%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n024.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 06:58:45 PM UTC 2026
% Result : Theorem 9.38s 2.21s
% Output : Refutation 9.38s
% Verified :
% SZS Type : Refutation
% Derivation depth : 50
% Number of leaves : 132
% Syntax : Number of formulae : 222 ( 22 unt; 0 typ; 0 def)
% Number of atoms : 11557 (5504 equ; 0 cnn)
% Maximal formula atoms : 530 ( 52 avg)
% Number of connectives : 26976 (3061 ~;2847 |;2425 &;17724 @)
% ( 507 <=>; 315 =>; 0 <=; 0 <~>)
% Maximal formula depth : 184 ( 15 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 2069 (2069 >; 0 *; 0 +; 0 <<)
% Number of symbols : 727 ( 723 usr; 294 con; 0-4 aty)
% ( 97 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 5911 ( 211 ^;4332 !;1368 ?;5911 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
term: $tType ).
thf(type_def_7,type,
subst: $tType ).
thf(type_def_8,type,
d_term: $tType ).
thf(type_def_9,type,
d_subst: $tType ).
thf(func_def_0,type,
term: $tType ).
thf(func_def_1,type,
subst: $tType ).
thf(func_def_2,type,
one: term ).
thf(func_def_3,type,
ap: term > term > term ).
thf(func_def_4,type,
lam: term > term ).
thf(func_def_5,type,
sub: term > subst > term ).
thf(func_def_6,type,
id: subst ).
thf(func_def_7,type,
sh: subst ).
thf(func_def_8,type,
push: term > subst > subst ).
thf(func_def_9,type,
comp: subst > subst > subst ).
thf(func_def_10,type,
var: term > $o ).
thf(func_def_18,type,
hoasap: subst > term > subst > term > term ).
thf(func_def_73,type,
hoaslam: subst > ( subst > term > term ) > term ).
thf(func_def_86,type,
hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
thf(func_def_107,type,
hoasvar: subst > term > subst > $o ).
thf(func_def_121,type,
pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
thf(func_def_124,type,
d_term: $tType ).
thf(func_def_125,type,
d_subst: $tType ).
thf(func_def_126,type,
d2term: d_term > term ).
thf(func_def_127,type,
d2subst: d_subst > subst ).
thf(func_def_128,type,
d_one: d_term ).
thf(func_def_129,type,
d_id: d_subst ).
thf(func_def_142,type,
sP0: ( subst > term > subst > $o ) > $o ).
thf(func_def_145,type,
sP3: ( subst > term > subst > $o ) > $o ).
thf(func_def_147,type,
sP5: ( term > $o ) > $o ).
thf(func_def_148,type,
sP6: ( term > $o ) > $o ).
thf(func_def_151,type,
sP9: ( term > $o ) > $o ).
thf(func_def_152,type,
sP10: ( term > $o ) > $o ).
thf(func_def_155,type,
sP13: ( subst > term > subst > $o ) > $o ).
thf(func_def_156,type,
sP14: ( subst > term > subst > $o ) > $o ).
thf(func_def_158,type,
sP16: ( term > $o ) > $o ).
thf(func_def_159,type,
sP17: ( subst > term > subst > $o ) > $o ).
thf(func_def_160,type,
sP18: ( subst > term > subst > $o ) > $o ).
thf(func_def_165,type,
sP23: ( subst > term > subst > $o ) > $o ).
thf(func_def_168,type,
sP26: ( subst > term > subst > $o ) > $o ).
thf(func_def_170,type,
sP28: ( subst > term > subst > $o ) > $o ).
thf(func_def_172,type,
sP30: ( subst > term > subst > $o ) > $o ).
thf(func_def_173,type,
sP31: ( term > $o ) > $o ).
thf(func_def_174,type,
sP32: ( subst > term > subst > $o ) > $o ).
thf(func_def_175,type,
sP33: ( subst > term > subst > $o ) > $o ).
thf(func_def_176,type,
sP34: ( subst > term > subst > $o ) > $o ).
thf(func_def_181,type,
sP39: ( term > $o ) > $o ).
thf(func_def_182,type,
sP40: ( term > $o ) > $o ).
thf(func_def_185,type,
sP43: ( subst > term > subst > $o ) > $o ).
thf(func_def_188,type,
sP46: ( subst > term > subst > $o ) > $o ).
thf(func_def_189,type,
sP47: ( subst > term > subst > $o ) > $o ).
thf(func_def_190,type,
sP48: ( term > $o ) > $o ).
thf(func_def_196,type,
sP54: ( term > $o ) > $o ).
thf(func_def_201,type,
sK59: subst > term > subst > $o ).
thf(func_def_202,type,
sK60: term ).
thf(func_def_203,type,
sK61: term > $o ).
thf(func_def_204,type,
sK62: ( subst > term > term ) > term ).
thf(func_def_205,type,
sK63: ( subst > term > term ) > subst ).
thf(func_def_206,type,
sK64: ( subst > term > term ) > term ).
thf(func_def_207,type,
sK65: ( subst > term > term ) > subst ).
thf(func_def_208,type,
sK66: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_209,type,
sK67: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_210,type,
sK68: ( term > $o ) > term > term ).
thf(func_def_211,type,
sK69: subst > term > subst > $o ).
thf(func_def_212,type,
sK70: term ).
thf(func_def_213,type,
sK71: ( subst > term > term ) > subst ).
thf(func_def_214,type,
sK72: ( subst > term > term ) > subst ).
thf(func_def_215,type,
sK73: ( subst > term > term ) > term ).
thf(func_def_216,type,
sK74: ( subst > term > term ) > term ).
thf(func_def_217,type,
sK75: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_218,type,
sK76: term > ( subst > term > subst > $o ) > term ).
thf(func_def_219,type,
sK77: term ).
thf(func_def_220,type,
sK78: subst > term > term ).
thf(func_def_221,type,
sK79: subst > term > term ).
thf(func_def_222,type,
sK80: subst ).
thf(func_def_223,type,
sK81: term ).
thf(func_def_224,type,
sK82: term ).
thf(func_def_225,type,
sK83: term ).
thf(func_def_226,type,
sK84: subst ).
thf(func_def_227,type,
sK85: term ).
thf(func_def_228,type,
sK86: subst ).
thf(func_def_229,type,
sK87: ( subst > term > term ) > subst ).
thf(func_def_230,type,
sK88: ( subst > term > term ) > subst ).
thf(func_def_231,type,
sK89: ( subst > term > term ) > term ).
thf(func_def_232,type,
sK90: ( subst > term > term ) > subst ).
thf(func_def_233,type,
sK91: ( subst > term > term ) > term ).
thf(func_def_234,type,
sK92: ( subst > term > term ) > subst ).
thf(func_def_235,type,
sK93: subst ).
thf(func_def_236,type,
sK94: term > $o ).
thf(func_def_237,type,
sK95: term ).
thf(func_def_238,type,
sK96: term > term ).
thf(func_def_239,type,
sK97: ( term > $o ) > term ).
thf(func_def_240,type,
sK98: ( term > $o ) > subst > term ).
thf(func_def_241,type,
sK99: ( term > $o ) > term ).
thf(func_def_242,type,
sK100: ( term > $o ) > term ).
thf(func_def_243,type,
sK101: subst > term > subst > $o ).
thf(func_def_244,type,
sK102: term ).
thf(func_def_245,type,
sK103: ( subst > term > term ) > term ).
thf(func_def_246,type,
sK104: ( subst > term > term ) > subst ).
thf(func_def_247,type,
sK105: ( subst > term > term ) > subst ).
thf(func_def_248,type,
sK106: ( subst > term > term ) > term ).
thf(func_def_249,type,
sK107: term > ( subst > term > subst > $o ) > term ).
thf(func_def_250,type,
sK108: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_251,type,
sK109: term ).
thf(func_def_252,type,
sK110: ( subst > term > subst > $o ) > term > term ).
thf(func_def_253,type,
sK111: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_254,type,
sK112: term ).
thf(func_def_255,type,
sK113: subst > term > subst > $o ).
thf(func_def_256,type,
sK114: ( subst > term > term ) > term ).
thf(func_def_257,type,
sK115: ( subst > term > term ) > term ).
thf(func_def_258,type,
sK116: ( subst > term > term ) > subst ).
thf(func_def_259,type,
sK117: ( subst > term > term ) > subst ).
thf(func_def_260,type,
sK118: ( term > $o ) > term ).
thf(func_def_261,type,
sK119: ( term > $o ) > term ).
thf(func_def_262,type,
sK120: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_263,type,
sK121: term ).
thf(func_def_264,type,
sK122: term ).
thf(func_def_265,type,
sK123: term > $o ).
thf(func_def_266,type,
sK124: term > term ).
thf(func_def_267,type,
sK125: subst > term > subst > $o ).
thf(func_def_268,type,
sK126: ( term > $o ) > term ).
thf(func_def_269,type,
sK127: subst > term > subst > $o ).
thf(func_def_270,type,
sK128: term ).
thf(func_def_271,type,
sK129: ( subst > term > term ) > term ).
thf(func_def_272,type,
sK130: ( subst > term > term ) > term ).
thf(func_def_273,type,
sK131: ( subst > term > term ) > subst ).
thf(func_def_274,type,
sK132: ( subst > term > term ) > subst ).
thf(func_def_275,type,
sK133: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_276,type,
sK134: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_277,type,
sK135: term > ( term > $o ) > term ).
thf(func_def_278,type,
sK136: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_279,type,
sK137: term > $o ).
thf(func_def_280,type,
sK138: subst > term > subst > $o ).
thf(func_def_281,type,
sK139: term ).
thf(func_def_282,type,
sK140: ( subst > term > term ) > term ).
thf(func_def_283,type,
sK141: ( subst > term > term ) > subst ).
thf(func_def_284,type,
sK142: ( subst > term > term ) > subst ).
thf(func_def_285,type,
sK143: ( subst > term > term ) > term ).
thf(func_def_286,type,
sK144: ( term > $o ) > term ).
thf(func_def_287,type,
sK145: ( term > $o ) > term ).
thf(func_def_288,type,
sK146: ( subst > term > subst > $o ) > term ).
thf(func_def_289,type,
sK147: ( subst > term > subst > $o ) > term ).
thf(func_def_290,type,
sK148: ( subst > term > subst > $o ) > term ).
thf(func_def_291,type,
sK149: ( subst > term > subst > $o ) > term ).
thf(func_def_292,type,
sK150: subst > term > subst > $o ).
thf(func_def_293,type,
sK151: term ).
thf(func_def_294,type,
sK152: term ).
thf(func_def_295,type,
sK153: term > ( subst > term > subst > $o ) > term ).
thf(func_def_296,type,
sK154: ( subst > term > subst > $o ) > subst ).
thf(func_def_297,type,
sK155: ( subst > term > subst > $o ) > term ).
thf(func_def_298,type,
sK156: ( subst > term > subst > $o ) > subst ).
thf(func_def_299,type,
sK157: ( subst > term > subst > $o ) > subst ).
thf(func_def_300,type,
sK158: ( subst > term > subst > $o ) > term ).
thf(func_def_301,type,
sK159: ( subst > term > subst > $o ) > subst ).
thf(func_def_302,type,
sK160: ( subst > term > subst > $o ) > subst ).
thf(func_def_303,type,
sK161: ( subst > term > subst > $o ) > subst ).
thf(func_def_304,type,
sK162: ( subst > term > subst > $o ) > term > term ).
thf(func_def_305,type,
sK163: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_306,type,
sK164: term ).
thf(func_def_307,type,
sK165: subst > term > subst > $o ).
thf(func_def_308,type,
sK166: ( subst > term > term ) > term ).
thf(func_def_309,type,
sK167: ( subst > term > term ) > subst ).
thf(func_def_310,type,
sK168: ( subst > term > term ) > subst ).
thf(func_def_311,type,
sK169: ( subst > term > term ) > term ).
thf(func_def_312,type,
sK170: ( subst > term > term ) > subst ).
thf(func_def_313,type,
sK171: ( subst > term > term ) > term ).
thf(func_def_314,type,
sK172: ( subst > term > term ) > subst ).
thf(func_def_315,type,
sK173: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_316,type,
sK174: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_317,type,
sK175: subst > term > subst > $o ).
thf(func_def_318,type,
sK176: term ).
thf(func_def_319,type,
sK177: ( subst > term > term ) > term ).
thf(func_def_320,type,
sK178: ( subst > term > term ) > subst ).
thf(func_def_321,type,
sK179: ( subst > term > term ) > subst ).
thf(func_def_322,type,
sK180: ( subst > term > term ) > term ).
thf(func_def_323,type,
sK181: term > ( subst > term > subst > $o ) > term ).
thf(func_def_324,type,
sK182: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_325,type,
sK183: term > $o ).
thf(func_def_326,type,
sK184: term ).
thf(func_def_327,type,
sK185: term > term ).
thf(func_def_328,type,
sK186: ( term > $o ) > term ).
thf(func_def_329,type,
sK187: ( term > $o ) > term ).
thf(func_def_330,type,
sK188: ( term > $o ) > term ).
thf(func_def_331,type,
sK189: ( term > $o ) > term ).
thf(func_def_332,type,
sK190: ( term > $o ) > term ).
thf(func_def_333,type,
sK191: ( term > $o ) > term ).
thf(func_def_334,type,
sK192: ( term > $o ) > term ).
thf(func_def_335,type,
sK193: ( term > $o ) > term ).
thf(func_def_336,type,
sK194: term ).
thf(func_def_337,type,
sK195: term > $o ).
thf(func_def_338,type,
sK196: term > term ).
thf(func_def_339,type,
sK197: ( subst > term > subst > $o ) > term ).
thf(func_def_340,type,
sK198: ( subst > term > subst > $o ) > subst ).
thf(func_def_341,type,
sK199: ( subst > term > subst > $o ) > subst ).
thf(func_def_342,type,
sK200: ( subst > term > subst > $o ) > subst ).
thf(func_def_343,type,
sK201: ( subst > term > subst > $o ) > subst ).
thf(func_def_344,type,
sK202: ( subst > term > subst > $o ) > term ).
thf(func_def_345,type,
sK203: ( subst > term > subst > $o ) > subst ).
thf(func_def_346,type,
sK204: ( subst > term > subst > $o ) > subst ).
thf(func_def_347,type,
sK205: subst > term > subst > $o ).
thf(func_def_348,type,
sK206: term ).
thf(func_def_349,type,
sK207: term ).
thf(func_def_350,type,
sK208: term ).
thf(func_def_351,type,
sK209: subst > term > subst > $o ).
thf(func_def_352,type,
sK210: ( subst > term > subst > $o ) > term ).
thf(func_def_353,type,
sK211: ( subst > term > subst > $o ) > term ).
thf(func_def_354,type,
sK212: ( subst > term > subst > $o ) > subst ).
thf(func_def_355,type,
sK213: ( subst > term > subst > $o ) > subst ).
thf(func_def_356,type,
sK214: ( subst > term > subst > $o ) > subst ).
thf(func_def_357,type,
sK215: ( subst > term > subst > $o ) > subst ).
thf(func_def_358,type,
sK216: ( subst > term > subst > $o ) > subst ).
thf(func_def_359,type,
sK217: ( subst > term > subst > $o ) > term ).
thf(func_def_360,type,
sK218: ( subst > term > subst > $o ) > subst ).
thf(func_def_361,type,
sK219: ( subst > term > subst > $o ) > subst ).
thf(func_def_362,type,
sK220: ( subst > term > subst > $o ) > term ).
thf(func_def_363,type,
sK221: ( subst > term > subst > $o ) > subst ).
thf(func_def_364,type,
sK222: ( subst > term > subst > $o ) > subst ).
thf(func_def_365,type,
sK223: ( subst > term > subst > $o ) > subst ).
thf(func_def_366,type,
sK224: ( subst > term > subst > $o ) > subst ).
thf(func_def_367,type,
sK225: ( subst > term > subst > $o ) > subst ).
thf(func_def_368,type,
sK226: ( subst > term > subst > $o ) > term ).
thf(func_def_369,type,
sK227: ( subst > term > subst > $o ) > term > term ).
thf(func_def_370,type,
sK228: term ).
thf(func_def_371,type,
sK229: subst > term > subst > $o ).
thf(func_def_372,type,
sK230: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_373,type,
sK231: ( subst > term > term ) > term ).
thf(func_def_374,type,
sK232: ( subst > term > term ) > subst ).
thf(func_def_375,type,
sK233: ( subst > term > term ) > subst ).
thf(func_def_376,type,
sK234: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_377,type,
sK235: ( subst > term > subst > $o ) > term ).
thf(func_def_378,type,
sK236: ( subst > term > subst > $o ) > term ).
thf(func_def_379,type,
sK237: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_380,type,
sK238: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_381,type,
sK239: ( subst > term > term ) > term ).
thf(func_def_382,type,
sK240: ( subst > term > term ) > subst ).
thf(func_def_383,type,
sK241: ( subst > term > term ) > subst ).
thf(func_def_384,type,
sK242: ( term > $o ) > term ).
thf(func_def_385,type,
sK243: ( term > $o ) > term ).
thf(func_def_386,type,
sK244: ( subst > term > subst > $o ) > term ).
thf(func_def_387,type,
sK245: ( subst > term > subst > $o ) > term ).
thf(func_def_388,type,
sK246: term ).
thf(func_def_389,type,
sK247: subst > term > subst > $o ).
thf(func_def_390,type,
sK248: term > $o ).
thf(func_def_391,type,
sK249: ( subst > term > subst > $o ) > subst ).
thf(func_def_392,type,
sK250: ( subst > term > subst > $o ) > subst ).
thf(func_def_393,type,
sK251: ( subst > term > subst > $o ) > term ).
thf(func_def_394,type,
sK252: ( subst > term > subst > $o ) > subst ).
thf(func_def_395,type,
sK253: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_396,type,
sK254: ( subst > term > subst > $o ) > subst ).
thf(func_def_397,type,
sK255: ( subst > term > subst > $o ) > term ).
thf(func_def_398,type,
sK256: ( subst > term > subst > $o ) > subst ).
thf(func_def_399,type,
sK257: ( subst > term > subst > $o ) > subst ).
thf(func_def_400,type,
sK258: ( term > $o ) > term > term ).
thf(func_def_401,type,
sK259: ( subst > term > term ) > term ).
thf(func_def_402,type,
sK260: ( subst > term > term ) > subst ).
thf(func_def_403,type,
sK261: ( subst > term > term ) > subst ).
thf(func_def_404,type,
sK262: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_405,type,
sK263: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_406,type,
sK264: subst > term > subst > $o ).
thf(func_def_407,type,
sK265: term ).
thf(func_def_408,type,
sK266: ( subst > term > subst > $o ) > term ).
thf(func_def_409,type,
sK267: ( subst > term > subst > $o ) > subst ).
thf(func_def_410,type,
sK268: ( subst > term > subst > $o ) > subst ).
thf(func_def_411,type,
sK269: ( subst > term > subst > $o ) > subst ).
thf(func_def_412,type,
sK270: term > ( subst > term > subst > $o ) > term ).
thf(func_def_413,type,
sK271: ( subst > term > subst > $o ) > subst ).
thf(func_def_414,type,
sK272: ( subst > term > subst > $o ) > subst ).
thf(func_def_415,type,
sK273: ( subst > term > subst > $o ) > term ).
thf(func_def_416,type,
sK274: ( subst > term > subst > $o ) > subst ).
thf(func_def_417,type,
sK275: ( subst > term > term ) > subst ).
thf(func_def_418,type,
sK276: ( subst > term > term ) > term ).
thf(func_def_419,type,
sK277: ( subst > term > term ) > subst ).
thf(func_def_420,type,
sK278: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_421,type,
sK279: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_422,type,
sK280: term ).
thf(func_def_423,type,
sK281: subst > term > subst > $o ).
thf(func_def_424,type,
sK282: ( subst > term > term ) > term ).
thf(func_def_425,type,
sK283: ( subst > term > term ) > subst ).
thf(func_def_426,type,
sK284: ( subst > term > term ) > subst ).
thf(func_def_427,type,
sK285: ( subst > term > term ) > term ).
thf(func_def_428,type,
sK286: ( subst > term > subst > $o ) > term > term ).
thf(func_def_429,type,
sK287: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_430,type,
sK288: subst > term > subst > $o ).
thf(func_def_431,type,
sK289: term ).
thf(func_def_432,type,
sK290: term > $o ).
thf(func_def_433,type,
sK291: term ).
thf(func_def_434,type,
sK292: ( subst > term > subst > $o ) > term ).
thf(func_def_435,type,
sK293: ( subst > term > subst > $o ) > subst ).
thf(func_def_436,type,
sK294: ( subst > term > subst > $o ) > subst ).
thf(func_def_437,type,
sK295: ( subst > term > subst > $o ) > subst ).
thf(func_def_438,type,
sK296: ( term > $o ) > term > term ).
thf(func_def_439,type,
sK297: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_440,type,
sK298: ( subst > term > subst > $o ) > subst ).
thf(func_def_441,type,
sK299: ( subst > term > subst > $o ) > subst ).
thf(func_def_442,type,
sK300: ( subst > term > subst > $o ) > term ).
thf(func_def_443,type,
sK301: ( subst > term > subst > $o ) > subst ).
thf(func_def_444,type,
sK302: ( subst > term > term ) > term ).
thf(func_def_445,type,
sK303: ( subst > term > term ) > subst ).
thf(func_def_446,type,
sK304: ( subst > term > term ) > subst ).
thf(func_def_447,type,
sK305: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_448,type,
sK306: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_449,type,
sK307: subst > term > subst > $o ).
thf(func_def_450,type,
sK308: term ).
thf(func_def_451,type,
sK309: ( subst > term > term ) > term ).
thf(func_def_452,type,
sK310: ( subst > term > term ) > term > subst ).
thf(func_def_453,type,
sK311: ( subst > term > term ) > term > term ).
thf(func_def_454,type,
sK312: ( subst > term > term ) > term > subst ).
thf(func_def_455,type,
sK313: ( subst > term > term ) > term > term ).
thf(func_def_456,type,
sK314: ( subst > term > term ) > term > subst ).
thf(func_def_457,type,
sK315: ( subst > term > term ) > term > subst ).
thf(func_def_458,type,
sK316: term > ( subst > term > subst > $o ) > term ).
thf(func_def_459,type,
sK317: ( subst > term > subst > $o ) > term ).
thf(func_def_460,type,
sK318: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_461,type,
sK319: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_462,type,
sK320: ( term > $o ) > term ).
thf(func_def_463,type,
sK321: ( term > $o ) > term ).
thf(func_def_464,type,
sK322: subst > term > subst > $o ).
thf(func_def_465,type,
sK323: ( term > $o ) > term ).
thf(func_def_466,type,
sK324: term > $o ).
thf(func_def_467,type,
sK325: term ).
thf(func_def_468,type,
sK326: term > term ).
thf(func_def_469,type,
sK327: term ).
thf(func_def_470,type,
sK328: subst > term > subst > $o ).
thf(func_def_471,type,
sK329: term ).
thf(func_def_472,type,
sK330: ( subst > term > subst > $o ) > subst ).
thf(func_def_473,type,
sK331: ( subst > term > subst > $o ) > subst ).
thf(func_def_474,type,
sK332: ( subst > term > subst > $o ) > subst ).
thf(func_def_475,type,
sK333: ( subst > term > subst > $o ) > term ).
thf(func_def_476,type,
sK334: ( subst > term > subst > $o ) > term ).
thf(func_def_477,type,
sK335: ( subst > term > subst > $o ) > subst ).
thf(func_def_478,type,
sK336: ( subst > term > subst > $o ) > subst ).
thf(func_def_479,type,
sK337: ( subst > term > subst > $o ) > subst ).
thf(func_def_480,type,
sK338: ( subst > term > subst > $o ) > term ).
thf(func_def_481,type,
sK339: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_482,type,
sK340: ( term > $o ) > term > term ).
thf(func_def_483,type,
sK341: ( term > $o ) > ( subst > term > subst > $o ) > term ).
thf(func_def_484,type,
sK342: subst > term > subst > $o ).
thf(func_def_485,type,
sK343: term ).
thf(func_def_486,type,
sK344: term > $o ).
thf(func_def_487,type,
sK345: ( subst > term > term ) > term ).
thf(func_def_488,type,
sK346: ( subst > term > term ) > subst ).
thf(func_def_489,type,
sK347: ( subst > term > term ) > term ).
thf(func_def_490,type,
sK348: ( subst > term > term ) > subst ).
thf(func_def_491,type,
sK349: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_492,type,
sK350: ( subst > term > term ) > subst ).
thf(func_def_493,type,
sK351: ( subst > term > term ) > term ).
thf(func_def_494,type,
sK352: ( subst > term > term ) > subst ).
thf(func_def_495,type,
sK353: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_496,type,
sK354: ( subst > term > subst > $o ) > term ).
thf(func_def_497,type,
sK355: ( subst > term > subst > $o ) > term ).
thf(func_def_498,type,
sK356: ( term > $o ) > term ).
thf(func_def_499,type,
sK357: ( term > $o ) > term ).
thf(func_def_500,type,
sK358: subst > term > subst > $o ).
thf(func_def_501,type,
sK359: term ).
thf(func_def_502,type,
sK360: ( subst > term > subst > $o ) > subst ).
thf(func_def_503,type,
sK361: ( subst > term > subst > $o ) > term ).
thf(func_def_504,type,
sK362: ( subst > term > subst > $o ) > subst ).
thf(func_def_505,type,
sK363: ( subst > term > subst > $o ) > subst ).
thf(func_def_506,type,
sK364: ( subst > term > subst > $o ) > term ).
thf(func_def_507,type,
sK365: ( subst > term > subst > $o ) > subst ).
thf(func_def_508,type,
sK366: ( subst > term > subst > $o ) > subst ).
thf(func_def_509,type,
sK367: ( subst > term > subst > $o ) > subst ).
thf(func_def_510,type,
sK368: ( subst > term > subst > $o ) > term ).
thf(func_def_511,type,
sK369: ( subst > term > term ) > ( subst > term > subst > $o ) > term ).
thf(func_def_512,type,
sK370: ( subst > term > term ) > subst ).
thf(func_def_513,type,
sK371: ( subst > term > term ) > subst ).
thf(func_def_514,type,
sK372: ( subst > term > term ) > term ).
thf(func_def_515,type,
sK373: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_516,type,
sK374: ( subst > term > subst > $o ) > term ).
thf(func_def_517,type,
sK375: ( subst > term > subst > $o ) > term ).
thf(func_def_518,type,
sK376: subst > ( term > $o ) > term ).
thf(func_def_519,type,
sK377: ( term > $o ) > term ).
thf(func_def_520,type,
sK378: term ).
thf(func_def_521,type,
sK379: term ).
thf(func_def_522,type,
sK380: term > $o ).
thf(func_def_523,type,
sK381: subst ).
thf(func_def_524,type,
sK382: term > term ).
thf(func_def_525,type,
sK383: term ).
thf(func_def_526,type,
sK384: term > $o ).
thf(func_def_527,type,
sK385: term > term ).
thf(func_def_528,type,
sK386: ( term > $o ) > term ).
thf(func_def_529,type,
sK387: ( term > $o ) > term ).
thf(func_def_530,type,
sK388: ( term > $o ) > term ).
thf(func_def_531,type,
sK389: ( term > $o ) > term ).
thf(func_def_532,type,
sK390: ( term > $o ) > term ).
thf(func_def_533,type,
sK391: ( term > $o ) > term ).
thf(func_def_534,type,
sK392: ( term > $o ) > term ).
thf(func_def_535,type,
sK393: ( term > $o ) > term ).
thf(func_def_536,type,
sK394: term > $o ).
thf(func_def_537,type,
sK395: term ).
thf(func_def_538,type,
sK396: subst ).
thf(func_def_539,type,
sK397: term ).
thf(func_def_540,type,
sK398: term ).
thf(func_def_541,type,
sK399: subst ).
thf(func_def_542,type,
sK400: subst ).
thf(func_def_543,type,
sK401: term > $o ).
thf(func_def_544,type,
sK402: term ).
thf(func_def_545,type,
sK403: subst ).
thf(func_def_546,type,
sK404: term > term ).
thf(func_def_547,type,
sK405: ( term > $o ) > term ).
thf(func_def_548,type,
sK406: subst > ( term > $o ) > term ).
thf(func_def_549,type,
sK407: ( term > $o ) > term ).
thf(func_def_550,type,
sK408: ( term > $o ) > term ).
thf(func_def_551,type,
sK409: ( term > $o ) > term ).
thf(func_def_552,type,
sK410: ( term > $o ) > term ).
thf(func_def_553,type,
sK411: term > $o ).
thf(func_def_554,type,
sK412: term ).
thf(func_def_555,type,
sK413: subst > term > subst > $o ).
thf(func_def_556,type,
sK414: term ).
thf(func_def_557,type,
sK415: ( subst > term > subst > $o ) > subst ).
thf(func_def_558,type,
sK416: ( subst > term > subst > $o ) > subst ).
thf(func_def_559,type,
sK417: ( subst > term > subst > $o ) > term ).
thf(func_def_560,type,
sK418: ( subst > term > subst > $o ) > subst ).
thf(func_def_561,type,
sK419: ( subst > term > subst > $o ) > subst ).
thf(func_def_562,type,
sK420: ( subst > term > subst > $o ) > subst ).
thf(func_def_563,type,
sK421: ( subst > term > subst > $o ) > term ).
thf(func_def_564,type,
sK422: ( subst > term > subst > $o ) > subst ).
thf(func_def_565,type,
sK423: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_566,type,
sK424: ( subst > term > subst > $o ) > term ).
thf(func_def_567,type,
sK425: ( subst > term > subst > $o ) > term ).
thf(func_def_568,type,
sK426: term ).
thf(func_def_569,type,
sK427: term > $o ).
thf(func_def_570,type,
sK428: subst > term > subst > $o ).
thf(func_def_571,type,
sK429: ( subst > term > term ) > term ).
thf(func_def_572,type,
sK430: ( subst > term > term ) > subst ).
thf(func_def_573,type,
sK431: ( subst > term > term ) > term ).
thf(func_def_574,type,
sK432: ( subst > term > term ) > subst ).
thf(func_def_575,type,
sK433: term > subst > term > term ).
thf(func_def_576,type,
sK434: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_577,type,
sK435: ( subst > term > subst > $o ) > ( term > $o ) > term ).
thf(func_def_578,type,
sK436: ( term > $o ) > term > term ).
thf(func_def_579,type,
sK437: term ).
thf(func_def_580,type,
sK438: term ).
thf(func_def_581,type,
sK439: ( subst > term > term ) > subst ).
thf(func_def_582,type,
sK440: ( subst > term > term ) > term ).
thf(func_def_583,type,
sK441: subst > term > subst > $o ).
thf(func_def_584,type,
sK442: term ).
thf(func_def_585,type,
sK443: term ).
thf(func_def_586,type,
sK444: ( subst > term > subst > $o ) > subst ).
thf(func_def_587,type,
sK445: ( subst > term > subst > $o ) > subst ).
thf(func_def_588,type,
sK446: ( subst > term > subst > $o ) > subst ).
thf(func_def_589,type,
sK447: ( subst > term > subst > $o ) > term ).
thf(func_def_590,type,
sK448: ( subst > term > subst > $o ) > term ).
thf(func_def_591,type,
sK449: ( subst > term > subst > $o ) > subst ).
thf(func_def_592,type,
sK450: ( subst > term > subst > $o ) > subst ).
thf(func_def_593,type,
sK451: ( subst > term > subst > $o ) > subst ).
thf(func_def_594,type,
sK452: ( subst > term > subst > $o ) > term ).
thf(func_def_595,type,
sK453: ( subst > term > subst > $o ) > term ).
thf(func_def_596,type,
sK454: term ).
thf(func_def_597,type,
sK455: term ).
thf(func_def_598,type,
sK456: term ).
thf(func_def_599,type,
sK457: term ).
thf(func_def_600,type,
sK458: term ).
thf(func_def_601,type,
sK459: subst ).
thf(func_def_602,type,
sK460: ( subst > term > term ) > subst ).
thf(func_def_603,type,
sK461: ( subst > term > term ) > subst ).
thf(func_def_604,type,
sK462: ( subst > term > term ) > term ).
thf(func_def_605,type,
sK463: term ).
thf(func_def_606,type,
sK464: term ).
thf(func_def_607,type,
sK465: subst > term > term ).
thf(func_def_608,type,
sK466: term ).
thf(func_def_609,type,
sK467: term ).
thf(func_def_610,type,
sK468: term ).
thf(func_def_611,type,
sK469: term ).
thf(func_def_612,type,
sK470: term ).
thf(func_def_613,type,
sK471: term ).
thf(func_def_614,type,
sK472: term ).
thf(func_def_615,type,
sK473: term ).
thf(func_def_616,type,
sK474: term ).
thf(func_def_617,type,
sK475: term ).
thf(func_def_618,type,
sK476: term ).
thf(func_def_619,type,
sK477: term ).
thf(func_def_620,type,
sK478: term ).
thf(func_def_621,type,
sK479: ( subst > term > subst > $o ) > subst > term > term ).
thf(func_def_622,type,
sK480: subst > term > subst > $o ).
thf(func_def_623,type,
sK481: term ).
thf(func_def_624,type,
sK482: ( subst > term > term ) > term ).
thf(func_def_625,type,
sK483: ( subst > term > term ) > term ).
thf(func_def_626,type,
sK484: ( subst > term > term ) > subst ).
thf(func_def_627,type,
sK485: ( subst > term > term ) > subst ).
thf(func_def_628,type,
sK486: term ).
thf(func_def_629,type,
sK487: ( term > $o ) > term > subst > term > $o ).
thf(func_def_630,type,
sK488: subst ).
thf(func_def_631,type,
sK489: term ).
thf(func_def_632,type,
sK490: term > $o ).
thf(func_def_633,type,
sK491: ( term > $o ) > term ).
thf(func_def_634,type,
sK492: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_635,type,
sK493: subst > term > subst > $o ).
thf(func_def_636,type,
sK494: ( term > $o ) > term ).
thf(func_def_637,type,
sK495: subst > term > subst > $o ).
thf(func_def_638,type,
sK496: ( term > $o ) > term ).
thf(func_def_639,type,
sK497: ( subst > term > subst > $o ) > term > $o ).
thf(func_def_640,type,
sK498: term ).
thf(func_def_641,type,
sK499: ( subst > term > term ) > term ).
thf(func_def_642,type,
sK500: ( subst > term > term ) > subst ).
thf(func_def_643,type,
sK501: term > subst > term > term ).
thf(func_def_644,type,
sK502: ( term > $o ) > term ).
thf(func_def_645,type,
sK503: ( term > $o ) > term ).
thf(func_def_646,type,
sK504: term ).
thf(func_def_647,type,
sK505: term > $o ).
thf(func_def_648,type,
sK506: term > $o ).
thf(func_def_649,type,
sK507: term ).
thf(func_def_650,type,
sK508: subst ).
thf(func_def_651,type,
sK509: ( term > $o ) > term ).
thf(func_def_652,type,
sK510: subst > term > ( term > $o ) > term > $o ).
thf(func_def_653,type,
sK511: subst ).
thf(func_def_654,type,
sK512: subst ).
thf(func_def_655,type,
sK513: ( subst > term > term ) > term ).
thf(func_def_656,type,
sK514: ( subst > term > term ) > subst ).
thf(func_def_657,type,
sK515: ( subst > term > term ) > subst ).
thf(func_def_658,type,
sK516: ( subst > term > term ) > subst ).
thf(func_def_659,type,
sK517: ( subst > term > term ) > term ).
thf(func_def_660,type,
sK518: ( subst > term > term ) > subst ).
thf(func_def_661,type,
sK519: term ).
thf(func_def_662,type,
sK520: subst ).
thf(func_def_663,type,
sK521: subst > term > term ).
thf(func_def_664,type,
sK522: subst > term > term ).
thf(func_def_665,type,
sK523: term ).
thf(func_def_666,type,
sK524: term ).
thf(func_def_667,type,
sK525: term ).
thf(func_def_668,type,
sK526: term ).
thf(func_def_669,type,
sK527: term ).
thf(func_def_670,type,
sK528: subst ).
thf(func_def_671,type,
sK529: term > $o ).
thf(func_def_672,type,
sK530: term > $o ).
thf(func_def_673,type,
sK531: ( term > $o ) > ( term > $o ) > subst > term > term ).
thf(func_def_674,type,
sK532: term ).
thf(func_def_675,type,
sK533: subst ).
thf(func_def_676,type,
sK534: term ).
thf(func_def_677,type,
sK535: term > $o ).
thf(func_def_678,type,
sK536: term > $o ).
thf(func_def_679,type,
sK537: subst > d_subst ).
thf(func_def_680,type,
sK538: term ).
thf(func_def_681,type,
sK539: term ).
thf(func_def_682,type,
sK540: term > $o ).
thf(func_def_683,type,
sK541: term ).
thf(func_def_684,type,
sK542: term > $o ).
thf(func_def_685,type,
sK543: subst ).
thf(func_def_686,type,
sK544: subst > ( term > $o ) > term > ( term > $o ) > term ).
thf(func_def_687,type,
sK545: subst ).
thf(func_def_688,type,
sK546: term ).
thf(func_def_689,type,
sK547: subst ).
thf(func_def_690,type,
sK548: subst ).
thf(func_def_691,type,
sK549: subst ).
thf(func_def_692,type,
sK550: subst ).
thf(func_def_693,type,
sK551: subst ).
thf(func_def_694,type,
sK552: term ).
thf(func_def_695,type,
sK553: term ).
thf(func_def_696,type,
sK554: term > subst > term > term ).
thf(func_def_697,type,
sK555: subst ).
thf(func_def_698,type,
sK556: term ).
thf(func_def_699,type,
sK557: subst ).
thf(func_def_700,type,
sK558: term ).
thf(func_def_701,type,
sK559: term ).
thf(func_def_702,type,
sK560: term > $o ).
thf(func_def_703,type,
sK561: term ).
thf(func_def_704,type,
sK562: subst ).
thf(func_def_705,type,
sK563: term > $o ).
thf(func_def_706,type,
sK564: term ).
thf(func_def_707,type,
sK565: term ).
thf(func_def_708,type,
sK566: term ).
thf(func_def_709,type,
sK567: term ).
thf(func_def_710,type,
sK568: term > subst > term > term ).
thf(func_def_711,type,
sK569: term ).
thf(func_def_712,type,
sK570: subst ).
thf(func_def_713,type,
sK571: term ).
thf(func_def_714,type,
sK572: term ).
thf(func_def_715,type,
sK573: term ).
thf(func_def_716,type,
sK574: term ).
thf(func_def_717,type,
sK575: subst ).
thf(func_def_718,type,
sK576: term > d_term ).
thf(func_def_719,type,
sK577: term ).
thf(func_def_720,type,
sK578: subst ).
thf(func_def_721,type,
sK579: term ).
thf(func_def_722,type,
sK580: term ).
thf(func_def_723,type,
sK581: term ).
thf(func_def_724,type,
sK582: term ).
thf(func_def_725,type,
sK583: term ).
thf(func_def_726,type,
sK584: term ).
thf(func_def_727,type,
sK585: term ).
thf(func_def_728,type,
sK586: term ).
thf(func_def_729,type,
sK587: term ).
thf(func_def_730,type,
sK588: ( subst > term > term ) > subst ).
thf(func_def_731,type,
sK589: ( subst > term > term ) > term ).
thf(func_def_732,type,
sK590: term > subst > term > term ).
thf(func_def_736,type,
sK591: term ).
thf(func_def_737,type,
sK592: term > $o ).
thf(func_def_738,type,
sK593: term > $o ).
thf(func_def_739,type,
sK594: subst ).
thf(func_def_740,type,
sK595: ( term > $o ) > term > subst > ( term > $o ) > term ).
thf(func_def_741,type,
ph596:
!>[X0: $tType] : X0 ).
thf(f4114,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f4087]) ).
thf(f4087,plain,
$true = $false,
inference(superposition,[status(thm)],[f4080,f1938]) ).
thf(f1938,plain,
! [X0: term] :
( $true
= ( sK530 @ X0 ) ),
inference(superposition,[status(thm)],[f1918,f1880]) ).
thf(f1880,plain,
! [X0: term,X1: term] : ( X0 = X1 ),
inference(superposition,[status(thm)],[f1874,f1874]) ).
thf(f1874,plain,
! [X250: term] :
( ( d2term @ d_one )
= X250 ),
inference(forward_demodulation,[status(thm)],[f1064,f1230]) ).
thf(f1230,plain,
! [X13: d_term] : ( d_one = X13 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f545,plain,
( ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ! [X0: d_subst] : ( d_id = X0 )
& ( hoasinduction_lem1_gthm = $true )
& ( ( shinj = $true )
| ( ( ( sub @ sK455 @ sh )
= ( sub @ sK454 @ sh ) )
& ( sK455 != sK454 ) ) )
& ( ! [X3: term,X4: term] :
( ( ( sub @ X3 @ sh )
!= ( sub @ X4 @ sh ) )
| ( X3 = X4 ) )
| ( shinj != $true ) )
& ( lamnotvar = $true )
& ( ( ( sub @ sK456 @ id )
!= sK456 )
| ! [X6: term] :
( ( sub @ X6 @ id )
= X6 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ! [X7: term] :
( ( sub @ X7 @ id )
= X7 )
& ( ( sub @ sK457 @ id )
!= sK457 ) ) )
& ( ( $true = sP58 )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( $true != sP58 ) )
& ( ( axvarid = $true )
| ( ( sub @ sK458 @ id )
!= sK458 ) )
& ( ! [X10: term] :
( ( sub @ X10 @ id )
= X10 )
| ( axvarid != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP57 ) )
& ( ( $true = sP57 )
| ( hoasinduction_lem3aa != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP56 ) )
& ( ( $true = sP56 )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( axidr = $true )
| ( ( comp @ sK459 @ id )
!= sK459 ) )
& ( ! [X12: subst] :
( ( comp @ X12 @ id )
= X12 )
| ( axidr != $true ) )
& ( ( $true = sP55 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP55 ) )
& ! [X13: d_term] : ( d_one = X13 )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP53 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP53 != $true ) )
& ( axapp = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP51 != $true ) )
& ( ( sP51 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( hoasapnotvar = $true )
& ( ! [X14: term,X15: term,X16: subst > term > term] :
( ( ( lam @ ( X16 @ sh @ one ) )
!= ( ap @ ( sub @ X15 @ id ) @ X14 ) )
| ( ( X16 @ ( comp @ ( sK461 @ X16 ) @ ( sK460 @ X16 ) ) @ ( sub @ ( sK462 @ X16 ) @ ( sK460 @ X16 ) ) )
!= ( sub @ ( X16 @ ( sK461 @ X16 ) @ ( sK462 @ X16 ) ) @ ( sK460 @ X16 ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoaslamnotap = $true )
| ( ( ( ap @ ( sub @ sK464 @ id ) @ sK463 )
= ( lam @ ( sK465 @ sh @ one ) ) )
& ! [X23: subst,X24: subst,X25: term] :
( ( sK465 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( sK465 @ X24 @ X25 ) @ X23 ) ) ) )
& ( axclos = $true )
& ( ( ( sK466 != sK469 )
& ( ( ap @ sK466 @ sK467 )
= ( ap @ sK469 @ sK468 ) ) )
| ! [X30: term,X31: term,X32: term,X33: term] :
( ( X32 = X33 )
| ( ( ap @ ( sub @ X32 @ id ) @ X30 )
!= ( ap @ ( sub @ X33 @ id ) @ X31 ) ) )
| ( sK470
!= ( sub @ sK470 @ id ) )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X35: term,X36: term,X37: term,X38: term] :
( ( X35 = X38 )
| ( ( ap @ X35 @ X36 )
!= ( ap @ X38 @ X37 ) ) )
& ( sK474 != sK473 )
& ( ( ap @ ( sub @ sK474 @ id ) @ sK472 )
= ( ap @ ( sub @ sK473 @ id ) @ sK471 ) )
& ! [X43: term] :
( ( sub @ X43 @ id )
= X43 ) ) )
& ( ulamvar1 != $true )
& ( ! [X44: term,X45: term,X46: term,X47: term] :
( ( ( ap @ ( sub @ X44 @ id ) @ X46 )
!= ( ap @ ( sub @ X47 @ id ) @ X45 ) )
| ( X44 = X47 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ( ( ( ap @ ( sub @ sK478 @ id ) @ sK476 )
= ( ap @ ( sub @ sK475 @ id ) @ sK477 ) )
& ( sK478 != sK475 ) ) )
& ( ! [X52: subst > term > subst > $o,X53: term] :
( ( ! [X55: term] :
( ( $true
= ( X52 @ id @ ( sK479 @ X52 @ id @ X55 ) @ id ) )
| ( $true
!= ( X52 @ id @ X55 @ id ) ) )
& ( $true
!= ( X52 @ id @ ( lam @ ( sK479 @ X52 @ sh @ one ) ) @ id ) )
& ! [X56: term,X57: subst,X58: subst] :
( ( sub @ ( sK479 @ X52 @ X58 @ X56 ) @ X57 )
= ( sK479 @ X52 @ ( comp @ X58 @ X57 ) @ ( sub @ X56 @ X57 ) ) ) )
| ( $true
= ( sP46 @ X52 ) )
| ( $true
= ( X52 @ id @ X53 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
| ( ! [X61: subst > term > term] :
( ( ( ( sK480 @ id @ ( X61 @ id @ ( sK482 @ X61 ) ) @ id )
!= $true )
& ( $true
= ( sK480 @ id @ ( sK482 @ X61 ) @ id ) ) )
| ( $true
= ( sK480 @ id @ ( lam @ ( X61 @ sh @ one ) ) @ id ) )
| ( ( X61 @ ( comp @ ( sK485 @ X61 ) @ ( sK484 @ X61 ) ) @ ( sub @ ( sK483 @ X61 ) @ ( sK484 @ X61 ) ) )
!= ( sub @ ( X61 @ ( sK485 @ X61 ) @ ( sK483 @ X61 ) ) @ ( sK484 @ X61 ) ) ) )
& ( $true
!= ( sP46 @ sK480 ) )
& ( ( sK480 @ id @ sK481 @ id )
!= $true ) ) )
& ( ! [X66: term] :
( ( sub @ X66 @ id )
= X66 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ( sK486
!= ( sub @ sK486 @ id ) ) )
& ( induction2_gthm = $true )
& ( pushprop = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X68: subst,X69: term,X70: term > $o,X72: term] :
( ( sK487 @ X70 @ X69 @ X68 @ X72 )
= ( X70 @ ( sub @ X72 @ ( push @ X69 @ X68 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ! [X76: term > $o] :
( ( X76 @ ( sK491 @ X76 ) )
!= ( sK490 @ ( sub @ ( sK491 @ X76 ) @ ( push @ sK489 @ sK488 ) ) ) ) )
& ! [X78: d_term,X79: d_term] :
( ( X78 = X79 )
| ( ( d2term @ X79 )
!= ( d2term @ X78 ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X80: subst > term > subst > $o,X82: term] :
( ( sK492 @ X80 @ X82 )
= ( X80 @ id @ X82 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ! [X84: term > $o] :
( ( sK493 @ id @ ( sK494 @ X84 ) @ id )
!= ( X84 @ ( sK494 @ X84 ) ) ) )
& ( ( $true = sP45 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP45 ) )
& ( ( $true = sP42 )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( $true != sP42 ) )
& ( hoasapnotvar_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem0 = $true )
| ! [X87: term > $o] :
( ( X87 @ ( sK496 @ X87 ) )
!= ( sK495 @ id @ ( sK496 @ X87 ) @ id ) ) )
& ( ! [X89: subst > term > subst > $o,X91: term] :
( ( X89 @ id @ X91 @ id )
= ( sK497 @ X89 @ X91 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ! [X93: subst > term > term] :
( ( sub @ sK498 @ ( push @ ( sK499 @ X93 ) @ ( sK500 @ X93 ) ) )
!= ( X93 @ ( sK500 @ X93 ) @ ( sK499 @ X93 ) ) ) )
& ( ! [X96: term,X98: term,X99: subst] :
( ( sub @ X96 @ ( push @ X98 @ X99 ) )
= ( sK501 @ X96 @ X99 @ X98 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( lamnotap != $true )
& ( one
= ( d2term @ d_one ) )
& ( axassoc = $true )
& ( ( induction2 = $true )
| ( $true != sP41 ) )
& ( ( $true = sP41 )
| ( induction2 != $true ) )
& ( ! [X100: term,X101: term > $o] :
( ( ( ( X101 @ ( sK502 @ X101 ) )
= $true )
& ( $true
!= ( X101 @ ( lam @ ( sK502 @ X101 ) ) ) ) )
| ( ( X101 @ X100 )
= $true )
| ( ( sP39 @ X101 )
= $true )
| ( ( $true
!= ( X101 @ ( sK503 @ X101 ) ) )
& ( $true
= ( var @ ( sK503 @ X101 ) ) ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ( ! [X106: term] :
( ( $true
!= ( sK505 @ X106 ) )
| ( $true
= ( sK505 @ ( lam @ X106 ) ) ) )
& ( $true
!= ( sK505 @ sK504 ) )
& ( ( sP39 @ sK505 )
!= $true )
& ! [X107: term] :
( ( $true
= ( sK505 @ X107 ) )
| ( $true
!= ( var @ X107 ) ) ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ! [X111: term > $o] :
( ( sK506 @ ( sub @ ( sK509 @ X111 ) @ ( push @ sK507 @ sK508 ) ) )
!= ( X111 @ ( sK509 @ X111 ) ) ) )
& ( ! [X113: term > $o,X114: term,X115: subst,X117: term] :
( ( X113 @ ( sub @ X117 @ ( push @ X114 @ X115 ) ) )
= ( sK510 @ X115 @ X114 @ X113 @ X117 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_lthm = $true )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP38 ) )
& ( ( $true = sP38 )
| ( hoasinduction_lthm != $true ) )
& ( ( ! [X118: subst] :
( ( comp @ X118 @ id )
= X118 )
& ! [X119: subst] :
( ( comp @ id @ X119 )
= X119 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ( ( comp @ sK511 @ id )
!= sK511 )
| ( sK512
!= ( comp @ id @ sK512 ) ) )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3v2 != $true ) )
& ( ! [X122: term,X123: subst,X124: subst > term > term,X125: subst > term > term] :
( ( ( X124 @ X123 @ X122 )
= ( X125 @ X123 @ X122 ) )
| ( ( lam @ ( X124 @ sh @ one ) )
!= ( lam @ ( X125 @ sh @ one ) ) )
| ( ( X124 @ ( comp @ ( sK515 @ X124 ) @ ( sK514 @ X124 ) ) @ ( sub @ ( sK513 @ X124 ) @ ( sK514 @ X124 ) ) )
!= ( sub @ ( X124 @ ( sK515 @ X124 ) @ ( sK513 @ X124 ) ) @ ( sK514 @ X124 ) ) )
| ( ( X125 @ ( comp @ ( sK516 @ X125 ) @ ( sK518 @ X125 ) ) @ ( sub @ ( sK517 @ X125 ) @ ( sK518 @ X125 ) ) )
!= ( sub @ ( X125 @ ( sK516 @ X125 ) @ ( sK517 @ X125 ) ) @ ( sK518 @ X125 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ( ( ( sK522 @ sK520 @ sK519 )
!= ( sK521 @ sK520 @ sK519 ) )
& ( ( lam @ ( sK522 @ sh @ one ) )
= ( lam @ ( sK521 @ sh @ one ) ) )
& ! [X136: term,X137: subst,X138: subst] :
( ( sK521 @ ( comp @ X138 @ X137 ) @ ( sub @ X136 @ X137 ) )
= ( sub @ ( sK521 @ X138 @ X136 ) @ X137 ) )
& ! [X139: subst,X140: term,X141: subst] :
( ( sK522 @ ( comp @ X139 @ X141 ) @ ( sub @ X140 @ X141 ) )
= ( sub @ ( sK522 @ X139 @ X140 ) @ X141 ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP27 )
| ( hoasinduction_lem3 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP27 ) )
& ( ( hoasapinj2 = $true )
| ( ( ( ap @ ( sub @ sK523 @ id ) @ sK526 )
= ( ap @ ( sub @ sK524 @ id ) @ sK525 ) )
& ( sK526 != sK525 ) ) )
& ( ! [X146: term,X147: term,X148: term,X149: term] :
( ( ( ap @ ( sub @ X147 @ id ) @ X148 )
!= ( ap @ ( sub @ X146 @ id ) @ X149 ) )
| ( X148 = X149 ) )
| ( hoasapinj2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ( ( $true
= ( sK530 @ sK527 ) )
& ! [X154: term] :
( ( sK530 @ ( sub @ X154 @ ( push @ sK527 @ sK528 ) ) )
= ( sK529 @ X154 ) )
& ( ( sK529 @ one )
!= $true ) ) )
& ( ! [X155: term,X156: subst,X157: term > $o,X158: term > $o] :
( ( $true
!= ( X158 @ X155 ) )
| ( ( X158 @ ( sub @ ( sK531 @ X158 @ X157 @ X156 @ X155 ) @ ( push @ X155 @ X156 ) ) )
!= ( X157 @ ( sK531 @ X158 @ X157 @ X156 @ X155 ) ) )
| ( $true
= ( X157 @ one ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( hoasapinj2_gthm = $true )
& ( ( pushprop_lem1 = $true )
| ( ( ( sK536 @ sK534 )
= $true )
& ( $true
!= ( sK535 @ ( sub @ sK534 @ ( push @ sK532 @ sK533 ) ) ) ) ) )
& ( ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
!= ( X169 @ X167 ) )
| ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) )
| ( pushprop_lem1 != $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( $true = sP25 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP25 ) )
& ( termmset_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP24 ) )
& ( ( $true = sP24 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ! [X170: subst] :
( ( d2subst @ ( sK537 @ X170 ) )
= X170 )
& ( axvarshift = $true )
& ( hoasapnotvar_lthm = $true )
& ( ! [X172: term,X173: term] :
( ( X172 = X173 )
| ( ( lam @ X172 )
!= ( lam @ X173 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ( ( sK539 != sK538 )
& ( ( lam @ sK539 )
= ( lam @ sK538 ) ) ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP22 ) )
& ( ( $true = sP22 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X176: d_subst,X177: d_subst] :
( ( X176 = X177 )
| ( ( d2subst @ X177 )
!= ( d2subst @ X176 ) ) )
& ( ( $true = sP21 )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( $true != sP21 ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ( $true
= ( sK542 @ sK541 ) )
& ( $true
!= ( sK540 @ one ) )
& ! [X182: term] :
( ( sK542 @ ( sub @ X182 @ ( push @ sK541 @ sK543 ) ) )
= ( sK540 @ X182 ) )
& ! [X183: subst,X184: term] :
( ( sub @ one @ ( push @ X184 @ X183 ) )
= X184 ) ) )
& ( ! [X185: term > $o,X186: term,X187: term > $o,X188: subst] :
( ( $true
!= ( X187 @ X186 ) )
| ( $true
= ( X185 @ one ) )
| ( ( X185 @ ( sK544 @ X188 @ X187 @ X186 @ X185 ) )
!= ( X187 @ ( sub @ ( sK544 @ X188 @ X187 @ X186 @ X185 ) @ ( push @ X186 @ X188 ) ) ) ) )
| ( ( sub @ one @ ( push @ sK546 @ sK545 ) )
!= sK546 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( hoaslamnotvar = $true )
& ( ( $true = sP15 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP15 ) )
& ( ( sP12 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP12 != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ( ( ( comp @ id @ sK547 )
!= sK547 )
| ( sK548
!= ( comp @ sK548 @ id ) ) )
& ! [X194: subst] :
( ( comp @ X194 @ id )
= X194 )
& ! [X195: subst] :
( ( comp @ id @ X195 )
= X195 ) ) )
& ( ( ! [X196: subst] :
( ( comp @ id @ X196 )
= X196 )
& ! [X197: subst] :
( ( comp @ X197 @ id )
= X197 ) )
| ( ( comp @ sK549 @ id )
!= sK549 )
| ( sK550
!= ( comp @ id @ sK550 ) )
| ( substmonoid_lthm != $true ) )
& ( ( axshiftcons = $true )
| ( sK551
!= ( comp @ sh @ ( push @ sK552 @ sK551 ) ) ) )
& ( ! [X202: subst,X203: term] :
( ( comp @ sh @ ( push @ X203 @ X202 ) )
= X202 )
| ( axshiftcons != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ! [X205: subst > term > term] :
( ( sub @ sK553 @ ( push @ one @ sh ) )
!= ( X205 @ sh @ one ) ) )
& ( ! [X206: term] :
( ( sK554 @ X206 @ sh @ one )
= ( sub @ X206 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ! [X208: term,X209: term > $o,X210: term,X211: subst,X212: term > $o] :
( ( $true
!= ( X212 @ X208 ) )
| ( $true
= ( X209 @ ( sub @ X208 @ ( push @ X210 @ X211 ) ) ) ) )
| ( sK556
!= ( sub @ one @ ( push @ sK556 @ sK555 ) ) )
| ( ( comp @ sh @ ( push @ sK558 @ sK557 ) )
!= sK557 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ( ( sK563 @ sK559 )
= $true )
& ( $true
!= ( sK560 @ ( sub @ sK559 @ ( push @ sK561 @ sK562 ) ) ) )
& ! [X222: subst,X223: term] :
( ( sub @ one @ ( push @ X223 @ X222 ) )
= X223 )
& ! [X224: subst,X225: term] :
( ( comp @ sh @ ( push @ X225 @ X224 ) )
= X224 ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ulamvarsh = $true )
& ( ( induction2lem_lthm = $true )
| ( $true != sP8 ) )
& ( ( $true = sP8 )
| ( induction2lem_lthm != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP4 != $true ) )
& ( ( sP4 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ulamvarind = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X226: term,X227: term,X228: term,X229: term] :
( ( ( ap @ X229 @ X228 )
!= ( ap @ X226 @ X227 ) )
| ( X226 = X229 ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ( ( ( ap @ sK567 @ sK566 )
= ( ap @ sK564 @ sK565 ) )
& ( sK564 != sK567 ) ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X234: term] :
( ( sK568 @ X234 @ sh @ one )
= ( sub @ X234 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ! [X237: subst > term > term] :
( ( X237 @ sh @ one )
!= ( sub @ sK569 @ ( push @ one @ sh ) ) ) )
& ( ( axidl = $true )
| ( ( comp @ id @ sK570 )
!= sK570 ) )
& ( ! [X239: subst] :
( ( comp @ id @ X239 )
= X239 )
| ( axidl != $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( apinj2 = $true )
| ( ( ( ap @ sK573 @ sK574 )
= ( ap @ sK571 @ sK572 ) )
& ( sK574 != sK572 ) ) )
& ( ! [X244: term,X245: term,X246: term,X247: term] :
( ( ( ap @ X246 @ X247 )
!= ( ap @ X244 @ X245 ) )
| ( X245 = X247 ) )
| ( apinj2 != $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 ) ) ) ) ) ) )
& ( ! [X248: subst] :
( ( push @ ( sub @ one @ X248 ) @ ( comp @ sh @ X248 ) )
= X248 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ( sK575
!= ( push @ ( sub @ one @ sK575 ) @ ( comp @ sh @ sK575 ) ) ) )
& ! [X250: term] :
( ( d2term @ ( sK576 @ X250 ) )
= X250 )
& ( ( axvarcons = $true )
| ( sK577
!= ( sub @ one @ ( push @ sK577 @ sK578 ) ) ) )
& ( ! [X254: term,X255: subst] :
( ( sub @ one @ ( push @ X254 @ X255 ) )
= X254 )
| ( axvarcons != $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ! [X256: term,X257: term,X258: term,X259: term] :
( ( ( ap @ X257 @ X256 )
!= ( ap @ X259 @ X258 ) )
| ( X256 = X258 ) )
& ( ( ap @ ( sub @ sK582 @ id ) @ sK580 )
= ( ap @ ( sub @ sK581 @ id ) @ sK579 ) )
& ( sK579 != sK580 ) ) )
& ( ( ( ( ap @ sK586 @ sK585 )
= ( ap @ sK584 @ sK583 ) )
& ( sK585 != sK583 ) )
| ! [X268: term,X269: term,X270: term,X271: term] :
( ( ( ap @ ( sub @ X271 @ id ) @ X269 )
!= ( ap @ ( sub @ X270 @ id ) @ X268 ) )
| ( X268 = X269 ) )
| ( hoasapinj2_lthm != $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ! [X273: subst > term > term] :
( ( X273 @ ( sK588 @ X273 ) @ ( sK589 @ X273 ) )
!= ( sub @ sK587 @ ( push @ ( sK589 @ X273 ) @ ( sK588 @ X273 ) ) ) ) )
& ( ! [X276: term,X278: subst,X279: term] :
( ( sK590 @ X276 @ X278 @ X279 )
= ( sub @ X276 @ ( push @ X279 @ X278 ) ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( pushprop_gthm = $true )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP2 ) )
& ( ( $true = sP2 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( hoasapinj1_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( $true = sP1 )
| ( hoasinduction_lem2 != $true ) )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP1 ) )
& ( pushprop_lem0_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( axmap = $true )
& ( induction2lem_gthm = $true )
& ( axabs = $true )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( d2subst @ d_id )
= id ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK454,sK455,sK456,sK457,sK458,sK459,sK460,sK461,sK462,sK463,sK464,sK465,sK466,sK467,sK468,sK469,sK470,sK471,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,sK487,sK488,sK489,sK490,sK491,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,sK509,sK510,sK511,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,sK521,sK522,sK523,sK524,sK525,sK526,sK527,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,sK536,sK537,sK538,sK539,sK540,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,sK562,sK563,sK564,sK565,sK566,sK567,sK568,sK569,sK570,sK571,sK572,sK573,sK574,sK575,sK576,sK577,sK578,sK579,sK580,sK581,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590])],[f473,f544,f543,f542,f541,f540,f539,f538,f537,f536,f535,f534,f533,f532,f531,f530,f529,f528,f527,f526,f525,f524,f523,f522,f521,f520,f519,f518,f517,f516,f515,f514,f513,f512,f511,f510,f509,f508,f507,f506,f505,f504,f503,f502,f501,f500,f499,f498,f497,f496,f495,f494,f493,f492,f491,f490,f489,f488,f487,f486,f485,f484,f483,f482,f481,f480,f479,f478,f477,f476,f475,f474]) ).
thf(f474,plain,
( ? [X1: term,X2: term] :
( ( ( sub @ X1 @ sh )
= ( sub @ X2 @ sh ) )
& ( X1 != X2 ) )
=> ( ( ( sub @ sK455 @ sh )
= ( sub @ sK454 @ sh ) )
& ( sK455 != sK454 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f475,plain,
( ? [X5: term] :
( ( sub @ X5 @ id )
!= X5 )
=> ( ( sub @ sK456 @ id )
!= sK456 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f476,plain,
( ? [X8: term] :
( ( sub @ X8 @ id )
!= X8 )
=> ( ( sub @ sK457 @ id )
!= sK457 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f477,plain,
( ? [X9: term] :
( ( sub @ X9 @ id )
!= X9 )
=> ( ( sub @ sK458 @ id )
!= sK458 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f478,plain,
( ? [X11: subst] :
( ( comp @ X11 @ id )
!= X11 )
=> ( ( comp @ sK459 @ id )
!= sK459 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f479,plain,
! [X16: subst > term > term] :
( ? [X17: subst,X18: subst,X19: term] :
( ( sub @ ( X16 @ X18 @ X19 ) @ X17 )
!= ( X16 @ ( comp @ X18 @ X17 ) @ ( sub @ X19 @ X17 ) ) )
=> ( ( X16 @ ( comp @ ( sK461 @ X16 ) @ ( sK460 @ X16 ) ) @ ( sub @ ( sK462 @ X16 ) @ ( sK460 @ X16 ) ) )
!= ( sub @ ( X16 @ ( sK461 @ X16 ) @ ( sK462 @ X16 ) ) @ ( sK460 @ X16 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f480,plain,
( ? [X20: term,X21: term,X22: subst > term > term] :
( ( ( lam @ ( X22 @ sh @ one ) )
= ( ap @ ( sub @ X21 @ id ) @ X20 ) )
& ! [X23: subst,X24: subst,X25: term] :
( ( sub @ ( X22 @ X24 @ X25 ) @ X23 )
= ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) ) ) )
=> ( ( ( ap @ ( sub @ sK464 @ id ) @ sK463 )
= ( lam @ ( sK465 @ sh @ one ) ) )
& ! [X25: term,X24: subst,X23: subst] :
( ( sK465 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) )
= ( sub @ ( sK465 @ X24 @ X25 ) @ X23 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f481,plain,
( ? [X26: term,X27: term,X28: term,X29: term] :
( ( X26 != X29 )
& ( ( ap @ X26 @ X27 )
= ( ap @ X29 @ X28 ) ) )
=> ( ( sK466 != sK469 )
& ( ( ap @ sK466 @ sK467 )
= ( ap @ sK469 @ sK468 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f482,plain,
( ? [X34: term] :
( ( sub @ X34 @ id )
!= X34 )
=> ( sK470
!= ( sub @ sK470 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f483,plain,
( ? [X39: term,X40: term,X41: term,X42: term] :
( ( X41 != X42 )
& ( ( ap @ ( sub @ X41 @ id ) @ X39 )
= ( ap @ ( sub @ X42 @ id ) @ X40 ) ) )
=> ( ( sK474 != sK473 )
& ( ( ap @ ( sub @ sK474 @ id ) @ sK472 )
= ( ap @ ( sub @ sK473 @ id ) @ sK471 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f484,plain,
( ? [X48: term,X49: term,X50: term,X51: term] :
( ( ( ap @ ( sub @ X48 @ id ) @ X50 )
= ( ap @ ( sub @ X51 @ id ) @ X49 ) )
& ( X48 != X51 ) )
=> ( ( ( ap @ ( sub @ sK478 @ id ) @ sK476 )
= ( ap @ ( sub @ sK475 @ id ) @ sK477 ) )
& ( sK478 != sK475 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f485,plain,
! [X52: subst > term > subst > $o] :
( ? [X54: subst > term > term] :
( ! [X55: term] :
( ( ( X52 @ id @ ( X54 @ id @ X55 ) @ id )
= $true )
| ( $true
!= ( X52 @ id @ X55 @ id ) ) )
& ( $true
!= ( X52 @ id @ ( lam @ ( X54 @ sh @ one ) ) @ id ) )
& ! [X56: term,X57: subst,X58: subst] :
( ( X54 @ ( comp @ X58 @ X57 ) @ ( sub @ X56 @ X57 ) )
= ( sub @ ( X54 @ X58 @ X56 ) @ X57 ) ) )
=> ( ! [X55: term] :
( ( $true
= ( X52 @ id @ ( sK479 @ X52 @ id @ X55 ) @ id ) )
| ( $true
!= ( X52 @ id @ X55 @ id ) ) )
& ( $true
!= ( X52 @ id @ ( lam @ ( sK479 @ X52 @ sh @ one ) ) @ id ) )
& ! [X58: subst,X57: subst,X56: term] :
( ( sub @ ( sK479 @ X52 @ X58 @ X56 ) @ X57 )
= ( sK479 @ X52 @ ( comp @ X58 @ X57 ) @ ( sub @ X56 @ X57 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f486,plain,
( ? [X59: subst > term > subst > $o,X60: term] :
( ! [X61: subst > term > term] :
( ? [X62: term] :
( ( $true
!= ( X59 @ id @ ( X61 @ id @ X62 ) @ id ) )
& ( $true
= ( X59 @ id @ X62 @ id ) ) )
| ( $true
= ( X59 @ id @ ( lam @ ( X61 @ sh @ one ) ) @ id ) )
| ? [X63: term,X64: subst,X65: subst] :
( ( X61 @ ( comp @ X65 @ X64 ) @ ( sub @ X63 @ X64 ) )
!= ( sub @ ( X61 @ X65 @ X63 ) @ X64 ) ) )
& ( $true
!= ( sP46 @ X59 ) )
& ( ( X59 @ id @ X60 @ id )
!= $true ) )
=> ( ! [X61: subst > term > term] :
( ? [X62: term] :
( ( $true
!= ( sK480 @ id @ ( X61 @ id @ X62 ) @ id ) )
& ( $true
= ( sK480 @ id @ X62 @ id ) ) )
| ( $true
= ( sK480 @ id @ ( lam @ ( X61 @ sh @ one ) ) @ id ) )
| ? [X63: term,X64: subst,X65: subst] :
( ( X61 @ ( comp @ X65 @ X64 ) @ ( sub @ X63 @ X64 ) )
!= ( sub @ ( X61 @ X65 @ X63 ) @ X64 ) ) )
& ( $true
!= ( sP46 @ sK480 ) )
& ( ( sK480 @ id @ sK481 @ id )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f487,plain,
! [X61: subst > term > term] :
( ? [X62: term] :
( ( $true
!= ( sK480 @ id @ ( X61 @ id @ X62 ) @ id ) )
& ( $true
= ( sK480 @ id @ X62 @ id ) ) )
=> ( ( ( sK480 @ id @ ( X61 @ id @ ( sK482 @ X61 ) ) @ id )
!= $true )
& ( $true
= ( sK480 @ id @ ( sK482 @ X61 ) @ id ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f488,plain,
! [X61: subst > term > term] :
( ? [X63: term,X64: subst,X65: subst] :
( ( X61 @ ( comp @ X65 @ X64 ) @ ( sub @ X63 @ X64 ) )
!= ( sub @ ( X61 @ X65 @ X63 ) @ X64 ) )
=> ( ( X61 @ ( comp @ ( sK485 @ X61 ) @ ( sK484 @ X61 ) ) @ ( sub @ ( sK483 @ X61 ) @ ( sK484 @ X61 ) ) )
!= ( sub @ ( X61 @ ( sK485 @ X61 ) @ ( sK483 @ X61 ) ) @ ( sK484 @ X61 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f489,plain,
( ? [X67: term] :
( ( sub @ X67 @ id )
!= X67 )
=> ( sK486
!= ( sub @ sK486 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f490,plain,
! [X68: subst,X69: term,X70: term > $o] :
( ? [X71: term > $o] :
! [X72: term] :
( ( X70 @ ( sub @ X72 @ ( push @ X69 @ X68 ) ) )
= ( X71 @ X72 ) )
=> ! [X72: term] :
( ( sK487 @ X70 @ X69 @ X68 @ X72 )
= ( X70 @ ( sub @ X72 @ ( push @ X69 @ X68 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f491,plain,
( ? [X73: subst,X74: term,X75: term > $o] :
! [X76: term > $o] :
? [X77: term] :
( ( X75 @ ( sub @ X77 @ ( push @ X74 @ X73 ) ) )
!= ( X76 @ X77 ) )
=> ! [X76: term > $o] :
? [X77: term] :
( ( sK490 @ ( sub @ X77 @ ( push @ sK489 @ sK488 ) ) )
!= ( X76 @ X77 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f492,plain,
! [X76: term > $o] :
( ? [X77: term] :
( ( sK490 @ ( sub @ X77 @ ( push @ sK489 @ sK488 ) ) )
!= ( X76 @ X77 ) )
=> ( ( X76 @ ( sK491 @ X76 ) )
!= ( sK490 @ ( sub @ ( sK491 @ X76 ) @ ( push @ sK489 @ sK488 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f493,plain,
! [X80: subst > term > subst > $o] :
( ? [X81: term > $o] :
! [X82: term] :
( ( X81 @ X82 )
= ( X80 @ id @ X82 @ id ) )
=> ! [X82: term] :
( ( sK492 @ X80 @ X82 )
= ( X80 @ id @ X82 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f494,plain,
( ? [X83: subst > term > subst > $o] :
! [X84: term > $o] :
? [X85: term] :
( ( X84 @ X85 )
!= ( X83 @ id @ X85 @ id ) )
=> ! [X84: term > $o] :
? [X85: term] :
( ( X84 @ X85 )
!= ( sK493 @ id @ X85 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f495,plain,
! [X84: term > $o] :
( ? [X85: term] :
( ( X84 @ X85 )
!= ( sK493 @ id @ X85 @ id ) )
=> ( ( sK493 @ id @ ( sK494 @ X84 ) @ id )
!= ( X84 @ ( sK494 @ X84 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f496,plain,
( ? [X86: subst > term > subst > $o] :
! [X87: term > $o] :
? [X88: term] :
( ( X86 @ id @ X88 @ id )
!= ( X87 @ X88 ) )
=> ! [X87: term > $o] :
? [X88: term] :
( ( X87 @ X88 )
!= ( sK495 @ id @ X88 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f497,plain,
! [X87: term > $o] :
( ? [X88: term] :
( ( X87 @ X88 )
!= ( sK495 @ id @ X88 @ id ) )
=> ( ( X87 @ ( sK496 @ X87 ) )
!= ( sK495 @ id @ ( sK496 @ X87 ) @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f498,plain,
! [X89: subst > term > subst > $o] :
( ? [X90: term > $o] :
! [X91: term] :
( ( X89 @ id @ X91 @ id )
= ( X90 @ X91 ) )
=> ! [X91: term] :
( ( X89 @ id @ X91 @ id )
= ( sK497 @ X89 @ X91 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f499,plain,
( ? [X92: term] :
! [X93: subst > term > term] :
? [X94: term,X95: subst] :
( ( X93 @ X95 @ X94 )
!= ( sub @ X92 @ ( push @ X94 @ X95 ) ) )
=> ! [X93: subst > term > term] :
? [X95: subst,X94: term] :
( ( X93 @ X95 @ X94 )
!= ( sub @ sK498 @ ( push @ X94 @ X95 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f500,plain,
! [X93: subst > term > term] :
( ? [X95: subst,X94: term] :
( ( X93 @ X95 @ X94 )
!= ( sub @ sK498 @ ( push @ X94 @ X95 ) ) )
=> ( ( sub @ sK498 @ ( push @ ( sK499 @ X93 ) @ ( sK500 @ X93 ) ) )
!= ( X93 @ ( sK500 @ X93 ) @ ( sK499 @ X93 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f501,plain,
! [X96: term] :
( ? [X97: subst > term > term] :
! [X98: term,X99: subst] :
( ( sub @ X96 @ ( push @ X98 @ X99 ) )
= ( X97 @ X99 @ X98 ) )
=> ! [X99: subst,X98: term] :
( ( sub @ X96 @ ( push @ X98 @ X99 ) )
= ( sK501 @ X96 @ X99 @ X98 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f502,plain,
! [X101: term > $o] :
( ? [X102: term] :
( ( $true
= ( X101 @ X102 ) )
& ( $true
!= ( X101 @ ( lam @ X102 ) ) ) )
=> ( ( ( X101 @ ( sK502 @ X101 ) )
= $true )
& ( $true
!= ( X101 @ ( lam @ ( sK502 @ X101 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f503,plain,
! [X101: term > $o] :
( ? [X103: term] :
( ( $true
!= ( X101 @ X103 ) )
& ( $true
= ( var @ X103 ) ) )
=> ( ( $true
!= ( X101 @ ( sK503 @ X101 ) ) )
& ( $true
= ( var @ ( sK503 @ X101 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f504,plain,
( ? [X104: term,X105: term > $o] :
( ! [X106: term] :
( ( $true
!= ( X105 @ X106 ) )
| ( $true
= ( X105 @ ( lam @ X106 ) ) ) )
& ( $true
!= ( X105 @ X104 ) )
& ( $true
!= ( sP39 @ X105 ) )
& ! [X107: term] :
( ( $true
= ( X105 @ X107 ) )
| ( $true
!= ( var @ X107 ) ) ) )
=> ( ! [X106: term] :
( ( $true
!= ( sK505 @ X106 ) )
| ( $true
= ( sK505 @ ( lam @ X106 ) ) ) )
& ( $true
!= ( sK505 @ sK504 ) )
& ( ( sP39 @ sK505 )
!= $true )
& ! [X107: term] :
( ( $true
= ( sK505 @ X107 ) )
| ( $true
!= ( var @ X107 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f505,plain,
( ? [X108: term > $o,X109: term,X110: subst] :
! [X111: term > $o] :
? [X112: term] :
( ( X108 @ ( sub @ X112 @ ( push @ X109 @ X110 ) ) )
!= ( X111 @ X112 ) )
=> ! [X111: term > $o] :
? [X112: term] :
( ( sK506 @ ( sub @ X112 @ ( push @ sK507 @ sK508 ) ) )
!= ( X111 @ X112 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f506,plain,
! [X111: term > $o] :
( ? [X112: term] :
( ( sK506 @ ( sub @ X112 @ ( push @ sK507 @ sK508 ) ) )
!= ( X111 @ X112 ) )
=> ( ( sK506 @ ( sub @ ( sK509 @ X111 ) @ ( push @ sK507 @ sK508 ) ) )
!= ( X111 @ ( sK509 @ X111 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f507,plain,
! [X113: term > $o,X114: term,X115: subst] :
( ? [X116: term > $o] :
! [X117: term] :
( ( X113 @ ( sub @ X117 @ ( push @ X114 @ X115 ) ) )
= ( X116 @ X117 ) )
=> ! [X117: term] :
( ( X113 @ ( sub @ X117 @ ( push @ X114 @ X115 ) ) )
= ( sK510 @ X115 @ X114 @ X113 @ X117 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f508,plain,
( ? [X120: subst] :
( ( comp @ X120 @ id )
!= X120 )
=> ( ( comp @ sK511 @ id )
!= sK511 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f509,plain,
( ? [X121: subst] :
( ( comp @ id @ X121 )
!= X121 )
=> ( sK512
!= ( comp @ id @ sK512 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f510,plain,
! [X124: subst > term > term] :
( ? [X126: term,X127: subst,X128: subst] :
( ( X124 @ ( comp @ X128 @ X127 ) @ ( sub @ X126 @ X127 ) )
!= ( sub @ ( X124 @ X128 @ X126 ) @ X127 ) )
=> ( ( X124 @ ( comp @ ( sK515 @ X124 ) @ ( sK514 @ X124 ) ) @ ( sub @ ( sK513 @ X124 ) @ ( sK514 @ X124 ) ) )
!= ( sub @ ( X124 @ ( sK515 @ X124 ) @ ( sK513 @ X124 ) ) @ ( sK514 @ X124 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f511,plain,
! [X125: subst > term > term] :
( ? [X129: subst,X130: term,X131: subst] :
( ( sub @ ( X125 @ X129 @ X130 ) @ X131 )
!= ( X125 @ ( comp @ X129 @ X131 ) @ ( sub @ X130 @ X131 ) ) )
=> ( ( X125 @ ( comp @ ( sK516 @ X125 ) @ ( sK518 @ X125 ) ) @ ( sub @ ( sK517 @ X125 ) @ ( sK518 @ X125 ) ) )
!= ( sub @ ( X125 @ ( sK516 @ X125 ) @ ( sK517 @ X125 ) ) @ ( sK518 @ X125 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f512,plain,
( ? [X132: term,X133: subst,X134: subst > term > term,X135: subst > term > term] :
( ( ( X135 @ X133 @ X132 )
!= ( X134 @ X133 @ X132 ) )
& ( ( lam @ ( X134 @ sh @ one ) )
= ( lam @ ( X135 @ sh @ one ) ) )
& ! [X136: term,X137: subst,X138: subst] :
( ( X134 @ ( comp @ X138 @ X137 ) @ ( sub @ X136 @ X137 ) )
= ( sub @ ( X134 @ X138 @ X136 ) @ X137 ) )
& ! [X139: subst,X140: term,X141: subst] :
( ( X135 @ ( comp @ X139 @ X141 ) @ ( sub @ X140 @ X141 ) )
= ( sub @ ( X135 @ X139 @ X140 ) @ X141 ) ) )
=> ( ( ( sK522 @ sK520 @ sK519 )
!= ( sK521 @ sK520 @ sK519 ) )
& ( ( lam @ ( sK522 @ sh @ one ) )
= ( lam @ ( sK521 @ sh @ one ) ) )
& ! [X138: subst,X137: subst,X136: term] :
( ( sK521 @ ( comp @ X138 @ X137 ) @ ( sub @ X136 @ X137 ) )
= ( sub @ ( sK521 @ X138 @ X136 ) @ X137 ) )
& ! [X141: subst,X140: term,X139: subst] :
( ( sK522 @ ( comp @ X139 @ X141 ) @ ( sub @ X140 @ X141 ) )
= ( sub @ ( sK522 @ X139 @ X140 ) @ X141 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f513,plain,
( ? [X142: term,X143: term,X144: term,X145: term] :
( ( ( ap @ ( sub @ X142 @ id ) @ X145 )
= ( ap @ ( sub @ X143 @ id ) @ X144 ) )
& ( X144 != X145 ) )
=> ( ( ( ap @ ( sub @ sK523 @ id ) @ sK526 )
= ( ap @ ( sub @ sK524 @ id ) @ sK525 ) )
& ( sK526 != sK525 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f514,plain,
( ? [X150: term,X151: subst,X152: term > $o,X153: term > $o] :
( ( $true
= ( X153 @ X150 ) )
& ! [X154: term] :
( ( X152 @ X154 )
= ( X153 @ ( sub @ X154 @ ( push @ X150 @ X151 ) ) ) )
& ( $true
!= ( X152 @ one ) ) )
=> ( ( $true
= ( sK530 @ sK527 ) )
& ! [X154: term] :
( ( sK530 @ ( sub @ X154 @ ( push @ sK527 @ sK528 ) ) )
= ( sK529 @ X154 ) )
& ( ( sK529 @ one )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f515,plain,
! [X155: term,X156: subst,X157: term > $o,X158: term > $o] :
( ? [X159: term] :
( ( X157 @ X159 )
!= ( X158 @ ( sub @ X159 @ ( push @ X155 @ X156 ) ) ) )
=> ( ( X158 @ ( sub @ ( sK531 @ X158 @ X157 @ X156 @ X155 ) @ ( push @ X155 @ X156 ) ) )
!= ( X157 @ ( sK531 @ X158 @ X157 @ X156 @ X155 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f516,plain,
( ? [X160: term,X161: subst,X162: term,X163: term > $o,X164: term > $o] :
( ( $true
= ( X164 @ X162 ) )
& ( $true
!= ( X163 @ ( sub @ X162 @ ( push @ X160 @ X161 ) ) ) ) )
=> ( ( ( sK536 @ sK534 )
= $true )
& ( $true
!= ( sK535 @ ( sub @ sK534 @ ( push @ sK532 @ sK533 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f517,plain,
! [X170: subst] :
( ? [X171: d_subst] :
( ( d2subst @ X171 )
= X170 )
=> ( ( d2subst @ ( sK537 @ X170 ) )
= X170 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f518,plain,
( ? [X174: term,X175: term] :
( ( X174 != X175 )
& ( ( lam @ X174 )
= ( lam @ X175 ) ) )
=> ( ( sK539 != sK538 )
& ( ( lam @ sK539 )
= ( lam @ sK538 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f519,plain,
( ? [X178: term > $o,X179: term,X180: term > $o,X181: subst] :
( ( $true
= ( X180 @ X179 ) )
& ( $true
!= ( X178 @ one ) )
& ! [X182: term] :
( ( X180 @ ( sub @ X182 @ ( push @ X179 @ X181 ) ) )
= ( X178 @ X182 ) ) )
=> ( ( $true
= ( sK542 @ sK541 ) )
& ( $true
!= ( sK540 @ one ) )
& ! [X182: term] :
( ( sK542 @ ( sub @ X182 @ ( push @ sK541 @ sK543 ) ) )
= ( sK540 @ X182 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f520,plain,
! [X185: term > $o,X186: term,X187: term > $o,X188: subst] :
( ? [X189: term] :
( ( X187 @ ( sub @ X189 @ ( push @ X186 @ X188 ) ) )
!= ( X185 @ X189 ) )
=> ( ( X185 @ ( sK544 @ X188 @ X187 @ X186 @ X185 ) )
!= ( X187 @ ( sub @ ( sK544 @ X188 @ X187 @ X186 @ X185 ) @ ( push @ X186 @ X188 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f521,plain,
( ? [X190: subst,X191: term] :
( ( sub @ one @ ( push @ X191 @ X190 ) )
!= X191 )
=> ( ( sub @ one @ ( push @ sK546 @ sK545 ) )
!= sK546 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f522,plain,
( ? [X192: subst] :
( ( comp @ id @ X192 )
!= X192 )
=> ( ( comp @ id @ sK547 )
!= sK547 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f523,plain,
( ? [X193: subst] :
( ( comp @ X193 @ id )
!= X193 )
=> ( sK548
!= ( comp @ sK548 @ id ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f524,plain,
( ? [X198: subst] :
( ( comp @ X198 @ id )
!= X198 )
=> ( ( comp @ sK549 @ id )
!= sK549 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f525,plain,
( ? [X199: subst] :
( ( comp @ id @ X199 )
!= X199 )
=> ( sK550
!= ( comp @ id @ sK550 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f526,plain,
( ? [X200: subst,X201: term] :
( ( comp @ sh @ ( push @ X201 @ X200 ) )
!= X200 )
=> ( sK551
!= ( comp @ sh @ ( push @ sK552 @ sK551 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f527,plain,
( ? [X204: term] :
! [X205: subst > term > term] :
( ( sub @ X204 @ ( push @ one @ sh ) )
!= ( X205 @ sh @ one ) )
=> ! [X205: subst > term > term] :
( ( sub @ sK553 @ ( push @ one @ sh ) )
!= ( X205 @ sh @ one ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f528,plain,
! [X206: term] :
( ? [X207: subst > term > term] :
( ( sub @ X206 @ ( push @ one @ sh ) )
= ( X207 @ sh @ one ) )
=> ( ( sK554 @ X206 @ sh @ one )
= ( sub @ X206 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f529,plain,
( ? [X213: subst,X214: term] :
( ( sub @ one @ ( push @ X214 @ X213 ) )
!= X214 )
=> ( sK556
!= ( sub @ one @ ( push @ sK556 @ sK555 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f530,plain,
( ? [X215: subst,X216: term] :
( ( comp @ sh @ ( push @ X216 @ X215 ) )
!= X215 )
=> ( ( comp @ sh @ ( push @ sK558 @ sK557 ) )
!= sK557 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f531,plain,
( ? [X217: term,X218: term > $o,X219: term,X220: subst,X221: term > $o] :
( ( $true
= ( X221 @ X217 ) )
& ( $true
!= ( X218 @ ( sub @ X217 @ ( push @ X219 @ X220 ) ) ) ) )
=> ( ( ( sK563 @ sK559 )
= $true )
& ( $true
!= ( sK560 @ ( sub @ sK559 @ ( push @ sK561 @ sK562 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f532,plain,
( ? [X230: term,X231: term,X232: term,X233: term] :
( ( ( ap @ X233 @ X232 )
= ( ap @ X230 @ X231 ) )
& ( X230 != X233 ) )
=> ( ( ( ap @ sK567 @ sK566 )
= ( ap @ sK564 @ sK565 ) )
& ( sK564 != sK567 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f533,plain,
! [X234: term] :
( ? [X235: subst > term > term] :
( ( X235 @ sh @ one )
= ( sub @ X234 @ ( push @ one @ sh ) ) )
=> ( ( sK568 @ X234 @ sh @ one )
= ( sub @ X234 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f534,plain,
( ? [X236: term] :
! [X237: subst > term > term] :
( ( X237 @ sh @ one )
!= ( sub @ X236 @ ( push @ one @ sh ) ) )
=> ! [X237: subst > term > term] :
( ( X237 @ sh @ one )
!= ( sub @ sK569 @ ( push @ one @ sh ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f535,plain,
( ? [X238: subst] :
( ( comp @ id @ X238 )
!= X238 )
=> ( ( comp @ id @ sK570 )
!= sK570 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f536,plain,
( ? [X240: term,X241: term,X242: term,X243: term] :
( ( ( ap @ X242 @ X243 )
= ( ap @ X240 @ X241 ) )
& ( X241 != X243 ) )
=> ( ( ( ap @ sK573 @ sK574 )
= ( ap @ sK571 @ sK572 ) )
& ( sK574 != sK572 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f537,plain,
( ? [X249: subst] :
( ( push @ ( sub @ one @ X249 ) @ ( comp @ sh @ X249 ) )
!= X249 )
=> ( sK575
!= ( push @ ( sub @ one @ sK575 ) @ ( comp @ sh @ sK575 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f538,plain,
! [X250: term] :
( ? [X251: d_term] :
( ( d2term @ X251 )
= X250 )
=> ( ( d2term @ ( sK576 @ X250 ) )
= X250 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f539,plain,
( ? [X252: term,X253: subst] :
( ( sub @ one @ ( push @ X252 @ X253 ) )
!= X252 )
=> ( sK577
!= ( sub @ one @ ( push @ sK577 @ sK578 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f540,plain,
( ? [X260: term,X261: term,X262: term,X263: term] :
( ( ( ap @ ( sub @ X262 @ id ) @ X260 )
= ( ap @ ( sub @ X263 @ id ) @ X261 ) )
& ( X260 != X261 ) )
=> ( ( ( ap @ ( sub @ sK582 @ id ) @ sK580 )
= ( ap @ ( sub @ sK581 @ id ) @ sK579 ) )
& ( sK579 != sK580 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f541,plain,
( ? [X264: term,X265: term,X266: term,X267: term] :
( ( ( ap @ X265 @ X264 )
= ( ap @ X267 @ X266 ) )
& ( X264 != X266 ) )
=> ( ( ( ap @ sK586 @ sK585 )
= ( ap @ sK584 @ sK583 ) )
& ( sK585 != sK583 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f542,plain,
( ? [X272: term] :
! [X273: subst > term > term] :
? [X274: subst,X275: term] :
( ( X273 @ X274 @ X275 )
!= ( sub @ X272 @ ( push @ X275 @ X274 ) ) )
=> ! [X273: subst > term > term] :
? [X275: term,X274: subst] :
( ( sub @ sK587 @ ( push @ X275 @ X274 ) )
!= ( X273 @ X274 @ X275 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f543,plain,
! [X273: subst > term > term] :
( ? [X275: term,X274: subst] :
( ( sub @ sK587 @ ( push @ X275 @ X274 ) )
!= ( X273 @ X274 @ X275 ) )
=> ( ( X273 @ ( sK588 @ X273 ) @ ( sK589 @ X273 ) )
!= ( sub @ sK587 @ ( push @ ( sK589 @ X273 ) @ ( sK588 @ X273 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f544,plain,
! [X276: term] :
( ? [X277: subst > term > term] :
! [X278: subst,X279: term] :
( ( sub @ X276 @ ( push @ X279 @ X278 ) )
= ( X277 @ X278 @ X279 ) )
=> ! [X279: term,X278: subst] :
( ( sK590 @ X276 @ X278 @ X279 )
= ( sub @ X276 @ ( push @ X279 @ X278 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f473,plain,
( ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ! [X0: d_subst] : ( d_id = X0 )
& ( hoasinduction_lem1_gthm = $true )
& ( ( shinj = $true )
| ? [X1: term,X2: term] :
( ( ( sub @ X1 @ sh )
= ( sub @ X2 @ sh ) )
& ( X1 != X2 ) ) )
& ( ! [X3: term,X4: term] :
( ( ( sub @ X3 @ sh )
!= ( sub @ X4 @ sh ) )
| ( X3 = X4 ) )
| ( shinj != $true ) )
& ( lamnotvar = $true )
& ( ? [X5: term] :
( ( sub @ X5 @ id )
!= X5 )
| ! [X6: term] :
( ( sub @ X6 @ id )
= X6 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ! [X7: term] :
( ( sub @ X7 @ id )
= X7 )
& ? [X8: term] :
( ( sub @ X8 @ id )
!= X8 ) ) )
& ( ( $true = sP58 )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( $true != sP58 ) )
& ( ( axvarid = $true )
| ? [X9: term] :
( ( sub @ X9 @ id )
!= X9 ) )
& ( ! [X10: term] :
( ( sub @ X10 @ id )
= X10 )
| ( axvarid != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP57 ) )
& ( ( $true = sP57 )
| ( hoasinduction_lem3aa != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP56 ) )
& ( ( $true = sP56 )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( axidr = $true )
| ? [X11: subst] :
( ( comp @ X11 @ id )
!= X11 ) )
& ( ! [X12: subst] :
( ( comp @ X12 @ id )
= X12 )
| ( axidr != $true ) )
& ( ( $true = sP55 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP55 ) )
& ! [X13: d_term] : ( d_one = X13 )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP53 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP53 != $true ) )
& ( axapp = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP51 != $true ) )
& ( ( sP51 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( hoasapnotvar = $true )
& ( ! [X14: term,X15: term,X16: subst > term > term] :
( ( ( lam @ ( X16 @ sh @ one ) )
!= ( ap @ ( sub @ X15 @ id ) @ X14 ) )
| ? [X17: subst,X18: subst,X19: term] :
( ( sub @ ( X16 @ X18 @ X19 ) @ X17 )
!= ( X16 @ ( comp @ X18 @ X17 ) @ ( sub @ X19 @ X17 ) ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X20: term,X21: term,X22: subst > term > term] :
( ( ( lam @ ( X22 @ sh @ one ) )
= ( ap @ ( sub @ X21 @ id ) @ X20 ) )
& ! [X23: subst,X24: subst,X25: term] :
( ( sub @ ( X22 @ X24 @ X25 ) @ X23 )
= ( X22 @ ( comp @ X24 @ X23 ) @ ( sub @ X25 @ X23 ) ) ) ) )
& ( axclos = $true )
& ( ? [X26: term,X27: term,X28: term,X29: term] :
( ( X26 != X29 )
& ( ( ap @ X26 @ X27 )
= ( ap @ X29 @ X28 ) ) )
| ! [X30: term,X31: term,X32: term,X33: term] :
( ( X32 = X33 )
| ( ( ap @ ( sub @ X32 @ id ) @ X30 )
!= ( ap @ ( sub @ X33 @ id ) @ X31 ) ) )
| ? [X34: term] :
( ( sub @ X34 @ id )
!= X34 )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X35: term,X36: term,X37: term,X38: term] :
( ( X35 = X38 )
| ( ( ap @ X35 @ X36 )
!= ( ap @ X38 @ X37 ) ) )
& ? [X39: term,X40: term,X41: term,X42: term] :
( ( X41 != X42 )
& ( ( ap @ ( sub @ X41 @ id ) @ X39 )
= ( ap @ ( sub @ X42 @ id ) @ X40 ) ) )
& ! [X43: term] :
( ( sub @ X43 @ id )
= X43 ) ) )
& ( ulamvar1 != $true )
& ( ! [X44: term,X45: term,X46: term,X47: term] :
( ( ( ap @ ( sub @ X44 @ id ) @ X46 )
!= ( ap @ ( sub @ X47 @ id ) @ X45 ) )
| ( X44 = X47 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X48: term,X49: term,X50: term,X51: term] :
( ( ( ap @ ( sub @ X48 @ id ) @ X50 )
= ( ap @ ( sub @ X51 @ id ) @ X49 ) )
& ( X48 != X51 ) ) )
& ( ! [X52: subst > term > subst > $o,X53: term] :
( ? [X54: subst > term > term] :
( ! [X55: term] :
( ( ( X52 @ id @ ( X54 @ id @ X55 ) @ id )
= $true )
| ( $true
!= ( X52 @ id @ X55 @ id ) ) )
& ( $true
!= ( X52 @ id @ ( lam @ ( X54 @ sh @ one ) ) @ id ) )
& ! [X56: term,X57: subst,X58: subst] :
( ( X54 @ ( comp @ X58 @ X57 ) @ ( sub @ X56 @ X57 ) )
= ( sub @ ( X54 @ X58 @ X56 ) @ X57 ) ) )
| ( $true
= ( sP46 @ X52 ) )
| ( $true
= ( X52 @ id @ X53 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X59: subst > term > subst > $o,X60: term] :
( ! [X61: subst > term > term] :
( ? [X62: term] :
( ( $true
!= ( X59 @ id @ ( X61 @ id @ X62 ) @ id ) )
& ( $true
= ( X59 @ id @ X62 @ id ) ) )
| ( $true
= ( X59 @ id @ ( lam @ ( X61 @ sh @ one ) ) @ id ) )
| ? [X63: term,X64: subst,X65: subst] :
( ( X61 @ ( comp @ X65 @ X64 ) @ ( sub @ X63 @ X64 ) )
!= ( sub @ ( X61 @ X65 @ X63 ) @ X64 ) ) )
& ( $true
!= ( sP46 @ X59 ) )
& ( ( X59 @ id @ X60 @ id )
!= $true ) ) )
& ( ! [X66: term] :
( ( sub @ X66 @ id )
= X66 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X67: term] :
( ( sub @ X67 @ id )
!= X67 ) )
& ( induction2_gthm = $true )
& ( pushprop = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X68: subst,X69: term,X70: term > $o] :
? [X71: term > $o] :
! [X72: term] :
( ( X70 @ ( sub @ X72 @ ( push @ X69 @ X68 ) ) )
= ( X71 @ X72 ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X73: subst,X74: term,X75: term > $o] :
! [X76: term > $o] :
? [X77: term] :
( ( X75 @ ( sub @ X77 @ ( push @ X74 @ X73 ) ) )
!= ( X76 @ X77 ) ) )
& ! [X78: d_term,X79: d_term] :
( ( X78 = X79 )
| ( ( d2term @ X79 )
!= ( d2term @ X78 ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X80: subst > term > subst > $o] :
? [X81: term > $o] :
! [X82: term] :
( ( X81 @ X82 )
= ( X80 @ id @ X82 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X83: subst > term > subst > $o] :
! [X84: term > $o] :
? [X85: term] :
( ( X84 @ X85 )
!= ( X83 @ id @ X85 @ id ) ) )
& ( ( $true = sP45 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP45 ) )
& ( ( $true = sP42 )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( $true != sP42 ) )
& ( hoasapnotvar_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X86: subst > term > subst > $o] :
! [X87: term > $o] :
? [X88: term] :
( ( X86 @ id @ X88 @ id )
!= ( X87 @ X88 ) ) )
& ( ! [X89: subst > term > subst > $o] :
? [X90: term > $o] :
! [X91: term] :
( ( X89 @ id @ X91 @ id )
= ( X90 @ X91 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X92: term] :
! [X93: subst > term > term] :
? [X94: term,X95: subst] :
( ( X93 @ X95 @ X94 )
!= ( sub @ X92 @ ( push @ X94 @ X95 ) ) ) )
& ( ! [X96: term] :
? [X97: subst > term > term] :
! [X98: term,X99: subst] :
( ( sub @ X96 @ ( push @ X98 @ X99 ) )
= ( X97 @ X99 @ X98 ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( lamnotap != $true )
& ( one
= ( d2term @ d_one ) )
& ( axassoc = $true )
& ( ( induction2 = $true )
| ( $true != sP41 ) )
& ( ( $true = sP41 )
| ( induction2 != $true ) )
& ( ! [X100: term,X101: term > $o] :
( ? [X102: term] :
( ( $true
= ( X101 @ X102 ) )
& ( $true
!= ( X101 @ ( lam @ X102 ) ) ) )
| ( ( X101 @ X100 )
= $true )
| ( ( sP39 @ X101 )
= $true )
| ? [X103: term] :
( ( $true
!= ( X101 @ X103 ) )
& ( $true
= ( var @ X103 ) ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X104: term,X105: term > $o] :
( ! [X106: term] :
( ( $true
!= ( X105 @ X106 ) )
| ( $true
= ( X105 @ ( lam @ X106 ) ) ) )
& ( $true
!= ( X105 @ X104 ) )
& ( $true
!= ( sP39 @ X105 ) )
& ! [X107: term] :
( ( $true
= ( X105 @ X107 ) )
| ( $true
!= ( var @ X107 ) ) ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X108: term > $o,X109: term,X110: subst] :
! [X111: term > $o] :
? [X112: term] :
( ( X108 @ ( sub @ X112 @ ( push @ X109 @ X110 ) ) )
!= ( X111 @ X112 ) ) )
& ( ! [X113: term > $o,X114: term,X115: subst] :
? [X116: term > $o] :
! [X117: term] :
( ( X113 @ ( sub @ X117 @ ( push @ X114 @ X115 ) ) )
= ( X116 @ X117 ) )
| ( pushprop_lem0_lthm != $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_lthm = $true )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP38 ) )
& ( ( $true = sP38 )
| ( hoasinduction_lthm != $true ) )
& ( ( ! [X118: subst] :
( ( comp @ X118 @ id )
= X118 )
& ! [X119: subst] :
( ( comp @ id @ X119 )
= X119 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X120: subst] :
( ( comp @ X120 @ id )
!= X120 )
| ? [X121: subst] :
( ( comp @ id @ X121 )
!= X121 ) )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3v2 != $true ) )
& ( ! [X122: term,X123: subst,X124: subst > term > term,X125: subst > term > term] :
( ( ( X124 @ X123 @ X122 )
= ( X125 @ X123 @ X122 ) )
| ( ( lam @ ( X124 @ sh @ one ) )
!= ( lam @ ( X125 @ sh @ one ) ) )
| ? [X126: term,X127: subst,X128: subst] :
( ( X124 @ ( comp @ X128 @ X127 ) @ ( sub @ X126 @ X127 ) )
!= ( sub @ ( X124 @ X128 @ X126 ) @ X127 ) )
| ? [X129: subst,X130: term,X131: subst] :
( ( sub @ ( X125 @ X129 @ X130 ) @ X131 )
!= ( X125 @ ( comp @ X129 @ X131 ) @ ( sub @ X130 @ X131 ) ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X132: term,X133: subst,X134: subst > term > term,X135: subst > term > term] :
( ( ( X135 @ X133 @ X132 )
!= ( X134 @ X133 @ X132 ) )
& ( ( lam @ ( X134 @ sh @ one ) )
= ( lam @ ( X135 @ sh @ one ) ) )
& ! [X136: term,X137: subst,X138: subst] :
( ( X134 @ ( comp @ X138 @ X137 ) @ ( sub @ X136 @ X137 ) )
= ( sub @ ( X134 @ X138 @ X136 ) @ X137 ) )
& ! [X139: subst,X140: term,X141: subst] :
( ( X135 @ ( comp @ X139 @ X141 ) @ ( sub @ X140 @ X141 ) )
= ( sub @ ( X135 @ X139 @ X140 ) @ X141 ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP27 )
| ( hoasinduction_lem3 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP27 ) )
& ( ( hoasapinj2 = $true )
| ? [X142: term,X143: term,X144: term,X145: term] :
( ( ( ap @ ( sub @ X142 @ id ) @ X145 )
= ( ap @ ( sub @ X143 @ id ) @ X144 ) )
& ( X144 != X145 ) ) )
& ( ! [X146: term,X147: term,X148: term,X149: term] :
( ( ( ap @ ( sub @ X147 @ id ) @ X148 )
!= ( ap @ ( sub @ X146 @ id ) @ X149 ) )
| ( X148 = X149 ) )
| ( hoasapinj2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X150: term,X151: subst,X152: term > $o,X153: term > $o] :
( ( $true
= ( X153 @ X150 ) )
& ! [X154: term] :
( ( X152 @ X154 )
= ( X153 @ ( sub @ X154 @ ( push @ X150 @ X151 ) ) ) )
& ( $true
!= ( X152 @ one ) ) ) )
& ( ! [X155: term,X156: subst,X157: term > $o,X158: term > $o] :
( ( $true
!= ( X158 @ X155 ) )
| ? [X159: term] :
( ( X157 @ X159 )
!= ( X158 @ ( sub @ X159 @ ( push @ X155 @ X156 ) ) ) )
| ( $true
= ( X157 @ one ) ) )
| ( pushprop_lem1v2 != $true ) )
& ( hoasapinj2_gthm = $true )
& ( ( pushprop_lem1 = $true )
| ? [X160: term,X161: subst,X162: term,X163: term > $o,X164: term > $o] :
( ( $true
= ( X164 @ X162 ) )
& ( $true
!= ( X163 @ ( sub @ X162 @ ( push @ X160 @ X161 ) ) ) ) ) )
& ( ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
!= ( X169 @ X167 ) )
| ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) )
| ( pushprop_lem1 != $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( $true = sP25 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP25 ) )
& ( termmset_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP24 ) )
& ( ( $true = sP24 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ! [X170: subst] :
? [X171: d_subst] :
( ( d2subst @ X171 )
= X170 )
& ( axvarshift = $true )
& ( hoasapnotvar_lthm = $true )
& ( ! [X172: term,X173: term] :
( ( X172 = X173 )
| ( ( lam @ X172 )
!= ( lam @ X173 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X174: term,X175: term] :
( ( X174 != X175 )
& ( ( lam @ X174 )
= ( lam @ X175 ) ) ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP22 ) )
& ( ( $true = sP22 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X176: d_subst,X177: d_subst] :
( ( X176 = X177 )
| ( ( d2subst @ X177 )
!= ( d2subst @ X176 ) ) )
& ( ( $true = sP21 )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( $true != sP21 ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X178: term > $o,X179: term,X180: term > $o,X181: subst] :
( ( $true
= ( X180 @ X179 ) )
& ( $true
!= ( X178 @ one ) )
& ! [X182: term] :
( ( X180 @ ( sub @ X182 @ ( push @ X179 @ X181 ) ) )
= ( X178 @ X182 ) ) )
& ! [X183: subst,X184: term] :
( ( sub @ one @ ( push @ X184 @ X183 ) )
= X184 ) ) )
& ( ! [X185: term > $o,X186: term,X187: term > $o,X188: subst] :
( ( $true
!= ( X187 @ X186 ) )
| ( $true
= ( X185 @ one ) )
| ? [X189: term] :
( ( X187 @ ( sub @ X189 @ ( push @ X186 @ X188 ) ) )
!= ( X185 @ X189 ) ) )
| ? [X190: subst,X191: term] :
( ( sub @ one @ ( push @ X191 @ X190 ) )
!= X191 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( hoaslamnotvar = $true )
& ( ( $true = sP15 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP15 ) )
& ( ( sP12 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP12 != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ( ? [X192: subst] :
( ( comp @ id @ X192 )
!= X192 )
| ? [X193: subst] :
( ( comp @ X193 @ id )
!= X193 ) )
& ! [X194: subst] :
( ( comp @ X194 @ id )
= X194 )
& ! [X195: subst] :
( ( comp @ id @ X195 )
= X195 ) ) )
& ( ( ! [X196: subst] :
( ( comp @ id @ X196 )
= X196 )
& ! [X197: subst] :
( ( comp @ X197 @ id )
= X197 ) )
| ? [X198: subst] :
( ( comp @ X198 @ id )
!= X198 )
| ? [X199: subst] :
( ( comp @ id @ X199 )
!= X199 )
| ( substmonoid_lthm != $true ) )
& ( ( axshiftcons = $true )
| ? [X200: subst,X201: term] :
( ( comp @ sh @ ( push @ X201 @ X200 ) )
!= X200 ) )
& ( ! [X202: subst,X203: term] :
( ( comp @ sh @ ( push @ X203 @ X202 ) )
= X202 )
| ( axshiftcons != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X204: term] :
! [X205: subst > term > term] :
( ( sub @ X204 @ ( push @ one @ sh ) )
!= ( X205 @ sh @ one ) ) )
& ( ! [X206: term] :
? [X207: subst > term > term] :
( ( sub @ X206 @ ( push @ one @ sh ) )
= ( X207 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ! [X208: term,X209: term > $o,X210: term,X211: subst,X212: term > $o] :
( ( $true
!= ( X212 @ X208 ) )
| ( $true
= ( X209 @ ( sub @ X208 @ ( push @ X210 @ X211 ) ) ) ) )
| ? [X213: subst,X214: term] :
( ( sub @ one @ ( push @ X214 @ X213 ) )
!= X214 )
| ? [X215: subst,X216: term] :
( ( comp @ sh @ ( push @ X216 @ X215 ) )
!= X215 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ? [X217: term,X218: term > $o,X219: term,X220: subst,X221: term > $o] :
( ( $true
= ( X221 @ X217 ) )
& ( $true
!= ( X218 @ ( sub @ X217 @ ( push @ X219 @ X220 ) ) ) ) )
& ! [X222: subst,X223: term] :
( ( sub @ one @ ( push @ X223 @ X222 ) )
= X223 )
& ! [X224: subst,X225: term] :
( ( comp @ sh @ ( push @ X225 @ X224 ) )
= X224 ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ulamvarsh = $true )
& ( ( induction2lem_lthm = $true )
| ( $true != sP8 ) )
& ( ( $true = sP8 )
| ( induction2lem_lthm != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP4 != $true ) )
& ( ( sP4 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ulamvarind = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X226: term,X227: term,X228: term,X229: term] :
( ( ( ap @ X229 @ X228 )
!= ( ap @ X226 @ X227 ) )
| ( X226 = X229 ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X230: term,X231: term,X232: term,X233: term] :
( ( ( ap @ X233 @ X232 )
= ( ap @ X230 @ X231 ) )
& ( X230 != X233 ) ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X234: term] :
? [X235: subst > term > term] :
( ( X235 @ sh @ one )
= ( sub @ X234 @ ( push @ one @ sh ) ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X236: term] :
! [X237: subst > term > term] :
( ( X237 @ sh @ one )
!= ( sub @ X236 @ ( push @ one @ sh ) ) ) )
& ( ( axidl = $true )
| ? [X238: subst] :
( ( comp @ id @ X238 )
!= X238 ) )
& ( ! [X239: subst] :
( ( comp @ id @ X239 )
= X239 )
| ( axidl != $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( apinj2 = $true )
| ? [X240: term,X241: term,X242: term,X243: term] :
( ( ( ap @ X242 @ X243 )
= ( ap @ X240 @ X241 ) )
& ( X241 != X243 ) ) )
& ( ! [X244: term,X245: term,X246: term,X247: term] :
( ( ( ap @ X246 @ X247 )
!= ( ap @ X244 @ X245 ) )
| ( X245 = X247 ) )
| ( apinj2 != $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 ) ) ) ) ) ) )
& ( ! [X248: subst] :
( ( push @ ( sub @ one @ X248 ) @ ( comp @ sh @ X248 ) )
= X248 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X249: subst] :
( ( push @ ( sub @ one @ X249 ) @ ( comp @ sh @ X249 ) )
!= X249 ) )
& ! [X250: term] :
? [X251: d_term] :
( ( d2term @ X251 )
= X250 )
& ( ( axvarcons = $true )
| ? [X252: term,X253: subst] :
( ( sub @ one @ ( push @ X252 @ X253 ) )
!= X252 ) )
& ( ! [X254: term,X255: subst] :
( ( sub @ one @ ( push @ X254 @ X255 ) )
= X254 )
| ( axvarcons != $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ! [X256: term,X257: term,X258: term,X259: term] :
( ( ( ap @ X257 @ X256 )
!= ( ap @ X259 @ X258 ) )
| ( X256 = X258 ) )
& ? [X260: term,X261: term,X262: term,X263: term] :
( ( ( ap @ ( sub @ X262 @ id ) @ X260 )
= ( ap @ ( sub @ X263 @ id ) @ X261 ) )
& ( X260 != X261 ) ) ) )
& ( ? [X264: term,X265: term,X266: term,X267: term] :
( ( ( ap @ X265 @ X264 )
= ( ap @ X267 @ X266 ) )
& ( X264 != X266 ) )
| ! [X268: term,X269: term,X270: term,X271: term] :
( ( ( ap @ ( sub @ X271 @ id ) @ X269 )
!= ( ap @ ( sub @ X270 @ id ) @ X268 ) )
| ( X268 = X269 ) )
| ( hoasapinj2_lthm != $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X272: term] :
! [X273: subst > term > term] :
? [X274: subst,X275: term] :
( ( X273 @ X274 @ X275 )
!= ( sub @ X272 @ ( push @ X275 @ X274 ) ) ) )
& ( ! [X276: term] :
? [X277: subst > term > term] :
! [X278: subst,X279: term] :
( ( sub @ X276 @ ( push @ X279 @ X278 ) )
= ( X277 @ X278 @ X279 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( pushprop_gthm = $true )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP2 ) )
& ( ( $true = sP2 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( hoasapinj1_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( $true = sP1 )
| ( hoasinduction_lem2 != $true ) )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP1 ) )
& ( pushprop_lem0_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( axmap = $true )
& ( induction2lem_gthm = $true )
& ( axabs = $true )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( d2subst @ d_id )
= id ) ),
inference(rectify,[status(thm)],[f472]) ).
thf(f472,plain,
( ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ! [X156: d_subst] : ( d_id = X156 )
& ( hoasinduction_lem1_gthm = $true )
& ( ( shinj = $true )
| ? [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
= ( sub @ X185 @ sh ) )
& ( X185 != X186 ) ) )
& ( ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) )
| ( shinj != $true ) )
& ( lamnotvar = $true )
& ( ? [X208: term] :
( ( sub @ X208 @ id )
!= X208 )
| ! [X209: term] :
( ( sub @ X209 @ id )
= X209 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ! [X208: term] :
( ( sub @ X208 @ id )
= X208 )
& ? [X209: term] :
( ( sub @ X209 @ id )
!= X209 ) ) )
& ( ( $true = sP58 )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( $true != sP58 ) )
& ( ( axvarid = $true )
| ? [X265: term] :
( ( sub @ X265 @ id )
!= X265 ) )
& ( ! [X265: term] :
( ( sub @ X265 @ id )
= X265 )
| ( axvarid != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP57 ) )
& ( ( $true = sP57 )
| ( hoasinduction_lem3aa != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP56 ) )
& ( ( $true = sP56 )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( axidr = $true )
| ? [X393: subst] :
( ( comp @ X393 @ id )
!= X393 ) )
& ( ! [X393: subst] :
( ( comp @ X393 @ id )
= X393 )
| ( axidr != $true ) )
& ( ( $true = sP55 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP55 ) )
& ! [X141: d_term] : ( d_one = X141 )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP53 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP53 != $true ) )
& ( axapp = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP51 != $true ) )
& ( ( sP51 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( hoasapnotvar = $true )
& ( ! [X378: term,X379: term,X380: subst > term > term] :
( ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( lam @ ( X380 @ sh @ one ) ) )
| ? [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
!= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X378: term,X379: term,X380: subst > term > term] :
( ( ( ap @ ( sub @ X379 @ id ) @ X378 )
= ( lam @ ( X380 @ sh @ one ) ) )
& ! [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) ) ) )
& ( axclos = $true )
& ( ? [X399: term,X400: term,X401: term,X402: term] :
( ( X399 != X402 )
& ( ( ap @ X399 @ X400 )
= ( ap @ X402 @ X401 ) ) )
| ! [X403: term,X404: term,X405: term,X406: term] :
( ( X405 = X406 )
| ( ( ap @ ( sub @ X405 @ id ) @ X403 )
!= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) )
| ? [X398: term] :
( ( sub @ X398 @ id )
!= X398 )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X399: term,X400: term,X401: term,X402: term] :
( ( X399 = X402 )
| ( ( ap @ X399 @ X400 )
!= ( ap @ X402 @ X401 ) ) )
& ? [X403: term,X404: term,X405: term,X406: term] :
( ( X405 != X406 )
& ( ( ap @ ( sub @ X405 @ id ) @ X403 )
= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) )
& ! [X398: term] :
( ( sub @ X398 @ id )
= X398 ) ) )
& ( ulamvar1 != $true )
& ( ! [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
!= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
| ( X394 = X397 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
& ( X394 != X397 ) ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ? [X58: subst > term > term] :
( ! [X59: term] :
( ( $true
= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
| ( $true
!= ( X54 @ id @ X59 @ id ) ) )
& ( $true
!= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
& ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) )
| ( $true
= ( sP46 @ X54 ) )
| ( $true
= ( X54 @ id @ X55 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X54: subst > term > subst > $o,X55: term] :
( ! [X58: subst > term > term] :
( ? [X59: term] :
( ( $true
!= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
& ( $true
= ( X54 @ id @ X59 @ id ) ) )
| ( $true
= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
| ? [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
!= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) )
& ( $true
!= ( sP46 @ X54 ) )
& ( $true
!= ( X54 @ id @ X55 @ id ) ) ) )
& ( ! [X276: term] :
( ( sub @ X276 @ id )
= X276 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X276: term] :
( ( sub @ X276 @ id )
!= X276 ) )
& ( induction2_gthm = $true )
& ( pushprop = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X21: subst,X22: term,X23: term > $o] :
? [X24: term > $o] :
! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X21: subst,X22: term,X23: term > $o] :
! [X24: term > $o] :
? [X25: term] :
( ( X24 @ X25 )
!= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) ) )
& ! [X337: d_term,X338: d_term] :
( ( X337 = X338 )
| ( ( d2term @ X337 )
!= ( d2term @ X338 ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X187: subst > term > subst > $o] :
? [X188: term > $o] :
! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X187: subst > term > subst > $o] :
! [X188: term > $o] :
? [X189: term] :
( ( X188 @ X189 )
!= ( X187 @ id @ X189 @ id ) ) )
& ( ( $true = sP45 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP45 ) )
& ( ( $true = sP42 )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( $true != sP42 ) )
& ( hoasapnotvar_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X138: subst > term > subst > $o] :
! [X139: term > $o] :
? [X140: term] :
( ( X138 @ id @ X140 @ id )
!= ( X139 @ X140 ) ) )
& ( ! [X138: subst > term > subst > $o] :
? [X139: term > $o] :
! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X95: term] :
! [X96: subst > term > term] :
? [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
!= ( sub @ X95 @ ( push @ X97 @ X98 ) ) ) )
& ( ! [X95: term] :
? [X96: subst > term > term] :
! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( lamnotap != $true )
& ( one
= ( d2term @ d_one ) )
& ( axassoc = $true )
& ( ( induction2 = $true )
| ( $true != sP41 ) )
& ( ( $true = sP41 )
| ( induction2 != $true ) )
& ( ! [X429: term,X430: term > $o] :
( ? [X432: term] :
( ( $true
= ( X430 @ X432 ) )
& ( $true
!= ( X430 @ ( lam @ X432 ) ) ) )
| ( $true
= ( X430 @ X429 ) )
| ( $true
= ( sP39 @ X430 ) )
| ? [X431: term] :
( ( $true
!= ( X430 @ X431 ) )
& ( $true
= ( var @ X431 ) ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X429: term,X430: term > $o] :
( ! [X432: term] :
( ( $true
!= ( X430 @ X432 ) )
| ( $true
= ( X430 @ ( lam @ X432 ) ) ) )
& ( $true
!= ( X430 @ X429 ) )
& ( $true
!= ( sP39 @ X430 ) )
& ! [X431: term] :
( ( $true
= ( X430 @ X431 ) )
| ( $true
!= ( var @ X431 ) ) ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X180: term > $o,X181: term,X182: subst] :
! [X183: term > $o] :
? [X184: term] :
( ( X183 @ X184 )
!= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) ) )
& ( ! [X180: term > $o,X181: term,X182: subst] :
? [X183: term > $o] :
! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) )
| ( pushprop_lem0_lthm != $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_lthm = $true )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP38 ) )
& ( ( $true = sP38 )
| ( hoasinduction_lthm != $true ) )
& ( ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X33: subst] :
( ( comp @ X33 @ id )
!= X33 )
| ? [X34: subst] :
( ( comp @ id @ X34 )
!= X34 ) )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3v2 != $true ) )
& ( ! [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
= ( X268 @ X267 @ X266 ) )
| ( ( lam @ ( X268 @ sh @ one ) )
!= ( lam @ ( X269 @ sh @ one ) ) )
| ? [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
!= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
| ? [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
!= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
!= ( X268 @ X267 @ X266 ) )
& ( ( lam @ ( X268 @ sh @ one ) )
= ( lam @ ( X269 @ sh @ one ) ) )
& ! [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
& ! [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP27 )
| ( hoasinduction_lem3 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP27 ) )
& ( ( hoasapinj2 = $true )
| ? [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
& ( X174 != X175 ) ) )
& ( ! [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
| ( X174 = X175 ) )
| ( hoasapinj2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( $true
= ( X243 @ X240 ) )
& ! [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
= ( X242 @ X244 ) )
& ( ( X242 @ one )
!= $true ) ) )
& ( ! [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( $true
!= ( X243 @ X240 ) )
| ? [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
!= ( X242 @ X244 ) )
| ( ( X242 @ one )
= $true ) )
| ( pushprop_lem1v2 != $true ) )
& ( hoasapinj2_gthm = $true )
& ( ( pushprop_lem1 = $true )
| ? [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
= ( X169 @ X167 ) )
& ( $true
!= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) ) )
& ( ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
!= ( X169 @ X167 ) )
| ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) )
| ( pushprop_lem1 != $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( $true = sP25 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP25 ) )
& ( termmset_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP24 ) )
& ( ( $true = sP24 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ( axvarshift = $true )
& ( hoasapnotvar_lthm = $true )
& ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X0: term,X1: term] :
( ( X0 != X1 )
& ( ( lam @ X1 )
= ( lam @ X0 ) ) ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP22 ) )
& ( ( $true = sP22 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X312: d_subst,X313: d_subst] :
( ( X312 = X313 )
| ( ( d2subst @ X312 )
!= ( d2subst @ X313 ) ) )
& ( ( $true = sP21 )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( $true != sP21 ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ( ( X30 @ X29 )
= $true )
& ( $true
!= ( X28 @ one ) )
& ! [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
= ( X28 @ X32 ) ) )
& ! [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
= X27 ) ) )
& ( ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ( ( X30 @ X29 )
!= $true )
| ( $true
= ( X28 @ one ) )
| ? [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
!= ( X28 @ X32 ) ) )
| ? [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
!= X27 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( hoaslamnotvar = $true )
& ( ( $true = sP15 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP15 ) )
& ( ( sP12 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP12 != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ( ? [X523: subst] :
( ( comp @ id @ X523 )
!= X523 )
| ? [X524: subst] :
( ( comp @ X524 @ id )
!= X524 ) )
& ! [X522: subst] :
( ( comp @ X522 @ id )
= X522 )
& ! [X521: subst] :
( ( comp @ id @ X521 )
= X521 ) ) )
& ( ( ! [X523: subst] :
( ( comp @ id @ X523 )
= X523 )
& ! [X524: subst] :
( ( comp @ X524 @ id )
= X524 ) )
| ? [X522: subst] :
( ( comp @ X522 @ id )
!= X522 )
| ? [X521: subst] :
( ( comp @ id @ X521 )
!= X521 )
| ( substmonoid_lthm != $true ) )
& ( ( axshiftcons = $true )
| ? [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
!= X198 ) )
& ( ! [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
= X198 )
| ( axshiftcons != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X263: term] :
! [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
!= ( X264 @ sh @ one ) ) )
& ( ! [X263: term] :
? [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
= ( X264 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ! [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
!= ( X336 @ X332 ) )
| ( $true
= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) ) )
| ? [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
!= X329 )
| ? [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
!= X330 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ? [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
= ( X336 @ X332 ) )
& ( $true
!= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) ) )
& ! [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
= X329 )
& ! [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
= X330 ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ulamvarsh = $true )
& ( ( induction2lem_lthm = $true )
| ( $true != sP8 ) )
& ( ( $true = sP8 )
| ( induction2lem_lthm != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP4 != $true ) )
& ( ( sP4 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ulamvarind = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
= ( ap @ X179 @ X178 ) )
& ( X176 != X179 ) ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X519: term] :
? [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
= ( X520 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X519: term] :
! [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
!= ( X520 @ sh @ one ) ) )
& ( ( axidl = $true )
| ? [X223: subst] :
( ( comp @ id @ X223 )
!= X223 ) )
& ( ! [X223: subst] :
( ( comp @ id @ X223 )
= X223 )
| ( axidl != $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( apinj2 = $true )
| ? [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
= ( ap @ X425 @ X426 ) )
& ( X426 != X428 ) ) )
& ( ! [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
!= ( ap @ X425 @ X426 ) )
| ( X426 = X428 ) )
| ( apinj2 != $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 ) ) ) ) ) ) )
& ( ! [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
= X384 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
!= X384 ) )
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& ( ( axvarcons = $true )
| ? [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
!= X245 ) )
& ( ! [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
= X245 )
| ( axvarcons != $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ! [X190: term,X191: term,X192: term,X193: term] :
( ( ( ap @ X191 @ X190 )
!= ( ap @ X193 @ X192 ) )
| ( X190 = X192 ) )
& ? [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
& ( X194 != X195 ) ) ) )
& ( ? [X190: term,X191: term,X192: term,X193: term] :
( ( ( ap @ X191 @ X190 )
= ( ap @ X193 @ X192 ) )
& ( X190 != X192 ) )
| ! [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
!= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
| ( X194 = X195 ) )
| ( hoasapinj2_lthm != $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X314: term] :
! [X315: subst > term > term] :
? [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
!= ( X315 @ X316 @ X317 ) ) )
& ( ! [X314: term] :
? [X315: subst > term > term] :
! [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
= ( X315 @ X316 @ X317 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( pushprop_gthm = $true )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP2 ) )
& ( ( $true = sP2 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( hoasapinj1_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( $true = sP1 )
| ( hoasinduction_lem2 != $true ) )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP1 ) )
& ( pushprop_lem0_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( axmap = $true )
& ( induction2lem_gthm = $true )
& ( axabs = $true )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( d2subst @ d_id )
= id ) ),
inference(flattening,[status(thm)],[f471]) ).
thf(f471,plain,
( ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ! [X156: d_subst] : ( d_id = X156 )
& ( hoasinduction_lem1_gthm = $true )
& ( ( shinj = $true )
| ? [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
= ( sub @ X185 @ sh ) )
& ( X185 != X186 ) ) )
& ( ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) )
| ( shinj != $true ) )
& ( lamnotvar = $true )
& ( ? [X208: term] :
( ( sub @ X208 @ id )
!= X208 )
| ! [X209: term] :
( ( sub @ X209 @ id )
= X209 )
| ( termmset_lthm != $true ) )
& ( ( termmset_lthm = $true )
| ( ! [X208: term] :
( ( sub @ X208 @ id )
= X208 )
& ? [X209: term] :
( ( sub @ X209 @ id )
!= X209 ) ) )
& ( ( $true = sP58 )
| ( hoasinduction_lem3v2a != $true ) )
& ( ( hoasinduction_lem3v2a = $true )
| ( $true != sP58 ) )
& ( ( axvarid = $true )
| ? [X265: term] :
( ( sub @ X265 @ id )
!= X265 ) )
& ( ! [X265: term] :
( ( sub @ X265 @ id )
= X265 )
| ( axvarid != $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3aa = $true )
| ( $true != sP57 ) )
& ( ( $true = sP57 )
| ( hoasinduction_lem3aa != $true ) )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
| ( $true != sP56 ) )
& ( ( $true = sP56 )
| ( hoaslaminj_lthm != $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( axidr = $true )
| ? [X393: subst] :
( ( comp @ X393 @ id )
!= X393 ) )
& ( ! [X393: subst] :
( ( comp @ X393 @ id )
= X393 )
| ( axidr != $true ) )
& ( ( $true = sP55 )
| ( induction2lem != $true ) )
& ( ( induction2lem = $true )
| ( $true != sP55 ) )
& ! [X141: d_term] : ( d_one = X141 )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP53 = $true )
| ( hoasinduction_lem3a_lthm != $true ) )
& ( ( hoasinduction_lem3a_lthm = $true )
| ( sP53 != $true ) )
& ( axapp = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
| ( sP51 != $true ) )
& ( ( sP51 = $true )
| ( hoasinduction_no_psi_cond_lthm != $true ) )
& ( hoasapnotvar = $true )
& ( ! [X378: term,X379: term,X380: subst > term > term] :
( ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( lam @ ( X380 @ sh @ one ) ) )
| ? [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
!= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) ) )
| ( hoaslamnotap != $true ) )
& ( ( hoaslamnotap = $true )
| ? [X378: term,X379: term,X380: subst > term > term] :
( ( ( ap @ ( sub @ X379 @ id ) @ X378 )
= ( lam @ ( X380 @ sh @ one ) ) )
& ! [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) ) ) )
& ( axclos = $true )
& ( ? [X399: term,X400: term,X401: term,X402: term] :
( ( X399 != X402 )
& ( ( ap @ X399 @ X400 )
= ( ap @ X402 @ X401 ) ) )
| ! [X403: term,X404: term,X405: term,X406: term] :
( ( X405 = X406 )
| ( ( ap @ ( sub @ X405 @ id ) @ X403 )
!= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) )
| ? [X398: term] :
( ( sub @ X398 @ id )
!= X398 )
| ( hoasapinj1_lthm != $true ) )
& ( ( hoasapinj1_lthm = $true )
| ( ! [X399: term,X400: term,X401: term,X402: term] :
( ( X399 = X402 )
| ( ( ap @ X399 @ X400 )
!= ( ap @ X402 @ X401 ) ) )
& ? [X403: term,X404: term,X405: term,X406: term] :
( ( X405 != X406 )
& ( ( ap @ ( sub @ X405 @ id ) @ X403 )
= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) )
& ! [X398: term] :
( ( sub @ X398 @ id )
= X398 ) ) )
& ( ulamvar1 != $true )
& ( ! [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
!= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
| ( X394 = X397 ) )
| ( hoasapinj1 != $true ) )
& ( ( hoasapinj1 = $true )
| ? [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
& ( X394 != X397 ) ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ? [X58: subst > term > term] :
( ! [X59: term] :
( ( $true
= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
| ( $true
!= ( X54 @ id @ X59 @ id ) ) )
& ( $true
!= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
& ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) )
| ( $true
= ( sP46 @ X54 ) )
| ( $true
= ( X54 @ id @ X55 @ id ) ) )
| ( hoasinduction_no_psi_cond != $true ) )
& ( ( hoasinduction_no_psi_cond = $true )
| ? [X54: subst > term > subst > $o,X55: term] :
( ! [X58: subst > term > term] :
( ? [X59: term] :
( ( $true
!= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
& ( $true
= ( X54 @ id @ X59 @ id ) ) )
| ( $true
= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
| ? [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
!= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) )
& ( $true
!= ( sP46 @ X54 ) )
& ( $true
!= ( X54 @ id @ X55 @ id ) ) ) )
& ( ! [X276: term] :
( ( sub @ X276 @ id )
= X276 )
| ( termmset != $true ) )
& ( ( termmset = $true )
| ? [X276: term] :
( ( sub @ X276 @ id )
!= X276 ) )
& ( induction2_gthm = $true )
& ( pushprop = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X21: subst,X22: term,X23: term > $o] :
? [X24: term > $o] :
! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) )
| ( pushprop_lem0 != $true ) )
& ( ( pushprop_lem0 = $true )
| ? [X21: subst,X22: term,X23: term > $o] :
! [X24: term > $o] :
? [X25: term] :
( ( X24 @ X25 )
!= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) ) )
& ! [X337: d_term,X338: d_term] :
( ( X337 = X338 )
| ( ( d2term @ X337 )
!= ( d2term @ X338 ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X187: subst > term > subst > $o] :
? [X188: term > $o] :
! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
| ( hoasinduction_lem0_lthm != $true ) )
& ( ( hoasinduction_lem0_lthm = $true )
| ? [X187: subst > term > subst > $o] :
! [X188: term > $o] :
? [X189: term] :
( ( X188 @ X189 )
!= ( X187 @ id @ X189 @ id ) ) )
& ( ( $true = sP45 )
| ( hoasinduction_lem3_lthm != $true ) )
& ( ( hoasinduction_lem3_lthm = $true )
| ( $true != sP45 ) )
& ( ( $true = sP42 )
| ( hoasinduction_lem3a != $true ) )
& ( ( hoasinduction_lem3a = $true )
| ( $true != sP42 ) )
& ( hoasapnotvar_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem0 = $true )
| ? [X138: subst > term > subst > $o] :
! [X139: term > $o] :
? [X140: term] :
( ( X138 @ id @ X140 @ id )
!= ( X139 @ X140 ) ) )
& ( ! [X138: subst > term > subst > $o] :
? [X139: term > $o] :
! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) )
| ( hoasinduction_lem0 != $true ) )
& ( ( hoasinduction_lem3v2_f = $true )
| ? [X95: term] :
! [X96: subst > term > term] :
? [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
!= ( sub @ X95 @ ( push @ X97 @ X98 ) ) ) )
& ( ! [X95: term] :
? [X96: subst > term > term] :
! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) )
| ( hoasinduction_lem3v2_f != $true ) )
& ( lamnotap != $true )
& ( one
= ( d2term @ d_one ) )
& ( axassoc = $true )
& ( ( induction2 = $true )
| ( $true != sP41 ) )
& ( ( $true = sP41 )
| ( induction2 != $true ) )
& ( ! [X429: term,X430: term > $o] :
( ? [X432: term] :
( ( $true
= ( X430 @ X432 ) )
& ( $true
!= ( X430 @ ( lam @ X432 ) ) ) )
| ( $true
= ( X430 @ X429 ) )
| ( $true
= ( sP39 @ X430 ) )
| ? [X431: term] :
( ( $true
!= ( X430 @ X431 ) )
& ( $true
= ( var @ X431 ) ) ) )
| ( induction != $true ) )
& ( ( induction = $true )
| ? [X429: term,X430: term > $o] :
( ! [X432: term] :
( ( $true
!= ( X430 @ X432 ) )
| ( $true
= ( X430 @ ( lam @ X432 ) ) ) )
& ( $true
!= ( X430 @ X429 ) )
& ( $true
!= ( sP39 @ X430 ) )
& ! [X431: term] :
( ( $true
= ( X430 @ X431 ) )
| ( $true
!= ( var @ X431 ) ) ) ) )
& ( ( pushprop_lem0_lthm = $true )
| ? [X180: term > $o,X181: term,X182: subst] :
! [X183: term > $o] :
? [X184: term] :
( ( X183 @ X184 )
!= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) ) )
& ( ! [X180: term > $o,X181: term,X182: subst] :
? [X183: term > $o] :
! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) )
| ( pushprop_lem0_lthm != $true ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_lthm = $true )
& ( ( hoasinduction_lthm = $true )
| ( $true != sP38 ) )
& ( ( $true = sP38 )
| ( hoasinduction_lthm != $true ) )
& ( ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) )
| ( substmonoid != $true ) )
& ( ( substmonoid = $true )
| ? [X33: subst] :
( ( comp @ X33 @ id )
!= X33 )
| ? [X34: subst] :
( ( comp @ id @ X34 )
!= X34 ) )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
| ( $true != sP29 ) )
& ( ( $true = sP29 )
| ( hoasinduction_lem3v2 != $true ) )
& ( ! [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
= ( X268 @ X267 @ X266 ) )
| ( ( lam @ ( X268 @ sh @ one ) )
!= ( lam @ ( X269 @ sh @ one ) ) )
| ? [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
!= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
| ? [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
!= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) )
| ( hoaslaminj != $true ) )
& ( ( hoaslaminj = $true )
| ? [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
!= ( X268 @ X267 @ X266 ) )
& ( ( lam @ ( X268 @ sh @ one ) )
= ( lam @ ( X269 @ sh @ one ) ) )
& ! [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
& ! [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP27 )
| ( hoasinduction_lem3 != $true ) )
& ( ( hoasinduction_lem3 = $true )
| ( $true != sP27 ) )
& ( ( hoasapinj2 = $true )
| ? [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
& ( X174 != X175 ) ) )
& ( ! [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
| ( X174 = X175 ) )
| ( hoasapinj2 != $true ) )
& ( ( pushprop_lem1v2 = $true )
| ? [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( $true
= ( X243 @ X240 ) )
& ! [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
= ( X242 @ X244 ) )
& ( ( X242 @ one )
!= $true ) ) )
& ( ! [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( $true
!= ( X243 @ X240 ) )
| ? [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
!= ( X242 @ X244 ) )
| ( ( X242 @ one )
= $true ) )
| ( pushprop_lem1v2 != $true ) )
& ( hoasapinj2_gthm = $true )
& ( ( pushprop_lem1 = $true )
| ? [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
= ( X169 @ X167 ) )
& ( $true
!= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) ) )
& ( ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
!= ( X169 @ X167 ) )
| ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) )
| ( pushprop_lem1 != $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( $true = sP25 )
| ( hoasinduction_lem3aa_lthm != $true ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
| ( $true != sP25 ) )
& ( termmset_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
| ( $true != sP24 ) )
& ( ( $true = sP24 )
| ( hoasinduction_lem3v2_lthm != $true ) )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ( axvarshift = $true )
& ( hoasapnotvar_lthm = $true )
& ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
| ( laminj != $true ) )
& ( ( laminj = $true )
| ? [X0: term,X1: term] :
( ( X0 != X1 )
& ( ( lam @ X1 )
= ( lam @ X0 ) ) ) )
& ( ( hoasinduction_lem3aaa = $true )
| ( $true != sP22 ) )
& ( ( $true = sP22 )
| ( hoasinduction_lem3aaa != $true ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X312: d_subst,X313: d_subst] :
( ( X312 = X313 )
| ( ( d2subst @ X312 )
!= ( d2subst @ X313 ) ) )
& ( ( $true = sP21 )
| ( hoasinduction_lthm_3 != $true ) )
& ( ( hoasinduction_lthm_3 = $true )
| ( $true != sP21 ) )
& ( ( pushprop_lem1v2_lthm = $true )
| ( ? [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ( ( X30 @ X29 )
= $true )
& ( $true
!= ( X28 @ one ) )
& ! [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
= ( X28 @ X32 ) ) )
& ! [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
= X27 ) ) )
& ( ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ( ( X30 @ X29 )
!= $true )
| ( $true
= ( X28 @ one ) )
| ? [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
!= ( X28 @ X32 ) ) )
| ? [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
!= X27 )
| ( pushprop_lem1v2_lthm != $true ) )
& ( hoaslamnotvar = $true )
& ( ( $true = sP15 )
| ( hoasinduction != $true ) )
& ( ( hoasinduction = $true )
| ( $true != sP15 ) )
& ( ( sP12 = $true )
| ( induction2_lthm != $true ) )
& ( ( induction2_lthm = $true )
| ( sP12 != $true ) )
& ( ( substmonoid_lthm = $true )
| ( ( ? [X523: subst] :
( ( comp @ id @ X523 )
!= X523 )
| ? [X524: subst] :
( ( comp @ X524 @ id )
!= X524 ) )
& ! [X522: subst] :
( ( comp @ X522 @ id )
= X522 )
& ! [X521: subst] :
( ( comp @ id @ X521 )
= X521 ) ) )
& ( ( ! [X523: subst] :
( ( comp @ id @ X523 )
= X523 )
& ! [X524: subst] :
( ( comp @ X524 @ id )
= X524 ) )
| ? [X522: subst] :
( ( comp @ X522 @ id )
!= X522 )
| ? [X521: subst] :
( ( comp @ id @ X521 )
!= X521 )
| ( substmonoid_lthm != $true ) )
& ( ( axshiftcons = $true )
| ? [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
!= X198 ) )
& ( ! [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
= X198 )
| ( axshiftcons != $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
| ? [X263: term] :
! [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
!= ( X264 @ sh @ one ) ) )
& ( ! [X263: term] :
? [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
= ( X264 @ sh @ one ) )
| ( hoasinduction_lem3b_lthm != $true ) )
& ( ! [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
!= ( X336 @ X332 ) )
| ( $true
= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) ) )
| ? [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
!= X329 )
| ? [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
!= X330 )
| ( pushprop_lem1_lthm != $true ) )
& ( ( pushprop_lem1_lthm = $true )
| ( ? [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
= ( X336 @ X332 ) )
& ( $true
!= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) ) )
& ! [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
= X329 )
& ! [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
= X330 ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ulamvarsh = $true )
& ( ( induction2lem_lthm = $true )
| ( $true != sP8 ) )
& ( ( $true = sP8 )
| ( induction2lem_lthm != $true ) )
& ( ( hoasinduction_lem2v2 = $true )
| ( sP4 != $true ) )
& ( ( sP4 = $true )
| ( hoasinduction_lem2v2 != $true ) )
& ( ulamvarind = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) )
| ( apinj1 != $true ) )
& ( ( apinj1 = $true )
| ? [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
= ( ap @ X179 @ X178 ) )
& ( X176 != X179 ) ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X519: term] :
? [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
= ( X520 @ sh @ one ) )
| ( hoasinduction_lem3b != $true ) )
& ( ( hoasinduction_lem3b = $true )
| ? [X519: term] :
! [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
!= ( X520 @ sh @ one ) ) )
& ( ( axidl = $true )
| ? [X223: subst] :
( ( comp @ id @ X223 )
!= X223 ) )
& ( ! [X223: subst] :
( ( comp @ id @ X223 )
= X223 )
| ( axidl != $true ) )
& ( hoaslaminj_gthm = $true )
& ( ( apinj2 = $true )
| ? [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
= ( ap @ X425 @ X426 ) )
& ( X426 != X428 ) ) )
& ( ! [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
!= ( ap @ X425 @ X426 ) )
| ( X426 = X428 ) )
| ( apinj2 != $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 ) ) ) ) ) ) )
& ( ! [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
= X384 )
| ( axscons != $true ) )
& ( ( axscons = $true )
| ? [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
!= X384 ) )
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& ( ( axvarcons = $true )
| ? [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
!= X245 ) )
& ( ! [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
= X245 )
| ( axvarcons != $true ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
| ( ! [X190: term,X191: term,X192: term,X193: term] :
( ( ( ap @ X191 @ X190 )
!= ( ap @ X193 @ X192 ) )
| ( X190 = X192 ) )
& ? [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
& ( X194 != X195 ) ) ) )
& ( ? [X190: term,X191: term,X192: term,X193: term] :
( ( ( ap @ X191 @ X190 )
= ( ap @ X193 @ X192 ) )
& ( X190 != X192 ) )
| ! [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
!= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
| ( X194 = X195 ) )
| ( hoasapinj2_lthm != $true ) )
& ( pushprop_lem2v2 = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
| ? [X314: term] :
! [X315: subst > term > term] :
? [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
!= ( X315 @ X316 @ X317 ) ) )
& ( ! [X314: term] :
? [X315: subst > term > term] :
! [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
= ( X315 @ X316 @ X317 ) )
| ( hoasinduction_lem3v2_f_lthm != $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( pushprop_gthm = $true )
& ( ( hoasinduction_lem3v2a_lthm = $true )
| ( $true != sP2 ) )
& ( ( $true = sP2 )
| ( hoasinduction_lem3v2a_lthm != $true ) )
& ( hoasapinj1_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( $true = sP1 )
| ( hoasinduction_lem2 != $true ) )
& ( ( hoasinduction_lem2 = $true )
| ( $true != sP1 ) )
& ( pushprop_lem0_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( axmap = $true )
& ( induction2lem_gthm = $true )
& ( axabs = $true )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( d2subst @ d_id )
= id ) ),
inference(nnf_transformation,[status(thm)],[f73]) ).
thf(f73,plain,
( ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ! [X156: d_subst] : ( d_id = X156 )
& ( hoasinduction_lem1_gthm = $true )
& ( ( shinj = $true )
<=> ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) ) )
& ( lamnotvar = $true )
& ( ( ? [X208: term] :
( ( sub @ X208 @ id )
!= X208 )
| ! [X209: term] :
( ( sub @ X209 @ id )
= X209 ) )
<=> ( termmset_lthm = $true ) )
& ( ( $true = sP58 )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( ( axvarid = $true )
<=> ! [X265: term] :
( ( sub @ X265 @ id )
= X265 ) )
& ( hoaslamnotvar_lthm = $true )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ( $true = sP57 ) )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( $true = sP56 ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( axidr = $true )
<=> ! [X393: subst] :
( ( comp @ X393 @ id )
= X393 ) )
& ( ( $true = sP55 )
<=> ( induction2lem = $true ) )
& ! [X141: d_term] : ( d_one = X141 )
& ( hoasinduction_lem3_gthm = $true )
& ( ( sP53 = $true )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( axapp = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( sP51 = $true ) )
& ( hoasapnotvar = $true )
& ( ! [X378: term,X379: term,X380: subst > term > term] :
( ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( lam @ ( X380 @ sh @ one ) ) )
| ? [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
!= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) ) )
<=> ( hoaslamnotap = $true ) )
& ( axclos = $true )
& ( ( ? [X399: term,X400: term,X401: term,X402: term] :
( ( X399 != X402 )
& ( ( ap @ X399 @ X400 )
= ( ap @ X402 @ X401 ) ) )
| ! [X403: term,X404: term,X405: term,X406: term] :
( ( X405 = X406 )
| ( ( ap @ ( sub @ X405 @ id ) @ X403 )
!= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) )
| ? [X398: term] :
( ( sub @ X398 @ id )
!= X398 ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( ulamvar1 != $true )
& ( ! [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
!= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
| ( X394 = X397 ) )
<=> ( hoasapinj1 = $true ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ? [X58: subst > term > term] :
( ! [X59: term] :
( ( $true
= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
| ( $true
!= ( X54 @ id @ X59 @ id ) ) )
& ( $true
!= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
& ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) )
| ( $true
= ( sP46 @ X54 ) )
| ( $true
= ( X54 @ id @ X55 @ id ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( ! [X276: term] :
( ( sub @ X276 @ id )
= X276 )
<=> ( termmset = $true ) )
& ( induction2_gthm = $true )
& ( pushprop = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X21: subst,X22: term,X23: term > $o] :
? [X24: term > $o] :
! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ! [X337: d_term,X338: d_term] :
( ( X337 = X338 )
| ( ( d2term @ X337 )
!= ( d2term @ X338 ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X187: subst > term > subst > $o] :
? [X188: term > $o] :
! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( $true = sP45 )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( ( $true = sP42 )
<=> ( hoasinduction_lem3a = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X138: subst > term > subst > $o] :
? [X139: term > $o] :
! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) ) )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X95: term] :
? [X96: subst > term > term] :
! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) ) )
& ( lamnotap != $true )
& ( one
= ( d2term @ d_one ) )
& ( axassoc = $true )
& ( ( induction2 = $true )
<=> ( $true = sP41 ) )
& ( ! [X429: term,X430: term > $o] :
( ? [X432: term] :
( ( $true
= ( X430 @ X432 ) )
& ( $true
!= ( X430 @ ( lam @ X432 ) ) ) )
| ( $true
= ( X430 @ X429 ) )
| ( $true
= ( sP39 @ X430 ) )
| ? [X431: term] :
( ( $true
!= ( X430 @ X431 ) )
& ( $true
= ( var @ X431 ) ) ) )
<=> ( induction = $true ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X180: term > $o,X181: term,X182: subst] :
? [X183: term > $o] :
! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_lthm = $true )
& ( ( hoasinduction_lthm = $true )
<=> ( $true = sP38 ) )
& ( ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) )
<=> ( substmonoid = $true ) )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ( $true = sP29 ) )
& ( ! [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
= ( X268 @ X267 @ X266 ) )
| ( ( lam @ ( X268 @ sh @ one ) )
!= ( lam @ ( X269 @ sh @ one ) ) )
| ? [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
!= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
| ? [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
!= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lem1 = $true )
& ( ( $true = sP27 )
<=> ( hoasinduction_lem3 = $true ) )
& ( ( hoasapinj2 = $true )
<=> ! [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
| ( X174 = X175 ) ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( $true
!= ( X243 @ X240 ) )
| ? [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
!= ( X242 @ X244 ) )
| ( ( X242 @ one )
= $true ) ) )
& ( hoasapinj2_gthm = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
!= ( X169 @ X167 ) )
| ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( $true = sP25 )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( termmset_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( $true = sP24 ) )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ( axvarshift = $true )
& ( hoasapnotvar_lthm = $true )
& ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
<=> ( laminj = $true ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ( $true = sP22 ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X312: d_subst,X313: d_subst] :
( ( X312 = X313 )
| ( ( d2subst @ X312 )
!= ( d2subst @ X313 ) ) )
& ( ( $true = sP21 )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ( ( X30 @ X29 )
!= $true )
| ( $true
= ( X28 @ one ) )
| ? [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
!= ( X28 @ X32 ) ) )
| ? [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
!= X27 ) ) )
& ( hoaslamnotvar = $true )
& ( ( $true = sP15 )
<=> ( hoasinduction = $true ) )
& ( ( sP12 = $true )
<=> ( induction2_lthm = $true ) )
& ( ( substmonoid_lthm = $true )
<=> ( ( ! [X523: subst] :
( ( comp @ id @ X523 )
= X523 )
& ! [X524: subst] :
( ( comp @ X524 @ id )
= X524 ) )
| ? [X522: subst] :
( ( comp @ X522 @ id )
!= X522 )
| ? [X521: subst] :
( ( comp @ id @ X521 )
!= X521 ) ) )
& ( ( axshiftcons = $true )
<=> ! [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
= X198 ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X263: term] :
? [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
= ( X264 @ sh @ one ) ) )
& ( ( ! [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
!= ( X336 @ X332 ) )
| ( $true
= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) ) )
| ? [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
!= X329 )
| ? [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
!= X330 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ulamvarsh = $true )
& ( ( induction2lem_lthm = $true )
<=> ( $true = sP8 ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ( sP4 = $true ) )
& ( ulamvarind = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) )
<=> ( apinj1 = $true ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X519: term] :
? [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
= ( X520 @ sh @ one ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( ( axidl = $true )
<=> ! [X223: subst] :
( ( comp @ id @ X223 )
= X223 ) )
& ( hoaslaminj_gthm = $true )
& ( ( apinj2 = $true )
<=> ! [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
!= ( ap @ X425 @ X426 ) )
| ( X426 = X428 ) ) )
& ( 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 ) ) ) ) ) ) )
& ( ! [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
= X384 )
<=> ( axscons = $true ) )
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& ( ( axvarcons = $true )
<=> ! [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
= X245 ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ? [X190: term,X191: term,X192: term,X193: term] :
( ( ( ap @ X191 @ X190 )
= ( ap @ X193 @ X192 ) )
& ( X190 != X192 ) )
| ! [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
!= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
| ( X194 = X195 ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X314: term] :
? [X315: subst > term > term] :
! [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
= ( X315 @ X316 @ X317 ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( pushprop_gthm = $true )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( $true = sP2 ) )
& ( hoasapinj1_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ( $true = sP1 )
<=> ( hoasinduction_lem2 = $true ) )
& ( pushprop_lem0_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( axmap = $true )
& ( induction2lem_gthm = $true )
& ( axabs = $true )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( d2subst @ d_id )
= id ) ),
inference(definition_folding,[status(thm)],[f13,f72,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14]) ).
thf(f14,plain,
! [X210: subst > term > subst > $o] :
( ( $true
= ( sP0 @ X210 ) )
<=> ? [X221: term,X222: term] :
( ( $true
= ( X210 @ id @ X221 @ id ) )
& ( $true
!= ( X210 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id ) )
& ( $true
= ( X210 @ id @ X222 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f15,plain,
( ( $true = sP1 )
<=> ! [X210: subst > term > subst > $o,X211: term,X212: term] :
( ? [X217: subst,X218: subst,X219: subst,X220: term] :
( ( ( X210 @ ( comp @ X219 @ X218 ) @ ( sub @ X220 @ X218 ) @ X217 )
= $true )
& ( $true
!= ( X210 @ X219 @ X220 @ ( comp @ X218 @ X217 ) ) ) )
| ? [X213: term,X214: subst,X215: subst,X216: subst] :
( ( $true
= ( X210 @ X215 @ X213 @ ( comp @ X216 @ X214 ) ) )
& ( ( X210 @ ( comp @ X215 @ X216 ) @ ( sub @ X213 @ X216 ) @ X214 )
!= $true ) )
| ( ( X210 @ id @ ( ap @ X212 @ X211 ) @ id )
= $true )
| ( $true
!= ( X210 @ id @ X212 @ id ) )
| ( $true
!= ( X210 @ id @ X211 @ id ) )
| ( $true
= ( sP0 @ X210 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f16,plain,
( ( $true = sP2 )
<=> ( ! [X530: term,X531: term > $o,X532: subst > term > subst > $o] :
( ? [X534: subst > term > term] :
( ! [X535: term] :
( ( ( X532 @ id @ ( X534 @ id @ X535 ) @ id )
= $true )
| ( $true
!= ( X532 @ id @ X535 @ id ) ) )
& ! [X536: subst,X537: term,X538: subst] :
( ( X534 @ ( comp @ X538 @ X536 ) @ ( sub @ X537 @ X536 ) )
= ( sub @ ( X534 @ X538 @ X537 ) @ X536 ) )
& ( $true
!= ( X532 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id ) ) )
| ? [X533: term] :
( ( X532 @ id @ X533 @ id )
!= ( X531 @ X533 ) )
| ? [X539: term] :
( ( $true
!= ( X531 @ ( sub @ X530 @ ( push @ X539 @ id ) ) ) )
& ( $true
= ( X531 @ X539 ) ) )
| ( $true
= ( X531 @ ( lam @ X530 ) ) ) )
| ? [X529: term] :
( ( sub @ X529 @ id )
!= X529 )
| ? [X525: term] :
! [X526: subst > term > term] :
? [X527: subst,X528: term] :
( ( X526 @ X527 @ X528 )
!= ( sub @ X525 @ ( push @ X528 @ X527 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f17,plain,
! [X125: subst > term > subst > $o] :
( ( $true
= ( sP3 @ X125 ) )
<=> ? [X127: term,X128: term] :
( ( ( X125 @ id @ ( ap @ ( sub @ X127 @ id ) @ X128 ) @ id )
!= $true )
& ( $true
= ( X125 @ id @ X128 @ id ) )
& ( $true
= ( X125 @ id @ X127 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f18,plain,
( ( sP4 = $true )
<=> ! [X123: term > $o,X124: term,X125: subst > term > subst > $o,X126: term] :
( ( ( X123 @ ( ap @ X126 @ X124 ) )
= $true )
| ? [X129: subst,X130: subst,X131: term,X132: subst] :
( ( $true
!= ( X125 @ X129 @ X131 @ ( comp @ X132 @ X130 ) ) )
& ( $true
= ( X125 @ ( comp @ X129 @ X132 ) @ ( sub @ X131 @ X132 ) @ X130 ) ) )
| ( $true
= ( sP3 @ X125 ) )
| ? [X133: subst,X134: subst,X135: term,X136: subst] :
( ( $true
!= ( X125 @ ( comp @ X136 @ X134 ) @ ( sub @ X135 @ X134 ) @ X133 ) )
& ( ( X125 @ X136 @ X135 @ ( comp @ X134 @ X133 ) )
= $true ) )
| ( $true
!= ( X123 @ X126 ) )
| ( $true
!= ( X123 @ X124 ) )
| ? [X137: term] :
( ( X125 @ id @ X137 @ id )
!= ( X123 @ X137 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f19,plain,
! [X13: term > $o] :
( ( $true
= ( sP5 @ X13 ) )
<=> ? [X16: term,X17: term] :
( ( $true
!= ( X13 @ ( ap @ X17 @ X16 ) ) )
& ( $true
= ( X13 @ X16 ) )
& ( $true
= ( X13 @ X17 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f20,plain,
! [X7: term > $o] :
( ( $true
= ( sP6 @ X7 ) )
<=> ! [X11: term,X12: term] :
( ( $true
!= ( X7 @ X11 ) )
| ( $true
!= ( X7 @ X12 ) )
| ( $true
= ( X7 @ ( ap @ X11 @ X12 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f21,plain,
( ( $true = sP7 )
<=> ! [X13: term > $o,X14: term,X15: subst] :
( ( $true
= ( sP5 @ X13 ) )
| ( $true
= ( X13 @ ( sub @ X14 @ X15 ) ) )
| ? [X18: term] :
( ( $true
!= ( X13 @ ( lam @ X18 ) ) )
& ! [X19: term] :
( ( $true
= ( X13 @ ( sub @ X18 @ ( push @ X19 @ id ) ) ) )
| ( ( X13 @ X19 )
!= $true ) ) )
| ? [X20: term] :
( ( $true
= ( var @ X20 ) )
& ( $true
!= ( X13 @ ( sub @ X20 @ X15 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f22,plain,
( ( $true = sP8 )
<=> ( ? [X7: term > $o,X8: term] :
( ( ( X7 @ X8 )
!= $true )
& ! [X9: term] :
( ( $true
!= ( X7 @ X9 ) )
| ( ( X7 @ ( lam @ X9 ) )
= $true ) )
& ! [X10: term] :
( ( $true
= ( X7 @ X10 ) )
| ( ( var @ X10 )
!= $true ) )
& ( $true
= ( sP6 @ X7 ) ) )
| ? [X2: subst,X3: term] :
( ( sub @ one @ ( push @ X3 @ X2 ) )
!= X3 )
| ? [X4: term,X5: subst] :
( ( comp @ sh @ ( push @ X4 @ X5 ) )
!= X5 )
| ? [X6: subst] :
( ( comp @ X6 @ id )
!= X6 )
| ( $true = sP7 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f23,plain,
! [X226: term > $o] :
( ( $true
= ( sP9 @ X226 ) )
<=> ! [X231: term,X232: term] :
( ( $true
!= ( X226 @ X232 ) )
| ( ( X226 @ X231 )
!= $true )
| ( $true
= ( X226 @ ( ap @ X232 @ X231 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f24,plain,
! [X234: term > $o] :
( ( $true
= ( sP10 @ X234 ) )
<=> ? [X237: term,X238: term] :
( ( $true
= ( X234 @ X238 ) )
& ( $true
= ( X234 @ X237 ) )
& ( $true
!= ( X234 @ ( ap @ X238 @ X237 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f25,plain,
( ( $true = sP11 )
<=> ! [X233: term,X234: term > $o] :
( ( $true
= ( sP10 @ X234 ) )
| ? [X235: term] :
( ( $true
!= ( X234 @ ( lam @ X235 ) ) )
& ! [X236: term] :
( ( $true
!= ( X234 @ X236 ) )
| ( $true
= ( X234 @ ( sub @ X235 @ ( push @ X236 @ id ) ) ) ) ) )
| ( $true
= ( X234 @ X233 ) )
| ? [X239: term] :
( ( $true
!= ( X234 @ X239 ) )
& ( $true
= ( var @ X239 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f26,plain,
( ( sP12 = $true )
<=> ( ? [X224: term] :
( ( sub @ X224 @ id )
!= X224 )
| ( $true = sP11 )
| ? [X225: term,X226: term > $o,X227: subst] :
( ! [X228: term] :
( ( $true
!= ( var @ X228 ) )
| ( $true
= ( X226 @ ( sub @ X228 @ X227 ) ) ) )
& ( $true
!= ( X226 @ ( sub @ X225 @ X227 ) ) )
& ! [X229: term] :
( ? [X230: term] :
( ( $true
!= ( X226 @ ( sub @ X229 @ ( push @ X230 @ id ) ) ) )
& ( $true
= ( X226 @ X230 ) ) )
| ( $true
= ( X226 @ ( lam @ X229 ) ) ) )
& ( $true
= ( sP9 @ X226 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f27,plain,
! [X294: subst > term > subst > $o] :
( ( $true
= ( sP13 @ X294 ) )
<=> ? [X296: term,X297: term] :
( ( ( X294 @ id @ X296 @ id )
= $true )
& ( $true
!= ( X294 @ id @ ( ap @ ( sub @ X297 @ id ) @ X296 ) @ id ) )
& ( ( X294 @ id @ X297 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f28,plain,
! [X294: subst > term > subst > $o] :
( ( $true
= ( sP14 @ X294 ) )
<=> ? [X302: subst > term > term] :
( ! [X303: term] :
( ( $true
= ( X294 @ id @ ( X302 @ id @ X303 ) @ id ) )
| ( $true
!= ( X294 @ id @ X303 @ id ) ) )
& ! [X304: subst,X305: subst,X306: term] :
( ( X302 @ ( comp @ X304 @ X305 ) @ ( sub @ X306 @ X305 ) )
= ( sub @ ( X302 @ X304 @ X306 ) @ X305 ) )
& ( $true
!= ( X294 @ id @ ( lam @ ( X302 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f29,plain,
( ( $true = sP15 )
<=> ! [X294: subst > term > subst > $o,X295: term] :
( ( $true
= ( sP14 @ X294 ) )
| ? [X307: subst,X308: term,X309: subst,X310: subst] :
( ( $true
!= ( X294 @ X310 @ X308 @ ( comp @ X309 @ X307 ) ) )
& ( $true
= ( X294 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) @ X307 ) ) )
| ? [X298: term,X299: subst,X300: subst,X301: subst] :
( ( $true
!= ( X294 @ ( comp @ X299 @ X301 ) @ ( sub @ X298 @ X301 ) @ X300 ) )
& ( $true
= ( X294 @ X299 @ X298 @ ( comp @ X301 @ X300 ) ) ) )
| ( $true
= ( sP13 @ X294 ) )
| ? [X311: term] :
( ( ( var @ ( sub @ X311 @ id ) )
= $true )
& ( $true
!= ( X294 @ id @ X311 @ id ) ) )
| ( $true
= ( X294 @ id @ X295 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f30,plain,
! [X342: term > $o] :
( ( $true
= ( sP16 @ X342 ) )
<=> ! [X344: term,X345: term] :
( ( ( X342 @ X344 )
!= $true )
| ( $true
= ( X342 @ ( ap @ X345 @ X344 ) ) )
| ( $true
!= ( X342 @ X345 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f31,plain,
! [X360: subst > term > subst > $o] :
( ( $true
= ( sP17 @ X360 ) )
<=> ? [X371: term,X372: term] :
( ( $true
= ( X360 @ id @ X371 @ id ) )
& ( $true
= ( X360 @ id @ X372 @ id ) )
& ( $true
!= ( X360 @ id @ ( ap @ ( sub @ X372 @ id ) @ X371 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f32,plain,
! [X360: subst > term > subst > $o] :
( ( $true
= ( sP18 @ X360 ) )
<=> ? [X366: subst > term > term] :
( ( $true
!= ( X360 @ id @ ( lam @ ( X366 @ sh @ one ) ) @ id ) )
& ! [X370: term] :
( ( $true
= ( X360 @ id @ ( X366 @ id @ X370 ) @ id ) )
| ( $true
!= ( X360 @ id @ X370 @ id ) ) )
& ! [X367: subst,X368: term,X369: subst] :
( ( sub @ ( X366 @ X369 @ X368 ) @ X367 )
= ( X366 @ ( comp @ X369 @ X367 ) @ ( sub @ X368 @ X367 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f33,plain,
( ( sP19 = $true )
<=> ? [X350: subst > term > subst > $o,X351: term,X352: term > $o] :
( ! [X355: subst > term > term] :
( ( $true
= ( X350 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id ) )
| ? [X356: term] :
( ( $true
= ( X350 @ id @ X356 @ id ) )
& ( $true
!= ( X350 @ id @ ( X355 @ id @ X356 ) @ id ) ) )
| ? [X357: subst,X358: term,X359: subst] :
( ( X355 @ ( comp @ X357 @ X359 ) @ ( sub @ X358 @ X359 ) )
!= ( sub @ ( X355 @ X357 @ X358 ) @ X359 ) ) )
& ! [X354: term] :
( ( $true
= ( X352 @ ( sub @ X351 @ ( push @ X354 @ id ) ) ) )
| ( $true
!= ( X352 @ X354 ) ) )
& ( $true
!= ( X352 @ ( lam @ X351 ) ) )
& ! [X353: term] :
( ( X352 @ X353 )
= ( X350 @ id @ X353 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f34,plain,
( ( sP20 = $true )
<=> ! [X360: subst > term > subst > $o,X361: term] :
( ? [X374: subst,X375: subst,X376: subst,X377: term] :
( ( $true
!= ( X360 @ ( comp @ X374 @ X375 ) @ ( sub @ X377 @ X375 ) @ X376 ) )
& ( $true
= ( X360 @ X374 @ X377 @ ( comp @ X375 @ X376 ) ) ) )
| ? [X373: term] :
( ( $true
= ( var @ ( sub @ X373 @ id ) ) )
& ( $true
!= ( X360 @ id @ X373 @ id ) ) )
| ( $true
= ( sP18 @ X360 ) )
| ( $true
= ( X360 @ id @ X361 @ id ) )
| ? [X362: subst,X363: subst,X364: subst,X365: term] :
( ( $true
= ( X360 @ ( comp @ X362 @ X363 ) @ ( sub @ X365 @ X363 ) @ X364 ) )
& ( $true
!= ( X360 @ X362 @ X365 @ ( comp @ X363 @ X364 ) ) ) )
| ( $true
= ( sP17 @ X360 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f35,plain,
( ( $true = sP21 )
<=> ( ? [X339: subst > term > subst > $o] :
! [X340: term > $o] :
? [X341: term] :
( ( X339 @ id @ X341 @ id )
!= ( X340 @ X341 ) )
| ( sP20 = $true )
| ? [X342: term > $o,X343: term] :
( ! [X348: term] :
( ( ( X342 @ X348 )
= $true )
| ( $true
!= ( var @ X348 ) ) )
& ( ( X342 @ X343 )
!= $true )
& ( $true
= ( sP16 @ X342 ) )
& ! [X346: term] :
( ( $true
= ( X342 @ ( lam @ X346 ) ) )
| ? [X347: term] :
( ( $true
!= ( X342 @ ( sub @ X346 @ ( push @ X347 @ id ) ) ) )
& ( $true
= ( X342 @ X347 ) ) ) ) )
| ? [X349: term] :
( ( sub @ X349 @ id )
!= X349 )
| ( sP19 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f36,plain,
( ( $true = sP22 )
<=> ! [X142: subst > term > subst > $o,X143: term] :
( ? [X153: term] :
( ( $true
= ( X142 @ id @ X153 @ id ) )
& ( $true
!= ( X142 @ id @ ( sub @ X143 @ ( push @ X153 @ id ) ) @ id ) ) )
| ? [X144: term,X145: subst > term > term] :
( ( ( X142 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id )
!= $true )
& ! [X149: term] :
( ( $true
= ( X142 @ id @ ( X145 @ id @ X149 ) @ id ) )
| ( $true
!= ( X142 @ id @ X149 @ id ) ) )
& ! [X146: subst,X147: term,X148: subst] :
( ( X145 @ ( comp @ X146 @ X148 ) @ ( sub @ X147 @ X148 ) )
= ( sub @ X144 @ ( push @ ( sub @ X147 @ X148 ) @ ( comp @ X146 @ X148 ) ) ) )
& ! [X150: term,X151: subst,X152: subst] :
( ( sub @ ( X145 @ X152 @ X150 ) @ X151 )
= ( sub @ ( sub @ X144 @ ( push @ X150 @ X152 ) ) @ X151 ) ) )
| ( ( X142 @ id @ ( lam @ ( sub @ X143 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f37,plain,
! [X36: subst > term > subst > $o] :
( ( $true
= ( sP23 @ X36 ) )
<=> ? [X44: subst > term > term] :
( ( $true
!= ( X36 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id ) )
& ! [X45: term,X46: subst,X47: subst] :
( ( sub @ ( X44 @ X47 @ X45 ) @ X46 )
= ( X44 @ ( comp @ X47 @ X46 ) @ ( sub @ X45 @ X46 ) ) )
& ! [X48: term] :
( ( $true
!= ( X36 @ id @ X48 @ id ) )
| ( $true
= ( X36 @ id @ ( X44 @ id @ X48 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f38,plain,
( ( $true = sP24 )
<=> ( ? [X35: term] :
( ( sub @ X35 @ id )
!= X35 )
| ! [X36: subst > term > subst > $o,X37: term,X38: term > $o] :
( ? [X39: term,X40: subst,X41: subst,X42: subst] :
( ( ( X36 @ X42 @ X39 @ ( comp @ X41 @ X40 ) )
!= $true )
& ( $true
= ( X36 @ ( comp @ X42 @ X41 ) @ ( sub @ X39 @ X41 ) @ X40 ) ) )
| ? [X49: term] :
( ( $true
= ( X38 @ X49 ) )
& ( $true
!= ( X38 @ ( sub @ X37 @ ( push @ X49 @ id ) ) ) ) )
| ( $true
= ( sP23 @ X36 ) )
| ( ( X38 @ ( lam @ X37 ) )
= $true )
| ? [X43: term] :
( ( X36 @ id @ X43 @ id )
!= ( X38 @ X43 ) )
| ? [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
!= ( X36 @ ( comp @ X53 @ X50 ) @ ( sub @ X52 @ X50 ) @ X51 ) )
& ( ( X36 @ X53 @ X52 @ ( comp @ X50 @ X51 ) )
= $true ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f39,plain,
( ( $true = sP25 )
<=> ! [X200: term,X201: subst > term > subst > $o] :
( ? [X207: term] :
( ( $true
= ( X201 @ id @ X207 @ id ) )
& ( $true
!= ( X201 @ id @ ( sub @ X200 @ ( push @ X207 @ id ) ) @ id ) ) )
| ? [X202: subst > term > term] :
( ( $true
!= ( X201 @ id @ ( lam @ ( X202 @ sh @ one ) ) @ id ) )
& ! [X204: term,X205: subst,X206: subst] :
( ( sub @ ( X202 @ X206 @ X204 ) @ X205 )
= ( X202 @ ( comp @ X206 @ X205 ) @ ( sub @ X204 @ X205 ) ) )
& ! [X203: term] :
( ( $true
= ( X201 @ id @ ( X202 @ id @ X203 ) @ id ) )
| ( $true
!= ( X201 @ id @ X203 @ id ) ) ) )
| ( ( X201 @ id @ ( lam @ ( sub @ X200 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f40,plain,
! [X247: subst > term > subst > $o] :
( ( ( sP26 @ X247 )
= $true )
<=> ? [X253: subst > term > term] :
( ! [X254: subst,X255: term,X256: subst] :
( ( sub @ ( X253 @ X256 @ X255 ) @ X254 )
= ( X253 @ ( comp @ X256 @ X254 ) @ ( sub @ X255 @ X254 ) ) )
& ( $true
!= ( X247 @ id @ ( lam @ ( X253 @ sh @ one ) ) @ id ) )
& ! [X257: term] :
( ( $true
!= ( X247 @ id @ X257 @ id ) )
| ( $true
= ( X247 @ id @ ( X253 @ id @ X257 ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f41,plain,
( ( $true = sP27 )
<=> ! [X247: subst > term > subst > $o,X248: term] :
( ? [X258: term,X259: subst,X260: subst,X261: subst] :
( ( $true
= ( X247 @ X261 @ X258 @ ( comp @ X260 @ X259 ) ) )
& ( ( X247 @ ( comp @ X261 @ X260 ) @ ( sub @ X258 @ X260 ) @ X259 )
!= $true ) )
| ( ( X247 @ id @ ( lam @ X248 ) @ id )
= $true )
| ? [X262: term] :
( ( ( X247 @ id @ ( sub @ X248 @ ( push @ X262 @ id ) ) @ id )
!= $true )
& ( $true
= ( X247 @ id @ X262 @ id ) ) )
| ( ( sP26 @ X247 )
= $true )
| ? [X249: subst,X250: subst,X251: term,X252: subst] :
( ( $true
= ( X247 @ ( comp @ X252 @ X249 ) @ ( sub @ X251 @ X249 ) @ X250 ) )
& ( $true
!= ( X247 @ X252 @ X251 @ ( comp @ X249 @ X250 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f42,plain,
! [X408: subst > term > subst > $o] :
( ( $true
= ( sP28 @ X408 ) )
<=> ? [X410: subst > term > term] :
( ( $true
!= ( X408 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
& ! [X411: term,X412: subst,X413: subst] :
( ( sub @ ( X410 @ X412 @ X411 ) @ X413 )
= ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) ) )
& ! [X414: term] :
( ( $true
= ( X408 @ id @ ( X410 @ id @ X414 ) @ id ) )
| ( $true
!= ( X408 @ id @ X414 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f43,plain,
( ( $true = sP29 )
<=> ! [X407: term,X408: subst > term > subst > $o,X409: term > $o] :
( ? [X416: subst,X417: subst,X418: term,X419: subst] :
( ( $true
= ( X408 @ ( comp @ X419 @ X417 ) @ ( sub @ X418 @ X417 ) @ X416 ) )
& ( $true
!= ( X408 @ X419 @ X418 @ ( comp @ X417 @ X416 ) ) ) )
| ( $true
= ( sP28 @ X408 ) )
| ( ( X409 @ ( lam @ X407 ) )
= $true )
| ? [X420: term] :
( ( X409 @ X420 )
!= ( X408 @ id @ X420 @ id ) )
| ? [X421: subst,X422: term,X423: subst,X424: subst] :
( ( $true
!= ( X408 @ ( comp @ X424 @ X421 ) @ ( sub @ X422 @ X421 ) @ X423 ) )
& ( $true
= ( X408 @ X424 @ X422 @ ( comp @ X421 @ X423 ) ) ) )
| ? [X415: term] :
( ( $true
!= ( X409 @ ( sub @ X407 @ ( push @ X415 @ id ) ) ) )
& ( $true
= ( X409 @ X415 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f44,plain,
! [X472: subst > term > subst > $o] :
( ( ( sP30 @ X472 )
= $true )
<=> ! [X483: term,X484: term] :
( ( $true
= ( X472 @ id @ ( ap @ ( sub @ X483 @ id ) @ X484 ) @ id ) )
| ( $true
!= ( X472 @ id @ X483 @ id ) )
| ( ( X472 @ id @ X484 @ id )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f45,plain,
! [X466: term > $o] :
( ( ( sP31 @ X466 )
= $true )
<=> ! [X470: term,X471: term] :
( ( $true
= ( X466 @ ( ap @ X470 @ X471 ) ) )
| ( $true
!= ( X466 @ X470 ) )
| ( $true
!= ( X466 @ X471 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f46,plain,
! [X486: subst > term > subst > $o] :
( ( $true
= ( sP32 @ X486 ) )
<=> ! [X495: subst > term > term] :
( ( $true
= ( X486 @ id @ ( lam @ ( X495 @ sh @ one ) ) @ id ) )
| ? [X496: term] :
( ( $true
!= ( X486 @ id @ ( X495 @ id @ X496 ) @ id ) )
& ( $true
= ( X486 @ id @ X496 @ id ) ) )
| ? [X497: term,X498: subst,X499: subst] :
( ( X495 @ ( comp @ X498 @ X499 ) @ ( sub @ X497 @ X499 ) )
!= ( sub @ ( X495 @ X498 @ X497 ) @ X499 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f47,plain,
! [X502: subst > term > subst > $o] :
( ( $true
= ( sP33 @ X502 ) )
<=> ? [X516: term,X517: term] :
( ( ( X502 @ id @ X517 @ id )
= $true )
& ( ( X502 @ id @ X516 @ id )
= $true )
& ( $true
!= ( X502 @ id @ ( ap @ ( sub @ X516 @ id ) @ X517 ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f48,plain,
! [X502: subst > term > subst > $o] :
( ( $true
= ( sP34 @ X502 ) )
<=> ? [X507: subst > term > term] :
( ! [X508: term] :
( ( $true
= ( X502 @ id @ ( X507 @ id @ X508 ) @ id ) )
| ( ( X502 @ id @ X508 @ id )
!= $true ) )
& ! [X509: term,X510: subst,X511: subst] :
( ( X507 @ ( comp @ X511 @ X510 ) @ ( sub @ X509 @ X510 ) )
= ( sub @ ( X507 @ X511 @ X509 ) @ X510 ) )
& ( $true
!= ( X502 @ id @ ( lam @ ( X507 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f49,plain,
( ( $true = sP35 )
<=> ? [X485: term,X486: subst > term > subst > $o] :
( ! [X491: subst,X492: term,X493: subst,X494: subst] :
( ( ( X486 @ ( comp @ X493 @ X494 ) @ ( sub @ X492 @ X494 ) @ X491 )
= $true )
| ( $true
!= ( X486 @ X493 @ X492 @ ( comp @ X494 @ X491 ) ) ) )
& ( $true
!= ( X486 @ id @ ( lam @ X485 ) @ id ) )
& ! [X487: subst,X488: subst,X489: subst,X490: term] :
( ( $true
!= ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X487 ) )
| ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X487 ) )
= $true ) )
& ( $true
= ( sP32 @ X486 ) )
& ! [X500: term] :
( ( ( X486 @ id @ ( sub @ X485 @ ( push @ X500 @ id ) ) @ id )
= $true )
| ( $true
!= ( X486 @ id @ X500 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f50,plain,
( ( $true = sP36 )
<=> ! [X501: term,X502: subst > term > subst > $o] :
( ( $true
= ( sP34 @ X502 ) )
| ? [X518: term] :
( ( ( X502 @ id @ X518 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X518 @ id ) ) ) )
| ? [X512: term,X513: subst,X514: subst,X515: subst] :
( ( ( X502 @ X515 @ X512 @ ( comp @ X514 @ X513 ) )
= $true )
& ( $true
!= ( X502 @ ( comp @ X515 @ X514 ) @ ( sub @ X512 @ X514 ) @ X513 ) ) )
| ? [X503: subst,X504: subst,X505: term,X506: subst] :
( ( $true
= ( X502 @ ( comp @ X504 @ X506 ) @ ( sub @ X505 @ X506 ) @ X503 ) )
& ( $true
!= ( X502 @ X504 @ X505 @ ( comp @ X506 @ X503 ) ) ) )
| ( $true
= ( sP33 @ X502 ) )
| ( ( X502 @ id @ X501 @ id )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f51,plain,
( ( sP37 = $true )
<=> ? [X472: subst > term > subst > $o,X473: term,X474: term] :
( ( $true
= ( X472 @ id @ X473 @ id ) )
& ( ( sP30 @ X472 )
= $true )
& ! [X479: term,X480: subst,X481: subst,X482: subst] :
( ( $true
!= ( X472 @ ( comp @ X480 @ X481 ) @ ( sub @ X479 @ X481 ) @ X482 ) )
| ( $true
= ( X472 @ X480 @ X479 @ ( comp @ X481 @ X482 ) ) ) )
& ( $true
!= ( X472 @ id @ ( ap @ X473 @ X474 ) @ id ) )
& ! [X475: subst,X476: term,X477: subst,X478: subst] :
( ( $true
= ( X472 @ ( comp @ X478 @ X477 ) @ ( sub @ X476 @ X477 ) @ X475 ) )
| ( $true
!= ( X472 @ X478 @ X476 @ ( comp @ X477 @ X475 ) ) ) )
& ( $true
= ( X472 @ id @ X474 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f52,plain,
( ( $true = sP38 )
<=> ( ( $true = sP36 )
| ( $true = sP35 )
| ? [X465: term,X466: term > $o] :
( ( $true
!= ( X466 @ X465 ) )
& ( ( sP31 @ X466 )
= $true )
& ! [X467: term] :
( ( $true
= ( X466 @ ( lam @ X467 ) ) )
| ? [X468: term] :
( ( $true
= ( X466 @ X468 ) )
& ( ( X466 @ ( sub @ X467 @ ( push @ X468 @ id ) ) )
!= $true ) ) )
& ! [X469: term] :
( ( $true
!= ( var @ X469 ) )
| ( ( X466 @ X469 )
= $true ) ) )
| ( sP37 = $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f53,plain,
! [X430: term > $o] :
( ( $true
= ( sP39 @ X430 ) )
<=> ? [X433: term,X434: term] :
( ( $true
= ( X430 @ X433 ) )
& ( ( X430 @ ( ap @ X433 @ X434 ) )
!= $true )
& ( $true
= ( X430 @ X434 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f54,plain,
! [X88: term > $o] :
( ( $true
= ( sP40 @ X88 ) )
<=> ? [X93: term,X94: term] :
( ( $true
= ( X88 @ X93 ) )
& ( $true
!= ( X88 @ ( ap @ X93 @ X94 ) ) )
& ( $true
= ( X88 @ X94 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f55,plain,
( ( $true = sP41 )
<=> ! [X88: term > $o,X89: term] :
( ? [X91: term] :
( ( $true
!= ( X88 @ ( lam @ X91 ) ) )
& ! [X92: term] :
( ( $true
!= ( X88 @ X92 ) )
| ( $true
= ( X88 @ ( sub @ X91 @ ( push @ X92 @ id ) ) ) ) ) )
| ( $true
= ( X88 @ X89 ) )
| ? [X90: term] :
( ( ( X88 @ X90 )
!= $true )
& ( $true
= ( var @ X90 ) ) )
| ( $true
= ( sP40 @ X88 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f56,plain,
( ( $true = sP42 )
<=> ! [X157: subst > term > subst > $o,X158: term] :
( ? [X159: term] :
( ( ( X157 @ id @ ( sub @ X158 @ ( push @ X159 @ id ) ) @ id )
!= $true )
& ( $true
= ( X157 @ id @ X159 @ id ) ) )
| ( $true
= ( X157 @ id @ ( lam @ X158 ) @ id ) )
| ? [X160: subst > term > term] :
( ! [X162: term,X163: subst,X164: subst] :
( ( sub @ ( X160 @ X163 @ X162 ) @ X164 )
= ( X160 @ ( comp @ X163 @ X164 ) @ ( sub @ X162 @ X164 ) ) )
& ( ( X157 @ id @ ( lam @ ( X160 @ sh @ one ) ) @ id )
!= $true )
& ! [X161: term] :
( ( ( X157 @ id @ X161 @ id )
!= $true )
| ( $true
= ( X157 @ id @ ( X160 @ id @ X161 ) @ id ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f57,plain,
! [X72: subst > term > subst > $o] :
( ( $true
= ( sP43 @ X72 ) )
<=> ? [X83: subst > term > term] :
( ! [X84: subst,X85: term,X86: subst] :
( ( X83 @ ( comp @ X86 @ X84 ) @ ( sub @ X85 @ X84 ) )
= ( sub @ ( X83 @ X86 @ X85 ) @ X84 ) )
& ! [X87: term] :
( ( ( X72 @ id @ ( X83 @ id @ X87 ) @ id )
= $true )
| ( $true
!= ( X72 @ id @ X87 @ id ) ) )
& ( $true
!= ( X72 @ id @ ( lam @ ( X83 @ sh @ one ) ) @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f58,plain,
( ( $true = sP44 )
<=> ? [X64: term,X65: subst > term > subst > $o] :
( ( $true
!= ( X65 @ id @ ( lam @ ( sub @ X64 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X66: term] :
( ( $true
!= ( X65 @ id @ X66 @ id ) )
| ( $true
= ( X65 @ id @ ( sub @ X64 @ ( push @ X66 @ id ) ) @ id ) ) )
& ! [X67: subst > term > term] :
( ? [X71: term] :
( ( $true
= ( X65 @ id @ X71 @ id ) )
& ( $true
!= ( X65 @ id @ ( X67 @ id @ X71 ) @ id ) ) )
| ? [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X67 @ X69 @ X70 ) @ X68 )
!= ( X67 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
| ( $true
= ( X65 @ id @ ( lam @ ( X67 @ sh @ one ) ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f59,plain,
( ( $true = sP45 )
<=> ( ( $true = sP44 )
| ? [X63: term] :
( ( sub @ X63 @ id )
!= X63 )
| ! [X72: subst > term > subst > $o,X73: term] :
( ( $true
= ( X72 @ id @ ( lam @ X73 ) @ id ) )
| ( $true
= ( sP43 @ X72 ) )
| ? [X74: term] :
( ( ( X72 @ id @ X74 @ id )
= $true )
& ( $true
!= ( X72 @ id @ ( sub @ X73 @ ( push @ X74 @ id ) ) @ id ) ) )
| ? [X75: subst,X76: term,X77: subst,X78: subst] :
( ( ( X72 @ ( comp @ X78 @ X77 ) @ ( sub @ X76 @ X77 ) @ X75 )
!= $true )
& ( $true
= ( X72 @ X78 @ X76 @ ( comp @ X77 @ X75 ) ) ) )
| ? [X79: term,X80: subst,X81: subst,X82: subst] :
( ( ( X72 @ ( comp @ X80 @ X82 ) @ ( sub @ X79 @ X82 ) @ X81 )
= $true )
& ( $true
!= ( X72 @ X80 @ X79 @ ( comp @ X82 @ X81 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f60,plain,
! [X54: subst > term > subst > $o] :
( ( $true
= ( sP46 @ X54 ) )
<=> ? [X56: term,X57: term] :
( ( $true
!= ( X54 @ id @ ( ap @ ( sub @ X56 @ id ) @ X57 ) @ id ) )
& ( ( X54 @ id @ X56 @ id )
= $true )
& ( $true
= ( X54 @ id @ X57 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f61,plain,
! [X456: subst > term > subst > $o] :
( ( $true
= ( sP47 @ X456 ) )
<=> ? [X463: term,X464: term] :
( ( ( X456 @ id @ X464 @ id )
= $true )
& ( ( X456 @ id @ ( ap @ ( sub @ X463 @ id ) @ X464 ) @ id )
!= $true )
& ( $true
= ( X456 @ id @ X463 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f62,plain,
! [X439: term > $o] :
( ( $true
= ( sP48 @ X439 ) )
<=> ! [X443: term,X444: term] :
( ( ( X439 @ X443 )
!= $true )
| ( $true
!= ( X439 @ X444 ) )
| ( ( X439 @ ( ap @ X444 @ X443 ) )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f63,plain,
( ( $true = sP49 )
<=> ? [X446: term > $o,X447: subst > term > subst > $o,X448: term] :
( ! [X451: subst > term > term] :
( ( $true
= ( X447 @ id @ ( lam @ ( X451 @ sh @ one ) ) @ id ) )
| ? [X452: term,X453: subst,X454: subst] :
( ( sub @ ( X451 @ X453 @ X452 ) @ X454 )
!= ( X451 @ ( comp @ X453 @ X454 ) @ ( sub @ X452 @ X454 ) ) )
| ? [X455: term] :
( ( $true
= ( X447 @ id @ X455 @ id ) )
& ( $true
!= ( X447 @ id @ ( X451 @ id @ X455 ) @ id ) ) ) )
& ! [X450: term] :
( ( ( X446 @ ( sub @ X448 @ ( push @ X450 @ id ) ) )
= $true )
| ( $true
!= ( X446 @ X450 ) ) )
& ! [X449: term] :
( ( X447 @ id @ X449 @ id )
= ( X446 @ X449 ) )
& ( $true
!= ( X446 @ ( lam @ X448 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f64,plain,
( ( $true = sP50 )
<=> ! [X456: subst > term > subst > $o,X457: term] :
( ? [X458: subst > term > term] :
( ! [X459: term] :
( ( $true
= ( X456 @ id @ ( X458 @ id @ X459 ) @ id ) )
| ( $true
!= ( X456 @ id @ X459 @ id ) ) )
& ! [X460: term,X461: subst,X462: subst] :
( ( X458 @ ( comp @ X462 @ X461 ) @ ( sub @ X460 @ X461 ) )
= ( sub @ ( X458 @ X462 @ X460 ) @ X461 ) )
& ( $true
!= ( X456 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X456 @ id @ X457 @ id ) )
| ( $true
= ( sP47 @ X456 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f65,plain,
( ( sP51 = $true )
<=> ( ? [X445: term] :
( ( sub @ X445 @ id )
!= X445 )
| ? [X438: term,X439: term > $o] :
( ! [X440: term] :
( ( $true
!= ( var @ X440 ) )
| ( $true
= ( X439 @ X440 ) ) )
& ! [X441: term] :
( ? [X442: term] :
( ( $true
!= ( X439 @ ( sub @ X441 @ ( push @ X442 @ id ) ) ) )
& ( $true
= ( X439 @ X442 ) ) )
| ( ( X439 @ ( lam @ X441 ) )
= $true ) )
& ( $true
= ( sP48 @ X439 ) )
& ( $true
!= ( X439 @ X438 ) ) )
| ( $true = sP49 )
| ( $true = sP50 )
| ? [X435: subst > term > subst > $o] :
! [X436: term > $o] :
? [X437: term] :
( ( X436 @ X437 )
!= ( X435 @ id @ X437 @ id ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f66,plain,
( ( $true = sP52 )
<=> ? [X278: term,X279: subst > term > subst > $o] :
( ! [X285: term] :
( ( $true
= ( X279 @ id @ ( sub @ X278 @ ( push @ X285 @ id ) ) @ id ) )
| ( $true
!= ( X279 @ id @ X285 @ id ) ) )
& ( $true
!= ( X279 @ id @ ( lam @ ( sub @ X278 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X280: subst > term > term] :
( ? [X281: term] :
( ( $true
!= ( X279 @ id @ ( X280 @ id @ X281 ) @ id ) )
& ( ( X279 @ id @ X281 @ id )
= $true ) )
| ? [X282: term,X283: subst,X284: subst] :
( ( sub @ ( X280 @ X284 @ X282 ) @ X283 )
!= ( X280 @ ( comp @ X284 @ X283 ) @ ( sub @ X282 @ X283 ) ) )
| ( $true
= ( X279 @ id @ ( lam @ ( X280 @ sh @ one ) ) @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f67,plain,
( ( sP53 = $true )
<=> ( ! [X286: subst > term > subst > $o,X287: term] :
( ? [X288: term] :
( ( $true
= ( X286 @ id @ X288 @ id ) )
& ( ( X286 @ id @ ( sub @ X287 @ ( push @ X288 @ id ) ) @ id )
!= $true ) )
| ( $true
= ( X286 @ id @ ( lam @ X287 ) @ id ) )
| ? [X289: subst > term > term] :
( ! [X290: term] :
( ( ( X286 @ id @ X290 @ id )
!= $true )
| ( $true
= ( X286 @ id @ ( X289 @ id @ X290 ) @ id ) ) )
& ! [X291: subst,X292: subst,X293: term] :
( ( sub @ ( X289 @ X292 @ X293 ) @ X291 )
= ( X289 @ ( comp @ X292 @ X291 ) @ ( sub @ X293 @ X291 ) ) )
& ( $true
!= ( X286 @ id @ ( lam @ ( X289 @ sh @ one ) ) @ id ) ) ) )
| ( $true = sP52 )
| ? [X277: term] :
( ( sub @ X277 @ id )
!= X277 ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f68,plain,
! [X386: term > $o] :
( ( $true
= ( sP54 @ X386 ) )
<=> ? [X391: term,X392: term] :
( ( $true
= ( X386 @ X392 ) )
& ( $true
= ( X386 @ X391 ) )
& ( ( X386 @ ( ap @ X392 @ X391 ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f69,plain,
( ( $true = sP55 )
<=> ! [X385: subst,X386: term > $o,X387: term] :
( ? [X388: term] :
( ! [X389: term] :
( ( $true
= ( X386 @ ( sub @ X388 @ ( push @ X389 @ id ) ) ) )
| ( ( X386 @ X389 )
!= $true ) )
& ( $true
!= ( X386 @ ( lam @ X388 ) ) ) )
| ( $true
= ( X386 @ ( sub @ X387 @ X385 ) ) )
| ? [X390: term] :
( ( $true
= ( var @ X390 ) )
& ( $true
!= ( X386 @ ( sub @ X390 @ X385 ) ) ) )
| ( $true
= ( sP54 @ X386 ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f70,plain,
( ( $true = sP56 )
<=> ( ? [X103: term,X104: term] :
( ( ( lam @ X103 )
= ( lam @ X104 ) )
& ( X103 != X104 ) )
| ? [X99: term,X100: subst] :
( ( sub @ one @ ( push @ X99 @ X100 ) )
!= X99 )
| ? [X101: term,X102: subst] :
( ( comp @ sh @ ( push @ X101 @ X102 ) )
!= X102 )
| ! [X105: term,X106: subst > term > term,X107: subst > term > term,X108: subst] :
( ? [X109: subst,X110: subst,X111: term] :
( ( sub @ ( X107 @ X109 @ X111 ) @ X110 )
!= ( X107 @ ( comp @ X109 @ X110 ) @ ( sub @ X111 @ X110 ) ) )
| ( ( lam @ ( X106 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ? [X112: subst,X113: term,X114: subst] :
( ( X106 @ ( comp @ X114 @ X112 ) @ ( sub @ X113 @ X112 ) )
!= ( sub @ ( X106 @ X114 @ X113 ) @ X112 ) )
| ( ( X106 @ X108 @ X105 )
= ( X107 @ X108 @ X105 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f71,plain,
( ( $true = sP57 )
<=> ! [X115: subst > term > subst > $o,X116: term] :
( ? [X117: subst > term > term] :
( ! [X118: subst,X119: subst,X120: term] :
( ( sub @ ( X117 @ X118 @ X120 ) @ X119 )
= ( X117 @ ( comp @ X118 @ X119 ) @ ( sub @ X120 @ X119 ) ) )
& ! [X121: term] :
( ( ( X115 @ id @ X121 @ id )
!= $true )
| ( $true
= ( X115 @ id @ ( X117 @ id @ X121 ) @ id ) ) )
& ( $true
!= ( X115 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X115 @ id @ ( lam @ ( sub @ X116 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X122: term] :
( ( $true
!= ( X115 @ id @ ( sub @ X116 @ ( push @ X122 @ id ) ) @ id ) )
& ( $true
= ( X115 @ id @ X122 @ id ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f72,plain,
( ( $true = sP58 )
<=> ! [X318: subst > term > subst > $o,X319: term,X320: term > $o] :
( ? [X321: subst > term > term] :
( ! [X325: term] :
( ( $true
= ( X318 @ id @ ( X321 @ id @ X325 ) @ id ) )
| ( ( X318 @ id @ X325 @ id )
!= $true ) )
& ( ( X318 @ id @ ( lam @ ( X321 @ sh @ one ) ) @ id )
!= $true )
& ! [X322: subst,X323: term,X324: subst] :
( ( sub @ ( X321 @ X322 @ X323 ) @ X324 )
= ( X321 @ ( comp @ X322 @ X324 ) @ ( sub @ X323 @ X324 ) ) ) )
| ? [X326: term] :
( ( X318 @ id @ X326 @ id )
!= ( X320 @ X326 ) )
| ? [X327: term] :
( ( $true
!= ( X320 @ ( sub @ X319 @ ( push @ X327 @ id ) ) ) )
& ( $true
= ( X320 @ X327 ) ) )
| ( $true
= ( X320 @ ( lam @ X319 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).
thf(f13,plain,
( ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ! [X156: d_subst] : ( d_id = X156 )
& ( hoasinduction_lem1_gthm = $true )
& ( ( shinj = $true )
<=> ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) ) )
& ( lamnotvar = $true )
& ( ( ? [X208: term] :
( ( sub @ X208 @ id )
!= X208 )
| ! [X209: term] :
( ( sub @ X209 @ id )
= X209 ) )
<=> ( termmset_lthm = $true ) )
& ( ! [X318: subst > term > subst > $o,X319: term,X320: term > $o] :
( ? [X321: subst > term > term] :
( ! [X325: term] :
( ( $true
= ( X318 @ id @ ( X321 @ id @ X325 ) @ id ) )
| ( ( X318 @ id @ X325 @ id )
!= $true ) )
& ( ( X318 @ id @ ( lam @ ( X321 @ sh @ one ) ) @ id )
!= $true )
& ! [X322: subst,X323: term,X324: subst] :
( ( sub @ ( X321 @ X322 @ X323 ) @ X324 )
= ( X321 @ ( comp @ X322 @ X324 ) @ ( sub @ X323 @ X324 ) ) ) )
| ? [X326: term] :
( ( X318 @ id @ X326 @ id )
!= ( X320 @ X326 ) )
| ? [X327: term] :
( ( $true
!= ( X320 @ ( sub @ X319 @ ( push @ X327 @ id ) ) ) )
& ( $true
= ( X320 @ X327 ) ) )
| ( $true
= ( X320 @ ( lam @ X319 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( ( axvarid = $true )
<=> ! [X265: term] :
( ( sub @ X265 @ id )
= X265 ) )
& ( hoaslamnotvar_lthm = $true )
& ( apnotvar = $true )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X115: subst > term > subst > $o,X116: term] :
( ? [X117: subst > term > term] :
( ! [X118: subst,X119: subst,X120: term] :
( ( sub @ ( X117 @ X118 @ X120 ) @ X119 )
= ( X117 @ ( comp @ X118 @ X119 ) @ ( sub @ X120 @ X119 ) ) )
& ! [X121: term] :
( ( ( X115 @ id @ X121 @ id )
!= $true )
| ( $true
= ( X115 @ id @ ( X117 @ id @ X121 ) @ id ) ) )
& ( $true
!= ( X115 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X115 @ id @ ( lam @ ( sub @ X116 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X122: term] :
( ( $true
!= ( X115 @ id @ ( sub @ X116 @ ( push @ X122 @ id ) ) @ id ) )
& ( $true
= ( X115 @ id @ X122 @ id ) ) ) ) )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ? [X103: term,X104: term] :
( ( ( lam @ X103 )
= ( lam @ X104 ) )
& ( X103 != X104 ) )
| ? [X99: term,X100: subst] :
( ( sub @ one @ ( push @ X99 @ X100 ) )
!= X99 )
| ? [X101: term,X102: subst] :
( ( comp @ sh @ ( push @ X101 @ X102 ) )
!= X102 )
| ! [X105: term,X106: subst > term > term,X107: subst > term > term,X108: subst] :
( ? [X109: subst,X110: subst,X111: term] :
( ( sub @ ( X107 @ X109 @ X111 ) @ X110 )
!= ( X107 @ ( comp @ X109 @ X110 ) @ ( sub @ X111 @ X110 ) ) )
| ( ( lam @ ( X106 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ? [X112: subst,X113: term,X114: subst] :
( ( X106 @ ( comp @ X114 @ X112 ) @ ( sub @ X113 @ X112 ) )
!= ( sub @ ( X106 @ X114 @ X113 ) @ X112 ) )
| ( ( X106 @ X108 @ X105 )
= ( X107 @ X108 @ X105 ) ) ) ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( axidr = $true )
<=> ! [X393: subst] :
( ( comp @ X393 @ id )
= X393 ) )
& ( ! [X385: subst,X386: term > $o,X387: term] :
( ? [X388: term] :
( ! [X389: term] :
( ( $true
= ( X386 @ ( sub @ X388 @ ( push @ X389 @ id ) ) ) )
| ( ( X386 @ X389 )
!= $true ) )
& ( $true
!= ( X386 @ ( lam @ X388 ) ) ) )
| ( $true
= ( X386 @ ( sub @ X387 @ X385 ) ) )
| ? [X390: term] :
( ( $true
= ( var @ X390 ) )
& ( $true
!= ( X386 @ ( sub @ X390 @ X385 ) ) ) )
| ? [X391: term,X392: term] :
( ( $true
= ( X386 @ X392 ) )
& ( $true
= ( X386 @ X391 ) )
& ( ( X386 @ ( ap @ X392 @ X391 ) )
!= $true ) ) )
<=> ( induction2lem = $true ) )
& ! [X141: d_term] : ( d_one = X141 )
& ( hoasinduction_lem3_gthm = $true )
& ( ( ! [X286: subst > term > subst > $o,X287: term] :
( ? [X288: term] :
( ( $true
= ( X286 @ id @ X288 @ id ) )
& ( ( X286 @ id @ ( sub @ X287 @ ( push @ X288 @ id ) ) @ id )
!= $true ) )
| ( $true
= ( X286 @ id @ ( lam @ X287 ) @ id ) )
| ? [X289: subst > term > term] :
( ! [X290: term] :
( ( ( X286 @ id @ X290 @ id )
!= $true )
| ( $true
= ( X286 @ id @ ( X289 @ id @ X290 ) @ id ) ) )
& ! [X291: subst,X292: subst,X293: term] :
( ( sub @ ( X289 @ X292 @ X293 ) @ X291 )
= ( X289 @ ( comp @ X292 @ X291 ) @ ( sub @ X293 @ X291 ) ) )
& ( $true
!= ( X286 @ id @ ( lam @ ( X289 @ sh @ one ) ) @ id ) ) ) )
| ? [X278: term,X279: subst > term > subst > $o] :
( ! [X285: term] :
( ( $true
= ( X279 @ id @ ( sub @ X278 @ ( push @ X285 @ id ) ) @ id ) )
| ( $true
!= ( X279 @ id @ X285 @ id ) ) )
& ( $true
!= ( X279 @ id @ ( lam @ ( sub @ X278 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X280: subst > term > term] :
( ? [X281: term] :
( ( $true
!= ( X279 @ id @ ( X280 @ id @ X281 ) @ id ) )
& ( ( X279 @ id @ X281 @ id )
= $true ) )
| ? [X282: term,X283: subst,X284: subst] :
( ( sub @ ( X280 @ X284 @ X282 ) @ X283 )
!= ( X280 @ ( comp @ X284 @ X283 ) @ ( sub @ X282 @ X283 ) ) )
| ( $true
= ( X279 @ id @ ( lam @ ( X280 @ sh @ one ) ) @ id ) ) ) )
| ? [X277: term] :
( ( sub @ X277 @ id )
!= X277 ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( axapp = $true )
& ( ( hoasinduction_no_psi_cond_lthm = $true )
<=> ( ? [X445: term] :
( ( sub @ X445 @ id )
!= X445 )
| ? [X438: term,X439: term > $o] :
( ! [X440: term] :
( ( $true
!= ( var @ X440 ) )
| ( $true
= ( X439 @ X440 ) ) )
& ! [X441: term] :
( ? [X442: term] :
( ( $true
!= ( X439 @ ( sub @ X441 @ ( push @ X442 @ id ) ) ) )
& ( $true
= ( X439 @ X442 ) ) )
| ( ( X439 @ ( lam @ X441 ) )
= $true ) )
& ! [X443: term,X444: term] :
( ( ( X439 @ X443 )
!= $true )
| ( $true
!= ( X439 @ X444 ) )
| ( ( X439 @ ( ap @ X444 @ X443 ) )
= $true ) )
& ( $true
!= ( X439 @ X438 ) ) )
| ? [X446: term > $o,X447: subst > term > subst > $o,X448: term] :
( ! [X451: subst > term > term] :
( ( $true
= ( X447 @ id @ ( lam @ ( X451 @ sh @ one ) ) @ id ) )
| ? [X452: term,X453: subst,X454: subst] :
( ( sub @ ( X451 @ X453 @ X452 ) @ X454 )
!= ( X451 @ ( comp @ X453 @ X454 ) @ ( sub @ X452 @ X454 ) ) )
| ? [X455: term] :
( ( $true
= ( X447 @ id @ X455 @ id ) )
& ( $true
!= ( X447 @ id @ ( X451 @ id @ X455 ) @ id ) ) ) )
& ! [X450: term] :
( ( ( X446 @ ( sub @ X448 @ ( push @ X450 @ id ) ) )
= $true )
| ( $true
!= ( X446 @ X450 ) ) )
& ! [X449: term] :
( ( X447 @ id @ X449 @ id )
= ( X446 @ X449 ) )
& ( $true
!= ( X446 @ ( lam @ X448 ) ) ) )
| ! [X456: subst > term > subst > $o,X457: term] :
( ? [X458: subst > term > term] :
( ! [X459: term] :
( ( $true
= ( X456 @ id @ ( X458 @ id @ X459 ) @ id ) )
| ( $true
!= ( X456 @ id @ X459 @ id ) ) )
& ! [X460: term,X461: subst,X462: subst] :
( ( X458 @ ( comp @ X462 @ X461 ) @ ( sub @ X460 @ X461 ) )
= ( sub @ ( X458 @ X462 @ X460 ) @ X461 ) )
& ( $true
!= ( X456 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X456 @ id @ X457 @ id ) )
| ? [X463: term,X464: term] :
( ( ( X456 @ id @ X464 @ id )
= $true )
& ( ( X456 @ id @ ( ap @ ( sub @ X463 @ id ) @ X464 ) @ id )
!= $true )
& ( $true
= ( X456 @ id @ X463 @ id ) ) ) )
| ? [X435: subst > term > subst > $o] :
! [X436: term > $o] :
? [X437: term] :
( ( X436 @ X437 )
!= ( X435 @ id @ X437 @ id ) ) ) )
& ( hoasapnotvar = $true )
& ( ! [X378: term,X379: term,X380: subst > term > term] :
( ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( lam @ ( X380 @ sh @ one ) ) )
| ? [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
!= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) ) )
<=> ( hoaslamnotap = $true ) )
& ( axclos = $true )
& ( ( ? [X399: term,X400: term,X401: term,X402: term] :
( ( X399 != X402 )
& ( ( ap @ X399 @ X400 )
= ( ap @ X402 @ X401 ) ) )
| ! [X403: term,X404: term,X405: term,X406: term] :
( ( X405 = X406 )
| ( ( ap @ ( sub @ X405 @ id ) @ X403 )
!= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) )
| ? [X398: term] :
( ( sub @ X398 @ id )
!= X398 ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( ulamvar1 != $true )
& ( ! [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
!= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
| ( X394 = X397 ) )
<=> ( hoasapinj1 = $true ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ? [X58: subst > term > term] :
( ! [X59: term] :
( ( $true
= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
| ( $true
!= ( X54 @ id @ X59 @ id ) ) )
& ( $true
!= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
& ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) )
| ? [X56: term,X57: term] :
( ( $true
!= ( X54 @ id @ ( ap @ ( sub @ X56 @ id ) @ X57 ) @ id ) )
& ( ( X54 @ id @ X56 @ id )
= $true )
& ( $true
= ( X54 @ id @ X57 @ id ) ) )
| ( $true
= ( X54 @ id @ X55 @ id ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( ! [X276: term] :
( ( sub @ X276 @ id )
= X276 )
<=> ( termmset = $true ) )
& ( induction2_gthm = $true )
& ( pushprop = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( ! [X21: subst,X22: term,X23: term > $o] :
? [X24: term > $o] :
! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ! [X337: d_term,X338: d_term] :
( ( X337 = X338 )
| ( ( d2term @ X337 )
!= ( d2term @ X338 ) ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X187: subst > term > subst > $o] :
? [X188: term > $o] :
! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( ( ? [X64: term,X65: subst > term > subst > $o] :
( ( $true
!= ( X65 @ id @ ( lam @ ( sub @ X64 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X66: term] :
( ( $true
!= ( X65 @ id @ X66 @ id ) )
| ( $true
= ( X65 @ id @ ( sub @ X64 @ ( push @ X66 @ id ) ) @ id ) ) )
& ! [X67: subst > term > term] :
( ? [X71: term] :
( ( $true
= ( X65 @ id @ X71 @ id ) )
& ( $true
!= ( X65 @ id @ ( X67 @ id @ X71 ) @ id ) ) )
| ? [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X67 @ X69 @ X70 ) @ X68 )
!= ( X67 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
| ( $true
= ( X65 @ id @ ( lam @ ( X67 @ sh @ one ) ) @ id ) ) ) )
| ? [X63: term] :
( ( sub @ X63 @ id )
!= X63 )
| ! [X72: subst > term > subst > $o,X73: term] :
( ( $true
= ( X72 @ id @ ( lam @ X73 ) @ id ) )
| ? [X83: subst > term > term] :
( ! [X84: subst,X85: term,X86: subst] :
( ( X83 @ ( comp @ X86 @ X84 ) @ ( sub @ X85 @ X84 ) )
= ( sub @ ( X83 @ X86 @ X85 ) @ X84 ) )
& ! [X87: term] :
( ( ( X72 @ id @ ( X83 @ id @ X87 ) @ id )
= $true )
| ( $true
!= ( X72 @ id @ X87 @ id ) ) )
& ( $true
!= ( X72 @ id @ ( lam @ ( X83 @ sh @ one ) ) @ id ) ) )
| ? [X74: term] :
( ( ( X72 @ id @ X74 @ id )
= $true )
& ( $true
!= ( X72 @ id @ ( sub @ X73 @ ( push @ X74 @ id ) ) @ id ) ) )
| ? [X75: subst,X76: term,X77: subst,X78: subst] :
( ( ( X72 @ ( comp @ X78 @ X77 ) @ ( sub @ X76 @ X77 ) @ X75 )
!= $true )
& ( $true
= ( X72 @ X78 @ X76 @ ( comp @ X77 @ X75 ) ) ) )
| ? [X79: term,X80: subst,X81: subst,X82: subst] :
( ( ( X72 @ ( comp @ X80 @ X82 ) @ ( sub @ X79 @ X82 ) @ X81 )
= $true )
& ( $true
!= ( X72 @ X80 @ X79 @ ( comp @ X82 @ X81 ) ) ) ) ) )
<=> ( hoasinduction_lem3_lthm = $true ) )
& ( ! [X157: subst > term > subst > $o,X158: term] :
( ? [X159: term] :
( ( ( X157 @ id @ ( sub @ X158 @ ( push @ X159 @ id ) ) @ id )
!= $true )
& ( $true
= ( X157 @ id @ X159 @ id ) ) )
| ( $true
= ( X157 @ id @ ( lam @ X158 ) @ id ) )
| ? [X160: subst > term > term] :
( ! [X162: term,X163: subst,X164: subst] :
( ( sub @ ( X160 @ X163 @ X162 ) @ X164 )
= ( X160 @ ( comp @ X163 @ X164 ) @ ( sub @ X162 @ X164 ) ) )
& ( ( X157 @ id @ ( lam @ ( X160 @ sh @ one ) ) @ id )
!= $true )
& ! [X161: term] :
( ( ( X157 @ id @ X161 @ id )
!= $true )
| ( $true
= ( X157 @ id @ ( X160 @ id @ X161 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X138: subst > term > subst > $o] :
? [X139: term > $o] :
! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) ) )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X95: term] :
? [X96: subst > term > term] :
! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) ) )
& ( lamnotap != $true )
& ( one
= ( d2term @ d_one ) )
& ( axassoc = $true )
& ( ( induction2 = $true )
<=> ! [X88: term > $o,X89: term] :
( ? [X91: term] :
( ( $true
!= ( X88 @ ( lam @ X91 ) ) )
& ! [X92: term] :
( ( $true
!= ( X88 @ X92 ) )
| ( $true
= ( X88 @ ( sub @ X91 @ ( push @ X92 @ id ) ) ) ) ) )
| ( $true
= ( X88 @ X89 ) )
| ? [X90: term] :
( ( ( X88 @ X90 )
!= $true )
& ( $true
= ( var @ X90 ) ) )
| ? [X93: term,X94: term] :
( ( $true
= ( X88 @ X93 ) )
& ( $true
!= ( X88 @ ( ap @ X93 @ X94 ) ) )
& ( $true
= ( X88 @ X94 ) ) ) ) )
& ( ! [X429: term,X430: term > $o] :
( ? [X432: term] :
( ( $true
= ( X430 @ X432 ) )
& ( $true
!= ( X430 @ ( lam @ X432 ) ) ) )
| ( $true
= ( X430 @ X429 ) )
| ? [X433: term,X434: term] :
( ( $true
= ( X430 @ X433 ) )
& ( ( X430 @ ( ap @ X433 @ X434 ) )
!= $true )
& ( $true
= ( X430 @ X434 ) ) )
| ? [X431: term] :
( ( $true
!= ( X430 @ X431 ) )
& ( $true
= ( var @ X431 ) ) ) )
<=> ( induction = $true ) )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X180: term > $o,X181: term,X182: subst] :
? [X183: term > $o] :
! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) ) )
& ( hoasinduction_lem1v2 = $true )
& ( pushprop_lthm = $true )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X501: term,X502: subst > term > subst > $o] :
( ? [X507: subst > term > term] :
( ! [X508: term] :
( ( $true
= ( X502 @ id @ ( X507 @ id @ X508 ) @ id ) )
| ( ( X502 @ id @ X508 @ id )
!= $true ) )
& ! [X509: term,X510: subst,X511: subst] :
( ( X507 @ ( comp @ X511 @ X510 ) @ ( sub @ X509 @ X510 ) )
= ( sub @ ( X507 @ X511 @ X509 ) @ X510 ) )
& ( $true
!= ( X502 @ id @ ( lam @ ( X507 @ sh @ one ) ) @ id ) ) )
| ? [X518: term] :
( ( ( X502 @ id @ X518 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X518 @ id ) ) ) )
| ? [X512: term,X513: subst,X514: subst,X515: subst] :
( ( ( X502 @ X515 @ X512 @ ( comp @ X514 @ X513 ) )
= $true )
& ( $true
!= ( X502 @ ( comp @ X515 @ X514 ) @ ( sub @ X512 @ X514 ) @ X513 ) ) )
| ? [X503: subst,X504: subst,X505: term,X506: subst] :
( ( $true
= ( X502 @ ( comp @ X504 @ X506 ) @ ( sub @ X505 @ X506 ) @ X503 ) )
& ( $true
!= ( X502 @ X504 @ X505 @ ( comp @ X506 @ X503 ) ) ) )
| ? [X516: term,X517: term] :
( ( ( X502 @ id @ X517 @ id )
= $true )
& ( ( X502 @ id @ X516 @ id )
= $true )
& ( $true
!= ( X502 @ id @ ( ap @ ( sub @ X516 @ id ) @ X517 ) @ id ) ) )
| ( ( X502 @ id @ X501 @ id )
= $true ) )
| ? [X485: term,X486: subst > term > subst > $o] :
( ! [X491: subst,X492: term,X493: subst,X494: subst] :
( ( ( X486 @ ( comp @ X493 @ X494 ) @ ( sub @ X492 @ X494 ) @ X491 )
= $true )
| ( $true
!= ( X486 @ X493 @ X492 @ ( comp @ X494 @ X491 ) ) ) )
& ( $true
!= ( X486 @ id @ ( lam @ X485 ) @ id ) )
& ! [X487: subst,X488: subst,X489: subst,X490: term] :
( ( $true
!= ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X487 ) )
| ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X487 ) )
= $true ) )
& ! [X495: subst > term > term] :
( ( $true
= ( X486 @ id @ ( lam @ ( X495 @ sh @ one ) ) @ id ) )
| ? [X496: term] :
( ( $true
!= ( X486 @ id @ ( X495 @ id @ X496 ) @ id ) )
& ( $true
= ( X486 @ id @ X496 @ id ) ) )
| ? [X497: term,X498: subst,X499: subst] :
( ( X495 @ ( comp @ X498 @ X499 ) @ ( sub @ X497 @ X499 ) )
!= ( sub @ ( X495 @ X498 @ X497 ) @ X499 ) ) )
& ! [X500: term] :
( ( ( X486 @ id @ ( sub @ X485 @ ( push @ X500 @ id ) ) @ id )
= $true )
| ( $true
!= ( X486 @ id @ X500 @ id ) ) ) )
| ? [X465: term,X466: term > $o] :
( ( $true
!= ( X466 @ X465 ) )
& ! [X470: term,X471: term] :
( ( $true
= ( X466 @ ( ap @ X470 @ X471 ) ) )
| ( $true
!= ( X466 @ X470 ) )
| ( $true
!= ( X466 @ X471 ) ) )
& ! [X467: term] :
( ( $true
= ( X466 @ ( lam @ X467 ) ) )
| ? [X468: term] :
( ( $true
= ( X466 @ X468 ) )
& ( ( X466 @ ( sub @ X467 @ ( push @ X468 @ id ) ) )
!= $true ) ) )
& ! [X469: term] :
( ( $true
!= ( var @ X469 ) )
| ( ( X466 @ X469 )
= $true ) ) )
| ? [X472: subst > term > subst > $o,X473: term,X474: term] :
( ( $true
= ( X472 @ id @ X473 @ id ) )
& ! [X483: term,X484: term] :
( ( $true
= ( X472 @ id @ ( ap @ ( sub @ X483 @ id ) @ X484 ) @ id ) )
| ( $true
!= ( X472 @ id @ X483 @ id ) )
| ( ( X472 @ id @ X484 @ id )
!= $true ) )
& ! [X479: term,X480: subst,X481: subst,X482: subst] :
( ( $true
!= ( X472 @ ( comp @ X480 @ X481 ) @ ( sub @ X479 @ X481 ) @ X482 ) )
| ( $true
= ( X472 @ X480 @ X479 @ ( comp @ X481 @ X482 ) ) ) )
& ( $true
!= ( X472 @ id @ ( ap @ X473 @ X474 ) @ id ) )
& ! [X475: subst,X476: term,X477: subst,X478: subst] :
( ( $true
= ( X472 @ ( comp @ X478 @ X477 ) @ ( sub @ X476 @ X477 ) @ X475 ) )
| ( $true
!= ( X472 @ X478 @ X476 @ ( comp @ X477 @ X475 ) ) ) )
& ( $true
= ( X472 @ id @ X474 @ id ) ) ) ) )
& ( ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) )
<=> ( substmonoid = $true ) )
& ( pushprop_lem1_gthm = $true )
& ( hoasinduction_lem1v2_gthm = $true )
& ( pushprop_lem2v2_lthm = $true )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X407: term,X408: subst > term > subst > $o,X409: term > $o] :
( ? [X416: subst,X417: subst,X418: term,X419: subst] :
( ( $true
= ( X408 @ ( comp @ X419 @ X417 ) @ ( sub @ X418 @ X417 ) @ X416 ) )
& ( $true
!= ( X408 @ X419 @ X418 @ ( comp @ X417 @ X416 ) ) ) )
| ? [X410: subst > term > term] :
( ( $true
!= ( X408 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
& ! [X411: term,X412: subst,X413: subst] :
( ( sub @ ( X410 @ X412 @ X411 ) @ X413 )
= ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) ) )
& ! [X414: term] :
( ( $true
= ( X408 @ id @ ( X410 @ id @ X414 ) @ id ) )
| ( $true
!= ( X408 @ id @ X414 @ id ) ) ) )
| ( ( X409 @ ( lam @ X407 ) )
= $true )
| ? [X420: term] :
( ( X409 @ X420 )
!= ( X408 @ id @ X420 @ id ) )
| ? [X421: subst,X422: term,X423: subst,X424: subst] :
( ( $true
!= ( X408 @ ( comp @ X424 @ X421 ) @ ( sub @ X422 @ X421 ) @ X423 ) )
& ( $true
= ( X408 @ X424 @ X422 @ ( comp @ X421 @ X423 ) ) ) )
| ? [X415: term] :
( ( $true
!= ( X409 @ ( sub @ X407 @ ( push @ X415 @ id ) ) ) )
& ( $true
= ( X409 @ X415 ) ) ) ) )
& ( ! [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
= ( X268 @ X267 @ X266 ) )
| ( ( lam @ ( X268 @ sh @ one ) )
!= ( lam @ ( X269 @ sh @ one ) ) )
| ? [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
!= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
| ? [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
!= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lem1 = $true )
& ( ! [X247: subst > term > subst > $o,X248: term] :
( ? [X258: term,X259: subst,X260: subst,X261: subst] :
( ( $true
= ( X247 @ X261 @ X258 @ ( comp @ X260 @ X259 ) ) )
& ( ( X247 @ ( comp @ X261 @ X260 ) @ ( sub @ X258 @ X260 ) @ X259 )
!= $true ) )
| ( ( X247 @ id @ ( lam @ X248 ) @ id )
= $true )
| ? [X262: term] :
( ( ( X247 @ id @ ( sub @ X248 @ ( push @ X262 @ id ) ) @ id )
!= $true )
& ( $true
= ( X247 @ id @ X262 @ id ) ) )
| ? [X253: subst > term > term] :
( ! [X254: subst,X255: term,X256: subst] :
( ( sub @ ( X253 @ X256 @ X255 ) @ X254 )
= ( X253 @ ( comp @ X256 @ X254 ) @ ( sub @ X255 @ X254 ) ) )
& ( $true
!= ( X247 @ id @ ( lam @ ( X253 @ sh @ one ) ) @ id ) )
& ! [X257: term] :
( ( $true
!= ( X247 @ id @ X257 @ id ) )
| ( $true
= ( X247 @ id @ ( X253 @ id @ X257 ) @ id ) ) ) )
| ? [X249: subst,X250: subst,X251: term,X252: subst] :
( ( $true
= ( X247 @ ( comp @ X252 @ X249 ) @ ( sub @ X251 @ X249 ) @ X250 ) )
& ( $true
!= ( X247 @ X252 @ X251 @ ( comp @ X249 @ X250 ) ) ) ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( ( hoasapinj2 = $true )
<=> ! [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
| ( X174 = X175 ) ) )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( $true
!= ( X243 @ X240 ) )
| ? [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
!= ( X242 @ X244 ) )
| ( ( X242 @ one )
= $true ) ) )
& ( hoasapinj2_gthm = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
!= ( X169 @ X167 ) )
| ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ! [X200: term,X201: subst > term > subst > $o] :
( ? [X207: term] :
( ( $true
= ( X201 @ id @ X207 @ id ) )
& ( $true
!= ( X201 @ id @ ( sub @ X200 @ ( push @ X207 @ id ) ) @ id ) ) )
| ? [X202: subst > term > term] :
( ( $true
!= ( X201 @ id @ ( lam @ ( X202 @ sh @ one ) ) @ id ) )
& ! [X204: term,X205: subst,X206: subst] :
( ( sub @ ( X202 @ X206 @ X204 ) @ X205 )
= ( X202 @ ( comp @ X206 @ X205 ) @ ( sub @ X204 @ X205 ) ) )
& ! [X203: term] :
( ( $true
= ( X201 @ id @ ( X202 @ id @ X203 ) @ id ) )
| ( $true
!= ( X201 @ id @ X203 @ id ) ) ) )
| ( ( X201 @ id @ ( lam @ ( sub @ X200 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( termmset_gthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ? [X35: term] :
( ( sub @ X35 @ id )
!= X35 )
| ! [X36: subst > term > subst > $o,X37: term,X38: term > $o] :
( ? [X39: term,X40: subst,X41: subst,X42: subst] :
( ( ( X36 @ X42 @ X39 @ ( comp @ X41 @ X40 ) )
!= $true )
& ( $true
= ( X36 @ ( comp @ X42 @ X41 ) @ ( sub @ X39 @ X41 ) @ X40 ) ) )
| ? [X49: term] :
( ( $true
= ( X38 @ X49 ) )
& ( $true
!= ( X38 @ ( sub @ X37 @ ( push @ X49 @ id ) ) ) ) )
| ? [X44: subst > term > term] :
( ( $true
!= ( X36 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id ) )
& ! [X45: term,X46: subst,X47: subst] :
( ( sub @ ( X44 @ X47 @ X45 ) @ X46 )
= ( X44 @ ( comp @ X47 @ X46 ) @ ( sub @ X45 @ X46 ) ) )
& ! [X48: term] :
( ( $true
!= ( X36 @ id @ X48 @ id ) )
| ( $true
= ( X36 @ id @ ( X44 @ id @ X48 ) @ id ) ) ) )
| ( ( X38 @ ( lam @ X37 ) )
= $true )
| ? [X43: term] :
( ( X36 @ id @ X43 @ id )
!= ( X38 @ X43 ) )
| ? [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
!= ( X36 @ ( comp @ X53 @ X50 ) @ ( sub @ X52 @ X50 ) @ X51 ) )
& ( ( X36 @ X53 @ X52 @ ( comp @ X50 @ X51 ) )
= $true ) ) ) ) )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ( axvarshift = $true )
& ( hoasapnotvar_lthm = $true )
& ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
<=> ( laminj = $true ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X142: subst > term > subst > $o,X143: term] :
( ? [X153: term] :
( ( $true
= ( X142 @ id @ X153 @ id ) )
& ( $true
!= ( X142 @ id @ ( sub @ X143 @ ( push @ X153 @ id ) ) @ id ) ) )
| ? [X144: term,X145: subst > term > term] :
( ( ( X142 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id )
!= $true )
& ! [X149: term] :
( ( $true
= ( X142 @ id @ ( X145 @ id @ X149 ) @ id ) )
| ( $true
!= ( X142 @ id @ X149 @ id ) ) )
& ! [X146: subst,X147: term,X148: subst] :
( ( X145 @ ( comp @ X146 @ X148 ) @ ( sub @ X147 @ X148 ) )
= ( sub @ X144 @ ( push @ ( sub @ X147 @ X148 ) @ ( comp @ X146 @ X148 ) ) ) )
& ! [X150: term,X151: subst,X152: subst] :
( ( sub @ ( X145 @ X152 @ X150 ) @ X151 )
= ( sub @ ( sub @ X144 @ ( push @ X150 @ X152 ) ) @ X151 ) ) )
| ( ( X142 @ id @ ( lam @ ( sub @ X143 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ! [X312: d_subst,X313: d_subst] :
( ( X312 = X313 )
| ( ( d2subst @ X312 )
!= ( d2subst @ X313 ) ) )
& ( ( ? [X339: subst > term > subst > $o] :
! [X340: term > $o] :
? [X341: term] :
( ( X339 @ id @ X341 @ id )
!= ( X340 @ X341 ) )
| ! [X360: subst > term > subst > $o,X361: term] :
( ? [X374: subst,X375: subst,X376: subst,X377: term] :
( ( $true
!= ( X360 @ ( comp @ X374 @ X375 ) @ ( sub @ X377 @ X375 ) @ X376 ) )
& ( $true
= ( X360 @ X374 @ X377 @ ( comp @ X375 @ X376 ) ) ) )
| ? [X373: term] :
( ( $true
= ( var @ ( sub @ X373 @ id ) ) )
& ( $true
!= ( X360 @ id @ X373 @ id ) ) )
| ? [X366: subst > term > term] :
( ( $true
!= ( X360 @ id @ ( lam @ ( X366 @ sh @ one ) ) @ id ) )
& ! [X370: term] :
( ( $true
= ( X360 @ id @ ( X366 @ id @ X370 ) @ id ) )
| ( $true
!= ( X360 @ id @ X370 @ id ) ) )
& ! [X367: subst,X368: term,X369: subst] :
( ( sub @ ( X366 @ X369 @ X368 ) @ X367 )
= ( X366 @ ( comp @ X369 @ X367 ) @ ( sub @ X368 @ X367 ) ) ) )
| ( $true
= ( X360 @ id @ X361 @ id ) )
| ? [X362: subst,X363: subst,X364: subst,X365: term] :
( ( $true
= ( X360 @ ( comp @ X362 @ X363 ) @ ( sub @ X365 @ X363 ) @ X364 ) )
& ( $true
!= ( X360 @ X362 @ X365 @ ( comp @ X363 @ X364 ) ) ) )
| ? [X371: term,X372: term] :
( ( $true
= ( X360 @ id @ X371 @ id ) )
& ( $true
= ( X360 @ id @ X372 @ id ) )
& ( $true
!= ( X360 @ id @ ( ap @ ( sub @ X372 @ id ) @ X371 ) @ id ) ) ) )
| ? [X342: term > $o,X343: term] :
( ! [X348: term] :
( ( ( X342 @ X348 )
= $true )
| ( $true
!= ( var @ X348 ) ) )
& ( ( X342 @ X343 )
!= $true )
& ! [X344: term,X345: term] :
( ( ( X342 @ X344 )
!= $true )
| ( $true
= ( X342 @ ( ap @ X345 @ X344 ) ) )
| ( $true
!= ( X342 @ X345 ) ) )
& ! [X346: term] :
( ( $true
= ( X342 @ ( lam @ X346 ) ) )
| ? [X347: term] :
( ( $true
!= ( X342 @ ( sub @ X346 @ ( push @ X347 @ id ) ) ) )
& ( $true
= ( X342 @ X347 ) ) ) ) )
| ? [X349: term] :
( ( sub @ X349 @ id )
!= X349 )
| ? [X350: subst > term > subst > $o,X351: term,X352: term > $o] :
( ! [X355: subst > term > term] :
( ( $true
= ( X350 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id ) )
| ? [X356: term] :
( ( $true
= ( X350 @ id @ X356 @ id ) )
& ( $true
!= ( X350 @ id @ ( X355 @ id @ X356 ) @ id ) ) )
| ? [X357: subst,X358: term,X359: subst] :
( ( X355 @ ( comp @ X357 @ X359 ) @ ( sub @ X358 @ X359 ) )
!= ( sub @ ( X355 @ X357 @ X358 ) @ X359 ) ) )
& ! [X354: term] :
( ( $true
= ( X352 @ ( sub @ X351 @ ( push @ X354 @ id ) ) ) )
| ( $true
!= ( X352 @ X354 ) ) )
& ( $true
!= ( X352 @ ( lam @ X351 ) ) )
& ! [X353: term] :
( ( X352 @ X353 )
= ( X350 @ id @ X353 @ id ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ( ( X30 @ X29 )
!= $true )
| ( $true
= ( X28 @ one ) )
| ? [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
!= ( X28 @ X32 ) ) )
| ? [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
!= X27 ) ) )
& ( hoaslamnotvar = $true )
& ( ! [X294: subst > term > subst > $o,X295: term] :
( ? [X302: subst > term > term] :
( ! [X303: term] :
( ( $true
= ( X294 @ id @ ( X302 @ id @ X303 ) @ id ) )
| ( $true
!= ( X294 @ id @ X303 @ id ) ) )
& ! [X304: subst,X305: subst,X306: term] :
( ( X302 @ ( comp @ X304 @ X305 ) @ ( sub @ X306 @ X305 ) )
= ( sub @ ( X302 @ X304 @ X306 ) @ X305 ) )
& ( $true
!= ( X294 @ id @ ( lam @ ( X302 @ sh @ one ) ) @ id ) ) )
| ? [X307: subst,X308: term,X309: subst,X310: subst] :
( ( $true
!= ( X294 @ X310 @ X308 @ ( comp @ X309 @ X307 ) ) )
& ( $true
= ( X294 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) @ X307 ) ) )
| ? [X298: term,X299: subst,X300: subst,X301: subst] :
( ( $true
!= ( X294 @ ( comp @ X299 @ X301 ) @ ( sub @ X298 @ X301 ) @ X300 ) )
& ( $true
= ( X294 @ X299 @ X298 @ ( comp @ X301 @ X300 ) ) ) )
| ? [X296: term,X297: term] :
( ( ( X294 @ id @ X296 @ id )
= $true )
& ( $true
!= ( X294 @ id @ ( ap @ ( sub @ X297 @ id ) @ X296 ) @ id ) )
& ( ( X294 @ id @ X297 @ id )
= $true ) )
| ? [X311: term] :
( ( ( var @ ( sub @ X311 @ id ) )
= $true )
& ( $true
!= ( X294 @ id @ X311 @ id ) ) )
| ( $true
= ( X294 @ id @ X295 @ id ) ) )
<=> ( hoasinduction = $true ) )
& ( ( ? [X224: term] :
( ( sub @ X224 @ id )
!= X224 )
| ! [X233: term,X234: term > $o] :
( ? [X237: term,X238: term] :
( ( $true
= ( X234 @ X238 ) )
& ( $true
= ( X234 @ X237 ) )
& ( $true
!= ( X234 @ ( ap @ X238 @ X237 ) ) ) )
| ? [X235: term] :
( ( $true
!= ( X234 @ ( lam @ X235 ) ) )
& ! [X236: term] :
( ( $true
!= ( X234 @ X236 ) )
| ( $true
= ( X234 @ ( sub @ X235 @ ( push @ X236 @ id ) ) ) ) ) )
| ( $true
= ( X234 @ X233 ) )
| ? [X239: term] :
( ( $true
!= ( X234 @ X239 ) )
& ( $true
= ( var @ X239 ) ) ) )
| ? [X225: term,X226: term > $o,X227: subst] :
( ! [X228: term] :
( ( $true
!= ( var @ X228 ) )
| ( $true
= ( X226 @ ( sub @ X228 @ X227 ) ) ) )
& ( $true
!= ( X226 @ ( sub @ X225 @ X227 ) ) )
& ! [X229: term] :
( ? [X230: term] :
( ( $true
!= ( X226 @ ( sub @ X229 @ ( push @ X230 @ id ) ) ) )
& ( $true
= ( X226 @ X230 ) ) )
| ( $true
= ( X226 @ ( lam @ X229 ) ) ) )
& ! [X231: term,X232: term] :
( ( $true
!= ( X226 @ X232 ) )
| ( ( X226 @ X231 )
!= $true )
| ( $true
= ( X226 @ ( ap @ X232 @ X231 ) ) ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( ( substmonoid_lthm = $true )
<=> ( ( ! [X523: subst] :
( ( comp @ id @ X523 )
= X523 )
& ! [X524: subst] :
( ( comp @ X524 @ id )
= X524 ) )
| ? [X522: subst] :
( ( comp @ X522 @ id )
!= X522 )
| ? [X521: subst] :
( ( comp @ id @ X521 )
!= X521 ) ) )
& ( ( axshiftcons = $true )
<=> ! [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
= X198 ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X263: term] :
? [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
= ( X264 @ sh @ one ) ) )
& ( ( ! [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
!= ( X336 @ X332 ) )
| ( $true
= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) ) )
| ? [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
!= X329 )
| ? [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
!= X330 ) )
<=> ( pushprop_lem1_lthm = $true ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ulamvarsh = $true )
& ( ( induction2lem_lthm = $true )
<=> ( ? [X7: term > $o,X8: term] :
( ( ( X7 @ X8 )
!= $true )
& ! [X9: term] :
( ( $true
!= ( X7 @ X9 ) )
| ( ( X7 @ ( lam @ X9 ) )
= $true ) )
& ! [X10: term] :
( ( $true
= ( X7 @ X10 ) )
| ( ( var @ X10 )
!= $true ) )
& ! [X11: term,X12: term] :
( ( $true
!= ( X7 @ X11 ) )
| ( $true
!= ( X7 @ X12 ) )
| ( $true
= ( X7 @ ( ap @ X11 @ X12 ) ) ) ) )
| ? [X2: subst,X3: term] :
( ( sub @ one @ ( push @ X3 @ X2 ) )
!= X3 )
| ? [X4: term,X5: subst] :
( ( comp @ sh @ ( push @ X4 @ X5 ) )
!= X5 )
| ? [X6: subst] :
( ( comp @ X6 @ id )
!= X6 )
| ! [X13: term > $o,X14: term,X15: subst] :
( ? [X16: term,X17: term] :
( ( $true
!= ( X13 @ ( ap @ X17 @ X16 ) ) )
& ( $true
= ( X13 @ X16 ) )
& ( $true
= ( X13 @ X17 ) ) )
| ( $true
= ( X13 @ ( sub @ X14 @ X15 ) ) )
| ? [X18: term] :
( ( $true
!= ( X13 @ ( lam @ X18 ) ) )
& ! [X19: term] :
( ( $true
= ( X13 @ ( sub @ X18 @ ( push @ X19 @ id ) ) ) )
| ( ( X13 @ X19 )
!= $true ) ) )
| ? [X20: term] :
( ( $true
= ( var @ X20 ) )
& ( $true
!= ( X13 @ ( sub @ X20 @ X15 ) ) ) ) ) ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X123: term > $o,X124: term,X125: subst > term > subst > $o,X126: term] :
( ( ( X123 @ ( ap @ X126 @ X124 ) )
= $true )
| ? [X129: subst,X130: subst,X131: term,X132: subst] :
( ( $true
!= ( X125 @ X129 @ X131 @ ( comp @ X132 @ X130 ) ) )
& ( $true
= ( X125 @ ( comp @ X129 @ X132 ) @ ( sub @ X131 @ X132 ) @ X130 ) ) )
| ? [X127: term,X128: term] :
( ( ( X125 @ id @ ( ap @ ( sub @ X127 @ id ) @ X128 ) @ id )
!= $true )
& ( $true
= ( X125 @ id @ X128 @ id ) )
& ( $true
= ( X125 @ id @ X127 @ id ) ) )
| ? [X133: subst,X134: subst,X135: term,X136: subst] :
( ( $true
!= ( X125 @ ( comp @ X136 @ X134 ) @ ( sub @ X135 @ X134 ) @ X133 ) )
& ( ( X125 @ X136 @ X135 @ ( comp @ X134 @ X133 ) )
= $true ) )
| ( $true
!= ( X123 @ X126 ) )
| ( $true
!= ( X123 @ X124 ) )
| ? [X137: term] :
( ( X125 @ id @ X137 @ id )
!= ( X123 @ X137 ) ) ) )
& ( ulamvarind = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) )
<=> ( apinj1 = $true ) )
& ( pushprop_lthm_orig = $true )
& ( ! [X519: term] :
? [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
= ( X520 @ sh @ one ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( ( axidl = $true )
<=> ! [X223: subst] :
( ( comp @ id @ X223 )
= X223 ) )
& ( hoaslaminj_gthm = $true )
& ( ( apinj2 = $true )
<=> ! [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
!= ( ap @ X425 @ X426 ) )
| ( X426 = X428 ) ) )
& ( 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 ) ) ) ) ) ) )
& ( ! [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
= X384 )
<=> ( axscons = $true ) )
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& ( ( axvarcons = $true )
<=> ! [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
= X245 ) )
& ( hoaslamnotvar_gthm = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ? [X190: term,X191: term,X192: term,X193: term] :
( ( ( ap @ X191 @ X190 )
= ( ap @ X193 @ X192 ) )
& ( X190 != X192 ) )
| ! [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
!= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
| ( X194 = X195 ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X314: term] :
? [X315: subst > term > term] :
! [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
= ( X315 @ X316 @ X317 ) ) )
& ( hoasinduction_lem2_lthm = $true )
& ( pushprop_gthm = $true )
& ( ( hoasinduction_lem3v2a_lthm = $true )
<=> ( ! [X530: term,X531: term > $o,X532: subst > term > subst > $o] :
( ? [X534: subst > term > term] :
( ! [X535: term] :
( ( ( X532 @ id @ ( X534 @ id @ X535 ) @ id )
= $true )
| ( $true
!= ( X532 @ id @ X535 @ id ) ) )
& ! [X536: subst,X537: term,X538: subst] :
( ( X534 @ ( comp @ X538 @ X536 ) @ ( sub @ X537 @ X536 ) )
= ( sub @ ( X534 @ X538 @ X537 ) @ X536 ) )
& ( $true
!= ( X532 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id ) ) )
| ? [X533: term] :
( ( X532 @ id @ X533 @ id )
!= ( X531 @ X533 ) )
| ? [X539: term] :
( ( $true
!= ( X531 @ ( sub @ X530 @ ( push @ X539 @ id ) ) ) )
& ( $true
= ( X531 @ X539 ) ) )
| ( $true
= ( X531 @ ( lam @ X530 ) ) ) )
| ? [X529: term] :
( ( sub @ X529 @ id )
!= X529 )
| ? [X525: term] :
! [X526: subst > term > term] :
? [X527: subst,X528: term] :
( ( X526 @ X527 @ X528 )
!= ( sub @ X525 @ ( push @ X528 @ X527 ) ) ) ) )
& ( hoasapinj1_gthm = $true )
& ( hoaslamnotap_gthm = $true )
& ( hoasinduction_lem3b_gthm = $true )
& ( ! [X210: subst > term > subst > $o,X211: term,X212: term] :
( ? [X217: subst,X218: subst,X219: subst,X220: term] :
( ( ( X210 @ ( comp @ X219 @ X218 ) @ ( sub @ X220 @ X218 ) @ X217 )
= $true )
& ( $true
!= ( X210 @ X219 @ X220 @ ( comp @ X218 @ X217 ) ) ) )
| ? [X213: term,X214: subst,X215: subst,X216: subst] :
( ( $true
= ( X210 @ X215 @ X213 @ ( comp @ X216 @ X214 ) ) )
& ( ( X210 @ ( comp @ X215 @ X216 ) @ ( sub @ X213 @ X216 ) @ X214 )
!= $true ) )
| ( ( X210 @ id @ ( ap @ X212 @ X211 ) @ id )
= $true )
| ( $true
!= ( X210 @ id @ X212 @ id ) )
| ( $true
!= ( X210 @ id @ X211 @ id ) )
| ? [X221: term,X222: term] :
( ( $true
= ( X210 @ id @ X221 @ id ) )
& ( $true
!= ( X210 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id ) )
& ( $true
= ( X210 @ id @ X222 @ id ) ) ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( pushprop_lem0_gthm = $true )
& ( pushprop_lem1v2_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( axmap = $true )
& ( induction2lem_gthm = $true )
& ( axabs = $true )
& ( substmonoid_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ( d2subst @ d_id )
= id ) ),
inference(flattening,[status(thm)],[f12]) ).
thf(f12,plain,
( ( axmap = $true )
& ( ( ! [X13: term > $o,X14: term,X15: subst] :
( ? [X16: term,X17: term] :
( ( $true
!= ( X13 @ ( ap @ X17 @ X16 ) ) )
& ( $true
= ( X13 @ X16 ) )
& ( $true
= ( X13 @ X17 ) ) )
| ( $true
= ( X13 @ ( sub @ X14 @ X15 ) ) )
| ? [X18: term] :
( ( $true
!= ( X13 @ ( lam @ X18 ) ) )
& ! [X19: term] :
( ( $true
= ( X13 @ ( sub @ X18 @ ( push @ X19 @ id ) ) ) )
| ( ( X13 @ X19 )
!= $true ) ) )
| ? [X20: term] :
( ( $true
= ( var @ X20 ) )
& ( $true
!= ( X13 @ ( sub @ X20 @ X15 ) ) ) ) )
| ? [X7: term > $o,X8: term] :
( ( ( X7 @ X8 )
!= $true )
& ! [X9: term] :
( ( $true
!= ( X7 @ X9 ) )
| ( ( X7 @ ( lam @ X9 ) )
= $true ) )
& ! [X10: term] :
( ( $true
= ( X7 @ X10 ) )
| ( ( var @ X10 )
!= $true ) )
& ! [X11: term,X12: term] :
( ( $true
!= ( X7 @ X11 ) )
| ( $true
!= ( X7 @ X12 ) )
| ( $true
= ( X7 @ ( ap @ X11 @ X12 ) ) ) ) )
| ? [X6: subst] :
( ( comp @ X6 @ id )
!= X6 )
| ? [X4: term,X5: subst] :
( ( comp @ sh @ ( push @ X4 @ X5 ) )
!= X5 )
| ? [X2: subst,X3: term] :
( ( sub @ one @ ( push @ X3 @ X2 ) )
!= X3 ) )
<=> ( induction2lem_lthm = $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ulamvarind = $true )
& ( ( axvarcons = $true )
<=> ! [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
= X245 ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X72: subst > term > subst > $o,X73: term] :
( ( $true
= ( X72 @ id @ ( lam @ X73 ) @ id ) )
| ? [X83: subst > term > term] :
( ! [X84: subst,X85: term,X86: subst] :
( ( X83 @ ( comp @ X86 @ X84 ) @ ( sub @ X85 @ X84 ) )
= ( sub @ ( X83 @ X86 @ X85 ) @ X84 ) )
& ! [X87: term] :
( ( ( X72 @ id @ ( X83 @ id @ X87 ) @ id )
= $true )
| ( $true
!= ( X72 @ id @ X87 @ id ) ) )
& ( $true
!= ( X72 @ id @ ( lam @ ( X83 @ sh @ one ) ) @ id ) ) )
| ? [X74: term] :
( ( ( X72 @ id @ X74 @ id )
= $true )
& ( $true
!= ( X72 @ id @ ( sub @ X73 @ ( push @ X74 @ id ) ) @ id ) ) )
| ? [X75: subst,X76: term,X77: subst,X78: subst] :
( ( ( X72 @ ( comp @ X78 @ X77 ) @ ( sub @ X76 @ X77 ) @ X75 )
!= $true )
& ( $true
= ( X72 @ X78 @ X76 @ ( comp @ X77 @ X75 ) ) ) )
| ? [X79: term,X80: subst,X81: subst,X82: subst] :
( ( ( X72 @ ( comp @ X80 @ X82 ) @ ( sub @ X79 @ X82 ) @ X81 )
= $true )
& ( $true
!= ( X72 @ X80 @ X79 @ ( comp @ X82 @ X81 ) ) ) ) )
| ? [X64: term,X65: subst > term > subst > $o] :
( ( $true
!= ( X65 @ id @ ( lam @ ( sub @ X64 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X66: term] :
( ( $true
!= ( X65 @ id @ X66 @ id ) )
| ( $true
= ( X65 @ id @ ( sub @ X64 @ ( push @ X66 @ id ) ) @ id ) ) )
& ! [X67: subst > term > term] :
( ? [X71: term] :
( ( $true
= ( X65 @ id @ X71 @ id ) )
& ( $true
!= ( X65 @ id @ ( X67 @ id @ X71 ) @ id ) ) )
| ? [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X67 @ X69 @ X70 ) @ X68 )
!= ( X67 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
| ( $true
= ( X65 @ id @ ( lam @ ( X67 @ sh @ one ) ) @ id ) ) ) )
| ? [X63: term] :
( ( sub @ X63 @ id )
!= X63 ) ) )
& ( hoasapinj2_gthm = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X314: term] :
? [X315: subst > term > term] :
! [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
= ( X315 @ X316 @ X317 ) ) )
& ( pushprop_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X105: term,X106: subst > term > term,X107: subst > term > term,X108: subst] :
( ? [X109: subst,X110: subst,X111: term] :
( ( sub @ ( X107 @ X109 @ X111 ) @ X110 )
!= ( X107 @ ( comp @ X109 @ X110 ) @ ( sub @ X111 @ X110 ) ) )
| ( ( lam @ ( X106 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ? [X112: subst,X113: term,X114: subst] :
( ( X106 @ ( comp @ X114 @ X112 ) @ ( sub @ X113 @ X112 ) )
!= ( sub @ ( X106 @ X114 @ X113 ) @ X112 ) )
| ( ( X106 @ X108 @ X105 )
= ( X107 @ X108 @ X105 ) ) )
| ? [X103: term,X104: term] :
( ( ( lam @ X103 )
= ( lam @ X104 ) )
& ( X103 != X104 ) )
| ? [X101: term,X102: subst] :
( ( comp @ sh @ ( push @ X101 @ X102 ) )
!= X102 )
| ? [X99: term,X100: subst] :
( ( sub @ one @ ( push @ X99 @ X100 ) )
!= X99 ) ) )
& ( ( d2subst @ d_id )
= id )
& ( ! [X318: subst > term > subst > $o,X319: term,X320: term > $o] :
( ? [X321: subst > term > term] :
( ! [X325: term] :
( ( $true
= ( X318 @ id @ ( X321 @ id @ X325 ) @ id ) )
| ( ( X318 @ id @ X325 @ id )
!= $true ) )
& ( ( X318 @ id @ ( lam @ ( X321 @ sh @ one ) ) @ id )
!= $true )
& ! [X322: subst,X323: term,X324: subst] :
( ( sub @ ( X321 @ X322 @ X323 ) @ X324 )
= ( X321 @ ( comp @ X322 @ X324 ) @ ( sub @ X323 @ X324 ) ) ) )
| ? [X326: term] :
( ( X318 @ id @ X326 @ id )
!= ( X320 @ X326 ) )
| ? [X327: term] :
( ( $true
!= ( X320 @ ( sub @ X319 @ ( push @ X327 @ id ) ) ) )
& ( $true
= ( X320 @ X327 ) ) )
| ( $true
= ( X320 @ ( lam @ X319 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X142: subst > term > subst > $o,X143: term] :
( ? [X153: term] :
( ( $true
= ( X142 @ id @ X153 @ id ) )
& ( $true
!= ( X142 @ id @ ( sub @ X143 @ ( push @ X153 @ id ) ) @ id ) ) )
| ? [X144: term,X145: subst > term > term] :
( ( ( X142 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id )
!= $true )
& ! [X149: term] :
( ( $true
= ( X142 @ id @ ( X145 @ id @ X149 ) @ id ) )
| ( $true
!= ( X142 @ id @ X149 @ id ) ) )
& ! [X146: subst,X147: term,X148: subst] :
( ( X145 @ ( comp @ X146 @ X148 ) @ ( sub @ X147 @ X148 ) )
= ( sub @ X144 @ ( push @ ( sub @ X147 @ X148 ) @ ( comp @ X146 @ X148 ) ) ) )
& ! [X150: term,X151: subst,X152: subst] :
( ( sub @ ( X145 @ X152 @ X150 ) @ X151 )
= ( sub @ ( sub @ X144 @ ( push @ X150 @ X152 ) ) @ X151 ) ) )
| ( ( X142 @ id @ ( lam @ ( sub @ X143 @ ( push @ one @ sh ) ) ) @ id )
= $true ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X123: term > $o,X124: term,X125: subst > term > subst > $o,X126: term] :
( ( ( X123 @ ( ap @ X126 @ X124 ) )
= $true )
| ? [X129: subst,X130: subst,X131: term,X132: subst] :
( ( $true
!= ( X125 @ X129 @ X131 @ ( comp @ X132 @ X130 ) ) )
& ( $true
= ( X125 @ ( comp @ X129 @ X132 ) @ ( sub @ X131 @ X132 ) @ X130 ) ) )
| ? [X127: term,X128: term] :
( ( ( X125 @ id @ ( ap @ ( sub @ X127 @ id ) @ X128 ) @ id )
!= $true )
& ( $true
= ( X125 @ id @ X128 @ id ) )
& ( $true
= ( X125 @ id @ X127 @ id ) ) )
| ? [X133: subst,X134: subst,X135: term,X136: subst] :
( ( $true
!= ( X125 @ ( comp @ X136 @ X134 ) @ ( sub @ X135 @ X134 ) @ X133 ) )
& ( ( X125 @ X136 @ X135 @ ( comp @ X134 @ X133 ) )
= $true ) )
| ( $true
!= ( X123 @ X126 ) )
| ( $true
!= ( X123 @ X124 ) )
| ? [X137: term] :
( ( X125 @ id @ X137 @ id )
!= ( X123 @ X137 ) ) ) )
& ( pushprop_lthm = $true )
& ( ( axidr = $true )
<=> ! [X393: subst] :
( ( comp @ X393 @ id )
= X393 ) )
& ( ( axvarid = $true )
<=> ! [X265: term] :
( ( sub @ X265 @ id )
= X265 ) )
& ( ( ! [X530: term,X531: term > $o,X532: subst > term > subst > $o] :
( ? [X534: subst > term > term] :
( ! [X535: term] :
( ( ( X532 @ id @ ( X534 @ id @ X535 ) @ id )
= $true )
| ( $true
!= ( X532 @ id @ X535 @ id ) ) )
& ! [X536: subst,X537: term,X538: subst] :
( ( X534 @ ( comp @ X538 @ X536 ) @ ( sub @ X537 @ X536 ) )
= ( sub @ ( X534 @ X538 @ X537 ) @ X536 ) )
& ( $true
!= ( X532 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id ) ) )
| ? [X533: term] :
( ( X532 @ id @ X533 @ id )
!= ( X531 @ X533 ) )
| ? [X539: term] :
( ( $true
!= ( X531 @ ( sub @ X530 @ ( push @ X539 @ id ) ) ) )
& ( $true
= ( X531 @ X539 ) ) )
| ( $true
= ( X531 @ ( lam @ X530 ) ) ) )
| ? [X529: term] :
( ( sub @ X529 @ id )
!= X529 )
| ? [X525: term] :
! [X526: subst > term > term] :
? [X527: subst,X528: term] :
( ( X526 @ X527 @ X528 )
!= ( sub @ X525 @ ( push @ X528 @ X527 ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( termmset_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( ! [X360: subst > term > subst > $o,X361: term] :
( ? [X374: subst,X375: subst,X376: subst,X377: term] :
( ( $true
!= ( X360 @ ( comp @ X374 @ X375 ) @ ( sub @ X377 @ X375 ) @ X376 ) )
& ( $true
= ( X360 @ X374 @ X377 @ ( comp @ X375 @ X376 ) ) ) )
| ? [X373: term] :
( ( $true
= ( var @ ( sub @ X373 @ id ) ) )
& ( $true
!= ( X360 @ id @ X373 @ id ) ) )
| ? [X366: subst > term > term] :
( ( $true
!= ( X360 @ id @ ( lam @ ( X366 @ sh @ one ) ) @ id ) )
& ! [X370: term] :
( ( $true
= ( X360 @ id @ ( X366 @ id @ X370 ) @ id ) )
| ( $true
!= ( X360 @ id @ X370 @ id ) ) )
& ! [X367: subst,X368: term,X369: subst] :
( ( sub @ ( X366 @ X369 @ X368 ) @ X367 )
= ( X366 @ ( comp @ X369 @ X367 ) @ ( sub @ X368 @ X367 ) ) ) )
| ( $true
= ( X360 @ id @ X361 @ id ) )
| ? [X362: subst,X363: subst,X364: subst,X365: term] :
( ( $true
= ( X360 @ ( comp @ X362 @ X363 ) @ ( sub @ X365 @ X363 ) @ X364 ) )
& ( $true
!= ( X360 @ X362 @ X365 @ ( comp @ X363 @ X364 ) ) ) )
| ? [X371: term,X372: term] :
( ( $true
= ( X360 @ id @ X371 @ id ) )
& ( $true
= ( X360 @ id @ X372 @ id ) )
& ( $true
!= ( X360 @ id @ ( ap @ ( sub @ X372 @ id ) @ X371 ) @ id ) ) ) )
| ? [X350: subst > term > subst > $o,X351: term,X352: term > $o] :
( ! [X355: subst > term > term] :
( ( $true
= ( X350 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id ) )
| ? [X356: term] :
( ( $true
= ( X350 @ id @ X356 @ id ) )
& ( $true
!= ( X350 @ id @ ( X355 @ id @ X356 ) @ id ) ) )
| ? [X357: subst,X358: term,X359: subst] :
( ( X355 @ ( comp @ X357 @ X359 ) @ ( sub @ X358 @ X359 ) )
!= ( sub @ ( X355 @ X357 @ X358 ) @ X359 ) ) )
& ! [X354: term] :
( ( $true
= ( X352 @ ( sub @ X351 @ ( push @ X354 @ id ) ) ) )
| ( $true
!= ( X352 @ X354 ) ) )
& ( $true
!= ( X352 @ ( lam @ X351 ) ) )
& ! [X353: term] :
( ( X352 @ X353 )
= ( X350 @ id @ X353 @ id ) ) )
| ? [X349: term] :
( ( sub @ X349 @ id )
!= X349 )
| ? [X342: term > $o,X343: term] :
( ! [X348: term] :
( ( ( X342 @ X348 )
= $true )
| ( $true
!= ( var @ X348 ) ) )
& ( ( X342 @ X343 )
!= $true )
& ! [X344: term,X345: term] :
( ( ( X342 @ X344 )
!= $true )
| ( $true
= ( X342 @ ( ap @ X345 @ X344 ) ) )
| ( $true
!= ( X342 @ X345 ) ) )
& ! [X346: term] :
( ( $true
= ( X342 @ ( lam @ X346 ) ) )
| ? [X347: term] :
( ( $true
!= ( X342 @ ( sub @ X346 @ ( push @ X347 @ id ) ) ) )
& ( $true
= ( X342 @ X347 ) ) ) ) )
| ? [X339: subst > term > subst > $o] :
! [X340: term > $o] :
? [X341: term] :
( ( X339 @ id @ X341 @ id )
!= ( X340 @ X341 ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& ( ( ? [X208: term] :
( ( sub @ X208 @ id )
!= X208 )
| ! [X209: term] :
( ( sub @ X209 @ id )
= X209 ) )
<=> ( termmset_lthm = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( hoasinduction_lem1_gthm = $true )
& ! [X312: d_subst,X313: d_subst] :
( ( X312 = X313 )
| ( ( d2subst @ X312 )
!= ( d2subst @ X313 ) ) )
& ( ! [X429: term,X430: term > $o] :
( ? [X432: term] :
( ( $true
= ( X430 @ X432 ) )
& ( $true
!= ( X430 @ ( lam @ X432 ) ) ) )
| ( $true
= ( X430 @ X429 ) )
| ? [X433: term,X434: term] :
( ( $true
= ( X430 @ X433 ) )
& ( ( X430 @ ( ap @ X433 @ X434 ) )
!= $true )
& ( $true
= ( X430 @ X434 ) ) )
| ? [X431: term] :
( ( $true
!= ( X430 @ X431 ) )
& ( $true
= ( var @ X431 ) ) ) )
<=> ( induction = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ( hoasinduction_lem3v2_f = $true )
<=> ! [X95: term] :
? [X96: subst > term > term] :
! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) ) )
& ( hoaslamnotap_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_lthm = $true )
& ! [X156: d_subst] : ( d_id = X156 )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X407: term,X408: subst > term > subst > $o,X409: term > $o] :
( ? [X416: subst,X417: subst,X418: term,X419: subst] :
( ( $true
= ( X408 @ ( comp @ X419 @ X417 ) @ ( sub @ X418 @ X417 ) @ X416 ) )
& ( $true
!= ( X408 @ X419 @ X418 @ ( comp @ X417 @ X416 ) ) ) )
| ? [X410: subst > term > term] :
( ( $true
!= ( X408 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
& ! [X411: term,X412: subst,X413: subst] :
( ( sub @ ( X410 @ X412 @ X411 ) @ X413 )
= ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) ) )
& ! [X414: term] :
( ( $true
= ( X408 @ id @ ( X410 @ id @ X414 ) @ id ) )
| ( $true
!= ( X408 @ id @ X414 @ id ) ) ) )
| ( ( X409 @ ( lam @ X407 ) )
= $true )
| ? [X420: term] :
( ( X409 @ X420 )
!= ( X408 @ id @ X420 @ id ) )
| ? [X421: subst,X422: term,X423: subst,X424: subst] :
( ( $true
!= ( X408 @ ( comp @ X424 @ X421 ) @ ( sub @ X422 @ X421 ) @ X423 ) )
& ( $true
= ( X408 @ X424 @ X422 @ ( comp @ X421 @ X423 ) ) ) )
| ? [X415: term] :
( ( $true
!= ( X409 @ ( sub @ X407 @ ( push @ X415 @ id ) ) ) )
& ( $true
= ( X409 @ X415 ) ) ) ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ? [X58: subst > term > term] :
( ! [X59: term] :
( ( $true
= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
| ( $true
!= ( X54 @ id @ X59 @ id ) ) )
& ( $true
!= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
& ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) )
| ? [X56: term,X57: term] :
( ( $true
!= ( X54 @ id @ ( ap @ ( sub @ X56 @ id ) @ X57 ) @ id ) )
& ( ( X54 @ id @ X56 @ id )
= $true )
& ( $true
= ( X54 @ id @ X57 @ id ) ) )
| ( $true
= ( X54 @ id @ X55 @ id ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
<=> ( laminj = $true ) )
& ( hoasapnotvar = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( substmonoid_lthm = $true )
<=> ( ( ! [X523: subst] :
( ( comp @ id @ X523 )
= X523 )
& ! [X524: subst] :
( ( comp @ X524 @ id )
= X524 ) )
| ? [X522: subst] :
( ( comp @ X522 @ id )
!= X522 )
| ? [X521: subst] :
( ( comp @ id @ X521 )
!= X521 ) ) )
& ( ! [X210: subst > term > subst > $o,X211: term,X212: term] :
( ? [X217: subst,X218: subst,X219: subst,X220: term] :
( ( ( X210 @ ( comp @ X219 @ X218 ) @ ( sub @ X220 @ X218 ) @ X217 )
= $true )
& ( $true
!= ( X210 @ X219 @ X220 @ ( comp @ X218 @ X217 ) ) ) )
| ? [X213: term,X214: subst,X215: subst,X216: subst] :
( ( $true
= ( X210 @ X215 @ X213 @ ( comp @ X216 @ X214 ) ) )
& ( ( X210 @ ( comp @ X215 @ X216 ) @ ( sub @ X213 @ X216 ) @ X214 )
!= $true ) )
| ( ( X210 @ id @ ( ap @ X212 @ X211 ) @ id )
= $true )
| ( $true
!= ( X210 @ id @ X212 @ id ) )
| ( $true
!= ( X210 @ id @ X211 @ id ) )
| ? [X221: term,X222: term] :
( ( $true
= ( X210 @ id @ X221 @ id ) )
& ( $true
!= ( X210 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id ) )
& ( $true
= ( X210 @ id @ X222 @ id ) ) ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( induction2_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( $true
!= ( X243 @ X240 ) )
| ? [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
!= ( X242 @ X244 ) )
| ( ( X242 @ one )
= $true ) ) )
& ( ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) )
<=> ( apinj1 = $true ) )
& ( ulamvar1 != $true )
& ( ( ! [X233: term,X234: term > $o] :
( ? [X237: term,X238: term] :
( ( $true
= ( X234 @ X238 ) )
& ( $true
= ( X234 @ X237 ) )
& ( $true
!= ( X234 @ ( ap @ X238 @ X237 ) ) ) )
| ? [X235: term] :
( ( $true
!= ( X234 @ ( lam @ X235 ) ) )
& ! [X236: term] :
( ( $true
!= ( X234 @ X236 ) )
| ( $true
= ( X234 @ ( sub @ X235 @ ( push @ X236 @ id ) ) ) ) ) )
| ( $true
= ( X234 @ X233 ) )
| ? [X239: term] :
( ( $true
!= ( X234 @ X239 ) )
& ( $true
= ( var @ X239 ) ) ) )
| ? [X225: term,X226: term > $o,X227: subst] :
( ! [X228: term] :
( ( $true
!= ( var @ X228 ) )
| ( $true
= ( X226 @ ( sub @ X228 @ X227 ) ) ) )
& ( $true
!= ( X226 @ ( sub @ X225 @ X227 ) ) )
& ! [X229: term] :
( ? [X230: term] :
( ( $true
!= ( X226 @ ( sub @ X229 @ ( push @ X230 @ id ) ) ) )
& ( $true
= ( X226 @ X230 ) ) )
| ( $true
= ( X226 @ ( lam @ X229 ) ) ) )
& ! [X231: term,X232: term] :
( ( $true
!= ( X226 @ X232 ) )
| ( ( X226 @ X231 )
!= $true )
| ( $true
= ( X226 @ ( ap @ X232 @ X231 ) ) ) ) )
| ? [X224: term] :
( ( sub @ X224 @ id )
!= X224 ) )
<=> ( induction2_lthm = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( lamnotvar = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( induction2 = $true )
<=> ! [X88: term > $o,X89: term] :
( ? [X91: term] :
( ( $true
!= ( X88 @ ( lam @ X91 ) ) )
& ! [X92: term] :
( ( $true
!= ( X88 @ X92 ) )
| ( $true
= ( X88 @ ( sub @ X91 @ ( push @ X92 @ id ) ) ) ) ) )
| ( $true
= ( X88 @ X89 ) )
| ? [X90: term] :
( ( ( X88 @ X90 )
!= $true )
& ( $true
= ( var @ X90 ) ) )
| ? [X93: term,X94: term] :
( ( $true
= ( X88 @ X93 ) )
& ( $true
!= ( X88 @ ( ap @ X93 @ X94 ) ) )
& ( $true
= ( X88 @ X94 ) ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( axapp = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X501: term,X502: subst > term > subst > $o] :
( ? [X507: subst > term > term] :
( ! [X508: term] :
( ( $true
= ( X502 @ id @ ( X507 @ id @ X508 ) @ id ) )
| ( ( X502 @ id @ X508 @ id )
!= $true ) )
& ! [X509: term,X510: subst,X511: subst] :
( ( X507 @ ( comp @ X511 @ X510 ) @ ( sub @ X509 @ X510 ) )
= ( sub @ ( X507 @ X511 @ X509 ) @ X510 ) )
& ( $true
!= ( X502 @ id @ ( lam @ ( X507 @ sh @ one ) ) @ id ) ) )
| ? [X518: term] :
( ( ( X502 @ id @ X518 @ id )
!= $true )
& ( $true
= ( var @ ( sub @ X518 @ id ) ) ) )
| ? [X512: term,X513: subst,X514: subst,X515: subst] :
( ( ( X502 @ X515 @ X512 @ ( comp @ X514 @ X513 ) )
= $true )
& ( $true
!= ( X502 @ ( comp @ X515 @ X514 ) @ ( sub @ X512 @ X514 ) @ X513 ) ) )
| ? [X503: subst,X504: subst,X505: term,X506: subst] :
( ( $true
= ( X502 @ ( comp @ X504 @ X506 ) @ ( sub @ X505 @ X506 ) @ X503 ) )
& ( $true
!= ( X502 @ X504 @ X505 @ ( comp @ X506 @ X503 ) ) ) )
| ? [X516: term,X517: term] :
( ( ( X502 @ id @ X517 @ id )
= $true )
& ( ( X502 @ id @ X516 @ id )
= $true )
& ( $true
!= ( X502 @ id @ ( ap @ ( sub @ X516 @ id ) @ X517 ) @ id ) ) )
| ( ( X502 @ id @ X501 @ id )
= $true ) )
| ? [X485: term,X486: subst > term > subst > $o] :
( ! [X491: subst,X492: term,X493: subst,X494: subst] :
( ( ( X486 @ ( comp @ X493 @ X494 ) @ ( sub @ X492 @ X494 ) @ X491 )
= $true )
| ( $true
!= ( X486 @ X493 @ X492 @ ( comp @ X494 @ X491 ) ) ) )
& ( $true
!= ( X486 @ id @ ( lam @ X485 ) @ id ) )
& ! [X487: subst,X488: subst,X489: subst,X490: term] :
( ( $true
!= ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X487 ) )
| ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X487 ) )
= $true ) )
& ! [X495: subst > term > term] :
( ( $true
= ( X486 @ id @ ( lam @ ( X495 @ sh @ one ) ) @ id ) )
| ? [X496: term] :
( ( $true
!= ( X486 @ id @ ( X495 @ id @ X496 ) @ id ) )
& ( $true
= ( X486 @ id @ X496 @ id ) ) )
| ? [X497: term,X498: subst,X499: subst] :
( ( X495 @ ( comp @ X498 @ X499 ) @ ( sub @ X497 @ X499 ) )
!= ( sub @ ( X495 @ X498 @ X497 ) @ X499 ) ) )
& ! [X500: term] :
( ( ( X486 @ id @ ( sub @ X485 @ ( push @ X500 @ id ) ) @ id )
= $true )
| ( $true
!= ( X486 @ id @ X500 @ id ) ) ) )
| ? [X472: subst > term > subst > $o,X473: term,X474: term] :
( ( $true
= ( X472 @ id @ X473 @ id ) )
& ! [X483: term,X484: term] :
( ( $true
= ( X472 @ id @ ( ap @ ( sub @ X483 @ id ) @ X484 ) @ id ) )
| ( $true
!= ( X472 @ id @ X483 @ id ) )
| ( ( X472 @ id @ X484 @ id )
!= $true ) )
& ! [X479: term,X480: subst,X481: subst,X482: subst] :
( ( $true
!= ( X472 @ ( comp @ X480 @ X481 ) @ ( sub @ X479 @ X481 ) @ X482 ) )
| ( $true
= ( X472 @ X480 @ X479 @ ( comp @ X481 @ X482 ) ) ) )
& ( $true
!= ( X472 @ id @ ( ap @ X473 @ X474 ) @ id ) )
& ! [X475: subst,X476: term,X477: subst,X478: subst] :
( ( $true
= ( X472 @ ( comp @ X478 @ X477 ) @ ( sub @ X476 @ X477 ) @ X475 ) )
| ( $true
!= ( X472 @ X478 @ X476 @ ( comp @ X477 @ X475 ) ) ) )
& ( $true
= ( X472 @ id @ X474 @ id ) ) )
| ? [X465: term,X466: term > $o] :
( ( $true
!= ( X466 @ X465 ) )
& ! [X470: term,X471: term] :
( ( $true
= ( X466 @ ( ap @ X470 @ X471 ) ) )
| ( $true
!= ( X466 @ X470 ) )
| ( $true
!= ( X466 @ X471 ) ) )
& ! [X467: term] :
( ( $true
= ( X466 @ ( lam @ X467 ) ) )
| ? [X468: term] :
( ( $true
= ( X466 @ X468 ) )
& ( ( X466 @ ( sub @ X467 @ ( push @ X468 @ id ) ) )
!= $true ) ) )
& ! [X469: term] :
( ( $true
!= ( var @ X469 ) )
| ( ( X466 @ X469 )
= $true ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoaslaminj_gthm = $true )
& ( ( pushprop_lem0_lthm = $true )
<=> ! [X180: term > $o,X181: term,X182: subst] :
? [X183: term > $o] :
! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ? [X190: term,X191: term,X192: term,X193: term] :
( ( ( ap @ X191 @ X190 )
= ( ap @ X193 @ X192 ) )
& ( X190 != X192 ) )
| ! [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
!= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
| ( X194 = X195 ) ) ) )
& ( hoaslamnotvar = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
| ( X174 = X175 ) ) )
& ( hoasapinj1_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( ! [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
= ( X268 @ X267 @ X266 ) )
| ( ( lam @ ( X268 @ sh @ one ) )
!= ( lam @ ( X269 @ sh @ one ) ) )
| ? [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
!= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
| ? [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
!= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ulamvarsh = $true )
& ( ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) )
<=> ( substmonoid = $true ) )
& ( pushprop = $true )
& ( ! [X157: subst > term > subst > $o,X158: term] :
( ? [X159: term] :
( ( ( X157 @ id @ ( sub @ X158 @ ( push @ X159 @ id ) ) @ id )
!= $true )
& ( $true
= ( X157 @ id @ X159 @ id ) ) )
| ( $true
= ( X157 @ id @ ( lam @ X158 ) @ id ) )
| ? [X160: subst > term > term] :
( ! [X162: term,X163: subst,X164: subst] :
( ( sub @ ( X160 @ X163 @ X162 ) @ X164 )
= ( X160 @ ( comp @ X163 @ X164 ) @ ( sub @ X162 @ X164 ) ) )
& ( ( X157 @ id @ ( lam @ ( X160 @ sh @ one ) ) @ id )
!= $true )
& ! [X161: term] :
( ( ( X157 @ id @ X161 @ id )
!= $true )
| ( $true
= ( X157 @ id @ ( X160 @ id @ X161 ) @ id ) ) ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ! [X337: d_term,X338: d_term] :
( ( X337 = X338 )
| ( ( d2term @ X337 )
!= ( d2term @ X338 ) ) )
& ( ! [X294: subst > term > subst > $o,X295: term] :
( ? [X302: subst > term > term] :
( ! [X303: term] :
( ( $true
= ( X294 @ id @ ( X302 @ id @ X303 ) @ id ) )
| ( $true
!= ( X294 @ id @ X303 @ id ) ) )
& ! [X304: subst,X305: subst,X306: term] :
( ( X302 @ ( comp @ X304 @ X305 ) @ ( sub @ X306 @ X305 ) )
= ( sub @ ( X302 @ X304 @ X306 ) @ X305 ) )
& ( $true
!= ( X294 @ id @ ( lam @ ( X302 @ sh @ one ) ) @ id ) ) )
| ? [X307: subst,X308: term,X309: subst,X310: subst] :
( ( $true
!= ( X294 @ X310 @ X308 @ ( comp @ X309 @ X307 ) ) )
& ( $true
= ( X294 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) @ X307 ) ) )
| ? [X298: term,X299: subst,X300: subst,X301: subst] :
( ( $true
!= ( X294 @ ( comp @ X299 @ X301 ) @ ( sub @ X298 @ X301 ) @ X300 ) )
& ( $true
= ( X294 @ X299 @ X298 @ ( comp @ X301 @ X300 ) ) ) )
| ? [X296: term,X297: term] :
( ( ( X294 @ id @ X296 @ id )
= $true )
& ( $true
!= ( X294 @ id @ ( ap @ ( sub @ X297 @ id ) @ X296 ) @ id ) )
& ( ( X294 @ id @ X297 @ id )
= $true ) )
| ? [X311: term] :
( ( ( var @ ( sub @ X311 @ id ) )
= $true )
& ( $true
!= ( X294 @ id @ X311 @ id ) ) )
| ( $true
= ( X294 @ id @ X295 @ id ) ) )
<=> ( hoasinduction = $true ) )
& ( ( pushprop_lem1v2_lthm = $true )
<=> ( ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ( ( X30 @ X29 )
!= $true )
| ( $true
= ( X28 @ one ) )
| ? [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
!= ( X28 @ X32 ) ) )
| ? [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
!= X27 ) ) )
& ( ( hoasinduction_lem0 = $true )
<=> ! [X138: subst > term > subst > $o] :
? [X139: term > $o] :
! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) ) )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
!= ( X336 @ X332 ) )
| ( $true
= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) ) )
| ? [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
!= X330 )
| ? [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
!= X329 ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X385: subst,X386: term > $o,X387: term] :
( ? [X388: term] :
( ! [X389: term] :
( ( $true
= ( X386 @ ( sub @ X388 @ ( push @ X389 @ id ) ) ) )
| ( ( X386 @ X389 )
!= $true ) )
& ( $true
!= ( X386 @ ( lam @ X388 ) ) ) )
| ( $true
= ( X386 @ ( sub @ X387 @ X385 ) ) )
| ? [X390: term] :
( ( $true
= ( var @ X390 ) )
& ( $true
!= ( X386 @ ( sub @ X390 @ X385 ) ) ) )
| ? [X391: term,X392: term] :
( ( $true
= ( X386 @ X392 ) )
& ( $true
= ( X386 @ X391 ) )
& ( ( X386 @ ( ap @ X392 @ X391 ) )
!= $true ) ) )
<=> ( induction2lem = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( lamnotap != $true )
& ( substmonoid_gthm = $true )
& ( hoasinduction_lem2_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 ) ) ) ) ) ) )
& ( ! [X247: subst > term > subst > $o,X248: term] :
( ? [X258: term,X259: subst,X260: subst,X261: subst] :
( ( $true
= ( X247 @ X261 @ X258 @ ( comp @ X260 @ X259 ) ) )
& ( ( X247 @ ( comp @ X261 @ X260 ) @ ( sub @ X258 @ X260 ) @ X259 )
!= $true ) )
| ( ( X247 @ id @ ( lam @ X248 ) @ id )
= $true )
| ? [X262: term] :
( ( ( X247 @ id @ ( sub @ X248 @ ( push @ X262 @ id ) ) @ id )
!= $true )
& ( $true
= ( X247 @ id @ X262 @ id ) ) )
| ? [X253: subst > term > term] :
( ! [X254: subst,X255: term,X256: subst] :
( ( sub @ ( X253 @ X256 @ X255 ) @ X254 )
= ( X253 @ ( comp @ X256 @ X254 ) @ ( sub @ X255 @ X254 ) ) )
& ( $true
!= ( X247 @ id @ ( lam @ ( X253 @ sh @ one ) ) @ id ) )
& ! [X257: term] :
( ( $true
!= ( X247 @ id @ X257 @ id ) )
| ( $true
= ( X247 @ id @ ( X253 @ id @ X257 ) @ id ) ) ) )
| ? [X249: subst,X250: subst,X251: term,X252: subst] :
( ( $true
= ( X247 @ ( comp @ X252 @ X249 ) @ ( sub @ X251 @ X249 ) @ X250 ) )
& ( $true
!= ( X247 @ X252 @ X251 @ ( comp @ X249 @ X250 ) ) ) ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( axclos = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( induction2lem_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X187: subst > term > subst > $o] :
? [X188: term > $o] :
! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( apnotvar = $true )
& ( ! [X21: subst,X22: term,X23: term > $o] :
? [X24: term > $o] :
! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) )
<=> ( pushprop_lem0 = $true ) )
& ( ! [X378: term,X379: term,X380: subst > term > term] :
( ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( lam @ ( X380 @ sh @ one ) ) )
| ? [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
!= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) ) )
<=> ( hoaslamnotap = $true ) )
& ( ( ! [X286: subst > term > subst > $o,X287: term] :
( ? [X288: term] :
( ( $true
= ( X286 @ id @ X288 @ id ) )
& ( ( X286 @ id @ ( sub @ X287 @ ( push @ X288 @ id ) ) @ id )
!= $true ) )
| ( $true
= ( X286 @ id @ ( lam @ X287 ) @ id ) )
| ? [X289: subst > term > term] :
( ! [X290: term] :
( ( ( X286 @ id @ X290 @ id )
!= $true )
| ( $true
= ( X286 @ id @ ( X289 @ id @ X290 ) @ id ) ) )
& ! [X291: subst,X292: subst,X293: term] :
( ( sub @ ( X289 @ X292 @ X293 ) @ X291 )
= ( X289 @ ( comp @ X292 @ X291 ) @ ( sub @ X293 @ X291 ) ) )
& ( $true
!= ( X286 @ id @ ( lam @ ( X289 @ sh @ one ) ) @ id ) ) ) )
| ? [X278: term,X279: subst > term > subst > $o] :
( ! [X285: term] :
( ( $true
= ( X279 @ id @ ( sub @ X278 @ ( push @ X285 @ id ) ) @ id ) )
| ( $true
!= ( X279 @ id @ X285 @ id ) ) )
& ( $true
!= ( X279 @ id @ ( lam @ ( sub @ X278 @ ( push @ one @ sh ) ) ) @ id ) )
& ! [X280: subst > term > term] :
( ? [X281: term] :
( ( $true
!= ( X279 @ id @ ( X280 @ id @ X281 ) @ id ) )
& ( ( X279 @ id @ X281 @ id )
= $true ) )
| ? [X282: term,X283: subst,X284: subst] :
( ( sub @ ( X280 @ X284 @ X282 ) @ X283 )
!= ( X280 @ ( comp @ X284 @ X283 ) @ ( sub @ X282 @ X283 ) ) )
| ( $true
= ( X279 @ id @ ( lam @ ( X280 @ sh @ one ) ) @ id ) ) ) )
| ? [X277: term] :
( ( sub @ X277 @ id )
!= X277 ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ! [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
!= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
| ( X394 = X397 ) )
<=> ( hoasapinj1 = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
!= ( X169 @ X167 ) )
| ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) ) )
& ( ! [X200: term,X201: subst > term > subst > $o] :
( ? [X207: term] :
( ( $true
= ( X201 @ id @ X207 @ id ) )
& ( $true
!= ( X201 @ id @ ( sub @ X200 @ ( push @ X207 @ id ) ) @ id ) ) )
| ? [X202: subst > term > term] :
( ( $true
!= ( X201 @ id @ ( lam @ ( X202 @ sh @ one ) ) @ id ) )
& ! [X204: term,X205: subst,X206: subst] :
( ( sub @ ( X202 @ X206 @ X204 ) @ X205 )
= ( X202 @ ( comp @ X206 @ X205 ) @ ( sub @ X204 @ X205 ) ) )
& ! [X203: term] :
( ( $true
= ( X201 @ id @ ( X202 @ id @ X203 ) @ id ) )
| ( $true
!= ( X201 @ id @ X203 @ id ) ) ) )
| ( ( X201 @ id @ ( lam @ ( sub @ X200 @ ( push @ one @ sh ) ) ) @ id )
= $true ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ( apinj2 = $true )
<=> ! [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
!= ( ap @ X425 @ X426 ) )
| ( X426 = X428 ) ) )
& ( axvarshift = $true )
& ( pushprop_lem0_gthm = $true )
& ( axabs = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ! [X519: term] :
? [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
= ( X520 @ sh @ one ) )
<=> ( hoasinduction_lem3b = $true ) )
& ( pushprop_lthm_orig = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ! [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
= X384 )
<=> ( axscons = $true ) )
& ( ( ! [X456: subst > term > subst > $o,X457: term] :
( ? [X458: subst > term > term] :
( ! [X459: term] :
( ( $true
= ( X456 @ id @ ( X458 @ id @ X459 ) @ id ) )
| ( $true
!= ( X456 @ id @ X459 @ id ) ) )
& ! [X460: term,X461: subst,X462: subst] :
( ( X458 @ ( comp @ X462 @ X461 ) @ ( sub @ X460 @ X461 ) )
= ( sub @ ( X458 @ X462 @ X460 ) @ X461 ) )
& ( $true
!= ( X456 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X456 @ id @ X457 @ id ) )
| ? [X463: term,X464: term] :
( ( ( X456 @ id @ X464 @ id )
= $true )
& ( ( X456 @ id @ ( ap @ ( sub @ X463 @ id ) @ X464 ) @ id )
!= $true )
& ( $true
= ( X456 @ id @ X463 @ id ) ) ) )
| ? [X446: term > $o,X447: subst > term > subst > $o,X448: term] :
( ! [X451: subst > term > term] :
( ( $true
= ( X447 @ id @ ( lam @ ( X451 @ sh @ one ) ) @ id ) )
| ? [X452: term,X453: subst,X454: subst] :
( ( sub @ ( X451 @ X453 @ X452 ) @ X454 )
!= ( X451 @ ( comp @ X453 @ X454 ) @ ( sub @ X452 @ X454 ) ) )
| ? [X455: term] :
( ( $true
= ( X447 @ id @ X455 @ id ) )
& ( $true
!= ( X447 @ id @ ( X451 @ id @ X455 ) @ id ) ) ) )
& ! [X450: term] :
( ( ( X446 @ ( sub @ X448 @ ( push @ X450 @ id ) ) )
= $true )
| ( $true
!= ( X446 @ X450 ) ) )
& ! [X449: term] :
( ( X447 @ id @ X449 @ id )
= ( X446 @ X449 ) )
& ( $true
!= ( X446 @ ( lam @ X448 ) ) ) )
| ? [X445: term] :
( ( sub @ X445 @ id )
!= X445 )
| ? [X438: term,X439: term > $o] :
( ! [X440: term] :
( ( $true
!= ( var @ X440 ) )
| ( $true
= ( X439 @ X440 ) ) )
& ! [X441: term] :
( ? [X442: term] :
( ( $true
!= ( X439 @ ( sub @ X441 @ ( push @ X442 @ id ) ) ) )
& ( $true
= ( X439 @ X442 ) ) )
| ( ( X439 @ ( lam @ X441 ) )
= $true ) )
& ! [X443: term,X444: term] :
( ( ( X439 @ X443 )
!= $true )
| ( $true
!= ( X439 @ X444 ) )
| ( ( X439 @ ( ap @ X444 @ X443 ) )
= $true ) )
& ( $true
!= ( X439 @ X438 ) ) )
| ? [X435: subst > term > subst > $o] :
! [X436: term > $o] :
? [X437: term] :
( ( X436 @ X437 )
!= ( X435 @ id @ X437 @ id ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X263: term] :
? [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
= ( X264 @ sh @ one ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ? [X35: term] :
( ( sub @ X35 @ id )
!= X35 )
| ! [X36: subst > term > subst > $o,X37: term,X38: term > $o] :
( ? [X39: term,X40: subst,X41: subst,X42: subst] :
( ( ( X36 @ X42 @ X39 @ ( comp @ X41 @ X40 ) )
!= $true )
& ( $true
= ( X36 @ ( comp @ X42 @ X41 ) @ ( sub @ X39 @ X41 ) @ X40 ) ) )
| ? [X49: term] :
( ( $true
= ( X38 @ X49 ) )
& ( $true
!= ( X38 @ ( sub @ X37 @ ( push @ X49 @ id ) ) ) ) )
| ? [X44: subst > term > term] :
( ( $true
!= ( X36 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id ) )
& ! [X45: term,X46: subst,X47: subst] :
( ( sub @ ( X44 @ X47 @ X45 ) @ X46 )
= ( X44 @ ( comp @ X47 @ X46 ) @ ( sub @ X45 @ X46 ) ) )
& ! [X48: term] :
( ( $true
!= ( X36 @ id @ X48 @ id ) )
| ( $true
= ( X36 @ id @ ( X44 @ id @ X48 ) @ id ) ) ) )
| ( ( X38 @ ( lam @ X37 ) )
= $true )
| ? [X43: term] :
( ( X36 @ id @ X43 @ id )
!= ( X38 @ X43 ) )
| ? [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
!= ( X36 @ ( comp @ X53 @ X50 ) @ ( sub @ X52 @ X50 ) @ X51 ) )
& ( ( X36 @ X53 @ X52 @ ( comp @ X50 @ X51 ) )
= $true ) ) ) ) )
& ( axassoc = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ( pushprop_lem1v2_gthm = $true )
& ! [X141: d_term] : ( d_one = X141 )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( axshiftcons = $true )
<=> ! [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
= X198 ) )
& ( ! [X276: term] :
( ( sub @ X276 @ id )
= X276 )
<=> ( termmset = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( ! [X403: term,X404: term,X405: term,X406: term] :
( ( X405 = X406 )
| ( ( ap @ ( sub @ X405 @ id ) @ X403 )
!= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) )
| ? [X399: term,X400: term,X401: term,X402: term] :
( ( X399 != X402 )
& ( ( ap @ X399 @ X400 )
= ( ap @ X402 @ X401 ) ) )
| ? [X398: term] :
( ( sub @ X398 @ id )
!= X398 ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( ( shinj = $true )
<=> ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) ) )
& ( ( axidl = $true )
<=> ! [X223: subst] :
( ( comp @ id @ X223 )
= X223 ) )
& ( ( hoasinduction_lem3aa = $true )
<=> ! [X115: subst > term > subst > $o,X116: term] :
( ? [X117: subst > term > term] :
( ! [X118: subst,X119: subst,X120: term] :
( ( sub @ ( X117 @ X118 @ X120 ) @ X119 )
= ( X117 @ ( comp @ X118 @ X119 ) @ ( sub @ X120 @ X119 ) ) )
& ! [X121: term] :
( ( ( X115 @ id @ X121 @ id )
!= $true )
| ( $true
= ( X115 @ id @ ( X117 @ id @ X121 ) @ id ) ) )
& ( $true
!= ( X115 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X115 @ id @ ( lam @ ( sub @ X116 @ ( push @ one @ sh ) ) ) @ id ) )
| ? [X122: term] :
( ( $true
!= ( X115 @ id @ ( sub @ X116 @ ( push @ X122 @ id ) ) @ id ) )
& ( $true
= ( X115 @ id @ X122 @ id ) ) ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3b_gthm = $true ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( ( axmap = $true )
& ( ( ! [X2: subst,X3: term] :
( ( sub @ one @ ( push @ X3 @ X2 ) )
= X3 )
=> ( ! [X4: term,X5: subst] :
( ( comp @ sh @ ( push @ X4 @ X5 ) )
= X5 )
=> ( ! [X6: subst] :
( ( comp @ X6 @ id )
= X6 )
=> ( ! [X7: term > $o,X8: term] :
( ~ ! [X11: term,X12: term] :
( ( $true
!= ( X7 @ X11 ) )
| ( $true
!= ( X7 @ X12 ) )
| ( $true
= ( X7 @ ( ap @ X11 @ X12 ) ) ) )
| ~ ! [X9: term] :
( ( $true
!= ( X7 @ X9 ) )
| ( ( X7 @ ( lam @ X9 ) )
= $true ) )
| ~ ! [X10: term] :
( ( $true
= ( X7 @ X10 ) )
| ( ( var @ X10 )
!= $true ) )
| ( ( X7 @ X8 )
= $true ) )
=> ! [X13: term > $o,X14: term,X15: subst] :
( ( $true
= ( X13 @ ( sub @ X14 @ X15 ) ) )
| ~ ! [X20: term] :
( ( $true
!= ( var @ X20 ) )
| ( $true
= ( X13 @ ( sub @ X20 @ X15 ) ) ) )
| ~ ! [X18: term] :
( ( $true
= ( X13 @ ( lam @ X18 ) ) )
| ~ ! [X19: term] :
( ( $true
= ( X13 @ ( sub @ X18 @ ( push @ X19 @ id ) ) ) )
| ( ( X13 @ X19 )
!= $true ) ) )
| ~ ! [X16: term,X17: term] :
( ( $true
= ( X13 @ ( ap @ X17 @ X16 ) ) )
| ( $true
!= ( X13 @ X16 ) )
| ( $true
!= ( X13 @ X17 ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( ulamvarind = $true )
& ( ( axvarcons = $true )
<=> ! [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
= X245 ) )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X63: term] :
( ( sub @ X63 @ id )
= X63 )
=> ( ! [X64: term,X65: subst > term > subst > $o] :
( ( $true
= ( X65 @ id @ ( lam @ ( sub @ X64 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X67: subst > term > term] :
( ~ ! [X71: term] :
( ( $true
!= ( X65 @ id @ X71 @ id ) )
| ( $true
= ( X65 @ id @ ( X67 @ id @ X71 ) @ id ) ) )
| ~ ! [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X67 @ X69 @ X70 ) @ X68 )
= ( X67 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
| ( $true
= ( X65 @ id @ ( lam @ ( X67 @ sh @ one ) ) @ id ) ) )
| ~ ! [X66: term] :
( ( $true
!= ( X65 @ id @ X66 @ id ) )
| ( $true
= ( X65 @ id @ ( sub @ X64 @ ( push @ X66 @ id ) ) @ id ) ) ) )
=> ! [X72: subst > term > subst > $o,X73: term] :
( ( $true
= ( X72 @ id @ ( lam @ X73 ) @ id ) )
| ~ ! [X79: term,X80: subst,X81: subst,X82: subst] :
( ( ( X72 @ ( comp @ X80 @ X82 ) @ ( sub @ X79 @ X82 ) @ X81 )
!= $true )
| ( $true
= ( X72 @ X80 @ X79 @ ( comp @ X82 @ X81 ) ) ) )
| ~ ! [X75: subst,X76: term,X77: subst,X78: subst] :
( ( ( X72 @ ( comp @ X78 @ X77 ) @ ( sub @ X76 @ X77 ) @ X75 )
= $true )
| ( $true
!= ( X72 @ X78 @ X76 @ ( comp @ X77 @ X75 ) ) ) )
| ~ ! [X83: subst > term > term] :
( ~ ! [X84: subst,X85: term,X86: subst] :
( ( X83 @ ( comp @ X86 @ X84 ) @ ( sub @ X85 @ X84 ) )
= ( sub @ ( X83 @ X86 @ X85 ) @ X84 ) )
| ( $true
= ( X72 @ id @ ( lam @ ( X83 @ sh @ one ) ) @ id ) )
| ~ ! [X87: term] :
( ( ( X72 @ id @ ( X83 @ id @ X87 ) @ id )
= $true )
| ( $true
!= ( X72 @ id @ X87 @ id ) ) ) )
| ~ ! [X74: term] :
( ( $true
= ( X72 @ id @ ( sub @ X73 @ ( push @ X74 @ id ) ) @ id ) )
| ( ( X72 @ id @ X74 @ id )
!= $true ) ) ) ) ) )
& ( hoasapinj2_gthm = $true )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X314: term] :
~ ! [X315: subst > term > term] :
~ ! [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
= ( X315 @ X316 @ X317 ) ) )
& ( pushprop_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X99: term,X100: subst] :
( ( sub @ one @ ( push @ X99 @ X100 ) )
= X99 )
=> ( ! [X101: term,X102: subst] :
( ( comp @ sh @ ( push @ X101 @ X102 ) )
= X102 )
=> ( ! [X103: term,X104: term] :
( ( X103 = X104 )
| ( ( lam @ X103 )
!= ( lam @ X104 ) ) )
=> ! [X105: term,X106: subst > term > term,X107: subst > term > term,X108: subst] :
( ~ ! [X109: subst,X110: subst,X111: term] :
( ( sub @ ( X107 @ X109 @ X111 ) @ X110 )
= ( X107 @ ( comp @ X109 @ X110 ) @ ( sub @ X111 @ X110 ) ) )
| ~ ! [X112: subst,X113: term,X114: subst] :
( ( X106 @ ( comp @ X114 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X106 @ X114 @ X113 ) @ X112 ) )
| ( ( X106 @ X108 @ X105 )
= ( X107 @ X108 @ X105 ) )
| ( ( lam @ ( X106 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) ) ) ) ) ) )
& ( ( d2subst @ d_id )
= id )
& ( ! [X318: subst > term > subst > $o,X319: term,X320: term > $o] :
( ~ ! [X327: term] :
( ( $true
!= ( X320 @ X327 ) )
| ( $true
= ( X320 @ ( sub @ X319 @ ( push @ X327 @ id ) ) ) ) )
| ~ ! [X326: term] :
( ( X318 @ id @ X326 @ id )
= ( X320 @ X326 ) )
| ~ ! [X321: subst > term > term] :
( ~ ! [X325: term] :
( ( $true
= ( X318 @ id @ ( X321 @ id @ X325 ) @ id ) )
| ( ( X318 @ id @ X325 @ id )
!= $true ) )
| ( ( X318 @ id @ ( lam @ ( X321 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X322: subst,X323: term,X324: subst] :
( ( sub @ ( X321 @ X322 @ X323 ) @ X324 )
= ( X321 @ ( comp @ X322 @ X324 ) @ ( sub @ X323 @ X324 ) ) ) )
| ( $true
= ( X320 @ ( lam @ X319 ) ) ) )
<=> ( hoasinduction_lem3v2a = $true ) )
& ( ( hoasinduction_lem3aaa = $true )
<=> ! [X142: subst > term > subst > $o,X143: term] :
( ( ( X142 @ id @ ( lam @ ( sub @ X143 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X144: term,X145: subst > term > term] :
( ~ ! [X146: subst,X147: term,X148: subst] :
( ( X145 @ ( comp @ X146 @ X148 ) @ ( sub @ X147 @ X148 ) )
= ( sub @ X144 @ ( push @ ( sub @ X147 @ X148 ) @ ( comp @ X146 @ X148 ) ) ) )
| ( ( X142 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X149: term] :
( ( $true
= ( X142 @ id @ ( X145 @ id @ X149 ) @ id ) )
| ( $true
!= ( X142 @ id @ X149 @ id ) ) )
| ~ ! [X150: term,X151: subst,X152: subst] :
( ( sub @ ( X145 @ X152 @ X150 ) @ X151 )
= ( sub @ ( sub @ X144 @ ( push @ X150 @ X152 ) ) @ X151 ) ) )
| ~ ! [X153: term] :
( ( $true
!= ( X142 @ id @ X153 @ id ) )
| ( $true
= ( X142 @ id @ ( sub @ X143 @ ( push @ X153 @ id ) ) @ id ) ) ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X123: term > $o,X124: term,X125: subst > term > subst > $o,X126: term] :
( ( ( X123 @ ( ap @ X126 @ X124 ) )
= $true )
| ~ ! [X137: term] :
( ( X125 @ id @ X137 @ id )
= ( X123 @ X137 ) )
| ~ ! [X129: subst,X130: subst,X131: term,X132: subst] :
( ( $true
!= ( X125 @ ( comp @ X129 @ X132 ) @ ( sub @ X131 @ X132 ) @ X130 ) )
| ( $true
= ( X125 @ X129 @ X131 @ ( comp @ X132 @ X130 ) ) ) )
| ~ ! [X127: term,X128: term] :
( ( $true
!= ( X125 @ id @ X127 @ id ) )
| ( ( X125 @ id @ ( ap @ ( sub @ X127 @ id ) @ X128 ) @ id )
= $true )
| ( $true
!= ( X125 @ id @ X128 @ id ) ) )
| ( $true
!= ( X123 @ X124 ) )
| ( $true
!= ( X123 @ X126 ) )
| ~ ! [X133: subst,X134: subst,X135: term,X136: subst] :
( ( ( X125 @ X136 @ X135 @ ( comp @ X134 @ X133 ) )
!= $true )
| ( $true
= ( X125 @ ( comp @ X136 @ X134 ) @ ( sub @ X135 @ X134 ) @ X133 ) ) ) ) )
& ( pushprop_lthm = $true )
& ( ( axidr = $true )
<=> ! [X393: subst] :
( ( comp @ X393 @ id )
= X393 ) )
& ( ( axvarid = $true )
<=> ! [X265: term] :
( ( sub @ X265 @ id )
= X265 ) )
& ( ( ! [X525: term] :
~ ! [X526: subst > term > term] :
~ ! [X527: subst,X528: term] :
( ( X526 @ X527 @ X528 )
= ( sub @ X525 @ ( push @ X528 @ X527 ) ) )
=> ( ! [X529: term] :
( ( sub @ X529 @ id )
= X529 )
=> ! [X530: term,X531: term > $o,X532: subst > term > subst > $o] :
( ~ ! [X539: term] :
( ( $true
= ( X531 @ ( sub @ X530 @ ( push @ X539 @ id ) ) ) )
| ( $true
!= ( X531 @ X539 ) ) )
| ( $true
= ( X531 @ ( lam @ X530 ) ) )
| ~ ! [X534: subst > term > term] :
( ( $true
= ( X532 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id ) )
| ~ ! [X535: term] :
( ( ( X532 @ id @ ( X534 @ id @ X535 ) @ id )
= $true )
| ( $true
!= ( X532 @ id @ X535 @ id ) ) )
| ~ ! [X536: subst,X537: term,X538: subst] :
( ( X534 @ ( comp @ X538 @ X536 ) @ ( sub @ X537 @ X536 ) )
= ( sub @ ( X534 @ X538 @ X537 ) @ X536 ) ) )
| ~ ! [X533: term] :
( ( X532 @ id @ X533 @ id )
= ( X531 @ X533 ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) )
& ( termmset_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( ! [X339: subst > term > subst > $o] :
~ ! [X340: term > $o] :
~ ! [X341: term] :
( ( X339 @ id @ X341 @ id )
= ( X340 @ X341 ) )
=> ( ! [X342: term > $o,X343: term] :
( ~ ! [X344: term,X345: term] :
( ( ( X342 @ X344 )
!= $true )
| ( $true
= ( X342 @ ( ap @ X345 @ X344 ) ) )
| ( $true
!= ( X342 @ X345 ) ) )
| ( ( X342 @ X343 )
= $true )
| ~ ! [X346: term] :
( ~ ! [X347: term] :
( ( $true
= ( X342 @ ( sub @ X346 @ ( push @ X347 @ id ) ) ) )
| ( $true
!= ( X342 @ X347 ) ) )
| ( $true
= ( X342 @ ( lam @ X346 ) ) ) )
| ~ ! [X348: term] :
( ( ( X342 @ X348 )
= $true )
| ( $true
!= ( var @ X348 ) ) ) )
=> ( ! [X349: term] :
( ( sub @ X349 @ id )
= X349 )
=> ( ! [X350: subst > term > subst > $o,X351: term,X352: term > $o] :
( ( $true
= ( X352 @ ( lam @ X351 ) ) )
| ~ ! [X353: term] :
( ( X352 @ X353 )
= ( X350 @ id @ X353 @ id ) )
| ~ ! [X354: term] :
( ( $true
= ( X352 @ ( sub @ X351 @ ( push @ X354 @ id ) ) ) )
| ( $true
!= ( X352 @ X354 ) ) )
| ~ ! [X355: subst > term > term] :
( ~ ! [X357: subst,X358: term,X359: subst] :
( ( X355 @ ( comp @ X357 @ X359 ) @ ( sub @ X358 @ X359 ) )
= ( sub @ ( X355 @ X357 @ X358 ) @ X359 ) )
| ~ ! [X356: term] :
( ( $true
= ( X350 @ id @ ( X355 @ id @ X356 ) @ id ) )
| ( $true
!= ( X350 @ id @ X356 @ id ) ) )
| ( $true
= ( X350 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id ) ) ) )
=> ! [X360: subst > term > subst > $o,X361: term] :
( ~ ! [X374: subst,X375: subst,X376: subst,X377: term] :
( ( $true
!= ( X360 @ X374 @ X377 @ ( comp @ X375 @ X376 ) ) )
| ( $true
= ( X360 @ ( comp @ X374 @ X375 ) @ ( sub @ X377 @ X375 ) @ X376 ) ) )
| ~ ! [X373: term] :
( ( $true
= ( X360 @ id @ X373 @ id ) )
| ( $true
!= ( var @ ( sub @ X373 @ id ) ) ) )
| ( $true
= ( X360 @ id @ X361 @ id ) )
| ~ ! [X371: term,X372: term] :
( ( $true
!= ( X360 @ id @ X372 @ id ) )
| ( $true
!= ( X360 @ id @ X371 @ id ) )
| ( $true
= ( X360 @ id @ ( ap @ ( sub @ X372 @ id ) @ X371 ) @ id ) ) )
| ~ ! [X362: subst,X363: subst,X364: subst,X365: term] :
( ( $true
!= ( X360 @ ( comp @ X362 @ X363 ) @ ( sub @ X365 @ X363 ) @ X364 ) )
| ( $true
= ( X360 @ X362 @ X365 @ ( comp @ X363 @ X364 ) ) ) )
| ~ ! [X366: subst > term > term] :
( ( $true
= ( X360 @ id @ ( lam @ ( X366 @ sh @ one ) ) @ id ) )
| ~ ! [X367: subst,X368: term,X369: subst] :
( ( sub @ ( X366 @ X369 @ X368 ) @ X367 )
= ( X366 @ ( comp @ X369 @ X367 ) @ ( sub @ X368 @ X367 ) ) )
| ~ ! [X370: term] :
( ( $true
= ( X360 @ id @ ( X366 @ id @ X370 ) @ id ) )
| ( $true
!= ( X360 @ id @ X370 @ id ) ) ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& ( ( ! [X208: term] :
( ( sub @ X208 @ id )
= X208 )
=> ! [X209: term] :
( ( sub @ X209 @ id )
= X209 ) )
<=> ( termmset_lthm = $true ) )
& ( hoasinduction_lem3v2_gthm = $true )
& ( hoasinduction_lem1_gthm = $true )
& ! [X312: d_subst,X313: d_subst] :
( ( ( d2subst @ X312 )
= ( d2subst @ X313 ) )
=> ( X312 = X313 ) )
& ( ( induction = $true )
<=> ! [X429: term,X430: term > $o] :
( ~ ! [X432: term] :
( ( $true
= ( X430 @ ( lam @ X432 ) ) )
| ( $true
!= ( X430 @ X432 ) ) )
| ~ ! [X431: term] :
( ( $true
= ( X430 @ X431 ) )
| ( $true
!= ( var @ X431 ) ) )
| ~ ! [X433: term,X434: term] :
( ( $true
!= ( X430 @ X433 ) )
| ( $true
!= ( X430 @ X434 ) )
| ( ( X430 @ ( ap @ X433 @ X434 ) )
= $true ) )
| ( $true
= ( X430 @ X429 ) ) ) )
& ( pushprop_lem3v2_lthm = $true )
& ( ! [X95: term] :
~ ! [X96: subst > term > term] :
~ ! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( hoaslamnotap_gthm = $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2_lthm = $true )
& ! [X156: d_subst] : ( d_id = X156 )
& ( ( hoasinduction_lem3v2 = $true )
<=> ! [X407: term,X408: subst > term > subst > $o,X409: term > $o] :
( ( ( X409 @ ( lam @ X407 ) )
= $true )
| ~ ! [X410: subst > term > term] :
( ~ ! [X414: term] :
( ( $true
= ( X408 @ id @ ( X410 @ id @ X414 ) @ id ) )
| ( $true
!= ( X408 @ id @ X414 @ id ) ) )
| ~ ! [X411: term,X412: subst,X413: subst] :
( ( sub @ ( X410 @ X412 @ X411 ) @ X413 )
= ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) ) )
| ( $true
= ( X408 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) ) )
| ~ ! [X415: term] :
( ( $true
= ( X409 @ ( sub @ X407 @ ( push @ X415 @ id ) ) ) )
| ( $true
!= ( X409 @ X415 ) ) )
| ~ ! [X420: term] :
( ( X409 @ X420 )
= ( X408 @ id @ X420 @ id ) )
| ~ ! [X421: subst,X422: term,X423: subst,X424: subst] :
( ( $true
!= ( X408 @ X424 @ X422 @ ( comp @ X421 @ X423 ) ) )
| ( $true
= ( X408 @ ( comp @ X424 @ X421 ) @ ( sub @ X422 @ X421 ) @ X423 ) ) )
| ~ ! [X416: subst,X417: subst,X418: term,X419: subst] :
( ( $true
= ( X408 @ X419 @ X418 @ ( comp @ X417 @ X416 ) ) )
| ( $true
!= ( X408 @ ( comp @ X419 @ X417 ) @ ( sub @ X418 @ X417 ) @ X416 ) ) ) ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ( $true
= ( X54 @ id @ X55 @ id ) )
| ~ ! [X58: subst > term > term] :
( ~ ! [X59: term] :
( ( $true
= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
| ( $true
!= ( X54 @ id @ X59 @ id ) ) )
| ( $true
= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
| ~ ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) )
| ~ ! [X56: term,X57: term] :
( ( $true
= ( X54 @ id @ ( ap @ ( sub @ X56 @ id ) @ X57 ) @ id ) )
| ( $true
!= ( X54 @ id @ X57 @ id ) )
| ( ( X54 @ id @ X56 @ id )
!= $true ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( hoasapnotvar_lthm = $true )
& ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
<=> ( laminj = $true ) )
& ( hoasapnotvar = $true )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X521: subst] :
( ( comp @ id @ X521 )
= X521 )
=> ( ! [X522: subst] :
( ( comp @ X522 @ id )
= X522 )
=> ( ! [X523: subst] :
( ( comp @ id @ X523 )
= X523 )
& ! [X524: subst] :
( ( comp @ X524 @ id )
= X524 ) ) ) ) )
& ( ( hoasinduction_lem2 = $true )
<=> ! [X210: subst > term > subst > $o,X211: term,X212: term] :
( ( $true
!= ( X210 @ id @ X212 @ id ) )
| ( ( X210 @ id @ ( ap @ X212 @ X211 ) @ id )
= $true )
| ( $true
!= ( X210 @ id @ X211 @ id ) )
| ~ ! [X217: subst,X218: subst,X219: subst,X220: term] :
( ( ( X210 @ ( comp @ X219 @ X218 ) @ ( sub @ X220 @ X218 ) @ X217 )
!= $true )
| ( $true
= ( X210 @ X219 @ X220 @ ( comp @ X218 @ X217 ) ) ) )
| ~ ! [X213: term,X214: subst,X215: subst,X216: subst] :
( ( $true
!= ( X210 @ X215 @ X213 @ ( comp @ X216 @ X214 ) ) )
| ( ( X210 @ ( comp @ X215 @ X216 ) @ ( sub @ X213 @ X216 ) @ X214 )
= $true ) )
| ~ ! [X221: term,X222: term] :
( ( $true
!= ( X210 @ id @ X222 @ id ) )
| ( $true
= ( X210 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id ) )
| ( $true
!= ( X210 @ id @ X221 @ id ) ) ) ) )
& ( induction2_gthm = $true )
& ( ( pushprop_lem1v2 = $true )
<=> ! [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( $true
!= ( X243 @ X240 ) )
| ~ ! [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
= ( X242 @ X244 ) )
| ( ( X242 @ one )
= $true ) ) )
& ( ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) )
<=> ( apinj1 = $true ) )
& ( ulamvar1 != $true )
& ( ( ! [X224: term] :
( ( sub @ X224 @ id )
= X224 )
=> ( ! [X225: term,X226: term > $o,X227: subst] :
( ~ ! [X229: term] :
( ~ ! [X230: term] :
( ( $true
= ( X226 @ ( sub @ X229 @ ( push @ X230 @ id ) ) ) )
| ( $true
!= ( X226 @ X230 ) ) )
| ( $true
= ( X226 @ ( lam @ X229 ) ) ) )
| ( $true
= ( X226 @ ( sub @ X225 @ X227 ) ) )
| ~ ! [X228: term] :
( ( $true
!= ( var @ X228 ) )
| ( $true
= ( X226 @ ( sub @ X228 @ X227 ) ) ) )
| ~ ! [X231: term,X232: term] :
( ( $true
!= ( X226 @ X232 ) )
| ( ( X226 @ X231 )
!= $true )
| ( $true
= ( X226 @ ( ap @ X232 @ X231 ) ) ) ) )
=> ! [X233: term,X234: term > $o] :
( ( $true
= ( X234 @ X233 ) )
| ~ ! [X237: term,X238: term] :
( ( $true
!= ( X234 @ X237 ) )
| ( $true
= ( X234 @ ( ap @ X238 @ X237 ) ) )
| ( $true
!= ( X234 @ X238 ) ) )
| ~ ! [X239: term] :
( ( $true
= ( X234 @ X239 ) )
| ( $true
!= ( var @ X239 ) ) )
| ~ ! [X235: term] :
( ( $true
= ( X234 @ ( lam @ X235 ) ) )
| ~ ! [X236: term] :
( ( $true
!= ( X234 @ X236 ) )
| ( $true
= ( X234 @ ( sub @ X235 @ ( push @ X236 @ id ) ) ) ) ) ) ) ) )
<=> ( induction2_lthm = $true ) )
& ( ( d2subst @ d_id )
= sh )
& ( lamnotvar = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( induction2 = $true )
<=> ! [X88: term > $o,X89: term] :
( ~ ! [X93: term,X94: term] :
( ( $true
!= ( X88 @ X93 ) )
| ( $true
!= ( X88 @ X94 ) )
| ( $true
= ( X88 @ ( ap @ X93 @ X94 ) ) ) )
| ~ ! [X90: term] :
( ( $true
!= ( var @ X90 ) )
| ( ( X88 @ X90 )
= $true ) )
| ~ ! [X91: term] :
( ( $true
= ( X88 @ ( lam @ X91 ) ) )
| ~ ! [X92: term] :
( ( $true
!= ( X88 @ X92 ) )
| ( $true
= ( X88 @ ( sub @ X91 @ ( push @ X92 @ id ) ) ) ) ) )
| ( $true
= ( X88 @ X89 ) ) ) )
& ( hoasinduction_lem3a_gthm = $true )
& ( axapp = $true )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X465: term,X466: term > $o] :
( ( $true
= ( X466 @ X465 ) )
| ~ ! [X469: term] :
( ( $true
!= ( var @ X469 ) )
| ( ( X466 @ X469 )
= $true ) )
| ~ ! [X470: term,X471: term] :
( ( $true
= ( X466 @ ( ap @ X470 @ X471 ) ) )
| ( $true
!= ( X466 @ X470 ) )
| ( $true
!= ( X466 @ X471 ) ) )
| ~ ! [X467: term] :
( ~ ! [X468: term] :
( ( ( X466 @ ( sub @ X467 @ ( push @ X468 @ id ) ) )
= $true )
| ( $true
!= ( X466 @ X468 ) ) )
| ( $true
= ( X466 @ ( lam @ X467 ) ) ) ) )
=> ( ! [X472: subst > term > subst > $o,X473: term,X474: term] :
( ( $true
!= ( X472 @ id @ X473 @ id ) )
| ~ ! [X483: term,X484: term] :
( ( $true
= ( X472 @ id @ ( ap @ ( sub @ X483 @ id ) @ X484 ) @ id ) )
| ( $true
!= ( X472 @ id @ X483 @ id ) )
| ( ( X472 @ id @ X484 @ id )
!= $true ) )
| ~ ! [X479: term,X480: subst,X481: subst,X482: subst] :
( ( $true
!= ( X472 @ ( comp @ X480 @ X481 ) @ ( sub @ X479 @ X481 ) @ X482 ) )
| ( $true
= ( X472 @ X480 @ X479 @ ( comp @ X481 @ X482 ) ) ) )
| ( $true
= ( X472 @ id @ ( ap @ X473 @ X474 ) @ id ) )
| ( $true
!= ( X472 @ id @ X474 @ id ) )
| ~ ! [X475: subst,X476: term,X477: subst,X478: subst] :
( ( $true
= ( X472 @ ( comp @ X478 @ X477 ) @ ( sub @ X476 @ X477 ) @ X475 ) )
| ( $true
!= ( X472 @ X478 @ X476 @ ( comp @ X477 @ X475 ) ) ) ) )
=> ( ! [X485: term,X486: subst > term > subst > $o] :
( ~ ! [X487: subst,X488: subst,X489: subst,X490: term] :
( ( $true
!= ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X487 ) )
| ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X487 ) )
= $true ) )
| ~ ! [X500: term] :
( ( ( X486 @ id @ ( sub @ X485 @ ( push @ X500 @ id ) ) @ id )
= $true )
| ( $true
!= ( X486 @ id @ X500 @ id ) ) )
| ~ ! [X491: subst,X492: term,X493: subst,X494: subst] :
( ( ( X486 @ ( comp @ X493 @ X494 ) @ ( sub @ X492 @ X494 ) @ X491 )
= $true )
| ( $true
!= ( X486 @ X493 @ X492 @ ( comp @ X494 @ X491 ) ) ) )
| ( $true
= ( X486 @ id @ ( lam @ X485 ) @ id ) )
| ~ ! [X495: subst > term > term] :
( ( $true
= ( X486 @ id @ ( lam @ ( X495 @ sh @ one ) ) @ id ) )
| ~ ! [X496: term] :
( ( $true
= ( X486 @ id @ ( X495 @ id @ X496 ) @ id ) )
| ( $true
!= ( X486 @ id @ X496 @ id ) ) )
| ~ ! [X497: term,X498: subst,X499: subst] :
( ( X495 @ ( comp @ X498 @ X499 ) @ ( sub @ X497 @ X499 ) )
= ( sub @ ( X495 @ X498 @ X497 ) @ X499 ) ) ) )
=> ! [X501: term,X502: subst > term > subst > $o] :
( ~ ! [X518: term] :
( ( ( X502 @ id @ X518 @ id )
= $true )
| ( $true
!= ( var @ ( sub @ X518 @ id ) ) ) )
| ~ ! [X512: term,X513: subst,X514: subst,X515: subst] :
( ( ( X502 @ X515 @ X512 @ ( comp @ X514 @ X513 ) )
!= $true )
| ( $true
= ( X502 @ ( comp @ X515 @ X514 ) @ ( sub @ X512 @ X514 ) @ X513 ) ) )
| ~ ! [X507: subst > term > term] :
( ~ ! [X509: term,X510: subst,X511: subst] :
( ( X507 @ ( comp @ X511 @ X510 ) @ ( sub @ X509 @ X510 ) )
= ( sub @ ( X507 @ X511 @ X509 ) @ X510 ) )
| ~ ! [X508: term] :
( ( $true
= ( X502 @ id @ ( X507 @ id @ X508 ) @ id ) )
| ( ( X502 @ id @ X508 @ id )
!= $true ) )
| ( $true
= ( X502 @ id @ ( lam @ ( X507 @ sh @ one ) ) @ id ) ) )
| ~ ! [X503: subst,X504: subst,X505: term,X506: subst] :
( ( $true
!= ( X502 @ ( comp @ X504 @ X506 ) @ ( sub @ X505 @ X506 ) @ X503 ) )
| ( $true
= ( X502 @ X504 @ X505 @ ( comp @ X506 @ X503 ) ) ) )
| ~ ! [X516: term,X517: term] :
( ( ( X502 @ id @ X516 @ id )
!= $true )
| ( $true
= ( X502 @ id @ ( ap @ ( sub @ X516 @ id ) @ X517 ) @ id ) )
| ( ( X502 @ id @ X517 @ id )
!= $true ) )
| ( ( X502 @ id @ X501 @ id )
= $true ) ) ) ) ) )
& ( pushprop_lem2v2 = $true )
& ( hoaslaminj_gthm = $true )
& ( ! [X180: term > $o,X181: term,X182: subst] :
~ ! [X183: term > $o] :
~ ! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X190: term,X191: term,X192: term,X193: term] :
( ( X190 = X192 )
| ( ( ap @ X191 @ X190 )
!= ( ap @ X193 @ X192 ) ) )
=> ! [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
!= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
| ( X194 = X195 ) ) ) )
& ( hoaslamnotvar = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
| ( X174 = X175 ) ) )
& ( hoasapinj1_gthm = $true )
& ( hoasinduction_gthm = $true )
& ( ( hoaslaminj = $true )
<=> ! [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
= ( X268 @ X267 @ X266 ) )
| ( ( lam @ ( X268 @ sh @ one ) )
!= ( lam @ ( X269 @ sh @ one ) ) )
| ~ ! [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
| ~ ! [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) ) )
& ( ulamvarsh = $true )
& ( ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) )
<=> ( substmonoid = $true ) )
& ( pushprop = $true )
& ( ! [X157: subst > term > subst > $o,X158: term] :
( ( $true
= ( X157 @ id @ ( lam @ X158 ) @ id ) )
| ~ ! [X160: subst > term > term] :
( ~ ! [X162: term,X163: subst,X164: subst] :
( ( sub @ ( X160 @ X163 @ X162 ) @ X164 )
= ( X160 @ ( comp @ X163 @ X164 ) @ ( sub @ X162 @ X164 ) ) )
| ( ( X157 @ id @ ( lam @ ( X160 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X161: term] :
( ( ( X157 @ id @ X161 @ id )
!= $true )
| ( $true
= ( X157 @ id @ ( X160 @ id @ X161 ) @ id ) ) ) )
| ~ ! [X159: term] :
( ( $true
!= ( X157 @ id @ X159 @ id ) )
| ( ( X157 @ id @ ( sub @ X158 @ ( push @ X159 @ id ) ) @ id )
= $true ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ! [X337: d_term,X338: d_term] :
( ( ( d2term @ X337 )
= ( d2term @ X338 ) )
=> ( X337 = X338 ) )
& ( ! [X294: subst > term > subst > $o,X295: term] :
( ~ ! [X311: term] :
( ( $true
= ( X294 @ id @ X311 @ id ) )
| ( ( var @ ( sub @ X311 @ id ) )
!= $true ) )
| ~ ! [X307: subst,X308: term,X309: subst,X310: subst] :
( ( $true
= ( X294 @ X310 @ X308 @ ( comp @ X309 @ X307 ) ) )
| ( $true
!= ( X294 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) @ X307 ) ) )
| ~ ! [X296: term,X297: term] :
( ( ( X294 @ id @ X297 @ id )
!= $true )
| ( ( X294 @ id @ X296 @ id )
!= $true )
| ( $true
= ( X294 @ id @ ( ap @ ( sub @ X297 @ id ) @ X296 ) @ id ) ) )
| ~ ! [X302: subst > term > term] :
( ( $true
= ( X294 @ id @ ( lam @ ( X302 @ sh @ one ) ) @ id ) )
| ~ ! [X303: term] :
( ( $true
= ( X294 @ id @ ( X302 @ id @ X303 ) @ id ) )
| ( $true
!= ( X294 @ id @ X303 @ id ) ) )
| ~ ! [X304: subst,X305: subst,X306: term] :
( ( X302 @ ( comp @ X304 @ X305 ) @ ( sub @ X306 @ X305 ) )
= ( sub @ ( X302 @ X304 @ X306 ) @ X305 ) ) )
| ( $true
= ( X294 @ id @ X295 @ id ) )
| ~ ! [X298: term,X299: subst,X300: subst,X301: subst] :
( ( $true
= ( X294 @ ( comp @ X299 @ X301 ) @ ( sub @ X298 @ X301 ) @ X300 ) )
| ( $true
!= ( X294 @ X299 @ X298 @ ( comp @ X301 @ X300 ) ) ) ) )
<=> ( hoasinduction = $true ) )
& ( ( ! [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
= X27 )
=> ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ( ( X30 @ X29 )
!= $true )
| ( $true
= ( X28 @ one ) )
| ~ ! [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
= ( X28 @ X32 ) ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( ! [X138: subst > term > subst > $o] :
~ ! [X139: term > $o] :
~ ! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
= X329 )
=> ( ! [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
= X330 )
=> ! [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
!= ( X336 @ X332 ) )
| ( $true
= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) ) ) ) ) )
& ( hoasinduction_lem1_lthm = $true )
& ( ( induction2lem = $true )
<=> ! [X385: subst,X386: term > $o,X387: term] :
( ~ ! [X391: term,X392: term] :
( ( $true
!= ( X386 @ X391 ) )
| ( ( X386 @ ( ap @ X392 @ X391 ) )
= $true )
| ( $true
!= ( X386 @ X392 ) ) )
| ~ ! [X388: term] :
( ~ ! [X389: term] :
( ( $true
= ( X386 @ ( sub @ X388 @ ( push @ X389 @ id ) ) ) )
| ( ( X386 @ X389 )
!= $true ) )
| ( $true
= ( X386 @ ( lam @ X388 ) ) ) )
| ( $true
= ( X386 @ ( sub @ X387 @ X385 ) ) )
| ~ ! [X390: term] :
( ( $true
= ( X386 @ ( sub @ X390 @ X385 ) ) )
| ( $true
!= ( var @ X390 ) ) ) ) )
& ( hoasapnotvar_gthm = $true )
& ( lamnotap != $true )
& ( substmonoid_gthm = $true )
& ( hoasinduction_lem2_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 ) ) ) ) ) ) )
& ( ! [X247: subst > term > subst > $o,X248: term] :
( ~ ! [X249: subst,X250: subst,X251: term,X252: subst] :
( ( $true
= ( X247 @ X252 @ X251 @ ( comp @ X249 @ X250 ) ) )
| ( $true
!= ( X247 @ ( comp @ X252 @ X249 ) @ ( sub @ X251 @ X249 ) @ X250 ) ) )
| ~ ! [X258: term,X259: subst,X260: subst,X261: subst] :
( ( ( X247 @ ( comp @ X261 @ X260 ) @ ( sub @ X258 @ X260 ) @ X259 )
= $true )
| ( $true
!= ( X247 @ X261 @ X258 @ ( comp @ X260 @ X259 ) ) ) )
| ~ ! [X262: term] :
( ( $true
!= ( X247 @ id @ X262 @ id ) )
| ( ( X247 @ id @ ( sub @ X248 @ ( push @ X262 @ id ) ) @ id )
= $true ) )
| ( ( X247 @ id @ ( lam @ X248 ) @ id )
= $true )
| ~ ! [X253: subst > term > term] :
( ( $true
= ( X247 @ id @ ( lam @ ( X253 @ sh @ one ) ) @ id ) )
| ~ ! [X257: term] :
( ( $true
!= ( X247 @ id @ X257 @ id ) )
| ( $true
= ( X247 @ id @ ( X253 @ id @ X257 ) @ id ) ) )
| ~ ! [X254: subst,X255: term,X256: subst] :
( ( sub @ ( X253 @ X256 @ X255 ) @ X254 )
= ( X253 @ ( comp @ X256 @ X254 ) @ ( sub @ X255 @ X254 ) ) ) ) )
<=> ( hoasinduction_lem3 = $true ) )
& ( hoasinduction_lem2_gthm = $true )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( axclos = $true )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( induction2lem_gthm = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X187: subst > term > subst > $o] :
~ ! [X188: term > $o] :
~ ! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( apnotvar = $true )
& ( ( pushprop_lem0 = $true )
<=> ! [X21: subst,X22: term,X23: term > $o] :
~ ! [X24: term > $o] :
~ ! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) ) )
& ( ( hoaslamnotap = $true )
<=> ! [X378: term,X379: term,X380: subst > term > term] :
( ~ ! [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( lam @ ( X380 @ sh @ one ) ) ) ) )
& ( ( ! [X277: term] :
( ( sub @ X277 @ id )
= X277 )
=> ( ! [X278: term,X279: subst > term > subst > $o] :
( ~ ! [X280: subst > term > term] :
( ~ ! [X281: term] :
( ( $true
= ( X279 @ id @ ( X280 @ id @ X281 ) @ id ) )
| ( ( X279 @ id @ X281 @ id )
!= $true ) )
| ~ ! [X282: term,X283: subst,X284: subst] :
( ( sub @ ( X280 @ X284 @ X282 ) @ X283 )
= ( X280 @ ( comp @ X284 @ X283 ) @ ( sub @ X282 @ X283 ) ) )
| ( $true
= ( X279 @ id @ ( lam @ ( X280 @ sh @ one ) ) @ id ) ) )
| ~ ! [X285: term] :
( ( $true
= ( X279 @ id @ ( sub @ X278 @ ( push @ X285 @ id ) ) @ id ) )
| ( $true
!= ( X279 @ id @ X285 @ id ) ) )
| ( $true
= ( X279 @ id @ ( lam @ ( sub @ X278 @ ( push @ one @ sh ) ) ) @ id ) ) )
=> ! [X286: subst > term > subst > $o,X287: term] :
( ~ ! [X288: term] :
( ( ( X286 @ id @ ( sub @ X287 @ ( push @ X288 @ id ) ) @ id )
= $true )
| ( $true
!= ( X286 @ id @ X288 @ id ) ) )
| ( $true
= ( X286 @ id @ ( lam @ X287 ) @ id ) )
| ~ ! [X289: subst > term > term] :
( ( $true
= ( X286 @ id @ ( lam @ ( X289 @ sh @ one ) ) @ id ) )
| ~ ! [X291: subst,X292: subst,X293: term] :
( ( sub @ ( X289 @ X292 @ X293 ) @ X291 )
= ( X289 @ ( comp @ X292 @ X291 ) @ ( sub @ X293 @ X291 ) ) )
| ~ ! [X290: term] :
( ( ( X286 @ id @ X290 @ id )
!= $true )
| ( $true
= ( X286 @ id @ ( X289 @ id @ X290 ) @ id ) ) ) ) ) ) )
<=> ( hoasinduction_lem3a_lthm = $true ) )
& ( hoasinduction_lem1 = $true )
& ( ! [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
!= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
| ( X394 = X397 ) )
<=> ( hoasapinj1 = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
!= ( X169 @ X167 ) )
| ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) ) ) )
& ( ( hoasinduction_lem3aa_lthm = $true )
<=> ! [X200: term,X201: subst > term > subst > $o] :
( ( ( X201 @ id @ ( lam @ ( sub @ X200 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X202: subst > term > term] :
( ( $true
= ( X201 @ id @ ( lam @ ( X202 @ sh @ one ) ) @ id ) )
| ~ ! [X203: term] :
( ( $true
= ( X201 @ id @ ( X202 @ id @ X203 ) @ id ) )
| ( $true
!= ( X201 @ id @ X203 @ id ) ) )
| ~ ! [X204: term,X205: subst,X206: subst] :
( ( sub @ ( X202 @ X206 @ X204 ) @ X205 )
= ( X202 @ ( comp @ X206 @ X205 ) @ ( sub @ X204 @ X205 ) ) ) )
| ~ ! [X207: term] :
( ( $true
= ( X201 @ id @ ( sub @ X200 @ ( push @ X207 @ id ) ) @ id ) )
| ( $true
!= ( X201 @ id @ X207 @ id ) ) ) ) )
& ( ( apinj2 = $true )
<=> ! [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
!= ( ap @ X425 @ X426 ) )
| ( X426 = X428 ) ) )
& ( axvarshift = $true )
& ( pushprop_lem0_gthm = $true )
& ( axabs = $true )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X519: term] :
~ ! [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
!= ( X520 @ sh @ one ) ) )
& ( pushprop_lthm_orig = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ! [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
= X384 )
<=> ( axscons = $true ) )
& ( ( ! [X435: subst > term > subst > $o] :
~ ! [X436: term > $o] :
~ ! [X437: term] :
( ( X436 @ X437 )
= ( X435 @ id @ X437 @ id ) )
=> ( ! [X438: term,X439: term > $o] :
( ~ ! [X443: term,X444: term] :
( ( ( X439 @ X443 )
!= $true )
| ( $true
!= ( X439 @ X444 ) )
| ( ( X439 @ ( ap @ X444 @ X443 ) )
= $true ) )
| ( $true
= ( X439 @ X438 ) )
| ~ ! [X440: term] :
( ( $true
!= ( var @ X440 ) )
| ( $true
= ( X439 @ X440 ) ) )
| ~ ! [X441: term] :
( ( ( X439 @ ( lam @ X441 ) )
= $true )
| ~ ! [X442: term] :
( ( $true
= ( X439 @ ( sub @ X441 @ ( push @ X442 @ id ) ) ) )
| ( $true
!= ( X439 @ X442 ) ) ) ) )
=> ( ! [X445: term] :
( ( sub @ X445 @ id )
= X445 )
=> ( ! [X446: term > $o,X447: subst > term > subst > $o,X448: term] :
( ~ ! [X450: term] :
( ( ( X446 @ ( sub @ X448 @ ( push @ X450 @ id ) ) )
= $true )
| ( $true
!= ( X446 @ X450 ) ) )
| ~ ! [X451: subst > term > term] :
( ( $true
= ( X447 @ id @ ( lam @ ( X451 @ sh @ one ) ) @ id ) )
| ~ ! [X455: term] :
( ( $true
= ( X447 @ id @ ( X451 @ id @ X455 ) @ id ) )
| ( $true
!= ( X447 @ id @ X455 @ id ) ) )
| ~ ! [X452: term,X453: subst,X454: subst] :
( ( sub @ ( X451 @ X453 @ X452 ) @ X454 )
= ( X451 @ ( comp @ X453 @ X454 ) @ ( sub @ X452 @ X454 ) ) ) )
| ~ ! [X449: term] :
( ( X447 @ id @ X449 @ id )
= ( X446 @ X449 ) )
| ( $true
= ( X446 @ ( lam @ X448 ) ) ) )
=> ! [X456: subst > term > subst > $o,X457: term] :
( ~ ! [X458: subst > term > term] :
( ~ ! [X459: term] :
( ( $true
= ( X456 @ id @ ( X458 @ id @ X459 ) @ id ) )
| ( $true
!= ( X456 @ id @ X459 @ id ) ) )
| ( $true
= ( X456 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id ) )
| ~ ! [X460: term,X461: subst,X462: subst] :
( ( X458 @ ( comp @ X462 @ X461 ) @ ( sub @ X460 @ X461 ) )
= ( sub @ ( X458 @ X462 @ X460 ) @ X461 ) ) )
| ~ ! [X463: term,X464: term] :
( ( ( X456 @ id @ ( ap @ ( sub @ X463 @ id ) @ X464 ) @ id )
= $true )
| ( ( X456 @ id @ X464 @ id )
!= $true )
| ( $true
!= ( X456 @ id @ X463 @ id ) ) )
| ( $true
= ( X456 @ id @ X457 @ id ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X263: term] :
~ ! [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
!= ( X264 @ sh @ one ) ) )
& ( hoaslamnotap_lthm = $true )
& ( ( ! [X35: term] :
( ( sub @ X35 @ id )
= X35 )
=> ! [X36: subst > term > subst > $o,X37: term,X38: term > $o] :
( ~ ! [X43: term] :
( ( X36 @ id @ X43 @ id )
= ( X38 @ X43 ) )
| ( ( X38 @ ( lam @ X37 ) )
= $true )
| ~ ! [X50: subst,X51: subst,X52: term,X53: subst] :
( ( $true
= ( X36 @ ( comp @ X53 @ X50 ) @ ( sub @ X52 @ X50 ) @ X51 ) )
| ( ( X36 @ X53 @ X52 @ ( comp @ X50 @ X51 ) )
!= $true ) )
| ~ ! [X39: term,X40: subst,X41: subst,X42: subst] :
( ( $true
!= ( X36 @ ( comp @ X42 @ X41 ) @ ( sub @ X39 @ X41 ) @ X40 ) )
| ( ( X36 @ X42 @ X39 @ ( comp @ X41 @ X40 ) )
= $true ) )
| ~ ! [X44: subst > term > term] :
( ( $true
= ( X36 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id ) )
| ~ ! [X48: term] :
( ( $true
!= ( X36 @ id @ X48 @ id ) )
| ( $true
= ( X36 @ id @ ( X44 @ id @ X48 ) @ id ) ) )
| ~ ! [X45: term,X46: subst,X47: subst] :
( ( sub @ ( X44 @ X47 @ X45 ) @ X46 )
= ( X44 @ ( comp @ X47 @ X46 ) @ ( sub @ X45 @ X46 ) ) ) )
| ~ ! [X49: term] :
( ( $true
= ( X38 @ ( sub @ X37 @ ( push @ X49 @ id ) ) ) )
| ( $true
!= ( X38 @ X49 ) ) ) ) )
<=> ( hoasinduction_lem3v2_lthm = $true ) )
& ( axassoc = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ( pushprop_lem1v2_gthm = $true )
& ! [X141: d_term] : ( d_one = X141 )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( ( axshiftcons = $true )
<=> ! [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
= X198 ) )
& ( ! [X276: term] :
( ( sub @ X276 @ id )
= X276 )
<=> ( termmset = $true ) )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ( ! [X398: term] :
( ( sub @ X398 @ id )
= X398 )
=> ( ! [X399: term,X400: term,X401: term,X402: term] :
( ( ( ap @ X399 @ X400 )
!= ( ap @ X402 @ X401 ) )
| ( X399 = X402 ) )
=> ! [X403: term,X404: term,X405: term,X406: term] :
( ( X405 = X406 )
| ( ( ap @ ( sub @ X405 @ id ) @ X403 )
!= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) ) ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( ( shinj = $true )
<=> ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) ) )
& ( ( axidl = $true )
<=> ! [X223: subst] :
( ( comp @ id @ X223 )
= X223 ) )
& ( ! [X115: subst > term > subst > $o,X116: term] :
( ~ ! [X122: term] :
( ( $true
= ( X115 @ id @ ( sub @ X116 @ ( push @ X122 @ id ) ) @ id ) )
| ( $true
!= ( X115 @ id @ X122 @ id ) ) )
| ( $true
= ( X115 @ id @ ( lam @ ( sub @ X116 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X117: subst > term > term] :
( ~ ! [X118: subst,X119: subst,X120: term] :
( ( sub @ ( X117 @ X118 @ X120 ) @ X119 )
= ( X117 @ ( comp @ X118 @ X119 ) @ ( sub @ X120 @ X119 ) ) )
| ~ ! [X121: term] :
( ( ( X115 @ id @ X121 @ id )
!= $true )
| ( $true
= ( X115 @ id @ ( X117 @ id @ X121 ) @ id ) ) )
| ( $true
= ( X115 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id ) ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3b_gthm = $true ) ),
inference(flattening,[status(thm)],[f10]) ).
thf(f10,plain,
( ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
<=> ( laminj = $true ) )
& ( ( ! [X2: subst,X3: term] :
( ( sub @ one @ ( push @ X3 @ X2 ) )
= X3 )
=> ( ! [X4: term,X5: subst] :
( ( comp @ sh @ ( push @ X4 @ X5 ) )
= X5 )
=> ( ! [X6: subst] :
( ( comp @ X6 @ id )
= X6 )
=> ( ! [X7: term > $o,X8: term] :
( ~ ! [X9: term] :
( ( ( X7 @ ( lam @ X9 ) )
= $true )
| ( $true
!= ( X7 @ X9 ) ) )
| ~ ! [X10: term] :
( ( ( var @ X10 )
!= $true )
| ( $true
= ( X7 @ X10 ) ) )
| ~ ! [X11: term,X12: term] :
( ( $true
= ( X7 @ ( ap @ X11 @ X12 ) ) )
| ( $true
!= ( X7 @ X11 ) )
| ( $true
!= ( X7 @ X12 ) ) )
| ( ( X7 @ X8 )
= $true ) )
=> ! [X13: term > $o,X14: term,X15: subst] :
( ~ ! [X16: term,X17: term] :
( ( $true
= ( X13 @ ( ap @ X17 @ X16 ) ) )
| ( $true
!= ( X13 @ X17 ) )
| ( $true
!= ( X13 @ X16 ) ) )
| ~ ! [X18: term] :
( ( $true
= ( X13 @ ( lam @ X18 ) ) )
| ~ ! [X19: term] :
( ( ( X13 @ X19 )
!= $true )
| ( $true
= ( X13 @ ( sub @ X18 @ ( push @ X19 @ id ) ) ) ) ) )
| ( $true
= ( X13 @ ( sub @ X14 @ X15 ) ) )
| ~ ! [X20: term] :
( ( $true
!= ( var @ X20 ) )
| ( $true
= ( X13 @ ( sub @ X20 @ X15 ) ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( ( pushprop_lem0 = $true )
<=> ! [X21: subst,X22: term,X23: term > $o] :
~ ! [X24: term > $o] :
~ ! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) ) )
& ( ( ! [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
= X27 )
=> ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ~ ! [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
= ( X28 @ X32 ) )
| ( ( X30 @ X29 )
!= $true )
| ( $true
= ( X28 @ one ) ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) )
<=> ( substmonoid = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X35: term] :
( ( sub @ X35 @ id )
= X35 )
=> ! [X36: subst > term > subst > $o,X37: term,X38: term > $o] :
( ~ ! [X39: term,X40: subst,X41: subst,X42: subst] :
( ( ( X36 @ X42 @ X39 @ ( comp @ X41 @ X40 ) )
= $true )
| ( $true
!= ( X36 @ ( comp @ X42 @ X41 ) @ ( sub @ X39 @ X41 ) @ X40 ) ) )
| ~ ! [X43: term] :
( ( X36 @ id @ X43 @ id )
= ( X38 @ X43 ) )
| ( ( X38 @ ( lam @ X37 ) )
= $true )
| ~ ! [X44: subst > term > term] :
( ~ ! [X45: term,X46: subst,X47: subst] :
( ( sub @ ( X44 @ X47 @ X45 ) @ X46 )
= ( X44 @ ( comp @ X47 @ X46 ) @ ( sub @ X45 @ X46 ) ) )
| ( $true
= ( X36 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id ) )
| ~ ! [X48: term] :
( ( $true
!= ( X36 @ id @ X48 @ id ) )
| ( $true
= ( X36 @ id @ ( X44 @ id @ X48 ) @ id ) ) ) )
| ~ ! [X49: term] :
( ( $true
!= ( X38 @ X49 ) )
| ( $true
= ( X38 @ ( sub @ X37 @ ( push @ X49 @ id ) ) ) ) )
| ~ ! [X50: subst,X51: subst,X52: term,X53: subst] :
( ( ( X36 @ X53 @ X52 @ ( comp @ X50 @ X51 ) )
!= $true )
| ( $true
= ( X36 @ ( comp @ X53 @ X50 ) @ ( sub @ X52 @ X50 ) @ X51 ) ) ) ) ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ~ ! [X56: term,X57: term] :
( ( $true
!= ( X54 @ id @ X57 @ id ) )
| ( ( X54 @ id @ X56 @ id )
!= $true )
| ( $true
= ( X54 @ id @ ( ap @ ( sub @ X56 @ id ) @ X57 ) @ id ) ) )
| ( $true
= ( X54 @ id @ X55 @ id ) )
| ~ ! [X58: subst > term > term] :
( ( $true
= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
| ~ ! [X59: term] :
( ( $true
= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
| ( $true
!= ( X54 @ id @ X59 @ id ) ) )
| ~ ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( substmonoid_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X63: term] :
( ( sub @ X63 @ id )
= X63 )
=> ( ! [X64: term,X65: subst > term > subst > $o] :
( ( $true
= ( X65 @ id @ ( lam @ ( sub @ X64 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X66: term] :
( ( $true
= ( X65 @ id @ ( sub @ X64 @ ( push @ X66 @ id ) ) @ id ) )
| ( $true
!= ( X65 @ id @ X66 @ id ) ) )
| ~ ! [X67: subst > term > term] :
( ( $true
= ( X65 @ id @ ( lam @ ( X67 @ sh @ one ) ) @ id ) )
| ~ ! [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X67 @ X69 @ X70 ) @ X68 )
= ( X67 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
| ~ ! [X71: term] :
( ( $true
!= ( X65 @ id @ X71 @ id ) )
| ( $true
= ( X65 @ id @ ( X67 @ id @ X71 ) @ id ) ) ) ) )
=> ! [X72: subst > term > subst > $o,X73: term] :
( ~ ! [X74: term] :
( ( $true
= ( X72 @ id @ ( sub @ X73 @ ( push @ X74 @ id ) ) @ id ) )
| ( ( X72 @ id @ X74 @ id )
!= $true ) )
| ~ ! [X75: subst,X76: term,X77: subst,X78: subst] :
( ( ( X72 @ ( comp @ X78 @ X77 ) @ ( sub @ X76 @ X77 ) @ X75 )
= $true )
| ( $true
!= ( X72 @ X78 @ X76 @ ( comp @ X77 @ X75 ) ) ) )
| ~ ! [X79: term,X80: subst,X81: subst,X82: subst] :
( ( ( X72 @ ( comp @ X80 @ X82 ) @ ( sub @ X79 @ X82 ) @ X81 )
!= $true )
| ( $true
= ( X72 @ X80 @ X79 @ ( comp @ X82 @ X81 ) ) ) )
| ~ ! [X83: subst > term > term] :
( ~ ! [X84: subst,X85: term,X86: subst] :
( ( X83 @ ( comp @ X86 @ X84 ) @ ( sub @ X85 @ X84 ) )
= ( sub @ ( X83 @ X86 @ X85 ) @ X84 ) )
| ( $true
= ( X72 @ id @ ( lam @ ( X83 @ sh @ one ) ) @ id ) )
| ~ ! [X87: term] :
( ( ( X72 @ id @ ( X83 @ id @ X87 ) @ id )
= $true )
| ( $true
!= ( X72 @ id @ X87 @ id ) ) ) )
| ( $true
= ( X72 @ id @ ( lam @ X73 ) @ id ) ) ) ) ) )
& ( ( induction2 = $true )
<=> ! [X88: term > $o,X89: term] :
( ~ ! [X90: term] :
( ( ( X88 @ X90 )
= $true )
| ( $true
!= ( var @ X90 ) ) )
| ~ ! [X91: term] :
( ( $true
= ( X88 @ ( lam @ X91 ) ) )
| ~ ! [X92: term] :
( ( $true
!= ( X88 @ X92 ) )
| ( $true
= ( X88 @ ( sub @ X91 @ ( push @ X92 @ id ) ) ) ) ) )
| ( $true
= ( X88 @ X89 ) )
| ~ ! [X93: term,X94: term] :
( ( $true
!= ( X88 @ X94 ) )
| ( $true
= ( X88 @ ( ap @ X93 @ X94 ) ) )
| ( $true
!= ( X88 @ X93 ) ) ) ) )
& ( ulamvar1 != $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ! [X95: term] :
~ ! [X96: subst > term > term] :
~ ! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X99: term,X100: subst] :
( ( sub @ one @ ( push @ X99 @ X100 ) )
= X99 )
=> ( ! [X101: term,X102: subst] :
( ( comp @ sh @ ( push @ X101 @ X102 ) )
= X102 )
=> ( ! [X103: term,X104: term] :
( ( X103 = X104 )
| ( ( lam @ X103 )
!= ( lam @ X104 ) ) )
=> ! [X105: term,X106: subst > term > term,X107: subst > term > term,X108: subst] :
( ~ ! [X109: subst,X110: subst,X111: term] :
( ( sub @ ( X107 @ X109 @ X111 ) @ X110 )
= ( X107 @ ( comp @ X109 @ X110 ) @ ( sub @ X111 @ X110 ) ) )
| ~ ! [X112: subst,X113: term,X114: subst] :
( ( X106 @ ( comp @ X114 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X106 @ X114 @ X113 ) @ X112 ) )
| ( ( X106 @ X108 @ X105 )
= ( X107 @ X108 @ X105 ) )
| ( ( lam @ ( X106 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X115: subst > term > subst > $o,X116: term] :
( ( $true
= ( X115 @ id @ ( lam @ ( sub @ X116 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X117: subst > term > term] :
( ( $true
= ( X115 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id ) )
| ~ ! [X118: subst,X119: subst,X120: term] :
( ( sub @ ( X117 @ X118 @ X120 ) @ X119 )
= ( X117 @ ( comp @ X118 @ X119 ) @ ( sub @ X120 @ X119 ) ) )
| ~ ! [X121: term] :
( ( $true
= ( X115 @ id @ ( X117 @ id @ X121 ) @ id ) )
| ( ( X115 @ id @ X121 @ id )
!= $true ) ) )
| ~ ! [X122: term] :
( ( $true
= ( X115 @ id @ ( sub @ X116 @ ( push @ X122 @ id ) ) @ id ) )
| ( $true
!= ( X115 @ id @ X122 @ id ) ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X123: term > $o,X124: term,X125: subst > term > subst > $o,X126: term] :
( ~ ! [X127: term,X128: term] :
( ( $true
!= ( X125 @ id @ X128 @ id ) )
| ( $true
!= ( X125 @ id @ X127 @ id ) )
| ( ( X125 @ id @ ( ap @ ( sub @ X127 @ id ) @ X128 ) @ id )
= $true ) )
| ( $true
!= ( X123 @ X124 ) )
| ( $true
!= ( X123 @ X126 ) )
| ~ ! [X129: subst,X130: subst,X131: term,X132: subst] :
( ( $true
!= ( X125 @ ( comp @ X129 @ X132 ) @ ( sub @ X131 @ X132 ) @ X130 ) )
| ( $true
= ( X125 @ X129 @ X131 @ ( comp @ X132 @ X130 ) ) ) )
| ~ ! [X133: subst,X134: subst,X135: term,X136: subst] :
( ( $true
= ( X125 @ ( comp @ X136 @ X134 ) @ ( sub @ X135 @ X134 ) @ X133 ) )
| ( ( X125 @ X136 @ X135 @ ( comp @ X134 @ X133 ) )
!= $true ) )
| ( ( X123 @ ( ap @ X126 @ X124 ) )
= $true )
| ~ ! [X137: term] :
( ( X125 @ id @ X137 @ id )
= ( X123 @ X137 ) ) ) )
& ( axapp = $true )
& ( ! [X138: subst > term > subst > $o] :
~ ! [X139: term > $o] :
~ ! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoasapinj1_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ! [X141: d_term] : ( d_one = X141 )
& ( hoasinduction_lem3a_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ! [X142: subst > term > subst > $o,X143: term] :
( ( ( X142 @ id @ ( lam @ ( sub @ X143 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X144: term,X145: subst > term > term] :
( ~ ! [X146: subst,X147: term,X148: subst] :
( ( X145 @ ( comp @ X146 @ X148 ) @ ( sub @ X147 @ X148 ) )
= ( sub @ X144 @ ( push @ ( sub @ X147 @ X148 ) @ ( comp @ X146 @ X148 ) ) ) )
| ~ ! [X149: term] :
( ( $true
= ( X142 @ id @ ( X145 @ id @ X149 ) @ id ) )
| ( $true
!= ( X142 @ id @ X149 @ id ) ) )
| ( ( X142 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X150: term,X151: subst,X152: subst] :
( ( sub @ ( X145 @ X152 @ X150 ) @ X151 )
= ( sub @ ( sub @ X144 @ ( push @ X150 @ X152 ) ) @ X151 ) ) )
| ~ ! [X153: term] :
( ( $true
!= ( X142 @ id @ X153 @ id ) )
| ( $true
= ( X142 @ id @ ( sub @ X143 @ ( push @ X153 @ id ) ) @ id ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ! [X156: d_subst] : ( d_id = X156 )
& ( ! [X157: subst > term > subst > $o,X158: term] :
( ~ ! [X159: term] :
( ( $true
!= ( X157 @ id @ X159 @ id ) )
| ( ( X157 @ id @ ( sub @ X158 @ ( push @ X159 @ id ) ) @ id )
= $true ) )
| ( $true
= ( X157 @ id @ ( lam @ X158 ) @ id ) )
| ~ ! [X160: subst > term > term] :
( ~ ! [X161: term] :
( ( ( X157 @ id @ X161 @ id )
!= $true )
| ( $true
= ( X157 @ id @ ( X160 @ id @ X161 ) @ id ) ) )
| ( ( X157 @ id @ ( lam @ ( X160 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X162: term,X163: subst,X164: subst] :
( ( sub @ ( X160 @ X163 @ X162 ) @ X164 )
= ( X160 @ ( comp @ X163 @ X164 ) @ ( sub @ X162 @ X164 ) ) ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasapnotvar_lthm = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) )
| ( $true
!= ( X169 @ X167 ) ) ) )
& ( hoasapnotvar = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_gthm = $true )
& ( axabs = $true )
& ( induction2_gthm = $true )
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& ( hoaslaminj_gthm = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
| ( X174 = X175 ) ) )
& ( ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) )
<=> ( apinj1 = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem1 = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2 = $true )
& ( ! [X180: term > $o,X181: term,X182: subst] :
~ ! [X183: term > $o] :
~ ! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( pushprop = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( shinj = $true )
<=> ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) ) )
& ( ! [X187: subst > term > subst > $o] :
~ ! [X188: term > $o] :
~ ! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( induction2lem_gthm = $true )
& ( termmset_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X190: term,X191: term,X192: term,X193: term] :
( ( X190 = X192 )
| ( ( ap @ X191 @ X190 )
!= ( ap @ X193 @ X192 ) ) )
=> ! [X194: term,X195: term,X196: term,X197: term] :
( ( ( ap @ ( sub @ X197 @ id ) @ X195 )
!= ( ap @ ( sub @ X196 @ id ) @ X194 ) )
| ( X194 = X195 ) ) ) )
& ( axmap = $true )
& ( axclos = $true )
& ( one
= ( d2term @ d_one ) )
& ( ( axshiftcons = $true )
<=> ! [X198: subst,X199: term] :
( ( comp @ sh @ ( push @ X199 @ X198 ) )
= X198 ) )
& ( hoasapnotvar_gthm = $true )
& ( axvarshift = $true )
& ( ! [X200: term,X201: subst > term > subst > $o] :
( ( ( X201 @ id @ ( lam @ ( sub @ X200 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X202: subst > term > term] :
( ( $true
= ( X201 @ id @ ( lam @ ( X202 @ sh @ one ) ) @ id ) )
| ~ ! [X203: term] :
( ( $true
= ( X201 @ id @ ( X202 @ id @ X203 ) @ id ) )
| ( $true
!= ( X201 @ id @ X203 @ id ) ) )
| ~ ! [X204: term,X205: subst,X206: subst] :
( ( sub @ ( X202 @ X206 @ X204 ) @ X205 )
= ( X202 @ ( comp @ X206 @ X205 ) @ ( sub @ X204 @ X205 ) ) ) )
| ~ ! [X207: term] :
( ( $true
!= ( X201 @ id @ X207 @ id ) )
| ( $true
= ( X201 @ id @ ( sub @ X200 @ ( push @ X207 @ id ) ) @ id ) ) ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ( ! [X208: term] :
( ( sub @ X208 @ id )
= X208 )
=> ! [X209: term] :
( ( sub @ X209 @ id )
= X209 ) )
<=> ( termmset_lthm = $true ) )
& ( ! [X210: subst > term > subst > $o,X211: term,X212: term] :
( ( $true
!= ( X210 @ id @ X212 @ id ) )
| ( $true
!= ( X210 @ id @ X211 @ id ) )
| ~ ! [X213: term,X214: subst,X215: subst,X216: subst] :
( ( $true
!= ( X210 @ X215 @ X213 @ ( comp @ X216 @ X214 ) ) )
| ( ( X210 @ ( comp @ X215 @ X216 ) @ ( sub @ X213 @ X216 ) @ X214 )
= $true ) )
| ~ ! [X217: subst,X218: subst,X219: subst,X220: term] :
( ( $true
= ( X210 @ X219 @ X220 @ ( comp @ X218 @ X217 ) ) )
| ( ( X210 @ ( comp @ X219 @ X218 ) @ ( sub @ X220 @ X218 ) @ X217 )
!= $true ) )
| ~ ! [X221: term,X222: term] :
( ( $true
= ( X210 @ id @ ( ap @ ( sub @ X221 @ id ) @ X222 ) @ id ) )
| ( $true
!= ( X210 @ id @ X221 @ id ) )
| ( $true
!= ( X210 @ id @ X222 @ id ) ) )
| ( ( X210 @ id @ ( ap @ X212 @ X211 ) @ id )
= $true ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( ( axidl = $true )
<=> ! [X223: subst] :
( ( comp @ id @ X223 )
= X223 ) )
& ( ( induction2_lthm = $true )
<=> ( ! [X224: term] :
( ( sub @ X224 @ id )
= X224 )
=> ( ! [X225: term,X226: term > $o,X227: subst] :
( ~ ! [X228: term] :
( ( $true
!= ( var @ X228 ) )
| ( $true
= ( X226 @ ( sub @ X228 @ X227 ) ) ) )
| ( $true
= ( X226 @ ( sub @ X225 @ X227 ) ) )
| ~ ! [X229: term] :
( ~ ! [X230: term] :
( ( $true
= ( X226 @ ( sub @ X229 @ ( push @ X230 @ id ) ) ) )
| ( $true
!= ( X226 @ X230 ) ) )
| ( $true
= ( X226 @ ( lam @ X229 ) ) ) )
| ~ ! [X231: term,X232: term] :
( ( ( X226 @ X231 )
!= $true )
| ( $true
!= ( X226 @ X232 ) )
| ( $true
= ( X226 @ ( ap @ X232 @ X231 ) ) ) ) )
=> ! [X233: term,X234: term > $o] :
( ( $true
= ( X234 @ X233 ) )
| ~ ! [X235: term] :
( ( $true
= ( X234 @ ( lam @ X235 ) ) )
| ~ ! [X236: term] :
( ( $true
!= ( X234 @ X236 ) )
| ( $true
= ( X234 @ ( sub @ X235 @ ( push @ X236 @ id ) ) ) ) ) )
| ~ ! [X237: term,X238: term] :
( ( $true
!= ( X234 @ X238 ) )
| ( $true
= ( X234 @ ( ap @ X238 @ X237 ) ) )
| ( $true
!= ( X234 @ X237 ) ) )
| ~ ! [X239: term] :
( ( $true
= ( X234 @ X239 ) )
| ( $true
!= ( var @ X239 ) ) ) ) ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( ! [X240: term,X241: subst,X242: term > $o,X243: term > $o] :
( ( ( X242 @ one )
= $true )
| ( $true
!= ( X243 @ X240 ) )
| ~ ! [X244: term] :
( ( X243 @ ( sub @ X244 @ ( push @ X240 @ X241 ) ) )
= ( X242 @ X244 ) ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1_gthm = $true )
& ( ( axvarcons = $true )
<=> ! [X245: term,X246: subst] :
( ( sub @ one @ ( push @ X245 @ X246 ) )
= X245 ) )
& ( pushprop_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X247: subst > term > subst > $o,X248: term] :
( ( ( X247 @ id @ ( lam @ X248 ) @ id )
= $true )
| ~ ! [X249: subst,X250: subst,X251: term,X252: subst] :
( ( $true
!= ( X247 @ ( comp @ X252 @ X249 ) @ ( sub @ X251 @ X249 ) @ X250 ) )
| ( $true
= ( X247 @ X252 @ X251 @ ( comp @ X249 @ X250 ) ) ) )
| ~ ! [X253: subst > term > term] :
( ( $true
= ( X247 @ id @ ( lam @ ( X253 @ sh @ one ) ) @ id ) )
| ~ ! [X254: subst,X255: term,X256: subst] :
( ( sub @ ( X253 @ X256 @ X255 ) @ X254 )
= ( X253 @ ( comp @ X256 @ X254 ) @ ( sub @ X255 @ X254 ) ) )
| ~ ! [X257: term] :
( ( $true
= ( X247 @ id @ ( X253 @ id @ X257 ) @ id ) )
| ( $true
!= ( X247 @ id @ X257 @ id ) ) ) )
| ~ ! [X258: term,X259: subst,X260: subst,X261: subst] :
( ( $true
!= ( X247 @ X261 @ X258 @ ( comp @ X260 @ X259 ) ) )
| ( ( X247 @ ( comp @ X261 @ X260 ) @ ( sub @ X258 @ X260 ) @ X259 )
= $true ) )
| ~ ! [X262: term] :
( ( ( X247 @ id @ ( sub @ X248 @ ( push @ X262 @ id ) ) @ id )
= $true )
| ( $true
!= ( X247 @ id @ X262 @ id ) ) ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3b_lthm = $true )
<=> ! [X263: term] :
~ ! [X264: subst > term > term] :
( ( sub @ X263 @ ( push @ one @ sh ) )
!= ( X264 @ sh @ one ) ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( ( axvarid = $true )
<=> ! [X265: term] :
( ( sub @ X265 @ id )
= X265 ) )
& ( ( hoaslaminj = $true )
<=> ! [X266: term,X267: subst,X268: subst > term > term,X269: subst > term > term] :
( ( ( X269 @ X267 @ X266 )
= ( X268 @ X267 @ X266 ) )
| ( ( lam @ ( X268 @ sh @ one ) )
!= ( lam @ ( X269 @ sh @ one ) ) )
| ~ ! [X273: term,X274: subst,X275: subst] :
( ( sub @ ( X268 @ X275 @ X273 ) @ X274 )
= ( X268 @ ( comp @ X275 @ X274 ) @ ( sub @ X273 @ X274 ) ) )
| ~ ! [X270: subst,X271: term,X272: subst] :
( ( X269 @ ( comp @ X270 @ X272 ) @ ( sub @ X271 @ X272 ) )
= ( sub @ ( X269 @ X270 @ X271 ) @ X272 ) ) ) )
& ( ! [X276: term] :
( ( sub @ X276 @ id )
= X276 )
<=> ( termmset = $true ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X277: term] :
( ( sub @ X277 @ id )
= X277 )
=> ( ! [X278: term,X279: subst > term > subst > $o] :
( ( $true
= ( X279 @ id @ ( lam @ ( sub @ X278 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X280: subst > term > term] :
( ( $true
= ( X279 @ id @ ( lam @ ( X280 @ sh @ one ) ) @ id ) )
| ~ ! [X281: term] :
( ( $true
= ( X279 @ id @ ( X280 @ id @ X281 ) @ id ) )
| ( ( X279 @ id @ X281 @ id )
!= $true ) )
| ~ ! [X282: term,X283: subst,X284: subst] :
( ( sub @ ( X280 @ X284 @ X282 ) @ X283 )
= ( X280 @ ( comp @ X284 @ X283 ) @ ( sub @ X282 @ X283 ) ) ) )
| ~ ! [X285: term] :
( ( $true
= ( X279 @ id @ ( sub @ X278 @ ( push @ X285 @ id ) ) @ id ) )
| ( $true
!= ( X279 @ id @ X285 @ id ) ) ) )
=> ! [X286: subst > term > subst > $o,X287: term] :
( ~ ! [X288: term] :
( ( ( X286 @ id @ ( sub @ X287 @ ( push @ X288 @ id ) ) @ id )
= $true )
| ( $true
!= ( X286 @ id @ X288 @ id ) ) )
| ~ ! [X289: subst > term > term] :
( ( $true
= ( X286 @ id @ ( lam @ ( X289 @ sh @ one ) ) @ id ) )
| ~ ! [X290: term] :
( ( $true
= ( X286 @ id @ ( X289 @ id @ X290 ) @ id ) )
| ( ( X286 @ id @ X290 @ id )
!= $true ) )
| ~ ! [X291: subst,X292: subst,X293: term] :
( ( sub @ ( X289 @ X292 @ X293 ) @ X291 )
= ( X289 @ ( comp @ X292 @ X291 ) @ ( sub @ X293 @ X291 ) ) ) )
| ( $true
= ( X286 @ id @ ( lam @ X287 ) @ id ) ) ) ) ) )
& ( ! [X294: subst > term > subst > $o,X295: term] :
( ~ ! [X296: term,X297: term] :
( ( ( X294 @ id @ X297 @ id )
!= $true )
| ( $true
= ( X294 @ id @ ( ap @ ( sub @ X297 @ id ) @ X296 ) @ id ) )
| ( ( X294 @ id @ X296 @ id )
!= $true ) )
| ~ ! [X298: term,X299: subst,X300: subst,X301: subst] :
( ( $true
= ( X294 @ ( comp @ X299 @ X301 ) @ ( sub @ X298 @ X301 ) @ X300 ) )
| ( $true
!= ( X294 @ X299 @ X298 @ ( comp @ X301 @ X300 ) ) ) )
| ~ ! [X302: subst > term > term] :
( ~ ! [X303: term] :
( ( $true
!= ( X294 @ id @ X303 @ id ) )
| ( $true
= ( X294 @ id @ ( X302 @ id @ X303 ) @ id ) ) )
| ( $true
= ( X294 @ id @ ( lam @ ( X302 @ sh @ one ) ) @ id ) )
| ~ ! [X304: subst,X305: subst,X306: term] :
( ( X302 @ ( comp @ X304 @ X305 ) @ ( sub @ X306 @ X305 ) )
= ( sub @ ( X302 @ X304 @ X306 ) @ X305 ) ) )
| ~ ! [X307: subst,X308: term,X309: subst,X310: subst] :
( ( $true
= ( X294 @ X310 @ X308 @ ( comp @ X309 @ X307 ) ) )
| ( $true
!= ( X294 @ ( comp @ X310 @ X309 ) @ ( sub @ X308 @ X309 ) @ X307 ) ) )
| ~ ! [X311: term] :
( ( ( var @ ( sub @ X311 @ id ) )
!= $true )
| ( $true
= ( X294 @ id @ X311 @ id ) ) )
| ( $true
= ( X294 @ id @ X295 @ id ) ) )
<=> ( hoasinduction = $true ) )
& ! [X312: d_subst,X313: d_subst] :
( ( ( d2subst @ X312 )
= ( d2subst @ X313 ) )
=> ( X312 = X313 ) )
& ( ( hoasinduction_lem3v2_f_lthm = $true )
<=> ! [X314: term] :
~ ! [X315: subst > term > term] :
~ ! [X316: subst,X317: term] :
( ( sub @ X314 @ ( push @ X317 @ X316 ) )
= ( X315 @ X316 @ X317 ) ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X318: subst > term > subst > $o,X319: term,X320: term > $o] :
( ~ ! [X321: subst > term > term] :
( ~ ! [X322: subst,X323: term,X324: subst] :
( ( sub @ ( X321 @ X322 @ X323 ) @ X324 )
= ( X321 @ ( comp @ X322 @ X324 ) @ ( sub @ X323 @ X324 ) ) )
| ~ ! [X325: term] :
( ( $true
= ( X318 @ id @ ( X321 @ id @ X325 ) @ id ) )
| ( ( X318 @ id @ X325 @ id )
!= $true ) )
| ( ( X318 @ id @ ( lam @ ( X321 @ sh @ one ) ) @ id )
= $true ) )
| ( $true
= ( X320 @ ( lam @ X319 ) ) )
| ~ ! [X326: term] :
( ( X318 @ id @ X326 @ id )
= ( X320 @ X326 ) )
| ~ ! [X327: term] :
( ( $true
!= ( X320 @ X327 ) )
| ( $true
= ( X320 @ ( sub @ X319 @ ( push @ X327 @ id ) ) ) ) ) ) )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X328: subst,X329: term] :
( ( sub @ one @ ( push @ X329 @ X328 ) )
= X329 )
=> ( ! [X330: subst,X331: term] :
( ( comp @ sh @ ( push @ X331 @ X330 ) )
= X330 )
=> ! [X332: term,X333: term > $o,X334: term,X335: subst,X336: term > $o] :
( ( $true
= ( X333 @ ( sub @ X332 @ ( push @ X334 @ X335 ) ) ) )
| ( $true
!= ( X336 @ X332 ) ) ) ) ) )
& ! [X337: d_term,X338: d_term] :
( ( ( d2term @ X337 )
= ( d2term @ X338 ) )
=> ( X337 = X338 ) )
& ( pushprop_lthm_orig = $true )
& ( apnotvar = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( ! [X339: subst > term > subst > $o] :
~ ! [X340: term > $o] :
~ ! [X341: term] :
( ( X339 @ id @ X341 @ id )
= ( X340 @ X341 ) )
=> ( ! [X342: term > $o,X343: term] :
( ~ ! [X344: term,X345: term] :
( ( $true
= ( X342 @ ( ap @ X345 @ X344 ) ) )
| ( ( X342 @ X344 )
!= $true )
| ( $true
!= ( X342 @ X345 ) ) )
| ( ( X342 @ X343 )
= $true )
| ~ ! [X346: term] :
( ~ ! [X347: term] :
( ( $true
!= ( X342 @ X347 ) )
| ( $true
= ( X342 @ ( sub @ X346 @ ( push @ X347 @ id ) ) ) ) )
| ( $true
= ( X342 @ ( lam @ X346 ) ) ) )
| ~ ! [X348: term] :
( ( $true
!= ( var @ X348 ) )
| ( ( X342 @ X348 )
= $true ) ) )
=> ( ! [X349: term] :
( ( sub @ X349 @ id )
= X349 )
=> ( ! [X350: subst > term > subst > $o,X351: term,X352: term > $o] :
( ~ ! [X353: term] :
( ( X352 @ X353 )
= ( X350 @ id @ X353 @ id ) )
| ~ ! [X354: term] :
( ( $true
= ( X352 @ ( sub @ X351 @ ( push @ X354 @ id ) ) ) )
| ( $true
!= ( X352 @ X354 ) ) )
| ( $true
= ( X352 @ ( lam @ X351 ) ) )
| ~ ! [X355: subst > term > term] :
( ( $true
= ( X350 @ id @ ( lam @ ( X355 @ sh @ one ) ) @ id ) )
| ~ ! [X356: term] :
( ( $true
= ( X350 @ id @ ( X355 @ id @ X356 ) @ id ) )
| ( $true
!= ( X350 @ id @ X356 @ id ) ) )
| ~ ! [X357: subst,X358: term,X359: subst] :
( ( X355 @ ( comp @ X357 @ X359 ) @ ( sub @ X358 @ X359 ) )
= ( sub @ ( X355 @ X357 @ X358 ) @ X359 ) ) ) )
=> ! [X360: subst > term > subst > $o,X361: term] :
( ~ ! [X362: subst,X363: subst,X364: subst,X365: term] :
( ( $true
!= ( X360 @ ( comp @ X362 @ X363 ) @ ( sub @ X365 @ X363 ) @ X364 ) )
| ( $true
= ( X360 @ X362 @ X365 @ ( comp @ X363 @ X364 ) ) ) )
| ~ ! [X366: subst > term > term] :
( ~ ! [X367: subst,X368: term,X369: subst] :
( ( sub @ ( X366 @ X369 @ X368 ) @ X367 )
= ( X366 @ ( comp @ X369 @ X367 ) @ ( sub @ X368 @ X367 ) ) )
| ( $true
= ( X360 @ id @ ( lam @ ( X366 @ sh @ one ) ) @ id ) )
| ~ ! [X370: term] :
( ( $true
!= ( X360 @ id @ X370 @ id ) )
| ( $true
= ( X360 @ id @ ( X366 @ id @ X370 ) @ id ) ) ) )
| ~ ! [X371: term,X372: term] :
( ( $true
= ( X360 @ id @ ( ap @ ( sub @ X372 @ id ) @ X371 ) @ id ) )
| ( $true
!= ( X360 @ id @ X371 @ id ) )
| ( $true
!= ( X360 @ id @ X372 @ id ) ) )
| ( $true
= ( X360 @ id @ X361 @ id ) )
| ~ ! [X373: term] :
( ( $true
!= ( var @ ( sub @ X373 @ id ) ) )
| ( $true
= ( X360 @ id @ X373 @ id ) ) )
| ~ ! [X374: subst,X375: subst,X376: subst,X377: term] :
( ( $true
!= ( X360 @ X374 @ X377 @ ( comp @ X375 @ X376 ) ) )
| ( $true
= ( X360 @ ( comp @ X374 @ X375 ) @ ( sub @ X377 @ X375 ) @ X376 ) ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ( hoaslamnotap = $true )
<=> ! [X378: term,X379: term,X380: subst > term > term] :
( ~ ! [X381: subst,X382: subst,X383: term] :
( ( X380 @ ( comp @ X382 @ X381 ) @ ( sub @ X383 @ X381 ) )
= ( sub @ ( X380 @ X382 @ X383 ) @ X381 ) )
| ( ( ap @ ( sub @ X379 @ id ) @ X378 )
!= ( lam @ ( X380 @ sh @ one ) ) ) ) )
& ( ! [X384: subst] :
( ( push @ ( sub @ one @ X384 ) @ ( comp @ sh @ X384 ) )
= X384 )
<=> ( axscons = $true ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X385: subst,X386: term > $o,X387: term] :
( ~ ! [X388: term] :
( ~ ! [X389: term] :
( ( ( X386 @ X389 )
!= $true )
| ( $true
= ( X386 @ ( sub @ X388 @ ( push @ X389 @ id ) ) ) ) )
| ( $true
= ( X386 @ ( lam @ X388 ) ) ) )
| ( $true
= ( X386 @ ( sub @ X387 @ X385 ) ) )
| ~ ! [X390: term] :
( ( $true
!= ( var @ X390 ) )
| ( $true
= ( X386 @ ( sub @ X390 @ X385 ) ) ) )
| ~ ! [X391: term,X392: term] :
( ( $true
!= ( X386 @ X391 ) )
| ( $true
!= ( X386 @ X392 ) )
| ( ( X386 @ ( ap @ X392 @ X391 ) )
= $true ) ) )
<=> ( induction2lem = $true ) )
& ( ( axidr = $true )
<=> ! [X393: subst] :
( ( comp @ X393 @ id )
= X393 ) )
& ( ! [X394: term,X395: term,X396: term,X397: term] :
( ( ( ap @ ( sub @ X397 @ id ) @ X395 )
!= ( ap @ ( sub @ X394 @ id ) @ X396 ) )
| ( X394 = X397 ) )
<=> ( hoasapinj1 = $true ) )
& ( axassoc = $true )
& ( ( ! [X398: term] :
( ( sub @ X398 @ id )
= X398 )
=> ( ! [X399: term,X400: term,X401: term,X402: term] :
( ( ( ap @ X399 @ X400 )
!= ( ap @ X402 @ X401 ) )
| ( X399 = X402 ) )
=> ! [X403: term,X404: term,X405: term,X406: term] :
( ( X405 = X406 )
| ( ( ap @ ( sub @ X405 @ id ) @ X403 )
!= ( ap @ ( sub @ X406 @ id ) @ X404 ) ) ) ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( ulamvarind = $true )
& ( lamnotap != $true )
& ( pushprop_lthm = $true )
& ( ! [X407: term,X408: subst > term > subst > $o,X409: term > $o] :
( ~ ! [X410: subst > term > term] :
( ~ ! [X411: term,X412: subst,X413: subst] :
( ( sub @ ( X410 @ X412 @ X411 ) @ X413 )
= ( X410 @ ( comp @ X412 @ X413 ) @ ( sub @ X411 @ X413 ) ) )
| ( $true
= ( X408 @ id @ ( lam @ ( X410 @ sh @ one ) ) @ id ) )
| ~ ! [X414: term] :
( ( $true
= ( X408 @ id @ ( X410 @ id @ X414 ) @ id ) )
| ( $true
!= ( X408 @ id @ X414 @ id ) ) ) )
| ~ ! [X415: term] :
( ( $true
!= ( X409 @ X415 ) )
| ( $true
= ( X409 @ ( sub @ X407 @ ( push @ X415 @ id ) ) ) ) )
| ~ ! [X416: subst,X417: subst,X418: term,X419: subst] :
( ( $true
= ( X408 @ X419 @ X418 @ ( comp @ X417 @ X416 ) ) )
| ( $true
!= ( X408 @ ( comp @ X419 @ X417 ) @ ( sub @ X418 @ X417 ) @ X416 ) ) )
| ~ ! [X420: term] :
( ( X409 @ X420 )
= ( X408 @ id @ X420 @ id ) )
| ( ( X409 @ ( lam @ X407 ) )
= $true )
| ~ ! [X421: subst,X422: term,X423: subst,X424: subst] :
( ( $true
!= ( X408 @ X424 @ X422 @ ( comp @ X421 @ X423 ) ) )
| ( $true
= ( X408 @ ( comp @ X424 @ X421 ) @ ( sub @ X422 @ X421 ) @ X423 ) ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ( d2subst @ d_id )
= id )
& ( pushprop_lem0_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ( apinj2 = $true )
<=> ! [X425: term,X426: term,X427: term,X428: term] :
( ( ( ap @ X427 @ X428 )
!= ( ap @ X425 @ X426 ) )
| ( X426 = X428 ) ) )
& ( ! [X429: term,X430: term > $o] :
( ( $true
= ( X430 @ X429 ) )
| ~ ! [X431: term] :
( ( $true
= ( X430 @ X431 ) )
| ( $true
!= ( var @ X431 ) ) )
| ~ ! [X432: term] :
( ( $true
!= ( X430 @ X432 ) )
| ( $true
= ( X430 @ ( lam @ X432 ) ) ) )
| ~ ! [X433: term,X434: term] :
( ( $true
!= ( X430 @ X433 ) )
| ( ( X430 @ ( ap @ X433 @ X434 ) )
= $true )
| ( $true
!= ( X430 @ X434 ) ) ) )
<=> ( induction = $true ) )
& ( ( ! [X435: subst > term > subst > $o] :
~ ! [X436: term > $o] :
~ ! [X437: term] :
( ( X436 @ X437 )
= ( X435 @ id @ X437 @ id ) )
=> ( ! [X438: term,X439: term > $o] :
( ~ ! [X440: term] :
( ( $true
!= ( var @ X440 ) )
| ( $true
= ( X439 @ X440 ) ) )
| ( $true
= ( X439 @ X438 ) )
| ~ ! [X441: term] :
( ( ( X439 @ ( lam @ X441 ) )
= $true )
| ~ ! [X442: term] :
( ( $true
!= ( X439 @ X442 ) )
| ( $true
= ( X439 @ ( sub @ X441 @ ( push @ X442 @ id ) ) ) ) ) )
| ~ ! [X443: term,X444: term] :
( ( ( X439 @ X443 )
!= $true )
| ( $true
!= ( X439 @ X444 ) )
| ( ( X439 @ ( ap @ X444 @ X443 ) )
= $true ) ) )
=> ( ! [X445: term] :
( ( sub @ X445 @ id )
= X445 )
=> ( ! [X446: term > $o,X447: subst > term > subst > $o,X448: term] :
( ~ ! [X449: term] :
( ( X447 @ id @ X449 @ id )
= ( X446 @ X449 ) )
| ~ ! [X450: term] :
( ( $true
!= ( X446 @ X450 ) )
| ( ( X446 @ ( sub @ X448 @ ( push @ X450 @ id ) ) )
= $true ) )
| ~ ! [X451: subst > term > term] :
( ~ ! [X452: term,X453: subst,X454: subst] :
( ( sub @ ( X451 @ X453 @ X452 ) @ X454 )
= ( X451 @ ( comp @ X453 @ X454 ) @ ( sub @ X452 @ X454 ) ) )
| ( $true
= ( X447 @ id @ ( lam @ ( X451 @ sh @ one ) ) @ id ) )
| ~ ! [X455: term] :
( ( $true
!= ( X447 @ id @ X455 @ id ) )
| ( $true
= ( X447 @ id @ ( X451 @ id @ X455 ) @ id ) ) ) )
| ( $true
= ( X446 @ ( lam @ X448 ) ) ) )
=> ! [X456: subst > term > subst > $o,X457: term] :
( ( $true
= ( X456 @ id @ X457 @ id ) )
| ~ ! [X458: subst > term > term] :
( ( $true
= ( X456 @ id @ ( lam @ ( X458 @ sh @ one ) ) @ id ) )
| ~ ! [X459: term] :
( ( $true
= ( X456 @ id @ ( X458 @ id @ X459 ) @ id ) )
| ( $true
!= ( X456 @ id @ X459 @ id ) ) )
| ~ ! [X460: term,X461: subst,X462: subst] :
( ( X458 @ ( comp @ X462 @ X461 ) @ ( sub @ X460 @ X461 ) )
= ( sub @ ( X458 @ X462 @ X460 ) @ X461 ) ) )
| ~ ! [X463: term,X464: term] :
( ( ( X456 @ id @ X464 @ id )
!= $true )
| ( ( X456 @ id @ ( ap @ ( sub @ X463 @ id ) @ X464 ) @ id )
= $true )
| ( $true
!= ( X456 @ id @ X463 @ id ) ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X465: term,X466: term > $o] :
( ~ ! [X467: term] :
( ~ ! [X468: term] :
( ( $true
!= ( X466 @ X468 ) )
| ( ( X466 @ ( sub @ X467 @ ( push @ X468 @ id ) ) )
= $true ) )
| ( $true
= ( X466 @ ( lam @ X467 ) ) ) )
| ( $true
= ( X466 @ X465 ) )
| ~ ! [X469: term] :
( ( $true
!= ( var @ X469 ) )
| ( ( X466 @ X469 )
= $true ) )
| ~ ! [X470: term,X471: term] :
( ( $true
= ( X466 @ ( ap @ X470 @ X471 ) ) )
| ( $true
!= ( X466 @ X471 ) )
| ( $true
!= ( X466 @ X470 ) ) ) )
=> ( ! [X472: subst > term > subst > $o,X473: term,X474: term] :
( ( $true
!= ( X472 @ id @ X473 @ id ) )
| ( $true
= ( X472 @ id @ ( ap @ X473 @ X474 ) @ id ) )
| ~ ! [X475: subst,X476: term,X477: subst,X478: subst] :
( ( $true
= ( X472 @ ( comp @ X478 @ X477 ) @ ( sub @ X476 @ X477 ) @ X475 ) )
| ( $true
!= ( X472 @ X478 @ X476 @ ( comp @ X477 @ X475 ) ) ) )
| ( $true
!= ( X472 @ id @ X474 @ id ) )
| ~ ! [X479: term,X480: subst,X481: subst,X482: subst] :
( ( $true
= ( X472 @ X480 @ X479 @ ( comp @ X481 @ X482 ) ) )
| ( $true
!= ( X472 @ ( comp @ X480 @ X481 ) @ ( sub @ X479 @ X481 ) @ X482 ) ) )
| ~ ! [X483: term,X484: term] :
( ( $true
= ( X472 @ id @ ( ap @ ( sub @ X483 @ id ) @ X484 ) @ id ) )
| ( ( X472 @ id @ X484 @ id )
!= $true )
| ( $true
!= ( X472 @ id @ X483 @ id ) ) ) )
=> ( ! [X485: term,X486: subst > term > subst > $o] :
( ~ ! [X487: subst,X488: subst,X489: subst,X490: term] :
( ( $true
!= ( X486 @ ( comp @ X488 @ X489 ) @ ( sub @ X490 @ X489 ) @ X487 ) )
| ( ( X486 @ X488 @ X490 @ ( comp @ X489 @ X487 ) )
= $true ) )
| ( $true
= ( X486 @ id @ ( lam @ X485 ) @ id ) )
| ~ ! [X491: subst,X492: term,X493: subst,X494: subst] :
( ( ( X486 @ ( comp @ X493 @ X494 ) @ ( sub @ X492 @ X494 ) @ X491 )
= $true )
| ( $true
!= ( X486 @ X493 @ X492 @ ( comp @ X494 @ X491 ) ) ) )
| ~ ! [X495: subst > term > term] :
( ~ ! [X496: term] :
( ( $true
!= ( X486 @ id @ X496 @ id ) )
| ( $true
= ( X486 @ id @ ( X495 @ id @ X496 ) @ id ) ) )
| ( $true
= ( X486 @ id @ ( lam @ ( X495 @ sh @ one ) ) @ id ) )
| ~ ! [X497: term,X498: subst,X499: subst] :
( ( X495 @ ( comp @ X498 @ X499 ) @ ( sub @ X497 @ X499 ) )
= ( sub @ ( X495 @ X498 @ X497 ) @ X499 ) ) )
| ~ ! [X500: term] :
( ( $true
!= ( X486 @ id @ X500 @ id ) )
| ( ( X486 @ id @ ( sub @ X485 @ ( push @ X500 @ id ) ) @ id )
= $true ) ) )
=> ! [X501: term,X502: subst > term > subst > $o] :
( ~ ! [X503: subst,X504: subst,X505: term,X506: subst] :
( ( $true
!= ( X502 @ ( comp @ X504 @ X506 ) @ ( sub @ X505 @ X506 ) @ X503 ) )
| ( $true
= ( X502 @ X504 @ X505 @ ( comp @ X506 @ X503 ) ) ) )
| ~ ! [X507: subst > term > term] :
( ( $true
= ( X502 @ id @ ( lam @ ( X507 @ sh @ one ) ) @ id ) )
| ~ ! [X508: term] :
( ( $true
= ( X502 @ id @ ( X507 @ id @ X508 ) @ id ) )
| ( ( X502 @ id @ X508 @ id )
!= $true ) )
| ~ ! [X509: term,X510: subst,X511: subst] :
( ( X507 @ ( comp @ X511 @ X510 ) @ ( sub @ X509 @ X510 ) )
= ( sub @ ( X507 @ X511 @ X509 ) @ X510 ) ) )
| ~ ! [X512: term,X513: subst,X514: subst,X515: subst] :
( ( ( X502 @ X515 @ X512 @ ( comp @ X514 @ X513 ) )
!= $true )
| ( $true
= ( X502 @ ( comp @ X515 @ X514 ) @ ( sub @ X512 @ X514 ) @ X513 ) ) )
| ( ( X502 @ id @ X501 @ id )
= $true )
| ~ ! [X516: term,X517: term] :
( ( ( X502 @ id @ X516 @ id )
!= $true )
| ( ( X502 @ id @ X517 @ id )
!= $true )
| ( $true
= ( X502 @ id @ ( ap @ ( sub @ X516 @ id ) @ X517 ) @ id ) ) )
| ~ ! [X518: term] :
( ( ( X502 @ id @ X518 @ id )
= $true )
| ( $true
!= ( var @ ( sub @ X518 @ id ) ) ) ) ) ) ) ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X519: term] :
~ ! [X520: subst > term > term] :
( ( sub @ X519 @ ( push @ one @ sh ) )
!= ( X520 @ sh @ one ) ) )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X521: subst] :
( ( comp @ id @ X521 )
= X521 )
=> ( ! [X522: subst] :
( ( comp @ X522 @ id )
= X522 )
=> ( ! [X523: subst] :
( ( comp @ id @ X523 )
= X523 )
& ! [X524: subst] :
( ( comp @ X524 @ id )
= X524 ) ) ) ) )
& ( hoaslamnotvar = $true )
& ( ( ! [X525: term] :
~ ! [X526: subst > term > term] :
~ ! [X527: subst,X528: term] :
( ( X526 @ X527 @ X528 )
= ( sub @ X525 @ ( push @ X528 @ X527 ) ) )
=> ( ! [X529: term] :
( ( sub @ X529 @ id )
= X529 )
=> ! [X530: term,X531: term > $o,X532: subst > term > subst > $o] :
( ( $true
= ( X531 @ ( lam @ X530 ) ) )
| ~ ! [X533: term] :
( ( X532 @ id @ X533 @ id )
= ( X531 @ X533 ) )
| ~ ! [X534: subst > term > term] :
( ~ ! [X535: term] :
( ( ( X532 @ id @ ( X534 @ id @ X535 ) @ id )
= $true )
| ( $true
!= ( X532 @ id @ X535 @ id ) ) )
| ( $true
= ( X532 @ id @ ( lam @ ( X534 @ sh @ one ) ) @ id ) )
| ~ ! [X536: subst,X537: term,X538: subst] :
( ( X534 @ ( comp @ X538 @ X536 ) @ ( sub @ X537 @ X536 ) )
= ( sub @ ( X534 @ X538 @ X537 ) @ X536 ) ) )
| ~ ! [X539: term] :
( ( $true
= ( X531 @ ( sub @ X530 @ ( push @ X539 @ id ) ) ) )
| ( $true
!= ( X531 @ X539 ) ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,plain,
( ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
<=> ( laminj = $true ) )
& ( ( ! [X2: subst,X3: term] :
( ( sub @ one @ ( push @ X3 @ X2 ) )
= X3 )
=> ( ! [X4: term,X5: subst] :
( ( comp @ sh @ ( push @ X4 @ X5 ) )
= X5 )
=> ( ! [X6: subst] :
( ( comp @ X6 @ id )
= X6 )
=> ( ! [X7: term > $o,X8: term] :
( ~ ! [X9: term] :
( ( ( X7 @ ( lam @ X9 ) )
= $true )
| ( $true
!= ( X7 @ X9 ) ) )
| ~ ! [X10: term] :
( ( ( var @ X10 )
!= $true )
| ( $true
= ( X7 @ X10 ) ) )
| ~ ! [X11: term,X12: term] :
( ( $true
= ( X7 @ ( ap @ X11 @ X12 ) ) )
| ( $true
!= ( X7 @ X11 ) )
| ( $true
!= ( X7 @ X12 ) ) )
| ( ( X7 @ X8 )
= $true ) )
=> ! [X13: term > $o,X14: term,X15: subst] :
( ~ ! [X16: term,X17: term] :
( ( $true
= ( X13 @ ( ap @ X17 @ X16 ) ) )
| ( $true
!= ( X13 @ X17 ) )
| ( $true
!= ( X13 @ X16 ) ) )
| ~ ! [X18: term] :
( ( $true
= ( X13 @ ( lam @ X18 ) ) )
| ~ ! [X19: term] :
( ( ( X13 @ X19 )
!= $true )
| ( $true
= ( X13 @ ( sub @ X18 @ ( push @ X19 @ id ) ) ) ) ) )
| ( $true
= ( X13 @ ( sub @ X14 @ X15 ) ) )
| ~ ! [X20: term] :
( ( $true
!= ( var @ X20 ) )
| ( $true
= ( X13 @ ( sub @ X20 @ X15 ) ) ) ) ) ) ) ) )
<=> ( induction2lem_lthm = $true ) )
& ( ( pushprop_lem0 = $true )
<=> ! [X21: subst,X22: term,X23: term > $o] :
~ ! [X24: term > $o] :
~ ! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) ) )
& ( ( ! [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
= X27 )
=> ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ~ ! [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
= ( X28 @ X32 ) )
| ( ( X30 @ X29 )
!= $true )
| ( $true
= ( X28 @ one ) ) ) )
<=> ( pushprop_lem1v2_lthm = $true ) )
& ( hoasinduction_lem2_lthm = $true )
& ( ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) )
<=> ( substmonoid = $true ) )
& ( pushprop_lem3v2_lthm = $true )
& ( lamnotvar = $true )
& ( ( hoasinduction_lem3v2_lthm = $true )
<=> ( ! [X35: term] :
( ( sub @ X35 @ id )
= X35 )
=> ! [X36: subst > term > subst > $o,X37: term,X38: term > $o] :
( ~ ! [X39: term,X40: subst,X41: subst,X42: subst] :
( ( ( X36 @ X42 @ X39 @ ( comp @ X41 @ X40 ) )
= $true )
| ( $true
!= ( X36 @ ( comp @ X42 @ X41 ) @ ( sub @ X39 @ X41 ) @ X40 ) ) )
| ~ ! [X43: term] :
( ( X36 @ id @ X43 @ id )
= ( X38 @ X43 ) )
| ( ( X38 @ ( lam @ X37 ) )
= $true )
| ~ ! [X44: subst > term > term] :
( ~ ! [X45: term,X46: subst,X47: subst] :
( ( sub @ ( X44 @ X47 @ X45 ) @ X46 )
= ( X44 @ ( comp @ X47 @ X46 ) @ ( sub @ X45 @ X46 ) ) )
| ( $true
= ( X36 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id ) )
| ~ ! [X48: term] :
( ( $true
!= ( X36 @ id @ X48 @ id ) )
| ( $true
= ( X36 @ id @ ( X44 @ id @ X48 ) @ id ) ) ) )
| ~ ! [X49: term] :
( ( $true
!= ( X38 @ X49 ) )
| ( $true
= ( X38 @ ( sub @ X37 @ ( push @ X49 @ id ) ) ) ) )
| ~ ! [X50: subst,X51: subst,X52: term,X53: subst] :
( ( ( X36 @ X53 @ X52 @ ( comp @ X50 @ X51 ) )
!= $true )
| ( $true
= ( X36 @ ( comp @ X53 @ X50 ) @ ( sub @ X52 @ X50 ) @ X51 ) ) ) ) ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ~ ! [X56: term,X57: term] :
( ( $true
!= ( X54 @ id @ X57 @ id ) )
| ( ( X54 @ id @ X56 @ id )
!= $true )
| ( $true
= ( X54 @ id @ ( ap @ ( sub @ X56 @ id ) @ X57 ) @ id ) ) )
| ( $true
= ( X54 @ id @ X55 @ id ) )
| ~ ! [X58: subst > term > term] :
( ( $true
= ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id ) )
| ~ ! [X59: term] :
( ( $true
= ( X54 @ id @ ( X58 @ id @ X59 ) @ id ) )
| ( $true
!= ( X54 @ id @ X59 @ id ) ) )
| ~ ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) ) )
<=> ( hoasinduction_no_psi_cond = $true ) )
& ( substmonoid_gthm = $true )
& ( hoaslamnotvar_gthm = $true )
& ( ( hoasinduction_lem3_lthm = $true )
<=> ( ! [X63: term] :
( ( sub @ X63 @ id )
= X63 )
=> ( ! [X64: term,X65: subst > term > subst > $o] :
( ( $true
= ( X65 @ id @ ( lam @ ( sub @ X64 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X66: term] :
( ( $true
= ( X65 @ id @ ( sub @ X64 @ ( push @ X66 @ id ) ) @ id ) )
| ( $true
!= ( X65 @ id @ X66 @ id ) ) )
| ~ ! [X67: subst > term > term] :
( ( $true
= ( X65 @ id @ ( lam @ ( X67 @ sh @ one ) ) @ id ) )
| ~ ! [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X67 @ X69 @ X70 ) @ X68 )
= ( X67 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
| ~ ! [X71: term] :
( ( $true
!= ( X65 @ id @ X71 @ id ) )
| ( $true
= ( X65 @ id @ ( X67 @ id @ X71 ) @ id ) ) ) ) )
=> ! [X72: subst > term > subst > $o,X73: term] :
( ~ ! [X74: term] :
( ( $true
= ( X72 @ id @ ( sub @ X73 @ ( push @ X74 @ id ) ) @ id ) )
| ( ( X72 @ id @ X74 @ id )
!= $true ) )
| ~ ! [X75: subst,X76: term,X77: subst,X78: subst] :
( ( ( X72 @ ( comp @ X78 @ X77 ) @ ( sub @ X76 @ X77 ) @ X75 )
= $true )
| ( $true
!= ( X72 @ X78 @ X76 @ ( comp @ X77 @ X75 ) ) ) )
| ~ ! [X79: term,X80: subst,X81: subst,X82: subst] :
( ( ( X72 @ ( comp @ X80 @ X82 ) @ ( sub @ X79 @ X82 ) @ X81 )
!= $true )
| ( $true
= ( X72 @ X80 @ X79 @ ( comp @ X82 @ X81 ) ) ) )
| ~ ! [X83: subst > term > term] :
( ~ ! [X84: subst,X85: term,X86: subst] :
( ( X83 @ ( comp @ X86 @ X84 ) @ ( sub @ X85 @ X84 ) )
= ( sub @ ( X83 @ X86 @ X85 ) @ X84 ) )
| ( $true
= ( X72 @ id @ ( lam @ ( X83 @ sh @ one ) ) @ id ) )
| ~ ! [X87: term] :
( ( ( X72 @ id @ ( X83 @ id @ X87 ) @ id )
= $true )
| ( $true
!= ( X72 @ id @ X87 @ id ) ) ) )
| ( $true
= ( X72 @ id @ ( lam @ X73 ) @ id ) ) ) ) ) )
& ( ( induction2 = $true )
<=> ! [X88: term > $o,X89: term] :
( ~ ! [X90: term] :
( ( ( X88 @ X90 )
= $true )
| ( $true
!= ( var @ X90 ) ) )
| ~ ! [X91: term] :
( ( $true
= ( X88 @ ( lam @ X91 ) ) )
| ~ ! [X92: term] :
( ( $true
!= ( X88 @ X92 ) )
| ( $true
= ( X88 @ ( sub @ X91 @ ( push @ X92 @ id ) ) ) ) ) )
| ( $true
= ( X88 @ X89 ) )
| ~ ! [X93: term,X94: term] :
( ( $true
!= ( X88 @ X94 ) )
| ( $true
= ( X88 @ ( ap @ X93 @ X94 ) ) )
| ( $true
!= ( X88 @ X93 ) ) ) ) )
& ( ulamvar1 != $true )
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem1v2_gthm = $true )
& ( ! [X95: term] :
~ ! [X96: subst > term > term] :
~ ! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) )
<=> ( hoasinduction_lem3v2_f = $true ) )
& ( hoasinduction_lem3_gthm = $true )
& ( ( hoaslaminj_lthm = $true )
<=> ( ! [X99: term,X100: subst] :
( ( sub @ one @ ( push @ X99 @ X100 ) )
= X99 )
=> ( ! [X101: term,X102: subst] :
( ( comp @ sh @ ( push @ X101 @ X102 ) )
= X102 )
=> ( ! [X103: term,X104: term] :
( ( X103 = X104 )
| ( ( lam @ X103 )
!= ( lam @ X104 ) ) )
=> ! [X105: term,X106: subst > term > term,X107: subst > term > term,X108: subst] :
( ~ ! [X109: subst,X110: subst,X111: term] :
( ( sub @ ( X107 @ X109 @ X111 ) @ X110 )
= ( X107 @ ( comp @ X109 @ X110 ) @ ( sub @ X111 @ X110 ) ) )
| ~ ! [X112: subst,X113: term,X114: subst] :
( ( X106 @ ( comp @ X114 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X106 @ X114 @ X113 ) @ X112 ) )
| ( ( X106 @ X108 @ X105 )
= ( X107 @ X108 @ X105 ) )
| ( ( lam @ ( X106 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) ) ) ) ) ) )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X115: subst > term > subst > $o,X116: term] :
( ( $true
= ( X115 @ id @ ( lam @ ( sub @ X116 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X117: subst > term > term] :
( ( $true
= ( X115 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id ) )
| ~ ! [X118: subst,X119: subst,X120: term] :
( ( sub @ ( X117 @ X118 @ X120 ) @ X119 )
= ( X117 @ ( comp @ X118 @ X119 ) @ ( sub @ X120 @ X119 ) ) )
| ~ ! [X121: term] :
( ( $true
= ( X115 @ id @ ( X117 @ id @ X121 ) @ id ) )
| ( ( X115 @ id @ X121 @ id )
!= $true ) ) )
| ~ ! [X122: term] :
( ( $true
= ( X115 @ id @ ( sub @ X116 @ ( push @ X122 @ id ) ) @ id ) )
| ( $true
!= ( X115 @ id @ X122 @ id ) ) ) )
<=> ( hoasinduction_lem3aa = $true ) )
& ( ( hoasinduction_lem2v2 = $true )
<=> ! [X123: term > $o,X124: term,X125: subst > term > subst > $o,X126: term] :
( ~ ! [X127: term,X128: term] :
( ( $true
!= ( X125 @ id @ X128 @ id ) )
| ( $true
!= ( X125 @ id @ X127 @ id ) )
| ( ( X125 @ id @ ( ap @ ( sub @ X127 @ id ) @ X128 ) @ id )
= $true ) )
| ( $true
!= ( X123 @ X124 ) )
| ( $true
!= ( X123 @ X126 ) )
| ~ ! [X129: subst,X130: subst,X131: term,X132: subst] :
( ( $true
!= ( X125 @ ( comp @ X129 @ X132 ) @ ( sub @ X131 @ X132 ) @ X130 ) )
| ( $true
= ( X125 @ X129 @ X131 @ ( comp @ X132 @ X130 ) ) ) )
| ~ ! [X133: subst,X134: subst,X135: term,X136: subst] :
( ( $true
= ( X125 @ ( comp @ X136 @ X134 ) @ ( sub @ X135 @ X134 ) @ X133 ) )
| ( ( X125 @ X136 @ X135 @ ( comp @ X134 @ X133 ) )
!= $true ) )
| ( ( X123 @ ( ap @ X126 @ X124 ) )
= $true )
| ~ ! [X137: term] :
( ( X125 @ id @ X137 @ id )
= ( X123 @ X137 ) ) ) )
& ( axapp = $true )
& ( ! [X138: subst > term > subst > $o] :
~ ! [X139: term > $o] :
~ ! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) )
<=> ( hoasinduction_lem0 = $true ) )
& ( pushprop_lem2v2_lthm = $true )
& ( hoasapinj1_gthm = $true )
& ( hoasapinj2_gthm = $true )
& ! [X141: d_term] : ( d_one = X141 )
& ( hoasinduction_lem3a_gthm = $true )
& ( pushprop_lem3v2 = $true )
& ( ! [X142: subst > term > subst > $o,X143: term] :
( ( ( X142 @ id @ ( lam @ ( sub @ X143 @ ( push @ one @ sh ) ) ) @ id )
= $true )
| ~ ! [X144: term,X145: subst > term > term] :
( ~ ! [X146: subst,X147: term,X148: subst] :
( ( X145 @ ( comp @ X146 @ X148 ) @ ( sub @ X147 @ X148 ) )
= ( sub @ X144 @ ( push @ ( sub @ X147 @ X148 ) @ ( comp @ X146 @ X148 ) ) ) )
| ~ ! [X149: term] :
( ( $true
= ( X142 @ id @ ( X145 @ id @ X149 ) @ id ) )
| ( $true
!= ( X142 @ id @ X149 @ id ) ) )
| ( ( X142 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X150: term,X151: subst,X152: subst] :
( ( sub @ ( X145 @ X152 @ X150 ) @ X151 )
= ( sub @ ( sub @ X144 @ ( push @ X150 @ X152 ) ) @ X151 ) ) )
| ~ ! [X153: term] :
( ( $true
!= ( X142 @ id @ X153 @ id ) )
| ( $true
= ( X142 @ id @ ( sub @ X143 @ ( push @ X153 @ id ) ) @ id ) ) ) )
<=> ( hoasinduction_lem3aaa = $true ) )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ! [X156: d_subst] : ( d_id = X156 )
& ( ! [X157: subst > term > subst > $o,X158: term] :
( ~ ! [X159: term] :
( ( $true
!= ( X157 @ id @ X159 @ id ) )
| ( ( X157 @ id @ ( sub @ X158 @ ( push @ X159 @ id ) ) @ id )
= $true ) )
| ( $true
= ( X157 @ id @ ( lam @ X158 ) @ id ) )
| ~ ! [X160: subst > term > term] :
( ~ ! [X161: term] :
( ( ( X157 @ id @ X161 @ id )
!= $true )
| ( $true
= ( X157 @ id @ ( X160 @ id @ X161 ) @ id ) ) )
| ( ( X157 @ id @ ( lam @ ( X160 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X162: term,X163: subst,X164: subst] :
( ( sub @ ( X160 @ X163 @ X162 ) @ X164 )
= ( X160 @ ( comp @ X163 @ X164 ) @ ( sub @ X162 @ X164 ) ) ) ) )
<=> ( hoasinduction_lem3a = $true ) )
& ( ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
!= $true )
& ( hoasapnotvar_lthm = $true )
& ( ( pushprop_lem1 = $true )
<=> ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( $true
= ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) ) )
| ( $true
!= ( X169 @ X167 ) ) ) )
& ( hoasapnotvar = $true )
& ( hoaslamnotap_lthm = $true )
& ( hoasinduction_gthm = $true )
& ( axabs = $true )
& ( induction2_gthm = $true )
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& ( hoaslaminj_gthm = $true )
& ( ( hoasapinj2 = $true )
<=> ! [X172: term,X173: term,X174: term,X175: term] :
( ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) )
| ( X174 = X175 ) ) )
& ( ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) )
<=> ( apinj1 = $true ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem1 = $true )
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& ( pushprop_lem2v2 = $true )
& ( ! [X180: term > $o,X181: term,X182: subst] :
~ ! [X183: term > $o] :
~ ! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) )
<=> ( pushprop_lem0_lthm = $true ) )
& ( pushprop = $true )
& ( hoasinduction_lem3v2_gthm = $true )
& ( ( shinj = $true )
<=> ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) ) )
& ( ! [X187: subst > term > subst > $o] :
~ ! [X188: term > $o] :
~ ! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
<=> ( hoasinduction_lem0_lthm = $true ) )
& ( hoaslamnotvar_lthm = $true )
& ( hoasinduction_p_and_p_prime
= ( ^ [Y0: subst > term > subst > $o,Y1: term > $o] :
( !! @ term
@ ^ [Y2: term] :
( ( Y0 @ id @ Y2 @ id )
= ( Y1 @ Y2 ) ) ) ) )
& ( induction2lem_gthm = $true )
& ( termmset_gthm = $true )
& ( ( hoasapinj2_lthm = $true )
<=> ( ! [X193: term,X194: term,X195: term,X196: term] :
( ( X193 = X195 )
| ( ( ap @ X196 @ X195 )
!= ( ap @ X194 @ X193 ) ) )
=> ! [X197: term,X198: term,X199: term,X200: term] :
( ( X197 = X198 )
| ( ( ap @ ( sub @ X199 @ id ) @ X197 )
!= ( ap @ ( sub @ X200 @ id ) @ X198 ) ) ) ) )
& ( axmap = $true )
& ( axclos = $true )
& ( one
= ( d2term @ d_one ) )
& ( ! [X201: subst,X202: term] :
( ( comp @ sh @ ( push @ X202 @ X201 ) )
= X201 )
<=> ( axshiftcons = $true ) )
& ( hoasapnotvar_gthm = $true )
& ( axvarshift = $true )
& ( ! [X203: term,X204: subst > term > subst > $o] :
( ( $true
= ( X204 @ id @ ( lam @ ( sub @ X203 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X205: subst > term > term] :
( ( ( X204 @ id @ ( lam @ ( X205 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X206: term] :
( ( $true
= ( X204 @ id @ ( X205 @ id @ X206 ) @ id ) )
| ( $true
!= ( X204 @ id @ X206 @ id ) ) )
| ~ ! [X207: term,X208: subst,X209: subst] :
( ( sub @ ( X205 @ X209 @ X207 ) @ X208 )
= ( X205 @ ( comp @ X209 @ X208 ) @ ( sub @ X207 @ X208 ) ) ) )
| ~ ! [X210: term] :
( ( $true
!= ( X204 @ id @ X210 @ id ) )
| ( $true
= ( X204 @ id @ ( sub @ X203 @ ( push @ X210 @ id ) ) @ id ) ) ) )
<=> ( hoasinduction_lem3aa_lthm = $true ) )
& ( ( ! [X211: term] :
( ( sub @ X211 @ id )
= X211 )
=> ! [X212: term] :
( ( sub @ X212 @ id )
= X212 ) )
<=> ( termmset_lthm = $true ) )
& ( ! [X213: subst > term > subst > $o,X214: term,X215: term] :
( ( $true
!= ( X213 @ id @ X215 @ id ) )
| ( $true
!= ( X213 @ id @ X214 @ id ) )
| ~ ! [X216: term,X217: subst,X218: subst,X219: subst] :
( ( $true
!= ( X213 @ X218 @ X216 @ ( comp @ X219 @ X217 ) ) )
| ( ( X213 @ ( comp @ X218 @ X219 ) @ ( sub @ X216 @ X219 ) @ X217 )
= $true ) )
| ~ ! [X220: subst,X221: subst,X222: subst,X223: term] :
( ( ( X213 @ X222 @ X223 @ ( comp @ X221 @ X220 ) )
= $true )
| ( ( X213 @ ( comp @ X222 @ X221 ) @ ( sub @ X223 @ X221 ) @ X220 )
!= $true ) )
| ~ ! [X224: term,X225: term] :
( ( $true
= ( X213 @ id @ ( ap @ ( sub @ X224 @ id ) @ X225 ) @ id ) )
| ( $true
!= ( X213 @ id @ X224 @ id ) )
| ( $true
!= ( X213 @ id @ X225 @ id ) ) )
| ( $true
= ( X213 @ id @ ( ap @ X215 @ X214 ) @ id ) ) )
<=> ( hoasinduction_lem2 = $true ) )
& ( ( axidl = $true )
<=> ! [X226: subst] :
( ( comp @ id @ X226 )
= X226 ) )
& ( ( induction2_lthm = $true )
<=> ( ! [X227: term] :
( ( sub @ X227 @ id )
= X227 )
=> ( ! [X228: term,X229: term > $o,X230: subst] :
( ~ ! [X231: term] :
( ( $true
!= ( var @ X231 ) )
| ( $true
= ( X229 @ ( sub @ X231 @ X230 ) ) ) )
| ( $true
= ( X229 @ ( sub @ X228 @ X230 ) ) )
| ~ ! [X232: term] :
( ~ ! [X233: term] :
( ( ( X229 @ ( sub @ X232 @ ( push @ X233 @ id ) ) )
= $true )
| ( $true
!= ( X229 @ X233 ) ) )
| ( $true
= ( X229 @ ( lam @ X232 ) ) ) )
| ~ ! [X234: term,X235: term] :
( ( $true
!= ( X229 @ X234 ) )
| ( ( X229 @ X235 )
!= $true )
| ( $true
= ( X229 @ ( ap @ X235 @ X234 ) ) ) ) )
=> ! [X236: term,X237: term > $o] :
( ( $true
= ( X237 @ X236 ) )
| ~ ! [X238: term] :
( ( ( X237 @ ( lam @ X238 ) )
= $true )
| ~ ! [X239: term] :
( ( $true
!= ( X237 @ X239 ) )
| ( $true
= ( X237 @ ( sub @ X238 @ ( push @ X239 @ id ) ) ) ) ) )
| ~ ! [X240: term,X241: term] :
( ( $true
!= ( X237 @ X241 ) )
| ( $true
= ( X237 @ ( ap @ X241 @ X240 ) ) )
| ( $true
!= ( X237 @ X240 ) ) )
| ~ ! [X242: term] :
( ( ( X237 @ X242 )
= $true )
| ( $true
!= ( var @ X242 ) ) ) ) ) ) )
& ( hoasinduction_lem3b_gthm = $true )
& ( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) )
& ( ! [X248: term,X249: subst,X250: term > $o,X251: term > $o] :
( ( $true
= ( X250 @ one ) )
| ( $true
!= ( X251 @ X248 ) )
| ~ ! [X252: term] :
( ( X251 @ ( sub @ X252 @ ( push @ X248 @ X249 ) ) )
= ( X250 @ X252 ) ) )
<=> ( pushprop_lem1v2 = $true ) )
& ( ulamvarsh = $true )
& ( hoasinduction_lem1_gthm = $true )
& ( ! [X253: term,X254: subst] :
( ( sub @ one @ ( push @ X253 @ X254 ) )
= X253 )
<=> ( axvarcons = $true ) )
& ( pushprop_gthm = $true )
& ( ( hoasinduction_lem3 = $true )
<=> ! [X255: subst > term > subst > $o,X256: term] :
( ( $true
= ( X255 @ id @ ( lam @ X256 ) @ id ) )
| ~ ! [X257: subst,X258: subst,X259: term,X260: subst] :
( ( $true
!= ( X255 @ ( comp @ X260 @ X257 ) @ ( sub @ X259 @ X257 ) @ X258 ) )
| ( $true
= ( X255 @ X260 @ X259 @ ( comp @ X257 @ X258 ) ) ) )
| ~ ! [X261: subst > term > term] :
( ( $true
= ( X255 @ id @ ( lam @ ( X261 @ sh @ one ) ) @ id ) )
| ~ ! [X262: subst,X263: term,X264: subst] :
( ( sub @ ( X261 @ X264 @ X263 ) @ X262 )
= ( X261 @ ( comp @ X264 @ X262 ) @ ( sub @ X263 @ X262 ) ) )
| ~ ! [X265: term] :
( ( $true
= ( X255 @ id @ ( X261 @ id @ X265 ) @ id ) )
| ( $true
!= ( X255 @ id @ X265 @ id ) ) ) )
| ~ ! [X266: term,X267: subst,X268: subst,X269: subst] :
( ( $true
!= ( X255 @ X269 @ X266 @ ( comp @ X268 @ X267 ) ) )
| ( $true
= ( X255 @ ( comp @ X269 @ X268 ) @ ( sub @ X266 @ X268 ) @ X267 ) ) )
| ~ ! [X270: term] :
( ( $true
= ( X255 @ id @ ( sub @ X256 @ ( push @ X270 @ id ) ) @ id ) )
| ( $true
!= ( X255 @ id @ X270 @ id ) ) ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X271: term] :
~ ! [X272: subst > term > term] :
( ( sub @ X271 @ ( push @ one @ sh ) )
!= ( X272 @ sh @ one ) )
<=> ( hoasinduction_lem3b_lthm = $true ) )
& ( hoasinduction_lem1v2_gthm = $true )
& ( pushprop_lem2v2_gthm = $true )
& ( ! [X273: term] :
( ( sub @ X273 @ id )
= X273 )
<=> ( axvarid = $true ) )
& ( ! [X274: term,X275: subst,X276: subst > term > term,X277: subst > term > term] :
( ~ ! [X278: subst,X279: term,X280: subst] :
( ( X277 @ ( comp @ X278 @ X280 ) @ ( sub @ X279 @ X280 ) )
= ( sub @ ( X277 @ X278 @ X279 ) @ X280 ) )
| ~ ! [X281: term,X282: subst,X283: subst] :
( ( sub @ ( X276 @ X283 @ X281 ) @ X282 )
= ( X276 @ ( comp @ X283 @ X282 ) @ ( sub @ X281 @ X282 ) ) )
| ( ( lam @ ( X277 @ sh @ one ) )
!= ( lam @ ( X276 @ sh @ one ) ) )
| ( ( X276 @ X275 @ X274 )
= ( X277 @ X275 @ X274 ) ) )
<=> ( hoaslaminj = $true ) )
& ( ( termmset = $true )
<=> ! [X284: term] :
( ( sub @ X284 @ id )
= X284 ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( ( hoasinduction_lem3a_lthm = $true )
<=> ( ! [X285: term] :
( ( sub @ X285 @ id )
= X285 )
=> ( ! [X286: term,X287: subst > term > subst > $o] :
( ( $true
= ( X287 @ id @ ( lam @ ( sub @ X286 @ ( push @ one @ sh ) ) ) @ id ) )
| ~ ! [X288: subst > term > term] :
( ( $true
= ( X287 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id ) )
| ~ ! [X289: term] :
( ( $true
= ( X287 @ id @ ( X288 @ id @ X289 ) @ id ) )
| ( $true
!= ( X287 @ id @ X289 @ id ) ) )
| ~ ! [X290: term,X291: subst,X292: subst] :
( ( sub @ ( X288 @ X292 @ X290 ) @ X291 )
= ( X288 @ ( comp @ X292 @ X291 ) @ ( sub @ X290 @ X291 ) ) ) )
| ~ ! [X293: term] :
( ( ( X287 @ id @ ( sub @ X286 @ ( push @ X293 @ id ) ) @ id )
= $true )
| ( $true
!= ( X287 @ id @ X293 @ id ) ) ) )
=> ! [X294: subst > term > subst > $o,X295: term] :
( ~ ! [X296: term] :
( ( $true
= ( X294 @ id @ ( sub @ X295 @ ( push @ X296 @ id ) ) @ id ) )
| ( ( X294 @ id @ X296 @ id )
!= $true ) )
| ~ ! [X297: subst > term > term] :
( ( $true
= ( X294 @ id @ ( lam @ ( X297 @ sh @ one ) ) @ id ) )
| ~ ! [X298: term] :
( ( $true
= ( X294 @ id @ ( X297 @ id @ X298 ) @ id ) )
| ( ( X294 @ id @ X298 @ id )
!= $true ) )
| ~ ! [X299: subst,X300: subst,X301: term] :
( ( X297 @ ( comp @ X300 @ X299 ) @ ( sub @ X301 @ X299 ) )
= ( sub @ ( X297 @ X300 @ X301 ) @ X299 ) ) )
| ( $true
= ( X294 @ id @ ( lam @ X295 ) @ id ) ) ) ) ) )
& ( ! [X302: subst > term > subst > $o,X303: term] :
( ~ ! [X304: term,X305: term] :
( ( $true
!= ( X302 @ id @ X305 @ id ) )
| ( ( X302 @ id @ ( ap @ ( sub @ X305 @ id ) @ X304 ) @ id )
= $true )
| ( $true
!= ( X302 @ id @ X304 @ id ) ) )
| ~ ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( $true
= ( X302 @ ( comp @ X307 @ X309 ) @ ( sub @ X306 @ X309 ) @ X308 ) )
| ( $true
!= ( X302 @ X307 @ X306 @ ( comp @ X309 @ X308 ) ) ) )
| ~ ! [X310: subst > term > term] :
( ~ ! [X311: term] :
( ( $true
!= ( X302 @ id @ X311 @ id ) )
| ( $true
= ( X302 @ id @ ( X310 @ id @ X311 ) @ id ) ) )
| ( $true
= ( X302 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id ) )
| ~ ! [X312: subst,X313: subst,X314: term] :
( ( X310 @ ( comp @ X312 @ X313 ) @ ( sub @ X314 @ X313 ) )
= ( sub @ ( X310 @ X312 @ X314 ) @ X313 ) ) )
| ~ ! [X315: subst,X316: term,X317: subst,X318: subst] :
( ( $true
= ( X302 @ X318 @ X316 @ ( comp @ X317 @ X315 ) ) )
| ( ( X302 @ ( comp @ X318 @ X317 ) @ ( sub @ X316 @ X317 ) @ X315 )
!= $true ) )
| ~ ! [X319: term] :
( ( $true
!= ( var @ ( sub @ X319 @ id ) ) )
| ( $true
= ( X302 @ id @ X319 @ id ) ) )
| ( ( X302 @ id @ X303 @ id )
= $true ) )
<=> ( hoasinduction = $true ) )
& ! [X320: d_subst,X321: d_subst] :
( ( ( d2subst @ X320 )
= ( d2subst @ X321 ) )
=> ( X320 = X321 ) )
& ( ! [X322: term] :
~ ! [X323: subst > term > term] :
~ ! [X324: subst,X325: term] :
( ( X323 @ X324 @ X325 )
= ( sub @ X322 @ ( push @ X325 @ X324 ) ) )
<=> ( hoasinduction_lem3v2_f_lthm = $true ) )
& ( ( hoasinduction_lem3v2a = $true )
<=> ! [X326: subst > term > subst > $o,X327: term,X328: term > $o] :
( ~ ! [X329: subst > term > term] :
( ~ ! [X330: subst,X331: term,X332: subst] :
( ( X329 @ ( comp @ X330 @ X332 ) @ ( sub @ X331 @ X332 ) )
= ( sub @ ( X329 @ X330 @ X331 ) @ X332 ) )
| ~ ! [X333: term] :
( ( $true
= ( X326 @ id @ ( X329 @ id @ X333 ) @ id ) )
| ( ( X326 @ id @ X333 @ id )
!= $true ) )
| ( $true
= ( X326 @ id @ ( lam @ ( X329 @ sh @ one ) ) @ id ) ) )
| ( $true
= ( X328 @ ( lam @ X327 ) ) )
| ~ ! [X334: term] :
( ( X328 @ X334 )
= ( X326 @ id @ X334 @ id ) )
| ~ ! [X335: term] :
( ( $true
!= ( X328 @ X335 ) )
| ( $true
= ( X328 @ ( sub @ X327 @ ( push @ X335 @ id ) ) ) ) ) ) )
& ( ( pushprop_lem1_lthm = $true )
<=> ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
=> ( ! [X338: subst,X339: term] :
( ( comp @ sh @ ( push @ X339 @ X338 ) )
= X338 )
=> ! [X340: term,X341: term > $o,X342: term,X343: subst,X344: term > $o] :
( ( $true
= ( X341 @ ( sub @ X340 @ ( push @ X342 @ X343 ) ) ) )
| ( ( X344 @ X340 )
!= $true ) ) ) ) )
& ! [X345: d_term,X346: d_term] :
( ( ( d2term @ X346 )
= ( d2term @ X345 ) )
=> ( X345 = X346 ) )
& ( pushprop_lthm_orig = $true )
& ( apnotvar = $true )
& ( hoaslamnotap_gthm = $true )
& ( ( ! [X347: subst > term > subst > $o] :
~ ! [X348: term > $o] :
~ ! [X349: term] :
( ( X347 @ id @ X349 @ id )
= ( X348 @ X349 ) )
=> ( ! [X350: term > $o,X351: term] :
( ~ ! [X352: term,X353: term] :
( ( $true
= ( X350 @ ( ap @ X353 @ X352 ) ) )
| ( ( X350 @ X352 )
!= $true )
| ( $true
!= ( X350 @ X353 ) ) )
| ( $true
= ( X350 @ X351 ) )
| ~ ! [X354: term] :
( ~ ! [X355: term] :
( ( $true
!= ( X350 @ X355 ) )
| ( $true
= ( X350 @ ( sub @ X354 @ ( push @ X355 @ id ) ) ) ) )
| ( ( X350 @ ( lam @ X354 ) )
= $true ) )
| ~ ! [X356: term] :
( ( $true
!= ( var @ X356 ) )
| ( $true
= ( X350 @ X356 ) ) ) )
=> ( ! [X357: term] :
( ( sub @ X357 @ id )
= X357 )
=> ( ! [X358: subst > term > subst > $o,X359: term,X360: term > $o] :
( ~ ! [X361: term] :
( ( X358 @ id @ X361 @ id )
= ( X360 @ X361 ) )
| ~ ! [X362: term] :
( ( $true
= ( X360 @ ( sub @ X359 @ ( push @ X362 @ id ) ) ) )
| ( $true
!= ( X360 @ X362 ) ) )
| ( ( X360 @ ( lam @ X359 ) )
= $true )
| ~ ! [X363: subst > term > term] :
( ( ( X358 @ id @ ( lam @ ( X363 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X364: term] :
( ( ( X358 @ id @ ( X363 @ id @ X364 ) @ id )
= $true )
| ( $true
!= ( X358 @ id @ X364 @ id ) ) )
| ~ ! [X365: subst,X366: term,X367: subst] :
( ( sub @ ( X363 @ X365 @ X366 ) @ X367 )
= ( X363 @ ( comp @ X365 @ X367 ) @ ( sub @ X366 @ X367 ) ) ) ) )
=> ! [X368: subst > term > subst > $o,X369: term] :
( ~ ! [X370: subst,X371: subst,X372: subst,X373: term] :
( ( ( X368 @ ( comp @ X370 @ X371 ) @ ( sub @ X373 @ X371 ) @ X372 )
!= $true )
| ( $true
= ( X368 @ X370 @ X373 @ ( comp @ X371 @ X372 ) ) ) )
| ~ ! [X374: subst > term > term] :
( ~ ! [X375: subst,X376: term,X377: subst] :
( ( X374 @ ( comp @ X377 @ X375 ) @ ( sub @ X376 @ X375 ) )
= ( sub @ ( X374 @ X377 @ X376 ) @ X375 ) )
| ( $true
= ( X368 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id ) )
| ~ ! [X378: term] :
( ( $true
!= ( X368 @ id @ X378 @ id ) )
| ( ( X368 @ id @ ( X374 @ id @ X378 ) @ id )
= $true ) ) )
| ~ ! [X379: term,X380: term] :
( ( $true
= ( X368 @ id @ ( ap @ ( sub @ X380 @ id ) @ X379 ) @ id ) )
| ( $true
!= ( X368 @ id @ X379 @ id ) )
| ( ( X368 @ id @ X380 @ id )
!= $true ) )
| ( $true
= ( X368 @ id @ X369 @ id ) )
| ~ ! [X381: term] :
( ( $true
!= ( var @ ( sub @ X381 @ id ) ) )
| ( $true
= ( X368 @ id @ X381 @ id ) ) )
| ~ ! [X382: subst,X383: subst,X384: subst,X385: term] :
( ( $true
!= ( X368 @ X382 @ X385 @ ( comp @ X383 @ X384 ) ) )
| ( $true
= ( X368 @ ( comp @ X382 @ X383 ) @ ( sub @ X385 @ X383 ) @ X384 ) ) ) ) ) ) ) )
<=> ( hoasinduction_lthm_3 = $true ) )
& ( ! [X386: term,X387: term,X388: subst > term > term] :
( ~ ! [X389: subst,X390: subst,X391: term] :
( ( sub @ ( X388 @ X390 @ X391 ) @ X389 )
= ( X388 @ ( comp @ X390 @ X389 ) @ ( sub @ X391 @ X389 ) ) )
| ( ( ap @ ( sub @ X387 @ id ) @ X386 )
!= ( lam @ ( X388 @ sh @ one ) ) ) )
<=> ( hoaslamnotap = $true ) )
& ( ( axscons = $true )
<=> ! [X392: subst] :
( ( push @ ( sub @ one @ X392 ) @ ( comp @ sh @ X392 ) )
= X392 ) )
& ( ( var @ ( d2term @ d_one ) )
!= $true )
& ( hoaslam
= ( ^ [Y0: subst,Y1: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X395: subst,X396: term > $o,X397: term] :
( ~ ! [X398: term] :
( ~ ! [X399: term] :
( ( $true
!= ( X396 @ X399 ) )
| ( $true
= ( X396 @ ( sub @ X398 @ ( push @ X399 @ id ) ) ) ) )
| ( ( X396 @ ( lam @ X398 ) )
= $true ) )
| ( $true
= ( X396 @ ( sub @ X397 @ X395 ) ) )
| ~ ! [X400: term] :
( ( $true
!= ( var @ X400 ) )
| ( $true
= ( X396 @ ( sub @ X400 @ X395 ) ) ) )
| ~ ! [X401: term,X402: term] :
( ( $true
!= ( X396 @ X401 ) )
| ( $true
!= ( X396 @ X402 ) )
| ( $true
= ( X396 @ ( ap @ X402 @ X401 ) ) ) ) )
<=> ( induction2lem = $true ) )
& ( ( axidr = $true )
<=> ! [X403: subst] :
( ( comp @ X403 @ id )
= X403 ) )
& ( ! [X404: term,X405: term,X406: term,X407: term] :
( ( X404 = X407 )
| ( ( ap @ ( sub @ X404 @ id ) @ X406 )
!= ( ap @ ( sub @ X407 @ id ) @ X405 ) ) )
<=> ( hoasapinj1 = $true ) )
& ( axassoc = $true )
& ( ( ! [X408: term] :
( ( sub @ X408 @ id )
= X408 )
=> ( ! [X409: term,X410: term,X411: term,X412: term] :
( ( X409 = X412 )
| ( ( ap @ X412 @ X411 )
!= ( ap @ X409 @ X410 ) ) )
=> ! [X413: term,X414: term,X415: term,X416: term] :
( ( X415 = X416 )
| ( ( ap @ ( sub @ X416 @ id ) @ X414 )
!= ( ap @ ( sub @ X415 @ id ) @ X413 ) ) ) ) )
<=> ( hoasapinj1_lthm = $true ) )
& ( ulamvarind = $true )
& ( lamnotap != $true )
& ( pushprop_lthm = $true )
& ( ! [X417: term,X418: subst > term > subst > $o,X419: term > $o] :
( ~ ! [X420: subst > term > term] :
( ~ ! [X421: term,X422: subst,X423: subst] :
( ( sub @ ( X420 @ X422 @ X421 ) @ X423 )
= ( X420 @ ( comp @ X422 @ X423 ) @ ( sub @ X421 @ X423 ) ) )
| ( $true
= ( X418 @ id @ ( lam @ ( X420 @ sh @ one ) ) @ id ) )
| ~ ! [X424: term] :
( ( $true
= ( X418 @ id @ ( X420 @ id @ X424 ) @ id ) )
| ( $true
!= ( X418 @ id @ X424 @ id ) ) ) )
| ~ ! [X425: term] :
( ( $true
!= ( X419 @ X425 ) )
| ( $true
= ( X419 @ ( sub @ X417 @ ( push @ X425 @ id ) ) ) ) )
| ~ ! [X426: subst,X427: subst,X428: term,X429: subst] :
( ( $true
= ( X418 @ X429 @ X428 @ ( comp @ X427 @ X426 ) ) )
| ( ( X418 @ ( comp @ X429 @ X427 ) @ ( sub @ X428 @ X427 ) @ X426 )
!= $true ) )
| ~ ! [X430: term] :
( ( X418 @ id @ X430 @ id )
= ( X419 @ X430 ) )
| ( $true
= ( X419 @ ( lam @ X417 ) ) )
| ~ ! [X431: subst,X432: term,X433: subst,X434: subst] :
( ( ( X418 @ X434 @ X432 @ ( comp @ X431 @ X433 ) )
!= $true )
| ( $true
= ( X418 @ ( comp @ X434 @ X431 ) @ ( sub @ X432 @ X431 ) @ X433 ) ) ) )
<=> ( hoasinduction_lem3v2 = $true ) )
& ( ( d2subst @ d_id )
= id )
& ( pushprop_lem0_gthm = $true )
& ( hoasinduction_lem1v2 = $true )
& ( hoasinduction_lem1_lthm = $true )
& ( ! [X435: term,X436: term,X437: term,X438: term] :
( ( X436 = X438 )
| ( ( ap @ X435 @ X436 )
!= ( ap @ X437 @ X438 ) ) )
<=> ( apinj2 = $true ) )
& ( ! [X439: term,X440: term > $o] :
( ( ( X440 @ X439 )
= $true )
| ~ ! [X441: term] :
( ( $true
= ( X440 @ X441 ) )
| ( $true
!= ( var @ X441 ) ) )
| ~ ! [X442: term] :
( ( ( X440 @ X442 )
!= $true )
| ( $true
= ( X440 @ ( lam @ X442 ) ) ) )
| ~ ! [X443: term,X444: term] :
( ( $true
!= ( X440 @ X443 ) )
| ( $true
= ( X440 @ ( ap @ X443 @ X444 ) ) )
| ( $true
!= ( X440 @ X444 ) ) ) )
<=> ( induction = $true ) )
& ( ( ! [X445: subst > term > subst > $o] :
~ ! [X446: term > $o] :
~ ! [X447: term] :
( ( X446 @ X447 )
= ( X445 @ id @ X447 @ id ) )
=> ( ! [X448: term,X449: term > $o] :
( ~ ! [X450: term] :
( ( ( var @ X450 )
!= $true )
| ( $true
= ( X449 @ X450 ) ) )
| ( $true
= ( X449 @ X448 ) )
| ~ ! [X451: term] :
( ( $true
= ( X449 @ ( lam @ X451 ) ) )
| ~ ! [X452: term] :
( ( $true
!= ( X449 @ X452 ) )
| ( $true
= ( X449 @ ( sub @ X451 @ ( push @ X452 @ id ) ) ) ) ) )
| ~ ! [X453: term,X454: term] :
( ( $true
!= ( X449 @ X453 ) )
| ( $true
!= ( X449 @ X454 ) )
| ( $true
= ( X449 @ ( ap @ X454 @ X453 ) ) ) ) )
=> ( ! [X455: term] :
( ( sub @ X455 @ id )
= X455 )
=> ( ! [X456: term > $o,X457: subst > term > subst > $o,X458: term] :
( ~ ! [X459: term] :
( ( X456 @ X459 )
= ( X457 @ id @ X459 @ id ) )
| ~ ! [X460: term] :
( ( ( X456 @ X460 )
!= $true )
| ( $true
= ( X456 @ ( sub @ X458 @ ( push @ X460 @ id ) ) ) ) )
| ~ ! [X461: subst > term > term] :
( ~ ! [X462: term,X463: subst,X464: subst] :
( ( X461 @ ( comp @ X463 @ X464 ) @ ( sub @ X462 @ X464 ) )
= ( sub @ ( X461 @ X463 @ X462 ) @ X464 ) )
| ( $true
= ( X457 @ id @ ( lam @ ( X461 @ sh @ one ) ) @ id ) )
| ~ ! [X465: term] :
( ( $true
!= ( X457 @ id @ X465 @ id ) )
| ( ( X457 @ id @ ( X461 @ id @ X465 ) @ id )
= $true ) ) )
| ( $true
= ( X456 @ ( lam @ X458 ) ) ) )
=> ! [X466: subst > term > subst > $o,X467: term] :
( ( ( X466 @ id @ X467 @ id )
= $true )
| ~ ! [X468: subst > term > term] :
( ( ( X466 @ id @ ( lam @ ( X468 @ sh @ one ) ) @ id )
= $true )
| ~ ! [X469: term] :
( ( ( X466 @ id @ ( X468 @ id @ X469 ) @ id )
= $true )
| ( $true
!= ( X466 @ id @ X469 @ id ) ) )
| ~ ! [X470: term,X471: subst,X472: subst] :
( ( sub @ ( X468 @ X472 @ X470 ) @ X471 )
= ( X468 @ ( comp @ X472 @ X471 ) @ ( sub @ X470 @ X471 ) ) ) )
| ~ ! [X473: term,X474: term] :
( ( $true
!= ( X466 @ id @ X474 @ id ) )
| ( $true
= ( X466 @ id @ ( ap @ ( sub @ X473 @ id ) @ X474 ) @ id ) )
| ( $true
!= ( X466 @ id @ X473 @ id ) ) ) ) ) ) ) )
<=> ( hoasinduction_no_psi_cond_lthm = $true ) )
& ( hoasinduction_lem2v2_gthm = $true )
& ( hoasinduction_lem2_gthm = $true )
& ( pushprop_lem1_gthm = $true )
& ( ( d2subst @ d_id )
= sh )
& ( ( hoasinduction_lthm = $true )
<=> ( ! [X475: term,X476: term > $o] :
( ~ ! [X477: term] :
( ~ ! [X478: term] :
( ( $true
!= ( X476 @ X478 ) )
| ( $true
= ( X476 @ ( sub @ X477 @ ( push @ X478 @ id ) ) ) ) )
| ( ( X476 @ ( lam @ X477 ) )
= $true ) )
| ( $true
= ( X476 @ X475 ) )
| ~ ! [X479: term] :
( ( ( var @ X479 )
!= $true )
| ( ( X476 @ X479 )
= $true ) )
| ~ ! [X480: term,X481: term] :
( ( ( X476 @ ( ap @ X480 @ X481 ) )
= $true )
| ( $true
!= ( X476 @ X481 ) )
| ( $true
!= ( X476 @ X480 ) ) ) )
=> ( ! [X482: subst > term > subst > $o,X483: term,X484: term] :
( ( $true
!= ( X482 @ id @ X483 @ id ) )
| ( $true
= ( X482 @ id @ ( ap @ X483 @ X484 ) @ id ) )
| ~ ! [X485: subst,X486: term,X487: subst,X488: subst] :
( ( $true
= ( X482 @ ( comp @ X488 @ X487 ) @ ( sub @ X486 @ X487 ) @ X485 ) )
| ( $true
!= ( X482 @ X488 @ X486 @ ( comp @ X487 @ X485 ) ) ) )
| ( ( X482 @ id @ X484 @ id )
!= $true )
| ~ ! [X489: term,X490: subst,X491: subst,X492: subst] :
( ( ( X482 @ X490 @ X489 @ ( comp @ X491 @ X492 ) )
= $true )
| ( $true
!= ( X482 @ ( comp @ X490 @ X491 ) @ ( sub @ X489 @ X491 ) @ X492 ) ) )
| ~ ! [X493: term,X494: term] :
( ( ( X482 @ id @ ( ap @ ( sub @ X493 @ id ) @ X494 ) @ id )
= $true )
| ( ( X482 @ id @ X494 @ id )
!= $true )
| ( $true
!= ( X482 @ id @ X493 @ id ) ) ) )
=> ( ! [X495: term,X496: subst > term > subst > $o] :
( ~ ! [X497: subst,X498: subst,X499: subst,X500: term] :
( ( $true
!= ( X496 @ ( comp @ X498 @ X499 ) @ ( sub @ X500 @ X499 ) @ X497 ) )
| ( $true
= ( X496 @ X498 @ X500 @ ( comp @ X499 @ X497 ) ) ) )
| ( $true
= ( X496 @ id @ ( lam @ X495 ) @ id ) )
| ~ ! [X501: subst,X502: term,X503: subst,X504: subst] :
( ( $true
= ( X496 @ ( comp @ X503 @ X504 ) @ ( sub @ X502 @ X504 ) @ X501 ) )
| ( $true
!= ( X496 @ X503 @ X502 @ ( comp @ X504 @ X501 ) ) ) )
| ~ ! [X505: subst > term > term] :
( ~ ! [X506: term] :
( ( $true
!= ( X496 @ id @ X506 @ id ) )
| ( $true
= ( X496 @ id @ ( X505 @ id @ X506 ) @ id ) ) )
| ( $true
= ( X496 @ id @ ( lam @ ( X505 @ sh @ one ) ) @ id ) )
| ~ ! [X507: term,X508: subst,X509: subst] :
( ( X505 @ ( comp @ X508 @ X509 ) @ ( sub @ X507 @ X509 ) )
= ( sub @ ( X505 @ X508 @ X507 ) @ X509 ) ) )
| ~ ! [X510: term] :
( ( $true
!= ( X496 @ id @ X510 @ id ) )
| ( $true
= ( X496 @ id @ ( sub @ X495 @ ( push @ X510 @ id ) ) @ id ) ) ) )
=> ! [X511: term,X512: subst > term > subst > $o] :
( ~ ! [X513: subst,X514: subst,X515: term,X516: subst] :
( ( $true
!= ( X512 @ ( comp @ X514 @ X516 ) @ ( sub @ X515 @ X516 ) @ X513 ) )
| ( $true
= ( X512 @ X514 @ X515 @ ( comp @ X516 @ X513 ) ) ) )
| ~ ! [X517: subst > term > term] :
( ( $true
= ( X512 @ id @ ( lam @ ( X517 @ sh @ one ) ) @ id ) )
| ~ ! [X518: term] :
( ( $true
= ( X512 @ id @ ( X517 @ id @ X518 ) @ id ) )
| ( $true
!= ( X512 @ id @ X518 @ id ) ) )
| ~ ! [X519: term,X520: subst,X521: subst] :
( ( X517 @ ( comp @ X521 @ X520 ) @ ( sub @ X519 @ X520 ) )
= ( sub @ ( X517 @ X521 @ X519 ) @ X520 ) ) )
| ~ ! [X522: term,X523: subst,X524: subst,X525: subst] :
( ( ( X512 @ X525 @ X522 @ ( comp @ X524 @ X523 ) )
!= $true )
| ( ( X512 @ ( comp @ X525 @ X524 ) @ ( sub @ X522 @ X524 ) @ X523 )
= $true ) )
| ( $true
= ( X512 @ id @ X511 @ id ) )
| ~ ! [X526: term,X527: term] :
( ( $true
!= ( X512 @ id @ X526 @ id ) )
| ( $true
!= ( X512 @ id @ X527 @ id ) )
| ( $true
= ( X512 @ id @ ( ap @ ( sub @ X526 @ id ) @ X527 ) @ id ) ) )
| ~ ! [X528: term] :
( ( $true
= ( X512 @ id @ X528 @ id ) )
| ( ( var @ ( sub @ X528 @ id ) )
!= $true ) ) ) ) ) ) )
& ( ( hoasinduction_lem3b = $true )
<=> ! [X529: term] :
~ ! [X530: subst > term > term] :
( ( sub @ X529 @ ( push @ one @ sh ) )
!= ( X530 @ sh @ one ) ) )
& ( ( substmonoid_lthm = $true )
<=> ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
=> ( ! [X532: subst] :
( ( comp @ X532 @ id )
= X532 )
=> ( ! [X533: subst] :
( ( comp @ id @ X533 )
= X533 )
& ! [X534: subst] :
( ( comp @ X534 @ id )
= X534 ) ) ) ) )
& ( hoaslamnotvar = $true )
& ( ( ! [X535: term] :
~ ! [X536: subst > term > term] :
~ ! [X537: subst,X538: term] :
( ( sub @ X535 @ ( push @ X538 @ X537 ) )
= ( X536 @ X537 @ X538 ) )
=> ( ! [X539: term] :
( ( sub @ X539 @ id )
= X539 )
=> ! [X540: term,X541: term > $o,X542: subst > term > subst > $o] :
( ( ( X541 @ ( lam @ X540 ) )
= $true )
| ~ ! [X543: term] :
( ( X542 @ id @ X543 @ id )
= ( X541 @ X543 ) )
| ~ ! [X544: subst > term > term] :
( ~ ! [X545: term] :
( ( $true
= ( X542 @ id @ ( X544 @ id @ X545 ) @ id ) )
| ( $true
!= ( X542 @ id @ X545 @ id ) ) )
| ( $true
= ( X542 @ id @ ( lam @ ( X544 @ sh @ one ) ) @ id ) )
| ~ ! [X546: subst,X547: term,X548: subst] :
( ( sub @ ( X544 @ X548 @ X547 ) @ X546 )
= ( X544 @ ( comp @ X548 @ X546 ) @ ( sub @ X547 @ X546 ) ) ) )
| ~ ! [X549: term] :
( ( $true
= ( X541 @ ( sub @ X540 @ ( push @ X549 @ id ) ) ) )
| ( ( X541 @ X549 )
!= $true ) ) ) ) )
<=> ( hoasinduction_lem3v2a_lthm = $true ) ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
( ( ! [X0: term,X1: term] :
( ( X0 = X1 )
| ( ( lam @ X1 )
!= ( lam @ X0 ) ) )
<=> laminj )
& ( ( ! [X2: subst,X3: term] :
( ( sub @ one @ ( push @ X3 @ X2 ) )
= X3 )
=> ( ! [X4: term,X5: subst] :
( ( comp @ sh @ ( push @ X4 @ X5 ) )
= X5 )
=> ( ! [X6: subst] :
( ( comp @ X6 @ id )
= X6 )
=> ( ! [X7: term > $o,X8: term] :
( ~ ! [X9: term] :
( ( X7 @ ( lam @ X9 ) )
| ~ ( X7 @ X9 ) )
| ~ ! [X10: term] :
( ~ ( var @ X10 )
| ( X7 @ X10 ) )
| ~ ! [X11: term,X12: term] :
( ( X7 @ ( ap @ X11 @ X12 ) )
| ~ ( X7 @ X11 )
| ~ ( X7 @ X12 ) )
| ( X7 @ X8 ) )
=> ! [X13: term > $o,X14: term,X15: subst] :
( ~ ! [X16: term,X17: term] :
( ( X13 @ ( ap @ X17 @ X16 ) )
| ~ ( X13 @ X17 )
| ~ ( X13 @ X16 ) )
| ~ ! [X18: term] :
( ( X13 @ ( lam @ X18 ) )
| ~ ! [X19: term] :
( ~ ( X13 @ X19 )
| ( X13 @ ( sub @ X18 @ ( push @ X19 @ id ) ) ) ) )
| ( X13 @ ( sub @ X14 @ X15 ) )
| ~ ! [X20: term] :
( ~ ( var @ X20 )
| ( X13 @ ( sub @ X20 @ X15 ) ) ) ) ) ) ) )
<=> induction2lem_lthm )
& ( pushprop_lem0
<=> ! [X21: subst,X22: term,X23: term > $o] :
~ ! [X24: term > $o] :
~ ! [X25: term] :
( ( X24 @ X25 )
= ( X23 @ ( sub @ X25 @ ( push @ X22 @ X21 ) ) ) ) )
& ( ( ! [X26: subst,X27: term] :
( ( sub @ one @ ( push @ X27 @ X26 ) )
= X27 )
=> ! [X28: term > $o,X29: term,X30: term > $o,X31: subst] :
( ~ ! [X32: term] :
( ( X30 @ ( sub @ X32 @ ( push @ X29 @ X31 ) ) )
= ( X28 @ X32 ) )
| ~ ( X30 @ X29 )
| ( X28 @ one ) ) )
<=> pushprop_lem1v2_lthm )
& hoasinduction_lem2_lthm
& ( substmonoid
<=> ( ! [X33: subst] :
( ( comp @ X33 @ id )
= X33 )
& ! [X34: subst] :
( ( comp @ id @ X34 )
= X34 ) ) )
& pushprop_lem3v2_lthm
& lamnotvar
& ( hoasinduction_lem3v2_lthm
<=> ( ! [X35: term] :
( ( sub @ X35 @ id )
= X35 )
=> ! [X36: subst > term > subst > $o,X37: term,X38: term > $o] :
( ~ ! [X39: term,X40: subst,X41: subst,X42: subst] :
( ( X36 @ X42 @ X39 @ ( comp @ X41 @ X40 ) )
| ~ ( X36 @ ( comp @ X42 @ X41 ) @ ( sub @ X39 @ X41 ) @ X40 ) )
| ~ ! [X43: term] :
( ( X36 @ id @ X43 @ id )
= ( X38 @ X43 ) )
| ( X38 @ ( lam @ X37 ) )
| ~ ! [X44: subst > term > term] :
( ~ ! [X45: term,X46: subst,X47: subst] :
( ( sub @ ( X44 @ X47 @ X45 ) @ X46 )
= ( X44 @ ( comp @ X47 @ X46 ) @ ( sub @ X45 @ X46 ) ) )
| ( X36 @ id @ ( lam @ ( X44 @ sh @ one ) ) @ id )
| ~ ! [X48: term] :
( ~ ( X36 @ id @ X48 @ id )
| ( X36 @ id @ ( X44 @ id @ X48 ) @ id ) ) )
| ~ ! [X49: term] :
( ~ ( X38 @ X49 )
| ( X38 @ ( sub @ X37 @ ( push @ X49 @ id ) ) ) )
| ~ ! [X50: subst,X51: subst,X52: term,X53: subst] :
( ~ ( X36 @ X53 @ X52 @ ( comp @ X50 @ X51 ) )
| ( X36 @ ( comp @ X53 @ X50 ) @ ( sub @ X52 @ X50 ) @ X51 ) ) ) ) )
& ( ! [X54: subst > term > subst > $o,X55: term] :
( ~ ! [X56: term,X57: term] :
( ~ ( X54 @ id @ X57 @ id )
| ~ ( X54 @ id @ X56 @ id )
| ( X54 @ id @ ( ap @ ( sub @ X56 @ id ) @ X57 ) @ id ) )
| ( X54 @ id @ X55 @ id )
| ~ ! [X58: subst > term > term] :
( ( X54 @ id @ ( lam @ ( X58 @ sh @ one ) ) @ id )
| ~ ! [X59: term] :
( ( X54 @ id @ ( X58 @ id @ X59 ) @ id )
| ~ ( X54 @ id @ X59 @ id ) )
| ~ ! [X60: term,X61: subst,X62: subst] :
( ( X58 @ ( comp @ X62 @ X61 ) @ ( sub @ X60 @ X61 ) )
= ( sub @ ( X58 @ X62 @ X60 ) @ X61 ) ) ) )
<=> hoasinduction_no_psi_cond )
& substmonoid_gthm
& hoaslamnotvar_gthm
& ( hoasinduction_lem3_lthm
<=> ( ! [X63: term] :
( ( sub @ X63 @ id )
= X63 )
=> ( ! [X64: term,X65: subst > term > subst > $o] :
( ( X65 @ id @ ( lam @ ( sub @ X64 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X66: term] :
( ( X65 @ id @ ( sub @ X64 @ ( push @ X66 @ id ) ) @ id )
| ~ ( X65 @ id @ X66 @ id ) )
| ~ ! [X67: subst > term > term] :
( ( X65 @ id @ ( lam @ ( X67 @ sh @ one ) ) @ id )
| ~ ! [X68: subst,X69: subst,X70: term] :
( ( sub @ ( X67 @ X69 @ X70 ) @ X68 )
= ( X67 @ ( comp @ X69 @ X68 ) @ ( sub @ X70 @ X68 ) ) )
| ~ ! [X71: term] :
( ~ ( X65 @ id @ X71 @ id )
| ( X65 @ id @ ( X67 @ id @ X71 ) @ id ) ) ) )
=> ! [X72: subst > term > subst > $o,X73: term] :
( ~ ! [X74: term] :
( ( X72 @ id @ ( sub @ X73 @ ( push @ X74 @ id ) ) @ id )
| ~ ( X72 @ id @ X74 @ id ) )
| ~ ! [X75: subst,X76: term,X77: subst,X78: subst] :
( ( X72 @ ( comp @ X78 @ X77 ) @ ( sub @ X76 @ X77 ) @ X75 )
| ~ ( X72 @ X78 @ X76 @ ( comp @ X77 @ X75 ) ) )
| ~ ! [X79: term,X80: subst,X81: subst,X82: subst] :
( ~ ( X72 @ ( comp @ X80 @ X82 ) @ ( sub @ X79 @ X82 ) @ X81 )
| ( X72 @ X80 @ X79 @ ( comp @ X82 @ X81 ) ) )
| ~ ! [X83: subst > term > term] :
( ~ ! [X84: subst,X85: term,X86: subst] :
( ( X83 @ ( comp @ X86 @ X84 ) @ ( sub @ X85 @ X84 ) )
= ( sub @ ( X83 @ X86 @ X85 ) @ X84 ) )
| ( X72 @ id @ ( lam @ ( X83 @ sh @ one ) ) @ id )
| ~ ! [X87: term] :
( ( X72 @ id @ ( X83 @ id @ X87 ) @ id )
| ~ ( X72 @ id @ X87 @ id ) ) )
| ( X72 @ id @ ( lam @ X73 ) @ id ) ) ) ) )
& ( induction2
<=> ! [X88: term > $o,X89: term] :
( ~ ! [X90: term] :
( ( X88 @ X90 )
| ~ ( var @ X90 ) )
| ~ ! [X91: term] :
( ( X88 @ ( lam @ X91 ) )
| ~ ! [X92: term] :
( ~ ( X88 @ X92 )
| ( X88 @ ( sub @ X91 @ ( push @ X92 @ id ) ) ) ) )
| ( X88 @ X89 )
| ~ ! [X93: term,X94: term] :
( ~ ( X88 @ X94 )
| ( X88 @ ( ap @ X93 @ X94 ) )
| ~ ( X88 @ X93 ) ) ) )
& ~ ulamvar1
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& pushprop_lem1v2_gthm
& ( ! [X95: term] :
~ ! [X96: subst > term > term] :
~ ! [X97: term,X98: subst] :
( ( X96 @ X98 @ X97 )
= ( sub @ X95 @ ( push @ X97 @ X98 ) ) )
<=> hoasinduction_lem3v2_f )
& hoasinduction_lem3_gthm
& ( ( ! [X99: term,X100: subst] :
( ( sub @ one @ ( push @ X99 @ X100 ) )
= X99 )
=> ( ! [X101: term,X102: subst] :
( ( comp @ sh @ ( push @ X101 @ X102 ) )
= X102 )
=> ( ! [X103: term,X104: term] :
( ( X103 = X104 )
| ( ( lam @ X103 )
!= ( lam @ X104 ) ) )
=> ! [X105: term,X106: subst > term > term,X107: subst > term > term,X108: subst] :
( ( ( lam @ ( X106 @ sh @ one ) )
!= ( lam @ ( X107 @ sh @ one ) ) )
| ( ( X106 @ X108 @ X105 )
= ( X107 @ X108 @ X105 ) )
| ~ ! [X109: subst,X110: subst,X111: term] :
( ( sub @ ( X107 @ X109 @ X111 ) @ X110 )
= ( X107 @ ( comp @ X109 @ X110 ) @ ( sub @ X111 @ X110 ) ) )
| ~ ! [X112: subst,X113: term,X114: subst] :
( ( X106 @ ( comp @ X114 @ X112 ) @ ( sub @ X113 @ X112 ) )
= ( sub @ ( X106 @ X114 @ X113 ) @ X112 ) ) ) ) ) )
<=> hoaslaminj_lthm )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X115: subst > term > subst > $o,X116: term] :
( ( X115 @ id @ ( lam @ ( sub @ X116 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X117: subst > term > term] :
( ( X115 @ id @ ( lam @ ( X117 @ sh @ one ) ) @ id )
| ~ ! [X118: subst,X119: subst,X120: term] :
( ( sub @ ( X117 @ X118 @ X120 ) @ X119 )
= ( X117 @ ( comp @ X118 @ X119 ) @ ( sub @ X120 @ X119 ) ) )
| ~ ! [X121: term] :
( ( X115 @ id @ ( X117 @ id @ X121 ) @ id )
| ~ ( X115 @ id @ X121 @ id ) ) )
| ~ ! [X122: term] :
( ( X115 @ id @ ( sub @ X116 @ ( push @ X122 @ id ) ) @ id )
| ~ ( X115 @ id @ X122 @ id ) ) )
<=> hoasinduction_lem3aa )
& ( hoasinduction_lem2v2
<=> ! [X123: term > $o,X124: term,X125: subst > term > subst > $o,X126: term] :
( ~ ! [X127: term,X128: term] :
( ~ ( X125 @ id @ X128 @ id )
| ~ ( X125 @ id @ X127 @ id )
| ( X125 @ id @ ( ap @ ( sub @ X127 @ id ) @ X128 ) @ id ) )
| ~ ( X123 @ X124 )
| ~ ( X123 @ X126 )
| ~ ! [X129: subst,X130: subst,X131: term,X132: subst] :
( ~ ( X125 @ ( comp @ X129 @ X132 ) @ ( sub @ X131 @ X132 ) @ X130 )
| ( X125 @ X129 @ X131 @ ( comp @ X132 @ X130 ) ) )
| ~ ! [X133: subst,X134: subst,X135: term,X136: subst] :
( ( X125 @ ( comp @ X136 @ X134 ) @ ( sub @ X135 @ X134 ) @ X133 )
| ~ ( X125 @ X136 @ X135 @ ( comp @ X134 @ X133 ) ) )
| ( X123 @ ( ap @ X126 @ X124 ) )
| ~ ! [X137: term] :
( ( X125 @ id @ X137 @ id )
= ( X123 @ X137 ) ) ) )
& axapp
& ( ! [X138: subst > term > subst > $o] :
~ ! [X139: term > $o] :
~ ! [X140: term] :
( ( X138 @ id @ X140 @ id )
= ( X139 @ X140 ) )
<=> hoasinduction_lem0 )
& pushprop_lem2v2_lthm
& hoasapinj1_gthm
& hoasapinj2_gthm
& ! [X141: d_term] : ( d_one = X141 )
& hoasinduction_lem3a_gthm
& pushprop_lem3v2
& ( ! [X142: subst > term > subst > $o,X143: term] :
( ( X142 @ id @ ( lam @ ( sub @ X143 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X144: term,X145: subst > term > term] :
( ~ ! [X146: subst,X147: term,X148: subst] :
( ( X145 @ ( comp @ X146 @ X148 ) @ ( sub @ X147 @ X148 ) )
= ( sub @ X144 @ ( push @ ( sub @ X147 @ X148 ) @ ( comp @ X146 @ X148 ) ) ) )
| ~ ! [X149: term] :
( ( X142 @ id @ ( X145 @ id @ X149 ) @ id )
| ~ ( X142 @ id @ X149 @ id ) )
| ( X142 @ id @ ( lam @ ( X145 @ sh @ one ) ) @ id )
| ~ ! [X150: term,X151: subst,X152: subst] :
( ( sub @ ( X145 @ X152 @ X150 ) @ X151 )
= ( sub @ ( sub @ X144 @ ( push @ X150 @ X152 ) ) @ X151 ) ) )
| ~ ! [X153: term] :
( ~ ( X142 @ id @ X153 @ id )
| ( X142 @ id @ ( sub @ X143 @ ( push @ X153 @ id ) ) @ id ) ) )
<=> hoasinduction_lem3aaa )
& ! [X154: subst] :
? [X155: d_subst] :
( ( d2subst @ X155 )
= X154 )
& ! [X156: d_subst] : ( d_id = X156 )
& ( ! [X157: subst > term > subst > $o,X158: term] :
( ~ ! [X159: term] :
( ~ ( X157 @ id @ X159 @ id )
| ( X157 @ id @ ( sub @ X158 @ ( push @ X159 @ id ) ) @ id ) )
| ( X157 @ id @ ( lam @ X158 ) @ id )
| ~ ! [X160: subst > term > term] :
( ~ ! [X161: term] :
( ~ ( X157 @ id @ X161 @ id )
| ( X157 @ id @ ( X160 @ id @ X161 ) @ id ) )
| ( X157 @ id @ ( lam @ ( X160 @ sh @ one ) ) @ id )
| ~ ! [X162: term,X163: subst,X164: subst] :
( ( sub @ ( X160 @ X163 @ X162 ) @ X164 )
= ( X160 @ ( comp @ X163 @ X164 ) @ ( sub @ X162 @ X164 ) ) ) ) )
<=> hoasinduction_lem3a )
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& hoasapnotvar_lthm
& ( pushprop_lem1
<=> ! [X165: term,X166: subst,X167: term,X168: term > $o,X169: term > $o] :
( ( X168 @ ( sub @ X167 @ ( push @ X165 @ X166 ) ) )
| ~ ( X169 @ X167 ) ) )
& hoasapnotvar
& hoaslamnotap_lthm
& hoasinduction_gthm
& axabs
& induction2_gthm
& ! [X170: term] :
? [X171: d_term] :
( ( d2term @ X171 )
= X170 )
& hoaslaminj_gthm
& ( hoasapinj2
<=> ! [X172: term,X173: term,X174: term,X175: term] :
( ( X174 = X175 )
| ( ( ap @ ( sub @ X173 @ id ) @ X174 )
!= ( ap @ ( sub @ X172 @ id ) @ X175 ) ) ) )
& ( apinj1
<=> ! [X176: term,X177: term,X178: term,X179: term] :
( ( ( ap @ X176 @ X177 )
!= ( ap @ X179 @ X178 ) )
| ( X176 = X179 ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& hoasinduction_lem1
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& pushprop_lem2v2
& ( ! [X180: term > $o,X181: term,X182: subst] :
~ ! [X183: term > $o] :
~ ! [X184: term] :
( ( X183 @ X184 )
= ( X180 @ ( sub @ X184 @ ( push @ X181 @ X182 ) ) ) )
<=> pushprop_lem0_lthm )
& pushprop
& hoasinduction_lem3v2_gthm
& ( ! [X185: term,X186: term] :
( ( ( sub @ X186 @ sh )
!= ( sub @ X185 @ sh ) )
| ( X185 = X186 ) )
<=> shinj )
& ( ! [X187: subst > term > subst > $o] :
~ ! [X188: term > $o] :
~ ! [X189: term] :
( ( X188 @ X189 )
= ( X187 @ id @ X189 @ id ) )
<=> hoasinduction_lem0_lthm )
& hoaslamnotvar_lthm
& ( hoasinduction_p_and_p_prime
= ( ^ [X190: subst > term > subst > $o,X191: term > $o] :
! [X192: term] :
( ( X190 @ id @ X192 @ id )
= ( X191 @ X192 ) ) ) )
& induction2lem_gthm
& termmset_gthm
& ( hoasapinj2_lthm
<=> ( ! [X193: term,X194: term,X195: term,X196: term] :
( ( X193 = X195 )
| ( ( ap @ X196 @ X195 )
!= ( ap @ X194 @ X193 ) ) )
=> ! [X197: term,X198: term,X199: term,X200: term] :
( ( X197 = X198 )
| ( ( ap @ ( sub @ X199 @ id ) @ X197 )
!= ( ap @ ( sub @ X200 @ id ) @ X198 ) ) ) ) )
& axmap
& axclos
& ( one
= ( d2term @ d_one ) )
& ( ! [X201: subst,X202: term] :
( ( comp @ sh @ ( push @ X202 @ X201 ) )
= X201 )
<=> axshiftcons )
& hoasapnotvar_gthm
& axvarshift
& ( ! [X203: term,X204: subst > term > subst > $o] :
( ( X204 @ id @ ( lam @ ( sub @ X203 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X205: subst > term > term] :
( ( X204 @ id @ ( lam @ ( X205 @ sh @ one ) ) @ id )
| ~ ! [X206: term] :
( ( X204 @ id @ ( X205 @ id @ X206 ) @ id )
| ~ ( X204 @ id @ X206 @ id ) )
| ~ ! [X207: term,X208: subst,X209: subst] :
( ( sub @ ( X205 @ X209 @ X207 ) @ X208 )
= ( X205 @ ( comp @ X209 @ X208 ) @ ( sub @ X207 @ X208 ) ) ) )
| ~ ! [X210: term] :
( ~ ( X204 @ id @ X210 @ id )
| ( X204 @ id @ ( sub @ X203 @ ( push @ X210 @ id ) ) @ id ) ) )
<=> hoasinduction_lem3aa_lthm )
& ( ( ! [X211: term] :
( ( sub @ X211 @ id )
= X211 )
=> ! [X212: term] :
( ( sub @ X212 @ id )
= X212 ) )
<=> termmset_lthm )
& ( ! [X213: subst > term > subst > $o,X214: term,X215: term] :
( ~ ( X213 @ id @ X215 @ id )
| ~ ( X213 @ id @ X214 @ id )
| ~ ! [X216: term,X217: subst,X218: subst,X219: subst] :
( ~ ( X213 @ X218 @ X216 @ ( comp @ X219 @ X217 ) )
| ( X213 @ ( comp @ X218 @ X219 ) @ ( sub @ X216 @ X219 ) @ X217 ) )
| ~ ! [X220: subst,X221: subst,X222: subst,X223: term] :
( ( X213 @ X222 @ X223 @ ( comp @ X221 @ X220 ) )
| ~ ( X213 @ ( comp @ X222 @ X221 ) @ ( sub @ X223 @ X221 ) @ X220 ) )
| ~ ! [X224: term,X225: term] :
( ( X213 @ id @ ( ap @ ( sub @ X224 @ id ) @ X225 ) @ id )
| ~ ( X213 @ id @ X224 @ id )
| ~ ( X213 @ id @ X225 @ id ) )
| ( X213 @ id @ ( ap @ X215 @ X214 ) @ id ) )
<=> hoasinduction_lem2 )
& ( axidl
<=> ! [X226: subst] :
( ( comp @ id @ X226 )
= X226 ) )
& ( induction2_lthm
<=> ( ! [X227: term] :
( ( sub @ X227 @ id )
= X227 )
=> ( ! [X228: term,X229: term > $o,X230: subst] :
( ~ ! [X231: term] :
( ~ ( var @ X231 )
| ( X229 @ ( sub @ X231 @ X230 ) ) )
| ( X229 @ ( sub @ X228 @ X230 ) )
| ~ ! [X232: term] :
( ~ ! [X233: term] :
( ( X229 @ ( sub @ X232 @ ( push @ X233 @ id ) ) )
| ~ ( X229 @ X233 ) )
| ( X229 @ ( lam @ X232 ) ) )
| ~ ! [X234: term,X235: term] :
( ~ ( X229 @ X234 )
| ~ ( X229 @ X235 )
| ( X229 @ ( ap @ X235 @ X234 ) ) ) )
=> ! [X236: term,X237: term > $o] :
( ( X237 @ X236 )
| ~ ! [X238: term] :
( ( X237 @ ( lam @ X238 ) )
| ~ ! [X239: term] :
( ~ ( X237 @ X239 )
| ( X237 @ ( sub @ X238 @ ( push @ X239 @ id ) ) ) ) )
| ~ ! [X240: term,X241: term] :
( ~ ( X237 @ X241 )
| ( X237 @ ( ap @ X241 @ X240 ) )
| ~ ( X237 @ X240 ) )
| ~ ! [X242: term] :
( ( X237 @ X242 )
| ~ ( var @ X242 ) ) ) ) ) )
& hoasinduction_lem3b_gthm
& ( ( ^ [X243: term,X244: subst,X245: term > $o,X246: term > $o] :
! [X247: term] :
( ( X246 @ X247 )
= ( X245 @ ( sub @ X247 @ ( push @ X243 @ X244 ) ) ) ) )
= pushprop_p_and_p_prime )
& ( ! [X248: term,X249: subst,X250: term > $o,X251: term > $o] :
( ( X250 @ one )
| ~ ( X251 @ X248 )
| ~ ! [X252: term] :
( ( X251 @ ( sub @ X252 @ ( push @ X248 @ X249 ) ) )
= ( X250 @ X252 ) ) )
<=> pushprop_lem1v2 )
& ulamvarsh
& hoasinduction_lem1_gthm
& ( ! [X253: term,X254: subst] :
( ( sub @ one @ ( push @ X253 @ X254 ) )
= X253 )
<=> axvarcons )
& pushprop_gthm
& ( hoasinduction_lem3
<=> ! [X255: subst > term > subst > $o,X256: term] :
( ( X255 @ id @ ( lam @ X256 ) @ id )
| ~ ! [X257: subst,X258: subst,X259: term,X260: subst] :
( ~ ( X255 @ ( comp @ X260 @ X257 ) @ ( sub @ X259 @ X257 ) @ X258 )
| ( X255 @ X260 @ X259 @ ( comp @ X257 @ X258 ) ) )
| ~ ! [X261: subst > term > term] :
( ( X255 @ id @ ( lam @ ( X261 @ sh @ one ) ) @ id )
| ~ ! [X262: subst,X263: term,X264: subst] :
( ( sub @ ( X261 @ X264 @ X263 ) @ X262 )
= ( X261 @ ( comp @ X264 @ X262 ) @ ( sub @ X263 @ X262 ) ) )
| ~ ! [X265: term] :
( ( X255 @ id @ ( X261 @ id @ X265 ) @ id )
| ~ ( X255 @ id @ X265 @ id ) ) )
| ~ ! [X266: term,X267: subst,X268: subst,X269: subst] :
( ~ ( X255 @ X269 @ X266 @ ( comp @ X268 @ X267 ) )
| ( X255 @ ( comp @ X269 @ X268 ) @ ( sub @ X266 @ X268 ) @ X267 ) )
| ~ ! [X270: term] :
( ( X255 @ id @ ( sub @ X256 @ ( push @ X270 @ id ) ) @ id )
| ~ ( X255 @ id @ X270 @ id ) ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X271: term] :
~ ! [X272: subst > term > term] :
( ( sub @ X271 @ ( push @ one @ sh ) )
!= ( X272 @ sh @ one ) )
<=> hoasinduction_lem3b_lthm )
& hoasinduction_lem1v2_gthm
& pushprop_lem2v2_gthm
& ( ! [X273: term] :
( ( sub @ X273 @ id )
= X273 )
<=> axvarid )
& ( ! [X274: term,X275: subst,X276: subst > term > term,X277: subst > term > term] :
( ~ ! [X278: subst,X279: term,X280: subst] :
( ( X277 @ ( comp @ X278 @ X280 ) @ ( sub @ X279 @ X280 ) )
= ( sub @ ( X277 @ X278 @ X279 ) @ X280 ) )
| ~ ! [X281: term,X282: subst,X283: subst] :
( ( sub @ ( X276 @ X283 @ X281 ) @ X282 )
= ( X276 @ ( comp @ X283 @ X282 ) @ ( sub @ X281 @ X282 ) ) )
| ( ( lam @ ( X277 @ sh @ one ) )
!= ( lam @ ( X276 @ sh @ one ) ) )
| ( ( X276 @ X275 @ X274 )
= ( X277 @ X275 @ X274 ) ) )
<=> hoaslaminj )
& ( termmset
<=> ! [X284: term] :
( ( sub @ X284 @ id )
= X284 ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3a_lthm
<=> ( ! [X285: term] :
( ( sub @ X285 @ id )
= X285 )
=> ( ! [X286: term,X287: subst > term > subst > $o] :
( ( X287 @ id @ ( lam @ ( sub @ X286 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X288: subst > term > term] :
( ( X287 @ id @ ( lam @ ( X288 @ sh @ one ) ) @ id )
| ~ ! [X289: term] :
( ( X287 @ id @ ( X288 @ id @ X289 ) @ id )
| ~ ( X287 @ id @ X289 @ id ) )
| ~ ! [X290: term,X291: subst,X292: subst] :
( ( sub @ ( X288 @ X292 @ X290 ) @ X291 )
= ( X288 @ ( comp @ X292 @ X291 ) @ ( sub @ X290 @ X291 ) ) ) )
| ~ ! [X293: term] :
( ( X287 @ id @ ( sub @ X286 @ ( push @ X293 @ id ) ) @ id )
| ~ ( X287 @ id @ X293 @ id ) ) )
=> ! [X294: subst > term > subst > $o,X295: term] :
( ~ ! [X296: term] :
( ( X294 @ id @ ( sub @ X295 @ ( push @ X296 @ id ) ) @ id )
| ~ ( X294 @ id @ X296 @ id ) )
| ~ ! [X297: subst > term > term] :
( ( X294 @ id @ ( lam @ ( X297 @ sh @ one ) ) @ id )
| ~ ! [X298: term] :
( ( X294 @ id @ ( X297 @ id @ X298 ) @ id )
| ~ ( X294 @ id @ X298 @ id ) )
| ~ ! [X299: subst,X300: subst,X301: term] :
( ( X297 @ ( comp @ X300 @ X299 ) @ ( sub @ X301 @ X299 ) )
= ( sub @ ( X297 @ X300 @ X301 ) @ X299 ) ) )
| ( X294 @ id @ ( lam @ X295 ) @ id ) ) ) ) )
& ( ! [X302: subst > term > subst > $o,X303: term] :
( ~ ! [X304: term,X305: term] :
( ~ ( X302 @ id @ X305 @ id )
| ( X302 @ id @ ( ap @ ( sub @ X305 @ id ) @ X304 ) @ id )
| ~ ( X302 @ id @ X304 @ id ) )
| ~ ! [X306: term,X307: subst,X308: subst,X309: subst] :
( ( X302 @ ( comp @ X307 @ X309 ) @ ( sub @ X306 @ X309 ) @ X308 )
| ~ ( X302 @ X307 @ X306 @ ( comp @ X309 @ X308 ) ) )
| ~ ! [X310: subst > term > term] :
( ~ ! [X311: term] :
( ~ ( X302 @ id @ X311 @ id )
| ( X302 @ id @ ( X310 @ id @ X311 ) @ id ) )
| ( X302 @ id @ ( lam @ ( X310 @ sh @ one ) ) @ id )
| ~ ! [X312: subst,X313: subst,X314: term] :
( ( X310 @ ( comp @ X312 @ X313 ) @ ( sub @ X314 @ X313 ) )
= ( sub @ ( X310 @ X312 @ X314 ) @ X313 ) ) )
| ~ ! [X315: subst,X316: term,X317: subst,X318: subst] :
( ( X302 @ X318 @ X316 @ ( comp @ X317 @ X315 ) )
| ~ ( X302 @ ( comp @ X318 @ X317 ) @ ( sub @ X316 @ X317 ) @ X315 ) )
| ~ ! [X319: term] :
( ~ ( var @ ( sub @ X319 @ id ) )
| ( X302 @ id @ X319 @ id ) )
| ( X302 @ id @ X303 @ id ) )
<=> hoasinduction )
& ! [X320: d_subst,X321: d_subst] :
( ( ( d2subst @ X320 )
= ( d2subst @ X321 ) )
=> ( X320 = X321 ) )
& ( ! [X322: term] :
~ ! [X323: subst > term > term] :
~ ! [X324: subst,X325: term] :
( ( X323 @ X324 @ X325 )
= ( sub @ X322 @ ( push @ X325 @ X324 ) ) )
<=> hoasinduction_lem3v2_f_lthm )
& ( hoasinduction_lem3v2a
<=> ! [X326: subst > term > subst > $o,X327: term,X328: term > $o] :
( ~ ! [X329: subst > term > term] :
( ~ ! [X330: subst,X331: term,X332: subst] :
( ( X329 @ ( comp @ X330 @ X332 ) @ ( sub @ X331 @ X332 ) )
= ( sub @ ( X329 @ X330 @ X331 ) @ X332 ) )
| ~ ! [X333: term] :
( ( X326 @ id @ ( X329 @ id @ X333 ) @ id )
| ~ ( X326 @ id @ X333 @ id ) )
| ( X326 @ id @ ( lam @ ( X329 @ sh @ one ) ) @ id ) )
| ( X328 @ ( lam @ X327 ) )
| ~ ! [X334: term] :
( ( X328 @ X334 )
= ( X326 @ id @ X334 @ id ) )
| ~ ! [X335: term] :
( ~ ( X328 @ X335 )
| ( X328 @ ( sub @ X327 @ ( push @ X335 @ id ) ) ) ) ) )
& ( pushprop_lem1_lthm
<=> ( ! [X336: subst,X337: term] :
( ( sub @ one @ ( push @ X337 @ X336 ) )
= X337 )
=> ( ! [X338: subst,X339: term] :
( ( comp @ sh @ ( push @ X339 @ X338 ) )
= X338 )
=> ! [X340: term,X341: term > $o,X342: term,X343: subst,X344: term > $o] :
( ( X341 @ ( sub @ X340 @ ( push @ X342 @ X343 ) ) )
| ~ ( X344 @ X340 ) ) ) ) )
& ! [X345: d_term,X346: d_term] :
( ( ( d2term @ X346 )
= ( d2term @ X345 ) )
=> ( X345 = X346 ) )
& pushprop_lthm_orig
& apnotvar
& hoaslamnotap_gthm
& ( ( ! [X347: subst > term > subst > $o] :
~ ! [X348: term > $o] :
~ ! [X349: term] :
( ( X347 @ id @ X349 @ id )
= ( X348 @ X349 ) )
=> ( ! [X350: term > $o,X351: term] :
( ~ ! [X352: term,X353: term] :
( ( X350 @ ( ap @ X353 @ X352 ) )
| ~ ( X350 @ X352 )
| ~ ( X350 @ X353 ) )
| ( X350 @ X351 )
| ~ ! [X354: term] :
( ~ ! [X355: term] :
( ~ ( X350 @ X355 )
| ( X350 @ ( sub @ X354 @ ( push @ X355 @ id ) ) ) )
| ( X350 @ ( lam @ X354 ) ) )
| ~ ! [X356: term] :
( ~ ( var @ X356 )
| ( X350 @ X356 ) ) )
=> ( ! [X357: term] :
( ( sub @ X357 @ id )
= X357 )
=> ( ! [X358: subst > term > subst > $o,X359: term,X360: term > $o] :
( ~ ! [X361: term] :
( ( X358 @ id @ X361 @ id )
= ( X360 @ X361 ) )
| ~ ! [X362: term] :
( ( X360 @ ( sub @ X359 @ ( push @ X362 @ id ) ) )
| ~ ( X360 @ X362 ) )
| ( X360 @ ( lam @ X359 ) )
| ~ ! [X363: subst > term > term] :
( ( X358 @ id @ ( lam @ ( X363 @ sh @ one ) ) @ id )
| ~ ! [X364: term] :
( ( X358 @ id @ ( X363 @ id @ X364 ) @ id )
| ~ ( X358 @ id @ X364 @ id ) )
| ~ ! [X365: subst,X366: term,X367: subst] :
( ( sub @ ( X363 @ X365 @ X366 ) @ X367 )
= ( X363 @ ( comp @ X365 @ X367 ) @ ( sub @ X366 @ X367 ) ) ) ) )
=> ! [X368: subst > term > subst > $o,X369: term] :
( ~ ! [X370: subst,X371: subst,X372: subst,X373: term] :
( ~ ( X368 @ ( comp @ X370 @ X371 ) @ ( sub @ X373 @ X371 ) @ X372 )
| ( X368 @ X370 @ X373 @ ( comp @ X371 @ X372 ) ) )
| ~ ! [X374: subst > term > term] :
( ~ ! [X375: subst,X376: term,X377: subst] :
( ( X374 @ ( comp @ X377 @ X375 ) @ ( sub @ X376 @ X375 ) )
= ( sub @ ( X374 @ X377 @ X376 ) @ X375 ) )
| ( X368 @ id @ ( lam @ ( X374 @ sh @ one ) ) @ id )
| ~ ! [X378: term] :
( ~ ( X368 @ id @ X378 @ id )
| ( X368 @ id @ ( X374 @ id @ X378 ) @ id ) ) )
| ~ ! [X379: term,X380: term] :
( ( X368 @ id @ ( ap @ ( sub @ X380 @ id ) @ X379 ) @ id )
| ~ ( X368 @ id @ X379 @ id )
| ~ ( X368 @ id @ X380 @ id ) )
| ( X368 @ id @ X369 @ id )
| ~ ! [X381: term] :
( ~ ( var @ ( sub @ X381 @ id ) )
| ( X368 @ id @ X381 @ id ) )
| ~ ! [X382: subst,X383: subst,X384: subst,X385: term] :
( ~ ( X368 @ X382 @ X385 @ ( comp @ X383 @ X384 ) )
| ( X368 @ ( comp @ X382 @ X383 ) @ ( sub @ X385 @ X383 ) @ X384 ) ) ) ) ) ) )
<=> hoasinduction_lthm_3 )
& ( ! [X386: term,X387: term,X388: subst > term > term] :
( ~ ! [X389: subst,X390: subst,X391: term] :
( ( sub @ ( X388 @ X390 @ X391 ) @ X389 )
= ( X388 @ ( comp @ X390 @ X389 ) @ ( sub @ X391 @ X389 ) ) )
| ( ( ap @ ( sub @ X387 @ id ) @ X386 )
!= ( lam @ ( X388 @ sh @ one ) ) ) )
<=> hoaslamnotap )
& ( axscons
<=> ! [X392: subst] :
( ( push @ ( sub @ one @ X392 ) @ ( comp @ sh @ X392 ) )
= X392 ) )
& ~ ( var @ ( d2term @ d_one ) )
& ( hoaslam
= ( ^ [X393: subst,X394: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X395: subst,X396: term > $o,X397: term] :
( ~ ! [X398: term] :
( ~ ! [X399: term] :
( ~ ( X396 @ X399 )
| ( X396 @ ( sub @ X398 @ ( push @ X399 @ id ) ) ) )
| ( X396 @ ( lam @ X398 ) ) )
| ( X396 @ ( sub @ X397 @ X395 ) )
| ~ ! [X400: term] :
( ~ ( var @ X400 )
| ( X396 @ ( sub @ X400 @ X395 ) ) )
| ~ ! [X401: term,X402: term] :
( ~ ( X396 @ X401 )
| ~ ( X396 @ X402 )
| ( X396 @ ( ap @ X402 @ X401 ) ) ) )
<=> induction2lem )
& ( axidr
<=> ! [X403: subst] :
( ( comp @ X403 @ id )
= X403 ) )
& ( ! [X404: term,X405: term,X406: term,X407: term] :
( ( X404 = X407 )
| ( ( ap @ ( sub @ X404 @ id ) @ X406 )
!= ( ap @ ( sub @ X407 @ id ) @ X405 ) ) )
<=> hoasapinj1 )
& axassoc
& ( ( ! [X408: term] :
( ( sub @ X408 @ id )
= X408 )
=> ( ! [X409: term,X410: term,X411: term,X412: term] :
( ( X409 = X412 )
| ( ( ap @ X412 @ X411 )
!= ( ap @ X409 @ X410 ) ) )
=> ! [X413: term,X414: term,X415: term,X416: term] :
( ( X415 = X416 )
| ( ( ap @ ( sub @ X416 @ id ) @ X414 )
!= ( ap @ ( sub @ X415 @ id ) @ X413 ) ) ) ) )
<=> hoasapinj1_lthm )
& ulamvarind
& ~ lamnotap
& pushprop_lthm
& ( ! [X417: term,X418: subst > term > subst > $o,X419: term > $o] :
( ~ ! [X420: subst > term > term] :
( ~ ! [X421: term,X422: subst,X423: subst] :
( ( sub @ ( X420 @ X422 @ X421 ) @ X423 )
= ( X420 @ ( comp @ X422 @ X423 ) @ ( sub @ X421 @ X423 ) ) )
| ( X418 @ id @ ( lam @ ( X420 @ sh @ one ) ) @ id )
| ~ ! [X424: term] :
( ( X418 @ id @ ( X420 @ id @ X424 ) @ id )
| ~ ( X418 @ id @ X424 @ id ) ) )
| ~ ! [X425: term] :
( ~ ( X419 @ X425 )
| ( X419 @ ( sub @ X417 @ ( push @ X425 @ id ) ) ) )
| ~ ! [X426: subst,X427: subst,X428: term,X429: subst] :
( ( X418 @ X429 @ X428 @ ( comp @ X427 @ X426 ) )
| ~ ( X418 @ ( comp @ X429 @ X427 ) @ ( sub @ X428 @ X427 ) @ X426 ) )
| ~ ! [X430: term] :
( ( X418 @ id @ X430 @ id )
= ( X419 @ X430 ) )
| ( X419 @ ( lam @ X417 ) )
| ~ ! [X431: subst,X432: term,X433: subst,X434: subst] :
( ~ ( X418 @ X434 @ X432 @ ( comp @ X431 @ X433 ) )
| ( X418 @ ( comp @ X434 @ X431 ) @ ( sub @ X432 @ X431 ) @ X433 ) ) )
<=> hoasinduction_lem3v2 )
& ( ( d2subst @ d_id )
= id )
& pushprop_lem0_gthm
& hoasinduction_lem1v2
& hoasinduction_lem1_lthm
& ( ! [X435: term,X436: term,X437: term,X438: term] :
( ( X436 = X438 )
| ( ( ap @ X435 @ X436 )
!= ( ap @ X437 @ X438 ) ) )
<=> apinj2 )
& ( ! [X439: term,X440: term > $o] :
( ( X440 @ X439 )
| ~ ! [X441: term] :
( ( X440 @ X441 )
| ~ ( var @ X441 ) )
| ~ ! [X442: term] :
( ~ ( X440 @ X442 )
| ( X440 @ ( lam @ X442 ) ) )
| ~ ! [X443: term,X444: term] :
( ~ ( X440 @ X443 )
| ( X440 @ ( ap @ X443 @ X444 ) )
| ~ ( X440 @ X444 ) ) )
<=> induction )
& ( ( ! [X445: subst > term > subst > $o] :
~ ! [X446: term > $o] :
~ ! [X447: term] :
( ( X446 @ X447 )
= ( X445 @ id @ X447 @ id ) )
=> ( ! [X448: term,X449: term > $o] :
( ~ ! [X450: term] :
( ~ ( var @ X450 )
| ( X449 @ X450 ) )
| ( X449 @ X448 )
| ~ ! [X451: term] :
( ( X449 @ ( lam @ X451 ) )
| ~ ! [X452: term] :
( ~ ( X449 @ X452 )
| ( X449 @ ( sub @ X451 @ ( push @ X452 @ id ) ) ) ) )
| ~ ! [X453: term,X454: term] :
( ~ ( X449 @ X453 )
| ~ ( X449 @ X454 )
| ( X449 @ ( ap @ X454 @ X453 ) ) ) )
=> ( ! [X455: term] :
( ( sub @ X455 @ id )
= X455 )
=> ( ! [X456: term > $o,X457: subst > term > subst > $o,X458: term] :
( ~ ! [X459: term] :
( ( X456 @ X459 )
= ( X457 @ id @ X459 @ id ) )
| ~ ! [X460: term] :
( ~ ( X456 @ X460 )
| ( X456 @ ( sub @ X458 @ ( push @ X460 @ id ) ) ) )
| ~ ! [X461: subst > term > term] :
( ~ ! [X462: term,X463: subst,X464: subst] :
( ( X461 @ ( comp @ X463 @ X464 ) @ ( sub @ X462 @ X464 ) )
= ( sub @ ( X461 @ X463 @ X462 ) @ X464 ) )
| ( X457 @ id @ ( lam @ ( X461 @ sh @ one ) ) @ id )
| ~ ! [X465: term] :
( ~ ( X457 @ id @ X465 @ id )
| ( X457 @ id @ ( X461 @ id @ X465 ) @ id ) ) )
| ( X456 @ ( lam @ X458 ) ) )
=> ! [X466: subst > term > subst > $o,X467: term] :
( ( X466 @ id @ X467 @ id )
| ~ ! [X468: subst > term > term] :
( ( X466 @ id @ ( lam @ ( X468 @ sh @ one ) ) @ id )
| ~ ! [X469: term] :
( ( X466 @ id @ ( X468 @ id @ X469 ) @ id )
| ~ ( X466 @ id @ X469 @ id ) )
| ~ ! [X470: term,X471: subst,X472: subst] :
( ( sub @ ( X468 @ X472 @ X470 ) @ X471 )
= ( X468 @ ( comp @ X472 @ X471 ) @ ( sub @ X470 @ X471 ) ) ) )
| ~ ! [X473: term,X474: term] :
( ~ ( X466 @ id @ X474 @ id )
| ( X466 @ id @ ( ap @ ( sub @ X473 @ id ) @ X474 ) @ id )
| ~ ( X466 @ id @ X473 @ id ) ) ) ) ) ) )
<=> hoasinduction_no_psi_cond_lthm )
& hoasinduction_lem2v2_gthm
& hoasinduction_lem2_gthm
& pushprop_lem1_gthm
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lthm
<=> ( ! [X475: term,X476: term > $o] :
( ~ ! [X477: term] :
( ~ ! [X478: term] :
( ~ ( X476 @ X478 )
| ( X476 @ ( sub @ X477 @ ( push @ X478 @ id ) ) ) )
| ( X476 @ ( lam @ X477 ) ) )
| ( X476 @ X475 )
| ~ ! [X479: term] :
( ~ ( var @ X479 )
| ( X476 @ X479 ) )
| ~ ! [X480: term,X481: term] :
( ( X476 @ ( ap @ X480 @ X481 ) )
| ~ ( X476 @ X481 )
| ~ ( X476 @ X480 ) ) )
=> ( ! [X482: subst > term > subst > $o,X483: term,X484: term] :
( ~ ( X482 @ id @ X483 @ id )
| ( X482 @ id @ ( ap @ X483 @ X484 ) @ id )
| ~ ! [X485: subst,X486: term,X487: subst,X488: subst] :
( ( X482 @ ( comp @ X488 @ X487 ) @ ( sub @ X486 @ X487 ) @ X485 )
| ~ ( X482 @ X488 @ X486 @ ( comp @ X487 @ X485 ) ) )
| ~ ( X482 @ id @ X484 @ id )
| ~ ! [X489: term,X490: subst,X491: subst,X492: subst] :
( ( X482 @ X490 @ X489 @ ( comp @ X491 @ X492 ) )
| ~ ( X482 @ ( comp @ X490 @ X491 ) @ ( sub @ X489 @ X491 ) @ X492 ) )
| ~ ! [X493: term,X494: term] :
( ( X482 @ id @ ( ap @ ( sub @ X493 @ id ) @ X494 ) @ id )
| ~ ( X482 @ id @ X494 @ id )
| ~ ( X482 @ id @ X493 @ id ) ) )
=> ( ! [X495: term,X496: subst > term > subst > $o] :
( ~ ! [X497: subst,X498: subst,X499: subst,X500: term] :
( ~ ( X496 @ ( comp @ X498 @ X499 ) @ ( sub @ X500 @ X499 ) @ X497 )
| ( X496 @ X498 @ X500 @ ( comp @ X499 @ X497 ) ) )
| ( X496 @ id @ ( lam @ X495 ) @ id )
| ~ ! [X501: subst,X502: term,X503: subst,X504: subst] :
( ( X496 @ ( comp @ X503 @ X504 ) @ ( sub @ X502 @ X504 ) @ X501 )
| ~ ( X496 @ X503 @ X502 @ ( comp @ X504 @ X501 ) ) )
| ~ ! [X505: subst > term > term] :
( ~ ! [X506: term] :
( ~ ( X496 @ id @ X506 @ id )
| ( X496 @ id @ ( X505 @ id @ X506 ) @ id ) )
| ( X496 @ id @ ( lam @ ( X505 @ sh @ one ) ) @ id )
| ~ ! [X507: term,X508: subst,X509: subst] :
( ( X505 @ ( comp @ X508 @ X509 ) @ ( sub @ X507 @ X509 ) )
= ( sub @ ( X505 @ X508 @ X507 ) @ X509 ) ) )
| ~ ! [X510: term] :
( ~ ( X496 @ id @ X510 @ id )
| ( X496 @ id @ ( sub @ X495 @ ( push @ X510 @ id ) ) @ id ) ) )
=> ! [X511: term,X512: subst > term > subst > $o] :
( ~ ! [X513: subst,X514: subst,X515: term,X516: subst] :
( ~ ( X512 @ ( comp @ X514 @ X516 ) @ ( sub @ X515 @ X516 ) @ X513 )
| ( X512 @ X514 @ X515 @ ( comp @ X516 @ X513 ) ) )
| ~ ! [X517: subst > term > term] :
( ( X512 @ id @ ( lam @ ( X517 @ sh @ one ) ) @ id )
| ~ ! [X518: term] :
( ( X512 @ id @ ( X517 @ id @ X518 ) @ id )
| ~ ( X512 @ id @ X518 @ id ) )
| ~ ! [X519: term,X520: subst,X521: subst] :
( ( X517 @ ( comp @ X521 @ X520 ) @ ( sub @ X519 @ X520 ) )
= ( sub @ ( X517 @ X521 @ X519 ) @ X520 ) ) )
| ~ ! [X522: term,X523: subst,X524: subst,X525: subst] :
( ~ ( X512 @ X525 @ X522 @ ( comp @ X524 @ X523 ) )
| ( X512 @ ( comp @ X525 @ X524 ) @ ( sub @ X522 @ X524 ) @ X523 ) )
| ( X512 @ id @ X511 @ id )
| ~ ! [X526: term,X527: term] :
( ~ ( X512 @ id @ X526 @ id )
| ~ ( X512 @ id @ X527 @ id )
| ( X512 @ id @ ( ap @ ( sub @ X526 @ id ) @ X527 ) @ id ) )
| ~ ! [X528: term] :
( ( X512 @ id @ X528 @ id )
| ~ ( var @ ( sub @ X528 @ id ) ) ) ) ) ) ) )
& ( hoasinduction_lem3b
<=> ! [X529: term] :
~ ! [X530: subst > term > term] :
( ( sub @ X529 @ ( push @ one @ sh ) )
!= ( X530 @ sh @ one ) ) )
& ( substmonoid_lthm
<=> ( ! [X531: subst] :
( ( comp @ id @ X531 )
= X531 )
=> ( ! [X532: subst] :
( ( comp @ X532 @ id )
= X532 )
=> ( ! [X533: subst] :
( ( comp @ id @ X533 )
= X533 )
& ! [X534: subst] :
( ( comp @ X534 @ id )
= X534 ) ) ) ) )
& hoaslamnotvar
& ( ( ! [X535: term] :
~ ! [X536: subst > term > term] :
~ ! [X537: subst,X538: term] :
( ( sub @ X535 @ ( push @ X538 @ X537 ) )
= ( X536 @ X537 @ X538 ) )
=> ( ! [X539: term] :
( ( sub @ X539 @ id )
= X539 )
=> ! [X540: term,X541: term > $o,X542: subst > term > subst > $o] :
( ( X541 @ ( lam @ X540 ) )
| ~ ! [X543: term] :
( ( X542 @ id @ X543 @ id )
= ( X541 @ X543 ) )
| ~ ! [X544: subst > term > term] :
( ~ ! [X545: term] :
( ( X542 @ id @ ( X544 @ id @ X545 ) @ id )
| ~ ( X542 @ id @ X545 @ id ) )
| ( X542 @ id @ ( lam @ ( X544 @ sh @ one ) ) @ id )
| ~ ! [X546: subst,X547: term,X548: subst] :
( ( sub @ ( X544 @ X548 @ X547 ) @ X546 )
= ( X544 @ ( comp @ X548 @ X546 ) @ ( sub @ X547 @ X546 ) ) ) )
| ~ ! [X549: term] :
( ( X541 @ ( sub @ X540 @ ( push @ X549 @ id ) ) )
| ~ ( X541 @ X549 ) ) ) ) )
<=> hoasinduction_lem3v2a_lthm ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ! [X13: term,X15: term] :
( ( X13 = X15 )
| ( ( lam @ X13 )
!= ( lam @ X15 ) ) )
<=> laminj )
& ( ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X10: term > $o,X21: term] :
( ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ( X10 @ X13 ) )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ( X10 @ X21 ) )
=> ! [X10: term > $o,X16: term,X17: subst] :
( ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) ) ) ) ) ) )
<=> induction2lem_lthm )
& ( pushprop_lem0
<=> ! [X14: subst,X13: term,X10: term > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) )
& ( ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ! [X11: term > $o,X13: term,X10: term > $o,X14: subst] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
| ~ ( X10 @ X13 )
| ( X11 @ one ) ) )
<=> pushprop_lem1v2_lthm )
& hoasinduction_lem2_lthm
& ( substmonoid
<=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
& ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) ) )
& pushprop_lem3v2_lthm
& lamnotvar
& ( hoasinduction_lem3v2_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X10: subst > term > subst > $o,X48: term,X11: term > $o] :
( ~ ! [X13: term,X25: subst,X26: subst,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) ) )
| ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) ) ) ) )
& ( ! [X10: subst > term > subst > $o,X30: term] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ( X10 @ id @ X30 @ id )
| ~ ! [X18: subst > term > term] :
( ( 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 ) ) ) ) )
<=> hoasinduction_no_psi_cond )
& substmonoid_gthm
& hoaslamnotvar_gthm
& ( hoasinduction_lem3_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) ) )
=> ! [X10: subst > term > subst > $o,X27: term] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X25: subst,X13: term,X26: subst,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X13: term,X14: subst,X25: subst,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ( X10 @ id @ ( lam @ X27 ) @ id ) ) ) ) )
& ( induction2
<=> ! [X10: term > $o,X22: term] :
( ~ ! [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,X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X13 ) ) ) )
& ~ ulamvar1
& ( ( d2subst @ d_id )
= ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& pushprop_lem1v2_gthm
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X13: term,X14: subst] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
<=> hoasinduction_lem3v2_f )
& hoasinduction_lem3_gthm
& ( ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X13: term,X14: subst] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ( ! [X13: term,X15: term] :
( ( X13 = X15 )
| ( ( lam @ X13 )
!= ( lam @ X15 ) ) )
=> ! [X33: term,X31: subst > term > term,X18: subst > term > term,X32: subst] :
( ( ( 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 ) ) )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) ) ) )
<=> hoaslaminj_lthm )
& ( ( d2term @ d_one )
= ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) ) )
& ( ! [X10: subst > term > subst > $o,X42: term] :
( ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ 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 @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) )
<=> hoasinduction_lem3aa )
& ( hoasinduction_lem2v2
<=> ! [X11: term > $o,X45: term,X10: subst > term > subst > $o,X44: term] :
( ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ( X11 @ X45 )
| ~ ( X11 @ X44 )
| ~ ! [X14: subst,X25: subst,X13: term,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X11 @ ( ap @ X44 @ X45 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) ) )
& axapp
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0 )
& pushprop_lem2v2_lthm
& hoasapinj1_gthm
& hoasapinj2_gthm
& ! [X1: d_term] : ( d_one = X1 )
& hoasinduction_lem3a_gthm
& pushprop_lem3v2
& ( ! [X10: subst > term > subst > $o,X34: term] :
( ( X10 @ id @ ( lam @ ( sub @ X34 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X35: term,X18: subst > term > term] :
( ~ ! [X39: subst,X40: term,X41: subst] :
( ( X18 @ ( comp @ X39 @ X41 ) @ ( sub @ X40 @ X41 ) )
= ( sub @ X35 @ ( push @ ( sub @ X40 @ X41 ) @ ( comp @ X39 @ X41 ) ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X37: term,X38: subst,X36: subst] :
( ( sub @ ( X18 @ X36 @ X37 ) @ X38 )
= ( sub @ ( sub @ X35 @ ( push @ X37 @ X36 ) ) @ X38 ) ) )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X34 @ ( push @ X15 @ id ) ) @ id ) ) )
<=> hoasinduction_lem3aaa )
& ! [X4: subst] :
? [X5: d_subst] :
( ( d2subst @ X5 )
= X4 )
& ! [X5: d_subst] : ( d_id = X5 )
& ( ! [X10: subst > term > subst > $o,X43: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id ) )
| ( X10 @ id @ ( lam @ X43 ) @ 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 ) ) ) ) )
<=> hoasinduction_lem3a )
& ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
& hoasapnotvar_lthm
& ( pushprop_lem1
<=> ! [X15: term,X14: subst,X13: term,X26: term > $o,X10: term > $o] :
( ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) )
| ~ ( X10 @ X13 ) ) )
& hoasapnotvar
& hoaslamnotap_lthm
& hoasinduction_gthm
& axabs
& induction2_gthm
& ! [X0: term] :
? [X1: d_term] :
( ( d2term @ X1 )
= X0 )
& hoaslaminj_gthm
& ( hoasapinj2
<=> ! [X13: term,X15: term,X20: term,X19: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) )
& ( apinj1
<=> ! [X13: term,X19: term,X20: term,X15: term] :
( ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) )
| ( X13 = X15 ) ) )
& ( ( d2term @ d_one )
= ( lam @ ( d2term @ d_one ) ) )
& hoasinduction_lem1
& ( ( d2subst @ d_id )
= ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) ) )
& pushprop_lem2v2
& ( ! [X10: term > $o,X13: term,X14: subst] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) )
<=> pushprop_lem0_lthm )
& pushprop
& hoasinduction_lem3v2_gthm
& ( ! [X15: term,X13: term] :
( ( ( sub @ X13 @ sh )
!= ( sub @ X15 @ sh ) )
| ( X13 = X15 ) )
<=> shinj )
& ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
<=> hoasinduction_lem0_lthm )
& hoaslamnotvar_lthm
& ( hoasinduction_p_and_p_prime
= ( ^ [X10: subst > term > subst > $o,X11: term > $o] :
! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) ) ) )
& induction2lem_gthm
& termmset_gthm
& ( hoasapinj2_lthm
<=> ( ! [X20: term,X15: term,X19: term,X13: term] :
( ( X19 = X20 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
=> ! [X19: term,X20: term,X13: term,X15: term] :
( ( X19 = X20 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) ) )
& axmap
& axclos
& ( one
= ( d2term @ d_one ) )
& ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
<=> axshiftcons )
& hoasapnotvar_gthm
& axvarshift
& ( ! [X42: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id ) ) )
<=> hoasinduction_lem3aa_lthm )
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) )
<=> termmset_lthm )
& ( ! [X10: subst > term > subst > $o,X24: term,X23: term] :
( ~ ( X10 @ id @ X23 @ id )
| ~ ( X10 @ id @ X24 @ id )
| ~ ! [X13: term,X25: subst,X14: subst,X26: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X25: subst,X26: subst,X14: subst,X13: term] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id ) )
<=> hoasinduction_lem2 )
& ( axidl
<=> ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 ) )
& ( induction2_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X16: term,X10: term > $o,X17: subst] :
( ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) )
| ~ ( X10 @ X15 ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) ) ) )
=> ! [X22: term,X10: term > $o] :
( ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) ) ) ) ) )
& hoasinduction_lem3b_gthm
& ( ( ^ [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 )
& ( ! [X13: term,X14: subst,X11: term > $o,X10: term > $o] :
( ( X11 @ one )
| ~ ( X10 @ X13 )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ ( sub @ X12 @ ( push @ X13 @ X14 ) ) ) ) )
<=> pushprop_lem1v2 )
& ulamvarsh
& hoasinduction_lem1_gthm
& ( ! [X13: term,X14: subst] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
<=> axvarcons )
& pushprop_gthm
& ( hoasinduction_lem3
<=> ! [X10: subst > term > subst > $o,X27: term] :
( ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X26: subst,X25: subst,X13: term,X14: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X13: term,X25: subst,X26: subst,X14: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X27 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) ) )
& ( ( d2term @ d_one )
= ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) ) )
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) )
<=> hoasinduction_lem3b_lthm )
& hoasinduction_lem1v2_gthm
& pushprop_lem2v2_gthm
& ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
<=> axvarid )
& ( ! [X33: term,X32: subst,X18: subst > term > term,X31: subst > term > term] :
( ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X31 @ X14 @ X13 ) @ X25 )
= ( X31 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( lam @ ( X31 @ sh @ one ) ) )
| ( ( X18 @ X32 @ X33 )
= ( X31 @ X32 @ X33 ) ) )
<=> hoaslaminj )
& ( termmset
<=> ! [X13: term] :
( ( sub @ X13 @ id )
= X13 ) )
& ( ( d2term @ d_one )
= ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) ) )
& ( hoasinduction_lem3a_lthm
<=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X42: term,X10: subst > term > subst > $o] :
( ( X10 @ id @ ( lam @ ( sub @ X42 @ ( push @ one @ sh ) ) ) @ id )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X42 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) ) )
=> ! [X10: subst > term > subst > $o,X43: term] :
( ~ ! [X15: term] :
( ( X10 @ id @ ( sub @ X43 @ ( push @ X15 @ id ) ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ( X10 @ id @ ( lam @ X43 ) @ id ) ) ) ) )
& ( ! [X10: subst > term > subst > $o,X28: term] :
( ~ ! [X15: term,X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id ) )
| ~ ! [X13: term,X14: subst,X25: subst,X26: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X14: subst,X25: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X25: subst,X13: term,X26: subst,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 ) )
| ( X10 @ id @ X28 @ id ) )
<=> hoasinduction )
& ! [X6: d_subst,X7: d_subst] :
( ( ( d2subst @ X6 )
= ( d2subst @ X7 ) )
=> ( X6 = X7 ) )
& ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
<=> hoasinduction_lem3v2_f_lthm )
& ( hoasinduction_lem3v2a
<=> ! [X10: subst > term > subst > $o,X29: term,X11: term > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) ) ) )
& ( pushprop_lem1_lthm
<=> ( ! [X14: subst,X13: term] :
( ( sub @ one @ ( push @ X13 @ X14 ) )
= X13 )
=> ( ! [X14: subst,X13: term] :
( ( comp @ sh @ ( push @ X13 @ X14 ) )
= X14 )
=> ! [X13: term,X26: term > $o,X15: term,X14: subst,X10: term > $o] :
( ( X26 @ ( sub @ X13 @ ( push @ X15 @ X14 ) ) )
| ~ ( X10 @ X13 ) ) ) ) )
& ! [X2: d_term,X3: d_term] :
( ( ( d2term @ X2 )
= ( d2term @ X3 ) )
=> ( X2 = X3 ) )
& pushprop_lthm_orig
& apnotvar
& hoaslamnotap_gthm
& ( ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X10: term > $o,X22: term] :
( ~ ! [X15: term,X13: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 )
| ~ ( X10 @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X10: subst > term > subst > $o,X29: term,X11: term > $o] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) )
| ( X11 @ ( lam @ X29 ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X14: subst,X13: term,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) ) )
=> ! [X10: subst > term > subst > $o,X28: term] :
( ~ ! [X14: subst,X26: subst,X25: subst,X13: term] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [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 ) ) )
| ~ ! [X15: term,X13: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ X28 @ id )
| ~ ! [X13: term] :
( ~ ( var @ ( sub @ X13 @ id ) )
| ( X10 @ id @ X13 @ id ) )
| ~ ! [X14: subst,X26: subst,X25: subst,X13: term] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) ) ) ) ) ) )
<=> hoasinduction_lthm_3 )
& ( ! [X47: term,X46: term,X18: subst > term > term] :
( ~ ! [X25: subst,X14: subst,X13: term] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( ( lam @ ( X18 @ sh @ one ) )
!= ( ap @ ( sub @ X46 @ id ) @ X47 ) ) )
<=> hoaslamnotap )
& ( axscons
<=> ! [X14: subst] :
( ( push @ ( sub @ one @ X14 ) @ ( comp @ sh @ X14 ) )
= X14 ) )
& ~ ( var @ ( d2term @ d_one ) )
& ( hoaslam
= ( ^ [X8: subst,X9: subst > term > term] : ( d2term @ d_one ) ) )
& ( ! [X17: subst,X10: term > $o,X16: term] :
( ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ ( sub @ X16 @ X17 ) )
| ~ ! [X15: term] :
( ~ ( var @ X15 )
| ( X10 @ ( sub @ X15 @ X17 ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) ) ) )
<=> induction2lem )
& ( axidr
<=> ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 ) )
& ( ! [X13: term,X20: term,X19: term,X15: term] :
( ( X13 = X15 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) )
<=> hoasapinj1 )
& axassoc
& ( ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X15: term,X20: term,X19: term,X13: term] :
( ( X13 = X15 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
=> ! [X20: term,X19: term,X15: term,X13: term] :
( ( X13 = X15 )
| ( ( ap @ ( sub @ X13 @ id ) @ X19 )
!= ( ap @ ( sub @ X15 @ id ) @ X20 ) ) ) ) )
<=> hoasapinj1_lthm )
& ulamvarind
& ~ lamnotap
& pushprop_lthm
& ( ! [X48: term,X10: subst > term > subst > $o,X11: term > $o] :
( ~ ! [X18: subst > term > term] :
( ~ ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X48 @ ( push @ X15 @ id ) ) ) )
| ~ ! [X25: subst,X26: subst,X13: term,X14: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ( X11 @ ( lam @ X48 ) )
| ~ ! [X26: subst,X13: term,X25: subst,X14: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) ) )
<=> hoasinduction_lem3v2 )
& ( ( d2subst @ d_id )
= id )
& pushprop_lem0_gthm
& hoasinduction_lem1v2
& hoasinduction_lem1_lthm
& ( ! [X13: term,X19: term,X15: term,X20: term] :
( ( X19 = X20 )
| ( ( ap @ X13 @ X19 )
!= ( ap @ X15 @ X20 ) ) )
<=> apinj2 )
& ( ! [X21: term,X10: term > $o] :
( ( X10 @ X21 )
| ~ ! [X13: term] :
( ( X10 @ X13 )
| ~ ( var @ X13 ) )
| ~ ! [X13: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( lam @ X13 ) ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 ) ) )
<=> induction )
& ( ( ! [X10: subst > term > subst > $o] :
~ ! [X11: term > $o] :
~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
=> ( ! [X22: term,X10: term > $o] :
( ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ( X10 @ ( lam @ X13 ) )
| ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) ) )
| ~ ! [X15: term,X13: term] :
( ~ ( X10 @ X15 )
| ~ ( X10 @ X13 )
| ( X10 @ ( ap @ X13 @ X15 ) ) ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ( ! [X11: term > $o,X10: subst > term > subst > $o,X29: term] :
( ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X15: term] :
( ~ ( X11 @ X15 )
| ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term,X14: subst,X25: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) ) )
| ( X11 @ ( lam @ X29 ) ) )
=> ! [X10: subst > term > subst > $o,X30: term] :
( ( X10 @ id @ X30 @ id )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [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 ) ) ) )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) ) ) ) ) ) )
<=> hoasinduction_no_psi_cond_lthm )
& hoasinduction_lem2v2_gthm
& hoasinduction_lem2_gthm
& pushprop_lem1_gthm
& ( ( d2subst @ d_id )
= sh )
& ( hoasinduction_lthm
<=> ( ! [X22: term,X10: term > $o] :
( ~ ! [X13: term] :
( ~ ! [X15: term] :
( ~ ( X10 @ X15 )
| ( X10 @ ( sub @ X13 @ ( push @ X15 @ id ) ) ) )
| ( X10 @ ( lam @ X13 ) ) )
| ( X10 @ X22 )
| ~ ! [X13: term] :
( ~ ( var @ X13 )
| ( X10 @ X13 ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ ( ap @ X13 @ X15 ) )
| ~ ( X10 @ X15 )
| ~ ( X10 @ X13 ) ) )
=> ( ! [X10: subst > term > subst > $o,X23: term,X24: term] :
( ~ ( X10 @ id @ X23 @ id )
| ( X10 @ id @ ( ap @ X23 @ X24 ) @ id )
| ~ ! [X25: subst,X13: term,X26: subst,X14: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ( X10 @ id @ X24 @ id )
| ~ ! [X13: term,X14: subst,X26: subst,X25: subst] :
( ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ~ ! [X13: term,X15: term] :
( ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id )
| ~ ( X10 @ id @ X15 @ id )
| ~ ( X10 @ id @ X13 @ id ) ) )
=> ( ! [X27: term,X10: subst > term > subst > $o] :
( ~ ! [X25: subst,X14: subst,X26: subst,X13: term] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ( X10 @ id @ ( lam @ X27 ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst,X26: subst] :
( ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ~ ( X10 @ id @ X13 @ id )
| ( X10 @ id @ ( X18 @ id @ X13 ) @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [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 @ X27 @ ( push @ X15 @ id ) ) @ id ) ) )
=> ! [X28: term,X10: subst > term > subst > $o] :
( ~ ! [X25: subst,X14: subst,X13: term,X26: subst] :
( ~ ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 )
| ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) ) )
| ~ ! [X18: subst > term > term] :
( ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ~ ! [X13: term,X25: subst,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X13: term,X25: subst,X26: subst,X14: subst] :
( ~ ( X10 @ X14 @ X13 @ ( comp @ X26 @ X25 ) )
| ( X10 @ ( comp @ X14 @ X26 ) @ ( sub @ X13 @ X26 ) @ X25 ) )
| ( X10 @ id @ X28 @ id )
| ~ ! [X13: term,X15: term] :
( ~ ( X10 @ id @ X13 @ id )
| ~ ( X10 @ id @ X15 @ id )
| ( X10 @ id @ ( ap @ ( sub @ X13 @ id ) @ X15 ) @ id ) )
| ~ ! [X13: term] :
( ( X10 @ id @ X13 @ id )
| ~ ( var @ ( sub @ X13 @ id ) ) ) ) ) ) ) )
& ( hoasinduction_lem3b
<=> ! [X15: term] :
~ ! [X18: subst > term > term] :
( ( X18 @ sh @ one )
!= ( sub @ X15 @ ( push @ one @ sh ) ) ) )
& ( substmonoid_lthm
<=> ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 )
=> ( ! [X14: subst] :
( ( comp @ id @ X14 )
= X14 )
& ! [X14: subst] :
( ( comp @ X14 @ id )
= X14 ) ) ) ) )
& hoaslamnotvar
& ( ( ! [X15: term] :
~ ! [X18: subst > term > term] :
~ ! [X14: subst,X13: term] :
( ( sub @ X15 @ ( push @ X13 @ X14 ) )
= ( X18 @ X14 @ X13 ) )
=> ( ! [X13: term] :
( ( sub @ X13 @ id )
= X13 )
=> ! [X29: term,X11: term > $o,X10: subst > term > subst > $o] :
( ( X11 @ ( lam @ X29 ) )
| ~ ! [X12: term] :
( ( X11 @ X12 )
= ( X10 @ id @ X12 @ id ) )
| ~ ! [X18: subst > term > term] :
( ~ ! [X13: term] :
( ( X10 @ id @ ( X18 @ id @ X13 ) @ id )
| ~ ( X10 @ id @ X13 @ id ) )
| ( X10 @ id @ ( lam @ ( X18 @ sh @ one ) ) @ id )
| ~ ! [X25: subst,X13: term,X14: subst] :
( ( sub @ ( X18 @ X14 @ X13 ) @ X25 )
= ( X18 @ ( comp @ X14 @ X25 ) @ ( sub @ X13 @ X25 ) ) ) )
| ~ ! [X15: term] :
( ( X11 @ ( sub @ X29 @ ( push @ X15 @ id ) ) )
| ~ ( X11 @ X15 ) ) ) ) )
<=> hoasinduction_lem3v2a_lthm ) ),
file('/export/starexec/sandbox/tmp/tmp.Ab9xLNyFM0/DTF2THF_1970.p',alg444_1) ).
thf(f1064,plain,
! [X250: term] :
( ( d2term @ ( sK576 @ X250 ) )
= X250 ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1918,plain,
( $true
= ( sK530 @ sK527 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1917]) ).
thf(f1917,plain,
( ( $true
= ( sK530 @ sK527 ) )
| ( $true = $false ) ),
inference(backward_demodulation,[status(thm)],[f1138,f1906]) ).
thf(f1906,plain,
pushprop_lem1v2 = $false,
inference(trivial_inequality_removal,[status(thm)],[f1905]) ).
thf(f1905,plain,
( ( pushprop_lem1v2 = $false )
| ( $true = $false ) ),
inference(duplicate_literal_removal,[status(thm)],[f1899]) ).
thf(f1899,plain,
( ( pushprop_lem1v2 = $false )
| ( pushprop_lem1v2 = $false )
| ( $true = $false ) ),
inference(superposition,[status(thm)],[f1896,f1708]) ).
thf(f1708,plain,
( ( ( sK592 @ ( d2term @ d_one ) )
= $false )
| ( pushprop_lem1v2 = $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1706]) ).
thf(f1706,plain,
( ( pushprop_lem1v2 = $false )
| ( ( ( !! @ term
@ ^ [Y0: term] :
( ( sK592 @ Y0 )
= ( sK593 @ ( sub @ Y0 @ ( push @ sK591 @ sK594 ) ) ) ) )
=> ( sK592 @ ( d2term @ d_one ) ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1705]) ).
thf(f1705,plain,
( ( ( ( sK593 @ sK591 )
=> ( ( !! @ term
@ ^ [Y0: term] :
( ( sK592 @ Y0 )
= ( sK593 @ ( sub @ Y0 @ ( push @ sK591 @ sK594 ) ) ) ) )
=> ( sK592 @ ( d2term @ d_one ) ) ) )
= $false )
| ( pushprop_lem1v2 = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1704]) ).
thf(f1704,plain,
( ( pushprop_lem1v2 = $false )
| ( ( ^ [Y0: subst] :
( ( sK593 @ sK591 )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( sK592 @ Y1 )
= ( sK593 @ ( sub @ Y1 @ ( push @ sK591 @ Y0 ) ) ) ) )
=> ( sK592 @ ( d2term @ d_one ) ) ) )
@ sK594 )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f1703]) ).
thf(f1703,plain,
( ( pushprop_lem1v2 = $false )
| ( $false
= ( !! @ subst
@ ^ [Y0: subst] :
( ( sK593 @ sK591 )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( sK592 @ Y1 )
= ( sK593 @ ( sub @ Y1 @ ( push @ sK591 @ Y0 ) ) ) ) )
=> ( sK592 @ ( d2term @ d_one ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1702]) ).
thf(f1702,plain,
( ( pushprop_lem1v2 = $false )
| ( ( ^ [Y0: term > $o] :
( !! @ subst
@ ^ [Y1: subst] :
( ( Y0 @ sK591 )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( sK592 @ Y2 )
= ( Y0 @ ( sub @ Y2 @ ( push @ sK591 @ Y1 ) ) ) ) )
=> ( sK592 @ ( d2term @ d_one ) ) ) ) )
@ sK593 )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f1701]) ).
thf(f1701,plain,
( ( ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ subst
@ ^ [Y1: subst] :
( ( Y0 @ sK591 )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( sK592 @ Y2 )
= ( Y0 @ ( sub @ Y2 @ ( push @ sK591 @ Y1 ) ) ) ) )
=> ( sK592 @ ( d2term @ d_one ) ) ) ) ) )
= $false )
| ( pushprop_lem1v2 = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1700]) ).
thf(f1700,plain,
( ( pushprop_lem1v2 = $false )
| ( ( ^ [Y0: term > $o] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ subst
@ ^ [Y2: subst] :
( ( Y1 @ sK591 )
=> ( ( !! @ term
@ ^ [Y3: term] :
( ( Y0 @ Y3 )
= ( Y1 @ ( sub @ Y3 @ ( push @ sK591 @ Y2 ) ) ) ) )
=> ( Y0 @ ( d2term @ d_one ) ) ) ) ) )
@ sK592 )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f1699]) ).
thf(f1699,plain,
( ( pushprop_lem1v2 = $false )
| ( ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ subst
@ ^ [Y2: subst] :
( ( Y1 @ sK591 )
=> ( ( !! @ term
@ ^ [Y3: term] :
( ( Y0 @ Y3 )
= ( Y1 @ ( sub @ Y3 @ ( push @ sK591 @ Y2 ) ) ) ) )
=> ( Y0 @ ( d2term @ d_one ) ) ) ) ) ) )
= $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1698]) ).
thf(f1698,plain,
( ( pushprop_lem1v2 = $false )
| ( ( ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( Y1 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y3 ) ) ) ) )
=> ( Y1 @ ( d2term @ d_one ) ) ) ) ) ) )
@ sK591 )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f1697]) ).
thf(f1697,plain,
( ( ( !! @ term
@ ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( Y1 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y3 ) ) ) ) )
=> ( Y1 @ ( d2term @ d_one ) ) ) ) ) ) ) )
= $false )
| ( pushprop_lem1v2 = $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1695]) ).
thf(f1695,plain,
( pushprop_lem1v2
!= ( !! @ term
@ ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( Y1 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y3 ) ) ) ) )
=> ( Y1 @ ( d2term @ d_one ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1664]) ).
thf(f1664,plain,
( pushprop_lem1v2
!= ( !! @ term
@ ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( ^ [Y4: term,Y5: subst,Y6: term > $o,Y7: term > $o] :
( !! @ term
@ ^ [Y8: term] :
( ( Y7 @ Y8 )
= ( Y6 @ ( sub @ Y8 @ ( push @ Y4 @ Y5 ) ) ) ) )
@ Y0
@ Y3
@ Y2
@ Y1 )
=> ( Y1 @ ( d2term @ d_one ) ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1260,f1067,f1176]) ).
thf(f1176,plain,
( one
= ( d2term @ d_one ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1067,plain,
( pushprop_p_and_p_prime
= ( ^ [Y0: term,Y1: subst,Y2: term > $o,Y3: term > $o] :
( !! @ term
@ ^ [Y4: term] :
( ( Y3 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y1 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f1260,plain,
( pushprop_lem1v2
!= ( !! @ term
@ ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( pushprop_p_and_p_prime @ Y0 @ Y3 @ Y2 @ Y1 )
=> ( Y1 @ one ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( pushprop_lem1v2
!= ( !! @ term
@ ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( pushprop_p_and_p_prime @ Y0 @ Y3 @ Y2 @ Y1 )
=> ( Y1 @ one ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f8]) ).
thf(f8,plain,
( pushprop_lem1v2
!= ( !! @ term
@ ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( pushprop_p_and_p_prime @ Y0 @ Y3 @ Y2 @ Y1 )
=> ( Y1 @ one ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ! [X0: subst,X1: term > $o,X2: term > $o,X3: term] :
( ( ( X1 @ X3 )
=> ( ( pushprop_p_and_p_prime @ X3 @ X0 @ X1 @ X2 )
=> ( X2 @ one ) ) )
= pushprop_lem1v2 ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
~ ! [X14: subst,X10: term > $o,X11: term > $o,X13: term] :
( ( ( X10 @ X13 )
=> ( ( pushprop_p_and_p_prime @ X13 @ X14 @ X10 @ X11 )
=> ( X11 @ one ) ) )
= pushprop_lem1v2 ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
! [X14: subst,X10: term > $o,X11: term > $o,X13: term] :
( ( ( X10 @ X13 )
=> ( ( pushprop_p_and_p_prime @ X13 @ X14 @ X10 @ X11 )
=> ( X11 @ one ) ) )
= pushprop_lem1v2 ),
file('/export/starexec/sandbox/tmp/tmp.Ab9xLNyFM0/DTF2THF_1970.p',pushprop_lem1v2) ).
thf(f1896,plain,
! [X0: term] :
( ( $true
= ( sK592 @ X0 ) )
| ( pushprop_lem1v2 = $false ) ),
inference(trivial_inequality_removal,[status(thm)],[f1895]) ).
thf(f1895,plain,
! [X0: term] :
( ( $true = $false )
| ( $true
= ( sK592 @ X0 ) )
| ( pushprop_lem1v2 = $false ) ),
inference(duplicate_literal_removal,[status(thm)],[f1893]) ).
thf(f1893,plain,
! [X0: term] :
( ( pushprop_lem1v2 = $false )
| ( pushprop_lem1v2 = $false )
| ( $true = $false )
| ( $true
= ( sK592 @ X0 ) ) ),
inference(superposition,[status(thm)],[f1892,f1707]) ).
thf(f1707,plain,
( ( $true
= ( sK593 @ sK591 ) )
| ( pushprop_lem1v2 = $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1705]) ).
thf(f1892,plain,
! [X0: term,X1: term] :
( ( $false
= ( sK593 @ X0 ) )
| ( ( sK592 @ X1 )
= $true )
| ( pushprop_lem1v2 = $false ) ),
inference(superposition,[status(thm)],[f1876,f1874]) ).
thf(f1876,plain,
! [X1: term] :
( ( ( sK593 @ ( d2term @ d_one ) )
= $false )
| ( ( sK592 @ X1 )
= $true )
| ( pushprop_lem1v2 = $false ) ),
inference(backward_demodulation,[status(thm)],[f1713,f1874]) ).
thf(f1713,plain,
! [X1: term] :
( ( $false
= ( sK593 @ ( sub @ X1 @ ( push @ sK591 @ sK594 ) ) ) )
| ( ( sK592 @ X1 )
= $true )
| ( pushprop_lem1v2 = $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1712]) ).
thf(f1712,plain,
! [X1: term] :
( ( pushprop_lem1v2 = $false )
| ( ( sK592 @ X1 )
= ( sK593 @ ( sub @ X1 @ ( push @ sK591 @ sK594 ) ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1711]) ).
thf(f1711,plain,
! [X1: term] :
( ( ( ( sK592 @ X1 )
= ( sK593 @ ( sub @ X1 @ ( push @ sK591 @ sK594 ) ) ) )
= $true )
| ( pushprop_lem1v2 = $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1710]) ).
thf(f1710,plain,
! [X1: term] :
( ( ( ^ [Y0: term] :
( ( sK592 @ Y0 )
= ( sK593 @ ( sub @ Y0 @ ( push @ sK591 @ sK594 ) ) ) )
@ X1 )
= $true )
| ( pushprop_lem1v2 = $false ) ),
inference(pi_clausification,[status(thm)],[f1709]) ).
thf(f1709,plain,
( ( pushprop_lem1v2 = $false )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( sK592 @ Y0 )
= ( sK593 @ ( sub @ Y0 @ ( push @ sK591 @ sK594 ) ) ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f1706]) ).
thf(f1138,plain,
( ( $true
= ( sK530 @ sK527 ) )
| ( pushprop_lem1v2 = $true ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f4080,plain,
( ( sK530 @ ( d2term @ d_one ) )
= $false ),
inference(trivial_inequality_removal,[status(thm)],[f4079]) ).
thf(f4079,plain,
( ( $true = $false )
| ( ( sK530 @ ( d2term @ d_one ) )
= $false ) ),
inference(forward_demodulation,[status(thm)],[f4078,f2082]) ).
thf(f2082,plain,
! [X0: term] :
( ( sK529 @ X0 )
= $false ),
inference(trivial_inequality_removal,[status(thm)],[f2075]) ).
thf(f2075,plain,
! [X0: term] :
( ( ( sK529 @ X0 )
= $false )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1984,f2020]) ).
thf(f2020,plain,
! [X0: term > $o,X1: term] :
( ( $true
= ( X0 @ ( d2term @ d_one ) ) )
| ( $false
= ( X0 @ X1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2018]) ).
thf(f2018,plain,
! [X0: term > $o,X1: term] :
( ( $false
= ( X0 @ X1 ) )
| ( $true
= ( X0 @ ( d2term @ d_one ) ) )
| ( $true != $true ) ),
inference(equality_factoring,[status(thm)],[f2001]) ).
thf(f2001,plain,
! [X2: term > $o,X3: term > $o,X1: term] :
( ( $true
= ( X3 @ ( d2term @ d_one ) ) )
| ( $false
= ( X3 @ X1 ) )
| ( ( X2 @ ( d2term @ d_one ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f2000,f1874]) ).
thf(f2000,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( $false
= ( X3 @ X1 ) )
| ( $true
= ( X3 @ ( sub @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) @ ( push @ X1 @ X4 ) ) ) ) ),
inference(duplicate_literal_removal,[status(thm)],[f1999]) ).
thf(f1999,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( $true
= ( X3 @ ( sub @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) @ ( push @ X1 @ X4 ) ) ) )
| ( $false
= ( X3 @ X1 ) ) ),
inference(forward_demodulation,[status(thm)],[f1998,f1874]) ).
thf(f1998,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( $false
= ( X3 @ X1 ) )
| ( ( X2 @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) )
= $true )
| ( $true
= ( X3 @ ( sub @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) @ ( push @ X1 @ X4 ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1997]) ).
thf(f1997,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( $true
= ( X3 @ ( sub @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) @ ( push @ X1 @ X4 ) ) ) )
| ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( $true = $false )
| ( ( X2 @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) )
= $true )
| ( $false
= ( X3 @ X1 ) ) ),
inference(forward_demodulation,[status(thm)],[f1728,f1906]) ).
thf(f1728,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( pushprop_lem1v2 = $true )
| ( ( X2 @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) )
= $true )
| ( $true
= ( X3 @ ( sub @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) @ ( push @ X1 @ X4 ) ) ) )
| ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( $false
= ( X3 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1727]) ).
thf(f1727,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( pushprop_lem1v2 = $true )
| ( $false
= ( X3 @ X1 ) )
| ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( ( X2 @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) )
!= ( X3 @ ( sub @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) @ ( push @ X1 @ X4 ) ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1726]) ).
thf(f1726,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( ( ( X2 @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) )
= ( X3 @ ( sub @ ( sK595 @ X3 @ X1 @ X4 @ X2 ) @ ( push @ X1 @ X4 ) ) ) )
= $false )
| ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( pushprop_lem1v2 = $true )
| ( $false
= ( X3 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1725]) ).
thf(f1725,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( $false
= ( X3 @ X1 ) )
| ( ( ^ [Y0: term] :
( ( X2 @ Y0 )
= ( X3 @ ( sub @ Y0 @ ( push @ X1 @ X4 ) ) ) )
@ ( sK595 @ X3 @ X1 @ X4 @ X2 ) )
= $false )
| ( pushprop_lem1v2 = $true )
| ( ( X2 @ ( d2term @ d_one ) )
= $true ) ),
inference(sigma_clausification,[status(thm)],[f1724]) ).
thf(f1724,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( pushprop_lem1v2 = $true )
| ( ( X2 @ ( d2term @ d_one ) )
= $true )
| ( ( !! @ term
@ ^ [Y0: term] :
( ( X2 @ Y0 )
= ( X3 @ ( sub @ Y0 @ ( push @ X1 @ X4 ) ) ) ) )
= $false )
| ( $false
= ( X3 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1723]) ).
thf(f1723,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( pushprop_lem1v2 = $true )
| ( ( ( !! @ term
@ ^ [Y0: term] :
( ( X2 @ Y0 )
= ( X3 @ ( sub @ Y0 @ ( push @ X1 @ X4 ) ) ) ) )
=> ( X2 @ ( d2term @ d_one ) ) )
= $true )
| ( $false
= ( X3 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1722]) ).
thf(f1722,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( $true
= ( ( X3 @ X1 )
=> ( ( !! @ term
@ ^ [Y0: term] :
( ( X2 @ Y0 )
= ( X3 @ ( sub @ Y0 @ ( push @ X1 @ X4 ) ) ) ) )
=> ( X2 @ ( d2term @ d_one ) ) ) ) )
| ( pushprop_lem1v2 = $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1721]) ).
thf(f1721,plain,
! [X2: term > $o,X3: term > $o,X1: term,X4: subst] :
( ( $true
= ( ^ [Y0: subst] :
( ( X3 @ X1 )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( X2 @ Y1 )
= ( X3 @ ( sub @ Y1 @ ( push @ X1 @ Y0 ) ) ) ) )
=> ( X2 @ ( d2term @ d_one ) ) ) )
@ X4 ) )
| ( pushprop_lem1v2 = $true ) ),
inference(pi_clausification,[status(thm)],[f1720]) ).
thf(f1720,plain,
! [X2: term > $o,X3: term > $o,X1: term] :
( ( pushprop_lem1v2 = $true )
| ( $true
= ( !! @ subst
@ ^ [Y0: subst] :
( ( X3 @ X1 )
=> ( ( !! @ term
@ ^ [Y1: term] :
( ( X2 @ Y1 )
= ( X3 @ ( sub @ Y1 @ ( push @ X1 @ Y0 ) ) ) ) )
=> ( X2 @ ( d2term @ d_one ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1719]) ).
thf(f1719,plain,
! [X2: term > $o,X3: term > $o,X1: term] :
( ( pushprop_lem1v2 = $true )
| ( $true
= ( ^ [Y0: term > $o] :
( !! @ subst
@ ^ [Y1: subst] :
( ( Y0 @ X1 )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( X2 @ Y2 )
= ( Y0 @ ( sub @ Y2 @ ( push @ X1 @ Y1 ) ) ) ) )
=> ( X2 @ ( d2term @ d_one ) ) ) ) )
@ X3 ) ) ),
inference(pi_clausification,[status(thm)],[f1718]) ).
thf(f1718,plain,
! [X2: term > $o,X1: term] :
( ( $true
= ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ subst
@ ^ [Y1: subst] :
( ( Y0 @ X1 )
=> ( ( !! @ term
@ ^ [Y2: term] :
( ( X2 @ Y2 )
= ( Y0 @ ( sub @ Y2 @ ( push @ X1 @ Y1 ) ) ) ) )
=> ( X2 @ ( d2term @ d_one ) ) ) ) ) ) )
| ( pushprop_lem1v2 = $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1717]) ).
thf(f1717,plain,
! [X2: term > $o,X1: term] :
( ( pushprop_lem1v2 = $true )
| ( ( ^ [Y0: term > $o] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ subst
@ ^ [Y2: subst] :
( ( Y1 @ X1 )
=> ( ( !! @ term
@ ^ [Y3: term] :
( ( Y0 @ Y3 )
= ( Y1 @ ( sub @ Y3 @ ( push @ X1 @ Y2 ) ) ) ) )
=> ( Y0 @ ( d2term @ d_one ) ) ) ) ) )
@ X2 )
= $true ) ),
inference(pi_clausification,[status(thm)],[f1716]) ).
thf(f1716,plain,
! [X1: term] :
( ( ( !! @ ( term > $o )
@ ^ [Y0: term > $o] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ subst
@ ^ [Y2: subst] :
( ( Y1 @ X1 )
=> ( ( !! @ term
@ ^ [Y3: term] :
( ( Y0 @ Y3 )
= ( Y1 @ ( sub @ Y3 @ ( push @ X1 @ Y2 ) ) ) ) )
=> ( Y0 @ ( d2term @ d_one ) ) ) ) ) ) )
= $true )
| ( pushprop_lem1v2 = $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1715]) ).
thf(f1715,plain,
! [X1: term] :
( ( ( ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( Y1 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y3 ) ) ) ) )
=> ( Y1 @ ( d2term @ d_one ) ) ) ) ) ) )
@ X1 )
= $true )
| ( pushprop_lem1v2 = $true ) ),
inference(pi_clausification,[status(thm)],[f1696]) ).
thf(f1696,plain,
( ( pushprop_lem1v2 = $true )
| ( ( !! @ term
@ ^ [Y0: term] :
( !! @ ( term > $o )
@ ^ [Y1: term > $o] :
( !! @ ( term > $o )
@ ^ [Y2: term > $o] :
( !! @ subst
@ ^ [Y3: subst] :
( ( Y2 @ Y0 )
=> ( ( !! @ term
@ ^ [Y4: term] :
( ( Y1 @ Y4 )
= ( Y2 @ ( sub @ Y4 @ ( push @ Y0 @ Y3 ) ) ) ) )
=> ( Y1 @ ( d2term @ d_one ) ) ) ) ) ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f1695]) ).
thf(f1984,plain,
( $true
!= ( sK529 @ ( d2term @ d_one ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1983]) ).
thf(f1983,plain,
( ( $true
!= ( sK529 @ ( d2term @ d_one ) ) )
| ( $true = $false ) ),
inference(forward_demodulation,[status(thm)],[f1636,f1906]) ).
thf(f1636,plain,
( ( $true
!= ( sK529 @ ( d2term @ d_one ) ) )
| ( pushprop_lem1v2 = $true ) ),
inference(definition_unfolding,[status(thm)],[f1136,f1176]) ).
thf(f1136,plain,
( ( pushprop_lem1v2 = $true )
| ( ( sK529 @ one )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
thf(f4078,plain,
! [X154: term] :
( ( ( sK530 @ ( d2term @ d_one ) )
= $false )
| ( $true
= ( sK529 @ X154 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4077]) ).
thf(f4077,plain,
! [X154: term] :
( ( $true
= ( sK529 @ X154 ) )
| ( $true = $false )
| ( ( sK530 @ ( d2term @ d_one ) )
= $false ) ),
inference(forward_demodulation,[status(thm)],[f4076,f1906]) ).
thf(f4076,plain,
! [X154: term] :
( ( pushprop_lem1v2 = $true )
| ( $true
= ( sK529 @ X154 ) )
| ( ( sK530 @ ( d2term @ d_one ) )
= $false ) ),
inference(forward_demodulation,[status(thm)],[f1823,f1874]) ).
thf(f1823,plain,
! [X154: term] :
( ( ( sK530 @ ( sub @ X154 @ ( push @ sK527 @ sK528 ) ) )
= $false )
| ( pushprop_lem1v2 = $true )
| ( $true
= ( sK529 @ X154 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1137]) ).
thf(f1137,plain,
! [X154: term] :
( ( pushprop_lem1v2 = $true )
| ( ( sK530 @ ( sub @ X154 @ ( push @ sK527 @ sK528 ) ) )
= ( sK529 @ X154 ) ) ),
inference(cnf_transformation,[status(thm)],[f545]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.00/0.07 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.25 % Computer : n024.cluster.edu
% 0.07/0.25 % Model : x86_64 x86_64
% 0.07/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.25 % Memory : 8042.1875MB
% 0.07/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.25 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue May 5 03:58:56 EDT 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.07/0.31 % Domain : Software Verification
% 0.07/0.31 % Problem : Benchmark: ALG444^1, Active target: pushprop_lem1v2
% 0.07/0.31 % Version : Especial.
% 0.07/0.31 % English :
% 0.07/0.31
% 0.07/0.31 % Refs : [Kon26] Kondylidou (2026), Email to Geoff Sutcliffe
% 0.07/0.31 % Source : [Kon26]
% 0.07/0.31 % Names : ALG444^1__005__axclos.verify [Kon26]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : ? v9.3.0
% 0.07/0.31 % Syntax : Number of formulae : 132 ( 1 unt; 130 typ; 0 def)
% 0.07/0.31 % Number of atoms : 255 ( 128 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 248 ( 127 avg)
% 0.07/0.31 % Number of connectives : 2577 ( 333 ~; 306 |; 129 &;1713 @)
% 0.07/0.31 % ( 56 <=>; 40 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 151 ( 76 avg)
% 0.07/0.31 % Number of types : 5 ( 4 usr)
% 0.07/0.31 % Number of type conns : 224 ( 224 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 127 ( 126 usr; 113 con; 0-4 aty)
% 0.07/0.31 % Number of variables : 554 ( 8 ^; 544 !; 2 ?; 554 :)
% 0.07/0.31 % SPC : TH0_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments :
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 thf(term_type,type,
% 0.07/0.31 term: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(subst_type,type,
% 0.07/0.31 subst: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(one_decl,type,
% 0.07/0.31 one: term ).
% 0.07/0.31
% 0.07/0.31 thf(ap_decl,type,
% 0.07/0.31 ap: term > term > term ).
% 0.07/0.31
% 0.07/0.31 thf(lam_decl,type,
% 0.07/0.31 lam: term > term ).
% 0.07/0.31
% 0.07/0.31 thf(sub_decl,type,
% 0.07/0.31 sub: term > subst > term ).
% 0.07/0.31
% 0.07/0.31 thf(id_decl,type,
% 0.07/0.31 id: subst ).
% 0.07/0.31
% 0.07/0.31 thf(sh_decl,type,
% 0.07/0.31 sh: subst ).
% 0.07/0.31
% 0.07/0.31 thf(push_decl,type,
% 0.07/0.31 push: term > subst > subst ).
% 0.07/0.31
% 0.07/0.31 thf(comp_decl,type,
% 0.07/0.31 comp: subst > subst > subst ).
% 0.07/0.31
% 0.07/0.31 thf(var_decl,type,
% 0.07/0.31 var: term > $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem1v2_decl,type,
% 0.07/0.31 pushprop_lem1v2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem1_gthm_decl,type,
% 0.07/0.31 pushprop_lem1_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axmap_decl,type,
% 0.07/0.31 axmap: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem0_gthm_decl,type,
% 0.07/0.31 pushprop_lem0_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(shinj_decl,type,
% 0.07/0.31 shinj: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem1v2_decl,type,
% 0.07/0.31 hoasinduction_lem1v2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem1v2_gthm_decl,type,
% 0.07/0.31 hoasinduction_lem1v2_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasap_decl,type,
% 0.07/0.31 hoasap: subst > term > subst > term > term ).
% 0.07/0.31
% 0.07/0.31 thf(induction2lem_decl,type,
% 0.07/0.31 induction2lem: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3v2_f_decl,type,
% 0.07/0.31 hoasinduction_lem3v2_f: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axvarshift_decl,type,
% 0.07/0.31 axvarshift: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapinj2_decl,type,
% 0.07/0.31 hoasapinj2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapnotvar_gthm_decl,type,
% 0.07/0.31 hoasapnotvar_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapinj1_decl,type,
% 0.07/0.31 hoasapinj1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ulamvar1_decl,type,
% 0.07/0.31 ulamvar1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(induction2lem_lthm_decl,type,
% 0.07/0.31 induction2lem_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3v2_gthm_decl,type,
% 0.07/0.31 hoasinduction_lem3v2_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(apnotvar_decl,type,
% 0.07/0.31 apnotvar: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lthm_orig_decl,type,
% 0.07/0.31 pushprop_lthm_orig: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem3v2_f_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lthm_decl,type,
% 0.07/0.31 hoasinduction_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 0.07/0.31 hoasinduction_no_psi_cond_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslaminj_decl,type,
% 0.07/0.31 hoaslaminj: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3aaa_decl,type,
% 0.07/0.31 hoasinduction_lem3aaa: $o ).
% 0.07/0.31
% 0.07/0.31 thf(induction2lem_gthm_decl,type,
% 0.07/0.31 induction2lem_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3aa_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem3aa_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3_decl,type,
% 0.07/0.31 hoasinduction_lem3: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem2_decl,type,
% 0.07/0.31 hoasinduction_lem2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(termmset_lthm_decl,type,
% 0.07/0.31 termmset_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem1_decl,type,
% 0.07/0.31 hoasinduction_lem1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslamnotap_lthm_decl,type,
% 0.07/0.31 hoaslamnotap_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem1v2_lthm_decl,type,
% 0.07/0.31 pushprop_lem1v2_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapnotvar_decl,type,
% 0.07/0.31 hoasapnotvar: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem0_decl,type,
% 0.07/0.31 hoasinduction_lem0: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_decl,type,
% 0.07/0.31 hoasinduction: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_gthm_decl,type,
% 0.07/0.31 hoasinduction_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axapp_decl,type,
% 0.07/0.31 axapp: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslamnotvar_lthm_decl,type,
% 0.07/0.31 hoaslamnotvar_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem3v2_lthm_decl,type,
% 0.07/0.31 pushprop_lem3v2_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3b_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem3b_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ulamvarind_decl,type,
% 0.07/0.31 ulamvarind: $o ).
% 0.07/0.31
% 0.07/0.31 thf(induction_decl,type,
% 0.07/0.31 induction: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3a_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem3a_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(termmset_gthm_decl,type,
% 0.07/0.31 termmset_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3aa_decl,type,
% 0.07/0.31 hoasinduction_lem3aa: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem1v2_gthm_decl,type,
% 0.07/0.31 pushprop_lem1v2_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslamnotap_gthm_decl,type,
% 0.07/0.31 hoaslamnotap_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslamnotvar_gthm_decl,type,
% 0.07/0.31 hoaslamnotvar_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3b_gthm_decl,type,
% 0.07/0.31 hoasinduction_lem3b_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem2v2_decl,type,
% 0.07/0.31 pushprop_lem2v2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3a_gthm_decl,type,
% 0.07/0.31 hoasinduction_lem3a_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axclos_decl,type,
% 0.07/0.31 axclos: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axassoc_decl,type,
% 0.07/0.31 axassoc: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem2v2_decl,type,
% 0.07/0.31 hoasinduction_lem2v2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lthm_decl,type,
% 0.07/0.31 pushprop_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(apinj2_decl,type,
% 0.07/0.31 apinj2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(apinj1_decl,type,
% 0.07/0.31 apinj1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapinj2_lthm_decl,type,
% 0.07/0.31 hoasapinj2_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3v2a_decl,type,
% 0.07/0.31 hoasinduction_lem3v2a: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapinj1_lthm_decl,type,
% 0.07/0.31 hoasapinj1_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslaminj_lthm_decl,type,
% 0.07/0.31 hoaslaminj_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axvarcons_decl,type,
% 0.07/0.31 axvarcons: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslam_decl,type,
% 0.07/0.31 hoaslam: subst > ( subst > term > term ) > term ).
% 0.07/0.31
% 0.07/0.31 thf(axscons_decl,type,
% 0.07/0.31 axscons: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem2v2_gthm_decl,type,
% 0.07/0.31 hoasinduction_lem2v2_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axidr_decl,type,
% 0.07/0.31 axidr: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem1_decl,type,
% 0.07/0.31 pushprop_lem1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(laminj_decl,type,
% 0.07/0.31 laminj: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem3_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem0_decl,type,
% 0.07/0.31 pushprop_lem0: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_gthm_decl,type,
% 0.07/0.31 pushprop_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axabs_decl,type,
% 0.07/0.31 axabs: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem3v2a_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem2_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem2_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapinj2_gthm_decl,type,
% 0.07/0.31 hoasapinj2_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_p_and_p_prime_decl,type,
% 0.07/0.31 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem1_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem1_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(lamnotap_decl,type,
% 0.07/0.31 lamnotap: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapinj1_gthm_decl,type,
% 0.07/0.31 hoasapinj1_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslamnotvar_decl,type,
% 0.07/0.31 hoaslamnotvar: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axidl_decl,type,
% 0.07/0.31 axidl: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslaminj_gthm_decl,type,
% 0.07/0.31 hoaslaminj_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(induction2_lthm_decl,type,
% 0.07/0.31 induction2_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem0_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem0_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(substmonoid_lthm_decl,type,
% 0.07/0.31 substmonoid_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_decl,type,
% 0.07/0.31 pushprop: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3_gthm_decl,type,
% 0.07/0.31 hoasinduction_lem3_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem2_gthm_decl,type,
% 0.07/0.31 hoasinduction_lem2_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3b_decl,type,
% 0.07/0.31 hoasinduction_lem3b: $o ).
% 0.07/0.31
% 0.07/0.31 thf(substmonoid_decl,type,
% 0.07/0.31 substmonoid: $o ).
% 0.07/0.31
% 0.07/0.31 thf(lamnotvar_decl,type,
% 0.07/0.31 lamnotvar: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3a_decl,type,
% 0.07/0.31 hoasinduction_lem3a: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem1_gthm_decl,type,
% 0.07/0.31 hoasinduction_lem1_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_no_psi_cond_decl,type,
% 0.07/0.31 hoasinduction_no_psi_cond: $o ).
% 0.07/0.31
% 0.07/0.31 thf(induction2_gthm_decl,type,
% 0.07/0.31 induction2_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem2v2_lthm_decl,type,
% 0.07/0.31 pushprop_lem2v2_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasvar_decl,type,
% 0.07/0.31 hoasvar: subst > term > subst > $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoaslamnotap_decl,type,
% 0.07/0.31 hoaslamnotap: $o ).
% 0.07/0.31
% 0.07/0.31 thf(substmonoid_gthm_decl,type,
% 0.07/0.31 substmonoid_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ulamvarsh_decl,type,
% 0.07/0.31 ulamvarsh: $o ).
% 0.07/0.31
% 0.07/0.31 thf(induction2_decl,type,
% 0.07/0.31 induction2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem3v2_decl,type,
% 0.07/0.31 pushprop_lem3v2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem2v2_gthm_decl,type,
% 0.07/0.31 pushprop_lem2v2_gthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem1_lthm_decl,type,
% 0.07/0.31 pushprop_lem1_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3v2_decl,type,
% 0.07/0.31 hoasinduction_lem3v2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(axshiftcons_decl,type,
% 0.07/0.31 axshiftcons: $o ).
% 0.07/0.31
% 0.07/0.31 thf(termmset_decl,type,
% 0.07/0.31 termmset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_lem0_lthm_decl,type,
% 0.07/0.31 pushprop_lem0_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasapnotvar_lthm_decl,type,
% 0.07/0.31 hoasapnotvar_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lem3v2_lthm_decl,type,
% 0.07/0.31 hoasinduction_lem3v2_lthm: $o ).
% 0.07/0.31
% 0.07/0.31 thf(pushprop_p_and_p_prime_decl,type,
% 0.07/0.31 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 0.07/0.31
% 0.07/0.31 thf(axvarid_decl,type,
% 0.07/0.31 axvarid: $o ).
% 0.07/0.31
% 0.07/0.31 thf(hoasinduction_lthm_3_decl,type,
% 0.07/0.31 hoasinduction_lthm_3: $o ).
% 0.07/0.31
% 0.07/0.31 %----Types of the domains
% 0.07/0.31 thf(d_term_type,type,
% 0.07/0.31 d_term: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(d_subst_type,type,
% 0.07/0.31 d_subst: $tType ).
% 0.07/0.31
% 0.07/0.31 %----Types of the promotion functions
% 0.07/0.31 thf(d2term_decl,type,
% 0.07/0.31 d2term: d_term > term ).
% 0.07/0.31
% 0.07/0.31 thf(d2subst_decl,type,
% 0.07/0.31 d2subst: d_subst > subst ).
% 0.07/0.31
% 0.07/0.31 %----Types of the domain elements
% 0.07/0.31 thf(d_one_decl,type,
% 0.07/0.31 d_one: d_term ).
% 0.07/0.31
% 0.07/0.31 thf(d_id_decl,type,
% 0.07/0.31 d_id: d_subst ).
% 0.07/0.31
% 0.07/0.31 thf(alg444_1,axiom,
% 0.07/0.31 ( ! [T: term] :
% 0.07/0.31 ? [DT: d_term] :
% 0.07/0.31 ( T
% 0.07/0.31 = ( d2term @ DT ) )
% 0.07/0.31 & ! [DT: d_term] : ( DT = d_one )
% 0.07/0.31 & ! [DT1: d_term,DT2: d_term] :
% 0.07/0.31 ( ( ( d2term @ DT1 )
% 0.07/0.31 = ( d2term @ DT2 ) )
% 0.07/0.31 => ( DT1 = DT2 ) )
% 0.07/0.31 & ! [S: subst] :
% 0.07/0.31 ? [DS: d_subst] :
% 0.07/0.31 ( S
% 0.07/0.31 = ( d2subst @ DS ) )
% 0.07/0.31 & ! [DS: d_subst] : ( DS = d_id )
% 0.07/0.31 & ! [DS1: d_subst,DS2: d_subst] :
% 0.07/0.31 ( ( ( d2subst @ DS1 )
% 0.07/0.31 = ( d2subst @ DS2 ) )
% 0.07/0.31 => ( DS1 = DS2 ) )
% 0.07/0.31 & ( one
% 0.07/0.31 = ( d2term @ d_one ) )
% 0.07/0.31 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 0.07/0.31 = ( d2term @ d_one ) )
% 0.07/0.31 & ( ( lam @ ( d2term @ d_one ) )
% 0.07/0.31 = ( d2term @ d_one ) )
% 0.07/0.31 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.07/0.31 = ( d2term @ d_one ) )
% 0.07/0.31 & ( id
% 0.07/0.31 = ( d2subst @ d_id ) )
% 0.07/0.31 & ( sh
% 0.07/0.31 = ( d2subst @ d_id ) )
% 0.07/0.31 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.07/0.31 = ( d2subst @ d_id ) )
% 0.07/0.31 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 0.07/0.31 = ( d2subst @ d_id ) )
% 0.07/0.31 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 0.07/0.31 = ( d2term @ d_one ) )
% 0.07/0.31 & ( hoaslam
% 0.07/0.31 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 0.07/0.31 & ( hoasinduction_p_and_p_prime
% 0.07/0.31 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 0.07/0.31 ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ id @ X @ id ) ) ) )
% 0.07/0.31 & ( pushprop_p_and_p_prime
% 0.07/0.31 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 0.07/0.31 ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 0.07/0.31 & ~ ( var @ ( d2term @ d_one ) )
% 0.07/0.31 & ( pushprop_lem1v2
% 0.07/0.31 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ~ ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.07/0.31 | ( Q @ one ) ) )
% 0.07/0.31 & pushprop_lem1_gthm
% 0.07/0.31 & axmap
% 0.07/0.31 & pushprop_lem0_gthm
% 0.07/0.31 & ( shinj
% 0.07/0.31 <=> ! [A: term,B: term] :
% 0.07/0.31 ( ( ( sub @ A @ sh )
% 0.07/0.31 != ( sub @ B @ sh ) )
% 0.07/0.31 | ( A = B ) ) )
% 0.07/0.31 & hoasinduction_lem1v2
% 0.07/0.31 & hoasinduction_lem1v2_gthm
% 0.07/0.31 & ( induction2lem
% 0.07/0.31 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.07/0.31 ( ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ~ ( P @ B )
% 0.07/0.31 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ B )
% 0.07/0.31 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.07/0.31 | ( P @ ( lam @ A ) ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( var @ B )
% 0.07/0.31 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.07/0.31 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 0.07/0.31 & ( hoasinduction_lem3v2_f
% 0.07/0.31 <=> ! [B: term] :
% 0.07/0.31 ~ ! [F: subst > term > term] :
% 0.07/0.31 ~ ! [A: term,M: subst] :
% 0.07/0.31 ( ( sub @ B @ ( push @ A @ M ) )
% 0.07/0.31 = ( F @ M @ A ) ) )
% 0.07/0.31 & axvarshift
% 0.07/0.31 & ( hoasapinj2
% 0.07/0.31 <=> ! [A: term,B: term,C: term,D: term] :
% 0.07/0.31 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.07/0.31 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.07/0.31 | ( C = D ) ) )
% 0.07/0.31 & hoasapnotvar_gthm
% 0.07/0.31 & ( hoasapinj1
% 0.07/0.31 <=> ! [A: term,B: term,C: term,D: term] :
% 0.07/0.31 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.07/0.31 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.07/0.31 | ( A = B ) ) )
% 0.07/0.31 & ~ ulamvar1
% 0.07/0.31 & ( induction2lem_lthm
% 0.07/0.31 <=> ( ! [A: term,M: subst] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.07/0.31 => ( ! [A: term,M: subst] :
% 0.07/0.31 ( M
% 0.07/0.31 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.07/0.31 => ( ! [M: subst] :
% 0.07/0.31 ( M
% 0.07/0.31 = ( comp @ M @ id ) )
% 0.07/0.31 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 0.07/0.31 ( ~ ! [A: term] :
% 0.07/0.31 ( ~ ( var @ A )
% 0.07/0.31 | ( P @ A ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ~ ( P @ B )
% 0.07/0.31 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ( P @ ( lam @ A ) ) )
% 0.07/0.31 | ( P @ Bound_variable_2140 ) )
% 0.07/0.31 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.07/0.31 ( ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ~ ( P @ B )
% 0.07/0.31 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ B )
% 0.07/0.31 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.07/0.31 | ( P @ ( lam @ A ) ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( var @ B )
% 0.07/0.31 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.07/0.31 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 0.07/0.31 & hoasinduction_lem3v2_gthm
% 0.07/0.31 & apnotvar
% 0.07/0.31 & pushprop_lthm_orig
% 0.07/0.31 & ( hoasinduction_lem3v2_f_lthm
% 0.07/0.31 <=> ! [B: term] :
% 0.07/0.31 ~ ! [F: subst > term > term] :
% 0.07/0.31 ~ ! [A: term,M: subst] :
% 0.07/0.31 ( ( sub @ B @ ( push @ A @ M ) )
% 0.07/0.31 = ( F @ M @ A ) ) )
% 0.07/0.31 & ( hoasinduction_lthm
% 0.07/0.31 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.07/0.31 ( ~ ! [A: term] :
% 0.07/0.31 ( ~ ( var @ A )
% 0.07/0.31 | ( P @ A ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ~ ( P @ B )
% 0.07/0.31 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ B )
% 0.07/0.31 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.07/0.31 | ( P @ ( lam @ A ) ) )
% 0.07/0.31 | ( P @ Bound_variable_2367 ) )
% 0.07/0.31 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.31 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.31 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.07/0.31 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.07/0.31 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.07/0.31 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 0.07/0.31 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.31 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.31 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.31 | ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 0.07/0.31 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.31 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.31 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( var @ ( sub @ A @ id ) )
% 0.07/0.31 | ( P @ id @ A @ id ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.07/0.31 | ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 0.07/0.31 & ( hoasinduction_no_psi_cond_lthm
% 0.07/0.31 <=> ( ! [P: subst > term > subst > $o] :
% 0.07/0.31 ~ ! [Q: term > $o] :
% 0.07/0.31 ~ ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ id @ X @ id ) )
% 0.07/0.31 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.07/0.31 ( ~ ! [A: term] :
% 0.07/0.31 ( ~ ( var @ A )
% 0.07/0.31 | ( P @ A ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ~ ( P @ B )
% 0.07/0.31 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ B )
% 0.07/0.31 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.07/0.31 | ( P @ ( lam @ A ) ) )
% 0.07/0.31 | ( P @ Bound_variable_2367 ) )
% 0.07/0.31 => ( ! [A: term] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ A @ id ) )
% 0.07/0.31 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.07/0.31 ( ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ id @ X @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( Q @ B )
% 0.07/0.31 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.07/0.31 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.07/0.31 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.07/0.31 ( ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.07/0.31 | ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 0.07/0.31 & ( hoaslaminj
% 0.07/0.31 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.07/0.31 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ( ( lam @ ( F @ sh @ one ) )
% 0.07/0.31 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.07/0.31 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.07/0.31 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 0.07/0.31 & ( hoasinduction_lem3aaa
% 0.07/0.31 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 0.07/0.31 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 0.07/0.31 ( ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 0.07/0.31 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 0.07/0.31 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 0.07/0.31 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 0.07/0.31 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 0.07/0.31 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 0.07/0.31 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.07/0.31 & induction2lem_gthm
% 0.07/0.31 & ( hoasinduction_lem3aa_lthm
% 0.07/0.31 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.07/0.31 ( ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.07/0.31 & ( hoasinduction_lem3
% 0.07/0.31 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.31 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.31 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.31 | ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 0.07/0.31 & ( hoasinduction_lem2
% 0.07/0.31 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.31 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.31 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.07/0.31 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 0.07/0.31 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 0.07/0.31 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 0.07/0.31 & ( termmset_lthm
% 0.07/0.31 <=> ( ! [A: term] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ A @ id ) )
% 0.07/0.31 => ! [A: term] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ A @ id ) ) ) )
% 0.07/0.31 & hoasinduction_lem1
% 0.07/0.31 & hoaslamnotap_lthm
% 0.07/0.31 & ( pushprop_lem1v2_lthm
% 0.07/0.31 <=> ( ! [A: term,M: subst] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.07/0.31 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ~ ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 0.07/0.31 | ( Q @ one ) ) ) )
% 0.07/0.31 & hoasapnotvar
% 0.07/0.31 & ( hoasinduction_lem0
% 0.07/0.31 <=> ! [P: subst > term > subst > $o] :
% 0.07/0.31 ~ ! [Q: term > $o] :
% 0.07/0.31 ~ ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ id @ X @ id ) ) )
% 0.07/0.31 & ( hoasinduction
% 0.07/0.31 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.31 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.31 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( var @ ( sub @ A @ id ) )
% 0.07/0.31 | ( P @ id @ A @ id ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.07/0.31 | ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 0.07/0.31 & hoasinduction_gthm
% 0.07/0.31 & axapp
% 0.07/0.31 & hoaslamnotvar_lthm
% 0.07/0.31 & pushprop_lem3v2_lthm
% 0.07/0.31 & ( hoasinduction_lem3b_lthm
% 0.07/0.31 <=> ! [B: term] :
% 0.07/0.31 ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ( F @ sh @ one )
% 0.07/0.31 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.07/0.31 & ulamvarind
% 0.07/0.31 & ( induction
% 0.07/0.31 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 0.07/0.31 ( ~ ! [A: term] :
% 0.07/0.31 ( ~ ( var @ A )
% 0.07/0.31 | ( P @ A ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ~ ( P @ B )
% 0.07/0.31 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ( P @ ( lam @ A ) ) )
% 0.07/0.31 | ( P @ Bound_variable_2140 ) ) )
% 0.07/0.31 & ( hoasinduction_lem3a_lthm
% 0.07/0.31 <=> ( ! [A: term] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ A @ id ) )
% 0.07/0.31 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.07/0.31 ( ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.07/0.31 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.07/0.31 ( ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 0.07/0.31 & termmset_gthm
% 0.07/0.31 & ( hoasinduction_lem3aa
% 0.07/0.31 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.07/0.31 ( ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 0.07/0.31 & pushprop_lem1v2_gthm
% 0.07/0.31 & hoaslamnotap_gthm
% 0.07/0.31 & hoaslamnotvar_gthm
% 0.07/0.31 & hoasinduction_lem3b_gthm
% 0.07/0.31 & pushprop_lem2v2
% 0.07/0.31 & hoasinduction_lem3a_gthm
% 0.07/0.31 & axclos
% 0.07/0.31 & axassoc
% 0.07/0.31 & ( hoasinduction_lem2v2
% 0.07/0.31 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.31 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.31 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.31 | ~ ! [A: term,B: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.07/0.31 | ~ ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ id @ X @ id ) )
% 0.07/0.31 | ~ ( Q @ Bound_variable_2820 )
% 0.07/0.31 | ~ ( Q @ Bound_variable_2822 )
% 0.07/0.31 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 0.07/0.31 & pushprop_lthm
% 0.07/0.31 & ( apinj2
% 0.07/0.31 <=> ! [A: term,B: term,C: term,D: term] :
% 0.07/0.31 ( ( ( ap @ A @ C )
% 0.07/0.31 != ( ap @ B @ D ) )
% 0.07/0.31 | ( C = D ) ) )
% 0.07/0.31 & ( apinj1
% 0.07/0.31 <=> ! [A: term,B: term,C: term,D: term] :
% 0.07/0.31 ( ( ( ap @ A @ C )
% 0.07/0.31 != ( ap @ B @ D ) )
% 0.07/0.31 | ( A = B ) ) )
% 0.07/0.31 & ( hoasapinj2_lthm
% 0.07/0.31 <=> ( ! [A: term,B: term,C: term,D: term] :
% 0.07/0.31 ( ( ( ap @ A @ C )
% 0.07/0.31 != ( ap @ B @ D ) )
% 0.07/0.31 | ( C = D ) )
% 0.07/0.31 => ! [A: term,B: term,C: term,D: term] :
% 0.07/0.31 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.07/0.31 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.07/0.31 | ( C = D ) ) ) )
% 0.07/0.31 & ( hoasinduction_lem3v2a
% 0.07/0.31 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.07/0.31 ( ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [X: term] :
% 0.07/0.31 ( ( Q @ X )
% 0.07/0.31 = ( P @ id @ X @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( Q @ B )
% 0.07/0.31 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.07/0.31 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 0.07/0.31 & ( hoasapinj1_lthm
% 0.07/0.31 <=> ( ! [A: term] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ A @ id ) )
% 0.07/0.31 => ( ! [A: term,B: term,C: term,D: term] :
% 0.07/0.31 ( ( ( ap @ A @ C )
% 0.07/0.31 != ( ap @ B @ D ) )
% 0.07/0.31 | ( A = B ) )
% 0.07/0.31 => ! [A: term,B: term,C: term,D: term] :
% 0.07/0.31 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 0.07/0.31 != ( ap @ ( sub @ B @ id ) @ D ) )
% 0.07/0.31 | ( A = B ) ) ) ) )
% 0.07/0.31 & ( hoaslaminj_lthm
% 0.07/0.31 <=> ( ! [A: term,M: subst] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.07/0.31 => ( ! [A: term,M: subst] :
% 0.07/0.31 ( M
% 0.07/0.31 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.07/0.31 => ( ! [A: term,B: term] :
% 0.07/0.31 ( ( ( lam @ A )
% 0.07/0.31 != ( lam @ B ) )
% 0.07/0.31 | ( A = B ) )
% 0.07/0.31 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 0.07/0.31 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ( ( lam @ ( F @ sh @ one ) )
% 0.07/0.31 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 0.07/0.31 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 0.07/0.31 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 0.07/0.31 & ( axvarcons
% 0.07/0.31 <=> ! [A: term,M: subst] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 0.07/0.31 & ( axscons
% 0.07/0.31 <=> ! [M: subst] :
% 0.07/0.31 ( M
% 0.07/0.31 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 0.07/0.31 & hoasinduction_lem2v2_gthm
% 0.07/0.31 & ( axidr
% 0.07/0.31 <=> ! [M: subst] :
% 0.07/0.31 ( M
% 0.07/0.31 = ( comp @ M @ id ) ) )
% 0.07/0.31 & ( pushprop_lem1
% 0.07/0.31 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.07/0.31 ( ~ ( P @ A )
% 0.07/0.31 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 0.07/0.31 & ( laminj
% 0.07/0.31 <=> ! [A: term,B: term] :
% 0.07/0.31 ( ( ( lam @ A )
% 0.07/0.31 != ( lam @ B ) )
% 0.07/0.31 | ( A = B ) ) )
% 0.07/0.31 & ( hoasinduction_lem3_lthm
% 0.07/0.31 <=> ( ! [A: term] :
% 0.07/0.31 ( A
% 0.07/0.31 = ( sub @ A @ id ) )
% 0.07/0.31 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 0.07/0.31 ( ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.31 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 0.07/0.31 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.31 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.31 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.31 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.31 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.31 | ~ ! [F: subst > term > term] :
% 0.07/0.31 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.31 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.31 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.31 | ~ ! [A: term] :
% 0.07/0.31 ( ~ ( P @ id @ A @ id )
% 0.07/0.31 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.31 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.31 | ~ ! [B: term] :
% 0.07/0.31 ( ~ ( P @ id @ B @ id )
% 0.07/0.32 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 0.07/0.32 & ( pushprop_lem0
% 0.07/0.32 <=> ! [P: term > $o,A: term,M: subst] :
% 0.07/0.32 ~ ! [Q: term > $o] :
% 0.07/0.32 ~ ! [X: term] :
% 0.07/0.32 ( ( Q @ X )
% 0.07/0.32 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.07/0.32 & pushprop_gthm
% 0.07/0.32 & axabs
% 0.07/0.32 & ( hoasinduction_lem3v2a_lthm
% 0.07/0.32 <=> ( ! [B: term] :
% 0.07/0.32 ~ ! [F: subst > term > term] :
% 0.07/0.32 ~ ! [A: term,M: subst] :
% 0.07/0.32 ( ( sub @ B @ ( push @ A @ M ) )
% 0.07/0.32 = ( F @ M @ A ) )
% 0.07/0.32 => ( ! [A: term] :
% 0.07/0.32 ( A
% 0.07/0.32 = ( sub @ A @ id ) )
% 0.07/0.32 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.07/0.32 ( ~ ! [F: subst > term > term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.32 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.32 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.32 | ~ ! [X: term] :
% 0.07/0.32 ( ( Q @ X )
% 0.07/0.32 = ( P @ id @ X @ id ) )
% 0.07/0.32 | ~ ! [B: term] :
% 0.07/0.32 ( ~ ( Q @ B )
% 0.07/0.32 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.07/0.32 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 0.07/0.32 & hoasinduction_lem2_lthm
% 0.07/0.32 & hoasapinj2_gthm
% 0.07/0.32 & hoasinduction_lem1_lthm
% 0.07/0.32 & ~ lamnotap
% 0.07/0.32 & hoasapinj1_gthm
% 0.07/0.32 & hoaslamnotvar
% 0.07/0.32 & ( axidl
% 0.07/0.32 <=> ! [M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ id @ M ) ) )
% 0.07/0.32 & hoaslaminj_gthm
% 0.07/0.32 & ( induction2_lthm
% 0.07/0.32 <=> ( ! [A: term] :
% 0.07/0.32 ( A
% 0.07/0.32 = ( sub @ A @ id ) )
% 0.07/0.32 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 0.07/0.32 ( ~ ! [A: term,B: term] :
% 0.07/0.32 ( ~ ( P @ A )
% 0.07/0.32 | ~ ( P @ B )
% 0.07/0.32 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ! [B: term] :
% 0.07/0.32 ( ~ ( P @ B )
% 0.07/0.32 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.07/0.32 | ( P @ ( lam @ A ) ) )
% 0.07/0.32 | ~ ! [B: term] :
% 0.07/0.32 ( ~ ( var @ B )
% 0.07/0.32 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 0.07/0.32 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 0.07/0.32 => ! [P: term > $o,Bound_variable_2367: term] :
% 0.07/0.32 ( ~ ! [A: term] :
% 0.07/0.32 ( ~ ( var @ A )
% 0.07/0.32 | ( P @ A ) )
% 0.07/0.32 | ~ ! [A: term,B: term] :
% 0.07/0.32 ( ~ ( P @ A )
% 0.07/0.32 | ~ ( P @ B )
% 0.07/0.32 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ! [B: term] :
% 0.07/0.32 ( ~ ( P @ B )
% 0.07/0.32 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.07/0.32 | ( P @ ( lam @ A ) ) )
% 0.07/0.32 | ( P @ Bound_variable_2367 ) ) ) ) )
% 0.07/0.32 & ( hoasinduction_lem0_lthm
% 0.07/0.32 <=> ! [P: subst > term > subst > $o] :
% 0.07/0.32 ~ ! [Q: term > $o] :
% 0.07/0.32 ~ ! [X: term] :
% 0.07/0.32 ( ( Q @ X )
% 0.07/0.32 = ( P @ id @ X @ id ) ) )
% 0.07/0.32 & ( substmonoid_lthm
% 0.07/0.32 <=> ( ! [M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ id @ M ) )
% 0.07/0.32 => ( ! [M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ M @ id ) )
% 0.07/0.32 => ( ! [M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ id @ M ) )
% 0.07/0.32 & ! [M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ M @ id ) ) ) ) ) )
% 0.07/0.32 & pushprop
% 0.07/0.32 & hoasinduction_lem3_gthm
% 0.07/0.32 & hoasinduction_lem2_gthm
% 0.07/0.32 & ( hoasinduction_lem3b
% 0.07/0.32 <=> ! [B: term] :
% 0.07/0.32 ~ ! [F: subst > term > term] :
% 0.07/0.32 ( ( F @ sh @ one )
% 0.07/0.32 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 0.07/0.32 & ( substmonoid
% 0.07/0.32 <=> ( ! [M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ id @ M ) )
% 0.07/0.32 & ! [M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ M @ id ) ) ) )
% 0.07/0.32 & lamnotvar
% 0.07/0.32 & ( hoasinduction_lem3a
% 0.07/0.32 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 0.07/0.32 ( ~ ! [F: subst > term > term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.32 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.32 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.32 | ~ ! [B: term] :
% 0.07/0.32 ( ~ ( P @ id @ B @ id )
% 0.07/0.32 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 0.07/0.32 & hoasinduction_lem1_gthm
% 0.07/0.32 & ( hoasinduction_no_psi_cond
% 0.07/0.32 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 0.07/0.32 ( ~ ! [A: term,B: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ~ ( P @ id @ B @ id )
% 0.07/0.32 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.07/0.32 | ~ ! [F: subst > term > term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.32 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.32 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.32 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 0.07/0.32 & induction2_gthm
% 0.07/0.32 & pushprop_lem2v2_lthm
% 0.07/0.32 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 0.07/0.32 & ( hoaslamnotap
% 0.07/0.32 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.32 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.32 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.32 | ( ( lam @ ( F @ sh @ one ) )
% 0.07/0.32 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 0.07/0.32 & substmonoid_gthm
% 0.07/0.32 & ulamvarsh
% 0.07/0.32 & ( induction2
% 0.07/0.32 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 0.07/0.32 ( ~ ! [A: term] :
% 0.07/0.32 ( ~ ( var @ A )
% 0.07/0.32 | ( P @ A ) )
% 0.07/0.32 | ~ ! [A: term,B: term] :
% 0.07/0.32 ( ~ ( P @ A )
% 0.07/0.32 | ~ ( P @ B )
% 0.07/0.32 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ! [B: term] :
% 0.07/0.32 ( ~ ( P @ B )
% 0.07/0.32 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.07/0.32 | ( P @ ( lam @ A ) ) )
% 0.07/0.32 | ( P @ Bound_variable_2367 ) ) )
% 0.07/0.32 & pushprop_lem3v2
% 0.07/0.32 & pushprop_lem2v2_gthm
% 0.07/0.32 & ( pushprop_lem1_lthm
% 0.07/0.32 <=> ( ! [A: term,M: subst] :
% 0.07/0.32 ( A
% 0.07/0.32 = ( sub @ one @ ( push @ A @ M ) ) )
% 0.07/0.32 => ( ! [A: term,M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ sh @ ( push @ A @ M ) ) )
% 0.07/0.32 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 0.07/0.32 ( ~ ( P @ A )
% 0.07/0.32 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 0.07/0.32 & ( hoasinduction_lem3v2
% 0.07/0.32 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.32 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.32 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.32 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.32 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.32 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.32 | ~ ! [F: subst > term > term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.32 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.32 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.32 | ~ ! [X: term] :
% 0.07/0.32 ( ( Q @ X )
% 0.07/0.32 = ( P @ id @ X @ id ) )
% 0.07/0.32 | ~ ! [B: term] :
% 0.07/0.32 ( ~ ( Q @ B )
% 0.07/0.32 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.07/0.32 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 0.07/0.32 & ( axshiftcons
% 0.07/0.32 <=> ! [A: term,M: subst] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 0.07/0.32 & ( termmset
% 0.07/0.32 <=> ! [A: term] :
% 0.07/0.32 ( A
% 0.07/0.32 = ( sub @ A @ id ) ) )
% 0.07/0.32 & ( pushprop_lem0_lthm
% 0.07/0.32 <=> ! [P: term > $o,A: term,M: subst] :
% 0.07/0.32 ~ ! [Q: term > $o] :
% 0.07/0.32 ~ ! [X: term] :
% 0.07/0.32 ( ( Q @ X )
% 0.07/0.32 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 0.07/0.32 & hoasapnotvar_lthm
% 0.07/0.32 & ( hoasinduction_lem3v2_lthm
% 0.07/0.32 <=> ( ! [A: term] :
% 0.07/0.32 ( A
% 0.07/0.32 = ( sub @ A @ id ) )
% 0.07/0.32 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.32 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.32 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.32 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.32 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.32 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.32 | ~ ! [F: subst > term > term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.32 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.32 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.32 | ~ ! [X: term] :
% 0.07/0.32 ( ( Q @ X )
% 0.07/0.32 = ( P @ id @ X @ id ) )
% 0.07/0.32 | ~ ! [B: term] :
% 0.07/0.32 ( ~ ( Q @ B )
% 0.07/0.32 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 0.07/0.32 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 0.07/0.32 & ( axvarid
% 0.07/0.32 <=> ! [A: term] :
% 0.07/0.32 ( A
% 0.07/0.32 = ( sub @ A @ id ) ) )
% 0.07/0.32 & ( hoasinduction_lthm_3
% 0.07/0.32 <=> ( ! [P: subst > term > subst > $o] :
% 0.07/0.32 ~ ! [Q: term > $o] :
% 0.07/0.32 ~ ! [X: term] :
% 0.07/0.32 ( ( Q @ X )
% 0.07/0.32 = ( P @ id @ X @ id ) )
% 0.07/0.32 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 0.07/0.32 ( ~ ! [A: term] :
% 0.07/0.32 ( ~ ( var @ A )
% 0.07/0.32 | ( P @ A ) )
% 0.07/0.32 | ~ ! [A: term,B: term] :
% 0.07/0.32 ( ~ ( P @ A )
% 0.07/0.32 | ~ ( P @ B )
% 0.07/0.32 | ( P @ ( ap @ A @ B ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ! [B: term] :
% 0.07/0.32 ( ~ ( P @ B )
% 0.07/0.32 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 0.07/0.32 | ( P @ ( lam @ A ) ) )
% 0.07/0.32 | ( P @ Bound_variable_2367 ) )
% 0.07/0.32 => ( ! [A: term] :
% 0.07/0.32 ( A
% 0.07/0.32 = ( sub @ A @ id ) )
% 0.07/0.32 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 0.07/0.32 ( ~ ! [F: subst > term > term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.32 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.32 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.32 | ~ ! [X: term] :
% 0.07/0.32 ( ( Q @ X )
% 0.07/0.32 = ( P @ id @ X @ id ) )
% 0.07/0.32 | ~ ! [B: term] :
% 0.07/0.32 ( ~ ( Q @ B )
% 0.07/0.32 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 0.07/0.32 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 0.07/0.32 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.32 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 0.07/0.32 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 0.07/0.32 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 0.07/0.32 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 0.07/0.32 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ( var @ ( sub @ A @ id ) )
% 0.07/0.32 | ( P @ id @ A @ id ) )
% 0.07/0.32 | ~ ! [A: term,B: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ~ ( P @ id @ B @ id )
% 0.07/0.32 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 0.07/0.32 | ~ ! [F: subst > term > term] :
% 0.07/0.32 ( ~ ! [M: subst,A: term,N: subst] :
% 0.07/0.32 ( ( sub @ ( F @ M @ A ) @ N )
% 0.07/0.32 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 0.07/0.32 | ~ ! [A: term] :
% 0.07/0.32 ( ~ ( P @ id @ A @ id )
% 0.07/0.32 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 0.07/0.32 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 0.07/0.32 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(pushprop_lem1v2,conjecture,
% 0.07/0.32 ( pushprop_lem1v2
% 0.07/0.32 = ( ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 0.07/0.32 ( ( P @ A )
% 0.07/0.32 => ( ( pushprop_p_and_p_prime @ A @ M @ P @ Q )
% 0.07/0.32 => ( Q @ one ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 ------------------------
% 1.58/1.06 ---- Embedded in THF ---
% 1.58/1.06 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.58/1.06 %%% Generated on Tue May 05 03:58:57 EDT 2026
% 1.58/1.06 %%% using '$$dhol' embedding, version 1.3.0.
% 1.58/1.06 %%% Logic specification used:
% 1.58/1.06 %%% thf(spec, logic, $$dhol).
% 1.58/1.06
% 1.58/1.06 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Ab9xLNyFM0/DTF2DTF1970.p
% See solution above
% 1.58/1.06 ------------------------
% 1.58/1.08 ---- Cleaned THF ---
% 1.58/1.08 thf(term_type,type,
% 1.58/1.08 term: $tType ).
% 1.58/1.08
% 1.58/1.08 thf(subst_type,type,
% 1.58/1.08 subst: $tType ).
% 1.58/1.08
% 1.58/1.08 thf(one_decl,type,
% 1.58/1.08 one: term ).
% 1.58/1.08
% 1.58/1.08 thf(ap_decl,type,
% 1.58/1.08 ap: term > term > term ).
% 1.58/1.08
% 1.58/1.08 thf(lam_decl,type,
% 1.58/1.08 lam: term > term ).
% 1.58/1.08
% 1.58/1.08 thf(sub_decl,type,
% 1.58/1.08 sub: term > subst > term ).
% 1.58/1.08
% 1.58/1.08 thf(id_decl,type,
% 1.58/1.08 id: subst ).
% 1.58/1.08
% 1.58/1.08 thf(sh_decl,type,
% 1.58/1.08 sh: subst ).
% 1.58/1.08
% 1.58/1.08 thf(push_decl,type,
% 1.58/1.08 push: term > subst > subst ).
% 1.58/1.08
% 1.58/1.08 thf(comp_decl,type,
% 1.58/1.08 comp: subst > subst > subst ).
% 1.58/1.08
% 1.58/1.08 thf(var_decl,type,
% 1.58/1.08 var: term > $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem1v2_decl,type,
% 1.58/1.08 pushprop_lem1v2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem1_gthm_decl,type,
% 1.58/1.08 pushprop_lem1_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axmap_decl,type,
% 1.58/1.08 axmap: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem0_gthm_decl,type,
% 1.58/1.08 pushprop_lem0_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(shinj_decl,type,
% 1.58/1.08 shinj: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem1v2_decl,type,
% 1.58/1.08 hoasinduction_lem1v2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem1v2_gthm_decl,type,
% 1.58/1.08 hoasinduction_lem1v2_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasap_decl,type,
% 1.58/1.08 hoasap: subst > term > subst > term > term ).
% 1.58/1.08
% 1.58/1.08 thf(induction2lem_decl,type,
% 1.58/1.08 induction2lem: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3v2_f_decl,type,
% 1.58/1.08 hoasinduction_lem3v2_f: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axvarshift_decl,type,
% 1.58/1.08 axvarshift: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapinj2_decl,type,
% 1.58/1.08 hoasapinj2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapnotvar_gthm_decl,type,
% 1.58/1.08 hoasapnotvar_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapinj1_decl,type,
% 1.58/1.08 hoasapinj1: $o ).
% 1.58/1.08
% 1.58/1.08 thf(ulamvar1_decl,type,
% 1.58/1.08 ulamvar1: $o ).
% 1.58/1.08
% 1.58/1.08 thf(induction2lem_lthm_decl,type,
% 1.58/1.08 induction2lem_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3v2_gthm_decl,type,
% 1.58/1.08 hoasinduction_lem3v2_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(apnotvar_decl,type,
% 1.58/1.08 apnotvar: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lthm_orig_decl,type,
% 1.58/1.08 pushprop_lthm_orig: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3v2_f_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem3v2_f_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lthm_decl,type,
% 1.58/1.08 hoasinduction_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_no_psi_cond_lthm_decl,type,
% 1.58/1.08 hoasinduction_no_psi_cond_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslaminj_decl,type,
% 1.58/1.08 hoaslaminj: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3aaa_decl,type,
% 1.58/1.08 hoasinduction_lem3aaa: $o ).
% 1.58/1.08
% 1.58/1.08 thf(induction2lem_gthm_decl,type,
% 1.58/1.08 induction2lem_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3aa_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem3aa_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3_decl,type,
% 1.58/1.08 hoasinduction_lem3: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem2_decl,type,
% 1.58/1.08 hoasinduction_lem2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(termmset_lthm_decl,type,
% 1.58/1.08 termmset_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem1_decl,type,
% 1.58/1.08 hoasinduction_lem1: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslamnotap_lthm_decl,type,
% 1.58/1.08 hoaslamnotap_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem1v2_lthm_decl,type,
% 1.58/1.08 pushprop_lem1v2_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapnotvar_decl,type,
% 1.58/1.08 hoasapnotvar: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem0_decl,type,
% 1.58/1.08 hoasinduction_lem0: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_decl,type,
% 1.58/1.08 hoasinduction: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_gthm_decl,type,
% 1.58/1.08 hoasinduction_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axapp_decl,type,
% 1.58/1.08 axapp: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslamnotvar_lthm_decl,type,
% 1.58/1.08 hoaslamnotvar_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem3v2_lthm_decl,type,
% 1.58/1.08 pushprop_lem3v2_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3b_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem3b_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(ulamvarind_decl,type,
% 1.58/1.08 ulamvarind: $o ).
% 1.58/1.08
% 1.58/1.08 thf(induction_decl,type,
% 1.58/1.08 induction: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3a_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem3a_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(termmset_gthm_decl,type,
% 1.58/1.08 termmset_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3aa_decl,type,
% 1.58/1.08 hoasinduction_lem3aa: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem1v2_gthm_decl,type,
% 1.58/1.08 pushprop_lem1v2_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslamnotap_gthm_decl,type,
% 1.58/1.08 hoaslamnotap_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslamnotvar_gthm_decl,type,
% 1.58/1.08 hoaslamnotvar_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3b_gthm_decl,type,
% 1.58/1.08 hoasinduction_lem3b_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem2v2_decl,type,
% 1.58/1.08 pushprop_lem2v2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3a_gthm_decl,type,
% 1.58/1.08 hoasinduction_lem3a_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axclos_decl,type,
% 1.58/1.08 axclos: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axassoc_decl,type,
% 1.58/1.08 axassoc: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem2v2_decl,type,
% 1.58/1.08 hoasinduction_lem2v2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lthm_decl,type,
% 1.58/1.08 pushprop_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(apinj2_decl,type,
% 1.58/1.08 apinj2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(apinj1_decl,type,
% 1.58/1.08 apinj1: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapinj2_lthm_decl,type,
% 1.58/1.08 hoasapinj2_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3v2a_decl,type,
% 1.58/1.08 hoasinduction_lem3v2a: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapinj1_lthm_decl,type,
% 1.58/1.08 hoasapinj1_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslaminj_lthm_decl,type,
% 1.58/1.08 hoaslaminj_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axvarcons_decl,type,
% 1.58/1.08 axvarcons: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslam_decl,type,
% 1.58/1.08 hoaslam: subst > ( subst > term > term ) > term ).
% 1.58/1.08
% 1.58/1.08 thf(axscons_decl,type,
% 1.58/1.08 axscons: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem2v2_gthm_decl,type,
% 1.58/1.08 hoasinduction_lem2v2_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axidr_decl,type,
% 1.58/1.08 axidr: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem1_decl,type,
% 1.58/1.08 pushprop_lem1: $o ).
% 1.58/1.08
% 1.58/1.08 thf(laminj_decl,type,
% 1.58/1.08 laminj: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem3_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem0_decl,type,
% 1.58/1.08 pushprop_lem0: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_gthm_decl,type,
% 1.58/1.08 pushprop_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axabs_decl,type,
% 1.58/1.08 axabs: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3v2a_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem3v2a_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem2_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem2_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapinj2_gthm_decl,type,
% 1.58/1.08 hoasapinj2_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_p_and_p_prime_decl,type,
% 1.58/1.08 hoasinduction_p_and_p_prime: ( subst > term > subst > $o ) > ( term > $o ) > $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem1_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem1_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(lamnotap_decl,type,
% 1.58/1.08 lamnotap: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapinj1_gthm_decl,type,
% 1.58/1.08 hoasapinj1_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslamnotvar_decl,type,
% 1.58/1.08 hoaslamnotvar: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axidl_decl,type,
% 1.58/1.08 axidl: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslaminj_gthm_decl,type,
% 1.58/1.08 hoaslaminj_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(induction2_lthm_decl,type,
% 1.58/1.08 induction2_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem0_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem0_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(substmonoid_lthm_decl,type,
% 1.58/1.08 substmonoid_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_decl,type,
% 1.58/1.08 pushprop: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3_gthm_decl,type,
% 1.58/1.08 hoasinduction_lem3_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem2_gthm_decl,type,
% 1.58/1.08 hoasinduction_lem2_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3b_decl,type,
% 1.58/1.08 hoasinduction_lem3b: $o ).
% 1.58/1.08
% 1.58/1.08 thf(substmonoid_decl,type,
% 1.58/1.08 substmonoid: $o ).
% 1.58/1.08
% 1.58/1.08 thf(lamnotvar_decl,type,
% 1.58/1.08 lamnotvar: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3a_decl,type,
% 1.58/1.08 hoasinduction_lem3a: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem1_gthm_decl,type,
% 1.58/1.08 hoasinduction_lem1_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_no_psi_cond_decl,type,
% 1.58/1.08 hoasinduction_no_psi_cond: $o ).
% 1.58/1.08
% 1.58/1.08 thf(induction2_gthm_decl,type,
% 1.58/1.08 induction2_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem2v2_lthm_decl,type,
% 1.58/1.08 pushprop_lem2v2_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasvar_decl,type,
% 1.58/1.08 hoasvar: subst > term > subst > $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoaslamnotap_decl,type,
% 1.58/1.08 hoaslamnotap: $o ).
% 1.58/1.08
% 1.58/1.08 thf(substmonoid_gthm_decl,type,
% 1.58/1.08 substmonoid_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(ulamvarsh_decl,type,
% 1.58/1.08 ulamvarsh: $o ).
% 1.58/1.08
% 1.58/1.08 thf(induction2_decl,type,
% 1.58/1.08 induction2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem3v2_decl,type,
% 1.58/1.08 pushprop_lem3v2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem2v2_gthm_decl,type,
% 1.58/1.08 pushprop_lem2v2_gthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem1_lthm_decl,type,
% 1.58/1.08 pushprop_lem1_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3v2_decl,type,
% 1.58/1.08 hoasinduction_lem3v2: $o ).
% 1.58/1.08
% 1.58/1.08 thf(axshiftcons_decl,type,
% 1.58/1.08 axshiftcons: $o ).
% 1.58/1.08
% 1.58/1.08 thf(termmset_decl,type,
% 1.58/1.08 termmset: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem0_lthm_decl,type,
% 1.58/1.08 pushprop_lem0_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasapnotvar_lthm_decl,type,
% 1.58/1.08 hoasapnotvar_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lem3v2_lthm_decl,type,
% 1.58/1.08 hoasinduction_lem3v2_lthm: $o ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_p_and_p_prime_decl,type,
% 1.58/1.08 pushprop_p_and_p_prime: term > subst > ( term > $o ) > ( term > $o ) > $o ).
% 1.58/1.08
% 1.58/1.08 thf(axvarid_decl,type,
% 1.58/1.08 axvarid: $o ).
% 1.58/1.08
% 1.58/1.08 thf(hoasinduction_lthm_3_decl,type,
% 1.58/1.08 hoasinduction_lthm_3: $o ).
% 1.58/1.08
% 1.58/1.08 thf(d_term_type,type,
% 1.58/1.08 d_term: $tType ).
% 1.58/1.08
% 1.58/1.08 thf(d_subst_type,type,
% 1.58/1.08 d_subst: $tType ).
% 1.58/1.08
% 1.58/1.08 thf(d2term_decl,type,
% 1.58/1.08 d2term: d_term > term ).
% 1.58/1.08
% 1.58/1.08 thf(d2subst_decl,type,
% 1.58/1.08 d2subst: d_subst > subst ).
% 1.58/1.08
% 1.58/1.08 thf(d_one_decl,type,
% 1.58/1.08 d_one: d_term ).
% 1.58/1.08
% 1.58/1.08 thf(d_id_decl,type,
% 1.58/1.08 d_id: d_subst ).
% 1.58/1.08
% 1.58/1.08 thf(alg444_1,axiom,
% 1.58/1.08 ( ! [T: term] :
% 1.58/1.08 ? [DT: d_term] :
% 1.58/1.08 ( T
% 1.58/1.08 = ( d2term @ DT ) )
% 1.58/1.08 & ! [DT: d_term] : ( DT = d_one )
% 1.58/1.08 & ! [DT1: d_term,DT2: d_term] :
% 1.58/1.08 ( ( ( d2term @ DT1 )
% 1.58/1.08 = ( d2term @ DT2 ) )
% 1.58/1.08 => ( DT1 = DT2 ) )
% 1.58/1.08 & ! [S: subst] :
% 1.58/1.08 ? [DS: d_subst] :
% 1.58/1.08 ( S
% 1.58/1.08 = ( d2subst @ DS ) )
% 1.58/1.08 & ! [DS: d_subst] : ( DS = d_id )
% 1.58/1.08 & ! [DS1: d_subst,DS2: d_subst] :
% 1.58/1.08 ( ( ( d2subst @ DS1 )
% 1.58/1.08 = ( d2subst @ DS2 ) )
% 1.58/1.08 => ( DS1 = DS2 ) )
% 1.58/1.08 & ( one
% 1.58/1.08 = ( d2term @ d_one ) )
% 1.58/1.08 & ( ( ap @ ( d2term @ d_one ) @ ( d2term @ d_one ) )
% 1.58/1.08 = ( d2term @ d_one ) )
% 1.58/1.08 & ( ( lam @ ( d2term @ d_one ) )
% 1.58/1.08 = ( d2term @ d_one ) )
% 1.58/1.08 & ( ( sub @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 1.58/1.08 = ( d2term @ d_one ) )
% 1.58/1.08 & ( id
% 1.58/1.08 = ( d2subst @ d_id ) )
% 1.58/1.08 & ( sh
% 1.58/1.08 = ( d2subst @ d_id ) )
% 1.58/1.08 & ( ( push @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 1.58/1.08 = ( d2subst @ d_id ) )
% 1.58/1.08 & ( ( comp @ ( d2subst @ d_id ) @ ( d2subst @ d_id ) )
% 1.58/1.08 = ( d2subst @ d_id ) )
% 1.58/1.08 & ( ( hoasap @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) @ ( d2term @ d_one ) )
% 1.58/1.08 = ( d2term @ d_one ) )
% 1.58/1.08 & ( hoaslam
% 1.58/1.08 = ( ^ [Bound_variable_4913: subst,Bound_variable_4915: subst > term > term] : ( d2term @ d_one ) ) )
% 1.58/1.08 & ( hoasinduction_p_and_p_prime
% 1.58/1.08 = ( ^ [P: subst > term > subst > $o,Q: term > $o] :
% 1.58/1.08 ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) ) ) )
% 1.58/1.08 & ( pushprop_p_and_p_prime
% 1.58/1.08 = ( ^ [A: term,M: subst,P: term > $o,Q: term > $o] :
% 1.58/1.08 ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) ) )
% 1.58/1.08 & ~ ( var @ ( d2term @ d_one ) )
% 1.58/1.08 & ( pushprop_lem1v2
% 1.58/1.08 <=> ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 1.58/1.08 | ( Q @ one ) ) )
% 1.58/1.08 & pushprop_lem1_gthm
% 1.58/1.08 & axmap
% 1.58/1.08 & pushprop_lem0_gthm
% 1.58/1.08 & ( shinj
% 1.58/1.08 <=> ! [A: term,B: term] :
% 1.58/1.08 ( ( ( sub @ A @ sh )
% 1.58/1.08 != ( sub @ B @ sh ) )
% 1.58/1.08 | ( A = B ) ) )
% 1.58/1.08 & hoasinduction_lem1v2
% 1.58/1.08 & hoasinduction_lem1v2_gthm
% 1.58/1.08 & ( induction2lem
% 1.58/1.08 <=> ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 1.58/1.08 ( ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ B )
% 1.58/1.08 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( var @ B )
% 1.58/1.08 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 1.58/1.08 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) )
% 1.58/1.08 & ( hoasinduction_lem3v2_f
% 1.58/1.08 <=> ! [B: term] :
% 1.58/1.08 ~ ! [F: subst > term > term] :
% 1.58/1.08 ~ ! [A: term,M: subst] :
% 1.58/1.08 ( ( sub @ B @ ( push @ A @ M ) )
% 1.58/1.08 = ( F @ M @ A ) ) )
% 1.58/1.08 & axvarshift
% 1.58/1.08 & ( hoasapinj2
% 1.58/1.08 <=> ! [A: term,B: term,C: term,D: term] :
% 1.58/1.08 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 1.58/1.08 != ( ap @ ( sub @ B @ id ) @ D ) )
% 1.58/1.08 | ( C = D ) ) )
% 1.58/1.08 & hoasapnotvar_gthm
% 1.58/1.08 & ( hoasapinj1
% 1.58/1.08 <=> ! [A: term,B: term,C: term,D: term] :
% 1.58/1.08 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 1.58/1.08 != ( ap @ ( sub @ B @ id ) @ D ) )
% 1.58/1.08 | ( A = B ) ) )
% 1.58/1.08 & ~ ulamvar1
% 1.58/1.08 & ( induction2lem_lthm
% 1.58/1.08 <=> ( ! [A: term,M: subst] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ one @ ( push @ A @ M ) ) )
% 1.58/1.08 => ( ! [A: term,M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ sh @ ( push @ A @ M ) ) )
% 1.58/1.08 => ( ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ M @ id ) )
% 1.58/1.08 => ( ! [P: term > $o,Bound_variable_2140: term] :
% 1.58/1.08 ( ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ A )
% 1.58/1.08 | ( P @ A ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ( P @ Bound_variable_2140 ) )
% 1.58/1.08 => ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 1.58/1.08 ( ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ B )
% 1.58/1.08 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( var @ B )
% 1.58/1.08 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 1.58/1.08 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) ) ) ) ) ) )
% 1.58/1.08 & hoasinduction_lem3v2_gthm
% 1.58/1.08 & apnotvar
% 1.58/1.08 & pushprop_lthm_orig
% 1.58/1.08 & ( hoasinduction_lem3v2_f_lthm
% 1.58/1.08 <=> ! [B: term] :
% 1.58/1.08 ~ ! [F: subst > term > term] :
% 1.58/1.08 ~ ! [A: term,M: subst] :
% 1.58/1.08 ( ( sub @ B @ ( push @ A @ M ) )
% 1.58/1.08 = ( F @ M @ A ) ) )
% 1.58/1.08 & ( hoasinduction_lthm
% 1.58/1.08 <=> ( ! [P: term > $o,Bound_variable_2367: term] :
% 1.58/1.08 ( ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ A )
% 1.58/1.08 | ( P @ A ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ B )
% 1.58/1.08 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ( P @ Bound_variable_2367 ) )
% 1.58/1.08 => ( ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 1.58/1.08 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 1.58/1.08 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) )
% 1.58/1.08 => ( ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) )
% 1.58/1.08 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ ( sub @ A @ id ) )
% 1.58/1.08 | ( P @ id @ A @ id ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) )
% 1.58/1.08 & ( hoasinduction_no_psi_cond_lthm
% 1.58/1.08 <=> ( ! [P: subst > term > subst > $o] :
% 1.58/1.08 ~ ! [Q: term > $o] :
% 1.58/1.08 ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 1.58/1.08 ( ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ A )
% 1.58/1.08 | ( P @ A ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ B )
% 1.58/1.08 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ( P @ Bound_variable_2367 ) )
% 1.58/1.08 => ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( Q @ B )
% 1.58/1.08 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 1.58/1.08 => ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 1.58/1.08 ( ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ Bound_variable_3295 @ id ) ) ) ) ) ) )
% 1.58/1.08 & ( hoaslaminj
% 1.58/1.08 <=> ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 1.58/1.08 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ( ( lam @ ( F @ sh @ one ) )
% 1.58/1.08 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 1.58/1.08 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 1.58/1.08 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) )
% 1.58/1.08 & ( hoasinduction_lem3aaa
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Bound_variable_3171: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term,Bound_variable_3149: term] :
% 1.58/1.08 ( ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id )
% 1.58/1.08 | ~ ! [Bound_variable_3072: subst,Bound_variable_3074: term,Bound_variable_3076: subst] :
% 1.58/1.08 ( ( sub @ ( F @ Bound_variable_3072 @ Bound_variable_3074 ) @ Bound_variable_3076 )
% 1.58/1.08 = ( sub @ ( sub @ Bound_variable_3149 @ ( push @ Bound_variable_3074 @ Bound_variable_3072 ) ) @ Bound_variable_3076 ) )
% 1.58/1.08 | ~ ! [Bound_variable_3090: subst,Bound_variable_3092: term,Bound_variable_3094: subst] :
% 1.58/1.08 ( ( F @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) )
% 1.58/1.08 = ( sub @ Bound_variable_3149 @ ( push @ ( sub @ Bound_variable_3092 @ Bound_variable_3094 ) @ ( comp @ Bound_variable_3090 @ Bound_variable_3094 ) ) ) ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3171 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3171 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 1.58/1.08 & induction2lem_gthm
% 1.58/1.08 & ( hoasinduction_lem3aa_lthm
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 1.58/1.08 & ( hoasinduction_lem3
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) )
% 1.58/1.08 & ( hoasinduction_lem2
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Bound_variable_2999: term,Bound_variable_3001: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 1.58/1.08 | ~ ( P @ id @ Bound_variable_2999 @ id )
% 1.58/1.08 | ~ ( P @ id @ Bound_variable_3001 @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ Bound_variable_2999 @ Bound_variable_3001 ) @ id ) ) )
% 1.58/1.08 & ( termmset_lthm
% 1.58/1.08 <=> ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) ) ) )
% 1.58/1.08 & hoasinduction_lem1
% 1.58/1.08 & hoaslamnotap_lthm
% 1.58/1.08 & ( pushprop_lem1v2_lthm
% 1.58/1.08 <=> ( ! [A: term,M: subst] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ one @ ( push @ A @ M ) ) )
% 1.58/1.08 => ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) )
% 1.58/1.08 | ( Q @ one ) ) ) )
% 1.58/1.08 & hoasapnotvar
% 1.58/1.08 & ( hoasinduction_lem0
% 1.58/1.08 <=> ! [P: subst > term > subst > $o] :
% 1.58/1.08 ~ ! [Q: term > $o] :
% 1.58/1.08 ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) ) )
% 1.58/1.08 & ( hoasinduction
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ ( sub @ A @ id ) )
% 1.58/1.08 | ( P @ id @ A @ id ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ Bound_variable_3283 @ id ) ) )
% 1.58/1.08 & hoasinduction_gthm
% 1.58/1.08 & axapp
% 1.58/1.08 & hoaslamnotvar_lthm
% 1.58/1.08 & pushprop_lem3v2_lthm
% 1.58/1.08 & ( hoasinduction_lem3b_lthm
% 1.58/1.08 <=> ! [B: term] :
% 1.58/1.08 ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ( F @ sh @ one )
% 1.58/1.08 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 1.58/1.08 & ulamvarind
% 1.58/1.08 & ( induction
% 1.58/1.08 <=> ! [P: term > $o,Bound_variable_2140: term] :
% 1.58/1.08 ( ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ A )
% 1.58/1.08 | ( P @ A ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ( P @ Bound_variable_2140 ) ) )
% 1.58/1.08 & ( hoasinduction_lem3a_lthm
% 1.58/1.08 <=> ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 1.58/1.08 => ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) ) ) )
% 1.58/1.08 & termmset_gthm
% 1.58/1.08 & ( hoasinduction_lem3aa
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) ) )
% 1.58/1.08 & pushprop_lem1v2_gthm
% 1.58/1.08 & hoaslamnotap_gthm
% 1.58/1.08 & hoaslamnotvar_gthm
% 1.58/1.08 & hoasinduction_lem3b_gthm
% 1.58/1.08 & pushprop_lem2v2
% 1.58/1.08 & hoasinduction_lem3a_gthm
% 1.58/1.08 & axclos
% 1.58/1.08 & axassoc
% 1.58/1.08 & ( hoasinduction_lem2v2
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2820: term,Bound_variable_2822: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 | ~ ( Q @ Bound_variable_2820 )
% 1.58/1.08 | ~ ( Q @ Bound_variable_2822 )
% 1.58/1.08 | ( Q @ ( ap @ Bound_variable_2820 @ Bound_variable_2822 ) ) ) )
% 1.58/1.08 & pushprop_lthm
% 1.58/1.08 & ( apinj2
% 1.58/1.08 <=> ! [A: term,B: term,C: term,D: term] :
% 1.58/1.08 ( ( ( ap @ A @ C )
% 1.58/1.08 != ( ap @ B @ D ) )
% 1.58/1.08 | ( C = D ) ) )
% 1.58/1.08 & ( apinj1
% 1.58/1.08 <=> ! [A: term,B: term,C: term,D: term] :
% 1.58/1.08 ( ( ( ap @ A @ C )
% 1.58/1.08 != ( ap @ B @ D ) )
% 1.58/1.08 | ( A = B ) ) )
% 1.58/1.08 & ( hoasapinj2_lthm
% 1.58/1.08 <=> ( ! [A: term,B: term,C: term,D: term] :
% 1.58/1.08 ( ( ( ap @ A @ C )
% 1.58/1.08 != ( ap @ B @ D ) )
% 1.58/1.08 | ( C = D ) )
% 1.58/1.08 => ! [A: term,B: term,C: term,D: term] :
% 1.58/1.08 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 1.58/1.08 != ( ap @ ( sub @ B @ id ) @ D ) )
% 1.58/1.08 | ( C = D ) ) ) )
% 1.58/1.08 & ( hoasinduction_lem3v2a
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( Q @ B )
% 1.58/1.08 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) )
% 1.58/1.08 & ( hoasapinj1_lthm
% 1.58/1.08 <=> ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ( ! [A: term,B: term,C: term,D: term] :
% 1.58/1.08 ( ( ( ap @ A @ C )
% 1.58/1.08 != ( ap @ B @ D ) )
% 1.58/1.08 | ( A = B ) )
% 1.58/1.08 => ! [A: term,B: term,C: term,D: term] :
% 1.58/1.08 ( ( ( ap @ ( sub @ A @ id ) @ C )
% 1.58/1.08 != ( ap @ ( sub @ B @ id ) @ D ) )
% 1.58/1.08 | ( A = B ) ) ) ) )
% 1.58/1.08 & ( hoaslaminj_lthm
% 1.58/1.08 <=> ( ! [A: term,M: subst] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ one @ ( push @ A @ M ) ) )
% 1.58/1.08 => ( ! [A: term,M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ sh @ ( push @ A @ M ) ) )
% 1.58/1.08 => ( ! [A: term,B: term] :
% 1.58/1.08 ( ( ( lam @ A )
% 1.58/1.08 != ( lam @ B ) )
% 1.58/1.08 | ( A = B ) )
% 1.58/1.08 => ! [F: subst > term > term,Bound_variable_2547: subst > term > term,Bound_variable_2549: subst,Bound_variable_2551: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( Bound_variable_2547 @ M @ A ) @ N )
% 1.58/1.08 = ( Bound_variable_2547 @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ( ( lam @ ( F @ sh @ one ) )
% 1.58/1.08 != ( lam @ ( Bound_variable_2547 @ sh @ one ) ) )
% 1.58/1.08 | ( ( F @ Bound_variable_2549 @ Bound_variable_2551 )
% 1.58/1.08 = ( Bound_variable_2547 @ Bound_variable_2549 @ Bound_variable_2551 ) ) ) ) ) ) )
% 1.58/1.08 & ( axvarcons
% 1.58/1.08 <=> ! [A: term,M: subst] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ one @ ( push @ A @ M ) ) ) )
% 1.58/1.08 & ( axscons
% 1.58/1.08 <=> ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( push @ ( sub @ one @ M ) @ ( comp @ sh @ M ) ) ) )
% 1.58/1.08 & hoasinduction_lem2v2_gthm
% 1.58/1.08 & ( axidr
% 1.58/1.08 <=> ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ M @ id ) ) )
% 1.58/1.08 & ( pushprop_lem1
% 1.58/1.08 <=> ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) )
% 1.58/1.08 & ( laminj
% 1.58/1.08 <=> ! [A: term,B: term] :
% 1.58/1.08 ( ( ( lam @ A )
% 1.58/1.08 != ( lam @ B ) )
% 1.58/1.08 | ( A = B ) ) )
% 1.58/1.08 & ( hoasinduction_lem3_lthm
% 1.58/1.08 <=> ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ( ! [P: subst > term > subst > $o,Bound_variable_3046: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3046 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( sub @ Bound_variable_3046 @ ( push @ one @ sh ) ) ) @ id ) )
% 1.58/1.08 => ! [P: subst > term > subst > $o,Bound_variable_3211: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3211 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ Bound_variable_3211 ) @ id ) ) ) ) )
% 1.58/1.08 & ( pushprop_lem0
% 1.58/1.08 <=> ! [P: term > $o,A: term,M: subst] :
% 1.58/1.08 ~ ! [Q: term > $o] :
% 1.58/1.08 ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 1.58/1.08 & pushprop_gthm
% 1.58/1.08 & axabs
% 1.58/1.08 & ( hoasinduction_lem3v2a_lthm
% 1.58/1.08 <=> ( ! [B: term] :
% 1.58/1.08 ~ ! [F: subst > term > term] :
% 1.58/1.08 ~ ! [A: term,M: subst] :
% 1.58/1.08 ( ( sub @ B @ ( push @ A @ M ) )
% 1.58/1.08 = ( F @ M @ A ) )
% 1.58/1.08 => ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( Q @ B )
% 1.58/1.08 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( Q @ ( lam @ Bound_variable_2931 ) ) ) ) ) )
% 1.58/1.08 & hoasinduction_lem2_lthm
% 1.58/1.08 & hoasapinj2_gthm
% 1.58/1.08 & hoasinduction_lem1_lthm
% 1.58/1.08 & ~ lamnotap
% 1.58/1.08 & hoasapinj1_gthm
% 1.58/1.08 & hoaslamnotvar
% 1.58/1.08 & ( axidl
% 1.58/1.08 <=> ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ id @ M ) ) )
% 1.58/1.08 & hoaslaminj_gthm
% 1.58/1.08 & ( induction2_lthm
% 1.58/1.08 <=> ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ( ! [P: term > $o,Bound_variable_2340: term,Bound_variable_2342: subst] :
% 1.58/1.08 ( ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ B )
% 1.58/1.08 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( var @ B )
% 1.58/1.08 | ( P @ ( sub @ B @ Bound_variable_2342 ) ) )
% 1.58/1.08 | ( P @ ( sub @ Bound_variable_2340 @ Bound_variable_2342 ) ) )
% 1.58/1.08 => ! [P: term > $o,Bound_variable_2367: term] :
% 1.58/1.08 ( ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ A )
% 1.58/1.08 | ( P @ A ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ B )
% 1.58/1.08 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ( P @ Bound_variable_2367 ) ) ) ) )
% 1.58/1.08 & ( hoasinduction_lem0_lthm
% 1.58/1.08 <=> ! [P: subst > term > subst > $o] :
% 1.58/1.08 ~ ! [Q: term > $o] :
% 1.58/1.08 ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) ) )
% 1.58/1.08 & ( substmonoid_lthm
% 1.58/1.08 <=> ( ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ id @ M ) )
% 1.58/1.08 => ( ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ M @ id ) )
% 1.58/1.08 => ( ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ id @ M ) )
% 1.58/1.08 & ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ M @ id ) ) ) ) ) )
% 1.58/1.08 & pushprop
% 1.58/1.08 & hoasinduction_lem3_gthm
% 1.58/1.08 & hoasinduction_lem2_gthm
% 1.58/1.08 & ( hoasinduction_lem3b
% 1.58/1.08 <=> ! [B: term] :
% 1.58/1.08 ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ( F @ sh @ one )
% 1.58/1.08 != ( sub @ B @ ( push @ one @ sh ) ) ) )
% 1.58/1.08 & ( substmonoid
% 1.58/1.08 <=> ( ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ id @ M ) )
% 1.58/1.08 & ! [M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ M @ id ) ) ) )
% 1.58/1.08 & lamnotvar
% 1.58/1.08 & ( hoasinduction_lem3a
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Bound_variable_3232: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( sub @ Bound_variable_3232 @ ( push @ B @ id ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ Bound_variable_3232 ) @ id ) ) )
% 1.58/1.08 & hoasinduction_lem1_gthm
% 1.58/1.08 & ( hoasinduction_no_psi_cond
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Bound_variable_3295: term] :
% 1.58/1.08 ( ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ Bound_variable_3295 @ id ) ) )
% 1.58/1.08 & induction2_gthm
% 1.58/1.08 & pushprop_lem2v2_lthm
% 1.58/1.08 & ~ ( hoasvar @ ( d2subst @ d_id ) @ ( d2term @ d_one ) @ ( d2subst @ d_id ) )
% 1.58/1.08 & ( hoaslamnotap
% 1.58/1.08 <=> ! [F: subst > term > term,Bound_variable_2603: term,Bound_variable_2605: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ( ( lam @ ( F @ sh @ one ) )
% 1.58/1.08 != ( ap @ ( sub @ Bound_variable_2603 @ id ) @ Bound_variable_2605 ) ) ) )
% 1.58/1.08 & substmonoid_gthm
% 1.58/1.08 & ulamvarsh
% 1.58/1.08 & ( induction2
% 1.58/1.08 <=> ! [P: term > $o,Bound_variable_2367: term] :
% 1.58/1.08 ( ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ A )
% 1.58/1.08 | ( P @ A ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ B )
% 1.58/1.08 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ( P @ Bound_variable_2367 ) ) )
% 1.58/1.08 & pushprop_lem3v2
% 1.58/1.08 & pushprop_lem2v2_gthm
% 1.58/1.08 & ( pushprop_lem1_lthm
% 1.58/1.08 <=> ( ! [A: term,M: subst] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ one @ ( push @ A @ M ) ) )
% 1.58/1.08 => ( ! [A: term,M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ sh @ ( push @ A @ M ) ) )
% 1.58/1.08 => ! [P: term > $o,K: term > $o,A: term,M: subst,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ( K @ ( sub @ A @ ( push @ B @ M ) ) ) ) ) ) )
% 1.58/1.08 & ( hoasinduction_lem3v2
% 1.58/1.08 <=> ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( Q @ B )
% 1.58/1.08 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) )
% 1.58/1.08 & ( axshiftcons
% 1.58/1.08 <=> ! [A: term,M: subst] :
% 1.58/1.08 ( M
% 1.58/1.08 = ( comp @ sh @ ( push @ A @ M ) ) ) )
% 1.58/1.08 & ( termmset
% 1.58/1.08 <=> ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) ) )
% 1.58/1.08 & ( pushprop_lem0_lthm
% 1.58/1.08 <=> ! [P: term > $o,A: term,M: subst] :
% 1.58/1.08 ~ ! [Q: term > $o] :
% 1.58/1.08 ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ ( sub @ X @ ( push @ A @ M ) ) ) ) )
% 1.58/1.08 & hoasapnotvar_lthm
% 1.58/1.08 & ( hoasinduction_lem3v2_lthm
% 1.58/1.08 <=> ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2910: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( Q @ B )
% 1.58/1.08 | ( Q @ ( sub @ Bound_variable_2910 @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( Q @ ( lam @ Bound_variable_2910 ) ) ) ) )
% 1.58/1.08 & ( axvarid
% 1.58/1.08 <=> ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) ) )
% 1.58/1.08 & ( hoasinduction_lthm_3
% 1.58/1.08 <=> ( ! [P: subst > term > subst > $o] :
% 1.58/1.08 ~ ! [Q: term > $o] :
% 1.58/1.08 ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 => ( ! [P: term > $o,Bound_variable_2367: term] :
% 1.58/1.08 ( ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ A )
% 1.58/1.08 | ( P @ A ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ A )
% 1.58/1.08 | ~ ( P @ B )
% 1.58/1.08 | ( P @ ( ap @ A @ B ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ! [B: term] :
% 1.58/1.08 ( ~ ( P @ B )
% 1.58/1.08 | ( P @ ( sub @ A @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( P @ ( lam @ A ) ) )
% 1.58/1.08 | ( P @ Bound_variable_2367 ) )
% 1.58/1.08 => ( ! [A: term] :
% 1.58/1.08 ( A
% 1.58/1.08 = ( sub @ A @ id ) )
% 1.58/1.08 => ( ! [P: subst > term > subst > $o,Q: term > $o,Bound_variable_2931: term] :
% 1.58/1.08 ( ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ~ ! [X: term] :
% 1.58/1.08 ( ( Q @ X )
% 1.58/1.08 = ( P @ id @ X @ id ) )
% 1.58/1.08 | ~ ! [B: term] :
% 1.58/1.08 ( ~ ( Q @ B )
% 1.58/1.08 | ( Q @ ( sub @ Bound_variable_2931 @ ( push @ B @ id ) ) ) )
% 1.58/1.08 | ( Q @ ( lam @ Bound_variable_2931 ) ) )
% 1.58/1.08 => ! [P: subst > term > subst > $o,Bound_variable_3283: term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ M @ A @ ( comp @ K @ N ) )
% 1.58/1.08 | ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N ) )
% 1.58/1.08 | ~ ! [M: subst,A: term,N: subst,K: subst] :
% 1.58/1.08 ( ~ ( P @ ( comp @ M @ K ) @ ( sub @ A @ K ) @ N )
% 1.58/1.08 | ( P @ M @ A @ ( comp @ K @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( var @ ( sub @ A @ id ) )
% 1.58/1.08 | ( P @ id @ A @ id ) )
% 1.58/1.08 | ~ ! [A: term,B: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ~ ( P @ id @ B @ id )
% 1.58/1.08 | ( P @ id @ ( ap @ ( sub @ A @ id ) @ B ) @ id ) )
% 1.58/1.08 | ~ ! [F: subst > term > term] :
% 1.58/1.08 ( ~ ! [M: subst,A: term,N: subst] :
% 1.58/1.08 ( ( sub @ ( F @ M @ A ) @ N )
% 1.58/1.08 = ( F @ ( comp @ M @ N ) @ ( sub @ A @ N ) ) )
% 1.58/1.08 | ~ ! [A: term] :
% 1.58/1.08 ( ~ ( P @ id @ A @ id )
% 1.58/1.08 | ( P @ id @ ( F @ id @ A ) @ id ) )
% 1.58/1.08 | ( P @ id @ ( lam @ ( F @ sh @ one ) ) @ id ) )
% 1.58/1.08 | ( P @ id @ Bound_variable_3283 @ id ) ) ) ) ) ) ) ) ).
% 1.58/1.08
% 1.58/1.08 thf(pushprop_lem1v2,conjecture,
% 1.58/1.08 ( pushprop_lem1v2
% 1.58/1.08 = ( ! [P: term > $o,Q: term > $o,A: term,M: subst] :
% 1.58/1.08 ( ( P @ A )
% 1.58/1.08 => ( ( pushprop_p_and_p_prime @ A @ M @ P @ Q )
% 1.58/1.09 => ( Q @ one ) ) ) ) ) ).
% 1.58/1.09 ------------------------
% 1.58/1.10 % (2072)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_1970 for (2999ds/27Mi)
% 1.58/1.10 % (2076)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/18Mi)
% 1.58/1.10 % (2070)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/183Mi)
% 1.58/1.10 % (2071)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_1970 for (2999ds/4Mi)
% 1.58/1.10 % (2073)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/2Mi)
% 1.58/1.10 % (2074)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_1970 for (2999ds/2Mi)
% 1.58/1.10 % (2073)Instruction limit reached!
% 1.58/1.10 % (2073)------------------------------
% 1.58/1.10 % (2073)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.10 % (2073)Termination reason: Unknown
% 1.58/1.10 % (2074)Instruction limit reached!
% 1.58/1.10 % (2074)------------------------------
% 1.58/1.10 % (2074)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.10 % (2073)Termination phase: shuffling
% 1.58/1.10
% 1.58/1.10 % (2073)Memory used [KB]: 1151
% 1.58/1.10 % (2073)Time elapsed: 0.002 s
% 1.58/1.10 % (2073)Instructions burned: 2 (million)
% 1.58/1.10 % (2073)------------------------------
% 1.58/1.10 % (2073)------------------------------
% 1.58/1.10 % (2074)Termination reason: Unknown
% 1.58/1.10 % (2074)Termination phase: shuffling
% 1.58/1.10
% 1.58/1.10 % (2074)Memory used [KB]: 1151
% 1.58/1.10 % (2074)Time elapsed: 0.002 s
% 1.58/1.10 % (2074)Instructions burned: 2 (million)
% 1.58/1.10 % (2074)------------------------------
% 1.58/1.10 % (2074)------------------------------
% 1.58/1.10 % (2075)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_1970 for (2999ds/275Mi)
% 1.58/1.10 % (2071)Instruction limit reached!
% 1.58/1.10 % (2071)------------------------------
% 1.58/1.10 % (2071)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.10 % (2071)Termination reason: Unknown
% 1.58/1.10 % (2071)Termination phase: shuffling
% 1.58/1.10
% 1.58/1.10 % (2071)Memory used [KB]: 1279
% 1.58/1.10 % (2071)Time elapsed: 0.003 s
% 1.58/1.10 % (2071)Instructions burned: 5 (million)
% 1.58/1.10 % (2071)------------------------------
% 1.58/1.10 % (2071)------------------------------
% 1.58/1.11 % (2076)Instruction limit reached!
% 1.58/1.11 % (2076)------------------------------
% 1.58/1.11 % (2076)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.11 % (2076)Termination reason: Unknown
% 1.58/1.11 % (2076)Termination phase: shuffling
% 1.58/1.11
% 1.58/1.11 % (2076)Memory used [KB]: 2046
% 1.58/1.11 % (2076)Time elapsed: 0.007 s
% 1.58/1.11 % (2076)Instructions burned: 19 (million)
% 1.58/1.11 % (2076)------------------------------
% 1.58/1.11 % (2076)------------------------------
% 1.58/1.11 % (2072)Instruction limit reached!
% 1.58/1.11 % (2072)------------------------------
% 1.58/1.11 % (2072)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.11 % (2072)Termination reason: Unknown
% 1.58/1.11 % (2072)Termination phase: Property scanning
% 1.58/1.11
% 1.58/1.11 % (2072)Memory used [KB]: 2174
% 1.58/1.11 % (2072)Time elapsed: 0.010 s
% 1.58/1.11 % (2072)Instructions burned: 28 (million)
% 1.58/1.11 % (2072)------------------------------
% 1.58/1.11 % (2072)------------------------------
% 1.58/1.11 % (2077)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/3Mi)
% 1.58/1.11 % (2078)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_1970 for (2999ds/37Mi)
% 1.74/1.11 % (2077)Instruction limit reached!
% 1.74/1.11 % (2077)------------------------------
% 1.74/1.11 % (2077)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.11 % (2077)Termination reason: Unknown
% 1.74/1.11 % (2077)Termination phase: shuffling
% 1.74/1.11
% 1.74/1.11 % (2077)Memory used [KB]: 1279
% 1.74/1.11 % (2077)Time elapsed: 0.002 s
% 1.74/1.11 % (2077)Instructions burned: 3 (million)
% 1.74/1.11 % (2077)------------------------------
% 1.74/1.11 % (2077)------------------------------
% 1.74/1.11 % (2079)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_1970 for (2999ds/15Mi)
% 1.74/1.11 % (2075)Refutation not found, incomplete strategy
% 1.74/1.11 % (2075)------------------------------
% 1.74/1.11 % (2075)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.11 % (2075)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.11
% 1.74/1.11
% 1.74/1.11 % (2075)Memory used [KB]: 6652
% 1.74/1.11 % (2075)Time elapsed: 0.012 s
% 1.74/1.11 % (2075)Instructions burned: 37 (million)
% 1.74/1.11 % (2075)------------------------------
% 1.74/1.11 % (2075)------------------------------
% 1.74/1.12 % (2080)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_1970 for (2999ds/3Mi)
% 1.74/1.12 % (2080)Instruction limit reached!
% 1.74/1.12 % (2080)------------------------------
% 1.74/1.12 % (2080)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.12 % (2080)Termination reason: Unknown
% 1.74/1.12 % (2080)Termination phase: shuffling
% 1.74/1.12
% 1.74/1.12 % (2080)Memory used [KB]: 1279
% 1.74/1.12 % (2080)Time elapsed: 0.002 s
% 1.74/1.12 % (2080)Instructions burned: 3 (million)
% 1.74/1.12 % (2080)------------------------------
% 1.74/1.12 % (2080)------------------------------
% 1.74/1.12 % (2079)Instruction limit reached!
% 1.74/1.12 % (2079)------------------------------
% 1.74/1.12 % (2079)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.12 % (2079)Termination reason: Unknown
% 1.74/1.12 % (2079)Termination phase: shuffling
% 1.74/1.12
% 1.74/1.12 % (2079)Memory used [KB]: 1918
% 1.74/1.12 % (2079)Time elapsed: 0.006 s
% 1.74/1.12 % (2079)Instructions burned: 16 (million)
% 1.74/1.12 % (2079)------------------------------
% 1.74/1.12 % (2079)------------------------------
% 1.74/1.12 % (2081)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_1970 for (2999ds/1041Mi)
% 1.74/1.12 % (2078)Instruction limit reached!
% 1.74/1.12 % (2078)------------------------------
% 1.74/1.12 % (2078)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.12 % (2078)Termination reason: Unknown
% 1.74/1.12 % (2078)Termination phase: Property scanning
% 1.74/1.12
% 1.74/1.12 % (2078)Memory used [KB]: 1918
% 1.74/1.12 % (2078)Time elapsed: 0.011 s
% 1.74/1.12 % (2078)Instructions burned: 38 (million)
% 1.74/1.12 % (2078)------------------------------
% 1.74/1.12 % (2078)------------------------------
% 1.74/1.12 % (2082)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/7Mi)
% 1.74/1.12 % (2083)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/16Mi)
% 1.74/1.12 % (2082)Instruction limit reached!
% 1.74/1.12 % (2082)------------------------------
% 1.74/1.12 % (2082)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.12 % (2082)Termination reason: Unknown
% 1.74/1.12 % (2082)Termination phase: shuffling
% 1.74/1.12
% 1.74/1.12 % (2082)Memory used [KB]: 1407
% 1.74/1.12 % (2082)Time elapsed: 0.003 s
% 1.74/1.12 % (2082)Instructions burned: 7 (million)
% 1.74/1.12 % (2082)------------------------------
% 1.74/1.12 % (2082)------------------------------
% 1.74/1.13 % (2084)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_1970 for (2999ds/3Mi)
% 1.74/1.13 % (2085)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_1970 for (2999ds/3Mi)
% 1.74/1.13 % (2084)Instruction limit reached!
% 1.74/1.13 % (2084)------------------------------
% 1.74/1.13 % (2084)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.13 % (2084)Termination reason: Unknown
% 1.74/1.13 % (2084)Termination phase: shuffling
% 1.74/1.13
% 1.74/1.13 % (2084)Memory used [KB]: 1279
% 1.74/1.13 % (2084)Time elapsed: 0.002 s
% 1.74/1.13 % (2084)Instructions burned: 3 (million)
% 1.74/1.13 % (2084)------------------------------
% 1.74/1.13 % (2084)------------------------------
% 1.74/1.13 % (2085)Instruction limit reached!
% 1.74/1.13 % (2085)------------------------------
% 1.74/1.13 % (2085)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.13 % (2085)Termination reason: Unknown
% 1.74/1.13 % (2085)Termination phase: shuffling
% 1.74/1.13
% 1.74/1.13 % (2085)Memory used [KB]: 1279
% 1.74/1.13 % (2085)Time elapsed: 0.002 s
% 1.74/1.13 % (2085)Instructions burned: 3 (million)
% 1.74/1.13 % (2085)------------------------------
% 1.74/1.13 % (2085)------------------------------
% 1.74/1.13 % (2083)Instruction limit reached!
% 1.74/1.13 % (2083)------------------------------
% 1.74/1.13 % (2083)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.13 % (2083)Termination reason: Unknown
% 1.74/1.13 % (2083)Termination phase: shuffling
% 1.74/1.13
% 1.74/1.13 % (2083)Memory used [KB]: 1918
% 1.74/1.13 % (2083)Time elapsed: 0.006 s
% 1.74/1.13 % (2083)Instructions burned: 17 (million)
% 1.74/1.13 % (2083)------------------------------
% 1.74/1.13 % (2083)------------------------------
% 1.74/1.13 % (2086)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/7Mi)
% 1.74/1.13 % (2086)Instruction limit reached!
% 1.74/1.13 % (2086)------------------------------
% 1.74/1.13 % (2086)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.13 % (2086)Termination reason: Unknown
% 1.74/1.13 % (2086)Termination phase: shuffling
% 1.74/1.13
% 1.74/1.13 % (2086)Memory used [KB]: 1407
% 1.74/1.13 % (2086)Time elapsed: 0.003 s
% 1.74/1.13 % (2086)Instructions burned: 7 (million)
% 1.74/1.13 % (2086)------------------------------
% 1.74/1.13 % (2086)------------------------------
% 1.74/1.13 % (2087)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/3Mi)
% 1.74/1.13 % (2087)Instruction limit reached!
% 1.74/1.13 % (2087)------------------------------
% 1.74/1.13 % (2087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.13 % (2087)Termination reason: Unknown
% 1.74/1.13 % (2087)Termination phase: shuffling
% 1.74/1.13
% 1.74/1.13 % (2087)Memory used [KB]: 1279
% 1.74/1.13 % (2087)Time elapsed: 0.002 s
% 1.74/1.13 % (2087)Instructions burned: 3 (million)
% 1.74/1.13 % (2087)------------------------------
% 1.74/1.13 % (2087)------------------------------
% 1.74/1.14 % (2088)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/4Mi)
% 1.74/1.14 % (2089)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/18Mi)
% 1.74/1.14 % (2090)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/710Mi)
% 1.74/1.14 % (2088)Instruction limit reached!
% 1.74/1.14 % (2088)------------------------------
% 1.74/1.14 % (2088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.14 % (2088)Termination reason: Unknown
% 1.74/1.14 % (2088)Termination phase: shuffling
% 1.74/1.14
% 1.74/1.14 % (2088)Memory used [KB]: 1407
% 1.74/1.14 % (2088)Time elapsed: 0.003 s
% 1.74/1.14 % (2088)Instructions burned: 7 (million)
% 1.74/1.14 % (2088)------------------------------
% 1.74/1.14 % (2088)------------------------------
% 1.74/1.14 % (2091)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/6Mi)
% 1.74/1.14 % (2089)Instruction limit reached!
% 1.74/1.14 % (2089)------------------------------
% 1.74/1.14 % (2089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.14 % (2092)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_1970 for (2999ds/902Mi)
% 1.74/1.14 % (2089)Termination reason: Unknown
% 1.74/1.14 % (2089)Termination phase: shuffling
% 1.74/1.14
% 1.74/1.14 % (2089)Memory used [KB]: 2046
% 1.74/1.14 % (2089)Time elapsed: 0.007 s
% 1.74/1.14 % (2089)Instructions burned: 19 (million)
% 1.74/1.14 % (2089)------------------------------
% 1.74/1.14 % (2089)------------------------------
% 1.74/1.14 % (2091)Instruction limit reached!
% 1.74/1.14 % (2091)------------------------------
% 1.74/1.14 % (2091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.14 % (2091)Termination reason: Unknown
% 1.74/1.14 % (2091)Termination phase: shuffling
% 1.74/1.14
% 1.74/1.14 % (2091)Memory used [KB]: 1407
% 1.74/1.14 % (2091)Time elapsed: 0.003 s
% 1.74/1.14 % (2091)Instructions burned: 7 (million)
% 1.74/1.15 % (2091)------------------------------
% 1.74/1.15 % (2091)------------------------------
% 1.74/1.15 % (2093)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_1970 for (2999ds/21Mi)
% 1.74/1.15 % (2070)Instruction limit reached!
% 1.74/1.15 % (2070)------------------------------
% 1.74/1.15 % (2070)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.15 % (2070)Termination reason: Unknown
% 1.74/1.15 % (2070)Termination phase: Property scanning
% 1.74/1.15
% 1.74/1.15 % (2070)Memory used [KB]: 4477
% 1.74/1.15 % (2070)Time elapsed: 0.051 s
% 1.74/1.15 % (2070)Instructions burned: 186 (million)
% 1.74/1.15 % (2070)------------------------------
% 1.74/1.15 % (2070)------------------------------
% 1.74/1.15 % (2094)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/5Mi)
% 1.74/1.15 % (2095)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_1970 for (2999ds/6Mi)
% 1.74/1.16 % (2094)Instruction limit reached!
% 1.74/1.16 % (2094)------------------------------
% 1.74/1.16 % (2094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.16 % (2094)Termination reason: Unknown
% 1.74/1.16 % (2094)Termination phase: shuffling
% 1.74/1.16
% 1.74/1.16 % (2094)Memory used [KB]: 1407
% 1.74/1.16 % (2094)Time elapsed: 0.003 s
% 1.74/1.16 % (2094)Instructions burned: 7 (million)
% 1.74/1.16 % (2094)------------------------------
% 1.74/1.16 % (2094)------------------------------
% 1.74/1.16 % (2093)Instruction limit reached!
% 1.74/1.16 % (2093)------------------------------
% 1.74/1.16 % (2093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.16 % (2093)Termination reason: Unknown
% 1.74/1.16 % (2093)Termination phase: shuffling
% 1.74/1.16
% 1.74/1.16 % (2093)Memory used [KB]: 2046
% 1.74/1.16 % (2095)Instruction limit reached!
% 1.74/1.16 % (2095)------------------------------
% 1.74/1.16 % (2095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.16 % (2093)Time elapsed: 0.008 s
% 1.74/1.16 % (2093)Instructions burned: 22 (million)
% 1.74/1.16 % (2093)------------------------------
% 1.74/1.16 % (2093)------------------------------
% 1.74/1.16 % (2095)Termination reason: Unknown
% 1.74/1.16 % (2095)Termination phase: shuffling
% 1.74/1.16
% 1.74/1.16 % (2095)Memory used [KB]: 1407
% 1.74/1.16 % (2095)Time elapsed: 0.003 s
% 1.74/1.16 % (2095)Instructions burned: 7 (million)
% 1.74/1.16 % (2095)------------------------------
% 1.74/1.16 % (2095)------------------------------
% 1.74/1.16 % (2096)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/377Mi)
% 1.74/1.17 % (2097)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_1970 for (2999ds/779Mi)
% 1.74/1.17 % (2098)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_1970 for (2999ds/19Mi)
% 1.74/1.17 % (2099)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/879Mi)
% 1.74/1.17 % (2098)Instruction limit reached!
% 1.74/1.17 % (2098)------------------------------
% 1.74/1.17 % (2098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.17 % (2098)Termination reason: Unknown
% 1.74/1.17 % (2098)Termination phase: shuffling
% 1.74/1.17
% 1.74/1.17 % (2098)Memory used [KB]: 1535
% 1.74/1.17 % (2098)Time elapsed: 0.006 s
% 1.74/1.17 % (2098)Instructions burned: 19 (million)
% 1.74/1.17 % (2098)------------------------------
% 1.74/1.17 % (2098)------------------------------
% 1.74/1.18 % (2100)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/17Mi)
% 1.74/1.19 % (2100)Instruction limit reached!
% 1.74/1.19 % (2100)------------------------------
% 1.74/1.19 % (2100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.19 % (2100)Termination reason: Unknown
% 1.74/1.19 % (2100)Termination phase: shuffling
% 1.74/1.19
% 1.74/1.19 % (2100)Memory used [KB]: 2046
% 1.74/1.19 % (2100)Time elapsed: 0.007 s
% 1.74/1.19 % (2100)Instructions burned: 18 (million)
% 1.74/1.19 % (2100)------------------------------
% 1.74/1.19 % (2100)------------------------------
% 1.74/1.20 % (2101)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/3Mi)
% 1.74/1.20 % (2101)Instruction limit reached!
% 1.74/1.20 % (2101)------------------------------
% 1.74/1.20 % (2101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.20 % (2101)Termination reason: Unknown
% 1.74/1.20 % (2101)Termination phase: shuffling
% 1.74/1.20
% 1.74/1.20 % (2101)Memory used [KB]: 1407
% 1.74/1.20 % (2101)Time elapsed: 0.003 s
% 1.74/1.20 % (2101)Instructions burned: 6 (million)
% 1.74/1.20 % (2101)------------------------------
% 1.74/1.20 % (2101)------------------------------
% 1.74/1.20 % (2092)Refutation not found, incomplete strategy
% 1.74/1.20 % (2092)------------------------------
% 1.74/1.20 % (2092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.20 % (2092)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.20
% 1.74/1.20
% 1.74/1.20 % (2092)Memory used [KB]: 9210
% 1.74/1.20 % (2092)Time elapsed: 0.057 s
% 1.74/1.20 % (2092)Instructions burned: 196 (million)
% 1.74/1.20 % (2092)------------------------------
% 1.74/1.20 % (2092)------------------------------
% 1.74/1.21 % (2102)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/30Mi)
% 1.74/1.21 % (2103)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_1970 for (2999ds/127Mi)
% 1.74/1.22 % (2102)Instruction limit reached!
% 1.74/1.22 % (2102)------------------------------
% 1.74/1.22 % (2102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.22 % (2102)Termination reason: Unknown
% 1.74/1.22 % (2102)Termination phase: Property scanning
% 1.74/1.22
% 1.74/1.22 % (2102)Memory used [KB]: 2174
% 1.74/1.22 % (2102)Time elapsed: 0.011 s
% 1.74/1.22 % (2102)Instructions burned: 33 (million)
% 1.74/1.22 % (2102)------------------------------
% 1.74/1.22 % (2102)------------------------------
% 1.74/1.23 % (2104)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_1970 for (2998ds/100Mi)
% 1.74/1.25 % (2103)Instruction limit reached!
% 1.74/1.25 % (2103)------------------------------
% 1.74/1.25 % (2103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.25 % (2103)Termination reason: Unknown
% 1.74/1.25 % (2103)Termination phase: Property scanning
% 1.74/1.25
% 1.74/1.25 % (2103)Memory used [KB]: 4093
% 1.74/1.25 % (2103)Time elapsed: 0.036 s
% 1.74/1.25 % (2103)Instructions burned: 127 (million)
% 1.74/1.25 % (2103)------------------------------
% 1.74/1.25 % (2103)------------------------------
% 1.74/1.25 % (2104)Instruction limit reached!
% 1.74/1.25 % (2104)------------------------------
% 1.74/1.25 % (2104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.25 % (2104)Termination reason: Unknown
% 1.74/1.25 % (2104)Termination phase: Saturation
% 1.74/1.25
% 1.74/1.25 % (2104)Memory used [KB]: 6524
% 1.74/1.25 % (2104)Time elapsed: 0.027 s
% 1.74/1.25 % (2104)Instructions burned: 102 (million)
% 1.74/1.25 % (2104)------------------------------
% 1.74/1.25 % (2104)------------------------------
% 1.74/1.26 % (2105)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_1970 for (2998ds/3Mi)
% 1.74/1.26 % (2105)Instruction limit reached!
% 1.74/1.26 % (2105)------------------------------
% 1.74/1.26 % (2105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.26 % (2105)Termination reason: Unknown
% 1.74/1.26 % (2105)Termination phase: shuffling
% 1.74/1.26
% 1.74/1.26 % (2105)Memory used [KB]: 1407
% 1.74/1.26 % (2105)Time elapsed: 0.003 s
% 1.74/1.26 % (2105)Instructions burned: 6 (million)
% 1.74/1.26 % (2105)------------------------------
% 1.74/1.26 % (2105)------------------------------
% 1.74/1.26 % (2106)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_1970 for (2998ds/20Mi)
% 1.74/1.27 % (2107)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_1970 for (2998ds/86Mi)
% 1.74/1.27 % (2096)Instruction limit reached!
% 1.74/1.27 % (2096)------------------------------
% 1.74/1.27 % (2096)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.27 % (2096)Termination reason: Unknown
% 1.74/1.27 % (2096)Termination phase: Saturation
% 1.74/1.27
% 1.74/1.27 % (2096)Memory used [KB]: 10362
% 1.74/1.27 % (2096)Time elapsed: 0.109 s
% 1.74/1.27 % (2096)Instructions burned: 377 (million)
% 1.74/1.27 % (2096)------------------------------
% 1.74/1.27 % (2096)------------------------------
% 1.74/1.27 % (2106)Instruction limit reached!
% 1.74/1.27 % (2106)------------------------------
% 1.74/1.27 % (2106)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.27 % (2106)Termination reason: Unknown
% 1.74/1.27 % (2106)Termination phase: shuffling
% 1.74/1.27
% 1.74/1.27 % (2106)Memory used [KB]: 2174
% 1.74/1.27 % (2106)Time elapsed: 0.008 s
% 1.74/1.27 % (2106)Instructions burned: 22 (million)
% 1.74/1.27 % (2106)------------------------------
% 1.74/1.27 % (2106)------------------------------
% 1.74/1.28 % (2108)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_1970 for (2998ds/107Mi)
% 1.74/1.28 % (2109)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_1970 for (2998ds/39Mi)
% 1.74/1.29 % (2109)Instruction limit reached!
% 1.74/1.29 % (2109)------------------------------
% 1.74/1.29 % (2109)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.29 % (2109)Termination reason: Unknown
% 1.74/1.29 % (2109)Termination phase: Property scanning
% 1.74/1.29
% 1.74/1.29 % (2109)Memory used [KB]: 1918
% 1.74/1.29 % (2109)Time elapsed: 0.012 s
% 1.74/1.29 % (2109)Instructions burned: 42 (million)
% 1.74/1.29 % (2109)------------------------------
% 1.74/1.29 % (2109)------------------------------
% 1.74/1.29 % (2107)Instruction limit reached!
% 1.74/1.29 % (2107)------------------------------
% 1.74/1.29 % (2107)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.29 % (2107)Termination reason: Unknown
% 1.74/1.29 % (2107)Termination phase: Preprocessing 3
% 1.74/1.29
% 1.74/1.29 % (2107)Memory used [KB]: 4093
% 1.74/1.29 % (2107)Time elapsed: 0.026 s
% 1.74/1.29 % (2107)Instructions burned: 86 (million)
% 1.74/1.29 % (2107)------------------------------
% 1.74/1.29 % (2107)------------------------------
% 1.74/1.30 % (2110)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_1970 for (2998ds/448Mi)
% 1.74/1.30 % (2111)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_1970 for (2997ds/46Mi)
% 1.74/1.31 % (2108)Instruction limit reached!
% 1.74/1.31 % (2108)------------------------------
% 1.74/1.31 % (2108)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.31 % (2108)Termination reason: Unknown
% 1.74/1.31 % (2108)Termination phase: Property scanning
% 1.74/1.31
% 1.74/1.31 % (2108)Memory used [KB]: 4349
% 1.74/1.31 % (2108)Time elapsed: 0.032 s
% 1.74/1.31 % (2108)Instructions burned: 109 (million)
% 1.74/1.31 % (2108)------------------------------
% 1.74/1.31 % (2108)------------------------------
% 1.74/1.32 % (2111)Instruction limit reached!
% 1.74/1.32 % (2111)------------------------------
% 1.74/1.32 % (2111)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.32 % (2111)Termination reason: Unknown
% 1.74/1.32 % (2111)Termination phase: Preprocessing 3
% 1.74/1.32
% 1.74/1.32 % (2111)Memory used [KB]: 2430
% 1.74/1.32 % (2111)Time elapsed: 0.015 s
% 1.74/1.32 % (2111)Instructions burned: 47 (million)
% 1.74/1.32 % (2111)------------------------------
% 1.74/1.32 % (2111)------------------------------
% 1.74/1.32 % (2112)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_1970 for (2997ds/98Mi)
% 1.74/1.33 % (2113)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_1970 for (2997ds/507Mi)
% 1.74/1.34 % (2112)Instruction limit reached!
% 1.74/1.34 % (2112)------------------------------
% 1.74/1.34 % (2112)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.34 % (2112)Termination reason: Unknown
% 1.74/1.34 % (2112)Termination phase: Saturation
% 1.74/1.34
% 1.74/1.34 % (2112)Memory used [KB]: 2046
% 1.74/1.34 % (2112)Time elapsed: 0.026 s
% 1.74/1.34 % (2112)Instructions burned: 100 (million)
% 1.74/1.34 % (2112)------------------------------
% 1.74/1.34 % (2112)------------------------------
% 1.74/1.35 % (2090)Instruction limit reached!
% 1.74/1.35 % (2090)------------------------------
% 1.74/1.35 % (2090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.35 % (2090)Termination reason: Unknown
% 1.74/1.35 % (2090)Termination phase: Saturation
% 1.74/1.35
% 1.74/1.35 % (2090)Memory used [KB]: 10362
% 1.74/1.35 % (2090)Time elapsed: 0.211 s
% 1.74/1.35 % (2090)Instructions burned: 712 (million)
% 1.74/1.35 % (2090)------------------------------
% 1.74/1.35 % (2090)------------------------------
% 1.74/1.35 % (2097)Instruction limit reached!
% 1.74/1.35 % (2097)------------------------------
% 1.74/1.35 % (2097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.35 % (2097)Termination reason: Unknown
% 1.74/1.35 % (2097)Termination phase: Saturation
% 1.74/1.35
% 1.74/1.35 % (2097)Memory used [KB]: 6780
% 1.74/1.35 % (2097)Time elapsed: 0.188 s
% 1.74/1.35 % (2097)Instructions burned: 780 (million)
% 1.74/1.35 % (2097)------------------------------
% 1.74/1.35 % (2097)------------------------------
% 1.74/1.35 % (2114)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_1970 for (2997ds/149Mi)
% 1.74/1.36 % (2115)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_1970 for (2997ds/39Mi)
% 1.74/1.36 % (2116)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_1970 for (2997ds/6Mi)
% 1.74/1.36 % (2116)Instruction limit reached!
% 1.74/1.36 % (2116)------------------------------
% 1.74/1.36 % (2116)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.36 % (2116)Termination reason: Unknown
% 1.74/1.36 % (2116)Termination phase: shuffling
% 1.74/1.36
% 1.74/1.36 % (2116)Memory used [KB]: 1407
% 1.74/1.36 % (2116)Time elapsed: 0.003 s
% 1.74/1.36 % (2116)Instructions burned: 6 (million)
% 1.74/1.36 % (2116)------------------------------
% 1.74/1.36 % (2116)------------------------------
% 1.74/1.37 % (2115)Instruction limit reached!
% 1.74/1.37 % (2115)------------------------------
% 1.74/1.37 % (2115)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.37 % (2115)Termination reason: Unknown
% 1.74/1.37 % (2115)Termination phase: Preprocessing 1
% 1.74/1.37
% 1.74/1.37 % (2115)Memory used [KB]: 2302
% 1.74/1.37 % (2115)Time elapsed: 0.013 s
% 1.74/1.37 % (2115)Instructions burned: 40 (million)
% 1.74/1.37 % (2115)------------------------------
% 1.74/1.37 % (2115)------------------------------
% 1.74/1.37 % (2117)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_1970 for (2997ds/1087Mi)
% 1.74/1.37 % (2081)Instruction limit reached!
% 1.74/1.37 % (2081)------------------------------
% 1.74/1.37 % (2081)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.37 % (2081)Termination reason: Unknown
% 1.74/1.37 % (2081)Termination phase: Saturation
% 1.74/1.37
% 1.74/1.37 % (2081)Memory used [KB]: 8059
% 1.74/1.37 % (2081)Time elapsed: 0.255 s
% 1.74/1.37 % (2081)Instructions burned: 1044 (million)
% 1.74/1.37 % (2081)------------------------------
% 1.74/1.37 % (2081)------------------------------
% 1.74/1.38 % (2118)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_1970 for (2997ds/47Mi)
% 1.74/1.38 % (2119)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_1970 for (2997ds/44Mi)
% 1.74/1.39 % (2118)Instruction limit reached!
% 1.74/1.39 % (2118)------------------------------
% 1.74/1.39 % (2118)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.39 % (2118)Termination reason: Unknown
% 1.74/1.39 % (2118)Termination phase: Preprocessing 3
% 1.74/1.39
% 1.74/1.39 % (2118)Memory used [KB]: 2558
% 1.74/1.39 % (2118)Time elapsed: 0.015 s
% 1.74/1.39 % (2118)Instructions burned: 48 (million)
% 1.74/1.39 % (2118)------------------------------
% 1.74/1.39 % (2118)------------------------------
% 1.74/1.40 % (2119)Instruction limit reached!
% 1.74/1.40 % (2119)------------------------------
% 1.74/1.40 % (2119)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.40 % (2119)Termination reason: Unknown
% 1.74/1.40 % (2119)Termination phase: Property scanning
% 1.74/1.40
% 1.74/1.40 % (2119)Memory used [KB]: 1918
% 1.74/1.40 % (2119)Time elapsed: 0.013 s
% 1.74/1.40 % (2119)Instructions burned: 45 (million)
% 1.74/1.40 % (2119)------------------------------
% 1.74/1.40 % (2119)------------------------------
% 1.74/1.40 % (2114)Instruction limit reached!
% 1.74/1.40 % (2114)------------------------------
% 1.74/1.40 % (2114)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.40 % (2114)Termination reason: Unknown
% 1.74/1.40 % (2114)Termination phase: Property scanning
% 1.74/1.40
% 1.74/1.40 % (2114)Memory used [KB]: 7931
% 1.74/1.40 % (2114)Time elapsed: 0.043 s
% 1.74/1.40 % (2114)Instructions burned: 152 (million)
% 1.74/1.40 % (2114)------------------------------
% 1.74/1.40 % (2114)------------------------------
% 1.74/1.40 % (2120)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_1970 for (2996ds/432Mi)
% 1.74/1.40 % (2113)Refutation not found, incomplete strategy
% 1.74/1.40 % (2113)------------------------------
% 1.74/1.40 % (2113)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.40 % (2113)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.40
% 1.74/1.40
% 1.74/1.40 % (2113)Memory used [KB]: 9978
% 1.74/1.40 % (2113)Time elapsed: 0.078 s
% 1.74/1.40 % (2113)Instructions burned: 278 (million)
% 1.74/1.40 % (2113)------------------------------
% 1.74/1.40 % (2113)------------------------------
% 1.74/1.41 % (2121)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_1970 for (2996ds/62Mi)
% 1.74/1.41 % (2122)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_1970 for (2996ds/31Mi)
% 1.74/1.41 % (2110)Instruction limit reached!
% 1.74/1.41 % (2110)------------------------------
% 1.74/1.41 % (2110)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.41 % (2110)Termination reason: Unknown
% 1.74/1.41 % (2110)Termination phase: Saturation
% 1.74/1.41
% 1.74/1.41 % (2110)Memory used [KB]: 11513
% 1.74/1.41 % (2110)Time elapsed: 0.137 s
% 1.74/1.41 % (2110)Instructions burned: 448 (million)
% 1.74/1.41 % (2110)------------------------------
% 1.74/1.41 % (2110)------------------------------
% 1.74/1.41 % (2123)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_1970 for (2996ds/96Mi)
% 1.74/1.42 % (2122)Instruction limit reached!
% 1.74/1.42 % (2122)------------------------------
% 1.74/1.42 % (2122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.42 % (2122)Termination reason: Unknown
% 1.74/1.42 % (2122)Termination phase: Property scanning
% 1.74/1.42
% 1.74/1.42 % (2122)Memory used [KB]: 2174
% 1.74/1.42 % (2122)Time elapsed: 0.011 s
% 1.74/1.42 % (2122)Instructions burned: 33 (million)
% 1.74/1.42 % (2122)------------------------------
% 1.74/1.42 % (2122)------------------------------
% 1.74/1.42 % (2121)Instruction limit reached!
% 1.74/1.42 % (2121)------------------------------
% 1.74/1.42 % (2121)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.42 % (2121)Termination reason: Unknown
% 1.74/1.42 % (2121)Termination phase: Property scanning
% 1.74/1.42
% 1.74/1.42 % (2121)Memory used [KB]: 1918
% 1.74/1.42 % (2121)Time elapsed: 0.017 s
% 1.74/1.42 % (2121)Instructions burned: 64 (million)
% 1.74/1.42 % (2121)------------------------------
% 1.74/1.42 % (2121)------------------------------
% 1.74/1.42 % (2124)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_1970 for (2996ds/453Mi)
% 1.74/1.43 % (2125)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_1970 for (2996ds/71Mi)
% 1.74/1.43 % (2099)Instruction limit reached!
% 1.74/1.43 % (2099)------------------------------
% 1.74/1.43 % (2099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.43 % (2099)Termination reason: Unknown
% 1.74/1.43 % (2099)Termination phase: Saturation
% 1.74/1.43
% 1.74/1.43 % (2099)Memory used [KB]: 11769
% 1.74/1.43 % (2099)Time elapsed: 0.264 s
% 1.74/1.43 % (2099)Instructions burned: 881 (million)
% 1.74/1.43 % (2099)------------------------------
% 1.74/1.43 % (2099)------------------------------
% 1.74/1.43 % (2126)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_1970 for (2996ds/83Mi)
% 1.74/1.44 % (2127)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_1970 for (2996ds/11Mi)
% 1.74/1.44 % (2123)Instruction limit reached!
% 1.74/1.44 % (2123)------------------------------
% 1.74/1.44 % (2123)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.44 % (2123)Termination reason: Unknown
% 1.74/1.44 % (2123)Termination phase: Property scanning
% 1.74/1.44
% 1.74/1.44 % (2123)Memory used [KB]: 4221
% 1.74/1.44 % (2123)Time elapsed: 0.029 s
% 1.74/1.44 % (2123)Instructions burned: 97 (million)
% 1.74/1.44 % (2123)------------------------------
% 1.74/1.44 % (2123)------------------------------
% 1.74/1.44 % (2127)Instruction limit reached!
% 1.74/1.44 % (2127)------------------------------
% 1.74/1.44 % (2127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.44 % (2127)Termination reason: Unknown
% 1.74/1.44 % (2127)Termination phase: shuffling
% 1.74/1.44
% 1.74/1.44 % (2127)Memory used [KB]: 1535
% 1.74/1.44 % (2127)Time elapsed: 0.005 s
% 1.74/1.44 % (2127)Instructions burned: 13 (million)
% 1.74/1.44 % (2127)------------------------------
% 1.74/1.44 % (2127)------------------------------
% 1.74/1.44 % (2126)Refutation not found, incomplete strategy
% 1.74/1.44 % (2126)------------------------------
% 1.74/1.44 % (2126)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.44 % (2126)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.44
% 1.74/1.44
% 1.74/1.44 % (2126)Memory used [KB]: 6780
% 1.74/1.44 % (2126)Time elapsed: 0.015 s
% 1.74/1.44 % (2126)Instructions burned: 46 (million)
% 1.74/1.44 % (2126)------------------------------
% 1.74/1.44 % (2126)------------------------------
% 1.74/1.45 % (2125)Instruction limit reached!
% 1.74/1.45 % (2125)------------------------------
% 1.74/1.45 % (2125)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.45 % (2125)Termination reason: Unknown
% 1.74/1.45 % (2125)Termination phase: Preprocessing 3
% 1.74/1.45
% 1.74/1.45 % (2125)Memory used [KB]: 3454
% 1.74/1.45 % (2125)Time elapsed: 0.022 s
% 1.74/1.45 % (2125)Instructions burned: 72 (million)
% 1.74/1.45 % (2125)------------------------------
% 1.74/1.45 % (2125)------------------------------
% 1.74/1.45 % (2129)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_1970 for (2996ds/6Mi)
% 1.74/1.45 % (2128)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_1970 for (2996ds/41Mi)
% 1.74/1.45 % (2129)Instruction limit reached!
% 1.74/1.45 % (2129)------------------------------
% 1.74/1.45 % (2129)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.45 % (2129)Termination reason: Unknown
% 1.74/1.45 % (2129)Termination phase: shuffling
% 1.74/1.45
% 1.74/1.45 % (2129)Memory used [KB]: 1407
% 1.74/1.45 % (2129)Time elapsed: 0.003 s
% 1.74/1.45 % (2129)Instructions burned: 6 (million)
% 1.74/1.45 % (2129)------------------------------
% 1.74/1.45 % (2129)------------------------------
% 1.74/1.45 % (2130)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_1970 for (2996ds/150Mi)
% 1.74/1.46 % (2131)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_1970 for (2996ds/343Mi)
% 1.74/1.46 % (2128)Instruction limit reached!
% 1.74/1.46 % (2128)------------------------------
% 1.74/1.46 % (2128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.46 % (2128)Termination reason: Unknown
% 1.74/1.46 % (2128)Termination phase: Property scanning
% 1.74/1.46
% 1.74/1.46 % (2128)Memory used [KB]: 1918
% 1.74/1.46 % (2128)Time elapsed: 0.012 s
% 1.74/1.46 % (2128)Instructions burned: 42 (million)
% 1.74/1.46 % (2128)------------------------------
% 1.74/1.46 % (2128)------------------------------
% 1.74/1.46 % (2132)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_1970 for (2996ds/109Mi)
% 1.74/1.47 % (2131)Refutation not found, incomplete strategy
% 1.74/1.47 % (2131)------------------------------
% 1.74/1.47 % (2131)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.47 % (2131)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.47
% 1.74/1.47
% 1.74/1.47 % (2131)Memory used [KB]: 6652
% 1.74/1.47 % (2131)Time elapsed: 0.013 s
% 1.74/1.47 % (2131)Instructions burned: 41 (million)
% 1.74/1.47 % (2131)------------------------------
% 1.74/1.47 % (2131)------------------------------
% 1.74/1.47 % (2133)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_1970 for (2996ds/253Mi)
% 1.74/1.48 % (2134)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_1970 for (2996ds/140Mi)
% 1.74/1.49 % (2133)Refutation not found, incomplete strategy
% 1.74/1.49 % (2133)------------------------------
% 1.74/1.49 % (2133)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.49 % (2133)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.49
% 1.74/1.49
% 1.74/1.49 % (2133)Memory used [KB]: 6780
% 1.74/1.49 % (2133)Time elapsed: 0.013 s
% 1.74/1.49 % (2133)Instructions burned: 39 (million)
% 1.74/1.49 % (2133)------------------------------
% 1.74/1.49 % (2133)------------------------------
% 1.74/1.49 % (2132)Instruction limit reached!
% 1.74/1.49 % (2132)------------------------------
% 1.74/1.49 % (2132)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.49 % (2130)Instruction limit reached!
% 1.74/1.49 % (2130)------------------------------
% 1.74/1.49 % (2130)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.49 % (2130)Termination reason: Unknown
% 1.74/1.49 % (2130)Termination phase: Property scanning
% 1.74/1.49
% 1.74/1.49 % (2130)Memory used [KB]: 4093
% 1.74/1.49 % (2130)Time elapsed: 0.041 s
% 1.74/1.49 % (2130)Instructions burned: 151 (million)
% 1.74/1.49 % (2130)------------------------------
% 1.74/1.49 % (2130)------------------------------
% 1.74/1.49 % (2132)Termination reason: Unknown
% 1.74/1.49 % (2132)Termination phase: Property scanning
% 1.74/1.49
% 1.74/1.49 % (2132)Memory used [KB]: 4221
% 1.74/1.49 % (2132)Time elapsed: 0.032 s
% 1.74/1.49 % (2132)Instructions burned: 110 (million)
% 1.74/1.49 % (2132)------------------------------
% 1.74/1.49 % (2132)------------------------------
% 1.74/1.50 % (2135)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_1970 for (2996ds/614Mi)
% 1.74/1.50 % (2136)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_1970 for (2996ds/202Mi)
% 1.74/1.50 % (2137)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_1970 for (2996ds/339Mi)
% 1.74/1.51 % (2120)Instruction limit reached!
% 1.74/1.51 % (2120)------------------------------
% 1.74/1.51 % (2120)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.51 % (2120)Termination reason: Unknown
% 1.74/1.51 % (2120)Termination phase: Saturation
% 1.74/1.51
% 1.74/1.51 % (2120)Memory used [KB]: 2558
% 1.74/1.51 % (2120)Time elapsed: 0.109 s
% 1.74/1.51 % (2120)Instructions burned: 433 (million)
% 1.74/1.51 % (2120)------------------------------
% 1.74/1.51 % (2120)------------------------------
% 1.74/1.52 % (2134)Instruction limit reached!
% 1.74/1.52 % (2134)------------------------------
% 1.74/1.52 % (2134)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.52 % (2134)Termination reason: Unknown
% 1.74/1.52 % (2134)Termination phase: Function definition elimination
% 1.74/1.52
% 1.74/1.52 % (2134)Memory used [KB]: 4349
% 1.74/1.52 % (2134)Time elapsed: 0.040 s
% 1.74/1.52 % (2134)Instructions burned: 143 (million)
% 1.74/1.52 % (2134)------------------------------
% 1.74/1.52 % (2134)------------------------------
% 1.74/1.52 % (2138)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_1970 for (2995ds/283Mi)
% 1.74/1.53 % (2139)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_1970 for (2995ds/134Mi)
% 1.74/1.53 % (2124)Instruction limit reached!
% 1.74/1.53 % (2124)------------------------------
% 1.74/1.53 % (2124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.53 % (2124)Termination reason: Unknown
% 1.74/1.53 % (2124)Termination phase: Saturation
% 1.74/1.53
% 1.74/1.53 % (2124)Memory used [KB]: 6652
% 1.74/1.53 % (2124)Time elapsed: 0.109 s
% 1.74/1.53 % (2124)Instructions burned: 454 (million)
% 1.74/1.53 % (2124)------------------------------
% 1.74/1.53 % (2124)------------------------------
% 1.74/1.54 % (2140)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_1970 for (2995ds/298Mi)
% 1.74/1.56 % (2136)Instruction limit reached!
% 1.74/1.56 % (2136)------------------------------
% 1.74/1.56 % (2136)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.56 % (2136)Termination reason: Unknown
% 1.74/1.56 % (2136)Termination phase: Saturation
% 1.74/1.56
% 1.74/1.56 % (2136)Memory used [KB]: 4477
% 1.74/1.56 % (2136)Time elapsed: 0.055 s
% 1.74/1.56 % (2136)Instructions burned: 202 (million)
% 1.74/1.56 % (2136)------------------------------
% 1.74/1.56 % (2136)------------------------------
% 1.74/1.56 % (2139)Instruction limit reached!
% 1.74/1.56 % (2139)------------------------------
% 1.74/1.56 % (2139)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.56 % (2139)Termination reason: Unknown
% 1.74/1.56 % (2139)Termination phase: Property scanning
% 1.74/1.56
% 1.74/1.56 % (2139)Memory used [KB]: 4221
% 1.74/1.56 % (2139)Time elapsed: 0.038 s
% 1.74/1.56 % (2139)Instructions burned: 135 (million)
% 1.74/1.56 % (2139)------------------------------
% 1.74/1.56 % (2139)------------------------------
% 1.74/1.57 % (2141)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_1970 for (2995ds/192Mi)
% 1.74/1.57 % (2142)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_1970 for (2995ds/258Mi)
% 1.74/1.59 % (2138)Instruction limit reached!
% 1.74/1.59 % (2138)------------------------------
% 1.74/1.59 % (2138)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.59 % (2138)Termination reason: Unknown
% 1.74/1.59 % (2138)Termination phase: Saturation
% 1.74/1.59
% 1.74/1.59 % (2138)Memory used [KB]: 6780
% 1.74/1.59 % (2138)Time elapsed: 0.072 s
% 1.74/1.59 % (2138)Instructions burned: 284 (million)
% 1.74/1.59 % (2138)------------------------------
% 1.74/1.59 % (2138)------------------------------
% 1.74/1.60 % (2143)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_1970 for (2995ds/90Mi)
% 1.74/1.62 % (2140)Instruction limit reached!
% 1.74/1.62 % (2140)------------------------------
% 1.74/1.62 % (2140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.62 % (2140)Termination reason: Unknown
% 1.74/1.62 % (2140)Termination phase: Saturation
% 1.74/1.62
% 1.74/1.62 % (2140)Memory used [KB]: 6524
% 1.74/1.62 % (2140)Time elapsed: 0.075 s
% 1.74/1.62 % (2140)Instructions burned: 298 (million)
% 1.74/1.62 % (2140)------------------------------
% 1.74/1.62 % (2140)------------------------------
% 1.74/1.62 % (2141)Instruction limit reached!
% 1.74/1.62 % (2141)------------------------------
% 1.74/1.62 % (2141)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.62 % (2141)Termination reason: Unknown
% 1.74/1.62 % (2141)Termination phase: Property scanning
% 1.74/1.62
% 1.74/1.62 % (2141)Memory used [KB]: 4605
% 1.74/1.62 % (2141)Time elapsed: 0.053 s
% 1.74/1.62 % (2141)Instructions burned: 195 (million)
% 1.74/1.62 % (2141)------------------------------
% 1.74/1.62 % (2141)------------------------------
% 1.74/1.62 % (2137)Instruction limit reached!
% 1.74/1.62 % (2137)------------------------------
% 1.74/1.62 % (2137)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.62 % (2137)Termination reason: Unknown
% 1.74/1.62 % (2137)Termination phase: Saturation
% 1.74/1.62
% 1.74/1.62 % (2137)Memory used [KB]: 10362
% 1.74/1.62 % (2137)Time elapsed: 0.117 s
% 1.74/1.62 % (2137)Instructions burned: 340 (million)
% 1.74/1.62 % (2137)------------------------------
% 1.74/1.62 % (2137)------------------------------
% 1.74/1.63 % (2144)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_1970 for (2994ds/231Mi)
% 1.74/1.63 % (2143)Instruction limit reached!
% 1.74/1.63 % (2143)------------------------------
% 1.74/1.63 % (2143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.63 % (2143)Termination reason: Unknown
% 1.74/1.63 % (2143)Termination phase: Preprocessing 3
% 1.74/1.63
% 1.74/1.63 % (2143)Memory used [KB]: 4093
% 1.74/1.63 % (2143)Time elapsed: 0.028 s
% 1.74/1.63 % (2143)Instructions burned: 90 (million)
% 1.74/1.63 % (2143)------------------------------
% 1.74/1.63 % (2143)------------------------------
% 1.74/1.63 % (2145)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_1970 for (2994ds/458Mi)
% 1.74/1.64 % (2146)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_1970 for (2994ds/31Mi)
% 1.74/1.64 % (2147)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_1970 for (2994ds/297Mi)
% 1.74/1.64 % (2142)Instruction limit reached!
% 1.74/1.64 % (2142)------------------------------
% 1.74/1.64 % (2142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.64 % (2142)Termination reason: Unknown
% 1.74/1.64 % (2142)Termination phase: Property scanning
% 1.74/1.64
% 1.74/1.64 % (2142)Memory used [KB]: 7931
% 1.74/1.64 % (2142)Time elapsed: 0.070 s
% 1.74/1.64 % (2142)Instructions burned: 259 (million)
% 1.74/1.64 % (2142)------------------------------
% 1.74/1.64 % (2142)------------------------------
% 1.74/1.65 % (2146)Instruction limit reached!
% 1.74/1.65 % (2146)------------------------------
% 1.74/1.65 % (2146)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.65 % (2146)Termination reason: Unknown
% 1.74/1.65 % (2146)Termination phase: Property scanning
% 1.74/1.65
% 1.74/1.65 % (2146)Memory used [KB]: 2174
% 1.74/1.65 % (2146)Time elapsed: 0.014 s
% 1.74/1.65 % (2146)Instructions burned: 31 (million)
% 1.74/1.65 % (2146)------------------------------
% 1.74/1.65 % (2146)------------------------------
% 1.74/1.65 % (2148)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_1970 for (2994ds/987Mi)
% 1.74/1.66 % (2147)Refutation not found, incomplete strategy
% 1.74/1.66 % (2147)------------------------------
% 1.74/1.66 % (2147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.66 % (2147)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.66
% 1.74/1.66
% 1.74/1.66 % (2147)Memory used [KB]: 6524
% 1.74/1.66 % (2147)Time elapsed: 0.023 s
% 1.74/1.66 % (2147)Instructions burned: 83 (million)
% 1.74/1.66 % (2147)------------------------------
% 1.74/1.66 % (2147)------------------------------
% 1.74/1.67 % (2149)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_1970 for (2994ds/182Mi)
% 1.74/1.67 % (2150)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_1970 for (2994ds/182Mi)
% 1.74/1.68 % (2135)Instruction limit reached!
% 1.74/1.68 % (2135)------------------------------
% 1.74/1.68 % (2135)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.68 % (2135)Termination reason: Unknown
% 1.74/1.68 % (2135)Termination phase: Saturation
% 1.74/1.68
% 1.74/1.68 % (2135)Memory used [KB]: 12025
% 1.74/1.68 % (2135)Time elapsed: 0.181 s
% 1.74/1.68 % (2135)Instructions burned: 616 (million)
% 1.74/1.68 % (2135)------------------------------
% 1.74/1.68 % (2135)------------------------------
% 1.74/1.68 % (2144)Instruction limit reached!
% 1.74/1.68 % (2144)------------------------------
% 1.74/1.68 % (2144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.68 % (2144)Termination reason: Unknown
% 1.74/1.68 % (2144)Termination phase: Saturation
% 1.74/1.68
% 1.74/1.68 % (2144)Memory used [KB]: 7164
% 1.74/1.68 % (2144)Time elapsed: 0.058 s
% 1.74/1.68 % (2144)Instructions burned: 232 (million)
% 1.74/1.68 % (2144)------------------------------
% 1.74/1.68 % (2144)------------------------------
% 1.74/1.69 % (2151)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_1970 for (2994ds/154Mi)
% 1.74/1.69 % (2152)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_1970 for (2994ds/79Mi)
% 1.74/1.71 % (2117)Instruction limit reached!
% 1.74/1.71 % (2117)------------------------------
% 1.74/1.71 % (2117)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.71 % (2117)Termination reason: Unknown
% 1.74/1.71 % (2117)Termination phase: Saturation
% 1.74/1.71
% 1.74/1.71 % (2117)Memory used [KB]: 12537
% 1.74/1.71 % (2117)Time elapsed: 0.338 s
% 1.74/1.71 % (2117)Instructions burned: 1090 (million)
% 1.74/1.71 % (2117)------------------------------
% 1.74/1.71 % (2117)------------------------------
% 1.74/1.72 % (2152)Instruction limit reached!
% 1.74/1.72 % (2152)------------------------------
% 1.74/1.72 % (2152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.72 % (2152)Termination reason: Unknown
% 1.74/1.72 % (2152)Termination phase: Preprocessing 3
% 1.74/1.72
% 1.74/1.72 % (2152)Memory used [KB]: 3837
% 1.74/1.72 % (2152)Time elapsed: 0.024 s
% 1.74/1.72 % (2152)Instructions burned: 79 (million)
% 1.74/1.72 % (2152)------------------------------
% 1.74/1.72 % (2152)------------------------------
% 1.74/1.72 % (2153)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/216Mi)
% 1.74/1.72 % (2150)Instruction limit reached!
% 1.74/1.72 % (2150)------------------------------
% 1.74/1.72 % (2150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.72 % (2150)Termination reason: Unknown
% 1.74/1.72 % (2150)Termination phase: Saturation
% 1.74/1.72
% 1.74/1.72 % (2150)Memory used [KB]: 9210
% 1.74/1.72 % (2150)Time elapsed: 0.052 s
% 1.74/1.72 % (2150)Instructions burned: 183 (million)
% 1.74/1.72 % (2150)------------------------------
% 1.74/1.72 % (2150)------------------------------
% 1.74/1.72 % (2151)Instruction limit reached!
% 1.74/1.72 % (2151)------------------------------
% 1.74/1.72 % (2151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.72 % (2151)Termination reason: Unknown
% 1.74/1.72 % (2151)Termination phase: Saturation
% 1.74/1.72
% 1.74/1.72 % (2151)Memory used [KB]: 2814
% 1.74/1.72 % (2151)Time elapsed: 0.040 s
% 1.74/1.72 % (2151)Instructions burned: 157 (million)
% 1.74/1.72 % (2151)------------------------------
% 1.74/1.72 % (2151)------------------------------
% 1.74/1.72 % (2154)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/114Mi)
% 1.74/1.73 % (2155)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/922Mi)
% 1.74/1.73 % (2156)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/182Mi)
% 1.74/1.74 % (2149)Instruction limit reached!
% 1.74/1.74 % (2149)------------------------------
% 1.74/1.74 % (2149)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.74 % (2149)Termination reason: Unknown
% 1.74/1.74 % (2149)Termination phase: Property scanning
% 1.74/1.74
% 1.74/1.74 % (2149)Memory used [KB]: 7931
% 1.74/1.74 % (2149)Time elapsed: 0.074 s
% 1.74/1.74 % (2149)Instructions burned: 183 (million)
% 1.74/1.74 % (2149)------------------------------
% 1.74/1.74 % (2149)------------------------------
% 1.74/1.75 % (2153)Refutation not found, incomplete strategy
% 1.74/1.75 % (2153)------------------------------
% 1.74/1.75 % (2153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.75 % (2153)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.75
% 1.74/1.75
% 1.74/1.75 % (2153)Memory used [KB]: 6652
% 1.74/1.75 % (2153)Time elapsed: 0.032 s
% 1.74/1.75 % (2153)Instructions burned: 113 (million)
% 1.74/1.75 % (2153)------------------------------
% 1.74/1.75 % (2153)------------------------------
% 1.74/1.75 % (2157)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/164Mi)
% 1.74/1.76 % (2154)Instruction limit reached!
% 1.74/1.76 % (2154)------------------------------
% 1.74/1.76 % (2154)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.76 % (2154)Termination reason: Unknown
% 1.74/1.76 % (2154)Termination phase: Property scanning
% 1.74/1.76
% 1.74/1.76 % (2154)Memory used [KB]: 4221
% 1.74/1.76 % (2154)Time elapsed: 0.033 s
% 1.74/1.76 % (2154)Instructions burned: 115 (million)
% 1.74/1.76 % (2154)------------------------------
% 1.74/1.76 % (2154)------------------------------
% 1.74/1.76 % (2145)Instruction limit reached!
% 1.74/1.76 % (2145)------------------------------
% 1.74/1.76 % (2145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.76 % (2145)Termination reason: Unknown
% 1.74/1.76 % (2158)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/50Mi)
% 1.74/1.76 % (2145)Termination phase: Saturation
% 1.74/1.76
% 1.74/1.76 % (2145)Memory used [KB]: 10362
% 1.74/1.76 % (2145)Time elapsed: 0.133 s
% 1.74/1.76 % (2145)Instructions burned: 460 (million)
% 1.74/1.76 % (2145)------------------------------
% 1.74/1.76 % (2145)------------------------------
% 1.74/1.77 % (2159)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/368Mi)
% 1.74/1.77 % (2160)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/1021Mi)
% 1.74/1.77 % (2158)Instruction limit reached!
% 1.74/1.77 % (2158)------------------------------
% 1.74/1.77 % (2158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.77 % (2158)Termination reason: Unknown
% 1.74/1.77 % (2158)Termination phase: Property scanning
% 1.74/1.77
% 1.74/1.77 % (2158)Memory used [KB]: 1918
% 1.74/1.77 % (2158)Time elapsed: 0.014 s
% 1.74/1.77 % (2158)Instructions burned: 50 (million)
% 1.74/1.77 % (2158)------------------------------
% 1.74/1.77 % (2158)------------------------------
% 1.74/1.78 % (2156)Instruction limit reached!
% 1.74/1.78 % (2156)------------------------------
% 1.74/1.78 % (2156)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.78 % (2156)Termination reason: Unknown
% 1.74/1.78 % (2156)Termination phase: Saturation
% 1.74/1.78
% 1.74/1.78 % (2156)Memory used [KB]: 6524
% 1.74/1.78 % (2156)Time elapsed: 0.046 s
% 1.74/1.78 % (2156)Instructions burned: 183 (million)
% 1.74/1.78 % (2156)------------------------------
% 1.74/1.78 % (2156)------------------------------
% 1.74/1.78 % (2161)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/267Mi)
% 1.74/1.78 % (2157)Refutation not found, incomplete strategy
% 1.74/1.78 % (2157)------------------------------
% 1.74/1.78 % (2157)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.78 % (2157)Termination reason: Refutation not found, incomplete strategy
% 1.74/1.78
% 1.74/1.78
% 1.74/1.78 % (2157)Memory used [KB]: 2046
% 1.74/1.78 % (2157)Time elapsed: 0.031 s
% 1.74/1.78 % (2157)Instructions burned: 78 (million)
% 1.74/1.78 % (2157)------------------------------
% 1.74/1.78 % (2157)------------------------------
% 1.74/1.79 % (2162)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/204Mi)
% 1.74/1.80 % (2163)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_1970 for (2993ds/26Mi)
% 1.74/1.81 % (2163)Instruction limit reached!
% 1.74/1.81 % (2163)------------------------------
% 1.74/1.81 % (2163)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.81 % (2163)Termination reason: Unknown
% 1.74/1.81 % (2163)Termination phase: shuffling
% 1.74/1.81
% 1.74/1.81 % (2163)Memory used [KB]: 1791
% 1.74/1.81 % (2163)Time elapsed: 0.012 s
% 1.74/1.81 % (2163)Instructions burned: 27 (million)
% 1.74/1.81 % (2163)------------------------------
% 1.74/1.81 % (2163)------------------------------
% 1.74/1.83 % (2164)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_1970 for (2992ds/53Mi)
% 1.74/1.84 % (2162)Instruction limit reached!
% 1.74/1.84 % (2162)------------------------------
% 1.74/1.84 % (2162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.84 % (2162)Termination reason: Unknown
% 1.74/1.84 % (2162)Termination phase: Property scanning
% 1.74/1.84
% 1.74/1.84 % (2162)Memory used [KB]: 4349
% 1.74/1.84 % (2162)Time elapsed: 0.055 s
% 1.74/1.84 % (2162)Instructions burned: 207 (million)
% 1.74/1.84 % (2162)------------------------------
% 1.74/1.84 % (2162)------------------------------
% 1.74/1.85 % (2164)Instruction limit reached!
% 1.74/1.85 % (2164)------------------------------
% 1.74/1.85 % (2164)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.85 % (2164)Termination reason: Unknown
% 1.74/1.85 % (2164)Termination phase: Property scanning
% 1.74/1.85
% 1.74/1.85 % (2164)Memory used [KB]: 1918
% 1.74/1.85 % (2164)Time elapsed: 0.022 s
% 1.74/1.85 % (2164)Instructions burned: 53 (million)
% 1.74/1.85 % (2164)------------------------------
% 1.74/1.85 % (2164)------------------------------
% 1.74/1.85 % (2161)Instruction limit reached!
% 1.74/1.85 % (2161)------------------------------
% 1.74/1.85 % (2161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.85 % (2161)Termination reason: Unknown
% 1.74/1.85 % (2161)Termination phase: Saturation
% 1.74/1.85
% 1.74/1.85 % (2161)Memory used [KB]: 7547
% 1.74/1.85 % (2161)Time elapsed: 0.068 s
% 1.74/1.85 % (2161)Instructions burned: 270 (million)
% 1.74/1.85 % (2161)------------------------------
% 1.74/1.85 % (2161)------------------------------
% 1.74/1.85 % (2165)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_1970 for (2992ds/4436Mi)
% 1.74/1.86 % (2167)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_1970 for (2992ds/408Mi)
% 1.74/1.86 % (2166)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_1970 for (2992ds/1539Mi)
% 1.74/1.87 % (2159)Instruction limit reached!
% 1.74/1.87 % (2159)------------------------------
% 1.74/1.87 % (2159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.87 % (2159)Termination reason: Unknown
% 1.74/1.87 % (2159)Termination phase: Saturation
% 1.74/1.87
% 1.74/1.87 % (2159)Memory used [KB]: 11641
% 1.74/1.87 % (2159)Time elapsed: 0.107 s
% 1.74/1.87 % (2159)Instructions burned: 371 (million)
% 1.74/1.87 % (2159)------------------------------
% 1.74/1.87 % (2159)------------------------------
% 1.74/1.88 % (2168)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_1970 for (2992ds/5205Mi)
% 1.74/1.89 % (2148)Instruction limit reached!
% 1.74/1.89 % (2148)------------------------------
% 1.74/1.89 % (2148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.89 % (2148)Termination reason: Unknown
% 1.74/1.89 % (2148)Termination phase: Saturation
% 1.74/1.89
% 1.74/1.89 % (2148)Memory used [KB]: 7675
% 1.74/1.89 % (2148)Time elapsed: 0.241 s
% 1.74/1.89 % (2148)Instructions burned: 987 (million)
% 1.74/1.89 % (2148)------------------------------
% 1.74/1.89 % (2148)------------------------------
% 1.74/1.90 % (2169)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_1970 for (2992ds/1913Mi)
% 1.74/1.98 % (2167)Instruction limit reached!
% 1.74/1.98 % (2167)------------------------------
% 1.74/1.98 % (2167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.74/1.98 % (2167)Termination reason: Unknown
% 1.74/1.98 % (2167)Termination phase: Saturation
% 1.74/1.98
% 1.74/1.98 % (2167)Memory used [KB]: 11129
% 1.74/1.98 % (2167)Time elapsed: 0.120 s
% 1.74/1.98 % (2167)Instructions burned: 408 (million)
% 1.74/1.98 % (2167)------------------------------
% 1.74/1.98 % (2167)------------------------------
% 1.74/1.99 % (2170)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_1970 for (2991ds/4350Mi)
% 8.07/2.02 % (2155)Instruction limit reached!
% 8.07/2.02 % (2155)------------------------------
% 8.07/2.02 % (2155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.07/2.02 % (2155)Termination reason: Unknown
% 8.07/2.02 % (2155)Termination phase: Saturation
% 8.07/2.02
% 8.07/2.02 % (2155)Memory used [KB]: 12665
% 8.07/2.02 % (2155)Time elapsed: 0.286 s
% 8.07/2.02 % (2155)Instructions burned: 922 (million)
% 8.07/2.02 % (2155)------------------------------
% 8.07/2.02 % (2155)------------------------------
% 8.07/2.03 % (2171)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_1970 for (2990ds/68Mi)
% 8.07/2.05 % (2171)Instruction limit reached!
% 8.07/2.05 % (2171)------------------------------
% 8.07/2.05 % (2171)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.07/2.05 % (2171)Termination reason: Unknown
% 8.07/2.05 % (2171)Termination phase: Preprocessing 3
% 8.07/2.05
% 8.07/2.05 % (2171)Memory used [KB]: 3070
% 8.07/2.05 % (2171)Time elapsed: 0.022 s
% 8.07/2.05 % (2171)Instructions burned: 70 (million)
% 8.07/2.05 % (2171)------------------------------
% 8.07/2.05 % (2171)------------------------------
% 8.07/2.06 % (2172)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_1970 for (2990ds/358Mi)
% 8.07/2.07 % (2160)Instruction limit reached!
% 8.07/2.07 % (2160)------------------------------
% 8.07/2.07 % (2160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.07/2.07 % (2160)Termination reason: Unknown
% 8.07/2.07 % (2160)Termination phase: Saturation
% 8.07/2.07
% 8.07/2.07 % (2160)Memory used [KB]: 12537
% 8.07/2.07 % (2160)Time elapsed: 0.302 s
% 8.07/2.07 % (2160)Instructions burned: 1021 (million)
% 8.07/2.07 % (2160)------------------------------
% 8.07/2.07 % (2160)------------------------------
% 8.07/2.08 % (2173)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_1970 for (2990ds/31Mi)
% 8.07/2.09 % (2173)Instruction limit reached!
% 8.07/2.09 % (2173)------------------------------
% 8.07/2.09 % (2173)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.07/2.09 % (2173)Termination reason: Unknown
% 8.07/2.09 % (2173)Termination phase: Property scanning
% 8.07/2.09
% 8.07/2.09 % (2173)Memory used [KB]: 2174
% 8.07/2.09 % (2173)Time elapsed: 0.011 s
% 8.07/2.09 % (2173)Instructions burned: 31 (million)
% 8.07/2.09 % (2173)------------------------------
% 8.07/2.09 % (2173)------------------------------
% 8.07/2.10 % (2174)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_1970 for (2990ds/279Mi)
% 8.71/2.17 % (2174)Instruction limit reached!
% 8.71/2.17 % (2174)------------------------------
% 8.71/2.17 % (2174)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.17 % (2174)Termination reason: Unknown
% 8.71/2.17 % (2174)Termination phase: Saturation
% 8.71/2.17
% 8.71/2.17 % (2174)Memory used [KB]: 7547
% 8.71/2.17 % (2174)Time elapsed: 0.070 s
% 8.71/2.17 % (2174)Instructions burned: 280 (million)
% 8.71/2.17 % (2174)------------------------------
% 8.71/2.17 % (2174)------------------------------
% 8.71/2.17 % (2172)Instruction limit reached!
% 8.71/2.17 % (2172)------------------------------
% 8.71/2.17 % (2172)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.17 % (2172)Termination reason: Unknown
% 8.71/2.17 % (2172)Termination phase: Saturation
% 8.71/2.17
% 8.71/2.17 % (2172)Memory used [KB]: 9594
% 8.71/2.17 % (2172)Time elapsed: 0.114 s
% 8.71/2.17 % (2172)Instructions burned: 360 (million)
% 8.71/2.17 % (2172)------------------------------
% 8.71/2.17 % (2172)------------------------------
% 8.71/2.18 % (2169)First to succeed.
% 8.71/2.18 % (2175)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_1970 for (2989ds/258Mi)
% 8.71/2.18 % (2176)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_1970 for (2989ds/26Mi)
% 8.71/2.19 % (2176)Instruction limit reached!
% 8.71/2.19 % (2176)------------------------------
% 8.71/2.19 % (2176)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.19 % (2176)Termination reason: Unknown
% 8.71/2.19 % (2176)Termination phase: shuffling
% 8.71/2.19
% 8.71/2.19 % (2176)Memory used [KB]: 1791
% 8.71/2.19 % (2176)Time elapsed: 0.009 s
% 8.71/2.19 % (2176)Instructions burned: 28 (million)
% 8.71/2.19 % (2176)------------------------------
% 8.71/2.19 % (2176)------------------------------
% 9.38/2.20 % (2177)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_1970 for (2989ds/995Mi)
% 9.38/2.21 % (2169)Refutation found. Thanks to Tanya!
% 9.38/2.21 % SZS status Theorem for DTF2THF_1970
% 9.38/2.21 % SZS output start Proof for DTF2THF_1970
% 9.38/2.21 thf(type_def_5, type, term: $tType).
% 9.38/2.21 thf(type_def_7, type, subst: $tType).
% 9.38/2.21 thf(type_def_8, type, d_term: $tType).
% 9.38/2.21 thf(type_def_9, type, d_subst: $tType).
% 9.38/2.21 thf(func_def_0, type, term: $tType).
% 9.38/2.21 thf(func_def_1, type, subst: $tType).
% 9.38/2.21 thf(func_def_2, type, one: term).
% 9.38/2.21 thf(func_def_3, type, ap: term > term > term).
% 9.38/2.21 thf(func_def_4, type, lam: term > term).
% 9.38/2.21 thf(func_def_5, type, sub: term > subst > term).
% 9.38/2.21 thf(func_def_6, type, id: subst).
% 9.38/2.21 thf(func_def_7, type, sh: subst).
% 9.38/2.21 thf(func_def_8, type, push: term > subst > subst).
% 9.38/2.21 thf(func_def_9, type, comp: subst > subst > subst).
% 9.38/2.21 thf(func_def_10, type, var: term > $o).
% 9.38/2.21 thf(func_def_18, type, hoasap: subst > term > subst > term > term).
% 9.38/2.21 thf(func_def_73, type, hoaslam: subst > (subst > term > term) > term).
% 9.38/2.21 thf(func_def_86, type, hoasinduction_p_and_p_prime: (subst > term > subst > $o) > (term > $o) > $o).
% 9.38/2.21 thf(func_def_107, type, hoasvar: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_121, type, pushprop_p_and_p_prime: term > subst > (term > $o) > (term > $o) > $o).
% 9.38/2.21 thf(func_def_124, type, d_term: $tType).
% 9.38/2.21 thf(func_def_125, type, d_subst: $tType).
% 9.38/2.21 thf(func_def_126, type, d2term: d_term > term).
% 9.38/2.21 thf(func_def_127, type, d2subst: d_subst > subst).
% 9.38/2.21 thf(func_def_128, type, d_one: d_term).
% 9.38/2.21 thf(func_def_129, type, d_id: d_subst).
% 9.38/2.21 thf(func_def_142, type, sP0: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_145, type, sP3: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_147, type, sP5: (term > $o) > $o).
% 9.38/2.21 thf(func_def_148, type, sP6: (term > $o) > $o).
% 9.38/2.21 thf(func_def_151, type, sP9: (term > $o) > $o).
% 9.38/2.21 thf(func_def_152, type, sP10: (term > $o) > $o).
% 9.38/2.21 thf(func_def_155, type, sP13: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_156, type, sP14: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_158, type, sP16: (term > $o) > $o).
% 9.38/2.21 thf(func_def_159, type, sP17: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_160, type, sP18: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_165, type, sP23: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_168, type, sP26: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_170, type, sP28: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_172, type, sP30: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_173, type, sP31: (term > $o) > $o).
% 9.38/2.21 thf(func_def_174, type, sP32: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_175, type, sP33: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_176, type, sP34: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_181, type, sP39: (term > $o) > $o).
% 9.38/2.21 thf(func_def_182, type, sP40: (term > $o) > $o).
% 9.38/2.21 thf(func_def_185, type, sP43: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_188, type, sP46: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_189, type, sP47: (subst > term > subst > $o) > $o).
% 9.38/2.21 thf(func_def_190, type, sP48: (term > $o) > $o).
% 9.38/2.21 thf(func_def_196, type, sP54: (term > $o) > $o).
% 9.38/2.21 thf(func_def_201, type, sK59: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_202, type, sK60: term).
% 9.38/2.21 thf(func_def_203, type, sK61: term > $o).
% 9.38/2.21 thf(func_def_204, type, sK62: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_205, type, sK63: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_206, type, sK64: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_207, type, sK65: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_208, type, sK66: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_209, type, sK67: (term > $o) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_210, type, sK68: (term > $o) > term > term).
% 9.38/2.21 thf(func_def_211, type, sK69: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_212, type, sK70: term).
% 9.38/2.21 thf(func_def_213, type, sK71: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_214, type, sK72: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_215, type, sK73: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_216, type, sK74: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_217, type, sK75: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_218, type, sK76: term > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_219, type, sK77: term).
% 9.38/2.21 thf(func_def_220, type, sK78: subst > term > term).
% 9.38/2.21 thf(func_def_221, type, sK79: subst > term > term).
% 9.38/2.21 thf(func_def_222, type, sK80: subst).
% 9.38/2.21 thf(func_def_223, type, sK81: term).
% 9.38/2.21 thf(func_def_224, type, sK82: term).
% 9.38/2.21 thf(func_def_225, type, sK83: term).
% 9.38/2.21 thf(func_def_226, type, sK84: subst).
% 9.38/2.21 thf(func_def_227, type, sK85: term).
% 9.38/2.21 thf(func_def_228, type, sK86: subst).
% 9.38/2.21 thf(func_def_229, type, sK87: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_230, type, sK88: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_231, type, sK89: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_232, type, sK90: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_233, type, sK91: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_234, type, sK92: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_235, type, sK93: subst).
% 9.38/2.21 thf(func_def_236, type, sK94: term > $o).
% 9.38/2.21 thf(func_def_237, type, sK95: term).
% 9.38/2.21 thf(func_def_238, type, sK96: term > term).
% 9.38/2.21 thf(func_def_239, type, sK97: (term > $o) > term).
% 9.38/2.21 thf(func_def_240, type, sK98: (term > $o) > subst > term).
% 9.38/2.21 thf(func_def_241, type, sK99: (term > $o) > term).
% 9.38/2.21 thf(func_def_242, type, sK100: (term > $o) > term).
% 9.38/2.21 thf(func_def_243, type, sK101: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_244, type, sK102: term).
% 9.38/2.21 thf(func_def_245, type, sK103: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_246, type, sK104: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_247, type, sK105: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_248, type, sK106: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_249, type, sK107: term > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_250, type, sK108: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_251, type, sK109: term).
% 9.38/2.21 thf(func_def_252, type, sK110: (subst > term > subst > $o) > term > term).
% 9.38/2.21 thf(func_def_253, type, sK111: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_254, type, sK112: term).
% 9.38/2.21 thf(func_def_255, type, sK113: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_256, type, sK114: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_257, type, sK115: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_258, type, sK116: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_259, type, sK117: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_260, type, sK118: (term > $o) > term).
% 9.38/2.21 thf(func_def_261, type, sK119: (term > $o) > term).
% 9.38/2.21 thf(func_def_262, type, sK120: (subst > term > subst > $o) > term > $o).
% 9.38/2.21 thf(func_def_263, type, sK121: term).
% 9.38/2.21 thf(func_def_264, type, sK122: term).
% 9.38/2.21 thf(func_def_265, type, sK123: term > $o).
% 9.38/2.21 thf(func_def_266, type, sK124: term > term).
% 9.38/2.21 thf(func_def_267, type, sK125: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_268, type, sK126: (term > $o) > term).
% 9.38/2.21 thf(func_def_269, type, sK127: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_270, type, sK128: term).
% 9.38/2.21 thf(func_def_271, type, sK129: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_272, type, sK130: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_273, type, sK131: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_274, type, sK132: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_275, type, sK133: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_276, type, sK134: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_277, type, sK135: term > (term > $o) > term).
% 9.38/2.21 thf(func_def_278, type, sK136: (subst > term > subst > $o) > (term > $o) > term).
% 9.38/2.21 thf(func_def_279, type, sK137: term > $o).
% 9.38/2.21 thf(func_def_280, type, sK138: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_281, type, sK139: term).
% 9.38/2.21 thf(func_def_282, type, sK140: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_283, type, sK141: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_284, type, sK142: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_285, type, sK143: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_286, type, sK144: (term > $o) > term).
% 9.38/2.21 thf(func_def_287, type, sK145: (term > $o) > term).
% 9.38/2.21 thf(func_def_288, type, sK146: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_289, type, sK147: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_290, type, sK148: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_291, type, sK149: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_292, type, sK150: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_293, type, sK151: term).
% 9.38/2.21 thf(func_def_294, type, sK152: term).
% 9.38/2.21 thf(func_def_295, type, sK153: term > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_296, type, sK154: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_297, type, sK155: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_298, type, sK156: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_299, type, sK157: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_300, type, sK158: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_301, type, sK159: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_302, type, sK160: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_303, type, sK161: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_304, type, sK162: (subst > term > subst > $o) > term > term).
% 9.38/2.21 thf(func_def_305, type, sK163: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_306, type, sK164: term).
% 9.38/2.21 thf(func_def_307, type, sK165: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_308, type, sK166: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_309, type, sK167: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_310, type, sK168: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_311, type, sK169: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_312, type, sK170: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_313, type, sK171: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_314, type, sK172: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_315, type, sK173: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_316, type, sK174: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_317, type, sK175: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_318, type, sK176: term).
% 9.38/2.21 thf(func_def_319, type, sK177: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_320, type, sK178: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_321, type, sK179: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_322, type, sK180: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_323, type, sK181: term > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_324, type, sK182: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_325, type, sK183: term > $o).
% 9.38/2.21 thf(func_def_326, type, sK184: term).
% 9.38/2.21 thf(func_def_327, type, sK185: term > term).
% 9.38/2.21 thf(func_def_328, type, sK186: (term > $o) > term).
% 9.38/2.21 thf(func_def_329, type, sK187: (term > $o) > term).
% 9.38/2.21 thf(func_def_330, type, sK188: (term > $o) > term).
% 9.38/2.21 thf(func_def_331, type, sK189: (term > $o) > term).
% 9.38/2.21 thf(func_def_332, type, sK190: (term > $o) > term).
% 9.38/2.21 thf(func_def_333, type, sK191: (term > $o) > term).
% 9.38/2.21 thf(func_def_334, type, sK192: (term > $o) > term).
% 9.38/2.21 thf(func_def_335, type, sK193: (term > $o) > term).
% 9.38/2.21 thf(func_def_336, type, sK194: term).
% 9.38/2.21 thf(func_def_337, type, sK195: term > $o).
% 9.38/2.21 thf(func_def_338, type, sK196: term > term).
% 9.38/2.21 thf(func_def_339, type, sK197: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_340, type, sK198: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_341, type, sK199: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_342, type, sK200: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_343, type, sK201: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_344, type, sK202: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_345, type, sK203: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_346, type, sK204: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_347, type, sK205: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_348, type, sK206: term).
% 9.38/2.21 thf(func_def_349, type, sK207: term).
% 9.38/2.21 thf(func_def_350, type, sK208: term).
% 9.38/2.21 thf(func_def_351, type, sK209: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_352, type, sK210: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_353, type, sK211: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_354, type, sK212: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_355, type, sK213: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_356, type, sK214: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_357, type, sK215: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_358, type, sK216: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_359, type, sK217: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_360, type, sK218: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_361, type, sK219: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_362, type, sK220: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_363, type, sK221: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_364, type, sK222: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_365, type, sK223: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_366, type, sK224: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_367, type, sK225: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_368, type, sK226: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_369, type, sK227: (subst > term > subst > $o) > term > term).
% 9.38/2.21 thf(func_def_370, type, sK228: term).
% 9.38/2.21 thf(func_def_371, type, sK229: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_372, type, sK230: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_373, type, sK231: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_374, type, sK232: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_375, type, sK233: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_376, type, sK234: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_377, type, sK235: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_378, type, sK236: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_379, type, sK237: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_380, type, sK238: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_381, type, sK239: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_382, type, sK240: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_383, type, sK241: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_384, type, sK242: (term > $o) > term).
% 9.38/2.21 thf(func_def_385, type, sK243: (term > $o) > term).
% 9.38/2.21 thf(func_def_386, type, sK244: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_387, type, sK245: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_388, type, sK246: term).
% 9.38/2.21 thf(func_def_389, type, sK247: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_390, type, sK248: term > $o).
% 9.38/2.21 thf(func_def_391, type, sK249: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_392, type, sK250: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_393, type, sK251: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_394, type, sK252: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_395, type, sK253: (term > $o) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_396, type, sK254: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_397, type, sK255: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_398, type, sK256: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_399, type, sK257: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_400, type, sK258: (term > $o) > term > term).
% 9.38/2.21 thf(func_def_401, type, sK259: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_402, type, sK260: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_403, type, sK261: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_404, type, sK262: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_405, type, sK263: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_406, type, sK264: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_407, type, sK265: term).
% 9.38/2.21 thf(func_def_408, type, sK266: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_409, type, sK267: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_410, type, sK268: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_411, type, sK269: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_412, type, sK270: term > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_413, type, sK271: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_414, type, sK272: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_415, type, sK273: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_416, type, sK274: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_417, type, sK275: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_418, type, sK276: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_419, type, sK277: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_420, type, sK278: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_421, type, sK279: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_422, type, sK280: term).
% 9.38/2.21 thf(func_def_423, type, sK281: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_424, type, sK282: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_425, type, sK283: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_426, type, sK284: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_427, type, sK285: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_428, type, sK286: (subst > term > subst > $o) > term > term).
% 9.38/2.21 thf(func_def_429, type, sK287: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_430, type, sK288: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_431, type, sK289: term).
% 9.38/2.21 thf(func_def_432, type, sK290: term > $o).
% 9.38/2.21 thf(func_def_433, type, sK291: term).
% 9.38/2.21 thf(func_def_434, type, sK292: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_435, type, sK293: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_436, type, sK294: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_437, type, sK295: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_438, type, sK296: (term > $o) > term > term).
% 9.38/2.21 thf(func_def_439, type, sK297: (term > $o) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_440, type, sK298: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_441, type, sK299: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_442, type, sK300: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_443, type, sK301: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_444, type, sK302: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_445, type, sK303: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_446, type, sK304: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_447, type, sK305: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_448, type, sK306: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_449, type, sK307: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_450, type, sK308: term).
% 9.38/2.21 thf(func_def_451, type, sK309: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_452, type, sK310: (subst > term > term) > term > subst).
% 9.38/2.21 thf(func_def_453, type, sK311: (subst > term > term) > term > term).
% 9.38/2.21 thf(func_def_454, type, sK312: (subst > term > term) > term > subst).
% 9.38/2.21 thf(func_def_455, type, sK313: (subst > term > term) > term > term).
% 9.38/2.21 thf(func_def_456, type, sK314: (subst > term > term) > term > subst).
% 9.38/2.21 thf(func_def_457, type, sK315: (subst > term > term) > term > subst).
% 9.38/2.21 thf(func_def_458, type, sK316: term > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_459, type, sK317: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_460, type, sK318: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_461, type, sK319: (subst > term > subst > $o) > term > $o).
% 9.38/2.21 thf(func_def_462, type, sK320: (term > $o) > term).
% 9.38/2.21 thf(func_def_463, type, sK321: (term > $o) > term).
% 9.38/2.21 thf(func_def_464, type, sK322: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_465, type, sK323: (term > $o) > term).
% 9.38/2.21 thf(func_def_466, type, sK324: term > $o).
% 9.38/2.21 thf(func_def_467, type, sK325: term).
% 9.38/2.21 thf(func_def_468, type, sK326: term > term).
% 9.38/2.21 thf(func_def_469, type, sK327: term).
% 9.38/2.21 thf(func_def_470, type, sK328: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_471, type, sK329: term).
% 9.38/2.21 thf(func_def_472, type, sK330: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_473, type, sK331: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_474, type, sK332: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_475, type, sK333: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_476, type, sK334: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_477, type, sK335: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_478, type, sK336: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_479, type, sK337: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_480, type, sK338: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_481, type, sK339: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_482, type, sK340: (term > $o) > term > term).
% 9.38/2.21 thf(func_def_483, type, sK341: (term > $o) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_484, type, sK342: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_485, type, sK343: term).
% 9.38/2.21 thf(func_def_486, type, sK344: term > $o).
% 9.38/2.21 thf(func_def_487, type, sK345: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_488, type, sK346: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_489, type, sK347: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_490, type, sK348: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_491, type, sK349: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_492, type, sK350: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_493, type, sK351: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_494, type, sK352: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_495, type, sK353: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_496, type, sK354: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_497, type, sK355: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_498, type, sK356: (term > $o) > term).
% 9.38/2.21 thf(func_def_499, type, sK357: (term > $o) > term).
% 9.38/2.21 thf(func_def_500, type, sK358: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_501, type, sK359: term).
% 9.38/2.21 thf(func_def_502, type, sK360: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_503, type, sK361: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_504, type, sK362: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_505, type, sK363: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_506, type, sK364: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_507, type, sK365: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_508, type, sK366: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_509, type, sK367: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_510, type, sK368: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_511, type, sK369: (subst > term > term) > (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_512, type, sK370: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_513, type, sK371: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_514, type, sK372: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_515, type, sK373: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_516, type, sK374: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_517, type, sK375: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_518, type, sK376: subst > (term > $o) > term).
% 9.38/2.21 thf(func_def_519, type, sK377: (term > $o) > term).
% 9.38/2.21 thf(func_def_520, type, sK378: term).
% 9.38/2.21 thf(func_def_521, type, sK379: term).
% 9.38/2.21 thf(func_def_522, type, sK380: term > $o).
% 9.38/2.21 thf(func_def_523, type, sK381: subst).
% 9.38/2.21 thf(func_def_524, type, sK382: term > term).
% 9.38/2.21 thf(func_def_525, type, sK383: term).
% 9.38/2.21 thf(func_def_526, type, sK384: term > $o).
% 9.38/2.21 thf(func_def_527, type, sK385: term > term).
% 9.38/2.21 thf(func_def_528, type, sK386: (term > $o) > term).
% 9.38/2.21 thf(func_def_529, type, sK387: (term > $o) > term).
% 9.38/2.21 thf(func_def_530, type, sK388: (term > $o) > term).
% 9.38/2.21 thf(func_def_531, type, sK389: (term > $o) > term).
% 9.38/2.21 thf(func_def_532, type, sK390: (term > $o) > term).
% 9.38/2.21 thf(func_def_533, type, sK391: (term > $o) > term).
% 9.38/2.21 thf(func_def_534, type, sK392: (term > $o) > term).
% 9.38/2.21 thf(func_def_535, type, sK393: (term > $o) > term).
% 9.38/2.21 thf(func_def_536, type, sK394: term > $o).
% 9.38/2.21 thf(func_def_537, type, sK395: term).
% 9.38/2.21 thf(func_def_538, type, sK396: subst).
% 9.38/2.21 thf(func_def_539, type, sK397: term).
% 9.38/2.21 thf(func_def_540, type, sK398: term).
% 9.38/2.21 thf(func_def_541, type, sK399: subst).
% 9.38/2.21 thf(func_def_542, type, sK400: subst).
% 9.38/2.21 thf(func_def_543, type, sK401: term > $o).
% 9.38/2.21 thf(func_def_544, type, sK402: term).
% 9.38/2.21 thf(func_def_545, type, sK403: subst).
% 9.38/2.21 thf(func_def_546, type, sK404: term > term).
% 9.38/2.21 thf(func_def_547, type, sK405: (term > $o) > term).
% 9.38/2.21 thf(func_def_548, type, sK406: subst > (term > $o) > term).
% 9.38/2.21 thf(func_def_549, type, sK407: (term > $o) > term).
% 9.38/2.21 thf(func_def_550, type, sK408: (term > $o) > term).
% 9.38/2.21 thf(func_def_551, type, sK409: (term > $o) > term).
% 9.38/2.21 thf(func_def_552, type, sK410: (term > $o) > term).
% 9.38/2.21 thf(func_def_553, type, sK411: term > $o).
% 9.38/2.21 thf(func_def_554, type, sK412: term).
% 9.38/2.21 thf(func_def_555, type, sK413: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_556, type, sK414: term).
% 9.38/2.21 thf(func_def_557, type, sK415: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_558, type, sK416: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_559, type, sK417: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_560, type, sK418: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_561, type, sK419: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_562, type, sK420: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_563, type, sK421: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_564, type, sK422: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_565, type, sK423: (subst > term > subst > $o) > (term > $o) > term).
% 9.38/2.21 thf(func_def_566, type, sK424: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_567, type, sK425: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_568, type, sK426: term).
% 9.38/2.21 thf(func_def_569, type, sK427: term > $o).
% 9.38/2.21 thf(func_def_570, type, sK428: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_571, type, sK429: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_572, type, sK430: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_573, type, sK431: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_574, type, sK432: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_575, type, sK433: term > subst > term > term).
% 9.38/2.21 thf(func_def_576, type, sK434: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_577, type, sK435: (subst > term > subst > $o) > (term > $o) > term).
% 9.38/2.21 thf(func_def_578, type, sK436: (term > $o) > term > term).
% 9.38/2.21 thf(func_def_579, type, sK437: term).
% 9.38/2.21 thf(func_def_580, type, sK438: term).
% 9.38/2.21 thf(func_def_581, type, sK439: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_582, type, sK440: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_583, type, sK441: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_584, type, sK442: term).
% 9.38/2.21 thf(func_def_585, type, sK443: term).
% 9.38/2.21 thf(func_def_586, type, sK444: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_587, type, sK445: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_588, type, sK446: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_589, type, sK447: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_590, type, sK448: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_591, type, sK449: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_592, type, sK450: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_593, type, sK451: (subst > term > subst > $o) > subst).
% 9.38/2.21 thf(func_def_594, type, sK452: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_595, type, sK453: (subst > term > subst > $o) > term).
% 9.38/2.21 thf(func_def_596, type, sK454: term).
% 9.38/2.21 thf(func_def_597, type, sK455: term).
% 9.38/2.21 thf(func_def_598, type, sK456: term).
% 9.38/2.21 thf(func_def_599, type, sK457: term).
% 9.38/2.21 thf(func_def_600, type, sK458: term).
% 9.38/2.21 thf(func_def_601, type, sK459: subst).
% 9.38/2.21 thf(func_def_602, type, sK460: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_603, type, sK461: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_604, type, sK462: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_605, type, sK463: term).
% 9.38/2.21 thf(func_def_606, type, sK464: term).
% 9.38/2.21 thf(func_def_607, type, sK465: subst > term > term).
% 9.38/2.21 thf(func_def_608, type, sK466: term).
% 9.38/2.21 thf(func_def_609, type, sK467: term).
% 9.38/2.21 thf(func_def_610, type, sK468: term).
% 9.38/2.21 thf(func_def_611, type, sK469: term).
% 9.38/2.21 thf(func_def_612, type, sK470: term).
% 9.38/2.21 thf(func_def_613, type, sK471: term).
% 9.38/2.21 thf(func_def_614, type, sK472: term).
% 9.38/2.21 thf(func_def_615, type, sK473: term).
% 9.38/2.21 thf(func_def_616, type, sK474: term).
% 9.38/2.21 thf(func_def_617, type, sK475: term).
% 9.38/2.21 thf(func_def_618, type, sK476: term).
% 9.38/2.21 thf(func_def_619, type, sK477: term).
% 9.38/2.21 thf(func_def_620, type, sK478: term).
% 9.38/2.21 thf(func_def_621, type, sK479: (subst > term > subst > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_622, type, sK480: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_623, type, sK481: term).
% 9.38/2.21 thf(func_def_624, type, sK482: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_625, type, sK483: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_626, type, sK484: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_627, type, sK485: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_628, type, sK486: term).
% 9.38/2.21 thf(func_def_629, type, sK487: (term > $o) > term > subst > term > $o).
% 9.38/2.21 thf(func_def_630, type, sK488: subst).
% 9.38/2.21 thf(func_def_631, type, sK489: term).
% 9.38/2.21 thf(func_def_632, type, sK490: term > $o).
% 9.38/2.21 thf(func_def_633, type, sK491: (term > $o) > term).
% 9.38/2.21 thf(func_def_634, type, sK492: (subst > term > subst > $o) > term > $o).
% 9.38/2.21 thf(func_def_635, type, sK493: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_636, type, sK494: (term > $o) > term).
% 9.38/2.21 thf(func_def_637, type, sK495: subst > term > subst > $o).
% 9.38/2.21 thf(func_def_638, type, sK496: (term > $o) > term).
% 9.38/2.21 thf(func_def_639, type, sK497: (subst > term > subst > $o) > term > $o).
% 9.38/2.21 thf(func_def_640, type, sK498: term).
% 9.38/2.21 thf(func_def_641, type, sK499: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_642, type, sK500: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_643, type, sK501: term > subst > term > term).
% 9.38/2.21 thf(func_def_644, type, sK502: (term > $o) > term).
% 9.38/2.21 thf(func_def_645, type, sK503: (term > $o) > term).
% 9.38/2.21 thf(func_def_646, type, sK504: term).
% 9.38/2.21 thf(func_def_647, type, sK505: term > $o).
% 9.38/2.21 thf(func_def_648, type, sK506: term > $o).
% 9.38/2.21 thf(func_def_649, type, sK507: term).
% 9.38/2.21 thf(func_def_650, type, sK508: subst).
% 9.38/2.21 thf(func_def_651, type, sK509: (term > $o) > term).
% 9.38/2.21 thf(func_def_652, type, sK510: subst > term > (term > $o) > term > $o).
% 9.38/2.21 thf(func_def_653, type, sK511: subst).
% 9.38/2.21 thf(func_def_654, type, sK512: subst).
% 9.38/2.21 thf(func_def_655, type, sK513: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_656, type, sK514: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_657, type, sK515: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_658, type, sK516: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_659, type, sK517: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_660, type, sK518: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_661, type, sK519: term).
% 9.38/2.21 thf(func_def_662, type, sK520: subst).
% 9.38/2.21 thf(func_def_663, type, sK521: subst > term > term).
% 9.38/2.21 thf(func_def_664, type, sK522: subst > term > term).
% 9.38/2.21 thf(func_def_665, type, sK523: term).
% 9.38/2.21 thf(func_def_666, type, sK524: term).
% 9.38/2.21 thf(func_def_667, type, sK525: term).
% 9.38/2.21 thf(func_def_668, type, sK526: term).
% 9.38/2.21 thf(func_def_669, type, sK527: term).
% 9.38/2.21 thf(func_def_670, type, sK528: subst).
% 9.38/2.21 thf(func_def_671, type, sK529: term > $o).
% 9.38/2.21 thf(func_def_672, type, sK530: term > $o).
% 9.38/2.21 thf(func_def_673, type, sK531: (term > $o) > (term > $o) > subst > term > term).
% 9.38/2.21 thf(func_def_674, type, sK532: term).
% 9.38/2.21 thf(func_def_675, type, sK533: subst).
% 9.38/2.21 thf(func_def_676, type, sK534: term).
% 9.38/2.21 thf(func_def_677, type, sK535: term > $o).
% 9.38/2.21 thf(func_def_678, type, sK536: term > $o).
% 9.38/2.21 thf(func_def_679, type, sK537: subst > d_subst).
% 9.38/2.21 thf(func_def_680, type, sK538: term).
% 9.38/2.21 thf(func_def_681, type, sK539: term).
% 9.38/2.21 thf(func_def_682, type, sK540: term > $o).
% 9.38/2.21 thf(func_def_683, type, sK541: term).
% 9.38/2.21 thf(func_def_684, type, sK542: term > $o).
% 9.38/2.21 thf(func_def_685, type, sK543: subst).
% 9.38/2.21 thf(func_def_686, type, sK544: subst > (term > $o) > term > (term > $o) > term).
% 9.38/2.21 thf(func_def_687, type, sK545: subst).
% 9.38/2.21 thf(func_def_688, type, sK546: term).
% 9.38/2.21 thf(func_def_689, type, sK547: subst).
% 9.38/2.21 thf(func_def_690, type, sK548: subst).
% 9.38/2.21 thf(func_def_691, type, sK549: subst).
% 9.38/2.21 thf(func_def_692, type, sK550: subst).
% 9.38/2.21 thf(func_def_693, type, sK551: subst).
% 9.38/2.21 thf(func_def_694, type, sK552: term).
% 9.38/2.21 thf(func_def_695, type, sK553: term).
% 9.38/2.21 thf(func_def_696, type, sK554: term > subst > term > term).
% 9.38/2.21 thf(func_def_697, type, sK555: subst).
% 9.38/2.21 thf(func_def_698, type, sK556: term).
% 9.38/2.21 thf(func_def_699, type, sK557: subst).
% 9.38/2.21 thf(func_def_700, type, sK558: term).
% 9.38/2.21 thf(func_def_701, type, sK559: term).
% 9.38/2.21 thf(func_def_702, type, sK560: term > $o).
% 9.38/2.21 thf(func_def_703, type, sK561: term).
% 9.38/2.21 thf(func_def_704, type, sK562: subst).
% 9.38/2.21 thf(func_def_705, type, sK563: term > $o).
% 9.38/2.21 thf(func_def_706, type, sK564: term).
% 9.38/2.21 thf(func_def_707, type, sK565: term).
% 9.38/2.21 thf(func_def_708, type, sK566: term).
% 9.38/2.21 thf(func_def_709, type, sK567: term).
% 9.38/2.21 thf(func_def_710, type, sK568: term > subst > term > term).
% 9.38/2.21 thf(func_def_711, type, sK569: term).
% 9.38/2.21 thf(func_def_712, type, sK570: subst).
% 9.38/2.21 thf(func_def_713, type, sK571: term).
% 9.38/2.21 thf(func_def_714, type, sK572: term).
% 9.38/2.21 thf(func_def_715, type, sK573: term).
% 9.38/2.21 thf(func_def_716, type, sK574: term).
% 9.38/2.21 thf(func_def_717, type, sK575: subst).
% 9.38/2.21 thf(func_def_718, type, sK576: term > d_term).
% 9.38/2.21 thf(func_def_719, type, sK577: term).
% 9.38/2.21 thf(func_def_720, type, sK578: subst).
% 9.38/2.21 thf(func_def_721, type, sK579: term).
% 9.38/2.21 thf(func_def_722, type, sK580: term).
% 9.38/2.21 thf(func_def_723, type, sK581: term).
% 9.38/2.21 thf(func_def_724, type, sK582: term).
% 9.38/2.21 thf(func_def_725, type, sK583: term).
% 9.38/2.21 thf(func_def_726, type, sK584: term).
% 9.38/2.21 thf(func_def_727, type, sK585: term).
% 9.38/2.21 thf(func_def_728, type, sK586: term).
% 9.38/2.21 thf(func_def_729, type, sK587: term).
% 9.38/2.21 thf(func_def_730, type, sK588: (subst > term > term) > subst).
% 9.38/2.21 thf(func_def_731, type, sK589: (subst > term > term) > term).
% 9.38/2.21 thf(func_def_732, type, sK590: term > subst > term > term).
% 9.38/2.21 thf(func_def_736, type, sK591: term).
% 9.38/2.21 thf(func_def_737, type, sK592: term > $o).
% 9.38/2.21 thf(func_def_738, type, sK593: term > $o).
% 9.38/2.21 thf(func_def_739, type, sK594: subst).
% 9.38/2.21 thf(func_def_740, type, sK595: (term > $o) > term > subst > (term > $o) > term).
% 9.38/2.21 thf(func_def_741, type, ph596: !>[X0: $tType]:(X0)).
% 9.38/2.21 thf(f4114,plain,(
% 9.38/2.21 $false),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f4087])).
% 9.38/2.21 thf(f4087,plain,(
% 9.38/2.21 ($true = $false)),
% 9.38/2.21 inference(superposition,[status(thm)],[f4080,f1938])).
% 9.38/2.21 thf(f1938,plain,(
% 9.38/2.21 ( ! [X0 : term] : (($true = (sK530 @ X0))) )),
% 9.38/2.21 inference(superposition,[status(thm)],[f1918,f1880])).
% 9.38/2.21 thf(f1880,plain,(
% 9.38/2.21 ( ! [X0 : term,X1 : term] : ((X0 = X1)) )),
% 9.38/2.21 inference(superposition,[status(thm)],[f1874,f1874])).
% 9.38/2.21 thf(f1874,plain,(
% 9.38/2.21 ( ! [X250 : term] : (((d2term @ d_one) = X250)) )),
% 9.38/2.21 inference(forward_demodulation,[status(thm)],[f1064,f1230])).
% 9.38/2.21 thf(f1230,plain,(
% 9.38/2.21 ( ! [X13 : d_term] : ((d_one = X13)) )),
% 9.38/2.21 inference(cnf_transformation,[status(thm)],[f545])).
% 9.38/2.21 thf(f545,plain,(
% 9.38/2.21 (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ! [X0 : d_subst] : (d_id = X0) & (hoasinduction_lem1_gthm = $true) & ((shinj = $true) | (((sub @ sK455 @ sh) = (sub @ sK454 @ sh)) & (sK455 != sK454))) & (! [X3 : term,X4 : term] : (((sub @ X3 @ sh) != (sub @ X4 @ sh)) | (X3 = X4)) | (shinj != $true)) & (lamnotvar = $true) & (((sub @ sK456 @ id) != sK456) | ! [X6 : term] : ((sub @ X6 @ id) = X6) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X7 : term] : ((sub @ X7 @ id) = X7) & ((sub @ sK457 @ id) != sK457))) & (($true = sP58) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP58)) & ((axvarid = $true) | ((sub @ sK458 @ id) != sK458)) & (! [X10 : term] : ((sub @ X10 @ id) = X10) | (axvarid != $true)) & (hoaslamnotvar_lthm = $true) & (apnotvar = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP57)) & (($true = sP57) | (hoasinduction_lem3aa != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) | ($true != sP56)) & (($true = sP56) | (hoaslaminj_lthm != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((axidr = $true) | ((comp @ sK459 @ id) != sK459)) & (! [X12 : subst] : ((comp @ X12 @ id) = X12) | (axidr != $true)) & (($true = sP55) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP55)) & ! [X13 : d_term] : (d_one = X13) & (hoasinduction_lem3_gthm = $true) & ((sP53 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP53 != $true)) & (axapp = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP51 != $true)) & ((sP51 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoasapnotvar = $true) & (! [X14 : term,X15 : term,X16 : subst > term > term] : (((lam @ (X16 @ sh @ one)) != (ap @ (sub @ X15 @ id) @ X14)) | ((X16 @ (comp @ (sK461 @ X16) @ (sK460 @ X16)) @ (sub @ (sK462 @ X16) @ (sK460 @ X16))) != (sub @ (X16 @ (sK461 @ X16) @ (sK462 @ X16)) @ (sK460 @ X16)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | (((ap @ (sub @ sK464 @ id) @ sK463) = (lam @ (sK465 @ sh @ one))) & ! [X23 : subst,X24 : subst,X25 : term] : ((sK465 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (sK465 @ X24 @ X25) @ X23)))) & (axclos = $true) & (((sK466 != sK469) & ((ap @ sK466 @ sK467) = (ap @ sK469 @ sK468))) | ! [X30 : term,X31 : term,X32 : term,X33 : term] : ((X32 = X33) | ((ap @ (sub @ X32 @ id) @ X30) != (ap @ (sub @ X33 @ id) @ X31))) | (sK470 != (sub @ sK470 @ id)) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X35 : term,X36 : term,X37 : term,X38 : term] : ((X35 = X38) | ((ap @ X35 @ X36) != (ap @ X38 @ X37))) & ((sK474 != sK473) & ((ap @ (sub @ sK474 @ id) @ sK472) = (ap @ (sub @ sK473 @ id) @ sK471))) & ! [X43 : term] : ((sub @ X43 @ id) = X43))) & (ulamvar1 != $true) & (! [X44 : term,X45 : term,X46 : term,X47 : term] : (((ap @ (sub @ X44 @ id) @ X46) != (ap @ (sub @ X47 @ id) @ X45)) | (X44 = X47)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | (((ap @ (sub @ sK478 @ id) @ sK476) = (ap @ (sub @ sK475 @ id) @ sK477)) & (sK478 != sK475))) & (! [X52 : subst > term > subst > $o,X53 : term] : ((! [X55 : term] : (($true = (X52 @ id @ (sK479 @ X52 @ id @ X55) @ id)) | ($true != (X52 @ id @ X55 @ id))) & ($true != (X52 @ id @ (lam @ (sK479 @ X52 @ sh @ one)) @ id)) & ! [X56 : term,X57 : subst,X58 : subst] : ((sub @ (sK479 @ X52 @ X58 @ X56) @ X57) = (sK479 @ X52 @ (comp @ X58 @ X57) @ (sub @ X56 @ X57)))) | ($true = (sP46 @ X52)) | ($true = (X52 @ id @ X53 @ id))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | (! [X61 : subst > term > term] : ((((sK480 @ id @ (X61 @ id @ (sK482 @ X61)) @ id) != $true) & ($true = (sK480 @ id @ (sK482 @ X61) @ id))) | ($true = (sK480 @ id @ (lam @ (X61 @ sh @ one)) @ id)) | ((X61 @ (comp @ (sK485 @ X61) @ (sK484 @ X61)) @ (sub @ (sK483 @ X61) @ (sK484 @ X61))) != (sub @ (X61 @ (sK485 @ X61) @ (sK483 @ X61)) @ (sK484 @ X61)))) & ($true != (sP46 @ sK480)) & ((sK480 @ id @ sK481 @ id) != $true))) & (! [X66 : term] : ((sub @ X66 @ id) = X66) | (termmset != $true)) & ((termmset = $true) | (sK486 != (sub @ sK486 @ id))) & (induction2_gthm = $true) & (pushprop = $true) & (hoasinduction_lem2_gthm = $true) & (! [X68 : subst,X69 : term,X70 : term > $o] : ! [X72 : term] : ((sK487 @ X70 @ X69 @ X68 @ X72) = (X70 @ (sub @ X72 @ (push @ X69 @ X68)))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X76 : term > $o] : ((X76 @ (sK491 @ X76)) != (sK490 @ (sub @ (sK491 @ X76) @ (push @ sK489 @ sK488))))) & ! [X78 : d_term,X79 : d_term] : ((X78 = X79) | ((d2term @ X79) != (d2term @ X78))) & ((var @ (d2term @ d_one)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (pushprop_lem2v2_gthm = $true) & (! [X80 : subst > term > subst > $o] : ! [X82 : term] : ((sK492 @ X80 @ X82) = (X80 @ id @ X82 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X84 : term > $o] : ((sK493 @ id @ (sK494 @ X84) @ id) != (X84 @ (sK494 @ X84)))) & (($true = sP45) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP45)) & (($true = sP42) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP42)) & (hoasapnotvar_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lem0 = $true) | ! [X87 : term > $o] : ((X87 @ (sK496 @ X87)) != (sK495 @ id @ (sK496 @ X87) @ id))) & (! [X89 : subst > term > subst > $o] : ! [X91 : term] : ((X89 @ id @ X91 @ id) = (sK497 @ X89 @ X91)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X93 : subst > term > term] : ((sub @ sK498 @ (push @ (sK499 @ X93) @ (sK500 @ X93))) != (X93 @ (sK500 @ X93) @ (sK499 @ X93)))) & (! [X96 : term] : ! [X98 : term,X99 : subst] : ((sub @ X96 @ (push @ X98 @ X99)) = (sK501 @ X96 @ X99 @ X98)) | (hoasinduction_lem3v2_f != $true)) & (lamnotap != $true) & (one = (d2term @ d_one)) & (axassoc = $true) & ((induction2 = $true) | ($true != sP41)) & (($true = sP41) | (induction2 != $true)) & (! [X100 : term,X101 : term > $o] : ((((X101 @ (sK502 @ X101)) = $true) & ($true != (X101 @ (lam @ (sK502 @ X101))))) | ((X101 @ X100) = $true) | ((sP39 @ X101) = $true) | (($true != (X101 @ (sK503 @ X101))) & ($true = (var @ (sK503 @ X101))))) | (induction != $true)) & ((induction = $true) | (! [X106 : term] : (($true != (sK505 @ X106)) | ($true = (sK505 @ (lam @ X106)))) & ($true != (sK505 @ sK504)) & ((sP39 @ sK505) != $true) & ! [X107 : term] : (($true = (sK505 @ X107)) | ($true != (var @ X107))))) & ((pushprop_lem0_lthm = $true) | ! [X111 : term > $o] : ((sK506 @ (sub @ (sK509 @ X111) @ (push @ sK507 @ sK508))) != (X111 @ (sK509 @ X111)))) & (! [X113 : term > $o,X114 : term,X115 : subst] : ! [X117 : term] : ((X113 @ (sub @ X117 @ (push @ X114 @ X115))) = (sK510 @ X115 @ X114 @ X113 @ X117)) | (pushprop_lem0_lthm != $true)) & (hoasinduction_lem1v2 = $true) & (pushprop_lthm = $true) & ((hoasinduction_lthm = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lthm != $true)) & ((! [X118 : subst] : ((comp @ X118 @ id) = X118) & ! [X119 : subst] : ((comp @ id @ X119) = X119)) | (substmonoid != $true)) & ((substmonoid = $true) | ((comp @ sK511 @ id) != sK511) | (sK512 != (comp @ id @ sK512))) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2 != $true)) & (! [X122 : term,X123 : subst,X124 : subst > term > term,X125 : subst > term > term] : (((X124 @ X123 @ X122) = (X125 @ X123 @ X122)) | ((lam @ (X124 @ sh @ one)) != (lam @ (X125 @ sh @ one))) | ((X124 @ (comp @ (sK515 @ X124) @ (sK514 @ X124)) @ (sub @ (sK513 @ X124) @ (sK514 @ X124))) != (sub @ (X124 @ (sK515 @ X124) @ (sK513 @ X124)) @ (sK514 @ X124))) | ((X125 @ (comp @ (sK516 @ X125) @ (sK518 @ X125)) @ (sub @ (sK517 @ X125) @ (sK518 @ X125))) != (sub @ (X125 @ (sK516 @ X125) @ (sK517 @ X125)) @ (sK518 @ X125)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | (((sK522 @ sK520 @ sK519) != (sK521 @ sK520 @ sK519)) & ((lam @ (sK522 @ sh @ one)) = (lam @ (sK521 @ sh @ one))) & ! [X136 : term,X137 : subst,X138 : subst] : ((sK521 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137)) = (sub @ (sK521 @ X138 @ X136) @ X137)) & ! [X139 : subst,X140 : term,X141 : subst] : ((sK522 @ (comp @ X139 @ X141) @ (sub @ X140 @ X141)) = (sub @ (sK522 @ X139 @ X140) @ X141)))) & ((d2subst @ d_id) = sh) & (hoasinduction_lem1 = $true) & (($true = sP27) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP27)) & ((hoasapinj2 = $true) | (((ap @ (sub @ sK523 @ id) @ sK526) = (ap @ (sub @ sK524 @ id) @ sK525)) & (sK526 != sK525))) & (! [X146 : term,X147 : term,X148 : term,X149 : term] : (((ap @ (sub @ X147 @ id) @ X148) != (ap @ (sub @ X146 @ id) @ X149)) | (X148 = X149)) | (hoasapinj2 != $true)) & ((pushprop_lem1v2 = $true) | (($true = (sK530 @ sK527)) & ! [X154 : term] : ((sK530 @ (sub @ X154 @ (push @ sK527 @ sK528))) = (sK529 @ X154)) & ((sK529 @ one) != $true))) & (! [X155 : term,X156 : subst,X157 : term > $o,X158 : term > $o] : (($true != (X158 @ X155)) | ((X158 @ (sub @ (sK531 @ X158 @ X157 @ X156 @ X155) @ (push @ X155 @ X156))) != (X157 @ (sK531 @ X158 @ X157 @ X156 @ X155))) | ($true = (X157 @ one))) | (pushprop_lem1v2 != $true)) & (hoasapinj2_gthm = $true) & ((pushprop_lem1 = $true) | (((sK536 @ sK534) = $true) & ($true != (sK535 @ (sub @ sK534 @ (push @ sK532 @ sK533)))))) & (! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true != (X169 @ X167)) | ($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166))))) | (pushprop_lem1 != $true)) & (pushprop_lem3v2_lthm = $true) & (($true = sP25) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP25)) & (termmset_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP24)) & (($true = sP24) | (hoasinduction_lem3v2_lthm != $true)) & ! [X170 : subst] : ((d2subst @ (sK537 @ X170)) = X170) & (axvarshift = $true) & (hoasapnotvar_lthm = $true) & (! [X172 : term,X173 : term] : ((X172 = X173) | ((lam @ X172) != (lam @ X173))) | (laminj != $true)) & ((laminj = $true) | ((sK539 != sK538) & ((lam @ sK539) = (lam @ sK538)))) & ((hoasinduction_lem3aaa = $true) | ($true != sP22)) & (($true = sP22) | (hoasinduction_lem3aaa != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X176 : d_subst,X177 : d_subst] : ((X176 = X177) | ((d2subst @ X177) != (d2subst @ X176))) & (($true = sP21) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP21)) & ((pushprop_lem1v2_lthm = $true) | ((($true = (sK542 @ sK541)) & ($true != (sK540 @ one)) & ! [X182 : term] : ((sK542 @ (sub @ X182 @ (push @ sK541 @ sK543))) = (sK540 @ X182))) & ! [X183 : subst,X184 : term] : ((sub @ one @ (push @ X184 @ X183)) = X184))) & (! [X185 : term > $o,X186 : term,X187 : term > $o,X188 : subst] : (($true != (X187 @ X186)) | ($true = (X185 @ one)) | ((X185 @ (sK544 @ X188 @ X187 @ X186 @ X185)) != (X187 @ (sub @ (sK544 @ X188 @ X187 @ X186 @ X185) @ (push @ X186 @ X188))))) | ((sub @ one @ (push @ sK546 @ sK545)) != sK546) | (pushprop_lem1v2_lthm != $true)) & (hoaslamnotvar = $true) & (($true = sP15) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP15)) & ((sP12 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP12 != $true)) & ((substmonoid_lthm = $true) | ((((comp @ id @ sK547) != sK547) | (sK548 != (comp @ sK548 @ id))) & ! [X194 : subst] : ((comp @ X194 @ id) = X194) & ! [X195 : subst] : ((comp @ id @ X195) = X195))) & ((! [X196 : subst] : ((comp @ id @ X196) = X196) & ! [X197 : subst] : ((comp @ X197 @ id) = X197)) | ((comp @ sK549 @ id) != sK549) | (sK550 != (comp @ id @ sK550)) | (substmonoid_lthm != $true)) & ((axshiftcons = $true) | (sK551 != (comp @ sh @ (push @ sK552 @ sK551)))) & (! [X202 : subst,X203 : term] : ((comp @ sh @ (push @ X203 @ X202)) = X202) | (axshiftcons != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X205 : subst > term > term] : ((sub @ sK553 @ (push @ one @ sh)) != (X205 @ sh @ one))) & (! [X206 : term] : ((sK554 @ X206 @ sh @ one) = (sub @ X206 @ (push @ one @ sh))) | (hoasinduction_lem3b_lthm != $true)) & (! [X208 : term,X209 : term > $o,X210 : term,X211 : subst,X212 : term > $o] : (($true != (X212 @ X208)) | ($true = (X209 @ (sub @ X208 @ (push @ X210 @ X211))))) | (sK556 != (sub @ one @ (push @ sK556 @ sK555))) | ((comp @ sh @ (push @ sK558 @ sK557)) != sK557) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | ((((sK563 @ sK559) = $true) & ($true != (sK560 @ (sub @ sK559 @ (push @ sK561 @ sK562))))) & ! [X222 : subst,X223 : term] : ((sub @ one @ (push @ X223 @ X222)) = X223) & ! [X224 : subst,X225 : term] : ((comp @ sh @ (push @ X225 @ X224)) = X224))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (ulamvarsh = $true) & ((induction2lem_lthm = $true) | ($true != sP8)) & (($true = sP8) | (induction2lem_lthm != $true)) & ((hoasinduction_lem2v2 = $true) | (sP4 != $true)) & ((sP4 = $true) | (hoasinduction_lem2v2 != $true)) & (ulamvarind = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem1_lthm = $true) & (! [X226 : term,X227 : term,X228 : term,X229 : term] : (((ap @ X229 @ X228) != (ap @ X226 @ X227)) | (X226 = X229)) | (apinj1 != $true)) & ((apinj1 = $true) | (((ap @ sK567 @ sK566) = (ap @ sK564 @ sK565)) & (sK564 != sK567))) & (pushprop_lthm_orig = $true) & (! [X234 : term] : ((sK568 @ X234 @ sh @ one) = (sub @ X234 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X237 : subst > term > term] : ((X237 @ sh @ one) != (sub @ sK569 @ (push @ one @ sh)))) & ((axidl = $true) | ((comp @ id @ sK570) != sK570)) & (! [X239 : subst] : ((comp @ id @ X239) = X239) | (axidl != $true)) & (hoaslaminj_gthm = $true) & ((apinj2 = $true) | (((ap @ sK573 @ sK574) = (ap @ sK571 @ sK572)) & (sK574 != sK572))) & (! [X244 : term,X245 : term,X246 : term,X247 : term] : (((ap @ X246 @ X247) != (ap @ X244 @ X245)) | (X245 = X247)) | (apinj2 != $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)))))))))))))) & (! [X248 : subst] : ((push @ (sub @ one @ X248) @ (comp @ sh @ X248)) = X248) | (axscons != $true)) & ((axscons = $true) | (sK575 != (push @ (sub @ one @ sK575) @ (comp @ sh @ sK575)))) & ! [X250 : term] : ((d2term @ (sK576 @ X250)) = X250) & ((axvarcons = $true) | (sK577 != (sub @ one @ (push @ sK577 @ sK578)))) & (! [X254 : term,X255 : subst] : ((sub @ one @ (push @ X254 @ X255)) = X254) | (axvarcons != $true)) & (hoaslamnotvar_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasapinj2_lthm = $true) | (! [X256 : term,X257 : term,X258 : term,X259 : term] : (((ap @ X257 @ X256) != (ap @ X259 @ X258)) | (X256 = X258)) & (((ap @ (sub @ sK582 @ id) @ sK580) = (ap @ (sub @ sK581 @ id) @ sK579)) & (sK579 != sK580)))) & ((((ap @ sK586 @ sK585) = (ap @ sK584 @ sK583)) & (sK585 != sK583)) | ! [X268 : term,X269 : term,X270 : term,X271 : term] : (((ap @ (sub @ X271 @ id) @ X269) != (ap @ (sub @ X270 @ id) @ X268)) | (X268 = X269)) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X273 : subst > term > term] : ((X273 @ (sK588 @ X273) @ (sK589 @ X273)) != (sub @ sK587 @ (push @ (sK589 @ X273) @ (sK588 @ X273))))) & (! [X276 : term] : ! [X278 : subst,X279 : term] : ((sK590 @ X276 @ X278 @ X279) = (sub @ X276 @ (push @ X279 @ X278))) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoasinduction_lem2_lthm = $true) & (pushprop_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP2)) & (($true = sP2) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasapinj1_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (($true = sP1) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP1)) & (pushprop_lem0_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_gthm = $true) & (axmap = $true) & (induction2lem_gthm = $true) & (axabs = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ((d2subst @ d_id) = id)),
% 9.38/2.21 inference(skolemisation,[status(esa),new_symbols(skolem,[sK454,sK455,sK456,sK457,sK458,sK459,sK460,sK461,sK462,sK463,sK464,sK465,sK466,sK467,sK468,sK469,sK470,sK471,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,sK487,sK488,sK489,sK490,sK491,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,sK509,sK510,sK511,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,sK521,sK522,sK523,sK524,sK525,sK526,sK527,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,sK536,sK537,sK538,sK539,sK540,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,sK562,sK563,sK564,sK565,sK566,sK567,sK568,sK569,sK570,sK571,sK572,sK573,sK574,sK575,sK576,sK577,sK578,sK579,sK580,sK581,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590])],[f473,f544,f543,f542,f541,f540,f539,f538,f537,f536,f535,f534,f533,f532,f531,f530,f529,f528,f527,f526,f525,f524,f523,f522,f521,f520,f519,f518,f517,f516,f515,f514,f513,f512,f511,f510,f509,f508,f507,f506,f505,f504,f503,f502,f501,f500,f499,f498,f497,f496,f495,f494,f493,f492,f491,f490,f489,f488,f487,f486,f485,f484,f483,f482,f481,f480,f479,f478,f477,f476,f475,f474])).
% 9.38/2.21 thf(f474,plain,(
% 9.38/2.21 ? [X1 : term,X2 : term] : (((sub @ X1 @ sh) = (sub @ X2 @ sh)) & (X1 != X2)) => (((sub @ sK455 @ sh) = (sub @ sK454 @ sh)) & (sK455 != sK454))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f475,plain,(
% 9.38/2.21 ? [X5 : term] : ((sub @ X5 @ id) != X5) => ((sub @ sK456 @ id) != sK456)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f476,plain,(
% 9.38/2.21 ? [X8 : term] : ((sub @ X8 @ id) != X8) => ((sub @ sK457 @ id) != sK457)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f477,plain,(
% 9.38/2.21 ? [X9 : term] : ((sub @ X9 @ id) != X9) => ((sub @ sK458 @ id) != sK458)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f478,plain,(
% 9.38/2.21 ? [X11 : subst] : ((comp @ X11 @ id) != X11) => ((comp @ sK459 @ id) != sK459)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f479,plain,(
% 9.38/2.21 ! [X16 : subst > term > term] : (? [X17 : subst,X18 : subst,X19 : term] : ((sub @ (X16 @ X18 @ X19) @ X17) != (X16 @ (comp @ X18 @ X17) @ (sub @ X19 @ X17))) => ((X16 @ (comp @ (sK461 @ X16) @ (sK460 @ X16)) @ (sub @ (sK462 @ X16) @ (sK460 @ X16))) != (sub @ (X16 @ (sK461 @ X16) @ (sK462 @ X16)) @ (sK460 @ X16))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f480,plain,(
% 9.38/2.21 ? [X20 : term,X21 : term,X22 : subst > term > term] : (((lam @ (X22 @ sh @ one)) = (ap @ (sub @ X21 @ id) @ X20)) & ! [X23 : subst,X24 : subst,X25 : term] : ((sub @ (X22 @ X24 @ X25) @ X23) = (X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)))) => (((ap @ (sub @ sK464 @ id) @ sK463) = (lam @ (sK465 @ sh @ one))) & ! [X25 : term,X24 : subst,X23 : subst] : ((sK465 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23)) = (sub @ (sK465 @ X24 @ X25) @ X23)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f481,plain,(
% 9.38/2.21 ? [X26 : term,X27 : term,X28 : term,X29 : term] : ((X26 != X29) & ((ap @ X26 @ X27) = (ap @ X29 @ X28))) => ((sK466 != sK469) & ((ap @ sK466 @ sK467) = (ap @ sK469 @ sK468)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f482,plain,(
% 9.38/2.21 ? [X34 : term] : ((sub @ X34 @ id) != X34) => (sK470 != (sub @ sK470 @ id))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f483,plain,(
% 9.38/2.21 ? [X39 : term,X40 : term,X41 : term,X42 : term] : ((X41 != X42) & ((ap @ (sub @ X41 @ id) @ X39) = (ap @ (sub @ X42 @ id) @ X40))) => ((sK474 != sK473) & ((ap @ (sub @ sK474 @ id) @ sK472) = (ap @ (sub @ sK473 @ id) @ sK471)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f484,plain,(
% 9.38/2.21 ? [X48 : term,X49 : term,X50 : term,X51 : term] : (((ap @ (sub @ X48 @ id) @ X50) = (ap @ (sub @ X51 @ id) @ X49)) & (X48 != X51)) => (((ap @ (sub @ sK478 @ id) @ sK476) = (ap @ (sub @ sK475 @ id) @ sK477)) & (sK478 != sK475))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f485,plain,(
% 9.38/2.21 ! [X52 : subst > term > subst > $o] : (? [X54 : subst > term > term] : (! [X55 : term] : (((X52 @ id @ (X54 @ id @ X55) @ id) = $true) | ($true != (X52 @ id @ X55 @ id))) & ($true != (X52 @ id @ (lam @ (X54 @ sh @ one)) @ id)) & ! [X56 : term,X57 : subst,X58 : subst] : ((X54 @ (comp @ X58 @ X57) @ (sub @ X56 @ X57)) = (sub @ (X54 @ X58 @ X56) @ X57))) => (! [X55 : term] : (($true = (X52 @ id @ (sK479 @ X52 @ id @ X55) @ id)) | ($true != (X52 @ id @ X55 @ id))) & ($true != (X52 @ id @ (lam @ (sK479 @ X52 @ sh @ one)) @ id)) & ! [X58 : subst,X57 : subst,X56 : term] : ((sub @ (sK479 @ X52 @ X58 @ X56) @ X57) = (sK479 @ X52 @ (comp @ X58 @ X57) @ (sub @ X56 @ X57)))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f486,plain,(
% 9.38/2.21 ? [X59 : subst > term > subst > $o,X60 : term] : (! [X61 : subst > term > term] : (? [X62 : term] : (($true != (X59 @ id @ (X61 @ id @ X62) @ id)) & ($true = (X59 @ id @ X62 @ id))) | ($true = (X59 @ id @ (lam @ (X61 @ sh @ one)) @ id)) | ? [X63 : term,X64 : subst,X65 : subst] : ((X61 @ (comp @ X65 @ X64) @ (sub @ X63 @ X64)) != (sub @ (X61 @ X65 @ X63) @ X64))) & ($true != (sP46 @ X59)) & ((X59 @ id @ X60 @ id) != $true)) => (! [X61 : subst > term > term] : (? [X62 : term] : (($true != (sK480 @ id @ (X61 @ id @ X62) @ id)) & ($true = (sK480 @ id @ X62 @ id))) | ($true = (sK480 @ id @ (lam @ (X61 @ sh @ one)) @ id)) | ? [X63 : term,X64 : subst,X65 : subst] : ((X61 @ (comp @ X65 @ X64) @ (sub @ X63 @ X64)) != (sub @ (X61 @ X65 @ X63) @ X64))) & ($true != (sP46 @ sK480)) & ((sK480 @ id @ sK481 @ id) != $true))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f487,plain,(
% 9.38/2.21 ! [X61 : subst > term > term] : (? [X62 : term] : (($true != (sK480 @ id @ (X61 @ id @ X62) @ id)) & ($true = (sK480 @ id @ X62 @ id))) => (((sK480 @ id @ (X61 @ id @ (sK482 @ X61)) @ id) != $true) & ($true = (sK480 @ id @ (sK482 @ X61) @ id))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f488,plain,(
% 9.38/2.21 ! [X61 : subst > term > term] : (? [X63 : term,X64 : subst,X65 : subst] : ((X61 @ (comp @ X65 @ X64) @ (sub @ X63 @ X64)) != (sub @ (X61 @ X65 @ X63) @ X64)) => ((X61 @ (comp @ (sK485 @ X61) @ (sK484 @ X61)) @ (sub @ (sK483 @ X61) @ (sK484 @ X61))) != (sub @ (X61 @ (sK485 @ X61) @ (sK483 @ X61)) @ (sK484 @ X61))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f489,plain,(
% 9.38/2.21 ? [X67 : term] : ((sub @ X67 @ id) != X67) => (sK486 != (sub @ sK486 @ id))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f490,plain,(
% 9.38/2.21 ! [X68 : subst,X69 : term,X70 : term > $o] : (? [X71 : term > $o] : ! [X72 : term] : ((X70 @ (sub @ X72 @ (push @ X69 @ X68))) = (X71 @ X72)) => ! [X72 : term] : ((sK487 @ X70 @ X69 @ X68 @ X72) = (X70 @ (sub @ X72 @ (push @ X69 @ X68)))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f491,plain,(
% 9.38/2.21 ? [X73 : subst,X74 : term,X75 : term > $o] : ! [X76 : term > $o] : ? [X77 : term] : ((X75 @ (sub @ X77 @ (push @ X74 @ X73))) != (X76 @ X77)) => ! [X76 : term > $o] : ? [X77 : term] : ((sK490 @ (sub @ X77 @ (push @ sK489 @ sK488))) != (X76 @ X77))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f492,plain,(
% 9.38/2.21 ! [X76 : term > $o] : (? [X77 : term] : ((sK490 @ (sub @ X77 @ (push @ sK489 @ sK488))) != (X76 @ X77)) => ((X76 @ (sK491 @ X76)) != (sK490 @ (sub @ (sK491 @ X76) @ (push @ sK489 @ sK488)))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f493,plain,(
% 9.38/2.21 ! [X80 : subst > term > subst > $o] : (? [X81 : term > $o] : ! [X82 : term] : ((X81 @ X82) = (X80 @ id @ X82 @ id)) => ! [X82 : term] : ((sK492 @ X80 @ X82) = (X80 @ id @ X82 @ id)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f494,plain,(
% 9.38/2.21 ? [X83 : subst > term > subst > $o] : ! [X84 : term > $o] : ? [X85 : term] : ((X84 @ X85) != (X83 @ id @ X85 @ id)) => ! [X84 : term > $o] : ? [X85 : term] : ((X84 @ X85) != (sK493 @ id @ X85 @ id))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f495,plain,(
% 9.38/2.21 ! [X84 : term > $o] : (? [X85 : term] : ((X84 @ X85) != (sK493 @ id @ X85 @ id)) => ((sK493 @ id @ (sK494 @ X84) @ id) != (X84 @ (sK494 @ X84))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f496,plain,(
% 9.38/2.21 ? [X86 : subst > term > subst > $o] : ! [X87 : term > $o] : ? [X88 : term] : ((X86 @ id @ X88 @ id) != (X87 @ X88)) => ! [X87 : term > $o] : ? [X88 : term] : ((X87 @ X88) != (sK495 @ id @ X88 @ id))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f497,plain,(
% 9.38/2.21 ! [X87 : term > $o] : (? [X88 : term] : ((X87 @ X88) != (sK495 @ id @ X88 @ id)) => ((X87 @ (sK496 @ X87)) != (sK495 @ id @ (sK496 @ X87) @ id)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f498,plain,(
% 9.38/2.21 ! [X89 : subst > term > subst > $o] : (? [X90 : term > $o] : ! [X91 : term] : ((X89 @ id @ X91 @ id) = (X90 @ X91)) => ! [X91 : term] : ((X89 @ id @ X91 @ id) = (sK497 @ X89 @ X91)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f499,plain,(
% 9.38/2.21 ? [X92 : term] : ! [X93 : subst > term > term] : ? [X94 : term,X95 : subst] : ((X93 @ X95 @ X94) != (sub @ X92 @ (push @ X94 @ X95))) => ! [X93 : subst > term > term] : ? [X95 : subst,X94 : term] : ((X93 @ X95 @ X94) != (sub @ sK498 @ (push @ X94 @ X95)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f500,plain,(
% 9.38/2.21 ! [X93 : subst > term > term] : (? [X95 : subst,X94 : term] : ((X93 @ X95 @ X94) != (sub @ sK498 @ (push @ X94 @ X95))) => ((sub @ sK498 @ (push @ (sK499 @ X93) @ (sK500 @ X93))) != (X93 @ (sK500 @ X93) @ (sK499 @ X93))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f501,plain,(
% 9.38/2.21 ! [X96 : term] : (? [X97 : subst > term > term] : ! [X98 : term,X99 : subst] : ((sub @ X96 @ (push @ X98 @ X99)) = (X97 @ X99 @ X98)) => ! [X99 : subst,X98 : term] : ((sub @ X96 @ (push @ X98 @ X99)) = (sK501 @ X96 @ X99 @ X98)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f502,plain,(
% 9.38/2.21 ! [X101 : term > $o] : (? [X102 : term] : (($true = (X101 @ X102)) & ($true != (X101 @ (lam @ X102)))) => (((X101 @ (sK502 @ X101)) = $true) & ($true != (X101 @ (lam @ (sK502 @ X101))))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f503,plain,(
% 9.38/2.21 ! [X101 : term > $o] : (? [X103 : term] : (($true != (X101 @ X103)) & ($true = (var @ X103))) => (($true != (X101 @ (sK503 @ X101))) & ($true = (var @ (sK503 @ X101)))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f504,plain,(
% 9.38/2.21 ? [X104 : term,X105 : term > $o] : (! [X106 : term] : (($true != (X105 @ X106)) | ($true = (X105 @ (lam @ X106)))) & ($true != (X105 @ X104)) & ($true != (sP39 @ X105)) & ! [X107 : term] : (($true = (X105 @ X107)) | ($true != (var @ X107)))) => (! [X106 : term] : (($true != (sK505 @ X106)) | ($true = (sK505 @ (lam @ X106)))) & ($true != (sK505 @ sK504)) & ((sP39 @ sK505) != $true) & ! [X107 : term] : (($true = (sK505 @ X107)) | ($true != (var @ X107))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f505,plain,(
% 9.38/2.21 ? [X108 : term > $o,X109 : term,X110 : subst] : ! [X111 : term > $o] : ? [X112 : term] : ((X108 @ (sub @ X112 @ (push @ X109 @ X110))) != (X111 @ X112)) => ! [X111 : term > $o] : ? [X112 : term] : ((sK506 @ (sub @ X112 @ (push @ sK507 @ sK508))) != (X111 @ X112))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f506,plain,(
% 9.38/2.21 ! [X111 : term > $o] : (? [X112 : term] : ((sK506 @ (sub @ X112 @ (push @ sK507 @ sK508))) != (X111 @ X112)) => ((sK506 @ (sub @ (sK509 @ X111) @ (push @ sK507 @ sK508))) != (X111 @ (sK509 @ X111))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f507,plain,(
% 9.38/2.21 ! [X113 : term > $o,X114 : term,X115 : subst] : (? [X116 : term > $o] : ! [X117 : term] : ((X113 @ (sub @ X117 @ (push @ X114 @ X115))) = (X116 @ X117)) => ! [X117 : term] : ((X113 @ (sub @ X117 @ (push @ X114 @ X115))) = (sK510 @ X115 @ X114 @ X113 @ X117)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f508,plain,(
% 9.38/2.21 ? [X120 : subst] : ((comp @ X120 @ id) != X120) => ((comp @ sK511 @ id) != sK511)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f509,plain,(
% 9.38/2.21 ? [X121 : subst] : ((comp @ id @ X121) != X121) => (sK512 != (comp @ id @ sK512))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f510,plain,(
% 9.38/2.21 ! [X124 : subst > term > term] : (? [X126 : term,X127 : subst,X128 : subst] : ((X124 @ (comp @ X128 @ X127) @ (sub @ X126 @ X127)) != (sub @ (X124 @ X128 @ X126) @ X127)) => ((X124 @ (comp @ (sK515 @ X124) @ (sK514 @ X124)) @ (sub @ (sK513 @ X124) @ (sK514 @ X124))) != (sub @ (X124 @ (sK515 @ X124) @ (sK513 @ X124)) @ (sK514 @ X124))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f511,plain,(
% 9.38/2.21 ! [X125 : subst > term > term] : (? [X129 : subst,X130 : term,X131 : subst] : ((sub @ (X125 @ X129 @ X130) @ X131) != (X125 @ (comp @ X129 @ X131) @ (sub @ X130 @ X131))) => ((X125 @ (comp @ (sK516 @ X125) @ (sK518 @ X125)) @ (sub @ (sK517 @ X125) @ (sK518 @ X125))) != (sub @ (X125 @ (sK516 @ X125) @ (sK517 @ X125)) @ (sK518 @ X125))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f512,plain,(
% 9.38/2.21 ? [X132 : term,X133 : subst,X134 : subst > term > term,X135 : subst > term > term] : (((X135 @ X133 @ X132) != (X134 @ X133 @ X132)) & ((lam @ (X134 @ sh @ one)) = (lam @ (X135 @ sh @ one))) & ! [X136 : term,X137 : subst,X138 : subst] : ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137)) = (sub @ (X134 @ X138 @ X136) @ X137)) & ! [X139 : subst,X140 : term,X141 : subst] : ((X135 @ (comp @ X139 @ X141) @ (sub @ X140 @ X141)) = (sub @ (X135 @ X139 @ X140) @ X141))) => (((sK522 @ sK520 @ sK519) != (sK521 @ sK520 @ sK519)) & ((lam @ (sK522 @ sh @ one)) = (lam @ (sK521 @ sh @ one))) & ! [X138 : subst,X137 : subst,X136 : term] : ((sK521 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137)) = (sub @ (sK521 @ X138 @ X136) @ X137)) & ! [X141 : subst,X140 : term,X139 : subst] : ((sK522 @ (comp @ X139 @ X141) @ (sub @ X140 @ X141)) = (sub @ (sK522 @ X139 @ X140) @ X141)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f513,plain,(
% 9.38/2.21 ? [X142 : term,X143 : term,X144 : term,X145 : term] : (((ap @ (sub @ X142 @ id) @ X145) = (ap @ (sub @ X143 @ id) @ X144)) & (X144 != X145)) => (((ap @ (sub @ sK523 @ id) @ sK526) = (ap @ (sub @ sK524 @ id) @ sK525)) & (sK526 != sK525))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f514,plain,(
% 9.38/2.21 ? [X150 : term,X151 : subst,X152 : term > $o,X153 : term > $o] : (($true = (X153 @ X150)) & ! [X154 : term] : ((X152 @ X154) = (X153 @ (sub @ X154 @ (push @ X150 @ X151)))) & ($true != (X152 @ one))) => (($true = (sK530 @ sK527)) & ! [X154 : term] : ((sK530 @ (sub @ X154 @ (push @ sK527 @ sK528))) = (sK529 @ X154)) & ((sK529 @ one) != $true))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f515,plain,(
% 9.38/2.21 ! [X155 : term,X156 : subst,X157 : term > $o,X158 : term > $o] : (? [X159 : term] : ((X157 @ X159) != (X158 @ (sub @ X159 @ (push @ X155 @ X156)))) => ((X158 @ (sub @ (sK531 @ X158 @ X157 @ X156 @ X155) @ (push @ X155 @ X156))) != (X157 @ (sK531 @ X158 @ X157 @ X156 @ X155))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f516,plain,(
% 9.38/2.21 ? [X160 : term,X161 : subst,X162 : term,X163 : term > $o,X164 : term > $o] : (($true = (X164 @ X162)) & ($true != (X163 @ (sub @ X162 @ (push @ X160 @ X161))))) => (((sK536 @ sK534) = $true) & ($true != (sK535 @ (sub @ sK534 @ (push @ sK532 @ sK533)))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f517,plain,(
% 9.38/2.21 ! [X170 : subst] : (? [X171 : d_subst] : ((d2subst @ X171) = X170) => ((d2subst @ (sK537 @ X170)) = X170))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f518,plain,(
% 9.38/2.21 ? [X174 : term,X175 : term] : ((X174 != X175) & ((lam @ X174) = (lam @ X175))) => ((sK539 != sK538) & ((lam @ sK539) = (lam @ sK538)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f519,plain,(
% 9.38/2.21 ? [X178 : term > $o,X179 : term,X180 : term > $o,X181 : subst] : (($true = (X180 @ X179)) & ($true != (X178 @ one)) & ! [X182 : term] : ((X180 @ (sub @ X182 @ (push @ X179 @ X181))) = (X178 @ X182))) => (($true = (sK542 @ sK541)) & ($true != (sK540 @ one)) & ! [X182 : term] : ((sK542 @ (sub @ X182 @ (push @ sK541 @ sK543))) = (sK540 @ X182)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f520,plain,(
% 9.38/2.21 ! [X185 : term > $o,X186 : term,X187 : term > $o,X188 : subst] : (? [X189 : term] : ((X187 @ (sub @ X189 @ (push @ X186 @ X188))) != (X185 @ X189)) => ((X185 @ (sK544 @ X188 @ X187 @ X186 @ X185)) != (X187 @ (sub @ (sK544 @ X188 @ X187 @ X186 @ X185) @ (push @ X186 @ X188)))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f521,plain,(
% 9.38/2.21 ? [X190 : subst,X191 : term] : ((sub @ one @ (push @ X191 @ X190)) != X191) => ((sub @ one @ (push @ sK546 @ sK545)) != sK546)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f522,plain,(
% 9.38/2.21 ? [X192 : subst] : ((comp @ id @ X192) != X192) => ((comp @ id @ sK547) != sK547)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f523,plain,(
% 9.38/2.21 ? [X193 : subst] : ((comp @ X193 @ id) != X193) => (sK548 != (comp @ sK548 @ id))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f524,plain,(
% 9.38/2.21 ? [X198 : subst] : ((comp @ X198 @ id) != X198) => ((comp @ sK549 @ id) != sK549)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f525,plain,(
% 9.38/2.21 ? [X199 : subst] : ((comp @ id @ X199) != X199) => (sK550 != (comp @ id @ sK550))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f526,plain,(
% 9.38/2.21 ? [X200 : subst,X201 : term] : ((comp @ sh @ (push @ X201 @ X200)) != X200) => (sK551 != (comp @ sh @ (push @ sK552 @ sK551)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f527,plain,(
% 9.38/2.21 ? [X204 : term] : ! [X205 : subst > term > term] : ((sub @ X204 @ (push @ one @ sh)) != (X205 @ sh @ one)) => ! [X205 : subst > term > term] : ((sub @ sK553 @ (push @ one @ sh)) != (X205 @ sh @ one))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f528,plain,(
% 9.38/2.21 ! [X206 : term] : (? [X207 : subst > term > term] : ((sub @ X206 @ (push @ one @ sh)) = (X207 @ sh @ one)) => ((sK554 @ X206 @ sh @ one) = (sub @ X206 @ (push @ one @ sh))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f529,plain,(
% 9.38/2.21 ? [X213 : subst,X214 : term] : ((sub @ one @ (push @ X214 @ X213)) != X214) => (sK556 != (sub @ one @ (push @ sK556 @ sK555)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f530,plain,(
% 9.38/2.21 ? [X215 : subst,X216 : term] : ((comp @ sh @ (push @ X216 @ X215)) != X215) => ((comp @ sh @ (push @ sK558 @ sK557)) != sK557)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f531,plain,(
% 9.38/2.21 ? [X217 : term,X218 : term > $o,X219 : term,X220 : subst,X221 : term > $o] : (($true = (X221 @ X217)) & ($true != (X218 @ (sub @ X217 @ (push @ X219 @ X220))))) => (((sK563 @ sK559) = $true) & ($true != (sK560 @ (sub @ sK559 @ (push @ sK561 @ sK562)))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f532,plain,(
% 9.38/2.21 ? [X230 : term,X231 : term,X232 : term,X233 : term] : (((ap @ X233 @ X232) = (ap @ X230 @ X231)) & (X230 != X233)) => (((ap @ sK567 @ sK566) = (ap @ sK564 @ sK565)) & (sK564 != sK567))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f533,plain,(
% 9.38/2.21 ! [X234 : term] : (? [X235 : subst > term > term] : ((X235 @ sh @ one) = (sub @ X234 @ (push @ one @ sh))) => ((sK568 @ X234 @ sh @ one) = (sub @ X234 @ (push @ one @ sh))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f534,plain,(
% 9.38/2.21 ? [X236 : term] : ! [X237 : subst > term > term] : ((X237 @ sh @ one) != (sub @ X236 @ (push @ one @ sh))) => ! [X237 : subst > term > term] : ((X237 @ sh @ one) != (sub @ sK569 @ (push @ one @ sh)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f535,plain,(
% 9.38/2.21 ? [X238 : subst] : ((comp @ id @ X238) != X238) => ((comp @ id @ sK570) != sK570)),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f536,plain,(
% 9.38/2.21 ? [X240 : term,X241 : term,X242 : term,X243 : term] : (((ap @ X242 @ X243) = (ap @ X240 @ X241)) & (X241 != X243)) => (((ap @ sK573 @ sK574) = (ap @ sK571 @ sK572)) & (sK574 != sK572))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f537,plain,(
% 9.38/2.21 ? [X249 : subst] : ((push @ (sub @ one @ X249) @ (comp @ sh @ X249)) != X249) => (sK575 != (push @ (sub @ one @ sK575) @ (comp @ sh @ sK575)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f538,plain,(
% 9.38/2.21 ! [X250 : term] : (? [X251 : d_term] : ((d2term @ X251) = X250) => ((d2term @ (sK576 @ X250)) = X250))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f539,plain,(
% 9.38/2.21 ? [X252 : term,X253 : subst] : ((sub @ one @ (push @ X252 @ X253)) != X252) => (sK577 != (sub @ one @ (push @ sK577 @ sK578)))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f540,plain,(
% 9.38/2.21 ? [X260 : term,X261 : term,X262 : term,X263 : term] : (((ap @ (sub @ X262 @ id) @ X260) = (ap @ (sub @ X263 @ id) @ X261)) & (X260 != X261)) => (((ap @ (sub @ sK582 @ id) @ sK580) = (ap @ (sub @ sK581 @ id) @ sK579)) & (sK579 != sK580))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f541,plain,(
% 9.38/2.21 ? [X264 : term,X265 : term,X266 : term,X267 : term] : (((ap @ X265 @ X264) = (ap @ X267 @ X266)) & (X264 != X266)) => (((ap @ sK586 @ sK585) = (ap @ sK584 @ sK583)) & (sK585 != sK583))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f542,plain,(
% 9.38/2.21 ? [X272 : term] : ! [X273 : subst > term > term] : ? [X274 : subst,X275 : term] : ((X273 @ X274 @ X275) != (sub @ X272 @ (push @ X275 @ X274))) => ! [X273 : subst > term > term] : ? [X275 : term,X274 : subst] : ((sub @ sK587 @ (push @ X275 @ X274)) != (X273 @ X274 @ X275))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f543,plain,(
% 9.38/2.21 ! [X273 : subst > term > term] : (? [X275 : term,X274 : subst] : ((sub @ sK587 @ (push @ X275 @ X274)) != (X273 @ X274 @ X275)) => ((X273 @ (sK588 @ X273) @ (sK589 @ X273)) != (sub @ sK587 @ (push @ (sK589 @ X273) @ (sK588 @ X273)))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f544,plain,(
% 9.38/2.21 ! [X276 : term] : (? [X277 : subst > term > term] : ! [X278 : subst,X279 : term] : ((sub @ X276 @ (push @ X279 @ X278)) = (X277 @ X278 @ X279)) => ! [X279 : term,X278 : subst] : ((sK590 @ X276 @ X278 @ X279) = (sub @ X276 @ (push @ X279 @ X278))))),
% 9.38/2.21 introduced(definition,[],[choice_axiom])).
% 9.38/2.21 thf(f473,plain,(
% 9.38/2.21 (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ! [X0 : d_subst] : (d_id = X0) & (hoasinduction_lem1_gthm = $true) & ((shinj = $true) | ? [X1 : term,X2 : term] : (((sub @ X1 @ sh) = (sub @ X2 @ sh)) & (X1 != X2))) & (! [X3 : term,X4 : term] : (((sub @ X3 @ sh) != (sub @ X4 @ sh)) | (X3 = X4)) | (shinj != $true)) & (lamnotvar = $true) & (? [X5 : term] : ((sub @ X5 @ id) != X5) | ! [X6 : term] : ((sub @ X6 @ id) = X6) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X7 : term] : ((sub @ X7 @ id) = X7) & ? [X8 : term] : ((sub @ X8 @ id) != X8))) & (($true = sP58) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP58)) & ((axvarid = $true) | ? [X9 : term] : ((sub @ X9 @ id) != X9)) & (! [X10 : term] : ((sub @ X10 @ id) = X10) | (axvarid != $true)) & (hoaslamnotvar_lthm = $true) & (apnotvar = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP57)) & (($true = sP57) | (hoasinduction_lem3aa != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) | ($true != sP56)) & (($true = sP56) | (hoaslaminj_lthm != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((axidr = $true) | ? [X11 : subst] : ((comp @ X11 @ id) != X11)) & (! [X12 : subst] : ((comp @ X12 @ id) = X12) | (axidr != $true)) & (($true = sP55) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP55)) & ! [X13 : d_term] : (d_one = X13) & (hoasinduction_lem3_gthm = $true) & ((sP53 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP53 != $true)) & (axapp = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP51 != $true)) & ((sP51 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoasapnotvar = $true) & (! [X14 : term,X15 : term,X16 : subst > term > term] : (((lam @ (X16 @ sh @ one)) != (ap @ (sub @ X15 @ id) @ X14)) | ? [X17 : subst,X18 : subst,X19 : term] : ((sub @ (X16 @ X18 @ X19) @ X17) != (X16 @ (comp @ X18 @ X17) @ (sub @ X19 @ X17)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X20 : term,X21 : term,X22 : subst > term > term] : (((lam @ (X22 @ sh @ one)) = (ap @ (sub @ X21 @ id) @ X20)) & ! [X23 : subst,X24 : subst,X25 : term] : ((sub @ (X22 @ X24 @ X25) @ X23) = (X22 @ (comp @ X24 @ X23) @ (sub @ X25 @ X23))))) & (axclos = $true) & (? [X26 : term,X27 : term,X28 : term,X29 : term] : ((X26 != X29) & ((ap @ X26 @ X27) = (ap @ X29 @ X28))) | ! [X30 : term,X31 : term,X32 : term,X33 : term] : ((X32 = X33) | ((ap @ (sub @ X32 @ id) @ X30) != (ap @ (sub @ X33 @ id) @ X31))) | ? [X34 : term] : ((sub @ X34 @ id) != X34) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X35 : term,X36 : term,X37 : term,X38 : term] : ((X35 = X38) | ((ap @ X35 @ X36) != (ap @ X38 @ X37))) & ? [X39 : term,X40 : term,X41 : term,X42 : term] : ((X41 != X42) & ((ap @ (sub @ X41 @ id) @ X39) = (ap @ (sub @ X42 @ id) @ X40))) & ! [X43 : term] : ((sub @ X43 @ id) = X43))) & (ulamvar1 != $true) & (! [X44 : term,X45 : term,X46 : term,X47 : term] : (((ap @ (sub @ X44 @ id) @ X46) != (ap @ (sub @ X47 @ id) @ X45)) | (X44 = X47)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X48 : term,X49 : term,X50 : term,X51 : term] : (((ap @ (sub @ X48 @ id) @ X50) = (ap @ (sub @ X51 @ id) @ X49)) & (X48 != X51))) & (! [X52 : subst > term > subst > $o,X53 : term] : (? [X54 : subst > term > term] : (! [X55 : term] : (((X52 @ id @ (X54 @ id @ X55) @ id) = $true) | ($true != (X52 @ id @ X55 @ id))) & ($true != (X52 @ id @ (lam @ (X54 @ sh @ one)) @ id)) & ! [X56 : term,X57 : subst,X58 : subst] : ((X54 @ (comp @ X58 @ X57) @ (sub @ X56 @ X57)) = (sub @ (X54 @ X58 @ X56) @ X57))) | ($true = (sP46 @ X52)) | ($true = (X52 @ id @ X53 @ id))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X59 : subst > term > subst > $o,X60 : term] : (! [X61 : subst > term > term] : (? [X62 : term] : (($true != (X59 @ id @ (X61 @ id @ X62) @ id)) & ($true = (X59 @ id @ X62 @ id))) | ($true = (X59 @ id @ (lam @ (X61 @ sh @ one)) @ id)) | ? [X63 : term,X64 : subst,X65 : subst] : ((X61 @ (comp @ X65 @ X64) @ (sub @ X63 @ X64)) != (sub @ (X61 @ X65 @ X63) @ X64))) & ($true != (sP46 @ X59)) & ((X59 @ id @ X60 @ id) != $true))) & (! [X66 : term] : ((sub @ X66 @ id) = X66) | (termmset != $true)) & ((termmset = $true) | ? [X67 : term] : ((sub @ X67 @ id) != X67)) & (induction2_gthm = $true) & (pushprop = $true) & (hoasinduction_lem2_gthm = $true) & (! [X68 : subst,X69 : term,X70 : term > $o] : ? [X71 : term > $o] : ! [X72 : term] : ((X70 @ (sub @ X72 @ (push @ X69 @ X68))) = (X71 @ X72)) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X73 : subst,X74 : term,X75 : term > $o] : ! [X76 : term > $o] : ? [X77 : term] : ((X75 @ (sub @ X77 @ (push @ X74 @ X73))) != (X76 @ X77))) & ! [X78 : d_term,X79 : d_term] : ((X78 = X79) | ((d2term @ X79) != (d2term @ X78))) & ((var @ (d2term @ d_one)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (pushprop_lem2v2_gthm = $true) & (! [X80 : subst > term > subst > $o] : ? [X81 : term > $o] : ! [X82 : term] : ((X81 @ X82) = (X80 @ id @ X82 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X83 : subst > term > subst > $o] : ! [X84 : term > $o] : ? [X85 : term] : ((X84 @ X85) != (X83 @ id @ X85 @ id))) & (($true = sP45) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP45)) & (($true = sP42) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP42)) & (hoasapnotvar_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lem0 = $true) | ? [X86 : subst > term > subst > $o] : ! [X87 : term > $o] : ? [X88 : term] : ((X86 @ id @ X88 @ id) != (X87 @ X88))) & (! [X89 : subst > term > subst > $o] : ? [X90 : term > $o] : ! [X91 : term] : ((X89 @ id @ X91 @ id) = (X90 @ X91)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X92 : term] : ! [X93 : subst > term > term] : ? [X94 : term,X95 : subst] : ((X93 @ X95 @ X94) != (sub @ X92 @ (push @ X94 @ X95)))) & (! [X96 : term] : ? [X97 : subst > term > term] : ! [X98 : term,X99 : subst] : ((sub @ X96 @ (push @ X98 @ X99)) = (X97 @ X99 @ X98)) | (hoasinduction_lem3v2_f != $true)) & (lamnotap != $true) & (one = (d2term @ d_one)) & (axassoc = $true) & ((induction2 = $true) | ($true != sP41)) & (($true = sP41) | (induction2 != $true)) & (! [X100 : term,X101 : term > $o] : (? [X102 : term] : (($true = (X101 @ X102)) & ($true != (X101 @ (lam @ X102)))) | ((X101 @ X100) = $true) | ((sP39 @ X101) = $true) | ? [X103 : term] : (($true != (X101 @ X103)) & ($true = (var @ X103)))) | (induction != $true)) & ((induction = $true) | ? [X104 : term,X105 : term > $o] : (! [X106 : term] : (($true != (X105 @ X106)) | ($true = (X105 @ (lam @ X106)))) & ($true != (X105 @ X104)) & ($true != (sP39 @ X105)) & ! [X107 : term] : (($true = (X105 @ X107)) | ($true != (var @ X107))))) & ((pushprop_lem0_lthm = $true) | ? [X108 : term > $o,X109 : term,X110 : subst] : ! [X111 : term > $o] : ? [X112 : term] : ((X108 @ (sub @ X112 @ (push @ X109 @ X110))) != (X111 @ X112))) & (! [X113 : term > $o,X114 : term,X115 : subst] : ? [X116 : term > $o] : ! [X117 : term] : ((X113 @ (sub @ X117 @ (push @ X114 @ X115))) = (X116 @ X117)) | (pushprop_lem0_lthm != $true)) & (hoasinduction_lem1v2 = $true) & (pushprop_lthm = $true) & ((hoasinduction_lthm = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lthm != $true)) & ((! [X118 : subst] : ((comp @ X118 @ id) = X118) & ! [X119 : subst] : ((comp @ id @ X119) = X119)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X120 : subst] : ((comp @ X120 @ id) != X120) | ? [X121 : subst] : ((comp @ id @ X121) != X121)) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2 != $true)) & (! [X122 : term,X123 : subst,X124 : subst > term > term,X125 : subst > term > term] : (((X124 @ X123 @ X122) = (X125 @ X123 @ X122)) | ((lam @ (X124 @ sh @ one)) != (lam @ (X125 @ sh @ one))) | ? [X126 : term,X127 : subst,X128 : subst] : ((X124 @ (comp @ X128 @ X127) @ (sub @ X126 @ X127)) != (sub @ (X124 @ X128 @ X126) @ X127)) | ? [X129 : subst,X130 : term,X131 : subst] : ((sub @ (X125 @ X129 @ X130) @ X131) != (X125 @ (comp @ X129 @ X131) @ (sub @ X130 @ X131)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X132 : term,X133 : subst,X134 : subst > term > term,X135 : subst > term > term] : (((X135 @ X133 @ X132) != (X134 @ X133 @ X132)) & ((lam @ (X134 @ sh @ one)) = (lam @ (X135 @ sh @ one))) & ! [X136 : term,X137 : subst,X138 : subst] : ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137)) = (sub @ (X134 @ X138 @ X136) @ X137)) & ! [X139 : subst,X140 : term,X141 : subst] : ((X135 @ (comp @ X139 @ X141) @ (sub @ X140 @ X141)) = (sub @ (X135 @ X139 @ X140) @ X141)))) & ((d2subst @ d_id) = sh) & (hoasinduction_lem1 = $true) & (($true = sP27) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP27)) & ((hoasapinj2 = $true) | ? [X142 : term,X143 : term,X144 : term,X145 : term] : (((ap @ (sub @ X142 @ id) @ X145) = (ap @ (sub @ X143 @ id) @ X144)) & (X144 != X145))) & (! [X146 : term,X147 : term,X148 : term,X149 : term] : (((ap @ (sub @ X147 @ id) @ X148) != (ap @ (sub @ X146 @ id) @ X149)) | (X148 = X149)) | (hoasapinj2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X150 : term,X151 : subst,X152 : term > $o,X153 : term > $o] : (($true = (X153 @ X150)) & ! [X154 : term] : ((X152 @ X154) = (X153 @ (sub @ X154 @ (push @ X150 @ X151)))) & ($true != (X152 @ one)))) & (! [X155 : term,X156 : subst,X157 : term > $o,X158 : term > $o] : (($true != (X158 @ X155)) | ? [X159 : term] : ((X157 @ X159) != (X158 @ (sub @ X159 @ (push @ X155 @ X156)))) | ($true = (X157 @ one))) | (pushprop_lem1v2 != $true)) & (hoasapinj2_gthm = $true) & ((pushprop_lem1 = $true) | ? [X160 : term,X161 : subst,X162 : term,X163 : term > $o,X164 : term > $o] : (($true = (X164 @ X162)) & ($true != (X163 @ (sub @ X162 @ (push @ X160 @ X161)))))) & (! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true != (X169 @ X167)) | ($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166))))) | (pushprop_lem1 != $true)) & (pushprop_lem3v2_lthm = $true) & (($true = sP25) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP25)) & (termmset_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP24)) & (($true = sP24) | (hoasinduction_lem3v2_lthm != $true)) & ! [X170 : subst] : ? [X171 : d_subst] : ((d2subst @ X171) = X170) & (axvarshift = $true) & (hoasapnotvar_lthm = $true) & (! [X172 : term,X173 : term] : ((X172 = X173) | ((lam @ X172) != (lam @ X173))) | (laminj != $true)) & ((laminj = $true) | ? [X174 : term,X175 : term] : ((X174 != X175) & ((lam @ X174) = (lam @ X175)))) & ((hoasinduction_lem3aaa = $true) | ($true != sP22)) & (($true = sP22) | (hoasinduction_lem3aaa != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X176 : d_subst,X177 : d_subst] : ((X176 = X177) | ((d2subst @ X177) != (d2subst @ X176))) & (($true = sP21) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP21)) & ((pushprop_lem1v2_lthm = $true) | (? [X178 : term > $o,X179 : term,X180 : term > $o,X181 : subst] : (($true = (X180 @ X179)) & ($true != (X178 @ one)) & ! [X182 : term] : ((X180 @ (sub @ X182 @ (push @ X179 @ X181))) = (X178 @ X182))) & ! [X183 : subst,X184 : term] : ((sub @ one @ (push @ X184 @ X183)) = X184))) & (! [X185 : term > $o,X186 : term,X187 : term > $o,X188 : subst] : (($true != (X187 @ X186)) | ($true = (X185 @ one)) | ? [X189 : term] : ((X187 @ (sub @ X189 @ (push @ X186 @ X188))) != (X185 @ X189))) | ? [X190 : subst,X191 : term] : ((sub @ one @ (push @ X191 @ X190)) != X191) | (pushprop_lem1v2_lthm != $true)) & (hoaslamnotvar = $true) & (($true = sP15) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP15)) & ((sP12 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP12 != $true)) & ((substmonoid_lthm = $true) | ((? [X192 : subst] : ((comp @ id @ X192) != X192) | ? [X193 : subst] : ((comp @ X193 @ id) != X193)) & ! [X194 : subst] : ((comp @ X194 @ id) = X194) & ! [X195 : subst] : ((comp @ id @ X195) = X195))) & ((! [X196 : subst] : ((comp @ id @ X196) = X196) & ! [X197 : subst] : ((comp @ X197 @ id) = X197)) | ? [X198 : subst] : ((comp @ X198 @ id) != X198) | ? [X199 : subst] : ((comp @ id @ X199) != X199) | (substmonoid_lthm != $true)) & ((axshiftcons = $true) | ? [X200 : subst,X201 : term] : ((comp @ sh @ (push @ X201 @ X200)) != X200)) & (! [X202 : subst,X203 : term] : ((comp @ sh @ (push @ X203 @ X202)) = X202) | (axshiftcons != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X204 : term] : ! [X205 : subst > term > term] : ((sub @ X204 @ (push @ one @ sh)) != (X205 @ sh @ one))) & (! [X206 : term] : ? [X207 : subst > term > term] : ((sub @ X206 @ (push @ one @ sh)) = (X207 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & (! [X208 : term,X209 : term > $o,X210 : term,X211 : subst,X212 : term > $o] : (($true != (X212 @ X208)) | ($true = (X209 @ (sub @ X208 @ (push @ X210 @ X211))))) | ? [X213 : subst,X214 : term] : ((sub @ one @ (push @ X214 @ X213)) != X214) | ? [X215 : subst,X216 : term] : ((comp @ sh @ (push @ X216 @ X215)) != X215) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X217 : term,X218 : term > $o,X219 : term,X220 : subst,X221 : term > $o] : (($true = (X221 @ X217)) & ($true != (X218 @ (sub @ X217 @ (push @ X219 @ X220))))) & ! [X222 : subst,X223 : term] : ((sub @ one @ (push @ X223 @ X222)) = X223) & ! [X224 : subst,X225 : term] : ((comp @ sh @ (push @ X225 @ X224)) = X224))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (ulamvarsh = $true) & ((induction2lem_lthm = $true) | ($true != sP8)) & (($true = sP8) | (induction2lem_lthm != $true)) & ((hoasinduction_lem2v2 = $true) | (sP4 != $true)) & ((sP4 = $true) | (hoasinduction_lem2v2 != $true)) & (ulamvarind = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem1_lthm = $true) & (! [X226 : term,X227 : term,X228 : term,X229 : term] : (((ap @ X229 @ X228) != (ap @ X226 @ X227)) | (X226 = X229)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X230 : term,X231 : term,X232 : term,X233 : term] : (((ap @ X233 @ X232) = (ap @ X230 @ X231)) & (X230 != X233))) & (pushprop_lthm_orig = $true) & (! [X234 : term] : ? [X235 : subst > term > term] : ((X235 @ sh @ one) = (sub @ X234 @ (push @ one @ sh))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X236 : term] : ! [X237 : subst > term > term] : ((X237 @ sh @ one) != (sub @ X236 @ (push @ one @ sh)))) & ((axidl = $true) | ? [X238 : subst] : ((comp @ id @ X238) != X238)) & (! [X239 : subst] : ((comp @ id @ X239) = X239) | (axidl != $true)) & (hoaslaminj_gthm = $true) & ((apinj2 = $true) | ? [X240 : term,X241 : term,X242 : term,X243 : term] : (((ap @ X242 @ X243) = (ap @ X240 @ X241)) & (X241 != X243))) & (! [X244 : term,X245 : term,X246 : term,X247 : term] : (((ap @ X246 @ X247) != (ap @ X244 @ X245)) | (X245 = X247)) | (apinj2 != $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)))))))))))))) & (! [X248 : subst] : ((push @ (sub @ one @ X248) @ (comp @ sh @ X248)) = X248) | (axscons != $true)) & ((axscons = $true) | ? [X249 : subst] : ((push @ (sub @ one @ X249) @ (comp @ sh @ X249)) != X249)) & ! [X250 : term] : ? [X251 : d_term] : ((d2term @ X251) = X250) & ((axvarcons = $true) | ? [X252 : term,X253 : subst] : ((sub @ one @ (push @ X252 @ X253)) != X252)) & (! [X254 : term,X255 : subst] : ((sub @ one @ (push @ X254 @ X255)) = X254) | (axvarcons != $true)) & (hoaslamnotvar_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasapinj2_lthm = $true) | (! [X256 : term,X257 : term,X258 : term,X259 : term] : (((ap @ X257 @ X256) != (ap @ X259 @ X258)) | (X256 = X258)) & ? [X260 : term,X261 : term,X262 : term,X263 : term] : (((ap @ (sub @ X262 @ id) @ X260) = (ap @ (sub @ X263 @ id) @ X261)) & (X260 != X261)))) & (? [X264 : term,X265 : term,X266 : term,X267 : term] : (((ap @ X265 @ X264) = (ap @ X267 @ X266)) & (X264 != X266)) | ! [X268 : term,X269 : term,X270 : term,X271 : term] : (((ap @ (sub @ X271 @ id) @ X269) != (ap @ (sub @ X270 @ id) @ X268)) | (X268 = X269)) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X272 : term] : ! [X273 : subst > term > term] : ? [X274 : subst,X275 : term] : ((X273 @ X274 @ X275) != (sub @ X272 @ (push @ X275 @ X274)))) & (! [X276 : term] : ? [X277 : subst > term > term] : ! [X278 : subst,X279 : term] : ((sub @ X276 @ (push @ X279 @ X278)) = (X277 @ X278 @ X279)) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoasinduction_lem2_lthm = $true) & (pushprop_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP2)) & (($true = sP2) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasapinj1_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (($true = sP1) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP1)) & (pushprop_lem0_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_gthm = $true) & (axmap = $true) & (induction2lem_gthm = $true) & (axabs = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ((d2subst @ d_id) = id)),
% 9.38/2.21 inference(rectify,[status(thm)],[f472])).
% 9.38/2.21 thf(f472,plain,(
% 9.38/2.21 (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ! [X156 : d_subst] : (d_id = X156) & (hoasinduction_lem1_gthm = $true) & ((shinj = $true) | ? [X185 : term,X186 : term] : (((sub @ X186 @ sh) = (sub @ X185 @ sh)) & (X185 != X186))) & (! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186)) | (shinj != $true)) & (lamnotvar = $true) & (? [X208 : term] : ((sub @ X208 @ id) != X208) | ! [X209 : term] : ((sub @ X209 @ id) = X209) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X208 : term] : ((sub @ X208 @ id) = X208) & ? [X209 : term] : ((sub @ X209 @ id) != X209))) & (($true = sP58) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP58)) & ((axvarid = $true) | ? [X265 : term] : ((sub @ X265 @ id) != X265)) & (! [X265 : term] : ((sub @ X265 @ id) = X265) | (axvarid != $true)) & (hoaslamnotvar_lthm = $true) & (apnotvar = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP57)) & (($true = sP57) | (hoasinduction_lem3aa != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) | ($true != sP56)) & (($true = sP56) | (hoaslaminj_lthm != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((axidr = $true) | ? [X393 : subst] : ((comp @ X393 @ id) != X393)) & (! [X393 : subst] : ((comp @ X393 @ id) = X393) | (axidr != $true)) & (($true = sP55) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP55)) & ! [X141 : d_term] : (d_one = X141) & (hoasinduction_lem3_gthm = $true) & ((sP53 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP53 != $true)) & (axapp = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP51 != $true)) & ((sP51 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoasapnotvar = $true) & (! [X378 : term,X379 : term,X380 : subst > term > term] : (((ap @ (sub @ X379 @ id) @ X378) != (lam @ (X380 @ sh @ one))) | ? [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) != (sub @ (X380 @ X382 @ X383) @ X381))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X378 : term,X379 : term,X380 : subst > term > term] : (((ap @ (sub @ X379 @ id) @ X378) = (lam @ (X380 @ sh @ one))) & ! [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) = (sub @ (X380 @ X382 @ X383) @ X381)))) & (axclos = $true) & (? [X399 : term,X400 : term,X401 : term,X402 : term] : ((X399 != X402) & ((ap @ X399 @ X400) = (ap @ X402 @ X401))) | ! [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 = X406) | ((ap @ (sub @ X405 @ id) @ X403) != (ap @ (sub @ X406 @ id) @ X404))) | ? [X398 : term] : ((sub @ X398 @ id) != X398) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X399 : term,X400 : term,X401 : term,X402 : term] : ((X399 = X402) | ((ap @ X399 @ X400) != (ap @ X402 @ X401))) & ? [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 != X406) & ((ap @ (sub @ X405 @ id) @ X403) = (ap @ (sub @ X406 @ id) @ X404))) & ! [X398 : term] : ((sub @ X398 @ id) = X398))) & (ulamvar1 != $true) & (! [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) != (ap @ (sub @ X394 @ id) @ X396)) | (X394 = X397)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) = (ap @ (sub @ X394 @ id) @ X396)) & (X394 != X397))) & (! [X54 : subst > term > subst > $o,X55 : term] : (? [X58 : subst > term > term] : (! [X59 : term] : (($true = (X54 @ id @ (X58 @ id @ X59) @ id)) | ($true != (X54 @ id @ X59 @ id))) & ($true != (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) & ! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61))) | ($true = (sP46 @ X54)) | ($true = (X54 @ id @ X55 @ id))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X54 : subst > term > subst > $o,X55 : term] : (! [X58 : subst > term > term] : (? [X59 : term] : (($true != (X54 @ id @ (X58 @ id @ X59) @ id)) & ($true = (X54 @ id @ X59 @ id))) | ($true = (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) | ? [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) != (sub @ (X58 @ X62 @ X60) @ X61))) & ($true != (sP46 @ X54)) & ($true != (X54 @ id @ X55 @ id)))) & (! [X276 : term] : ((sub @ X276 @ id) = X276) | (termmset != $true)) & ((termmset = $true) | ? [X276 : term] : ((sub @ X276 @ id) != X276)) & (induction2_gthm = $true) & (pushprop = $true) & (hoasinduction_lem2_gthm = $true) & (! [X21 : subst,X22 : term,X23 : term > $o] : ? [X24 : term > $o] : ! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21)))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X21 : subst,X22 : term,X23 : term > $o] : ! [X24 : term > $o] : ? [X25 : term] : ((X24 @ X25) != (X23 @ (sub @ X25 @ (push @ X22 @ X21))))) & ! [X337 : d_term,X338 : d_term] : ((X337 = X338) | ((d2term @ X337) != (d2term @ X338))) & ((var @ (d2term @ d_one)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (pushprop_lem2v2_gthm = $true) & (! [X187 : subst > term > subst > $o] : ? [X188 : term > $o] : ! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X187 : subst > term > subst > $o] : ! [X188 : term > $o] : ? [X189 : term] : ((X188 @ X189) != (X187 @ id @ X189 @ id))) & (($true = sP45) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP45)) & (($true = sP42) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP42)) & (hoasapnotvar_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lem0 = $true) | ? [X138 : subst > term > subst > $o] : ! [X139 : term > $o] : ? [X140 : term] : ((X138 @ id @ X140 @ id) != (X139 @ X140))) & (! [X138 : subst > term > subst > $o] : ? [X139 : term > $o] : ! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140)) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X95 : term] : ! [X96 : subst > term > term] : ? [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) != (sub @ X95 @ (push @ X97 @ X98)))) & (! [X95 : term] : ? [X96 : subst > term > term] : ! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98))) | (hoasinduction_lem3v2_f != $true)) & (lamnotap != $true) & (one = (d2term @ d_one)) & (axassoc = $true) & ((induction2 = $true) | ($true != sP41)) & (($true = sP41) | (induction2 != $true)) & (! [X429 : term,X430 : term > $o] : (? [X432 : term] : (($true = (X430 @ X432)) & ($true != (X430 @ (lam @ X432)))) | ($true = (X430 @ X429)) | ($true = (sP39 @ X430)) | ? [X431 : term] : (($true != (X430 @ X431)) & ($true = (var @ X431)))) | (induction != $true)) & ((induction = $true) | ? [X429 : term,X430 : term > $o] : (! [X432 : term] : (($true != (X430 @ X432)) | ($true = (X430 @ (lam @ X432)))) & ($true != (X430 @ X429)) & ($true != (sP39 @ X430)) & ! [X431 : term] : (($true = (X430 @ X431)) | ($true != (var @ X431))))) & ((pushprop_lem0_lthm = $true) | ? [X180 : term > $o,X181 : term,X182 : subst] : ! [X183 : term > $o] : ? [X184 : term] : ((X183 @ X184) != (X180 @ (sub @ X184 @ (push @ X181 @ X182))))) & (! [X180 : term > $o,X181 : term,X182 : subst] : ? [X183 : term > $o] : ! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182)))) | (pushprop_lem0_lthm != $true)) & (hoasinduction_lem1v2 = $true) & (pushprop_lthm = $true) & ((hoasinduction_lthm = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lthm != $true)) & ((! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X33 : subst] : ((comp @ X33 @ id) != X33) | ? [X34 : subst] : ((comp @ id @ X34) != X34)) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2 != $true)) & (! [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) = (X268 @ X267 @ X266)) | ((lam @ (X268 @ sh @ one)) != (lam @ (X269 @ sh @ one))) | ? [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) != (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) | ? [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) != (sub @ (X269 @ X270 @ X271) @ X272))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) != (X268 @ X267 @ X266)) & ((lam @ (X268 @ sh @ one)) = (lam @ (X269 @ sh @ one))) & ! [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) = (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) & ! [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) = (sub @ (X269 @ X270 @ X271) @ X272)))) & ((d2subst @ d_id) = sh) & (hoasinduction_lem1 = $true) & (($true = sP27) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP27)) & ((hoasapinj2 = $true) | ? [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) = (ap @ (sub @ X172 @ id) @ X175)) & (X174 != X175))) & (! [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)) | (X174 = X175)) | (hoasapinj2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (($true = (X243 @ X240)) & ! [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) = (X242 @ X244)) & ((X242 @ one) != $true))) & (! [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (($true != (X243 @ X240)) | ? [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) != (X242 @ X244)) | ((X242 @ one) = $true)) | (pushprop_lem1v2 != $true)) & (hoasapinj2_gthm = $true) & ((pushprop_lem1 = $true) | ? [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true = (X169 @ X167)) & ($true != (X168 @ (sub @ X167 @ (push @ X165 @ X166)))))) & (! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true != (X169 @ X167)) | ($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166))))) | (pushprop_lem1 != $true)) & (pushprop_lem3v2_lthm = $true) & (($true = sP25) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP25)) & (termmset_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP24)) & (($true = sP24) | (hoasinduction_lem3v2_lthm != $true)) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & (axvarshift = $true) & (hoasapnotvar_lthm = $true) & (! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) | (laminj != $true)) & ((laminj = $true) | ? [X0 : term,X1 : term] : ((X0 != X1) & ((lam @ X1) = (lam @ X0)))) & ((hoasinduction_lem3aaa = $true) | ($true != sP22)) & (($true = sP22) | (hoasinduction_lem3aaa != $true)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X312 : d_subst,X313 : d_subst] : ((X312 = X313) | ((d2subst @ X312) != (d2subst @ X313))) & (($true = sP21) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP21)) & ((pushprop_lem1v2_lthm = $true) | (? [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (((X30 @ X29) = $true) & ($true != (X28 @ one)) & ! [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) = (X28 @ X32))) & ! [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) = X27))) & (! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (((X30 @ X29) != $true) | ($true = (X28 @ one)) | ? [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) != (X28 @ X32))) | ? [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) != X27) | (pushprop_lem1v2_lthm != $true)) & (hoaslamnotvar = $true) & (($true = sP15) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP15)) & ((sP12 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP12 != $true)) & ((substmonoid_lthm = $true) | ((? [X523 : subst] : ((comp @ id @ X523) != X523) | ? [X524 : subst] : ((comp @ X524 @ id) != X524)) & ! [X522 : subst] : ((comp @ X522 @ id) = X522) & ! [X521 : subst] : ((comp @ id @ X521) = X521))) & ((! [X523 : subst] : ((comp @ id @ X523) = X523) & ! [X524 : subst] : ((comp @ X524 @ id) = X524)) | ? [X522 : subst] : ((comp @ X522 @ id) != X522) | ? [X521 : subst] : ((comp @ id @ X521) != X521) | (substmonoid_lthm != $true)) & ((axshiftcons = $true) | ? [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) != X198)) & (! [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) = X198) | (axshiftcons != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X263 : term] : ! [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) != (X264 @ sh @ one))) & (! [X263 : term] : ? [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) = (X264 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true)) & (! [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true != (X336 @ X332)) | ($true = (X333 @ (sub @ X332 @ (push @ X334 @ X335))))) | ? [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) != X329) | ? [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) != X330) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true = (X336 @ X332)) & ($true != (X333 @ (sub @ X332 @ (push @ X334 @ X335))))) & ! [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) = X329) & ! [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) = X330))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (ulamvarsh = $true) & ((induction2lem_lthm = $true) | ($true != sP8)) & (($true = sP8) | (induction2lem_lthm != $true)) & ((hoasinduction_lem2v2 = $true) | (sP4 != $true)) & ((sP4 = $true) | (hoasinduction_lem2v2 != $true)) & (ulamvarind = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem1_lthm = $true) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) = (ap @ X179 @ X178)) & (X176 != X179))) & (pushprop_lthm_orig = $true) & (! [X519 : term] : ? [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) = (X520 @ sh @ one)) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X519 : term] : ! [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) != (X520 @ sh @ one))) & ((axidl = $true) | ? [X223 : subst] : ((comp @ id @ X223) != X223)) & (! [X223 : subst] : ((comp @ id @ X223) = X223) | (axidl != $true)) & (hoaslaminj_gthm = $true) & ((apinj2 = $true) | ? [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) = (ap @ X425 @ X426)) & (X426 != X428))) & (! [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) != (ap @ X425 @ X426)) | (X426 = X428)) | (apinj2 != $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)))))))))))))) & (! [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) = X384) | (axscons != $true)) & ((axscons = $true) | ? [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) != X384)) & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & ((axvarcons = $true) | ? [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) != X245)) & (! [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) = X245) | (axvarcons != $true)) & (hoaslamnotvar_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasapinj2_lthm = $true) | (! [X190 : term,X191 : term,X192 : term,X193 : term] : (((ap @ X191 @ X190) != (ap @ X193 @ X192)) | (X190 = X192)) & ? [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) = (ap @ (sub @ X196 @ id) @ X194)) & (X194 != X195)))) & (? [X190 : term,X191 : term,X192 : term,X193 : term] : (((ap @ X191 @ X190) = (ap @ X193 @ X192)) & (X190 != X192)) | ! [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) != (ap @ (sub @ X196 @ id) @ X194)) | (X194 = X195)) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X314 : term] : ! [X315 : subst > term > term] : ? [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) != (X315 @ X316 @ X317))) & (! [X314 : term] : ? [X315 : subst > term > term] : ! [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) = (X315 @ X316 @ X317)) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoasinduction_lem2_lthm = $true) & (pushprop_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP2)) & (($true = sP2) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasapinj1_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (($true = sP1) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP1)) & (pushprop_lem0_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_gthm = $true) & (axmap = $true) & (induction2lem_gthm = $true) & (axabs = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ((d2subst @ d_id) = id)),
% 9.38/2.21 inference(flattening,[status(thm)],[f471])).
% 9.38/2.21 thf(f471,plain,(
% 9.38/2.21 (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ! [X156 : d_subst] : (d_id = X156) & (hoasinduction_lem1_gthm = $true) & (((shinj = $true) | ? [X185 : term,X186 : term] : (((sub @ X186 @ sh) = (sub @ X185 @ sh)) & (X185 != X186))) & (! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186)) | (shinj != $true))) & (lamnotvar = $true) & (((? [X208 : term] : ((sub @ X208 @ id) != X208) | ! [X209 : term] : ((sub @ X209 @ id) = X209)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X208 : term] : ((sub @ X208 @ id) = X208) & ? [X209 : term] : ((sub @ X209 @ id) != X209)))) & ((($true = sP58) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP58))) & (((axvarid = $true) | ? [X265 : term] : ((sub @ X265 @ id) != X265)) & (! [X265 : term] : ((sub @ X265 @ id) = X265) | (axvarid != $true))) & (hoaslamnotvar_lthm = $true) & (apnotvar = $true) & (((hoasinduction_lem3aa = $true) | ($true != sP57)) & (($true = sP57) | (hoasinduction_lem3aa != $true))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (((hoaslaminj_lthm = $true) | ($true != sP56)) & (($true = sP56) | (hoaslaminj_lthm != $true))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (((axidr = $true) | ? [X393 : subst] : ((comp @ X393 @ id) != X393)) & (! [X393 : subst] : ((comp @ X393 @ id) = X393) | (axidr != $true))) & ((($true = sP55) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP55))) & ! [X141 : d_term] : (d_one = X141) & (hoasinduction_lem3_gthm = $true) & (((sP53 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP53 != $true))) & (axapp = $true) & (((hoasinduction_no_psi_cond_lthm = $true) | (sP51 != $true)) & ((sP51 = $true) | (hoasinduction_no_psi_cond_lthm != $true))) & (hoasapnotvar = $true) & ((! [X378 : term,X379 : term,X380 : subst > term > term] : (((ap @ (sub @ X379 @ id) @ X378) != (lam @ (X380 @ sh @ one))) | ? [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) != (sub @ (X380 @ X382 @ X383) @ X381))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X378 : term,X379 : term,X380 : subst > term > term] : (((ap @ (sub @ X379 @ id) @ X378) = (lam @ (X380 @ sh @ one))) & ! [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) = (sub @ (X380 @ X382 @ X383) @ X381))))) & (axclos = $true) & (((? [X399 : term,X400 : term,X401 : term,X402 : term] : ((X399 != X402) & ((ap @ X399 @ X400) = (ap @ X402 @ X401))) | ! [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 = X406) | ((ap @ (sub @ X405 @ id) @ X403) != (ap @ (sub @ X406 @ id) @ X404))) | ? [X398 : term] : ((sub @ X398 @ id) != X398)) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X399 : term,X400 : term,X401 : term,X402 : term] : ((X399 = X402) | ((ap @ X399 @ X400) != (ap @ X402 @ X401))) & ? [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 != X406) & ((ap @ (sub @ X405 @ id) @ X403) = (ap @ (sub @ X406 @ id) @ X404))) & ! [X398 : term] : ((sub @ X398 @ id) = X398)))) & (ulamvar1 != $true) & ((! [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) != (ap @ (sub @ X394 @ id) @ X396)) | (X394 = X397)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) = (ap @ (sub @ X394 @ id) @ X396)) & (X394 != X397)))) & ((! [X54 : subst > term > subst > $o,X55 : term] : (? [X58 : subst > term > term] : (! [X59 : term] : (($true = (X54 @ id @ (X58 @ id @ X59) @ id)) | ($true != (X54 @ id @ X59 @ id))) & ($true != (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) & ! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61))) | ($true = (sP46 @ X54)) | ($true = (X54 @ id @ X55 @ id))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X54 : subst > term > subst > $o,X55 : term] : (! [X58 : subst > term > term] : (? [X59 : term] : (($true != (X54 @ id @ (X58 @ id @ X59) @ id)) & ($true = (X54 @ id @ X59 @ id))) | ($true = (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) | ? [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) != (sub @ (X58 @ X62 @ X60) @ X61))) & ($true != (sP46 @ X54)) & ($true != (X54 @ id @ X55 @ id))))) & ((! [X276 : term] : ((sub @ X276 @ id) = X276) | (termmset != $true)) & ((termmset = $true) | ? [X276 : term] : ((sub @ X276 @ id) != X276))) & (induction2_gthm = $true) & (pushprop = $true) & (hoasinduction_lem2_gthm = $true) & ((! [X21 : subst,X22 : term,X23 : term > $o] : ? [X24 : term > $o] : ! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21)))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X21 : subst,X22 : term,X23 : term > $o] : ! [X24 : term > $o] : ? [X25 : term] : ((X24 @ X25) != (X23 @ (sub @ X25 @ (push @ X22 @ X21)))))) & ! [X337 : d_term,X338 : d_term] : ((X337 = X338) | ((d2term @ X337) != (d2term @ X338))) & ((var @ (d2term @ d_one)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (pushprop_lem2v2_gthm = $true) & ((! [X187 : subst > term > subst > $o] : ? [X188 : term > $o] : ! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X187 : subst > term > subst > $o] : ! [X188 : term > $o] : ? [X189 : term] : ((X188 @ X189) != (X187 @ id @ X189 @ id)))) & ((($true = sP45) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP45))) & ((($true = sP42) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP42))) & (hoasapnotvar_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (((hoasinduction_lem0 = $true) | ? [X138 : subst > term > subst > $o] : ! [X139 : term > $o] : ? [X140 : term] : ((X138 @ id @ X140 @ id) != (X139 @ X140))) & (! [X138 : subst > term > subst > $o] : ? [X139 : term > $o] : ! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140)) | (hoasinduction_lem0 != $true))) & (((hoasinduction_lem3v2_f = $true) | ? [X95 : term] : ! [X96 : subst > term > term] : ? [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) != (sub @ X95 @ (push @ X97 @ X98)))) & (! [X95 : term] : ? [X96 : subst > term > term] : ! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98))) | (hoasinduction_lem3v2_f != $true))) & (lamnotap != $true) & (one = (d2term @ d_one)) & (axassoc = $true) & (((induction2 = $true) | ($true != sP41)) & (($true = sP41) | (induction2 != $true))) & ((! [X429 : term,X430 : term > $o] : (? [X432 : term] : (($true = (X430 @ X432)) & ($true != (X430 @ (lam @ X432)))) | ($true = (X430 @ X429)) | ($true = (sP39 @ X430)) | ? [X431 : term] : (($true != (X430 @ X431)) & ($true = (var @ X431)))) | (induction != $true)) & ((induction = $true) | ? [X429 : term,X430 : term > $o] : (! [X432 : term] : (($true != (X430 @ X432)) | ($true = (X430 @ (lam @ X432)))) & ($true != (X430 @ X429)) & ($true != (sP39 @ X430)) & ! [X431 : term] : (($true = (X430 @ X431)) | ($true != (var @ X431)))))) & (((pushprop_lem0_lthm = $true) | ? [X180 : term > $o,X181 : term,X182 : subst] : ! [X183 : term > $o] : ? [X184 : term] : ((X183 @ X184) != (X180 @ (sub @ X184 @ (push @ X181 @ X182))))) & (! [X180 : term > $o,X181 : term,X182 : subst] : ? [X183 : term > $o] : ! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182)))) | (pushprop_lem0_lthm != $true))) & (hoasinduction_lem1v2 = $true) & (pushprop_lthm = $true) & (((hoasinduction_lthm = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lthm != $true))) & (((! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X33 : subst] : ((comp @ X33 @ id) != X33) | ? [X34 : subst] : ((comp @ id @ X34) != X34)))) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (((hoasinduction_lem3v2 = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2 != $true))) & ((! [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) = (X268 @ X267 @ X266)) | ((lam @ (X268 @ sh @ one)) != (lam @ (X269 @ sh @ one))) | ? [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) != (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) | ? [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) != (sub @ (X269 @ X270 @ X271) @ X272))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) != (X268 @ X267 @ X266)) & ((lam @ (X268 @ sh @ one)) = (lam @ (X269 @ sh @ one))) & ! [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) = (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) & ! [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) = (sub @ (X269 @ X270 @ X271) @ X272))))) & ((d2subst @ d_id) = sh) & (hoasinduction_lem1 = $true) & ((($true = sP27) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP27))) & (((hoasapinj2 = $true) | ? [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) = (ap @ (sub @ X172 @ id) @ X175)) & (X174 != X175))) & (! [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)) | (X174 = X175)) | (hoasapinj2 != $true))) & (((pushprop_lem1v2 = $true) | ? [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (($true = (X243 @ X240)) & ! [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) = (X242 @ X244)) & ((X242 @ one) != $true))) & (! [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (($true != (X243 @ X240)) | ? [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) != (X242 @ X244)) | ((X242 @ one) = $true)) | (pushprop_lem1v2 != $true))) & (hoasapinj2_gthm = $true) & (((pushprop_lem1 = $true) | ? [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true = (X169 @ X167)) & ($true != (X168 @ (sub @ X167 @ (push @ X165 @ X166)))))) & (! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true != (X169 @ X167)) | ($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166))))) | (pushprop_lem1 != $true))) & (pushprop_lem3v2_lthm = $true) & ((($true = sP25) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP25))) & (termmset_gthm = $true) & (((hoasinduction_lem3v2_lthm = $true) | ($true != sP24)) & (($true = sP24) | (hoasinduction_lem3v2_lthm != $true))) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & (axvarshift = $true) & (hoasapnotvar_lthm = $true) & ((! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) | (laminj != $true)) & ((laminj = $true) | ? [X0 : term,X1 : term] : ((X0 != X1) & ((lam @ X1) = (lam @ X0))))) & (((hoasinduction_lem3aaa = $true) | ($true != sP22)) & (($true = sP22) | (hoasinduction_lem3aaa != $true))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X312 : d_subst,X313 : d_subst] : ((X312 = X313) | ((d2subst @ X312) != (d2subst @ X313))) & ((($true = sP21) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP21))) & (((pushprop_lem1v2_lthm = $true) | (? [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (((X30 @ X29) = $true) & ($true != (X28 @ one)) & ! [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) = (X28 @ X32))) & ! [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) = X27))) & ((! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (((X30 @ X29) != $true) | ($true = (X28 @ one)) | ? [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) != (X28 @ X32))) | ? [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) != X27)) | (pushprop_lem1v2_lthm != $true))) & (hoaslamnotvar = $true) & ((($true = sP15) | (hoasinduction != $true)) & ((hoasinduction = $true) | ($true != sP15))) & (((sP12 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP12 != $true))) & (((substmonoid_lthm = $true) | ((? [X523 : subst] : ((comp @ id @ X523) != X523) | ? [X524 : subst] : ((comp @ X524 @ id) != X524)) & ! [X522 : subst] : ((comp @ X522 @ id) = X522) & ! [X521 : subst] : ((comp @ id @ X521) = X521))) & (((! [X523 : subst] : ((comp @ id @ X523) = X523) & ! [X524 : subst] : ((comp @ X524 @ id) = X524)) | ? [X522 : subst] : ((comp @ X522 @ id) != X522) | ? [X521 : subst] : ((comp @ id @ X521) != X521)) | (substmonoid_lthm != $true))) & (((axshiftcons = $true) | ? [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) != X198)) & (! [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) = X198) | (axshiftcons != $true))) & (((hoasinduction_lem3b_lthm = $true) | ? [X263 : term] : ! [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) != (X264 @ sh @ one))) & (! [X263 : term] : ? [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) = (X264 @ sh @ one)) | (hoasinduction_lem3b_lthm != $true))) & (((! [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true != (X336 @ X332)) | ($true = (X333 @ (sub @ X332 @ (push @ X334 @ X335))))) | ? [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) != X329) | ? [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) != X330)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true = (X336 @ X332)) & ($true != (X333 @ (sub @ X332 @ (push @ X334 @ X335))))) & ! [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) = X329) & ! [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) = X330)))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (ulamvarsh = $true) & (((induction2lem_lthm = $true) | ($true != sP8)) & (($true = sP8) | (induction2lem_lthm != $true))) & (((hoasinduction_lem2v2 = $true) | (sP4 != $true)) & ((sP4 = $true) | (hoasinduction_lem2v2 != $true))) & (ulamvarind = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem1_lthm = $true) & ((! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) = (ap @ X179 @ X178)) & (X176 != X179)))) & (pushprop_lthm_orig = $true) & ((! [X519 : term] : ? [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) = (X520 @ sh @ one)) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X519 : term] : ! [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) != (X520 @ sh @ one)))) & (((axidl = $true) | ? [X223 : subst] : ((comp @ id @ X223) != X223)) & (! [X223 : subst] : ((comp @ id @ X223) = X223) | (axidl != $true))) & (hoaslaminj_gthm = $true) & (((apinj2 = $true) | ? [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) = (ap @ X425 @ X426)) & (X426 != X428))) & (! [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) != (ap @ X425 @ X426)) | (X426 = X428)) | (apinj2 != $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)))))))))))))) & ((! [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) = X384) | (axscons != $true)) & ((axscons = $true) | ? [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) != X384))) & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & (((axvarcons = $true) | ? [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) != X245)) & (! [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) = X245) | (axvarcons != $true))) & (hoaslamnotvar_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((hoasapinj2_lthm = $true) | (! [X190 : term,X191 : term,X192 : term,X193 : term] : (((ap @ X191 @ X190) != (ap @ X193 @ X192)) | (X190 = X192)) & ? [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) = (ap @ (sub @ X196 @ id) @ X194)) & (X194 != X195)))) & ((? [X190 : term,X191 : term,X192 : term,X193 : term] : (((ap @ X191 @ X190) = (ap @ X193 @ X192)) & (X190 != X192)) | ! [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) != (ap @ (sub @ X196 @ id) @ X194)) | (X194 = X195))) | (hoasapinj2_lthm != $true))) & (pushprop_lem2v2 = $true) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X314 : term] : ! [X315 : subst > term > term] : ? [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) != (X315 @ X316 @ X317))) & (! [X314 : term] : ? [X315 : subst > term > term] : ! [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) = (X315 @ X316 @ X317)) | (hoasinduction_lem3v2_f_lthm != $true))) & (hoasinduction_lem2_lthm = $true) & (pushprop_gthm = $true) & (((hoasinduction_lem3v2a_lthm = $true) | ($true != sP2)) & (($true = sP2) | (hoasinduction_lem3v2a_lthm != $true))) & (hoasapinj1_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((($true = sP1) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP1))) & (pushprop_lem0_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_gthm = $true) & (axmap = $true) & (induction2lem_gthm = $true) & (axabs = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ((d2subst @ d_id) = id)),
% 9.38/2.21 inference(nnf_transformation,[status(thm)],[f73])).
% 9.38/2.21 thf(f73,plain,(
% 9.38/2.21 (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ! [X156 : d_subst] : (d_id = X156) & (hoasinduction_lem1_gthm = $true) & ((shinj = $true) <=> ! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186))) & (lamnotvar = $true) & ((? [X208 : term] : ((sub @ X208 @ id) != X208) | ! [X209 : term] : ((sub @ X209 @ id) = X209)) <=> (termmset_lthm = $true)) & (($true = sP58) <=> (hoasinduction_lem3v2a = $true)) & ((axvarid = $true) <=> ! [X265 : term] : ((sub @ X265 @ id) = X265)) & (hoaslamnotvar_lthm = $true) & (apnotvar = $true) & ((hoasinduction_lem3aa = $true) <=> ($true = sP57)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> ($true = sP56)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((axidr = $true) <=> ! [X393 : subst] : ((comp @ X393 @ id) = X393)) & (($true = sP55) <=> (induction2lem = $true)) & ! [X141 : d_term] : (d_one = X141) & (hoasinduction_lem3_gthm = $true) & ((sP53 = $true) <=> (hoasinduction_lem3a_lthm = $true)) & (axapp = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (sP51 = $true)) & (hoasapnotvar = $true) & (! [X378 : term,X379 : term,X380 : subst > term > term] : (((ap @ (sub @ X379 @ id) @ X378) != (lam @ (X380 @ sh @ one))) | ? [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) != (sub @ (X380 @ X382 @ X383) @ X381))) <=> (hoaslamnotap = $true)) & (axclos = $true) & ((? [X399 : term,X400 : term,X401 : term,X402 : term] : ((X399 != X402) & ((ap @ X399 @ X400) = (ap @ X402 @ X401))) | ! [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 = X406) | ((ap @ (sub @ X405 @ id) @ X403) != (ap @ (sub @ X406 @ id) @ X404))) | ? [X398 : term] : ((sub @ X398 @ id) != X398)) <=> (hoasapinj1_lthm = $true)) & (ulamvar1 != $true) & (! [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) != (ap @ (sub @ X394 @ id) @ X396)) | (X394 = X397)) <=> (hoasapinj1 = $true)) & (! [X54 : subst > term > subst > $o,X55 : term] : (? [X58 : subst > term > term] : (! [X59 : term] : (($true = (X54 @ id @ (X58 @ id @ X59) @ id)) | ($true != (X54 @ id @ X59 @ id))) & ($true != (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) & ! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61))) | ($true = (sP46 @ X54)) | ($true = (X54 @ id @ X55 @ id))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X276 : term] : ((sub @ X276 @ id) = X276) <=> (termmset = $true)) & (induction2_gthm = $true) & (pushprop = $true) & (hoasinduction_lem2_gthm = $true) & (! [X21 : subst,X22 : term,X23 : term > $o] : ? [X24 : term > $o] : ! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21)))) <=> (pushprop_lem0 = $true)) & ! [X337 : d_term,X338 : d_term] : ((X337 = X338) | ((d2term @ X337) != (d2term @ X338))) & ((var @ (d2term @ d_one)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (pushprop_lem2v2_gthm = $true) & (! [X187 : subst > term > subst > $o] : ? [X188 : term > $o] : ! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (($true = sP45) <=> (hoasinduction_lem3_lthm = $true)) & (($true = sP42) <=> (hoasinduction_lem3a = $true)) & (hoasapnotvar_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lem0 = $true) <=> ! [X138 : subst > term > subst > $o] : ? [X139 : term > $o] : ! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X95 : term] : ? [X96 : subst > term > term] : ! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98)))) & (lamnotap != $true) & (one = (d2term @ d_one)) & (axassoc = $true) & ((induction2 = $true) <=> ($true = sP41)) & (! [X429 : term,X430 : term > $o] : (? [X432 : term] : (($true = (X430 @ X432)) & ($true != (X430 @ (lam @ X432)))) | ($true = (X430 @ X429)) | ($true = (sP39 @ X430)) | ? [X431 : term] : (($true != (X430 @ X431)) & ($true = (var @ X431)))) <=> (induction = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X180 : term > $o,X181 : term,X182 : subst] : ? [X183 : term > $o] : ! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182))))) & (hoasinduction_lem1v2 = $true) & (pushprop_lthm = $true) & ((hoasinduction_lthm = $true) <=> ($true = sP38)) & ((! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34)) <=> (substmonoid = $true)) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ($true = sP29)) & (! [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) = (X268 @ X267 @ X266)) | ((lam @ (X268 @ sh @ one)) != (lam @ (X269 @ sh @ one))) | ? [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) != (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) | ? [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) != (sub @ (X269 @ X270 @ X271) @ X272))) <=> (hoaslaminj = $true)) & ((d2subst @ d_id) = sh) & (hoasinduction_lem1 = $true) & (($true = sP27) <=> (hoasinduction_lem3 = $true)) & ((hoasapinj2 = $true) <=> ! [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)) | (X174 = X175))) & ((pushprop_lem1v2 = $true) <=> ! [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (($true != (X243 @ X240)) | ? [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) != (X242 @ X244)) | ((X242 @ one) = $true))) & (hoasapinj2_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true != (X169 @ X167)) | ($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166)))))) & (pushprop_lem3v2_lthm = $true) & (($true = sP25) <=> (hoasinduction_lem3aa_lthm = $true)) & (termmset_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> ($true = sP24)) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & (axvarshift = $true) & (hoasapnotvar_lthm = $true) & (! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) <=> (laminj = $true)) & ((hoasinduction_lem3aaa = $true) <=> ($true = sP22)) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X312 : d_subst,X313 : d_subst] : ((X312 = X313) | ((d2subst @ X312) != (d2subst @ X313))) & (($true = sP21) <=> (hoasinduction_lthm_3 = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (((X30 @ X29) != $true) | ($true = (X28 @ one)) | ? [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) != (X28 @ X32))) | ? [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) != X27))) & (hoaslamnotvar = $true) & (($true = sP15) <=> (hoasinduction = $true)) & ((sP12 = $true) <=> (induction2_lthm = $true)) & ((substmonoid_lthm = $true) <=> ((! [X523 : subst] : ((comp @ id @ X523) = X523) & ! [X524 : subst] : ((comp @ X524 @ id) = X524)) | ? [X522 : subst] : ((comp @ X522 @ id) != X522) | ? [X521 : subst] : ((comp @ id @ X521) != X521))) & ((axshiftcons = $true) <=> ! [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) = X198)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X263 : term] : ? [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) = (X264 @ sh @ one))) & ((! [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true != (X336 @ X332)) | ($true = (X333 @ (sub @ X332 @ (push @ X334 @ X335))))) | ? [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) != X329) | ? [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) != X330)) <=> (pushprop_lem1_lthm = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (ulamvarsh = $true) & ((induction2lem_lthm = $true) <=> ($true = sP8)) & ((hoasinduction_lem2v2 = $true) <=> (sP4 = $true)) & (ulamvarind = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem1_lthm = $true) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179)) <=> (apinj1 = $true)) & (pushprop_lthm_orig = $true) & (! [X519 : term] : ? [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) = (X520 @ sh @ one)) <=> (hoasinduction_lem3b = $true)) & ((axidl = $true) <=> ! [X223 : subst] : ((comp @ id @ X223) = X223)) & (hoaslaminj_gthm = $true) & ((apinj2 = $true) <=> ! [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) != (ap @ X425 @ X426)) | (X426 = X428))) & (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)))))))))))))) & (! [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) = X384) <=> (axscons = $true)) & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & ((axvarcons = $true) <=> ! [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) = X245)) & (hoaslamnotvar_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (? [X190 : term,X191 : term,X192 : term,X193 : term] : (((ap @ X191 @ X190) = (ap @ X193 @ X192)) & (X190 != X192)) | ! [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) != (ap @ (sub @ X196 @ id) @ X194)) | (X194 = X195)))) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X314 : term] : ? [X315 : subst > term > term] : ! [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) = (X315 @ X316 @ X317))) & (hoasinduction_lem2_lthm = $true) & (pushprop_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> ($true = sP2)) & (hoasapinj1_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (($true = sP1) <=> (hoasinduction_lem2 = $true)) & (pushprop_lem0_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_gthm = $true) & (axmap = $true) & (induction2lem_gthm = $true) & (axabs = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ((d2subst @ d_id) = id)),
% 9.38/2.21 inference(definition_folding,[status(thm)],[f13,f72,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14])).
% 9.38/2.21 thf(f14,plain,(
% 9.38/2.21 ! [X210 : subst > term > subst > $o] : (($true = (sP0 @ X210)) <=> ? [X221 : term,X222 : term] : (($true = (X210 @ id @ X221 @ id)) & ($true != (X210 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id)) & ($true = (X210 @ id @ X222 @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f15,plain,(
% 9.38/2.21 ($true = sP1) <=> ! [X210 : subst > term > subst > $o,X211 : term,X212 : term] : (? [X217 : subst,X218 : subst,X219 : subst,X220 : term] : (((X210 @ (comp @ X219 @ X218) @ (sub @ X220 @ X218) @ X217) = $true) & ($true != (X210 @ X219 @ X220 @ (comp @ X218 @ X217)))) | ? [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (($true = (X210 @ X215 @ X213 @ (comp @ X216 @ X214))) & ((X210 @ (comp @ X215 @ X216) @ (sub @ X213 @ X216) @ X214) != $true)) | ((X210 @ id @ (ap @ X212 @ X211) @ id) = $true) | ($true != (X210 @ id @ X212 @ id)) | ($true != (X210 @ id @ X211 @ id)) | ($true = (sP0 @ X210)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f16,plain,(
% 9.38/2.21 ($true = sP2) <=> (! [X530 : term,X531 : term > $o,X532 : subst > term > subst > $o] : (? [X534 : subst > term > term] : (! [X535 : term] : (((X532 @ id @ (X534 @ id @ X535) @ id) = $true) | ($true != (X532 @ id @ X535 @ id))) & ! [X536 : subst,X537 : term,X538 : subst] : ((X534 @ (comp @ X538 @ X536) @ (sub @ X537 @ X536)) = (sub @ (X534 @ X538 @ X537) @ X536)) & ($true != (X532 @ id @ (lam @ (X534 @ sh @ one)) @ id))) | ? [X533 : term] : ((X532 @ id @ X533 @ id) != (X531 @ X533)) | ? [X539 : term] : (($true != (X531 @ (sub @ X530 @ (push @ X539 @ id)))) & ($true = (X531 @ X539))) | ($true = (X531 @ (lam @ X530)))) | ? [X529 : term] : ((sub @ X529 @ id) != X529) | ? [X525 : term] : ! [X526 : subst > term > term] : ? [X527 : subst,X528 : term] : ((X526 @ X527 @ X528) != (sub @ X525 @ (push @ X528 @ X527))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f17,plain,(
% 9.38/2.21 ! [X125 : subst > term > subst > $o] : (($true = (sP3 @ X125)) <=> ? [X127 : term,X128 : term] : (((X125 @ id @ (ap @ (sub @ X127 @ id) @ X128) @ id) != $true) & ($true = (X125 @ id @ X128 @ id)) & ($true = (X125 @ id @ X127 @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f18,plain,(
% 9.38/2.21 (sP4 = $true) <=> ! [X123 : term > $o,X124 : term,X125 : subst > term > subst > $o,X126 : term] : (((X123 @ (ap @ X126 @ X124)) = $true) | ? [X129 : subst,X130 : subst,X131 : term,X132 : subst] : (($true != (X125 @ X129 @ X131 @ (comp @ X132 @ X130))) & ($true = (X125 @ (comp @ X129 @ X132) @ (sub @ X131 @ X132) @ X130))) | ($true = (sP3 @ X125)) | ? [X133 : subst,X134 : subst,X135 : term,X136 : subst] : (($true != (X125 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134) @ X133)) & ((X125 @ X136 @ X135 @ (comp @ X134 @ X133)) = $true)) | ($true != (X123 @ X126)) | ($true != (X123 @ X124)) | ? [X137 : term] : ((X125 @ id @ X137 @ id) != (X123 @ X137)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f19,plain,(
% 9.38/2.21 ! [X13 : term > $o] : (($true = (sP5 @ X13)) <=> ? [X16 : term,X17 : term] : (($true != (X13 @ (ap @ X17 @ X16))) & ($true = (X13 @ X16)) & ($true = (X13 @ X17))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f20,plain,(
% 9.38/2.21 ! [X7 : term > $o] : (($true = (sP6 @ X7)) <=> ! [X11 : term,X12 : term] : (($true != (X7 @ X11)) | ($true != (X7 @ X12)) | ($true = (X7 @ (ap @ X11 @ X12)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f21,plain,(
% 9.38/2.21 ($true = sP7) <=> ! [X13 : term > $o,X14 : term,X15 : subst] : (($true = (sP5 @ X13)) | ($true = (X13 @ (sub @ X14 @ X15))) | ? [X18 : term] : (($true != (X13 @ (lam @ X18))) & ! [X19 : term] : (($true = (X13 @ (sub @ X18 @ (push @ X19 @ id)))) | ((X13 @ X19) != $true))) | ? [X20 : term] : (($true = (var @ X20)) & ($true != (X13 @ (sub @ X20 @ X15)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f22,plain,(
% 9.38/2.21 ($true = sP8) <=> (? [X7 : term > $o,X8 : term] : (((X7 @ X8) != $true) & ! [X9 : term] : (($true != (X7 @ X9)) | ((X7 @ (lam @ X9)) = $true)) & ! [X10 : term] : (($true = (X7 @ X10)) | ((var @ X10) != $true)) & ($true = (sP6 @ X7))) | ? [X2 : subst,X3 : term] : ((sub @ one @ (push @ X3 @ X2)) != X3) | ? [X4 : term,X5 : subst] : ((comp @ sh @ (push @ X4 @ X5)) != X5) | ? [X6 : subst] : ((comp @ X6 @ id) != X6) | ($true = sP7))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f23,plain,(
% 9.38/2.21 ! [X226 : term > $o] : (($true = (sP9 @ X226)) <=> ! [X231 : term,X232 : term] : (($true != (X226 @ X232)) | ((X226 @ X231) != $true) | ($true = (X226 @ (ap @ X232 @ X231)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f24,plain,(
% 9.38/2.21 ! [X234 : term > $o] : (($true = (sP10 @ X234)) <=> ? [X237 : term,X238 : term] : (($true = (X234 @ X238)) & ($true = (X234 @ X237)) & ($true != (X234 @ (ap @ X238 @ X237)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f25,plain,(
% 9.38/2.21 ($true = sP11) <=> ! [X233 : term,X234 : term > $o] : (($true = (sP10 @ X234)) | ? [X235 : term] : (($true != (X234 @ (lam @ X235))) & ! [X236 : term] : (($true != (X234 @ X236)) | ($true = (X234 @ (sub @ X235 @ (push @ X236 @ id)))))) | ($true = (X234 @ X233)) | ? [X239 : term] : (($true != (X234 @ X239)) & ($true = (var @ X239))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f26,plain,(
% 9.38/2.21 (sP12 = $true) <=> (? [X224 : term] : ((sub @ X224 @ id) != X224) | ($true = sP11) | ? [X225 : term,X226 : term > $o,X227 : subst] : (! [X228 : term] : (($true != (var @ X228)) | ($true = (X226 @ (sub @ X228 @ X227)))) & ($true != (X226 @ (sub @ X225 @ X227))) & ! [X229 : term] : (? [X230 : term] : (($true != (X226 @ (sub @ X229 @ (push @ X230 @ id)))) & ($true = (X226 @ X230))) | ($true = (X226 @ (lam @ X229)))) & ($true = (sP9 @ X226))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f27,plain,(
% 9.38/2.21 ! [X294 : subst > term > subst > $o] : (($true = (sP13 @ X294)) <=> ? [X296 : term,X297 : term] : (((X294 @ id @ X296 @ id) = $true) & ($true != (X294 @ id @ (ap @ (sub @ X297 @ id) @ X296) @ id)) & ((X294 @ id @ X297 @ id) = $true)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f28,plain,(
% 9.38/2.21 ! [X294 : subst > term > subst > $o] : (($true = (sP14 @ X294)) <=> ? [X302 : subst > term > term] : (! [X303 : term] : (($true = (X294 @ id @ (X302 @ id @ X303) @ id)) | ($true != (X294 @ id @ X303 @ id))) & ! [X304 : subst,X305 : subst,X306 : term] : ((X302 @ (comp @ X304 @ X305) @ (sub @ X306 @ X305)) = (sub @ (X302 @ X304 @ X306) @ X305)) & ($true != (X294 @ id @ (lam @ (X302 @ sh @ one)) @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f29,plain,(
% 9.38/2.21 ($true = sP15) <=> ! [X294 : subst > term > subst > $o,X295 : term] : (($true = (sP14 @ X294)) | ? [X307 : subst,X308 : term,X309 : subst,X310 : subst] : (($true != (X294 @ X310 @ X308 @ (comp @ X309 @ X307))) & ($true = (X294 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309) @ X307))) | ? [X298 : term,X299 : subst,X300 : subst,X301 : subst] : (($true != (X294 @ (comp @ X299 @ X301) @ (sub @ X298 @ X301) @ X300)) & ($true = (X294 @ X299 @ X298 @ (comp @ X301 @ X300)))) | ($true = (sP13 @ X294)) | ? [X311 : term] : (((var @ (sub @ X311 @ id)) = $true) & ($true != (X294 @ id @ X311 @ id))) | ($true = (X294 @ id @ X295 @ id)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f30,plain,(
% 9.38/2.21 ! [X342 : term > $o] : (($true = (sP16 @ X342)) <=> ! [X344 : term,X345 : term] : (((X342 @ X344) != $true) | ($true = (X342 @ (ap @ X345 @ X344))) | ($true != (X342 @ X345))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f31,plain,(
% 9.38/2.21 ! [X360 : subst > term > subst > $o] : (($true = (sP17 @ X360)) <=> ? [X371 : term,X372 : term] : (($true = (X360 @ id @ X371 @ id)) & ($true = (X360 @ id @ X372 @ id)) & ($true != (X360 @ id @ (ap @ (sub @ X372 @ id) @ X371) @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f32,plain,(
% 9.38/2.21 ! [X360 : subst > term > subst > $o] : (($true = (sP18 @ X360)) <=> ? [X366 : subst > term > term] : (($true != (X360 @ id @ (lam @ (X366 @ sh @ one)) @ id)) & ! [X370 : term] : (($true = (X360 @ id @ (X366 @ id @ X370) @ id)) | ($true != (X360 @ id @ X370 @ id))) & ! [X367 : subst,X368 : term,X369 : subst] : ((sub @ (X366 @ X369 @ X368) @ X367) = (X366 @ (comp @ X369 @ X367) @ (sub @ X368 @ X367)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f33,plain,(
% 9.38/2.21 (sP19 = $true) <=> ? [X350 : subst > term > subst > $o,X351 : term,X352 : term > $o] : (! [X355 : subst > term > term] : (($true = (X350 @ id @ (lam @ (X355 @ sh @ one)) @ id)) | ? [X356 : term] : (($true = (X350 @ id @ X356 @ id)) & ($true != (X350 @ id @ (X355 @ id @ X356) @ id))) | ? [X357 : subst,X358 : term,X359 : subst] : ((X355 @ (comp @ X357 @ X359) @ (sub @ X358 @ X359)) != (sub @ (X355 @ X357 @ X358) @ X359))) & ! [X354 : term] : (($true = (X352 @ (sub @ X351 @ (push @ X354 @ id)))) | ($true != (X352 @ X354))) & ($true != (X352 @ (lam @ X351))) & ! [X353 : term] : ((X352 @ X353) = (X350 @ id @ X353 @ id)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f34,plain,(
% 9.38/2.21 (sP20 = $true) <=> ! [X360 : subst > term > subst > $o,X361 : term] : (? [X374 : subst,X375 : subst,X376 : subst,X377 : term] : (($true != (X360 @ (comp @ X374 @ X375) @ (sub @ X377 @ X375) @ X376)) & ($true = (X360 @ X374 @ X377 @ (comp @ X375 @ X376)))) | ? [X373 : term] : (($true = (var @ (sub @ X373 @ id))) & ($true != (X360 @ id @ X373 @ id))) | ($true = (sP18 @ X360)) | ($true = (X360 @ id @ X361 @ id)) | ? [X362 : subst,X363 : subst,X364 : subst,X365 : term] : (($true = (X360 @ (comp @ X362 @ X363) @ (sub @ X365 @ X363) @ X364)) & ($true != (X360 @ X362 @ X365 @ (comp @ X363 @ X364)))) | ($true = (sP17 @ X360)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f35,plain,(
% 9.38/2.21 ($true = sP21) <=> (? [X339 : subst > term > subst > $o] : ! [X340 : term > $o] : ? [X341 : term] : ((X339 @ id @ X341 @ id) != (X340 @ X341)) | (sP20 = $true) | ? [X342 : term > $o,X343 : term] : (! [X348 : term] : (((X342 @ X348) = $true) | ($true != (var @ X348))) & ((X342 @ X343) != $true) & ($true = (sP16 @ X342)) & ! [X346 : term] : (($true = (X342 @ (lam @ X346))) | ? [X347 : term] : (($true != (X342 @ (sub @ X346 @ (push @ X347 @ id)))) & ($true = (X342 @ X347))))) | ? [X349 : term] : ((sub @ X349 @ id) != X349) | (sP19 = $true))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f36,plain,(
% 9.38/2.21 ($true = sP22) <=> ! [X142 : subst > term > subst > $o,X143 : term] : (? [X153 : term] : (($true = (X142 @ id @ X153 @ id)) & ($true != (X142 @ id @ (sub @ X143 @ (push @ X153 @ id)) @ id))) | ? [X144 : term,X145 : subst > term > term] : (((X142 @ id @ (lam @ (X145 @ sh @ one)) @ id) != $true) & ! [X149 : term] : (($true = (X142 @ id @ (X145 @ id @ X149) @ id)) | ($true != (X142 @ id @ X149 @ id))) & ! [X146 : subst,X147 : term,X148 : subst] : ((X145 @ (comp @ X146 @ X148) @ (sub @ X147 @ X148)) = (sub @ X144 @ (push @ (sub @ X147 @ X148) @ (comp @ X146 @ X148)))) & ! [X150 : term,X151 : subst,X152 : subst] : ((sub @ (X145 @ X152 @ X150) @ X151) = (sub @ (sub @ X144 @ (push @ X150 @ X152)) @ X151))) | ((X142 @ id @ (lam @ (sub @ X143 @ (push @ one @ sh))) @ id) = $true))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f37,plain,(
% 9.38/2.21 ! [X36 : subst > term > subst > $o] : (($true = (sP23 @ X36)) <=> ? [X44 : subst > term > term] : (($true != (X36 @ id @ (lam @ (X44 @ sh @ one)) @ id)) & ! [X45 : term,X46 : subst,X47 : subst] : ((sub @ (X44 @ X47 @ X45) @ X46) = (X44 @ (comp @ X47 @ X46) @ (sub @ X45 @ X46))) & ! [X48 : term] : (($true != (X36 @ id @ X48 @ id)) | ($true = (X36 @ id @ (X44 @ id @ X48) @ id)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f38,plain,(
% 9.38/2.21 ($true = sP24) <=> (? [X35 : term] : ((sub @ X35 @ id) != X35) | ! [X36 : subst > term > subst > $o,X37 : term,X38 : term > $o] : (? [X39 : term,X40 : subst,X41 : subst,X42 : subst] : (((X36 @ X42 @ X39 @ (comp @ X41 @ X40)) != $true) & ($true = (X36 @ (comp @ X42 @ X41) @ (sub @ X39 @ X41) @ X40))) | ? [X49 : term] : (($true = (X38 @ X49)) & ($true != (X38 @ (sub @ X37 @ (push @ X49 @ id))))) | ($true = (sP23 @ X36)) | ((X38 @ (lam @ X37)) = $true) | ? [X43 : term] : ((X36 @ id @ X43 @ id) != (X38 @ X43)) | ? [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true != (X36 @ (comp @ X53 @ X50) @ (sub @ X52 @ X50) @ X51)) & ((X36 @ X53 @ X52 @ (comp @ X50 @ X51)) = $true))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f39,plain,(
% 9.38/2.21 ($true = sP25) <=> ! [X200 : term,X201 : subst > term > subst > $o] : (? [X207 : term] : (($true = (X201 @ id @ X207 @ id)) & ($true != (X201 @ id @ (sub @ X200 @ (push @ X207 @ id)) @ id))) | ? [X202 : subst > term > term] : (($true != (X201 @ id @ (lam @ (X202 @ sh @ one)) @ id)) & ! [X204 : term,X205 : subst,X206 : subst] : ((sub @ (X202 @ X206 @ X204) @ X205) = (X202 @ (comp @ X206 @ X205) @ (sub @ X204 @ X205))) & ! [X203 : term] : (($true = (X201 @ id @ (X202 @ id @ X203) @ id)) | ($true != (X201 @ id @ X203 @ id)))) | ((X201 @ id @ (lam @ (sub @ X200 @ (push @ one @ sh))) @ id) = $true))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f40,plain,(
% 9.38/2.21 ! [X247 : subst > term > subst > $o] : (((sP26 @ X247) = $true) <=> ? [X253 : subst > term > term] : (! [X254 : subst,X255 : term,X256 : subst] : ((sub @ (X253 @ X256 @ X255) @ X254) = (X253 @ (comp @ X256 @ X254) @ (sub @ X255 @ X254))) & ($true != (X247 @ id @ (lam @ (X253 @ sh @ one)) @ id)) & ! [X257 : term] : (($true != (X247 @ id @ X257 @ id)) | ($true = (X247 @ id @ (X253 @ id @ X257) @ id)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f41,plain,(
% 9.38/2.21 ($true = sP27) <=> ! [X247 : subst > term > subst > $o,X248 : term] : (? [X258 : term,X259 : subst,X260 : subst,X261 : subst] : (($true = (X247 @ X261 @ X258 @ (comp @ X260 @ X259))) & ((X247 @ (comp @ X261 @ X260) @ (sub @ X258 @ X260) @ X259) != $true)) | ((X247 @ id @ (lam @ X248) @ id) = $true) | ? [X262 : term] : (((X247 @ id @ (sub @ X248 @ (push @ X262 @ id)) @ id) != $true) & ($true = (X247 @ id @ X262 @ id))) | ((sP26 @ X247) = $true) | ? [X249 : subst,X250 : subst,X251 : term,X252 : subst] : (($true = (X247 @ (comp @ X252 @ X249) @ (sub @ X251 @ X249) @ X250)) & ($true != (X247 @ X252 @ X251 @ (comp @ X249 @ X250)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f42,plain,(
% 9.38/2.21 ! [X408 : subst > term > subst > $o] : (($true = (sP28 @ X408)) <=> ? [X410 : subst > term > term] : (($true != (X408 @ id @ (lam @ (X410 @ sh @ one)) @ id)) & ! [X411 : term,X412 : subst,X413 : subst] : ((sub @ (X410 @ X412 @ X411) @ X413) = (X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413))) & ! [X414 : term] : (($true = (X408 @ id @ (X410 @ id @ X414) @ id)) | ($true != (X408 @ id @ X414 @ id)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f43,plain,(
% 9.38/2.21 ($true = sP29) <=> ! [X407 : term,X408 : subst > term > subst > $o,X409 : term > $o] : (? [X416 : subst,X417 : subst,X418 : term,X419 : subst] : (($true = (X408 @ (comp @ X419 @ X417) @ (sub @ X418 @ X417) @ X416)) & ($true != (X408 @ X419 @ X418 @ (comp @ X417 @ X416)))) | ($true = (sP28 @ X408)) | ((X409 @ (lam @ X407)) = $true) | ? [X420 : term] : ((X409 @ X420) != (X408 @ id @ X420 @ id)) | ? [X421 : subst,X422 : term,X423 : subst,X424 : subst] : (($true != (X408 @ (comp @ X424 @ X421) @ (sub @ X422 @ X421) @ X423)) & ($true = (X408 @ X424 @ X422 @ (comp @ X421 @ X423)))) | ? [X415 : term] : (($true != (X409 @ (sub @ X407 @ (push @ X415 @ id)))) & ($true = (X409 @ X415))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f44,plain,(
% 9.38/2.21 ! [X472 : subst > term > subst > $o] : (((sP30 @ X472) = $true) <=> ! [X483 : term,X484 : term] : (($true = (X472 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) | ($true != (X472 @ id @ X483 @ id)) | ((X472 @ id @ X484 @ id) != $true)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f45,plain,(
% 9.38/2.21 ! [X466 : term > $o] : (((sP31 @ X466) = $true) <=> ! [X470 : term,X471 : term] : (($true = (X466 @ (ap @ X470 @ X471))) | ($true != (X466 @ X470)) | ($true != (X466 @ X471))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f46,plain,(
% 9.38/2.21 ! [X486 : subst > term > subst > $o] : (($true = (sP32 @ X486)) <=> ! [X495 : subst > term > term] : (($true = (X486 @ id @ (lam @ (X495 @ sh @ one)) @ id)) | ? [X496 : term] : (($true != (X486 @ id @ (X495 @ id @ X496) @ id)) & ($true = (X486 @ id @ X496 @ id))) | ? [X497 : term,X498 : subst,X499 : subst] : ((X495 @ (comp @ X498 @ X499) @ (sub @ X497 @ X499)) != (sub @ (X495 @ X498 @ X497) @ X499))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f47,plain,(
% 9.38/2.21 ! [X502 : subst > term > subst > $o] : (($true = (sP33 @ X502)) <=> ? [X516 : term,X517 : term] : (((X502 @ id @ X517 @ id) = $true) & ((X502 @ id @ X516 @ id) = $true) & ($true != (X502 @ id @ (ap @ (sub @ X516 @ id) @ X517) @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f48,plain,(
% 9.38/2.21 ! [X502 : subst > term > subst > $o] : (($true = (sP34 @ X502)) <=> ? [X507 : subst > term > term] : (! [X508 : term] : (($true = (X502 @ id @ (X507 @ id @ X508) @ id)) | ((X502 @ id @ X508 @ id) != $true)) & ! [X509 : term,X510 : subst,X511 : subst] : ((X507 @ (comp @ X511 @ X510) @ (sub @ X509 @ X510)) = (sub @ (X507 @ X511 @ X509) @ X510)) & ($true != (X502 @ id @ (lam @ (X507 @ sh @ one)) @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f49,plain,(
% 9.38/2.21 ($true = sP35) <=> ? [X485 : term,X486 : subst > term > subst > $o] : (! [X491 : subst,X492 : term,X493 : subst,X494 : subst] : (((X486 @ (comp @ X493 @ X494) @ (sub @ X492 @ X494) @ X491) = $true) | ($true != (X486 @ X493 @ X492 @ (comp @ X494 @ X491)))) & ($true != (X486 @ id @ (lam @ X485) @ id)) & ! [X487 : subst,X488 : subst,X489 : subst,X490 : term] : (($true != (X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X487)) | ((X486 @ X488 @ X490 @ (comp @ X489 @ X487)) = $true)) & ($true = (sP32 @ X486)) & ! [X500 : term] : (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id) = $true) | ($true != (X486 @ id @ X500 @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f50,plain,(
% 9.38/2.21 ($true = sP36) <=> ! [X501 : term,X502 : subst > term > subst > $o] : (($true = (sP34 @ X502)) | ? [X518 : term] : (((X502 @ id @ X518 @ id) != $true) & ($true = (var @ (sub @ X518 @ id)))) | ? [X512 : term,X513 : subst,X514 : subst,X515 : subst] : (((X502 @ X515 @ X512 @ (comp @ X514 @ X513)) = $true) & ($true != (X502 @ (comp @ X515 @ X514) @ (sub @ X512 @ X514) @ X513))) | ? [X503 : subst,X504 : subst,X505 : term,X506 : subst] : (($true = (X502 @ (comp @ X504 @ X506) @ (sub @ X505 @ X506) @ X503)) & ($true != (X502 @ X504 @ X505 @ (comp @ X506 @ X503)))) | ($true = (sP33 @ X502)) | ((X502 @ id @ X501 @ id) = $true))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f51,plain,(
% 9.38/2.21 (sP37 = $true) <=> ? [X472 : subst > term > subst > $o,X473 : term,X474 : term] : (($true = (X472 @ id @ X473 @ id)) & ((sP30 @ X472) = $true) & ! [X479 : term,X480 : subst,X481 : subst,X482 : subst] : (($true != (X472 @ (comp @ X480 @ X481) @ (sub @ X479 @ X481) @ X482)) | ($true = (X472 @ X480 @ X479 @ (comp @ X481 @ X482)))) & ($true != (X472 @ id @ (ap @ X473 @ X474) @ id)) & ! [X475 : subst,X476 : term,X477 : subst,X478 : subst] : (($true = (X472 @ (comp @ X478 @ X477) @ (sub @ X476 @ X477) @ X475)) | ($true != (X472 @ X478 @ X476 @ (comp @ X477 @ X475)))) & ($true = (X472 @ id @ X474 @ id)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f52,plain,(
% 9.38/2.21 ($true = sP38) <=> (($true = sP36) | ($true = sP35) | ? [X465 : term,X466 : term > $o] : (($true != (X466 @ X465)) & ((sP31 @ X466) = $true) & ! [X467 : term] : (($true = (X466 @ (lam @ X467))) | ? [X468 : term] : (($true = (X466 @ X468)) & ((X466 @ (sub @ X467 @ (push @ X468 @ id))) != $true))) & ! [X469 : term] : (($true != (var @ X469)) | ((X466 @ X469) = $true))) | (sP37 = $true))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f53,plain,(
% 9.38/2.21 ! [X430 : term > $o] : (($true = (sP39 @ X430)) <=> ? [X433 : term,X434 : term] : (($true = (X430 @ X433)) & ((X430 @ (ap @ X433 @ X434)) != $true) & ($true = (X430 @ X434))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f54,plain,(
% 9.38/2.21 ! [X88 : term > $o] : (($true = (sP40 @ X88)) <=> ? [X93 : term,X94 : term] : (($true = (X88 @ X93)) & ($true != (X88 @ (ap @ X93 @ X94))) & ($true = (X88 @ X94))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f55,plain,(
% 9.38/2.21 ($true = sP41) <=> ! [X88 : term > $o,X89 : term] : (? [X91 : term] : (($true != (X88 @ (lam @ X91))) & ! [X92 : term] : (($true != (X88 @ X92)) | ($true = (X88 @ (sub @ X91 @ (push @ X92 @ id)))))) | ($true = (X88 @ X89)) | ? [X90 : term] : (((X88 @ X90) != $true) & ($true = (var @ X90))) | ($true = (sP40 @ X88)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f56,plain,(
% 9.38/2.21 ($true = sP42) <=> ! [X157 : subst > term > subst > $o,X158 : term] : (? [X159 : term] : (((X157 @ id @ (sub @ X158 @ (push @ X159 @ id)) @ id) != $true) & ($true = (X157 @ id @ X159 @ id))) | ($true = (X157 @ id @ (lam @ X158) @ id)) | ? [X160 : subst > term > term] : (! [X162 : term,X163 : subst,X164 : subst] : ((sub @ (X160 @ X163 @ X162) @ X164) = (X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164))) & ((X157 @ id @ (lam @ (X160 @ sh @ one)) @ id) != $true) & ! [X161 : term] : (((X157 @ id @ X161 @ id) != $true) | ($true = (X157 @ id @ (X160 @ id @ X161) @ id)))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f57,plain,(
% 9.38/2.21 ! [X72 : subst > term > subst > $o] : (($true = (sP43 @ X72)) <=> ? [X83 : subst > term > term] : (! [X84 : subst,X85 : term,X86 : subst] : ((X83 @ (comp @ X86 @ X84) @ (sub @ X85 @ X84)) = (sub @ (X83 @ X86 @ X85) @ X84)) & ! [X87 : term] : (((X72 @ id @ (X83 @ id @ X87) @ id) = $true) | ($true != (X72 @ id @ X87 @ id))) & ($true != (X72 @ id @ (lam @ (X83 @ sh @ one)) @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f58,plain,(
% 9.38/2.21 ($true = sP44) <=> ? [X64 : term,X65 : subst > term > subst > $o] : (($true != (X65 @ id @ (lam @ (sub @ X64 @ (push @ one @ sh))) @ id)) & ! [X66 : term] : (($true != (X65 @ id @ X66 @ id)) | ($true = (X65 @ id @ (sub @ X64 @ (push @ X66 @ id)) @ id))) & ! [X67 : subst > term > term] : (? [X71 : term] : (($true = (X65 @ id @ X71 @ id)) & ($true != (X65 @ id @ (X67 @ id @ X71) @ id))) | ? [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X67 @ X69 @ X70) @ X68) != (X67 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) | ($true = (X65 @ id @ (lam @ (X67 @ sh @ one)) @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f59,plain,(
% 9.38/2.21 ($true = sP45) <=> (($true = sP44) | ? [X63 : term] : ((sub @ X63 @ id) != X63) | ! [X72 : subst > term > subst > $o,X73 : term] : (($true = (X72 @ id @ (lam @ X73) @ id)) | ($true = (sP43 @ X72)) | ? [X74 : term] : (((X72 @ id @ X74 @ id) = $true) & ($true != (X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) | ? [X75 : subst,X76 : term,X77 : subst,X78 : subst] : (((X72 @ (comp @ X78 @ X77) @ (sub @ X76 @ X77) @ X75) != $true) & ($true = (X72 @ X78 @ X76 @ (comp @ X77 @ X75)))) | ? [X79 : term,X80 : subst,X81 : subst,X82 : subst] : (((X72 @ (comp @ X80 @ X82) @ (sub @ X79 @ X82) @ X81) = $true) & ($true != (X72 @ X80 @ X79 @ (comp @ X82 @ X81))))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f60,plain,(
% 9.38/2.21 ! [X54 : subst > term > subst > $o] : (($true = (sP46 @ X54)) <=> ? [X56 : term,X57 : term] : (($true != (X54 @ id @ (ap @ (sub @ X56 @ id) @ X57) @ id)) & ((X54 @ id @ X56 @ id) = $true) & ($true = (X54 @ id @ X57 @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f61,plain,(
% 9.38/2.21 ! [X456 : subst > term > subst > $o] : (($true = (sP47 @ X456)) <=> ? [X463 : term,X464 : term] : (((X456 @ id @ X464 @ id) = $true) & ((X456 @ id @ (ap @ (sub @ X463 @ id) @ X464) @ id) != $true) & ($true = (X456 @ id @ X463 @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f62,plain,(
% 9.38/2.21 ! [X439 : term > $o] : (($true = (sP48 @ X439)) <=> ! [X443 : term,X444 : term] : (((X439 @ X443) != $true) | ($true != (X439 @ X444)) | ((X439 @ (ap @ X444 @ X443)) = $true)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f63,plain,(
% 9.38/2.21 ($true = sP49) <=> ? [X446 : term > $o,X447 : subst > term > subst > $o,X448 : term] : (! [X451 : subst > term > term] : (($true = (X447 @ id @ (lam @ (X451 @ sh @ one)) @ id)) | ? [X452 : term,X453 : subst,X454 : subst] : ((sub @ (X451 @ X453 @ X452) @ X454) != (X451 @ (comp @ X453 @ X454) @ (sub @ X452 @ X454))) | ? [X455 : term] : (($true = (X447 @ id @ X455 @ id)) & ($true != (X447 @ id @ (X451 @ id @ X455) @ id)))) & ! [X450 : term] : (((X446 @ (sub @ X448 @ (push @ X450 @ id))) = $true) | ($true != (X446 @ X450))) & ! [X449 : term] : ((X447 @ id @ X449 @ id) = (X446 @ X449)) & ($true != (X446 @ (lam @ X448))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f64,plain,(
% 9.38/2.21 ($true = sP50) <=> ! [X456 : subst > term > subst > $o,X457 : term] : (? [X458 : subst > term > term] : (! [X459 : term] : (($true = (X456 @ id @ (X458 @ id @ X459) @ id)) | ($true != (X456 @ id @ X459 @ id))) & ! [X460 : term,X461 : subst,X462 : subst] : ((X458 @ (comp @ X462 @ X461) @ (sub @ X460 @ X461)) = (sub @ (X458 @ X462 @ X460) @ X461)) & ($true != (X456 @ id @ (lam @ (X458 @ sh @ one)) @ id))) | ($true = (X456 @ id @ X457 @ id)) | ($true = (sP47 @ X456)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f65,plain,(
% 9.38/2.21 (sP51 = $true) <=> (? [X445 : term] : ((sub @ X445 @ id) != X445) | ? [X438 : term,X439 : term > $o] : (! [X440 : term] : (($true != (var @ X440)) | ($true = (X439 @ X440))) & ! [X441 : term] : (? [X442 : term] : (($true != (X439 @ (sub @ X441 @ (push @ X442 @ id)))) & ($true = (X439 @ X442))) | ((X439 @ (lam @ X441)) = $true)) & ($true = (sP48 @ X439)) & ($true != (X439 @ X438))) | ($true = sP49) | ($true = sP50) | ? [X435 : subst > term > subst > $o] : ! [X436 : term > $o] : ? [X437 : term] : ((X436 @ X437) != (X435 @ id @ X437 @ id)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f66,plain,(
% 9.38/2.21 ($true = sP52) <=> ? [X278 : term,X279 : subst > term > subst > $o] : (! [X285 : term] : (($true = (X279 @ id @ (sub @ X278 @ (push @ X285 @ id)) @ id)) | ($true != (X279 @ id @ X285 @ id))) & ($true != (X279 @ id @ (lam @ (sub @ X278 @ (push @ one @ sh))) @ id)) & ! [X280 : subst > term > term] : (? [X281 : term] : (($true != (X279 @ id @ (X280 @ id @ X281) @ id)) & ((X279 @ id @ X281 @ id) = $true)) | ? [X282 : term,X283 : subst,X284 : subst] : ((sub @ (X280 @ X284 @ X282) @ X283) != (X280 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283))) | ($true = (X279 @ id @ (lam @ (X280 @ sh @ one)) @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f67,plain,(
% 9.38/2.21 (sP53 = $true) <=> (! [X286 : subst > term > subst > $o,X287 : term] : (? [X288 : term] : (($true = (X286 @ id @ X288 @ id)) & ((X286 @ id @ (sub @ X287 @ (push @ X288 @ id)) @ id) != $true)) | ($true = (X286 @ id @ (lam @ X287) @ id)) | ? [X289 : subst > term > term] : (! [X290 : term] : (((X286 @ id @ X290 @ id) != $true) | ($true = (X286 @ id @ (X289 @ id @ X290) @ id))) & ! [X291 : subst,X292 : subst,X293 : term] : ((sub @ (X289 @ X292 @ X293) @ X291) = (X289 @ (comp @ X292 @ X291) @ (sub @ X293 @ X291))) & ($true != (X286 @ id @ (lam @ (X289 @ sh @ one)) @ id)))) | ($true = sP52) | ? [X277 : term] : ((sub @ X277 @ id) != X277))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f68,plain,(
% 9.38/2.21 ! [X386 : term > $o] : (($true = (sP54 @ X386)) <=> ? [X391 : term,X392 : term] : (($true = (X386 @ X392)) & ($true = (X386 @ X391)) & ((X386 @ (ap @ X392 @ X391)) != $true)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f69,plain,(
% 9.38/2.21 ($true = sP55) <=> ! [X385 : subst,X386 : term > $o,X387 : term] : (? [X388 : term] : (! [X389 : term] : (($true = (X386 @ (sub @ X388 @ (push @ X389 @ id)))) | ((X386 @ X389) != $true)) & ($true != (X386 @ (lam @ X388)))) | ($true = (X386 @ (sub @ X387 @ X385))) | ? [X390 : term] : (($true = (var @ X390)) & ($true != (X386 @ (sub @ X390 @ X385)))) | ($true = (sP54 @ X386)))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f70,plain,(
% 9.38/2.21 ($true = sP56) <=> (? [X103 : term,X104 : term] : (((lam @ X103) = (lam @ X104)) & (X103 != X104)) | ? [X99 : term,X100 : subst] : ((sub @ one @ (push @ X99 @ X100)) != X99) | ? [X101 : term,X102 : subst] : ((comp @ sh @ (push @ X101 @ X102)) != X102) | ! [X105 : term,X106 : subst > term > term,X107 : subst > term > term,X108 : subst] : (? [X109 : subst,X110 : subst,X111 : term] : ((sub @ (X107 @ X109 @ X111) @ X110) != (X107 @ (comp @ X109 @ X110) @ (sub @ X111 @ X110))) | ((lam @ (X106 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ? [X112 : subst,X113 : term,X114 : subst] : ((X106 @ (comp @ X114 @ X112) @ (sub @ X113 @ X112)) != (sub @ (X106 @ X114 @ X113) @ X112)) | ((X106 @ X108 @ X105) = (X107 @ X108 @ X105))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f71,plain,(
% 9.38/2.21 ($true = sP57) <=> ! [X115 : subst > term > subst > $o,X116 : term] : (? [X117 : subst > term > term] : (! [X118 : subst,X119 : subst,X120 : term] : ((sub @ (X117 @ X118 @ X120) @ X119) = (X117 @ (comp @ X118 @ X119) @ (sub @ X120 @ X119))) & ! [X121 : term] : (((X115 @ id @ X121 @ id) != $true) | ($true = (X115 @ id @ (X117 @ id @ X121) @ id))) & ($true != (X115 @ id @ (lam @ (X117 @ sh @ one)) @ id))) | ($true = (X115 @ id @ (lam @ (sub @ X116 @ (push @ one @ sh))) @ id)) | ? [X122 : term] : (($true != (X115 @ id @ (sub @ X116 @ (push @ X122 @ id)) @ id)) & ($true = (X115 @ id @ X122 @ id))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f72,plain,(
% 9.38/2.21 ($true = sP58) <=> ! [X318 : subst > term > subst > $o,X319 : term,X320 : term > $o] : (? [X321 : subst > term > term] : (! [X325 : term] : (($true = (X318 @ id @ (X321 @ id @ X325) @ id)) | ((X318 @ id @ X325 @ id) != $true)) & ((X318 @ id @ (lam @ (X321 @ sh @ one)) @ id) != $true) & ! [X322 : subst,X323 : term,X324 : subst] : ((sub @ (X321 @ X322 @ X323) @ X324) = (X321 @ (comp @ X322 @ X324) @ (sub @ X323 @ X324)))) | ? [X326 : term] : ((X318 @ id @ X326 @ id) != (X320 @ X326)) | ? [X327 : term] : (($true != (X320 @ (sub @ X319 @ (push @ X327 @ id)))) & ($true = (X320 @ X327))) | ($true = (X320 @ (lam @ X319))))),
% 9.38/2.21 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 9.38/2.21 thf(f13,plain,(
% 9.38/2.21 (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ! [X156 : d_subst] : (d_id = X156) & (hoasinduction_lem1_gthm = $true) & ((shinj = $true) <=> ! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186))) & (lamnotvar = $true) & ((? [X208 : term] : ((sub @ X208 @ id) != X208) | ! [X209 : term] : ((sub @ X209 @ id) = X209)) <=> (termmset_lthm = $true)) & (! [X318 : subst > term > subst > $o,X319 : term,X320 : term > $o] : (? [X321 : subst > term > term] : (! [X325 : term] : (($true = (X318 @ id @ (X321 @ id @ X325) @ id)) | ((X318 @ id @ X325 @ id) != $true)) & ((X318 @ id @ (lam @ (X321 @ sh @ one)) @ id) != $true) & ! [X322 : subst,X323 : term,X324 : subst] : ((sub @ (X321 @ X322 @ X323) @ X324) = (X321 @ (comp @ X322 @ X324) @ (sub @ X323 @ X324)))) | ? [X326 : term] : ((X318 @ id @ X326 @ id) != (X320 @ X326)) | ? [X327 : term] : (($true != (X320 @ (sub @ X319 @ (push @ X327 @ id)))) & ($true = (X320 @ X327))) | ($true = (X320 @ (lam @ X319)))) <=> (hoasinduction_lem3v2a = $true)) & ((axvarid = $true) <=> ! [X265 : term] : ((sub @ X265 @ id) = X265)) & (hoaslamnotvar_lthm = $true) & (apnotvar = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X115 : subst > term > subst > $o,X116 : term] : (? [X117 : subst > term > term] : (! [X118 : subst,X119 : subst,X120 : term] : ((sub @ (X117 @ X118 @ X120) @ X119) = (X117 @ (comp @ X118 @ X119) @ (sub @ X120 @ X119))) & ! [X121 : term] : (((X115 @ id @ X121 @ id) != $true) | ($true = (X115 @ id @ (X117 @ id @ X121) @ id))) & ($true != (X115 @ id @ (lam @ (X117 @ sh @ one)) @ id))) | ($true = (X115 @ id @ (lam @ (sub @ X116 @ (push @ one @ sh))) @ id)) | ? [X122 : term] : (($true != (X115 @ id @ (sub @ X116 @ (push @ X122 @ id)) @ id)) & ($true = (X115 @ id @ X122 @ id))))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (? [X103 : term,X104 : term] : (((lam @ X103) = (lam @ X104)) & (X103 != X104)) | ? [X99 : term,X100 : subst] : ((sub @ one @ (push @ X99 @ X100)) != X99) | ? [X101 : term,X102 : subst] : ((comp @ sh @ (push @ X101 @ X102)) != X102) | ! [X105 : term,X106 : subst > term > term,X107 : subst > term > term,X108 : subst] : (? [X109 : subst,X110 : subst,X111 : term] : ((sub @ (X107 @ X109 @ X111) @ X110) != (X107 @ (comp @ X109 @ X110) @ (sub @ X111 @ X110))) | ((lam @ (X106 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ? [X112 : subst,X113 : term,X114 : subst] : ((X106 @ (comp @ X114 @ X112) @ (sub @ X113 @ X112)) != (sub @ (X106 @ X114 @ X113) @ X112)) | ((X106 @ X108 @ X105) = (X107 @ X108 @ X105))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((axidr = $true) <=> ! [X393 : subst] : ((comp @ X393 @ id) = X393)) & (! [X385 : subst,X386 : term > $o,X387 : term] : (? [X388 : term] : (! [X389 : term] : (($true = (X386 @ (sub @ X388 @ (push @ X389 @ id)))) | ((X386 @ X389) != $true)) & ($true != (X386 @ (lam @ X388)))) | ($true = (X386 @ (sub @ X387 @ X385))) | ? [X390 : term] : (($true = (var @ X390)) & ($true != (X386 @ (sub @ X390 @ X385)))) | ? [X391 : term,X392 : term] : (($true = (X386 @ X392)) & ($true = (X386 @ X391)) & ((X386 @ (ap @ X392 @ X391)) != $true))) <=> (induction2lem = $true)) & ! [X141 : d_term] : (d_one = X141) & (hoasinduction_lem3_gthm = $true) & ((! [X286 : subst > term > subst > $o,X287 : term] : (? [X288 : term] : (($true = (X286 @ id @ X288 @ id)) & ((X286 @ id @ (sub @ X287 @ (push @ X288 @ id)) @ id) != $true)) | ($true = (X286 @ id @ (lam @ X287) @ id)) | ? [X289 : subst > term > term] : (! [X290 : term] : (((X286 @ id @ X290 @ id) != $true) | ($true = (X286 @ id @ (X289 @ id @ X290) @ id))) & ! [X291 : subst,X292 : subst,X293 : term] : ((sub @ (X289 @ X292 @ X293) @ X291) = (X289 @ (comp @ X292 @ X291) @ (sub @ X293 @ X291))) & ($true != (X286 @ id @ (lam @ (X289 @ sh @ one)) @ id)))) | ? [X278 : term,X279 : subst > term > subst > $o] : (! [X285 : term] : (($true = (X279 @ id @ (sub @ X278 @ (push @ X285 @ id)) @ id)) | ($true != (X279 @ id @ X285 @ id))) & ($true != (X279 @ id @ (lam @ (sub @ X278 @ (push @ one @ sh))) @ id)) & ! [X280 : subst > term > term] : (? [X281 : term] : (($true != (X279 @ id @ (X280 @ id @ X281) @ id)) & ((X279 @ id @ X281 @ id) = $true)) | ? [X282 : term,X283 : subst,X284 : subst] : ((sub @ (X280 @ X284 @ X282) @ X283) != (X280 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283))) | ($true = (X279 @ id @ (lam @ (X280 @ sh @ one)) @ id)))) | ? [X277 : term] : ((sub @ X277 @ id) != X277)) <=> (hoasinduction_lem3a_lthm = $true)) & (axapp = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X445 : term] : ((sub @ X445 @ id) != X445) | ? [X438 : term,X439 : term > $o] : (! [X440 : term] : (($true != (var @ X440)) | ($true = (X439 @ X440))) & ! [X441 : term] : (? [X442 : term] : (($true != (X439 @ (sub @ X441 @ (push @ X442 @ id)))) & ($true = (X439 @ X442))) | ((X439 @ (lam @ X441)) = $true)) & ! [X443 : term,X444 : term] : (((X439 @ X443) != $true) | ($true != (X439 @ X444)) | ((X439 @ (ap @ X444 @ X443)) = $true)) & ($true != (X439 @ X438))) | ? [X446 : term > $o,X447 : subst > term > subst > $o,X448 : term] : (! [X451 : subst > term > term] : (($true = (X447 @ id @ (lam @ (X451 @ sh @ one)) @ id)) | ? [X452 : term,X453 : subst,X454 : subst] : ((sub @ (X451 @ X453 @ X452) @ X454) != (X451 @ (comp @ X453 @ X454) @ (sub @ X452 @ X454))) | ? [X455 : term] : (($true = (X447 @ id @ X455 @ id)) & ($true != (X447 @ id @ (X451 @ id @ X455) @ id)))) & ! [X450 : term] : (((X446 @ (sub @ X448 @ (push @ X450 @ id))) = $true) | ($true != (X446 @ X450))) & ! [X449 : term] : ((X447 @ id @ X449 @ id) = (X446 @ X449)) & ($true != (X446 @ (lam @ X448)))) | ! [X456 : subst > term > subst > $o,X457 : term] : (? [X458 : subst > term > term] : (! [X459 : term] : (($true = (X456 @ id @ (X458 @ id @ X459) @ id)) | ($true != (X456 @ id @ X459 @ id))) & ! [X460 : term,X461 : subst,X462 : subst] : ((X458 @ (comp @ X462 @ X461) @ (sub @ X460 @ X461)) = (sub @ (X458 @ X462 @ X460) @ X461)) & ($true != (X456 @ id @ (lam @ (X458 @ sh @ one)) @ id))) | ($true = (X456 @ id @ X457 @ id)) | ? [X463 : term,X464 : term] : (((X456 @ id @ X464 @ id) = $true) & ((X456 @ id @ (ap @ (sub @ X463 @ id) @ X464) @ id) != $true) & ($true = (X456 @ id @ X463 @ id)))) | ? [X435 : subst > term > subst > $o] : ! [X436 : term > $o] : ? [X437 : term] : ((X436 @ X437) != (X435 @ id @ X437 @ id)))) & (hoasapnotvar = $true) & (! [X378 : term,X379 : term,X380 : subst > term > term] : (((ap @ (sub @ X379 @ id) @ X378) != (lam @ (X380 @ sh @ one))) | ? [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) != (sub @ (X380 @ X382 @ X383) @ X381))) <=> (hoaslamnotap = $true)) & (axclos = $true) & ((? [X399 : term,X400 : term,X401 : term,X402 : term] : ((X399 != X402) & ((ap @ X399 @ X400) = (ap @ X402 @ X401))) | ! [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 = X406) | ((ap @ (sub @ X405 @ id) @ X403) != (ap @ (sub @ X406 @ id) @ X404))) | ? [X398 : term] : ((sub @ X398 @ id) != X398)) <=> (hoasapinj1_lthm = $true)) & (ulamvar1 != $true) & (! [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) != (ap @ (sub @ X394 @ id) @ X396)) | (X394 = X397)) <=> (hoasapinj1 = $true)) & (! [X54 : subst > term > subst > $o,X55 : term] : (? [X58 : subst > term > term] : (! [X59 : term] : (($true = (X54 @ id @ (X58 @ id @ X59) @ id)) | ($true != (X54 @ id @ X59 @ id))) & ($true != (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) & ! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61))) | ? [X56 : term,X57 : term] : (($true != (X54 @ id @ (ap @ (sub @ X56 @ id) @ X57) @ id)) & ((X54 @ id @ X56 @ id) = $true) & ($true = (X54 @ id @ X57 @ id))) | ($true = (X54 @ id @ X55 @ id))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X276 : term] : ((sub @ X276 @ id) = X276) <=> (termmset = $true)) & (induction2_gthm = $true) & (pushprop = $true) & (hoasinduction_lem2_gthm = $true) & (! [X21 : subst,X22 : term,X23 : term > $o] : ? [X24 : term > $o] : ! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21)))) <=> (pushprop_lem0 = $true)) & ! [X337 : d_term,X338 : d_term] : ((X337 = X338) | ((d2term @ X337) != (d2term @ X338))) & ((var @ (d2term @ d_one)) != $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (pushprop_lem2v2_gthm = $true) & (! [X187 : subst > term > subst > $o] : ? [X188 : term > $o] : ! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & ((? [X64 : term,X65 : subst > term > subst > $o] : (($true != (X65 @ id @ (lam @ (sub @ X64 @ (push @ one @ sh))) @ id)) & ! [X66 : term] : (($true != (X65 @ id @ X66 @ id)) | ($true = (X65 @ id @ (sub @ X64 @ (push @ X66 @ id)) @ id))) & ! [X67 : subst > term > term] : (? [X71 : term] : (($true = (X65 @ id @ X71 @ id)) & ($true != (X65 @ id @ (X67 @ id @ X71) @ id))) | ? [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X67 @ X69 @ X70) @ X68) != (X67 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) | ($true = (X65 @ id @ (lam @ (X67 @ sh @ one)) @ id)))) | ? [X63 : term] : ((sub @ X63 @ id) != X63) | ! [X72 : subst > term > subst > $o,X73 : term] : (($true = (X72 @ id @ (lam @ X73) @ id)) | ? [X83 : subst > term > term] : (! [X84 : subst,X85 : term,X86 : subst] : ((X83 @ (comp @ X86 @ X84) @ (sub @ X85 @ X84)) = (sub @ (X83 @ X86 @ X85) @ X84)) & ! [X87 : term] : (((X72 @ id @ (X83 @ id @ X87) @ id) = $true) | ($true != (X72 @ id @ X87 @ id))) & ($true != (X72 @ id @ (lam @ (X83 @ sh @ one)) @ id))) | ? [X74 : term] : (((X72 @ id @ X74 @ id) = $true) & ($true != (X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) | ? [X75 : subst,X76 : term,X77 : subst,X78 : subst] : (((X72 @ (comp @ X78 @ X77) @ (sub @ X76 @ X77) @ X75) != $true) & ($true = (X72 @ X78 @ X76 @ (comp @ X77 @ X75)))) | ? [X79 : term,X80 : subst,X81 : subst,X82 : subst] : (((X72 @ (comp @ X80 @ X82) @ (sub @ X79 @ X82) @ X81) = $true) & ($true != (X72 @ X80 @ X79 @ (comp @ X82 @ X81)))))) <=> (hoasinduction_lem3_lthm = $true)) & (! [X157 : subst > term > subst > $o,X158 : term] : (? [X159 : term] : (((X157 @ id @ (sub @ X158 @ (push @ X159 @ id)) @ id) != $true) & ($true = (X157 @ id @ X159 @ id))) | ($true = (X157 @ id @ (lam @ X158) @ id)) | ? [X160 : subst > term > term] : (! [X162 : term,X163 : subst,X164 : subst] : ((sub @ (X160 @ X163 @ X162) @ X164) = (X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164))) & ((X157 @ id @ (lam @ (X160 @ sh @ one)) @ id) != $true) & ! [X161 : term] : (((X157 @ id @ X161 @ id) != $true) | ($true = (X157 @ id @ (X160 @ id @ X161) @ id))))) <=> (hoasinduction_lem3a = $true)) & (hoasapnotvar_gthm = $true) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((hoasinduction_lem0 = $true) <=> ! [X138 : subst > term > subst > $o] : ? [X139 : term > $o] : ! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X95 : term] : ? [X96 : subst > term > term] : ! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98)))) & (lamnotap != $true) & (one = (d2term @ d_one)) & (axassoc = $true) & ((induction2 = $true) <=> ! [X88 : term > $o,X89 : term] : (? [X91 : term] : (($true != (X88 @ (lam @ X91))) & ! [X92 : term] : (($true != (X88 @ X92)) | ($true = (X88 @ (sub @ X91 @ (push @ X92 @ id)))))) | ($true = (X88 @ X89)) | ? [X90 : term] : (((X88 @ X90) != $true) & ($true = (var @ X90))) | ? [X93 : term,X94 : term] : (($true = (X88 @ X93)) & ($true != (X88 @ (ap @ X93 @ X94))) & ($true = (X88 @ X94))))) & (! [X429 : term,X430 : term > $o] : (? [X432 : term] : (($true = (X430 @ X432)) & ($true != (X430 @ (lam @ X432)))) | ($true = (X430 @ X429)) | ? [X433 : term,X434 : term] : (($true = (X430 @ X433)) & ((X430 @ (ap @ X433 @ X434)) != $true) & ($true = (X430 @ X434))) | ? [X431 : term] : (($true != (X430 @ X431)) & ($true = (var @ X431)))) <=> (induction = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X180 : term > $o,X181 : term,X182 : subst] : ? [X183 : term > $o] : ! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182))))) & (hoasinduction_lem1v2 = $true) & (pushprop_lthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X501 : term,X502 : subst > term > subst > $o] : (? [X507 : subst > term > term] : (! [X508 : term] : (($true = (X502 @ id @ (X507 @ id @ X508) @ id)) | ((X502 @ id @ X508 @ id) != $true)) & ! [X509 : term,X510 : subst,X511 : subst] : ((X507 @ (comp @ X511 @ X510) @ (sub @ X509 @ X510)) = (sub @ (X507 @ X511 @ X509) @ X510)) & ($true != (X502 @ id @ (lam @ (X507 @ sh @ one)) @ id))) | ? [X518 : term] : (((X502 @ id @ X518 @ id) != $true) & ($true = (var @ (sub @ X518 @ id)))) | ? [X512 : term,X513 : subst,X514 : subst,X515 : subst] : (((X502 @ X515 @ X512 @ (comp @ X514 @ X513)) = $true) & ($true != (X502 @ (comp @ X515 @ X514) @ (sub @ X512 @ X514) @ X513))) | ? [X503 : subst,X504 : subst,X505 : term,X506 : subst] : (($true = (X502 @ (comp @ X504 @ X506) @ (sub @ X505 @ X506) @ X503)) & ($true != (X502 @ X504 @ X505 @ (comp @ X506 @ X503)))) | ? [X516 : term,X517 : term] : (((X502 @ id @ X517 @ id) = $true) & ((X502 @ id @ X516 @ id) = $true) & ($true != (X502 @ id @ (ap @ (sub @ X516 @ id) @ X517) @ id))) | ((X502 @ id @ X501 @ id) = $true)) | ? [X485 : term,X486 : subst > term > subst > $o] : (! [X491 : subst,X492 : term,X493 : subst,X494 : subst] : (((X486 @ (comp @ X493 @ X494) @ (sub @ X492 @ X494) @ X491) = $true) | ($true != (X486 @ X493 @ X492 @ (comp @ X494 @ X491)))) & ($true != (X486 @ id @ (lam @ X485) @ id)) & ! [X487 : subst,X488 : subst,X489 : subst,X490 : term] : (($true != (X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X487)) | ((X486 @ X488 @ X490 @ (comp @ X489 @ X487)) = $true)) & ! [X495 : subst > term > term] : (($true = (X486 @ id @ (lam @ (X495 @ sh @ one)) @ id)) | ? [X496 : term] : (($true != (X486 @ id @ (X495 @ id @ X496) @ id)) & ($true = (X486 @ id @ X496 @ id))) | ? [X497 : term,X498 : subst,X499 : subst] : ((X495 @ (comp @ X498 @ X499) @ (sub @ X497 @ X499)) != (sub @ (X495 @ X498 @ X497) @ X499))) & ! [X500 : term] : (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id) = $true) | ($true != (X486 @ id @ X500 @ id)))) | ? [X465 : term,X466 : term > $o] : (($true != (X466 @ X465)) & ! [X470 : term,X471 : term] : (($true = (X466 @ (ap @ X470 @ X471))) | ($true != (X466 @ X470)) | ($true != (X466 @ X471))) & ! [X467 : term] : (($true = (X466 @ (lam @ X467))) | ? [X468 : term] : (($true = (X466 @ X468)) & ((X466 @ (sub @ X467 @ (push @ X468 @ id))) != $true))) & ! [X469 : term] : (($true != (var @ X469)) | ((X466 @ X469) = $true))) | ? [X472 : subst > term > subst > $o,X473 : term,X474 : term] : (($true = (X472 @ id @ X473 @ id)) & ! [X483 : term,X484 : term] : (($true = (X472 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) | ($true != (X472 @ id @ X483 @ id)) | ((X472 @ id @ X484 @ id) != $true)) & ! [X479 : term,X480 : subst,X481 : subst,X482 : subst] : (($true != (X472 @ (comp @ X480 @ X481) @ (sub @ X479 @ X481) @ X482)) | ($true = (X472 @ X480 @ X479 @ (comp @ X481 @ X482)))) & ($true != (X472 @ id @ (ap @ X473 @ X474) @ id)) & ! [X475 : subst,X476 : term,X477 : subst,X478 : subst] : (($true = (X472 @ (comp @ X478 @ X477) @ (sub @ X476 @ X477) @ X475)) | ($true != (X472 @ X478 @ X476 @ (comp @ X477 @ X475)))) & ($true = (X472 @ id @ X474 @ id))))) & ((! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34)) <=> (substmonoid = $true)) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X407 : term,X408 : subst > term > subst > $o,X409 : term > $o] : (? [X416 : subst,X417 : subst,X418 : term,X419 : subst] : (($true = (X408 @ (comp @ X419 @ X417) @ (sub @ X418 @ X417) @ X416)) & ($true != (X408 @ X419 @ X418 @ (comp @ X417 @ X416)))) | ? [X410 : subst > term > term] : (($true != (X408 @ id @ (lam @ (X410 @ sh @ one)) @ id)) & ! [X411 : term,X412 : subst,X413 : subst] : ((sub @ (X410 @ X412 @ X411) @ X413) = (X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413))) & ! [X414 : term] : (($true = (X408 @ id @ (X410 @ id @ X414) @ id)) | ($true != (X408 @ id @ X414 @ id)))) | ((X409 @ (lam @ X407)) = $true) | ? [X420 : term] : ((X409 @ X420) != (X408 @ id @ X420 @ id)) | ? [X421 : subst,X422 : term,X423 : subst,X424 : subst] : (($true != (X408 @ (comp @ X424 @ X421) @ (sub @ X422 @ X421) @ X423)) & ($true = (X408 @ X424 @ X422 @ (comp @ X421 @ X423)))) | ? [X415 : term] : (($true != (X409 @ (sub @ X407 @ (push @ X415 @ id)))) & ($true = (X409 @ X415))))) & (! [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) = (X268 @ X267 @ X266)) | ((lam @ (X268 @ sh @ one)) != (lam @ (X269 @ sh @ one))) | ? [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) != (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) | ? [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) != (sub @ (X269 @ X270 @ X271) @ X272))) <=> (hoaslaminj = $true)) & ((d2subst @ d_id) = sh) & (hoasinduction_lem1 = $true) & (! [X247 : subst > term > subst > $o,X248 : term] : (? [X258 : term,X259 : subst,X260 : subst,X261 : subst] : (($true = (X247 @ X261 @ X258 @ (comp @ X260 @ X259))) & ((X247 @ (comp @ X261 @ X260) @ (sub @ X258 @ X260) @ X259) != $true)) | ((X247 @ id @ (lam @ X248) @ id) = $true) | ? [X262 : term] : (((X247 @ id @ (sub @ X248 @ (push @ X262 @ id)) @ id) != $true) & ($true = (X247 @ id @ X262 @ id))) | ? [X253 : subst > term > term] : (! [X254 : subst,X255 : term,X256 : subst] : ((sub @ (X253 @ X256 @ X255) @ X254) = (X253 @ (comp @ X256 @ X254) @ (sub @ X255 @ X254))) & ($true != (X247 @ id @ (lam @ (X253 @ sh @ one)) @ id)) & ! [X257 : term] : (($true != (X247 @ id @ X257 @ id)) | ($true = (X247 @ id @ (X253 @ id @ X257) @ id)))) | ? [X249 : subst,X250 : subst,X251 : term,X252 : subst] : (($true = (X247 @ (comp @ X252 @ X249) @ (sub @ X251 @ X249) @ X250)) & ($true != (X247 @ X252 @ X251 @ (comp @ X249 @ X250))))) <=> (hoasinduction_lem3 = $true)) & ((hoasapinj2 = $true) <=> ! [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)) | (X174 = X175))) & ((pushprop_lem1v2 = $true) <=> ! [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (($true != (X243 @ X240)) | ? [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) != (X242 @ X244)) | ((X242 @ one) = $true))) & (hoasapinj2_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true != (X169 @ X167)) | ($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166)))))) & (pushprop_lem3v2_lthm = $true) & (! [X200 : term,X201 : subst > term > subst > $o] : (? [X207 : term] : (($true = (X201 @ id @ X207 @ id)) & ($true != (X201 @ id @ (sub @ X200 @ (push @ X207 @ id)) @ id))) | ? [X202 : subst > term > term] : (($true != (X201 @ id @ (lam @ (X202 @ sh @ one)) @ id)) & ! [X204 : term,X205 : subst,X206 : subst] : ((sub @ (X202 @ X206 @ X204) @ X205) = (X202 @ (comp @ X206 @ X205) @ (sub @ X204 @ X205))) & ! [X203 : term] : (($true = (X201 @ id @ (X202 @ id @ X203) @ id)) | ($true != (X201 @ id @ X203 @ id)))) | ((X201 @ id @ (lam @ (sub @ X200 @ (push @ one @ sh))) @ id) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & (termmset_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X35 : term] : ((sub @ X35 @ id) != X35) | ! [X36 : subst > term > subst > $o,X37 : term,X38 : term > $o] : (? [X39 : term,X40 : subst,X41 : subst,X42 : subst] : (((X36 @ X42 @ X39 @ (comp @ X41 @ X40)) != $true) & ($true = (X36 @ (comp @ X42 @ X41) @ (sub @ X39 @ X41) @ X40))) | ? [X49 : term] : (($true = (X38 @ X49)) & ($true != (X38 @ (sub @ X37 @ (push @ X49 @ id))))) | ? [X44 : subst > term > term] : (($true != (X36 @ id @ (lam @ (X44 @ sh @ one)) @ id)) & ! [X45 : term,X46 : subst,X47 : subst] : ((sub @ (X44 @ X47 @ X45) @ X46) = (X44 @ (comp @ X47 @ X46) @ (sub @ X45 @ X46))) & ! [X48 : term] : (($true != (X36 @ id @ X48 @ id)) | ($true = (X36 @ id @ (X44 @ id @ X48) @ id)))) | ((X38 @ (lam @ X37)) = $true) | ? [X43 : term] : ((X36 @ id @ X43 @ id) != (X38 @ X43)) | ? [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true != (X36 @ (comp @ X53 @ X50) @ (sub @ X52 @ X50) @ X51)) & ((X36 @ X53 @ X52 @ (comp @ X50 @ X51)) = $true))))) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & (axvarshift = $true) & (hoasapnotvar_lthm = $true) & (! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) <=> (laminj = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X142 : subst > term > subst > $o,X143 : term] : (? [X153 : term] : (($true = (X142 @ id @ X153 @ id)) & ($true != (X142 @ id @ (sub @ X143 @ (push @ X153 @ id)) @ id))) | ? [X144 : term,X145 : subst > term > term] : (((X142 @ id @ (lam @ (X145 @ sh @ one)) @ id) != $true) & ! [X149 : term] : (($true = (X142 @ id @ (X145 @ id @ X149) @ id)) | ($true != (X142 @ id @ X149 @ id))) & ! [X146 : subst,X147 : term,X148 : subst] : ((X145 @ (comp @ X146 @ X148) @ (sub @ X147 @ X148)) = (sub @ X144 @ (push @ (sub @ X147 @ X148) @ (comp @ X146 @ X148)))) & ! [X150 : term,X151 : subst,X152 : subst] : ((sub @ (X145 @ X152 @ X150) @ X151) = (sub @ (sub @ X144 @ (push @ X150 @ X152)) @ X151))) | ((X142 @ id @ (lam @ (sub @ X143 @ (push @ one @ sh))) @ id) = $true))) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & ! [X312 : d_subst,X313 : d_subst] : ((X312 = X313) | ((d2subst @ X312) != (d2subst @ X313))) & ((? [X339 : subst > term > subst > $o] : ! [X340 : term > $o] : ? [X341 : term] : ((X339 @ id @ X341 @ id) != (X340 @ X341)) | ! [X360 : subst > term > subst > $o,X361 : term] : (? [X374 : subst,X375 : subst,X376 : subst,X377 : term] : (($true != (X360 @ (comp @ X374 @ X375) @ (sub @ X377 @ X375) @ X376)) & ($true = (X360 @ X374 @ X377 @ (comp @ X375 @ X376)))) | ? [X373 : term] : (($true = (var @ (sub @ X373 @ id))) & ($true != (X360 @ id @ X373 @ id))) | ? [X366 : subst > term > term] : (($true != (X360 @ id @ (lam @ (X366 @ sh @ one)) @ id)) & ! [X370 : term] : (($true = (X360 @ id @ (X366 @ id @ X370) @ id)) | ($true != (X360 @ id @ X370 @ id))) & ! [X367 : subst,X368 : term,X369 : subst] : ((sub @ (X366 @ X369 @ X368) @ X367) = (X366 @ (comp @ X369 @ X367) @ (sub @ X368 @ X367)))) | ($true = (X360 @ id @ X361 @ id)) | ? [X362 : subst,X363 : subst,X364 : subst,X365 : term] : (($true = (X360 @ (comp @ X362 @ X363) @ (sub @ X365 @ X363) @ X364)) & ($true != (X360 @ X362 @ X365 @ (comp @ X363 @ X364)))) | ? [X371 : term,X372 : term] : (($true = (X360 @ id @ X371 @ id)) & ($true = (X360 @ id @ X372 @ id)) & ($true != (X360 @ id @ (ap @ (sub @ X372 @ id) @ X371) @ id)))) | ? [X342 : term > $o,X343 : term] : (! [X348 : term] : (((X342 @ X348) = $true) | ($true != (var @ X348))) & ((X342 @ X343) != $true) & ! [X344 : term,X345 : term] : (((X342 @ X344) != $true) | ($true = (X342 @ (ap @ X345 @ X344))) | ($true != (X342 @ X345))) & ! [X346 : term] : (($true = (X342 @ (lam @ X346))) | ? [X347 : term] : (($true != (X342 @ (sub @ X346 @ (push @ X347 @ id)))) & ($true = (X342 @ X347))))) | ? [X349 : term] : ((sub @ X349 @ id) != X349) | ? [X350 : subst > term > subst > $o,X351 : term,X352 : term > $o] : (! [X355 : subst > term > term] : (($true = (X350 @ id @ (lam @ (X355 @ sh @ one)) @ id)) | ? [X356 : term] : (($true = (X350 @ id @ X356 @ id)) & ($true != (X350 @ id @ (X355 @ id @ X356) @ id))) | ? [X357 : subst,X358 : term,X359 : subst] : ((X355 @ (comp @ X357 @ X359) @ (sub @ X358 @ X359)) != (sub @ (X355 @ X357 @ X358) @ X359))) & ! [X354 : term] : (($true = (X352 @ (sub @ X351 @ (push @ X354 @ id)))) | ($true != (X352 @ X354))) & ($true != (X352 @ (lam @ X351))) & ! [X353 : term] : ((X352 @ X353) = (X350 @ id @ X353 @ id)))) <=> (hoasinduction_lthm_3 = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (((X30 @ X29) != $true) | ($true = (X28 @ one)) | ? [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) != (X28 @ X32))) | ? [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) != X27))) & (hoaslamnotvar = $true) & (! [X294 : subst > term > subst > $o,X295 : term] : (? [X302 : subst > term > term] : (! [X303 : term] : (($true = (X294 @ id @ (X302 @ id @ X303) @ id)) | ($true != (X294 @ id @ X303 @ id))) & ! [X304 : subst,X305 : subst,X306 : term] : ((X302 @ (comp @ X304 @ X305) @ (sub @ X306 @ X305)) = (sub @ (X302 @ X304 @ X306) @ X305)) & ($true != (X294 @ id @ (lam @ (X302 @ sh @ one)) @ id))) | ? [X307 : subst,X308 : term,X309 : subst,X310 : subst] : (($true != (X294 @ X310 @ X308 @ (comp @ X309 @ X307))) & ($true = (X294 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309) @ X307))) | ? [X298 : term,X299 : subst,X300 : subst,X301 : subst] : (($true != (X294 @ (comp @ X299 @ X301) @ (sub @ X298 @ X301) @ X300)) & ($true = (X294 @ X299 @ X298 @ (comp @ X301 @ X300)))) | ? [X296 : term,X297 : term] : (((X294 @ id @ X296 @ id) = $true) & ($true != (X294 @ id @ (ap @ (sub @ X297 @ id) @ X296) @ id)) & ((X294 @ id @ X297 @ id) = $true)) | ? [X311 : term] : (((var @ (sub @ X311 @ id)) = $true) & ($true != (X294 @ id @ X311 @ id))) | ($true = (X294 @ id @ X295 @ id))) <=> (hoasinduction = $true)) & ((? [X224 : term] : ((sub @ X224 @ id) != X224) | ! [X233 : term,X234 : term > $o] : (? [X237 : term,X238 : term] : (($true = (X234 @ X238)) & ($true = (X234 @ X237)) & ($true != (X234 @ (ap @ X238 @ X237)))) | ? [X235 : term] : (($true != (X234 @ (lam @ X235))) & ! [X236 : term] : (($true != (X234 @ X236)) | ($true = (X234 @ (sub @ X235 @ (push @ X236 @ id)))))) | ($true = (X234 @ X233)) | ? [X239 : term] : (($true != (X234 @ X239)) & ($true = (var @ X239)))) | ? [X225 : term,X226 : term > $o,X227 : subst] : (! [X228 : term] : (($true != (var @ X228)) | ($true = (X226 @ (sub @ X228 @ X227)))) & ($true != (X226 @ (sub @ X225 @ X227))) & ! [X229 : term] : (? [X230 : term] : (($true != (X226 @ (sub @ X229 @ (push @ X230 @ id)))) & ($true = (X226 @ X230))) | ($true = (X226 @ (lam @ X229)))) & ! [X231 : term,X232 : term] : (($true != (X226 @ X232)) | ((X226 @ X231) != $true) | ($true = (X226 @ (ap @ X232 @ X231)))))) <=> (induction2_lthm = $true)) & ((substmonoid_lthm = $true) <=> ((! [X523 : subst] : ((comp @ id @ X523) = X523) & ! [X524 : subst] : ((comp @ X524 @ id) = X524)) | ? [X522 : subst] : ((comp @ X522 @ id) != X522) | ? [X521 : subst] : ((comp @ id @ X521) != X521))) & ((axshiftcons = $true) <=> ! [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) = X198)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X263 : term] : ? [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) = (X264 @ sh @ one))) & ((! [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true != (X336 @ X332)) | ($true = (X333 @ (sub @ X332 @ (push @ X334 @ X335))))) | ? [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) != X329) | ? [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) != X330)) <=> (pushprop_lem1_lthm = $true)) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (ulamvarsh = $true) & ((induction2lem_lthm = $true) <=> (? [X7 : term > $o,X8 : term] : (((X7 @ X8) != $true) & ! [X9 : term] : (($true != (X7 @ X9)) | ((X7 @ (lam @ X9)) = $true)) & ! [X10 : term] : (($true = (X7 @ X10)) | ((var @ X10) != $true)) & ! [X11 : term,X12 : term] : (($true != (X7 @ X11)) | ($true != (X7 @ X12)) | ($true = (X7 @ (ap @ X11 @ X12))))) | ? [X2 : subst,X3 : term] : ((sub @ one @ (push @ X3 @ X2)) != X3) | ? [X4 : term,X5 : subst] : ((comp @ sh @ (push @ X4 @ X5)) != X5) | ? [X6 : subst] : ((comp @ X6 @ id) != X6) | ! [X13 : term > $o,X14 : term,X15 : subst] : (? [X16 : term,X17 : term] : (($true != (X13 @ (ap @ X17 @ X16))) & ($true = (X13 @ X16)) & ($true = (X13 @ X17))) | ($true = (X13 @ (sub @ X14 @ X15))) | ? [X18 : term] : (($true != (X13 @ (lam @ X18))) & ! [X19 : term] : (($true = (X13 @ (sub @ X18 @ (push @ X19 @ id)))) | ((X13 @ X19) != $true))) | ? [X20 : term] : (($true = (var @ X20)) & ($true != (X13 @ (sub @ X20 @ X15))))))) & ((hoasinduction_lem2v2 = $true) <=> ! [X123 : term > $o,X124 : term,X125 : subst > term > subst > $o,X126 : term] : (((X123 @ (ap @ X126 @ X124)) = $true) | ? [X129 : subst,X130 : subst,X131 : term,X132 : subst] : (($true != (X125 @ X129 @ X131 @ (comp @ X132 @ X130))) & ($true = (X125 @ (comp @ X129 @ X132) @ (sub @ X131 @ X132) @ X130))) | ? [X127 : term,X128 : term] : (((X125 @ id @ (ap @ (sub @ X127 @ id) @ X128) @ id) != $true) & ($true = (X125 @ id @ X128 @ id)) & ($true = (X125 @ id @ X127 @ id))) | ? [X133 : subst,X134 : subst,X135 : term,X136 : subst] : (($true != (X125 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134) @ X133)) & ((X125 @ X136 @ X135 @ (comp @ X134 @ X133)) = $true)) | ($true != (X123 @ X126)) | ($true != (X123 @ X124)) | ? [X137 : term] : ((X125 @ id @ X137 @ id) != (X123 @ X137)))) & (ulamvarind = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (hoasinduction_lem1_lthm = $true) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179)) <=> (apinj1 = $true)) & (pushprop_lthm_orig = $true) & (! [X519 : term] : ? [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) = (X520 @ sh @ one)) <=> (hoasinduction_lem3b = $true)) & ((axidl = $true) <=> ! [X223 : subst] : ((comp @ id @ X223) = X223)) & (hoaslaminj_gthm = $true) & ((apinj2 = $true) <=> ! [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) != (ap @ X425 @ X426)) | (X426 = X428))) & (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)))))))))))))) & (! [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) = X384) <=> (axscons = $true)) & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & ((axvarcons = $true) <=> ! [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) = X245)) & (hoaslamnotvar_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (? [X190 : term,X191 : term,X192 : term,X193 : term] : (((ap @ X191 @ X190) = (ap @ X193 @ X192)) & (X190 != X192)) | ! [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) != (ap @ (sub @ X196 @ id) @ X194)) | (X194 = X195)))) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X314 : term] : ? [X315 : subst > term > term] : ! [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) = (X315 @ X316 @ X317))) & (hoasinduction_lem2_lthm = $true) & (pushprop_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X530 : term,X531 : term > $o,X532 : subst > term > subst > $o] : (? [X534 : subst > term > term] : (! [X535 : term] : (((X532 @ id @ (X534 @ id @ X535) @ id) = $true) | ($true != (X532 @ id @ X535 @ id))) & ! [X536 : subst,X537 : term,X538 : subst] : ((X534 @ (comp @ X538 @ X536) @ (sub @ X537 @ X536)) = (sub @ (X534 @ X538 @ X537) @ X536)) & ($true != (X532 @ id @ (lam @ (X534 @ sh @ one)) @ id))) | ? [X533 : term] : ((X532 @ id @ X533 @ id) != (X531 @ X533)) | ? [X539 : term] : (($true != (X531 @ (sub @ X530 @ (push @ X539 @ id)))) & ($true = (X531 @ X539))) | ($true = (X531 @ (lam @ X530)))) | ? [X529 : term] : ((sub @ X529 @ id) != X529) | ? [X525 : term] : ! [X526 : subst > term > term] : ? [X527 : subst,X528 : term] : ((X526 @ X527 @ X528) != (sub @ X525 @ (push @ X528 @ X527))))) & (hoasapinj1_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X210 : subst > term > subst > $o,X211 : term,X212 : term] : (? [X217 : subst,X218 : subst,X219 : subst,X220 : term] : (((X210 @ (comp @ X219 @ X218) @ (sub @ X220 @ X218) @ X217) = $true) & ($true != (X210 @ X219 @ X220 @ (comp @ X218 @ X217)))) | ? [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (($true = (X210 @ X215 @ X213 @ (comp @ X216 @ X214))) & ((X210 @ (comp @ X215 @ X216) @ (sub @ X213 @ X216) @ X214) != $true)) | ((X210 @ id @ (ap @ X212 @ X211) @ id) = $true) | ($true != (X210 @ id @ X212 @ id)) | ($true != (X210 @ id @ X211 @ id)) | ? [X221 : term,X222 : term] : (($true = (X210 @ id @ X221 @ id)) & ($true != (X210 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id)) & ($true = (X210 @ id @ X222 @ id)))) <=> (hoasinduction_lem2 = $true)) & (pushprop_lem0_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_gthm = $true) & (axmap = $true) & (induction2lem_gthm = $true) & (axabs = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2 = $true) & ((d2subst @ d_id) = id)),
% 9.38/2.21 inference(flattening,[status(thm)],[f12])).
% 9.38/2.21 thf(f12,plain,(
% 9.38/2.21 (axmap = $true) & (((((! [X13 : term > $o,X14 : term,X15 : subst] : (? [X16 : term,X17 : term] : (($true != (X13 @ (ap @ X17 @ X16))) & ($true = (X13 @ X16)) & ($true = (X13 @ X17))) | ($true = (X13 @ (sub @ X14 @ X15))) | ? [X18 : term] : (($true != (X13 @ (lam @ X18))) & ! [X19 : term] : (($true = (X13 @ (sub @ X18 @ (push @ X19 @ id)))) | ((X13 @ X19) != $true))) | ? [X20 : term] : (($true = (var @ X20)) & ($true != (X13 @ (sub @ X20 @ X15))))) | ? [X7 : term > $o,X8 : term] : (((X7 @ X8) != $true) & ! [X9 : term] : (($true != (X7 @ X9)) | ((X7 @ (lam @ X9)) = $true)) & ! [X10 : term] : (($true = (X7 @ X10)) | ((var @ X10) != $true)) & ! [X11 : term,X12 : term] : (($true != (X7 @ X11)) | ($true != (X7 @ X12)) | ($true = (X7 @ (ap @ X11 @ X12)))))) | ? [X6 : subst] : ((comp @ X6 @ id) != X6)) | ? [X4 : term,X5 : subst] : ((comp @ sh @ (push @ X4 @ X5)) != X5)) | ? [X2 : subst,X3 : term] : ((sub @ one @ (push @ X3 @ X2)) != X3)) <=> (induction2lem_lthm = $true)) & (hoaslamnotvar_lthm = $true) & (ulamvarind = $true) & ((axvarcons = $true) <=> ! [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) = X245)) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X72 : subst > term > subst > $o,X73 : term] : (($true = (X72 @ id @ (lam @ X73) @ id)) | ? [X83 : subst > term > term] : (! [X84 : subst,X85 : term,X86 : subst] : ((X83 @ (comp @ X86 @ X84) @ (sub @ X85 @ X84)) = (sub @ (X83 @ X86 @ X85) @ X84)) & ! [X87 : term] : (((X72 @ id @ (X83 @ id @ X87) @ id) = $true) | ($true != (X72 @ id @ X87 @ id))) & ($true != (X72 @ id @ (lam @ (X83 @ sh @ one)) @ id))) | ? [X74 : term] : (((X72 @ id @ X74 @ id) = $true) & ($true != (X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) | ? [X75 : subst,X76 : term,X77 : subst,X78 : subst] : (((X72 @ (comp @ X78 @ X77) @ (sub @ X76 @ X77) @ X75) != $true) & ($true = (X72 @ X78 @ X76 @ (comp @ X77 @ X75)))) | ? [X79 : term,X80 : subst,X81 : subst,X82 : subst] : (((X72 @ (comp @ X80 @ X82) @ (sub @ X79 @ X82) @ X81) = $true) & ($true != (X72 @ X80 @ X79 @ (comp @ X82 @ X81))))) | ? [X64 : term,X65 : subst > term > subst > $o] : (($true != (X65 @ id @ (lam @ (sub @ X64 @ (push @ one @ sh))) @ id)) & ! [X66 : term] : (($true != (X65 @ id @ X66 @ id)) | ($true = (X65 @ id @ (sub @ X64 @ (push @ X66 @ id)) @ id))) & ! [X67 : subst > term > term] : (? [X71 : term] : (($true = (X65 @ id @ X71 @ id)) & ($true != (X65 @ id @ (X67 @ id @ X71) @ id))) | ? [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X67 @ X69 @ X70) @ X68) != (X67 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) | ($true = (X65 @ id @ (lam @ (X67 @ sh @ one)) @ id))))) | ? [X63 : term] : ((sub @ X63 @ id) != X63))) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X314 : term] : ? [X315 : subst > term > term] : ! [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) = (X315 @ X316 @ X317))) & (pushprop_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (((! [X105 : term,X106 : subst > term > term,X107 : subst > term > term,X108 : subst] : (? [X109 : subst,X110 : subst,X111 : term] : ((sub @ (X107 @ X109 @ X111) @ X110) != (X107 @ (comp @ X109 @ X110) @ (sub @ X111 @ X110))) | ((lam @ (X106 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ? [X112 : subst,X113 : term,X114 : subst] : ((X106 @ (comp @ X114 @ X112) @ (sub @ X113 @ X112)) != (sub @ (X106 @ X114 @ X113) @ X112)) | ((X106 @ X108 @ X105) = (X107 @ X108 @ X105))) | ? [X103 : term,X104 : term] : (((lam @ X103) = (lam @ X104)) & (X103 != X104))) | ? [X101 : term,X102 : subst] : ((comp @ sh @ (push @ X101 @ X102)) != X102)) | ? [X99 : term,X100 : subst] : ((sub @ one @ (push @ X99 @ X100)) != X99))) & ((d2subst @ d_id) = id) & (! [X318 : subst > term > subst > $o,X319 : term,X320 : term > $o] : (? [X321 : subst > term > term] : (! [X325 : term] : (($true = (X318 @ id @ (X321 @ id @ X325) @ id)) | ((X318 @ id @ X325 @ id) != $true)) & ((X318 @ id @ (lam @ (X321 @ sh @ one)) @ id) != $true) & ! [X322 : subst,X323 : term,X324 : subst] : ((sub @ (X321 @ X322 @ X323) @ X324) = (X321 @ (comp @ X322 @ X324) @ (sub @ X323 @ X324)))) | ? [X326 : term] : ((X318 @ id @ X326 @ id) != (X320 @ X326)) | ? [X327 : term] : (($true != (X320 @ (sub @ X319 @ (push @ X327 @ id)))) & ($true = (X320 @ X327))) | ($true = (X320 @ (lam @ X319)))) <=> (hoasinduction_lem3v2a = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X142 : subst > term > subst > $o,X143 : term] : (? [X153 : term] : (($true = (X142 @ id @ X153 @ id)) & ($true != (X142 @ id @ (sub @ X143 @ (push @ X153 @ id)) @ id))) | ? [X144 : term,X145 : subst > term > term] : (((X142 @ id @ (lam @ (X145 @ sh @ one)) @ id) != $true) & ! [X149 : term] : (($true = (X142 @ id @ (X145 @ id @ X149) @ id)) | ($true != (X142 @ id @ X149 @ id))) & ! [X146 : subst,X147 : term,X148 : subst] : ((X145 @ (comp @ X146 @ X148) @ (sub @ X147 @ X148)) = (sub @ X144 @ (push @ (sub @ X147 @ X148) @ (comp @ X146 @ X148)))) & ! [X150 : term,X151 : subst,X152 : subst] : ((sub @ (X145 @ X152 @ X150) @ X151) = (sub @ (sub @ X144 @ (push @ X150 @ X152)) @ X151))) | ((X142 @ id @ (lam @ (sub @ X143 @ (push @ one @ sh))) @ id) = $true))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X123 : term > $o,X124 : term,X125 : subst > term > subst > $o,X126 : term] : (((X123 @ (ap @ X126 @ X124)) = $true) | ? [X129 : subst,X130 : subst,X131 : term,X132 : subst] : (($true != (X125 @ X129 @ X131 @ (comp @ X132 @ X130))) & ($true = (X125 @ (comp @ X129 @ X132) @ (sub @ X131 @ X132) @ X130))) | ? [X127 : term,X128 : term] : (((X125 @ id @ (ap @ (sub @ X127 @ id) @ X128) @ id) != $true) & ($true = (X125 @ id @ X128 @ id)) & ($true = (X125 @ id @ X127 @ id))) | ? [X133 : subst,X134 : subst,X135 : term,X136 : subst] : (($true != (X125 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134) @ X133)) & ((X125 @ X136 @ X135 @ (comp @ X134 @ X133)) = $true)) | ($true != (X123 @ X126)) | ($true != (X123 @ X124)) | ? [X137 : term] : ((X125 @ id @ X137 @ id) != (X123 @ X137)))) & (pushprop_lthm = $true) & ((axidr = $true) <=> ! [X393 : subst] : ((comp @ X393 @ id) = X393)) & ((axvarid = $true) <=> ! [X265 : term] : ((sub @ X265 @ id) = X265)) & (((! [X530 : term,X531 : term > $o,X532 : subst > term > subst > $o] : (? [X534 : subst > term > term] : (! [X535 : term] : (((X532 @ id @ (X534 @ id @ X535) @ id) = $true) | ($true != (X532 @ id @ X535 @ id))) & ! [X536 : subst,X537 : term,X538 : subst] : ((X534 @ (comp @ X538 @ X536) @ (sub @ X537 @ X536)) = (sub @ (X534 @ X538 @ X537) @ X536)) & ($true != (X532 @ id @ (lam @ (X534 @ sh @ one)) @ id))) | ? [X533 : term] : ((X532 @ id @ X533 @ id) != (X531 @ X533)) | ? [X539 : term] : (($true != (X531 @ (sub @ X530 @ (push @ X539 @ id)))) & ($true = (X531 @ X539))) | ($true = (X531 @ (lam @ X530)))) | ? [X529 : term] : ((sub @ X529 @ id) != X529)) | ? [X525 : term] : ! [X526 : subst > term > term] : ? [X527 : subst,X528 : term] : ((X526 @ X527 @ X528) != (sub @ X525 @ (push @ X528 @ X527)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (termmset_gthm = $true) & (hoaslamnotvar_gthm = $true) & (((((! [X360 : subst > term > subst > $o,X361 : term] : (? [X374 : subst,X375 : subst,X376 : subst,X377 : term] : (($true != (X360 @ (comp @ X374 @ X375) @ (sub @ X377 @ X375) @ X376)) & ($true = (X360 @ X374 @ X377 @ (comp @ X375 @ X376)))) | ? [X373 : term] : (($true = (var @ (sub @ X373 @ id))) & ($true != (X360 @ id @ X373 @ id))) | ? [X366 : subst > term > term] : (($true != (X360 @ id @ (lam @ (X366 @ sh @ one)) @ id)) & ! [X370 : term] : (($true = (X360 @ id @ (X366 @ id @ X370) @ id)) | ($true != (X360 @ id @ X370 @ id))) & ! [X367 : subst,X368 : term,X369 : subst] : ((sub @ (X366 @ X369 @ X368) @ X367) = (X366 @ (comp @ X369 @ X367) @ (sub @ X368 @ X367)))) | ($true = (X360 @ id @ X361 @ id)) | ? [X362 : subst,X363 : subst,X364 : subst,X365 : term] : (($true = (X360 @ (comp @ X362 @ X363) @ (sub @ X365 @ X363) @ X364)) & ($true != (X360 @ X362 @ X365 @ (comp @ X363 @ X364)))) | ? [X371 : term,X372 : term] : (($true = (X360 @ id @ X371 @ id)) & ($true = (X360 @ id @ X372 @ id)) & ($true != (X360 @ id @ (ap @ (sub @ X372 @ id) @ X371) @ id)))) | ? [X350 : subst > term > subst > $o,X351 : term,X352 : term > $o] : (! [X355 : subst > term > term] : (($true = (X350 @ id @ (lam @ (X355 @ sh @ one)) @ id)) | ? [X356 : term] : (($true = (X350 @ id @ X356 @ id)) & ($true != (X350 @ id @ (X355 @ id @ X356) @ id))) | ? [X357 : subst,X358 : term,X359 : subst] : ((X355 @ (comp @ X357 @ X359) @ (sub @ X358 @ X359)) != (sub @ (X355 @ X357 @ X358) @ X359))) & ! [X354 : term] : (($true = (X352 @ (sub @ X351 @ (push @ X354 @ id)))) | ($true != (X352 @ X354))) & ($true != (X352 @ (lam @ X351))) & ! [X353 : term] : ((X352 @ X353) = (X350 @ id @ X353 @ id)))) | ? [X349 : term] : ((sub @ X349 @ id) != X349)) | ? [X342 : term > $o,X343 : term] : (! [X348 : term] : (((X342 @ X348) = $true) | ($true != (var @ X348))) & ((X342 @ X343) != $true) & ! [X344 : term,X345 : term] : (((X342 @ X344) != $true) | ($true = (X342 @ (ap @ X345 @ X344))) | ($true != (X342 @ X345))) & ! [X346 : term] : (($true = (X342 @ (lam @ X346))) | ? [X347 : term] : (($true != (X342 @ (sub @ X346 @ (push @ X347 @ id)))) & ($true = (X342 @ X347)))))) | ? [X339 : subst > term > subst > $o] : ! [X340 : term > $o] : ? [X341 : term] : ((X339 @ id @ X341 @ id) != (X340 @ X341))) <=> (hoasinduction_lthm_3 = $true)) & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & ((? [X208 : term] : ((sub @ X208 @ id) != X208) | ! [X209 : term] : ((sub @ X209 @ id) = X209)) <=> (termmset_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & ! [X312 : d_subst,X313 : d_subst] : ((X312 = X313) | ((d2subst @ X312) != (d2subst @ X313))) & (! [X429 : term,X430 : term > $o] : (? [X432 : term] : (($true = (X430 @ X432)) & ($true != (X430 @ (lam @ X432)))) | ($true = (X430 @ X429)) | ? [X433 : term,X434 : term] : (($true = (X430 @ X433)) & ((X430 @ (ap @ X433 @ X434)) != $true) & ($true = (X430 @ X434))) | ? [X431 : term] : (($true != (X430 @ X431)) & ($true = (var @ X431)))) <=> (induction = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X95 : term] : ? [X96 : subst > term > term] : ! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98)))) & (hoaslamnotap_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem2v2_lthm = $true) & ! [X156 : d_subst] : (d_id = X156) & ((hoasinduction_lem3v2 = $true) <=> ! [X407 : term,X408 : subst > term > subst > $o,X409 : term > $o] : (? [X416 : subst,X417 : subst,X418 : term,X419 : subst] : (($true = (X408 @ (comp @ X419 @ X417) @ (sub @ X418 @ X417) @ X416)) & ($true != (X408 @ X419 @ X418 @ (comp @ X417 @ X416)))) | ? [X410 : subst > term > term] : (($true != (X408 @ id @ (lam @ (X410 @ sh @ one)) @ id)) & ! [X411 : term,X412 : subst,X413 : subst] : ((sub @ (X410 @ X412 @ X411) @ X413) = (X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413))) & ! [X414 : term] : (($true = (X408 @ id @ (X410 @ id @ X414) @ id)) | ($true != (X408 @ id @ X414 @ id)))) | ((X409 @ (lam @ X407)) = $true) | ? [X420 : term] : ((X409 @ X420) != (X408 @ id @ X420 @ id)) | ? [X421 : subst,X422 : term,X423 : subst,X424 : subst] : (($true != (X408 @ (comp @ X424 @ X421) @ (sub @ X422 @ X421) @ X423)) & ($true = (X408 @ X424 @ X422 @ (comp @ X421 @ X423)))) | ? [X415 : term] : (($true != (X409 @ (sub @ X407 @ (push @ X415 @ id)))) & ($true = (X409 @ X415))))) & (! [X54 : subst > term > subst > $o,X55 : term] : (? [X58 : subst > term > term] : (! [X59 : term] : (($true = (X54 @ id @ (X58 @ id @ X59) @ id)) | ($true != (X54 @ id @ X59 @ id))) & ($true != (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) & ! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61))) | ? [X56 : term,X57 : term] : (($true != (X54 @ id @ (ap @ (sub @ X56 @ id) @ X57) @ id)) & ((X54 @ id @ X56 @ id) = $true) & ($true = (X54 @ id @ X57 @ id))) | ($true = (X54 @ id @ X55 @ id))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasapnotvar_lthm = $true) & (! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) <=> (laminj = $true)) & (hoasapnotvar = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((substmonoid_lthm = $true) <=> (((! [X523 : subst] : ((comp @ id @ X523) = X523) & ! [X524 : subst] : ((comp @ X524 @ id) = X524)) | ? [X522 : subst] : ((comp @ X522 @ id) != X522)) | ? [X521 : subst] : ((comp @ id @ X521) != X521))) & (! [X210 : subst > term > subst > $o,X211 : term,X212 : term] : (? [X217 : subst,X218 : subst,X219 : subst,X220 : term] : (((X210 @ (comp @ X219 @ X218) @ (sub @ X220 @ X218) @ X217) = $true) & ($true != (X210 @ X219 @ X220 @ (comp @ X218 @ X217)))) | ? [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (($true = (X210 @ X215 @ X213 @ (comp @ X216 @ X214))) & ((X210 @ (comp @ X215 @ X216) @ (sub @ X213 @ X216) @ X214) != $true)) | ((X210 @ id @ (ap @ X212 @ X211) @ id) = $true) | ($true != (X210 @ id @ X212 @ id)) | ($true != (X210 @ id @ X211 @ id)) | ? [X221 : term,X222 : term] : (($true = (X210 @ id @ X221 @ id)) & ($true != (X210 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id)) & ($true = (X210 @ id @ X222 @ id)))) <=> (hoasinduction_lem2 = $true)) & (induction2_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (($true != (X243 @ X240)) | ? [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) != (X242 @ X244)) | ((X242 @ one) = $true))) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179)) <=> (apinj1 = $true)) & (ulamvar1 != $true) & (((! [X233 : term,X234 : term > $o] : (? [X237 : term,X238 : term] : (($true = (X234 @ X238)) & ($true = (X234 @ X237)) & ($true != (X234 @ (ap @ X238 @ X237)))) | ? [X235 : term] : (($true != (X234 @ (lam @ X235))) & ! [X236 : term] : (($true != (X234 @ X236)) | ($true = (X234 @ (sub @ X235 @ (push @ X236 @ id)))))) | ($true = (X234 @ X233)) | ? [X239 : term] : (($true != (X234 @ X239)) & ($true = (var @ X239)))) | ? [X225 : term,X226 : term > $o,X227 : subst] : (! [X228 : term] : (($true != (var @ X228)) | ($true = (X226 @ (sub @ X228 @ X227)))) & ($true != (X226 @ (sub @ X225 @ X227))) & ! [X229 : term] : (? [X230 : term] : (($true != (X226 @ (sub @ X229 @ (push @ X230 @ id)))) & ($true = (X226 @ X230))) | ($true = (X226 @ (lam @ X229)))) & ! [X231 : term,X232 : term] : (($true != (X226 @ X232)) | ((X226 @ X231) != $true) | ($true = (X226 @ (ap @ X232 @ X231)))))) | ? [X224 : term] : ((sub @ X224 @ id) != X224)) <=> (induction2_lthm = $true)) & ((d2subst @ d_id) = sh) & (lamnotvar = $true) & (pushprop_lem1_gthm = $true) & ((induction2 = $true) <=> ! [X88 : term > $o,X89 : term] : (? [X91 : term] : (($true != (X88 @ (lam @ X91))) & ! [X92 : term] : (($true != (X88 @ X92)) | ($true = (X88 @ (sub @ X91 @ (push @ X92 @ id)))))) | ($true = (X88 @ X89)) | ? [X90 : term] : (((X88 @ X90) != $true) & ($true = (var @ X90))) | ? [X93 : term,X94 : term] : (($true = (X88 @ X93)) & ($true != (X88 @ (ap @ X93 @ X94))) & ($true = (X88 @ X94))))) & (hoasinduction_lem3a_gthm = $true) & (axapp = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lthm = $true) <=> (((! [X501 : term,X502 : subst > term > subst > $o] : (? [X507 : subst > term > term] : (! [X508 : term] : (($true = (X502 @ id @ (X507 @ id @ X508) @ id)) | ((X502 @ id @ X508 @ id) != $true)) & ! [X509 : term,X510 : subst,X511 : subst] : ((X507 @ (comp @ X511 @ X510) @ (sub @ X509 @ X510)) = (sub @ (X507 @ X511 @ X509) @ X510)) & ($true != (X502 @ id @ (lam @ (X507 @ sh @ one)) @ id))) | ? [X518 : term] : (((X502 @ id @ X518 @ id) != $true) & ($true = (var @ (sub @ X518 @ id)))) | ? [X512 : term,X513 : subst,X514 : subst,X515 : subst] : (((X502 @ X515 @ X512 @ (comp @ X514 @ X513)) = $true) & ($true != (X502 @ (comp @ X515 @ X514) @ (sub @ X512 @ X514) @ X513))) | ? [X503 : subst,X504 : subst,X505 : term,X506 : subst] : (($true = (X502 @ (comp @ X504 @ X506) @ (sub @ X505 @ X506) @ X503)) & ($true != (X502 @ X504 @ X505 @ (comp @ X506 @ X503)))) | ? [X516 : term,X517 : term] : (((X502 @ id @ X517 @ id) = $true) & ((X502 @ id @ X516 @ id) = $true) & ($true != (X502 @ id @ (ap @ (sub @ X516 @ id) @ X517) @ id))) | ((X502 @ id @ X501 @ id) = $true)) | ? [X485 : term,X486 : subst > term > subst > $o] : (! [X491 : subst,X492 : term,X493 : subst,X494 : subst] : (((X486 @ (comp @ X493 @ X494) @ (sub @ X492 @ X494) @ X491) = $true) | ($true != (X486 @ X493 @ X492 @ (comp @ X494 @ X491)))) & ($true != (X486 @ id @ (lam @ X485) @ id)) & ! [X487 : subst,X488 : subst,X489 : subst,X490 : term] : (($true != (X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X487)) | ((X486 @ X488 @ X490 @ (comp @ X489 @ X487)) = $true)) & ! [X495 : subst > term > term] : (($true = (X486 @ id @ (lam @ (X495 @ sh @ one)) @ id)) | ? [X496 : term] : (($true != (X486 @ id @ (X495 @ id @ X496) @ id)) & ($true = (X486 @ id @ X496 @ id))) | ? [X497 : term,X498 : subst,X499 : subst] : ((X495 @ (comp @ X498 @ X499) @ (sub @ X497 @ X499)) != (sub @ (X495 @ X498 @ X497) @ X499))) & ! [X500 : term] : (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id) = $true) | ($true != (X486 @ id @ X500 @ id))))) | ? [X472 : subst > term > subst > $o,X473 : term,X474 : term] : (($true = (X472 @ id @ X473 @ id)) & ! [X483 : term,X484 : term] : (($true = (X472 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) | ($true != (X472 @ id @ X483 @ id)) | ((X472 @ id @ X484 @ id) != $true)) & ! [X479 : term,X480 : subst,X481 : subst,X482 : subst] : (($true != (X472 @ (comp @ X480 @ X481) @ (sub @ X479 @ X481) @ X482)) | ($true = (X472 @ X480 @ X479 @ (comp @ X481 @ X482)))) & ($true != (X472 @ id @ (ap @ X473 @ X474) @ id)) & ! [X475 : subst,X476 : term,X477 : subst,X478 : subst] : (($true = (X472 @ (comp @ X478 @ X477) @ (sub @ X476 @ X477) @ X475)) | ($true != (X472 @ X478 @ X476 @ (comp @ X477 @ X475)))) & ($true = (X472 @ id @ X474 @ id)))) | ? [X465 : term,X466 : term > $o] : (($true != (X466 @ X465)) & ! [X470 : term,X471 : term] : (($true = (X466 @ (ap @ X470 @ X471))) | ($true != (X466 @ X470)) | ($true != (X466 @ X471))) & ! [X467 : term] : (($true = (X466 @ (lam @ X467))) | ? [X468 : term] : (($true = (X466 @ X468)) & ((X466 @ (sub @ X467 @ (push @ X468 @ id))) != $true))) & ! [X469 : term] : (($true != (var @ X469)) | ((X466 @ X469) = $true))))) & (pushprop_lem2v2 = $true) & (hoaslaminj_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X180 : term > $o,X181 : term,X182 : subst] : ? [X183 : term > $o] : ! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182))))) & ((hoasapinj2_lthm = $true) <=> (? [X190 : term,X191 : term,X192 : term,X193 : term] : (((ap @ X191 @ X190) = (ap @ X193 @ X192)) & (X190 != X192)) | ! [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) != (ap @ (sub @ X196 @ id) @ X194)) | (X194 = X195)))) & (hoaslamnotvar = $true) & ((hoasapinj2 = $true) <=> ! [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)) | (X174 = X175))) & (hoasapinj1_gthm = $true) & (hoasinduction_gthm = $true) & (! [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) = (X268 @ X267 @ X266)) | ((lam @ (X268 @ sh @ one)) != (lam @ (X269 @ sh @ one))) | ? [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) != (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) | ? [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) != (sub @ (X269 @ X270 @ X271) @ X272))) <=> (hoaslaminj = $true)) & (ulamvarsh = $true) & ((! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34)) <=> (substmonoid = $true)) & (pushprop = $true) & (! [X157 : subst > term > subst > $o,X158 : term] : (? [X159 : term] : (((X157 @ id @ (sub @ X158 @ (push @ X159 @ id)) @ id) != $true) & ($true = (X157 @ id @ X159 @ id))) | ($true = (X157 @ id @ (lam @ X158) @ id)) | ? [X160 : subst > term > term] : (! [X162 : term,X163 : subst,X164 : subst] : ((sub @ (X160 @ X163 @ X162) @ X164) = (X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164))) & ((X157 @ id @ (lam @ (X160 @ sh @ one)) @ id) != $true) & ! [X161 : term] : (((X157 @ id @ X161 @ id) != $true) | ($true = (X157 @ id @ (X160 @ id @ X161) @ id))))) <=> (hoasinduction_lem3a = $true)) & ! [X337 : d_term,X338 : d_term] : ((X337 = X338) | ((d2term @ X337) != (d2term @ X338))) & (! [X294 : subst > term > subst > $o,X295 : term] : (? [X302 : subst > term > term] : (! [X303 : term] : (($true = (X294 @ id @ (X302 @ id @ X303) @ id)) | ($true != (X294 @ id @ X303 @ id))) & ! [X304 : subst,X305 : subst,X306 : term] : ((X302 @ (comp @ X304 @ X305) @ (sub @ X306 @ X305)) = (sub @ (X302 @ X304 @ X306) @ X305)) & ($true != (X294 @ id @ (lam @ (X302 @ sh @ one)) @ id))) | ? [X307 : subst,X308 : term,X309 : subst,X310 : subst] : (($true != (X294 @ X310 @ X308 @ (comp @ X309 @ X307))) & ($true = (X294 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309) @ X307))) | ? [X298 : term,X299 : subst,X300 : subst,X301 : subst] : (($true != (X294 @ (comp @ X299 @ X301) @ (sub @ X298 @ X301) @ X300)) & ($true = (X294 @ X299 @ X298 @ (comp @ X301 @ X300)))) | ? [X296 : term,X297 : term] : (((X294 @ id @ X296 @ id) = $true) & ($true != (X294 @ id @ (ap @ (sub @ X297 @ id) @ X296) @ id)) & ((X294 @ id @ X297 @ id) = $true)) | ? [X311 : term] : (((var @ (sub @ X311 @ id)) = $true) & ($true != (X294 @ id @ X311 @ id))) | ($true = (X294 @ id @ X295 @ id))) <=> (hoasinduction = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (((X30 @ X29) != $true) | ($true = (X28 @ one)) | ? [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) != (X28 @ X32))) | ? [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) != X27))) & ((hoasinduction_lem0 = $true) <=> ! [X138 : subst > term > subst > $o] : ? [X139 : term > $o] : ! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140))) & ((pushprop_lem1_lthm = $true) <=> ((! [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true != (X336 @ X332)) | ($true = (X333 @ (sub @ X332 @ (push @ X334 @ X335))))) | ? [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) != X330)) | ? [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) != X329))) & (hoasinduction_lem1_lthm = $true) & (! [X385 : subst,X386 : term > $o,X387 : term] : (? [X388 : term] : (! [X389 : term] : (($true = (X386 @ (sub @ X388 @ (push @ X389 @ id)))) | ((X386 @ X389) != $true)) & ($true != (X386 @ (lam @ X388)))) | ($true = (X386 @ (sub @ X387 @ X385))) | ? [X390 : term] : (($true = (var @ X390)) & ($true != (X386 @ (sub @ X390 @ X385)))) | ? [X391 : term,X392 : term] : (($true = (X386 @ X392)) & ($true = (X386 @ X391)) & ((X386 @ (ap @ X392 @ X391)) != $true))) <=> (induction2lem = $true)) & (hoasapnotvar_gthm = $true) & (lamnotap != $true) & (substmonoid_gthm = $true) & (hoasinduction_lem2_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)))))))))))))) & (! [X247 : subst > term > subst > $o,X248 : term] : (? [X258 : term,X259 : subst,X260 : subst,X261 : subst] : (($true = (X247 @ X261 @ X258 @ (comp @ X260 @ X259))) & ((X247 @ (comp @ X261 @ X260) @ (sub @ X258 @ X260) @ X259) != $true)) | ((X247 @ id @ (lam @ X248) @ id) = $true) | ? [X262 : term] : (((X247 @ id @ (sub @ X248 @ (push @ X262 @ id)) @ id) != $true) & ($true = (X247 @ id @ X262 @ id))) | ? [X253 : subst > term > term] : (! [X254 : subst,X255 : term,X256 : subst] : ((sub @ (X253 @ X256 @ X255) @ X254) = (X253 @ (comp @ X256 @ X254) @ (sub @ X255 @ X254))) & ($true != (X247 @ id @ (lam @ (X253 @ sh @ one)) @ id)) & ! [X257 : term] : (($true != (X247 @ id @ X257 @ id)) | ($true = (X247 @ id @ (X253 @ id @ X257) @ id)))) | ? [X249 : subst,X250 : subst,X251 : term,X252 : subst] : (($true = (X247 @ (comp @ X252 @ X249) @ (sub @ X251 @ X249) @ X250)) & ($true != (X247 @ X252 @ X251 @ (comp @ X249 @ X250))))) <=> (hoasinduction_lem3 = $true)) & (hoasinduction_lem2_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (axclos = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (induction2lem_gthm = $true) & (one = (d2term @ d_one)) & (! [X187 : subst > term > subst > $o] : ? [X188 : term > $o] : ! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (apnotvar = $true) & (! [X21 : subst,X22 : term,X23 : term > $o] : ? [X24 : term > $o] : ! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21)))) <=> (pushprop_lem0 = $true)) & (! [X378 : term,X379 : term,X380 : subst > term > term] : (((ap @ (sub @ X379 @ id) @ X378) != (lam @ (X380 @ sh @ one))) | ? [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) != (sub @ (X380 @ X382 @ X383) @ X381))) <=> (hoaslamnotap = $true)) & (((! [X286 : subst > term > subst > $o,X287 : term] : (? [X288 : term] : (($true = (X286 @ id @ X288 @ id)) & ((X286 @ id @ (sub @ X287 @ (push @ X288 @ id)) @ id) != $true)) | ($true = (X286 @ id @ (lam @ X287) @ id)) | ? [X289 : subst > term > term] : (! [X290 : term] : (((X286 @ id @ X290 @ id) != $true) | ($true = (X286 @ id @ (X289 @ id @ X290) @ id))) & ! [X291 : subst,X292 : subst,X293 : term] : ((sub @ (X289 @ X292 @ X293) @ X291) = (X289 @ (comp @ X292 @ X291) @ (sub @ X293 @ X291))) & ($true != (X286 @ id @ (lam @ (X289 @ sh @ one)) @ id)))) | ? [X278 : term,X279 : subst > term > subst > $o] : (! [X285 : term] : (($true = (X279 @ id @ (sub @ X278 @ (push @ X285 @ id)) @ id)) | ($true != (X279 @ id @ X285 @ id))) & ($true != (X279 @ id @ (lam @ (sub @ X278 @ (push @ one @ sh))) @ id)) & ! [X280 : subst > term > term] : (? [X281 : term] : (($true != (X279 @ id @ (X280 @ id @ X281) @ id)) & ((X279 @ id @ X281 @ id) = $true)) | ? [X282 : term,X283 : subst,X284 : subst] : ((sub @ (X280 @ X284 @ X282) @ X283) != (X280 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283))) | ($true = (X279 @ id @ (lam @ (X280 @ sh @ one)) @ id))))) | ? [X277 : term] : ((sub @ X277 @ id) != X277)) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem1 = $true) & (! [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) != (ap @ (sub @ X394 @ id) @ X396)) | (X394 = X397)) <=> (hoasapinj1 = $true)) & (hoasinduction_lem3_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true != (X169 @ X167)) | ($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166)))))) & (! [X200 : term,X201 : subst > term > subst > $o] : (? [X207 : term] : (($true = (X201 @ id @ X207 @ id)) & ($true != (X201 @ id @ (sub @ X200 @ (push @ X207 @ id)) @ id))) | ? [X202 : subst > term > term] : (($true != (X201 @ id @ (lam @ (X202 @ sh @ one)) @ id)) & ! [X204 : term,X205 : subst,X206 : subst] : ((sub @ (X202 @ X206 @ X204) @ X205) = (X202 @ (comp @ X206 @ X205) @ (sub @ X204 @ X205))) & ! [X203 : term] : (($true = (X201 @ id @ (X202 @ id @ X203) @ id)) | ($true != (X201 @ id @ X203 @ id)))) | ((X201 @ id @ (lam @ (sub @ X200 @ (push @ one @ sh))) @ id) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & ((apinj2 = $true) <=> ! [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) != (ap @ X425 @ X426)) | (X426 = X428))) & (axvarshift = $true) & (pushprop_lem0_gthm = $true) & (axabs = $true) & ((var @ (d2term @ d_one)) != $true) & (! [X519 : term] : ? [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) = (X520 @ sh @ one)) <=> (hoasinduction_lem3b = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) = X384) <=> (axscons = $true)) & (((((! [X456 : subst > term > subst > $o,X457 : term] : (? [X458 : subst > term > term] : (! [X459 : term] : (($true = (X456 @ id @ (X458 @ id @ X459) @ id)) | ($true != (X456 @ id @ X459 @ id))) & ! [X460 : term,X461 : subst,X462 : subst] : ((X458 @ (comp @ X462 @ X461) @ (sub @ X460 @ X461)) = (sub @ (X458 @ X462 @ X460) @ X461)) & ($true != (X456 @ id @ (lam @ (X458 @ sh @ one)) @ id))) | ($true = (X456 @ id @ X457 @ id)) | ? [X463 : term,X464 : term] : (((X456 @ id @ X464 @ id) = $true) & ((X456 @ id @ (ap @ (sub @ X463 @ id) @ X464) @ id) != $true) & ($true = (X456 @ id @ X463 @ id)))) | ? [X446 : term > $o,X447 : subst > term > subst > $o,X448 : term] : (! [X451 : subst > term > term] : (($true = (X447 @ id @ (lam @ (X451 @ sh @ one)) @ id)) | ? [X452 : term,X453 : subst,X454 : subst] : ((sub @ (X451 @ X453 @ X452) @ X454) != (X451 @ (comp @ X453 @ X454) @ (sub @ X452 @ X454))) | ? [X455 : term] : (($true = (X447 @ id @ X455 @ id)) & ($true != (X447 @ id @ (X451 @ id @ X455) @ id)))) & ! [X450 : term] : (((X446 @ (sub @ X448 @ (push @ X450 @ id))) = $true) | ($true != (X446 @ X450))) & ! [X449 : term] : ((X447 @ id @ X449 @ id) = (X446 @ X449)) & ($true != (X446 @ (lam @ X448))))) | ? [X445 : term] : ((sub @ X445 @ id) != X445)) | ? [X438 : term,X439 : term > $o] : (! [X440 : term] : (($true != (var @ X440)) | ($true = (X439 @ X440))) & ! [X441 : term] : (? [X442 : term] : (($true != (X439 @ (sub @ X441 @ (push @ X442 @ id)))) & ($true = (X439 @ X442))) | ((X439 @ (lam @ X441)) = $true)) & ! [X443 : term,X444 : term] : (((X439 @ X443) != $true) | ($true != (X439 @ X444)) | ((X439 @ (ap @ X444 @ X443)) = $true)) & ($true != (X439 @ X438)))) | ? [X435 : subst > term > subst > $o] : ! [X436 : term > $o] : ? [X437 : term] : ((X436 @ X437) != (X435 @ id @ X437 @ id))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X263 : term] : ? [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) = (X264 @ sh @ one))) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X35 : term] : ((sub @ X35 @ id) != X35) | ! [X36 : subst > term > subst > $o,X37 : term,X38 : term > $o] : (? [X39 : term,X40 : subst,X41 : subst,X42 : subst] : (((X36 @ X42 @ X39 @ (comp @ X41 @ X40)) != $true) & ($true = (X36 @ (comp @ X42 @ X41) @ (sub @ X39 @ X41) @ X40))) | ? [X49 : term] : (($true = (X38 @ X49)) & ($true != (X38 @ (sub @ X37 @ (push @ X49 @ id))))) | ? [X44 : subst > term > term] : (($true != (X36 @ id @ (lam @ (X44 @ sh @ one)) @ id)) & ! [X45 : term,X46 : subst,X47 : subst] : ((sub @ (X44 @ X47 @ X45) @ X46) = (X44 @ (comp @ X47 @ X46) @ (sub @ X45 @ X46))) & ! [X48 : term] : (($true != (X36 @ id @ X48 @ id)) | ($true = (X36 @ id @ (X44 @ id @ X48) @ id)))) | ((X38 @ (lam @ X37)) = $true) | ? [X43 : term] : ((X36 @ id @ X43 @ id) != (X38 @ X43)) | ? [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true != (X36 @ (comp @ X53 @ X50) @ (sub @ X52 @ X50) @ X51)) & ((X36 @ X53 @ X52 @ (comp @ X50 @ X51)) = $true))))) & (axassoc = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & (pushprop_lem1v2_gthm = $true) & ! [X141 : d_term] : (d_one = X141) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((axshiftcons = $true) <=> ! [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) = X198)) & (! [X276 : term] : ((sub @ X276 @ id) = X276) <=> (termmset = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((! [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 = X406) | ((ap @ (sub @ X405 @ id) @ X403) != (ap @ (sub @ X406 @ id) @ X404))) | ? [X399 : term,X400 : term,X401 : term,X402 : term] : ((X399 != X402) & ((ap @ X399 @ X400) = (ap @ X402 @ X401)))) | ? [X398 : term] : ((sub @ X398 @ id) != X398)) <=> (hoasapinj1_lthm = $true)) & ((shinj = $true) <=> ! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186))) & ((axidl = $true) <=> ! [X223 : subst] : ((comp @ id @ X223) = X223)) & ((hoasinduction_lem3aa = $true) <=> ! [X115 : subst > term > subst > $o,X116 : term] : (? [X117 : subst > term > term] : (! [X118 : subst,X119 : subst,X120 : term] : ((sub @ (X117 @ X118 @ X120) @ X119) = (X117 @ (comp @ X118 @ X119) @ (sub @ X120 @ X119))) & ! [X121 : term] : (((X115 @ id @ X121 @ id) != $true) | ($true = (X115 @ id @ (X117 @ id @ X121) @ id))) & ($true != (X115 @ id @ (lam @ (X117 @ sh @ one)) @ id))) | ($true = (X115 @ id @ (lam @ (sub @ X116 @ (push @ one @ sh))) @ id)) | ? [X122 : term] : (($true != (X115 @ id @ (sub @ X116 @ (push @ X122 @ id)) @ id)) & ($true = (X115 @ id @ X122 @ id))))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem3b_gthm = $true)),
% 9.38/2.21 inference(ennf_transformation,[status(thm)],[f11])).
% 9.38/2.21 thf(f11,plain,(
% 9.38/2.21 (axmap = $true) & ((! [X2 : subst,X3 : term] : ((sub @ one @ (push @ X3 @ X2)) = X3) => (! [X4 : term,X5 : subst] : ((comp @ sh @ (push @ X4 @ X5)) = X5) => (! [X6 : subst] : ((comp @ X6 @ id) = X6) => (! [X7 : term > $o,X8 : term] : (~! [X11 : term,X12 : term] : (($true != (X7 @ X11)) | ($true != (X7 @ X12)) | ($true = (X7 @ (ap @ X11 @ X12)))) | ~! [X9 : term] : (($true != (X7 @ X9)) | ((X7 @ (lam @ X9)) = $true)) | ~! [X10 : term] : (($true = (X7 @ X10)) | ((var @ X10) != $true)) | ((X7 @ X8) = $true)) => ! [X13 : term > $o,X14 : term,X15 : subst] : (($true = (X13 @ (sub @ X14 @ X15))) | ~! [X20 : term] : (($true != (var @ X20)) | ($true = (X13 @ (sub @ X20 @ X15)))) | ~! [X18 : term] : (($true = (X13 @ (lam @ X18))) | ~! [X19 : term] : (($true = (X13 @ (sub @ X18 @ (push @ X19 @ id)))) | ((X13 @ X19) != $true))) | ~! [X16 : term,X17 : term] : (($true = (X13 @ (ap @ X17 @ X16))) | ($true != (X13 @ X16)) | ($true != (X13 @ X17)))))))) <=> (induction2lem_lthm = $true)) & (hoaslamnotvar_lthm = $true) & (ulamvarind = $true) & ((axvarcons = $true) <=> ! [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) = X245)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X63 : term] : ((sub @ X63 @ id) = X63) => (! [X64 : term,X65 : subst > term > subst > $o] : (($true = (X65 @ id @ (lam @ (sub @ X64 @ (push @ one @ sh))) @ id)) | ~! [X67 : subst > term > term] : (~! [X71 : term] : (($true != (X65 @ id @ X71 @ id)) | ($true = (X65 @ id @ (X67 @ id @ X71) @ id))) | ~! [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X67 @ X69 @ X70) @ X68) = (X67 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) | ($true = (X65 @ id @ (lam @ (X67 @ sh @ one)) @ id))) | ~! [X66 : term] : (($true != (X65 @ id @ X66 @ id)) | ($true = (X65 @ id @ (sub @ X64 @ (push @ X66 @ id)) @ id)))) => ! [X72 : subst > term > subst > $o,X73 : term] : (($true = (X72 @ id @ (lam @ X73) @ id)) | ~! [X79 : term,X80 : subst,X81 : subst,X82 : subst] : (((X72 @ (comp @ X80 @ X82) @ (sub @ X79 @ X82) @ X81) != $true) | ($true = (X72 @ X80 @ X79 @ (comp @ X82 @ X81)))) | ~! [X75 : subst,X76 : term,X77 : subst,X78 : subst] : (((X72 @ (comp @ X78 @ X77) @ (sub @ X76 @ X77) @ X75) = $true) | ($true != (X72 @ X78 @ X76 @ (comp @ X77 @ X75)))) | ~! [X83 : subst > term > term] : (~! [X84 : subst,X85 : term,X86 : subst] : ((X83 @ (comp @ X86 @ X84) @ (sub @ X85 @ X84)) = (sub @ (X83 @ X86 @ X85) @ X84)) | ($true = (X72 @ id @ (lam @ (X83 @ sh @ one)) @ id)) | ~! [X87 : term] : (((X72 @ id @ (X83 @ id @ X87) @ id) = $true) | ($true != (X72 @ id @ X87 @ id)))) | ~! [X74 : term] : (($true = (X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id)) | ((X72 @ id @ X74 @ id) != $true)))))) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X314 : term] : ~! [X315 : subst > term > term] : ~! [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) = (X315 @ X316 @ X317))) & (pushprop_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X99 : term,X100 : subst] : ((sub @ one @ (push @ X99 @ X100)) = X99) => (! [X101 : term,X102 : subst] : ((comp @ sh @ (push @ X101 @ X102)) = X102) => (! [X103 : term,X104 : term] : ((X103 = X104) | ((lam @ X103) != (lam @ X104))) => ! [X105 : term,X106 : subst > term > term,X107 : subst > term > term,X108 : subst] : (~! [X109 : subst,X110 : subst,X111 : term] : ((sub @ (X107 @ X109 @ X111) @ X110) = (X107 @ (comp @ X109 @ X110) @ (sub @ X111 @ X110))) | ~! [X112 : subst,X113 : term,X114 : subst] : ((X106 @ (comp @ X114 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X106 @ X114 @ X113) @ X112)) | ((X106 @ X108 @ X105) = (X107 @ X108 @ X105)) | ((lam @ (X106 @ sh @ one)) != (lam @ (X107 @ sh @ one)))))))) & ((d2subst @ d_id) = id) & (! [X318 : subst > term > subst > $o,X319 : term,X320 : term > $o] : (~! [X327 : term] : (($true != (X320 @ X327)) | ($true = (X320 @ (sub @ X319 @ (push @ X327 @ id))))) | ~! [X326 : term] : ((X318 @ id @ X326 @ id) = (X320 @ X326)) | ~! [X321 : subst > term > term] : (~! [X325 : term] : (($true = (X318 @ id @ (X321 @ id @ X325) @ id)) | ((X318 @ id @ X325 @ id) != $true)) | ((X318 @ id @ (lam @ (X321 @ sh @ one)) @ id) = $true) | ~! [X322 : subst,X323 : term,X324 : subst] : ((sub @ (X321 @ X322 @ X323) @ X324) = (X321 @ (comp @ X322 @ X324) @ (sub @ X323 @ X324)))) | ($true = (X320 @ (lam @ X319)))) <=> (hoasinduction_lem3v2a = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X142 : subst > term > subst > $o,X143 : term] : (((X142 @ id @ (lam @ (sub @ X143 @ (push @ one @ sh))) @ id) = $true) | ~! [X144 : term,X145 : subst > term > term] : (~! [X146 : subst,X147 : term,X148 : subst] : ((X145 @ (comp @ X146 @ X148) @ (sub @ X147 @ X148)) = (sub @ X144 @ (push @ (sub @ X147 @ X148) @ (comp @ X146 @ X148)))) | ((X142 @ id @ (lam @ (X145 @ sh @ one)) @ id) = $true) | ~! [X149 : term] : (($true = (X142 @ id @ (X145 @ id @ X149) @ id)) | ($true != (X142 @ id @ X149 @ id))) | ~! [X150 : term,X151 : subst,X152 : subst] : ((sub @ (X145 @ X152 @ X150) @ X151) = (sub @ (sub @ X144 @ (push @ X150 @ X152)) @ X151))) | ~! [X153 : term] : (($true != (X142 @ id @ X153 @ id)) | ($true = (X142 @ id @ (sub @ X143 @ (push @ X153 @ id)) @ id))))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X123 : term > $o,X124 : term,X125 : subst > term > subst > $o,X126 : term] : (((X123 @ (ap @ X126 @ X124)) = $true) | ~! [X137 : term] : ((X125 @ id @ X137 @ id) = (X123 @ X137)) | ~! [X129 : subst,X130 : subst,X131 : term,X132 : subst] : (($true != (X125 @ (comp @ X129 @ X132) @ (sub @ X131 @ X132) @ X130)) | ($true = (X125 @ X129 @ X131 @ (comp @ X132 @ X130)))) | ~! [X127 : term,X128 : term] : (($true != (X125 @ id @ X127 @ id)) | ((X125 @ id @ (ap @ (sub @ X127 @ id) @ X128) @ id) = $true) | ($true != (X125 @ id @ X128 @ id))) | ($true != (X123 @ X124)) | ($true != (X123 @ X126)) | ~! [X133 : subst,X134 : subst,X135 : term,X136 : subst] : (((X125 @ X136 @ X135 @ (comp @ X134 @ X133)) != $true) | ($true = (X125 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134) @ X133))))) & (pushprop_lthm = $true) & ((axidr = $true) <=> ! [X393 : subst] : ((comp @ X393 @ id) = X393)) & ((axvarid = $true) <=> ! [X265 : term] : ((sub @ X265 @ id) = X265)) & ((! [X525 : term] : ~! [X526 : subst > term > term] : ~! [X527 : subst,X528 : term] : ((X526 @ X527 @ X528) = (sub @ X525 @ (push @ X528 @ X527))) => (! [X529 : term] : ((sub @ X529 @ id) = X529) => ! [X530 : term,X531 : term > $o,X532 : subst > term > subst > $o] : (~! [X539 : term] : (($true = (X531 @ (sub @ X530 @ (push @ X539 @ id)))) | ($true != (X531 @ X539))) | ($true = (X531 @ (lam @ X530))) | ~! [X534 : subst > term > term] : (($true = (X532 @ id @ (lam @ (X534 @ sh @ one)) @ id)) | ~! [X535 : term] : (((X532 @ id @ (X534 @ id @ X535) @ id) = $true) | ($true != (X532 @ id @ X535 @ id))) | ~! [X536 : subst,X537 : term,X538 : subst] : ((X534 @ (comp @ X538 @ X536) @ (sub @ X537 @ X536)) = (sub @ (X534 @ X538 @ X537) @ X536))) | ~! [X533 : term] : ((X532 @ id @ X533 @ id) = (X531 @ X533))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (termmset_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((! [X339 : subst > term > subst > $o] : ~! [X340 : term > $o] : ~! [X341 : term] : ((X339 @ id @ X341 @ id) = (X340 @ X341)) => (! [X342 : term > $o,X343 : term] : (~! [X344 : term,X345 : term] : (((X342 @ X344) != $true) | ($true = (X342 @ (ap @ X345 @ X344))) | ($true != (X342 @ X345))) | ((X342 @ X343) = $true) | ~! [X346 : term] : (~! [X347 : term] : (($true = (X342 @ (sub @ X346 @ (push @ X347 @ id)))) | ($true != (X342 @ X347))) | ($true = (X342 @ (lam @ X346)))) | ~! [X348 : term] : (((X342 @ X348) = $true) | ($true != (var @ X348)))) => (! [X349 : term] : ((sub @ X349 @ id) = X349) => (! [X350 : subst > term > subst > $o,X351 : term,X352 : term > $o] : (($true = (X352 @ (lam @ X351))) | ~! [X353 : term] : ((X352 @ X353) = (X350 @ id @ X353 @ id)) | ~! [X354 : term] : (($true = (X352 @ (sub @ X351 @ (push @ X354 @ id)))) | ($true != (X352 @ X354))) | ~! [X355 : subst > term > term] : (~! [X357 : subst,X358 : term,X359 : subst] : ((X355 @ (comp @ X357 @ X359) @ (sub @ X358 @ X359)) = (sub @ (X355 @ X357 @ X358) @ X359)) | ~! [X356 : term] : (($true = (X350 @ id @ (X355 @ id @ X356) @ id)) | ($true != (X350 @ id @ X356 @ id))) | ($true = (X350 @ id @ (lam @ (X355 @ sh @ one)) @ id)))) => ! [X360 : subst > term > subst > $o,X361 : term] : (~! [X374 : subst,X375 : subst,X376 : subst,X377 : term] : (($true != (X360 @ X374 @ X377 @ (comp @ X375 @ X376))) | ($true = (X360 @ (comp @ X374 @ X375) @ (sub @ X377 @ X375) @ X376))) | ~! [X373 : term] : (($true = (X360 @ id @ X373 @ id)) | ($true != (var @ (sub @ X373 @ id)))) | ($true = (X360 @ id @ X361 @ id)) | ~! [X371 : term,X372 : term] : (($true != (X360 @ id @ X372 @ id)) | ($true != (X360 @ id @ X371 @ id)) | ($true = (X360 @ id @ (ap @ (sub @ X372 @ id) @ X371) @ id))) | ~! [X362 : subst,X363 : subst,X364 : subst,X365 : term] : (($true != (X360 @ (comp @ X362 @ X363) @ (sub @ X365 @ X363) @ X364)) | ($true = (X360 @ X362 @ X365 @ (comp @ X363 @ X364)))) | ~! [X366 : subst > term > term] : (($true = (X360 @ id @ (lam @ (X366 @ sh @ one)) @ id)) | ~! [X367 : subst,X368 : term,X369 : subst] : ((sub @ (X366 @ X369 @ X368) @ X367) = (X366 @ (comp @ X369 @ X367) @ (sub @ X368 @ X367))) | ~! [X370 : term] : (($true = (X360 @ id @ (X366 @ id @ X370) @ id)) | ($true != (X360 @ id @ X370 @ id))))))))) <=> (hoasinduction_lthm_3 = $true)) & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & ((! [X208 : term] : ((sub @ X208 @ id) = X208) => ! [X209 : term] : ((sub @ X209 @ id) = X209)) <=> (termmset_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & ! [X312 : d_subst,X313 : d_subst] : (((d2subst @ X312) = (d2subst @ X313)) => (X312 = X313)) & ((induction = $true) <=> ! [X429 : term,X430 : term > $o] : (~! [X432 : term] : (($true = (X430 @ (lam @ X432))) | ($true != (X430 @ X432))) | ~! [X431 : term] : (($true = (X430 @ X431)) | ($true != (var @ X431))) | ~! [X433 : term,X434 : term] : (($true != (X430 @ X433)) | ($true != (X430 @ X434)) | ((X430 @ (ap @ X433 @ X434)) = $true)) | ($true = (X430 @ X429)))) & (pushprop_lem3v2_lthm = $true) & (! [X95 : term] : ~! [X96 : subst > term > term] : ~! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98))) <=> (hoasinduction_lem3v2_f = $true)) & (hoaslamnotap_gthm = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem2v2_lthm = $true) & ! [X156 : d_subst] : (d_id = X156) & ((hoasinduction_lem3v2 = $true) <=> ! [X407 : term,X408 : subst > term > subst > $o,X409 : term > $o] : (((X409 @ (lam @ X407)) = $true) | ~! [X410 : subst > term > term] : (~! [X414 : term] : (($true = (X408 @ id @ (X410 @ id @ X414) @ id)) | ($true != (X408 @ id @ X414 @ id))) | ~! [X411 : term,X412 : subst,X413 : subst] : ((sub @ (X410 @ X412 @ X411) @ X413) = (X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413))) | ($true = (X408 @ id @ (lam @ (X410 @ sh @ one)) @ id))) | ~! [X415 : term] : (($true = (X409 @ (sub @ X407 @ (push @ X415 @ id)))) | ($true != (X409 @ X415))) | ~! [X420 : term] : ((X409 @ X420) = (X408 @ id @ X420 @ id)) | ~! [X421 : subst,X422 : term,X423 : subst,X424 : subst] : (($true != (X408 @ X424 @ X422 @ (comp @ X421 @ X423))) | ($true = (X408 @ (comp @ X424 @ X421) @ (sub @ X422 @ X421) @ X423))) | ~! [X416 : subst,X417 : subst,X418 : term,X419 : subst] : (($true = (X408 @ X419 @ X418 @ (comp @ X417 @ X416))) | ($true != (X408 @ (comp @ X419 @ X417) @ (sub @ X418 @ X417) @ X416))))) & (! [X54 : subst > term > subst > $o,X55 : term] : (($true = (X54 @ id @ X55 @ id)) | ~! [X58 : subst > term > term] : (~! [X59 : term] : (($true = (X54 @ id @ (X58 @ id @ X59) @ id)) | ($true != (X54 @ id @ X59 @ id))) | ($true = (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) | ~! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61))) | ~! [X56 : term,X57 : term] : (($true = (X54 @ id @ (ap @ (sub @ X56 @ id) @ X57) @ id)) | ($true != (X54 @ id @ X57 @ id)) | ((X54 @ id @ X56 @ id) != $true))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasapnotvar_lthm = $true) & (! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) <=> (laminj = $true)) & (hoasapnotvar = $true) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & ((substmonoid_lthm = $true) <=> (! [X521 : subst] : ((comp @ id @ X521) = X521) => (! [X522 : subst] : ((comp @ X522 @ id) = X522) => (! [X523 : subst] : ((comp @ id @ X523) = X523) & ! [X524 : subst] : ((comp @ X524 @ id) = X524))))) & ((hoasinduction_lem2 = $true) <=> ! [X210 : subst > term > subst > $o,X211 : term,X212 : term] : (($true != (X210 @ id @ X212 @ id)) | ((X210 @ id @ (ap @ X212 @ X211) @ id) = $true) | ($true != (X210 @ id @ X211 @ id)) | ~! [X217 : subst,X218 : subst,X219 : subst,X220 : term] : (((X210 @ (comp @ X219 @ X218) @ (sub @ X220 @ X218) @ X217) != $true) | ($true = (X210 @ X219 @ X220 @ (comp @ X218 @ X217)))) | ~! [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (($true != (X210 @ X215 @ X213 @ (comp @ X216 @ X214))) | ((X210 @ (comp @ X215 @ X216) @ (sub @ X213 @ X216) @ X214) = $true)) | ~! [X221 : term,X222 : term] : (($true != (X210 @ id @ X222 @ id)) | ($true = (X210 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id)) | ($true != (X210 @ id @ X221 @ id))))) & (induction2_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (($true != (X243 @ X240)) | ~! [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) = (X242 @ X244)) | ((X242 @ one) = $true))) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179)) <=> (apinj1 = $true)) & (ulamvar1 != $true) & ((! [X224 : term] : ((sub @ X224 @ id) = X224) => (! [X225 : term,X226 : term > $o,X227 : subst] : (~! [X229 : term] : (~! [X230 : term] : (($true = (X226 @ (sub @ X229 @ (push @ X230 @ id)))) | ($true != (X226 @ X230))) | ($true = (X226 @ (lam @ X229)))) | ($true = (X226 @ (sub @ X225 @ X227))) | ~! [X228 : term] : (($true != (var @ X228)) | ($true = (X226 @ (sub @ X228 @ X227)))) | ~! [X231 : term,X232 : term] : (($true != (X226 @ X232)) | ((X226 @ X231) != $true) | ($true = (X226 @ (ap @ X232 @ X231))))) => ! [X233 : term,X234 : term > $o] : (($true = (X234 @ X233)) | ~! [X237 : term,X238 : term] : (($true != (X234 @ X237)) | ($true = (X234 @ (ap @ X238 @ X237))) | ($true != (X234 @ X238))) | ~! [X239 : term] : (($true = (X234 @ X239)) | ($true != (var @ X239))) | ~! [X235 : term] : (($true = (X234 @ (lam @ X235))) | ~! [X236 : term] : (($true != (X234 @ X236)) | ($true = (X234 @ (sub @ X235 @ (push @ X236 @ id))))))))) <=> (induction2_lthm = $true)) & ((d2subst @ d_id) = sh) & (lamnotvar = $true) & (pushprop_lem1_gthm = $true) & ((induction2 = $true) <=> ! [X88 : term > $o,X89 : term] : (~! [X93 : term,X94 : term] : (($true != (X88 @ X93)) | ($true != (X88 @ X94)) | ($true = (X88 @ (ap @ X93 @ X94)))) | ~! [X90 : term] : (($true != (var @ X90)) | ((X88 @ X90) = $true)) | ~! [X91 : term] : (($true = (X88 @ (lam @ X91))) | ~! [X92 : term] : (($true != (X88 @ X92)) | ($true = (X88 @ (sub @ X91 @ (push @ X92 @ id)))))) | ($true = (X88 @ X89)))) & (hoasinduction_lem3a_gthm = $true) & (axapp = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & ((hoasinduction_lthm = $true) <=> (! [X465 : term,X466 : term > $o] : (($true = (X466 @ X465)) | ~! [X469 : term] : (($true != (var @ X469)) | ((X466 @ X469) = $true)) | ~! [X470 : term,X471 : term] : (($true = (X466 @ (ap @ X470 @ X471))) | ($true != (X466 @ X470)) | ($true != (X466 @ X471))) | ~! [X467 : term] : (~! [X468 : term] : (((X466 @ (sub @ X467 @ (push @ X468 @ id))) = $true) | ($true != (X466 @ X468))) | ($true = (X466 @ (lam @ X467))))) => (! [X472 : subst > term > subst > $o,X473 : term,X474 : term] : (($true != (X472 @ id @ X473 @ id)) | ~! [X483 : term,X484 : term] : (($true = (X472 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) | ($true != (X472 @ id @ X483 @ id)) | ((X472 @ id @ X484 @ id) != $true)) | ~! [X479 : term,X480 : subst,X481 : subst,X482 : subst] : (($true != (X472 @ (comp @ X480 @ X481) @ (sub @ X479 @ X481) @ X482)) | ($true = (X472 @ X480 @ X479 @ (comp @ X481 @ X482)))) | ($true = (X472 @ id @ (ap @ X473 @ X474) @ id)) | ($true != (X472 @ id @ X474 @ id)) | ~! [X475 : subst,X476 : term,X477 : subst,X478 : subst] : (($true = (X472 @ (comp @ X478 @ X477) @ (sub @ X476 @ X477) @ X475)) | ($true != (X472 @ X478 @ X476 @ (comp @ X477 @ X475))))) => (! [X485 : term,X486 : subst > term > subst > $o] : (~! [X487 : subst,X488 : subst,X489 : subst,X490 : term] : (($true != (X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X487)) | ((X486 @ X488 @ X490 @ (comp @ X489 @ X487)) = $true)) | ~! [X500 : term] : (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id) = $true) | ($true != (X486 @ id @ X500 @ id))) | ~! [X491 : subst,X492 : term,X493 : subst,X494 : subst] : (((X486 @ (comp @ X493 @ X494) @ (sub @ X492 @ X494) @ X491) = $true) | ($true != (X486 @ X493 @ X492 @ (comp @ X494 @ X491)))) | ($true = (X486 @ id @ (lam @ X485) @ id)) | ~! [X495 : subst > term > term] : (($true = (X486 @ id @ (lam @ (X495 @ sh @ one)) @ id)) | ~! [X496 : term] : (($true = (X486 @ id @ (X495 @ id @ X496) @ id)) | ($true != (X486 @ id @ X496 @ id))) | ~! [X497 : term,X498 : subst,X499 : subst] : ((X495 @ (comp @ X498 @ X499) @ (sub @ X497 @ X499)) = (sub @ (X495 @ X498 @ X497) @ X499)))) => ! [X501 : term,X502 : subst > term > subst > $o] : (~! [X518 : term] : (((X502 @ id @ X518 @ id) = $true) | ($true != (var @ (sub @ X518 @ id)))) | ~! [X512 : term,X513 : subst,X514 : subst,X515 : subst] : (((X502 @ X515 @ X512 @ (comp @ X514 @ X513)) != $true) | ($true = (X502 @ (comp @ X515 @ X514) @ (sub @ X512 @ X514) @ X513))) | ~! [X507 : subst > term > term] : (~! [X509 : term,X510 : subst,X511 : subst] : ((X507 @ (comp @ X511 @ X510) @ (sub @ X509 @ X510)) = (sub @ (X507 @ X511 @ X509) @ X510)) | ~! [X508 : term] : (($true = (X502 @ id @ (X507 @ id @ X508) @ id)) | ((X502 @ id @ X508 @ id) != $true)) | ($true = (X502 @ id @ (lam @ (X507 @ sh @ one)) @ id))) | ~! [X503 : subst,X504 : subst,X505 : term,X506 : subst] : (($true != (X502 @ (comp @ X504 @ X506) @ (sub @ X505 @ X506) @ X503)) | ($true = (X502 @ X504 @ X505 @ (comp @ X506 @ X503)))) | ~! [X516 : term,X517 : term] : (((X502 @ id @ X516 @ id) != $true) | ($true = (X502 @ id @ (ap @ (sub @ X516 @ id) @ X517) @ id)) | ((X502 @ id @ X517 @ id) != $true)) | ((X502 @ id @ X501 @ id) = $true)))))) & (pushprop_lem2v2 = $true) & (hoaslaminj_gthm = $true) & (! [X180 : term > $o,X181 : term,X182 : subst] : ~! [X183 : term > $o] : ~! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182)))) <=> (pushprop_lem0_lthm = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X190 : term,X191 : term,X192 : term,X193 : term] : ((X190 = X192) | ((ap @ X191 @ X190) != (ap @ X193 @ X192))) => ! [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) != (ap @ (sub @ X196 @ id) @ X194)) | (X194 = X195)))) & (hoaslamnotvar = $true) & ((hoasapinj2 = $true) <=> ! [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)) | (X174 = X175))) & (hoasapinj1_gthm = $true) & (hoasinduction_gthm = $true) & ((hoaslaminj = $true) <=> ! [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) = (X268 @ X267 @ X266)) | ((lam @ (X268 @ sh @ one)) != (lam @ (X269 @ sh @ one))) | ~! [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) = (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) | ~! [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) = (sub @ (X269 @ X270 @ X271) @ X272)))) & (ulamvarsh = $true) & ((! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34)) <=> (substmonoid = $true)) & (pushprop = $true) & (! [X157 : subst > term > subst > $o,X158 : term] : (($true = (X157 @ id @ (lam @ X158) @ id)) | ~! [X160 : subst > term > term] : (~! [X162 : term,X163 : subst,X164 : subst] : ((sub @ (X160 @ X163 @ X162) @ X164) = (X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164))) | ((X157 @ id @ (lam @ (X160 @ sh @ one)) @ id) = $true) | ~! [X161 : term] : (((X157 @ id @ X161 @ id) != $true) | ($true = (X157 @ id @ (X160 @ id @ X161) @ id)))) | ~! [X159 : term] : (($true != (X157 @ id @ X159 @ id)) | ((X157 @ id @ (sub @ X158 @ (push @ X159 @ id)) @ id) = $true))) <=> (hoasinduction_lem3a = $true)) & ! [X337 : d_term,X338 : d_term] : (((d2term @ X337) = (d2term @ X338)) => (X337 = X338)) & (! [X294 : subst > term > subst > $o,X295 : term] : (~! [X311 : term] : (($true = (X294 @ id @ X311 @ id)) | ((var @ (sub @ X311 @ id)) != $true)) | ~! [X307 : subst,X308 : term,X309 : subst,X310 : subst] : (($true = (X294 @ X310 @ X308 @ (comp @ X309 @ X307))) | ($true != (X294 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309) @ X307))) | ~! [X296 : term,X297 : term] : (((X294 @ id @ X297 @ id) != $true) | ((X294 @ id @ X296 @ id) != $true) | ($true = (X294 @ id @ (ap @ (sub @ X297 @ id) @ X296) @ id))) | ~! [X302 : subst > term > term] : (($true = (X294 @ id @ (lam @ (X302 @ sh @ one)) @ id)) | ~! [X303 : term] : (($true = (X294 @ id @ (X302 @ id @ X303) @ id)) | ($true != (X294 @ id @ X303 @ id))) | ~! [X304 : subst,X305 : subst,X306 : term] : ((X302 @ (comp @ X304 @ X305) @ (sub @ X306 @ X305)) = (sub @ (X302 @ X304 @ X306) @ X305))) | ($true = (X294 @ id @ X295 @ id)) | ~! [X298 : term,X299 : subst,X300 : subst,X301 : subst] : (($true = (X294 @ (comp @ X299 @ X301) @ (sub @ X298 @ X301) @ X300)) | ($true != (X294 @ X299 @ X298 @ (comp @ X301 @ X300))))) <=> (hoasinduction = $true)) & ((! [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) = X27) => ! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (((X30 @ X29) != $true) | ($true = (X28 @ one)) | ~! [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) = (X28 @ X32)))) <=> (pushprop_lem1v2_lthm = $true)) & (! [X138 : subst > term > subst > $o] : ~! [X139 : term > $o] : ~! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140)) <=> (hoasinduction_lem0 = $true)) & ((pushprop_lem1_lthm = $true) <=> (! [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) = X329) => (! [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) = X330) => ! [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true != (X336 @ X332)) | ($true = (X333 @ (sub @ X332 @ (push @ X334 @ X335)))))))) & (hoasinduction_lem1_lthm = $true) & ((induction2lem = $true) <=> ! [X385 : subst,X386 : term > $o,X387 : term] : (~! [X391 : term,X392 : term] : (($true != (X386 @ X391)) | ((X386 @ (ap @ X392 @ X391)) = $true) | ($true != (X386 @ X392))) | ~! [X388 : term] : (~! [X389 : term] : (($true = (X386 @ (sub @ X388 @ (push @ X389 @ id)))) | ((X386 @ X389) != $true)) | ($true = (X386 @ (lam @ X388)))) | ($true = (X386 @ (sub @ X387 @ X385))) | ~! [X390 : term] : (($true = (X386 @ (sub @ X390 @ X385))) | ($true != (var @ X390))))) & (hoasapnotvar_gthm = $true) & (lamnotap != $true) & (substmonoid_gthm = $true) & (hoasinduction_lem2_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)))))))))))))) & (! [X247 : subst > term > subst > $o,X248 : term] : (~! [X249 : subst,X250 : subst,X251 : term,X252 : subst] : (($true = (X247 @ X252 @ X251 @ (comp @ X249 @ X250))) | ($true != (X247 @ (comp @ X252 @ X249) @ (sub @ X251 @ X249) @ X250))) | ~! [X258 : term,X259 : subst,X260 : subst,X261 : subst] : (((X247 @ (comp @ X261 @ X260) @ (sub @ X258 @ X260) @ X259) = $true) | ($true != (X247 @ X261 @ X258 @ (comp @ X260 @ X259)))) | ~! [X262 : term] : (($true != (X247 @ id @ X262 @ id)) | ((X247 @ id @ (sub @ X248 @ (push @ X262 @ id)) @ id) = $true)) | ((X247 @ id @ (lam @ X248) @ id) = $true) | ~! [X253 : subst > term > term] : (($true = (X247 @ id @ (lam @ (X253 @ sh @ one)) @ id)) | ~! [X257 : term] : (($true != (X247 @ id @ X257 @ id)) | ($true = (X247 @ id @ (X253 @ id @ X257) @ id))) | ~! [X254 : subst,X255 : term,X256 : subst] : ((sub @ (X253 @ X256 @ X255) @ X254) = (X253 @ (comp @ X256 @ X254) @ (sub @ X255 @ X254))))) <=> (hoasinduction_lem3 = $true)) & (hoasinduction_lem2_gthm = $true) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (axclos = $true) & ((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) != $true) & (induction2lem_gthm = $true) & (one = (d2term @ d_one)) & (! [X187 : subst > term > subst > $o] : ~! [X188 : term > $o] : ~! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (apnotvar = $true) & ((pushprop_lem0 = $true) <=> ! [X21 : subst,X22 : term,X23 : term > $o] : ~! [X24 : term > $o] : ~! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21))))) & ((hoaslamnotap = $true) <=> ! [X378 : term,X379 : term,X380 : subst > term > term] : (~! [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) = (sub @ (X380 @ X382 @ X383) @ X381)) | ((ap @ (sub @ X379 @ id) @ X378) != (lam @ (X380 @ sh @ one))))) & ((! [X277 : term] : ((sub @ X277 @ id) = X277) => (! [X278 : term,X279 : subst > term > subst > $o] : (~! [X280 : subst > term > term] : (~! [X281 : term] : (($true = (X279 @ id @ (X280 @ id @ X281) @ id)) | ((X279 @ id @ X281 @ id) != $true)) | ~! [X282 : term,X283 : subst,X284 : subst] : ((sub @ (X280 @ X284 @ X282) @ X283) = (X280 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283))) | ($true = (X279 @ id @ (lam @ (X280 @ sh @ one)) @ id))) | ~! [X285 : term] : (($true = (X279 @ id @ (sub @ X278 @ (push @ X285 @ id)) @ id)) | ($true != (X279 @ id @ X285 @ id))) | ($true = (X279 @ id @ (lam @ (sub @ X278 @ (push @ one @ sh))) @ id))) => ! [X286 : subst > term > subst > $o,X287 : term] : (~! [X288 : term] : (((X286 @ id @ (sub @ X287 @ (push @ X288 @ id)) @ id) = $true) | ($true != (X286 @ id @ X288 @ id))) | ($true = (X286 @ id @ (lam @ X287) @ id)) | ~! [X289 : subst > term > term] : (($true = (X286 @ id @ (lam @ (X289 @ sh @ one)) @ id)) | ~! [X291 : subst,X292 : subst,X293 : term] : ((sub @ (X289 @ X292 @ X293) @ X291) = (X289 @ (comp @ X292 @ X291) @ (sub @ X293 @ X291))) | ~! [X290 : term] : (((X286 @ id @ X290 @ id) != $true) | ($true = (X286 @ id @ (X289 @ id @ X290) @ id))))))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem1 = $true) & (! [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) != (ap @ (sub @ X394 @ id) @ X396)) | (X394 = X397)) <=> (hoasapinj1 = $true)) & (hoasinduction_lem3_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true != (X169 @ X167)) | ($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166)))))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X200 : term,X201 : subst > term > subst > $o] : (((X201 @ id @ (lam @ (sub @ X200 @ (push @ one @ sh))) @ id) = $true) | ~! [X202 : subst > term > term] : (($true = (X201 @ id @ (lam @ (X202 @ sh @ one)) @ id)) | ~! [X203 : term] : (($true = (X201 @ id @ (X202 @ id @ X203) @ id)) | ($true != (X201 @ id @ X203 @ id))) | ~! [X204 : term,X205 : subst,X206 : subst] : ((sub @ (X202 @ X206 @ X204) @ X205) = (X202 @ (comp @ X206 @ X205) @ (sub @ X204 @ X205)))) | ~! [X207 : term] : (($true = (X201 @ id @ (sub @ X200 @ (push @ X207 @ id)) @ id)) | ($true != (X201 @ id @ X207 @ id))))) & ((apinj2 = $true) <=> ! [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) != (ap @ X425 @ X426)) | (X426 = X428))) & (axvarshift = $true) & (pushprop_lem0_gthm = $true) & (axabs = $true) & ((var @ (d2term @ d_one)) != $true) & ((hoasinduction_lem3b = $true) <=> ! [X519 : term] : ~! [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) != (X520 @ sh @ one))) & (pushprop_lthm_orig = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) = X384) <=> (axscons = $true)) & ((! [X435 : subst > term > subst > $o] : ~! [X436 : term > $o] : ~! [X437 : term] : ((X436 @ X437) = (X435 @ id @ X437 @ id)) => (! [X438 : term,X439 : term > $o] : (~! [X443 : term,X444 : term] : (((X439 @ X443) != $true) | ($true != (X439 @ X444)) | ((X439 @ (ap @ X444 @ X443)) = $true)) | ($true = (X439 @ X438)) | ~! [X440 : term] : (($true != (var @ X440)) | ($true = (X439 @ X440))) | ~! [X441 : term] : (((X439 @ (lam @ X441)) = $true) | ~! [X442 : term] : (($true = (X439 @ (sub @ X441 @ (push @ X442 @ id)))) | ($true != (X439 @ X442))))) => (! [X445 : term] : ((sub @ X445 @ id) = X445) => (! [X446 : term > $o,X447 : subst > term > subst > $o,X448 : term] : (~! [X450 : term] : (((X446 @ (sub @ X448 @ (push @ X450 @ id))) = $true) | ($true != (X446 @ X450))) | ~! [X451 : subst > term > term] : (($true = (X447 @ id @ (lam @ (X451 @ sh @ one)) @ id)) | ~! [X455 : term] : (($true = (X447 @ id @ (X451 @ id @ X455) @ id)) | ($true != (X447 @ id @ X455 @ id))) | ~! [X452 : term,X453 : subst,X454 : subst] : ((sub @ (X451 @ X453 @ X452) @ X454) = (X451 @ (comp @ X453 @ X454) @ (sub @ X452 @ X454)))) | ~! [X449 : term] : ((X447 @ id @ X449 @ id) = (X446 @ X449)) | ($true = (X446 @ (lam @ X448)))) => ! [X456 : subst > term > subst > $o,X457 : term] : (~! [X458 : subst > term > term] : (~! [X459 : term] : (($true = (X456 @ id @ (X458 @ id @ X459) @ id)) | ($true != (X456 @ id @ X459 @ id))) | ($true = (X456 @ id @ (lam @ (X458 @ sh @ one)) @ id)) | ~! [X460 : term,X461 : subst,X462 : subst] : ((X458 @ (comp @ X462 @ X461) @ (sub @ X460 @ X461)) = (sub @ (X458 @ X462 @ X460) @ X461))) | ~! [X463 : term,X464 : term] : (((X456 @ id @ (ap @ (sub @ X463 @ id) @ X464) @ id) = $true) | ((X456 @ id @ X464 @ id) != $true) | ($true != (X456 @ id @ X463 @ id))) | ($true = (X456 @ id @ X457 @ id))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X263 : term] : ~! [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) != (X264 @ sh @ one))) & (hoaslamnotap_lthm = $true) & ((! [X35 : term] : ((sub @ X35 @ id) = X35) => ! [X36 : subst > term > subst > $o,X37 : term,X38 : term > $o] : (~! [X43 : term] : ((X36 @ id @ X43 @ id) = (X38 @ X43)) | ((X38 @ (lam @ X37)) = $true) | ~! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true = (X36 @ (comp @ X53 @ X50) @ (sub @ X52 @ X50) @ X51)) | ((X36 @ X53 @ X52 @ (comp @ X50 @ X51)) != $true)) | ~! [X39 : term,X40 : subst,X41 : subst,X42 : subst] : (($true != (X36 @ (comp @ X42 @ X41) @ (sub @ X39 @ X41) @ X40)) | ((X36 @ X42 @ X39 @ (comp @ X41 @ X40)) = $true)) | ~! [X44 : subst > term > term] : (($true = (X36 @ id @ (lam @ (X44 @ sh @ one)) @ id)) | ~! [X48 : term] : (($true != (X36 @ id @ X48 @ id)) | ($true = (X36 @ id @ (X44 @ id @ X48) @ id))) | ~! [X45 : term,X46 : subst,X47 : subst] : ((sub @ (X44 @ X47 @ X45) @ X46) = (X44 @ (comp @ X47 @ X46) @ (sub @ X45 @ X46)))) | ~! [X49 : term] : (($true = (X38 @ (sub @ X37 @ (push @ X49 @ id)))) | ($true != (X38 @ X49))))) <=> (hoasinduction_lem3v2_lthm = $true)) & (axassoc = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & (pushprop_lem1v2_gthm = $true) & ! [X141 : d_term] : (d_one = X141) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & ((axshiftcons = $true) <=> ! [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) = X198)) & (! [X276 : term] : ((sub @ X276 @ id) = X276) <=> (termmset = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((! [X398 : term] : ((sub @ X398 @ id) = X398) => (! [X399 : term,X400 : term,X401 : term,X402 : term] : (((ap @ X399 @ X400) != (ap @ X402 @ X401)) | (X399 = X402)) => ! [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 = X406) | ((ap @ (sub @ X405 @ id) @ X403) != (ap @ (sub @ X406 @ id) @ X404))))) <=> (hoasapinj1_lthm = $true)) & ((shinj = $true) <=> ! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186))) & ((axidl = $true) <=> ! [X223 : subst] : ((comp @ id @ X223) = X223)) & (! [X115 : subst > term > subst > $o,X116 : term] : (~! [X122 : term] : (($true = (X115 @ id @ (sub @ X116 @ (push @ X122 @ id)) @ id)) | ($true != (X115 @ id @ X122 @ id))) | ($true = (X115 @ id @ (lam @ (sub @ X116 @ (push @ one @ sh))) @ id)) | ~! [X117 : subst > term > term] : (~! [X118 : subst,X119 : subst,X120 : term] : ((sub @ (X117 @ X118 @ X120) @ X119) = (X117 @ (comp @ X118 @ X119) @ (sub @ X120 @ X119))) | ~! [X121 : term] : (((X115 @ id @ X121 @ id) != $true) | ($true = (X115 @ id @ (X117 @ id @ X121) @ id))) | ($true = (X115 @ id @ (lam @ (X117 @ sh @ one)) @ id)))) <=> (hoasinduction_lem3aa = $true)) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (hoasinduction_lem3b_gthm = $true)),
% 9.38/2.21 inference(flattening,[status(thm)],[f10])).
% 9.38/2.21 thf(f10,plain,(
% 9.38/2.21 (! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) <=> (laminj = $true)) & ((! [X2 : subst,X3 : term] : ((sub @ one @ (push @ X3 @ X2)) = X3) => (! [X4 : term,X5 : subst] : ((comp @ sh @ (push @ X4 @ X5)) = X5) => (! [X6 : subst] : ((comp @ X6 @ id) = X6) => (! [X7 : term > $o,X8 : term] : (~! [X9 : term] : (((X7 @ (lam @ X9)) = $true) | ~($true = (X7 @ X9))) | ~! [X10 : term] : (~((var @ X10) = $true) | ($true = (X7 @ X10))) | ~! [X11 : term,X12 : term] : (($true = (X7 @ (ap @ X11 @ X12))) | ~($true = (X7 @ X11)) | ~($true = (X7 @ X12))) | ((X7 @ X8) = $true)) => ! [X13 : term > $o,X14 : term,X15 : subst] : (~! [X16 : term,X17 : term] : (($true = (X13 @ (ap @ X17 @ X16))) | ~($true = (X13 @ X17)) | ~($true = (X13 @ X16))) | ~! [X18 : term] : (($true = (X13 @ (lam @ X18))) | ~! [X19 : term] : (~((X13 @ X19) = $true) | ($true = (X13 @ (sub @ X18 @ (push @ X19 @ id)))))) | ($true = (X13 @ (sub @ X14 @ X15))) | ~! [X20 : term] : (~($true = (var @ X20)) | ($true = (X13 @ (sub @ X20 @ X15))))))))) <=> (induction2lem_lthm = $true)) & ((pushprop_lem0 = $true) <=> ! [X21 : subst,X22 : term,X23 : term > $o] : ~! [X24 : term > $o] : ~! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21))))) & ((! [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) = X27) => ! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (~! [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) = (X28 @ X32)) | ~((X30 @ X29) = $true) | ($true = (X28 @ one)))) <=> (pushprop_lem1v2_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & ((! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34)) <=> (substmonoid = $true)) & (pushprop_lem3v2_lthm = $true) & (lamnotvar = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X35 : term] : ((sub @ X35 @ id) = X35) => ! [X36 : subst > term > subst > $o,X37 : term,X38 : term > $o] : (~! [X39 : term,X40 : subst,X41 : subst,X42 : subst] : (((X36 @ X42 @ X39 @ (comp @ X41 @ X40)) = $true) | ~($true = (X36 @ (comp @ X42 @ X41) @ (sub @ X39 @ X41) @ X40))) | ~! [X43 : term] : ((X36 @ id @ X43 @ id) = (X38 @ X43)) | ((X38 @ (lam @ X37)) = $true) | ~! [X44 : subst > term > term] : (~! [X45 : term,X46 : subst,X47 : subst] : ((sub @ (X44 @ X47 @ X45) @ X46) = (X44 @ (comp @ X47 @ X46) @ (sub @ X45 @ X46))) | ($true = (X36 @ id @ (lam @ (X44 @ sh @ one)) @ id)) | ~! [X48 : term] : (~($true = (X36 @ id @ X48 @ id)) | ($true = (X36 @ id @ (X44 @ id @ X48) @ id)))) | ~! [X49 : term] : (~($true = (X38 @ X49)) | ($true = (X38 @ (sub @ X37 @ (push @ X49 @ id))))) | ~! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (~((X36 @ X53 @ X52 @ (comp @ X50 @ X51)) = $true) | ($true = (X36 @ (comp @ X53 @ X50) @ (sub @ X52 @ X50) @ X51)))))) & (! [X54 : subst > term > subst > $o,X55 : term] : (~! [X56 : term,X57 : term] : (~($true = (X54 @ id @ X57 @ id)) | ~((X54 @ id @ X56 @ id) = $true) | ($true = (X54 @ id @ (ap @ (sub @ X56 @ id) @ X57) @ id))) | ($true = (X54 @ id @ X55 @ id)) | ~! [X58 : subst > term > term] : (($true = (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) | ~! [X59 : term] : (($true = (X54 @ id @ (X58 @ id @ X59) @ id)) | ~($true = (X54 @ id @ X59 @ id))) | ~! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61)))) <=> (hoasinduction_no_psi_cond = $true)) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X63 : term] : ((sub @ X63 @ id) = X63) => (! [X64 : term,X65 : subst > term > subst > $o] : (($true = (X65 @ id @ (lam @ (sub @ X64 @ (push @ one @ sh))) @ id)) | ~! [X66 : term] : (($true = (X65 @ id @ (sub @ X64 @ (push @ X66 @ id)) @ id)) | ~($true = (X65 @ id @ X66 @ id))) | ~! [X67 : subst > term > term] : (($true = (X65 @ id @ (lam @ (X67 @ sh @ one)) @ id)) | ~! [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X67 @ X69 @ X70) @ X68) = (X67 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) | ~! [X71 : term] : (~($true = (X65 @ id @ X71 @ id)) | ($true = (X65 @ id @ (X67 @ id @ X71) @ id))))) => ! [X72 : subst > term > subst > $o,X73 : term] : (~! [X74 : term] : (($true = (X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id)) | ~((X72 @ id @ X74 @ id) = $true)) | ~! [X75 : subst,X76 : term,X77 : subst,X78 : subst] : (((X72 @ (comp @ X78 @ X77) @ (sub @ X76 @ X77) @ X75) = $true) | ~($true = (X72 @ X78 @ X76 @ (comp @ X77 @ X75)))) | ~! [X79 : term,X80 : subst,X81 : subst,X82 : subst] : (~((X72 @ (comp @ X80 @ X82) @ (sub @ X79 @ X82) @ X81) = $true) | ($true = (X72 @ X80 @ X79 @ (comp @ X82 @ X81)))) | ~! [X83 : subst > term > term] : (~! [X84 : subst,X85 : term,X86 : subst] : ((X83 @ (comp @ X86 @ X84) @ (sub @ X85 @ X84)) = (sub @ (X83 @ X86 @ X85) @ X84)) | ($true = (X72 @ id @ (lam @ (X83 @ sh @ one)) @ id)) | ~! [X87 : term] : (((X72 @ id @ (X83 @ id @ X87) @ id) = $true) | ~($true = (X72 @ id @ X87 @ id)))) | ($true = (X72 @ id @ (lam @ X73) @ id)))))) & ((induction2 = $true) <=> ! [X88 : term > $o,X89 : term] : (~! [X90 : term] : (((X88 @ X90) = $true) | ~($true = (var @ X90))) | ~! [X91 : term] : (($true = (X88 @ (lam @ X91))) | ~! [X92 : term] : (~($true = (X88 @ X92)) | ($true = (X88 @ (sub @ X91 @ (push @ X92 @ id)))))) | ($true = (X88 @ X89)) | ~! [X93 : term,X94 : term] : (~($true = (X88 @ X94)) | ($true = (X88 @ (ap @ X93 @ X94))) | ~($true = (X88 @ X93))))) & ~(ulamvar1 = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem1v2_gthm = $true) & (! [X95 : term] : ~! [X96 : subst > term > term] : ~! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem3_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X99 : term,X100 : subst] : ((sub @ one @ (push @ X99 @ X100)) = X99) => (! [X101 : term,X102 : subst] : ((comp @ sh @ (push @ X101 @ X102)) = X102) => (! [X103 : term,X104 : term] : ((X103 = X104) | ((lam @ X103) != (lam @ X104))) => ! [X105 : term,X106 : subst > term > term,X107 : subst > term > term,X108 : subst] : (~! [X109 : subst,X110 : subst,X111 : term] : ((sub @ (X107 @ X109 @ X111) @ X110) = (X107 @ (comp @ X109 @ X110) @ (sub @ X111 @ X110))) | ~! [X112 : subst,X113 : term,X114 : subst] : ((X106 @ (comp @ X114 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X106 @ X114 @ X113) @ X112)) | ((X106 @ X108 @ X105) = (X107 @ X108 @ X105)) | ((lam @ (X106 @ sh @ one)) != (lam @ (X107 @ sh @ one)))))))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X115 : subst > term > subst > $o,X116 : term] : (($true = (X115 @ id @ (lam @ (sub @ X116 @ (push @ one @ sh))) @ id)) | ~! [X117 : subst > term > term] : (($true = (X115 @ id @ (lam @ (X117 @ sh @ one)) @ id)) | ~! [X118 : subst,X119 : subst,X120 : term] : ((sub @ (X117 @ X118 @ X120) @ X119) = (X117 @ (comp @ X118 @ X119) @ (sub @ X120 @ X119))) | ~! [X121 : term] : (($true = (X115 @ id @ (X117 @ id @ X121) @ id)) | ~((X115 @ id @ X121 @ id) = $true))) | ~! [X122 : term] : (($true = (X115 @ id @ (sub @ X116 @ (push @ X122 @ id)) @ id)) | ~($true = (X115 @ id @ X122 @ id)))) <=> (hoasinduction_lem3aa = $true)) & ((hoasinduction_lem2v2 = $true) <=> ! [X123 : term > $o,X124 : term,X125 : subst > term > subst > $o,X126 : term] : (~! [X127 : term,X128 : term] : (~($true = (X125 @ id @ X128 @ id)) | ~($true = (X125 @ id @ X127 @ id)) | ((X125 @ id @ (ap @ (sub @ X127 @ id) @ X128) @ id) = $true)) | ~($true = (X123 @ X124)) | ~($true = (X123 @ X126)) | ~! [X129 : subst,X130 : subst,X131 : term,X132 : subst] : (~($true = (X125 @ (comp @ X129 @ X132) @ (sub @ X131 @ X132) @ X130)) | ($true = (X125 @ X129 @ X131 @ (comp @ X132 @ X130)))) | ~! [X133 : subst,X134 : subst,X135 : term,X136 : subst] : (($true = (X125 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134) @ X133)) | ~((X125 @ X136 @ X135 @ (comp @ X134 @ X133)) = $true)) | ((X123 @ (ap @ X126 @ X124)) = $true) | ~! [X137 : term] : ((X125 @ id @ X137 @ id) = (X123 @ X137)))) & (axapp = $true) & (! [X138 : subst > term > subst > $o] : ~! [X139 : term > $o] : ~! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140)) <=> (hoasinduction_lem0 = $true)) & (pushprop_lem2v2_lthm = $true) & (hoasapinj1_gthm = $true) & (hoasapinj2_gthm = $true) & ! [X141 : d_term] : (d_one = X141) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X142 : subst > term > subst > $o,X143 : term] : (((X142 @ id @ (lam @ (sub @ X143 @ (push @ one @ sh))) @ id) = $true) | ~! [X144 : term,X145 : subst > term > term] : (~! [X146 : subst,X147 : term,X148 : subst] : ((X145 @ (comp @ X146 @ X148) @ (sub @ X147 @ X148)) = (sub @ X144 @ (push @ (sub @ X147 @ X148) @ (comp @ X146 @ X148)))) | ~! [X149 : term] : (($true = (X142 @ id @ (X145 @ id @ X149) @ id)) | ~($true = (X142 @ id @ X149 @ id))) | ((X142 @ id @ (lam @ (X145 @ sh @ one)) @ id) = $true) | ~! [X150 : term,X151 : subst,X152 : subst] : ((sub @ (X145 @ X152 @ X150) @ X151) = (sub @ (sub @ X144 @ (push @ X150 @ X152)) @ X151))) | ~! [X153 : term] : (~($true = (X142 @ id @ X153 @ id)) | ($true = (X142 @ id @ (sub @ X143 @ (push @ X153 @ id)) @ id)))) <=> (hoasinduction_lem3aaa = $true)) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & ! [X156 : d_subst] : (d_id = X156) & (! [X157 : subst > term > subst > $o,X158 : term] : (~! [X159 : term] : (~($true = (X157 @ id @ X159 @ id)) | ((X157 @ id @ (sub @ X158 @ (push @ X159 @ id)) @ id) = $true)) | ($true = (X157 @ id @ (lam @ X158) @ id)) | ~! [X160 : subst > term > term] : (~! [X161 : term] : (~((X157 @ id @ X161 @ id) = $true) | ($true = (X157 @ id @ (X160 @ id @ X161) @ id))) | ((X157 @ id @ (lam @ (X160 @ sh @ one)) @ id) = $true) | ~! [X162 : term,X163 : subst,X164 : subst] : ((sub @ (X160 @ X163 @ X162) @ X164) = (X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164))))) <=> (hoasinduction_lem3a = $true)) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1 = $true) <=> ! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166)))) | ~($true = (X169 @ X167)))) & (hoasapnotvar = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_gthm = $true) & (axabs = $true) & (induction2_gthm = $true) & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & (hoaslaminj_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)) | (X174 = X175))) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179)) <=> (apinj1 = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (hoasinduction_lem1 = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lem2v2 = $true) & (! [X180 : term > $o,X181 : term,X182 : subst] : ~! [X183 : term > $o] : ~! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182)))) <=> (pushprop_lem0_lthm = $true)) & (pushprop = $true) & (hoasinduction_lem3v2_gthm = $true) & ((shinj = $true) <=> ! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186))) & (! [X187 : subst > term > subst > $o] : ~! [X188 : term > $o] : ~! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (hoaslamnotvar_lthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (induction2lem_gthm = $true) & (termmset_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X190 : term,X191 : term,X192 : term,X193 : term] : ((X190 = X192) | ((ap @ X191 @ X190) != (ap @ X193 @ X192))) => ! [X194 : term,X195 : term,X196 : term,X197 : term] : (((ap @ (sub @ X197 @ id) @ X195) != (ap @ (sub @ X196 @ id) @ X194)) | (X194 = X195)))) & (axmap = $true) & (axclos = $true) & (one = (d2term @ d_one)) & ((axshiftcons = $true) <=> ! [X198 : subst,X199 : term] : ((comp @ sh @ (push @ X199 @ X198)) = X198)) & (hoasapnotvar_gthm = $true) & (axvarshift = $true) & (! [X200 : term,X201 : subst > term > subst > $o] : (((X201 @ id @ (lam @ (sub @ X200 @ (push @ one @ sh))) @ id) = $true) | ~! [X202 : subst > term > term] : (($true = (X201 @ id @ (lam @ (X202 @ sh @ one)) @ id)) | ~! [X203 : term] : (($true = (X201 @ id @ (X202 @ id @ X203) @ id)) | ~($true = (X201 @ id @ X203 @ id))) | ~! [X204 : term,X205 : subst,X206 : subst] : ((sub @ (X202 @ X206 @ X204) @ X205) = (X202 @ (comp @ X206 @ X205) @ (sub @ X204 @ X205)))) | ~! [X207 : term] : (~($true = (X201 @ id @ X207 @ id)) | ($true = (X201 @ id @ (sub @ X200 @ (push @ X207 @ id)) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((! [X208 : term] : ((sub @ X208 @ id) = X208) => ! [X209 : term] : ((sub @ X209 @ id) = X209)) <=> (termmset_lthm = $true)) & (! [X210 : subst > term > subst > $o,X211 : term,X212 : term] : (~($true = (X210 @ id @ X212 @ id)) | ~($true = (X210 @ id @ X211 @ id)) | ~! [X213 : term,X214 : subst,X215 : subst,X216 : subst] : (~($true = (X210 @ X215 @ X213 @ (comp @ X216 @ X214))) | ((X210 @ (comp @ X215 @ X216) @ (sub @ X213 @ X216) @ X214) = $true)) | ~! [X217 : subst,X218 : subst,X219 : subst,X220 : term] : (($true = (X210 @ X219 @ X220 @ (comp @ X218 @ X217))) | ~((X210 @ (comp @ X219 @ X218) @ (sub @ X220 @ X218) @ X217) = $true)) | ~! [X221 : term,X222 : term] : (($true = (X210 @ id @ (ap @ (sub @ X221 @ id) @ X222) @ id)) | ~($true = (X210 @ id @ X221 @ id)) | ~($true = (X210 @ id @ X222 @ id))) | ((X210 @ id @ (ap @ X212 @ X211) @ id) = $true)) <=> (hoasinduction_lem2 = $true)) & ((axidl = $true) <=> ! [X223 : subst] : ((comp @ id @ X223) = X223)) & ((induction2_lthm = $true) <=> (! [X224 : term] : ((sub @ X224 @ id) = X224) => (! [X225 : term,X226 : term > $o,X227 : subst] : (~! [X228 : term] : (~($true = (var @ X228)) | ($true = (X226 @ (sub @ X228 @ X227)))) | ($true = (X226 @ (sub @ X225 @ X227))) | ~! [X229 : term] : (~! [X230 : term] : (($true = (X226 @ (sub @ X229 @ (push @ X230 @ id)))) | ~($true = (X226 @ X230))) | ($true = (X226 @ (lam @ X229)))) | ~! [X231 : term,X232 : term] : (~((X226 @ X231) = $true) | ~($true = (X226 @ X232)) | ($true = (X226 @ (ap @ X232 @ X231))))) => ! [X233 : term,X234 : term > $o] : (($true = (X234 @ X233)) | ~! [X235 : term] : (($true = (X234 @ (lam @ X235))) | ~! [X236 : term] : (~($true = (X234 @ X236)) | ($true = (X234 @ (sub @ X235 @ (push @ X236 @ id)))))) | ~! [X237 : term,X238 : term] : (~($true = (X234 @ X238)) | ($true = (X234 @ (ap @ X238 @ X237))) | ~($true = (X234 @ X237))) | ~! [X239 : term] : (($true = (X234 @ X239)) | ~($true = (var @ X239))))))) & (hoasinduction_lem3b_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (! [X240 : term,X241 : subst,X242 : term > $o,X243 : term > $o] : (((X242 @ one) = $true) | ~($true = (X243 @ X240)) | ~! [X244 : term] : ((X243 @ (sub @ X244 @ (push @ X240 @ X241))) = (X242 @ X244))) <=> (pushprop_lem1v2 = $true)) & (ulamvarsh = $true) & (hoasinduction_lem1_gthm = $true) & ((axvarcons = $true) <=> ! [X245 : term,X246 : subst] : ((sub @ one @ (push @ X245 @ X246)) = X245)) & (pushprop_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X247 : subst > term > subst > $o,X248 : term] : (((X247 @ id @ (lam @ X248) @ id) = $true) | ~! [X249 : subst,X250 : subst,X251 : term,X252 : subst] : (~($true = (X247 @ (comp @ X252 @ X249) @ (sub @ X251 @ X249) @ X250)) | ($true = (X247 @ X252 @ X251 @ (comp @ X249 @ X250)))) | ~! [X253 : subst > term > term] : (($true = (X247 @ id @ (lam @ (X253 @ sh @ one)) @ id)) | ~! [X254 : subst,X255 : term,X256 : subst] : ((sub @ (X253 @ X256 @ X255) @ X254) = (X253 @ (comp @ X256 @ X254) @ (sub @ X255 @ X254))) | ~! [X257 : term] : (($true = (X247 @ id @ (X253 @ id @ X257) @ id)) | ~($true = (X247 @ id @ X257 @ id)))) | ~! [X258 : term,X259 : subst,X260 : subst,X261 : subst] : (~($true = (X247 @ X261 @ X258 @ (comp @ X260 @ X259))) | ((X247 @ (comp @ X261 @ X260) @ (sub @ X258 @ X260) @ X259) = $true)) | ~! [X262 : term] : (((X247 @ id @ (sub @ X248 @ (push @ X262 @ id)) @ id) = $true) | ~($true = (X247 @ id @ X262 @ id))))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X263 : term] : ~! [X264 : subst > term > term] : ((sub @ X263 @ (push @ one @ sh)) != (X264 @ sh @ one))) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((axvarid = $true) <=> ! [X265 : term] : ((sub @ X265 @ id) = X265)) & ((hoaslaminj = $true) <=> ! [X266 : term,X267 : subst,X268 : subst > term > term,X269 : subst > term > term] : (((X269 @ X267 @ X266) = (X268 @ X267 @ X266)) | ((lam @ (X268 @ sh @ one)) != (lam @ (X269 @ sh @ one))) | ~! [X273 : term,X274 : subst,X275 : subst] : ((sub @ (X268 @ X275 @ X273) @ X274) = (X268 @ (comp @ X275 @ X274) @ (sub @ X273 @ X274))) | ~! [X270 : subst,X271 : term,X272 : subst] : ((X269 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272)) = (sub @ (X269 @ X270 @ X271) @ X272)))) & (! [X276 : term] : ((sub @ X276 @ id) = X276) <=> (termmset = $true)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X277 : term] : ((sub @ X277 @ id) = X277) => (! [X278 : term,X279 : subst > term > subst > $o] : (($true = (X279 @ id @ (lam @ (sub @ X278 @ (push @ one @ sh))) @ id)) | ~! [X280 : subst > term > term] : (($true = (X279 @ id @ (lam @ (X280 @ sh @ one)) @ id)) | ~! [X281 : term] : (($true = (X279 @ id @ (X280 @ id @ X281) @ id)) | ~((X279 @ id @ X281 @ id) = $true)) | ~! [X282 : term,X283 : subst,X284 : subst] : ((sub @ (X280 @ X284 @ X282) @ X283) = (X280 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283)))) | ~! [X285 : term] : (($true = (X279 @ id @ (sub @ X278 @ (push @ X285 @ id)) @ id)) | ~($true = (X279 @ id @ X285 @ id)))) => ! [X286 : subst > term > subst > $o,X287 : term] : (~! [X288 : term] : (((X286 @ id @ (sub @ X287 @ (push @ X288 @ id)) @ id) = $true) | ~($true = (X286 @ id @ X288 @ id))) | ~! [X289 : subst > term > term] : (($true = (X286 @ id @ (lam @ (X289 @ sh @ one)) @ id)) | ~! [X290 : term] : (($true = (X286 @ id @ (X289 @ id @ X290) @ id)) | ~((X286 @ id @ X290 @ id) = $true)) | ~! [X291 : subst,X292 : subst,X293 : term] : ((sub @ (X289 @ X292 @ X293) @ X291) = (X289 @ (comp @ X292 @ X291) @ (sub @ X293 @ X291)))) | ($true = (X286 @ id @ (lam @ X287) @ id)))))) & (! [X294 : subst > term > subst > $o,X295 : term] : (~! [X296 : term,X297 : term] : (~((X294 @ id @ X297 @ id) = $true) | ($true = (X294 @ id @ (ap @ (sub @ X297 @ id) @ X296) @ id)) | ~((X294 @ id @ X296 @ id) = $true)) | ~! [X298 : term,X299 : subst,X300 : subst,X301 : subst] : (($true = (X294 @ (comp @ X299 @ X301) @ (sub @ X298 @ X301) @ X300)) | ~($true = (X294 @ X299 @ X298 @ (comp @ X301 @ X300)))) | ~! [X302 : subst > term > term] : (~! [X303 : term] : (~($true = (X294 @ id @ X303 @ id)) | ($true = (X294 @ id @ (X302 @ id @ X303) @ id))) | ($true = (X294 @ id @ (lam @ (X302 @ sh @ one)) @ id)) | ~! [X304 : subst,X305 : subst,X306 : term] : ((X302 @ (comp @ X304 @ X305) @ (sub @ X306 @ X305)) = (sub @ (X302 @ X304 @ X306) @ X305))) | ~! [X307 : subst,X308 : term,X309 : subst,X310 : subst] : (($true = (X294 @ X310 @ X308 @ (comp @ X309 @ X307))) | ~($true = (X294 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309) @ X307))) | ~! [X311 : term] : (~((var @ (sub @ X311 @ id)) = $true) | ($true = (X294 @ id @ X311 @ id))) | ($true = (X294 @ id @ X295 @ id))) <=> (hoasinduction = $true)) & ! [X312 : d_subst,X313 : d_subst] : (((d2subst @ X312) = (d2subst @ X313)) => (X312 = X313)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X314 : term] : ~! [X315 : subst > term > term] : ~! [X316 : subst,X317 : term] : ((sub @ X314 @ (push @ X317 @ X316)) = (X315 @ X316 @ X317))) & ((hoasinduction_lem3v2a = $true) <=> ! [X318 : subst > term > subst > $o,X319 : term,X320 : term > $o] : (~! [X321 : subst > term > term] : (~! [X322 : subst,X323 : term,X324 : subst] : ((sub @ (X321 @ X322 @ X323) @ X324) = (X321 @ (comp @ X322 @ X324) @ (sub @ X323 @ X324))) | ~! [X325 : term] : (($true = (X318 @ id @ (X321 @ id @ X325) @ id)) | ~((X318 @ id @ X325 @ id) = $true)) | ((X318 @ id @ (lam @ (X321 @ sh @ one)) @ id) = $true)) | ($true = (X320 @ (lam @ X319))) | ~! [X326 : term] : ((X318 @ id @ X326 @ id) = (X320 @ X326)) | ~! [X327 : term] : (~($true = (X320 @ X327)) | ($true = (X320 @ (sub @ X319 @ (push @ X327 @ id))))))) & ((pushprop_lem1_lthm = $true) <=> (! [X328 : subst,X329 : term] : ((sub @ one @ (push @ X329 @ X328)) = X329) => (! [X330 : subst,X331 : term] : ((comp @ sh @ (push @ X331 @ X330)) = X330) => ! [X332 : term,X333 : term > $o,X334 : term,X335 : subst,X336 : term > $o] : (($true = (X333 @ (sub @ X332 @ (push @ X334 @ X335)))) | ~($true = (X336 @ X332)))))) & ! [X337 : d_term,X338 : d_term] : (((d2term @ X337) = (d2term @ X338)) => (X337 = X338)) & (pushprop_lthm_orig = $true) & (apnotvar = $true) & (hoaslamnotap_gthm = $true) & ((! [X339 : subst > term > subst > $o] : ~! [X340 : term > $o] : ~! [X341 : term] : ((X339 @ id @ X341 @ id) = (X340 @ X341)) => (! [X342 : term > $o,X343 : term] : (~! [X344 : term,X345 : term] : (($true = (X342 @ (ap @ X345 @ X344))) | ~((X342 @ X344) = $true) | ~($true = (X342 @ X345))) | ((X342 @ X343) = $true) | ~! [X346 : term] : (~! [X347 : term] : (~($true = (X342 @ X347)) | ($true = (X342 @ (sub @ X346 @ (push @ X347 @ id))))) | ($true = (X342 @ (lam @ X346)))) | ~! [X348 : term] : (~($true = (var @ X348)) | ((X342 @ X348) = $true))) => (! [X349 : term] : ((sub @ X349 @ id) = X349) => (! [X350 : subst > term > subst > $o,X351 : term,X352 : term > $o] : (~! [X353 : term] : ((X352 @ X353) = (X350 @ id @ X353 @ id)) | ~! [X354 : term] : (($true = (X352 @ (sub @ X351 @ (push @ X354 @ id)))) | ~($true = (X352 @ X354))) | ($true = (X352 @ (lam @ X351))) | ~! [X355 : subst > term > term] : (($true = (X350 @ id @ (lam @ (X355 @ sh @ one)) @ id)) | ~! [X356 : term] : (($true = (X350 @ id @ (X355 @ id @ X356) @ id)) | ~($true = (X350 @ id @ X356 @ id))) | ~! [X357 : subst,X358 : term,X359 : subst] : ((X355 @ (comp @ X357 @ X359) @ (sub @ X358 @ X359)) = (sub @ (X355 @ X357 @ X358) @ X359)))) => ! [X360 : subst > term > subst > $o,X361 : term] : (~! [X362 : subst,X363 : subst,X364 : subst,X365 : term] : (~($true = (X360 @ (comp @ X362 @ X363) @ (sub @ X365 @ X363) @ X364)) | ($true = (X360 @ X362 @ X365 @ (comp @ X363 @ X364)))) | ~! [X366 : subst > term > term] : (~! [X367 : subst,X368 : term,X369 : subst] : ((sub @ (X366 @ X369 @ X368) @ X367) = (X366 @ (comp @ X369 @ X367) @ (sub @ X368 @ X367))) | ($true = (X360 @ id @ (lam @ (X366 @ sh @ one)) @ id)) | ~! [X370 : term] : (~($true = (X360 @ id @ X370 @ id)) | ($true = (X360 @ id @ (X366 @ id @ X370) @ id)))) | ~! [X371 : term,X372 : term] : (($true = (X360 @ id @ (ap @ (sub @ X372 @ id) @ X371) @ id)) | ~($true = (X360 @ id @ X371 @ id)) | ~($true = (X360 @ id @ X372 @ id))) | ($true = (X360 @ id @ X361 @ id)) | ~! [X373 : term] : (~($true = (var @ (sub @ X373 @ id))) | ($true = (X360 @ id @ X373 @ id))) | ~! [X374 : subst,X375 : subst,X376 : subst,X377 : term] : (~($true = (X360 @ X374 @ X377 @ (comp @ X375 @ X376))) | ($true = (X360 @ (comp @ X374 @ X375) @ (sub @ X377 @ X375) @ X376)))))))) <=> (hoasinduction_lthm_3 = $true)) & ((hoaslamnotap = $true) <=> ! [X378 : term,X379 : term,X380 : subst > term > term] : (~! [X381 : subst,X382 : subst,X383 : term] : ((X380 @ (comp @ X382 @ X381) @ (sub @ X383 @ X381)) = (sub @ (X380 @ X382 @ X383) @ X381)) | ((ap @ (sub @ X379 @ id) @ X378) != (lam @ (X380 @ sh @ one))))) & (! [X384 : subst] : ((push @ (sub @ one @ X384) @ (comp @ sh @ X384)) = X384) <=> (axscons = $true)) & ~((var @ (d2term @ d_one)) = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X385 : subst,X386 : term > $o,X387 : term] : (~! [X388 : term] : (~! [X389 : term] : (~((X386 @ X389) = $true) | ($true = (X386 @ (sub @ X388 @ (push @ X389 @ id))))) | ($true = (X386 @ (lam @ X388)))) | ($true = (X386 @ (sub @ X387 @ X385))) | ~! [X390 : term] : (~($true = (var @ X390)) | ($true = (X386 @ (sub @ X390 @ X385)))) | ~! [X391 : term,X392 : term] : (~($true = (X386 @ X391)) | ~($true = (X386 @ X392)) | ((X386 @ (ap @ X392 @ X391)) = $true))) <=> (induction2lem = $true)) & ((axidr = $true) <=> ! [X393 : subst] : ((comp @ X393 @ id) = X393)) & (! [X394 : term,X395 : term,X396 : term,X397 : term] : (((ap @ (sub @ X397 @ id) @ X395) != (ap @ (sub @ X394 @ id) @ X396)) | (X394 = X397)) <=> (hoasapinj1 = $true)) & (axassoc = $true) & ((! [X398 : term] : ((sub @ X398 @ id) = X398) => (! [X399 : term,X400 : term,X401 : term,X402 : term] : (((ap @ X399 @ X400) != (ap @ X402 @ X401)) | (X399 = X402)) => ! [X403 : term,X404 : term,X405 : term,X406 : term] : ((X405 = X406) | ((ap @ (sub @ X405 @ id) @ X403) != (ap @ (sub @ X406 @ id) @ X404))))) <=> (hoasapinj1_lthm = $true)) & (ulamvarind = $true) & ~(lamnotap = $true) & (pushprop_lthm = $true) & (! [X407 : term,X408 : subst > term > subst > $o,X409 : term > $o] : (~! [X410 : subst > term > term] : (~! [X411 : term,X412 : subst,X413 : subst] : ((sub @ (X410 @ X412 @ X411) @ X413) = (X410 @ (comp @ X412 @ X413) @ (sub @ X411 @ X413))) | ($true = (X408 @ id @ (lam @ (X410 @ sh @ one)) @ id)) | ~! [X414 : term] : (($true = (X408 @ id @ (X410 @ id @ X414) @ id)) | ~($true = (X408 @ id @ X414 @ id)))) | ~! [X415 : term] : (~($true = (X409 @ X415)) | ($true = (X409 @ (sub @ X407 @ (push @ X415 @ id))))) | ~! [X416 : subst,X417 : subst,X418 : term,X419 : subst] : (($true = (X408 @ X419 @ X418 @ (comp @ X417 @ X416))) | ~($true = (X408 @ (comp @ X419 @ X417) @ (sub @ X418 @ X417) @ X416))) | ~! [X420 : term] : ((X409 @ X420) = (X408 @ id @ X420 @ id)) | ((X409 @ (lam @ X407)) = $true) | ~! [X421 : subst,X422 : term,X423 : subst,X424 : subst] : (~($true = (X408 @ X424 @ X422 @ (comp @ X421 @ X423))) | ($true = (X408 @ (comp @ X424 @ X421) @ (sub @ X422 @ X421) @ X423)))) <=> (hoasinduction_lem3v2 = $true)) & ((d2subst @ d_id) = id) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1_lthm = $true) & ((apinj2 = $true) <=> ! [X425 : term,X426 : term,X427 : term,X428 : term] : (((ap @ X427 @ X428) != (ap @ X425 @ X426)) | (X426 = X428))) & (! [X429 : term,X430 : term > $o] : (($true = (X430 @ X429)) | ~! [X431 : term] : (($true = (X430 @ X431)) | ~($true = (var @ X431))) | ~! [X432 : term] : (~($true = (X430 @ X432)) | ($true = (X430 @ (lam @ X432)))) | ~! [X433 : term,X434 : term] : (~($true = (X430 @ X433)) | ((X430 @ (ap @ X433 @ X434)) = $true) | ~($true = (X430 @ X434)))) <=> (induction = $true)) & ((! [X435 : subst > term > subst > $o] : ~! [X436 : term > $o] : ~! [X437 : term] : ((X436 @ X437) = (X435 @ id @ X437 @ id)) => (! [X438 : term,X439 : term > $o] : (~! [X440 : term] : (~($true = (var @ X440)) | ($true = (X439 @ X440))) | ($true = (X439 @ X438)) | ~! [X441 : term] : (((X439 @ (lam @ X441)) = $true) | ~! [X442 : term] : (~($true = (X439 @ X442)) | ($true = (X439 @ (sub @ X441 @ (push @ X442 @ id)))))) | ~! [X443 : term,X444 : term] : (~((X439 @ X443) = $true) | ~($true = (X439 @ X444)) | ((X439 @ (ap @ X444 @ X443)) = $true))) => (! [X445 : term] : ((sub @ X445 @ id) = X445) => (! [X446 : term > $o,X447 : subst > term > subst > $o,X448 : term] : (~! [X449 : term] : ((X447 @ id @ X449 @ id) = (X446 @ X449)) | ~! [X450 : term] : (~($true = (X446 @ X450)) | ((X446 @ (sub @ X448 @ (push @ X450 @ id))) = $true)) | ~! [X451 : subst > term > term] : (~! [X452 : term,X453 : subst,X454 : subst] : ((sub @ (X451 @ X453 @ X452) @ X454) = (X451 @ (comp @ X453 @ X454) @ (sub @ X452 @ X454))) | ($true = (X447 @ id @ (lam @ (X451 @ sh @ one)) @ id)) | ~! [X455 : term] : (~($true = (X447 @ id @ X455 @ id)) | ($true = (X447 @ id @ (X451 @ id @ X455) @ id)))) | ($true = (X446 @ (lam @ X448)))) => ! [X456 : subst > term > subst > $o,X457 : term] : (($true = (X456 @ id @ X457 @ id)) | ~! [X458 : subst > term > term] : (($true = (X456 @ id @ (lam @ (X458 @ sh @ one)) @ id)) | ~! [X459 : term] : (($true = (X456 @ id @ (X458 @ id @ X459) @ id)) | ~($true = (X456 @ id @ X459 @ id))) | ~! [X460 : term,X461 : subst,X462 : subst] : ((X458 @ (comp @ X462 @ X461) @ (sub @ X460 @ X461)) = (sub @ (X458 @ X462 @ X460) @ X461))) | ~! [X463 : term,X464 : term] : (~((X456 @ id @ X464 @ id) = $true) | ((X456 @ id @ (ap @ (sub @ X463 @ id) @ X464) @ id) = $true) | ~($true = (X456 @ id @ X463 @ id)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1_gthm = $true) & ((d2subst @ d_id) = sh) & ((hoasinduction_lthm = $true) <=> (! [X465 : term,X466 : term > $o] : (~! [X467 : term] : (~! [X468 : term] : (~($true = (X466 @ X468)) | ((X466 @ (sub @ X467 @ (push @ X468 @ id))) = $true)) | ($true = (X466 @ (lam @ X467)))) | ($true = (X466 @ X465)) | ~! [X469 : term] : (~($true = (var @ X469)) | ((X466 @ X469) = $true)) | ~! [X470 : term,X471 : term] : (($true = (X466 @ (ap @ X470 @ X471))) | ~($true = (X466 @ X471)) | ~($true = (X466 @ X470)))) => (! [X472 : subst > term > subst > $o,X473 : term,X474 : term] : (~($true = (X472 @ id @ X473 @ id)) | ($true = (X472 @ id @ (ap @ X473 @ X474) @ id)) | ~! [X475 : subst,X476 : term,X477 : subst,X478 : subst] : (($true = (X472 @ (comp @ X478 @ X477) @ (sub @ X476 @ X477) @ X475)) | ~($true = (X472 @ X478 @ X476 @ (comp @ X477 @ X475)))) | ~($true = (X472 @ id @ X474 @ id)) | ~! [X479 : term,X480 : subst,X481 : subst,X482 : subst] : (($true = (X472 @ X480 @ X479 @ (comp @ X481 @ X482))) | ~($true = (X472 @ (comp @ X480 @ X481) @ (sub @ X479 @ X481) @ X482))) | ~! [X483 : term,X484 : term] : (($true = (X472 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) | ~((X472 @ id @ X484 @ id) = $true) | ~($true = (X472 @ id @ X483 @ id)))) => (! [X485 : term,X486 : subst > term > subst > $o] : (~! [X487 : subst,X488 : subst,X489 : subst,X490 : term] : (~($true = (X486 @ (comp @ X488 @ X489) @ (sub @ X490 @ X489) @ X487)) | ((X486 @ X488 @ X490 @ (comp @ X489 @ X487)) = $true)) | ($true = (X486 @ id @ (lam @ X485) @ id)) | ~! [X491 : subst,X492 : term,X493 : subst,X494 : subst] : (((X486 @ (comp @ X493 @ X494) @ (sub @ X492 @ X494) @ X491) = $true) | ~($true = (X486 @ X493 @ X492 @ (comp @ X494 @ X491)))) | ~! [X495 : subst > term > term] : (~! [X496 : term] : (~($true = (X486 @ id @ X496 @ id)) | ($true = (X486 @ id @ (X495 @ id @ X496) @ id))) | ($true = (X486 @ id @ (lam @ (X495 @ sh @ one)) @ id)) | ~! [X497 : term,X498 : subst,X499 : subst] : ((X495 @ (comp @ X498 @ X499) @ (sub @ X497 @ X499)) = (sub @ (X495 @ X498 @ X497) @ X499))) | ~! [X500 : term] : (~($true = (X486 @ id @ X500 @ id)) | ((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id) = $true))) => ! [X501 : term,X502 : subst > term > subst > $o] : (~! [X503 : subst,X504 : subst,X505 : term,X506 : subst] : (~($true = (X502 @ (comp @ X504 @ X506) @ (sub @ X505 @ X506) @ X503)) | ($true = (X502 @ X504 @ X505 @ (comp @ X506 @ X503)))) | ~! [X507 : subst > term > term] : (($true = (X502 @ id @ (lam @ (X507 @ sh @ one)) @ id)) | ~! [X508 : term] : (($true = (X502 @ id @ (X507 @ id @ X508) @ id)) | ~((X502 @ id @ X508 @ id) = $true)) | ~! [X509 : term,X510 : subst,X511 : subst] : ((X507 @ (comp @ X511 @ X510) @ (sub @ X509 @ X510)) = (sub @ (X507 @ X511 @ X509) @ X510))) | ~! [X512 : term,X513 : subst,X514 : subst,X515 : subst] : (~((X502 @ X515 @ X512 @ (comp @ X514 @ X513)) = $true) | ($true = (X502 @ (comp @ X515 @ X514) @ (sub @ X512 @ X514) @ X513))) | ((X502 @ id @ X501 @ id) = $true) | ~! [X516 : term,X517 : term] : (~((X502 @ id @ X516 @ id) = $true) | ~((X502 @ id @ X517 @ id) = $true) | ($true = (X502 @ id @ (ap @ (sub @ X516 @ id) @ X517) @ id))) | ~! [X518 : term] : (((X502 @ id @ X518 @ id) = $true) | ~($true = (var @ (sub @ X518 @ id))))))))) & ((hoasinduction_lem3b = $true) <=> ! [X519 : term] : ~! [X520 : subst > term > term] : ((sub @ X519 @ (push @ one @ sh)) != (X520 @ sh @ one))) & ((substmonoid_lthm = $true) <=> (! [X521 : subst] : ((comp @ id @ X521) = X521) => (! [X522 : subst] : ((comp @ X522 @ id) = X522) => (! [X523 : subst] : ((comp @ id @ X523) = X523) & ! [X524 : subst] : ((comp @ X524 @ id) = X524))))) & (hoaslamnotvar = $true) & ((! [X525 : term] : ~! [X526 : subst > term > term] : ~! [X527 : subst,X528 : term] : ((X526 @ X527 @ X528) = (sub @ X525 @ (push @ X528 @ X527))) => (! [X529 : term] : ((sub @ X529 @ id) = X529) => ! [X530 : term,X531 : term > $o,X532 : subst > term > subst > $o] : (($true = (X531 @ (lam @ X530))) | ~! [X533 : term] : ((X532 @ id @ X533 @ id) = (X531 @ X533)) | ~! [X534 : subst > term > term] : (~! [X535 : term] : (((X532 @ id @ (X534 @ id @ X535) @ id) = $true) | ~($true = (X532 @ id @ X535 @ id))) | ($true = (X532 @ id @ (lam @ (X534 @ sh @ one)) @ id)) | ~! [X536 : subst,X537 : term,X538 : subst] : ((X534 @ (comp @ X538 @ X536) @ (sub @ X537 @ X536)) = (sub @ (X534 @ X538 @ X537) @ X536))) | ~! [X539 : term] : (($true = (X531 @ (sub @ X530 @ (push @ X539 @ id)))) | ~($true = (X531 @ X539)))))) <=> (hoasinduction_lem3v2a_lthm = $true))),
% 9.38/2.21 inference(rectify,[status(thm)],[f6])).
% 9.38/2.21 thf(f6,plain,(
% 9.38/2.21 (! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) <=> (laminj = $true)) & ((! [X2 : subst,X3 : term] : ((sub @ one @ (push @ X3 @ X2)) = X3) => (! [X4 : term,X5 : subst] : ((comp @ sh @ (push @ X4 @ X5)) = X5) => (! [X6 : subst] : ((comp @ X6 @ id) = X6) => (! [X7 : term > $o,X8 : term] : (~! [X9 : term] : (((X7 @ (lam @ X9)) = $true) | ~($true = (X7 @ X9))) | ~! [X10 : term] : (~((var @ X10) = $true) | ($true = (X7 @ X10))) | ~! [X11 : term,X12 : term] : (($true = (X7 @ (ap @ X11 @ X12))) | ~($true = (X7 @ X11)) | ~($true = (X7 @ X12))) | ((X7 @ X8) = $true)) => ! [X13 : term > $o,X14 : term,X15 : subst] : (~! [X16 : term,X17 : term] : (($true = (X13 @ (ap @ X17 @ X16))) | ~($true = (X13 @ X17)) | ~($true = (X13 @ X16))) | ~! [X18 : term] : (($true = (X13 @ (lam @ X18))) | ~! [X19 : term] : (~((X13 @ X19) = $true) | ($true = (X13 @ (sub @ X18 @ (push @ X19 @ id)))))) | ($true = (X13 @ (sub @ X14 @ X15))) | ~! [X20 : term] : (~($true = (var @ X20)) | ($true = (X13 @ (sub @ X20 @ X15))))))))) <=> (induction2lem_lthm = $true)) & ((pushprop_lem0 = $true) <=> ! [X21 : subst,X22 : term,X23 : term > $o] : ~! [X24 : term > $o] : ~! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21))))) & ((! [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) = X27) => ! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (~! [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) = (X28 @ X32)) | ~((X30 @ X29) = $true) | ($true = (X28 @ one)))) <=> (pushprop_lem1v2_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & ((! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34)) <=> (substmonoid = $true)) & (pushprop_lem3v2_lthm = $true) & (lamnotvar = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X35 : term] : ((sub @ X35 @ id) = X35) => ! [X36 : subst > term > subst > $o,X37 : term,X38 : term > $o] : (~! [X39 : term,X40 : subst,X41 : subst,X42 : subst] : (((X36 @ X42 @ X39 @ (comp @ X41 @ X40)) = $true) | ~($true = (X36 @ (comp @ X42 @ X41) @ (sub @ X39 @ X41) @ X40))) | ~! [X43 : term] : ((X36 @ id @ X43 @ id) = (X38 @ X43)) | ((X38 @ (lam @ X37)) = $true) | ~! [X44 : subst > term > term] : (~! [X45 : term,X46 : subst,X47 : subst] : ((sub @ (X44 @ X47 @ X45) @ X46) = (X44 @ (comp @ X47 @ X46) @ (sub @ X45 @ X46))) | ($true = (X36 @ id @ (lam @ (X44 @ sh @ one)) @ id)) | ~! [X48 : term] : (~($true = (X36 @ id @ X48 @ id)) | ($true = (X36 @ id @ (X44 @ id @ X48) @ id)))) | ~! [X49 : term] : (~($true = (X38 @ X49)) | ($true = (X38 @ (sub @ X37 @ (push @ X49 @ id))))) | ~! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (~((X36 @ X53 @ X52 @ (comp @ X50 @ X51)) = $true) | ($true = (X36 @ (comp @ X53 @ X50) @ (sub @ X52 @ X50) @ X51)))))) & (! [X54 : subst > term > subst > $o,X55 : term] : (~! [X56 : term,X57 : term] : (~($true = (X54 @ id @ X57 @ id)) | ~((X54 @ id @ X56 @ id) = $true) | ($true = (X54 @ id @ (ap @ (sub @ X56 @ id) @ X57) @ id))) | ($true = (X54 @ id @ X55 @ id)) | ~! [X58 : subst > term > term] : (($true = (X54 @ id @ (lam @ (X58 @ sh @ one)) @ id)) | ~! [X59 : term] : (($true = (X54 @ id @ (X58 @ id @ X59) @ id)) | ~($true = (X54 @ id @ X59 @ id))) | ~! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61)))) <=> (hoasinduction_no_psi_cond = $true)) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X63 : term] : ((sub @ X63 @ id) = X63) => (! [X64 : term,X65 : subst > term > subst > $o] : (($true = (X65 @ id @ (lam @ (sub @ X64 @ (push @ one @ sh))) @ id)) | ~! [X66 : term] : (($true = (X65 @ id @ (sub @ X64 @ (push @ X66 @ id)) @ id)) | ~($true = (X65 @ id @ X66 @ id))) | ~! [X67 : subst > term > term] : (($true = (X65 @ id @ (lam @ (X67 @ sh @ one)) @ id)) | ~! [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X67 @ X69 @ X70) @ X68) = (X67 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) | ~! [X71 : term] : (~($true = (X65 @ id @ X71 @ id)) | ($true = (X65 @ id @ (X67 @ id @ X71) @ id))))) => ! [X72 : subst > term > subst > $o,X73 : term] : (~! [X74 : term] : (($true = (X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id)) | ~((X72 @ id @ X74 @ id) = $true)) | ~! [X75 : subst,X76 : term,X77 : subst,X78 : subst] : (((X72 @ (comp @ X78 @ X77) @ (sub @ X76 @ X77) @ X75) = $true) | ~($true = (X72 @ X78 @ X76 @ (comp @ X77 @ X75)))) | ~! [X79 : term,X80 : subst,X81 : subst,X82 : subst] : (~((X72 @ (comp @ X80 @ X82) @ (sub @ X79 @ X82) @ X81) = $true) | ($true = (X72 @ X80 @ X79 @ (comp @ X82 @ X81)))) | ~! [X83 : subst > term > term] : (~! [X84 : subst,X85 : term,X86 : subst] : ((X83 @ (comp @ X86 @ X84) @ (sub @ X85 @ X84)) = (sub @ (X83 @ X86 @ X85) @ X84)) | ($true = (X72 @ id @ (lam @ (X83 @ sh @ one)) @ id)) | ~! [X87 : term] : (((X72 @ id @ (X83 @ id @ X87) @ id) = $true) | ~($true = (X72 @ id @ X87 @ id)))) | ($true = (X72 @ id @ (lam @ X73) @ id)))))) & ((induction2 = $true) <=> ! [X88 : term > $o,X89 : term] : (~! [X90 : term] : (((X88 @ X90) = $true) | ~($true = (var @ X90))) | ~! [X91 : term] : (($true = (X88 @ (lam @ X91))) | ~! [X92 : term] : (~($true = (X88 @ X92)) | ($true = (X88 @ (sub @ X91 @ (push @ X92 @ id)))))) | ($true = (X88 @ X89)) | ~! [X93 : term,X94 : term] : (~($true = (X88 @ X94)) | ($true = (X88 @ (ap @ X93 @ X94))) | ~($true = (X88 @ X93))))) & ~(ulamvar1 = $true) & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & (pushprop_lem1v2_gthm = $true) & (! [X95 : term] : ~! [X96 : subst > term > term] : ~! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem3_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X99 : term,X100 : subst] : ((sub @ one @ (push @ X99 @ X100)) = X99) => (! [X101 : term,X102 : subst] : ((comp @ sh @ (push @ X101 @ X102)) = X102) => (! [X103 : term,X104 : term] : ((X103 = X104) | ((lam @ X103) != (lam @ X104))) => ! [X105 : term,X106 : subst > term > term,X107 : subst > term > term,X108 : subst] : (~! [X109 : subst,X110 : subst,X111 : term] : ((sub @ (X107 @ X109 @ X111) @ X110) = (X107 @ (comp @ X109 @ X110) @ (sub @ X111 @ X110))) | ~! [X112 : subst,X113 : term,X114 : subst] : ((X106 @ (comp @ X114 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X106 @ X114 @ X113) @ X112)) | ((X106 @ X108 @ X105) = (X107 @ X108 @ X105)) | ((lam @ (X106 @ sh @ one)) != (lam @ (X107 @ sh @ one)))))))) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X115 : subst > term > subst > $o,X116 : term] : (($true = (X115 @ id @ (lam @ (sub @ X116 @ (push @ one @ sh))) @ id)) | ~! [X117 : subst > term > term] : (($true = (X115 @ id @ (lam @ (X117 @ sh @ one)) @ id)) | ~! [X118 : subst,X119 : subst,X120 : term] : ((sub @ (X117 @ X118 @ X120) @ X119) = (X117 @ (comp @ X118 @ X119) @ (sub @ X120 @ X119))) | ~! [X121 : term] : (($true = (X115 @ id @ (X117 @ id @ X121) @ id)) | ~((X115 @ id @ X121 @ id) = $true))) | ~! [X122 : term] : (($true = (X115 @ id @ (sub @ X116 @ (push @ X122 @ id)) @ id)) | ~($true = (X115 @ id @ X122 @ id)))) <=> (hoasinduction_lem3aa = $true)) & ((hoasinduction_lem2v2 = $true) <=> ! [X123 : term > $o,X124 : term,X125 : subst > term > subst > $o,X126 : term] : (~! [X127 : term,X128 : term] : (~($true = (X125 @ id @ X128 @ id)) | ~($true = (X125 @ id @ X127 @ id)) | ((X125 @ id @ (ap @ (sub @ X127 @ id) @ X128) @ id) = $true)) | ~($true = (X123 @ X124)) | ~($true = (X123 @ X126)) | ~! [X129 : subst,X130 : subst,X131 : term,X132 : subst] : (~($true = (X125 @ (comp @ X129 @ X132) @ (sub @ X131 @ X132) @ X130)) | ($true = (X125 @ X129 @ X131 @ (comp @ X132 @ X130)))) | ~! [X133 : subst,X134 : subst,X135 : term,X136 : subst] : (($true = (X125 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134) @ X133)) | ~((X125 @ X136 @ X135 @ (comp @ X134 @ X133)) = $true)) | ((X123 @ (ap @ X126 @ X124)) = $true) | ~! [X137 : term] : ((X125 @ id @ X137 @ id) = (X123 @ X137)))) & (axapp = $true) & (! [X138 : subst > term > subst > $o] : ~! [X139 : term > $o] : ~! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140)) <=> (hoasinduction_lem0 = $true)) & (pushprop_lem2v2_lthm = $true) & (hoasapinj1_gthm = $true) & (hoasapinj2_gthm = $true) & ! [X141 : d_term] : (d_one = X141) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X142 : subst > term > subst > $o,X143 : term] : (((X142 @ id @ (lam @ (sub @ X143 @ (push @ one @ sh))) @ id) = $true) | ~! [X144 : term,X145 : subst > term > term] : (~! [X146 : subst,X147 : term,X148 : subst] : ((X145 @ (comp @ X146 @ X148) @ (sub @ X147 @ X148)) = (sub @ X144 @ (push @ (sub @ X147 @ X148) @ (comp @ X146 @ X148)))) | ~! [X149 : term] : (($true = (X142 @ id @ (X145 @ id @ X149) @ id)) | ~($true = (X142 @ id @ X149 @ id))) | ((X142 @ id @ (lam @ (X145 @ sh @ one)) @ id) = $true) | ~! [X150 : term,X151 : subst,X152 : subst] : ((sub @ (X145 @ X152 @ X150) @ X151) = (sub @ (sub @ X144 @ (push @ X150 @ X152)) @ X151))) | ~! [X153 : term] : (~($true = (X142 @ id @ X153 @ id)) | ($true = (X142 @ id @ (sub @ X143 @ (push @ X153 @ id)) @ id)))) <=> (hoasinduction_lem3aaa = $true)) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & ! [X156 : d_subst] : (d_id = X156) & (! [X157 : subst > term > subst > $o,X158 : term] : (~! [X159 : term] : (~($true = (X157 @ id @ X159 @ id)) | ((X157 @ id @ (sub @ X158 @ (push @ X159 @ id)) @ id) = $true)) | ($true = (X157 @ id @ (lam @ X158) @ id)) | ~! [X160 : subst > term > term] : (~! [X161 : term] : (~((X157 @ id @ X161 @ id) = $true) | ($true = (X157 @ id @ (X160 @ id @ X161) @ id))) | ((X157 @ id @ (lam @ (X160 @ sh @ one)) @ id) = $true) | ~! [X162 : term,X163 : subst,X164 : subst] : ((sub @ (X160 @ X163 @ X162) @ X164) = (X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164))))) <=> (hoasinduction_lem3a = $true)) & ~((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1 = $true) <=> ! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : (($true = (X168 @ (sub @ X167 @ (push @ X165 @ X166)))) | ~($true = (X169 @ X167)))) & (hoasapnotvar = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_gthm = $true) & (axabs = $true) & (induction2_gthm = $true) & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & (hoaslaminj_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X172 : term,X173 : term,X174 : term,X175 : term] : (((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)) | (X174 = X175))) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179)) <=> (apinj1 = $true)) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & (hoasinduction_lem1 = $true) & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & (pushprop_lem2v2 = $true) & (! [X180 : term > $o,X181 : term,X182 : subst] : ~! [X183 : term > $o] : ~! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182)))) <=> (pushprop_lem0_lthm = $true)) & (pushprop = $true) & (hoasinduction_lem3v2_gthm = $true) & ((shinj = $true) <=> ! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186))) & (! [X187 : subst > term > subst > $o] : ~! [X188 : term > $o] : ~! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) <=> (hoasinduction_lem0_lthm = $true)) & (hoaslamnotvar_lthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (induction2lem_gthm = $true) & (termmset_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X193 : term,X194 : term,X195 : term,X196 : term] : ((X193 = X195) | ((ap @ X196 @ X195) != (ap @ X194 @ X193))) => ! [X197 : term,X198 : term,X199 : term,X200 : term] : ((X197 = X198) | ((ap @ (sub @ X199 @ id) @ X197) != (ap @ (sub @ X200 @ id) @ X198))))) & (axmap = $true) & (axclos = $true) & (one = (d2term @ d_one)) & (! [X201 : subst,X202 : term] : ((comp @ sh @ (push @ X202 @ X201)) = X201) <=> (axshiftcons = $true)) & (hoasapnotvar_gthm = $true) & (axvarshift = $true) & (! [X203 : term,X204 : subst > term > subst > $o] : (($true = (X204 @ id @ (lam @ (sub @ X203 @ (push @ one @ sh))) @ id)) | ~! [X205 : subst > term > term] : (((X204 @ id @ (lam @ (X205 @ sh @ one)) @ id) = $true) | ~! [X206 : term] : (($true = (X204 @ id @ (X205 @ id @ X206) @ id)) | ~($true = (X204 @ id @ X206 @ id))) | ~! [X207 : term,X208 : subst,X209 : subst] : ((sub @ (X205 @ X209 @ X207) @ X208) = (X205 @ (comp @ X209 @ X208) @ (sub @ X207 @ X208)))) | ~! [X210 : term] : (~($true = (X204 @ id @ X210 @ id)) | ($true = (X204 @ id @ (sub @ X203 @ (push @ X210 @ id)) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((! [X211 : term] : ((sub @ X211 @ id) = X211) => ! [X212 : term] : ((sub @ X212 @ id) = X212)) <=> (termmset_lthm = $true)) & (! [X213 : subst > term > subst > $o,X214 : term,X215 : term] : (~($true = (X213 @ id @ X215 @ id)) | ~($true = (X213 @ id @ X214 @ id)) | ~! [X216 : term,X217 : subst,X218 : subst,X219 : subst] : (~($true = (X213 @ X218 @ X216 @ (comp @ X219 @ X217))) | ((X213 @ (comp @ X218 @ X219) @ (sub @ X216 @ X219) @ X217) = $true)) | ~! [X220 : subst,X221 : subst,X222 : subst,X223 : term] : (((X213 @ X222 @ X223 @ (comp @ X221 @ X220)) = $true) | ~((X213 @ (comp @ X222 @ X221) @ (sub @ X223 @ X221) @ X220) = $true)) | ~! [X224 : term,X225 : term] : (($true = (X213 @ id @ (ap @ (sub @ X224 @ id) @ X225) @ id)) | ~($true = (X213 @ id @ X224 @ id)) | ~($true = (X213 @ id @ X225 @ id))) | ($true = (X213 @ id @ (ap @ X215 @ X214) @ id))) <=> (hoasinduction_lem2 = $true)) & ((axidl = $true) <=> ! [X226 : subst] : ((comp @ id @ X226) = X226)) & ((induction2_lthm = $true) <=> (! [X227 : term] : ((sub @ X227 @ id) = X227) => (! [X228 : term,X229 : term > $o,X230 : subst] : (~! [X231 : term] : (~($true = (var @ X231)) | ($true = (X229 @ (sub @ X231 @ X230)))) | ($true = (X229 @ (sub @ X228 @ X230))) | ~! [X232 : term] : (~! [X233 : term] : (((X229 @ (sub @ X232 @ (push @ X233 @ id))) = $true) | ~($true = (X229 @ X233))) | ($true = (X229 @ (lam @ X232)))) | ~! [X234 : term,X235 : term] : (~($true = (X229 @ X234)) | ~((X229 @ X235) = $true) | ($true = (X229 @ (ap @ X235 @ X234))))) => ! [X236 : term,X237 : term > $o] : (($true = (X237 @ X236)) | ~! [X238 : term] : (((X237 @ (lam @ X238)) = $true) | ~! [X239 : term] : (~($true = (X237 @ X239)) | ($true = (X237 @ (sub @ X238 @ (push @ X239 @ id)))))) | ~! [X240 : term,X241 : term] : (~($true = (X237 @ X241)) | ($true = (X237 @ (ap @ X241 @ X240))) | ~($true = (X237 @ X240))) | ~! [X242 : term] : (((X237 @ X242) = $true) | ~($true = (var @ X242))))))) & (hoasinduction_lem3b_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (! [X248 : term,X249 : subst,X250 : term > $o,X251 : term > $o] : (($true = (X250 @ one)) | ~($true = (X251 @ X248)) | ~! [X252 : term] : ((X251 @ (sub @ X252 @ (push @ X248 @ X249))) = (X250 @ X252))) <=> (pushprop_lem1v2 = $true)) & (ulamvarsh = $true) & (hoasinduction_lem1_gthm = $true) & (! [X253 : term,X254 : subst] : ((sub @ one @ (push @ X253 @ X254)) = X253) <=> (axvarcons = $true)) & (pushprop_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X255 : subst > term > subst > $o,X256 : term] : (($true = (X255 @ id @ (lam @ X256) @ id)) | ~! [X257 : subst,X258 : subst,X259 : term,X260 : subst] : (~($true = (X255 @ (comp @ X260 @ X257) @ (sub @ X259 @ X257) @ X258)) | ($true = (X255 @ X260 @ X259 @ (comp @ X257 @ X258)))) | ~! [X261 : subst > term > term] : (($true = (X255 @ id @ (lam @ (X261 @ sh @ one)) @ id)) | ~! [X262 : subst,X263 : term,X264 : subst] : ((sub @ (X261 @ X264 @ X263) @ X262) = (X261 @ (comp @ X264 @ X262) @ (sub @ X263 @ X262))) | ~! [X265 : term] : (($true = (X255 @ id @ (X261 @ id @ X265) @ id)) | ~($true = (X255 @ id @ X265 @ id)))) | ~! [X266 : term,X267 : subst,X268 : subst,X269 : subst] : (~($true = (X255 @ X269 @ X266 @ (comp @ X268 @ X267))) | ($true = (X255 @ (comp @ X269 @ X268) @ (sub @ X266 @ X268) @ X267))) | ~! [X270 : term] : (($true = (X255 @ id @ (sub @ X256 @ (push @ X270 @ id)) @ id)) | ~($true = (X255 @ id @ X270 @ id))))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X271 : term] : ~! [X272 : subst > term > term] : ((sub @ X271 @ (push @ one @ sh)) != (X272 @ sh @ one)) <=> (hoasinduction_lem3b_lthm = $true)) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (! [X273 : term] : ((sub @ X273 @ id) = X273) <=> (axvarid = $true)) & (! [X274 : term,X275 : subst,X276 : subst > term > term,X277 : subst > term > term] : (~! [X278 : subst,X279 : term,X280 : subst] : ((X277 @ (comp @ X278 @ X280) @ (sub @ X279 @ X280)) = (sub @ (X277 @ X278 @ X279) @ X280)) | ~! [X281 : term,X282 : subst,X283 : subst] : ((sub @ (X276 @ X283 @ X281) @ X282) = (X276 @ (comp @ X283 @ X282) @ (sub @ X281 @ X282))) | ((lam @ (X277 @ sh @ one)) != (lam @ (X276 @ sh @ one))) | ((X276 @ X275 @ X274) = (X277 @ X275 @ X274))) <=> (hoaslaminj = $true)) & ((termmset = $true) <=> ! [X284 : term] : ((sub @ X284 @ id) = X284)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X285 : term] : ((sub @ X285 @ id) = X285) => (! [X286 : term,X287 : subst > term > subst > $o] : (($true = (X287 @ id @ (lam @ (sub @ X286 @ (push @ one @ sh))) @ id)) | ~! [X288 : subst > term > term] : (($true = (X287 @ id @ (lam @ (X288 @ sh @ one)) @ id)) | ~! [X289 : term] : (($true = (X287 @ id @ (X288 @ id @ X289) @ id)) | ~($true = (X287 @ id @ X289 @ id))) | ~! [X290 : term,X291 : subst,X292 : subst] : ((sub @ (X288 @ X292 @ X290) @ X291) = (X288 @ (comp @ X292 @ X291) @ (sub @ X290 @ X291)))) | ~! [X293 : term] : (((X287 @ id @ (sub @ X286 @ (push @ X293 @ id)) @ id) = $true) | ~($true = (X287 @ id @ X293 @ id)))) => ! [X294 : subst > term > subst > $o,X295 : term] : (~! [X296 : term] : (($true = (X294 @ id @ (sub @ X295 @ (push @ X296 @ id)) @ id)) | ~((X294 @ id @ X296 @ id) = $true)) | ~! [X297 : subst > term > term] : (($true = (X294 @ id @ (lam @ (X297 @ sh @ one)) @ id)) | ~! [X298 : term] : (($true = (X294 @ id @ (X297 @ id @ X298) @ id)) | ~((X294 @ id @ X298 @ id) = $true)) | ~! [X299 : subst,X300 : subst,X301 : term] : ((X297 @ (comp @ X300 @ X299) @ (sub @ X301 @ X299)) = (sub @ (X297 @ X300 @ X301) @ X299))) | ($true = (X294 @ id @ (lam @ X295) @ id)))))) & (! [X302 : subst > term > subst > $o,X303 : term] : (~! [X304 : term,X305 : term] : (~($true = (X302 @ id @ X305 @ id)) | ((X302 @ id @ (ap @ (sub @ X305 @ id) @ X304) @ id) = $true) | ~($true = (X302 @ id @ X304 @ id))) | ~! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : (($true = (X302 @ (comp @ X307 @ X309) @ (sub @ X306 @ X309) @ X308)) | ~($true = (X302 @ X307 @ X306 @ (comp @ X309 @ X308)))) | ~! [X310 : subst > term > term] : (~! [X311 : term] : (~($true = (X302 @ id @ X311 @ id)) | ($true = (X302 @ id @ (X310 @ id @ X311) @ id))) | ($true = (X302 @ id @ (lam @ (X310 @ sh @ one)) @ id)) | ~! [X312 : subst,X313 : subst,X314 : term] : ((X310 @ (comp @ X312 @ X313) @ (sub @ X314 @ X313)) = (sub @ (X310 @ X312 @ X314) @ X313))) | ~! [X315 : subst,X316 : term,X317 : subst,X318 : subst] : (($true = (X302 @ X318 @ X316 @ (comp @ X317 @ X315))) | ~((X302 @ (comp @ X318 @ X317) @ (sub @ X316 @ X317) @ X315) = $true)) | ~! [X319 : term] : (~($true = (var @ (sub @ X319 @ id))) | ($true = (X302 @ id @ X319 @ id))) | ((X302 @ id @ X303 @ id) = $true)) <=> (hoasinduction = $true)) & ! [X320 : d_subst,X321 : d_subst] : (((d2subst @ X320) = (d2subst @ X321)) => (X320 = X321)) & (! [X322 : term] : ~! [X323 : subst > term > term] : ~! [X324 : subst,X325 : term] : ((X323 @ X324 @ X325) = (sub @ X322 @ (push @ X325 @ X324))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X326 : subst > term > subst > $o,X327 : term,X328 : term > $o] : (~! [X329 : subst > term > term] : (~! [X330 : subst,X331 : term,X332 : subst] : ((X329 @ (comp @ X330 @ X332) @ (sub @ X331 @ X332)) = (sub @ (X329 @ X330 @ X331) @ X332)) | ~! [X333 : term] : (($true = (X326 @ id @ (X329 @ id @ X333) @ id)) | ~((X326 @ id @ X333 @ id) = $true)) | ($true = (X326 @ id @ (lam @ (X329 @ sh @ one)) @ id))) | ($true = (X328 @ (lam @ X327))) | ~! [X334 : term] : ((X328 @ X334) = (X326 @ id @ X334 @ id)) | ~! [X335 : term] : (~($true = (X328 @ X335)) | ($true = (X328 @ (sub @ X327 @ (push @ X335 @ id))))))) & ((pushprop_lem1_lthm = $true) <=> (! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) => (! [X338 : subst,X339 : term] : ((comp @ sh @ (push @ X339 @ X338)) = X338) => ! [X340 : term,X341 : term > $o,X342 : term,X343 : subst,X344 : term > $o] : (($true = (X341 @ (sub @ X340 @ (push @ X342 @ X343)))) | ~((X344 @ X340) = $true))))) & ! [X345 : d_term,X346 : d_term] : (((d2term @ X346) = (d2term @ X345)) => (X345 = X346)) & (pushprop_lthm_orig = $true) & (apnotvar = $true) & (hoaslamnotap_gthm = $true) & ((! [X347 : subst > term > subst > $o] : ~! [X348 : term > $o] : ~! [X349 : term] : ((X347 @ id @ X349 @ id) = (X348 @ X349)) => (! [X350 : term > $o,X351 : term] : (~! [X352 : term,X353 : term] : (($true = (X350 @ (ap @ X353 @ X352))) | ~((X350 @ X352) = $true) | ~($true = (X350 @ X353))) | ($true = (X350 @ X351)) | ~! [X354 : term] : (~! [X355 : term] : (~($true = (X350 @ X355)) | ($true = (X350 @ (sub @ X354 @ (push @ X355 @ id))))) | ((X350 @ (lam @ X354)) = $true)) | ~! [X356 : term] : (~($true = (var @ X356)) | ($true = (X350 @ X356)))) => (! [X357 : term] : ((sub @ X357 @ id) = X357) => (! [X358 : subst > term > subst > $o,X359 : term,X360 : term > $o] : (~! [X361 : term] : ((X358 @ id @ X361 @ id) = (X360 @ X361)) | ~! [X362 : term] : (($true = (X360 @ (sub @ X359 @ (push @ X362 @ id)))) | ~($true = (X360 @ X362))) | ((X360 @ (lam @ X359)) = $true) | ~! [X363 : subst > term > term] : (((X358 @ id @ (lam @ (X363 @ sh @ one)) @ id) = $true) | ~! [X364 : term] : (((X358 @ id @ (X363 @ id @ X364) @ id) = $true) | ~($true = (X358 @ id @ X364 @ id))) | ~! [X365 : subst,X366 : term,X367 : subst] : ((sub @ (X363 @ X365 @ X366) @ X367) = (X363 @ (comp @ X365 @ X367) @ (sub @ X366 @ X367))))) => ! [X368 : subst > term > subst > $o,X369 : term] : (~! [X370 : subst,X371 : subst,X372 : subst,X373 : term] : (~((X368 @ (comp @ X370 @ X371) @ (sub @ X373 @ X371) @ X372) = $true) | ($true = (X368 @ X370 @ X373 @ (comp @ X371 @ X372)))) | ~! [X374 : subst > term > term] : (~! [X375 : subst,X376 : term,X377 : subst] : ((X374 @ (comp @ X377 @ X375) @ (sub @ X376 @ X375)) = (sub @ (X374 @ X377 @ X376) @ X375)) | ($true = (X368 @ id @ (lam @ (X374 @ sh @ one)) @ id)) | ~! [X378 : term] : (~($true = (X368 @ id @ X378 @ id)) | ((X368 @ id @ (X374 @ id @ X378) @ id) = $true))) | ~! [X379 : term,X380 : term] : (($true = (X368 @ id @ (ap @ (sub @ X380 @ id) @ X379) @ id)) | ~($true = (X368 @ id @ X379 @ id)) | ~((X368 @ id @ X380 @ id) = $true)) | ($true = (X368 @ id @ X369 @ id)) | ~! [X381 : term] : (~($true = (var @ (sub @ X381 @ id))) | ($true = (X368 @ id @ X381 @ id))) | ~! [X382 : subst,X383 : subst,X384 : subst,X385 : term] : (~($true = (X368 @ X382 @ X385 @ (comp @ X383 @ X384))) | ($true = (X368 @ (comp @ X382 @ X383) @ (sub @ X385 @ X383) @ X384)))))))) <=> (hoasinduction_lthm_3 = $true)) & (! [X386 : term,X387 : term,X388 : subst > term > term] : (~! [X389 : subst,X390 : subst,X391 : term] : ((sub @ (X388 @ X390 @ X391) @ X389) = (X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389))) | ((ap @ (sub @ X387 @ id) @ X386) != (lam @ (X388 @ sh @ one)))) <=> (hoaslamnotap = $true)) & ((axscons = $true) <=> ! [X392 : subst] : ((push @ (sub @ one @ X392) @ (comp @ sh @ X392)) = X392)) & ~((var @ (d2term @ d_one)) = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X395 : subst,X396 : term > $o,X397 : term] : (~! [X398 : term] : (~! [X399 : term] : (~($true = (X396 @ X399)) | ($true = (X396 @ (sub @ X398 @ (push @ X399 @ id))))) | ((X396 @ (lam @ X398)) = $true)) | ($true = (X396 @ (sub @ X397 @ X395))) | ~! [X400 : term] : (~($true = (var @ X400)) | ($true = (X396 @ (sub @ X400 @ X395)))) | ~! [X401 : term,X402 : term] : (~($true = (X396 @ X401)) | ~($true = (X396 @ X402)) | ($true = (X396 @ (ap @ X402 @ X401))))) <=> (induction2lem = $true)) & ((axidr = $true) <=> ! [X403 : subst] : ((comp @ X403 @ id) = X403)) & (! [X404 : term,X405 : term,X406 : term,X407 : term] : ((X404 = X407) | ((ap @ (sub @ X404 @ id) @ X406) != (ap @ (sub @ X407 @ id) @ X405))) <=> (hoasapinj1 = $true)) & (axassoc = $true) & ((! [X408 : term] : ((sub @ X408 @ id) = X408) => (! [X409 : term,X410 : term,X411 : term,X412 : term] : ((X409 = X412) | ((ap @ X412 @ X411) != (ap @ X409 @ X410))) => ! [X413 : term,X414 : term,X415 : term,X416 : term] : ((X415 = X416) | ((ap @ (sub @ X416 @ id) @ X414) != (ap @ (sub @ X415 @ id) @ X413))))) <=> (hoasapinj1_lthm = $true)) & (ulamvarind = $true) & ~(lamnotap = $true) & (pushprop_lthm = $true) & (! [X417 : term,X418 : subst > term > subst > $o,X419 : term > $o] : (~! [X420 : subst > term > term] : (~! [X421 : term,X422 : subst,X423 : subst] : ((sub @ (X420 @ X422 @ X421) @ X423) = (X420 @ (comp @ X422 @ X423) @ (sub @ X421 @ X423))) | ($true = (X418 @ id @ (lam @ (X420 @ sh @ one)) @ id)) | ~! [X424 : term] : (($true = (X418 @ id @ (X420 @ id @ X424) @ id)) | ~($true = (X418 @ id @ X424 @ id)))) | ~! [X425 : term] : (~($true = (X419 @ X425)) | ($true = (X419 @ (sub @ X417 @ (push @ X425 @ id))))) | ~! [X426 : subst,X427 : subst,X428 : term,X429 : subst] : (($true = (X418 @ X429 @ X428 @ (comp @ X427 @ X426))) | ~((X418 @ (comp @ X429 @ X427) @ (sub @ X428 @ X427) @ X426) = $true)) | ~! [X430 : term] : ((X418 @ id @ X430 @ id) = (X419 @ X430)) | ($true = (X419 @ (lam @ X417))) | ~! [X431 : subst,X432 : term,X433 : subst,X434 : subst] : (~((X418 @ X434 @ X432 @ (comp @ X431 @ X433)) = $true) | ($true = (X418 @ (comp @ X434 @ X431) @ (sub @ X432 @ X431) @ X433)))) <=> (hoasinduction_lem3v2 = $true)) & ((d2subst @ d_id) = id) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1_lthm = $true) & (! [X435 : term,X436 : term,X437 : term,X438 : term] : ((X436 = X438) | ((ap @ X435 @ X436) != (ap @ X437 @ X438))) <=> (apinj2 = $true)) & (! [X439 : term,X440 : term > $o] : (((X440 @ X439) = $true) | ~! [X441 : term] : (($true = (X440 @ X441)) | ~($true = (var @ X441))) | ~! [X442 : term] : (~((X440 @ X442) = $true) | ($true = (X440 @ (lam @ X442)))) | ~! [X443 : term,X444 : term] : (~($true = (X440 @ X443)) | ($true = (X440 @ (ap @ X443 @ X444))) | ~($true = (X440 @ X444)))) <=> (induction = $true)) & ((! [X445 : subst > term > subst > $o] : ~! [X446 : term > $o] : ~! [X447 : term] : ((X446 @ X447) = (X445 @ id @ X447 @ id)) => (! [X448 : term,X449 : term > $o] : (~! [X450 : term] : (~((var @ X450) = $true) | ($true = (X449 @ X450))) | ($true = (X449 @ X448)) | ~! [X451 : term] : (($true = (X449 @ (lam @ X451))) | ~! [X452 : term] : (~($true = (X449 @ X452)) | ($true = (X449 @ (sub @ X451 @ (push @ X452 @ id)))))) | ~! [X453 : term,X454 : term] : (~($true = (X449 @ X453)) | ~($true = (X449 @ X454)) | ($true = (X449 @ (ap @ X454 @ X453))))) => (! [X455 : term] : ((sub @ X455 @ id) = X455) => (! [X456 : term > $o,X457 : subst > term > subst > $o,X458 : term] : (~! [X459 : term] : ((X456 @ X459) = (X457 @ id @ X459 @ id)) | ~! [X460 : term] : (~((X456 @ X460) = $true) | ($true = (X456 @ (sub @ X458 @ (push @ X460 @ id))))) | ~! [X461 : subst > term > term] : (~! [X462 : term,X463 : subst,X464 : subst] : ((X461 @ (comp @ X463 @ X464) @ (sub @ X462 @ X464)) = (sub @ (X461 @ X463 @ X462) @ X464)) | ($true = (X457 @ id @ (lam @ (X461 @ sh @ one)) @ id)) | ~! [X465 : term] : (~($true = (X457 @ id @ X465 @ id)) | ((X457 @ id @ (X461 @ id @ X465) @ id) = $true))) | ($true = (X456 @ (lam @ X458)))) => ! [X466 : subst > term > subst > $o,X467 : term] : (((X466 @ id @ X467 @ id) = $true) | ~! [X468 : subst > term > term] : (((X466 @ id @ (lam @ (X468 @ sh @ one)) @ id) = $true) | ~! [X469 : term] : (((X466 @ id @ (X468 @ id @ X469) @ id) = $true) | ~($true = (X466 @ id @ X469 @ id))) | ~! [X470 : term,X471 : subst,X472 : subst] : ((sub @ (X468 @ X472 @ X470) @ X471) = (X468 @ (comp @ X472 @ X471) @ (sub @ X470 @ X471)))) | ~! [X473 : term,X474 : term] : (~($true = (X466 @ id @ X474 @ id)) | ($true = (X466 @ id @ (ap @ (sub @ X473 @ id) @ X474) @ id)) | ~($true = (X466 @ id @ X473 @ id)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1_gthm = $true) & ((d2subst @ d_id) = sh) & ((hoasinduction_lthm = $true) <=> (! [X475 : term,X476 : term > $o] : (~! [X477 : term] : (~! [X478 : term] : (~($true = (X476 @ X478)) | ($true = (X476 @ (sub @ X477 @ (push @ X478 @ id))))) | ((X476 @ (lam @ X477)) = $true)) | ($true = (X476 @ X475)) | ~! [X479 : term] : (~((var @ X479) = $true) | ((X476 @ X479) = $true)) | ~! [X480 : term,X481 : term] : (((X476 @ (ap @ X480 @ X481)) = $true) | ~($true = (X476 @ X481)) | ~($true = (X476 @ X480)))) => (! [X482 : subst > term > subst > $o,X483 : term,X484 : term] : (~($true = (X482 @ id @ X483 @ id)) | ($true = (X482 @ id @ (ap @ X483 @ X484) @ id)) | ~! [X485 : subst,X486 : term,X487 : subst,X488 : subst] : (($true = (X482 @ (comp @ X488 @ X487) @ (sub @ X486 @ X487) @ X485)) | ~($true = (X482 @ X488 @ X486 @ (comp @ X487 @ X485)))) | ~((X482 @ id @ X484 @ id) = $true) | ~! [X489 : term,X490 : subst,X491 : subst,X492 : subst] : (((X482 @ X490 @ X489 @ (comp @ X491 @ X492)) = $true) | ~($true = (X482 @ (comp @ X490 @ X491) @ (sub @ X489 @ X491) @ X492))) | ~! [X493 : term,X494 : term] : (((X482 @ id @ (ap @ (sub @ X493 @ id) @ X494) @ id) = $true) | ~((X482 @ id @ X494 @ id) = $true) | ~($true = (X482 @ id @ X493 @ id)))) => (! [X495 : term,X496 : subst > term > subst > $o] : (~! [X497 : subst,X498 : subst,X499 : subst,X500 : term] : (~($true = (X496 @ (comp @ X498 @ X499) @ (sub @ X500 @ X499) @ X497)) | ($true = (X496 @ X498 @ X500 @ (comp @ X499 @ X497)))) | ($true = (X496 @ id @ (lam @ X495) @ id)) | ~! [X501 : subst,X502 : term,X503 : subst,X504 : subst] : (($true = (X496 @ (comp @ X503 @ X504) @ (sub @ X502 @ X504) @ X501)) | ~($true = (X496 @ X503 @ X502 @ (comp @ X504 @ X501)))) | ~! [X505 : subst > term > term] : (~! [X506 : term] : (~($true = (X496 @ id @ X506 @ id)) | ($true = (X496 @ id @ (X505 @ id @ X506) @ id))) | ($true = (X496 @ id @ (lam @ (X505 @ sh @ one)) @ id)) | ~! [X507 : term,X508 : subst,X509 : subst] : ((X505 @ (comp @ X508 @ X509) @ (sub @ X507 @ X509)) = (sub @ (X505 @ X508 @ X507) @ X509))) | ~! [X510 : term] : (~($true = (X496 @ id @ X510 @ id)) | ($true = (X496 @ id @ (sub @ X495 @ (push @ X510 @ id)) @ id)))) => ! [X511 : term,X512 : subst > term > subst > $o] : (~! [X513 : subst,X514 : subst,X515 : term,X516 : subst] : (~($true = (X512 @ (comp @ X514 @ X516) @ (sub @ X515 @ X516) @ X513)) | ($true = (X512 @ X514 @ X515 @ (comp @ X516 @ X513)))) | ~! [X517 : subst > term > term] : (($true = (X512 @ id @ (lam @ (X517 @ sh @ one)) @ id)) | ~! [X518 : term] : (($true = (X512 @ id @ (X517 @ id @ X518) @ id)) | ~($true = (X512 @ id @ X518 @ id))) | ~! [X519 : term,X520 : subst,X521 : subst] : ((X517 @ (comp @ X521 @ X520) @ (sub @ X519 @ X520)) = (sub @ (X517 @ X521 @ X519) @ X520))) | ~! [X522 : term,X523 : subst,X524 : subst,X525 : subst] : (~((X512 @ X525 @ X522 @ (comp @ X524 @ X523)) = $true) | ((X512 @ (comp @ X525 @ X524) @ (sub @ X522 @ X524) @ X523) = $true)) | ($true = (X512 @ id @ X511 @ id)) | ~! [X526 : term,X527 : term] : (~($true = (X512 @ id @ X526 @ id)) | ~($true = (X512 @ id @ X527 @ id)) | ($true = (X512 @ id @ (ap @ (sub @ X526 @ id) @ X527) @ id))) | ~! [X528 : term] : (($true = (X512 @ id @ X528 @ id)) | ~((var @ (sub @ X528 @ id)) = $true))))))) & ((hoasinduction_lem3b = $true) <=> ! [X529 : term] : ~! [X530 : subst > term > term] : ((sub @ X529 @ (push @ one @ sh)) != (X530 @ sh @ one))) & ((substmonoid_lthm = $true) <=> (! [X531 : subst] : ((comp @ id @ X531) = X531) => (! [X532 : subst] : ((comp @ X532 @ id) = X532) => (! [X533 : subst] : ((comp @ id @ X533) = X533) & ! [X534 : subst] : ((comp @ X534 @ id) = X534))))) & (hoaslamnotvar = $true) & ((! [X535 : term] : ~! [X536 : subst > term > term] : ~! [X537 : subst,X538 : term] : ((sub @ X535 @ (push @ X538 @ X537)) = (X536 @ X537 @ X538)) => (! [X539 : term] : ((sub @ X539 @ id) = X539) => ! [X540 : term,X541 : term > $o,X542 : subst > term > subst > $o] : (((X541 @ (lam @ X540)) = $true) | ~! [X543 : term] : ((X542 @ id @ X543 @ id) = (X541 @ X543)) | ~! [X544 : subst > term > term] : (~! [X545 : term] : (($true = (X542 @ id @ (X544 @ id @ X545) @ id)) | ~($true = (X542 @ id @ X545 @ id))) | ($true = (X542 @ id @ (lam @ (X544 @ sh @ one)) @ id)) | ~! [X546 : subst,X547 : term,X548 : subst] : ((sub @ (X544 @ X548 @ X547) @ X546) = (X544 @ (comp @ X548 @ X546) @ (sub @ X547 @ X546)))) | ~! [X549 : term] : (($true = (X541 @ (sub @ X540 @ (push @ X549 @ id)))) | ~((X541 @ X549) = $true))))) <=> (hoasinduction_lem3v2a_lthm = $true))),
% 9.38/2.21 inference(fool_elimination,[status(thm)],[f5])).
% 9.38/2.21 thf(f5,plain,(
% 9.38/2.21 (! [X0 : term,X1 : term] : ((X0 = X1) | ((lam @ X1) != (lam @ X0))) <=> laminj) & ((! [X2 : subst,X3 : term] : ((sub @ one @ (push @ X3 @ X2)) = X3) => (! [X4 : term,X5 : subst] : ((comp @ sh @ (push @ X4 @ X5)) = X5) => (! [X6 : subst] : ((comp @ X6 @ id) = X6) => (! [X7 : term > $o,X8 : term] : (~! [X9 : term] : ((X7 @ (lam @ X9)) | ~(X7 @ X9)) | ~! [X10 : term] : (~(var @ X10) | (X7 @ X10)) | ~! [X11 : term,X12 : term] : ((X7 @ (ap @ X11 @ X12)) | ~(X7 @ X11) | ~(X7 @ X12)) | (X7 @ X8)) => ! [X13 : term > $o,X14 : term,X15 : subst] : (~! [X16 : term,X17 : term] : ((X13 @ (ap @ X17 @ X16)) | ~(X13 @ X17) | ~(X13 @ X16)) | ~! [X18 : term] : ((X13 @ (lam @ X18)) | ~! [X19 : term] : (~(X13 @ X19) | (X13 @ (sub @ X18 @ (push @ X19 @ id))))) | (X13 @ (sub @ X14 @ X15)) | ~! [X20 : term] : (~(var @ X20) | (X13 @ (sub @ X20 @ X15)))))))) <=> induction2lem_lthm) & (pushprop_lem0 <=> ! [X21 : subst,X22 : term,X23 : term > $o] : ~! [X24 : term > $o] : ~! [X25 : term] : ((X24 @ X25) = (X23 @ (sub @ X25 @ (push @ X22 @ X21))))) & ((! [X26 : subst,X27 : term] : ((sub @ one @ (push @ X27 @ X26)) = X27) => ! [X28 : term > $o,X29 : term,X30 : term > $o,X31 : subst] : (~! [X32 : term] : ((X30 @ (sub @ X32 @ (push @ X29 @ X31))) = (X28 @ X32)) | ~(X30 @ X29) | (X28 @ one))) <=> pushprop_lem1v2_lthm) & hoasinduction_lem2_lthm & (substmonoid <=> (! [X33 : subst] : ((comp @ X33 @ id) = X33) & ! [X34 : subst] : ((comp @ id @ X34) = X34))) & pushprop_lem3v2_lthm & lamnotvar & (hoasinduction_lem3v2_lthm <=> (! [X35 : term] : ((sub @ X35 @ id) = X35) => ! [X36 : subst > term > subst > $o,X37 : term,X38 : term > $o] : (~! [X39 : term,X40 : subst,X41 : subst,X42 : subst] : ((X36 @ X42 @ X39 @ (comp @ X41 @ X40)) | ~(X36 @ (comp @ X42 @ X41) @ (sub @ X39 @ X41) @ X40)) | ~! [X43 : term] : ((X36 @ id @ X43 @ id) = (X38 @ X43)) | (X38 @ (lam @ X37)) | ~! [X44 : subst > term > term] : (~! [X45 : term,X46 : subst,X47 : subst] : ((sub @ (X44 @ X47 @ X45) @ X46) = (X44 @ (comp @ X47 @ X46) @ (sub @ X45 @ X46))) | (X36 @ id @ (lam @ (X44 @ sh @ one)) @ id) | ~! [X48 : term] : (~(X36 @ id @ X48 @ id) | (X36 @ id @ (X44 @ id @ X48) @ id))) | ~! [X49 : term] : (~(X38 @ X49) | (X38 @ (sub @ X37 @ (push @ X49 @ id)))) | ~! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (~(X36 @ X53 @ X52 @ (comp @ X50 @ X51)) | (X36 @ (comp @ X53 @ X50) @ (sub @ X52 @ X50) @ X51))))) & (! [X54 : subst > term > subst > $o,X55 : term] : (~! [X56 : term,X57 : term] : (~(X54 @ id @ X57 @ id) | ~(X54 @ id @ X56 @ id) | (X54 @ id @ (ap @ (sub @ X56 @ id) @ X57) @ id)) | (X54 @ id @ X55 @ id) | ~! [X58 : subst > term > term] : ((X54 @ id @ (lam @ (X58 @ sh @ one)) @ id) | ~! [X59 : term] : ((X54 @ id @ (X58 @ id @ X59) @ id) | ~(X54 @ id @ X59 @ id)) | ~! [X60 : term,X61 : subst,X62 : subst] : ((X58 @ (comp @ X62 @ X61) @ (sub @ X60 @ X61)) = (sub @ (X58 @ X62 @ X60) @ X61)))) <=> hoasinduction_no_psi_cond) & substmonoid_gthm & hoaslamnotvar_gthm & (hoasinduction_lem3_lthm <=> (! [X63 : term] : ((sub @ X63 @ id) = X63) => (! [X64 : term,X65 : subst > term > subst > $o] : ((X65 @ id @ (lam @ (sub @ X64 @ (push @ one @ sh))) @ id) | ~! [X66 : term] : ((X65 @ id @ (sub @ X64 @ (push @ X66 @ id)) @ id) | ~(X65 @ id @ X66 @ id)) | ~! [X67 : subst > term > term] : ((X65 @ id @ (lam @ (X67 @ sh @ one)) @ id) | ~! [X68 : subst,X69 : subst,X70 : term] : ((sub @ (X67 @ X69 @ X70) @ X68) = (X67 @ (comp @ X69 @ X68) @ (sub @ X70 @ X68))) | ~! [X71 : term] : (~(X65 @ id @ X71 @ id) | (X65 @ id @ (X67 @ id @ X71) @ id)))) => ! [X72 : subst > term > subst > $o,X73 : term] : (~! [X74 : term] : ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id) | ~(X72 @ id @ X74 @ id)) | ~! [X75 : subst,X76 : term,X77 : subst,X78 : subst] : ((X72 @ (comp @ X78 @ X77) @ (sub @ X76 @ X77) @ X75) | ~(X72 @ X78 @ X76 @ (comp @ X77 @ X75))) | ~! [X79 : term,X80 : subst,X81 : subst,X82 : subst] : (~(X72 @ (comp @ X80 @ X82) @ (sub @ X79 @ X82) @ X81) | (X72 @ X80 @ X79 @ (comp @ X82 @ X81))) | ~! [X83 : subst > term > term] : (~! [X84 : subst,X85 : term,X86 : subst] : ((X83 @ (comp @ X86 @ X84) @ (sub @ X85 @ X84)) = (sub @ (X83 @ X86 @ X85) @ X84)) | (X72 @ id @ (lam @ (X83 @ sh @ one)) @ id) | ~! [X87 : term] : ((X72 @ id @ (X83 @ id @ X87) @ id) | ~(X72 @ id @ X87 @ id))) | (X72 @ id @ (lam @ X73) @ id))))) & (induction2 <=> ! [X88 : term > $o,X89 : term] : (~! [X90 : term] : ((X88 @ X90) | ~(var @ X90)) | ~! [X91 : term] : ((X88 @ (lam @ X91)) | ~! [X92 : term] : (~(X88 @ X92) | (X88 @ (sub @ X91 @ (push @ X92 @ id))))) | (X88 @ X89) | ~! [X93 : term,X94 : term] : (~(X88 @ X94) | (X88 @ (ap @ X93 @ X94)) | ~(X88 @ X93)))) & ~ulamvar1 & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & pushprop_lem1v2_gthm & (! [X95 : term] : ~! [X96 : subst > term > term] : ~! [X97 : term,X98 : subst] : ((X96 @ X98 @ X97) = (sub @ X95 @ (push @ X97 @ X98))) <=> hoasinduction_lem3v2_f) & hoasinduction_lem3_gthm & ((! [X99 : term,X100 : subst] : ((sub @ one @ (push @ X99 @ X100)) = X99) => (! [X101 : term,X102 : subst] : ((comp @ sh @ (push @ X101 @ X102)) = X102) => (! [X103 : term,X104 : term] : ((X103 = X104) | ((lam @ X103) != (lam @ X104))) => ! [X105 : term,X106 : subst > term > term,X107 : subst > term > term,X108 : subst] : (((lam @ (X106 @ sh @ one)) != (lam @ (X107 @ sh @ one))) | ((X106 @ X108 @ X105) = (X107 @ X108 @ X105)) | ~! [X109 : subst,X110 : subst,X111 : term] : ((sub @ (X107 @ X109 @ X111) @ X110) = (X107 @ (comp @ X109 @ X110) @ (sub @ X111 @ X110))) | ~! [X112 : subst,X113 : term,X114 : subst] : ((X106 @ (comp @ X114 @ X112) @ (sub @ X113 @ X112)) = (sub @ (X106 @ X114 @ X113) @ X112)))))) <=> hoaslaminj_lthm) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X115 : subst > term > subst > $o,X116 : term] : ((X115 @ id @ (lam @ (sub @ X116 @ (push @ one @ sh))) @ id) | ~! [X117 : subst > term > term] : ((X115 @ id @ (lam @ (X117 @ sh @ one)) @ id) | ~! [X118 : subst,X119 : subst,X120 : term] : ((sub @ (X117 @ X118 @ X120) @ X119) = (X117 @ (comp @ X118 @ X119) @ (sub @ X120 @ X119))) | ~! [X121 : term] : ((X115 @ id @ (X117 @ id @ X121) @ id) | ~(X115 @ id @ X121 @ id))) | ~! [X122 : term] : ((X115 @ id @ (sub @ X116 @ (push @ X122 @ id)) @ id) | ~(X115 @ id @ X122 @ id))) <=> hoasinduction_lem3aa) & (hoasinduction_lem2v2 <=> ! [X123 : term > $o,X124 : term,X125 : subst > term > subst > $o,X126 : term] : (~! [X127 : term,X128 : term] : (~(X125 @ id @ X128 @ id) | ~(X125 @ id @ X127 @ id) | (X125 @ id @ (ap @ (sub @ X127 @ id) @ X128) @ id)) | ~(X123 @ X124) | ~(X123 @ X126) | ~! [X129 : subst,X130 : subst,X131 : term,X132 : subst] : (~(X125 @ (comp @ X129 @ X132) @ (sub @ X131 @ X132) @ X130) | (X125 @ X129 @ X131 @ (comp @ X132 @ X130))) | ~! [X133 : subst,X134 : subst,X135 : term,X136 : subst] : ((X125 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134) @ X133) | ~(X125 @ X136 @ X135 @ (comp @ X134 @ X133))) | (X123 @ (ap @ X126 @ X124)) | ~! [X137 : term] : ((X125 @ id @ X137 @ id) = (X123 @ X137)))) & axapp & (! [X138 : subst > term > subst > $o] : ~! [X139 : term > $o] : ~! [X140 : term] : ((X138 @ id @ X140 @ id) = (X139 @ X140)) <=> hoasinduction_lem0) & pushprop_lem2v2_lthm & hoasapinj1_gthm & hoasapinj2_gthm & ! [X141 : d_term] : (d_one = X141) & hoasinduction_lem3a_gthm & pushprop_lem3v2 & (! [X142 : subst > term > subst > $o,X143 : term] : ((X142 @ id @ (lam @ (sub @ X143 @ (push @ one @ sh))) @ id) | ~! [X144 : term,X145 : subst > term > term] : (~! [X146 : subst,X147 : term,X148 : subst] : ((X145 @ (comp @ X146 @ X148) @ (sub @ X147 @ X148)) = (sub @ X144 @ (push @ (sub @ X147 @ X148) @ (comp @ X146 @ X148)))) | ~! [X149 : term] : ((X142 @ id @ (X145 @ id @ X149) @ id) | ~(X142 @ id @ X149 @ id)) | (X142 @ id @ (lam @ (X145 @ sh @ one)) @ id) | ~! [X150 : term,X151 : subst,X152 : subst] : ((sub @ (X145 @ X152 @ X150) @ X151) = (sub @ (sub @ X144 @ (push @ X150 @ X152)) @ X151))) | ~! [X153 : term] : (~(X142 @ id @ X153 @ id) | (X142 @ id @ (sub @ X143 @ (push @ X153 @ id)) @ id))) <=> hoasinduction_lem3aaa) & ! [X154 : subst] : ? [X155 : d_subst] : ((d2subst @ X155) = X154) & ! [X156 : d_subst] : (d_id = X156) & (! [X157 : subst > term > subst > $o,X158 : term] : (~! [X159 : term] : (~(X157 @ id @ X159 @ id) | (X157 @ id @ (sub @ X158 @ (push @ X159 @ id)) @ id)) | (X157 @ id @ (lam @ X158) @ id) | ~! [X160 : subst > term > term] : (~! [X161 : term] : (~(X157 @ id @ X161 @ id) | (X157 @ id @ (X160 @ id @ X161) @ id)) | (X157 @ id @ (lam @ (X160 @ sh @ one)) @ id) | ~! [X162 : term,X163 : subst,X164 : subst] : ((sub @ (X160 @ X163 @ X162) @ X164) = (X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164))))) <=> hoasinduction_lem3a) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & hoasapnotvar_lthm & (pushprop_lem1 <=> ! [X165 : term,X166 : subst,X167 : term,X168 : term > $o,X169 : term > $o] : ((X168 @ (sub @ X167 @ (push @ X165 @ X166))) | ~(X169 @ X167))) & hoasapnotvar & hoaslamnotap_lthm & hoasinduction_gthm & axabs & induction2_gthm & ! [X170 : term] : ? [X171 : d_term] : ((d2term @ X171) = X170) & hoaslaminj_gthm & (hoasapinj2 <=> ! [X172 : term,X173 : term,X174 : term,X175 : term] : ((X174 = X175) | ((ap @ (sub @ X173 @ id) @ X174) != (ap @ (sub @ X172 @ id) @ X175)))) & (apinj1 <=> ! [X176 : term,X177 : term,X178 : term,X179 : term] : (((ap @ X176 @ X177) != (ap @ X179 @ X178)) | (X176 = X179))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & hoasinduction_lem1 & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & pushprop_lem2v2 & (! [X180 : term > $o,X181 : term,X182 : subst] : ~! [X183 : term > $o] : ~! [X184 : term] : ((X183 @ X184) = (X180 @ (sub @ X184 @ (push @ X181 @ X182)))) <=> pushprop_lem0_lthm) & pushprop & hoasinduction_lem3v2_gthm & (! [X185 : term,X186 : term] : (((sub @ X186 @ sh) != (sub @ X185 @ sh)) | (X185 = X186)) <=> shinj) & (! [X187 : subst > term > subst > $o] : ~! [X188 : term > $o] : ~! [X189 : term] : ((X188 @ X189) = (X187 @ id @ X189 @ id)) <=> hoasinduction_lem0_lthm) & hoaslamnotvar_lthm & (hoasinduction_p_and_p_prime = (^[X190 : subst > term > subst > $o, X191 : term > $o] : (! [X192 : term] : ((X190 @ id @ X192 @ id) = (X191 @ X192))))) & induction2lem_gthm & termmset_gthm & (hoasapinj2_lthm <=> (! [X193 : term,X194 : term,X195 : term,X196 : term] : ((X193 = X195) | ((ap @ X196 @ X195) != (ap @ X194 @ X193))) => ! [X197 : term,X198 : term,X199 : term,X200 : term] : ((X197 = X198) | ((ap @ (sub @ X199 @ id) @ X197) != (ap @ (sub @ X200 @ id) @ X198))))) & axmap & axclos & (one = (d2term @ d_one)) & (! [X201 : subst,X202 : term] : ((comp @ sh @ (push @ X202 @ X201)) = X201) <=> axshiftcons) & hoasapnotvar_gthm & axvarshift & (! [X203 : term,X204 : subst > term > subst > $o] : ((X204 @ id @ (lam @ (sub @ X203 @ (push @ one @ sh))) @ id) | ~! [X205 : subst > term > term] : ((X204 @ id @ (lam @ (X205 @ sh @ one)) @ id) | ~! [X206 : term] : ((X204 @ id @ (X205 @ id @ X206) @ id) | ~(X204 @ id @ X206 @ id)) | ~! [X207 : term,X208 : subst,X209 : subst] : ((sub @ (X205 @ X209 @ X207) @ X208) = (X205 @ (comp @ X209 @ X208) @ (sub @ X207 @ X208)))) | ~! [X210 : term] : (~(X204 @ id @ X210 @ id) | (X204 @ id @ (sub @ X203 @ (push @ X210 @ id)) @ id))) <=> hoasinduction_lem3aa_lthm) & ((! [X211 : term] : ((sub @ X211 @ id) = X211) => ! [X212 : term] : ((sub @ X212 @ id) = X212)) <=> termmset_lthm) & (! [X213 : subst > term > subst > $o,X214 : term,X215 : term] : (~(X213 @ id @ X215 @ id) | ~(X213 @ id @ X214 @ id) | ~! [X216 : term,X217 : subst,X218 : subst,X219 : subst] : (~(X213 @ X218 @ X216 @ (comp @ X219 @ X217)) | (X213 @ (comp @ X218 @ X219) @ (sub @ X216 @ X219) @ X217)) | ~! [X220 : subst,X221 : subst,X222 : subst,X223 : term] : ((X213 @ X222 @ X223 @ (comp @ X221 @ X220)) | ~(X213 @ (comp @ X222 @ X221) @ (sub @ X223 @ X221) @ X220)) | ~! [X224 : term,X225 : term] : ((X213 @ id @ (ap @ (sub @ X224 @ id) @ X225) @ id) | ~(X213 @ id @ X224 @ id) | ~(X213 @ id @ X225 @ id)) | (X213 @ id @ (ap @ X215 @ X214) @ id)) <=> hoasinduction_lem2) & (axidl <=> ! [X226 : subst] : ((comp @ id @ X226) = X226)) & (induction2_lthm <=> (! [X227 : term] : ((sub @ X227 @ id) = X227) => (! [X228 : term,X229 : term > $o,X230 : subst] : (~! [X231 : term] : (~(var @ X231) | (X229 @ (sub @ X231 @ X230))) | (X229 @ (sub @ X228 @ X230)) | ~! [X232 : term] : (~! [X233 : term] : ((X229 @ (sub @ X232 @ (push @ X233 @ id))) | ~(X229 @ X233)) | (X229 @ (lam @ X232))) | ~! [X234 : term,X235 : term] : (~(X229 @ X234) | ~(X229 @ X235) | (X229 @ (ap @ X235 @ X234)))) => ! [X236 : term,X237 : term > $o] : ((X237 @ X236) | ~! [X238 : term] : ((X237 @ (lam @ X238)) | ~! [X239 : term] : (~(X237 @ X239) | (X237 @ (sub @ X238 @ (push @ X239 @ id))))) | ~! [X240 : term,X241 : term] : (~(X237 @ X241) | (X237 @ (ap @ X241 @ X240)) | ~(X237 @ X240)) | ~! [X242 : term] : ((X237 @ X242) | ~(var @ X242)))))) & hoasinduction_lem3b_gthm & ((^[X243 : term, X244 : subst, X245 : term > $o, X246 : term > $o] : (! [X247 : term] : ((X246 @ X247) = (X245 @ (sub @ X247 @ (push @ X243 @ X244)))))) = pushprop_p_and_p_prime) & (! [X248 : term,X249 : subst,X250 : term > $o,X251 : term > $o] : ((X250 @ one) | ~(X251 @ X248) | ~! [X252 : term] : ((X251 @ (sub @ X252 @ (push @ X248 @ X249))) = (X250 @ X252))) <=> pushprop_lem1v2) & ulamvarsh & hoasinduction_lem1_gthm & (! [X253 : term,X254 : subst] : ((sub @ one @ (push @ X253 @ X254)) = X253) <=> axvarcons) & pushprop_gthm & (hoasinduction_lem3 <=> ! [X255 : subst > term > subst > $o,X256 : term] : ((X255 @ id @ (lam @ X256) @ id) | ~! [X257 : subst,X258 : subst,X259 : term,X260 : subst] : (~(X255 @ (comp @ X260 @ X257) @ (sub @ X259 @ X257) @ X258) | (X255 @ X260 @ X259 @ (comp @ X257 @ X258))) | ~! [X261 : subst > term > term] : ((X255 @ id @ (lam @ (X261 @ sh @ one)) @ id) | ~! [X262 : subst,X263 : term,X264 : subst] : ((sub @ (X261 @ X264 @ X263) @ X262) = (X261 @ (comp @ X264 @ X262) @ (sub @ X263 @ X262))) | ~! [X265 : term] : ((X255 @ id @ (X261 @ id @ X265) @ id) | ~(X255 @ id @ X265 @ id))) | ~! [X266 : term,X267 : subst,X268 : subst,X269 : subst] : (~(X255 @ X269 @ X266 @ (comp @ X268 @ X267)) | (X255 @ (comp @ X269 @ X268) @ (sub @ X266 @ X268) @ X267)) | ~! [X270 : term] : ((X255 @ id @ (sub @ X256 @ (push @ X270 @ id)) @ id) | ~(X255 @ id @ X270 @ id)))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X271 : term] : ~! [X272 : subst > term > term] : ((sub @ X271 @ (push @ one @ sh)) != (X272 @ sh @ one)) <=> hoasinduction_lem3b_lthm) & hoasinduction_lem1v2_gthm & pushprop_lem2v2_gthm & (! [X273 : term] : ((sub @ X273 @ id) = X273) <=> axvarid) & (! [X274 : term,X275 : subst,X276 : subst > term > term,X277 : subst > term > term] : (~! [X278 : subst,X279 : term,X280 : subst] : ((X277 @ (comp @ X278 @ X280) @ (sub @ X279 @ X280)) = (sub @ (X277 @ X278 @ X279) @ X280)) | ~! [X281 : term,X282 : subst,X283 : subst] : ((sub @ (X276 @ X283 @ X281) @ X282) = (X276 @ (comp @ X283 @ X282) @ (sub @ X281 @ X282))) | ((lam @ (X277 @ sh @ one)) != (lam @ (X276 @ sh @ one))) | ((X276 @ X275 @ X274) = (X277 @ X275 @ X274))) <=> hoaslaminj) & (termmset <=> ! [X284 : term] : ((sub @ X284 @ id) = X284)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (hoasinduction_lem3a_lthm <=> (! [X285 : term] : ((sub @ X285 @ id) = X285) => (! [X286 : term,X287 : subst > term > subst > $o] : ((X287 @ id @ (lam @ (sub @ X286 @ (push @ one @ sh))) @ id) | ~! [X288 : subst > term > term] : ((X287 @ id @ (lam @ (X288 @ sh @ one)) @ id) | ~! [X289 : term] : ((X287 @ id @ (X288 @ id @ X289) @ id) | ~(X287 @ id @ X289 @ id)) | ~! [X290 : term,X291 : subst,X292 : subst] : ((sub @ (X288 @ X292 @ X290) @ X291) = (X288 @ (comp @ X292 @ X291) @ (sub @ X290 @ X291)))) | ~! [X293 : term] : ((X287 @ id @ (sub @ X286 @ (push @ X293 @ id)) @ id) | ~(X287 @ id @ X293 @ id))) => ! [X294 : subst > term > subst > $o,X295 : term] : (~! [X296 : term] : ((X294 @ id @ (sub @ X295 @ (push @ X296 @ id)) @ id) | ~(X294 @ id @ X296 @ id)) | ~! [X297 : subst > term > term] : ((X294 @ id @ (lam @ (X297 @ sh @ one)) @ id) | ~! [X298 : term] : ((X294 @ id @ (X297 @ id @ X298) @ id) | ~(X294 @ id @ X298 @ id)) | ~! [X299 : subst,X300 : subst,X301 : term] : ((X297 @ (comp @ X300 @ X299) @ (sub @ X301 @ X299)) = (sub @ (X297 @ X300 @ X301) @ X299))) | (X294 @ id @ (lam @ X295) @ id))))) & (! [X302 : subst > term > subst > $o,X303 : term] : (~! [X304 : term,X305 : term] : (~(X302 @ id @ X305 @ id) | (X302 @ id @ (ap @ (sub @ X305 @ id) @ X304) @ id) | ~(X302 @ id @ X304 @ id)) | ~! [X306 : term,X307 : subst,X308 : subst,X309 : subst] : ((X302 @ (comp @ X307 @ X309) @ (sub @ X306 @ X309) @ X308) | ~(X302 @ X307 @ X306 @ (comp @ X309 @ X308))) | ~! [X310 : subst > term > term] : (~! [X311 : term] : (~(X302 @ id @ X311 @ id) | (X302 @ id @ (X310 @ id @ X311) @ id)) | (X302 @ id @ (lam @ (X310 @ sh @ one)) @ id) | ~! [X312 : subst,X313 : subst,X314 : term] : ((X310 @ (comp @ X312 @ X313) @ (sub @ X314 @ X313)) = (sub @ (X310 @ X312 @ X314) @ X313))) | ~! [X315 : subst,X316 : term,X317 : subst,X318 : subst] : ((X302 @ X318 @ X316 @ (comp @ X317 @ X315)) | ~(X302 @ (comp @ X318 @ X317) @ (sub @ X316 @ X317) @ X315)) | ~! [X319 : term] : (~(var @ (sub @ X319 @ id)) | (X302 @ id @ X319 @ id)) | (X302 @ id @ X303 @ id)) <=> hoasinduction) & ! [X320 : d_subst,X321 : d_subst] : (((d2subst @ X320) = (d2subst @ X321)) => (X320 = X321)) & (! [X322 : term] : ~! [X323 : subst > term > term] : ~! [X324 : subst,X325 : term] : ((X323 @ X324 @ X325) = (sub @ X322 @ (push @ X325 @ X324))) <=> hoasinduction_lem3v2_f_lthm) & (hoasinduction_lem3v2a <=> ! [X326 : subst > term > subst > $o,X327 : term,X328 : term > $o] : (~! [X329 : subst > term > term] : (~! [X330 : subst,X331 : term,X332 : subst] : ((X329 @ (comp @ X330 @ X332) @ (sub @ X331 @ X332)) = (sub @ (X329 @ X330 @ X331) @ X332)) | ~! [X333 : term] : ((X326 @ id @ (X329 @ id @ X333) @ id) | ~(X326 @ id @ X333 @ id)) | (X326 @ id @ (lam @ (X329 @ sh @ one)) @ id)) | (X328 @ (lam @ X327)) | ~! [X334 : term] : ((X328 @ X334) = (X326 @ id @ X334 @ id)) | ~! [X335 : term] : (~(X328 @ X335) | (X328 @ (sub @ X327 @ (push @ X335 @ id)))))) & (pushprop_lem1_lthm <=> (! [X336 : subst,X337 : term] : ((sub @ one @ (push @ X337 @ X336)) = X337) => (! [X338 : subst,X339 : term] : ((comp @ sh @ (push @ X339 @ X338)) = X338) => ! [X340 : term,X341 : term > $o,X342 : term,X343 : subst,X344 : term > $o] : ((X341 @ (sub @ X340 @ (push @ X342 @ X343))) | ~(X344 @ X340))))) & ! [X345 : d_term,X346 : d_term] : (((d2term @ X346) = (d2term @ X345)) => (X345 = X346)) & pushprop_lthm_orig & apnotvar & hoaslamnotap_gthm & ((! [X347 : subst > term > subst > $o] : ~! [X348 : term > $o] : ~! [X349 : term] : ((X347 @ id @ X349 @ id) = (X348 @ X349)) => (! [X350 : term > $o,X351 : term] : (~! [X352 : term,X353 : term] : ((X350 @ (ap @ X353 @ X352)) | ~(X350 @ X352) | ~(X350 @ X353)) | (X350 @ X351) | ~! [X354 : term] : (~! [X355 : term] : (~(X350 @ X355) | (X350 @ (sub @ X354 @ (push @ X355 @ id)))) | (X350 @ (lam @ X354))) | ~! [X356 : term] : (~(var @ X356) | (X350 @ X356))) => (! [X357 : term] : ((sub @ X357 @ id) = X357) => (! [X358 : subst > term > subst > $o,X359 : term,X360 : term > $o] : (~! [X361 : term] : ((X358 @ id @ X361 @ id) = (X360 @ X361)) | ~! [X362 : term] : ((X360 @ (sub @ X359 @ (push @ X362 @ id))) | ~(X360 @ X362)) | (X360 @ (lam @ X359)) | ~! [X363 : subst > term > term] : ((X358 @ id @ (lam @ (X363 @ sh @ one)) @ id) | ~! [X364 : term] : ((X358 @ id @ (X363 @ id @ X364) @ id) | ~(X358 @ id @ X364 @ id)) | ~! [X365 : subst,X366 : term,X367 : subst] : ((sub @ (X363 @ X365 @ X366) @ X367) = (X363 @ (comp @ X365 @ X367) @ (sub @ X366 @ X367))))) => ! [X368 : subst > term > subst > $o,X369 : term] : (~! [X370 : subst,X371 : subst,X372 : subst,X373 : term] : (~(X368 @ (comp @ X370 @ X371) @ (sub @ X373 @ X371) @ X372) | (X368 @ X370 @ X373 @ (comp @ X371 @ X372))) | ~! [X374 : subst > term > term] : (~! [X375 : subst,X376 : term,X377 : subst] : ((X374 @ (comp @ X377 @ X375) @ (sub @ X376 @ X375)) = (sub @ (X374 @ X377 @ X376) @ X375)) | (X368 @ id @ (lam @ (X374 @ sh @ one)) @ id) | ~! [X378 : term] : (~(X368 @ id @ X378 @ id) | (X368 @ id @ (X374 @ id @ X378) @ id))) | ~! [X379 : term,X380 : term] : ((X368 @ id @ (ap @ (sub @ X380 @ id) @ X379) @ id) | ~(X368 @ id @ X379 @ id) | ~(X368 @ id @ X380 @ id)) | (X368 @ id @ X369 @ id) | ~! [X381 : term] : (~(var @ (sub @ X381 @ id)) | (X368 @ id @ X381 @ id)) | ~! [X382 : subst,X383 : subst,X384 : subst,X385 : term] : (~(X368 @ X382 @ X385 @ (comp @ X383 @ X384)) | (X368 @ (comp @ X382 @ X383) @ (sub @ X385 @ X383) @ X384))))))) <=> hoasinduction_lthm_3) & (! [X386 : term,X387 : term,X388 : subst > term > term] : (~! [X389 : subst,X390 : subst,X391 : term] : ((sub @ (X388 @ X390 @ X391) @ X389) = (X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389))) | ((ap @ (sub @ X387 @ id) @ X386) != (lam @ (X388 @ sh @ one)))) <=> hoaslamnotap) & (axscons <=> ! [X392 : subst] : ((push @ (sub @ one @ X392) @ (comp @ sh @ X392)) = X392)) & ~(var @ (d2term @ d_one)) & (hoaslam = (^[X393 : subst, X394 : subst > term > term] : (d2term @ d_one))) & (! [X395 : subst,X396 : term > $o,X397 : term] : (~! [X398 : term] : (~! [X399 : term] : (~(X396 @ X399) | (X396 @ (sub @ X398 @ (push @ X399 @ id)))) | (X396 @ (lam @ X398))) | (X396 @ (sub @ X397 @ X395)) | ~! [X400 : term] : (~(var @ X400) | (X396 @ (sub @ X400 @ X395))) | ~! [X401 : term,X402 : term] : (~(X396 @ X401) | ~(X396 @ X402) | (X396 @ (ap @ X402 @ X401)))) <=> induction2lem) & (axidr <=> ! [X403 : subst] : ((comp @ X403 @ id) = X403)) & (! [X404 : term,X405 : term,X406 : term,X407 : term] : ((X404 = X407) | ((ap @ (sub @ X404 @ id) @ X406) != (ap @ (sub @ X407 @ id) @ X405))) <=> hoasapinj1) & axassoc & ((! [X408 : term] : ((sub @ X408 @ id) = X408) => (! [X409 : term,X410 : term,X411 : term,X412 : term] : ((X409 = X412) | ((ap @ X412 @ X411) != (ap @ X409 @ X410))) => ! [X413 : term,X414 : term,X415 : term,X416 : term] : ((X415 = X416) | ((ap @ (sub @ X416 @ id) @ X414) != (ap @ (sub @ X415 @ id) @ X413))))) <=> hoasapinj1_lthm) & ulamvarind & ~lamnotap & pushprop_lthm & (! [X417 : term,X418 : subst > term > subst > $o,X419 : term > $o] : (~! [X420 : subst > term > term] : (~! [X421 : term,X422 : subst,X423 : subst] : ((sub @ (X420 @ X422 @ X421) @ X423) = (X420 @ (comp @ X422 @ X423) @ (sub @ X421 @ X423))) | (X418 @ id @ (lam @ (X420 @ sh @ one)) @ id) | ~! [X424 : term] : ((X418 @ id @ (X420 @ id @ X424) @ id) | ~(X418 @ id @ X424 @ id))) | ~! [X425 : term] : (~(X419 @ X425) | (X419 @ (sub @ X417 @ (push @ X425 @ id)))) | ~! [X426 : subst,X427 : subst,X428 : term,X429 : subst] : ((X418 @ X429 @ X428 @ (comp @ X427 @ X426)) | ~(X418 @ (comp @ X429 @ X427) @ (sub @ X428 @ X427) @ X426)) | ~! [X430 : term] : ((X418 @ id @ X430 @ id) = (X419 @ X430)) | (X419 @ (lam @ X417)) | ~! [X431 : subst,X432 : term,X433 : subst,X434 : subst] : (~(X418 @ X434 @ X432 @ (comp @ X431 @ X433)) | (X418 @ (comp @ X434 @ X431) @ (sub @ X432 @ X431) @ X433))) <=> hoasinduction_lem3v2) & ((d2subst @ d_id) = id) & pushprop_lem0_gthm & hoasinduction_lem1v2 & hoasinduction_lem1_lthm & (! [X435 : term,X436 : term,X437 : term,X438 : term] : ((X436 = X438) | ((ap @ X435 @ X436) != (ap @ X437 @ X438))) <=> apinj2) & (! [X439 : term,X440 : term > $o] : ((X440 @ X439) | ~! [X441 : term] : ((X440 @ X441) | ~(var @ X441)) | ~! [X442 : term] : (~(X440 @ X442) | (X440 @ (lam @ X442))) | ~! [X443 : term,X444 : term] : (~(X440 @ X443) | (X440 @ (ap @ X443 @ X444)) | ~(X440 @ X444))) <=> induction) & ((! [X445 : subst > term > subst > $o] : ~! [X446 : term > $o] : ~! [X447 : term] : ((X446 @ X447) = (X445 @ id @ X447 @ id)) => (! [X448 : term,X449 : term > $o] : (~! [X450 : term] : (~(var @ X450) | (X449 @ X450)) | (X449 @ X448) | ~! [X451 : term] : ((X449 @ (lam @ X451)) | ~! [X452 : term] : (~(X449 @ X452) | (X449 @ (sub @ X451 @ (push @ X452 @ id))))) | ~! [X453 : term,X454 : term] : (~(X449 @ X453) | ~(X449 @ X454) | (X449 @ (ap @ X454 @ X453)))) => (! [X455 : term] : ((sub @ X455 @ id) = X455) => (! [X456 : term > $o,X457 : subst > term > subst > $o,X458 : term] : (~! [X459 : term] : ((X456 @ X459) = (X457 @ id @ X459 @ id)) | ~! [X460 : term] : (~(X456 @ X460) | (X456 @ (sub @ X458 @ (push @ X460 @ id)))) | ~! [X461 : subst > term > term] : (~! [X462 : term,X463 : subst,X464 : subst] : ((X461 @ (comp @ X463 @ X464) @ (sub @ X462 @ X464)) = (sub @ (X461 @ X463 @ X462) @ X464)) | (X457 @ id @ (lam @ (X461 @ sh @ one)) @ id) | ~! [X465 : term] : (~(X457 @ id @ X465 @ id) | (X457 @ id @ (X461 @ id @ X465) @ id))) | (X456 @ (lam @ X458))) => ! [X466 : subst > term > subst > $o,X467 : term] : ((X466 @ id @ X467 @ id) | ~! [X468 : subst > term > term] : ((X466 @ id @ (lam @ (X468 @ sh @ one)) @ id) | ~! [X469 : term] : ((X466 @ id @ (X468 @ id @ X469) @ id) | ~(X466 @ id @ X469 @ id)) | ~! [X470 : term,X471 : subst,X472 : subst] : ((sub @ (X468 @ X472 @ X470) @ X471) = (X468 @ (comp @ X472 @ X471) @ (sub @ X470 @ X471)))) | ~! [X473 : term,X474 : term] : (~(X466 @ id @ X474 @ id) | (X466 @ id @ (ap @ (sub @ X473 @ id) @ X474) @ id) | ~(X466 @ id @ X473 @ id))))))) <=> hoasinduction_no_psi_cond_lthm) & hoasinduction_lem2v2_gthm & hoasinduction_lem2_gthm & pushprop_lem1_gthm & ((d2subst @ d_id) = sh) & (hoasinduction_lthm <=> (! [X475 : term,X476 : term > $o] : (~! [X477 : term] : (~! [X478 : term] : (~(X476 @ X478) | (X476 @ (sub @ X477 @ (push @ X478 @ id)))) | (X476 @ (lam @ X477))) | (X476 @ X475) | ~! [X479 : term] : (~(var @ X479) | (X476 @ X479)) | ~! [X480 : term,X481 : term] : ((X476 @ (ap @ X480 @ X481)) | ~(X476 @ X481) | ~(X476 @ X480))) => (! [X482 : subst > term > subst > $o,X483 : term,X484 : term] : (~(X482 @ id @ X483 @ id) | (X482 @ id @ (ap @ X483 @ X484) @ id) | ~! [X485 : subst,X486 : term,X487 : subst,X488 : subst] : ((X482 @ (comp @ X488 @ X487) @ (sub @ X486 @ X487) @ X485) | ~(X482 @ X488 @ X486 @ (comp @ X487 @ X485))) | ~(X482 @ id @ X484 @ id) | ~! [X489 : term,X490 : subst,X491 : subst,X492 : subst] : ((X482 @ X490 @ X489 @ (comp @ X491 @ X492)) | ~(X482 @ (comp @ X490 @ X491) @ (sub @ X489 @ X491) @ X492)) | ~! [X493 : term,X494 : term] : ((X482 @ id @ (ap @ (sub @ X493 @ id) @ X494) @ id) | ~(X482 @ id @ X494 @ id) | ~(X482 @ id @ X493 @ id))) => (! [X495 : term,X496 : subst > term > subst > $o] : (~! [X497 : subst,X498 : subst,X499 : subst,X500 : term] : (~(X496 @ (comp @ X498 @ X499) @ (sub @ X500 @ X499) @ X497) | (X496 @ X498 @ X500 @ (comp @ X499 @ X497))) | (X496 @ id @ (lam @ X495) @ id) | ~! [X501 : subst,X502 : term,X503 : subst,X504 : subst] : ((X496 @ (comp @ X503 @ X504) @ (sub @ X502 @ X504) @ X501) | ~(X496 @ X503 @ X502 @ (comp @ X504 @ X501))) | ~! [X505 : subst > term > term] : (~! [X506 : term] : (~(X496 @ id @ X506 @ id) | (X496 @ id @ (X505 @ id @ X506) @ id)) | (X496 @ id @ (lam @ (X505 @ sh @ one)) @ id) | ~! [X507 : term,X508 : subst,X509 : subst] : ((X505 @ (comp @ X508 @ X509) @ (sub @ X507 @ X509)) = (sub @ (X505 @ X508 @ X507) @ X509))) | ~! [X510 : term] : (~(X496 @ id @ X510 @ id) | (X496 @ id @ (sub @ X495 @ (push @ X510 @ id)) @ id))) => ! [X511 : term,X512 : subst > term > subst > $o] : (~! [X513 : subst,X514 : subst,X515 : term,X516 : subst] : (~(X512 @ (comp @ X514 @ X516) @ (sub @ X515 @ X516) @ X513) | (X512 @ X514 @ X515 @ (comp @ X516 @ X513))) | ~! [X517 : subst > term > term] : ((X512 @ id @ (lam @ (X517 @ sh @ one)) @ id) | ~! [X518 : term] : ((X512 @ id @ (X517 @ id @ X518) @ id) | ~(X512 @ id @ X518 @ id)) | ~! [X519 : term,X520 : subst,X521 : subst] : ((X517 @ (comp @ X521 @ X520) @ (sub @ X519 @ X520)) = (sub @ (X517 @ X521 @ X519) @ X520))) | ~! [X522 : term,X523 : subst,X524 : subst,X525 : subst] : (~(X512 @ X525 @ X522 @ (comp @ X524 @ X523)) | (X512 @ (comp @ X525 @ X524) @ (sub @ X522 @ X524) @ X523)) | (X512 @ id @ X511 @ id) | ~! [X526 : term,X527 : term] : (~(X512 @ id @ X526 @ id) | ~(X512 @ id @ X527 @ id) | (X512 @ id @ (ap @ (sub @ X526 @ id) @ X527) @ id)) | ~! [X528 : term] : ((X512 @ id @ X528 @ id) | ~(var @ (sub @ X528 @ id)))))))) & (hoasinduction_lem3b <=> ! [X529 : term] : ~! [X530 : subst > term > term] : ((sub @ X529 @ (push @ one @ sh)) != (X530 @ sh @ one))) & (substmonoid_lthm <=> (! [X531 : subst] : ((comp @ id @ X531) = X531) => (! [X532 : subst] : ((comp @ X532 @ id) = X532) => (! [X533 : subst] : ((comp @ id @ X533) = X533) & ! [X534 : subst] : ((comp @ X534 @ id) = X534))))) & hoaslamnotvar & ((! [X535 : term] : ~! [X536 : subst > term > term] : ~! [X537 : subst,X538 : term] : ((sub @ X535 @ (push @ X538 @ X537)) = (X536 @ X537 @ X538)) => (! [X539 : term] : ((sub @ X539 @ id) = X539) => ! [X540 : term,X541 : term > $o,X542 : subst > term > subst > $o] : ((X541 @ (lam @ X540)) | ~! [X543 : term] : ((X542 @ id @ X543 @ id) = (X541 @ X543)) | ~! [X544 : subst > term > term] : (~! [X545 : term] : ((X542 @ id @ (X544 @ id @ X545) @ id) | ~(X542 @ id @ X545 @ id)) | (X542 @ id @ (lam @ (X544 @ sh @ one)) @ id) | ~! [X546 : subst,X547 : term,X548 : subst] : ((sub @ (X544 @ X548 @ X547) @ X546) = (X544 @ (comp @ X548 @ X546) @ (sub @ X547 @ X546)))) | ~! [X549 : term] : ((X541 @ (sub @ X540 @ (push @ X549 @ id))) | ~(X541 @ X549))))) <=> hoasinduction_lem3v2a_lthm)),
% 9.38/2.21 inference(rectify,[status(thm)],[f1])).
% 9.38/2.21 thf(f1,axiom,(
% 9.38/2.21 (! [X13 : term,X15 : term] : ((X13 = X15) | ((lam @ X13) != (lam @ X15))) <=> laminj) & ((! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X10 : term > $o,X21 : term] : (~! [X13 : term] : ((X10 @ (lam @ X13)) | ~(X10 @ X13)) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X13 : term,X15 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | (X10 @ X21)) => ! [X10 : term > $o,X16 : term,X17 : subst] : (~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13) | ~(X10 @ X15)) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id))))) | (X10 @ (sub @ X16 @ X17)) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17)))))))) <=> induction2lem_lthm) & (pushprop_lem0 <=> ! [X14 : subst,X13 : term,X10 : term > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14))))) & ((! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => ! [X11 : term > $o,X13 : term,X10 : term > $o,X14 : subst] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) | ~(X10 @ X13) | (X11 @ one))) <=> pushprop_lem1v2_lthm) & hoasinduction_lem2_lthm & (substmonoid <=> (! [X14 : subst] : ((comp @ X14 @ id) = X14) & ! [X14 : subst] : ((comp @ id @ X14) = X14))) & pushprop_lem3v2_lthm & lamnotvar & (hoasinduction_lem3v2_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X10 : subst > term > subst > $o,X48 : term,X11 : term > $o] : (~! [X13 : term,X25 : subst,X26 : subst,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X48)) | ~! [X18 : subst > term > term] : (~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X48 @ (push @ X15 @ id)))) | ~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25))))) & (! [X10 : subst > term > subst > $o,X30 : term] : (~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | (X10 @ id @ X30 @ id) | ~! [X18 : subst > term > term] : ((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))))) <=> hoasinduction_no_psi_cond) & substmonoid_gthm & hoaslamnotvar_gthm & (hoasinduction_lem3_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)))) => ! [X10 : subst > term > subst > $o,X27 : term] : (~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X25 : subst,X13 : term,X26 : subst,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X13 : term,X14 : subst,X25 : subst,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : (~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | (X10 @ id @ (lam @ X27) @ id))))) & (induction2 <=> ! [X10 : term > $o,X22 : term] : (~! [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,X15 : term] : (~(X10 @ X15) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X13)))) & ~ulamvar1 & ((d2subst @ d_id) = (push @ (d2term @ d_one) @ (d2subst @ d_id))) & pushprop_lem1v2_gthm & (! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X13 : term,X14 : subst] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) <=> hoasinduction_lem3v2_f) & hoasinduction_lem3_gthm & ((! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X13 : term,X14 : subst] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => (! [X13 : term,X15 : term] : ((X13 = X15) | ((lam @ X13) != (lam @ X15))) => ! [X33 : term,X31 : subst > term > term,X18 : subst > term > term,X32 : subst] : (((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))) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))))))) <=> hoaslaminj_lthm) & ((d2term @ d_one) = (sub @ (d2term @ d_one) @ (d2subst @ d_id))) & (! [X10 : subst > term > subst > $o,X42 : term] : ((X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ 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 @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id))) <=> hoasinduction_lem3aa) & (hoasinduction_lem2v2 <=> ! [X11 : term > $o,X45 : term,X10 : subst > term > subst > $o,X44 : term] : (~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~(X11 @ X45) | ~(X11 @ X44) | ~! [X14 : subst,X25 : subst,X13 : term,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X11 @ (ap @ X44 @ X45)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)))) & axapp & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0) & pushprop_lem2v2_lthm & hoasapinj1_gthm & hoasapinj2_gthm & ! [X1 : d_term] : (d_one = X1) & hoasinduction_lem3a_gthm & pushprop_lem3v2 & (! [X10 : subst > term > subst > $o,X34 : term] : ((X10 @ id @ (lam @ (sub @ X34 @ (push @ one @ sh))) @ id) | ~! [X35 : term,X18 : subst > term > term] : (~! [X39 : subst,X40 : term,X41 : subst] : ((X18 @ (comp @ X39 @ X41) @ (sub @ X40 @ X41)) = (sub @ X35 @ (push @ (sub @ X40 @ X41) @ (comp @ X39 @ X41)))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X37 : term,X38 : subst,X36 : subst] : ((sub @ (X18 @ X36 @ X37) @ X38) = (sub @ (sub @ X35 @ (push @ X37 @ X36)) @ X38))) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X34 @ (push @ X15 @ id)) @ id))) <=> hoasinduction_lem3aaa) & ! [X4 : subst] : ? [X5 : d_subst] : ((d2subst @ X5) = X4) & ! [X5 : d_subst] : (d_id = X5) & (! [X10 : subst > term > subst > $o,X43 : term] : (~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id)) | (X10 @ id @ (lam @ X43) @ 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))))) <=> hoasinduction_lem3a) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & hoasapnotvar_lthm & (pushprop_lem1 <=> ! [X15 : term,X14 : subst,X13 : term,X26 : term > $o,X10 : term > $o] : ((X26 @ (sub @ X13 @ (push @ X15 @ X14))) | ~(X10 @ X13))) & hoasapnotvar & hoaslamnotap_lthm & hoasinduction_gthm & axabs & induction2_gthm & ! [X0 : term] : ? [X1 : d_term] : ((d2term @ X1) = X0) & hoaslaminj_gthm & (hoasapinj2 <=> ! [X13 : term,X15 : term,X20 : term,X19 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20)))) & (apinj1 <=> ! [X13 : term,X19 : term,X20 : term,X15 : term] : (((ap @ X13 @ X19) != (ap @ X15 @ X20)) | (X13 = X15))) & ((d2term @ d_one) = (lam @ (d2term @ d_one))) & hoasinduction_lem1 & ((d2subst @ d_id) = (comp @ (d2subst @ d_id) @ (d2subst @ d_id))) & pushprop_lem2v2 & (! [X10 : term > $o,X13 : term,X14 : subst] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14)))) <=> pushprop_lem0_lthm) & pushprop & hoasinduction_lem3v2_gthm & (! [X15 : term,X13 : term] : (((sub @ X13 @ sh) != (sub @ X15 @ sh)) | (X13 = X15)) <=> shinj) & (! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) <=> hoasinduction_lem0_lthm) & hoaslamnotvar_lthm & (hoasinduction_p_and_p_prime = (^[X10 : subst > term > subst > $o, X11 : term > $o] : (! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id))))) & induction2lem_gthm & termmset_gthm & (hoasapinj2_lthm <=> (! [X20 : term,X15 : term,X19 : term,X13 : term] : ((X19 = X20) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) => ! [X19 : term,X20 : term,X13 : term,X15 : term] : ((X19 = X20) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))))) & axmap & axclos & (one = (d2term @ d_one)) & (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) <=> axshiftcons) & hoasapnotvar_gthm & axvarshift & (! [X42 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id))) <=> hoasinduction_lem3aa_lthm) & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X13 : term] : ((sub @ X13 @ id) = X13)) <=> termmset_lthm) & (! [X10 : subst > term > subst > $o,X24 : term,X23 : term] : (~(X10 @ id @ X23 @ id) | ~(X10 @ id @ X24 @ id) | ~! [X13 : term,X25 : subst,X14 : subst,X26 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X25 : subst,X26 : subst,X14 : subst,X13 : term] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id)) | (X10 @ id @ (ap @ X23 @ X24) @ id)) <=> hoasinduction_lem2) & (axidl <=> ! [X14 : subst] : ((comp @ id @ X14) = X14)) & (induction2_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X16 : term,X10 : term > $o,X17 : subst] : (~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | (X10 @ (sub @ X16 @ X17)) | ~! [X13 : term] : (~! [X15 : term] : ((X10 @ (sub @ X13 @ (push @ X15 @ id))) | ~(X10 @ X15)) | (X10 @ (lam @ X13))) | ~! [X15 : term,X13 : term] : (~(X10 @ X15) | ~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)))) => ! [X22 : term,X10 : term > $o] : ((X10 @ X22) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id))))) | ~! [X15 : term,X13 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15)) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)))))) & hoasinduction_lem3b_gthm & ((^[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) & (! [X13 : term,X14 : subst,X11 : term > $o,X10 : term > $o] : ((X11 @ one) | ~(X10 @ X13) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ (sub @ X12 @ (push @ X13 @ X14))))) <=> pushprop_lem1v2) & ulamvarsh & hoasinduction_lem1_gthm & (! [X13 : term,X14 : subst] : ((sub @ one @ (push @ X13 @ X14)) = X13) <=> axvarcons) & pushprop_gthm & (hoasinduction_lem3 <=> ! [X10 : subst > term > subst > $o,X27 : term] : ((X10 @ id @ (lam @ X27) @ id) | ~! [X26 : subst,X25 : subst,X13 : term,X14 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X13 : term,X25 : subst,X26 : subst,X14 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X15 : term] : ((X10 @ id @ (sub @ X27 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)))) & ((d2term @ d_one) = (hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) & (! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh))) <=> hoasinduction_lem3b_lthm) & hoasinduction_lem1v2_gthm & pushprop_lem2v2_gthm & (! [X13 : term] : ((sub @ X13 @ id) = X13) <=> axvarid) & (! [X33 : term,X32 : subst,X18 : subst > term > term,X31 : subst > term > term] : (~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X31 @ X14 @ X13) @ X25) = (X31 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (lam @ (X31 @ sh @ one))) | ((X18 @ X32 @ X33) = (X31 @ X32 @ X33))) <=> hoaslaminj) & (termmset <=> ! [X13 : term] : ((sub @ X13 @ id) = X13)) & ((d2term @ d_one) = (ap @ (d2term @ d_one) @ (d2term @ d_one))) & (hoasinduction_lem3a_lthm <=> (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X42 : term,X10 : subst > term > subst > $o] : ((X10 @ id @ (lam @ (sub @ X42 @ (push @ one @ sh))) @ id) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X15 : term] : ((X10 @ id @ (sub @ X42 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id))) => ! [X10 : subst > term > subst > $o,X43 : term] : (~! [X15 : term] : ((X10 @ id @ (sub @ X43 @ (push @ X15 @ id)) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | (X10 @ id @ (lam @ X43) @ id))))) & (! [X10 : subst > term > subst > $o,X28 : term] : (~! [X15 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id)) | ~! [X13 : term,X14 : subst,X25 : subst,X26 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X14 : subst,X25 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X25 : subst,X13 : term,X26 : subst,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)) | (X10 @ id @ X28 @ id)) <=> hoasinduction) & ! [X6 : d_subst,X7 : d_subst] : (((d2subst @ X6) = (d2subst @ X7)) => (X6 = X7)) & (! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) <=> hoasinduction_lem3v2_f_lthm) & (hoasinduction_lem3v2a <=> ! [X10 : subst > term > subst > $o,X29 : term,X11 : term > $o] : (~! [X18 : subst > term > term] : (~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id)) | (X11 @ (lam @ X29)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))))) & (pushprop_lem1_lthm <=> (! [X14 : subst,X13 : term] : ((sub @ one @ (push @ X13 @ X14)) = X13) => (! [X14 : subst,X13 : term] : ((comp @ sh @ (push @ X13 @ X14)) = X14) => ! [X13 : term,X26 : term > $o,X15 : term,X14 : subst,X10 : term > $o] : ((X26 @ (sub @ X13 @ (push @ X15 @ X14))) | ~(X10 @ X13))))) & ! [X2 : d_term,X3 : d_term] : (((d2term @ X2) = (d2term @ X3)) => (X2 = X3)) & pushprop_lthm_orig & apnotvar & hoaslamnotap_gthm & ((! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X10 : term > $o,X22 : term] : (~! [X15 : term,X13 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15) | ~(X10 @ X13)) | (X10 @ X22) | ~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X10 : subst > term > subst > $o,X29 : term,X11 : term > $o] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15)) | (X11 @ (lam @ X29)) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X14 : subst,X13 : term,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))))) => ! [X10 : subst > term > subst > $o,X28 : term] : (~! [X14 : subst,X26 : subst,X25 : subst,X13 : term] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [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))) | ~! [X15 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ X28 @ id) | ~! [X13 : term] : (~(var @ (sub @ X13 @ id)) | (X10 @ id @ X13 @ id)) | ~! [X14 : subst,X26 : subst,X25 : subst,X13 : term] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25))))))) <=> hoasinduction_lthm_3) & (! [X47 : term,X46 : term,X18 : subst > term > term] : (~! [X25 : subst,X14 : subst,X13 : term] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | ((lam @ (X18 @ sh @ one)) != (ap @ (sub @ X46 @ id) @ X47))) <=> hoaslamnotap) & (axscons <=> ! [X14 : subst] : ((push @ (sub @ one @ X14) @ (comp @ sh @ X14)) = X14)) & ~(var @ (d2term @ d_one)) & (hoaslam = (^[X8 : subst, X9 : subst > term > term] : (d2term @ d_one))) & (! [X17 : subst,X10 : term > $o,X16 : term] : (~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | (X10 @ (sub @ X16 @ X17)) | ~! [X15 : term] : (~(var @ X15) | (X10 @ (sub @ X15 @ X17))) | ~! [X15 : term,X13 : term] : (~(X10 @ X15) | ~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)))) <=> induction2lem) & (axidr <=> ! [X14 : subst] : ((comp @ X14 @ id) = X14)) & (! [X13 : term,X20 : term,X19 : term,X15 : term] : ((X13 = X15) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))) <=> hoasapinj1) & axassoc & ((! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X15 : term,X20 : term,X19 : term,X13 : term] : ((X13 = X15) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) => ! [X20 : term,X19 : term,X15 : term,X13 : term] : ((X13 = X15) | ((ap @ (sub @ X13 @ id) @ X19) != (ap @ (sub @ X15 @ id) @ X20))))) <=> hoasapinj1_lthm) & ulamvarind & ~lamnotap & pushprop_lthm & (! [X48 : term,X10 : subst > term > subst > $o,X11 : term > $o] : (~! [X18 : subst > term > term] : (~! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id))) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X48 @ (push @ X15 @ id)))) | ~! [X25 : subst,X26 : subst,X13 : term,X14 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | (X11 @ (lam @ X48)) | ~! [X26 : subst,X13 : term,X25 : subst,X14 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25))) <=> hoasinduction_lem3v2) & ((d2subst @ d_id) = id) & pushprop_lem0_gthm & hoasinduction_lem1v2 & hoasinduction_lem1_lthm & (! [X13 : term,X19 : term,X15 : term,X20 : term] : ((X19 = X20) | ((ap @ X13 @ X19) != (ap @ X15 @ X20))) <=> apinj2) & (! [X21 : term,X10 : term > $o] : ((X10 @ X21) | ~! [X13 : term] : ((X10 @ X13) | ~(var @ X13)) | ~! [X13 : term] : (~(X10 @ X13) | (X10 @ (lam @ X13))) | ~! [X13 : term,X15 : term] : (~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15))) <=> induction) & ((! [X10 : subst > term > subst > $o] : ~! [X11 : term > $o] : ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) => (! [X22 : term,X10 : term > $o] : (~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | (X10 @ X22) | ~! [X13 : term] : ((X10 @ (lam @ X13)) | ~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id))))) | ~! [X15 : term,X13 : term] : (~(X10 @ X15) | ~(X10 @ X13) | (X10 @ (ap @ X13 @ X15)))) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => (! [X11 : term > $o,X10 : subst > term > subst > $o,X29 : term] : (~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X15 : term] : (~(X11 @ X15) | (X11 @ (sub @ X29 @ (push @ X15 @ id)))) | ~! [X18 : subst > term > term] : (~! [X13 : term,X14 : subst,X25 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25))) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id))) | (X11 @ (lam @ X29))) => ! [X10 : subst > term > subst > $o,X30 : term] : ((X10 @ id @ X30 @ id) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [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)))) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X13 @ id))))))) <=> hoasinduction_no_psi_cond_lthm) & hoasinduction_lem2v2_gthm & hoasinduction_lem2_gthm & pushprop_lem1_gthm & ((d2subst @ d_id) = sh) & (hoasinduction_lthm <=> (! [X22 : term,X10 : term > $o] : (~! [X13 : term] : (~! [X15 : term] : (~(X10 @ X15) | (X10 @ (sub @ X13 @ (push @ X15 @ id)))) | (X10 @ (lam @ X13))) | (X10 @ X22) | ~! [X13 : term] : (~(var @ X13) | (X10 @ X13)) | ~! [X13 : term,X15 : term] : ((X10 @ (ap @ X13 @ X15)) | ~(X10 @ X15) | ~(X10 @ X13))) => (! [X10 : subst > term > subst > $o,X23 : term,X24 : term] : (~(X10 @ id @ X23 @ id) | (X10 @ id @ (ap @ X23 @ X24) @ id) | ~! [X25 : subst,X13 : term,X26 : subst,X14 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~(X10 @ id @ X24 @ id) | ~! [X13 : term,X14 : subst,X26 : subst,X25 : subst] : ((X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | ~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | ~! [X13 : term,X15 : term] : ((X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id) | ~(X10 @ id @ X15 @ id) | ~(X10 @ id @ X13 @ id))) => (! [X27 : term,X10 : subst > term > subst > $o] : (~! [X25 : subst,X14 : subst,X26 : subst,X13 : term] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | (X10 @ id @ (lam @ X27) @ id) | ~! [X25 : subst,X13 : term,X14 : subst,X26 : subst] : ((X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | ~(X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : (~! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (X18 @ id @ X13) @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [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 @ X27 @ (push @ X15 @ id)) @ id))) => ! [X28 : term,X10 : subst > term > subst > $o] : (~! [X25 : subst,X14 : subst,X13 : term,X26 : subst] : (~(X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25) | (X10 @ X14 @ X13 @ (comp @ X26 @ X25))) | ~! [X18 : subst > term > term] : ((X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~! [X13 : term,X25 : subst,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X13 : term,X25 : subst,X26 : subst,X14 : subst] : (~(X10 @ X14 @ X13 @ (comp @ X26 @ X25)) | (X10 @ (comp @ X14 @ X26) @ (sub @ X13 @ X26) @ X25)) | (X10 @ id @ X28 @ id) | ~! [X13 : term,X15 : term] : (~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X15 @ id) | (X10 @ id @ (ap @ (sub @ X13 @ id) @ X15) @ id)) | ~! [X13 : term] : ((X10 @ id @ X13 @ id) | ~(var @ (sub @ X13 @ id)))))))) & (hoasinduction_lem3b <=> ! [X15 : term] : ~! [X18 : subst > term > term] : ((X18 @ sh @ one) != (sub @ X15 @ (push @ one @ sh)))) & (substmonoid_lthm <=> (! [X14 : subst] : ((comp @ id @ X14) = X14) => (! [X14 : subst] : ((comp @ X14 @ id) = X14) => (! [X14 : subst] : ((comp @ id @ X14) = X14) & ! [X14 : subst] : ((comp @ X14 @ id) = X14))))) & hoaslamnotvar & ((! [X15 : term] : ~! [X18 : subst > term > term] : ~! [X14 : subst,X13 : term] : ((sub @ X15 @ (push @ X13 @ X14)) = (X18 @ X14 @ X13)) => (! [X13 : term] : ((sub @ X13 @ id) = X13) => ! [X29 : term,X11 : term > $o,X10 : subst > term > subst > $o] : ((X11 @ (lam @ X29)) | ~! [X12 : term] : ((X11 @ X12) = (X10 @ id @ X12 @ id)) | ~! [X18 : subst > term > term] : (~! [X13 : term] : ((X10 @ id @ (X18 @ id @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | (X10 @ id @ (lam @ (X18 @ sh @ one)) @ id) | ~! [X25 : subst,X13 : term,X14 : subst] : ((sub @ (X18 @ X14 @ X13) @ X25) = (X18 @ (comp @ X14 @ X25) @ (sub @ X13 @ X25)))) | ~! [X15 : term] : ((X11 @ (sub @ X29 @ (push @ X15 @ id))) | ~(X11 @ X15))))) <=> hoasinduction_lem3v2a_lthm)),
% 9.38/2.21 file('/export/starexec/sandbox/tmp/tmp.Ab9xLNyFM0/DTF2THF_1970.p',alg444_1)).
% 9.38/2.21 thf(f1064,plain,(
% 9.38/2.21 ( ! [X250 : term] : (((d2term @ (sK576 @ X250)) = X250)) )),
% 9.38/2.21 inference(cnf_transformation,[status(thm)],[f545])).
% 9.38/2.21 thf(f1918,plain,(
% 9.38/2.21 ($true = (sK530 @ sK527))),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f1917])).
% 9.38/2.21 thf(f1917,plain,(
% 9.38/2.21 ($true = (sK530 @ sK527)) | ($true = $false)),
% 9.38/2.21 inference(backward_demodulation,[status(thm)],[f1138,f1906])).
% 9.38/2.21 thf(f1906,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false)),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f1905])).
% 9.38/2.21 thf(f1905,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | ($true = $false)),
% 9.38/2.21 inference(duplicate_literal_removal,[status(thm)],[f1899])).
% 9.38/2.21 thf(f1899,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | (pushprop_lem1v2 = $false) | ($true = $false)),
% 9.38/2.21 inference(superposition,[status(thm)],[f1896,f1708])).
% 9.38/2.21 thf(f1708,plain,(
% 9.38/2.21 ((sK592 @ (d2term @ d_one)) = $false) | (pushprop_lem1v2 = $false)),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1706])).
% 9.38/2.21 thf(f1706,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | (((!! @ term @ (^[Y0 : term]: ((sK592 @ Y0) = (sK593 @ (sub @ Y0 @ (push @ sK591 @ sK594)))))) => (sK592 @ (d2term @ d_one))) = $false)),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1705])).
% 9.38/2.21 thf(f1705,plain,(
% 9.38/2.21 (((sK593 @ sK591) => ((!! @ term @ (^[Y0 : term]: ((sK592 @ Y0) = (sK593 @ (sub @ Y0 @ (push @ sK591 @ sK594)))))) => (sK592 @ (d2term @ d_one)))) = $false) | (pushprop_lem1v2 = $false)),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1704])).
% 9.38/2.21 thf(f1704,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | (((^[Y0 : subst]: ((sK593 @ sK591) => ((!! @ term @ (^[Y1 : term]: ((sK592 @ Y1) = (sK593 @ (sub @ Y1 @ (push @ sK591 @ Y0)))))) => (sK592 @ (d2term @ d_one))))) @ sK594) = $false)),
% 9.38/2.21 inference(sigma_clausification,[status(thm)],[f1703])).
% 9.38/2.21 thf(f1703,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | ($false = (!! @ subst @ (^[Y0 : subst]: ((sK593 @ sK591) => ((!! @ term @ (^[Y1 : term]: ((sK592 @ Y1) = (sK593 @ (sub @ Y1 @ (push @ sK591 @ Y0)))))) => (sK592 @ (d2term @ d_one)))))))),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1702])).
% 9.38/2.21 thf(f1702,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | (((^[Y0 : term > $o]: (!! @ subst @ (^[Y1 : subst]: ((Y0 @ sK591) => ((!! @ term @ (^[Y2 : term]: ((sK592 @ Y2) = (Y0 @ (sub @ Y2 @ (push @ sK591 @ Y1)))))) => (sK592 @ (d2term @ d_one))))))) @ sK593) = $false)),
% 9.38/2.21 inference(sigma_clausification,[status(thm)],[f1701])).
% 9.38/2.21 thf(f1701,plain,(
% 9.38/2.21 ((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ subst @ (^[Y1 : subst]: ((Y0 @ sK591) => ((!! @ term @ (^[Y2 : term]: ((sK592 @ Y2) = (Y0 @ (sub @ Y2 @ (push @ sK591 @ Y1)))))) => (sK592 @ (d2term @ d_one)))))))) = $false) | (pushprop_lem1v2 = $false)),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1700])).
% 9.38/2.21 thf(f1700,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | (((^[Y0 : term > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ subst @ (^[Y2 : subst]: ((Y1 @ sK591) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ Y3) = (Y1 @ (sub @ Y3 @ (push @ sK591 @ Y2)))))) => (Y0 @ (d2term @ d_one))))))))) @ sK592) = $false)),
% 9.38/2.21 inference(sigma_clausification,[status(thm)],[f1699])).
% 9.38/2.21 thf(f1699,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | ((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ subst @ (^[Y2 : subst]: ((Y1 @ sK591) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ Y3) = (Y1 @ (sub @ Y3 @ (push @ sK591 @ Y2)))))) => (Y0 @ (d2term @ d_one)))))))))) = $false)),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1698])).
% 9.38/2.21 thf(f1698,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | (((^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => ((!! @ term @ (^[Y4 : term]: ((Y1 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y3)))))) => (Y1 @ (d2term @ d_one))))))))))) @ sK591) = $false)),
% 9.38/2.21 inference(sigma_clausification,[status(thm)],[f1697])).
% 9.38/2.21 thf(f1697,plain,(
% 9.38/2.21 ((!! @ term @ (^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => ((!! @ term @ (^[Y4 : term]: ((Y1 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y3)))))) => (Y1 @ (d2term @ d_one)))))))))))) = $false) | (pushprop_lem1v2 = $false)),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1695])).
% 9.38/2.21 thf(f1695,plain,(
% 9.38/2.21 (pushprop_lem1v2 != (!! @ term @ (^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => ((!! @ term @ (^[Y4 : term]: ((Y1 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y3)))))) => (Y1 @ (d2term @ d_one)))))))))))))),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1664])).
% 9.38/2.21 thf(f1664,plain,(
% 9.38/2.21 (pushprop_lem1v2 != (!! @ term @ (^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y7 @ Y8) = (Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))))))))))))) @ Y0 @ Y3 @ Y2 @ Y1) => (Y1 @ (d2term @ d_one)))))))))))))),
% 9.38/2.21 inference(definition_unfolding,[status(thm)],[f1260,f1067,f1176])).
% 9.38/2.21 thf(f1176,plain,(
% 9.38/2.21 (one = (d2term @ d_one))),
% 9.38/2.21 inference(cnf_transformation,[status(thm)],[f545])).
% 9.38/2.21 thf(f1067,plain,(
% 9.38/2.21 (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))))))))))))))),
% 9.38/2.21 inference(cnf_transformation,[status(thm)],[f545])).
% 9.38/2.21 thf(f1260,plain,(
% 9.38/2.21 (pushprop_lem1v2 != (!! @ term @ (^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => ((pushprop_p_and_p_prime @ Y0 @ Y3 @ Y2 @ Y1) => (Y1 @ one))))))))))))),
% 9.38/2.21 inference(cnf_transformation,[status(thm)],[f9])).
% 9.38/2.21 thf(f9,plain,(
% 9.38/2.21 (pushprop_lem1v2 != (!! @ term @ (^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => ((pushprop_p_and_p_prime @ Y0 @ Y3 @ Y2 @ Y1) => (Y1 @ one))))))))))))),
% 9.38/2.21 inference(flattening,[status(thm)],[f8])).
% 9.38/2.21 thf(f8,plain,(
% 9.38/2.21 ~(pushprop_lem1v2 = (!! @ term @ (^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => ((pushprop_p_and_p_prime @ Y0 @ Y3 @ Y2 @ Y1) => (Y1 @ one))))))))))))),
% 9.38/2.21 inference(fool_elimination,[status(thm)],[f7])).
% 9.38/2.21 thf(f7,plain,(
% 9.38/2.21 ~(! [X0 : subst,X1 : term > $o,X2 : term > $o,X3 : term] : ((X1 @ X3) => ((pushprop_p_and_p_prime @ X3 @ X0 @ X1 @ X2) => (X2 @ one))) = pushprop_lem1v2)),
% 9.38/2.21 inference(rectify,[status(thm)],[f3])).
% 9.38/2.21 thf(f3,negated_conjecture,(
% 9.38/2.21 ~(! [X14 : subst,X10 : term > $o,X11 : term > $o,X13 : term] : ((X10 @ X13) => ((pushprop_p_and_p_prime @ X13 @ X14 @ X10 @ X11) => (X11 @ one))) = pushprop_lem1v2)),
% 9.38/2.21 inference(negated_conjecture,[status(cth)],[f2])).
% 9.38/2.21 thf(f2,conjecture,(
% 9.38/2.21 (! [X14 : subst,X10 : term > $o,X11 : term > $o,X13 : term] : ((X10 @ X13) => ((pushprop_p_and_p_prime @ X13 @ X14 @ X10 @ X11) => (X11 @ one))) = pushprop_lem1v2)),
% 9.38/2.21 file('/export/starexec/sandbox/tmp/tmp.Ab9xLNyFM0/DTF2THF_1970.p',pushprop_lem1v2)).
% 9.38/2.21 thf(f1896,plain,(
% 9.38/2.21 ( ! [X0 : term] : (($true = (sK592 @ X0)) | (pushprop_lem1v2 = $false)) )),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f1895])).
% 9.38/2.21 thf(f1895,plain,(
% 9.38/2.21 ( ! [X0 : term] : (($true = $false) | ($true = (sK592 @ X0)) | (pushprop_lem1v2 = $false)) )),
% 9.38/2.21 inference(duplicate_literal_removal,[status(thm)],[f1893])).
% 9.38/2.21 thf(f1893,plain,(
% 9.38/2.21 ( ! [X0 : term] : ((pushprop_lem1v2 = $false) | (pushprop_lem1v2 = $false) | ($true = $false) | ($true = (sK592 @ X0))) )),
% 9.38/2.21 inference(superposition,[status(thm)],[f1892,f1707])).
% 9.38/2.21 thf(f1707,plain,(
% 9.38/2.21 ($true = (sK593 @ sK591)) | (pushprop_lem1v2 = $false)),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1705])).
% 9.38/2.21 thf(f1892,plain,(
% 9.38/2.21 ( ! [X0 : term,X1 : term] : (($false = (sK593 @ X0)) | ((sK592 @ X1) = $true) | (pushprop_lem1v2 = $false)) )),
% 9.38/2.21 inference(superposition,[status(thm)],[f1876,f1874])).
% 9.38/2.21 thf(f1876,plain,(
% 9.38/2.21 ( ! [X1 : term] : (((sK593 @ (d2term @ d_one)) = $false) | ((sK592 @ X1) = $true) | (pushprop_lem1v2 = $false)) )),
% 9.38/2.21 inference(backward_demodulation,[status(thm)],[f1713,f1874])).
% 9.38/2.21 thf(f1713,plain,(
% 9.38/2.21 ( ! [X1 : term] : (($false = (sK593 @ (sub @ X1 @ (push @ sK591 @ sK594)))) | ((sK592 @ X1) = $true) | (pushprop_lem1v2 = $false)) )),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1712])).
% 9.38/2.21 thf(f1712,plain,(
% 9.38/2.21 ( ! [X1 : term] : ((pushprop_lem1v2 = $false) | ((sK592 @ X1) = (sK593 @ (sub @ X1 @ (push @ sK591 @ sK594))))) )),
% 9.38/2.21 inference(equality_proxy_clausification,[status(thm)],[f1711])).
% 9.38/2.21 thf(f1711,plain,(
% 9.38/2.21 ( ! [X1 : term] : ((((sK592 @ X1) = (sK593 @ (sub @ X1 @ (push @ sK591 @ sK594)))) = $true) | (pushprop_lem1v2 = $false)) )),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1710])).
% 9.38/2.21 thf(f1710,plain,(
% 9.38/2.21 ( ! [X1 : term] : ((((^[Y0 : term]: ((sK592 @ Y0) = (sK593 @ (sub @ Y0 @ (push @ sK591 @ sK594))))) @ X1) = $true) | (pushprop_lem1v2 = $false)) )),
% 9.38/2.21 inference(pi_clausification,[status(thm)],[f1709])).
% 9.38/2.21 thf(f1709,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $false) | ((!! @ term @ (^[Y0 : term]: ((sK592 @ Y0) = (sK593 @ (sub @ Y0 @ (push @ sK591 @ sK594)))))) = $true)),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1706])).
% 9.38/2.21 thf(f1138,plain,(
% 9.38/2.21 ($true = (sK530 @ sK527)) | (pushprop_lem1v2 = $true)),
% 9.38/2.21 inference(cnf_transformation,[status(thm)],[f545])).
% 9.38/2.21 thf(f4080,plain,(
% 9.38/2.21 ((sK530 @ (d2term @ d_one)) = $false)),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f4079])).
% 9.38/2.21 thf(f4079,plain,(
% 9.38/2.21 ($true = $false) | ((sK530 @ (d2term @ d_one)) = $false)),
% 9.38/2.21 inference(forward_demodulation,[status(thm)],[f4078,f2082])).
% 9.38/2.21 thf(f2082,plain,(
% 9.38/2.21 ( ! [X0 : term] : (((sK529 @ X0) = $false)) )),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f2075])).
% 9.38/2.21 thf(f2075,plain,(
% 9.38/2.21 ( ! [X0 : term] : (((sK529 @ X0) = $false) | ($true != $true)) )),
% 9.38/2.21 inference(superposition,[status(thm)],[f1984,f2020])).
% 9.38/2.21 thf(f2020,plain,(
% 9.38/2.21 ( ! [X0 : term > $o,X1 : term] : (($true = (X0 @ (d2term @ d_one))) | ($false = (X0 @ X1))) )),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f2018])).
% 9.38/2.21 thf(f2018,plain,(
% 9.38/2.21 ( ! [X0 : term > $o,X1 : term] : (($false = (X0 @ X1)) | ($true = (X0 @ (d2term @ d_one))) | ($true != $true)) )),
% 9.38/2.21 inference(equality_factoring,[status(thm)],[f2001])).
% 9.38/2.21 thf(f2001,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term] : (($true = (X3 @ (d2term @ d_one))) | ($false = (X3 @ X1)) | ((X2 @ (d2term @ d_one)) = $true)) )),
% 9.38/2.21 inference(forward_demodulation,[status(thm)],[f2000,f1874])).
% 9.38/2.21 thf(f2000,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : (((X2 @ (d2term @ d_one)) = $true) | ($false = (X3 @ X1)) | ($true = (X3 @ (sub @ (sK595 @ X3 @ X1 @ X4 @ X2) @ (push @ X1 @ X4))))) )),
% 9.38/2.21 inference(duplicate_literal_removal,[status(thm)],[f1999])).
% 9.38/2.21 thf(f1999,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : (((X2 @ (d2term @ d_one)) = $true) | ((X2 @ (d2term @ d_one)) = $true) | ($true = (X3 @ (sub @ (sK595 @ X3 @ X1 @ X4 @ X2) @ (push @ X1 @ X4)))) | ($false = (X3 @ X1))) )),
% 9.38/2.21 inference(forward_demodulation,[status(thm)],[f1998,f1874])).
% 9.38/2.21 thf(f1998,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : (((X2 @ (d2term @ d_one)) = $true) | ($false = (X3 @ X1)) | ((X2 @ (sK595 @ X3 @ X1 @ X4 @ X2)) = $true) | ($true = (X3 @ (sub @ (sK595 @ X3 @ X1 @ X4 @ X2) @ (push @ X1 @ X4))))) )),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f1997])).
% 9.38/2.21 thf(f1997,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : (($true = (X3 @ (sub @ (sK595 @ X3 @ X1 @ X4 @ X2) @ (push @ X1 @ X4)))) | ((X2 @ (d2term @ d_one)) = $true) | ($true = $false) | ((X2 @ (sK595 @ X3 @ X1 @ X4 @ X2)) = $true) | ($false = (X3 @ X1))) )),
% 9.38/2.21 inference(forward_demodulation,[status(thm)],[f1728,f1906])).
% 9.38/2.21 thf(f1728,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : ((pushprop_lem1v2 = $true) | ((X2 @ (sK595 @ X3 @ X1 @ X4 @ X2)) = $true) | ($true = (X3 @ (sub @ (sK595 @ X3 @ X1 @ X4 @ X2) @ (push @ X1 @ X4)))) | ((X2 @ (d2term @ d_one)) = $true) | ($false = (X3 @ X1))) )),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1727])).
% 9.38/2.21 thf(f1727,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : ((pushprop_lem1v2 = $true) | ($false = (X3 @ X1)) | ((X2 @ (d2term @ d_one)) = $true) | ((X2 @ (sK595 @ X3 @ X1 @ X4 @ X2)) != (X3 @ (sub @ (sK595 @ X3 @ X1 @ X4 @ X2) @ (push @ X1 @ X4))))) )),
% 9.38/2.21 inference(equality_proxy_clausification,[status(thm)],[f1726])).
% 9.38/2.21 thf(f1726,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : ((((X2 @ (sK595 @ X3 @ X1 @ X4 @ X2)) = (X3 @ (sub @ (sK595 @ X3 @ X1 @ X4 @ X2) @ (push @ X1 @ X4)))) = $false) | ((X2 @ (d2term @ d_one)) = $true) | (pushprop_lem1v2 = $true) | ($false = (X3 @ X1))) )),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1725])).
% 9.38/2.21 thf(f1725,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : (($false = (X3 @ X1)) | (((^[Y0 : term]: ((X2 @ Y0) = (X3 @ (sub @ Y0 @ (push @ X1 @ X4))))) @ (sK595 @ X3 @ X1 @ X4 @ X2)) = $false) | (pushprop_lem1v2 = $true) | ((X2 @ (d2term @ d_one)) = $true)) )),
% 9.38/2.21 inference(sigma_clausification,[status(thm)],[f1724])).
% 9.38/2.21 thf(f1724,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : ((pushprop_lem1v2 = $true) | ((X2 @ (d2term @ d_one)) = $true) | ((!! @ term @ (^[Y0 : term]: ((X2 @ Y0) = (X3 @ (sub @ Y0 @ (push @ X1 @ X4)))))) = $false) | ($false = (X3 @ X1))) )),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1723])).
% 9.38/2.21 thf(f1723,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : ((pushprop_lem1v2 = $true) | (((!! @ term @ (^[Y0 : term]: ((X2 @ Y0) = (X3 @ (sub @ Y0 @ (push @ X1 @ X4)))))) => (X2 @ (d2term @ d_one))) = $true) | ($false = (X3 @ X1))) )),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1722])).
% 9.38/2.21 thf(f1722,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : (($true = ((X3 @ X1) => ((!! @ term @ (^[Y0 : term]: ((X2 @ Y0) = (X3 @ (sub @ Y0 @ (push @ X1 @ X4)))))) => (X2 @ (d2term @ d_one))))) | (pushprop_lem1v2 = $true)) )),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1721])).
% 9.38/2.21 thf(f1721,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term,X4 : subst] : (($true = ((^[Y0 : subst]: ((X3 @ X1) => ((!! @ term @ (^[Y1 : term]: ((X2 @ Y1) = (X3 @ (sub @ Y1 @ (push @ X1 @ Y0)))))) => (X2 @ (d2term @ d_one))))) @ X4)) | (pushprop_lem1v2 = $true)) )),
% 9.38/2.21 inference(pi_clausification,[status(thm)],[f1720])).
% 9.38/2.21 thf(f1720,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term] : ((pushprop_lem1v2 = $true) | ($true = (!! @ subst @ (^[Y0 : subst]: ((X3 @ X1) => ((!! @ term @ (^[Y1 : term]: ((X2 @ Y1) = (X3 @ (sub @ Y1 @ (push @ X1 @ Y0)))))) => (X2 @ (d2term @ d_one)))))))) )),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1719])).
% 9.38/2.21 thf(f1719,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X3 : term > $o,X1 : term] : ((pushprop_lem1v2 = $true) | ($true = ((^[Y0 : term > $o]: (!! @ subst @ (^[Y1 : subst]: ((Y0 @ X1) => ((!! @ term @ (^[Y2 : term]: ((X2 @ Y2) = (Y0 @ (sub @ Y2 @ (push @ X1 @ Y1)))))) => (X2 @ (d2term @ d_one))))))) @ X3))) )),
% 9.38/2.21 inference(pi_clausification,[status(thm)],[f1718])).
% 9.38/2.21 thf(f1718,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X1 : term] : (($true = (!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ subst @ (^[Y1 : subst]: ((Y0 @ X1) => ((!! @ term @ (^[Y2 : term]: ((X2 @ Y2) = (Y0 @ (sub @ Y2 @ (push @ X1 @ Y1)))))) => (X2 @ (d2term @ d_one))))))))) | (pushprop_lem1v2 = $true)) )),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1717])).
% 9.38/2.21 thf(f1717,plain,(
% 9.38/2.21 ( ! [X2 : term > $o,X1 : term] : ((pushprop_lem1v2 = $true) | (((^[Y0 : term > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ subst @ (^[Y2 : subst]: ((Y1 @ X1) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ Y3) = (Y1 @ (sub @ Y3 @ (push @ X1 @ Y2)))))) => (Y0 @ (d2term @ d_one))))))))) @ X2) = $true)) )),
% 9.38/2.21 inference(pi_clausification,[status(thm)],[f1716])).
% 9.38/2.21 thf(f1716,plain,(
% 9.38/2.21 ( ! [X1 : term] : (((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ subst @ (^[Y2 : subst]: ((Y1 @ X1) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ Y3) = (Y1 @ (sub @ Y3 @ (push @ X1 @ Y2)))))) => (Y0 @ (d2term @ d_one)))))))))) = $true) | (pushprop_lem1v2 = $true)) )),
% 9.38/2.21 inference(beta_eta_normalization,[status(thm)],[f1715])).
% 9.38/2.21 thf(f1715,plain,(
% 9.38/2.21 ( ! [X1 : term] : ((((^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => ((!! @ term @ (^[Y4 : term]: ((Y1 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y3)))))) => (Y1 @ (d2term @ d_one))))))))))) @ X1) = $true) | (pushprop_lem1v2 = $true)) )),
% 9.38/2.21 inference(pi_clausification,[status(thm)],[f1696])).
% 9.38/2.21 thf(f1696,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $true) | ((!! @ term @ (^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ subst @ (^[Y3 : subst]: ((Y2 @ Y0) => ((!! @ term @ (^[Y4 : term]: ((Y1 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y3)))))) => (Y1 @ (d2term @ d_one)))))))))))) = $true)),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1695])).
% 9.38/2.21 thf(f1984,plain,(
% 9.38/2.21 ($true != (sK529 @ (d2term @ d_one)))),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f1983])).
% 9.38/2.21 thf(f1983,plain,(
% 9.38/2.21 ($true != (sK529 @ (d2term @ d_one))) | ($true = $false)),
% 9.38/2.21 inference(forward_demodulation,[status(thm)],[f1636,f1906])).
% 9.38/2.21 thf(f1636,plain,(
% 9.38/2.21 ($true != (sK529 @ (d2term @ d_one))) | (pushprop_lem1v2 = $true)),
% 9.38/2.21 inference(definition_unfolding,[status(thm)],[f1136,f1176])).
% 9.38/2.21 thf(f1136,plain,(
% 9.38/2.21 (pushprop_lem1v2 = $true) | ((sK529 @ one) != $true)),
% 9.38/2.21 inference(cnf_transformation,[status(thm)],[f545])).
% 9.38/2.21 thf(f4078,plain,(
% 9.38/2.21 ( ! [X154 : term] : (((sK530 @ (d2term @ d_one)) = $false) | ($true = (sK529 @ X154))) )),
% 9.38/2.21 inference(trivial_inequality_removal,[status(thm)],[f4077])).
% 9.38/2.21 thf(f4077,plain,(
% 9.38/2.21 ( ! [X154 : term] : (($true = (sK529 @ X154)) | ($true = $false) | ((sK530 @ (d2term @ d_one)) = $false)) )),
% 9.38/2.21 inference(forward_demodulation,[status(thm)],[f4076,f1906])).
% 9.38/2.21 thf(f4076,plain,(
% 9.38/2.21 ( ! [X154 : term] : ((pushprop_lem1v2 = $true) | ($true = (sK529 @ X154)) | ((sK530 @ (d2term @ d_one)) = $false)) )),
% 9.38/2.21 inference(forward_demodulation,[status(thm)],[f1823,f1874])).
% 9.38/2.21 thf(f1823,plain,(
% 9.38/2.21 ( ! [X154 : term] : (((sK530 @ (sub @ X154 @ (push @ sK527 @ sK528))) = $false) | (pushprop_lem1v2 = $true) | ($true = (sK529 @ X154))) )),
% 9.38/2.21 inference(binary_proxy_clausification,[status(thm)],[f1137])).
% 9.38/2.21 thf(f1137,plain,(
% 9.38/2.21 ( ! [X154 : term] : ((pushprop_lem1v2 = $true) | ((sK530 @ (sub @ X154 @ (push @ sK527 @ sK528))) = (sK529 @ X154))) )),
% 9.38/2.21 inference(cnf_transformation,[status(thm)],[f545])).
% 9.38/2.21 % SZS output end Proof for DTF2THF_1970
% 9.38/2.21 % (2169)------------------------------
% 9.38/2.21 % (2169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.38/2.21 % (2169)Termination reason: Refutation
% 9.38/2.21
% 9.38/2.21 % (2169)Memory used [KB]: 6780
% 9.38/2.21 % (2169)Time elapsed: 0.308 s
% 9.38/2.21 % (2169)Instructions burned: 910 (million)
% 9.38/2.21 % (2169)------------------------------
% 9.38/2.21 % (2169)------------------------------
% 9.38/2.21 % (2069)Success in time 1.113 s
%------------------------------------------------------------------------------