2.43/1.71 % (2072)Time elapsed: 0.029 s
% 2.43/1.71 % (2072)Instructions burned: 101 (million)
% 2.43/1.71 % (2072)------------------------------
% 2.43/1.71 % (2072)------------------------------
% 2.43/1.72 % (2040)Instruction limit reached!
% 2.43/1.72 % (2040)------------------------------
% 2.43/1.72 % (2040)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.72 % (2040)Termination reason: Unknown
% 2.43/1.72 % (2040)Termination phase: Saturation
% 2.43/1.72
% 2.43/1.72 % (2040)Memory used [KB]: 7803
% 2.43/1.72 % (2040)Time elapsed: 0.266 s
% 2.43/1.72 % (2040)Instructions burned: 1043 (million)
% 2.43/1.72 % (2040)------------------------------
% 2.43/1.72 % (2040)------------------------------
% 2.43/1.72 % (2074)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_1662 for (2997ds/149Mi)
% 2.43/1.72 % (2050)Instruction limit reached!
% 2.43/1.72 % (2050)------------------------------
% 2.43/1.72 % (2050)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.72 % (2050)Termination reason: Unknown
% 2.43/1.72 % (2050)Termination phase: Saturation
% 2.43/1.72
% 2.43/1.72 % (2050)Memory used [KB]: 11385
% 2.43/1.72 % (2050)Time elapsed: 0.232 s
% 2.43/1.72 % (2050)Instructions burned: 712 (million)
% 2.43/1.72 % (2050)------------------------------
% 2.43/1.72 % (2050)------------------------------
% 2.43/1.72 % (2057)Instruction limit reached!
% 2.43/1.72 % (2057)------------------------------
% 2.43/1.72 % (2057)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.72 % (2057)Termination reason: Unknown
% 2.43/1.72 % (2057)Termination phase: Saturation
% 2.43/1.72
% 2.43/1.72 % (2057)Memory used [KB]: 7036
% 2.43/1.72 % (2057)Time elapsed: 0.209 s
% 2.43/1.72 % (2057)Instructions burned: 782 (million)
% 2.43/1.72 % (2057)------------------------------
% 2.43/1.72 % (2057)------------------------------
% 2.43/1.73 % (2075)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/39Mi)
% 2.43/1.73 % (2076)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/6Mi)
% 2.43/1.73 % (2076)Instruction limit reached!
% 2.43/1.73 % (2076)------------------------------
% 2.43/1.73 % (2076)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.73 % (2076)Termination reason: Unknown
% 2.43/1.73 % (2076)Termination phase: shuffling
% 2.43/1.73
% 2.43/1.73 % (2076)Memory used [KB]: 1407
% 2.43/1.73 % (2076)Time elapsed: 0.003 s
% 2.43/1.73 % (2076)Instructions burned: 6 (million)
% 2.43/1.73 % (2076)------------------------------
% 2.43/1.73 % (2076)------------------------------
% 2.43/1.73 % (2070)Refutation not found, incomplete strategy
% 2.43/1.73 % (2070)------------------------------
% 2.43/1.73 % (2070)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.73 % (2070)Termination reason: Refutation not found, incomplete strategy
% 2.43/1.73
% 2.43/1.73
% 2.43/1.73 % (2070)Memory used [KB]: 7036
% 2.43/1.73 % (2070)Time elapsed: 0.065 s
% 2.43/1.73 % (2070)Instructions burned: 214 (million)
% 2.43/1.73 % (2070)------------------------------
% 2.43/1.73 % (2070)------------------------------
% 2.43/1.74 % (2077)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/1087Mi)
% 2.43/1.74 % (2075)Instruction limit reached!
% 2.43/1.74 % (2075)------------------------------
% 2.43/1.74 % (2075)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.74 % (2075)Termination reason: Unknown
% 2.43/1.74 % (2075)Termination phase: Preprocessing 1
% 2.43/1.74
% 2.43/1.74 % (2075)Memory used [KB]: 2302
% 2.43/1.74 % (2075)Time elapsed: 0.013 s
% 2.43/1.74 % (2075)Instructions burned: 40 (million)
% 2.43/1.74 % (2075)------------------------------
% 2.43/1.74 % (2075)------------------------------
% 2.43/1.74 % (2078)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/47Mi)
% 2.43/1.75 % (2079)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/44Mi)
% 2.43/1.75 % (2080)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/432Mi)
% 2.43/1.76 % (2078)Instruction limit reached!
% 2.43/1.76 % (2078)------------------------------
% 2.43/1.76 % (2078)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.76 % (2078)Termination reason: Unknown
% 2.43/1.76 % (2078)Termination phase: Naming
% 2.43/1.76
% 2.43/1.76 % (2078)Memory used [KB]: 2814
% 2.43/1.76 % (2078)Time elapsed: 0.016 s
% 2.43/1.76 % (2078)Instructions burned: 49 (million)
% 2.43/1.76 % (2078)------------------------------
% 2.43/1.76 % (2078)------------------------------
% 2.43/1.76 % (2079)Instruction limit reached!
% 2.43/1.76 % (2079)------------------------------
% 2.43/1.76 % (2079)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.76 % (2079)Termination reason: Unknown
% 2.43/1.76 % (2079)Termination phase: Property scanning
% 2.43/1.76
% 2.43/1.76 % (2079)Memory used [KB]: 1918
% 2.43/1.76 % (2079)Time elapsed: 0.014 s
% 2.43/1.76 % (2079)Instructions burned: 46 (million)
% 2.43/1.76 % (2079)------------------------------
% 2.43/1.76 % (2079)------------------------------
% 2.43/1.77 % (2074)Instruction limit reached!
% 2.43/1.77 % (2074)------------------------------
% 2.43/1.77 % (2074)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.77 % (2074)Termination reason: Unknown
% 2.43/1.77 % (2074)Termination phase: Property scanning
% 2.43/1.77
% 2.43/1.77 % (2074)Memory used [KB]: 7931
% 2.43/1.77 % (2074)Time elapsed: 0.048 s
% 2.43/1.77 % (2074)Instructions burned: 152 (million)
% 2.43/1.77 % (2074)------------------------------
% 2.43/1.77 % (2074)------------------------------
% 2.43/1.77 % (2081)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/62Mi)
% 2.43/1.77 % (2082)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/31Mi)
% 2.43/1.78 % (2083)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/96Mi)
% 2.43/1.78 % (2082)Instruction limit reached!
% 2.43/1.78 % (2082)------------------------------
% 2.43/1.78 % (2082)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.78 % (2082)Termination reason: Unknown
% 2.43/1.78 % (2082)Termination phase: Property scanning
% 2.43/1.78
% 2.43/1.78 % (2082)Memory used [KB]: 2174
% 2.43/1.78 % (2082)Time elapsed: 0.011 s
% 2.43/1.78 % (2082)Instructions burned: 31 (million)
% 2.43/1.78 % (2082)------------------------------
% 2.43/1.78 % (2082)------------------------------
% 2.43/1.78 % (2073)Refutation not found, incomplete strategy
% 2.43/1.78 % (2073)------------------------------
% 2.43/1.78 % (2073)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.78 % (2073)Termination reason: Refutation not found, incomplete strategy
% 2.43/1.78
% 2.43/1.78
% 2.43/1.78 % (2073)Memory used [KB]: 9850
% 2.43/1.78 % (2073)Time elapsed: 0.079 s
% 2.43/1.78 % (2073)Instructions burned: 274 (million)
% 2.43/1.78 % (2073)------------------------------
% 2.43/1.78 % (2073)------------------------------
% 2.43/1.78 % (2081)Instruction limit reached!
% 2.43/1.78 % (2081)------------------------------
% 2.43/1.78 % (2081)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.78 % (2081)Termination reason: Unknown
% 2.43/1.78 % (2081)Termination phase: Property scanning
% 2.43/1.78
% 2.43/1.78 % (2081)Memory used [KB]: 2046
% 2.43/1.78 % (2081)Time elapsed: 0.017 s
% 2.43/1.78 % (2081)Instructions burned: 62 (million)
% 2.43/1.78 % (2081)------------------------------
% 2.43/1.78 % (2081)------------------------------
% 2.43/1.79 % (2084)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/453Mi)
% 2.43/1.79 % (2085)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/71Mi)
% 2.43/1.79 % (2086)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/83Mi)
% 2.43/1.81 % (2083)Instruction limit reached!
% 2.43/1.81 % (2083)------------------------------
% 2.43/1.81 % (2083)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.81 % (2083)Termination reason: Unknown
% 2.43/1.81 % (2083)Termination phase: Clausification
% 2.43/1.81
% 2.43/1.81 % (2083)Memory used [KB]: 4349
% 2.43/1.81 % (2083)Time elapsed: 0.031 s
% 2.43/1.81 % (2083)Instructions burned: 96 (million)
% 2.43/1.81 % (2083)------------------------------
% 2.43/1.81 % (2083)------------------------------
% 2.43/1.81 % (2085)Instruction limit reached!
% 2.43/1.81 % (2085)------------------------------
% 2.43/1.81 % (2085)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.81 % (2085)Termination reason: Unknown
% 2.43/1.81 % (2085)Termination phase: Preprocessing 3
% 2.43/1.81
% 2.43/1.81 % (2085)Memory used [KB]: 3326
% 2.43/1.81 % (2085)Time elapsed: 0.022 s
% 2.43/1.81 % (2085)Instructions burned: 72 (million)
% 2.43/1.81 % (2085)------------------------------
% 2.43/1.81 % (2085)------------------------------
% 2.43/1.82 % (2086)Instruction limit reached!
% 2.43/1.82 % (2086)------------------------------
% 2.43/1.82 % (2086)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.82 % (2086)Termination reason: Unknown
% 2.43/1.82 % (2086)Termination phase: Saturation
% 2.43/1.82
% 2.43/1.82 % (2086)Memory used [KB]: 7036
% 2.43/1.82 % (2086)Time elapsed: 0.026 s
% 2.43/1.82 % (2086)Instructions burned: 86 (million)
% 2.43/1.82 % (2086)------------------------------
% 2.43/1.82 % (2086)------------------------------
% 2.43/1.82 % (2087)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/11Mi)
% 2.43/1.82 % (2088)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/41Mi)
% 2.43/1.82 % (2087)Instruction limit reached!
% 2.43/1.82 % (2087)------------------------------
% 2.43/1.82 % (2087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.82 % (2087)Termination reason: Unknown
% 2.43/1.82 % (2087)Termination phase: shuffling
% 2.43/1.82
% 2.43/1.82 % (2087)Memory used [KB]: 1535
% 2.43/1.82 % (2087)Time elapsed: 0.005 s
% 2.43/1.82 % (2087)Instructions burned: 11 (million)
% 2.43/1.82 % (2087)------------------------------
% 2.43/1.82 % (2087)------------------------------
% 2.43/1.82 % (2059)Instruction limit reached!
% 2.43/1.82 % (2059)------------------------------
% 2.43/1.82 % (2059)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.82 % (2059)Termination reason: Unknown
% 2.43/1.82 % (2059)Termination phase: Saturation
% 2.43/1.82
% 2.43/1.82 % (2059)Memory used [KB]: 11769
% 2.43/1.82 % (2059)Time elapsed: 0.290 s
% 2.43/1.82 % (2059)Instructions burned: 880 (million)
% 2.43/1.82 % (2059)------------------------------
% 2.43/1.82 % (2059)------------------------------
% 2.43/1.83 % (2089)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/6Mi)
% 2.43/1.83 % (2089)Instruction limit reached!
% 2.43/1.83 % (2089)------------------------------
% 2.43/1.83 % (2089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.83 % (2089)Termination reason: Unknown
% 2.43/1.83 % (2089)Termination phase: shuffling
% 2.43/1.83
% 2.43/1.83 % (2089)Memory used [KB]: 1407
% 2.43/1.83 % (2089)Time elapsed: 0.003 s
% 2.43/1.83 % (2089)Instructions burned: 7 (million)
% 2.43/1.83 % (2089)------------------------------
% 2.43/1.83 % (2089)------------------------------
% 2.43/1.83 % (2090)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/150Mi)
% 2.43/1.83 % (2088)Instruction limit reached!
% 2.43/1.83 % (2088)------------------------------
% 2.43/1.83 % (2088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.83 % (2088)Termination reason: Unknown
% 2.43/1.83 % (2088)Termination phase: Property scanning
% 2.43/1.83
% 2.43/1.83 % (2088)Memory used [KB]: 1918
% 2.43/1.83 % (2088)Time elapsed: 0.012 s
% 2.43/1.83 % (2088)Instructions burned: 43 (million)
% 2.43/1.83 % (2088)------------------------------
% 2.43/1.83 % (2088)------------------------------
% 2.43/1.83 % (2091)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_1662 for (2996ds/343Mi)
% 2.43/1.84 % (2092)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/109Mi)
% 2.43/1.84 % (2093)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/253Mi)
% 2.43/1.86 % (2080)Instruction limit reached!
% 2.43/1.86 % (2080)------------------------------
% 2.43/1.86 % (2080)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.86 % (2080)Termination reason: Unknown
% 2.43/1.86 % (2080)Termination phase: Saturation
% 2.43/1.86
% 2.43/1.86 % (2080)Memory used [KB]: 2558
% 2.43/1.86 % (2080)Time elapsed: 0.110 s
% 2.43/1.86 % (2080)Instructions burned: 433 (million)
% 2.43/1.86 % (2080)------------------------------
% 2.43/1.86 % (2080)------------------------------
% 2.43/1.86 % (2091)Refutation not found, incomplete strategy
% 2.43/1.86 % (2091)------------------------------
% 2.43/1.86 % (2091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.86 % (2091)Termination reason: Refutation not found, incomplete strategy
% 2.43/1.86
% 2.43/1.86
% 2.43/1.86 % (2091)Memory used [KB]: 7036
% 2.43/1.86 % (2091)Time elapsed: 0.029 s
% 2.43/1.86 % (2091)Instructions burned: 89 (million)
% 2.43/1.86 % (2091)------------------------------
% 2.43/1.86 % (2091)------------------------------
% 2.43/1.86 % (2093)Refutation not found, incomplete strategy
% 2.43/1.86 % (2093)------------------------------
% 2.43/1.86 % (2093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.86 % (2093)Termination reason: Refutation not found, incomplete strategy
% 2.43/1.86
% 2.43/1.86
% 2.43/1.86 % (2093)Memory used [KB]: 6780
% 2.43/1.86 % (2093)Time elapsed: 0.042 s
% 2.43/1.86 % (2093)Instructions burned: 67 (million)
% 2.43/1.86 % (2093)------------------------------
% 2.43/1.86 % (2093)------------------------------
% 2.43/1.87 % (2094)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/140Mi)
% 2.43/1.87 % (2092)Instruction limit reached!
% 2.43/1.87 % (2092)------------------------------
% 2.43/1.87 % (2092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.87 % (2092)Termination reason: Unknown
% 2.43/1.87 % (2092)Termination phase: Property scanning
% 2.43/1.87
% 2.43/1.87 % (2092)Memory used [KB]: 4221
% 2.43/1.87 % (2092)Time elapsed: 0.032 s
% 2.43/1.87 % (2092)Instructions burned: 110 (million)
% 2.43/1.87 % (2092)------------------------------
% 2.43/1.87 % (2092)------------------------------
% 2.43/1.87 % (2097)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/202Mi)
% 2.43/1.87 % (2095)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/614Mi)
% 2.43/1.88 % (2090)Instruction limit reached!
% 2.43/1.88 % (2090)------------------------------
% 2.43/1.88 % (2090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.88 % (2090)Termination reason: Unknown
% 2.43/1.88 % (2090)Termination phase: Property scanning
% 2.43/1.88
% 2.43/1.88 % (2090)Memory used [KB]: 4221
% 2.43/1.88 % (2090)Time elapsed: 0.046 s
% 2.43/1.88 % (2090)Instructions burned: 152 (million)
% 2.43/1.88 % (2090)------------------------------
% 2.43/1.88 % (2090)------------------------------
% 2.43/1.88 % (2098)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/339Mi)
% 2.43/1.89 % (2099)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/283Mi)
% 2.43/1.89 % (2077)First to succeed.
% 2.43/1.91 % (2094)Instruction limit reached!
% 2.43/1.91 % (2094)------------------------------
% 2.43/1.91 % (2094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.91 % (2094)Termination reason: Unknown
% 2.43/1.91 % (2094)Termination phase: Function definition elimination
% 2.43/1.91
% 2.43/1.91 % (2094)Memory used [KB]: 4221
% 2.43/1.91 % (2094)Time elapsed: 0.040 s
% 2.43/1.91 % (2094)Instructions burned: 143 (million)
% 2.43/1.91 % (2094)------------------------------
% 2.43/1.91 % (2094)------------------------------
% 2.43/1.91 % (2084)Instruction limit reached!
% 2.43/1.91 % (2084)------------------------------
% 2.43/1.91 % (2084)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.91 % (2084)Termination reason: Unknown
% 2.43/1.91 % (2084)Termination phase: Saturation
% 2.43/1.91
% 2.43/1.91 % (2084)Memory used [KB]: 6652
% 2.43/1.91 % (2084)Time elapsed: 0.122 s
% 2.43/1.91 % (2084)Instructions burned: 455 (million)
% 2.43/1.91 % (2084)------------------------------
% 2.43/1.91 % (2084)------------------------------
% 2.43/1.92 % (2100)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/134Mi)
% 2.43/1.92 % (2101)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_1662 for (2995ds/298Mi)
% 2.43/1.93 % (2097)Instruction limit reached!
% 2.43/1.93 % (2097)------------------------------
% 2.43/1.93 % (2097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.93 % (2097)Termination reason: Unknown
% 2.43/1.93 % (2097)Termination phase: Saturation
% 2.43/1.93
% 2.43/1.93 % (2097)Memory used [KB]: 4477
% 2.43/1.93 % (2097)Time elapsed: 0.055 s
% 2.43/1.93 % (2097)Instructions burned: 204 (million)
% 2.43/1.93 % (2097)------------------------------
% 2.43/1.93 % (2097)------------------------------
% 2.43/1.93 % (2077)Refutation found. Thanks to Tanya!
% 2.43/1.93 % SZS status Theorem for DTF2THF_1662
% 2.43/1.93 % SZS output start Proof for DTF2THF_1662
% 2.43/1.93 thf(type_def_5, type, term: $tType).
% 2.43/1.93 thf(type_def_7, type, subst: $tType).
% 2.43/1.93 thf(type_def_8, type, d_term: $tType).
% 2.43/1.93 thf(type_def_9, type, d_subst: $tType).
% 2.43/1.93 thf(func_def_0, type, term: $tType).
% 2.43/1.93 thf(func_def_1, type, subst: $tType).
% 2.43/1.93 thf(func_def_2, type, one: term).
% 2.43/1.93 thf(func_def_3, type, ap: term > term > term).
% 2.43/1.93 thf(func_def_4, type, lam: term > term).
% 2.43/1.93 thf(func_def_5, type, sub: term > subst > term).
% 2.43/1.93 thf(func_def_6, type, id: subst).
% 2.43/1.93 thf(func_def_7, type, sh: subst).
% 2.43/1.93 thf(func_def_8, type, push: term > subst > subst).
% 2.43/1.93 thf(func_def_9, type, comp: subst > subst > subst).
% 2.43/1.93 thf(func_def_10, type, var: term > $o).
% 2.43/1.93 thf(func_def_18, type, hoasap: subst > term > subst > term > term).
% 2.43/1.93 thf(func_def_73, type, hoaslam: subst > (subst > term > term) > term).
% 2.43/1.93 thf(func_def_86, type, hoasinduction_p_and_p_prime: (subst > term > subst > $o) > (term > $o) > $o).
% 2.43/1.93 thf(func_def_107, type, hoasvar: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_121, type, pushprop_p_and_p_prime: term > subst > (term > $o) > (term > $o) > $o).
% 2.43/1.93 thf(func_def_124, type, d_term: $tType).
% 2.43/1.93 thf(func_def_125, type, d_subst: $tType).
% 2.43/1.93 thf(func_def_126, type, d2term: d_term > term).
% 2.43/1.93 thf(func_def_127, type, d2subst: d_subst > subst).
% 2.43/1.93 thf(func_def_128, type, d_one: d_term).
% 2.43/1.93 thf(func_def_129, type, d_id: d_subst).
% 2.43/1.93 thf(func_def_142, type, sP1: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_144, type, sP3: (term > $o) > $o).
% 2.43/1.93 thf(func_def_145, type, sP4: (term > $o) > $o).
% 2.43/1.93 thf(func_def_146, type, sP5: (term > $o) > $o).
% 2.43/1.93 thf(func_def_147, type, sP6: (term > $o) > subst > $o).
% 2.43/1.93 thf(func_def_148, type, sP7: (term > $o) > $o).
% 2.43/1.93 thf(func_def_151, type, sP10: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_153, type, sP12: (term > $o) > $o).
% 2.43/1.93 thf(func_def_154, type, sP13: (term > $o) > $o).
% 2.43/1.93 thf(func_def_156, type, sP15: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_157, type, sP16: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_158, type, sP17: (subst > term > subst > $o) > term > $o).
% 2.43/1.93 thf(func_def_160, type, sP19: (term > $o) > $o).
% 2.43/1.93 thf(func_def_161, type, sP20: (term > $o) > $o).
% 2.43/1.93 thf(func_def_162, type, sP21: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_163, type, sP22: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_164, type, sP23: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_165, type, sP24: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_166, type, sP25: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_170, type, sP29: (term > $o) > $o).
% 2.43/1.93 thf(func_def_171, type, sP30: (term > $o) > $o).
% 2.43/1.93 thf(func_def_175, type, sP34: (term > $o) > $o).
% 2.43/1.93 thf(func_def_176, type, sP35: (term > $o) > $o).
% 2.43/1.93 thf(func_def_177, type, sP36: (term > $o) > $o).
% 2.43/1.93 thf(func_def_178, type, sP37: (term > $o) > $o).
% 2.43/1.93 thf(func_def_181, type, sP40: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_182, type, sP41: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_183, type, sP42: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_184, type, sP43: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_185, type, sP44: (term > $o) > $o).
% 2.43/1.93 thf(func_def_186, type, sP45: (term > $o) > $o).
% 2.43/1.93 thf(func_def_187, type, sP46: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_188, type, sP47: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_189, type, sP48: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_190, type, sP49: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_191, type, sP50: (subst > term > subst > $o) > term > $o).
% 2.43/1.93 thf(func_def_196, type, sP55: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_197, type, sP56: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_198, type, sP57: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_200, type, sP59: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_201, type, sP60: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_202, type, sP61: term > (term > $o) > $o).
% 2.43/1.93 thf(func_def_204, type, sP63: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_205, type, sP64: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_208, type, sP67: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_209, type, sP68: (subst > term > subst > $o) > term > $o).
% 2.43/1.93 thf(func_def_210, type, sP69: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_211, type, sP70: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_216, type, sP75: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_218, type, sP77: (term > $o) > $o).
% 2.43/1.93 thf(func_def_219, type, sP78: (term > $o) > $o).
% 2.43/1.93 thf(func_def_220, type, sP79: (term > $o) > $o).
% 2.43/1.93 thf(func_def_221, type, sP80: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_222, type, sP81: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_223, type, sP82: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_229, type, sP88: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_230, type, sP89: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_231, type, sP90: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_232, type, sP91: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_234, type, sP93: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_236, type, sP95: (term > $o) > $o).
% 2.43/1.93 thf(func_def_237, type, sP96: (term > $o) > $o).
% 2.43/1.93 thf(func_def_239, type, sP98: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_241, type, sP100: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_242, type, sP101: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_244, type, sP103: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_245, type, sP104: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_246, type, sP105: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_248, type, sP107: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_249, type, sP108: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_251, type, sP110: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_252, type, sP111: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_253, type, sP112: (subst > term > subst > $o) > $o).
% 2.43/1.93 thf(func_def_255, type, sK114: term > $o).
% 2.43/1.93 thf(func_def_256, type, sK115: term).
% 2.43/1.93 thf(func_def_257, type, sK116: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_258, type, sK117: term).
% 2.43/1.93 thf(func_def_259, type, sK118: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_260, type, sK119: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_261, type, sK120: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_262, type, sK121: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_263, type, sK122: (subst > term > subst > $o) > (term > $o) > term).
% 2.43/1.93 thf(func_def_264, type, sK123: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_265, type, sK124: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_266, type, sK125: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_267, type, sK126: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_268, type, sK127: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_269, type, sK128: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_270, type, sK129: term > $o).
% 2.43/1.93 thf(func_def_271, type, sK130: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_272, type, sK131: term).
% 2.43/1.93 thf(func_def_273, type, sK132: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_274, type, sK133: (subst > term > subst > $o) > (term > $o) > term).
% 2.43/1.93 thf(func_def_275, type, sK134: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_276, type, sK135: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_277, type, sK136: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_278, type, sK137: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_279, type, sK138: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_280, type, sK139: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_281, type, sK140: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_282, type, sK141: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_283, type, sK142: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_284, type, sK143: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_285, type, sK144: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_286, type, sK145: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_287, type, sK146: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_288, type, sK147: term).
% 2.43/1.93 thf(func_def_289, type, sK148: term).
% 2.43/1.93 thf(func_def_290, type, sK149: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_291, type, sK150: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_292, type, sK151: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_293, type, sK152: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_294, type, sK153: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_295, type, sK154: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_296, type, sK155: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_297, type, sK156: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_298, type, sK157: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_299, type, sK158: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_300, type, sK159: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_301, type, sK160: term).
% 2.43/1.93 thf(func_def_302, type, sK161: term > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_303, type, sK162: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_304, type, sK163: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_305, type, sK164: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_306, type, sK165: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_307, type, sK166: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_308, type, sK167: term > $o).
% 2.43/1.93 thf(func_def_309, type, sK168: term).
% 2.43/1.93 thf(func_def_310, type, sK169: (term > $o) > term).
% 2.43/1.93 thf(func_def_311, type, sK170: (term > $o) > term).
% 2.43/1.93 thf(func_def_312, type, sK171: (term > $o) > term).
% 2.43/1.93 thf(func_def_313, type, sK172: (term > $o) > term).
% 2.43/1.93 thf(func_def_314, type, sK173: term).
% 2.43/1.93 thf(func_def_315, type, sK174: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_316, type, sK175: (subst > term > subst > $o) > term > term).
% 2.43/1.93 thf(func_def_317, type, sK176: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_318, type, sK177: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_319, type, sK178: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_320, type, sK179: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_321, type, sK180: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_322, type, sK181: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_323, type, sK182: term).
% 2.43/1.93 thf(func_def_324, type, sK183: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_325, type, sK184: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_326, type, sK185: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_327, type, sK186: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_328, type, sK187: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_329, type, sK188: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_330, type, sK189: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_331, type, sK190: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_332, type, sK191: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_333, type, sK192: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_334, type, sK193: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_335, type, sK194: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_336, type, sK195: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_337, type, sK196: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_338, type, sK197: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_339, type, sK198: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_340, type, sK199: term).
% 2.43/1.93 thf(func_def_341, type, sK200: term).
% 2.43/1.93 thf(func_def_342, type, sK201: term).
% 2.43/1.93 thf(func_def_343, type, sK202: term).
% 2.43/1.93 thf(func_def_344, type, sK203: term).
% 2.43/1.93 thf(func_def_345, type, sK204: term).
% 2.43/1.93 thf(func_def_346, type, sK205: term).
% 2.43/1.93 thf(func_def_347, type, sK206: term).
% 2.43/1.93 thf(func_def_348, type, sK207: term).
% 2.43/1.93 thf(func_def_349, type, sK208: term).
% 2.43/1.93 thf(func_def_350, type, sK209: term > $o).
% 2.43/1.93 thf(func_def_351, type, sK210: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_352, type, sK211: (subst > term > subst > $o) > (term > $o) > term).
% 2.43/1.93 thf(func_def_353, type, sK212: term > $o).
% 2.43/1.93 thf(func_def_354, type, sK213: term).
% 2.43/1.93 thf(func_def_355, type, sK214: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_356, type, sK215: term).
% 2.43/1.93 thf(func_def_357, type, sK216: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_358, type, sK217: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_359, type, sK218: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_360, type, sK219: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_361, type, sK220: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_362, type, sK221: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_363, type, sK222: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_364, type, sK223: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_365, type, sK224: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_366, type, sK225: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_367, type, sK226: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_368, type, sK227: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_369, type, sK228: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_370, type, sK229: (term > $o) > term).
% 2.43/1.93 thf(func_def_371, type, sK230: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_372, type, sK231: (term > $o) > term).
% 2.43/1.93 thf(func_def_373, type, sK232: (term > $o) > term).
% 2.43/1.93 thf(func_def_374, type, sK233: (term > $o) > term).
% 2.43/1.93 thf(func_def_375, type, sK234: term).
% 2.43/1.93 thf(func_def_376, type, sK235: term > $o).
% 2.43/1.93 thf(func_def_377, type, sK236: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_378, type, sK237: (subst > term > subst > $o) > (term > $o) > term).
% 2.43/1.93 thf(func_def_379, type, sK238: (term > $o) > term > term).
% 2.43/1.93 thf(func_def_380, type, sK239: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_381, type, sK240: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_382, type, sK241: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_383, type, sK242: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_384, type, sK243: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_385, type, sK244: term).
% 2.43/1.93 thf(func_def_386, type, sK245: term).
% 2.43/1.93 thf(func_def_387, type, sK246: term).
% 2.43/1.93 thf(func_def_388, type, sK247: term).
% 2.43/1.93 thf(func_def_389, type, sK248: term).
% 2.43/1.93 thf(func_def_390, type, sK249: term).
% 2.43/1.93 thf(func_def_391, type, sK250: term).
% 2.43/1.93 thf(func_def_392, type, sK251: term).
% 2.43/1.93 thf(func_def_393, type, sK252: (subst > term > subst > $o) > term > term).
% 2.43/1.93 thf(func_def_394, type, sK253: term).
% 2.43/1.93 thf(func_def_395, type, sK254: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_396, type, sK255: term).
% 2.43/1.93 thf(func_def_397, type, sK256: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_398, type, sK257: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_399, type, sK258: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_400, type, sK259: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_401, type, sK260: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_402, type, sK261: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_403, type, sK262: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_404, type, sK263: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_405, type, sK264: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_406, type, sK265: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_407, type, sK266: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_408, type, sK267: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_409, type, sK268: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_410, type, sK269: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_411, type, sK270: (subst > term > subst > $o) > term > term).
% 2.43/1.93 thf(func_def_412, type, sK271: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_413, type, sK272: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_414, type, sK273: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_415, type, sK274: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_416, type, sK275: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_417, type, sK276: (subst > term > subst > $o) > term > term).
% 2.43/1.93 thf(func_def_418, type, sK277: term).
% 2.43/1.93 thf(func_def_419, type, sK278: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_420, type, sK279: term).
% 2.43/1.93 thf(func_def_421, type, sK280: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_422, type, sK281: (subst > term > subst > $o) > term > term).
% 2.43/1.93 thf(func_def_423, type, sK282: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_424, type, sK283: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_425, type, sK284: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_426, type, sK285: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_427, type, sK286: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_428, type, sK287: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_429, type, sK288: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_430, type, sK289: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_431, type, sK290: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_432, type, sK291: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_433, type, sK292: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_434, type, sK293: term).
% 2.43/1.93 thf(func_def_435, type, sK294: term > $o).
% 2.43/1.93 thf(func_def_436, type, sK295: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_437, type, sK296: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_438, type, sK297: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_439, type, sK298: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_440, type, sK299: (term > $o) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_441, type, sK300: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_442, type, sK301: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_443, type, sK302: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_444, type, sK303: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_445, type, sK304: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_446, type, sK305: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_447, type, sK306: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_448, type, sK307: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_449, type, sK308: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_450, type, sK309: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_451, type, sK310: term).
% 2.43/1.93 thf(func_def_452, type, sK311: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_453, type, sK312: term > $o).
% 2.43/1.93 thf(func_def_454, type, sK313: (term > $o) > term > term).
% 2.43/1.93 thf(func_def_455, type, sK314: (term > $o) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_456, type, sK315: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_457, type, sK316: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_458, type, sK317: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_459, type, sK318: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_460, type, sK319: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_461, type, sK320: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_462, type, sK321: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_463, type, sK322: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_464, type, sK323: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_465, type, sK324: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_466, type, sK325: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_467, type, sK326: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_468, type, sK327: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_469, type, sK328: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_470, type, sK329: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_471, type, sK330: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_472, type, sK331: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_473, type, sK332: term).
% 2.43/1.93 thf(func_def_474, type, sK333: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_475, type, sK334: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_476, type, sK335: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_477, type, sK336: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_478, type, sK337: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_479, type, sK338: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_480, type, sK339: term).
% 2.43/1.93 thf(func_def_481, type, sK340: term).
% 2.43/1.93 thf(func_def_482, type, sK341: (term > $o) > term).
% 2.43/1.93 thf(func_def_483, type, sK342: term > $o).
% 2.43/1.93 thf(func_def_484, type, sK343: term).
% 2.43/1.93 thf(func_def_485, type, sK344: term).
% 2.43/1.93 thf(func_def_486, type, sK345: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_487, type, sK346: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_488, type, sK347: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_489, type, sK348: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_490, type, sK349: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_491, type, sK350: (subst > term > subst > $o) > term > term).
% 2.43/1.93 thf(func_def_492, type, sK351: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_493, type, sK352: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_494, type, sK353: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_495, type, sK354: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_496, type, sK355: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_497, type, sK356: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_498, type, sK357: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_499, type, sK358: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_500, type, sK359: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_501, type, sK360: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_502, type, sK361: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_503, type, sK362: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_504, type, sK363: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_505, type, sK364: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_506, type, sK365: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_507, type, sK366: (term > $o) > term).
% 2.43/1.93 thf(func_def_508, type, sK367: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_509, type, sK368: (term > $o) > term).
% 2.43/1.93 thf(func_def_510, type, sK369: (term > $o) > term).
% 2.43/1.93 thf(func_def_511, type, sK370: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_512, type, sK371: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_513, type, sK372: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_514, type, sK373: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_515, type, sK374: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_516, type, sK375: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_517, type, sK376: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_518, type, sK377: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_519, type, sK378: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_520, type, sK379: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_521, type, sK380: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_522, type, sK381: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_523, type, sK382: (term > $o) > term).
% 2.43/1.93 thf(func_def_524, type, sK383: term).
% 2.43/1.93 thf(func_def_525, type, sK384: term > $o).
% 2.43/1.93 thf(func_def_526, type, sK385: term).
% 2.43/1.93 thf(func_def_527, type, sK386: term > $o).
% 2.43/1.93 thf(func_def_528, type, sK387: subst).
% 2.43/1.93 thf(func_def_529, type, sK388: subst > (term > $o) > term).
% 2.43/1.93 thf(func_def_530, type, sK389: (term > $o) > term).
% 2.43/1.93 thf(func_def_531, type, sK390: (term > $o) > term).
% 2.43/1.93 thf(func_def_532, type, sK391: (term > $o) > term).
% 2.43/1.93 thf(func_def_533, type, sK392: (term > $o) > term).
% 2.43/1.93 thf(func_def_534, type, sK393: (term > $o) > term).
% 2.43/1.93 thf(func_def_535, type, sK394: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_536, type, sK395: (term > $o) > term).
% 2.43/1.93 thf(func_def_537, type, sK396: subst).
% 2.43/1.93 thf(func_def_538, type, sK397: term).
% 2.43/1.93 thf(func_def_539, type, sK398: term).
% 2.43/1.93 thf(func_def_540, type, sK399: term).
% 2.43/1.93 thf(func_def_541, type, sK400: subst).
% 2.43/1.93 thf(func_def_542, type, sK401: term).
% 2.43/1.93 thf(func_def_543, type, sK402: term).
% 2.43/1.93 thf(func_def_544, type, sK403: subst > term > term).
% 2.43/1.93 thf(func_def_545, type, sK404: subst).
% 2.43/1.93 thf(func_def_546, type, sK405: subst > term > term).
% 2.43/1.93 thf(func_def_547, type, sK406: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_548, type, sK407: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_549, type, sK408: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_550, type, sK409: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_551, type, sK410: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_552, type, sK411: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_553, type, sK412: term).
% 2.43/1.93 thf(func_def_554, type, sK413: term > $o).
% 2.43/1.93 thf(func_def_555, type, sK414: (term > $o) > term).
% 2.43/1.93 thf(func_def_556, type, sK415: (term > $o) > term).
% 2.43/1.93 thf(func_def_557, type, sK416: (term > $o) > term).
% 2.43/1.93 thf(func_def_558, type, sK417: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_559, type, sK418: (term > $o) > term).
% 2.43/1.93 thf(func_def_560, type, sK419: (term > $o) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_561, type, sK420: (term > $o) > term > term).
% 2.43/1.93 thf(func_def_562, type, sK421: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_563, type, sK422: term).
% 2.43/1.93 thf(func_def_564, type, sK423: term > $o).
% 2.43/1.93 thf(func_def_565, type, sK424: (term > $o) > term).
% 2.43/1.93 thf(func_def_566, type, sK425: term > $o).
% 2.43/1.93 thf(func_def_567, type, sK426: term).
% 2.43/1.93 thf(func_def_568, type, sK427: term).
% 2.43/1.93 thf(func_def_569, type, sK428: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_570, type, sK429: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_571, type, sK430: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_572, type, sK431: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_573, type, sK432: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_574, type, sK433: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_575, type, sK434: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_576, type, sK435: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_577, type, sK436: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_578, type, sK437: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_579, type, sK438: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_580, type, sK439: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_581, type, sK440: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_582, type, sK441: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_583, type, sK442: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_584, type, sK443: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_585, type, sK444: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_586, type, sK445: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_587, type, sK446: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_588, type, sK447: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_589, type, sK448: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_590, type, sK449: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_591, type, sK450: (term > $o) > term).
% 2.43/1.93 thf(func_def_592, type, sK451: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_593, type, sK452: (term > $o) > term).
% 2.43/1.93 thf(func_def_594, type, sK453: (term > $o) > term).
% 2.43/1.93 thf(func_def_595, type, sK454: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_596, type, sK455: term).
% 2.43/1.93 thf(func_def_597, type, sK456: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_598, type, sK457: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_599, type, sK458: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_600, type, sK459: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_601, type, sK460: (subst > term > subst > $o) > term > term).
% 2.43/1.93 thf(func_def_602, type, sK461: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_603, type, sK462: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_604, type, sK463: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_605, type, sK464: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_606, type, sK465: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_607, type, sK466: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_608, type, sK467: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_609, type, sK468: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_610, type, sK469: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_611, type, sK470: term).
% 2.43/1.93 thf(func_def_612, type, sK471: term > $o).
% 2.43/1.93 thf(func_def_613, type, sK472: subst).
% 2.43/1.93 thf(func_def_614, type, sK473: subst > (term > $o) > term).
% 2.43/1.93 thf(func_def_615, type, sK474: (term > $o) > term).
% 2.43/1.93 thf(func_def_616, type, sK475: (term > $o) > term).
% 2.43/1.93 thf(func_def_617, type, sK476: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_618, type, sK477: (term > $o) > term).
% 2.43/1.93 thf(func_def_619, type, sK478: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_620, type, sK479: term).
% 2.43/1.93 thf(func_def_621, type, sK480: term > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_622, type, sK481: (subst > term > term) > term > subst).
% 2.43/1.93 thf(func_def_623, type, sK482: (subst > term > term) > term > subst).
% 2.43/1.93 thf(func_def_624, type, sK483: (subst > term > term) > term > term).
% 2.43/1.93 thf(func_def_625, type, sK484: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_626, type, sK485: (subst > term > term) > term > subst).
% 2.43/1.93 thf(func_def_627, type, sK486: (subst > term > term) > term > term).
% 2.43/1.93 thf(func_def_628, type, sK487: (subst > term > term) > term > subst).
% 2.43/1.93 thf(func_def_629, type, sK488: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_630, type, sK489: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_631, type, sK490: term).
% 2.43/1.93 thf(func_def_632, type, sK491: subst).
% 2.43/1.93 thf(func_def_633, type, sK492: term > $o).
% 2.43/1.93 thf(func_def_634, type, sK493: term).
% 2.43/1.93 thf(func_def_635, type, sK494: term > $o).
% 2.43/1.93 thf(func_def_636, type, sK495: (term > $o) > term).
% 2.43/1.93 thf(func_def_637, type, sK496: (term > $o) > term).
% 2.43/1.93 thf(func_def_638, type, sK497: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_639, type, sK498: (term > $o) > subst > term).
% 2.43/1.93 thf(func_def_640, type, sK499: (term > $o) > term).
% 2.43/1.93 thf(func_def_641, type, sK500: (term > $o) > term).
% 2.43/1.93 thf(func_def_642, type, sK501: (term > $o) > term).
% 2.43/1.93 thf(func_def_643, type, sK502: (term > $o) > term).
% 2.43/1.93 thf(func_def_644, type, sK503: term > (term > $o) > term).
% 2.43/1.93 thf(func_def_645, type, sK504: (term > $o) > term).
% 2.43/1.93 thf(func_def_646, type, sK505: term).
% 2.43/1.93 thf(func_def_647, type, sK506: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_648, type, sK507: (subst > term > subst > $o) > term > term).
% 2.43/1.93 thf(func_def_649, type, sK508: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_650, type, sK509: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_651, type, sK510: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_652, type, sK511: (subst > term > term) > (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_653, type, sK512: (subst > term > subst > $o) > subst > term > term).
% 2.43/1.93 thf(func_def_654, type, sK513: subst).
% 2.43/1.93 thf(func_def_655, type, sK514: subst).
% 2.43/1.93 thf(func_def_656, type, sK515: subst).
% 2.43/1.93 thf(func_def_657, type, sK516: subst).
% 2.43/1.93 thf(func_def_658, type, sK517: term > $o).
% 2.43/1.93 thf(func_def_659, type, sK518: term).
% 2.43/1.93 thf(func_def_660, type, sK519: subst).
% 2.43/1.93 thf(func_def_661, type, sK520: term).
% 2.43/1.93 thf(func_def_662, type, sK521: term > $o).
% 2.43/1.93 thf(func_def_663, type, sK522: subst).
% 2.43/1.93 thf(func_def_664, type, sK523: term).
% 2.43/1.93 thf(func_def_665, type, sK524: subst).
% 2.43/1.93 thf(func_def_666, type, sK525: term).
% 2.43/1.93 thf(func_def_667, type, sK526: term).
% 2.43/1.93 thf(func_def_668, type, sK527: term).
% 2.43/1.93 thf(func_def_669, type, sK528: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_670, type, sK529: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_671, type, sK530: term > subst > term > term).
% 2.43/1.93 thf(func_def_672, type, sK531: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_673, type, sK532: (term > $o) > term).
% 2.43/1.93 thf(func_def_674, type, sK533: (subst > term > subst > $o) > term > $o).
% 2.43/1.93 thf(func_def_675, type, sK534: term).
% 2.43/1.93 thf(func_def_676, type, sK535: term > subst > term > term).
% 2.43/1.93 thf(func_def_677, type, sK536: subst).
% 2.43/1.93 thf(func_def_678, type, sK537: term).
% 2.43/1.93 thf(func_def_679, type, sK538: term).
% 2.43/1.93 thf(func_def_680, type, sK539: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_681, type, sK540: term).
% 2.43/1.93 thf(func_def_682, type, sK541: term > $o).
% 2.43/1.93 thf(func_def_683, type, sK542: subst).
% 2.43/1.93 thf(func_def_684, type, sK543: (term > $o) > term).
% 2.43/1.93 thf(func_def_685, type, sK544: subst > (term > $o) > term > term > $o).
% 2.43/1.93 thf(func_def_686, type, sK545: subst).
% 2.43/1.93 thf(func_def_687, type, sK546: subst).
% 2.43/1.93 thf(func_def_688, type, sK547: term).
% 2.43/1.93 thf(func_def_689, type, sK548: term).
% 2.43/1.93 thf(func_def_690, type, sK549: (subst > term > subst > $o) > term > $o).
% 2.43/1.93 thf(func_def_691, type, sK550: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_692, type, sK551: (term > $o) > term).
% 2.43/1.93 thf(func_def_693, type, sK552: term).
% 2.43/1.93 thf(func_def_694, type, sK553: term).
% 2.43/1.93 thf(func_def_695, type, sK554: term).
% 2.43/1.93 thf(func_def_696, type, sK555: term).
% 2.43/1.93 thf(func_def_697, type, sK556: term).
% 2.43/1.93 thf(func_def_698, type, sK557: subst > d_subst).
% 2.43/1.93 thf(func_def_699, type, sK558: term).
% 2.43/1.93 thf(func_def_700, type, sK559: term > $o).
% 2.43/1.93 thf(func_def_701, type, sK560: term > $o).
% 2.43/1.93 thf(func_def_702, type, sK561: term).
% 2.43/1.93 thf(func_def_703, type, sK562: subst).
% 2.43/1.93 thf(func_def_704, type, sK563: term).
% 2.43/1.93 thf(func_def_705, type, sK564: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_706, type, sK565: (term > $o) > term).
% 2.43/1.93 thf(func_def_707, type, sK566: (subst > term > subst > $o) > term > $o).
% 2.43/1.93 thf(func_def_708, type, sK567: term).
% 2.43/1.93 thf(func_def_709, type, sK568: term).
% 2.43/1.93 thf(func_def_710, type, sK569: term).
% 2.43/1.93 thf(func_def_711, type, sK570: term).
% 2.43/1.93 thf(func_def_712, type, sK571: term > subst > term > term).
% 2.43/1.93 thf(func_def_713, type, sK572: term).
% 2.43/1.93 thf(func_def_714, type, sK573: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_715, type, sK574: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_716, type, sK575: term).
% 2.43/1.93 thf(func_def_717, type, sK576: term > d_term).
% 2.43/1.93 thf(func_def_718, type, sK577: subst).
% 2.43/1.93 thf(func_def_719, type, sK578: term).
% 2.43/1.93 thf(func_def_720, type, sK579: term).
% 2.43/1.93 thf(func_def_721, type, sK580: subst).
% 2.43/1.93 thf(func_def_722, type, sK581: subst).
% 2.43/1.93 thf(func_def_723, type, sK582: term).
% 2.43/1.93 thf(func_def_724, type, sK583: term).
% 2.43/1.93 thf(func_def_725, type, sK584: term).
% 2.43/1.93 thf(func_def_726, type, sK585: term).
% 2.43/1.93 thf(func_def_727, type, sK586: subst).
% 2.43/1.93 thf(func_def_728, type, sK587: subst).
% 2.43/1.93 thf(func_def_729, type, sK588: term).
% 2.43/1.93 thf(func_def_730, type, sK589: (subst > term > subst > $o) > term > $o).
% 2.43/1.93 thf(func_def_731, type, sK590: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_732, type, sK591: (term > $o) > term).
% 2.43/1.93 thf(func_def_733, type, sK592: term).
% 2.43/1.93 thf(func_def_734, type, sK593: term).
% 2.43/1.93 thf(func_def_735, type, sK594: term).
% 2.43/1.93 thf(func_def_736, type, sK595: term).
% 2.43/1.93 thf(func_def_737, type, sK596: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_738, type, sK597: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_739, type, sK598: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_740, type, sK599: term).
% 2.43/1.93 thf(func_def_741, type, sK600: subst > term > term).
% 2.43/1.93 thf(func_def_742, type, sK601: term).
% 2.43/1.93 thf(func_def_743, type, sK602: term).
% 2.43/1.93 thf(func_def_744, type, sK603: term > $o).
% 2.43/1.93 thf(func_def_745, type, sK604: term > $o).
% 2.43/1.93 thf(func_def_746, type, sK605: subst).
% 2.43/1.93 thf(func_def_747, type, sK606: subst > (term > $o) > (term > $o) > term > term).
% 2.43/1.93 thf(func_def_748, type, sK607: term > subst > term > term).
% 2.43/1.93 thf(func_def_749, type, sK608: term).
% 2.43/1.93 thf(func_def_750, type, sK609: subst).
% 2.43/1.93 thf(func_def_751, type, sK610: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_752, type, sK611: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_753, type, sK612: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_754, type, sK613: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_755, type, sK614: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_756, type, sK615: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_757, type, sK616: term).
% 2.43/1.93 thf(func_def_758, type, sK617: subst > term > term).
% 2.43/1.93 thf(func_def_759, type, sK618: subst).
% 2.43/1.93 thf(func_def_760, type, sK619: subst > term > term).
% 2.43/1.93 thf(func_def_761, type, sK620: subst > term > (term > $o) > term > $o).
% 2.43/1.93 thf(func_def_762, type, sK621: term > $o).
% 2.43/1.93 thf(func_def_763, type, sK622: term).
% 2.43/1.93 thf(func_def_764, type, sK623: subst).
% 2.43/1.93 thf(func_def_765, type, sK624: (term > $o) > term).
% 2.43/1.93 thf(func_def_766, type, sK625: term).
% 2.43/1.93 thf(func_def_767, type, sK626: term).
% 2.43/1.93 thf(func_def_768, type, sK627: term).
% 2.43/1.93 thf(func_def_769, type, sK628: term).
% 2.43/1.93 thf(func_def_770, type, sK629: term).
% 2.43/1.93 thf(func_def_771, type, sK630: term > subst > (term > $o) > (term > $o) > term).
% 2.43/1.93 thf(func_def_772, type, sK631: term).
% 2.43/1.93 thf(func_def_773, type, sK632: subst).
% 2.43/1.93 thf(func_def_774, type, sK633: term > $o).
% 2.43/1.93 thf(func_def_775, type, sK634: term > $o).
% 2.43/1.93 thf(func_def_776, type, sK635: subst).
% 2.43/1.93 thf(func_def_777, type, sK636: term).
% 2.43/1.93 thf(func_def_778, type, sK637: subst).
% 2.43/1.93 thf(func_def_779, type, sK638: term > subst > term > term).
% 2.43/1.93 thf(func_def_780, type, sK639: term).
% 2.43/1.93 thf(func_def_781, type, sK640: (subst > term > term) > term).
% 2.43/1.93 thf(func_def_782, type, sK641: (subst > term > term) > subst).
% 2.43/1.93 thf(func_def_783, type, sK642: term).
% 2.43/1.93 thf(func_def_784, type, sK643: term).
% 2.43/1.93 thf(func_def_785, type, sK644: term).
% 2.43/1.93 thf(func_def_786, type, sK645: term).
% 2.43/1.93 thf(func_def_787, type, sK646: subst > term > subst > $o).
% 2.43/1.93 thf(func_def_788, type, sK647: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_789, type, sK648: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_790, type, sK649: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_791, type, sK650: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_792, type, sK651: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_793, type, sK652: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_794, type, sK653: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_795, type, sK654: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_796, type, sK655: (subst > term > subst > $o) > subst).
% 2.43/1.93 thf(func_def_797, type, sK656: (subst > term > subst > $o) > term).
% 2.43/1.93 thf(func_def_800, type, ph658: !>[X0: $tType]:(X0)).
% 2.43/1.93 thf(f4958,plain,(
% 2.43/1.93 $false),
% 2.43/1.93 inference(avatar_sat_refutation,[status(thm)],[f2942,f3332,f4957])).
% 2.43/1.93 thf(f4957,plain,(
% 2.43/1.93 spl657_345),
% 2.43/1.93 inference(avatar_contradiction_clause,[status(thm)],[f4956])).
% 2.43/1.93 thf(f4956,plain,(
% 2.43/1.93 $false | spl657_345),
% 2.43/1.93 inference(trivial_inequality_removal,[status(thm)],[f4955])).
% 2.43/1.93 thf(f4955,plain,(
% 2.43/1.93 ($true != $true) | spl657_345),
% 2.43/1.93 inference(superposition,[status(thm)],[f3331,f4954])).
% 2.43/1.93 thf(f4954,plain,(
% 2.43/1.93 ( ! [X0 : subst > term > subst > $o] : (($true = (sP110 @ X0))) )),
% 2.43/1.93 inference(subsumption_resolution,[status(thm)],[f1530,f3880])).
% 2.43/1.93 thf(f3880,plain,(
% 2.43/1.93 ( ! [X0 : term] : (((var @ X0) != $true)) )),
% 2.43/1.93 inference(superposition,[status(thm)],[f1395,f3872])).
% 2.43/1.93 thf(f3872,plain,(
% 2.43/1.93 ( ! [X124 : term] : (((d2term @ d_one) = X124)) )),
% 2.43/1.93 inference(forward_demodulation,[status(thm)],[f1393,f1474])).
% 2.43/1.93 thf(f1474,plain,(
% 2.43/1.93 ( ! [X62 : d_term] : ((d_one = X62)) )),
% 2.43/1.93 inference(cnf_transformation,[status(thm)],[f749])).
% 2.43/1.93 thf(f749,plain,(
% 2.43/1.93 ((pushprop_lem1_lthm = $true) | ((((sK517 @ (sub @ sK518 @ (push @ sK520 @ sK519))) != $true) & ((sK521 @ sK518) = $true)) & ! [X5 : subst,X6 : term] : ((sub @ one @ (push @ X6 @ X5)) = X6) & ! [X7 : subst,X8 : term] : ((comp @ sh @ (push @ X8 @ X7)) = X7))) & (! [X9 : term > $o,X10 : term,X11 : subst,X12 : term,X13 : term > $o] : (((X9 @ (sub @ X10 @ (push @ X12 @ X11))) = $true) | ((X13 @ X10) != $true)) | (sK523 != (sub @ one @ (push @ sK523 @ sK522))) | ((comp @ sh @ (push @ sK525 @ sK524)) != sK524) | (pushprop_lem1_lthm != $true)) & ! [X18 : d_subst,X19 : d_subst] : (((d2subst @ X18) != (d2subst @ X19)) | (X18 = X19)) & ((hoasinduction_lem2v2 = $true) | ($true != sP109)) & (($true = sP109) | (hoasinduction_lem2v2 != $true)) & (((sub @ sK526 @ id) != sK526) | ! [X22 : subst > term > term] : ((sub @ sK527 @ (push @ (sK528 @ X22) @ (sK529 @ X22))) != (X22 @ (sK529 @ X22) @ (sK528 @ X22))) | ($true = sP106) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (! [X25 : term] : ((sub @ X25 @ id) = X25) & ! [X26 : term] : ! [X28 : term,X29 : subst] : ((sub @ X26 @ (push @ X28 @ X29)) = (sK530 @ X26 @ X29 @ X28)) & ($true != sP106))) & ((hoasinduction_lem0 = $true) | ! [X31 : term > $o] : ((X31 @ (sK532 @ X31)) != (sK531 @ id @ (sK532 @ X31) @ id))) & (! [X33 : subst > term > subst > $o] : ! [X35 : term] : ((sK533 @ X33 @ X35) = (X33 @ id @ X35 @ id)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X37 : subst > term > term] : ((X37 @ sh @ one) != (sub @ sK534 @ (push @ one @ sh)))) & (! [X38 : term] : ((sK535 @ X38 @ sh @ one) = (sub @ X38 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & (induction2lem_gthm = $true) & (one = (d2term @ d_one)) & (hoasinduction_lem3b_gthm = $true) & (! [X40 : subst,X41 : term] : ((comp @ sh @ (push @ X41 @ X40)) = X40) | (axshiftcons != $true)) & ((axshiftcons = $true) | ((comp @ sh @ (push @ sK537 @ sK536)) != sK536)) & (! [X44 : term,X45 : subst > term > subst > $o] : (((sP103 @ X45) = $true) | ((X45 @ id @ X44 @ id) = $true) | ((sP104 @ X45) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | (($true != (sP103 @ sK539)) & ((sK539 @ id @ sK538 @ id) != $true) & ($true != (sP104 @ sK539)))) & (termmset_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem2 = $true) | ($true != sP102)) & (($true = sP102) | (hoasinduction_lem2 != $true)) & (hoasinduction_lem1_lthm = $true) & (($true = sP99) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP99)) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0_lthm = $true) | ! [X51 : term > $o] : ((X51 @ (sK543 @ X51)) != (sK541 @ (sub @ (sK543 @ X51) @ (push @ sK540 @ sK542))))) & (! [X53 : term,X54 : term > $o,X55 : subst] : ! [X57 : term] : ((X54 @ (sub @ X57 @ (push @ X53 @ X55))) = (sK544 @ X55 @ X54 @ X53 @ X57)) | (pushprop_lem0_lthm != $true)) & (hoasapnotvar_lthm = $true) & ((! [X58 : subst] : ((comp @ id @ X58) = X58) & ! [X59 : subst] : ((comp @ X59 @ id) = X59)) | (substmonoid != $true)) & ((substmonoid = $true) | (sK545 != (comp @ id @ sK545)) | ((comp @ sK546 @ id) != sK546)) & (hoaslamnotap_lthm = $true) & (pushprop_lthm = $true) & ! [X62 : d_term] : (d_one = X62) & (lamnotvar = $true) & ((induction = $true) | (sP97 != $true)) & ((sP97 = $true) | (induction != $true)) & (! [X63 : term,X64 : term] : (((sub @ X63 @ sh) != (sub @ X64 @ sh)) | (X63 = X64)) | (shinj != $true)) & ((shinj = $true) | (((sub @ sK547 @ sh) = (sub @ sK548 @ sh)) & (sK548 != sK547))) & (hoasinduction_lem1v2_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X67 : subst > term > subst > $o] : ! [X69 : term] : ((X67 @ id @ X69 @ id) = (sK549 @ X67 @ X69)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X71 : term > $o] : ((sK550 @ id @ (sK551 @ X71) @ id) != (X71 @ (sK551 @ X71)))) & ((sP94 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP94 != $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction = $true) | (sP92 != $true)) & ((sP92 = $true) | (hoasinduction != $true)) & ((sP87 = $true) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (sP87 != $true)) & (lamnotap != $true) & ((apinj2 = $true) | (((ap @ sK554 @ sK552) = (ap @ sK555 @ sK553)) & (sK553 != sK552))) & (! [X77 : term,X78 : term,X79 : term,X80 : term] : (((ap @ X80 @ X78) != (ap @ X79 @ X77)) | (X77 = X78)) | (apinj2 != $true)) & ((termmset = $true) | ((sub @ sK556 @ id) != sK556)) & (! [X82 : term] : ((sub @ X82 @ id) = X82) | (termmset != $true)) & (hoasinduction_lem3v2_gthm = $true) & (ulamvar1 != $true) & (axapp = $true) & ! [X83 : subst] : ((d2subst @ (sK557 @ X83)) = X83) & (! [X85 : term,X86 : term > $o,X87 : term > $o,X88 : term,X89 : subst] : (((X87 @ (sub @ X85 @ (push @ X88 @ X89))) = $true) | ((X86 @ X85) != $true)) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | (((sK560 @ (sub @ sK558 @ (push @ sK561 @ sK562))) != $true) & ((sK559 @ sK558) = $true))) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ! [X95 : d_subst] : (d_id = X95) & ! [X96 : d_term,X97 : d_term] : ((X96 = X97) | ((d2term @ X97) != (d2term @ X96))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((sK563 != (sub @ sK563 @ id)) | (sP84 = $true) | ! [X100 : term > $o] : ((sK564 @ id @ (sK565 @ X100) @ id) != (X100 @ (sK565 @ X100))) | (sP83 = $true) | (sP85 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X102 : term] : ((sub @ X102 @ id) = X102) & (sP84 != $true) & ! [X103 : subst > term > subst > $o] : ! [X105 : term] : ((sK566 @ X103 @ X105) = (X103 @ id @ X105 @ id)) & (sP83 != $true) & (sP85 != $true))) & (axclos = $true) & ((laminj = $true) | ((sK567 != sK568) & ((lam @ sK567) = (lam @ sK568)))) & (! [X108 : term,X109 : term] : ((X108 = X109) | ((lam @ X109) != (lam @ X108))) | (laminj != $true)) & ((sP76 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP76 != $true)) & ((hoasapinj2_lthm = $true) | (sP74 != $true)) & ((sP74 = $true) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ((sP72 != $true) & ! [X110 : term] : ((sub @ X110 @ id) = X110) & (sP71 != $true))) & ((sP72 = $true) | (sK569 != (sub @ sK569 @ id)) | (sP71 = $true) | (hoasinduction_lem3_lthm != $true)) & (hoaslamnotap_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lthm_orig = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) | (! [X112 : term] : ((sub @ X112 @ id) = X112) & ($true != sP65) & ($true != sP66))) & (((sub @ sK570 @ id) != sK570) | ($true = sP65) | ($true = sP66) | (hoasinduction_lem3a_lthm != $true)) & (! [X114 : term] : ! [X116 : term,X117 : subst] : ((sub @ X114 @ (push @ X116 @ X117)) = (sK571 @ X114 @ X117 @ X116)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X119 : subst > term > term] : ((X119 @ (sK574 @ X119) @ (sK573 @ X119)) != (sub @ sK572 @ (push @ (sK573 @ X119) @ (sK574 @ X119))))) & ((hoasinduction_lem3v2 = $true) | (sP62 != $true)) & ((sP62 = $true) | (hoasinduction_lem3v2 != $true)) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | (! [X122 : term] : ((sub @ X122 @ id) = X122) & (sP58 != $true))) & (((sub @ sK575 @ id) != sK575) | (sP58 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lthm = $true) | ((sP54 != $true) & ($true != sP53) & (sP52 != $true) & (sP51 != $true))) & ((sP54 = $true) | ($true = sP53) | (sP52 = $true) | (sP51 = $true) | (hoasinduction_lthm != $true)) & ((var @ (d2term @ d_one)) != $true) & (hoasapinj1_gthm = $true) & ! [X124 : term] : ((d2term @ (sK576 @ X124)) = X124) & ((induction2lem_lthm = $true) | (! [X126 : subst,X127 : term] : ((comp @ sh @ (push @ X127 @ X126)) = X126) & ! [X128 : term,X129 : subst] : ((sub @ one @ (push @ X128 @ X129)) = X128) & (sP39 != $true) & (sP38 != $true) & ! [X130 : subst] : ((comp @ X130 @ id) = X130))) & ((sK577 != (comp @ sh @ (push @ sK578 @ sK577))) | (sK579 != (sub @ one @ (push @ sK579 @ sK580))) | (sP39 = $true) | (sP38 = $true) | (sK581 != (comp @ sK581 @ id)) | (induction2lem_lthm != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj2 = $true) | ((sK583 != sK585) & ((ap @ (sub @ sK582 @ id) @ sK583) = (ap @ (sub @ sK584 @ id) @ sK585)))) & (! [X140 : term,X141 : term,X142 : term,X143 : term] : ((X141 = X143) | ((ap @ (sub @ X142 @ id) @ X143) != (ap @ (sub @ X140 @ id) @ X141))) | (hoasapinj2 != $true)) & (hoasinduction_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (! [X144 : subst] : ((comp @ id @ X144) = X144) | (axidl != $true)) & ((axidl = $true) | (sK586 != (comp @ id @ sK586))) & (hoaslamnotvar_gthm = $true) & ((hoaslaminj_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoaslaminj_lthm != $true)) & (($true = sP31) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP31)) & (! [X146 : subst,X147 : term] : ((sub @ one @ (push @ X147 @ X146)) = X147) | (axvarcons != $true)) & ((axvarcons = $true) | (sK588 != (sub @ one @ (push @ sK588 @ sK587)))) & (ulamvarsh = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lthm_3 = $true) | ((sP26 != $true) & ($true != sP28) & ! [X150 : subst > term > subst > $o] : ! [X152 : term] : ((sK589 @ X150 @ X152) = (X150 @ id @ X152 @ id)) & ! [X153 : term] : ((sub @ X153 @ id) = X153) & (sP27 != $true))) & ((sP26 = $true) | ($true = sP28) | ! [X155 : term > $o] : ((X155 @ (sK591 @ X155)) != (sK590 @ id @ (sK591 @ X155) @ id)) | (sK592 != (sub @ sK592 @ id)) | (sP27 = $true) | (hoasinduction_lthm_3 != $true)) & (apnotvar = $true) & ((hoasinduction_lem3 = $true) | (sP18 != $true)) & ((sP18 = $true) | (hoasinduction_lem3 != $true)) & (! [X158 : term] : ((sub @ X158 @ id) = X158) | (axvarid != $true)) & ((axvarid = $true) | ((sub @ sK593 @ id) != sK593)) & (hoasinduction_lem2_gthm = $true) & ((termmset_lthm = $true) | (! [X160 : term] : ((sub @ X160 @ id) = X160) & (sK594 != (sub @ sK594 @ id)))) & ((sK595 != (sub @ sK595 @ id)) | ! [X163 : term] : ((sub @ X163 @ id) = X163) | (termmset_lthm != $true)) & (! [X164 : term,X165 : subst > term > term,X166 : term] : (((ap @ (sub @ X166 @ id) @ X164) != (lam @ (X165 @ sh @ one))) | ((sub @ (X165 @ (sK598 @ X165) @ (sK596 @ X165)) @ (sK597 @ X165)) != (X165 @ (comp @ (sK598 @ X165) @ (sK597 @ X165)) @ (sub @ (sK596 @ X165) @ (sK597 @ X165))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | (((ap @ (sub @ sK601 @ id) @ sK599) = (lam @ (sK600 @ sh @ one))) & ! [X173 : term,X174 : subst,X175 : subst] : ((sub @ (sK600 @ X175 @ X173) @ X174) = (sK600 @ (comp @ X175 @ X174) @ (sub @ X173 @ X174))))) & (hoasinduction_lem1v2 = $true) & (pushprop_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop = $true) & (axmap = $true) & ((pushprop_lem1v2 = $true) | (((sK604 @ sK602) = $true) & ! [X180 : term] : ((sK603 @ X180) = (sK604 @ (sub @ X180 @ (push @ sK602 @ sK605)))) & ($true != (sK603 @ one)))) & (! [X181 : term,X182 : term > $o,X183 : term > $o,X184 : subst] : (((X183 @ X181) != $true) | ((X182 @ (sK606 @ X184 @ X183 @ X182 @ X181)) != (X183 @ (sub @ (sK606 @ X184 @ X183 @ X182 @ X181) @ (push @ X181 @ X184)))) | ($true = (X182 @ one))) | (pushprop_lem1v2 != $true)) & (! [X186 : term] : ((sK607 @ X186 @ sh @ one) = (sub @ X186 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X189 : subst > term > term] : ((X189 @ sh @ one) != (sub @ sK608 @ (push @ one @ sh)))) & (! [X190 : subst] : ((push @ (sub @ one @ X190) @ (comp @ sh @ X190)) = X190) | (axscons != $true)) & ((axscons = $true) | (sK609 != (push @ (sub @ one @ sK609) @ (comp @ sh @ sK609)))) & (hoasinduction_lem3a_gthm = $true) & ((induction2lem = $true) | ($true != sP14)) & (($true = sP14) | (induction2lem != $true)) & (pushprop_lem0_gthm = $true) & (axassoc = $true) & (hoaslamnotvar = $true) & (! [X192 : term,X193 : subst > term > term,X194 : subst,X195 : subst > term > term] : (((sub @ (X193 @ (sK610 @ X193) @ (sK611 @ X193)) @ (sK612 @ X193)) != (X193 @ (comp @ (sK610 @ X193) @ (sK612 @ X193)) @ (sub @ (sK611 @ X193) @ (sK612 @ X193)))) | ((sub @ (X195 @ (sK613 @ X195) @ (sK614 @ X195)) @ (sK615 @ X195)) != (X195 @ (comp @ (sK613 @ X195) @ (sK615 @ X195)) @ (sub @ (sK614 @ X195) @ (sK615 @ X195)))) | ((X193 @ X194 @ X192) = (X195 @ X194 @ X192)) | ((lam @ (X195 @ sh @ one)) != (lam @ (X193 @ sh @ one)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | (! [X206 : subst,X207 : term,X208 : subst] : ((sK617 @ (comp @ X206 @ X208) @ (sub @ X207 @ X208)) = (sub @ (sK617 @ X206 @ X207) @ X208)) & ! [X209 : subst,X210 : term,X211 : subst] : ((sK619 @ (comp @ X209 @ X211) @ (sub @ X210 @ X211)) = (sub @ (sK619 @ X209 @ X210) @ X211)) & ((sK619 @ sK618 @ sK616) != (sK617 @ sK618 @ sK616)) & ((lam @ (sK617 @ sh @ one)) = (lam @ (sK619 @ sh @ one))))) & ((d2subst @ d_id) = sh) & (axvarshift = $true) & ((hoasinduction_lem3aaa = $true) | (sP11 != $true)) & ((sP11 = $true) | (hoasinduction_lem3aaa != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & (! [X212 : term > $o,X213 : term,X214 : subst] : ! [X216 : term] : ((sK620 @ X214 @ X213 @ X212 @ X216) = (X212 @ (sub @ X216 @ (push @ X213 @ X214)))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X220 : term > $o] : ((sK621 @ (sub @ (sK624 @ X220) @ (push @ sK622 @ sK623))) != (X220 @ (sK624 @ X220)))) & (ulamvarind = $true) & ((induction2_lthm = $true) | ((sP9 != $true) & ! [X222 : term] : ((sub @ X222 @ id) = X222) & (sP8 != $true))) & ((sP9 = $true) | (sK625 != (sub @ sK625 @ id)) | (sP8 = $true) | (induction2_lthm != $true)) & (axabs = $true) & ((apinj1 = $true) | ((sK627 != sK629) & ((ap @ sK629 @ sK628) = (ap @ sK627 @ sK626)))) & (! [X228 : term,X229 : term,X230 : term,X231 : term] : ((X229 = X231) | ((ap @ X229 @ X228) != (ap @ X231 @ X230))) | (apinj1 != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X232 : term > $o,X233 : term > $o,X234 : subst,X235 : term] : (((X233 @ X235) != $true) | ((X233 @ (sub @ (sK630 @ X235 @ X234 @ X233 @ X232) @ (push @ X235 @ X234))) != (X232 @ (sK630 @ X235 @ X234 @ X233 @ X232))) | ((X232 @ one) = $true)) | ((sub @ one @ (push @ sK631 @ sK632)) != sK631) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | ((((sK634 @ sK636) = $true) & ! [X243 : term] : ((sK633 @ X243) = (sK634 @ (sub @ X243 @ (push @ sK636 @ sK635)))) & ($true != (sK633 @ one))) & ! [X244 : term,X245 : subst] : ((sub @ one @ (push @ X244 @ X245)) = X244))) & ((d2subst @ d_id) = id) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X246 : subst] : ((comp @ X246 @ id) = X246) | (axidr != $true)) & ((axidr = $true) | (sK637 != (comp @ sK637 @ id))) & (! [X248 : term] : ! [X250 : term,X251 : subst] : ((sK638 @ X248 @ X251 @ X250) = (sub @ X248 @ (push @ X250 @ X251))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X253 : subst > term > term] : ((X253 @ (sK641 @ X253) @ (sK640 @ X253)) != (sub @ sK639 @ (push @ (sK640 @ X253) @ (sK641 @ X253))))) & ((hoasapinj1 = $true) | ((sK643 != sK644) & ((ap @ (sub @ sK643 @ id) @ sK642) = (ap @ (sub @ sK644 @ id) @ sK645)))) & (! [X260 : term,X261 : term,X262 : term,X263 : term] : ((X261 = X262) | ((ap @ (sub @ X261 @ id) @ X260) != (ap @ (sub @ X262 @ id) @ X263))) | (hoasapinj1 != $true)) & (pushprop_lem2v2 = $true) & (($true = sP2) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP2)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (($true = sP0) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ($true != sP0))),
% 2.43/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[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,sK591,sK592,sK593,sK594,sK595,sK596,sK597,sK598,sK599,sK600,sK601,sK602,sK603,sK604,sK605,sK606,sK607,sK608,sK609,sK610,sK611,sK612,sK613,sK614,sK615,sK616,sK617,sK618,sK619,sK620,sK621,sK622,sK623,sK624,sK625,sK626,sK627,sK628,sK629,sK630,sK631,sK632,sK633,sK634,sK635,sK636,sK637,sK638,sK639,sK640,sK641,sK642,sK643,sK644,sK645])],[f673,f748,f747,f746,f745,f744,f743,f742,f741,f740,f739,f738,f737,f736,f735,f734,f733,f732,f731,f730,f729,f728,f727,f726,f725,f724,f723,f722,f721,f720,f719,f718,f717,f716,f715,f714,f713,f712,f711,f710,f709,f708,f707,f706,f705,f704,f703,f702,f701,f700,f699,f698,f697,f696,f695,f694,f693,f692,f691,f690,f689,f688,f687,f686,f685,f684,f683,f682,f681,f680,f679,f678,f677,f676,f675,f674])).
% 2.43/1.93 thf(f674,plain,(
% 2.43/1.93 ? [X0 : term > $o,X1 : term,X2 : subst,X3 : term,X4 : term > $o] : (($true != (X0 @ (sub @ X1 @ (push @ X3 @ X2)))) & ($true = (X4 @ X1))) => (((sK517 @ (sub @ sK518 @ (push @ sK520 @ sK519))) != $true) & ((sK521 @ sK518) = $true))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f675,plain,(
% 2.43/1.93 ? [X14 : subst,X15 : term] : ((sub @ one @ (push @ X15 @ X14)) != X15) => (sK523 != (sub @ one @ (push @ sK523 @ sK522)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f676,plain,(
% 2.43/1.93 ? [X16 : subst,X17 : term] : ((comp @ sh @ (push @ X17 @ X16)) != X16) => ((comp @ sh @ (push @ sK525 @ sK524)) != sK524)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f677,plain,(
% 2.43/1.93 ? [X20 : term] : ((sub @ X20 @ id) != X20) => ((sub @ sK526 @ id) != sK526)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f678,plain,(
% 2.43/1.93 ? [X21 : term] : ! [X22 : subst > term > term] : ? [X23 : term,X24 : subst] : ((sub @ X21 @ (push @ X23 @ X24)) != (X22 @ X24 @ X23)) => ! [X22 : subst > term > term] : ? [X24 : subst,X23 : term] : ((sub @ sK527 @ (push @ X23 @ X24)) != (X22 @ X24 @ X23))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f679,plain,(
% 2.43/1.93 ! [X22 : subst > term > term] : (? [X24 : subst,X23 : term] : ((sub @ sK527 @ (push @ X23 @ X24)) != (X22 @ X24 @ X23)) => ((sub @ sK527 @ (push @ (sK528 @ X22) @ (sK529 @ X22))) != (X22 @ (sK529 @ X22) @ (sK528 @ X22))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f680,plain,(
% 2.43/1.93 ! [X26 : term] : (? [X27 : subst > term > term] : ! [X28 : term,X29 : subst] : ((sub @ X26 @ (push @ X28 @ X29)) = (X27 @ X29 @ X28)) => ! [X29 : subst,X28 : term] : ((sub @ X26 @ (push @ X28 @ X29)) = (sK530 @ X26 @ X29 @ X28)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f681,plain,(
% 2.43/1.93 ? [X30 : subst > term > subst > $o] : ! [X31 : term > $o] : ? [X32 : term] : ((X31 @ X32) != (X30 @ id @ X32 @ id)) => ! [X31 : term > $o] : ? [X32 : term] : ((X31 @ X32) != (sK531 @ id @ X32 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f682,plain,(
% 2.43/1.93 ! [X31 : term > $o] : (? [X32 : term] : ((X31 @ X32) != (sK531 @ id @ X32 @ id)) => ((X31 @ (sK532 @ X31)) != (sK531 @ id @ (sK532 @ X31) @ id)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f683,plain,(
% 2.43/1.93 ! [X33 : subst > term > subst > $o] : (? [X34 : term > $o] : ! [X35 : term] : ((X33 @ id @ X35 @ id) = (X34 @ X35)) => ! [X35 : term] : ((sK533 @ X33 @ X35) = (X33 @ id @ X35 @ id)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f684,plain,(
% 2.43/1.93 ? [X36 : term] : ! [X37 : subst > term > term] : ((sub @ X36 @ (push @ one @ sh)) != (X37 @ sh @ one)) => ! [X37 : subst > term > term] : ((X37 @ sh @ one) != (sub @ sK534 @ (push @ one @ sh)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f685,plain,(
% 2.43/1.93 ! [X38 : term] : (? [X39 : subst > term > term] : ((X39 @ sh @ one) = (sub @ X38 @ (push @ one @ sh))) => ((sK535 @ X38 @ sh @ one) = (sub @ X38 @ (push @ one @ sh))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f686,plain,(
% 2.43/1.93 ? [X42 : subst,X43 : term] : ((comp @ sh @ (push @ X43 @ X42)) != X42) => ((comp @ sh @ (push @ sK537 @ sK536)) != sK536)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f687,plain,(
% 2.43/1.93 ? [X46 : term,X47 : subst > term > subst > $o] : (((sP103 @ X47) != $true) & ($true != (X47 @ id @ X46 @ id)) & ($true != (sP104 @ X47))) => (($true != (sP103 @ sK539)) & ((sK539 @ id @ sK538 @ id) != $true) & ($true != (sP104 @ sK539)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f688,plain,(
% 2.43/1.93 ? [X48 : term,X49 : term > $o,X50 : subst] : ! [X51 : term > $o] : ? [X52 : term] : ((X51 @ X52) != (X49 @ (sub @ X52 @ (push @ X48 @ X50)))) => ! [X51 : term > $o] : ? [X52 : term] : ((sK541 @ (sub @ X52 @ (push @ sK540 @ sK542))) != (X51 @ X52))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f689,plain,(
% 2.43/1.93 ! [X51 : term > $o] : (? [X52 : term] : ((sK541 @ (sub @ X52 @ (push @ sK540 @ sK542))) != (X51 @ X52)) => ((X51 @ (sK543 @ X51)) != (sK541 @ (sub @ (sK543 @ X51) @ (push @ sK540 @ sK542)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f690,plain,(
% 2.43/1.93 ! [X53 : term,X54 : term > $o,X55 : subst] : (? [X56 : term > $o] : ! [X57 : term] : ((X54 @ (sub @ X57 @ (push @ X53 @ X55))) = (X56 @ X57)) => ! [X57 : term] : ((X54 @ (sub @ X57 @ (push @ X53 @ X55))) = (sK544 @ X55 @ X54 @ X53 @ X57)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f691,plain,(
% 2.43/1.93 ? [X60 : subst] : ((comp @ id @ X60) != X60) => (sK545 != (comp @ id @ sK545))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f692,plain,(
% 2.43/1.93 ? [X61 : subst] : ((comp @ X61 @ id) != X61) => ((comp @ sK546 @ id) != sK546)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f693,plain,(
% 2.43/1.93 ? [X65 : term,X66 : term] : (((sub @ X65 @ sh) = (sub @ X66 @ sh)) & (X65 != X66)) => (((sub @ sK547 @ sh) = (sub @ sK548 @ sh)) & (sK548 != sK547))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f694,plain,(
% 2.43/1.93 ! [X67 : subst > term > subst > $o] : (? [X68 : term > $o] : ! [X69 : term] : ((X67 @ id @ X69 @ id) = (X68 @ X69)) => ! [X69 : term] : ((X67 @ id @ X69 @ id) = (sK549 @ X67 @ X69)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f695,plain,(
% 2.43/1.93 ? [X70 : subst > term > subst > $o] : ! [X71 : term > $o] : ? [X72 : term] : ((X70 @ id @ X72 @ id) != (X71 @ X72)) => ! [X71 : term > $o] : ? [X72 : term] : ((X71 @ X72) != (sK550 @ id @ X72 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f696,plain,(
% 2.43/1.93 ! [X71 : term > $o] : (? [X72 : term] : ((X71 @ X72) != (sK550 @ id @ X72 @ id)) => ((sK550 @ id @ (sK551 @ X71) @ id) != (X71 @ (sK551 @ X71))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f697,plain,(
% 2.43/1.93 ? [X73 : term,X74 : term,X75 : term,X76 : term] : (((ap @ X75 @ X73) = (ap @ X76 @ X74)) & (X73 != X74)) => (((ap @ sK554 @ sK552) = (ap @ sK555 @ sK553)) & (sK553 != sK552))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f698,plain,(
% 2.43/1.93 ? [X81 : term] : ((sub @ X81 @ id) != X81) => ((sub @ sK556 @ id) != sK556)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f699,plain,(
% 2.43/1.93 ! [X83 : subst] : (? [X84 : d_subst] : ((d2subst @ X84) = X83) => ((d2subst @ (sK557 @ X83)) = X83))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f700,plain,(
% 2.43/1.93 ? [X90 : term,X91 : term > $o,X92 : term > $o,X93 : term,X94 : subst] : (($true != (X92 @ (sub @ X90 @ (push @ X93 @ X94)))) & ((X91 @ X90) = $true)) => (((sK560 @ (sub @ sK558 @ (push @ sK561 @ sK562))) != $true) & ((sK559 @ sK558) = $true))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f701,plain,(
% 2.43/1.93 ? [X98 : term] : ((sub @ X98 @ id) != X98) => (sK563 != (sub @ sK563 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f702,plain,(
% 2.43/1.93 ? [X99 : subst > term > subst > $o] : ! [X100 : term > $o] : ? [X101 : term] : ((X100 @ X101) != (X99 @ id @ X101 @ id)) => ! [X100 : term > $o] : ? [X101 : term] : ((X100 @ X101) != (sK564 @ id @ X101 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f703,plain,(
% 2.43/1.93 ! [X100 : term > $o] : (? [X101 : term] : ((X100 @ X101) != (sK564 @ id @ X101 @ id)) => ((sK564 @ id @ (sK565 @ X100) @ id) != (X100 @ (sK565 @ X100))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f704,plain,(
% 2.43/1.93 ! [X103 : subst > term > subst > $o] : (? [X104 : term > $o] : ! [X105 : term] : ((X104 @ X105) = (X103 @ id @ X105 @ id)) => ! [X105 : term] : ((sK566 @ X103 @ X105) = (X103 @ id @ X105 @ id)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f705,plain,(
% 2.43/1.93 ? [X106 : term,X107 : term] : ((X106 != X107) & ((lam @ X106) = (lam @ X107))) => ((sK567 != sK568) & ((lam @ sK567) = (lam @ sK568)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f706,plain,(
% 2.43/1.93 ? [X111 : term] : ((sub @ X111 @ id) != X111) => (sK569 != (sub @ sK569 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f707,plain,(
% 2.43/1.93 ? [X113 : term] : ((sub @ X113 @ id) != X113) => ((sub @ sK570 @ id) != sK570)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f708,plain,(
% 2.43/1.93 ! [X114 : term] : (? [X115 : subst > term > term] : ! [X116 : term,X117 : subst] : ((sub @ X114 @ (push @ X116 @ X117)) = (X115 @ X117 @ X116)) => ! [X117 : subst,X116 : term] : ((sub @ X114 @ (push @ X116 @ X117)) = (sK571 @ X114 @ X117 @ X116)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f709,plain,(
% 2.43/1.93 ? [X118 : term] : ! [X119 : subst > term > term] : ? [X120 : term,X121 : subst] : ((X119 @ X121 @ X120) != (sub @ X118 @ (push @ X120 @ X121))) => ! [X119 : subst > term > term] : ? [X121 : subst,X120 : term] : ((sub @ sK572 @ (push @ X120 @ X121)) != (X119 @ X121 @ X120))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f710,plain,(
% 2.43/1.93 ! [X119 : subst > term > term] : (? [X121 : subst,X120 : term] : ((sub @ sK572 @ (push @ X120 @ X121)) != (X119 @ X121 @ X120)) => ((X119 @ (sK574 @ X119) @ (sK573 @ X119)) != (sub @ sK572 @ (push @ (sK573 @ X119) @ (sK574 @ X119)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f711,plain,(
% 2.43/1.93 ? [X123 : term] : ((sub @ X123 @ id) != X123) => ((sub @ sK575 @ id) != sK575)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f712,plain,(
% 2.43/1.93 ! [X124 : term] : (? [X125 : d_term] : ((d2term @ X125) = X124) => ((d2term @ (sK576 @ X124)) = X124))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f713,plain,(
% 2.43/1.93 ? [X131 : subst,X132 : term] : ((comp @ sh @ (push @ X132 @ X131)) != X131) => (sK577 != (comp @ sh @ (push @ sK578 @ sK577)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f714,plain,(
% 2.43/1.93 ? [X133 : term,X134 : subst] : ((sub @ one @ (push @ X133 @ X134)) != X133) => (sK579 != (sub @ one @ (push @ sK579 @ sK580)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f715,plain,(
% 2.43/1.93 ? [X135 : subst] : ((comp @ X135 @ id) != X135) => (sK581 != (comp @ sK581 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f716,plain,(
% 2.43/1.93 ? [X136 : term,X137 : term,X138 : term,X139 : term] : ((X137 != X139) & ((ap @ (sub @ X136 @ id) @ X137) = (ap @ (sub @ X138 @ id) @ X139))) => ((sK583 != sK585) & ((ap @ (sub @ sK582 @ id) @ sK583) = (ap @ (sub @ sK584 @ id) @ sK585)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f717,plain,(
% 2.43/1.93 ? [X145 : subst] : ((comp @ id @ X145) != X145) => (sK586 != (comp @ id @ sK586))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f718,plain,(
% 2.43/1.93 ? [X148 : subst,X149 : term] : ((sub @ one @ (push @ X149 @ X148)) != X149) => (sK588 != (sub @ one @ (push @ sK588 @ sK587)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f719,plain,(
% 2.43/1.93 ! [X150 : subst > term > subst > $o] : (? [X151 : term > $o] : ! [X152 : term] : ((X151 @ X152) = (X150 @ id @ X152 @ id)) => ! [X152 : term] : ((sK589 @ X150 @ X152) = (X150 @ id @ X152 @ id)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f720,plain,(
% 2.43/1.93 ? [X154 : subst > term > subst > $o] : ! [X155 : term > $o] : ? [X156 : term] : ((X154 @ id @ X156 @ id) != (X155 @ X156)) => ! [X155 : term > $o] : ? [X156 : term] : ((X155 @ X156) != (sK590 @ id @ X156 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f721,plain,(
% 2.43/1.93 ! [X155 : term > $o] : (? [X156 : term] : ((X155 @ X156) != (sK590 @ id @ X156 @ id)) => ((X155 @ (sK591 @ X155)) != (sK590 @ id @ (sK591 @ X155) @ id)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f722,plain,(
% 2.43/1.93 ? [X157 : term] : ((sub @ X157 @ id) != X157) => (sK592 != (sub @ sK592 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f723,plain,(
% 2.43/1.93 ? [X159 : term] : ((sub @ X159 @ id) != X159) => ((sub @ sK593 @ id) != sK593)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f724,plain,(
% 2.43/1.93 ? [X161 : term] : ((sub @ X161 @ id) != X161) => (sK594 != (sub @ sK594 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f725,plain,(
% 2.43/1.93 ? [X162 : term] : ((sub @ X162 @ id) != X162) => (sK595 != (sub @ sK595 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f726,plain,(
% 2.43/1.93 ! [X165 : subst > term > term] : (? [X167 : term,X168 : subst,X169 : subst] : ((X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168)) != (sub @ (X165 @ X169 @ X167) @ X168)) => ((sub @ (X165 @ (sK598 @ X165) @ (sK596 @ X165)) @ (sK597 @ X165)) != (X165 @ (comp @ (sK598 @ X165) @ (sK597 @ X165)) @ (sub @ (sK596 @ X165) @ (sK597 @ X165)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f727,plain,(
% 2.43/1.93 ? [X170 : term,X171 : subst > term > term,X172 : term] : (((lam @ (X171 @ sh @ one)) = (ap @ (sub @ X172 @ id) @ X170)) & ! [X173 : term,X174 : subst,X175 : subst] : ((sub @ (X171 @ X175 @ X173) @ X174) = (X171 @ (comp @ X175 @ X174) @ (sub @ X173 @ X174)))) => (((ap @ (sub @ sK601 @ id) @ sK599) = (lam @ (sK600 @ sh @ one))) & ! [X175 : subst,X174 : subst,X173 : term] : ((sub @ (sK600 @ X175 @ X173) @ X174) = (sK600 @ (comp @ X175 @ X174) @ (sub @ X173 @ X174))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f728,plain,(
% 2.43/1.93 ? [X176 : term,X177 : term > $o,X178 : term > $o,X179 : subst] : (($true = (X178 @ X176)) & ! [X180 : term] : ((X178 @ (sub @ X180 @ (push @ X176 @ X179))) = (X177 @ X180)) & ((X177 @ one) != $true)) => (((sK604 @ sK602) = $true) & ! [X180 : term] : ((sK603 @ X180) = (sK604 @ (sub @ X180 @ (push @ sK602 @ sK605)))) & ($true != (sK603 @ one)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f729,plain,(
% 2.43/1.93 ! [X181 : term,X182 : term > $o,X183 : term > $o,X184 : subst] : (? [X185 : term] : ((X182 @ X185) != (X183 @ (sub @ X185 @ (push @ X181 @ X184)))) => ((X182 @ (sK606 @ X184 @ X183 @ X182 @ X181)) != (X183 @ (sub @ (sK606 @ X184 @ X183 @ X182 @ X181) @ (push @ X181 @ X184)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f730,plain,(
% 2.43/1.93 ! [X186 : term] : (? [X187 : subst > term > term] : ((X187 @ sh @ one) = (sub @ X186 @ (push @ one @ sh))) => ((sK607 @ X186 @ sh @ one) = (sub @ X186 @ (push @ one @ sh))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f731,plain,(
% 2.43/1.93 ? [X188 : term] : ! [X189 : subst > term > term] : ((sub @ X188 @ (push @ one @ sh)) != (X189 @ sh @ one)) => ! [X189 : subst > term > term] : ((X189 @ sh @ one) != (sub @ sK608 @ (push @ one @ sh)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f732,plain,(
% 2.43/1.93 ? [X191 : subst] : ((push @ (sub @ one @ X191) @ (comp @ sh @ X191)) != X191) => (sK609 != (push @ (sub @ one @ sK609) @ (comp @ sh @ sK609)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f733,plain,(
% 2.43/1.93 ! [X193 : subst > term > term] : (? [X196 : subst,X197 : term,X198 : subst] : ((sub @ (X193 @ X196 @ X197) @ X198) != (X193 @ (comp @ X196 @ X198) @ (sub @ X197 @ X198))) => ((sub @ (X193 @ (sK610 @ X193) @ (sK611 @ X193)) @ (sK612 @ X193)) != (X193 @ (comp @ (sK610 @ X193) @ (sK612 @ X193)) @ (sub @ (sK611 @ X193) @ (sK612 @ X193)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f734,plain,(
% 2.43/1.93 ! [X195 : subst > term > term] : (? [X199 : subst,X200 : term,X201 : subst] : ((X195 @ (comp @ X199 @ X201) @ (sub @ X200 @ X201)) != (sub @ (X195 @ X199 @ X200) @ X201)) => ((sub @ (X195 @ (sK613 @ X195) @ (sK614 @ X195)) @ (sK615 @ X195)) != (X195 @ (comp @ (sK613 @ X195) @ (sK615 @ X195)) @ (sub @ (sK614 @ X195) @ (sK615 @ X195)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f735,plain,(
% 2.43/1.93 ? [X202 : term,X203 : subst > term > term,X204 : subst,X205 : subst > term > term] : (! [X206 : subst,X207 : term,X208 : subst] : ((sub @ (X203 @ X206 @ X207) @ X208) = (X203 @ (comp @ X206 @ X208) @ (sub @ X207 @ X208))) & ! [X209 : subst,X210 : term,X211 : subst] : ((sub @ (X205 @ X209 @ X210) @ X211) = (X205 @ (comp @ X209 @ X211) @ (sub @ X210 @ X211))) & ((X203 @ X204 @ X202) != (X205 @ X204 @ X202)) & ((lam @ (X205 @ sh @ one)) = (lam @ (X203 @ sh @ one)))) => (! [X208 : subst,X207 : term,X206 : subst] : ((sK617 @ (comp @ X206 @ X208) @ (sub @ X207 @ X208)) = (sub @ (sK617 @ X206 @ X207) @ X208)) & ! [X211 : subst,X210 : term,X209 : subst] : ((sK619 @ (comp @ X209 @ X211) @ (sub @ X210 @ X211)) = (sub @ (sK619 @ X209 @ X210) @ X211)) & ((sK619 @ sK618 @ sK616) != (sK617 @ sK618 @ sK616)) & ((lam @ (sK617 @ sh @ one)) = (lam @ (sK619 @ sh @ one))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f736,plain,(
% 2.43/1.93 ! [X212 : term > $o,X213 : term,X214 : subst] : (? [X215 : term > $o] : ! [X216 : term] : ((X212 @ (sub @ X216 @ (push @ X213 @ X214))) = (X215 @ X216)) => ! [X216 : term] : ((sK620 @ X214 @ X213 @ X212 @ X216) = (X212 @ (sub @ X216 @ (push @ X213 @ X214)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f737,plain,(
% 2.43/1.93 ? [X217 : term > $o,X218 : term,X219 : subst] : ! [X220 : term > $o] : ? [X221 : term] : ((X220 @ X221) != (X217 @ (sub @ X221 @ (push @ X218 @ X219)))) => ! [X220 : term > $o] : ? [X221 : term] : ((sK621 @ (sub @ X221 @ (push @ sK622 @ sK623))) != (X220 @ X221))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f738,plain,(
% 2.43/1.93 ! [X220 : term > $o] : (? [X221 : term] : ((sK621 @ (sub @ X221 @ (push @ sK622 @ sK623))) != (X220 @ X221)) => ((sK621 @ (sub @ (sK624 @ X220) @ (push @ sK622 @ sK623))) != (X220 @ (sK624 @ X220))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f739,plain,(
% 2.43/1.93 ? [X223 : term] : ((sub @ X223 @ id) != X223) => (sK625 != (sub @ sK625 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f740,plain,(
% 2.43/1.93 ? [X224 : term,X225 : term,X226 : term,X227 : term] : ((X225 != X227) & ((ap @ X225 @ X224) = (ap @ X227 @ X226))) => ((sK627 != sK629) & ((ap @ sK629 @ sK628) = (ap @ sK627 @ sK626)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f741,plain,(
% 2.43/1.93 ! [X232 : term > $o,X233 : term > $o,X234 : subst,X235 : term] : (? [X236 : term] : ((X233 @ (sub @ X236 @ (push @ X235 @ X234))) != (X232 @ X236)) => ((X233 @ (sub @ (sK630 @ X235 @ X234 @ X233 @ X232) @ (push @ X235 @ X234))) != (X232 @ (sK630 @ X235 @ X234 @ X233 @ X232))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f742,plain,(
% 2.43/1.93 ? [X237 : term,X238 : subst] : ((sub @ one @ (push @ X237 @ X238)) != X237) => ((sub @ one @ (push @ sK631 @ sK632)) != sK631)),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f743,plain,(
% 2.43/1.93 ? [X239 : term > $o,X240 : term > $o,X241 : subst,X242 : term] : (((X240 @ X242) = $true) & ! [X243 : term] : ((X240 @ (sub @ X243 @ (push @ X242 @ X241))) = (X239 @ X243)) & ($true != (X239 @ one))) => (((sK634 @ sK636) = $true) & ! [X243 : term] : ((sK633 @ X243) = (sK634 @ (sub @ X243 @ (push @ sK636 @ sK635)))) & ($true != (sK633 @ one)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f744,plain,(
% 2.43/1.93 ? [X247 : subst] : ((comp @ X247 @ id) != X247) => (sK637 != (comp @ sK637 @ id))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f745,plain,(
% 2.43/1.93 ! [X248 : term] : (? [X249 : subst > term > term] : ! [X250 : term,X251 : subst] : ((X249 @ X251 @ X250) = (sub @ X248 @ (push @ X250 @ X251))) => ! [X251 : subst,X250 : term] : ((sK638 @ X248 @ X251 @ X250) = (sub @ X248 @ (push @ X250 @ X251))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f746,plain,(
% 2.43/1.93 ? [X252 : term] : ! [X253 : subst > term > term] : ? [X254 : term,X255 : subst] : ((X253 @ X255 @ X254) != (sub @ X252 @ (push @ X254 @ X255))) => ! [X253 : subst > term > term] : ? [X255 : subst,X254 : term] : ((X253 @ X255 @ X254) != (sub @ sK639 @ (push @ X254 @ X255)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f747,plain,(
% 2.43/1.93 ! [X253 : subst > term > term] : (? [X255 : subst,X254 : term] : ((X253 @ X255 @ X254) != (sub @ sK639 @ (push @ X254 @ X255))) => ((X253 @ (sK641 @ X253) @ (sK640 @ X253)) != (sub @ sK639 @ (push @ (sK640 @ X253) @ (sK641 @ X253)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f748,plain,(
% 2.43/1.93 ? [X256 : term,X257 : term,X258 : term,X259 : term] : ((X257 != X258) & ((ap @ (sub @ X257 @ id) @ X256) = (ap @ (sub @ X258 @ id) @ X259))) => ((sK643 != sK644) & ((ap @ (sub @ sK643 @ id) @ sK642) = (ap @ (sub @ sK644 @ id) @ sK645)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f673,plain,(
% 2.43/1.93 ((pushprop_lem1_lthm = $true) | (? [X0 : term > $o,X1 : term,X2 : subst,X3 : term,X4 : term > $o] : (($true != (X0 @ (sub @ X1 @ (push @ X3 @ X2)))) & ($true = (X4 @ X1))) & ! [X5 : subst,X6 : term] : ((sub @ one @ (push @ X6 @ X5)) = X6) & ! [X7 : subst,X8 : term] : ((comp @ sh @ (push @ X8 @ X7)) = X7))) & (! [X9 : term > $o,X10 : term,X11 : subst,X12 : term,X13 : term > $o] : (((X9 @ (sub @ X10 @ (push @ X12 @ X11))) = $true) | ((X13 @ X10) != $true)) | ? [X14 : subst,X15 : term] : ((sub @ one @ (push @ X15 @ X14)) != X15) | ? [X16 : subst,X17 : term] : ((comp @ sh @ (push @ X17 @ X16)) != X16) | (pushprop_lem1_lthm != $true)) & ! [X18 : d_subst,X19 : d_subst] : (((d2subst @ X18) != (d2subst @ X19)) | (X18 = X19)) & ((hoasinduction_lem2v2 = $true) | ($true != sP109)) & (($true = sP109) | (hoasinduction_lem2v2 != $true)) & (? [X20 : term] : ((sub @ X20 @ id) != X20) | ? [X21 : term] : ! [X22 : subst > term > term] : ? [X23 : term,X24 : subst] : ((sub @ X21 @ (push @ X23 @ X24)) != (X22 @ X24 @ X23)) | ($true = sP106) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (! [X25 : term] : ((sub @ X25 @ id) = X25) & ! [X26 : term] : ? [X27 : subst > term > term] : ! [X28 : term,X29 : subst] : ((sub @ X26 @ (push @ X28 @ X29)) = (X27 @ X29 @ X28)) & ($true != sP106))) & ((hoasinduction_lem0 = $true) | ? [X30 : subst > term > subst > $o] : ! [X31 : term > $o] : ? [X32 : term] : ((X31 @ X32) != (X30 @ id @ X32 @ id))) & (! [X33 : subst > term > subst > $o] : ? [X34 : term > $o] : ! [X35 : term] : ((X33 @ id @ X35 @ id) = (X34 @ X35)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X36 : term] : ! [X37 : subst > term > term] : ((sub @ X36 @ (push @ one @ sh)) != (X37 @ sh @ one))) & (! [X38 : term] : ? [X39 : subst > term > term] : ((X39 @ sh @ one) = (sub @ X38 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & (induction2lem_gthm = $true) & (one = (d2term @ d_one)) & (hoasinduction_lem3b_gthm = $true) & (! [X40 : subst,X41 : term] : ((comp @ sh @ (push @ X41 @ X40)) = X40) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X42 : subst,X43 : term] : ((comp @ sh @ (push @ X43 @ X42)) != X42)) & (! [X44 : term,X45 : subst > term > subst > $o] : (((sP103 @ X45) = $true) | ((X45 @ id @ X44 @ id) = $true) | ((sP104 @ X45) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X46 : term,X47 : subst > term > subst > $o] : (((sP103 @ X47) != $true) & ($true != (X47 @ id @ X46 @ id)) & ($true != (sP104 @ X47)))) & (termmset_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem2 = $true) | ($true != sP102)) & (($true = sP102) | (hoasinduction_lem2 != $true)) & (hoasinduction_lem1_lthm = $true) & (($true = sP99) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP99)) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0_lthm = $true) | ? [X48 : term,X49 : term > $o,X50 : subst] : ! [X51 : term > $o] : ? [X52 : term] : ((X51 @ X52) != (X49 @ (sub @ X52 @ (push @ X48 @ X50))))) & (! [X53 : term,X54 : term > $o,X55 : subst] : ? [X56 : term > $o] : ! [X57 : term] : ((X54 @ (sub @ X57 @ (push @ X53 @ X55))) = (X56 @ X57)) | (pushprop_lem0_lthm != $true)) & (hoasapnotvar_lthm = $true) & ((! [X58 : subst] : ((comp @ id @ X58) = X58) & ! [X59 : subst] : ((comp @ X59 @ id) = X59)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X60 : subst] : ((comp @ id @ X60) != X60) | ? [X61 : subst] : ((comp @ X61 @ id) != X61)) & (hoaslamnotap_lthm = $true) & (pushprop_lthm = $true) & ! [X62 : d_term] : (d_one = X62) & (lamnotvar = $true) & ((induction = $true) | (sP97 != $true)) & ((sP97 = $true) | (induction != $true)) & (! [X63 : term,X64 : term] : (((sub @ X63 @ sh) != (sub @ X64 @ sh)) | (X63 = X64)) | (shinj != $true)) & ((shinj = $true) | ? [X65 : term,X66 : term] : (((sub @ X65 @ sh) = (sub @ X66 @ sh)) & (X65 != X66))) & (hoasinduction_lem1v2_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X67 : subst > term > subst > $o] : ? [X68 : term > $o] : ! [X69 : term] : ((X67 @ id @ X69 @ id) = (X68 @ X69)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X70 : subst > term > subst > $o] : ! [X71 : term > $o] : ? [X72 : term] : ((X70 @ id @ X72 @ id) != (X71 @ X72))) & ((sP94 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP94 != $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction = $true) | (sP92 != $true)) & ((sP92 = $true) | (hoasinduction != $true)) & ((sP87 = $true) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (sP87 != $true)) & (lamnotap != $true) & ((apinj2 = $true) | ? [X73 : term,X74 : term,X75 : term,X76 : term] : (((ap @ X75 @ X73) = (ap @ X76 @ X74)) & (X73 != X74))) & (! [X77 : term,X78 : term,X79 : term,X80 : term] : (((ap @ X80 @ X78) != (ap @ X79 @ X77)) | (X77 = X78)) | (apinj2 != $true)) & ((termmset = $true) | ? [X81 : term] : ((sub @ X81 @ id) != X81)) & (! [X82 : term] : ((sub @ X82 @ id) = X82) | (termmset != $true)) & (hoasinduction_lem3v2_gthm = $true) & (ulamvar1 != $true) & (axapp = $true) & ! [X83 : subst] : ? [X84 : d_subst] : ((d2subst @ X84) = X83) & (! [X85 : term,X86 : term > $o,X87 : term > $o,X88 : term,X89 : subst] : (((X87 @ (sub @ X85 @ (push @ X88 @ X89))) = $true) | ((X86 @ X85) != $true)) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X90 : term,X91 : term > $o,X92 : term > $o,X93 : term,X94 : subst] : (($true != (X92 @ (sub @ X90 @ (push @ X93 @ X94)))) & ((X91 @ X90) = $true))) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ! [X95 : d_subst] : (d_id = X95) & ! [X96 : d_term,X97 : d_term] : ((X96 = X97) | ((d2term @ X97) != (d2term @ X96))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (? [X98 : term] : ((sub @ X98 @ id) != X98) | (sP84 = $true) | ? [X99 : subst > term > subst > $o] : ! [X100 : term > $o] : ? [X101 : term] : ((X100 @ X101) != (X99 @ id @ X101 @ id)) | (sP83 = $true) | (sP85 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X102 : term] : ((sub @ X102 @ id) = X102) & (sP84 != $true) & ! [X103 : subst > term > subst > $o] : ? [X104 : term > $o] : ! [X105 : term] : ((X104 @ X105) = (X103 @ id @ X105 @ id)) & (sP83 != $true) & (sP85 != $true))) & (axclos = $true) & ((laminj = $true) | ? [X106 : term,X107 : term] : ((X106 != X107) & ((lam @ X106) = (lam @ X107)))) & (! [X108 : term,X109 : term] : ((X108 = X109) | ((lam @ X109) != (lam @ X108))) | (laminj != $true)) & ((sP76 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP76 != $true)) & ((hoasapinj2_lthm = $true) | (sP74 != $true)) & ((sP74 = $true) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ((sP72 != $true) & ! [X110 : term] : ((sub @ X110 @ id) = X110) & (sP71 != $true))) & ((sP72 = $true) | ? [X111 : term] : ((sub @ X111 @ id) != X111) | (sP71 = $true) | (hoasinduction_lem3_lthm != $true)) & (hoaslamnotap_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lthm_orig = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) | (! [X112 : term] : ((sub @ X112 @ id) = X112) & ($true != sP65) & ($true != sP66))) & (? [X113 : term] : ((sub @ X113 @ id) != X113) | ($true = sP65) | ($true = sP66) | (hoasinduction_lem3a_lthm != $true)) & (! [X114 : term] : ? [X115 : subst > term > term] : ! [X116 : term,X117 : subst] : ((sub @ X114 @ (push @ X116 @ X117)) = (X115 @ X117 @ X116)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X118 : term] : ! [X119 : subst > term > term] : ? [X120 : term,X121 : subst] : ((X119 @ X121 @ X120) != (sub @ X118 @ (push @ X120 @ X121)))) & ((hoasinduction_lem3v2 = $true) | (sP62 != $true)) & ((sP62 = $true) | (hoasinduction_lem3v2 != $true)) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | (! [X122 : term] : ((sub @ X122 @ id) = X122) & (sP58 != $true))) & (? [X123 : term] : ((sub @ X123 @ id) != X123) | (sP58 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lthm = $true) | ((sP54 != $true) & ($true != sP53) & (sP52 != $true) & (sP51 != $true))) & ((sP54 = $true) | ($true = sP53) | (sP52 = $true) | (sP51 = $true) | (hoasinduction_lthm != $true)) & ((var @ (d2term @ d_one)) != $true) & (hoasapinj1_gthm = $true) & ! [X124 : term] : ? [X125 : d_term] : ((d2term @ X125) = X124) & ((induction2lem_lthm = $true) | (! [X126 : subst,X127 : term] : ((comp @ sh @ (push @ X127 @ X126)) = X126) & ! [X128 : term,X129 : subst] : ((sub @ one @ (push @ X128 @ X129)) = X128) & (sP39 != $true) & (sP38 != $true) & ! [X130 : subst] : ((comp @ X130 @ id) = X130))) & (? [X131 : subst,X132 : term] : ((comp @ sh @ (push @ X132 @ X131)) != X131) | ? [X133 : term,X134 : subst] : ((sub @ one @ (push @ X133 @ X134)) != X133) | (sP39 = $true) | (sP38 = $true) | ? [X135 : subst] : ((comp @ X135 @ id) != X135) | (induction2lem_lthm != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj2 = $true) | ? [X136 : term,X137 : term,X138 : term,X139 : term] : ((X137 != X139) & ((ap @ (sub @ X136 @ id) @ X137) = (ap @ (sub @ X138 @ id) @ X139)))) & (! [X140 : term,X141 : term,X142 : term,X143 : term] : ((X141 = X143) | ((ap @ (sub @ X142 @ id) @ X143) != (ap @ (sub @ X140 @ id) @ X141))) | (hoasapinj2 != $true)) & (hoasinduction_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (! [X144 : subst] : ((comp @ id @ X144) = X144) | (axidl != $true)) & ((axidl = $true) | ? [X145 : subst] : ((comp @ id @ X145) != X145)) & (hoaslamnotvar_gthm = $true) & ((hoaslaminj_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoaslaminj_lthm != $true)) & (($true = sP31) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP31)) & (! [X146 : subst,X147 : term] : ((sub @ one @ (push @ X147 @ X146)) = X147) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X148 : subst,X149 : term] : ((sub @ one @ (push @ X149 @ X148)) != X149)) & (ulamvarsh = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lthm_3 = $true) | ((sP26 != $true) & ($true != sP28) & ! [X150 : subst > term > subst > $o] : ? [X151 : term > $o] : ! [X152 : term] : ((X151 @ X152) = (X150 @ id @ X152 @ id)) & ! [X153 : term] : ((sub @ X153 @ id) = X153) & (sP27 != $true))) & ((sP26 = $true) | ($true = sP28) | ? [X154 : subst > term > subst > $o] : ! [X155 : term > $o] : ? [X156 : term] : ((X154 @ id @ X156 @ id) != (X155 @ X156)) | ? [X157 : term] : ((sub @ X157 @ id) != X157) | (sP27 = $true) | (hoasinduction_lthm_3 != $true)) & (apnotvar = $true) & ((hoasinduction_lem3 = $true) | (sP18 != $true)) & ((sP18 = $true) | (hoasinduction_lem3 != $true)) & (! [X158 : term] : ((sub @ X158 @ id) = X158) | (axvarid != $true)) & ((axvarid = $true) | ? [X159 : term] : ((sub @ X159 @ id) != X159)) & (hoasinduction_lem2_gthm = $true) & ((termmset_lthm = $true) | (! [X160 : term] : ((sub @ X160 @ id) = X160) & ? [X161 : term] : ((sub @ X161 @ id) != X161))) & (? [X162 : term] : ((sub @ X162 @ id) != X162) | ! [X163 : term] : ((sub @ X163 @ id) = X163) | (termmset_lthm != $true)) & (! [X164 : term,X165 : subst > term > term,X166 : term] : (((ap @ (sub @ X166 @ id) @ X164) != (lam @ (X165 @ sh @ one))) | ? [X167 : term,X168 : subst,X169 : subst] : ((X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168)) != (sub @ (X165 @ X169 @ X167) @ X168))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X170 : term,X171 : subst > term > term,X172 : term] : (((lam @ (X171 @ sh @ one)) = (ap @ (sub @ X172 @ id) @ X170)) & ! [X173 : term,X174 : subst,X175 : subst] : ((sub @ (X171 @ X175 @ X173) @ X174) = (X171 @ (comp @ X175 @ X174) @ (sub @ X173 @ X174))))) & (hoasinduction_lem1v2 = $true) & (pushprop_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop = $true) & (axmap = $true) & ((pushprop_lem1v2 = $true) | ? [X176 : term,X177 : term > $o,X178 : term > $o,X179 : subst] : (($true = (X178 @ X176)) & ! [X180 : term] : ((X178 @ (sub @ X180 @ (push @ X176 @ X179))) = (X177 @ X180)) & ((X177 @ one) != $true))) & (! [X181 : term,X182 : term > $o,X183 : term > $o,X184 : subst] : (((X183 @ X181) != $true) | ? [X185 : term] : ((X182 @ X185) != (X183 @ (sub @ X185 @ (push @ X181 @ X184)))) | ($true = (X182 @ one))) | (pushprop_lem1v2 != $true)) & (! [X186 : term] : ? [X187 : subst > term > term] : ((X187 @ sh @ one) = (sub @ X186 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X188 : term] : ! [X189 : subst > term > term] : ((sub @ X188 @ (push @ one @ sh)) != (X189 @ sh @ one))) & (! [X190 : subst] : ((push @ (sub @ one @ X190) @ (comp @ sh @ X190)) = X190) | (axscons != $true)) & ((axscons = $true) | ? [X191 : subst] : ((push @ (sub @ one @ X191) @ (comp @ sh @ X191)) != X191)) & (hoasinduction_lem3a_gthm = $true) & ((induction2lem = $true) | ($true != sP14)) & (($true = sP14) | (induction2lem != $true)) & (pushprop_lem0_gthm = $true) & (axassoc = $true) & (hoaslamnotvar = $true) & (! [X192 : term,X193 : subst > term > term,X194 : subst,X195 : subst > term > term] : (? [X196 : subst,X197 : term,X198 : subst] : ((sub @ (X193 @ X196 @ X197) @ X198) != (X193 @ (comp @ X196 @ X198) @ (sub @ X197 @ X198))) | ? [X199 : subst,X200 : term,X201 : subst] : ((X195 @ (comp @ X199 @ X201) @ (sub @ X200 @ X201)) != (sub @ (X195 @ X199 @ X200) @ X201)) | ((X193 @ X194 @ X192) = (X195 @ X194 @ X192)) | ((lam @ (X195 @ sh @ one)) != (lam @ (X193 @ sh @ one)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X202 : term,X203 : subst > term > term,X204 : subst,X205 : subst > term > term] : (! [X206 : subst,X207 : term,X208 : subst] : ((sub @ (X203 @ X206 @ X207) @ X208) = (X203 @ (comp @ X206 @ X208) @ (sub @ X207 @ X208))) & ! [X209 : subst,X210 : term,X211 : subst] : ((sub @ (X205 @ X209 @ X210) @ X211) = (X205 @ (comp @ X209 @ X211) @ (sub @ X210 @ X211))) & ((X203 @ X204 @ X202) != (X205 @ X204 @ X202)) & ((lam @ (X205 @ sh @ one)) = (lam @ (X203 @ sh @ one))))) & ((d2subst @ d_id) = sh) & (axvarshift = $true) & ((hoasinduction_lem3aaa = $true) | (sP11 != $true)) & ((sP11 = $true) | (hoasinduction_lem3aaa != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & (! [X212 : term > $o,X213 : term,X214 : subst] : ? [X215 : term > $o] : ! [X216 : term] : ((X212 @ (sub @ X216 @ (push @ X213 @ X214))) = (X215 @ X216)) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X217 : term > $o,X218 : term,X219 : subst] : ! [X220 : term > $o] : ? [X221 : term] : ((X220 @ X221) != (X217 @ (sub @ X221 @ (push @ X218 @ X219))))) & (ulamvarind = $true) & ((induction2_lthm = $true) | ((sP9 != $true) & ! [X222 : term] : ((sub @ X222 @ id) = X222) & (sP8 != $true))) & ((sP9 = $true) | ? [X223 : term] : ((sub @ X223 @ id) != X223) | (sP8 = $true) | (induction2_lthm != $true)) & (axabs = $true) & ((apinj1 = $true) | ? [X224 : term,X225 : term,X226 : term,X227 : term] : ((X225 != X227) & ((ap @ X225 @ X224) = (ap @ X227 @ X226)))) & (! [X228 : term,X229 : term,X230 : term,X231 : term] : ((X229 = X231) | ((ap @ X229 @ X228) != (ap @ X231 @ X230))) | (apinj1 != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X232 : term > $o,X233 : term > $o,X234 : subst,X235 : term] : (((X233 @ X235) != $true) | ? [X236 : term] : ((X233 @ (sub @ X236 @ (push @ X235 @ X234))) != (X232 @ X236)) | ((X232 @ one) = $true)) | ? [X237 : term,X238 : subst] : ((sub @ one @ (push @ X237 @ X238)) != X237) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X239 : term > $o,X240 : term > $o,X241 : subst,X242 : term] : (((X240 @ X242) = $true) & ! [X243 : term] : ((X240 @ (sub @ X243 @ (push @ X242 @ X241))) = (X239 @ X243)) & ($true != (X239 @ one))) & ! [X244 : term,X245 : subst] : ((sub @ one @ (push @ X244 @ X245)) = X244))) & ((d2subst @ d_id) = id) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X246 : subst] : ((comp @ X246 @ id) = X246) | (axidr != $true)) & ((axidr = $true) | ? [X247 : subst] : ((comp @ X247 @ id) != X247)) & (! [X248 : term] : ? [X249 : subst > term > term] : ! [X250 : term,X251 : subst] : ((X249 @ X251 @ X250) = (sub @ X248 @ (push @ X250 @ X251))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X252 : term] : ! [X253 : subst > term > term] : ? [X254 : term,X255 : subst] : ((X253 @ X255 @ X254) != (sub @ X252 @ (push @ X254 @ X255)))) & ((hoasapinj1 = $true) | ? [X256 : term,X257 : term,X258 : term,X259 : term] : ((X257 != X258) & ((ap @ (sub @ X257 @ id) @ X256) = (ap @ (sub @ X258 @ id) @ X259)))) & (! [X260 : term,X261 : term,X262 : term,X263 : term] : ((X261 = X262) | ((ap @ (sub @ X261 @ id) @ X260) != (ap @ (sub @ X262 @ id) @ X263))) | (hoasapinj1 != $true)) & (pushprop_lem2v2 = $true) & (($true = sP2) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP2)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (($true = sP0) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ($true != sP0))),
% 2.43/1.93 inference(rectify,[status(thm)],[f672])).
% 2.43/1.93 thf(f672,plain,(
% 2.43/1.93 ((pushprop_lem1_lthm = $true) | (? [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true != (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) & ((X18 @ X15) = $true)) & ! [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) = X11) & ! [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) = X12))) & (! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true = (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) | ((X18 @ X15) != $true)) | ? [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) != X11) | ? [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) != X12) | (pushprop_lem1_lthm != $true)) & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) != (d2subst @ X55)) | (X55 = X56)) & ((hoasinduction_lem2v2 = $true) | ($true != sP109)) & (($true = sP109) | (hoasinduction_lem2v2 != $true)) & (? [X281 : term] : ((sub @ X281 @ id) != X281) | ? [X277 : term] : ! [X278 : subst > term > term] : ? [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) != (X278 @ X280 @ X279)) | ($true = sP106) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (! [X281 : term] : ((sub @ X281 @ id) = X281) & ! [X277 : term] : ? [X278 : subst > term > term] : ! [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) = (X278 @ X280 @ X279)) & ($true != sP106))) & ((hoasinduction_lem0 = $true) | ? [X7 : subst > term > subst > $o] : ! [X8 : term > $o] : ? [X9 : term] : ((X8 @ X9) != (X7 @ id @ X9 @ id))) & (! [X7 : subst > term > subst > $o] : ? [X8 : term > $o] : ! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X338 : term] : ! [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) != (X339 @ sh @ one))) & (! [X338 : term] : ? [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) = (X339 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & (induction2lem_gthm = $true) & (one = (d2term @ d_one)) & (hoasinduction_lem3b_gthm = $true) & (! [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) = X273) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) != X273)) & (! [X105 : term,X106 : subst > term > subst > $o] : (($true = (sP103 @ X106)) | ($true = (X106 @ id @ X105 @ id)) | ($true = (sP104 @ X106))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X105 : term,X106 : subst > term > subst > $o] : (($true != (sP103 @ X106)) & ($true != (X106 @ id @ X105 @ id)) & ($true != (sP104 @ X106)))) & (termmset_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem2 = $true) | ($true != sP102)) & (($true = sP102) | (hoasinduction_lem2 != $true)) & (hoasinduction_lem1_lthm = $true) & (($true = sP99) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP99)) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0_lthm = $true) | ? [X513 : term,X514 : term > $o,X515 : subst] : ! [X516 : term > $o] : ? [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) != (X516 @ X517))) & (! [X513 : term,X514 : term > $o,X515 : subst] : ? [X516 : term > $o] : ! [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) = (X516 @ X517)) | (pushprop_lem0_lthm != $true)) & (hoasapnotvar_lthm = $true) & ((! [X59 : subst] : ((comp @ id @ X59) = X59) & ! [X58 : subst] : ((comp @ X58 @ id) = X58)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X59 : subst] : ((comp @ id @ X59) != X59) | ? [X58 : subst] : ((comp @ X58 @ id) != X58)) & (hoaslamnotap_lthm = $true) & (pushprop_lthm = $true) & ! [X96 : d_term] : (d_one = X96) & (lamnotvar = $true) & ((induction = $true) | (sP97 != $true)) & ((sP97 = $true) | (induction != $true)) & (! [X275 : term,X276 : term] : (((sub @ X276 @ sh) != (sub @ X275 @ sh)) | (X275 = X276)) | (shinj != $true)) & ((shinj = $true) | ? [X275 : term,X276 : term] : (((sub @ X276 @ sh) = (sub @ X275 @ sh)) & (X275 != X276))) & (hoasinduction_lem1v2_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X93 : subst > term > subst > $o] : ? [X94 : term > $o] : ! [X95 : term] : ((X94 @ X95) = (X93 @ id @ X95 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X93 : subst > term > subst > $o] : ! [X94 : term > $o] : ? [X95 : term] : ((X94 @ X95) != (X93 @ id @ X95 @ id))) & ((sP94 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP94 != $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction = $true) | (sP92 != $true)) & ((sP92 = $true) | (hoasinduction != $true)) & ((sP87 = $true) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (sP87 != $true)) & (lamnotap != $true) & ((apinj2 = $true) | ? [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) = (ap @ X100 @ X98)) & (X97 != X98))) & (! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) != (ap @ X100 @ X98)) | (X97 = X98)) | (apinj2 != $true)) & ((termmset = $true) | ? [X57 : term] : ((sub @ X57 @ id) != X57)) & (! [X57 : term] : ((sub @ X57 @ id) = X57) | (termmset != $true)) & (hoasinduction_lem3v2_gthm = $true) & (ulamvar1 != $true) & (axapp = $true) & ! [X382 : subst] : ? [X383 : d_subst] : ((d2subst @ X383) = X382) & (! [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (($true = (X267 @ (sub @ X265 @ (push @ X268 @ X269)))) | ((X266 @ X265) != $true)) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (($true != (X267 @ (sub @ X265 @ (push @ X268 @ X269)))) & ((X266 @ X265) = $true))) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ! [X270 : d_subst] : (d_id = X270) & ! [X23 : d_term,X24 : d_term] : ((X23 = X24) | ((d2term @ X23) != (d2term @ X24))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (? [X394 : term] : ((sub @ X394 @ id) != X394) | (sP84 = $true) | ? [X384 : subst > term > subst > $o] : ! [X385 : term > $o] : ? [X386 : term] : ((X384 @ id @ X386 @ id) != (X385 @ X386)) | (sP83 = $true) | (sP85 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X394 : term] : ((sub @ X394 @ id) = X394) & (sP84 != $true) & ! [X384 : subst > term > subst > $o] : ? [X385 : term > $o] : ! [X386 : term] : ((X384 @ id @ X386 @ id) = (X385 @ X386)) & (sP83 != $true) & (sP85 != $true))) & (axclos = $true) & ((laminj = $true) | ? [X167 : term,X168 : term] : ((X167 != X168) & ((lam @ X168) = (lam @ X167)))) & (! [X167 : term,X168 : term] : ((X167 = X168) | ((lam @ X168) != (lam @ X167))) | (laminj != $true)) & ((sP76 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP76 != $true)) & ((hoasapinj2_lthm = $true) | (sP74 != $true)) & ((sP74 = $true) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ((sP72 != $true) & ! [X114 : term] : ((sub @ X114 @ id) = X114) & (sP71 != $true))) & ((sP72 = $true) | ? [X114 : term] : ((sub @ X114 @ id) != X114) | (sP71 = $true) | (hoasinduction_lem3_lthm != $true)) & (hoaslamnotap_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lthm_orig = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) | (! [X60 : term] : ((sub @ X60 @ id) = X60) & ($true != sP65) & ($true != sP66))) & (? [X60 : term] : ((sub @ X60 @ id) != X60) | ($true = sP65) | ($true = sP66) | (hoasinduction_lem3a_lthm != $true)) & (! [X101 : term] : ? [X102 : subst > term > term] : ! [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) = (X102 @ X104 @ X103)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X101 : term] : ! [X102 : subst > term > term] : ? [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) != (X102 @ X104 @ X103))) & ((hoasinduction_lem3v2 = $true) | (sP62 != $true)) & ((sP62 = $true) | (hoasinduction_lem3v2 != $true)) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | (! [X349 : term] : ((sub @ X349 @ id) = X349) & (sP58 != $true))) & (? [X349 : term] : ((sub @ X349 @ id) != X349) | (sP58 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lthm = $true) | ((sP54 != $true) & ($true != sP53) & (sP52 != $true) & (sP51 != $true))) & ((sP54 = $true) | ($true = sP53) | (sP52 = $true) | (sP51 = $true) | (hoasinduction_lthm != $true)) & ((var @ (d2term @ d_one)) != $true) & (hoasapinj1_gthm = $true) & ! [X271 : term] : ? [X272 : d_term] : ((d2term @ X272) = X271) & ((induction2lem_lthm = $true) | (! [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) = X171) & ! [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) = X169) & (sP39 != $true) & (sP38 != $true) & ! [X173 : subst] : ((comp @ X173 @ id) = X173))) & (? [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) != X171) | ? [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) != X169) | (sP39 = $true) | (sP38 = $true) | ? [X173 : subst] : ((comp @ X173 @ id) != X173) | (induction2lem_lthm != $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj2 = $true) | ? [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 != X22) & ((ap @ (sub @ X19 @ id) @ X20) = (ap @ (sub @ X21 @ id) @ X22)))) & (! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22))) | (hoasapinj2 != $true)) & (hoasinduction_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (! [X531 : subst] : ((comp @ id @ X531) = X531) | (axidl != $true)) & ((axidl = $true) | ? [X531 : subst] : ((comp @ id @ X531) != X531)) & (hoaslamnotvar_gthm = $true) & ((hoaslaminj_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoaslaminj_lthm != $true)) & (($true = sP31) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP31)) & (! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) != X337)) & (ulamvarsh = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lthm_3 = $true) | ((sP26 != $true) & ($true != sP28) & ! [X292 : subst > term > subst > $o] : ? [X293 : term > $o] : ! [X294 : term] : ((X292 @ id @ X294 @ id) = (X293 @ X294)) & ! [X302 : term] : ((sub @ X302 @ id) = X302) & (sP27 != $true))) & ((sP26 = $true) | ($true = sP28) | ? [X292 : subst > term > subst > $o] : ! [X293 : term > $o] : ? [X294 : term] : ((X292 @ id @ X294 @ id) != (X293 @ X294)) | ? [X302 : term] : ((sub @ X302 @ id) != X302) | (sP27 = $true) | (hoasinduction_lthm_3 != $true)) & (apnotvar = $true) & ((hoasinduction_lem3 = $true) | (sP18 != $true)) & ((sP18 = $true) | (hoasinduction_lem3 != $true)) & (! [X264 : term] : ((sub @ X264 @ id) = X264) | (axvarid != $true)) & ((axvarid = $true) | ? [X264 : term] : ((sub @ X264 @ id) != X264)) & (hoasinduction_lem2_gthm = $true) & ((termmset_lthm = $true) | (! [X488 : term] : ((sub @ X488 @ id) = X488) & ? [X489 : term] : ((sub @ X489 @ id) != X489))) & (? [X488 : term] : ((sub @ X488 @ id) != X488) | ! [X489 : term] : ((sub @ X489 @ id) = X489) | (termmset_lthm != $true)) & (! [X258 : term,X259 : subst > term > term,X260 : term] : (((lam @ (X259 @ sh @ one)) != (ap @ (sub @ X260 @ id) @ X258)) | ? [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) != (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X258 : term,X259 : subst > term > term,X260 : term] : (((lam @ (X259 @ sh @ one)) = (ap @ (sub @ X260 @ id) @ X258)) & ! [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) = (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262))))) & (hoasinduction_lem1v2 = $true) & (pushprop_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop = $true) & (axmap = $true) & ((pushprop_lem1v2 = $true) | ? [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (($true = (X217 @ X215)) & ! [X219 : term] : ((X216 @ X219) = (X217 @ (sub @ X219 @ (push @ X215 @ X218)))) & ((X216 @ one) != $true))) & (! [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (($true != (X217 @ X215)) | ? [X219 : term] : ((X216 @ X219) != (X217 @ (sub @ X219 @ (push @ X215 @ X218)))) | ((X216 @ one) = $true)) | (pushprop_lem1v2 != $true)) & (! [X212 : term] : ? [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) = (X213 @ sh @ one)) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X212 : term] : ! [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) != (X213 @ sh @ one))) & (! [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) = X335) | (axscons != $true)) & ((axscons = $true) | ? [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) != X335)) & (hoasinduction_lem3a_gthm = $true) & ((induction2lem = $true) | ($true != sP14)) & (($true = sP14) | (induction2lem != $true)) & (pushprop_lem0_gthm = $true) & (axassoc = $true) & (hoaslamnotvar = $true) & (! [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (? [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) != (sub @ (X496 @ X499 @ X500) @ X501)) | ? [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) != (sub @ (X498 @ X502 @ X503) @ X504)) | ((X498 @ X497 @ X495) = (X496 @ X497 @ X495)) | ((lam @ (X498 @ sh @ one)) != (lam @ (X496 @ sh @ one)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (! [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) = (sub @ (X496 @ X499 @ X500) @ X501)) & ! [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X498 @ X502 @ X503) @ X504)) & ((X498 @ X497 @ X495) != (X496 @ X497 @ X495)) & ((lam @ (X498 @ sh @ one)) = (lam @ (X496 @ sh @ one))))) & ((d2subst @ d_id) = sh) & (axvarshift = $true) & ((hoasinduction_lem3aaa = $true) | (sP11 != $true)) & ((sP11 = $true) | (hoasinduction_lem3aaa != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & (! [X490 : term > $o,X491 : term,X492 : subst] : ? [X493 : term > $o] : ! [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) = (X493 @ X494)) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X490 : term > $o,X491 : term,X492 : subst] : ! [X493 : term > $o] : ? [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) != (X493 @ X494))) & (ulamvarind = $true) & ((induction2_lthm = $true) | ((sP9 != $true) & ! [X151 : term] : ((sub @ X151 @ id) = X151) & (sP8 != $true))) & ((sP9 = $true) | ? [X151 : term] : ((sub @ X151 @ id) != X151) | (sP8 = $true) | (induction2_lthm != $true)) & (axabs = $true) & ((apinj1 = $true) | ? [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 != X487) & ((ap @ X485 @ X484) = (ap @ X487 @ X486)))) & (! [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 = X487) | ((ap @ X485 @ X484) != (ap @ X487 @ X486))) | (apinj1 != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (((X3 @ X5) != $true) | ? [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) != (X2 @ X6)) | ((X2 @ one) = $true)) | ? [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) != X0) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (((X3 @ X5) = $true) & ! [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) = (X2 @ X6)) & ((X2 @ one) != $true)) & ! [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) = X0))) & ((d2subst @ d_id) = id) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X214 : subst] : ((comp @ X214 @ id) = X214) | (axidr != $true)) & ((axidr = $true) | ? [X214 : subst] : ((comp @ X214 @ id) != X214)) & (! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) = (sub @ X147 @ (push @ X149 @ X150))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X147 : term] : ! [X148 : subst > term > term] : ? [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) != (sub @ X147 @ (push @ X149 @ X150)))) & ((hoasapinj1 = $true) | ? [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 != X380) & ((ap @ (sub @ X379 @ id) @ X378) = (ap @ (sub @ X380 @ id) @ X381)))) & (! [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 = X380) | ((ap @ (sub @ X379 @ id) @ X378) != (ap @ (sub @ X380 @ id) @ X381))) | (hoasapinj1 != $true)) & (pushprop_lem2v2 = $true) & (($true = sP2) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP2)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (($true = sP0) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ($true != sP0))),
% 2.43/1.93 inference(flattening,[status(thm)],[f671])).
% 2.43/1.93 thf(f671,plain,(
% 2.43/1.93 (((pushprop_lem1_lthm = $true) | (? [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true != (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) & ((X18 @ X15) = $true)) & ! [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) = X11) & ! [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) = X12))) & ((! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true = (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) | ((X18 @ X15) != $true)) | ? [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) != X11) | ? [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) != X12)) | (pushprop_lem1_lthm != $true))) & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) != (d2subst @ X55)) | (X55 = X56)) & (((hoasinduction_lem2v2 = $true) | ($true != sP109)) & (($true = sP109) | (hoasinduction_lem2v2 != $true))) & (((? [X281 : term] : ((sub @ X281 @ id) != X281) | ? [X277 : term] : ! [X278 : subst > term > term] : ? [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) != (X278 @ X280 @ X279)) | ($true = sP106)) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (! [X281 : term] : ((sub @ X281 @ id) = X281) & ! [X277 : term] : ? [X278 : subst > term > term] : ! [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) = (X278 @ X280 @ X279)) & ($true != sP106)))) & (((hoasinduction_lem0 = $true) | ? [X7 : subst > term > subst > $o] : ! [X8 : term > $o] : ? [X9 : term] : ((X8 @ X9) != (X7 @ id @ X9 @ id))) & (! [X7 : subst > term > subst > $o] : ? [X8 : term > $o] : ! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id)) | (hoasinduction_lem0 != $true))) & (((hoasinduction_lem3b_lthm = $true) | ? [X338 : term] : ! [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) != (X339 @ sh @ one))) & (! [X338 : term] : ? [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) = (X339 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true))) & (induction2lem_gthm = $true) & (one = (d2term @ d_one)) & (hoasinduction_lem3b_gthm = $true) & ((! [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) = X273) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) != X273))) & ((! [X105 : term,X106 : subst > term > subst > $o] : (($true = (sP103 @ X106)) | ($true = (X106 @ id @ X105 @ id)) | ($true = (sP104 @ X106))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X105 : term,X106 : subst > term > subst > $o] : (($true != (sP103 @ X106)) & ($true != (X106 @ id @ X105 @ id)) & ($true != (sP104 @ X106))))) & (termmset_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (((hoasinduction_lem2 = $true) | ($true != sP102)) & (($true = sP102) | (hoasinduction_lem2 != $true))) & (hoasinduction_lem1_lthm = $true) & ((($true = sP99) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP99))) & (hoasinduction_lem1_gthm = $true) & (((pushprop_lem0_lthm = $true) | ? [X513 : term,X514 : term > $o,X515 : subst] : ! [X516 : term > $o] : ? [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) != (X516 @ X517))) & (! [X513 : term,X514 : term > $o,X515 : subst] : ? [X516 : term > $o] : ! [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) = (X516 @ X517)) | (pushprop_lem0_lthm != $true))) & (hoasapnotvar_lthm = $true) & (((! [X59 : subst] : ((comp @ id @ X59) = X59) & ! [X58 : subst] : ((comp @ X58 @ id) = X58)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X59 : subst] : ((comp @ id @ X59) != X59) | ? [X58 : subst] : ((comp @ X58 @ id) != X58)))) & (hoaslamnotap_lthm = $true) & (pushprop_lthm = $true) & ! [X96 : d_term] : (d_one = X96) & (lamnotvar = $true) & (((induction = $true) | (sP97 != $true)) & ((sP97 = $true) | (induction != $true))) & ((! [X275 : term,X276 : term] : (((sub @ X276 @ sh) != (sub @ X275 @ sh)) | (X275 = X276)) | (shinj != $true)) & ((shinj = $true) | ? [X275 : term,X276 : term] : (((sub @ X276 @ sh) = (sub @ X275 @ sh)) & (X275 != X276)))) & (hoasinduction_lem1v2_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((! [X93 : subst > term > subst > $o] : ? [X94 : term > $o] : ! [X95 : term] : ((X94 @ X95) = (X93 @ id @ X95 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X93 : subst > term > subst > $o] : ! [X94 : term > $o] : ? [X95 : term] : ((X94 @ X95) != (X93 @ id @ X95 @ id)))) & (((sP94 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP94 != $true))) & (hoaslamnotvar_lthm = $true) & (((hoasinduction = $true) | (sP92 != $true)) & ((sP92 = $true) | (hoasinduction != $true))) & (((sP87 = $true) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (sP87 != $true))) & (lamnotap != $true) & (((apinj2 = $true) | ? [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) = (ap @ X100 @ X98)) & (X97 != X98))) & (! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) != (ap @ X100 @ X98)) | (X97 = X98)) | (apinj2 != $true))) & (((termmset = $true) | ? [X57 : term] : ((sub @ X57 @ id) != X57)) & (! [X57 : term] : ((sub @ X57 @ id) = X57) | (termmset != $true))) & (hoasinduction_lem3v2_gthm = $true) & (ulamvar1 != $true) & (axapp = $true) & ! [X382 : subst] : ? [X383 : d_subst] : ((d2subst @ X383) = X382) & ((! [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (($true = (X267 @ (sub @ X265 @ (push @ X268 @ X269)))) | ((X266 @ X265) != $true)) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (($true != (X267 @ (sub @ X265 @ (push @ X268 @ X269)))) & ((X266 @ X265) = $true)))) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ! [X270 : d_subst] : (d_id = X270) & ! [X23 : d_term,X24 : d_term] : ((X23 = X24) | ((d2term @ X23) != (d2term @ X24))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (((? [X394 : term] : ((sub @ X394 @ id) != X394) | (sP84 = $true) | ? [X384 : subst > term > subst > $o] : ! [X385 : term > $o] : ? [X386 : term] : ((X384 @ id @ X386 @ id) != (X385 @ X386)) | (sP83 = $true) | (sP85 = $true)) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X394 : term] : ((sub @ X394 @ id) = X394) & (sP84 != $true) & ! [X384 : subst > term > subst > $o] : ? [X385 : term > $o] : ! [X386 : term] : ((X384 @ id @ X386 @ id) = (X385 @ X386)) & (sP83 != $true) & (sP85 != $true)))) & (axclos = $true) & (((laminj = $true) | ? [X167 : term,X168 : term] : ((X167 != X168) & ((lam @ X168) = (lam @ X167)))) & (! [X167 : term,X168 : term] : ((X167 = X168) | ((lam @ X168) != (lam @ X167))) | (laminj != $true))) & (((sP76 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP76 != $true))) & (((hoasapinj2_lthm = $true) | (sP74 != $true)) & ((sP74 = $true) | (hoasapinj2_lthm != $true))) & (pushprop_lem2v2_gthm = $true) & (((hoasinduction_lem3_lthm = $true) | ((sP72 != $true) & ! [X114 : term] : ((sub @ X114 @ id) = X114) & (sP71 != $true))) & (((sP72 = $true) | ? [X114 : term] : ((sub @ X114 @ id) != X114) | (sP71 = $true)) | (hoasinduction_lem3_lthm != $true))) & (hoaslamnotap_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lthm_orig = $true) & (hoasapnotvar = $true) & (((hoasinduction_lem3a_lthm = $true) | (! [X60 : term] : ((sub @ X60 @ id) = X60) & ($true != sP65) & ($true != sP66))) & ((? [X60 : term] : ((sub @ X60 @ id) != X60) | ($true = sP65) | ($true = sP66)) | (hoasinduction_lem3a_lthm != $true))) & ((! [X101 : term] : ? [X102 : subst > term > term] : ! [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) = (X102 @ X104 @ X103)) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X101 : term] : ! [X102 : subst > term > term] : ? [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) != (X102 @ X104 @ X103)))) & (((hoasinduction_lem3v2 = $true) | (sP62 != $true)) & ((sP62 = $true) | (hoasinduction_lem3v2 != $true))) & (hoaslaminj_gthm = $true) & (((hoasinduction_lem3v2_lthm = $true) | (! [X349 : term] : ((sub @ X349 @ id) = X349) & (sP58 != $true))) & ((? [X349 : term] : ((sub @ X349 @ id) != X349) | (sP58 = $true)) | (hoasinduction_lem3v2_lthm != $true))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (((hoasinduction_lthm = $true) | ((sP54 != $true) & ($true != sP53) & (sP52 != $true) & (sP51 != $true))) & (((sP54 = $true) | ($true = sP53) | (sP52 = $true) | (sP51 = $true)) | (hoasinduction_lthm != $true))) & ((var @ (d2term @ d_one)) != $true) & (hoasapinj1_gthm = $true) & ! [X271 : term] : ? [X272 : d_term] : ((d2term @ X272) = X271) & (((induction2lem_lthm = $true) | (! [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) = X171) & ! [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) = X169) & (sP39 != $true) & (sP38 != $true) & ! [X173 : subst] : ((comp @ X173 @ id) = X173))) & ((? [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) != X171) | ? [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) != X169) | (sP39 = $true) | (sP38 = $true) | ? [X173 : subst] : ((comp @ X173 @ id) != X173)) | (induction2lem_lthm != $true))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem1_gthm = $true) & (((hoasapinj2 = $true) | ? [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 != X22) & ((ap @ (sub @ X19 @ id) @ X20) = (ap @ (sub @ X21 @ id) @ X22)))) & (! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22))) | (hoasapinj2 != $true))) & (hoasinduction_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & ((! [X531 : subst] : ((comp @ id @ X531) = X531) | (axidl != $true)) & ((axidl = $true) | ? [X531 : subst] : ((comp @ id @ X531) != X531))) & (hoaslamnotvar_gthm = $true) & (((hoaslaminj_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoaslaminj_lthm != $true))) & ((($true = sP31) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP31))) & ((! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) != X337))) & (ulamvarsh = $true) & (hoasinduction_lem1 = $true) & (((hoasinduction_lthm_3 = $true) | ((sP26 != $true) & ($true != sP28) & ! [X292 : subst > term > subst > $o] : ? [X293 : term > $o] : ! [X294 : term] : ((X292 @ id @ X294 @ id) = (X293 @ X294)) & ! [X302 : term] : ((sub @ X302 @ id) = X302) & (sP27 != $true))) & (((sP26 = $true) | ($true = sP28) | ? [X292 : subst > term > subst > $o] : ! [X293 : term > $o] : ? [X294 : term] : ((X292 @ id @ X294 @ id) != (X293 @ X294)) | ? [X302 : term] : ((sub @ X302 @ id) != X302) | (sP27 = $true)) | (hoasinduction_lthm_3 != $true))) & (apnotvar = $true) & (((hoasinduction_lem3 = $true) | (sP18 != $true)) & ((sP18 = $true) | (hoasinduction_lem3 != $true))) & ((! [X264 : term] : ((sub @ X264 @ id) = X264) | (axvarid != $true)) & ((axvarid = $true) | ? [X264 : term] : ((sub @ X264 @ id) != X264))) & (hoasinduction_lem2_gthm = $true) & (((termmset_lthm = $true) | (! [X488 : term] : ((sub @ X488 @ id) = X488) & ? [X489 : term] : ((sub @ X489 @ id) != X489))) & ((? [X488 : term] : ((sub @ X488 @ id) != X488) | ! [X489 : term] : ((sub @ X489 @ id) = X489)) | (termmset_lthm != $true))) & ((! [X258 : term,X259 : subst > term > term,X260 : term] : (((lam @ (X259 @ sh @ one)) != (ap @ (sub @ X260 @ id) @ X258)) | ? [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) != (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X258 : term,X259 : subst > term > term,X260 : term] : (((lam @ (X259 @ sh @ one)) = (ap @ (sub @ X260 @ id) @ X258)) & ! [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) = (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262)))))) & (hoasinduction_lem1v2 = $true) & (pushprop_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop = $true) & (axmap = $true) & (((pushprop_lem1v2 = $true) | ? [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (($true = (X217 @ X215)) & ! [X219 : term] : ((X216 @ X219) = (X217 @ (sub @ X219 @ (push @ X215 @ X218)))) & ((X216 @ one) != $true))) & (! [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (($true != (X217 @ X215)) | ? [X219 : term] : ((X216 @ X219) != (X217 @ (sub @ X219 @ (push @ X215 @ X218)))) | ((X216 @ one) = $true)) | (pushprop_lem1v2 != $true))) & ((! [X212 : term] : ? [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) = (X213 @ sh @ one)) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X212 : term] : ! [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) != (X213 @ sh @ one)))) & ((! [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) = X335) | (axscons != $true)) & ((axscons = $true) | ? [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) != X335))) & (hoasinduction_lem3a_gthm = $true) & (((induction2lem = $true) | ($true != sP14)) & (($true = sP14) | (induction2lem != $true))) & (pushprop_lem0_gthm = $true) & (axassoc = $true) & (hoaslamnotvar = $true) & ((! [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (? [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) != (sub @ (X496 @ X499 @ X500) @ X501)) | ? [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) != (sub @ (X498 @ X502 @ X503) @ X504)) | ((X498 @ X497 @ X495) = (X496 @ X497 @ X495)) | ((lam @ (X498 @ sh @ one)) != (lam @ (X496 @ sh @ one)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (! [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) = (sub @ (X496 @ X499 @ X500) @ X501)) & ! [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X498 @ X502 @ X503) @ X504)) & ((X498 @ X497 @ X495) != (X496 @ X497 @ X495)) & ((lam @ (X498 @ sh @ one)) = (lam @ (X496 @ sh @ one)))))) & ((d2subst @ d_id) = sh) & (axvarshift = $true) & (((hoasinduction_lem3aaa = $true) | (sP11 != $true)) & ((sP11 = $true) | (hoasinduction_lem3aaa != $true))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & ((! [X490 : term > $o,X491 : term,X492 : subst] : ? [X493 : term > $o] : ! [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) = (X493 @ X494)) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X490 : term > $o,X491 : term,X492 : subst] : ! [X493 : term > $o] : ? [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) != (X493 @ X494)))) & (ulamvarind = $true) & (((induction2_lthm = $true) | ((sP9 != $true) & ! [X151 : term] : ((sub @ X151 @ id) = X151) & (sP8 != $true))) & (((sP9 = $true) | ? [X151 : term] : ((sub @ X151 @ id) != X151) | (sP8 = $true)) | (induction2_lthm != $true))) & (axabs = $true) & (((apinj1 = $true) | ? [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 != X487) & ((ap @ X485 @ X484) = (ap @ X487 @ X486)))) & (! [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 = X487) | ((ap @ X485 @ X484) != (ap @ X487 @ X486))) | (apinj1 != $true))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (((! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (((X3 @ X5) != $true) | ? [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) != (X2 @ X6)) | ((X2 @ one) = $true)) | ? [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) != X0)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (((X3 @ X5) = $true) & ! [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) = (X2 @ X6)) & ((X2 @ one) != $true)) & ! [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) = X0)))) & ((d2subst @ d_id) = id) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((! [X214 : subst] : ((comp @ X214 @ id) = X214) | (axidr != $true)) & ((axidr = $true) | ? [X214 : subst] : ((comp @ X214 @ id) != X214))) & ((! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) = (sub @ X147 @ (push @ X149 @ X150))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X147 : term] : ! [X148 : subst > term > term] : ? [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) != (sub @ X147 @ (push @ X149 @ X150))))) & (((hoasapinj1 = $true) | ? [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 != X380) & ((ap @ (sub @ X379 @ id) @ X378) = (ap @ (sub @ X380 @ id) @ X381)))) & (! [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 = X380) | ((ap @ (sub @ X379 @ id) @ X378) != (ap @ (sub @ X380 @ id) @ X381))) | (hoasapinj1 != $true))) & (pushprop_lem2v2 = $true) & ((($true = sP2) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP2))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((($true = sP0) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ($true != sP0)))),
% 2.43/1.93 inference(nnf_transformation,[status(thm)],[f125])).
% 2.43/1.93 thf(f125,plain,(
% 2.43/1.93 ((pushprop_lem1_lthm = $true) <=> (! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true = (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) | ((X18 @ X15) != $true)) | ? [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) != X11) | ? [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) != X12))) & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) != (d2subst @ X55)) | (X55 = X56)) & ((hoasinduction_lem2v2 = $true) <=> ($true = sP109)) & ((? [X281 : term] : ((sub @ X281 @ id) != X281) | ? [X277 : term] : ! [X278 : subst > term > term] : ? [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) != (X278 @ X280 @ X279)) | ($true = sP106)) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X7 : subst > term > subst > $o] : ? [X8 : term > $o] : ! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X338 : term] : ? [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) = (X339 @ sh @ one))) & (induction2lem_gthm = $true) & (one = (d2term @ d_one)) & (hoasinduction_lem3b_gthm = $true) & (! [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) = X273) <=> (axshiftcons = $true)) & (! [X105 : term,X106 : subst > term > subst > $o] : (($true = (sP103 @ X106)) | ($true = (X106 @ id @ X105 @ id)) | ($true = (sP104 @ X106))) <=> (hoasinduction_no_psi_cond = $true)) & (termmset_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem2 = $true) <=> ($true = sP102)) & (hoasinduction_lem1_lthm = $true) & (($true = sP99) <=> (hoasinduction_lem3aa_lthm = $true)) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X513 : term,X514 : term > $o,X515 : subst] : ? [X516 : term > $o] : ! [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) = (X516 @ X517))) & (hoasapnotvar_lthm = $true) & ((! [X59 : subst] : ((comp @ id @ X59) = X59) & ! [X58 : subst] : ((comp @ X58 @ id) = X58)) <=> (substmonoid = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lthm = $true) & ! [X96 : d_term] : (d_one = X96) & (lamnotvar = $true) & ((induction = $true) <=> (sP97 = $true)) & (! [X275 : term,X276 : term] : (((sub @ X276 @ sh) != (sub @ X275 @ sh)) | (X275 = X276)) <=> (shinj = $true)) & (hoasinduction_lem1v2_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X93 : subst > term > subst > $o] : ? [X94 : term > $o] : ! [X95 : term] : ((X94 @ X95) = (X93 @ id @ X95 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & ((sP94 = $true) <=> (hoasinduction_lem3aa = $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction = $true) <=> (sP92 = $true)) & ((sP87 = $true) <=> (hoasapinj1_lthm = $true)) & (lamnotap != $true) & ((apinj2 = $true) <=> ! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) != (ap @ X100 @ X98)) | (X97 = X98))) & ((termmset = $true) <=> ! [X57 : term] : ((sub @ X57 @ id) = X57)) & (hoasinduction_lem3v2_gthm = $true) & (ulamvar1 != $true) & (axapp = $true) & ! [X382 : subst] : ? [X383 : d_subst] : ((d2subst @ X383) = X382) & (! [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (($true = (X267 @ (sub @ X265 @ (push @ X268 @ X269)))) | ((X266 @ X265) != $true)) <=> (pushprop_lem1 = $true)) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ! [X270 : d_subst] : (d_id = X270) & ! [X23 : d_term,X24 : d_term] : ((X23 = X24) | ((d2term @ X23) != (d2term @ X24))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((? [X394 : term] : ((sub @ X394 @ id) != X394) | (sP84 = $true) | ? [X384 : subst > term > subst > $o] : ! [X385 : term > $o] : ? [X386 : term] : ((X384 @ id @ X386 @ id) != (X385 @ X386)) | (sP83 = $true) | (sP85 = $true)) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (axclos = $true) & ((laminj = $true) <=> ! [X167 : term,X168 : term] : ((X167 = X168) | ((lam @ X168) != (lam @ X167)))) & ((sP76 = $true) <=> (hoasinduction_lem3v2a = $true)) & ((hoasapinj2_lthm = $true) <=> (sP74 = $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> ((sP72 = $true) | ? [X114 : term] : ((sub @ X114 @ id) != X114) | (sP71 = $true))) & (hoaslamnotap_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lthm_orig = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (? [X60 : term] : ((sub @ X60 @ id) != X60) | ($true = sP65) | ($true = sP66))) & (! [X101 : term] : ? [X102 : subst > term > term] : ! [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) = (X102 @ X104 @ X103)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem3v2 = $true) <=> (sP62 = $true)) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X349 : term] : ((sub @ X349 @ id) != X349) | (sP58 = $true))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lthm = $true) <=> ((sP54 = $true) | ($true = sP53) | (sP52 = $true) | (sP51 = $true))) & ((var @ (d2term @ d_one)) != $true) & (hoasapinj1_gthm = $true) & ! [X271 : term] : ? [X272 : d_term] : ((d2term @ X272) = X271) & ((induction2lem_lthm = $true) <=> (? [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) != X171) | ? [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) != X169) | (sP39 = $true) | (sP38 = $true) | ? [X173 : subst] : ((comp @ X173 @ id) != X173))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22)))) & (hoasinduction_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (! [X531 : subst] : ((comp @ id @ X531) = X531) <=> (axidl = $true)) & (hoaslamnotvar_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (sP33 = $true)) & (($true = sP31) <=> (induction2 = $true)) & (! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) <=> (axvarcons = $true)) & (ulamvarsh = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lthm_3 = $true) <=> ((sP26 = $true) | ($true = sP28) | ? [X292 : subst > term > subst > $o] : ! [X293 : term > $o] : ? [X294 : term] : ((X292 @ id @ X294 @ id) != (X293 @ X294)) | ? [X302 : term] : ((sub @ X302 @ id) != X302) | (sP27 = $true))) & (apnotvar = $true) & ((hoasinduction_lem3 = $true) <=> (sP18 = $true)) & (! [X264 : term] : ((sub @ X264 @ id) = X264) <=> (axvarid = $true)) & (hoasinduction_lem2_gthm = $true) & ((termmset_lthm = $true) <=> (? [X488 : term] : ((sub @ X488 @ id) != X488) | ! [X489 : term] : ((sub @ X489 @ id) = X489))) & (! [X258 : term,X259 : subst > term > term,X260 : term] : (((lam @ (X259 @ sh @ one)) != (ap @ (sub @ X260 @ id) @ X258)) | ? [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) != (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262)))) <=> (hoaslamnotap = $true)) & (hoasinduction_lem1v2 = $true) & (pushprop_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop = $true) & (axmap = $true) & ((pushprop_lem1v2 = $true) <=> ! [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (($true != (X217 @ X215)) | ? [X219 : term] : ((X216 @ X219) != (X217 @ (sub @ X219 @ (push @ X215 @ X218)))) | ((X216 @ one) = $true))) & (! [X212 : term] : ? [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) = (X213 @ sh @ one)) <=> (hoasinduction_lem3b = $true)) & (! [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) = X335) <=> (axscons = $true)) & (hoasinduction_lem3a_gthm = $true) & ((induction2lem = $true) <=> ($true = sP14)) & (pushprop_lem0_gthm = $true) & (axassoc = $true) & (hoaslamnotvar = $true) & (! [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (? [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) != (sub @ (X496 @ X499 @ X500) @ X501)) | ? [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) != (sub @ (X498 @ X502 @ X503) @ X504)) | ((X498 @ X497 @ X495) = (X496 @ X497 @ X495)) | ((lam @ (X498 @ sh @ one)) != (lam @ (X496 @ sh @ one)))) <=> (hoaslaminj = $true)) & ((d2subst @ d_id) = sh) & (axvarshift = $true) & ((hoasinduction_lem3aaa = $true) <=> (sP11 = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & (! [X490 : term > $o,X491 : term,X492 : subst] : ? [X493 : term > $o] : ! [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) = (X493 @ X494)) <=> (pushprop_lem0 = $true)) & (ulamvarind = $true) & ((induction2_lthm = $true) <=> ((sP9 = $true) | ? [X151 : term] : ((sub @ X151 @ id) != X151) | (sP8 = $true))) & (axabs = $true) & ((apinj1 = $true) <=> ! [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 = X487) | ((ap @ X485 @ X484) != (ap @ X487 @ X486)))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (((X3 @ X5) != $true) | ? [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) != (X2 @ X6)) | ((X2 @ one) = $true)) | ? [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) != X0)) <=> (pushprop_lem1v2_lthm = $true)) & ((d2subst @ d_id) = id) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X214 : subst] : ((comp @ X214 @ id) = X214) <=> (axidr = $true)) & (! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) = (sub @ X147 @ (push @ X149 @ X150))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasapinj1 = $true) <=> ! [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 = X380) | ((ap @ (sub @ X379 @ id) @ X378) != (ap @ (sub @ X380 @ id) @ X381)))) & (pushprop_lem2v2 = $true) & (($true = sP2) <=> (hoasinduction_lem3a = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (($true = sP0) <=> (substmonoid_lthm = $true))),
% 2.43/1.93 inference(definition_folding,[status(thm)],[f14,f124,f123,f122,f121,f120,f119,f118,f117,f116,f115,f114,f113,f112,f111,f110,f109,f108,f107,f106,f105,f104,f103,f102,f101,f100,f99,f98,f97,f96,f95,f94,f93,f92,f91,f90,f89,f88,f87,f86,f85,f84,f83,f82,f81,f80,f79,f78,f77,f76,f75,f74,f73,f72,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15])).
% 2.43/1.93 thf(f15,plain,(
% 2.43/1.93 ($true = sP0) <=> (? [X332 : subst] : ((comp @ X332 @ id) != X332) | (! [X333 : subst] : ((comp @ X333 @ id) = X333) & ! [X334 : subst] : ((comp @ id @ X334) = X334)) | ? [X331 : subst] : ((comp @ id @ X331) != X331))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f16,plain,(
% 2.43/1.93 ! [X533 : subst > term > subst > $o] : (($true = (sP1 @ X533)) <=> ? [X534 : subst > term > term] : (((X533 @ id @ (lam @ (X534 @ sh @ one)) @ id) != $true) & ! [X535 : subst,X536 : subst,X537 : term] : ((sub @ (X534 @ X535 @ X537) @ X536) = (X534 @ (comp @ X535 @ X536) @ (sub @ X537 @ X536))) & ! [X538 : term] : (((X533 @ id @ X538 @ id) != $true) | ((X533 @ id @ (X534 @ id @ X538) @ id) = $true))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f17,plain,(
% 2.43/1.93 ($true = sP2) <=> ! [X532 : term,X533 : subst > term > subst > $o] : (($true = (X533 @ id @ (lam @ X532) @ id)) | ($true = (sP1 @ X533)) | ? [X539 : term] : (((X533 @ id @ X539 @ id) = $true) & ((X533 @ id @ (sub @ X532 @ (push @ X539 @ id)) @ id) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f18,plain,(
% 2.43/1.93 ! [X161 : term > $o] : (((sP3 @ X161) = $true) <=> ? [X165 : term] : (($true != (X161 @ (lam @ X165))) & ! [X166 : term] : (((X161 @ X166) != $true) | ($true = (X161 @ (sub @ X165 @ (push @ X166 @ id)))))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f19,plain,(
% 2.43/1.93 ! [X161 : term > $o] : (($true = (sP4 @ X161)) <=> ? [X163 : term,X164 : term] : (((X161 @ (ap @ X164 @ X163)) != $true) & ($true = (X161 @ X163)) & ((X161 @ X164) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f20,plain,(
% 2.43/1.93 ! [X154 : term > $o] : (((sP5 @ X154) = $true) <=> ! [X157 : term,X158 : term] : (($true != (X154 @ X158)) | ((X154 @ X157) != $true) | ((X154 @ (ap @ X158 @ X157)) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f21,plain,(
% 2.43/1.93 ! [X153 : subst,X154 : term > $o] : (((sP6 @ X154 @ X153) = $true) <=> ! [X159 : term] : (((var @ X159) != $true) | ((X154 @ (sub @ X159 @ X153)) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f22,plain,(
% 2.43/1.93 ! [X154 : term > $o] : (($true = (sP7 @ X154)) <=> ! [X155 : term] : (($true = (X154 @ (lam @ X155))) | ? [X156 : term] : (((X154 @ X156) = $true) & ((X154 @ (sub @ X155 @ (push @ X156 @ id))) != $true))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f23,plain,(
% 2.43/1.93 (sP8 = $true) <=> ! [X160 : term,X161 : term > $o] : (((sP3 @ X161) = $true) | ? [X162 : term] : (($true != (X161 @ X162)) & ($true = (var @ X162))) | ($true = (sP4 @ X161)) | ((X161 @ X160) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f24,plain,(
% 2.43/1.93 (sP9 = $true) <=> ? [X152 : term,X153 : subst,X154 : term > $o] : (($true = (sP7 @ X154)) & ((sP6 @ X154 @ X153) = $true) & ((sP5 @ X154) = $true) & ((X154 @ (sub @ X152 @ X153)) != $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f25,plain,(
% 2.43/1.93 ! [X25 : subst > term > subst > $o] : (((sP10 @ X25) = $true) <=> ? [X28 : term,X29 : subst > term > term] : (((X25 @ id @ (lam @ (X29 @ sh @ one)) @ id) != $true) & ! [X34 : subst,X35 : subst,X36 : term] : ((sub @ X28 @ (push @ (sub @ X36 @ X34) @ (comp @ X35 @ X34))) = (X29 @ (comp @ X35 @ X34) @ (sub @ X36 @ X34))) & ! [X30 : term] : (((X25 @ id @ (X29 @ id @ X30) @ id) = $true) | ((X25 @ id @ X30 @ id) != $true)) & ! [X31 : subst,X32 : term,X33 : subst] : ((sub @ (sub @ X28 @ (push @ X32 @ X33)) @ X31) = (sub @ (X29 @ X33 @ X32) @ X31))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f26,plain,(
% 2.43/1.93 (sP11 = $true) <=> ! [X25 : subst > term > subst > $o,X26 : term] : (? [X27 : term] : (($true = (X25 @ id @ X27 @ id)) & ($true != (X25 @ id @ (sub @ X26 @ (push @ X27 @ id)) @ id))) | ((sP10 @ X25) = $true) | ($true = (X25 @ id @ (lam @ (sub @ X26 @ (push @ one @ sh))) @ id)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f27,plain,(
% 2.43/1.93 ! [X140 : term > $o] : (((sP12 @ X140) = $true) <=> ? [X143 : term] : (! [X144 : term] : (((X140 @ X144) != $true) | ((X140 @ (sub @ X143 @ (push @ X144 @ id))) = $true)) & ($true != (X140 @ (lam @ X143)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f28,plain,(
% 2.43/1.93 ! [X140 : term > $o] : (((sP13 @ X140) = $true) <=> ? [X145 : term,X146 : term] : (((X140 @ X146) = $true) & ($true != (X140 @ (ap @ X145 @ X146))) & ((X140 @ X145) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f29,plain,(
% 2.43/1.93 ($true = sP14) <=> ! [X139 : term,X140 : term > $o,X141 : subst] : (((sP13 @ X140) = $true) | ? [X142 : term] : (((var @ X142) = $true) & ((X140 @ (sub @ X142 @ X141)) != $true)) | ((sP12 @ X140) = $true) | ($true = (X140 @ (sub @ X139 @ X141))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f30,plain,(
% 2.43/1.93 ! [X235 : subst > term > subst > $o] : (($true = (sP15 @ X235)) <=> ? [X241 : subst > term > term] : (((X235 @ id @ (lam @ (X241 @ sh @ one)) @ id) != $true) & ! [X245 : term] : (($true = (X235 @ id @ (X241 @ id @ X245) @ id)) | ((X235 @ id @ X245 @ id) != $true)) & ! [X242 : term,X243 : subst,X244 : subst] : ((X241 @ (comp @ X244 @ X243) @ (sub @ X242 @ X243)) = (sub @ (X241 @ X244 @ X242) @ X243))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f31,plain,(
% 2.43/1.93 ! [X235 : subst > term > subst > $o] : (($true = (sP16 @ X235)) <=> ? [X237 : subst,X238 : subst,X239 : subst,X240 : term] : (((X235 @ X238 @ X240 @ (comp @ X239 @ X237)) != $true) & ($true = (X235 @ (comp @ X238 @ X239) @ (sub @ X240 @ X239) @ X237))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f32,plain,(
% 2.43/1.93 ! [X236 : term,X235 : subst > term > subst > $o] : (((sP17 @ X235 @ X236) = $true) <=> ? [X246 : term] : (($true != (X235 @ id @ (sub @ X236 @ (push @ X246 @ id)) @ id)) & ((X235 @ id @ X246 @ id) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f33,plain,(
% 2.43/1.93 (sP18 = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term] : (((X235 @ id @ (lam @ X236) @ id) = $true) | ? [X247 : subst,X248 : subst,X249 : subst,X250 : term] : (((X235 @ X248 @ X250 @ (comp @ X249 @ X247)) = $true) & ($true != (X235 @ (comp @ X248 @ X249) @ (sub @ X250 @ X249) @ X247))) | ((sP17 @ X235 @ X236) = $true) | ($true = (sP16 @ X235)) | ($true = (sP15 @ X235)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f34,plain,(
% 2.43/1.93 ! [X295 : term > $o] : (((sP19 @ X295) = $true) <=> ! [X297 : term,X298 : term] : (((X295 @ X297) != $true) | ((X295 @ X298) != $true) | ($true = (X295 @ (ap @ X297 @ X298)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f35,plain,(
% 2.43/1.93 ! [X295 : term > $o] : (($true = (sP20 @ X295)) <=> ! [X300 : term] : (? [X301 : term] : (((X295 @ X301) = $true) & ($true != (X295 @ (sub @ X300 @ (push @ X301 @ id))))) | ((X295 @ (lam @ X300)) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f36,plain,(
% 2.43/1.93 ! [X303 : subst > term > subst > $o] : (($true = (sP21 @ X303)) <=> ! [X306 : subst > term > term] : (? [X307 : term] : (($true = (X303 @ id @ X307 @ id)) & ($true != (X303 @ id @ (X306 @ id @ X307) @ id))) | ? [X308 : term,X309 : subst,X310 : subst] : ((X306 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309)) != (sub @ (X306 @ X310 @ X308) @ X309)) | ($true = (X303 @ id @ (lam @ (X306 @ sh @ one)) @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f37,plain,(
% 2.43/1.93 ! [X314 : subst > term > subst > $o] : (((sP22 @ X314) = $true) <=> ? [X326 : subst > term > term] : (((X314 @ id @ (lam @ (X326 @ sh @ one)) @ id) != $true) & ! [X330 : term] : (((X314 @ id @ X330 @ id) != $true) | ((X314 @ id @ (X326 @ id @ X330) @ id) = $true)) & ! [X327 : subst,X328 : term,X329 : subst] : ((X326 @ (comp @ X327 @ X329) @ (sub @ X328 @ X329)) = (sub @ (X326 @ X327 @ X328) @ X329))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f38,plain,(
% 2.43/1.93 ! [X314 : subst > term > subst > $o] : (($true = (sP23 @ X314)) <=> ? [X320 : term,X321 : term] : (((X314 @ id @ (ap @ (sub @ X320 @ id) @ X321) @ id) != $true) & ($true = (X314 @ id @ X321 @ id)) & ((X314 @ id @ X320 @ id) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f39,plain,(
% 2.43/1.93 ! [X314 : subst > term > subst > $o] : (($true = (sP24 @ X314)) <=> ? [X315 : subst,X316 : subst,X317 : term,X318 : subst] : (($true != (X314 @ X318 @ X317 @ (comp @ X315 @ X316))) & ($true = (X314 @ (comp @ X318 @ X315) @ (sub @ X317 @ X315) @ X316))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f40,plain,(
% 2.43/1.93 ! [X314 : subst > term > subst > $o] : (($true = (sP25 @ X314)) <=> ? [X322 : subst,X323 : subst,X324 : term,X325 : subst] : (((X314 @ X322 @ X324 @ (comp @ X323 @ X325)) = $true) & ((X314 @ (comp @ X322 @ X323) @ (sub @ X324 @ X323) @ X325) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f41,plain,(
% 2.43/1.93 (sP26 = $true) <=> ! [X313 : term,X314 : subst > term > subst > $o] : (? [X319 : term] : (((X314 @ id @ X319 @ id) != $true) & ($true = (var @ (sub @ X319 @ id)))) | ((sP22 @ X314) = $true) | ((X314 @ id @ X313 @ id) = $true) | ($true = (sP25 @ X314)) | ($true = (sP24 @ X314)) | ($true = (sP23 @ X314)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f42,plain,(
% 2.43/1.93 (sP27 = $true) <=> ? [X295 : term > $o,X296 : term] : (((sP19 @ X295) = $true) & ! [X299 : term] : (($true != (var @ X299)) | ((X295 @ X299) = $true)) & ($true = (sP20 @ X295)) & ((X295 @ X296) != $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f43,plain,(
% 2.43/1.93 ($true = sP28) <=> ? [X303 : subst > term > subst > $o,X304 : term,X305 : term > $o] : (((X305 @ (lam @ X304)) != $true) & ! [X312 : term] : ((X303 @ id @ X312 @ id) = (X305 @ X312)) & ($true = (sP21 @ X303)) & ! [X311 : term] : (((X305 @ X311) != $true) | ((X305 @ (sub @ X304 @ (push @ X311 @ id))) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f44,plain,(
% 2.43/1.93 ! [X252 : term > $o] : (($true = (sP29 @ X252)) <=> ? [X255 : term] : (((X252 @ (lam @ X255)) != $true) & ! [X256 : term] : (($true != (X252 @ X256)) | ((X252 @ (sub @ X255 @ (push @ X256 @ id))) = $true))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f45,plain,(
% 2.43/1.93 ! [X252 : term > $o] : (((sP30 @ X252) = $true) <=> ? [X253 : term,X254 : term] : (($true != (X252 @ (ap @ X254 @ X253))) & ((X252 @ X254) = $true) & ($true = (X252 @ X253))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f46,plain,(
% 2.43/1.93 ($true = sP31) <=> ! [X251 : term,X252 : term > $o] : (? [X257 : term] : (($true != (X252 @ X257)) & ($true = (var @ X257))) | ((sP30 @ X252) = $true) | ($true = (sP29 @ X252)) | ($true = (X252 @ X251)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f47,plain,(
% 2.43/1.93 (sP32 = $true) <=> ! [X83 : term,X84 : subst > term > term,X85 : subst,X86 : subst > term > term] : (((lam @ (X84 @ sh @ one)) != (lam @ (X86 @ sh @ one))) | ? [X90 : subst,X91 : subst,X92 : term] : ((X84 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91)) != (sub @ (X84 @ X90 @ X92) @ X91)) | ((X84 @ X85 @ X83) = (X86 @ X85 @ X83)) | ? [X87 : subst,X88 : subst,X89 : term] : ((X86 @ (comp @ X87 @ X88) @ (sub @ X89 @ X88)) != (sub @ (X86 @ X87 @ X89) @ X88)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f48,plain,(
% 2.43/1.93 (sP33 = $true) <=> (? [X79 : subst,X80 : term] : ((comp @ sh @ (push @ X80 @ X79)) != X79) | ? [X81 : term,X82 : term] : (((lam @ X81) = (lam @ X82)) & (X81 != X82)) | (sP32 = $true) | ? [X77 : subst,X78 : term] : ((sub @ one @ (push @ X78 @ X77)) != X78))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f49,plain,(
% 2.43/1.93 ! [X181 : term > $o] : (((sP34 @ X181) = $true) <=> ? [X186 : term] : (! [X187 : term] : (((X181 @ (sub @ X186 @ (push @ X187 @ id))) = $true) | ((X181 @ X187) != $true)) & ($true != (X181 @ (lam @ X186)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f50,plain,(
% 2.43/1.93 ! [X181 : term > $o] : (((sP35 @ X181) = $true) <=> ? [X183 : term,X184 : term] : (($true = (X181 @ X183)) & ($true != (X181 @ (ap @ X184 @ X183))) & ((X181 @ X184) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f51,plain,(
% 2.43/1.93 ! [X175 : term > $o] : (($true = (sP36 @ X175)) <=> ! [X176 : term,X177 : term] : (($true = (X175 @ (ap @ X177 @ X176))) | ((X175 @ X177) != $true) | ((X175 @ X176) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f52,plain,(
% 2.43/1.93 ! [X175 : term > $o] : (((sP37 @ X175) = $true) <=> ! [X178 : term] : (((X175 @ X178) != $true) | ($true = (X175 @ (lam @ X178)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f53,plain,(
% 2.43/1.93 (sP38 = $true) <=> ! [X180 : term,X181 : term > $o,X182 : subst] : (((sP35 @ X181) = $true) | ? [X185 : term] : (((var @ X185) = $true) & ((X181 @ (sub @ X185 @ X182)) != $true)) | ((sP34 @ X181) = $true) | ((X181 @ (sub @ X180 @ X182)) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f54,plain,(
% 2.43/1.93 (sP39 = $true) <=> ? [X174 : term,X175 : term > $o] : (((X175 @ X174) != $true) & ! [X179 : term] : (($true = (X175 @ X179)) | ($true != (var @ X179))) & ($true = (sP36 @ X175)) & ((sP37 @ X175) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f55,plain,(
% 2.43/1.93 ! [X459 : subst > term > subst > $o] : (($true = (sP40 @ X459)) <=> ? [X462 : subst > term > term] : (! [X464 : subst,X465 : term,X466 : subst] : ((X462 @ (comp @ X466 @ X464) @ (sub @ X465 @ X464)) = (sub @ (X462 @ X466 @ X465) @ X464)) & ! [X463 : term] : (((X459 @ id @ (X462 @ id @ X463) @ id) = $true) | ((X459 @ id @ X463 @ id) != $true)) & ($true != (X459 @ id @ (lam @ (X462 @ sh @ one)) @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f56,plain,(
% 2.43/1.93 ! [X459 : subst > term > subst > $o] : (($true = (sP41 @ X459)) <=> ? [X460 : term,X461 : term] : (($true = (X459 @ id @ X460 @ id)) & ($true = (X459 @ id @ X461 @ id)) & ($true != (X459 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f57,plain,(
% 2.43/1.93 ! [X459 : subst > term > subst > $o] : (((sP42 @ X459) = $true) <=> ? [X475 : term] : (($true = (var @ (sub @ X475 @ id))) & ($true != (X459 @ id @ X475 @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f58,plain,(
% 2.43/1.93 ! [X459 : subst > term > subst > $o] : (((sP43 @ X459) = $true) <=> ? [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (((X459 @ (comp @ X469 @ X470) @ (sub @ X467 @ X470) @ X468) = $true) & ((X459 @ X469 @ X467 @ (comp @ X470 @ X468)) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f59,plain,(
% 2.43/1.93 ! [X422 : term > $o] : (((sP44 @ X422) = $true) <=> ! [X425 : term,X426 : term] : (($true != (X422 @ X425)) | ((X422 @ X426) != $true) | ((X422 @ (ap @ X426 @ X425)) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f60,plain,(
% 2.43/1.93 ! [X422 : term > $o] : (($true = (sP45 @ X422)) <=> ! [X427 : term] : (? [X428 : term] : (((X422 @ (sub @ X427 @ (push @ X428 @ id))) != $true) & ((X422 @ X428) = $true)) | ($true = (X422 @ (lam @ X427)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f61,plain,(
% 2.43/1.93 ! [X429 : subst > term > subst > $o] : (($true = (sP46 @ X429)) <=> ! [X440 : term,X441 : term] : (((X429 @ id @ X440 @ id) != $true) | ($true != (X429 @ id @ X441 @ id)) | ($true = (X429 @ id @ (ap @ (sub @ X440 @ id) @ X441) @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f62,plain,(
% 2.43/1.93 ! [X429 : subst > term > subst > $o] : (((sP47 @ X429) = $true) <=> ! [X432 : term,X433 : subst,X434 : subst,X435 : subst] : (((X429 @ X435 @ X432 @ (comp @ X434 @ X433)) = $true) | ((X429 @ (comp @ X435 @ X434) @ (sub @ X432 @ X434) @ X433) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f63,plain,(
% 2.43/1.93 ! [X443 : subst > term > subst > $o] : (((sP48 @ X443) = $true) <=> ! [X444 : subst > term > term] : (? [X445 : subst,X446 : term,X447 : subst] : ((X444 @ (comp @ X447 @ X445) @ (sub @ X446 @ X445)) != (sub @ (X444 @ X447 @ X446) @ X445)) | ($true = (X443 @ id @ (lam @ (X444 @ sh @ one)) @ id)) | ? [X448 : term] : (((X443 @ id @ X448 @ id) = $true) & ($true != (X443 @ id @ (X444 @ id @ X448) @ id)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f64,plain,(
% 2.43/1.93 ! [X443 : subst > term > subst > $o] : (((sP49 @ X443) = $true) <=> ! [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (($true != (X443 @ X457 @ X456 @ (comp @ X455 @ X454))) | ((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f65,plain,(
% 2.43/1.93 ! [X442 : term,X443 : subst > term > subst > $o] : (((sP50 @ X443 @ X442) = $true) <=> ! [X449 : term] : (((X443 @ id @ (sub @ X442 @ (push @ X449 @ id)) @ id) = $true) | ($true != (X443 @ id @ X449 @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f66,plain,(
% 2.43/1.93 (sP51 = $true) <=> ! [X458 : term,X459 : subst > term > subst > $o] : (? [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (((X459 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471) != $true) & ((X459 @ X473 @ X474 @ (comp @ X472 @ X471)) = $true)) | ($true = (sP40 @ X459)) | ((X459 @ id @ X458 @ id) = $true) | ((sP43 @ X459) = $true) | ($true = (sP41 @ X459)) | ((sP42 @ X459) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f67,plain,(
% 2.43/1.93 (sP52 = $true) <=> ? [X422 : term > $o,X423 : term] : (! [X424 : term] : (($true = (X422 @ X424)) | ((var @ X424) != $true)) & ((sP44 @ X422) = $true) & ($true = (sP45 @ X422)) & ((X422 @ X423) != $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f68,plain,(
% 2.43/1.93 ($true = sP53) <=> ? [X429 : subst > term > subst > $o,X430 : term,X431 : term] : (($true = (sP46 @ X429)) & ((X429 @ id @ X430 @ id) = $true) & ! [X436 : subst,X437 : subst,X438 : term,X439 : subst] : (($true != (X429 @ X437 @ X438 @ (comp @ X436 @ X439))) | ((X429 @ (comp @ X437 @ X436) @ (sub @ X438 @ X436) @ X439) = $true)) & ((X429 @ id @ X431 @ id) = $true) & ((X429 @ id @ (ap @ X431 @ X430) @ id) != $true) & ((sP47 @ X429) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f69,plain,(
% 2.43/1.93 (sP54 = $true) <=> ? [X442 : term,X443 : subst > term > subst > $o] : (! [X450 : subst,X451 : term,X452 : subst,X453 : subst] : (((X443 @ (comp @ X452 @ X453) @ (sub @ X451 @ X453) @ X450) != $true) | ($true = (X443 @ X452 @ X451 @ (comp @ X453 @ X450)))) & ((sP50 @ X443 @ X442) = $true) & ((X443 @ id @ (lam @ X442) @ id) != $true) & ((sP49 @ X443) = $true) & ((sP48 @ X443) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f70,plain,(
% 2.43/1.93 ! [X351 : subst > term > subst > $o] : (((sP55 @ X351) = $true) <=> ? [X362 : subst > term > term] : (! [X363 : term,X364 : subst,X365 : subst] : ((X362 @ (comp @ X365 @ X364) @ (sub @ X363 @ X364)) = (sub @ (X362 @ X365 @ X363) @ X364)) & ! [X366 : term] : (((X351 @ id @ (X362 @ id @ X366) @ id) = $true) | ($true != (X351 @ id @ X366 @ id))) & ((X351 @ id @ (lam @ (X362 @ sh @ one)) @ id) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f71,plain,(
% 2.43/1.93 ! [X351 : subst > term > subst > $o] : (((sP56 @ X351) = $true) <=> ? [X358 : subst,X359 : subst,X360 : term,X361 : subst] : (($true != (X351 @ X358 @ X360 @ (comp @ X359 @ X361))) & ($true = (X351 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359) @ X361))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f72,plain,(
% 2.43/1.93 ! [X351 : subst > term > subst > $o] : (((sP57 @ X351) = $true) <=> ? [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (((X351 @ (comp @ X356 @ X354) @ (sub @ X357 @ X354) @ X355) != $true) & ((X351 @ X356 @ X357 @ (comp @ X354 @ X355)) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f73,plain,(
% 2.43/1.93 (sP58 = $true) <=> ! [X350 : term,X351 : subst > term > subst > $o,X352 : term > $o] : (? [X367 : term] : (((X352 @ (sub @ X350 @ (push @ X367 @ id))) != $true) & ((X352 @ X367) = $true)) | ((sP55 @ X351) = $true) | ((sP57 @ X351) = $true) | ((sP56 @ X351) = $true) | ? [X353 : term] : ((X351 @ id @ X353 @ id) != (X352 @ X353)) | ((X352 @ (lam @ X350)) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f74,plain,(
% 2.43/1.93 ! [X188 : subst > term > subst > $o] : (((sP59 @ X188) = $true) <=> ? [X200 : subst > term > term] : (! [X204 : term] : (((X188 @ id @ X204 @ id) != $true) | ((X188 @ id @ (X200 @ id @ X204) @ id) = $true)) & ((X188 @ id @ (lam @ (X200 @ sh @ one)) @ id) != $true) & ! [X201 : subst,X202 : subst,X203 : term] : ((X200 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201)) = (sub @ (X200 @ X202 @ X203) @ X201))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f75,plain,(
% 2.43/1.93 ! [X188 : subst > term > subst > $o] : (($true = (sP60 @ X188)) <=> ? [X195 : subst,X196 : subst,X197 : subst,X198 : term] : (((X188 @ (comp @ X197 @ X196) @ (sub @ X198 @ X196) @ X195) = $true) & ((X188 @ X197 @ X198 @ (comp @ X196 @ X195)) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f76,plain,(
% 2.43/1.93 ! [X190 : term > $o,X189 : term] : (($true = (sP61 @ X189 @ X190)) <=> ? [X199 : term] : (((X190 @ X199) = $true) & ((X190 @ (sub @ X189 @ (push @ X199 @ id))) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f77,plain,(
% 2.43/1.93 (sP62 = $true) <=> ! [X188 : subst > term > subst > $o,X189 : term,X190 : term > $o] : (((X190 @ (lam @ X189)) = $true) | ? [X191 : subst,X192 : subst,X193 : subst,X194 : term] : (((X188 @ (comp @ X193 @ X191) @ (sub @ X194 @ X191) @ X192) != $true) & ((X188 @ X193 @ X194 @ (comp @ X191 @ X192)) = $true)) | ((sP59 @ X188) = $true) | ? [X205 : term] : ((X188 @ id @ X205 @ id) != (X190 @ X205)) | ($true = (sP61 @ X189 @ X190)) | ($true = (sP60 @ X188)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f78,plain,(
% 2.43/1.93 ! [X62 : subst > term > subst > $o] : (($true = (sP63 @ X62)) <=> ! [X63 : subst > term > term] : (((X62 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true) | ? [X64 : subst,X65 : subst,X66 : term] : ((X63 @ (comp @ X65 @ X64) @ (sub @ X66 @ X64)) != (sub @ (X63 @ X65 @ X66) @ X64)) | ? [X67 : term] : (($true != (X62 @ id @ (X63 @ id @ X67) @ id)) & ($true = (X62 @ id @ X67 @ id)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f79,plain,(
% 2.43/1.93 ! [X70 : subst > term > subst > $o] : (($true = (sP64 @ X70)) <=> ? [X72 : subst > term > term] : (! [X76 : term] : (($true != (X70 @ id @ X76 @ id)) | ((X70 @ id @ (X72 @ id @ X76) @ id) = $true)) & ($true != (X70 @ id @ (lam @ (X72 @ sh @ one)) @ id)) & ! [X73 : subst,X74 : term,X75 : subst] : ((sub @ (X72 @ X75 @ X74) @ X73) = (X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f80,plain,(
% 2.43/1.93 ($true = sP65) <=> ! [X69 : term,X70 : subst > term > subst > $o] : (($true = (sP64 @ X70)) | ? [X71 : term] : (($true = (X70 @ id @ X71 @ id)) & ($true != (X70 @ id @ (sub @ X69 @ (push @ X71 @ id)) @ id))) | ((X70 @ id @ (lam @ X69) @ id) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f81,plain,(
% 2.43/1.93 ($true = sP66) <=> ? [X61 : term,X62 : subst > term > subst > $o] : (! [X68 : term] : (((X62 @ id @ (sub @ X61 @ (push @ X68 @ id)) @ id) = $true) | ((X62 @ id @ X68 @ id) != $true)) & ((X62 @ id @ (lam @ (sub @ X61 @ (push @ one @ sh))) @ id) != $true) & ($true = (sP63 @ X62)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f82,plain,(
% 2.43/1.93 ! [X124 : subst > term > subst > $o] : (((sP67 @ X124) = $true) <=> ? [X125 : subst > term > term] : (! [X127 : term,X128 : subst,X129 : subst] : ((X125 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128)) = (sub @ (X125 @ X129 @ X127) @ X128)) & ! [X126 : term] : (((X124 @ id @ X126 @ id) != $true) | ((X124 @ id @ (X125 @ id @ X126) @ id) = $true)) & ($true != (X124 @ id @ (lam @ (X125 @ sh @ one)) @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f83,plain,(
% 2.43/1.93 ! [X123 : term,X124 : subst > term > subst > $o] : (($true = (sP68 @ X124 @ X123)) <=> ? [X138 : term] : (((X124 @ id @ (sub @ X123 @ (push @ X138 @ id)) @ id) != $true) & ((X124 @ id @ X138 @ id) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f84,plain,(
% 2.43/1.93 ! [X124 : subst > term > subst > $o] : (($true = (sP69 @ X124)) <=> ? [X134 : term,X135 : subst,X136 : subst,X137 : subst] : (((X124 @ X137 @ X134 @ (comp @ X136 @ X135)) = $true) & ($true != (X124 @ (comp @ X137 @ X136) @ (sub @ X134 @ X136) @ X135))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f85,plain,(
% 2.43/1.93 ! [X116 : subst > term > subst > $o] : (($true = (sP70 @ X116)) <=> ! [X117 : subst > term > term] : (((X116 @ id @ (lam @ (X117 @ sh @ one)) @ id) = $true) | ? [X119 : term,X120 : subst,X121 : subst] : ((X117 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)) != (sub @ (X117 @ X121 @ X119) @ X120)) | ? [X118 : term] : (($true = (X116 @ id @ X118 @ id)) & ($true != (X116 @ id @ (X117 @ id @ X118) @ id)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f86,plain,(
% 2.43/1.93 (sP71 = $true) <=> ! [X123 : term,X124 : subst > term > subst > $o] : (($true = (X124 @ id @ (lam @ X123) @ id)) | ? [X130 : term,X131 : subst,X132 : subst,X133 : subst] : (((X124 @ X132 @ X130 @ (comp @ X131 @ X133)) != $true) & ($true = (X124 @ (comp @ X132 @ X131) @ (sub @ X130 @ X131) @ X133))) | ($true = (sP69 @ X124)) | ($true = (sP68 @ X124 @ X123)) | ((sP67 @ X124) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f87,plain,(
% 2.43/1.93 (sP72 = $true) <=> ? [X115 : term,X116 : subst > term > subst > $o] : (! [X122 : term] : (((X116 @ id @ X122 @ id) != $true) | ((X116 @ id @ (sub @ X115 @ (push @ X122 @ id)) @ id) = $true)) & ((X116 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id) != $true) & ($true = (sP70 @ X116)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f88,plain,(
% 2.43/1.93 (sP73 = $true) <=> ! [X480 : term,X481 : term,X482 : term,X483 : term] : ((X482 = X483) | ((ap @ (sub @ X481 @ id) @ X483) != (ap @ (sub @ X480 @ id) @ X482)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f89,plain,(
% 2.43/1.93 (sP74 = $true) <=> (? [X476 : term,X477 : term,X478 : term,X479 : term] : (((ap @ X477 @ X476) = (ap @ X478 @ X479)) & (X476 != X479)) | (sP73 = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f90,plain,(
% 2.43/1.93 ! [X370 : subst > term > subst > $o] : (((sP75 @ X370) = $true) <=> ? [X371 : subst > term > term] : (($true != (X370 @ id @ (lam @ (X371 @ sh @ one)) @ id)) & ! [X372 : term] : (($true != (X370 @ id @ X372 @ id)) | ($true = (X370 @ id @ (X371 @ id @ X372) @ id))) & ! [X373 : term,X374 : subst,X375 : subst] : ((sub @ (X371 @ X375 @ X373) @ X374) = (X371 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f91,plain,(
% 2.43/1.93 (sP76 = $true) <=> ! [X368 : term,X369 : term > $o,X370 : subst > term > subst > $o] : (? [X376 : term] : ((X370 @ id @ X376 @ id) != (X369 @ X376)) | ? [X377 : term] : (((X369 @ (sub @ X368 @ (push @ X377 @ id))) != $true) & ((X369 @ X377) = $true)) | ((sP75 @ X370) = $true) | ($true = (X369 @ (lam @ X368))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f92,plain,(
% 2.43/1.93 ! [X388 : term > $o] : (((sP77 @ X388) = $true) <=> ! [X392 : term,X393 : term] : (((X388 @ X393) != $true) | ($true = (X388 @ (ap @ X393 @ X392))) | ($true != (X388 @ X392))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f93,plain,(
% 2.43/1.93 ! [X388 : term > $o] : (($true = (sP78 @ X388)) <=> ! [X389 : term] : (($true != (var @ X389)) | ($true = (X388 @ X389))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f94,plain,(
% 2.43/1.93 ! [X388 : term > $o] : (((sP79 @ X388) = $true) <=> ! [X390 : term] : (? [X391 : term] : (((X388 @ X391) = $true) & ((X388 @ (sub @ X390 @ (push @ X391 @ id))) != $true)) | ((X388 @ (lam @ X390)) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f95,plain,(
% 2.43/1.93 ! [X406 : subst > term > subst > $o] : (((sP80 @ X406) = $true) <=> ? [X407 : subst > term > term] : (! [X408 : term] : (((X406 @ id @ (X407 @ id @ X408) @ id) = $true) | ((X406 @ id @ X408 @ id) != $true)) & ($true != (X406 @ id @ (lam @ (X407 @ sh @ one)) @ id)) & ! [X409 : subst,X410 : subst,X411 : term] : ((sub @ (X407 @ X410 @ X411) @ X409) = (X407 @ (comp @ X410 @ X409) @ (sub @ X411 @ X409)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f96,plain,(
% 2.43/1.93 ! [X406 : subst > term > subst > $o] : (((sP81 @ X406) = $true) <=> ? [X412 : term,X413 : term] : (((X406 @ id @ (ap @ (sub @ X412 @ id) @ X413) @ id) != $true) & ($true = (X406 @ id @ X413 @ id)) & ((X406 @ id @ X412 @ id) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f97,plain,(
% 2.43/1.93 ! [X397 : subst > term > subst > $o] : (((sP82 @ X397) = $true) <=> ! [X399 : subst > term > term] : (? [X400 : subst,X401 : subst,X402 : term] : ((sub @ (X399 @ X400 @ X402) @ X401) != (X399 @ (comp @ X400 @ X401) @ (sub @ X402 @ X401))) | ? [X403 : term] : (($true != (X397 @ id @ (X399 @ id @ X403) @ id)) & ((X397 @ id @ X403 @ id) = $true)) | ($true = (X397 @ id @ (lam @ (X399 @ sh @ one)) @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f98,plain,(
% 2.43/1.93 (sP83 = $true) <=> ! [X405 : term,X406 : subst > term > subst > $o] : (($true = (X406 @ id @ X405 @ id)) | ((sP80 @ X406) = $true) | ((sP81 @ X406) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f99,plain,(
% 2.43/1.93 (sP84 = $true) <=> ? [X395 : term > $o,X396 : term,X397 : subst > term > subst > $o] : (! [X404 : term] : (((X395 @ (sub @ X396 @ (push @ X404 @ id))) = $true) | ((X395 @ X404) != $true)) & ! [X398 : term] : ((X397 @ id @ X398 @ id) = (X395 @ X398)) & ((sP82 @ X397) = $true) & ($true != (X395 @ (lam @ X396))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f100,plain,(
% 2.43/1.93 (sP85 = $true) <=> ? [X387 : term,X388 : term > $o] : (((X388 @ X387) != $true) & ((sP79 @ X388) = $true) & ($true = (sP78 @ X388)) & ((sP77 @ X388) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f101,plain,(
% 2.43/1.93 (sP86 = $true) <=> ! [X345 : term,X346 : term,X347 : term,X348 : term] : (((ap @ (sub @ X345 @ id) @ X347) != (ap @ (sub @ X348 @ id) @ X346)) | (X345 = X348))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f102,plain,(
% 2.43/1.93 (sP87 = $true) <=> (? [X341 : term,X342 : term,X343 : term,X344 : term] : ((X341 != X342) & ((ap @ X342 @ X343) = (ap @ X341 @ X344))) | (sP86 = $true) | ? [X340 : term] : ((sub @ X340 @ id) != X340))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f103,plain,(
% 2.43/1.93 ! [X37 : subst > term > subst > $o] : (($true = (sP88 @ X37)) <=> ? [X41 : subst > term > term] : (((X37 @ id @ (lam @ (X41 @ sh @ one)) @ id) != $true) & ! [X45 : term] : (($true = (X37 @ id @ (X41 @ id @ X45) @ id)) | ((X37 @ id @ X45 @ id) != $true)) & ! [X42 : term,X43 : subst,X44 : subst] : ((X41 @ (comp @ X43 @ X44) @ (sub @ X42 @ X44)) = (sub @ (X41 @ X43 @ X42) @ X44))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f104,plain,(
% 2.43/1.93 ! [X37 : subst > term > subst > $o] : (((sP89 @ X37) = $true) <=> ? [X39 : term,X40 : term] : (($true = (X37 @ id @ X40 @ id)) & ($true = (X37 @ id @ X39 @ id)) & ($true != (X37 @ id @ (ap @ (sub @ X40 @ id) @ X39) @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f105,plain,(
% 2.43/1.93 ! [X37 : subst > term > subst > $o] : (((sP90 @ X37) = $true) <=> ? [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true = (X37 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) & ((X37 @ X51 @ X52 @ (comp @ X50 @ X53)) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f106,plain,(
% 2.43/1.93 ! [X37 : subst > term > subst > $o] : (((sP91 @ X37) = $true) <=> ? [X54 : term] : (($true = (var @ (sub @ X54 @ id))) & ($true != (X37 @ id @ X54 @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f107,plain,(
% 2.43/1.93 (sP92 = $true) <=> ! [X37 : subst > term > subst > $o,X38 : term] : (? [X46 : subst,X47 : subst,X48 : subst,X49 : term] : (((X37 @ X46 @ X49 @ (comp @ X47 @ X48)) = $true) & ((X37 @ (comp @ X46 @ X47) @ (sub @ X49 @ X47) @ X48) != $true)) | ($true = (X37 @ id @ X38 @ id)) | ((sP89 @ X37) = $true) | ($true = (sP88 @ X37)) | ((sP91 @ X37) = $true) | ((sP90 @ X37) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f108,plain,(
% 2.43/1.93 ! [X415 : subst > term > subst > $o] : (((sP93 @ X415) = $true) <=> ? [X416 : subst > term > term] : (! [X417 : term] : (((X415 @ id @ (X416 @ id @ X417) @ id) = $true) | ((X415 @ id @ X417 @ id) != $true)) & ((X415 @ id @ (lam @ (X416 @ sh @ one)) @ id) != $true) & ! [X418 : term,X419 : subst,X420 : subst] : ((sub @ (X416 @ X419 @ X418) @ X420) = (X416 @ (comp @ X419 @ X420) @ (sub @ X418 @ X420)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f109,plain,(
% 2.43/1.93 (sP94 = $true) <=> ! [X414 : term,X415 : subst > term > subst > $o] : (((sP93 @ X415) = $true) | ($true = (X415 @ id @ (lam @ (sub @ X414 @ (push @ one @ sh))) @ id)) | ? [X421 : term] : (($true = (X415 @ id @ X421 @ id)) & ((X415 @ id @ (sub @ X414 @ (push @ X421 @ id)) @ id) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f110,plain,(
% 2.43/1.93 ! [X206 : term > $o] : (((sP95 @ X206) = $true) <=> ? [X209 : term,X210 : term] : (($true = (X206 @ X210)) & ((X206 @ (ap @ X209 @ X210)) != $true) & ($true = (X206 @ X209))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f111,plain,(
% 2.43/1.93 ! [X206 : term > $o] : (((sP96 @ X206) = $true) <=> ? [X208 : term] : (($true != (X206 @ (lam @ X208))) & ($true = (X206 @ X208))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f112,plain,(
% 2.43/1.93 (sP97 = $true) <=> ! [X206 : term > $o,X207 : term] : (((sP95 @ X206) = $true) | ? [X211 : term] : (($true = (var @ X211)) & ($true != (X206 @ X211))) | ((sP96 @ X206) = $true) | ((X206 @ X207) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f113,plain,(
% 2.43/1.93 ! [X505 : subst > term > subst > $o] : (($true = (sP98 @ X505)) <=> ? [X508 : subst > term > term] : (((X505 @ id @ (lam @ (X508 @ sh @ one)) @ id) != $true) & ! [X510 : subst,X511 : subst,X512 : term] : ((X508 @ (comp @ X511 @ X510) @ (sub @ X512 @ X510)) = (sub @ (X508 @ X511 @ X512) @ X510)) & ! [X509 : term] : (($true != (X505 @ id @ X509 @ id)) | ($true = (X505 @ id @ (X508 @ id @ X509) @ id)))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f114,plain,(
% 2.43/1.93 ($true = sP99) <=> ! [X505 : subst > term > subst > $o,X506 : term] : (($true = (sP98 @ X505)) | ? [X507 : term] : (($true = (X505 @ id @ X507 @ id)) & ((X505 @ id @ (sub @ X506 @ (push @ X507 @ id)) @ id) != $true)) | ((X505 @ id @ (lam @ (sub @ X506 @ (push @ one @ sh))) @ id) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f115,plain,(
% 2.43/1.93 ! [X518 : subst > term > subst > $o] : (($true = (sP100 @ X518)) <=> ? [X525 : term,X526 : term] : (($true = (X518 @ id @ X525 @ id)) & ($true = (X518 @ id @ X526 @ id)) & ((X518 @ id @ (ap @ (sub @ X525 @ id) @ X526) @ id) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f116,plain,(
% 2.43/1.93 ! [X518 : subst > term > subst > $o] : (((sP101 @ X518) = $true) <=> ? [X527 : subst,X528 : subst,X529 : term,X530 : subst] : (($true = (X518 @ X528 @ X529 @ (comp @ X530 @ X527))) & ($true != (X518 @ (comp @ X528 @ X530) @ (sub @ X529 @ X530) @ X527))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f117,plain,(
% 2.43/1.93 ($true = sP102) <=> ! [X518 : subst > term > subst > $o,X519 : term,X520 : term] : (((X518 @ id @ X519 @ id) != $true) | ((X518 @ id @ (ap @ X519 @ X520) @ id) = $true) | ($true != (X518 @ id @ X520 @ id)) | ? [X521 : subst,X522 : subst,X523 : term,X524 : subst] : (($true != (X518 @ X522 @ X523 @ (comp @ X524 @ X521))) & ($true = (X518 @ (comp @ X522 @ X524) @ (sub @ X523 @ X524) @ X521))) | ($true = (sP100 @ X518)) | ((sP101 @ X518) = $true))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f118,plain,(
% 2.43/1.93 ! [X106 : subst > term > subst > $o] : (($true = (sP103 @ X106)) <=> ? [X107 : subst > term > term] : (! [X108 : term] : (($true = (X106 @ id @ (X107 @ id @ X108) @ id)) | ($true != (X106 @ id @ X108 @ id))) & ((X106 @ id @ (lam @ (X107 @ sh @ one)) @ id) != $true) & ! [X109 : subst,X110 : term,X111 : subst] : ((X107 @ (comp @ X111 @ X109) @ (sub @ X110 @ X109)) = (sub @ (X107 @ X111 @ X110) @ X109))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f119,plain,(
% 2.43/1.93 ! [X106 : subst > term > subst > $o] : (($true = (sP104 @ X106)) <=> ? [X112 : term,X113 : term] : (((X106 @ id @ X113 @ id) = $true) & ((X106 @ id @ X112 @ id) = $true) & ($true != (X106 @ id @ (ap @ (sub @ X113 @ id) @ X112) @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f120,plain,(
% 2.43/1.93 ! [X283 : subst > term > subst > $o] : (((sP105 @ X283) = $true) <=> ? [X285 : subst > term > term] : (! [X286 : term] : (($true = (X283 @ id @ (X285 @ id @ X286) @ id)) | ($true != (X283 @ id @ X286 @ id))) & ($true != (X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) & ! [X287 : term,X288 : subst,X289 : subst] : ((X285 @ (comp @ X289 @ X288) @ (sub @ X287 @ X288)) = (sub @ (X285 @ X289 @ X287) @ X288))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f121,plain,(
% 2.43/1.93 ($true = sP106) <=> ! [X282 : term > $o,X283 : subst > term > subst > $o,X284 : term] : (((sP105 @ X283) = $true) | ((X282 @ (lam @ X284)) = $true) | ? [X291 : term] : (((X282 @ X291) = $true) & ($true != (X282 @ (sub @ X284 @ (push @ X291 @ id))))) | ? [X290 : term] : ((X282 @ X290) != (X283 @ id @ X290 @ id)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f122,plain,(
% 2.43/1.93 ! [X222 : subst > term > subst > $o] : (((sP107 @ X222) = $true) <=> ? [X233 : term,X234 : term] : (($true != (X222 @ id @ (ap @ (sub @ X234 @ id) @ X233) @ id)) & ((X222 @ id @ X234 @ id) = $true) & ($true = (X222 @ id @ X233 @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f123,plain,(
% 2.43/1.93 ! [X222 : subst > term > subst > $o] : (((sP108 @ X222) = $true) <=> ? [X224 : subst,X225 : term,X226 : subst,X227 : subst] : (((X222 @ (comp @ X224 @ X227) @ (sub @ X225 @ X227) @ X226) != $true) & ((X222 @ X224 @ X225 @ (comp @ X227 @ X226)) = $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f124,plain,(
% 2.43/1.93 ($true = sP109) <=> ! [X220 : term > $o,X221 : term,X222 : subst > term > subst > $o,X223 : term] : (((X220 @ X223) != $true) | ((sP107 @ X222) = $true) | ? [X228 : subst,X229 : subst,X230 : subst,X231 : term] : (($true != (X222 @ X230 @ X231 @ (comp @ X228 @ X229))) & ((X222 @ (comp @ X230 @ X228) @ (sub @ X231 @ X228) @ X229) = $true)) | ((X220 @ X221) != $true) | ((sP108 @ X222) = $true) | ((X220 @ (ap @ X221 @ X223)) = $true) | ? [X232 : term] : ((X220 @ X232) != (X222 @ id @ X232 @ id)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f14,plain,(
% 2.43/1.93 ((pushprop_lem1_lthm = $true) <=> (! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true = (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) | ((X18 @ X15) != $true)) | ? [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) != X11) | ? [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) != X12))) & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) != (d2subst @ X55)) | (X55 = X56)) & ((hoasinduction_lem2v2 = $true) <=> ! [X220 : term > $o,X221 : term,X222 : subst > term > subst > $o,X223 : term] : (((X220 @ X223) != $true) | ? [X233 : term,X234 : term] : (($true != (X222 @ id @ (ap @ (sub @ X234 @ id) @ X233) @ id)) & ((X222 @ id @ X234 @ id) = $true) & ($true = (X222 @ id @ X233 @ id))) | ? [X228 : subst,X229 : subst,X230 : subst,X231 : term] : (($true != (X222 @ X230 @ X231 @ (comp @ X228 @ X229))) & ((X222 @ (comp @ X230 @ X228) @ (sub @ X231 @ X228) @ X229) = $true)) | ((X220 @ X221) != $true) | ? [X224 : subst,X225 : term,X226 : subst,X227 : subst] : (((X222 @ (comp @ X224 @ X227) @ (sub @ X225 @ X227) @ X226) != $true) & ((X222 @ X224 @ X225 @ (comp @ X227 @ X226)) = $true)) | ((X220 @ (ap @ X221 @ X223)) = $true) | ? [X232 : term] : ((X220 @ X232) != (X222 @ id @ X232 @ id)))) & ((? [X281 : term] : ((sub @ X281 @ id) != X281) | ? [X277 : term] : ! [X278 : subst > term > term] : ? [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) != (X278 @ X280 @ X279)) | ! [X282 : term > $o,X283 : subst > term > subst > $o,X284 : term] : (? [X285 : subst > term > term] : (! [X286 : term] : (($true = (X283 @ id @ (X285 @ id @ X286) @ id)) | ($true != (X283 @ id @ X286 @ id))) & ($true != (X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) & ! [X287 : term,X288 : subst,X289 : subst] : ((X285 @ (comp @ X289 @ X288) @ (sub @ X287 @ X288)) = (sub @ (X285 @ X289 @ X287) @ X288))) | ((X282 @ (lam @ X284)) = $true) | ? [X291 : term] : (((X282 @ X291) = $true) & ($true != (X282 @ (sub @ X284 @ (push @ X291 @ id))))) | ? [X290 : term] : ((X282 @ X290) != (X283 @ id @ X290 @ id)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X7 : subst > term > subst > $o] : ? [X8 : term > $o] : ! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X338 : term] : ? [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) = (X339 @ sh @ one))) & (induction2lem_gthm = $true) & (one = (d2term @ d_one)) & (hoasinduction_lem3b_gthm = $true) & (! [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) = X273) <=> (axshiftcons = $true)) & (! [X105 : term,X106 : subst > term > subst > $o] : (? [X107 : subst > term > term] : (! [X108 : term] : (($true = (X106 @ id @ (X107 @ id @ X108) @ id)) | ($true != (X106 @ id @ X108 @ id))) & ((X106 @ id @ (lam @ (X107 @ sh @ one)) @ id) != $true) & ! [X109 : subst,X110 : term,X111 : subst] : ((X107 @ (comp @ X111 @ X109) @ (sub @ X110 @ X109)) = (sub @ (X107 @ X111 @ X110) @ X109))) | ($true = (X106 @ id @ X105 @ id)) | ? [X112 : term,X113 : term] : (((X106 @ id @ X113 @ id) = $true) & ((X106 @ id @ X112 @ id) = $true) & ($true != (X106 @ id @ (ap @ (sub @ X113 @ id) @ X112) @ id)))) <=> (hoasinduction_no_psi_cond = $true)) & (termmset_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X518 : subst > term > subst > $o,X519 : term,X520 : term] : (((X518 @ id @ X519 @ id) != $true) | ((X518 @ id @ (ap @ X519 @ X520) @ id) = $true) | ($true != (X518 @ id @ X520 @ id)) | ? [X521 : subst,X522 : subst,X523 : term,X524 : subst] : (($true != (X518 @ X522 @ X523 @ (comp @ X524 @ X521))) & ($true = (X518 @ (comp @ X522 @ X524) @ (sub @ X523 @ X524) @ X521))) | ? [X525 : term,X526 : term] : (($true = (X518 @ id @ X525 @ id)) & ($true = (X518 @ id @ X526 @ id)) & ((X518 @ id @ (ap @ (sub @ X525 @ id) @ X526) @ id) != $true)) | ? [X527 : subst,X528 : subst,X529 : term,X530 : subst] : (($true = (X518 @ X528 @ X529 @ (comp @ X530 @ X527))) & ($true != (X518 @ (comp @ X528 @ X530) @ (sub @ X529 @ X530) @ X527))))) & (hoasinduction_lem1_lthm = $true) & (! [X505 : subst > term > subst > $o,X506 : term] : (? [X508 : subst > term > term] : (((X505 @ id @ (lam @ (X508 @ sh @ one)) @ id) != $true) & ! [X510 : subst,X511 : subst,X512 : term] : ((X508 @ (comp @ X511 @ X510) @ (sub @ X512 @ X510)) = (sub @ (X508 @ X511 @ X512) @ X510)) & ! [X509 : term] : (($true != (X505 @ id @ X509 @ id)) | ($true = (X505 @ id @ (X508 @ id @ X509) @ id)))) | ? [X507 : term] : (($true = (X505 @ id @ X507 @ id)) & ((X505 @ id @ (sub @ X506 @ (push @ X507 @ id)) @ id) != $true)) | ((X505 @ id @ (lam @ (sub @ X506 @ (push @ one @ sh))) @ id) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X513 : term,X514 : term > $o,X515 : subst] : ? [X516 : term > $o] : ! [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) = (X516 @ X517))) & (hoasapnotvar_lthm = $true) & ((! [X59 : subst] : ((comp @ id @ X59) = X59) & ! [X58 : subst] : ((comp @ X58 @ id) = X58)) <=> (substmonoid = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lthm = $true) & ! [X96 : d_term] : (d_one = X96) & (lamnotvar = $true) & ((induction = $true) <=> ! [X206 : term > $o,X207 : term] : (? [X209 : term,X210 : term] : (($true = (X206 @ X210)) & ((X206 @ (ap @ X209 @ X210)) != $true) & ($true = (X206 @ X209))) | ? [X211 : term] : (($true = (var @ X211)) & ($true != (X206 @ X211))) | ? [X208 : term] : (($true != (X206 @ (lam @ X208))) & ($true = (X206 @ X208))) | ((X206 @ X207) = $true))) & (! [X275 : term,X276 : term] : (((sub @ X276 @ sh) != (sub @ X275 @ sh)) | (X275 = X276)) <=> (shinj = $true)) & (hoasinduction_lem1v2_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (! [X93 : subst > term > subst > $o] : ? [X94 : term > $o] : ! [X95 : term] : ((X94 @ X95) = (X93 @ id @ X95 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (! [X414 : term,X415 : subst > term > subst > $o] : (? [X416 : subst > term > term] : (! [X417 : term] : (((X415 @ id @ (X416 @ id @ X417) @ id) = $true) | ((X415 @ id @ X417 @ id) != $true)) & ((X415 @ id @ (lam @ (X416 @ sh @ one)) @ id) != $true) & ! [X418 : term,X419 : subst,X420 : subst] : ((sub @ (X416 @ X419 @ X418) @ X420) = (X416 @ (comp @ X419 @ X420) @ (sub @ X418 @ X420)))) | ($true = (X415 @ id @ (lam @ (sub @ X414 @ (push @ one @ sh))) @ id)) | ? [X421 : term] : (($true = (X415 @ id @ X421 @ id)) & ((X415 @ id @ (sub @ X414 @ (push @ X421 @ id)) @ id) != $true))) <=> (hoasinduction_lem3aa = $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction = $true) <=> ! [X37 : subst > term > subst > $o,X38 : term] : (? [X46 : subst,X47 : subst,X48 : subst,X49 : term] : (((X37 @ X46 @ X49 @ (comp @ X47 @ X48)) = $true) & ((X37 @ (comp @ X46 @ X47) @ (sub @ X49 @ X47) @ X48) != $true)) | ($true = (X37 @ id @ X38 @ id)) | ? [X39 : term,X40 : term] : (($true = (X37 @ id @ X40 @ id)) & ($true = (X37 @ id @ X39 @ id)) & ($true != (X37 @ id @ (ap @ (sub @ X40 @ id) @ X39) @ id))) | ? [X41 : subst > term > term] : (((X37 @ id @ (lam @ (X41 @ sh @ one)) @ id) != $true) & ! [X45 : term] : (($true = (X37 @ id @ (X41 @ id @ X45) @ id)) | ((X37 @ id @ X45 @ id) != $true)) & ! [X42 : term,X43 : subst,X44 : subst] : ((X41 @ (comp @ X43 @ X44) @ (sub @ X42 @ X44)) = (sub @ (X41 @ X43 @ X42) @ X44))) | ? [X54 : term] : (($true = (var @ (sub @ X54 @ id))) & ($true != (X37 @ id @ X54 @ id))) | ? [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true = (X37 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) & ((X37 @ X51 @ X52 @ (comp @ X50 @ X53)) != $true)))) & ((? [X341 : term,X342 : term,X343 : term,X344 : term] : ((X341 != X342) & ((ap @ X342 @ X343) = (ap @ X341 @ X344))) | ! [X345 : term,X346 : term,X347 : term,X348 : term] : (((ap @ (sub @ X345 @ id) @ X347) != (ap @ (sub @ X348 @ id) @ X346)) | (X345 = X348)) | ? [X340 : term] : ((sub @ X340 @ id) != X340)) <=> (hoasapinj1_lthm = $true)) & (lamnotap != $true) & ((apinj2 = $true) <=> ! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) != (ap @ X100 @ X98)) | (X97 = X98))) & ((termmset = $true) <=> ! [X57 : term] : ((sub @ X57 @ id) = X57)) & (hoasinduction_lem3v2_gthm = $true) & (ulamvar1 != $true) & (axapp = $true) & ! [X382 : subst] : ? [X383 : d_subst] : ((d2subst @ X383) = X382) & (! [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (($true = (X267 @ (sub @ X265 @ (push @ X268 @ X269)))) | ((X266 @ X265) != $true)) <=> (pushprop_lem1 = $true)) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ! [X270 : d_subst] : (d_id = X270) & ! [X23 : d_term,X24 : d_term] : ((X23 = X24) | ((d2term @ X23) != (d2term @ X24))) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((? [X394 : term] : ((sub @ X394 @ id) != X394) | ? [X395 : term > $o,X396 : term,X397 : subst > term > subst > $o] : (! [X404 : term] : (((X395 @ (sub @ X396 @ (push @ X404 @ id))) = $true) | ((X395 @ X404) != $true)) & ! [X398 : term] : ((X397 @ id @ X398 @ id) = (X395 @ X398)) & ! [X399 : subst > term > term] : (? [X400 : subst,X401 : subst,X402 : term] : ((sub @ (X399 @ X400 @ X402) @ X401) != (X399 @ (comp @ X400 @ X401) @ (sub @ X402 @ X401))) | ? [X403 : term] : (($true != (X397 @ id @ (X399 @ id @ X403) @ id)) & ((X397 @ id @ X403 @ id) = $true)) | ($true = (X397 @ id @ (lam @ (X399 @ sh @ one)) @ id))) & ($true != (X395 @ (lam @ X396)))) | ? [X384 : subst > term > subst > $o] : ! [X385 : term > $o] : ? [X386 : term] : ((X384 @ id @ X386 @ id) != (X385 @ X386)) | ! [X405 : term,X406 : subst > term > subst > $o] : (($true = (X406 @ id @ X405 @ id)) | ? [X407 : subst > term > term] : (! [X408 : term] : (((X406 @ id @ (X407 @ id @ X408) @ id) = $true) | ((X406 @ id @ X408 @ id) != $true)) & ($true != (X406 @ id @ (lam @ (X407 @ sh @ one)) @ id)) & ! [X409 : subst,X410 : subst,X411 : term] : ((sub @ (X407 @ X410 @ X411) @ X409) = (X407 @ (comp @ X410 @ X409) @ (sub @ X411 @ X409)))) | ? [X412 : term,X413 : term] : (((X406 @ id @ (ap @ (sub @ X412 @ id) @ X413) @ id) != $true) & ($true = (X406 @ id @ X413 @ id)) & ((X406 @ id @ X412 @ id) = $true))) | ? [X387 : term,X388 : term > $o] : (((X388 @ X387) != $true) & ! [X390 : term] : (? [X391 : term] : (((X388 @ X391) = $true) & ((X388 @ (sub @ X390 @ (push @ X391 @ id))) != $true)) | ((X388 @ (lam @ X390)) = $true)) & ! [X389 : term] : (($true != (var @ X389)) | ($true = (X388 @ X389))) & ! [X392 : term,X393 : term] : (((X388 @ X393) != $true) | ($true = (X388 @ (ap @ X393 @ X392))) | ($true != (X388 @ X392))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (axclos = $true) & ((laminj = $true) <=> ! [X167 : term,X168 : term] : ((X167 = X168) | ((lam @ X168) != (lam @ X167)))) & (! [X368 : term,X369 : term > $o,X370 : subst > term > subst > $o] : (? [X376 : term] : ((X370 @ id @ X376 @ id) != (X369 @ X376)) | ? [X377 : term] : (((X369 @ (sub @ X368 @ (push @ X377 @ id))) != $true) & ((X369 @ X377) = $true)) | ? [X371 : subst > term > term] : (($true != (X370 @ id @ (lam @ (X371 @ sh @ one)) @ id)) & ! [X372 : term] : (($true != (X370 @ id @ X372 @ id)) | ($true = (X370 @ id @ (X371 @ id @ X372) @ id))) & ! [X373 : term,X374 : subst,X375 : subst] : ((sub @ (X371 @ X375 @ X373) @ X374) = (X371 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))) | ($true = (X369 @ (lam @ X368)))) <=> (hoasinduction_lem3v2a = $true)) & ((hoasapinj2_lthm = $true) <=> (? [X476 : term,X477 : term,X478 : term,X479 : term] : (((ap @ X477 @ X476) = (ap @ X478 @ X479)) & (X476 != X479)) | ! [X480 : term,X481 : term,X482 : term,X483 : term] : ((X482 = X483) | ((ap @ (sub @ X481 @ id) @ X483) != (ap @ (sub @ X480 @ id) @ X482))))) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (? [X115 : term,X116 : subst > term > subst > $o] : (! [X122 : term] : (((X116 @ id @ X122 @ id) != $true) | ((X116 @ id @ (sub @ X115 @ (push @ X122 @ id)) @ id) = $true)) & ((X116 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id) != $true) & ! [X117 : subst > term > term] : (((X116 @ id @ (lam @ (X117 @ sh @ one)) @ id) = $true) | ? [X119 : term,X120 : subst,X121 : subst] : ((X117 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)) != (sub @ (X117 @ X121 @ X119) @ X120)) | ? [X118 : term] : (($true = (X116 @ id @ X118 @ id)) & ($true != (X116 @ id @ (X117 @ id @ X118) @ id))))) | ? [X114 : term] : ((sub @ X114 @ id) != X114) | ! [X123 : term,X124 : subst > term > subst > $o] : (($true = (X124 @ id @ (lam @ X123) @ id)) | ? [X130 : term,X131 : subst,X132 : subst,X133 : subst] : (((X124 @ X132 @ X130 @ (comp @ X131 @ X133)) != $true) & ($true = (X124 @ (comp @ X132 @ X131) @ (sub @ X130 @ X131) @ X133))) | ? [X134 : term,X135 : subst,X136 : subst,X137 : subst] : (((X124 @ X137 @ X134 @ (comp @ X136 @ X135)) = $true) & ($true != (X124 @ (comp @ X137 @ X136) @ (sub @ X134 @ X136) @ X135))) | ? [X138 : term] : (((X124 @ id @ (sub @ X123 @ (push @ X138 @ id)) @ id) != $true) & ((X124 @ id @ X138 @ id) = $true)) | ? [X125 : subst > term > term] : (! [X127 : term,X128 : subst,X129 : subst] : ((X125 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128)) = (sub @ (X125 @ X129 @ X127) @ X128)) & ! [X126 : term] : (((X124 @ id @ X126 @ id) != $true) | ((X124 @ id @ (X125 @ id @ X126) @ id) = $true)) & ($true != (X124 @ id @ (lam @ (X125 @ sh @ one)) @ id)))))) & (hoaslamnotap_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lthm_orig = $true) & (hoasapnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (? [X60 : term] : ((sub @ X60 @ id) != X60) | ! [X69 : term,X70 : subst > term > subst > $o] : (? [X72 : subst > term > term] : (! [X76 : term] : (($true != (X70 @ id @ X76 @ id)) | ((X70 @ id @ (X72 @ id @ X76) @ id) = $true)) & ($true != (X70 @ id @ (lam @ (X72 @ sh @ one)) @ id)) & ! [X73 : subst,X74 : term,X75 : subst] : ((sub @ (X72 @ X75 @ X74) @ X73) = (X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73)))) | ? [X71 : term] : (($true = (X70 @ id @ X71 @ id)) & ($true != (X70 @ id @ (sub @ X69 @ (push @ X71 @ id)) @ id))) | ((X70 @ id @ (lam @ X69) @ id) = $true)) | ? [X61 : term,X62 : subst > term > subst > $o] : (! [X68 : term] : (((X62 @ id @ (sub @ X61 @ (push @ X68 @ id)) @ id) = $true) | ((X62 @ id @ X68 @ id) != $true)) & ((X62 @ id @ (lam @ (sub @ X61 @ (push @ one @ sh))) @ id) != $true) & ! [X63 : subst > term > term] : (((X62 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true) | ? [X64 : subst,X65 : subst,X66 : term] : ((X63 @ (comp @ X65 @ X64) @ (sub @ X66 @ X64)) != (sub @ (X63 @ X65 @ X66) @ X64)) | ? [X67 : term] : (($true != (X62 @ id @ (X63 @ id @ X67) @ id)) & ($true = (X62 @ id @ X67 @ id))))))) & (! [X101 : term] : ? [X102 : subst > term > term] : ! [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) = (X102 @ X104 @ X103)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X188 : subst > term > subst > $o,X189 : term,X190 : term > $o] : (((X190 @ (lam @ X189)) = $true) | ? [X191 : subst,X192 : subst,X193 : subst,X194 : term] : (((X188 @ (comp @ X193 @ X191) @ (sub @ X194 @ X191) @ X192) != $true) & ((X188 @ X193 @ X194 @ (comp @ X191 @ X192)) = $true)) | ? [X200 : subst > term > term] : (! [X204 : term] : (((X188 @ id @ X204 @ id) != $true) | ((X188 @ id @ (X200 @ id @ X204) @ id) = $true)) & ((X188 @ id @ (lam @ (X200 @ sh @ one)) @ id) != $true) & ! [X201 : subst,X202 : subst,X203 : term] : ((X200 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201)) = (sub @ (X200 @ X202 @ X203) @ X201))) | ? [X205 : term] : ((X188 @ id @ X205 @ id) != (X190 @ X205)) | ? [X199 : term] : (((X190 @ X199) = $true) & ((X190 @ (sub @ X189 @ (push @ X199 @ id))) != $true)) | ? [X195 : subst,X196 : subst,X197 : subst,X198 : term] : (((X188 @ (comp @ X197 @ X196) @ (sub @ X198 @ X196) @ X195) = $true) & ((X188 @ X197 @ X198 @ (comp @ X196 @ X195)) != $true)))) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X349 : term] : ((sub @ X349 @ id) != X349) | ! [X350 : term,X351 : subst > term > subst > $o,X352 : term > $o] : (? [X367 : term] : (((X352 @ (sub @ X350 @ (push @ X367 @ id))) != $true) & ((X352 @ X367) = $true)) | ? [X362 : subst > term > term] : (! [X363 : term,X364 : subst,X365 : subst] : ((X362 @ (comp @ X365 @ X364) @ (sub @ X363 @ X364)) = (sub @ (X362 @ X365 @ X363) @ X364)) & ! [X366 : term] : (((X351 @ id @ (X362 @ id @ X366) @ id) = $true) | ($true != (X351 @ id @ X366 @ id))) & ((X351 @ id @ (lam @ (X362 @ sh @ one)) @ id) != $true)) | ? [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (((X351 @ (comp @ X356 @ X354) @ (sub @ X357 @ X354) @ X355) != $true) & ((X351 @ X356 @ X357 @ (comp @ X354 @ X355)) = $true)) | ? [X358 : subst,X359 : subst,X360 : term,X361 : subst] : (($true != (X351 @ X358 @ X360 @ (comp @ X359 @ X361))) & ($true = (X351 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359) @ X361))) | ? [X353 : term] : ((X351 @ id @ X353 @ id) != (X352 @ X353)) | ((X352 @ (lam @ X350)) = $true)))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lthm = $true) <=> (? [X442 : term,X443 : subst > term > subst > $o] : (! [X450 : subst,X451 : term,X452 : subst,X453 : subst] : (((X443 @ (comp @ X452 @ X453) @ (sub @ X451 @ X453) @ X450) != $true) | ($true = (X443 @ X452 @ X451 @ (comp @ X453 @ X450)))) & ! [X449 : term] : (((X443 @ id @ (sub @ X442 @ (push @ X449 @ id)) @ id) = $true) | ($true != (X443 @ id @ X449 @ id))) & ((X443 @ id @ (lam @ X442) @ id) != $true) & ! [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (($true != (X443 @ X457 @ X456 @ (comp @ X455 @ X454))) | ((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454) = $true)) & ! [X444 : subst > term > term] : (? [X445 : subst,X446 : term,X447 : subst] : ((X444 @ (comp @ X447 @ X445) @ (sub @ X446 @ X445)) != (sub @ (X444 @ X447 @ X446) @ X445)) | ($true = (X443 @ id @ (lam @ (X444 @ sh @ one)) @ id)) | ? [X448 : term] : (((X443 @ id @ X448 @ id) = $true) & ($true != (X443 @ id @ (X444 @ id @ X448) @ id))))) | ? [X429 : subst > term > subst > $o,X430 : term,X431 : term] : (! [X440 : term,X441 : term] : (((X429 @ id @ X440 @ id) != $true) | ($true != (X429 @ id @ X441 @ id)) | ($true = (X429 @ id @ (ap @ (sub @ X440 @ id) @ X441) @ id))) & ((X429 @ id @ X430 @ id) = $true) & ! [X436 : subst,X437 : subst,X438 : term,X439 : subst] : (($true != (X429 @ X437 @ X438 @ (comp @ X436 @ X439))) | ((X429 @ (comp @ X437 @ X436) @ (sub @ X438 @ X436) @ X439) = $true)) & ((X429 @ id @ X431 @ id) = $true) & ((X429 @ id @ (ap @ X431 @ X430) @ id) != $true) & ! [X432 : term,X433 : subst,X434 : subst,X435 : subst] : (((X429 @ X435 @ X432 @ (comp @ X434 @ X433)) = $true) | ((X429 @ (comp @ X435 @ X434) @ (sub @ X432 @ X434) @ X433) != $true))) | ? [X422 : term > $o,X423 : term] : (! [X424 : term] : (($true = (X422 @ X424)) | ((var @ X424) != $true)) & ! [X425 : term,X426 : term] : (($true != (X422 @ X425)) | ((X422 @ X426) != $true) | ((X422 @ (ap @ X426 @ X425)) = $true)) & ! [X427 : term] : (? [X428 : term] : (((X422 @ (sub @ X427 @ (push @ X428 @ id))) != $true) & ((X422 @ X428) = $true)) | ($true = (X422 @ (lam @ X427)))) & ((X422 @ X423) != $true)) | ! [X458 : term,X459 : subst > term > subst > $o] : (? [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (((X459 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471) != $true) & ((X459 @ X473 @ X474 @ (comp @ X472 @ X471)) = $true)) | ? [X462 : subst > term > term] : (! [X464 : subst,X465 : term,X466 : subst] : ((X462 @ (comp @ X466 @ X464) @ (sub @ X465 @ X464)) = (sub @ (X462 @ X466 @ X465) @ X464)) & ! [X463 : term] : (((X459 @ id @ (X462 @ id @ X463) @ id) = $true) | ((X459 @ id @ X463 @ id) != $true)) & ($true != (X459 @ id @ (lam @ (X462 @ sh @ one)) @ id))) | ((X459 @ id @ X458 @ id) = $true) | ? [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (((X459 @ (comp @ X469 @ X470) @ (sub @ X467 @ X470) @ X468) = $true) & ((X459 @ X469 @ X467 @ (comp @ X470 @ X468)) != $true)) | ? [X460 : term,X461 : term] : (($true = (X459 @ id @ X460 @ id)) & ($true = (X459 @ id @ X461 @ id)) & ($true != (X459 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id))) | ? [X475 : term] : (($true = (var @ (sub @ X475 @ id))) & ($true != (X459 @ id @ X475 @ id)))))) & ((var @ (d2term @ d_one)) != $true) & (hoasapinj1_gthm = $true) & ! [X271 : term] : ? [X272 : d_term] : ((d2term @ X272) = X271) & ((induction2lem_lthm = $true) <=> (? [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) != X171) | ? [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) != X169) | ? [X174 : term,X175 : term > $o] : (((X175 @ X174) != $true) & ! [X179 : term] : (($true = (X175 @ X179)) | ($true != (var @ X179))) & ! [X176 : term,X177 : term] : (($true = (X175 @ (ap @ X177 @ X176))) | ((X175 @ X177) != $true) | ((X175 @ X176) != $true)) & ! [X178 : term] : (((X175 @ X178) != $true) | ($true = (X175 @ (lam @ X178))))) | ! [X180 : term,X181 : term > $o,X182 : subst] : (? [X183 : term,X184 : term] : (($true = (X181 @ X183)) & ($true != (X181 @ (ap @ X184 @ X183))) & ((X181 @ X184) = $true)) | ? [X185 : term] : (((var @ X185) = $true) & ((X181 @ (sub @ X185 @ X182)) != $true)) | ? [X186 : term] : (! [X187 : term] : (((X181 @ (sub @ X186 @ (push @ X187 @ id))) = $true) | ((X181 @ X187) != $true)) & ($true != (X181 @ (lam @ X186)))) | ((X181 @ (sub @ X180 @ X182)) = $true)) | ? [X173 : subst] : ((comp @ X173 @ id) != X173))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem1_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22)))) & (hoasinduction_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (! [X531 : subst] : ((comp @ id @ X531) = X531) <=> (axidl = $true)) & (hoaslamnotvar_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (? [X79 : subst,X80 : term] : ((comp @ sh @ (push @ X80 @ X79)) != X79) | ? [X81 : term,X82 : term] : (((lam @ X81) = (lam @ X82)) & (X81 != X82)) | ! [X83 : term,X84 : subst > term > term,X85 : subst,X86 : subst > term > term] : (((lam @ (X84 @ sh @ one)) != (lam @ (X86 @ sh @ one))) | ? [X90 : subst,X91 : subst,X92 : term] : ((X84 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91)) != (sub @ (X84 @ X90 @ X92) @ X91)) | ((X84 @ X85 @ X83) = (X86 @ X85 @ X83)) | ? [X87 : subst,X88 : subst,X89 : term] : ((X86 @ (comp @ X87 @ X88) @ (sub @ X89 @ X88)) != (sub @ (X86 @ X87 @ X89) @ X88))) | ? [X77 : subst,X78 : term] : ((sub @ one @ (push @ X78 @ X77)) != X78))) & (! [X251 : term,X252 : term > $o] : (? [X257 : term] : (($true != (X252 @ X257)) & ($true = (var @ X257))) | ? [X253 : term,X254 : term] : (($true != (X252 @ (ap @ X254 @ X253))) & ((X252 @ X254) = $true) & ($true = (X252 @ X253))) | ? [X255 : term] : (((X252 @ (lam @ X255)) != $true) & ! [X256 : term] : (($true != (X252 @ X256)) | ((X252 @ (sub @ X255 @ (push @ X256 @ id))) = $true))) | ($true = (X252 @ X251))) <=> (induction2 = $true)) & (! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) <=> (axvarcons = $true)) & (ulamvarsh = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lthm_3 = $true) <=> (! [X313 : term,X314 : subst > term > subst > $o] : (? [X319 : term] : (((X314 @ id @ X319 @ id) != $true) & ($true = (var @ (sub @ X319 @ id)))) | ? [X326 : subst > term > term] : (((X314 @ id @ (lam @ (X326 @ sh @ one)) @ id) != $true) & ! [X330 : term] : (((X314 @ id @ X330 @ id) != $true) | ((X314 @ id @ (X326 @ id @ X330) @ id) = $true)) & ! [X327 : subst,X328 : term,X329 : subst] : ((X326 @ (comp @ X327 @ X329) @ (sub @ X328 @ X329)) = (sub @ (X326 @ X327 @ X328) @ X329))) | ((X314 @ id @ X313 @ id) = $true) | ? [X322 : subst,X323 : subst,X324 : term,X325 : subst] : (((X314 @ X322 @ X324 @ (comp @ X323 @ X325)) = $true) & ((X314 @ (comp @ X322 @ X323) @ (sub @ X324 @ X323) @ X325) != $true)) | ? [X315 : subst,X316 : subst,X317 : term,X318 : subst] : (($true != (X314 @ X318 @ X317 @ (comp @ X315 @ X316))) & ($true = (X314 @ (comp @ X318 @ X315) @ (sub @ X317 @ X315) @ X316))) | ? [X320 : term,X321 : term] : (((X314 @ id @ (ap @ (sub @ X320 @ id) @ X321) @ id) != $true) & ($true = (X314 @ id @ X321 @ id)) & ((X314 @ id @ X320 @ id) = $true))) | ? [X303 : subst > term > subst > $o,X304 : term,X305 : term > $o] : (((X305 @ (lam @ X304)) != $true) & ! [X312 : term] : ((X303 @ id @ X312 @ id) = (X305 @ X312)) & ! [X306 : subst > term > term] : (? [X307 : term] : (($true = (X303 @ id @ X307 @ id)) & ($true != (X303 @ id @ (X306 @ id @ X307) @ id))) | ? [X308 : term,X309 : subst,X310 : subst] : ((X306 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309)) != (sub @ (X306 @ X310 @ X308) @ X309)) | ($true = (X303 @ id @ (lam @ (X306 @ sh @ one)) @ id))) & ! [X311 : term] : (((X305 @ X311) != $true) | ((X305 @ (sub @ X304 @ (push @ X311 @ id))) = $true))) | ? [X292 : subst > term > subst > $o] : ! [X293 : term > $o] : ? [X294 : term] : ((X292 @ id @ X294 @ id) != (X293 @ X294)) | ? [X302 : term] : ((sub @ X302 @ id) != X302) | ? [X295 : term > $o,X296 : term] : (! [X297 : term,X298 : term] : (((X295 @ X297) != $true) | ((X295 @ X298) != $true) | ($true = (X295 @ (ap @ X297 @ X298)))) & ! [X299 : term] : (($true != (var @ X299)) | ((X295 @ X299) = $true)) & ! [X300 : term] : (? [X301 : term] : (((X295 @ X301) = $true) & ($true != (X295 @ (sub @ X300 @ (push @ X301 @ id))))) | ((X295 @ (lam @ X300)) = $true)) & ((X295 @ X296) != $true)))) & (apnotvar = $true) & ((hoasinduction_lem3 = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term] : (((X235 @ id @ (lam @ X236) @ id) = $true) | ? [X247 : subst,X248 : subst,X249 : subst,X250 : term] : (((X235 @ X248 @ X250 @ (comp @ X249 @ X247)) = $true) & ($true != (X235 @ (comp @ X248 @ X249) @ (sub @ X250 @ X249) @ X247))) | ? [X246 : term] : (($true != (X235 @ id @ (sub @ X236 @ (push @ X246 @ id)) @ id)) & ((X235 @ id @ X246 @ id) = $true)) | ? [X237 : subst,X238 : subst,X239 : subst,X240 : term] : (((X235 @ X238 @ X240 @ (comp @ X239 @ X237)) != $true) & ($true = (X235 @ (comp @ X238 @ X239) @ (sub @ X240 @ X239) @ X237))) | ? [X241 : subst > term > term] : (((X235 @ id @ (lam @ (X241 @ sh @ one)) @ id) != $true) & ! [X245 : term] : (($true = (X235 @ id @ (X241 @ id @ X245) @ id)) | ((X235 @ id @ X245 @ id) != $true)) & ! [X242 : term,X243 : subst,X244 : subst] : ((X241 @ (comp @ X244 @ X243) @ (sub @ X242 @ X243)) = (sub @ (X241 @ X244 @ X242) @ X243))))) & (! [X264 : term] : ((sub @ X264 @ id) = X264) <=> (axvarid = $true)) & (hoasinduction_lem2_gthm = $true) & ((termmset_lthm = $true) <=> (? [X488 : term] : ((sub @ X488 @ id) != X488) | ! [X489 : term] : ((sub @ X489 @ id) = X489))) & (! [X258 : term,X259 : subst > term > term,X260 : term] : (((lam @ (X259 @ sh @ one)) != (ap @ (sub @ X260 @ id) @ X258)) | ? [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) != (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262)))) <=> (hoaslamnotap = $true)) & (hoasinduction_lem1v2 = $true) & (pushprop_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop = $true) & (axmap = $true) & ((pushprop_lem1v2 = $true) <=> ! [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (($true != (X217 @ X215)) | ? [X219 : term] : ((X216 @ X219) != (X217 @ (sub @ X219 @ (push @ X215 @ X218)))) | ((X216 @ one) = $true))) & (! [X212 : term] : ? [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) = (X213 @ sh @ one)) <=> (hoasinduction_lem3b = $true)) & (! [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) = X335) <=> (axscons = $true)) & (hoasinduction_lem3a_gthm = $true) & ((induction2lem = $true) <=> ! [X139 : term,X140 : term > $o,X141 : subst] : (? [X145 : term,X146 : term] : (((X140 @ X146) = $true) & ($true != (X140 @ (ap @ X145 @ X146))) & ((X140 @ X145) = $true)) | ? [X142 : term] : (((var @ X142) = $true) & ((X140 @ (sub @ X142 @ X141)) != $true)) | ? [X143 : term] : (! [X144 : term] : (((X140 @ X144) != $true) | ((X140 @ (sub @ X143 @ (push @ X144 @ id))) = $true)) & ($true != (X140 @ (lam @ X143)))) | ($true = (X140 @ (sub @ X139 @ X141))))) & (pushprop_lem0_gthm = $true) & (axassoc = $true) & (hoaslamnotvar = $true) & (! [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (? [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) != (sub @ (X496 @ X499 @ X500) @ X501)) | ? [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) != (sub @ (X498 @ X502 @ X503) @ X504)) | ((X498 @ X497 @ X495) = (X496 @ X497 @ X495)) | ((lam @ (X498 @ sh @ one)) != (lam @ (X496 @ sh @ one)))) <=> (hoaslaminj = $true)) & ((d2subst @ d_id) = sh) & (axvarshift = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X25 : subst > term > subst > $o,X26 : term] : (? [X27 : term] : (($true = (X25 @ id @ X27 @ id)) & ($true != (X25 @ id @ (sub @ X26 @ (push @ X27 @ id)) @ id))) | ? [X28 : term,X29 : subst > term > term] : (((X25 @ id @ (lam @ (X29 @ sh @ one)) @ id) != $true) & ! [X34 : subst,X35 : subst,X36 : term] : ((sub @ X28 @ (push @ (sub @ X36 @ X34) @ (comp @ X35 @ X34))) = (X29 @ (comp @ X35 @ X34) @ (sub @ X36 @ X34))) & ! [X30 : term] : (((X25 @ id @ (X29 @ id @ X30) @ id) = $true) | ((X25 @ id @ X30 @ id) != $true)) & ! [X31 : subst,X32 : term,X33 : subst] : ((sub @ (sub @ X28 @ (push @ X32 @ X33)) @ X31) = (sub @ (X29 @ X33 @ X32) @ X31))) | ($true = (X25 @ id @ (lam @ (sub @ X26 @ (push @ one @ sh))) @ id)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & (! [X490 : term > $o,X491 : term,X492 : subst] : ? [X493 : term > $o] : ! [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) = (X493 @ X494)) <=> (pushprop_lem0 = $true)) & (ulamvarind = $true) & ((induction2_lthm = $true) <=> (? [X152 : term,X153 : subst,X154 : term > $o] : (! [X155 : term] : (($true = (X154 @ (lam @ X155))) | ? [X156 : term] : (((X154 @ X156) = $true) & ((X154 @ (sub @ X155 @ (push @ X156 @ id))) != $true))) & ! [X159 : term] : (((var @ X159) != $true) | ((X154 @ (sub @ X159 @ X153)) = $true)) & ! [X157 : term,X158 : term] : (($true != (X154 @ X158)) | ((X154 @ X157) != $true) | ((X154 @ (ap @ X158 @ X157)) = $true)) & ((X154 @ (sub @ X152 @ X153)) != $true)) | ? [X151 : term] : ((sub @ X151 @ id) != X151) | ! [X160 : term,X161 : term > $o] : (? [X165 : term] : (($true != (X161 @ (lam @ X165))) & ! [X166 : term] : (((X161 @ X166) != $true) | ($true = (X161 @ (sub @ X165 @ (push @ X166 @ id)))))) | ? [X162 : term] : (($true != (X161 @ X162)) & ($true = (var @ X162))) | ? [X163 : term,X164 : term] : (((X161 @ (ap @ X164 @ X163)) != $true) & ($true = (X161 @ X163)) & ((X161 @ X164) = $true)) | ((X161 @ X160) = $true)))) & (axabs = $true) & ((apinj1 = $true) <=> ! [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 = X487) | ((ap @ X485 @ X484) != (ap @ X487 @ X486)))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (((X3 @ X5) != $true) | ? [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) != (X2 @ X6)) | ((X2 @ one) = $true)) | ? [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) != X0)) <=> (pushprop_lem1v2_lthm = $true)) & ((d2subst @ d_id) = id) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X214 : subst] : ((comp @ X214 @ id) = X214) <=> (axidr = $true)) & (! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) = (sub @ X147 @ (push @ X149 @ X150))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasapinj1 = $true) <=> ! [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 = X380) | ((ap @ (sub @ X379 @ id) @ X378) != (ap @ (sub @ X380 @ id) @ X381)))) & (pushprop_lem2v2 = $true) & (! [X532 : term,X533 : subst > term > subst > $o] : (($true = (X533 @ id @ (lam @ X532) @ id)) | ? [X534 : subst > term > term] : (((X533 @ id @ (lam @ (X534 @ sh @ one)) @ id) != $true) & ! [X535 : subst,X536 : subst,X537 : term] : ((sub @ (X534 @ X535 @ X537) @ X536) = (X534 @ (comp @ X535 @ X536) @ (sub @ X537 @ X536))) & ! [X538 : term] : (((X533 @ id @ X538 @ id) != $true) | ((X533 @ id @ (X534 @ id @ X538) @ id) = $true))) | ? [X539 : term] : (((X533 @ id @ X539 @ id) = $true) & ((X533 @ id @ (sub @ X532 @ (push @ X539 @ id)) @ id) != $true))) <=> (hoasinduction_lem3a = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((? [X332 : subst] : ((comp @ X332 @ id) != X332) | (! [X333 : subst] : ((comp @ X333 @ id) = X333) & ! [X334 : subst] : ((comp @ id @ X334) = X334)) | ? [X331 : subst] : ((comp @ id @ X331) != X331)) <=> (substmonoid_lthm = $true))),
% 2.43/1.93 inference(flattening,[status(thm)],[f13])).
% 2.43/1.93 thf(f13,plain,(
% 2.43/1.93 (hoasinduction_lem1 = $true) & (lamnotap != $true) & (! [X531 : subst] : ((comp @ id @ X531) = X531) <=> (axidl = $true)) & ((hoasinduction_lthm_3 = $true) <=> ((((! [X313 : term,X314 : subst > term > subst > $o] : (? [X319 : term] : (((X314 @ id @ X319 @ id) != $true) & ($true = (var @ (sub @ X319 @ id)))) | ? [X326 : subst > term > term] : (((X314 @ id @ (lam @ (X326 @ sh @ one)) @ id) != $true) & ! [X330 : term] : (((X314 @ id @ X330 @ id) != $true) | ((X314 @ id @ (X326 @ id @ X330) @ id) = $true)) & ! [X327 : subst,X328 : term,X329 : subst] : ((X326 @ (comp @ X327 @ X329) @ (sub @ X328 @ X329)) = (sub @ (X326 @ X327 @ X328) @ X329))) | ((X314 @ id @ X313 @ id) = $true) | ? [X322 : subst,X323 : subst,X324 : term,X325 : subst] : (((X314 @ X322 @ X324 @ (comp @ X323 @ X325)) = $true) & ((X314 @ (comp @ X322 @ X323) @ (sub @ X324 @ X323) @ X325) != $true)) | ? [X315 : subst,X316 : subst,X317 : term,X318 : subst] : (($true != (X314 @ X318 @ X317 @ (comp @ X315 @ X316))) & ($true = (X314 @ (comp @ X318 @ X315) @ (sub @ X317 @ X315) @ X316))) | ? [X320 : term,X321 : term] : (((X314 @ id @ (ap @ (sub @ X320 @ id) @ X321) @ id) != $true) & ($true = (X314 @ id @ X321 @ id)) & ((X314 @ id @ X320 @ id) = $true))) | ? [X303 : subst > term > subst > $o,X304 : term,X305 : term > $o] : (((X305 @ (lam @ X304)) != $true) & ! [X312 : term] : ((X303 @ id @ X312 @ id) = (X305 @ X312)) & ! [X306 : subst > term > term] : (? [X307 : term] : (($true = (X303 @ id @ X307 @ id)) & ($true != (X303 @ id @ (X306 @ id @ X307) @ id))) | ? [X308 : term,X309 : subst,X310 : subst] : ((X306 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309)) != (sub @ (X306 @ X310 @ X308) @ X309)) | ($true = (X303 @ id @ (lam @ (X306 @ sh @ one)) @ id))) & ! [X311 : term] : (((X305 @ X311) != $true) | ((X305 @ (sub @ X304 @ (push @ X311 @ id))) = $true)))) | ? [X302 : term] : ((sub @ X302 @ id) != X302)) | ? [X295 : term > $o,X296 : term] : (! [X297 : term,X298 : term] : (((X295 @ X297) != $true) | ((X295 @ X298) != $true) | ($true = (X295 @ (ap @ X297 @ X298)))) & ! [X299 : term] : (($true != (var @ X299)) | ((X295 @ X299) = $true)) & ! [X300 : term] : (? [X301 : term] : (((X295 @ X301) = $true) & ($true != (X295 @ (sub @ X300 @ (push @ X301 @ id))))) | ((X295 @ (lam @ X300)) = $true)) & ((X295 @ X296) != $true))) | ? [X292 : subst > term > subst > $o] : ! [X293 : term > $o] : ? [X294 : term] : ((X292 @ id @ X294 @ id) != (X293 @ X294)))) & (pushprop_lem3v2 = $true) & (hoasinduction_gthm = $true) & (! [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (($true = (X267 @ (sub @ X265 @ (push @ X268 @ X269)))) | ((X266 @ X265) != $true)) <=> (pushprop_lem1 = $true)) & ! [X96 : d_term] : (d_one = X96) & (! [X368 : term,X369 : term > $o,X370 : subst > term > subst > $o] : (? [X376 : term] : ((X370 @ id @ X376 @ id) != (X369 @ X376)) | ? [X377 : term] : (((X369 @ (sub @ X368 @ (push @ X377 @ id))) != $true) & ((X369 @ X377) = $true)) | ? [X371 : subst > term > term] : (($true != (X370 @ id @ (lam @ (X371 @ sh @ one)) @ id)) & ! [X372 : term] : (($true != (X370 @ id @ X372 @ id)) | ($true = (X370 @ id @ (X371 @ id @ X372) @ id))) & ! [X373 : term,X374 : subst,X375 : subst] : ((sub @ (X371 @ X375 @ X373) @ X374) = (X371 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))) | ($true = (X369 @ (lam @ X368)))) <=> (hoasinduction_lem3v2a = $true)) & (ulamvar1 != $true) & (! [X147 : term] : ? [X148 : subst > term > term] : ! [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) = (sub @ X147 @ (push @ X149 @ X150))) <=> (hoasinduction_lem3v2_f = $true)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (((! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true = (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) | ((X18 @ X15) != $true)) | ? [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) != X12)) | ? [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) != X11)) <=> (pushprop_lem1_lthm = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_gthm = $true) & (induction2lem_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X220 : term > $o,X221 : term,X222 : subst > term > subst > $o,X223 : term] : (((X220 @ X223) != $true) | ? [X233 : term,X234 : term] : (($true != (X222 @ id @ (ap @ (sub @ X234 @ id) @ X233) @ id)) & ((X222 @ id @ X234 @ id) = $true) & ($true = (X222 @ id @ X233 @ id))) | ? [X228 : subst,X229 : subst,X230 : subst,X231 : term] : (($true != (X222 @ X230 @ X231 @ (comp @ X228 @ X229))) & ((X222 @ (comp @ X230 @ X228) @ (sub @ X231 @ X228) @ X229) = $true)) | ((X220 @ X221) != $true) | ? [X224 : subst,X225 : term,X226 : subst,X227 : subst] : (((X222 @ (comp @ X224 @ X227) @ (sub @ X225 @ X227) @ X226) != $true) & ((X222 @ X224 @ X225 @ (comp @ X227 @ X226)) = $true)) | ((X220 @ (ap @ X221 @ X223)) = $true) | ? [X232 : term] : ((X220 @ X232) != (X222 @ id @ X232 @ id)))) & ((apinj1 = $true) <=> ! [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 = X487) | ((ap @ X485 @ X484) != (ap @ X487 @ X486)))) & (! [X251 : term,X252 : term > $o] : (? [X257 : term] : (($true != (X252 @ X257)) & ($true = (var @ X257))) | ? [X253 : term,X254 : term] : (($true != (X252 @ (ap @ X254 @ X253))) & ((X252 @ X254) = $true) & ($true = (X252 @ X253))) | ? [X255 : term] : (((X252 @ (lam @ X255)) != $true) & ! [X256 : term] : (($true != (X252 @ X256)) | ((X252 @ (sub @ X255 @ (push @ X256 @ id))) = $true))) | ($true = (X252 @ X251))) <=> (induction2 = $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lthm_orig = $true) & (! [X93 : subst > term > subst > $o] : ? [X94 : term > $o] : ! [X95 : term] : ((X94 @ X95) = (X93 @ id @ X95 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (apnotvar = $true) & (pushprop_lem3v2_lthm = $true) & (! [X258 : term,X259 : subst > term > term,X260 : term] : (((lam @ (X259 @ sh @ one)) != (ap @ (sub @ X260 @ id) @ X258)) | ? [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) != (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262)))) <=> (hoaslamnotap = $true)) & (hoasapnotvar_gthm = $true) & (axassoc = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X338 : term] : ? [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) = (X339 @ sh @ one))) & ((termmset = $true) <=> ! [X57 : term] : ((sub @ X57 @ id) = X57)) & (! [X505 : subst > term > subst > $o,X506 : term] : (? [X508 : subst > term > term] : (((X505 @ id @ (lam @ (X508 @ sh @ one)) @ id) != $true) & ! [X510 : subst,X511 : subst,X512 : term] : ((X508 @ (comp @ X511 @ X510) @ (sub @ X512 @ X510)) = (sub @ (X508 @ X511 @ X512) @ X510)) & ! [X509 : term] : (($true != (X505 @ id @ X509 @ id)) | ($true = (X505 @ id @ (X508 @ id @ X509) @ id)))) | ? [X507 : term] : (($true = (X505 @ id @ X507 @ id)) & ((X505 @ id @ (sub @ X506 @ (push @ X507 @ id)) @ id) != $true)) | ((X505 @ id @ (lam @ (sub @ X506 @ (push @ one @ sh))) @ id) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & (hoasinduction_lem1v2 = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (axclos = $true) & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) != (d2subst @ X55)) | (X55 = X56)) & ((termmset_lthm = $true) <=> (? [X488 : term] : ((sub @ X488 @ id) != X488) | ! [X489 : term] : ((sub @ X489 @ id) = X489))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X532 : term,X533 : subst > term > subst > $o] : (($true = (X533 @ id @ (lam @ X532) @ id)) | ? [X534 : subst > term > term] : (((X533 @ id @ (lam @ (X534 @ sh @ one)) @ id) != $true) & ! [X535 : subst,X536 : subst,X537 : term] : ((sub @ (X534 @ X535 @ X537) @ X536) = (X534 @ (comp @ X535 @ X536) @ (sub @ X537 @ X536))) & ! [X538 : term] : (((X533 @ id @ X538 @ id) != $true) | ((X533 @ id @ (X534 @ id @ X538) @ id) = $true))) | ? [X539 : term] : (((X533 @ id @ X539 @ id) = $true) & ((X533 @ id @ (sub @ X532 @ (push @ X539 @ id)) @ id) != $true))) <=> (hoasinduction_lem3a = $true)) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) = X273) <=> (axshiftcons = $true)) & (! [X214 : subst] : ((comp @ X214 @ id) = X214) <=> (axidr = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X25 : subst > term > subst > $o,X26 : term] : (? [X27 : term] : (($true = (X25 @ id @ X27 @ id)) & ($true != (X25 @ id @ (sub @ X26 @ (push @ X27 @ id)) @ id))) | ? [X28 : term,X29 : subst > term > term] : (((X25 @ id @ (lam @ (X29 @ sh @ one)) @ id) != $true) & ! [X34 : subst,X35 : subst,X36 : term] : ((sub @ X28 @ (push @ (sub @ X36 @ X34) @ (comp @ X35 @ X34))) = (X29 @ (comp @ X35 @ X34) @ (sub @ X36 @ X34))) & ! [X30 : term] : (((X25 @ id @ (X29 @ id @ X30) @ id) = $true) | ((X25 @ id @ X30 @ id) != $true)) & ! [X31 : subst,X32 : term,X33 : subst] : ((sub @ (sub @ X28 @ (push @ X32 @ X33)) @ X31) = (sub @ (X29 @ X33 @ X32) @ X31))) | ($true = (X25 @ id @ (lam @ (sub @ X26 @ (push @ one @ sh))) @ id)))) & ((! [X59 : subst] : ((comp @ id @ X59) = X59) & ! [X58 : subst] : ((comp @ X58 @ id) = X58)) <=> (substmonoid = $true)) & ((hoasinduction_lthm = $true) <=> (((! [X458 : term,X459 : subst > term > subst > $o] : (? [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (((X459 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471) != $true) & ((X459 @ X473 @ X474 @ (comp @ X472 @ X471)) = $true)) | ? [X462 : subst > term > term] : (! [X464 : subst,X465 : term,X466 : subst] : ((X462 @ (comp @ X466 @ X464) @ (sub @ X465 @ X464)) = (sub @ (X462 @ X466 @ X465) @ X464)) & ! [X463 : term] : (((X459 @ id @ (X462 @ id @ X463) @ id) = $true) | ((X459 @ id @ X463 @ id) != $true)) & ($true != (X459 @ id @ (lam @ (X462 @ sh @ one)) @ id))) | ((X459 @ id @ X458 @ id) = $true) | ? [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (((X459 @ (comp @ X469 @ X470) @ (sub @ X467 @ X470) @ X468) = $true) & ((X459 @ X469 @ X467 @ (comp @ X470 @ X468)) != $true)) | ? [X460 : term,X461 : term] : (($true = (X459 @ id @ X460 @ id)) & ($true = (X459 @ id @ X461 @ id)) & ($true != (X459 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id))) | ? [X475 : term] : (($true = (var @ (sub @ X475 @ id))) & ($true != (X459 @ id @ X475 @ id)))) | ? [X442 : term,X443 : subst > term > subst > $o] : (! [X450 : subst,X451 : term,X452 : subst,X453 : subst] : (((X443 @ (comp @ X452 @ X453) @ (sub @ X451 @ X453) @ X450) != $true) | ($true = (X443 @ X452 @ X451 @ (comp @ X453 @ X450)))) & ! [X449 : term] : (((X443 @ id @ (sub @ X442 @ (push @ X449 @ id)) @ id) = $true) | ($true != (X443 @ id @ X449 @ id))) & ((X443 @ id @ (lam @ X442) @ id) != $true) & ! [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (($true != (X443 @ X457 @ X456 @ (comp @ X455 @ X454))) | ((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454) = $true)) & ! [X444 : subst > term > term] : (? [X445 : subst,X446 : term,X447 : subst] : ((X444 @ (comp @ X447 @ X445) @ (sub @ X446 @ X445)) != (sub @ (X444 @ X447 @ X446) @ X445)) | ($true = (X443 @ id @ (lam @ (X444 @ sh @ one)) @ id)) | ? [X448 : term] : (((X443 @ id @ X448 @ id) = $true) & ($true != (X443 @ id @ (X444 @ id @ X448) @ id)))))) | ? [X429 : subst > term > subst > $o,X430 : term,X431 : term] : (! [X440 : term,X441 : term] : (((X429 @ id @ X440 @ id) != $true) | ($true != (X429 @ id @ X441 @ id)) | ($true = (X429 @ id @ (ap @ (sub @ X440 @ id) @ X441) @ id))) & ((X429 @ id @ X430 @ id) = $true) & ! [X436 : subst,X437 : subst,X438 : term,X439 : subst] : (($true != (X429 @ X437 @ X438 @ (comp @ X436 @ X439))) | ((X429 @ (comp @ X437 @ X436) @ (sub @ X438 @ X436) @ X439) = $true)) & ((X429 @ id @ X431 @ id) = $true) & ((X429 @ id @ (ap @ X431 @ X430) @ id) != $true) & ! [X432 : term,X433 : subst,X434 : subst,X435 : subst] : (((X429 @ X435 @ X432 @ (comp @ X434 @ X433)) = $true) | ((X429 @ (comp @ X435 @ X434) @ (sub @ X432 @ X434) @ X433) != $true)))) | ? [X422 : term > $o,X423 : term] : (! [X424 : term] : (($true = (X422 @ X424)) | ((var @ X424) != $true)) & ! [X425 : term,X426 : term] : (($true != (X422 @ X425)) | ((X422 @ X426) != $true) | ((X422 @ (ap @ X426 @ X425)) = $true)) & ! [X427 : term] : (? [X428 : term] : (((X422 @ (sub @ X427 @ (push @ X428 @ id))) != $true) & ((X422 @ X428) = $true)) | ($true = (X422 @ (lam @ X427)))) & ((X422 @ X423) != $true)))) & (! [X212 : term] : ? [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) = (X213 @ sh @ one)) <=> (hoasinduction_lem3b = $true)) & (axmap = $true) & (hoasinduction_lem1v2_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (($true != (X217 @ X215)) | ? [X219 : term] : ((X216 @ X219) != (X217 @ (sub @ X219 @ (push @ X215 @ X218)))) | ((X216 @ one) = $true))) & (axvarshift = $true) & ! [X270 : d_subst] : (d_id = X270) & (! [X490 : term > $o,X491 : term,X492 : subst] : ? [X493 : term > $o] : ! [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) = (X493 @ X494)) <=> (pushprop_lem0 = $true)) & (pushprop = $true) & (hoasinduction_lem1_gthm = $true) & (hoaslamnotvar_lthm = $true) & ((induction2_lthm = $true) <=> ((! [X160 : term,X161 : term > $o] : (? [X165 : term] : (($true != (X161 @ (lam @ X165))) & ! [X166 : term] : (((X161 @ X166) != $true) | ($true = (X161 @ (sub @ X165 @ (push @ X166 @ id)))))) | ? [X162 : term] : (($true != (X161 @ X162)) & ($true = (var @ X162))) | ? [X163 : term,X164 : term] : (((X161 @ (ap @ X164 @ X163)) != $true) & ($true = (X161 @ X163)) & ((X161 @ X164) = $true)) | ((X161 @ X160) = $true)) | ? [X152 : term,X153 : subst,X154 : term > $o] : (! [X155 : term] : (($true = (X154 @ (lam @ X155))) | ? [X156 : term] : (((X154 @ X156) = $true) & ((X154 @ (sub @ X155 @ (push @ X156 @ id))) != $true))) & ! [X159 : term] : (((var @ X159) != $true) | ((X154 @ (sub @ X159 @ X153)) = $true)) & ! [X157 : term,X158 : term] : (($true != (X154 @ X158)) | ((X154 @ X157) != $true) | ((X154 @ (ap @ X158 @ X157)) = $true)) & ((X154 @ (sub @ X152 @ X153)) != $true))) | ? [X151 : term] : ((sub @ X151 @ id) != X151))) & (hoasapnotvar = $true) & (! [X105 : term,X106 : subst > term > subst > $o] : (? [X107 : subst > term > term] : (! [X108 : term] : (($true = (X106 @ id @ (X107 @ id @ X108) @ id)) | ($true != (X106 @ id @ X108 @ id))) & ((X106 @ id @ (lam @ (X107 @ sh @ one)) @ id) != $true) & ! [X109 : subst,X110 : term,X111 : subst] : ((X107 @ (comp @ X111 @ X109) @ (sub @ X110 @ X109)) = (sub @ (X107 @ X111 @ X110) @ X109))) | ($true = (X106 @ id @ X105 @ id)) | ? [X112 : term,X113 : term] : (((X106 @ id @ X113 @ id) = $true) & ((X106 @ id @ X112 @ id) = $true) & ($true != (X106 @ id @ (ap @ (sub @ X113 @ id) @ X112) @ id)))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X275 : term,X276 : term] : (((sub @ X276 @ sh) != (sub @ X275 @ sh)) | (X275 = X276)) <=> (shinj = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term] : (((X235 @ id @ (lam @ X236) @ id) = $true) | ? [X247 : subst,X248 : subst,X249 : subst,X250 : term] : (((X235 @ X248 @ X250 @ (comp @ X249 @ X247)) = $true) & ($true != (X235 @ (comp @ X248 @ X249) @ (sub @ X250 @ X249) @ X247))) | ? [X246 : term] : (($true != (X235 @ id @ (sub @ X236 @ (push @ X246 @ id)) @ id)) & ((X235 @ id @ X246 @ id) = $true)) | ? [X237 : subst,X238 : subst,X239 : subst,X240 : term] : (((X235 @ X238 @ X240 @ (comp @ X239 @ X237)) != $true) & ($true = (X235 @ (comp @ X238 @ X239) @ (sub @ X240 @ X239) @ X237))) | ? [X241 : subst > term > term] : (((X235 @ id @ (lam @ (X241 @ sh @ one)) @ id) != $true) & ! [X245 : term] : (($true = (X235 @ id @ (X241 @ id @ X245) @ id)) | ((X235 @ id @ X245 @ id) != $true)) & ! [X242 : term,X243 : subst,X244 : subst] : ((X241 @ (comp @ X244 @ X243) @ (sub @ X242 @ X243)) = (sub @ (X241 @ X244 @ X242) @ X243))))) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X188 : subst > term > subst > $o,X189 : term,X190 : term > $o] : (((X190 @ (lam @ X189)) = $true) | ? [X191 : subst,X192 : subst,X193 : subst,X194 : term] : (((X188 @ (comp @ X193 @ X191) @ (sub @ X194 @ X191) @ X192) != $true) & ((X188 @ X193 @ X194 @ (comp @ X191 @ X192)) = $true)) | ? [X200 : subst > term > term] : (! [X204 : term] : (((X188 @ id @ X204 @ id) != $true) | ((X188 @ id @ (X200 @ id @ X204) @ id) = $true)) & ((X188 @ id @ (lam @ (X200 @ sh @ one)) @ id) != $true) & ! [X201 : subst,X202 : subst,X203 : term] : ((X200 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201)) = (sub @ (X200 @ X202 @ X203) @ X201))) | ? [X205 : term] : ((X188 @ id @ X205 @ id) != (X190 @ X205)) | ? [X199 : term] : (((X190 @ X199) = $true) & ((X190 @ (sub @ X189 @ (push @ X199 @ id))) != $true)) | ? [X195 : subst,X196 : subst,X197 : subst,X198 : term] : (((X188 @ (comp @ X197 @ X196) @ (sub @ X198 @ X196) @ X195) = $true) & ((X188 @ X197 @ X198 @ (comp @ X196 @ X195)) != $true)))) & (! [X264 : term] : ((sub @ X264 @ id) = X264) <=> (axvarid = $true)) & (pushprop_gthm = $true) & (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) & (hoasinduction_lem2v2_gthm = $true) & (lamnotvar = $true) & (! [X101 : term] : ? [X102 : subst > term > term] : ! [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) = (X102 @ X104 @ X103)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X518 : subst > term > subst > $o,X519 : term,X520 : term] : (((X518 @ id @ X519 @ id) != $true) | ((X518 @ id @ (ap @ X519 @ X520) @ id) = $true) | ($true != (X518 @ id @ X520 @ id)) | ? [X521 : subst,X522 : subst,X523 : term,X524 : subst] : (($true != (X518 @ X522 @ X523 @ (comp @ X524 @ X521))) & ($true = (X518 @ (comp @ X522 @ X524) @ (sub @ X523 @ X524) @ X521))) | ? [X525 : term,X526 : term] : (($true = (X518 @ id @ X525 @ id)) & ($true = (X518 @ id @ X526 @ id)) & ((X518 @ id @ (ap @ (sub @ X525 @ id) @ X526) @ id) != $true)) | ? [X527 : subst,X528 : subst,X529 : term,X530 : subst] : (($true = (X518 @ X528 @ X529 @ (comp @ X530 @ X527))) & ($true != (X518 @ (comp @ X528 @ X530) @ (sub @ X529 @ X530) @ X527))))) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X282 : term > $o,X283 : subst > term > subst > $o,X284 : term] : (? [X285 : subst > term > term] : (! [X286 : term] : (($true = (X283 @ id @ (X285 @ id @ X286) @ id)) | ($true != (X283 @ id @ X286 @ id))) & ($true != (X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) & ! [X287 : term,X288 : subst,X289 : subst] : ((X285 @ (comp @ X289 @ X288) @ (sub @ X287 @ X288)) = (sub @ (X285 @ X289 @ X287) @ X288))) | ((X282 @ (lam @ X284)) = $true) | ? [X291 : term] : (((X282 @ X291) = $true) & ($true != (X282 @ (sub @ X284 @ (push @ X291 @ id))))) | ? [X290 : term] : ((X282 @ X290) != (X283 @ id @ X290 @ id))) | ? [X281 : term] : ((sub @ X281 @ id) != X281)) | ? [X277 : term] : ! [X278 : subst > term > term] : ? [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) != (X278 @ X280 @ X279)))) & (ulamvarind = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ((((! [X405 : term,X406 : subst > term > subst > $o] : (($true = (X406 @ id @ X405 @ id)) | ? [X407 : subst > term > term] : (! [X408 : term] : (((X406 @ id @ (X407 @ id @ X408) @ id) = $true) | ((X406 @ id @ X408 @ id) != $true)) & ($true != (X406 @ id @ (lam @ (X407 @ sh @ one)) @ id)) & ! [X409 : subst,X410 : subst,X411 : term] : ((sub @ (X407 @ X410 @ X411) @ X409) = (X407 @ (comp @ X410 @ X409) @ (sub @ X411 @ X409)))) | ? [X412 : term,X413 : term] : (((X406 @ id @ (ap @ (sub @ X412 @ id) @ X413) @ id) != $true) & ($true = (X406 @ id @ X413 @ id)) & ((X406 @ id @ X412 @ id) = $true))) | ? [X395 : term > $o,X396 : term,X397 : subst > term > subst > $o] : (! [X404 : term] : (((X395 @ (sub @ X396 @ (push @ X404 @ id))) = $true) | ((X395 @ X404) != $true)) & ! [X398 : term] : ((X397 @ id @ X398 @ id) = (X395 @ X398)) & ! [X399 : subst > term > term] : (? [X400 : subst,X401 : subst,X402 : term] : ((sub @ (X399 @ X400 @ X402) @ X401) != (X399 @ (comp @ X400 @ X401) @ (sub @ X402 @ X401))) | ? [X403 : term] : (($true != (X397 @ id @ (X399 @ id @ X403) @ id)) & ((X397 @ id @ X403 @ id) = $true)) | ($true = (X397 @ id @ (lam @ (X399 @ sh @ one)) @ id))) & ($true != (X395 @ (lam @ X396))))) | ? [X394 : term] : ((sub @ X394 @ id) != X394)) | ? [X387 : term,X388 : term > $o] : (((X388 @ X387) != $true) & ! [X390 : term] : (? [X391 : term] : (((X388 @ X391) = $true) & ((X388 @ (sub @ X390 @ (push @ X391 @ id))) != $true)) | ((X388 @ (lam @ X390)) = $true)) & ! [X389 : term] : (($true != (var @ X389)) | ($true = (X388 @ X389))) & ! [X392 : term,X393 : term] : (((X388 @ X393) != $true) | ($true = (X388 @ (ap @ X393 @ X392))) | ($true != (X388 @ X392))))) | ? [X384 : subst > term > subst > $o] : ! [X385 : term > $o] : ? [X386 : term] : ((X384 @ id @ X386 @ id) != (X385 @ X386)))) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axapp = $true) & ! [X23 : d_term,X24 : d_term] : ((X23 = X24) | ((d2term @ X23) != (d2term @ X24))) & ((pushprop_lem0_lthm = $true) <=> ! [X513 : term,X514 : term > $o,X515 : subst] : ? [X516 : term > $o] : ! [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) = (X516 @ X517))) & ((hoasapinj2_lthm = $true) <=> (? [X476 : term,X477 : term,X478 : term,X479 : term] : (((ap @ X477 @ X476) = (ap @ X478 @ X479)) & (X476 != X479)) | ! [X480 : term,X481 : term,X482 : term,X483 : term] : ((X482 = X483) | ((ap @ (sub @ X481 @ id) @ X483) != (ap @ (sub @ X480 @ id) @ X482))))) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X349 : term] : ((sub @ X349 @ id) != X349) | ! [X350 : term,X351 : subst > term > subst > $o,X352 : term > $o] : (? [X367 : term] : (((X352 @ (sub @ X350 @ (push @ X367 @ id))) != $true) & ((X352 @ X367) = $true)) | ? [X362 : subst > term > term] : (! [X363 : term,X364 : subst,X365 : subst] : ((X362 @ (comp @ X365 @ X364) @ (sub @ X363 @ X364)) = (sub @ (X362 @ X365 @ X363) @ X364)) & ! [X366 : term] : (((X351 @ id @ (X362 @ id @ X366) @ id) = $true) | ($true != (X351 @ id @ X366 @ id))) & ((X351 @ id @ (lam @ (X362 @ sh @ one)) @ id) != $true)) | ? [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (((X351 @ (comp @ X356 @ X354) @ (sub @ X357 @ X354) @ X355) != $true) & ((X351 @ X356 @ X357 @ (comp @ X354 @ X355)) = $true)) | ? [X358 : subst,X359 : subst,X360 : term,X361 : subst] : (($true != (X351 @ X358 @ X360 @ (comp @ X359 @ X361))) & ($true = (X351 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359) @ X361))) | ? [X353 : term] : ((X351 @ id @ X353 @ id) != (X352 @ X353)) | ((X352 @ (lam @ X350)) = $true)))) & (pushprop_lem0_gthm = $true) & (hoaslamnotvar = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (((((! [X180 : term,X181 : term > $o,X182 : subst] : (? [X183 : term,X184 : term] : (($true = (X181 @ X183)) & ($true != (X181 @ (ap @ X184 @ X183))) & ((X181 @ X184) = $true)) | ? [X185 : term] : (((var @ X185) = $true) & ((X181 @ (sub @ X185 @ X182)) != $true)) | ? [X186 : term] : (! [X187 : term] : (((X181 @ (sub @ X186 @ (push @ X187 @ id))) = $true) | ((X181 @ X187) != $true)) & ($true != (X181 @ (lam @ X186)))) | ((X181 @ (sub @ X180 @ X182)) = $true)) | ? [X174 : term,X175 : term > $o] : (((X175 @ X174) != $true) & ! [X179 : term] : (($true = (X175 @ X179)) | ($true != (var @ X179))) & ! [X176 : term,X177 : term] : (($true = (X175 @ (ap @ X177 @ X176))) | ((X175 @ X177) != $true) | ((X175 @ X176) != $true)) & ! [X178 : term] : (((X175 @ X178) != $true) | ($true = (X175 @ (lam @ X178)))))) | ? [X173 : subst] : ((comp @ X173 @ id) != X173)) | ? [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) != X171)) | ? [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) != X169)) <=> (induction2lem_lthm = $true)) & (substmonoid_gthm = $true) & ! [X271 : term] : ? [X272 : d_term] : ((d2term @ X272) = X271) & ((laminj = $true) <=> ! [X167 : term,X168 : term] : ((X167 = X168) | ((lam @ X168) != (lam @ X167)))) & (axabs = $true) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X123 : term,X124 : subst > term > subst > $o] : (($true = (X124 @ id @ (lam @ X123) @ id)) | ? [X130 : term,X131 : subst,X132 : subst,X133 : subst] : (((X124 @ X132 @ X130 @ (comp @ X131 @ X133)) != $true) & ($true = (X124 @ (comp @ X132 @ X131) @ (sub @ X130 @ X131) @ X133))) | ? [X134 : term,X135 : subst,X136 : subst,X137 : subst] : (((X124 @ X137 @ X134 @ (comp @ X136 @ X135)) = $true) & ($true != (X124 @ (comp @ X137 @ X136) @ (sub @ X134 @ X136) @ X135))) | ? [X138 : term] : (((X124 @ id @ (sub @ X123 @ (push @ X138 @ id)) @ id) != $true) & ((X124 @ id @ X138 @ id) = $true)) | ? [X125 : subst > term > term] : (! [X127 : term,X128 : subst,X129 : subst] : ((X125 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128)) = (sub @ (X125 @ X129 @ X127) @ X128)) & ! [X126 : term] : (((X124 @ id @ X126 @ id) != $true) | ((X124 @ id @ (X125 @ id @ X126) @ id) = $true)) & ($true != (X124 @ id @ (lam @ (X125 @ sh @ one)) @ id)))) | ? [X115 : term,X116 : subst > term > subst > $o] : (! [X122 : term] : (((X116 @ id @ X122 @ id) != $true) | ((X116 @ id @ (sub @ X115 @ (push @ X122 @ id)) @ id) = $true)) & ((X116 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id) != $true) & ! [X117 : subst > term > term] : (((X116 @ id @ (lam @ (X117 @ sh @ one)) @ id) = $true) | ? [X119 : term,X120 : subst,X121 : subst] : ((X117 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)) != (sub @ (X117 @ X121 @ X119) @ X120)) | ? [X118 : term] : (($true = (X116 @ id @ X118 @ id)) & ($true != (X116 @ id @ (X117 @ id @ X118) @ id)))))) | ? [X114 : term] : ((sub @ X114 @ id) != X114))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3v2_gthm = $true) & ((d2subst @ d_id) = sh) & (hoasapinj1_gthm = $true) & (! [X414 : term,X415 : subst > term > subst > $o] : (? [X416 : subst > term > term] : (! [X417 : term] : (((X415 @ id @ (X416 @ id @ X417) @ id) = $true) | ((X415 @ id @ X417 @ id) != $true)) & ((X415 @ id @ (lam @ (X416 @ sh @ one)) @ id) != $true) & ! [X418 : term,X419 : subst,X420 : subst] : ((sub @ (X416 @ X419 @ X418) @ X420) = (X416 @ (comp @ X419 @ X420) @ (sub @ X418 @ X420)))) | ($true = (X415 @ id @ (lam @ (sub @ X414 @ (push @ one @ sh))) @ id)) | ? [X421 : term] : (($true = (X415 @ id @ X421 @ id)) & ((X415 @ id @ (sub @ X414 @ (push @ X421 @ id)) @ id) != $true))) <=> (hoasinduction_lem3aa = $true)) & ((apinj2 = $true) <=> ! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) != (ap @ X100 @ X98)) | (X97 = X98))) & (! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) <=> (axvarcons = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X7 : subst > term > subst > $o] : ? [X8 : term > $o] : ! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id))) & (hoaslamnotap_gthm = $true) & ((hoasapinj1_lthm = $true) <=> ((! [X345 : term,X346 : term,X347 : term,X348 : term] : (((ap @ (sub @ X345 @ id) @ X347) != (ap @ (sub @ X348 @ id) @ X346)) | (X345 = X348)) | ? [X341 : term,X342 : term,X343 : term,X344 : term] : ((X341 != X342) & ((ap @ X342 @ X343) = (ap @ X341 @ X344)))) | ? [X340 : term] : ((sub @ X340 @ id) != X340))) & ((substmonoid_lthm = $true) <=> (((! [X333 : subst] : ((comp @ X333 @ id) = X333) & ! [X334 : subst] : ((comp @ id @ X334) = X334)) | ? [X332 : subst] : ((comp @ X332 @ id) != X332)) | ? [X331 : subst] : ((comp @ id @ X331) != X331))) & ((hoasapinj1 = $true) <=> ! [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 = X380) | ((ap @ (sub @ X379 @ id) @ X378) != (ap @ (sub @ X380 @ id) @ X381)))) & ! [X382 : subst] : ? [X383 : d_subst] : ((d2subst @ X383) = X382) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (((X3 @ X5) != $true) | ? [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) != (X2 @ X6)) | ((X2 @ one) = $true)) | ? [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) != X0)) <=> (pushprop_lem1v2_lthm = $true)) & (hoasinduction_lem1_lthm = $true) & ((d2subst @ d_id) = id) & (hoasinduction_lem2_lthm = $true) & (one = (d2term @ d_one)) & (pushprop_lem1_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X69 : term,X70 : subst > term > subst > $o] : (? [X72 : subst > term > term] : (! [X76 : term] : (($true != (X70 @ id @ X76 @ id)) | ((X70 @ id @ (X72 @ id @ X76) @ id) = $true)) & ($true != (X70 @ id @ (lam @ (X72 @ sh @ one)) @ id)) & ! [X73 : subst,X74 : term,X75 : subst] : ((sub @ (X72 @ X75 @ X74) @ X73) = (X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73)))) | ? [X71 : term] : (($true = (X70 @ id @ X71 @ id)) & ($true != (X70 @ id @ (sub @ X69 @ (push @ X71 @ id)) @ id))) | ((X70 @ id @ (lam @ X69) @ id) = $true)) | ? [X61 : term,X62 : subst > term > subst > $o] : (! [X68 : term] : (((X62 @ id @ (sub @ X61 @ (push @ X68 @ id)) @ id) = $true) | ((X62 @ id @ X68 @ id) != $true)) & ((X62 @ id @ (lam @ (sub @ X61 @ (push @ one @ sh))) @ id) != $true) & ! [X63 : subst > term > term] : (((X62 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true) | ? [X64 : subst,X65 : subst,X66 : term] : ((X63 @ (comp @ X65 @ X64) @ (sub @ X66 @ X64)) != (sub @ (X63 @ X65 @ X66) @ X64)) | ? [X67 : term] : (($true != (X62 @ id @ (X63 @ id @ X67) @ id)) & ($true = (X62 @ id @ X67 @ id)))))) | ? [X60 : term] : ((sub @ X60 @ id) != X60))) & ((induction2lem = $true) <=> ! [X139 : term,X140 : term > $o,X141 : subst] : (? [X145 : term,X146 : term] : (((X140 @ X146) = $true) & ($true != (X140 @ (ap @ X145 @ X146))) & ((X140 @ X145) = $true)) | ? [X142 : term] : (((var @ X142) = $true) & ((X140 @ (sub @ X142 @ X141)) != $true)) | ? [X143 : term] : (! [X144 : term] : (((X140 @ X144) != $true) | ((X140 @ (sub @ X143 @ (push @ X144 @ id))) = $true)) & ($true != (X140 @ (lam @ X143)))) | ($true = (X140 @ (sub @ X139 @ X141))))) & ((hoasinduction = $true) <=> ! [X37 : subst > term > subst > $o,X38 : term] : (? [X46 : subst,X47 : subst,X48 : subst,X49 : term] : (((X37 @ X46 @ X49 @ (comp @ X47 @ X48)) = $true) & ((X37 @ (comp @ X46 @ X47) @ (sub @ X49 @ X47) @ X48) != $true)) | ($true = (X37 @ id @ X38 @ id)) | ? [X39 : term,X40 : term] : (($true = (X37 @ id @ X40 @ id)) & ($true = (X37 @ id @ X39 @ id)) & ($true != (X37 @ id @ (ap @ (sub @ X40 @ id) @ X39) @ id))) | ? [X41 : subst > term > term] : (((X37 @ id @ (lam @ (X41 @ sh @ one)) @ id) != $true) & ! [X45 : term] : (($true = (X37 @ id @ (X41 @ id @ X45) @ id)) | ((X37 @ id @ X45 @ id) != $true)) & ! [X42 : term,X43 : subst,X44 : subst] : ((X41 @ (comp @ X43 @ X44) @ (sub @ X42 @ X44)) = (sub @ (X41 @ X43 @ X42) @ X44))) | ? [X54 : term] : (($true = (var @ (sub @ X54 @ id))) & ($true != (X37 @ id @ X54 @ id))) | ? [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true = (X37 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) & ((X37 @ X51 @ X52 @ (comp @ X50 @ X53)) != $true)))) & ((hoasapinj2 = $true) <=> ! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22)))) & (hoasapinj2_gthm = $true) & (termmset_gthm = $true) & (pushprop_lem2v2 = $true) & (induction2_gthm = $true) & (! [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) = X335) <=> (axscons = $true)) & (hoaslaminj_gthm = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (! [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (? [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) != (sub @ (X496 @ X499 @ X500) @ X501)) | ? [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) != (sub @ (X498 @ X502 @ X503) @ X504)) | ((X498 @ X497 @ X495) = (X496 @ X497 @ X495)) | ((lam @ (X498 @ sh @ one)) != (lam @ (X496 @ sh @ one)))) <=> (hoaslaminj = $true)) & (hoasinduction_lem3a_gthm = $true) & (ulamvarsh = $true) & ((((! [X83 : term,X84 : subst > term > term,X85 : subst,X86 : subst > term > term] : (((lam @ (X84 @ sh @ one)) != (lam @ (X86 @ sh @ one))) | ? [X90 : subst,X91 : subst,X92 : term] : ((X84 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91)) != (sub @ (X84 @ X90 @ X92) @ X91)) | ((X84 @ X85 @ X83) = (X86 @ X85 @ X83)) | ? [X87 : subst,X88 : subst,X89 : term] : ((X86 @ (comp @ X87 @ X88) @ (sub @ X89 @ X88)) != (sub @ (X86 @ X87 @ X89) @ X88))) | ? [X81 : term,X82 : term] : (((lam @ X81) = (lam @ X82)) & (X81 != X82))) | ? [X79 : subst,X80 : term] : ((comp @ sh @ (push @ X80 @ X79)) != X79)) | ? [X77 : subst,X78 : term] : ((sub @ one @ (push @ X78 @ X77)) != X78)) <=> (hoaslaminj_lthm = $true)) & ((induction = $true) <=> ! [X206 : term > $o,X207 : term] : (? [X209 : term,X210 : term] : (($true = (X206 @ X210)) & ((X206 @ (ap @ X209 @ X210)) != $true) & ($true = (X206 @ X209))) | ? [X211 : term] : (($true = (var @ X211)) & ($true != (X206 @ X211))) | ? [X208 : term] : (($true != (X206 @ (lam @ X208))) & ($true = (X206 @ X208))) | ((X206 @ X207) = $true))) & (pushprop_lem1v2_gthm = $true)),
% 2.43/1.93 inference(ennf_transformation,[status(thm)],[f10])).
% 2.43/1.93 thf(f10,plain,(
% 2.43/1.93 (hoasinduction_lem1 = $true) & (lamnotap != $true) & (! [X531 : subst] : ((comp @ id @ X531) = X531) <=> (axidl = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X292 : subst > term > subst > $o] : ~! [X293 : term > $o] : ~! [X294 : term] : ((X292 @ id @ X294 @ id) = (X293 @ X294)) => (! [X295 : term > $o,X296 : term] : (~! [X299 : term] : (($true != (var @ X299)) | ((X295 @ X299) = $true)) | ~! [X297 : term,X298 : term] : (((X295 @ X297) != $true) | ((X295 @ X298) != $true) | ($true = (X295 @ (ap @ X297 @ X298)))) | ((X295 @ X296) = $true) | ~! [X300 : term] : (((X295 @ (lam @ X300)) = $true) | ~! [X301 : term] : (((X295 @ X301) != $true) | ($true = (X295 @ (sub @ X300 @ (push @ X301 @ id))))))) => (! [X302 : term] : ((sub @ X302 @ id) = X302) => (! [X303 : subst > term > subst > $o,X304 : term,X305 : term > $o] : (((X305 @ (lam @ X304)) = $true) | ~! [X311 : term] : (((X305 @ X311) != $true) | ((X305 @ (sub @ X304 @ (push @ X311 @ id))) = $true)) | ~! [X306 : subst > term > term] : (($true = (X303 @ id @ (lam @ (X306 @ sh @ one)) @ id)) | ~! [X308 : term,X309 : subst,X310 : subst] : ((X306 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309)) = (sub @ (X306 @ X310 @ X308) @ X309)) | ~! [X307 : term] : (($true = (X303 @ id @ (X306 @ id @ X307) @ id)) | ($true != (X303 @ id @ X307 @ id)))) | ~! [X312 : term] : ((X303 @ id @ X312 @ id) = (X305 @ X312))) => ! [X313 : term,X314 : subst > term > subst > $o] : (~! [X326 : subst > term > term] : (((X314 @ id @ (lam @ (X326 @ sh @ one)) @ id) = $true) | ~! [X327 : subst,X328 : term,X329 : subst] : ((X326 @ (comp @ X327 @ X329) @ (sub @ X328 @ X329)) = (sub @ (X326 @ X327 @ X328) @ X329)) | ~! [X330 : term] : (((X314 @ id @ X330 @ id) != $true) | ((X314 @ id @ (X326 @ id @ X330) @ id) = $true))) | ~! [X319 : term] : (($true != (var @ (sub @ X319 @ id))) | ((X314 @ id @ X319 @ id) = $true)) | ~! [X322 : subst,X323 : subst,X324 : term,X325 : subst] : (((X314 @ (comp @ X322 @ X323) @ (sub @ X324 @ X323) @ X325) = $true) | ((X314 @ X322 @ X324 @ (comp @ X323 @ X325)) != $true)) | ~! [X320 : term,X321 : term] : (((X314 @ id @ (ap @ (sub @ X320 @ id) @ X321) @ id) = $true) | ((X314 @ id @ X320 @ id) != $true) | ($true != (X314 @ id @ X321 @ id))) | ((X314 @ id @ X313 @ id) = $true) | ~! [X315 : subst,X316 : subst,X317 : term,X318 : subst] : (($true = (X314 @ X318 @ X317 @ (comp @ X315 @ X316))) | ($true != (X314 @ (comp @ X318 @ X315) @ (sub @ X317 @ X315) @ X316))))))))) & (pushprop_lem3v2 = $true) & (hoasinduction_gthm = $true) & (! [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (($true = (X267 @ (sub @ X265 @ (push @ X268 @ X269)))) | ((X266 @ X265) != $true)) <=> (pushprop_lem1 = $true)) & ! [X96 : d_term] : (d_one = X96) & (! [X368 : term,X369 : term > $o,X370 : subst > term > subst > $o] : (~! [X371 : subst > term > term] : (($true = (X370 @ id @ (lam @ (X371 @ sh @ one)) @ id)) | ~! [X373 : term,X374 : subst,X375 : subst] : ((sub @ (X371 @ X375 @ X373) @ X374) = (X371 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374))) | ~! [X372 : term] : (($true != (X370 @ id @ X372 @ id)) | ($true = (X370 @ id @ (X371 @ id @ X372) @ id)))) | ~! [X377 : term] : (((X369 @ X377) != $true) | ((X369 @ (sub @ X368 @ (push @ X377 @ id))) = $true)) | ~! [X376 : term] : ((X370 @ id @ X376 @ id) = (X369 @ X376)) | ($true = (X369 @ (lam @ X368)))) <=> (hoasinduction_lem3v2a = $true)) & (ulamvar1 != $true) & (! [X147 : term] : ~! [X148 : subst > term > term] : ~! [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) = (sub @ X147 @ (push @ X149 @ X150))) <=> (hoasinduction_lem3v2_f = $true)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((! [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) = X11) => (! [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) = X12) => ! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true = (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) | ((X18 @ X15) != $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_gthm = $true) & (induction2lem_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X220 : term > $o,X221 : term,X222 : subst > term > subst > $o,X223 : term] : (~! [X224 : subst,X225 : term,X226 : subst,X227 : subst] : (((X222 @ (comp @ X224 @ X227) @ (sub @ X225 @ X227) @ X226) = $true) | ((X222 @ X224 @ X225 @ (comp @ X227 @ X226)) != $true)) | ~! [X232 : term] : ((X220 @ X232) = (X222 @ id @ X232 @ id)) | ((X220 @ X221) != $true) | ((X220 @ X223) != $true) | ~! [X228 : subst,X229 : subst,X230 : subst,X231 : term] : (((X222 @ (comp @ X230 @ X228) @ (sub @ X231 @ X228) @ X229) != $true) | ($true = (X222 @ X230 @ X231 @ (comp @ X228 @ X229)))) | ~! [X233 : term,X234 : term] : (((X222 @ id @ X234 @ id) != $true) | ($true = (X222 @ id @ (ap @ (sub @ X234 @ id) @ X233) @ id)) | ($true != (X222 @ id @ X233 @ id))) | ((X220 @ (ap @ X221 @ X223)) = $true))) & ((apinj1 = $true) <=> ! [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 = X487) | ((ap @ X485 @ X484) != (ap @ X487 @ X486)))) & ((induction2 = $true) <=> ! [X251 : term,X252 : term > $o] : (~! [X257 : term] : (($true != (var @ X257)) | ($true = (X252 @ X257))) | ~! [X253 : term,X254 : term] : (($true != (X252 @ X253)) | ((X252 @ X254) != $true) | ($true = (X252 @ (ap @ X254 @ X253)))) | ~! [X255 : term] : (((X252 @ (lam @ X255)) = $true) | ~! [X256 : term] : (($true != (X252 @ X256)) | ((X252 @ (sub @ X255 @ (push @ X256 @ id))) = $true))) | ($true = (X252 @ X251)))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X93 : subst > term > subst > $o] : ~! [X94 : term > $o] : ~! [X95 : term] : ((X94 @ X95) = (X93 @ id @ X95 @ id))) & (apnotvar = $true) & (pushprop_lem3v2_lthm = $true) & (! [X258 : term,X259 : subst > term > term,X260 : term] : (~! [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) = (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262))) | ((lam @ (X259 @ sh @ one)) != (ap @ (sub @ X260 @ id) @ X258))) <=> (hoaslamnotap = $true)) & (hoasapnotvar_gthm = $true) & (axassoc = $true) & (! [X338 : term] : ~! [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) != (X339 @ sh @ one)) <=> (hoasinduction_lem3b_lthm = $true)) & ((termmset = $true) <=> ! [X57 : term] : ((sub @ X57 @ id) = X57)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X505 : subst > term > subst > $o,X506 : term] : (~! [X507 : term] : (($true != (X505 @ id @ X507 @ id)) | ((X505 @ id @ (sub @ X506 @ (push @ X507 @ id)) @ id) = $true)) | ~! [X508 : subst > term > term] : (((X505 @ id @ (lam @ (X508 @ sh @ one)) @ id) = $true) | ~! [X509 : term] : (($true != (X505 @ id @ X509 @ id)) | ($true = (X505 @ id @ (X508 @ id @ X509) @ id))) | ~! [X510 : subst,X511 : subst,X512 : term] : ((X508 @ (comp @ X511 @ X510) @ (sub @ X512 @ X510)) = (sub @ (X508 @ X511 @ X512) @ X510))) | ((X505 @ id @ (lam @ (sub @ X506 @ (push @ one @ sh))) @ id) = $true))) & (hoasinduction_lem1v2 = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (axclos = $true) & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) = (d2subst @ X55)) => (X55 = X56)) & ((! [X488 : term] : ((sub @ X488 @ id) = X488) => ! [X489 : term] : ((sub @ X489 @ id) = X489)) <=> (termmset_lthm = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lem3a = $true) <=> ! [X532 : term,X533 : subst > term > subst > $o] : (~! [X539 : term] : (((X533 @ id @ X539 @ id) != $true) | ((X533 @ id @ (sub @ X532 @ (push @ X539 @ id)) @ id) = $true)) | ($true = (X533 @ id @ (lam @ X532) @ id)) | ~! [X534 : subst > term > term] : (((X533 @ id @ (lam @ (X534 @ sh @ one)) @ id) = $true) | ~! [X535 : subst,X536 : subst,X537 : term] : ((sub @ (X534 @ X535 @ X537) @ X536) = (X534 @ (comp @ X535 @ X536) @ (sub @ X537 @ X536))) | ~! [X538 : term] : (((X533 @ id @ X538 @ id) != $true) | ((X533 @ id @ (X534 @ id @ X538) @ id) = $true))))) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) = X273) <=> (axshiftcons = $true)) & (! [X214 : subst] : ((comp @ X214 @ id) = X214) <=> (axidr = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & (! [X25 : subst > term > subst > $o,X26 : term] : (~! [X27 : term] : (($true != (X25 @ id @ X27 @ id)) | ($true = (X25 @ id @ (sub @ X26 @ (push @ X27 @ id)) @ id))) | ~! [X28 : term,X29 : subst > term > term] : (~! [X30 : term] : (((X25 @ id @ (X29 @ id @ X30) @ id) = $true) | ((X25 @ id @ X30 @ id) != $true)) | ~! [X34 : subst,X35 : subst,X36 : term] : ((sub @ X28 @ (push @ (sub @ X36 @ X34) @ (comp @ X35 @ X34))) = (X29 @ (comp @ X35 @ X34) @ (sub @ X36 @ X34))) | ((X25 @ id @ (lam @ (X29 @ sh @ one)) @ id) = $true) | ~! [X31 : subst,X32 : term,X33 : subst] : ((sub @ (sub @ X28 @ (push @ X32 @ X33)) @ X31) = (sub @ (X29 @ X33 @ X32) @ X31))) | ($true = (X25 @ id @ (lam @ (sub @ X26 @ (push @ one @ sh))) @ id))) <=> (hoasinduction_lem3aaa = $true)) & ((! [X59 : subst] : ((comp @ id @ X59) = X59) & ! [X58 : subst] : ((comp @ X58 @ id) = X58)) <=> (substmonoid = $true)) & ((hoasinduction_lthm = $true) <=> (! [X422 : term > $o,X423 : term] : (((X422 @ X423) = $true) | ~! [X425 : term,X426 : term] : (($true != (X422 @ X425)) | ((X422 @ X426) != $true) | ((X422 @ (ap @ X426 @ X425)) = $true)) | ~! [X424 : term] : (($true = (X422 @ X424)) | ((var @ X424) != $true)) | ~! [X427 : term] : (($true = (X422 @ (lam @ X427))) | ~! [X428 : term] : (((X422 @ X428) != $true) | ((X422 @ (sub @ X427 @ (push @ X428 @ id))) = $true)))) => (! [X429 : subst > term > subst > $o,X430 : term,X431 : term] : (~! [X436 : subst,X437 : subst,X438 : term,X439 : subst] : (($true != (X429 @ X437 @ X438 @ (comp @ X436 @ X439))) | ((X429 @ (comp @ X437 @ X436) @ (sub @ X438 @ X436) @ X439) = $true)) | ~! [X432 : term,X433 : subst,X434 : subst,X435 : subst] : (((X429 @ X435 @ X432 @ (comp @ X434 @ X433)) = $true) | ((X429 @ (comp @ X435 @ X434) @ (sub @ X432 @ X434) @ X433) != $true)) | ((X429 @ id @ X430 @ id) != $true) | ((X429 @ id @ (ap @ X431 @ X430) @ id) = $true) | ((X429 @ id @ X431 @ id) != $true) | ~! [X440 : term,X441 : term] : (((X429 @ id @ X440 @ id) != $true) | ($true != (X429 @ id @ X441 @ id)) | ($true = (X429 @ id @ (ap @ (sub @ X440 @ id) @ X441) @ id)))) => (! [X442 : term,X443 : subst > term > subst > $o] : (~! [X449 : term] : (((X443 @ id @ (sub @ X442 @ (push @ X449 @ id)) @ id) = $true) | ($true != (X443 @ id @ X449 @ id))) | ~! [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (($true != (X443 @ X457 @ X456 @ (comp @ X455 @ X454))) | ((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454) = $true)) | ~! [X450 : subst,X451 : term,X452 : subst,X453 : subst] : (((X443 @ (comp @ X452 @ X453) @ (sub @ X451 @ X453) @ X450) != $true) | ($true = (X443 @ X452 @ X451 @ (comp @ X453 @ X450)))) | ((X443 @ id @ (lam @ X442) @ id) = $true) | ~! [X444 : subst > term > term] : (~! [X445 : subst,X446 : term,X447 : subst] : ((X444 @ (comp @ X447 @ X445) @ (sub @ X446 @ X445)) = (sub @ (X444 @ X447 @ X446) @ X445)) | ~! [X448 : term] : (((X443 @ id @ X448 @ id) != $true) | ($true = (X443 @ id @ (X444 @ id @ X448) @ id))) | ($true = (X443 @ id @ (lam @ (X444 @ sh @ one)) @ id)))) => ! [X458 : term,X459 : subst > term > subst > $o] : (~! [X462 : subst > term > term] : (~! [X464 : subst,X465 : term,X466 : subst] : ((X462 @ (comp @ X466 @ X464) @ (sub @ X465 @ X464)) = (sub @ (X462 @ X466 @ X465) @ X464)) | ($true = (X459 @ id @ (lam @ (X462 @ sh @ one)) @ id)) | ~! [X463 : term] : (((X459 @ id @ (X462 @ id @ X463) @ id) = $true) | ((X459 @ id @ X463 @ id) != $true))) | ~! [X460 : term,X461 : term] : (($true = (X459 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id)) | ($true != (X459 @ id @ X461 @ id)) | ($true != (X459 @ id @ X460 @ id))) | ((X459 @ id @ X458 @ id) = $true) | ~! [X475 : term] : (($true != (var @ (sub @ X475 @ id))) | ($true = (X459 @ id @ X475 @ id))) | ~! [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (((X459 @ X469 @ X467 @ (comp @ X470 @ X468)) = $true) | ((X459 @ (comp @ X469 @ X470) @ (sub @ X467 @ X470) @ X468) != $true)) | ~! [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (((X459 @ X473 @ X474 @ (comp @ X472 @ X471)) != $true) | ((X459 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471) = $true))))))) & (! [X212 : term] : ~! [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) != (X213 @ sh @ one)) <=> (hoasinduction_lem3b = $true)) & (axmap = $true) & (hoasinduction_lem1v2_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (((X216 @ one) = $true) | ($true != (X217 @ X215)) | ~! [X219 : term] : ((X216 @ X219) = (X217 @ (sub @ X219 @ (push @ X215 @ X218)))))) & (axvarshift = $true) & ! [X270 : d_subst] : (d_id = X270) & (! [X490 : term > $o,X491 : term,X492 : subst] : ~! [X493 : term > $o] : ~! [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) = (X493 @ X494)) <=> (pushprop_lem0 = $true)) & (pushprop = $true) & (hoasinduction_lem1_gthm = $true) & (hoaslamnotvar_lthm = $true) & ((induction2_lthm = $true) <=> (! [X151 : term] : ((sub @ X151 @ id) = X151) => (! [X152 : term,X153 : subst,X154 : term > $o] : (~! [X159 : term] : (((var @ X159) != $true) | ((X154 @ (sub @ X159 @ X153)) = $true)) | ((X154 @ (sub @ X152 @ X153)) = $true) | ~! [X155 : term] : (~! [X156 : term] : (((X154 @ (sub @ X155 @ (push @ X156 @ id))) = $true) | ((X154 @ X156) != $true)) | ($true = (X154 @ (lam @ X155)))) | ~! [X157 : term,X158 : term] : (($true != (X154 @ X158)) | ((X154 @ X157) != $true) | ((X154 @ (ap @ X158 @ X157)) = $true))) => ! [X160 : term,X161 : term > $o] : (((X161 @ X160) = $true) | ~! [X162 : term] : (($true != (var @ X162)) | ($true = (X161 @ X162))) | ~! [X163 : term,X164 : term] : (($true != (X161 @ X163)) | ((X161 @ X164) != $true) | ((X161 @ (ap @ X164 @ X163)) = $true)) | ~! [X165 : term] : (~! [X166 : term] : (((X161 @ X166) != $true) | ($true = (X161 @ (sub @ X165 @ (push @ X166 @ id))))) | ($true = (X161 @ (lam @ X165)))))))) & (hoasapnotvar = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X105 : term,X106 : subst > term > subst > $o] : (~! [X112 : term,X113 : term] : (((X106 @ id @ X113 @ id) != $true) | ((X106 @ id @ X112 @ id) != $true) | ($true = (X106 @ id @ (ap @ (sub @ X113 @ id) @ X112) @ id))) | ($true = (X106 @ id @ X105 @ id)) | ~! [X107 : subst > term > term] : (~! [X109 : subst,X110 : term,X111 : subst] : ((X107 @ (comp @ X111 @ X109) @ (sub @ X110 @ X109)) = (sub @ (X107 @ X111 @ X110) @ X109)) | ~! [X108 : term] : (($true = (X106 @ id @ (X107 @ id @ X108) @ id)) | ($true != (X106 @ id @ X108 @ id))) | ((X106 @ id @ (lam @ (X107 @ sh @ one)) @ id) = $true)))) & (! [X275 : term,X276 : term] : (((sub @ X276 @ sh) != (sub @ X275 @ sh)) | (X275 = X276)) <=> (shinj = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term] : (~! [X237 : subst,X238 : subst,X239 : subst,X240 : term] : (($true != (X235 @ (comp @ X238 @ X239) @ (sub @ X240 @ X239) @ X237)) | ((X235 @ X238 @ X240 @ (comp @ X239 @ X237)) = $true)) | ~! [X246 : term] : (($true = (X235 @ id @ (sub @ X236 @ (push @ X246 @ id)) @ id)) | ((X235 @ id @ X246 @ id) != $true)) | ~! [X241 : subst > term > term] : (((X235 @ id @ (lam @ (X241 @ sh @ one)) @ id) = $true) | ~! [X245 : term] : (($true = (X235 @ id @ (X241 @ id @ X245) @ id)) | ((X235 @ id @ X245 @ id) != $true)) | ~! [X242 : term,X243 : subst,X244 : subst] : ((X241 @ (comp @ X244 @ X243) @ (sub @ X242 @ X243)) = (sub @ (X241 @ X244 @ X242) @ X243))) | ~! [X247 : subst,X248 : subst,X249 : subst,X250 : term] : (((X235 @ X248 @ X250 @ (comp @ X249 @ X247)) != $true) | ($true = (X235 @ (comp @ X248 @ X249) @ (sub @ X250 @ X249) @ X247))) | ((X235 @ id @ (lam @ X236) @ id) = $true))) & (hoasapnotvar_lthm = $true) & (! [X188 : subst > term > subst > $o,X189 : term,X190 : term > $o] : (((X190 @ (lam @ X189)) = $true) | ~! [X200 : subst > term > term] : (~! [X201 : subst,X202 : subst,X203 : term] : ((X200 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201)) = (sub @ (X200 @ X202 @ X203) @ X201)) | ((X188 @ id @ (lam @ (X200 @ sh @ one)) @ id) = $true) | ~! [X204 : term] : (((X188 @ id @ X204 @ id) != $true) | ((X188 @ id @ (X200 @ id @ X204) @ id) = $true))) | ~! [X191 : subst,X192 : subst,X193 : subst,X194 : term] : (((X188 @ (comp @ X193 @ X191) @ (sub @ X194 @ X191) @ X192) = $true) | ((X188 @ X193 @ X194 @ (comp @ X191 @ X192)) != $true)) | ~! [X195 : subst,X196 : subst,X197 : subst,X198 : term] : (((X188 @ X197 @ X198 @ (comp @ X196 @ X195)) = $true) | ((X188 @ (comp @ X197 @ X196) @ (sub @ X198 @ X196) @ X195) != $true)) | ~! [X199 : term] : (((X190 @ (sub @ X189 @ (push @ X199 @ id))) = $true) | ((X190 @ X199) != $true)) | ~! [X205 : term] : ((X188 @ id @ X205 @ id) = (X190 @ X205))) <=> (hoasinduction_lem3v2 = $true)) & (! [X264 : term] : ((sub @ X264 @ id) = X264) <=> (axvarid = $true)) & (pushprop_gthm = $true) & (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) & (hoasinduction_lem2v2_gthm = $true) & (lamnotvar = $true) & (! [X101 : term] : ~! [X102 : subst > term > term] : ~! [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) = (X102 @ X104 @ X103)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X518 : subst > term > subst > $o,X519 : term,X520 : term] : (((X518 @ id @ (ap @ X519 @ X520) @ id) = $true) | ~! [X527 : subst,X528 : subst,X529 : term,X530 : subst] : (($true != (X518 @ X528 @ X529 @ (comp @ X530 @ X527))) | ($true = (X518 @ (comp @ X528 @ X530) @ (sub @ X529 @ X530) @ X527))) | ($true != (X518 @ id @ X520 @ id)) | ((X518 @ id @ X519 @ id) != $true) | ~! [X525 : term,X526 : term] : (($true != (X518 @ id @ X525 @ id)) | ((X518 @ id @ (ap @ (sub @ X525 @ id) @ X526) @ id) = $true) | ($true != (X518 @ id @ X526 @ id))) | ~! [X521 : subst,X522 : subst,X523 : term,X524 : subst] : (($true != (X518 @ (comp @ X522 @ X524) @ (sub @ X523 @ X524) @ X521)) | ($true = (X518 @ X522 @ X523 @ (comp @ X524 @ X521)))))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X277 : term] : ~! [X278 : subst > term > term] : ~! [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) = (X278 @ X280 @ X279)) => (! [X281 : term] : ((sub @ X281 @ id) = X281) => ! [X282 : term > $o,X283 : subst > term > subst > $o,X284 : term] : (~! [X291 : term] : (($true = (X282 @ (sub @ X284 @ (push @ X291 @ id)))) | ((X282 @ X291) != $true)) | ((X282 @ (lam @ X284)) = $true) | ~! [X285 : subst > term > term] : (~! [X287 : term,X288 : subst,X289 : subst] : ((X285 @ (comp @ X289 @ X288) @ (sub @ X287 @ X288)) = (sub @ (X285 @ X289 @ X287) @ X288)) | ~! [X286 : term] : (($true = (X283 @ id @ (X285 @ id @ X286) @ id)) | ($true != (X283 @ id @ X286 @ id))) | ($true = (X283 @ id @ (lam @ (X285 @ sh @ one)) @ id))) | ~! [X290 : term] : ((X282 @ X290) = (X283 @ id @ X290 @ id)))))) & (ulamvarind = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X384 : subst > term > subst > $o] : ~! [X385 : term > $o] : ~! [X386 : term] : ((X384 @ id @ X386 @ id) = (X385 @ X386)) => (! [X387 : term,X388 : term > $o] : (~! [X390 : term] : (~! [X391 : term] : (((X388 @ X391) != $true) | ((X388 @ (sub @ X390 @ (push @ X391 @ id))) = $true)) | ((X388 @ (lam @ X390)) = $true)) | ((X388 @ X387) = $true) | ~! [X392 : term,X393 : term] : (((X388 @ X393) != $true) | ($true = (X388 @ (ap @ X393 @ X392))) | ($true != (X388 @ X392))) | ~! [X389 : term] : (($true != (var @ X389)) | ($true = (X388 @ X389)))) => (! [X394 : term] : ((sub @ X394 @ id) = X394) => (! [X395 : term > $o,X396 : term,X397 : subst > term > subst > $o] : (~! [X398 : term] : ((X397 @ id @ X398 @ id) = (X395 @ X398)) | ($true = (X395 @ (lam @ X396))) | ~! [X399 : subst > term > term] : (~! [X400 : subst,X401 : subst,X402 : term] : ((sub @ (X399 @ X400 @ X402) @ X401) = (X399 @ (comp @ X400 @ X401) @ (sub @ X402 @ X401))) | ($true = (X397 @ id @ (lam @ (X399 @ sh @ one)) @ id)) | ~! [X403 : term] : (($true = (X397 @ id @ (X399 @ id @ X403) @ id)) | ((X397 @ id @ X403 @ id) != $true))) | ~! [X404 : term] : (((X395 @ (sub @ X396 @ (push @ X404 @ id))) = $true) | ((X395 @ X404) != $true))) => ! [X405 : term,X406 : subst > term > subst > $o] : (($true = (X406 @ id @ X405 @ id)) | ~! [X407 : subst > term > term] : (~! [X408 : term] : (((X406 @ id @ (X407 @ id @ X408) @ id) = $true) | ((X406 @ id @ X408 @ id) != $true)) | ($true = (X406 @ id @ (lam @ (X407 @ sh @ one)) @ id)) | ~! [X409 : subst,X410 : subst,X411 : term] : ((sub @ (X407 @ X410 @ X411) @ X409) = (X407 @ (comp @ X410 @ X409) @ (sub @ X411 @ X409)))) | ~! [X412 : term,X413 : term] : (($true != (X406 @ id @ X413 @ id)) | ((X406 @ id @ X412 @ id) != $true) | ((X406 @ id @ (ap @ (sub @ X412 @ id) @ X413) @ id) = $true)))))))) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axapp = $true) & ! [X23 : d_term,X24 : d_term] : (((d2term @ X23) = (d2term @ X24)) => (X23 = X24)) & (! [X513 : term,X514 : term > $o,X515 : subst] : ~! [X516 : term > $o] : ~! [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) = (X516 @ X517)) <=> (pushprop_lem0_lthm = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X476 : term,X477 : term,X478 : term,X479 : term] : ((X476 = X479) | ((ap @ X477 @ X476) != (ap @ X478 @ X479))) => ! [X480 : term,X481 : term,X482 : term,X483 : term] : ((X482 = X483) | ((ap @ (sub @ X481 @ id) @ X483) != (ap @ (sub @ X480 @ id) @ X482))))) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X349 : term] : ((sub @ X349 @ id) = X349) => ! [X350 : term,X351 : subst > term > subst > $o,X352 : term > $o] : (~! [X367 : term] : (((X352 @ X367) != $true) | ((X352 @ (sub @ X350 @ (push @ X367 @ id))) = $true)) | ~! [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (((X351 @ X356 @ X357 @ (comp @ X354 @ X355)) != $true) | ((X351 @ (comp @ X356 @ X354) @ (sub @ X357 @ X354) @ X355) = $true)) | ((X352 @ (lam @ X350)) = $true) | ~! [X353 : term] : ((X351 @ id @ X353 @ id) = (X352 @ X353)) | ~! [X358 : subst,X359 : subst,X360 : term,X361 : subst] : (($true != (X351 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359) @ X361)) | ($true = (X351 @ X358 @ X360 @ (comp @ X359 @ X361)))) | ~! [X362 : subst > term > term] : (~! [X366 : term] : (((X351 @ id @ (X362 @ id @ X366) @ id) = $true) | ($true != (X351 @ id @ X366 @ id))) | ~! [X363 : term,X364 : subst,X365 : subst] : ((X362 @ (comp @ X365 @ X364) @ (sub @ X363 @ X364)) = (sub @ (X362 @ X365 @ X363) @ X364)) | ((X351 @ id @ (lam @ (X362 @ sh @ one)) @ id) = $true))))) & (pushprop_lem0_gthm = $true) & (hoaslamnotvar = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) = X169) => (! [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) = X171) => (! [X173 : subst] : ((comp @ X173 @ id) = X173) => (! [X174 : term,X175 : term > $o] : (~! [X178 : term] : (((X175 @ X178) != $true) | ($true = (X175 @ (lam @ X178)))) | ~! [X176 : term,X177 : term] : (($true = (X175 @ (ap @ X177 @ X176))) | ((X175 @ X177) != $true) | ((X175 @ X176) != $true)) | ~! [X179 : term] : (($true = (X175 @ X179)) | ($true != (var @ X179))) | ((X175 @ X174) = $true)) => ! [X180 : term,X181 : term > $o,X182 : subst] : (~! [X185 : term] : (((var @ X185) != $true) | ((X181 @ (sub @ X185 @ X182)) = $true)) | ~! [X186 : term] : (($true = (X181 @ (lam @ X186))) | ~! [X187 : term] : (((X181 @ (sub @ X186 @ (push @ X187 @ id))) = $true) | ((X181 @ X187) != $true))) | ~! [X183 : term,X184 : term] : (($true = (X181 @ (ap @ X184 @ X183))) | ($true != (X181 @ X183)) | ((X181 @ X184) != $true)) | ((X181 @ (sub @ X180 @ X182)) = $true)))))) <=> (induction2lem_lthm = $true)) & (substmonoid_gthm = $true) & ! [X271 : term] : ? [X272 : d_term] : ((d2term @ X272) = X271) & ((laminj = $true) <=> ! [X167 : term,X168 : term] : ((X167 = X168) | ((lam @ X168) != (lam @ X167)))) & (axabs = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X114 : term] : ((sub @ X114 @ id) = X114) => (! [X115 : term,X116 : subst > term > subst > $o] : (((X116 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id) = $true) | ~! [X122 : term] : (((X116 @ id @ X122 @ id) != $true) | ((X116 @ id @ (sub @ X115 @ (push @ X122 @ id)) @ id) = $true)) | ~! [X117 : subst > term > term] : (~! [X118 : term] : (($true = (X116 @ id @ (X117 @ id @ X118) @ id)) | ($true != (X116 @ id @ X118 @ id))) | ~! [X119 : term,X120 : subst,X121 : subst] : ((X117 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)) = (sub @ (X117 @ X121 @ X119) @ X120)) | ((X116 @ id @ (lam @ (X117 @ sh @ one)) @ id) = $true))) => ! [X123 : term,X124 : subst > term > subst > $o] : (~! [X125 : subst > term > term] : (~! [X127 : term,X128 : subst,X129 : subst] : ((X125 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128)) = (sub @ (X125 @ X129 @ X127) @ X128)) | ($true = (X124 @ id @ (lam @ (X125 @ sh @ one)) @ id)) | ~! [X126 : term] : (((X124 @ id @ X126 @ id) != $true) | ((X124 @ id @ (X125 @ id @ X126) @ id) = $true))) | ($true = (X124 @ id @ (lam @ X123) @ id)) | ~! [X130 : term,X131 : subst,X132 : subst,X133 : subst] : (($true != (X124 @ (comp @ X132 @ X131) @ (sub @ X130 @ X131) @ X133)) | ((X124 @ X132 @ X130 @ (comp @ X131 @ X133)) = $true)) | ~! [X134 : term,X135 : subst,X136 : subst,X137 : subst] : (($true = (X124 @ (comp @ X137 @ X136) @ (sub @ X134 @ X136) @ X135)) | ((X124 @ X137 @ X134 @ (comp @ X136 @ X135)) != $true)) | ~! [X138 : term] : (((X124 @ id @ X138 @ id) != $true) | ((X124 @ id @ (sub @ X123 @ (push @ X138 @ id)) @ id) = $true)))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3v2_gthm = $true) & ((d2subst @ d_id) = sh) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X414 : term,X415 : subst > term > subst > $o] : (~! [X416 : subst > term > term] : (((X415 @ id @ (lam @ (X416 @ sh @ one)) @ id) = $true) | ~! [X418 : term,X419 : subst,X420 : subst] : ((sub @ (X416 @ X419 @ X418) @ X420) = (X416 @ (comp @ X419 @ X420) @ (sub @ X418 @ X420))) | ~! [X417 : term] : (((X415 @ id @ (X416 @ id @ X417) @ id) = $true) | ((X415 @ id @ X417 @ id) != $true))) | ($true = (X415 @ id @ (lam @ (sub @ X414 @ (push @ one @ sh))) @ id)) | ~! [X421 : term] : (((X415 @ id @ (sub @ X414 @ (push @ X421 @ id)) @ id) = $true) | ($true != (X415 @ id @ X421 @ id))))) & ((apinj2 = $true) <=> ! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) != (ap @ X100 @ X98)) | (X97 = X98))) & (! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) <=> (axvarcons = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X7 : subst > term > subst > $o] : ~! [X8 : term > $o] : ~! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id))) & (hoaslamnotap_gthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X340 : term] : ((sub @ X340 @ id) = X340) => (! [X341 : term,X342 : term,X343 : term,X344 : term] : ((X341 = X342) | ((ap @ X342 @ X343) != (ap @ X341 @ X344))) => ! [X345 : term,X346 : term,X347 : term,X348 : term] : (((ap @ (sub @ X345 @ id) @ X347) != (ap @ (sub @ X348 @ id) @ X346)) | (X345 = X348))))) & ((substmonoid_lthm = $true) <=> (! [X331 : subst] : ((comp @ id @ X331) = X331) => (! [X332 : subst] : ((comp @ X332 @ id) = X332) => (! [X333 : subst] : ((comp @ X333 @ id) = X333) & ! [X334 : subst] : ((comp @ id @ X334) = X334))))) & ((hoasapinj1 = $true) <=> ! [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 = X380) | ((ap @ (sub @ X379 @ id) @ X378) != (ap @ (sub @ X380 @ id) @ X381)))) & ! [X382 : subst] : ? [X383 : d_subst] : ((d2subst @ X383) = X382) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((! [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) = X0) => ! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (~! [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) = (X2 @ X6)) | ((X3 @ X5) != $true) | ((X2 @ one) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & (hoasinduction_lem1_lthm = $true) & ((d2subst @ d_id) = id) & (hoasinduction_lem2_lthm = $true) & (one = (d2term @ d_one)) & (pushprop_lem1_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X60 : term] : ((sub @ X60 @ id) = X60) => (! [X61 : term,X62 : subst > term > subst > $o] : (((X62 @ id @ (lam @ (sub @ X61 @ (push @ one @ sh))) @ id) = $true) | ~! [X68 : term] : (((X62 @ id @ (sub @ X61 @ (push @ X68 @ id)) @ id) = $true) | ((X62 @ id @ X68 @ id) != $true)) | ~! [X63 : subst > term > term] : (~! [X67 : term] : (($true = (X62 @ id @ (X63 @ id @ X67) @ id)) | ($true != (X62 @ id @ X67 @ id))) | ~! [X64 : subst,X65 : subst,X66 : term] : ((X63 @ (comp @ X65 @ X64) @ (sub @ X66 @ X64)) = (sub @ (X63 @ X65 @ X66) @ X64)) | ((X62 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true))) => ! [X69 : term,X70 : subst > term > subst > $o] : (~! [X71 : term] : (($true = (X70 @ id @ (sub @ X69 @ (push @ X71 @ id)) @ id)) | ($true != (X70 @ id @ X71 @ id))) | ~! [X72 : subst > term > term] : (~! [X76 : term] : (($true != (X70 @ id @ X76 @ id)) | ((X70 @ id @ (X72 @ id @ X76) @ id) = $true)) | ~! [X73 : subst,X74 : term,X75 : subst] : ((sub @ (X72 @ X75 @ X74) @ X73) = (X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73))) | ($true = (X70 @ id @ (lam @ (X72 @ sh @ one)) @ id))) | ((X70 @ id @ (lam @ X69) @ id) = $true))))) & (! [X139 : term,X140 : term > $o,X141 : subst] : (~! [X142 : term] : (((var @ X142) != $true) | ((X140 @ (sub @ X142 @ X141)) = $true)) | ~! [X145 : term,X146 : term] : (((X140 @ X146) != $true) | ($true = (X140 @ (ap @ X145 @ X146))) | ((X140 @ X145) != $true)) | ($true = (X140 @ (sub @ X139 @ X141))) | ~! [X143 : term] : (($true = (X140 @ (lam @ X143))) | ~! [X144 : term] : (((X140 @ X144) != $true) | ((X140 @ (sub @ X143 @ (push @ X144 @ id))) = $true)))) <=> (induction2lem = $true)) & (! [X37 : subst > term > subst > $o,X38 : term] : (~! [X41 : subst > term > term] : (~! [X42 : term,X43 : subst,X44 : subst] : ((X41 @ (comp @ X43 @ X44) @ (sub @ X42 @ X44)) = (sub @ (X41 @ X43 @ X42) @ X44)) | ~! [X45 : term] : (($true = (X37 @ id @ (X41 @ id @ X45) @ id)) | ((X37 @ id @ X45 @ id) != $true)) | ((X37 @ id @ (lam @ (X41 @ sh @ one)) @ id) = $true)) | ($true = (X37 @ id @ X38 @ id)) | ~! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (((X37 @ X51 @ X52 @ (comp @ X50 @ X53)) = $true) | ($true != (X37 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53))) | ~! [X54 : term] : (($true != (var @ (sub @ X54 @ id))) | ($true = (X37 @ id @ X54 @ id))) | ~! [X39 : term,X40 : term] : (($true = (X37 @ id @ (ap @ (sub @ X40 @ id) @ X39) @ id)) | ($true != (X37 @ id @ X40 @ id)) | ($true != (X37 @ id @ X39 @ id))) | ~! [X46 : subst,X47 : subst,X48 : subst,X49 : term] : (((X37 @ X46 @ X49 @ (comp @ X47 @ X48)) != $true) | ((X37 @ (comp @ X46 @ X47) @ (sub @ X49 @ X47) @ X48) = $true))) <=> (hoasinduction = $true)) & ((hoasapinj2 = $true) <=> ! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22)))) & (hoasapinj2_gthm = $true) & (termmset_gthm = $true) & (pushprop_lem2v2 = $true) & (induction2_gthm = $true) & (! [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) = X335) <=> (axscons = $true)) & (hoaslaminj_gthm = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & ((hoaslaminj = $true) <=> ! [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (~! [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X498 @ X502 @ X503) @ X504)) | ((X498 @ X497 @ X495) = (X496 @ X497 @ X495)) | ((lam @ (X498 @ sh @ one)) != (lam @ (X496 @ sh @ one))) | ~! [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) = (sub @ (X496 @ X499 @ X500) @ X501)))) & (hoasinduction_lem3a_gthm = $true) & (ulamvarsh = $true) & ((! [X77 : subst,X78 : term] : ((sub @ one @ (push @ X78 @ X77)) = X78) => (! [X79 : subst,X80 : term] : ((comp @ sh @ (push @ X80 @ X79)) = X79) => (! [X81 : term,X82 : term] : ((X81 = X82) | ((lam @ X81) != (lam @ X82))) => ! [X83 : term,X84 : subst > term > term,X85 : subst,X86 : subst > term > term] : (((X84 @ X85 @ X83) = (X86 @ X85 @ X83)) | ((lam @ (X84 @ sh @ one)) != (lam @ (X86 @ sh @ one))) | ~! [X90 : subst,X91 : subst,X92 : term] : ((X84 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91)) = (sub @ (X84 @ X90 @ X92) @ X91)) | ~! [X87 : subst,X88 : subst,X89 : term] : ((X86 @ (comp @ X87 @ X88) @ (sub @ X89 @ X88)) = (sub @ (X86 @ X87 @ X89) @ X88)))))) <=> (hoaslaminj_lthm = $true)) & ((induction = $true) <=> ! [X206 : term > $o,X207 : term] : (~! [X209 : term,X210 : term] : (((X206 @ (ap @ X209 @ X210)) = $true) | ($true != (X206 @ X209)) | ($true != (X206 @ X210))) | ((X206 @ X207) = $true) | ~! [X211 : term] : (($true != (var @ X211)) | ($true = (X206 @ X211))) | ~! [X208 : term] : (($true = (X206 @ (lam @ X208))) | ($true != (X206 @ X208))))) & (pushprop_lem1v2_gthm = $true)),
% 2.43/1.93 inference(flattening,[status(thm)],[f9])).
% 2.43/1.93 thf(f9,plain,(
% 2.43/1.93 ((pushprop_lem1v2_lthm = $true) <=> (! [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) = X0) => ! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (~((X3 @ X5) = $true) | ~! [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) = (X2 @ X6)) | ((X2 @ one) = $true)))) & ((hoasinduction_lem0 = $true) <=> ! [X7 : subst > term > subst > $o] : ~! [X8 : term > $o] : ~! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id))) & ((! [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) = X11) => (! [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) = X12) => ! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true = (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) | ~((X18 @ X15) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem3_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22)))) & (axclos = $true) & ! [X23 : d_term,X24 : d_term] : (((d2term @ X23) = (d2term @ X24)) => (X23 = X24)) & (! [X25 : subst > term > subst > $o,X26 : term] : (~! [X27 : term] : (~($true = (X25 @ id @ X27 @ id)) | ($true = (X25 @ id @ (sub @ X26 @ (push @ X27 @ id)) @ id))) | ($true = (X25 @ id @ (lam @ (sub @ X26 @ (push @ one @ sh))) @ id)) | ~! [X28 : term,X29 : subst > term > term] : (((X25 @ id @ (lam @ (X29 @ sh @ one)) @ id) = $true) | ~! [X30 : term] : (((X25 @ id @ (X29 @ id @ X30) @ id) = $true) | ~((X25 @ id @ X30 @ id) = $true)) | ~! [X31 : subst,X32 : term,X33 : subst] : ((sub @ (sub @ X28 @ (push @ X32 @ X33)) @ X31) = (sub @ (X29 @ X33 @ X32) @ X31)) | ~! [X34 : subst,X35 : subst,X36 : term] : ((sub @ X28 @ (push @ (sub @ X36 @ X34) @ (comp @ X35 @ X34))) = (X29 @ (comp @ X35 @ X34) @ (sub @ X36 @ X34))))) <=> (hoasinduction_lem3aaa = $true)) & (pushprop_gthm = $true) & ((hoasinduction = $true) <=> ! [X37 : subst > term > subst > $o,X38 : term] : (~! [X39 : term,X40 : term] : (~($true = (X37 @ id @ X40 @ id)) | ~($true = (X37 @ id @ X39 @ id)) | ($true = (X37 @ id @ (ap @ (sub @ X40 @ id) @ X39) @ id))) | ~! [X41 : subst > term > term] : (~! [X42 : term,X43 : subst,X44 : subst] : ((X41 @ (comp @ X43 @ X44) @ (sub @ X42 @ X44)) = (sub @ (X41 @ X43 @ X42) @ X44)) | ~! [X45 : term] : (($true = (X37 @ id @ (X41 @ id @ X45) @ id)) | ~((X37 @ id @ X45 @ id) = $true)) | ((X37 @ id @ (lam @ (X41 @ sh @ one)) @ id) = $true)) | ~! [X46 : subst,X47 : subst,X48 : subst,X49 : term] : (((X37 @ (comp @ X46 @ X47) @ (sub @ X49 @ X47) @ X48) = $true) | ~((X37 @ X46 @ X49 @ (comp @ X47 @ X48)) = $true)) | ~! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (~($true = (X37 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) | ((X37 @ X51 @ X52 @ (comp @ X50 @ X53)) = $true)) | ~! [X54 : term] : (~($true = (var @ (sub @ X54 @ id))) | ($true = (X37 @ id @ X54 @ id))) | ($true = (X37 @ id @ X38 @ id)))) & (ulamvarsh = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) = (d2subst @ X55)) => (X55 = X56)) & (pushprop_lem2v2_lthm = $true) & ((termmset = $true) <=> ! [X57 : term] : ((sub @ X57 @ id) = X57)) & (axvarshift = $true) & ((! [X59 : subst] : ((comp @ id @ X59) = X59) & ! [X58 : subst] : ((comp @ X58 @ id) = X58)) <=> (substmonoid = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lthm = $true) & (pushprop_lem1v2_gthm = $true) & ((! [X60 : term] : ((sub @ X60 @ id) = X60) => (! [X61 : term,X62 : subst > term > subst > $o] : (~! [X63 : subst > term > term] : (((X62 @ id @ (lam @ (X63 @ sh @ one)) @ id) = $true) | ~! [X64 : subst,X65 : subst,X66 : term] : ((X63 @ (comp @ X65 @ X64) @ (sub @ X66 @ X64)) = (sub @ (X63 @ X65 @ X66) @ X64)) | ~! [X67 : term] : (~($true = (X62 @ id @ X67 @ id)) | ($true = (X62 @ id @ (X63 @ id @ X67) @ id)))) | ~! [X68 : term] : (~((X62 @ id @ X68 @ id) = $true) | ((X62 @ id @ (sub @ X61 @ (push @ X68 @ id)) @ id) = $true)) | ((X62 @ id @ (lam @ (sub @ X61 @ (push @ one @ sh))) @ id) = $true)) => ! [X69 : term,X70 : subst > term > subst > $o] : (~! [X71 : term] : (~($true = (X70 @ id @ X71 @ id)) | ($true = (X70 @ id @ (sub @ X69 @ (push @ X71 @ id)) @ id))) | ~! [X72 : subst > term > term] : (~! [X73 : subst,X74 : term,X75 : subst] : ((sub @ (X72 @ X75 @ X74) @ X73) = (X72 @ (comp @ X75 @ X73) @ (sub @ X74 @ X73))) | ($true = (X70 @ id @ (lam @ (X72 @ sh @ one)) @ id)) | ~! [X76 : term] : (~($true = (X70 @ id @ X76 @ id)) | ((X70 @ id @ (X72 @ id @ X76) @ id) = $true))) | ((X70 @ id @ (lam @ X69) @ id) = $true)))) <=> (hoasinduction_lem3a_lthm = $true)) & ((! [X77 : subst,X78 : term] : ((sub @ one @ (push @ X78 @ X77)) = X78) => (! [X79 : subst,X80 : term] : ((comp @ sh @ (push @ X80 @ X79)) = X79) => (! [X81 : term,X82 : term] : ((X81 = X82) | ((lam @ X81) != (lam @ X82))) => ! [X83 : term,X84 : subst > term > term,X85 : subst,X86 : subst > term > term] : (((X84 @ X85 @ X83) = (X86 @ X85 @ X83)) | ((lam @ (X84 @ sh @ one)) != (lam @ (X86 @ sh @ one))) | ~! [X90 : subst,X91 : subst,X92 : term] : ((X84 @ (comp @ X90 @ X91) @ (sub @ X92 @ X91)) = (sub @ (X84 @ X90 @ X92) @ X91)) | ~! [X87 : subst,X88 : subst,X89 : term] : ((X86 @ (comp @ X87 @ X88) @ (sub @ X89 @ X88)) = (sub @ (X86 @ X87 @ X89) @ X88)))))) <=> (hoaslaminj_lthm = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X93 : subst > term > subst > $o] : ~! [X94 : term > $o] : ~! [X95 : term] : ((X94 @ X95) = (X93 @ id @ X95 @ id))) & (apnotvar = $true) & (pushprop = $true) & ((d2subst @ d_id) = sh) & ! [X96 : d_term] : (d_one = X96) & ((apinj2 = $true) <=> ! [X97 : term,X98 : term,X99 : term,X100 : term] : (((ap @ X99 @ X97) != (ap @ X100 @ X98)) | (X97 = X98))) & (ulamvarind = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (! [X101 : term] : ~! [X102 : subst > term > term] : ~! [X103 : term,X104 : subst] : ((sub @ X101 @ (push @ X103 @ X104)) = (X102 @ X104 @ X103)) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X105 : term,X106 : subst > term > subst > $o] : (~! [X107 : subst > term > term] : (~! [X108 : term] : (($true = (X106 @ id @ (X107 @ id @ X108) @ id)) | ~($true = (X106 @ id @ X108 @ id))) | ~! [X109 : subst,X110 : term,X111 : subst] : ((X107 @ (comp @ X111 @ X109) @ (sub @ X110 @ X109)) = (sub @ (X107 @ X111 @ X110) @ X109)) | ((X106 @ id @ (lam @ (X107 @ sh @ one)) @ id) = $true)) | ~! [X112 : term,X113 : term] : (~((X106 @ id @ X113 @ id) = $true) | ~((X106 @ id @ X112 @ id) = $true) | ($true = (X106 @ id @ (ap @ (sub @ X113 @ id) @ X112) @ id))) | ($true = (X106 @ id @ X105 @ id)))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X114 : term] : ((sub @ X114 @ id) = X114) => (! [X115 : term,X116 : subst > term > subst > $o] : (~! [X117 : subst > term > term] : (~! [X118 : term] : (($true = (X116 @ id @ (X117 @ id @ X118) @ id)) | ~($true = (X116 @ id @ X118 @ id))) | ~! [X119 : term,X120 : subst,X121 : subst] : ((X117 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)) = (sub @ (X117 @ X121 @ X119) @ X120)) | ((X116 @ id @ (lam @ (X117 @ sh @ one)) @ id) = $true)) | ((X116 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id) = $true) | ~! [X122 : term] : (~((X116 @ id @ X122 @ id) = $true) | ((X116 @ id @ (sub @ X115 @ (push @ X122 @ id)) @ id) = $true))) => ! [X123 : term,X124 : subst > term > subst > $o] : (($true = (X124 @ id @ (lam @ X123) @ id)) | ~! [X125 : subst > term > term] : (($true = (X124 @ id @ (lam @ (X125 @ sh @ one)) @ id)) | ~! [X126 : term] : (~((X124 @ id @ X126 @ id) = $true) | ((X124 @ id @ (X125 @ id @ X126) @ id) = $true)) | ~! [X127 : term,X128 : subst,X129 : subst] : ((X125 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128)) = (sub @ (X125 @ X129 @ X127) @ X128))) | ~! [X130 : term,X131 : subst,X132 : subst,X133 : subst] : (((X124 @ X132 @ X130 @ (comp @ X131 @ X133)) = $true) | ~($true = (X124 @ (comp @ X132 @ X131) @ (sub @ X130 @ X131) @ X133))) | ~! [X134 : term,X135 : subst,X136 : subst,X137 : subst] : (($true = (X124 @ (comp @ X137 @ X136) @ (sub @ X134 @ X136) @ X135)) | ~((X124 @ X137 @ X134 @ (comp @ X136 @ X135)) = $true)) | ~! [X138 : term] : (((X124 @ id @ (sub @ X123 @ (push @ X138 @ id)) @ id) = $true) | ~((X124 @ id @ X138 @ id) = $true)))))) & ((induction2lem = $true) <=> ! [X139 : term,X140 : term > $o,X141 : subst] : (~! [X142 : term] : (((X140 @ (sub @ X142 @ X141)) = $true) | ~((var @ X142) = $true)) | ~! [X143 : term] : (~! [X144 : term] : (((X140 @ (sub @ X143 @ (push @ X144 @ id))) = $true) | ~((X140 @ X144) = $true)) | ($true = (X140 @ (lam @ X143)))) | ($true = (X140 @ (sub @ X139 @ X141))) | ~! [X145 : term,X146 : term] : (~((X140 @ X145) = $true) | ~((X140 @ X146) = $true) | ($true = (X140 @ (ap @ X145 @ X146)))))) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & (! [X147 : term] : ~! [X148 : subst > term > term] : ~! [X149 : term,X150 : subst] : ((X148 @ X150 @ X149) = (sub @ X147 @ (push @ X149 @ X150))) <=> (hoasinduction_lem3v2_f = $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ((! [X151 : term] : ((sub @ X151 @ id) = X151) => (! [X152 : term,X153 : subst,X154 : term > $o] : (~! [X155 : term] : (($true = (X154 @ (lam @ X155))) | ~! [X156 : term] : (((X154 @ (sub @ X155 @ (push @ X156 @ id))) = $true) | ~((X154 @ X156) = $true))) | ~! [X157 : term,X158 : term] : (~($true = (X154 @ X158)) | ((X154 @ (ap @ X158 @ X157)) = $true) | ~((X154 @ X157) = $true)) | ((X154 @ (sub @ X152 @ X153)) = $true) | ~! [X159 : term] : (~((var @ X159) = $true) | ((X154 @ (sub @ X159 @ X153)) = $true))) => ! [X160 : term,X161 : term > $o] : (~! [X162 : term] : (($true = (X161 @ X162)) | ~($true = (var @ X162))) | ~! [X163 : term,X164 : term] : (((X161 @ (ap @ X164 @ X163)) = $true) | ~($true = (X161 @ X163)) | ~((X161 @ X164) = $true)) | ~! [X165 : term] : (~! [X166 : term] : (($true = (X161 @ (sub @ X165 @ (push @ X166 @ id)))) | ~((X161 @ X166) = $true)) | ($true = (X161 @ (lam @ X165)))) | ((X161 @ X160) = $true)))) <=> (induction2_lthm = $true)) & ((laminj = $true) <=> ! [X167 : term,X168 : term] : ((X167 = X168) | ((lam @ X168) != (lam @ X167)))) & (hoaslamnotvar_lthm = $true) & (pushprop_lem3v2_lthm = $true) & ~(ulamvar1 = $true) & ((! [X169 : term,X170 : subst] : ((sub @ one @ (push @ X169 @ X170)) = X169) => (! [X171 : subst,X172 : term] : ((comp @ sh @ (push @ X172 @ X171)) = X171) => (! [X173 : subst] : ((comp @ X173 @ id) = X173) => (! [X174 : term,X175 : term > $o] : (~! [X176 : term,X177 : term] : (~((X175 @ X176) = $true) | ~((X175 @ X177) = $true) | ($true = (X175 @ (ap @ X177 @ X176)))) | ((X175 @ X174) = $true) | ~! [X178 : term] : (($true = (X175 @ (lam @ X178))) | ~((X175 @ X178) = $true)) | ~! [X179 : term] : (($true = (X175 @ X179)) | ~($true = (var @ X179)))) => ! [X180 : term,X181 : term > $o,X182 : subst] : (((X181 @ (sub @ X180 @ X182)) = $true) | ~! [X183 : term,X184 : term] : (($true = (X181 @ (ap @ X184 @ X183))) | ~((X181 @ X184) = $true) | ~($true = (X181 @ X183))) | ~! [X185 : term] : (((X181 @ (sub @ X185 @ X182)) = $true) | ~((var @ X185) = $true)) | ~! [X186 : term] : (($true = (X181 @ (lam @ X186))) | ~! [X187 : term] : (~((X181 @ X187) = $true) | ((X181 @ (sub @ X186 @ (push @ X187 @ id))) = $true)))))))) <=> (induction2lem_lthm = $true)) & (! [X188 : subst > term > subst > $o,X189 : term,X190 : term > $o] : (~! [X191 : subst,X192 : subst,X193 : subst,X194 : term] : (~((X188 @ X193 @ X194 @ (comp @ X191 @ X192)) = $true) | ((X188 @ (comp @ X193 @ X191) @ (sub @ X194 @ X191) @ X192) = $true)) | ~! [X195 : subst,X196 : subst,X197 : subst,X198 : term] : (~((X188 @ (comp @ X197 @ X196) @ (sub @ X198 @ X196) @ X195) = $true) | ((X188 @ X197 @ X198 @ (comp @ X196 @ X195)) = $true)) | ~! [X199 : term] : (((X190 @ (sub @ X189 @ (push @ X199 @ id))) = $true) | ~((X190 @ X199) = $true)) | ((X190 @ (lam @ X189)) = $true) | ~! [X200 : subst > term > term] : (((X188 @ id @ (lam @ (X200 @ sh @ one)) @ id) = $true) | ~! [X201 : subst,X202 : subst,X203 : term] : ((X200 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201)) = (sub @ (X200 @ X202 @ X203) @ X201)) | ~! [X204 : term] : (~((X188 @ id @ X204 @ id) = $true) | ((X188 @ id @ (X200 @ id @ X204) @ id) = $true))) | ~! [X205 : term] : ((X188 @ id @ X205 @ id) = (X190 @ X205))) <=> (hoasinduction_lem3v2 = $true)) & (! [X206 : term > $o,X207 : term] : (((X206 @ X207) = $true) | ~! [X208 : term] : (($true = (X206 @ (lam @ X208))) | ~($true = (X206 @ X208))) | ~! [X209 : term,X210 : term] : (~($true = (X206 @ X209)) | ((X206 @ (ap @ X209 @ X210)) = $true) | ~($true = (X206 @ X210))) | ~! [X211 : term] : (~($true = (var @ X211)) | ($true = (X206 @ X211)))) <=> (induction = $true)) & (hoasinduction_lem1_gthm = $true) & (induction2lem_gthm = $true) & (! [X212 : term] : ~! [X213 : subst > term > term] : ((sub @ X212 @ (push @ one @ sh)) != (X213 @ sh @ one)) <=> (hoasinduction_lem3b = $true)) & (! [X214 : subst] : ((comp @ X214 @ id) = X214) <=> (axidr = $true)) & (termmset_gthm = $true) & (! [X215 : term,X216 : term > $o,X217 : term > $o,X218 : subst] : (((X216 @ one) = $true) | ~($true = (X217 @ X215)) | ~! [X219 : term] : ((X216 @ X219) = (X217 @ (sub @ X219 @ (push @ X215 @ X218))))) <=> (pushprop_lem1v2 = $true)) & ((d2subst @ d_id) = id) & ((hoasinduction_lem2v2 = $true) <=> ! [X220 : term > $o,X221 : term,X222 : subst > term > subst > $o,X223 : term] : (~((X220 @ X221) = $true) | ~! [X224 : subst,X225 : term,X226 : subst,X227 : subst] : (((X222 @ (comp @ X224 @ X227) @ (sub @ X225 @ X227) @ X226) = $true) | ~((X222 @ X224 @ X225 @ (comp @ X227 @ X226)) = $true)) | ~((X220 @ X223) = $true) | ~! [X228 : subst,X229 : subst,X230 : subst,X231 : term] : (($true = (X222 @ X230 @ X231 @ (comp @ X228 @ X229))) | ~((X222 @ (comp @ X230 @ X228) @ (sub @ X231 @ X228) @ X229) = $true)) | ((X220 @ (ap @ X221 @ X223)) = $true) | ~! [X232 : term] : ((X220 @ X232) = (X222 @ id @ X232 @ id)) | ~! [X233 : term,X234 : term] : (($true = (X222 @ id @ (ap @ (sub @ X234 @ id) @ X233) @ id)) | ~($true = (X222 @ id @ X233 @ id)) | ~((X222 @ id @ X234 @ id) = $true)))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X235 : subst > term > subst > $o,X236 : term] : (~! [X237 : subst,X238 : subst,X239 : subst,X240 : term] : (((X235 @ X238 @ X240 @ (comp @ X239 @ X237)) = $true) | ~($true = (X235 @ (comp @ X238 @ X239) @ (sub @ X240 @ X239) @ X237))) | ((X235 @ id @ (lam @ X236) @ id) = $true) | ~! [X241 : subst > term > term] : (~! [X242 : term,X243 : subst,X244 : subst] : ((X241 @ (comp @ X244 @ X243) @ (sub @ X242 @ X243)) = (sub @ (X241 @ X244 @ X242) @ X243)) | ((X235 @ id @ (lam @ (X241 @ sh @ one)) @ id) = $true) | ~! [X245 : term] : (($true = (X235 @ id @ (X241 @ id @ X245) @ id)) | ~((X235 @ id @ X245 @ id) = $true))) | ~! [X246 : term] : (~((X235 @ id @ X246 @ id) = $true) | ($true = (X235 @ id @ (sub @ X236 @ (push @ X246 @ id)) @ id))) | ~! [X247 : subst,X248 : subst,X249 : subst,X250 : term] : (($true = (X235 @ (comp @ X248 @ X249) @ (sub @ X250 @ X249) @ X247)) | ~((X235 @ X248 @ X250 @ (comp @ X249 @ X247)) = $true)))) & (hoasapnotvar_gthm = $true) & ~(lamnotap = $true) & (! [X251 : term,X252 : term > $o] : (($true = (X252 @ X251)) | ~! [X253 : term,X254 : term] : (~((X252 @ X254) = $true) | ~($true = (X252 @ X253)) | ($true = (X252 @ (ap @ X254 @ X253)))) | ~! [X255 : term] : (~! [X256 : term] : (~($true = (X252 @ X256)) | ((X252 @ (sub @ X255 @ (push @ X256 @ id))) = $true)) | ((X252 @ (lam @ X255)) = $true)) | ~! [X257 : term] : (($true = (X252 @ X257)) | ~($true = (var @ X257)))) <=> (induction2 = $true)) & (! [X258 : term,X259 : subst > term > term,X260 : term] : (~! [X261 : term,X262 : subst,X263 : subst] : ((sub @ (X259 @ X263 @ X261) @ X262) = (X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262))) | ((lam @ (X259 @ sh @ one)) != (ap @ (sub @ X260 @ id) @ X258))) <=> (hoaslamnotap = $true)) & (axmap = $true) & (one = (d2term @ d_one)) & (! [X264 : term] : ((sub @ X264 @ id) = X264) <=> (axvarid = $true)) & ((pushprop_lem1 = $true) <=> ! [X265 : term,X266 : term > $o,X267 : term > $o,X268 : term,X269 : subst] : (~((X266 @ X265) = $true) | ($true = (X267 @ (sub @ X265 @ (push @ X268 @ X269)))))) & (hoaslamnotvar = $true) & (pushprop_lem2v2_gthm = $true) & (hoaslaminj_gthm = $true) & (lamnotvar = $true) & ! [X270 : d_subst] : (d_id = X270) & ! [X271 : term] : ? [X272 : d_term] : ((d2term @ X272) = X271) & (! [X273 : subst,X274 : term] : ((comp @ sh @ (push @ X274 @ X273)) = X273) <=> (axshiftcons = $true)) & (! [X275 : term,X276 : term] : (((sub @ X276 @ sh) != (sub @ X275 @ sh)) | (X275 = X276)) <=> (shinj = $true)) & (pushprop_lthm_orig = $true) & ((! [X277 : term] : ~! [X278 : subst > term > term] : ~! [X279 : term,X280 : subst] : ((sub @ X277 @ (push @ X279 @ X280)) = (X278 @ X280 @ X279)) => (! [X281 : term] : ((sub @ X281 @ id) = X281) => ! [X282 : term > $o,X283 : subst > term > subst > $o,X284 : term] : (~! [X285 : subst > term > term] : (($true = (X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) | ~! [X286 : term] : (~($true = (X283 @ id @ X286 @ id)) | ($true = (X283 @ id @ (X285 @ id @ X286) @ id))) | ~! [X287 : term,X288 : subst,X289 : subst] : ((X285 @ (comp @ X289 @ X288) @ (sub @ X287 @ X288)) = (sub @ (X285 @ X289 @ X287) @ X288))) | ((X282 @ (lam @ X284)) = $true) | ~! [X290 : term] : ((X282 @ X290) = (X283 @ id @ X290 @ id)) | ~! [X291 : term] : (~((X282 @ X291) = $true) | ($true = (X282 @ (sub @ X284 @ (push @ X291 @ id)))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X292 : subst > term > subst > $o] : ~! [X293 : term > $o] : ~! [X294 : term] : ((X292 @ id @ X294 @ id) = (X293 @ X294)) => (! [X295 : term > $o,X296 : term] : (~! [X297 : term,X298 : term] : (~((X295 @ X297) = $true) | ($true = (X295 @ (ap @ X297 @ X298))) | ~((X295 @ X298) = $true)) | ~! [X299 : term] : (((X295 @ X299) = $true) | ~($true = (var @ X299))) | ~! [X300 : term] : (((X295 @ (lam @ X300)) = $true) | ~! [X301 : term] : (~((X295 @ X301) = $true) | ($true = (X295 @ (sub @ X300 @ (push @ X301 @ id)))))) | ((X295 @ X296) = $true)) => (! [X302 : term] : ((sub @ X302 @ id) = X302) => (! [X303 : subst > term > subst > $o,X304 : term,X305 : term > $o] : (~! [X306 : subst > term > term] : (($true = (X303 @ id @ (lam @ (X306 @ sh @ one)) @ id)) | ~! [X307 : term] : (($true = (X303 @ id @ (X306 @ id @ X307) @ id)) | ~($true = (X303 @ id @ X307 @ id))) | ~! [X308 : term,X309 : subst,X310 : subst] : ((X306 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309)) = (sub @ (X306 @ X310 @ X308) @ X309))) | ((X305 @ (lam @ X304)) = $true) | ~! [X311 : term] : (~((X305 @ X311) = $true) | ((X305 @ (sub @ X304 @ (push @ X311 @ id))) = $true)) | ~! [X312 : term] : ((X303 @ id @ X312 @ id) = (X305 @ X312))) => ! [X313 : term,X314 : subst > term > subst > $o] : (~! [X315 : subst,X316 : subst,X317 : term,X318 : subst] : (($true = (X314 @ X318 @ X317 @ (comp @ X315 @ X316))) | ~($true = (X314 @ (comp @ X318 @ X315) @ (sub @ X317 @ X315) @ X316))) | ~! [X319 : term] : (~($true = (var @ (sub @ X319 @ id))) | ((X314 @ id @ X319 @ id) = $true)) | ~! [X320 : term,X321 : term] : (~((X314 @ id @ X320 @ id) = $true) | ((X314 @ id @ (ap @ (sub @ X320 @ id) @ X321) @ id) = $true) | ~($true = (X314 @ id @ X321 @ id))) | ~! [X322 : subst,X323 : subst,X324 : term,X325 : subst] : (((X314 @ (comp @ X322 @ X323) @ (sub @ X324 @ X323) @ X325) = $true) | ~((X314 @ X322 @ X324 @ (comp @ X323 @ X325)) = $true)) | ~! [X326 : subst > term > term] : (~! [X327 : subst,X328 : term,X329 : subst] : ((X326 @ (comp @ X327 @ X329) @ (sub @ X328 @ X329)) = (sub @ (X326 @ X327 @ X328) @ X329)) | ((X314 @ id @ (lam @ (X326 @ sh @ one)) @ id) = $true) | ~! [X330 : term] : (~((X314 @ id @ X330 @ id) = $true) | ((X314 @ id @ (X326 @ id @ X330) @ id) = $true))) | ((X314 @ id @ X313 @ id) = $true)))))) <=> (hoasinduction_lthm_3 = $true)) & ((substmonoid_lthm = $true) <=> (! [X331 : subst] : ((comp @ id @ X331) = X331) => (! [X332 : subst] : ((comp @ X332 @ id) = X332) => (! [X333 : subst] : ((comp @ X333 @ id) = X333) & ! [X334 : subst] : ((comp @ id @ X334) = X334))))) & (hoasinduction_lem1_lthm = $true) & (axassoc = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_gthm = $true) & (induction2_gthm = $true) & (! [X335 : subst] : ((push @ (sub @ one @ X335) @ (comp @ sh @ X335)) = X335) <=> (axscons = $true)) & (! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) <=> (axvarcons = $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (! [X338 : term] : ~! [X339 : subst > term > term] : ((sub @ X338 @ (push @ one @ sh)) != (X339 @ sh @ one)) <=> (hoasinduction_lem3b_lthm = $true)) & ((hoasapinj1_lthm = $true) <=> (! [X340 : term] : ((sub @ X340 @ id) = X340) => (! [X341 : term,X342 : term,X343 : term,X344 : term] : ((X341 = X342) | ((ap @ X342 @ X343) != (ap @ X341 @ X344))) => ! [X345 : term,X346 : term,X347 : term,X348 : term] : (((ap @ (sub @ X345 @ id) @ X347) != (ap @ (sub @ X348 @ id) @ X346)) | (X345 = X348))))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X349 : term] : ((sub @ X349 @ id) = X349) => ! [X350 : term,X351 : subst > term > subst > $o,X352 : term > $o] : (~! [X353 : term] : ((X351 @ id @ X353 @ id) = (X352 @ X353)) | ((X352 @ (lam @ X350)) = $true) | ~! [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (~((X351 @ X356 @ X357 @ (comp @ X354 @ X355)) = $true) | ((X351 @ (comp @ X356 @ X354) @ (sub @ X357 @ X354) @ X355) = $true)) | ~! [X358 : subst,X359 : subst,X360 : term,X361 : subst] : (($true = (X351 @ X358 @ X360 @ (comp @ X359 @ X361))) | ~($true = (X351 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359) @ X361))) | ~! [X362 : subst > term > term] : (~! [X363 : term,X364 : subst,X365 : subst] : ((X362 @ (comp @ X365 @ X364) @ (sub @ X363 @ X364)) = (sub @ (X362 @ X365 @ X363) @ X364)) | ((X351 @ id @ (lam @ (X362 @ sh @ one)) @ id) = $true) | ~! [X366 : term] : (((X351 @ id @ (X362 @ id @ X366) @ id) = $true) | ~($true = (X351 @ id @ X366 @ id)))) | ~! [X367 : term] : (~((X352 @ X367) = $true) | ((X352 @ (sub @ X350 @ (push @ X367 @ id))) = $true))))) & (hoaslamnotap_gthm = $true) & (! [X368 : term,X369 : term > $o,X370 : subst > term > subst > $o] : (~! [X371 : subst > term > term] : (($true = (X370 @ id @ (lam @ (X371 @ sh @ one)) @ id)) | ~! [X372 : term] : (~($true = (X370 @ id @ X372 @ id)) | ($true = (X370 @ id @ (X371 @ id @ X372) @ id))) | ~! [X373 : term,X374 : subst,X375 : subst] : ((sub @ (X371 @ X375 @ X373) @ X374) = (X371 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))) | ~! [X376 : term] : ((X370 @ id @ X376 @ id) = (X369 @ X376)) | ~! [X377 : term] : (((X369 @ (sub @ X368 @ (push @ X377 @ id))) = $true) | ~((X369 @ X377) = $true)) | ($true = (X369 @ (lam @ X368)))) <=> (hoasinduction_lem3v2a = $true)) & (substmonoid_gthm = $true) & ((hoasapinj1 = $true) <=> ! [X378 : term,X379 : term,X380 : term,X381 : term] : ((X379 = X380) | ((ap @ (sub @ X379 @ id) @ X378) != (ap @ (sub @ X380 @ id) @ X381)))) & ! [X382 : subst] : ? [X383 : d_subst] : ((d2subst @ X383) = X382) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X384 : subst > term > subst > $o] : ~! [X385 : term > $o] : ~! [X386 : term] : ((X384 @ id @ X386 @ id) = (X385 @ X386)) => (! [X387 : term,X388 : term > $o] : (~! [X389 : term] : (($true = (X388 @ X389)) | ~($true = (var @ X389))) | ((X388 @ X387) = $true) | ~! [X390 : term] : (((X388 @ (lam @ X390)) = $true) | ~! [X391 : term] : (((X388 @ (sub @ X390 @ (push @ X391 @ id))) = $true) | ~((X388 @ X391) = $true))) | ~! [X392 : term,X393 : term] : (~($true = (X388 @ X392)) | ~((X388 @ X393) = $true) | ($true = (X388 @ (ap @ X393 @ X392))))) => (! [X394 : term] : ((sub @ X394 @ id) = X394) => (! [X395 : term > $o,X396 : term,X397 : subst > term > subst > $o] : (~! [X398 : term] : ((X397 @ id @ X398 @ id) = (X395 @ X398)) | ~! [X399 : subst > term > term] : (~! [X400 : subst,X401 : subst,X402 : term] : ((sub @ (X399 @ X400 @ X402) @ X401) = (X399 @ (comp @ X400 @ X401) @ (sub @ X402 @ X401))) | ~! [X403 : term] : (~((X397 @ id @ X403 @ id) = $true) | ($true = (X397 @ id @ (X399 @ id @ X403) @ id))) | ($true = (X397 @ id @ (lam @ (X399 @ sh @ one)) @ id))) | ($true = (X395 @ (lam @ X396))) | ~! [X404 : term] : (~((X395 @ X404) = $true) | ((X395 @ (sub @ X396 @ (push @ X404 @ id))) = $true))) => ! [X405 : term,X406 : subst > term > subst > $o] : (~! [X407 : subst > term > term] : (~! [X408 : term] : (((X406 @ id @ (X407 @ id @ X408) @ id) = $true) | ~((X406 @ id @ X408 @ id) = $true)) | ~! [X409 : subst,X410 : subst,X411 : term] : ((sub @ (X407 @ X410 @ X411) @ X409) = (X407 @ (comp @ X410 @ X409) @ (sub @ X411 @ X409))) | ($true = (X406 @ id @ (lam @ (X407 @ sh @ one)) @ id))) | ~! [X412 : term,X413 : term] : (((X406 @ id @ (ap @ (sub @ X412 @ id) @ X413) @ id) = $true) | ~((X406 @ id @ X412 @ id) = $true) | ~($true = (X406 @ id @ X413 @ id))) | ($true = (X406 @ id @ X405 @ id)))))))) & (pushprop_lem2v2 = $true) & (hoasapnotvar_lthm = $true) & (axabs = $true) & (! [X414 : term,X415 : subst > term > subst > $o] : (($true = (X415 @ id @ (lam @ (sub @ X414 @ (push @ one @ sh))) @ id)) | ~! [X416 : subst > term > term] : (~! [X417 : term] : (~((X415 @ id @ X417 @ id) = $true) | ((X415 @ id @ (X416 @ id @ X417) @ id) = $true)) | ((X415 @ id @ (lam @ (X416 @ sh @ one)) @ id) = $true) | ~! [X418 : term,X419 : subst,X420 : subst] : ((sub @ (X416 @ X419 @ X418) @ X420) = (X416 @ (comp @ X419 @ X420) @ (sub @ X418 @ X420)))) | ~! [X421 : term] : (~($true = (X415 @ id @ X421 @ id)) | ((X415 @ id @ (sub @ X414 @ (push @ X421 @ id)) @ id) = $true))) <=> (hoasinduction_lem3aa = $true)) & (hoasapnotvar = $true) & ((! [X422 : term > $o,X423 : term] : (~! [X424 : term] : (($true = (X422 @ X424)) | ~((var @ X424) = $true)) | ~! [X425 : term,X426 : term] : (~((X422 @ X426) = $true) | ((X422 @ (ap @ X426 @ X425)) = $true) | ~($true = (X422 @ X425))) | ~! [X427 : term] : (($true = (X422 @ (lam @ X427))) | ~! [X428 : term] : (((X422 @ (sub @ X427 @ (push @ X428 @ id))) = $true) | ~((X422 @ X428) = $true))) | ((X422 @ X423) = $true)) => (! [X429 : subst > term > subst > $o,X430 : term,X431 : term] : (~! [X432 : term,X433 : subst,X434 : subst,X435 : subst] : (((X429 @ X435 @ X432 @ (comp @ X434 @ X433)) = $true) | ~((X429 @ (comp @ X435 @ X434) @ (sub @ X432 @ X434) @ X433) = $true)) | ~((X429 @ id @ X430 @ id) = $true) | ~((X429 @ id @ X431 @ id) = $true) | ~! [X436 : subst,X437 : subst,X438 : term,X439 : subst] : (((X429 @ (comp @ X437 @ X436) @ (sub @ X438 @ X436) @ X439) = $true) | ~($true = (X429 @ X437 @ X438 @ (comp @ X436 @ X439)))) | ~! [X440 : term,X441 : term] : (($true = (X429 @ id @ (ap @ (sub @ X440 @ id) @ X441) @ id)) | ~((X429 @ id @ X440 @ id) = $true) | ~($true = (X429 @ id @ X441 @ id))) | ((X429 @ id @ (ap @ X431 @ X430) @ id) = $true)) => (! [X442 : term,X443 : subst > term > subst > $o] : (~! [X444 : subst > term > term] : (($true = (X443 @ id @ (lam @ (X444 @ sh @ one)) @ id)) | ~! [X445 : subst,X446 : term,X447 : subst] : ((X444 @ (comp @ X447 @ X445) @ (sub @ X446 @ X445)) = (sub @ (X444 @ X447 @ X446) @ X445)) | ~! [X448 : term] : (~((X443 @ id @ X448 @ id) = $true) | ($true = (X443 @ id @ (X444 @ id @ X448) @ id)))) | ~! [X449 : term] : (~($true = (X443 @ id @ X449 @ id)) | ((X443 @ id @ (sub @ X442 @ (push @ X449 @ id)) @ id) = $true)) | ~! [X450 : subst,X451 : term,X452 : subst,X453 : subst] : (($true = (X443 @ X452 @ X451 @ (comp @ X453 @ X450))) | ~((X443 @ (comp @ X452 @ X453) @ (sub @ X451 @ X453) @ X450) = $true)) | ~! [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454) = $true) | ~($true = (X443 @ X457 @ X456 @ (comp @ X455 @ X454)))) | ((X443 @ id @ (lam @ X442) @ id) = $true)) => ! [X458 : term,X459 : subst > term > subst > $o] : (~! [X460 : term,X461 : term] : (($true = (X459 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id)) | ~($true = (X459 @ id @ X460 @ id)) | ~($true = (X459 @ id @ X461 @ id))) | ~! [X462 : subst > term > term] : (($true = (X459 @ id @ (lam @ (X462 @ sh @ one)) @ id)) | ~! [X463 : term] : (((X459 @ id @ (X462 @ id @ X463) @ id) = $true) | ~((X459 @ id @ X463 @ id) = $true)) | ~! [X464 : subst,X465 : term,X466 : subst] : ((X462 @ (comp @ X466 @ X464) @ (sub @ X465 @ X464)) = (sub @ (X462 @ X466 @ X465) @ X464))) | ~! [X467 : term,X468 : subst,X469 : subst,X470 : subst] : (~((X459 @ (comp @ X469 @ X470) @ (sub @ X467 @ X470) @ X468) = $true) | ((X459 @ X469 @ X467 @ (comp @ X470 @ X468)) = $true)) | ~! [X471 : subst,X472 : subst,X473 : subst,X474 : term] : (~((X459 @ X473 @ X474 @ (comp @ X472 @ X471)) = $true) | ((X459 @ (comp @ X473 @ X472) @ (sub @ X474 @ X472) @ X471) = $true)) | ((X459 @ id @ X458 @ id) = $true) | ~! [X475 : term] : (~($true = (var @ (sub @ X475 @ id))) | ($true = (X459 @ id @ X475 @ id))))))) <=> (hoasinduction_lthm = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X476 : term,X477 : term,X478 : term,X479 : term] : ((X476 = X479) | ((ap @ X477 @ X476) != (ap @ X478 @ X479))) => ! [X480 : term,X481 : term,X482 : term,X483 : term] : ((X482 = X483) | ((ap @ (sub @ X481 @ id) @ X483) != (ap @ (sub @ X480 @ id) @ X482))))) & ((apinj1 = $true) <=> ! [X484 : term,X485 : term,X486 : term,X487 : term] : ((X485 = X487) | ((ap @ X485 @ X484) != (ap @ X487 @ X486)))) & (hoaslamnotvar_gthm = $true) & ((! [X488 : term] : ((sub @ X488 @ id) = X488) => ! [X489 : term] : ((sub @ X489 @ id) = X489)) <=> (termmset_lthm = $true)) & (axapp = $true) & ~((var @ (d2term @ d_one)) = $true) & (! [X490 : term > $o,X491 : term,X492 : subst] : ~! [X493 : term > $o] : ~! [X494 : term] : ((X490 @ (sub @ X494 @ (push @ X491 @ X492))) = (X493 @ X494)) <=> (pushprop_lem0 = $true)) & ((hoaslaminj = $true) <=> ! [X495 : term,X496 : subst > term > term,X497 : subst,X498 : subst > term > term] : (~! [X502 : subst,X503 : term,X504 : subst] : ((X498 @ (comp @ X502 @ X504) @ (sub @ X503 @ X504)) = (sub @ (X498 @ X502 @ X503) @ X504)) | ((X498 @ X497 @ X495) = (X496 @ X497 @ X495)) | ((lam @ (X498 @ sh @ one)) != (lam @ (X496 @ sh @ one))) | ~! [X499 : subst,X500 : term,X501 : subst] : ((X496 @ (comp @ X499 @ X501) @ (sub @ X500 @ X501)) = (sub @ (X496 @ X499 @ X500) @ X501)))) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X505 : subst > term > subst > $o,X506 : term] : (~! [X507 : term] : (((X505 @ id @ (sub @ X506 @ (push @ X507 @ id)) @ id) = $true) | ~($true = (X505 @ id @ X507 @ id))) | ((X505 @ id @ (lam @ (sub @ X506 @ (push @ one @ sh))) @ id) = $true) | ~! [X508 : subst > term > term] : (~! [X509 : term] : (($true = (X505 @ id @ (X508 @ id @ X509) @ id)) | ~($true = (X505 @ id @ X509 @ id))) | ((X505 @ id @ (lam @ (X508 @ sh @ one)) @ id) = $true) | ~! [X510 : subst,X511 : subst,X512 : term] : ((X508 @ (comp @ X511 @ X510) @ (sub @ X512 @ X510)) = (sub @ (X508 @ X511 @ X512) @ X510))))) & (hoasinduction_lem3a_gthm = $true) & (! [X513 : term,X514 : term > $o,X515 : subst] : ~! [X516 : term > $o] : ~! [X517 : term] : ((X514 @ (sub @ X517 @ (push @ X513 @ X515))) = (X516 @ X517)) <=> (pushprop_lem0_lthm = $true)) & (! [X518 : subst > term > subst > $o,X519 : term,X520 : term] : (~! [X521 : subst,X522 : subst,X523 : term,X524 : subst] : (~($true = (X518 @ (comp @ X522 @ X524) @ (sub @ X523 @ X524) @ X521)) | ($true = (X518 @ X522 @ X523 @ (comp @ X524 @ X521)))) | ~($true = (X518 @ id @ X520 @ id)) | ((X518 @ id @ (ap @ X519 @ X520) @ id) = $true) | ~! [X525 : term,X526 : term] : (~($true = (X518 @ id @ X525 @ id)) | ~($true = (X518 @ id @ X526 @ id)) | ((X518 @ id @ (ap @ (sub @ X525 @ id) @ X526) @ id) = $true)) | ~! [X527 : subst,X528 : subst,X529 : term,X530 : subst] : (($true = (X518 @ (comp @ X528 @ X530) @ (sub @ X529 @ X530) @ X527)) | ~($true = (X518 @ X528 @ X529 @ (comp @ X530 @ X527)))) | ~((X518 @ id @ X519 @ id) = $true)) <=> (hoasinduction_lem2 = $true)) & (! [X531 : subst] : ((comp @ id @ X531) = X531) <=> (axidl = $true)) & (! [X532 : term,X533 : subst > term > subst > $o] : (($true = (X533 @ id @ (lam @ X532) @ id)) | ~! [X534 : subst > term > term] : (((X533 @ id @ (lam @ (X534 @ sh @ one)) @ id) = $true) | ~! [X535 : subst,X536 : subst,X537 : term] : ((sub @ (X534 @ X535 @ X537) @ X536) = (X534 @ (comp @ X535 @ X536) @ (sub @ X537 @ X536))) | ~! [X538 : term] : (((X533 @ id @ (X534 @ id @ X538) @ id) = $true) | ~((X533 @ id @ X538 @ id) = $true))) | ~! [X539 : term] : (((X533 @ id @ (sub @ X532 @ (push @ X539 @ id)) @ id) = $true) | ~((X533 @ id @ X539 @ id) = $true))) <=> (hoasinduction_lem3a = $true)) & (hoasapinj1_gthm = $true)),
% 2.43/1.93 inference(rectify,[status(thm)],[f8])).
% 2.43/1.93 thf(f8,plain,(
% 2.43/1.93 ((pushprop_lem1v2_lthm = $true) <=> (! [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) = X0) => ! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (~((X3 @ X5) = $true) | ~! [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) = (X2 @ X6)) | ((X2 @ one) = $true)))) & ((hoasinduction_lem0 = $true) <=> ! [X7 : subst > term > subst > $o] : ~! [X8 : term > $o] : ~! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id))) & ((! [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) = X11) => (! [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) = X12) => ! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : (($true = (X14 @ (sub @ X15 @ (push @ X17 @ X16)))) | ~((X18 @ X15) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem3_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22)))) & (axclos = $true) & ! [X23 : d_term,X24 : d_term] : (((d2term @ X23) = (d2term @ X24)) => (X23 = X24)) & (! [X25 : subst > term > subst > $o,X26 : term] : (~! [X27 : term] : (~($true = (X25 @ id @ X27 @ id)) | ($true = (X25 @ id @ (sub @ X26 @ (push @ X27 @ id)) @ id))) | ($true = (X25 @ id @ (lam @ (sub @ X26 @ (push @ one @ sh))) @ id)) | ~! [X28 : term,X29 : subst > term > term] : (((X25 @ id @ (lam @ (X29 @ sh @ one)) @ id) = $true) | ~! [X30 : term] : (((X25 @ id @ (X29 @ id @ X30) @ id) = $true) | ~((X25 @ id @ X30 @ id) = $true)) | ~! [X31 : subst,X32 : term,X33 : subst] : ((sub @ (sub @ X28 @ (push @ X32 @ X33)) @ X31) = (sub @ (X29 @ X33 @ X32) @ X31)) | ~! [X34 : subst,X35 : subst,X36 : term] : ((sub @ X28 @ (push @ (sub @ X36 @ X34) @ (comp @ X35 @ X34))) = (X29 @ (comp @ X35 @ X34) @ (sub @ X36 @ X34))))) <=> (hoasinduction_lem3aaa = $true)) & (pushprop_gthm = $true) & ((hoasinduction = $true) <=> ! [X37 : subst > term > subst > $o,X38 : term] : (~! [X39 : term,X40 : term] : (~($true = (X37 @ id @ X40 @ id)) | ~($true = (X37 @ id @ X39 @ id)) | ($true = (X37 @ id @ (ap @ (sub @ X40 @ id) @ X39) @ id))) | ~! [X41 : subst > term > term] : (~! [X42 : term,X43 : subst,X44 : subst] : ((X41 @ (comp @ X43 @ X44) @ (sub @ X42 @ X44)) = (sub @ (X41 @ X43 @ X42) @ X44)) | ~! [X45 : term] : (($true = (X37 @ id @ (X41 @ id @ X45) @ id)) | ~((X37 @ id @ X45 @ id) = $true)) | ((X37 @ id @ (lam @ (X41 @ sh @ one)) @ id) = $true)) | ~! [X46 : subst,X47 : subst,X48 : subst,X49 : term] : (((X37 @ (comp @ X46 @ X47) @ (sub @ X49 @ X47) @ X48) = $true) | ~((X37 @ X46 @ X49 @ (comp @ X47 @ X48)) = $true)) | ~! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (~($true = (X37 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) | ((X37 @ X51 @ X52 @ (comp @ X50 @ X53)) = $true)) | ~! [X54 : term] : (~($true = (var @ (sub @ X54 @ id))) | ($true = (X37 @ id @ X54 @ id))) | ($true = (X37 @ id @ X38 @ id)))) & (ulamvarsh = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) = (d2subst @ X55)) => (X55 = X56)) & (pushprop_lem2v2_lthm = $true) & ((termmset = $true) <=> ! [X57 : term] : ((sub @ X57 @ id) = X57)) & (axvarshift = $true) & ((! [X59 : subst] : ((comp @ id @ X59) = X59) & ! [X58 : subst] : ((comp @ X58 @ id) = X58)) <=> (substmonoid = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lthm = $true) & (pushprop_lem1v2_gthm = $true) & ((! [X62 : term] : ((sub @ X62 @ id) = X62) => (! [X63 : term,X64 : subst > term > subst > $o] : (~! [X65 : subst > term > term] : (($true = (X64 @ id @ (lam @ (X65 @ sh @ one)) @ id)) | ~! [X66 : subst,X67 : subst,X68 : term] : ((X65 @ (comp @ X67 @ X66) @ (sub @ X68 @ X66)) = (sub @ (X65 @ X67 @ X68) @ X66)) | ~! [X69 : term] : (~($true = (X64 @ id @ X69 @ id)) | ((X64 @ id @ (X65 @ id @ X69) @ id) = $true))) | ~! [X70 : term] : (~($true = (X64 @ id @ X70 @ id)) | ((X64 @ id @ (sub @ X63 @ (push @ X70 @ id)) @ id) = $true)) | ((X64 @ id @ (lam @ (sub @ X63 @ (push @ one @ sh))) @ id) = $true)) => ! [X71 : term,X72 : subst > term > subst > $o] : (~! [X73 : term] : (~((X72 @ id @ X73 @ id) = $true) | ((X72 @ id @ (sub @ X71 @ (push @ X73 @ id)) @ id) = $true)) | ~! [X74 : subst > term > term] : (~! [X75 : subst,X76 : term,X77 : subst] : ((sub @ (X74 @ X77 @ X76) @ X75) = (X74 @ (comp @ X77 @ X75) @ (sub @ X76 @ X75))) | ((X72 @ id @ (lam @ (X74 @ sh @ one)) @ id) = $true) | ~! [X78 : term] : (~((X72 @ id @ X78 @ id) = $true) | ((X72 @ id @ (X74 @ id @ X78) @ id) = $true))) | ($true = (X72 @ id @ (lam @ X71) @ id))))) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoaslaminj_lthm = $true) <=> (! [X79 : subst,X80 : term] : ((sub @ one @ (push @ X80 @ X79)) = X80) => (! [X81 : subst,X82 : term] : ((comp @ sh @ (push @ X82 @ X81)) = X81) => (! [X83 : term,X84 : term] : (((lam @ X83) != (lam @ X84)) | (X83 = X84)) => ! [X85 : term,X86 : subst > term > term,X87 : subst,X88 : subst > term > term] : (((lam @ (X86 @ sh @ one)) != (lam @ (X88 @ sh @ one))) | ((X88 @ X87 @ X85) = (X86 @ X87 @ X85)) | ~! [X89 : subst,X90 : subst,X91 : term] : ((sub @ (X88 @ X89 @ X91) @ X90) = (X88 @ (comp @ X89 @ X90) @ (sub @ X91 @ X90))) | ~! [X92 : subst,X93 : subst,X94 : term] : ((X86 @ (comp @ X92 @ X93) @ (sub @ X94 @ X93)) = (sub @ (X86 @ X92 @ X94) @ X93))))))) & (! [X95 : subst > term > subst > $o] : ~! [X96 : term > $o] : ~! [X97 : term] : ((X95 @ id @ X97 @ id) = (X96 @ X97)) <=> (hoasinduction_lem0_lthm = $true)) & (apnotvar = $true) & (pushprop = $true) & ((d2subst @ d_id) = sh) & ! [X98 : d_term] : (d_one = X98) & (! [X99 : term,X100 : term,X101 : term,X102 : term] : ((X99 = X100) | ((ap @ X102 @ X100) != (ap @ X101 @ X99))) <=> (apinj2 = $true)) & (ulamvarind = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X108 : term] : ~! [X109 : subst > term > term] : ~! [X110 : term,X111 : subst] : ((sub @ X108 @ (push @ X110 @ X111)) = (X109 @ X111 @ X110))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X112 : term,X113 : subst > term > subst > $o] : (~! [X114 : subst > term > term] : (~! [X115 : term] : (((X113 @ id @ (X114 @ id @ X115) @ id) = $true) | ~((X113 @ id @ X115 @ id) = $true)) | ~! [X116 : subst,X117 : term,X118 : subst] : ((sub @ (X114 @ X118 @ X117) @ X116) = (X114 @ (comp @ X118 @ X116) @ (sub @ X117 @ X116))) | ((X113 @ id @ (lam @ (X114 @ sh @ one)) @ id) = $true)) | ~! [X119 : term,X120 : term] : (~($true = (X113 @ id @ X120 @ id)) | ~($true = (X113 @ id @ X119 @ id)) | ((X113 @ id @ (ap @ (sub @ X120 @ id) @ X119) @ id) = $true)) | ($true = (X113 @ id @ X112 @ id)))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X121 : term] : ((sub @ X121 @ id) = X121) => (! [X122 : term,X123 : subst > term > subst > $o] : (~! [X124 : subst > term > term] : (~! [X125 : term] : (((X123 @ id @ (X124 @ id @ X125) @ id) = $true) | ~($true = (X123 @ id @ X125 @ id))) | ~! [X126 : term,X127 : subst,X128 : subst] : ((X124 @ (comp @ X128 @ X127) @ (sub @ X126 @ X127)) = (sub @ (X124 @ X128 @ X126) @ X127)) | ((X123 @ id @ (lam @ (X124 @ sh @ one)) @ id) = $true)) | ((X123 @ id @ (lam @ (sub @ X122 @ (push @ one @ sh))) @ id) = $true) | ~! [X129 : term] : (~((X123 @ id @ X129 @ id) = $true) | ((X123 @ id @ (sub @ X122 @ (push @ X129 @ id)) @ id) = $true))) => ! [X130 : term,X131 : subst > term > subst > $o] : (($true = (X131 @ id @ (lam @ X130) @ id)) | ~! [X132 : subst > term > term] : (($true = (X131 @ id @ (lam @ (X132 @ sh @ one)) @ id)) | ~! [X133 : term] : (~($true = (X131 @ id @ X133 @ id)) | ((X131 @ id @ (X132 @ id @ X133) @ id) = $true)) | ~! [X134 : term,X135 : subst,X136 : subst] : ((sub @ (X132 @ X136 @ X134) @ X135) = (X132 @ (comp @ X136 @ X135) @ (sub @ X134 @ X135)))) | ~! [X137 : term,X138 : subst,X139 : subst,X140 : subst] : (((X131 @ X139 @ X137 @ (comp @ X138 @ X140)) = $true) | ~((X131 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X140) = $true)) | ~! [X141 : term,X142 : subst,X143 : subst,X144 : subst] : (($true = (X131 @ (comp @ X144 @ X143) @ (sub @ X141 @ X143) @ X142)) | ~($true = (X131 @ X144 @ X141 @ (comp @ X143 @ X142)))) | ~! [X145 : term] : (((X131 @ id @ (sub @ X130 @ (push @ X145 @ id)) @ id) = $true) | ~((X131 @ id @ X145 @ id) = $true)))))) & ((induction2lem = $true) <=> ! [X146 : term,X147 : term > $o,X148 : subst] : (~! [X149 : term] : (($true = (X147 @ (sub @ X149 @ X148))) | ~($true = (var @ X149))) | ~! [X150 : term] : (~! [X151 : term] : (((X147 @ (sub @ X150 @ (push @ X151 @ id))) = $true) | ~($true = (X147 @ X151))) | ((X147 @ (lam @ X150)) = $true)) | ((X147 @ (sub @ X146 @ X148)) = $true) | ~! [X152 : term,X153 : term] : (~((X147 @ X152) = $true) | ~($true = (X147 @ X153)) | ((X147 @ (ap @ X152 @ X153)) = $true)))) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X154 : term] : ~! [X155 : subst > term > term] : ~! [X156 : term,X157 : subst] : ((sub @ X154 @ (push @ X156 @ X157)) = (X155 @ X157 @ X156))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & ((! [X158 : term] : ((sub @ X158 @ id) = X158) => (! [X159 : term,X160 : subst,X161 : term > $o] : (~! [X162 : term] : (($true = (X161 @ (lam @ X162))) | ~! [X163 : term] : (($true = (X161 @ (sub @ X162 @ (push @ X163 @ id)))) | ~($true = (X161 @ X163)))) | ~! [X164 : term,X165 : term] : (~((X161 @ X165) = $true) | ((X161 @ (ap @ X165 @ X164)) = $true) | ~((X161 @ X164) = $true)) | ($true = (X161 @ (sub @ X159 @ X160))) | ~! [X166 : term] : (~((var @ X166) = $true) | ((X161 @ (sub @ X166 @ X160)) = $true))) => ! [X167 : term,X168 : term > $o] : (~! [X169 : term] : (($true = (X168 @ X169)) | ~((var @ X169) = $true)) | ~! [X170 : term,X171 : term] : (((X168 @ (ap @ X171 @ X170)) = $true) | ~((X168 @ X170) = $true) | ~($true = (X168 @ X171))) | ~! [X172 : term] : (~! [X173 : term] : (((X168 @ (sub @ X172 @ (push @ X173 @ id))) = $true) | ~((X168 @ X173) = $true)) | ((X168 @ (lam @ X172)) = $true)) | ((X168 @ X167) = $true)))) <=> (induction2_lthm = $true)) & (! [X174 : term,X175 : term] : ((X174 = X175) | ((lam @ X175) != (lam @ X174))) <=> (laminj = $true)) & (hoaslamnotvar_lthm = $true) & (pushprop_lem3v2_lthm = $true) & ~(ulamvar1 = $true) & ((! [X176 : term,X177 : subst] : ((sub @ one @ (push @ X176 @ X177)) = X176) => (! [X178 : subst,X179 : term] : ((comp @ sh @ (push @ X179 @ X178)) = X178) => (! [X180 : subst] : ((comp @ X180 @ id) = X180) => (! [X181 : term,X182 : term > $o] : (~! [X183 : term,X184 : term] : (~($true = (X182 @ X183)) | ~((X182 @ X184) = $true) | ($true = (X182 @ (ap @ X184 @ X183)))) | ((X182 @ X181) = $true) | ~! [X185 : term] : (((X182 @ (lam @ X185)) = $true) | ~($true = (X182 @ X185))) | ~! [X186 : term] : (($true = (X182 @ X186)) | ~($true = (var @ X186)))) => ! [X187 : term,X188 : term > $o,X189 : subst] : (($true = (X188 @ (sub @ X187 @ X189))) | ~! [X190 : term,X191 : term] : (($true = (X188 @ (ap @ X191 @ X190))) | ~($true = (X188 @ X191)) | ~($true = (X188 @ X190))) | ~! [X192 : term] : (($true = (X188 @ (sub @ X192 @ X189))) | ~((var @ X192) = $true)) | ~! [X193 : term] : (((X188 @ (lam @ X193)) = $true) | ~! [X194 : term] : (~((X188 @ X194) = $true) | ($true = (X188 @ (sub @ X193 @ (push @ X194 @ id))))))))))) <=> (induction2lem_lthm = $true)) & (! [X195 : subst > term > subst > $o,X196 : term,X197 : term > $o] : (~! [X198 : subst,X199 : subst,X200 : subst,X201 : term] : (~($true = (X195 @ X200 @ X201 @ (comp @ X198 @ X199))) | ($true = (X195 @ (comp @ X200 @ X198) @ (sub @ X201 @ X198) @ X199))) | ~! [X202 : subst,X203 : subst,X204 : subst,X205 : term] : (~($true = (X195 @ (comp @ X204 @ X203) @ (sub @ X205 @ X203) @ X202)) | ($true = (X195 @ X204 @ X205 @ (comp @ X203 @ X202)))) | ~! [X206 : term] : (((X197 @ (sub @ X196 @ (push @ X206 @ id))) = $true) | ~($true = (X197 @ X206))) | ((X197 @ (lam @ X196)) = $true) | ~! [X207 : subst > term > term] : (((X195 @ id @ (lam @ (X207 @ sh @ one)) @ id) = $true) | ~! [X208 : subst,X209 : subst,X210 : term] : ((X207 @ (comp @ X209 @ X208) @ (sub @ X210 @ X208)) = (sub @ (X207 @ X209 @ X210) @ X208)) | ~! [X211 : term] : (~($true = (X195 @ id @ X211 @ id)) | ((X195 @ id @ (X207 @ id @ X211) @ id) = $true))) | ~! [X212 : term] : ((X195 @ id @ X212 @ id) = (X197 @ X212))) <=> (hoasinduction_lem3v2 = $true)) & (! [X213 : term > $o,X214 : term] : (((X213 @ X214) = $true) | ~! [X215 : term] : (($true = (X213 @ (lam @ X215))) | ~($true = (X213 @ X215))) | ~! [X216 : term,X217 : term] : (~($true = (X213 @ X216)) | ($true = (X213 @ (ap @ X216 @ X217))) | ~($true = (X213 @ X217))) | ~! [X218 : term] : (~($true = (var @ X218)) | ($true = (X213 @ X218)))) <=> (induction = $true)) & (hoasinduction_lem1_gthm = $true) & (induction2lem_gthm = $true) & (! [X219 : term] : ~! [X220 : subst > term > term] : ((X220 @ sh @ one) != (sub @ X219 @ (push @ one @ sh))) <=> (hoasinduction_lem3b = $true)) & (! [X221 : subst] : ((comp @ X221 @ id) = X221) <=> (axidr = $true)) & (termmset_gthm = $true) & (! [X222 : term,X223 : term > $o,X224 : term > $o,X225 : subst] : (((X223 @ one) = $true) | ~($true = (X224 @ X222)) | ~! [X226 : term] : ((X223 @ X226) = (X224 @ (sub @ X226 @ (push @ X222 @ X225))))) <=> (pushprop_lem1v2 = $true)) & ((d2subst @ d_id) = id) & ((hoasinduction_lem2v2 = $true) <=> ! [X227 : term > $o,X228 : term,X229 : subst > term > subst > $o,X230 : term] : (~((X227 @ X228) = $true) | ~! [X231 : subst,X232 : term,X233 : subst,X234 : subst] : (((X229 @ (comp @ X231 @ X234) @ (sub @ X232 @ X234) @ X233) = $true) | ~((X229 @ X231 @ X232 @ (comp @ X234 @ X233)) = $true)) | ~((X227 @ X230) = $true) | ~! [X235 : subst,X236 : subst,X237 : subst,X238 : term] : (((X229 @ X237 @ X238 @ (comp @ X235 @ X236)) = $true) | ~((X229 @ (comp @ X237 @ X235) @ (sub @ X238 @ X235) @ X236) = $true)) | ((X227 @ (ap @ X228 @ X230)) = $true) | ~! [X239 : term] : ((X229 @ id @ X239 @ id) = (X227 @ X239)) | ~! [X240 : term,X241 : term] : (($true = (X229 @ id @ (ap @ (sub @ X241 @ id) @ X240) @ id)) | ~((X229 @ id @ X240 @ id) = $true) | ~((X229 @ id @ X241 @ id) = $true)))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X242 : subst > term > subst > $o,X243 : term] : (~! [X244 : subst,X245 : subst,X246 : subst,X247 : term] : (($true = (X242 @ X245 @ X247 @ (comp @ X246 @ X244))) | ~((X242 @ (comp @ X245 @ X246) @ (sub @ X247 @ X246) @ X244) = $true)) | ((X242 @ id @ (lam @ X243) @ id) = $true) | ~! [X248 : subst > term > term] : (~! [X249 : term,X250 : subst,X251 : subst] : ((X248 @ (comp @ X251 @ X250) @ (sub @ X249 @ X250)) = (sub @ (X248 @ X251 @ X249) @ X250)) | ((X242 @ id @ (lam @ (X248 @ sh @ one)) @ id) = $true) | ~! [X252 : term] : (((X242 @ id @ (X248 @ id @ X252) @ id) = $true) | ~((X242 @ id @ X252 @ id) = $true))) | ~! [X253 : term] : (~((X242 @ id @ X253 @ id) = $true) | ((X242 @ id @ (sub @ X243 @ (push @ X253 @ id)) @ id) = $true)) | ~! [X254 : subst,X255 : subst,X256 : subst,X257 : term] : (((X242 @ (comp @ X255 @ X256) @ (sub @ X257 @ X256) @ X254) = $true) | ~((X242 @ X255 @ X257 @ (comp @ X256 @ X254)) = $true)))) & (hoasapnotvar_gthm = $true) & ~(lamnotap = $true) & (! [X258 : term,X259 : term > $o] : (((X259 @ X258) = $true) | ~! [X260 : term,X261 : term] : (~((X259 @ X261) = $true) | ~($true = (X259 @ X260)) | ((X259 @ (ap @ X261 @ X260)) = $true)) | ~! [X262 : term] : (~! [X263 : term] : (~((X259 @ X263) = $true) | ($true = (X259 @ (sub @ X262 @ (push @ X263 @ id))))) | ((X259 @ (lam @ X262)) = $true)) | ~! [X264 : term] : (($true = (X259 @ X264)) | ~($true = (var @ X264)))) <=> (induction2 = $true)) & (! [X265 : term,X266 : subst > term > term,X267 : term] : (~! [X268 : term,X269 : subst,X270 : subst] : ((sub @ (X266 @ X270 @ X268) @ X269) = (X266 @ (comp @ X270 @ X269) @ (sub @ X268 @ X269))) | ((ap @ (sub @ X267 @ id) @ X265) != (lam @ (X266 @ sh @ one)))) <=> (hoaslamnotap = $true)) & (axmap = $true) & (one = (d2term @ d_one)) & (! [X271 : term] : ((sub @ X271 @ id) = X271) <=> (axvarid = $true)) & ((pushprop_lem1 = $true) <=> ! [X272 : term,X273 : term > $o,X274 : term > $o,X275 : term,X276 : subst] : (~((X273 @ X272) = $true) | ((X274 @ (sub @ X272 @ (push @ X275 @ X276))) = $true))) & (hoaslamnotvar = $true) & (pushprop_lem2v2_gthm = $true) & (hoaslaminj_gthm = $true) & (lamnotvar = $true) & ! [X277 : d_subst] : (d_id = X277) & ! [X278 : term] : ? [X279 : d_term] : ((d2term @ X279) = X278) & ((axshiftcons = $true) <=> ! [X280 : subst,X281 : term] : ((comp @ sh @ (push @ X281 @ X280)) = X280)) & ((shinj = $true) <=> ! [X282 : term,X283 : term] : ((X282 = X283) | ((sub @ X282 @ sh) != (sub @ X283 @ sh)))) & (pushprop_lthm_orig = $true) & ((! [X284 : term] : ~! [X285 : subst > term > term] : ~! [X286 : term,X287 : subst] : ((sub @ X284 @ (push @ X286 @ X287)) = (X285 @ X287 @ X286)) => (! [X288 : term] : ((sub @ X288 @ id) = X288) => ! [X289 : term > $o,X290 : subst > term > subst > $o,X291 : term] : (~! [X292 : subst > term > term] : (($true = (X290 @ id @ (lam @ (X292 @ sh @ one)) @ id)) | ~! [X293 : term] : (~($true = (X290 @ id @ X293 @ id)) | ((X290 @ id @ (X292 @ id @ X293) @ id) = $true)) | ~! [X294 : term,X295 : subst,X296 : subst] : ((X292 @ (comp @ X296 @ X295) @ (sub @ X294 @ X295)) = (sub @ (X292 @ X296 @ X294) @ X295))) | ((X289 @ (lam @ X291)) = $true) | ~! [X297 : term] : ((X289 @ X297) = (X290 @ id @ X297 @ id)) | ~! [X298 : term] : (~((X289 @ X298) = $true) | ((X289 @ (sub @ X291 @ (push @ X298 @ id))) = $true))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X299 : subst > term > subst > $o] : ~! [X300 : term > $o] : ~! [X301 : term] : ((X299 @ id @ X301 @ id) = (X300 @ X301)) => (! [X302 : term > $o,X303 : term] : (~! [X304 : term,X305 : term] : (~((X302 @ X304) = $true) | ($true = (X302 @ (ap @ X304 @ X305))) | ~((X302 @ X305) = $true)) | ~! [X306 : term] : (((X302 @ X306) = $true) | ~((var @ X306) = $true)) | ~! [X307 : term] : (($true = (X302 @ (lam @ X307))) | ~! [X308 : term] : (~((X302 @ X308) = $true) | ((X302 @ (sub @ X307 @ (push @ X308 @ id))) = $true))) | ($true = (X302 @ X303))) => (! [X309 : term] : ((sub @ X309 @ id) = X309) => (! [X310 : subst > term > subst > $o,X311 : term,X312 : term > $o] : (~! [X313 : subst > term > term] : (($true = (X310 @ id @ (lam @ (X313 @ sh @ one)) @ id)) | ~! [X314 : term] : (($true = (X310 @ id @ (X313 @ id @ X314) @ id)) | ~($true = (X310 @ id @ X314 @ id))) | ~! [X315 : term,X316 : subst,X317 : subst] : ((sub @ (X313 @ X317 @ X315) @ X316) = (X313 @ (comp @ X317 @ X316) @ (sub @ X315 @ X316)))) | ((X312 @ (lam @ X311)) = $true) | ~! [X318 : term] : (~($true = (X312 @ X318)) | ($true = (X312 @ (sub @ X311 @ (push @ X318 @ id))))) | ~! [X319 : term] : ((X310 @ id @ X319 @ id) = (X312 @ X319))) => ! [X320 : term,X321 : subst > term > subst > $o] : (~! [X322 : subst,X323 : subst,X324 : term,X325 : subst] : (($true = (X321 @ X325 @ X324 @ (comp @ X322 @ X323))) | ~((X321 @ (comp @ X325 @ X322) @ (sub @ X324 @ X322) @ X323) = $true)) | ~! [X326 : term] : (~($true = (var @ (sub @ X326 @ id))) | ((X321 @ id @ X326 @ id) = $true)) | ~! [X327 : term,X328 : term] : (~((X321 @ id @ X327 @ id) = $true) | ((X321 @ id @ (ap @ (sub @ X327 @ id) @ X328) @ id) = $true) | ~((X321 @ id @ X328 @ id) = $true)) | ~! [X329 : subst,X330 : subst,X331 : term,X332 : subst] : (((X321 @ (comp @ X329 @ X330) @ (sub @ X331 @ X330) @ X332) = $true) | ~((X321 @ X329 @ X331 @ (comp @ X330 @ X332)) = $true)) | ~! [X333 : subst > term > term] : (~! [X334 : subst,X335 : term,X336 : subst] : ((sub @ (X333 @ X334 @ X335) @ X336) = (X333 @ (comp @ X334 @ X336) @ (sub @ X335 @ X336))) | ((X321 @ id @ (lam @ (X333 @ sh @ one)) @ id) = $true) | ~! [X337 : term] : (~((X321 @ id @ X337 @ id) = $true) | ((X321 @ id @ (X333 @ id @ X337) @ id) = $true))) | ((X321 @ id @ X320 @ id) = $true)))))) <=> (hoasinduction_lthm_3 = $true)) & ((! [X338 : subst] : ((comp @ id @ X338) = X338) => (! [X339 : subst] : ((comp @ X339 @ id) = X339) => (! [X340 : subst] : ((comp @ X340 @ id) = X340) & ! [X341 : subst] : ((comp @ id @ X341) = X341)))) <=> (substmonoid_lthm = $true)) & (hoasinduction_lem1_lthm = $true) & (axassoc = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_gthm = $true) & (induction2_gthm = $true) & ((axscons = $true) <=> ! [X342 : subst] : ((push @ (sub @ one @ X342) @ (comp @ sh @ X342)) = X342)) & (! [X343 : subst,X344 : term] : ((sub @ one @ (push @ X344 @ X343)) = X344) <=> (axvarcons = $true)) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X345 : term] : ~! [X346 : subst > term > term] : ((X346 @ sh @ one) != (sub @ X345 @ (push @ one @ sh)))) & ((! [X347 : term] : ((sub @ X347 @ id) = X347) => (! [X348 : term,X349 : term,X350 : term,X351 : term] : ((X348 = X349) | ((ap @ X349 @ X350) != (ap @ X348 @ X351))) => ! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X352 = X355) | ((ap @ (sub @ X352 @ id) @ X354) != (ap @ (sub @ X355 @ id) @ X353))))) <=> (hoasapinj1_lthm = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X356 : term] : ((sub @ X356 @ id) = X356) => ! [X357 : term,X358 : subst > term > subst > $o,X359 : term > $o] : (~! [X360 : term] : ((X358 @ id @ X360 @ id) = (X359 @ X360)) | ((X359 @ (lam @ X357)) = $true) | ~! [X361 : subst,X362 : subst,X363 : subst,X364 : term] : (~((X358 @ X363 @ X364 @ (comp @ X361 @ X362)) = $true) | ($true = (X358 @ (comp @ X363 @ X361) @ (sub @ X364 @ X361) @ X362))) | ~! [X365 : subst,X366 : subst,X367 : term,X368 : subst] : (($true = (X358 @ X365 @ X367 @ (comp @ X366 @ X368))) | ~((X358 @ (comp @ X365 @ X366) @ (sub @ X367 @ X366) @ X368) = $true)) | ~! [X369 : subst > term > term] : (~! [X370 : term,X371 : subst,X372 : subst] : ((sub @ (X369 @ X372 @ X370) @ X371) = (X369 @ (comp @ X372 @ X371) @ (sub @ X370 @ X371))) | ($true = (X358 @ id @ (lam @ (X369 @ sh @ one)) @ id)) | ~! [X373 : term] : (((X358 @ id @ (X369 @ id @ X373) @ id) = $true) | ~((X358 @ id @ X373 @ id) = $true))) | ~! [X374 : term] : (~($true = (X359 @ X374)) | ((X359 @ (sub @ X357 @ (push @ X374 @ id))) = $true))))) & (hoaslamnotap_gthm = $true) & (! [X375 : term,X376 : term > $o,X377 : subst > term > subst > $o] : (~! [X378 : subst > term > term] : (((X377 @ id @ (lam @ (X378 @ sh @ one)) @ id) = $true) | ~! [X379 : term] : (~((X377 @ id @ X379 @ id) = $true) | ((X377 @ id @ (X378 @ id @ X379) @ id) = $true)) | ~! [X380 : term,X381 : subst,X382 : subst] : ((sub @ (X378 @ X382 @ X380) @ X381) = (X378 @ (comp @ X382 @ X381) @ (sub @ X380 @ X381)))) | ~! [X383 : term] : ((X377 @ id @ X383 @ id) = (X376 @ X383)) | ~! [X384 : term] : (((X376 @ (sub @ X375 @ (push @ X384 @ id))) = $true) | ~((X376 @ X384) = $true)) | ($true = (X376 @ (lam @ X375)))) <=> (hoasinduction_lem3v2a = $true)) & (substmonoid_gthm = $true) & ((hoasapinj1 = $true) <=> ! [X385 : term,X386 : term,X387 : term,X388 : term] : (((ap @ (sub @ X386 @ id) @ X385) != (ap @ (sub @ X387 @ id) @ X388)) | (X386 = X387))) & ! [X389 : subst] : ? [X390 : d_subst] : ((d2subst @ X390) = X389) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X394 : subst > term > subst > $o] : ~! [X395 : term > $o] : ~! [X396 : term] : ((X394 @ id @ X396 @ id) = (X395 @ X396)) => (! [X397 : term,X398 : term > $o] : (~! [X399 : term] : (($true = (X398 @ X399)) | ~((var @ X399) = $true)) | ($true = (X398 @ X397)) | ~! [X400 : term] : (((X398 @ (lam @ X400)) = $true) | ~! [X401 : term] : (($true = (X398 @ (sub @ X400 @ (push @ X401 @ id)))) | ~($true = (X398 @ X401)))) | ~! [X402 : term,X403 : term] : (~((X398 @ X402) = $true) | ~($true = (X398 @ X403)) | ($true = (X398 @ (ap @ X403 @ X402))))) => (! [X404 : term] : ((sub @ X404 @ id) = X404) => (! [X405 : term > $o,X406 : term,X407 : subst > term > subst > $o] : (~! [X408 : term] : ((X405 @ X408) = (X407 @ id @ X408 @ id)) | ~! [X409 : subst > term > term] : (~! [X410 : subst,X411 : subst,X412 : term] : ((sub @ (X409 @ X410 @ X412) @ X411) = (X409 @ (comp @ X410 @ X411) @ (sub @ X412 @ X411))) | ~! [X413 : term] : (~($true = (X407 @ id @ X413 @ id)) | ($true = (X407 @ id @ (X409 @ id @ X413) @ id))) | ($true = (X407 @ id @ (lam @ (X409 @ sh @ one)) @ id))) | ((X405 @ (lam @ X406)) = $true) | ~! [X414 : term] : (~((X405 @ X414) = $true) | ((X405 @ (sub @ X406 @ (push @ X414 @ id))) = $true))) => ! [X415 : term,X416 : subst > term > subst > $o] : (~! [X417 : subst > term > term] : (~! [X418 : term] : (((X416 @ id @ (X417 @ id @ X418) @ id) = $true) | ~((X416 @ id @ X418 @ id) = $true)) | ~! [X419 : subst,X420 : subst,X421 : term] : ((X417 @ (comp @ X420 @ X419) @ (sub @ X421 @ X419)) = (sub @ (X417 @ X420 @ X421) @ X419)) | ($true = (X416 @ id @ (lam @ (X417 @ sh @ one)) @ id))) | ~! [X422 : term,X423 : term] : (($true = (X416 @ id @ (ap @ (sub @ X422 @ id) @ X423) @ id)) | ~((X416 @ id @ X422 @ id) = $true) | ~((X416 @ id @ X423 @ id) = $true)) | ($true = (X416 @ id @ X415 @ id)))))))) & (pushprop_lem2v2 = $true) & (hoasapnotvar_lthm = $true) & (axabs = $true) & (! [X424 : term,X425 : subst > term > subst > $o] : (((X425 @ id @ (lam @ (sub @ X424 @ (push @ one @ sh))) @ id) = $true) | ~! [X426 : subst > term > term] : (~! [X427 : term] : (~((X425 @ id @ X427 @ id) = $true) | ((X425 @ id @ (X426 @ id @ X427) @ id) = $true)) | ($true = (X425 @ id @ (lam @ (X426 @ sh @ one)) @ id)) | ~! [X428 : term,X429 : subst,X430 : subst] : ((X426 @ (comp @ X429 @ X430) @ (sub @ X428 @ X430)) = (sub @ (X426 @ X429 @ X428) @ X430))) | ~! [X431 : term] : (~((X425 @ id @ X431 @ id) = $true) | ($true = (X425 @ id @ (sub @ X424 @ (push @ X431 @ id)) @ id)))) <=> (hoasinduction_lem3aa = $true)) & (hoasapnotvar = $true) & ((! [X432 : term > $o,X433 : term] : (~! [X434 : term] : (((X432 @ X434) = $true) | ~((var @ X434) = $true)) | ~! [X435 : term,X436 : term] : (~((X432 @ X436) = $true) | ((X432 @ (ap @ X436 @ X435)) = $true) | ~($true = (X432 @ X435))) | ~! [X437 : term] : (((X432 @ (lam @ X437)) = $true) | ~! [X438 : term] : (((X432 @ (sub @ X437 @ (push @ X438 @ id))) = $true) | ~((X432 @ X438) = $true))) | ((X432 @ X433) = $true)) => (! [X439 : subst > term > subst > $o,X440 : term,X441 : term] : (~! [X442 : term,X443 : subst,X444 : subst,X445 : subst] : (((X439 @ X445 @ X442 @ (comp @ X444 @ X443)) = $true) | ~((X439 @ (comp @ X445 @ X444) @ (sub @ X442 @ X444) @ X443) = $true)) | ~((X439 @ id @ X440 @ id) = $true) | ~((X439 @ id @ X441 @ id) = $true) | ~! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (((X439 @ (comp @ X447 @ X446) @ (sub @ X448 @ X446) @ X449) = $true) | ~((X439 @ X447 @ X448 @ (comp @ X446 @ X449)) = $true)) | ~! [X450 : term,X451 : term] : (((X439 @ id @ (ap @ (sub @ X450 @ id) @ X451) @ id) = $true) | ~((X439 @ id @ X450 @ id) = $true) | ~($true = (X439 @ id @ X451 @ id))) | ((X439 @ id @ (ap @ X441 @ X440) @ id) = $true)) => (! [X452 : term,X453 : subst > term > subst > $o] : (~! [X454 : subst > term > term] : (((X453 @ id @ (lam @ (X454 @ sh @ one)) @ id) = $true) | ~! [X455 : subst,X456 : term,X457 : subst] : ((X454 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455)) = (sub @ (X454 @ X457 @ X456) @ X455)) | ~! [X458 : term] : (~($true = (X453 @ id @ X458 @ id)) | ($true = (X453 @ id @ (X454 @ id @ X458) @ id)))) | ~! [X459 : term] : (~((X453 @ id @ X459 @ id) = $true) | ((X453 @ id @ (sub @ X452 @ (push @ X459 @ id)) @ id) = $true)) | ~! [X460 : subst,X461 : term,X462 : subst,X463 : subst] : (($true = (X453 @ X462 @ X461 @ (comp @ X463 @ X460))) | ~((X453 @ (comp @ X462 @ X463) @ (sub @ X461 @ X463) @ X460) = $true)) | ~! [X464 : subst,X465 : subst,X466 : term,X467 : subst] : (((X453 @ (comp @ X467 @ X465) @ (sub @ X466 @ X465) @ X464) = $true) | ~($true = (X453 @ X467 @ X466 @ (comp @ X465 @ X464)))) | ((X453 @ id @ (lam @ X452) @ id) = $true)) => ! [X468 : term,X469 : subst > term > subst > $o] : (~! [X470 : term,X471 : term] : (((X469 @ id @ (ap @ (sub @ X470 @ id) @ X471) @ id) = $true) | ~((X469 @ id @ X470 @ id) = $true) | ~((X469 @ id @ X471 @ id) = $true)) | ~! [X472 : subst > term > term] : (($true = (X469 @ id @ (lam @ (X472 @ sh @ one)) @ id)) | ~! [X473 : term] : (((X469 @ id @ (X472 @ id @ X473) @ id) = $true) | ~((X469 @ id @ X473 @ id) = $true)) | ~! [X474 : subst,X475 : term,X476 : subst] : ((sub @ (X472 @ X476 @ X475) @ X474) = (X472 @ (comp @ X476 @ X474) @ (sub @ X475 @ X474)))) | ~! [X477 : term,X478 : subst,X479 : subst,X480 : subst] : (~((X469 @ (comp @ X479 @ X480) @ (sub @ X477 @ X480) @ X478) = $true) | ((X469 @ X479 @ X477 @ (comp @ X480 @ X478)) = $true)) | ~! [X481 : subst,X482 : subst,X483 : subst,X484 : term] : (~($true = (X469 @ X483 @ X484 @ (comp @ X482 @ X481))) | ((X469 @ (comp @ X483 @ X482) @ (sub @ X484 @ X482) @ X481) = $true)) | ((X469 @ id @ X468 @ id) = $true) | ~! [X485 : term] : (~((var @ (sub @ X485 @ id)) = $true) | ($true = (X469 @ id @ X485 @ id))))))) <=> (hoasinduction_lthm = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X486 : term,X487 : term,X488 : term,X489 : term] : ((X486 = X489) | ((ap @ X488 @ X489) != (ap @ X487 @ X486))) => ! [X490 : term,X491 : term,X492 : term,X493 : term] : ((X492 = X493) | ((ap @ (sub @ X491 @ id) @ X493) != (ap @ (sub @ X490 @ id) @ X492))))) & (! [X494 : term,X495 : term,X496 : term,X497 : term] : (((ap @ X497 @ X496) != (ap @ X495 @ X494)) | (X495 = X497)) <=> (apinj1 = $true)) & (hoaslamnotvar_gthm = $true) & ((termmset_lthm = $true) <=> (! [X498 : term] : ((sub @ X498 @ id) = X498) => ! [X499 : term] : ((sub @ X499 @ id) = X499))) & (axapp = $true) & ~((var @ (d2term @ d_one)) = $true) & (! [X500 : term > $o,X501 : term,X502 : subst] : ~! [X503 : term > $o] : ~! [X504 : term] : ((X500 @ (sub @ X504 @ (push @ X501 @ X502))) = (X503 @ X504)) <=> (pushprop_lem0 = $true)) & (! [X505 : term,X506 : subst > term > term,X507 : subst,X508 : subst > term > term] : (~! [X509 : subst,X510 : term,X511 : subst] : ((X506 @ (comp @ X509 @ X511) @ (sub @ X510 @ X511)) = (sub @ (X506 @ X509 @ X510) @ X511)) | ~! [X512 : subst,X513 : term,X514 : subst] : ((sub @ (X508 @ X512 @ X513) @ X514) = (X508 @ (comp @ X512 @ X514) @ (sub @ X513 @ X514))) | ((X508 @ X507 @ X505) = (X506 @ X507 @ X505)) | ((lam @ (X506 @ sh @ one)) != (lam @ (X508 @ sh @ one)))) <=> (hoaslaminj = $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X515 : subst > term > subst > $o,X516 : term] : (~! [X517 : term] : (((X515 @ id @ (sub @ X516 @ (push @ X517 @ id)) @ id) = $true) | ~($true = (X515 @ id @ X517 @ id))) | ((X515 @ id @ (lam @ (sub @ X516 @ (push @ one @ sh))) @ id) = $true) | ~! [X518 : subst > term > term] : (~! [X519 : term] : (((X515 @ id @ (X518 @ id @ X519) @ id) = $true) | ~((X515 @ id @ X519 @ id) = $true)) | ((X515 @ id @ (lam @ (X518 @ sh @ one)) @ id) = $true) | ~! [X520 : subst,X521 : subst,X522 : term] : ((X518 @ (comp @ X521 @ X520) @ (sub @ X522 @ X520)) = (sub @ (X518 @ X521 @ X522) @ X520))))) & (hoasinduction_lem3a_gthm = $true) & (! [X523 : term,X524 : term > $o,X525 : subst] : ~! [X526 : term > $o] : ~! [X527 : term] : ((X524 @ (sub @ X527 @ (push @ X523 @ X525))) = (X526 @ X527)) <=> (pushprop_lem0_lthm = $true)) & (! [X528 : subst > term > subst > $o,X529 : term,X530 : term] : (~! [X531 : subst,X532 : subst,X533 : term,X534 : subst] : (~($true = (X528 @ (comp @ X532 @ X534) @ (sub @ X533 @ X534) @ X531)) | ((X528 @ X532 @ X533 @ (comp @ X534 @ X531)) = $true)) | ~((X528 @ id @ X530 @ id) = $true) | ((X528 @ id @ (ap @ X529 @ X530) @ id) = $true) | ~! [X535 : term,X536 : term] : (~((X528 @ id @ X535 @ id) = $true) | ~((X528 @ id @ X536 @ id) = $true) | ($true = (X528 @ id @ (ap @ (sub @ X535 @ id) @ X536) @ id))) | ~! [X537 : subst,X538 : subst,X539 : term,X540 : subst] : (($true = (X528 @ (comp @ X538 @ X540) @ (sub @ X539 @ X540) @ X537)) | ~((X528 @ X538 @ X539 @ (comp @ X540 @ X537)) = $true)) | ~((X528 @ id @ X529 @ id) = $true)) <=> (hoasinduction_lem2 = $true)) & ((axidl = $true) <=> ! [X541 : subst] : ((comp @ id @ X541) = X541)) & (! [X542 : term,X543 : subst > term > subst > $o] : (($true = (X543 @ id @ (lam @ X542) @ id)) | ~! [X544 : subst > term > term] : (($true = (X543 @ id @ (lam @ (X544 @ sh @ one)) @ id)) | ~! [X545 : subst,X546 : subst,X547 : term] : ((X544 @ (comp @ X545 @ X546) @ (sub @ X547 @ X546)) = (sub @ (X544 @ X545 @ X547) @ X546)) | ~! [X548 : term] : (($true = (X543 @ id @ (X544 @ id @ X548) @ id)) | ~($true = (X543 @ id @ X548 @ id)))) | ~! [X549 : term] : (((X543 @ id @ (sub @ X542 @ (push @ X549 @ id)) @ id) = $true) | ~((X543 @ id @ X549 @ id) = $true))) <=> (hoasinduction_lem3a = $true)) & (hoasapinj1_gthm = $true)),
% 2.43/1.93 inference(fool_elimination,[status(thm)],[f7])).
% 2.43/1.93 thf(f7,plain,(
% 2.43/1.93 (pushprop_lem1v2_lthm <=> (! [X0 : term,X1 : subst] : ((sub @ one @ (push @ X0 @ X1)) = X0) => ! [X2 : term > $o,X3 : term > $o,X4 : subst,X5 : term] : (~(X3 @ X5) | ~! [X6 : term] : ((X3 @ (sub @ X6 @ (push @ X5 @ X4))) = (X2 @ X6)) | (X2 @ one)))) & (! [X7 : subst > term > subst > $o] : ~! [X8 : term > $o] : ~! [X9 : term] : ((X8 @ X9) = (X7 @ id @ X9 @ id)) <=> hoasinduction_lem0) & ((! [X10 : subst,X11 : term] : ((sub @ one @ (push @ X11 @ X10)) = X11) => (! [X12 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X12)) = X12) => ! [X14 : term > $o,X15 : term,X16 : subst,X17 : term,X18 : term > $o] : ((X14 @ (sub @ X15 @ (push @ X17 @ X16))) | ~(X18 @ X15)))) <=> pushprop_lem1_lthm) & hoasinduction_lem3_gthm & (! [X19 : term,X20 : term,X21 : term,X22 : term] : ((X20 = X22) | ((ap @ (sub @ X19 @ id) @ X20) != (ap @ (sub @ X21 @ id) @ X22))) <=> hoasapinj2) & axclos & ! [X23 : d_term,X24 : d_term] : (((d2term @ X23) = (d2term @ X24)) => (X23 = X24)) & (! [X25 : subst > term > subst > $o,X26 : term] : (~! [X27 : term] : (~(X25 @ id @ X27 @ id) | (X25 @ id @ (sub @ X26 @ (push @ X27 @ id)) @ id)) | (X25 @ id @ (lam @ (sub @ X26 @ (push @ one @ sh))) @ id) | ~! [X28 : term,X29 : subst > term > term] : ((X25 @ id @ (lam @ (X29 @ sh @ one)) @ id) | ~! [X30 : term] : ((X25 @ id @ (X29 @ id @ X30) @ id) | ~(X25 @ id @ X30 @ id)) | ~! [X31 : subst,X32 : term,X33 : subst] : ((sub @ (sub @ X28 @ (push @ X32 @ X33)) @ X31) = (sub @ (X29 @ X33 @ X32) @ X31)) | ~! [X34 : subst,X35 : subst,X36 : term] : ((sub @ X28 @ (push @ (sub @ X36 @ X34) @ (comp @ X35 @ X34))) = (X29 @ (comp @ X35 @ X34) @ (sub @ X36 @ X34))))) <=> hoasinduction_lem3aaa) & pushprop_gthm & (hoasinduction <=> ! [X37 : subst > term > subst > $o,X38 : term] : (~! [X39 : term,X40 : term] : (~(X37 @ id @ X40 @ id) | ~(X37 @ id @ X39 @ id) | (X37 @ id @ (ap @ (sub @ X40 @ id) @ X39) @ id)) | ~! [X41 : subst > term > term] : (~! [X42 : term,X43 : subst,X44 : subst] : ((X41 @ (comp @ X43 @ X44) @ (sub @ X42 @ X44)) = (sub @ (X41 @ X43 @ X42) @ X44)) | ~! [X45 : term] : ((X37 @ id @ (X41 @ id @ X45) @ id) | ~(X37 @ id @ X45 @ id)) | (X37 @ id @ (lam @ (X41 @ sh @ one)) @ id)) | ~! [X46 : subst,X47 : subst,X48 : subst,X49 : term] : ((X37 @ (comp @ X46 @ X47) @ (sub @ X49 @ X47) @ X48) | ~(X37 @ X46 @ X49 @ (comp @ X47 @ X48))) | ~! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (~(X37 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53) | (X37 @ X51 @ X52 @ (comp @ X50 @ X53))) | ~! [X54 : term] : (~(var @ (sub @ X54 @ id)) | (X37 @ id @ X54 @ id)) | (X37 @ id @ X38 @ id))) & ulamvarsh & hoasapinj2_gthm & hoasinduction_lem3v2_gthm & ! [X55 : d_subst,X56 : d_subst] : (((d2subst @ X56) = (d2subst @ X55)) => (X55 = X56)) & pushprop_lem2v2_lthm & (termmset <=> ! [X57 : term] : ((sub @ X57 @ id) = X57)) & axvarshift & (substmonoid <=> (! [X58 : subst] : ((comp @ X58 @ id) = X58) & ! [X59 : subst] : ((comp @ id @ X59) = X59))) & ((^[X60 : subst, X61 : subst > term > term] : (d2term @ d_one)) = hoaslam) & pushprop_lthm & pushprop_lem1v2_gthm & ((! [X62 : term] : ((sub @ X62 @ id) = X62) => (! [X63 : term,X64 : subst > term > subst > $o] : (~! [X65 : subst > term > term] : ((X64 @ id @ (lam @ (X65 @ sh @ one)) @ id) | ~! [X66 : subst,X67 : subst,X68 : term] : ((X65 @ (comp @ X67 @ X66) @ (sub @ X68 @ X66)) = (sub @ (X65 @ X67 @ X68) @ X66)) | ~! [X69 : term] : (~(X64 @ id @ X69 @ id) | (X64 @ id @ (X65 @ id @ X69) @ id))) | ~! [X70 : term] : (~(X64 @ id @ X70 @ id) | (X64 @ id @ (sub @ X63 @ (push @ X70 @ id)) @ id)) | (X64 @ id @ (lam @ (sub @ X63 @ (push @ one @ sh))) @ id)) => ! [X71 : term,X72 : subst > term > subst > $o] : (~! [X73 : term] : (~(X72 @ id @ X73 @ id) | (X72 @ id @ (sub @ X71 @ (push @ X73 @ id)) @ id)) | ~! [X74 : subst > term > term] : (~! [X75 : subst,X76 : term,X77 : subst] : ((sub @ (X74 @ X77 @ X76) @ X75) = (X74 @ (comp @ X77 @ X75) @ (sub @ X76 @ X75))) | (X72 @ id @ (lam @ (X74 @ sh @ one)) @ id) | ~! [X78 : term] : (~(X72 @ id @ X78 @ id) | (X72 @ id @ (X74 @ id @ X78) @ id))) | (X72 @ id @ (lam @ X71) @ id)))) <=> hoasinduction_lem3a_lthm) & (hoaslaminj_lthm <=> (! [X79 : subst,X80 : term] : ((sub @ one @ (push @ X80 @ X79)) = X80) => (! [X81 : subst,X82 : term] : ((comp @ sh @ (push @ X82 @ X81)) = X81) => (! [X83 : term,X84 : term] : (((lam @ X83) != (lam @ X84)) | (X83 = X84)) => ! [X85 : term,X86 : subst > term > term,X87 : subst,X88 : subst > term > term] : (((lam @ (X86 @ sh @ one)) != (lam @ (X88 @ sh @ one))) | ((X88 @ X87 @ X85) = (X86 @ X87 @ X85)) | ~! [X89 : subst,X90 : subst,X91 : term] : ((sub @ (X88 @ X89 @ X91) @ X90) = (X88 @ (comp @ X89 @ X90) @ (sub @ X91 @ X90))) | ~! [X92 : subst,X93 : subst,X94 : term] : ((X86 @ (comp @ X92 @ X93) @ (sub @ X94 @ X93)) = (sub @ (X86 @ X92 @ X94) @ X93))))))) & (! [X95 : subst > term > subst > $o] : ~! [X96 : term > $o] : ~! [X97 : term] : ((X95 @ id @ X97 @ id) = (X96 @ X97)) <=> hoasinduction_lem0_lthm) & apnotvar & pushprop & ((d2subst @ d_id) = sh) & ! [X98 : d_term] : (d_one = X98) & (! [X99 : term,X100 : term,X101 : term,X102 : term] : ((X99 = X100) | ((ap @ X102 @ X100) != (ap @ X101 @ X99))) <=> apinj2) & ulamvarind & ((^[X103 : term, X104 : subst, X105 : term > $o, X106 : term > $o] : (! [X107 : term] : ((X106 @ X107) = (X105 @ (sub @ X107 @ (push @ X103 @ X104)))))) = pushprop_p_and_p_prime) & (hoasinduction_lem3v2_f_lthm <=> ! [X108 : term] : ~! [X109 : subst > term > term] : ~! [X110 : term,X111 : subst] : ((sub @ X108 @ (push @ X110 @ X111)) = (X109 @ X111 @ X110))) & (hoasinduction_no_psi_cond <=> ! [X112 : term,X113 : subst > term > subst > $o] : (~! [X114 : subst > term > term] : (~! [X115 : term] : ((X113 @ id @ (X114 @ id @ X115) @ id) | ~(X113 @ id @ X115 @ id)) | ~! [X116 : subst,X117 : term,X118 : subst] : ((sub @ (X114 @ X118 @ X117) @ X116) = (X114 @ (comp @ X118 @ X116) @ (sub @ X117 @ X116))) | (X113 @ id @ (lam @ (X114 @ sh @ one)) @ id)) | ~! [X119 : term,X120 : term] : (~(X113 @ id @ X120 @ id) | ~(X113 @ id @ X119 @ id) | (X113 @ id @ (ap @ (sub @ X120 @ id) @ X119) @ id)) | (X113 @ id @ X112 @ id))) & (hoasinduction_lem3_lthm <=> (! [X121 : term] : ((sub @ X121 @ id) = X121) => (! [X122 : term,X123 : subst > term > subst > $o] : (~! [X124 : subst > term > term] : (~! [X125 : term] : ((X123 @ id @ (X124 @ id @ X125) @ id) | ~(X123 @ id @ X125 @ id)) | ~! [X126 : term,X127 : subst,X128 : subst] : ((X124 @ (comp @ X128 @ X127) @ (sub @ X126 @ X127)) = (sub @ (X124 @ X128 @ X126) @ X127)) | (X123 @ id @ (lam @ (X124 @ sh @ one)) @ id)) | (X123 @ id @ (lam @ (sub @ X122 @ (push @ one @ sh))) @ id) | ~! [X129 : term] : (~(X123 @ id @ X129 @ id) | (X123 @ id @ (sub @ X122 @ (push @ X129 @ id)) @ id))) => ! [X130 : term,X131 : subst > term > subst > $o] : ((X131 @ id @ (lam @ X130) @ id) | ~! [X132 : subst > term > term] : ((X131 @ id @ (lam @ (X132 @ sh @ one)) @ id) | ~! [X133 : term] : (~(X131 @ id @ X133 @ id) | (X131 @ id @ (X132 @ id @ X133) @ id)) | ~! [X134 : term,X135 : subst,X136 : subst] : ((sub @ (X132 @ X136 @ X134) @ X135) = (X132 @ (comp @ X136 @ X135) @ (sub @ X134 @ X135)))) | ~! [X137 : term,X138 : subst,X139 : subst,X140 : subst] : ((X131 @ X139 @ X137 @ (comp @ X138 @ X140)) | ~(X131 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X140)) | ~! [X141 : term,X142 : subst,X143 : subst,X144 : subst] : ((X131 @ (comp @ X144 @ X143) @ (sub @ X141 @ X143) @ X142) | ~(X131 @ X144 @ X141 @ (comp @ X143 @ X142))) | ~! [X145 : term] : ((X131 @ id @ (sub @ X130 @ (push @ X145 @ id)) @ id) | ~(X131 @ id @ X145 @ id)))))) & (induction2lem <=> ! [X146 : term,X147 : term > $o,X148 : subst] : (~! [X149 : term] : ((X147 @ (sub @ X149 @ X148)) | ~(var @ X149)) | ~! [X150 : term] : (~! [X151 : term] : ((X147 @ (sub @ X150 @ (push @ X151 @ id))) | ~(X147 @ X151)) | (X147 @ (lam @ X150))) | (X147 @ (sub @ X146 @ X148)) | ~! [X152 : term,X153 : term] : (~(X147 @ X152) | ~(X147 @ X153) | (X147 @ (ap @ X152 @ X153))))) & pushprop_lem3v2 & hoasinduction_lem1v2 & (hoasinduction_lem3v2_f <=> ! [X154 : term] : ~! [X155 : subst > term > term] : ~! [X156 : term,X157 : subst] : ((sub @ X154 @ (push @ X156 @ X157)) = (X155 @ X157 @ X156))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ((! [X158 : term] : ((sub @ X158 @ id) = X158) => (! [X159 : term,X160 : subst,X161 : term > $o] : (~! [X162 : term] : ((X161 @ (lam @ X162)) | ~! [X163 : term] : ((X161 @ (sub @ X162 @ (push @ X163 @ id))) | ~(X161 @ X163))) | ~! [X164 : term,X165 : term] : (~(X161 @ X165) | (X161 @ (ap @ X165 @ X164)) | ~(X161 @ X164)) | (X161 @ (sub @ X159 @ X160)) | ~! [X166 : term] : (~(var @ X166) | (X161 @ (sub @ X166 @ X160)))) => ! [X167 : term,X168 : term > $o] : (~! [X169 : term] : ((X168 @ X169) | ~(var @ X169)) | ~! [X170 : term,X171 : term] : ((X168 @ (ap @ X171 @ X170)) | ~(X168 @ X170) | ~(X168 @ X171)) | ~! [X172 : term] : (~! [X173 : term] : ((X168 @ (sub @ X172 @ (push @ X173 @ id))) | ~(X168 @ X173)) | (X168 @ (lam @ X172))) | (X168 @ X167)))) <=> induction2_lthm) & (! [X174 : term,X175 : term] : ((X174 = X175) | ((lam @ X175) != (lam @ X174))) <=> laminj) & hoaslamnotvar_lthm & pushprop_lem3v2_lthm & ~ulamvar1 & ((! [X176 : term,X177 : subst] : ((sub @ one @ (push @ X176 @ X177)) = X176) => (! [X178 : subst,X179 : term] : ((comp @ sh @ (push @ X179 @ X178)) = X178) => (! [X180 : subst] : ((comp @ X180 @ id) = X180) => (! [X181 : term,X182 : term > $o] : (~! [X183 : term,X184 : term] : (~(X182 @ X183) | ~(X182 @ X184) | (X182 @ (ap @ X184 @ X183))) | (X182 @ X181) | ~! [X185 : term] : ((X182 @ (lam @ X185)) | ~(X182 @ X185)) | ~! [X186 : term] : ((X182 @ X186) | ~(var @ X186))) => ! [X187 : term,X188 : term > $o,X189 : subst] : ((X188 @ (sub @ X187 @ X189)) | ~! [X190 : term,X191 : term] : ((X188 @ (ap @ X191 @ X190)) | ~(X188 @ X191) | ~(X188 @ X190)) | ~! [X192 : term] : ((X188 @ (sub @ X192 @ X189)) | ~(var @ X192)) | ~! [X193 : term] : ((X188 @ (lam @ X193)) | ~! [X194 : term] : (~(X188 @ X194) | (X188 @ (sub @ X193 @ (push @ X194 @ id)))))))))) <=> induction2lem_lthm) & (! [X195 : subst > term > subst > $o,X196 : term,X197 : term > $o] : (~! [X198 : subst,X199 : subst,X200 : subst,X201 : term] : (~(X195 @ X200 @ X201 @ (comp @ X198 @ X199)) | (X195 @ (comp @ X200 @ X198) @ (sub @ X201 @ X198) @ X199)) | ~! [X202 : subst,X203 : subst,X204 : subst,X205 : term] : (~(X195 @ (comp @ X204 @ X203) @ (sub @ X205 @ X203) @ X202) | (X195 @ X204 @ X205 @ (comp @ X203 @ X202))) | ~! [X206 : term] : ((X197 @ (sub @ X196 @ (push @ X206 @ id))) | ~(X197 @ X206)) | (X197 @ (lam @ X196)) | ~! [X207 : subst > term > term] : ((X195 @ id @ (lam @ (X207 @ sh @ one)) @ id) | ~! [X208 : subst,X209 : subst,X210 : term] : ((X207 @ (comp @ X209 @ X208) @ (sub @ X210 @ X208)) = (sub @ (X207 @ X209 @ X210) @ X208)) | ~! [X211 : term] : (~(X195 @ id @ X211 @ id) | (X195 @ id @ (X207 @ id @ X211) @ id))) | ~! [X212 : term] : ((X195 @ id @ X212 @ id) = (X197 @ X212))) <=> hoasinduction_lem3v2) & (! [X213 : term > $o,X214 : term] : ((X213 @ X214) | ~! [X215 : term] : ((X213 @ (lam @ X215)) | ~(X213 @ X215)) | ~! [X216 : term,X217 : term] : (~(X213 @ X216) | (X213 @ (ap @ X216 @ X217)) | ~(X213 @ X217)) | ~! [X218 : term] : (~(var @ X218) | (X213 @ X218))) <=> induction) & hoasinduction_lem1_gthm & induction2lem_gthm & (! [X219 : term] : ~! [X220 : subst > term > term] : ((X220 @ sh @ one) != (sub @ X219 @ (push @ one @ sh))) <=> hoasinduction_lem3b) & (! [X221 : subst] : ((comp @ X221 @ id) = X221) <=> axidr) & termmset_gthm & (! [X222 : term,X223 : term > $o,X224 : term > $o,X225 : subst] : ((X223 @ one) | ~(X224 @ X222) | ~! [X226 : term] : ((X223 @ X226) = (X224 @ (sub @ X226 @ (push @ X222 @ X225))))) <=> pushprop_lem1v2) & ((d2subst @ d_id) = id) & (hoasinduction_lem2v2 <=> ! [X227 : term > $o,X228 : term,X229 : subst > term > subst > $o,X230 : term] : (~(X227 @ X228) | ~! [X231 : subst,X232 : term,X233 : subst,X234 : subst] : ((X229 @ (comp @ X231 @ X234) @ (sub @ X232 @ X234) @ X233) | ~(X229 @ X231 @ X232 @ (comp @ X234 @ X233))) | ~(X227 @ X230) | ~! [X235 : subst,X236 : subst,X237 : subst,X238 : term] : ((X229 @ X237 @ X238 @ (comp @ X235 @ X236)) | ~(X229 @ (comp @ X237 @ X235) @ (sub @ X238 @ X235) @ X236)) | (X227 @ (ap @ X228 @ X230)) | ~! [X239 : term] : ((X229 @ id @ X239 @ id) = (X227 @ X239)) | ~! [X240 : term,X241 : term] : ((X229 @ id @ (ap @ (sub @ X241 @ id) @ X240) @ id) | ~(X229 @ id @ X240 @ id) | ~(X229 @ id @ X241 @ id)))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & hoaslamnotap_lthm & (hoasinduction_lem3 <=> ! [X242 : subst > term > subst > $o,X243 : term] : (~! [X244 : subst,X245 : subst,X246 : subst,X247 : term] : ((X242 @ X245 @ X247 @ (comp @ X246 @ X244)) | ~(X242 @ (comp @ X245 @ X246) @ (sub @ X247 @ X246) @ X244)) | (X242 @ id @ (lam @ X243) @ id) | ~! [X248 : subst > term > term] : (~! [X249 : term,X250 : subst,X251 : subst] : ((X248 @ (comp @ X251 @ X250) @ (sub @ X249 @ X250)) = (sub @ (X248 @ X251 @ X249) @ X250)) | (X242 @ id @ (lam @ (X248 @ sh @ one)) @ id) | ~! [X252 : term] : ((X242 @ id @ (X248 @ id @ X252) @ id) | ~(X242 @ id @ X252 @ id))) | ~! [X253 : term] : (~(X242 @ id @ X253 @ id) | (X242 @ id @ (sub @ X243 @ (push @ X253 @ id)) @ id)) | ~! [X254 : subst,X255 : subst,X256 : subst,X257 : term] : ((X242 @ (comp @ X255 @ X256) @ (sub @ X257 @ X256) @ X254) | ~(X242 @ X255 @ X257 @ (comp @ X256 @ X254))))) & hoasapnotvar_gthm & ~lamnotap & (! [X258 : term,X259 : term > $o] : ((X259 @ X258) | ~! [X260 : term,X261 : term] : (~(X259 @ X261) | ~(X259 @ X260) | (X259 @ (ap @ X261 @ X260))) | ~! [X262 : term] : (~! [X263 : term] : (~(X259 @ X263) | (X259 @ (sub @ X262 @ (push @ X263 @ id)))) | (X259 @ (lam @ X262))) | ~! [X264 : term] : ((X259 @ X264) | ~(var @ X264))) <=> induction2) & (! [X265 : term,X266 : subst > term > term,X267 : term] : (~! [X268 : term,X269 : subst,X270 : subst] : ((sub @ (X266 @ X270 @ X268) @ X269) = (X266 @ (comp @ X270 @ X269) @ (sub @ X268 @ X269))) | ((ap @ (sub @ X267 @ id) @ X265) != (lam @ (X266 @ sh @ one)))) <=> hoaslamnotap) & axmap & (one = (d2term @ d_one)) & (! [X271 : term] : ((sub @ X271 @ id) = X271) <=> axvarid) & (pushprop_lem1 <=> ! [X272 : term,X273 : term > $o,X274 : term > $o,X275 : term,X276 : subst] : (~(X273 @ X272) | (X274 @ (sub @ X272 @ (push @ X275 @ X276))))) & hoaslamnotvar & pushprop_lem2v2_gthm & hoaslaminj_gthm & lamnotvar & ! [X277 : d_subst] : (d_id = X277) & ! [X278 : term] : ? [X279 : d_term] : ((d2term @ X279) = X278) & (axshiftcons <=> ! [X280 : subst,X281 : term] : ((comp @ sh @ (push @ X281 @ X280)) = X280)) & (shinj <=> ! [X282 : term,X283 : term] : ((X282 = X283) | ((sub @ X282 @ sh) != (sub @ X283 @ sh)))) & pushprop_lthm_orig & ((! [X284 : term] : ~! [X285 : subst > term > term] : ~! [X286 : term,X287 : subst] : ((sub @ X284 @ (push @ X286 @ X287)) = (X285 @ X287 @ X286)) => (! [X288 : term] : ((sub @ X288 @ id) = X288) => ! [X289 : term > $o,X290 : subst > term > subst > $o,X291 : term] : (~! [X292 : subst > term > term] : ((X290 @ id @ (lam @ (X292 @ sh @ one)) @ id) | ~! [X293 : term] : (~(X290 @ id @ X293 @ id) | (X290 @ id @ (X292 @ id @ X293) @ id)) | ~! [X294 : term,X295 : subst,X296 : subst] : ((X292 @ (comp @ X296 @ X295) @ (sub @ X294 @ X295)) = (sub @ (X292 @ X296 @ X294) @ X295))) | (X289 @ (lam @ X291)) | ~! [X297 : term] : ((X289 @ X297) = (X290 @ id @ X297 @ id)) | ~! [X298 : term] : (~(X289 @ X298) | (X289 @ (sub @ X291 @ (push @ X298 @ id))))))) <=> hoasinduction_lem3v2a_lthm) & hoasinduction_lem2_gthm & pushprop_lem1_gthm & hoasinduction_lem2v2_gthm & hoasinduction_lem3b_gthm & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X299 : subst > term > subst > $o] : ~! [X300 : term > $o] : ~! [X301 : term] : ((X299 @ id @ X301 @ id) = (X300 @ X301)) => (! [X302 : term > $o,X303 : term] : (~! [X304 : term,X305 : term] : (~(X302 @ X304) | (X302 @ (ap @ X304 @ X305)) | ~(X302 @ X305)) | ~! [X306 : term] : ((X302 @ X306) | ~(var @ X306)) | ~! [X307 : term] : ((X302 @ (lam @ X307)) | ~! [X308 : term] : (~(X302 @ X308) | (X302 @ (sub @ X307 @ (push @ X308 @ id))))) | (X302 @ X303)) => (! [X309 : term] : ((sub @ X309 @ id) = X309) => (! [X310 : subst > term > subst > $o,X311 : term,X312 : term > $o] : (~! [X313 : subst > term > term] : ((X310 @ id @ (lam @ (X313 @ sh @ one)) @ id) | ~! [X314 : term] : ((X310 @ id @ (X313 @ id @ X314) @ id) | ~(X310 @ id @ X314 @ id)) | ~! [X315 : term,X316 : subst,X317 : subst] : ((sub @ (X313 @ X317 @ X315) @ X316) = (X313 @ (comp @ X317 @ X316) @ (sub @ X315 @ X316)))) | (X312 @ (lam @ X311)) | ~! [X318 : term] : (~(X312 @ X318) | (X312 @ (sub @ X311 @ (push @ X318 @ id)))) | ~! [X319 : term] : ((X310 @ id @ X319 @ id) = (X312 @ X319))) => ! [X320 : term,X321 : subst > term > subst > $o] : (~! [X322 : subst,X323 : subst,X324 : term,X325 : subst] : ((X321 @ X325 @ X324 @ (comp @ X322 @ X323)) | ~(X321 @ (comp @ X325 @ X322) @ (sub @ X324 @ X322) @ X323)) | ~! [X326 : term] : (~(var @ (sub @ X326 @ id)) | (X321 @ id @ X326 @ id)) | ~! [X327 : term,X328 : term] : (~(X321 @ id @ X327 @ id) | (X321 @ id @ (ap @ (sub @ X327 @ id) @ X328) @ id) | ~(X321 @ id @ X328 @ id)) | ~! [X329 : subst,X330 : subst,X331 : term,X332 : subst] : ((X321 @ (comp @ X329 @ X330) @ (sub @ X331 @ X330) @ X332) | ~(X321 @ X329 @ X331 @ (comp @ X330 @ X332))) | ~! [X333 : subst > term > term] : (~! [X334 : subst,X335 : term,X336 : subst] : ((sub @ (X333 @ X334 @ X335) @ X336) = (X333 @ (comp @ X334 @ X336) @ (sub @ X335 @ X336))) | (X321 @ id @ (lam @ (X333 @ sh @ one)) @ id) | ~! [X337 : term] : (~(X321 @ id @ X337 @ id) | (X321 @ id @ (X333 @ id @ X337) @ id))) | (X321 @ id @ X320 @ id)))))) <=> hoasinduction_lthm_3) & ((! [X338 : subst] : ((comp @ id @ X338) = X338) => (! [X339 : subst] : ((comp @ X339 @ id) = X339) => (! [X340 : subst] : ((comp @ X340 @ id) = X340) & ! [X341 : subst] : ((comp @ id @ X341) = X341)))) <=> substmonoid_lthm) & hoasinduction_lem1_lthm & axassoc & hoasinduction_lem1v2_gthm & hoasinduction_gthm & induction2_gthm & (axscons <=> ! [X342 : subst] : ((push @ (sub @ one @ X342) @ (comp @ sh @ X342)) = X342)) & (! [X343 : subst,X344 : term] : ((sub @ one @ (push @ X344 @ X343)) = X344) <=> axvarcons) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (hoasinduction_lem3b_lthm <=> ! [X345 : term] : ~! [X346 : subst > term > term] : ((X346 @ sh @ one) != (sub @ X345 @ (push @ one @ sh)))) & ((! [X347 : term] : ((sub @ X347 @ id) = X347) => (! [X348 : term,X349 : term,X350 : term,X351 : term] : ((X348 = X349) | ((ap @ X349 @ X350) != (ap @ X348 @ X351))) => ! [X352 : term,X353 : term,X354 : term,X355 : term] : ((X352 = X355) | ((ap @ (sub @ X352 @ id) @ X354) != (ap @ (sub @ X355 @ id) @ X353))))) <=> hoasapinj1_lthm) & (hoasinduction_lem3v2_lthm <=> (! [X356 : term] : ((sub @ X356 @ id) = X356) => ! [X357 : term,X358 : subst > term > subst > $o,X359 : term > $o] : (~! [X360 : term] : ((X358 @ id @ X360 @ id) = (X359 @ X360)) | (X359 @ (lam @ X357)) | ~! [X361 : subst,X362 : subst,X363 : subst,X364 : term] : (~(X358 @ X363 @ X364 @ (comp @ X361 @ X362)) | (X358 @ (comp @ X363 @ X361) @ (sub @ X364 @ X361) @ X362)) | ~! [X365 : subst,X366 : subst,X367 : term,X368 : subst] : ((X358 @ X365 @ X367 @ (comp @ X366 @ X368)) | ~(X358 @ (comp @ X365 @ X366) @ (sub @ X367 @ X366) @ X368)) | ~! [X369 : subst > term > term] : (~! [X370 : term,X371 : subst,X372 : subst] : ((sub @ (X369 @ X372 @ X370) @ X371) = (X369 @ (comp @ X372 @ X371) @ (sub @ X370 @ X371))) | (X358 @ id @ (lam @ (X369 @ sh @ one)) @ id) | ~! [X373 : term] : ((X358 @ id @ (X369 @ id @ X373) @ id) | ~(X358 @ id @ X373 @ id))) | ~! [X374 : term] : (~(X359 @ X374) | (X359 @ (sub @ X357 @ (push @ X374 @ id))))))) & hoaslamnotap_gthm & (! [X375 : term,X376 : term > $o,X377 : subst > term > subst > $o] : (~! [X378 : subst > term > term] : ((X377 @ id @ (lam @ (X378 @ sh @ one)) @ id) | ~! [X379 : term] : (~(X377 @ id @ X379 @ id) | (X377 @ id @ (X378 @ id @ X379) @ id)) | ~! [X380 : term,X381 : subst,X382 : subst] : ((sub @ (X378 @ X382 @ X380) @ X381) = (X378 @ (comp @ X382 @ X381) @ (sub @ X380 @ X381)))) | ~! [X383 : term] : ((X377 @ id @ X383 @ id) = (X376 @ X383)) | ~! [X384 : term] : ((X376 @ (sub @ X375 @ (push @ X384 @ id))) | ~(X376 @ X384)) | (X376 @ (lam @ X375))) <=> hoasinduction_lem3v2a) & substmonoid_gthm & (hoasapinj1 <=> ! [X385 : term,X386 : term,X387 : term,X388 : term] : (((ap @ (sub @ X386 @ id) @ X385) != (ap @ (sub @ X387 @ id) @ X388)) | (X386 = X387))) & ! [X389 : subst] : ? [X390 : d_subst] : ((d2subst @ X390) = X389) & (hoasinduction_p_and_p_prime = (^[X391 : subst > term > subst > $o, X392 : term > $o] : (! [X393 : term] : ((X392 @ X393) = (X391 @ id @ X393 @ id))))) & hoasinduction_lem1 & pushprop_lem0_gthm & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoasinduction_no_psi_cond_lthm <=> (! [X394 : subst > term > subst > $o] : ~! [X395 : term > $o] : ~! [X396 : term] : ((X394 @ id @ X396 @ id) = (X395 @ X396)) => (! [X397 : term,X398 : term > $o] : (~! [X399 : term] : ((X398 @ X399) | ~(var @ X399)) | (X398 @ X397) | ~! [X400 : term] : ((X398 @ (lam @ X400)) | ~! [X401 : term] : ((X398 @ (sub @ X400 @ (push @ X401 @ id))) | ~(X398 @ X401))) | ~! [X402 : term,X403 : term] : (~(X398 @ X402) | ~(X398 @ X403) | (X398 @ (ap @ X403 @ X402)))) => (! [X404 : term] : ((sub @ X404 @ id) = X404) => (! [X405 : term > $o,X406 : term,X407 : subst > term > subst > $o] : (~! [X408 : term] : ((X405 @ X408) = (X407 @ id @ X408 @ id)) | ~! [X409 : subst > term > term] : (~! [X410 : subst,X411 : subst,X412 : term] : ((sub @ (X409 @ X410 @ X412) @ X411) = (X409 @ (comp @ X410 @ X411) @ (sub @ X412 @ X411))) | ~! [X413 : term] : (~(X407 @ id @ X413 @ id) | (X407 @ id @ (X409 @ id @ X413) @ id)) | (X407 @ id @ (lam @ (X409 @ sh @ one)) @ id)) | (X405 @ (lam @ X406)) | ~! [X414 : term] : (~(X405 @ X414) | (X405 @ (sub @ X406 @ (push @ X414 @ id))))) => ! [X415 : term,X416 : subst > term > subst > $o] : (~! [X417 : subst > term > term] : (~! [X418 : term] : ((X416 @ id @ (X417 @ id @ X418) @ id) | ~(X416 @ id @ X418 @ id)) | ~! [X419 : subst,X420 : subst,X421 : term] : ((X417 @ (comp @ X420 @ X419) @ (sub @ X421 @ X419)) = (sub @ (X417 @ X420 @ X421) @ X419)) | (X416 @ id @ (lam @ (X417 @ sh @ one)) @ id)) | ~! [X422 : term,X423 : term] : ((X416 @ id @ (ap @ (sub @ X422 @ id) @ X423) @ id) | ~(X416 @ id @ X422 @ id) | ~(X416 @ id @ X423 @ id)) | (X416 @ id @ X415 @ id))))))) & pushprop_lem2v2 & hoasapnotvar_lthm & axabs & (! [X424 : term,X425 : subst > term > subst > $o] : ((X425 @ id @ (lam @ (sub @ X424 @ (push @ one @ sh))) @ id) | ~! [X426 : subst > term > term] : (~! [X427 : term] : (~(X425 @ id @ X427 @ id) | (X425 @ id @ (X426 @ id @ X427) @ id)) | (X425 @ id @ (lam @ (X426 @ sh @ one)) @ id) | ~! [X428 : term,X429 : subst,X430 : subst] : ((X426 @ (comp @ X429 @ X430) @ (sub @ X428 @ X430)) = (sub @ (X426 @ X429 @ X428) @ X430))) | ~! [X431 : term] : (~(X425 @ id @ X431 @ id) | (X425 @ id @ (sub @ X424 @ (push @ X431 @ id)) @ id))) <=> hoasinduction_lem3aa) & hoasapnotvar & ((! [X432 : term > $o,X433 : term] : (~! [X434 : term] : ((X432 @ X434) | ~(var @ X434)) | ~! [X435 : term,X436 : term] : (~(X432 @ X436) | (X432 @ (ap @ X436 @ X435)) | ~(X432 @ X435)) | ~! [X437 : term] : ((X432 @ (lam @ X437)) | ~! [X438 : term] : ((X432 @ (sub @ X437 @ (push @ X438 @ id))) | ~(X432 @ X438))) | (X432 @ X433)) => (! [X439 : subst > term > subst > $o,X440 : term,X441 : term] : (~! [X442 : term,X443 : subst,X444 : subst,X445 : subst] : ((X439 @ X445 @ X442 @ (comp @ X444 @ X443)) | ~(X439 @ (comp @ X445 @ X444) @ (sub @ X442 @ X444) @ X443)) | ~(X439 @ id @ X440 @ id) | ~(X439 @ id @ X441 @ id) | ~! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : ((X439 @ (comp @ X447 @ X446) @ (sub @ X448 @ X446) @ X449) | ~(X439 @ X447 @ X448 @ (comp @ X446 @ X449))) | ~! [X450 : term,X451 : term] : ((X439 @ id @ (ap @ (sub @ X450 @ id) @ X451) @ id) | ~(X439 @ id @ X450 @ id) | ~(X439 @ id @ X451 @ id)) | (X439 @ id @ (ap @ X441 @ X440) @ id)) => (! [X452 : term,X453 : subst > term > subst > $o] : (~! [X454 : subst > term > term] : ((X453 @ id @ (lam @ (X454 @ sh @ one)) @ id) | ~! [X455 : subst,X456 : term,X457 : subst] : ((X454 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455)) = (sub @ (X454 @ X457 @ X456) @ X455)) | ~! [X458 : term] : (~(X453 @ id @ X458 @ id) | (X453 @ id @ (X454 @ id @ X458) @ id))) | ~! [X459 : term] : (~(X453 @ id @ X459 @ id) | (X453 @ id @ (sub @ X452 @ (push @ X459 @ id)) @ id)) | ~! [X460 : subst,X461 : term,X462 : subst,X463 : subst] : ((X453 @ X462 @ X461 @ (comp @ X463 @ X460)) | ~(X453 @ (comp @ X462 @ X463) @ (sub @ X461 @ X463) @ X460)) | ~! [X464 : subst,X465 : subst,X466 : term,X467 : subst] : ((X453 @ (comp @ X467 @ X465) @ (sub @ X466 @ X465) @ X464) | ~(X453 @ X467 @ X466 @ (comp @ X465 @ X464))) | (X453 @ id @ (lam @ X452) @ id)) => ! [X468 : term,X469 : subst > term > subst > $o] : (~! [X470 : term,X471 : term] : ((X469 @ id @ (ap @ (sub @ X470 @ id) @ X471) @ id) | ~(X469 @ id @ X470 @ id) | ~(X469 @ id @ X471 @ id)) | ~! [X472 : subst > term > term] : ((X469 @ id @ (lam @ (X472 @ sh @ one)) @ id) | ~! [X473 : term] : ((X469 @ id @ (X472 @ id @ X473) @ id) | ~(X469 @ id @ X473 @ id)) | ~! [X474 : subst,X475 : term,X476 : subst] : ((sub @ (X472 @ X476 @ X475) @ X474) = (X472 @ (comp @ X476 @ X474) @ (sub @ X475 @ X474)))) | ~! [X477 : term,X478 : subst,X479 : subst,X480 : subst] : (~(X469 @ (comp @ X479 @ X480) @ (sub @ X477 @ X480) @ X478) | (X469 @ X479 @ X477 @ (comp @ X480 @ X478))) | ~! [X481 : subst,X482 : subst,X483 : subst,X484 : term] : (~(X469 @ X483 @ X484 @ (comp @ X482 @ X481)) | (X469 @ (comp @ X483 @ X482) @ (sub @ X484 @ X482) @ X481)) | (X469 @ id @ X468 @ id) | ~! [X485 : term] : (~(var @ (sub @ X485 @ id)) | (X469 @ id @ X485 @ id)))))) <=> hoasinduction_lthm) & (hoasapinj2_lthm <=> (! [X486 : term,X487 : term,X488 : term,X489 : term] : ((X486 = X489) | ((ap @ X488 @ X489) != (ap @ X487 @ X486))) => ! [X490 : term,X491 : term,X492 : term,X493 : term] : ((X492 = X493) | ((ap @ (sub @ X491 @ id) @ X493) != (ap @ (sub @ X490 @ id) @ X492))))) & (! [X494 : term,X495 : term,X496 : term,X497 : term] : (((ap @ X497 @ X496) != (ap @ X495 @ X494)) | (X495 = X497)) <=> apinj1) & hoaslamnotvar_gthm & (termmset_lthm <=> (! [X498 : term] : ((sub @ X498 @ id) = X498) => ! [X499 : term] : ((sub @ X499 @ id) = X499))) & axapp & ~(var @ (d2term @ d_one)) & (! [X500 : term > $o,X501 : term,X502 : subst] : ~! [X503 : term > $o] : ~! [X504 : term] : ((X500 @ (sub @ X504 @ (push @ X501 @ X502))) = (X503 @ X504)) <=> pushprop_lem0) & (! [X505 : term,X506 : subst > term > term,X507 : subst,X508 : subst > term > term] : (~! [X509 : subst,X510 : term,X511 : subst] : ((X506 @ (comp @ X509 @ X511) @ (sub @ X510 @ X511)) = (sub @ (X506 @ X509 @ X510) @ X511)) | ~! [X512 : subst,X513 : term,X514 : subst] : ((sub @ (X508 @ X512 @ X513) @ X514) = (X508 @ (comp @ X512 @ X514) @ (sub @ X513 @ X514))) | ((X508 @ X507 @ X505) = (X506 @ X507 @ X505)) | ((lam @ (X506 @ sh @ one)) != (lam @ (X508 @ sh @ one)))) <=> hoaslaminj) & hoasinduction_lem2_lthm & (hoasinduction_lem3aa_lthm <=> ! [X515 : subst > term > subst > $o,X516 : term] : (~! [X517 : term] : ((X515 @ id @ (sub @ X516 @ (push @ X517 @ id)) @ id) | ~(X515 @ id @ X517 @ id)) | (X515 @ id @ (lam @ (sub @ X516 @ (push @ one @ sh))) @ id) | ~! [X518 : subst > term > term] : (~! [X519 : term] : ((X515 @ id @ (X518 @ id @ X519) @ id) | ~(X515 @ id @ X519 @ id)) | (X515 @ id @ (lam @ (X518 @ sh @ one)) @ id) | ~! [X520 : subst,X521 : subst,X522 : term] : ((X518 @ (comp @ X521 @ X520) @ (sub @ X522 @ X520)) = (sub @ (X518 @ X521 @ X522) @ X520))))) & hoasinduction_lem3a_gthm & (! [X523 : term,X524 : term > $o,X525 : subst] : ~! [X526 : term > $o] : ~! [X527 : term] : ((X524 @ (sub @ X527 @ (push @ X523 @ X525))) = (X526 @ X527)) <=> pushprop_lem0_lthm) & (! [X528 : subst > term > subst > $o,X529 : term,X530 : term] : (~! [X531 : subst,X532 : subst,X533 : term,X534 : subst] : (~(X528 @ (comp @ X532 @ X534) @ (sub @ X533 @ X534) @ X531) | (X528 @ X532 @ X533 @ (comp @ X534 @ X531))) | ~(X528 @ id @ X530 @ id) | (X528 @ id @ (ap @ X529 @ X530) @ id) | ~! [X535 : term,X536 : term] : (~(X528 @ id @ X535 @ id) | ~(X528 @ id @ X536 @ id) | (X528 @ id @ (ap @ (sub @ X535 @ id) @ X536) @ id)) | ~! [X537 : subst,X538 : subst,X539 : term,X540 : subst] : ((X528 @ (comp @ X538 @ X540) @ (sub @ X539 @ X540) @ X537) | ~(X528 @ X538 @ X539 @ (comp @ X540 @ X537))) | ~(X528 @ id @ X529 @ id)) <=> hoasinduction_lem2) & (axidl <=> ! [X541 : subst] : ((comp @ id @ X541) = X541)) & (! [X542 : term,X543 : subst > term > subst > $o] : ((X543 @ id @ (lam @ X542) @ id) | ~! [X544 : subst > term > term] : ((X543 @ id @ (lam @ (X544 @ sh @ one)) @ id) | ~! [X545 : subst,X546 : subst,X547 : term] : ((X544 @ (comp @ X545 @ X546) @ (sub @ X547 @ X546)) = (sub @ (X544 @ X545 @ X547) @ X546)) | ~! [X548 : term] : ((X543 @ id @ (X544 @ id @ X548) @ id) | ~(X543 @ id @ X548 @ id))) | ~! [X549 : term] : ((X543 @ id @ (sub @ X542 @ (push @ X549 @ id)) @ id) | ~(X543 @ id @ X549 @ id))) <=> hoasinduction_lem3a) & hoasapinj1_gthm),
% 2.43/1.93 inference(rectify,[status(thm)],[f1])).
% 2.43/1.93 thf(f1,axiom,(
% 2.43/1.93 (pushprop_lem1v2_lthm <=> (! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => ! [X11 : term > $o,X10 : term > $o,X14 : subst,X13 : term] : (~(X10 @ X13) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) | (X11 @ one)))) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0) & ((! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => ! [X26 : term > $o,X13 : term,X14 : subst,X15 : term,X10 : term > $o] : ((X26 @ (sub @ X13 @ (push @ X15 @ X14))) | ~(X10 @ X13)))) <=> pushprop_lem1_lthm) & hoasinduction_lem3_gthm & (! [X15 : term,X20 : term,X13 : term,X19 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))) <=> hoasapinj2) & axclos & ! [X3 : d_term,X2 : d_term] : (((d2term @ X2) = (d2term @ X3)) => (X2 = X3)) & (! [X10 : subst > term > subst > $o,X34 : term] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X34 @ (push @ X15 @ id)) @ id)) | (X10 @ id @ (lam @ (sub @ X34 @ (push @ one @ sh))) @ id) | ~! [X35 : term,X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X38 : subst,X37 : term,X36 : subst] : ((sub @ (X18 @ X36 @ X37) @ X38) = (sub @ (sub @ X35 @ (push @ X37 @ X36)) @ X38)) | ~! [X41 : subst,X39 : subst,X40 : term] : ((X18 @ (comp @ X39 @ X41) @ (sub @ X40 @ X41)) = (sub @ X35 @ (push @ (sub @ X40 @ X41) @ (comp @ X39 @ X41)))))) <=> hoasinduction_lem3aaa) & pushprop_gthm & (hoasinduction <=> ! [X10 : subst > term > subst > $o,X28 : term] : (~! [X15 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ 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)) | ~! [X14 : subst,X26 : subst,X25 : subst,X13 : term] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | (X10 @ id @ X28 @ id))) & ulamvarsh & hoasapinj2_gthm & hoasinduction_lem3v2_gthm & ! [X6 : d_subst,X7 : d_subst] : (((d2subst @ X6) = (d2subst @ X7)) => (X6 = X7)) & pushprop_lem2v2_lthm & (termmset <=> ! [X13 : term] : ((sub @ X13 @ id) = X13)) & axvarshift & (substmonoid <=> (! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X14 : subst] : ((comp @ id @ X14) = X14))) & ((^[X8 : subst, X9 : subst > term > term] : (d2term @ d_one)) = hoaslam) & pushprop_lthm & pushprop_lem1v2_gthm & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id)) => ! [X43 : term,X10 : subst > term > subst > $o] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id)) | ~! [X18 : subst > term > term] : (~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | (X10 @ id @ (lam @ X43) @ id)))) <=> hoasinduction_lem3a_lthm) & (hoaslaminj_lthm <=> (! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X13 : term,X15 : term] : (((lam @ X13) != (lam @ X15)) | (X13 = X15)) => ! [X33 : term,X31 : subst > term > term,X32 : subst,X18 : subst > term > term] : (((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33)) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))))))) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0_lthm) & apnotvar & pushprop & ((d2subst @ d_id) = sh) & ! [X1 : d_term] : (d_one = X1) & (! [X19 : term,X20 : term,X13 : term,X15 : term] : ((X19 = X20) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) <=> apinj2) & ulamvarind & ((^[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) & (hoasinduction_lem3v2_f_lthm <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13))) & (hoasinduction_no_psi_cond <=> ! [X30 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X15 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | (X10 @ id @ X30 @ id))) & (hoasinduction_lem3_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [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 @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id))) => ! [X27 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ X27) @ id) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X13 : term,X26 : subst,X14 : subst,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term,X25 : subst,X26 : subst,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)))))) & (induction2lem <=> ! [X16 : term,X10 : term > $o,X17 : subst] : (~! [X15 : term] : ((X10 @ (sub @ X15 @ X17)) | ~(var @ X15)) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | (X10 @ (sub @ X16 @ X17)) | ~! [X13 : term,X15 : term] : (~(X10 @ X13) | ~(X10 @ X15) | (X10 @ (ap @ X13 @ X15))))) & pushprop_lem3v2 & hoasinduction_lem1v2 & (hoasinduction_lem3v2_f <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X16 : term,X17 : subst,X10 : term > $o] : (~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15))) | ~! [X15 : term,X13 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | (X10 @ (sub @ X16 @ X17)) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17)))) => ! [X22 : term,X10 : term > $o] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15) | ~(X10 @ X13)) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | (X10 @ X22)))) <=> induction2_lthm) & (! [X15 : term,X13 : term] : ((X13 = X15) | ((lam @ X13) != (lam @ X15))) <=> laminj) & hoaslamnotvar_lthm & pushprop_lem3v2_lthm & ~ulamvar1 & ((! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X21 : term,X10 : term > $o] : (~! [X15 : term,X13 : term] : (~(X10 @ X15) | ~(X10 @ X13) | (X10 @ (ap @ X13 @ X15))) | (X10 @ X21) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~(X10 @ X13)) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13))) => ! [X16 : term,X10 : term > $o,X17 : subst] : ((X10 @ (sub @ X16 @ X17)) | ~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | ~! [X15 : term] : ((X10 @ (sub @ X15 @ X17)) | ~(var @ X15)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))))))))) <=> induction2lem_lthm) & (! [X10 : subst > term > subst > $o,X48 : term,X11 : term > $o] : (~! [X26 : subst,X25 : subst,X14 : subst,X13 : term] : (~(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))) | ~! [X15 : term] : ((X11 @ (sub @ X48 @ (push @ X15 @ id))) | ~(X11 @ X15)) | (X11 @ (lam @ X48)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))) <=> hoasinduction_lem3v2) & (! [X10 : term > $o,X21 : term] : ((X10 @ X21) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~(X10 @ X13)) | ~! [X13 : term,X15 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13))) <=> induction) & hoasinduction_lem1_gthm & induction2lem_gthm & (! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh))) <=> hoasinduction_lem3b) & (! [X14 : subst] : ((comp @ X14 @ id) = X14) <=> axidr) & termmset_gthm & (! [X13 : term,X11 : term > $o,X10 : term > $o,X14 : subst] : ((X11 @ one) | ~(X10 @ X13) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14))))) <=> pushprop_lem1v2) & ((d2subst @ d_id) = id) & (hoasinduction_lem2v2 <=> ! [X11 : term > $o,X44 : term,X10 : subst > term > subst > $o,X45 : term] : (~(X11 @ X44) | ~! [X14 : subst,X13 : term,X25 : subst,X26 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~(X11 @ X45) | ~! [X26 : subst,X25 : subst,X14 : subst,X13 : term] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X11 @ (ap @ X44 @ X45)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id)))) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & hoaslamnotap_lthm & (hoasinduction_lem3 <=> ! [X10 : subst > term > subst > $o,X27 : term] : (~! [X25 : subst,X14 : subst,X26 : subst,X13 : term] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X10 @ id @ (lam @ X27) @ id) | ~! [X18 : subst > term > term] : (~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id)) | ~! [X25 : subst,X14 : subst,X26 : subst,X13 : term] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))))) & hoasapnotvar_gthm & ~lamnotap & (! [X22 : term,X10 : term > $o] : ((X10 @ X22) | ~! [X15 : term,X13 : term] : (~(X10 @ X13) | ~(X10 @ X15) | (X10 @ (ap @ X13 @ X15))) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13))) <=> induction2) & (! [X47 : term,X18 : subst > term > term,X46 : term] : (~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (ap @ (sub @ X46 @ id) @ X47))) <=> hoaslamnotap) & axmap & (one = (d2term @ d_one)) & (! [X13 : term] : ((sub @ X13 @ id) = X13) <=> axvarid) & (pushprop_lem1 <=> ! [X13 : term,X10 : term > $o,X26 : term > $o,X15 : term,X14 : subst] : (~(X10 @ X13) | (X26 @ (sub @ X13 @ (push @ X15 @ X14))))) & hoaslamnotvar & pushprop_lem2v2_gthm & hoaslaminj_gthm & lamnotvar & ! [X5 : d_subst] : (d_id = X5) & ! [X0 : term] : ? [X1 : d_term] : ((d2term @ X1) = X0) & (axshiftcons <=> ! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14)) & (shinj <=> ! [X13 : term,X15 : term] : ((X13 = X15) | ((sub @ X13 @ sh) != (sub @ X15 @ sh)))) & pushprop_lthm_orig & ((! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X11 : term > $o,X10 : subst > term > subst > $o,X29 : term] : (~! [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,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X11 @ (lam @ X29)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id))))))) <=> hoasinduction_lem3v2a_lthm) & hoasinduction_lem2_gthm & pushprop_lem1_gthm & hoasinduction_lem2v2_gthm & hoasinduction_lem3b_gthm & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X10 : term > $o,X22 : term] : (~! [X13 : term,X15 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id))))) | (X10 @ X22)) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : subst > term > subst > $o,X29 : term,X11 : term > $o] : (~! [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)))) | (X11 @ (lam @ X29)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))) => ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : (~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | (X10 @ id @ X28 @ id)))))) <=> hoasinduction_lthm_3) & ((! [X14 : subst] : ((comp @ id @ X14) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X14 : subst] : ((comp @ id @ X14) = X14)))) <=> substmonoid_lthm) & hoasinduction_lem1_lthm & axassoc & hoasinduction_lem1v2_gthm & hoasinduction_gthm & induction2_gthm & (axscons <=> ! [X14 : subst] : ((push @ (sub @ one @ X14) @ (comp @ sh @ X14)) = X14)) & (! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) <=> axvarcons) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (hoasinduction_lem3b_lthm <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh)))) & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X15 : term,X13 : term,X19 : term,X20 : term] : ((X13 = X15) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) => ! [X15 : term,X19 : term,X20 : term,X13 : term] : ((X13 = X15) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))))) <=> hoasapinj1_lthm) & (hoasinduction_lem3v2_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)) | (X11 @ (lam @ X48)) | ~! [X26 : subst,X25 : subst,X14 : subst,X13 : term] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X14 : subst,X26 : subst,X13 : term,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ 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 @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X48 @ (push @ X15 @ id))))))) & hoaslamnotap_gthm & (! [X29 : term,X11 : term > $o,X10 : subst > term > subst > $o] : (~! [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,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | (X11 @ (lam @ X29))) <=> hoasinduction_lem3v2a) & substmonoid_gthm & (hoasapinj1 <=> ! [X19 : term,X13 : term,X15 : term,X20 : term] : (((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)) | (X13 = X15))) & ! [X4 : subst] : ? [X5 : d_subst] : ((d2subst @ X5) = X4) & (hoasinduction_p_and_p_prime = (^[X10 : subst > term > subst > $o, X11 : term > $o] : (! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))))) & hoasinduction_lem1 & pushprop_lem0_gthm & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (hoasinduction_no_psi_cond_lthm <=> (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X22 : term,X10 : term > $o] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | (X10 @ X22) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15))) | ~! [X15 : term,X13 : term] : (~(X10 @ X15) | ~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X11 : term > $o,X29 : term,X10 : subst > term > subst > $o] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X18 : subst > term > term] : (~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X11 @ (lam @ X29)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id))))) => ! [X30 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ X30 @ id))))))) & pushprop_lem2v2 & hoasapnotvar_lthm & axabs & (! [X42 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ 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,X14 : subst,X25 : subst] : ((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))) <=> hoasinduction_lem3aa) & hoasapnotvar & ((! [X10 : term > $o,X22 : term] : (~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X15 : term,X13 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15))) | (X10 @ X22)) => (! [X10 : subst > term > subst > $o,X24 : term,X23 : term] : (~! [X13 : term,X25 : subst,X26 : subst,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~(X10 @ id @ X24 @ id) | ~(X10 @ id @ X23 @ id) | ~! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (ap @ X23 @ X24) @ id)) => (! [X27 : term,X10 : subst > term > subst > $o] : (~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id)) | ~! [X25 : subst,X13 : term,X14 : subst,X26 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ (lam @ X27) @ id)) => ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X13 : term,X25 : subst,X14 : subst,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X25 : subst,X26 : subst,X14 : subst,X13 : term] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X10 @ id @ X28 @ id) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)))))) <=> hoasinduction_lthm) & (hoasapinj2_lthm <=> (! [X20 : term,X15 : term,X13 : term,X19 : term] : ((X19 = X20) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) => ! [X15 : term,X13 : term,X20 : term,X19 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))))) & (! [X20 : term,X15 : term,X19 : term,X13 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X13 = X15)) <=> apinj1) & hoaslamnotvar_gthm & (termmset_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X13 : term] : ((sub @ X13 @ id) = X13))) & axapp & ~(var @ (d2term @ d_one)) & (! [X10 : term > $o,X13 : term,X14 : subst] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) <=> pushprop_lem0) & (! [X33 : term,X31 : subst > term > term,X32 : subst,X18 : subst > term > term] : (~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33)) | ((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one)))) <=> hoaslaminj) & hoasinduction_lem2_lthm & (hoasinduction_lem3aa_lthm <=> ! [X10 : subst > term > subst > $o,X42 : term] : (~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))))) & hoasinduction_lem3a_gthm & (! [X13 : term,X10 : term > $o,X14 : subst] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) <=> pushprop_lem0_lthm) & (! [X10 : subst > term > subst > $o,X23 : term,X24 : term] : (~! [X25 : subst,X14 : subst,X13 : term,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~(X10 @ id @ X24 @ id) | (X10 @ id @ (ap @ X23 @ X24) @ id) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~! [X25 : subst,X14 : subst,X13 : term,X26 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~(X10 @ id @ X23 @ id)) <=> hoasinduction_lem2) & (axidl <=> ! [X14 : subst] : ((comp @ id @ X14) = X14)) & (! [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,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X15 : term] : ((X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id))) <=> hoasinduction_lem3a) & hoasapinj1_gthm),
% 2.43/1.93 file('/export/starexec/sandbox2/tmp/tmp.sO7hdZQLTS/DTF2THF_1662.p',alg444_1)).
% 2.43/1.93 thf(f1393,plain,(
% 2.43/1.93 ( ! [X124 : term] : (((d2term @ (sK576 @ X124)) = X124)) )),
% 2.43/1.93 inference(cnf_transformation,[status(thm)],[f749])).
% 2.43/1.93 thf(f1395,plain,(
% 2.43/1.93 ((var @ (d2term @ d_one)) != $true)),
% 2.43/1.93 inference(cnf_transformation,[status(thm)],[f749])).
% 2.43/1.93 thf(f1530,plain,(
% 2.43/1.93 ( ! [X0 : subst > term > subst > $o] : (((var @ (sK656 @ X0)) = $true) | ($true = (sP110 @ X0))) )),
% 2.43/1.93 inference(cnf_transformation,[status(thm)],[f766])).
% 2.43/1.93 thf(f766,plain,(
% 2.43/1.93 ! [X0 : subst > term > subst > $o] : ((($true = (sP110 @ X0)) | (((var @ (sK656 @ X0)) = $true) & ((X0 @ id @ (sK656 @ X0) @ id) != $true))) & (! [X2 : term] : (((var @ X2) != $true) | ((X0 @ id @ X2 @ id) = $true)) | ($true != (sP110 @ X0))))),
% 2.43/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK656])],[f764,f765])).
% 2.43/1.93 thf(f765,plain,(
% 2.43/1.93 ! [X0 : subst > term > subst > $o] : (? [X1 : term] : (((var @ X1) = $true) & ($true != (X0 @ id @ X1 @ id))) => (((var @ (sK656 @ X0)) = $true) & ((X0 @ id @ (sK656 @ X0) @ id) != $true)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f764,plain,(
% 2.43/1.93 ! [X0 : subst > term > subst > $o] : ((($true = (sP110 @ X0)) | ? [X1 : term] : (((var @ X1) = $true) & ($true != (X0 @ id @ X1 @ id)))) & (! [X2 : term] : (((var @ X2) != $true) | ((X0 @ id @ X2 @ id) = $true)) | ($true != (sP110 @ X0))))),
% 2.43/1.93 inference(rectify,[status(thm)],[f763])).
% 2.43/1.93 thf(f763,plain,(
% 2.43/1.93 ! [X0 : subst > term > subst > $o] : ((($true = (sP110 @ X0)) | ? [X10 : term] : (((var @ X10) = $true) & ($true != (X0 @ id @ X10 @ id)))) & (! [X10 : term] : (((var @ X10) != $true) | ($true = (X0 @ id @ X10 @ id))) | ($true != (sP110 @ X0))))),
% 2.43/1.93 inference(nnf_transformation,[status(thm)],[f126])).
% 2.43/1.93 thf(f126,plain,(
% 2.43/1.93 ! [X0 : subst > term > subst > $o] : (($true = (sP110 @ X0)) <=> ! [X10 : term] : (((var @ X10) != $true) | ($true = (X0 @ id @ X10 @ id))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f3331,plain,(
% 2.43/1.93 ($true != (sP110 @ sK646)) | spl657_345),
% 2.43/1.93 inference(avatar_component_clause,[status(thm)],[f3330])).
% 2.43/1.93 thf(f3330,definition,(
% 2.43/1.93 spl657_345 <=> ($true = (sP110 @ sK646))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[spl657_345])],[avatar_definition])).
% 2.43/1.93 thf(f3332,plain,(
% 2.43/1.93 ~spl657_345 | spl657_173),
% 2.43/1.93 inference(avatar_split_clause,[status(thm)],[f1521,f2586,f3330])).
% 2.43/1.93 thf(f2586,definition,(
% 2.43/1.93 spl657_173 <=> (sP113 = $true)),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[spl657_173])],[avatar_definition])).
% 2.43/1.93 thf(f1521,plain,(
% 2.43/1.93 ($true != (sP110 @ sK646)) | (sP113 = $true)),
% 2.43/1.93 inference(cnf_transformation,[status(thm)],[f754])).
% 2.43/1.93 thf(f754,plain,(
% 2.43/1.93 ((sP113 = $true) | (($true != (sP110 @ sK646)) & ! [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((sK646 @ X2 @ X3 @ (comp @ X1 @ X4)) != $true) | ((sK646 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4) = $true)) & ((sP112 @ sK646) != $true) & ($true != (sP111 @ sK646)))) & (! [X5 : subst > term > subst > $o] : (($true = (sP110 @ X5)) | (((X5 @ (sK648 @ X5) @ (sK649 @ X5) @ (comp @ (sK647 @ X5) @ (sK650 @ X5))) = $true) & ($true != (X5 @ (comp @ (sK648 @ X5) @ (sK647 @ X5)) @ (sub @ (sK649 @ X5) @ (sK647 @ X5)) @ (sK650 @ X5)))) | ($true = (sP112 @ X5)) | ($true = (sP111 @ X5))) | (sP113 != $true))),
% 2.43/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK646,sK647,sK648,sK649,sK650])],[f751,f753,f752])).
% 2.43/1.93 thf(f752,plain,(
% 2.43/1.93 ? [X0 : subst > term > subst > $o] : (($true != (sP110 @ X0)) & ! [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) != $true) | ($true = (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4))) & ($true != (sP112 @ X0)) & ($true != (sP111 @ X0))) => (($true != (sP110 @ sK646)) & ! [X4 : subst,X3 : term,X2 : subst,X1 : subst] : (((sK646 @ X2 @ X3 @ (comp @ X1 @ X4)) != $true) | ((sK646 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4) = $true)) & ((sP112 @ sK646) != $true) & ($true != (sP111 @ sK646)))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f753,plain,(
% 2.43/1.93 ! [X5 : subst > term > subst > $o] : (? [X6 : subst,X7 : subst,X8 : term,X9 : subst] : (($true = (X5 @ X7 @ X8 @ (comp @ X6 @ X9))) & ($true != (X5 @ (comp @ X7 @ X6) @ (sub @ X8 @ X6) @ X9))) => (((X5 @ (sK648 @ X5) @ (sK649 @ X5) @ (comp @ (sK647 @ X5) @ (sK650 @ X5))) = $true) & ($true != (X5 @ (comp @ (sK648 @ X5) @ (sK647 @ X5)) @ (sub @ (sK649 @ X5) @ (sK647 @ X5)) @ (sK650 @ X5)))))),
% 2.43/1.93 introduced(definition,[],[choice_axiom])).
% 2.43/1.93 thf(f751,plain,(
% 2.43/1.93 ((sP113 = $true) | ? [X0 : subst > term > subst > $o] : (($true != (sP110 @ X0)) & ! [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) != $true) | ($true = (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4))) & ($true != (sP112 @ X0)) & ($true != (sP111 @ X0)))) & (! [X5 : subst > term > subst > $o] : (($true = (sP110 @ X5)) | ? [X6 : subst,X7 : subst,X8 : term,X9 : subst] : (($true = (X5 @ X7 @ X8 @ (comp @ X6 @ X9))) & ($true != (X5 @ (comp @ X7 @ X6) @ (sub @ X8 @ X6) @ X9))) | ($true = (sP112 @ X5)) | ($true = (sP111 @ X5))) | (sP113 != $true))),
% 2.43/1.93 inference(rectify,[status(thm)],[f750])).
% 2.43/1.93 thf(f750,plain,(
% 2.43/1.93 ((sP113 = $true) | ? [X0 : subst > term > subst > $o] : (($true != (sP110 @ X0)) & ! [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) != $true) | ($true = (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4))) & ($true != (sP112 @ X0)) & ($true != (sP111 @ X0)))) & (! [X0 : subst > term > subst > $o] : (($true = (sP110 @ X0)) | ? [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) = $true) & ($true != (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4))) | ($true = (sP112 @ X0)) | ($true = (sP111 @ X0))) | (sP113 != $true))),
% 2.43/1.93 inference(nnf_transformation,[status(thm)],[f129])).
% 2.43/1.93 thf(f129,plain,(
% 2.43/1.93 (sP113 = $true) <=> ! [X0 : subst > term > subst > $o] : (($true = (sP110 @ X0)) | ? [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) = $true) & ($true != (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4))) | ($true = (sP112 @ X0)) | ($true = (sP111 @ X0)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f2942,plain,(
% 2.43/1.93 ~spl657_173),
% 2.43/1.93 inference(avatar_split_clause,[status(thm)],[f1899,f2586])).
% 2.43/1.93 thf(f1899,plain,(
% 2.43/1.93 (sP113 != $true)),
% 2.43/1.93 inference(trivial_inequality_removal,[status(thm)],[f1859])).
% 2.43/1.93 thf(f1859,plain,(
% 2.43/1.93 ($true != $true) | (sP113 != $true)),
% 2.43/1.93 inference(definition_unfolding,[status(thm)],[f1532,f1367])).
% 2.43/1.93 thf(f1367,plain,(
% 2.43/1.93 (hoasinduction_lem1 = $true)),
% 2.43/1.93 inference(cnf_transformation,[status(thm)],[f749])).
% 2.43/1.93 thf(f1532,plain,(
% 2.43/1.93 (hoasinduction_lem1 != $true) | (sP113 != $true)),
% 2.43/1.93 inference(cnf_transformation,[status(thm)],[f767])).
% 2.43/1.93 thf(f767,plain,(
% 2.43/1.93 ((hoasinduction_lem1 != $true) | (sP113 != $true)) & ((hoasinduction_lem1 = $true) | (sP113 = $true))),
% 2.43/1.93 inference(nnf_transformation,[status(thm)],[f130])).
% 2.43/1.93 thf(f130,plain,(
% 2.43/1.93 (sP113 = $true) <~> (hoasinduction_lem1 = $true)),
% 2.43/1.93 inference(definition_folding,[status(thm)],[f12,f129,f128,f127,f126])).
% 2.43/1.93 thf(f127,plain,(
% 2.43/1.93 ! [X0 : subst > term > subst > $o] : (($true = (sP111 @ X0)) <=> ? [X5 : subst,X6 : subst,X7 : term,X8 : subst] : (((X0 @ X8 @ X7 @ (comp @ X6 @ X5)) != $true) & ($true = (X0 @ (comp @ X8 @ X6) @ (sub @ X7 @ X6) @ X5))))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f128,plain,(
% 2.43/1.93 ! [X0 : subst > term > subst > $o] : (($true = (sP112 @ X0)) <=> ? [X9 : term] : (((hoasvar @ id @ X9 @ id) = $true) & ((X0 @ id @ X9 @ id) != $true)))),
% 2.43/1.93 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.43/1.93 thf(f12,plain,(
% 2.43/1.93 ! [X0 : subst > term > subst > $o] : (! [X10 : term] : (((var @ X10) != $true) | ($true = (X0 @ id @ X10 @ id))) | ? [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) = $true) & ($true != (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4))) | ? [X9 : term] : (((hoasvar @ id @ X9 @ id) = $true) & ((X0 @ id @ X9 @ id) != $true)) | ? [X5 : subst,X6 : subst,X7 : term,X8 : subst] : (((X0 @ X8 @ X7 @ (comp @ X6 @ X5)) != $true) & ($true = (X0 @ (comp @ X8 @ X6) @ (sub @ X7 @ X6) @ X5)))) <~> (hoasinduction_lem1 = $true)),
% 2.43/1.93 inference(flattening,[status(thm)],[f11])).
% 2.43/1.93 thf(f11,plain,(
% 2.43/1.93 (hoasinduction_lem1 = $true) <~> ! [X0 : subst > term > subst > $o] : (((! [X10 : term] : (((var @ X10) != $true) | ($true = (X0 @ id @ X10 @ id))) | ? [X9 : term] : (((hoasvar @ id @ X9 @ id) = $true) & ((X0 @ id @ X9 @ id) != $true))) | ? [X5 : subst,X6 : subst,X7 : term,X8 : subst] : (((X0 @ X8 @ X7 @ (comp @ X6 @ X5)) != $true) & ($true = (X0 @ (comp @ X8 @ X6) @ (sub @ X7 @ X6) @ X5)))) | ? [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) = $true) & ($true != (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4))))),
% 2.43/1.93 inference(ennf_transformation,[status(thm)],[f6])).
% 2.43/1.93 thf(f6,plain,(
% 2.43/1.93 ~((hoasinduction_lem1 = $true) <=> ! [X0 : subst > term > subst > $o] : (! [X1 : subst,X2 : subst,X3 : term,X4 : subst] : (((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) = $true) => ($true = (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4))) => (! [X5 : subst,X6 : subst,X7 : term,X8 : subst] : (($true = (X0 @ (comp @ X8 @ X6) @ (sub @ X7 @ X6) @ X5)) => ((X0 @ X8 @ X7 @ (comp @ X6 @ X5)) = $true)) => (! [X9 : term] : (((hoasvar @ id @ X9 @ id) = $true) => ((X0 @ id @ X9 @ id) = $true)) => ! [X10 : term] : (((var @ X10) = $true) => ($true = (X0 @ id @ X10 @ id)))))))),
% 2.43/1.93 inference(fool_elimination,[status(thm)],[f5])).
% 2.43/1.93 thf(f5,plain,(
% 2.43/1.93 ~(hoasinduction_lem1 = ! [X0 : subst > term > subst > $o] : (! [X1 : subst,X2 : subst,X3 : term,X4 : subst] : ((X0 @ X2 @ X3 @ (comp @ X1 @ X4)) => (X0 @ (comp @ X2 @ X1) @ (sub @ X3 @ X1) @ X4)) => (! [X5 : subst,X6 : subst,X7 : term,X8 : subst] : ((X0 @ (comp @ X8 @ X6) @ (sub @ X7 @ X6) @ X5) => (X0 @ X8 @ X7 @ (comp @ X6 @ X5))) => (! [X9 : term] : ((hoasvar @ id @ X9 @ id) => (X0 @ id @ X9 @ id)) => ! [X10 : term] : ((var @ X10) => (X0 @ id @ X10 @ id))))))),
% 2.43/1.93 inference(rectify,[status(thm)],[f3])).
% 2.43/1.93 thf(f3,negated_conjecture,(
% 2.43/1.93 ~(hoasinduction_lem1 = ! [X10 : subst > term > subst > $o] : (! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) => (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) => (! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) => (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) => (! [X13 : term] : ((hoasvar @ id @ X13 @ id) => (X10 @ id @ X13 @ id)) => ! [X13 : term] : ((var @ X13) => (X10 @ id @ X13 @ id))))))),
% 2.43/1.93 inference(negated_conjecture,[status(cth)],[f2])).
% 2.43/1.93 thf(f2,conjecture,(
% 2.43/1.93 (hoasinduction_lem1 = ! [X10 : subst > term > subst > $o] : (! [X26 : subst,X14 : subst,X13 : term,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) => (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) => (! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) => (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) => (! [X13 : term] : ((hoasvar @ id @ X13 @ id) => (X10 @ id @ X13 @ id)) => ! [X13 : term] : ((var @ X13) => (X10 @ id @ X13 @ id))))))),
% 2.43/1.93 file('/export/starexec/sandbox2/tmp/tmp.sO7hdZQLTS/DTF2THF_1662.p',hoasinduction_lem1)).
% 2.43/1.93 % SZS output end Proof for DTF2THF_1662
% 2.43/1.93 % (2077)------------------------------
% 2.43/1.93 % (2077)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.93 % (2077)Termination reason: Refutation
% 2.43/1.93
% 2.43/1.93 % (2077)Memory used [KB]: 12792
% 2.43/1.93 % (2077)Time elapsed: 0.191 s
% 2.43/1.93 % (2077)Instructions burned: 481 (million)
% 2.43/1.93 % (2077)------------------------------
% 2.43/1.93 % (2077)------------------------------
% 2.43/1.93 % (2028)Success in time 0.518 s
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