%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP043^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n025.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 : Wed Feb 25 08:55:05 AM UTC 2026
% Result : Theorem 4.83s 2.07s
% Output : Refutation 4.83s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 4
% Syntax : Number of formulae : 19 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 19 ( 18 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 140 ( 6 ~; 0 |; 0 &; 134 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 3 ( 2 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 266 ( 264 usr; 14 con; 0-3 aty)
% Number of variables : 8 ( 0 ^; 8 !; 0 ?; 8 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
list_nat: $tType ).
thf(type_def_7,type,
list_a: $tType ).
thf(type_def_8,type,
int: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
a: $tType ).
thf(type_def_11,type,
list_list_nat: $tType ).
thf(type_def_12,type,
list_list_a: $tType ).
thf(func_def_0,type,
list_list_nat: $tType ).
thf(func_def_1,type,
list_list_a: $tType ).
thf(func_def_2,type,
list_nat: $tType ).
thf(func_def_3,type,
list_a: $tType ).
thf(func_def_4,type,
nat: $tType ).
thf(func_def_5,type,
int: $tType ).
thf(func_def_6,type,
a: $tType ).
thf(func_def_7,type,
descar226543321ms_nat: list_nat > list_nat ).
thf(func_def_8,type,
descar1375166517sums_a: list_a > list_a ).
thf(func_def_9,type,
plus_plus_int: int > int > int ).
thf(func_def_10,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_11,type,
plus_plus_a: a > a > a ).
thf(func_def_12,type,
zero_zero_int: int ).
thf(func_def_13,type,
zero_zero_nat: nat ).
thf(func_def_14,type,
zero_zero_a: a ).
thf(func_def_15,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_16,type,
append_a: list_a > list_a > list_a ).
thf(func_def_17,type,
last_nat: list_nat > nat ).
thf(func_def_18,type,
last_a: list_a > a ).
thf(func_def_19,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(func_def_20,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(func_def_21,type,
cons_nat: nat > list_nat > list_nat ).
thf(func_def_22,type,
cons_a: a > list_a > list_a ).
thf(func_def_23,type,
nil_list_nat: list_list_nat ).
thf(func_def_24,type,
nil_list_a: list_list_a ).
thf(func_def_25,type,
nil_nat: list_nat ).
thf(func_def_26,type,
nil_a: list_a ).
thf(func_def_27,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_28,type,
nth_a: list_a > nat > a ).
thf(func_def_29,type,
replicate_nat: nat > nat > list_nat ).
thf(func_def_30,type,
replicate_a: nat > a > list_a ).
thf(func_def_31,type,
take_a: nat > list_a > list_a ).
thf(func_def_32,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_33,type,
size_size_list_a: list_a > nat ).
thf(func_def_34,type,
ord_less_int: int > int > $o ).
thf(func_def_35,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_36,type,
ord_less_a: a > a > $o ).
thf(func_def_37,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_38,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_39,type,
ord_less_eq_a: a > a > $o ).
thf(func_def_40,type,
p: nat ).
thf(func_def_41,type,
v: list_a > int ).
thf(func_def_42,type,
xa: a ).
thf(func_def_43,type,
xs: list_a ).
thf(func_def_44,type,
xs1: list_a ).
thf(func_def_45,type,
xs2: list_a ).
thf(func_def_46,type,
xsa: list_a ).
thf(func_def_60,type,
sK0:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_61,type,
sK1:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_63,type,
sK2: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_64,type,
sK3: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_65,type,
sK4: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_66,type,
sK5: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_67,type,
sK6: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_68,type,
sK7: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_69,type,
sK8: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_70,type,
sK9: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_71,type,
sK10: nat > ( nat > $o ) > nat ).
thf(func_def_72,type,
sK11: ( list_nat > list_a > $o ) > list_nat ).
thf(func_def_73,type,
sK12: ( list_nat > list_a > $o ) > nat ).
thf(func_def_74,type,
sK13: ( list_nat > list_a > $o ) > list_a ).
thf(func_def_75,type,
sK14: ( list_nat > list_a > $o ) > a ).
thf(func_def_76,type,
sK15: ( list_nat > list_a > $o ) > nat ).
thf(func_def_77,type,
sK16: ( list_nat > list_a > $o ) > list_nat ).
thf(func_def_78,type,
sK17: ( list_nat > list_a > $o ) > list_a ).
thf(func_def_79,type,
sK18: ( list_nat > list_a > $o ) > a ).
thf(func_def_80,type,
sK19: nat > nat ).
thf(func_def_81,type,
sK20: list_nat > nat ).
thf(func_def_82,type,
sK21: list_nat > nat ).
thf(func_def_83,type,
sK22: list_nat > list_nat ).
thf(func_def_84,type,
sK23: list_nat > nat ).
thf(func_def_85,type,
sK24: list_list_a > list_list_a ).
thf(func_def_86,type,
sK25: list_list_a > list_a ).
thf(func_def_87,type,
sK26: list_list_a > list_list_a ).
thf(func_def_88,type,
sK27: list_list_a > a ).
thf(func_def_89,type,
sK28: list_nat > nat ).
thf(func_def_90,type,
sK29: list_nat > list_nat ).
thf(func_def_91,type,
sK30: ( list_nat > $o ) > nat ).
thf(func_def_92,type,
sK31: ( list_nat > $o ) > nat ).
thf(func_def_93,type,
sK32: ( list_nat > $o ) > nat ).
thf(func_def_94,type,
sK33: ( list_nat > $o ) > list_nat ).
thf(func_def_95,type,
sK34: ( ( nat > nat ) > list_nat > list_nat > $o ) > nat > nat ).
thf(func_def_96,type,
sK35: ( ( nat > nat ) > list_nat > list_nat > $o ) > nat ).
thf(func_def_97,type,
sK36: ( ( nat > nat ) > list_nat > list_nat > $o ) > list_nat ).
thf(func_def_98,type,
sK37: ( ( nat > nat ) > list_nat > list_nat > $o ) > list_nat ).
thf(func_def_99,type,
sK38: ( ( nat > nat ) > list_nat > list_nat > $o ) > list_nat ).
thf(func_def_100,type,
sK39: ( ( nat > nat ) > list_nat > list_nat > $o ) > nat > nat ).
thf(func_def_101,type,
sK40: ( ( a > a > $o ) > list_a > $o ) > a ).
thf(func_def_102,type,
sK41: ( ( a > a > $o ) > list_a > $o ) > a > a > $o ).
thf(func_def_103,type,
sK42: ( ( a > a > $o ) > list_a > $o ) > a > a > $o ).
thf(func_def_104,type,
sK43: ( ( a > a > $o ) > list_a > $o ) > a ).
thf(func_def_105,type,
sK44: ( ( a > a > $o ) > list_a > $o ) > a ).
thf(func_def_106,type,
sK45: ( ( a > a > $o ) > list_a > $o ) > list_a ).
thf(func_def_107,type,
sK46: ( ( a > a > $o ) > list_a > $o ) > a > a > $o ).
thf(func_def_108,type,
sK47: ( ( a > nat ) > list_a > list_nat > $o ) > list_nat ).
thf(func_def_109,type,
sK48: ( ( a > nat ) > list_a > list_nat > $o ) > a > nat ).
thf(func_def_110,type,
sK49: ( ( a > nat ) > list_a > list_nat > $o ) > list_a ).
thf(func_def_111,type,
sK50: ( ( a > nat ) > list_a > list_nat > $o ) > a > nat ).
thf(func_def_112,type,
sK51: ( ( a > nat ) > list_a > list_nat > $o ) > list_nat ).
thf(func_def_113,type,
sK52: ( ( a > nat ) > list_a > list_nat > $o ) > a ).
thf(func_def_114,type,
sK53: ( list_a > $o ) > a ).
thf(func_def_115,type,
sK54: ( list_a > $o ) > list_a ).
thf(func_def_116,type,
sK55: ( list_a > $o ) > a ).
thf(func_def_117,type,
sK56: ( list_a > $o ) > a ).
thf(func_def_118,type,
sK57: ( list_nat > $o ) > nat ).
thf(func_def_119,type,
sK58: ( list_nat > $o ) > list_nat ).
thf(func_def_120,type,
sK59: ( ( a > a > $o ) > list_a > $o ) > a > a > $o ).
thf(func_def_121,type,
sK60: ( ( a > a > $o ) > list_a > $o ) > a ).
thf(func_def_122,type,
sK61: ( ( a > a > $o ) > list_a > $o ) > a > a > $o ).
thf(func_def_123,type,
sK62: ( ( a > a > $o ) > list_a > $o ) > list_a ).
thf(func_def_124,type,
sK63: list_a > a ).
thf(func_def_125,type,
sK64: list_a > list_a ).
thf(func_def_126,type,
sK65: list_a > a ).
thf(func_def_127,type,
sK66: list_a > a ).
thf(func_def_128,type,
sK67: ( list_a > $o ) > a ).
thf(func_def_129,type,
sK68: ( list_a > $o ) > list_a ).
thf(func_def_130,type,
sK69: ( ( nat > a ) > list_nat > list_a > $o ) > list_nat ).
thf(func_def_131,type,
sK70: ( ( nat > a ) > list_nat > list_a > $o ) > nat > a ).
thf(func_def_132,type,
sK71: ( ( nat > a ) > list_nat > list_a > $o ) > list_a ).
thf(func_def_133,type,
sK72: ( ( nat > a ) > list_nat > list_a > $o ) > nat ).
thf(func_def_134,type,
sK73: ( ( nat > a ) > list_nat > list_a > $o ) > list_a ).
thf(func_def_135,type,
sK74: ( ( nat > a ) > list_nat > list_a > $o ) > nat > a ).
thf(func_def_136,type,
sK75: ( ( nat > nat > $o ) > list_nat > $o ) > nat > nat > $o ).
thf(func_def_137,type,
sK76: ( ( nat > nat > $o ) > list_nat > $o ) > nat ).
thf(func_def_138,type,
sK77: ( ( nat > nat > $o ) > list_nat > $o ) > nat ).
thf(func_def_139,type,
sK78: ( ( nat > nat > $o ) > list_nat > $o ) > nat > nat > $o ).
thf(func_def_140,type,
sK79: ( ( nat > nat > $o ) > list_nat > $o ) > list_nat ).
thf(func_def_141,type,
sK80: ( ( nat > nat > $o ) > list_nat > $o ) > nat ).
thf(func_def_142,type,
sK81: ( ( nat > nat > $o ) > list_nat > $o ) > nat > nat > $o ).
thf(func_def_143,type,
sK82: ( list_a > list_a > $o ) > list_a ).
thf(func_def_144,type,
sK83: ( list_a > list_a > $o ) > list_a ).
thf(func_def_145,type,
sK84: ( list_a > list_a > $o ) > list_a ).
thf(func_def_146,type,
sK85: ( list_a > list_a > $o ) > a ).
thf(func_def_147,type,
sK86: ( list_a > list_a > $o ) > a ).
thf(func_def_148,type,
sK87: ( list_a > list_a > $o ) > a ).
thf(func_def_149,type,
sK88: ( list_a > list_a > $o ) > list_a ).
thf(func_def_150,type,
sK89: list_a > a ).
thf(func_def_151,type,
sK90: list_a > list_a ).
thf(func_def_152,type,
sK91: ( list_a > list_a > $o ) > a ).
thf(func_def_153,type,
sK92: ( list_a > list_a > $o ) > a ).
thf(func_def_154,type,
sK93: ( list_a > list_a > $o ) > list_a ).
thf(func_def_155,type,
sK94: ( list_a > list_a > $o ) > list_a ).
thf(func_def_156,type,
sK95: ( list_a > list_a > $o ) > list_a ).
thf(func_def_157,type,
sK96: ( list_a > list_a > $o ) > a ).
thf(func_def_158,type,
sK97: ( list_a > list_a > $o ) > list_a ).
thf(func_def_159,type,
sK98: list_a > list_a ).
thf(func_def_160,type,
sK99: list_a > a ).
thf(func_def_161,type,
sK100: ( list_a > $o ) > a ).
thf(func_def_162,type,
sK101: ( list_a > $o ) > list_a ).
thf(func_def_163,type,
sK102: ( list_nat > $o ) > list_nat ).
thf(func_def_164,type,
sK103: ( list_nat > $o ) > nat ).
thf(func_def_165,type,
sK104: ( list_nat > $o ) > nat ).
thf(func_def_166,type,
sK105: ( list_nat > $o ) > list_nat ).
thf(func_def_167,type,
sK106: ( list_nat > $o ) > nat ).
thf(func_def_168,type,
sK107: ( list_a > list_a > $o ) > list_a ).
thf(func_def_169,type,
sK108: ( list_a > list_a > $o ) > a ).
thf(func_def_170,type,
sK109: ( list_a > list_a > $o ) > list_a ).
thf(func_def_171,type,
sK110: ( list_a > list_a > $o ) > list_a ).
thf(func_def_172,type,
sK111: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_173,type,
sK112: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_174,type,
sK113: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_175,type,
sK114: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_176,type,
sK115: list_a > list_a > nat ).
thf(func_def_177,type,
sK116: ( list_a > $o ) > list_a ).
thf(func_def_178,type,
sK117: ( list_a > $o ) > a ).
thf(func_def_179,type,
sK118: ( list_a > $o ) > a ).
thf(func_def_180,type,
sK119: ( list_a > $o ) > a ).
thf(func_def_181,type,
sK120: list_nat > list_nat ).
thf(func_def_182,type,
sK121: list_nat > nat ).
thf(func_def_183,type,
sK122: ( list_nat > $o ) > nat ).
thf(func_def_184,type,
sK123: ( list_nat > $o ) > list_nat ).
thf(func_def_185,type,
sK124: ( list_nat > $o ) > nat ).
thf(func_def_186,type,
sK125: ( list_a > list_a > $o ) > list_a ).
thf(func_def_187,type,
sK126: ( list_a > list_a > $o ) > a ).
thf(func_def_188,type,
sK127: ( list_a > list_a > $o ) > list_a ).
thf(func_def_189,type,
sK128: ( list_a > list_a > $o ) > a ).
thf(func_def_190,type,
sK129: ( list_a > list_a > $o ) > list_a ).
thf(func_def_191,type,
sK130: ( list_a > list_a > $o ) > a ).
thf(func_def_192,type,
sK131: ( list_a > list_a > $o ) > a ).
thf(func_def_193,type,
sK132: ( list_a > list_a > $o ) > list_a ).
thf(func_def_194,type,
sK133: ( list_a > $o ) > a ).
thf(func_def_195,type,
sK134: ( list_a > $o ) > a ).
thf(func_def_196,type,
sK135: ( list_a > $o ) > list_a ).
thf(func_def_197,type,
sK136: ( ( a > a ) > list_a > list_a > $o ) > a > a ).
thf(func_def_198,type,
sK137: ( ( a > a ) > list_a > list_a > $o ) > list_a ).
thf(func_def_199,type,
sK138: ( ( a > a ) > list_a > list_a > $o ) > list_a ).
thf(func_def_200,type,
sK139: ( ( a > a ) > list_a > list_a > $o ) > a ).
thf(func_def_201,type,
sK140: ( ( a > a ) > list_a > list_a > $o ) > a > a ).
thf(func_def_202,type,
sK141: ( ( a > a ) > list_a > list_a > $o ) > list_a ).
thf(func_def_203,type,
sK142: list_list_nat > list_list_nat ).
thf(func_def_204,type,
sK143: list_list_nat > list_nat ).
thf(func_def_205,type,
sK144: list_list_nat > nat ).
thf(func_def_206,type,
sK145: list_list_nat > list_list_nat ).
thf(func_def_207,type,
sK146: ( ( nat > nat > $o ) > list_nat > $o ) > nat > nat > $o ).
thf(func_def_208,type,
sK147: ( ( nat > nat > $o ) > list_nat > $o ) > nat > nat > $o ).
thf(func_def_209,type,
sK148: ( ( nat > nat > $o ) > list_nat > $o ) > nat ).
thf(func_def_210,type,
sK149: ( ( nat > nat > $o ) > list_nat > $o ) > list_nat ).
thf(func_def_211,type,
sK150: ( list_a > $o ) > list_a ).
thf(func_def_212,type,
sK151: ( list_a > $o ) > a ).
thf(func_def_213,type,
sK152: ( list_a > $o ) > list_a ).
thf(func_def_214,type,
sK153: ( list_nat > $o ) > nat ).
thf(func_def_215,type,
sK154: ( list_nat > $o ) > list_nat ).
thf(func_def_216,type,
sK155: ( list_nat > $o ) > nat ).
thf(func_def_217,type,
sK156: ( list_nat > $o ) > nat ).
thf(func_def_218,type,
sK157: list_nat > nat ).
thf(func_def_219,type,
sK158: list_nat > list_nat ).
thf(func_def_220,type,
sK159: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_221,type,
sK160: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_222,type,
sK161: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_223,type,
sK162: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_224,type,
sK163: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_225,type,
sK164: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_226,type,
sK165: ( list_a > list_nat > $o ) > a ).
thf(func_def_227,type,
sK166: ( list_a > list_nat > $o ) > nat ).
thf(func_def_228,type,
sK167: ( list_a > list_nat > $o ) > list_nat ).
thf(func_def_229,type,
sK168: ( list_a > list_nat > $o ) > list_a ).
thf(func_def_230,type,
sK169: ( list_a > list_nat > $o ) > list_nat ).
thf(func_def_231,type,
sK170: ( list_a > list_nat > $o ) > nat ).
thf(func_def_232,type,
sK171: ( list_a > list_nat > $o ) > a ).
thf(func_def_233,type,
sK172: ( list_a > list_nat > $o ) > list_a ).
thf(func_def_234,type,
sK173: ( list_nat > $o ) > list_nat ).
thf(func_def_235,type,
sK174: ( list_nat > $o ) > nat ).
thf(func_def_236,type,
sK175: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_237,type,
sK176: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_238,type,
sK177: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_239,type,
sK178: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_240,type,
sK179: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_241,type,
sK180: ( list_nat > list_nat > $o ) > nat ).
thf(func_def_242,type,
sK181: ( list_nat > list_nat > $o ) > list_nat ).
thf(func_def_243,type,
sK182: list_a > list_a > nat > nat ).
thf(func_def_244,type,
sK183: list_nat > list_nat > list_nat ).
thf(func_def_245,type,
sK184: list_nat > list_nat > list_nat ).
thf(func_def_246,type,
sK185: list_nat > list_nat > list_nat ).
thf(func_def_247,type,
sK186: nat > nat ).
thf(func_def_248,type,
sK187: ( list_nat > $o ) > list_nat ).
thf(func_def_249,type,
sK188: ( list_nat > $o ) > nat ).
thf(func_def_250,type,
sK189: nat > list_a ).
thf(func_def_251,type,
sK190: list_a > a ).
thf(func_def_252,type,
sK191: list_a > list_a ).
thf(func_def_253,type,
sK192: list_a > a ).
thf(func_def_254,type,
sK193: list_a > a ).
thf(func_def_255,type,
sK194: list_a > list_a > list_a ).
thf(func_def_256,type,
sK195: list_a > list_a > list_a ).
thf(func_def_257,type,
sK196: list_a > list_a > list_a ).
thf(func_def_258,type,
sK197: ( nat > nat ) > nat ).
thf(func_def_259,type,
sK198: ( nat > nat ) > nat ).
thf(func_def_260,type,
sK199: ( list_a > $o ) > list_a ).
thf(func_def_261,type,
sK200: ( list_a > $o ) > a ).
thf(func_def_262,type,
sK201: nat > nat > nat ).
thf(func_def_263,type,
sK202: a > a ).
thf(func_def_264,type,
sK203: a > a ).
thf(func_def_265,type,
sK204: ( list_a > list_a > $o ) > list_a ).
thf(func_def_266,type,
sK205: ( list_a > list_a > $o ) > list_a ).
thf(func_def_267,type,
sK206: ( list_a > list_a > $o ) > a ).
thf(func_def_268,type,
sK207: ( list_a > list_a > $o ) > list_a ).
thf(func_def_269,type,
sK208: ( list_a > list_a > $o ) > a ).
thf(func_def_270,type,
sK209: ( list_a > list_a > $o ) > list_a ).
thf(func_def_271,type,
sK210: list_a > list_a ).
thf(func_def_272,type,
sK211: list_a > a ).
thf(func_def_273,type,
sK212: list_a > a ).
thf(func_def_274,type,
sK213: list_a > list_a ).
thf(func_def_275,type,
sK214: list_nat > nat ).
thf(func_def_276,type,
sK215: list_nat > nat ).
thf(func_def_277,type,
sK216: list_nat > nat ).
thf(func_def_278,type,
sK217: list_nat > list_nat ).
thf(func_def_279,type,
sK218: list_nat > nat ).
thf(func_def_280,type,
sK219: list_nat > list_nat ).
thf(func_def_281,type,
sK220: ( list_a > $o ) > list_a ).
thf(func_def_282,type,
sK221: ( list_a > $o ) > a ).
thf(func_def_283,type,
sK222: ( list_a > $o ) > a ).
thf(func_def_285,type,
ph224:
!>[X0: $tType] : X0 ).
thf(f2057,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2056]) ).
thf(f2056,plain,
( ( descar1375166517sums_a @ xsa )
!= ( descar1375166517sums_a @ xsa ) ),
inference(superposition,[status(thm)],[f2054,f1896]) ).
thf(f1896,plain,
( xsa
= ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ),
inference(definition_unfolding,[status(thm)],[f1494,f1853]) ).
thf(f1853,plain,
( xs1
= ( replicate_a @ p @ zero_zero_a ) ),
inference(cnf_transformation,[status(thm)],[f1001]) ).
thf(f1001,plain,
( xs1
= ( replicate_a @ p @ zero_zero_a ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,axiom,
( xs1
= ( replicate_a @ p @ zero_zero_a ) ),
file('/export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2THF_2527.p',fact_5_xs1__def) ).
thf(f1494,plain,
( xsa
= ( append_a @ xs1 @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ),
inference(cnf_transformation,[status(thm)],[f837]) ).
thf(f837,plain,
( xsa
= ( append_a @ xs1 @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,axiom,
( xsa
= ( append_a @ xs1 @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2THF_2527.p',fact_6_xs__decompose) ).
thf(f2054,plain,
( ( descar1375166517sums_a @ xsa )
!= ( descar1375166517sums_a @ ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ),
inference(superposition,[status(thm)],[f1785,f1645]) ).
thf(f1645,plain,
! [X0: nat,X1: list_a] :
( ( append_a @ ( replicate_a @ X0 @ zero_zero_a ) @ ( descar1375166517sums_a @ X1 ) )
= ( descar1375166517sums_a @ ( append_a @ ( replicate_a @ X0 @ zero_zero_a ) @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f969]) ).
thf(f969,plain,
! [X0: nat,X1: list_a] :
( ( append_a @ ( replicate_a @ X0 @ zero_zero_a ) @ ( descar1375166517sums_a @ X1 ) )
= ( descar1375166517sums_a @ ( append_a @ ( replicate_a @ X0 @ zero_zero_a ) @ X1 ) ) ),
inference(fool_elimination,[status(thm)],[f968]) ).
thf(f968,plain,
! [X0: nat,X1: list_a] :
( ( append_a @ ( replicate_a @ X0 @ zero_zero_a ) @ ( descar1375166517sums_a @ X1 ) )
= ( descar1375166517sums_a @ ( append_a @ ( replicate_a @ X0 @ zero_zero_a ) @ X1 ) ) ),
inference(rectify,[status(thm)],[f9]) ).
thf(f9,axiom,
! [X1: nat,X0: list_a] :
( ( descar1375166517sums_a @ ( append_a @ ( replicate_a @ X1 @ zero_zero_a ) @ X0 ) )
= ( append_a @ ( replicate_a @ X1 @ zero_zero_a ) @ ( descar1375166517sums_a @ X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2THF_2527.p',fact_8_psums__replicate__0__append) ).
thf(f1785,plain,
( ( descar1375166517sums_a @ xsa )
!= ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( descar1375166517sums_a @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1055]) ).
thf(f1055,plain,
( ( descar1375166517sums_a @ xsa )
!= ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( descar1375166517sums_a @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ),
inference(flattening,[status(thm)],[f413]) ).
thf(f413,plain,
( ( descar1375166517sums_a @ xsa )
!= ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( descar1375166517sums_a @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f353]) ).
thf(f353,negated_conjecture,
( ( descar1375166517sums_a @ xsa )
!= ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( descar1375166517sums_a @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f352]) ).
thf(f352,conjecture,
( ( descar1375166517sums_a @ xsa )
= ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( descar1375166517sums_a @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2THF_2527.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP043^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n025.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:24:28 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.47 ---- Original DTF file ---
% 0.17/0.47 thf(spec,logic,$$dhol).
% 0.17/0.47 %------------------------------------------------------------------------------
% 0.17/0.47 % File : ITP043^1 : TPTP v9.2.1. Released v7.5.0.
% 0.17/0.47 % Domain : Interactive Theorem Proving
% 0.17/0.47 % Problem : Sledgehammer Descartes_Sign_Rule problem prob_378__5868812_1
% 0.17/0.47 % Version : Especial.
% 0.17/0.47 % English :
% 0.17/0.47
% 0.17/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.17/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.17/0.47 % Source : [Des21]
% 0.17/0.47 % Names : Descartes_Sign_Rule/prob_378__5868812_1 [Des21]
% 0.17/0.47
% 0.17/0.47 % Status : Theorem
% 0.17/0.47 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.17/0.47 % Syntax : Number of formulae : 399 ( 204 unt; 47 typ; 0 def)
% 0.17/0.47 % Number of atoms : 793 ( 548 equ; 0 cnn)
% 0.17/0.47 % Maximal formula atoms : 5 ( 2 avg)
% 0.17/0.47 % Number of connectives : 2521 ( 104 ~; 15 |; 59 &;2048 @)
% 0.17/0.47 % ( 0 <=>; 295 =>; 0 <=; 0 <~>)
% 0.17/0.47 % Maximal formula depth : 17 ( 6 avg)
% 0.17/0.47 % Number of types : 8 ( 7 usr)
% 0.17/0.47 % Number of type conns : 160 ( 160 >; 0 *; 0 +; 0 <<)
% 0.17/0.47 % Number of symbols : 41 ( 40 usr; 13 con; 0-2 aty)
% 0.17/0.47 % Number of variables : 1002 ( 9 ^; 970 !; 23 ?;1002 :)
% 0.17/0.47 % SPC : TH0_THM_EQU_NAR
% 0.17/0.47
% 0.17/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:40:52.257
% 0.17/0.47 %------------------------------------------------------------------------------
% 0.17/0.47 % Could-be-implicit typings (7)
% 0.17/0.47 thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.17/0.47 list_list_nat: $tType ).
% 0.17/0.47
% 0.17/0.47 thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 0.17/0.47 list_list_a: $tType ).
% 0.17/0.47
% 0.17/0.47 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.47 list_nat: $tType ).
% 0.17/0.47
% 0.17/0.47 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 0.17/0.47 list_a: $tType ).
% 0.17/0.47
% 0.17/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.17/0.47 nat: $tType ).
% 0.17/0.47
% 0.17/0.47 thf(ty_n_t__Int__Oint,type,
% 0.17/0.47 int: $tType ).
% 0.17/0.47
% 0.17/0.47 thf(ty_n_tf__a,type,
% 0.17/0.47 a: $tType ).
% 0.17/0.47
% 0.17/0.47 % Explicit typings (40)
% 0.17/0.47 thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Opsums_001t__Nat__Onat,type,
% 0.17/0.47 descar226543321ms_nat: list_nat > list_nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Opsums_001tf__a,type,
% 0.17/0.47 descar1375166517sums_a: list_a > list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 0.17/0.47 plus_plus_int: int > int > int ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.17/0.47 plus_plus_nat: nat > nat > nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
% 0.17/0.47 plus_plus_a: a > a > a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.17/0.47 zero_zero_int: int ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.17/0.47 zero_zero_nat: nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
% 0.17/0.47 zero_zero_a: a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 0.17/0.47 append_nat: list_nat > list_nat > list_nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Oappend_001tf__a,type,
% 0.17/0.47 append_a: list_a > list_a > list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 0.17/0.47 last_nat: list_nat > nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olast_001tf__a,type,
% 0.17/0.47 last_a: list_a > a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.47 cons_list_nat: list_nat > list_list_nat > list_list_nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
% 0.17/0.47 cons_list_a: list_a > list_list_a > list_list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 0.17/0.47 cons_nat: nat > list_nat > list_nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olist_OCons_001tf__a,type,
% 0.17/0.47 cons_a: a > list_a > list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.47 nil_list_nat: list_list_nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
% 0.17/0.47 nil_list_a: list_list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 0.17/0.47 nil_nat: list_nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Olist_ONil_001tf__a,type,
% 0.17/0.47 nil_a: list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.17/0.47 nth_nat: list_nat > nat > nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Onth_001tf__a,type,
% 0.17/0.47 nth_a: list_a > nat > a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 0.17/0.47 replicate_nat: nat > nat > list_nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Oreplicate_001tf__a,type,
% 0.17/0.47 replicate_a: nat > a > list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_List_Otake_001tf__a,type,
% 0.17/0.47 take_a: nat > list_a > list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.47 size_size_list_nat: list_nat > nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 0.17/0.47 size_size_list_a: list_a > nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.17/0.47 ord_less_int: int > int > $o ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.17/0.47 ord_less_nat: nat > nat > $o ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
% 0.17/0.47 ord_less_a: a > a > $o ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.17/0.47 ord_less_eq_int: int > int > $o ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.17/0.47 ord_less_eq_nat: nat > nat > $o ).
% 0.17/0.47
% 0.17/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 0.17/0.47 ord_less_eq_a: a > a > $o ).
% 0.17/0.47
% 0.17/0.47 thf(sy_v_p____,type,
% 0.17/0.47 p: nat ).
% 0.17/0.47
% 0.17/0.47 thf(sy_v_v,type,
% 0.17/0.47 v: list_a > int ).
% 0.17/0.47
% 0.17/0.47 thf(sy_v_xa____,type,
% 0.17/0.47 xa: a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_v_xs,type,
% 0.17/0.47 xs: list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_v_xs1____,type,
% 0.17/0.47 xs1: list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_v_xs2____,type,
% 0.17/0.47 xs2: list_a ).
% 0.17/0.47
% 0.17/0.47 thf(sy_v_xsa____,type,
% 0.17/0.47 xsa: list_a ).
% 0.17/0.47
% 0.17/0.47 % Relevant facts (351)
% 0.17/0.47 thf(fact_0_nonneg_Ohyps_I1_J,axiom,
% 0.17/0.47 xsa != nil_a ).
% 0.17/0.47
% 0.17/0.47 % nonneg.hyps(1)
% 0.17/0.47 thf(fact_1_p__nz,axiom,
% 0.17/0.47 ( ( nth_a @ xsa @ p )
% 0.17/0.47 != zero_zero_a ) ).
% 0.17/0.47
% 0.17/0.47 % p_nz
% 0.17/0.47 thf(fact_2_nonneg_Ohyps_I2_J,axiom,
% 0.17/0.47 ( ( last_a @ xsa )
% 0.17/0.47 != zero_zero_a ) ).
% 0.17/0.47
% 0.17/0.47 % nonneg.hyps(2)
% 0.17/0.47 thf(fact_3_psums__0__Cons,axiom,
% 0.17/0.47 ! [Xs: list_nat] :
% 0.17/0.47 ( ( descar226543321ms_nat @ ( cons_nat @ zero_zero_nat @ Xs ) )
% 0.17/0.47 = ( cons_nat @ zero_zero_nat @ ( descar226543321ms_nat @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % psums_0_Cons
% 0.17/0.47 thf(fact_4_psums__0__Cons,axiom,
% 0.17/0.47 ! [Xs: list_a] :
% 0.17/0.47 ( ( descar1375166517sums_a @ ( cons_a @ zero_zero_a @ Xs ) )
% 0.17/0.47 = ( cons_a @ zero_zero_a @ ( descar1375166517sums_a @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % psums_0_Cons
% 0.17/0.47 thf(fact_5_xs1__def,axiom,
% 0.17/0.47 ( xs1
% 0.17/0.47 = ( replicate_a @ p @ zero_zero_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % xs1_def
% 0.17/0.47 thf(fact_6_xs__decompose,axiom,
% 0.17/0.47 ( xsa
% 0.17/0.47 = ( append_a @ xs1 @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % xs_decompose
% 0.17/0.47 thf(fact_7_psums__replicate__0__append,axiom,
% 0.17/0.47 ! [N: nat,Xs: list_nat] :
% 0.17/0.47 ( ( descar226543321ms_nat @ ( append_nat @ ( replicate_nat @ N @ zero_zero_nat ) @ Xs ) )
% 0.17/0.47 = ( append_nat @ ( replicate_nat @ N @ zero_zero_nat ) @ ( descar226543321ms_nat @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % psums_replicate_0_append
% 0.17/0.47 thf(fact_8_psums__replicate__0__append,axiom,
% 0.17/0.47 ! [N: nat,Xs: list_a] :
% 0.17/0.47 ( ( descar1375166517sums_a @ ( append_a @ ( replicate_a @ N @ zero_zero_a ) @ Xs ) )
% 0.17/0.47 = ( append_a @ ( replicate_a @ N @ zero_zero_a ) @ ( descar1375166517sums_a @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % psums_replicate_0_append
% 0.17/0.47 thf(fact_9_less__p__zero,axiom,
% 0.17/0.47 ! [Q: nat] :
% 0.17/0.47 ( ( ord_less_nat @ Q @ p )
% 0.17/0.47 => ( ( nth_a @ xsa @ Q )
% 0.17/0.47 = zero_zero_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % less_p_zero
% 0.17/0.47 thf(fact_10__092_060open_062take_Ap_Axs_A_061_Areplicate_Ap_A_I0_058_058_Ha_J_092_060close_062,axiom,
% 0.17/0.47 ( ( take_a @ p @ xsa )
% 0.17/0.47 = ( replicate_a @ p @ zero_zero_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % \<open>take p xs = replicate p (0::'a)\<close>
% 0.17/0.47 thf(fact_11_replicate__app__Cons__same,axiom,
% 0.17/0.47 ! [N: nat,X: nat,Xs: list_nat] :
% 0.17/0.47 ( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
% 0.17/0.47 = ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % replicate_app_Cons_same
% 0.17/0.47 thf(fact_12_replicate__app__Cons__same,axiom,
% 0.17/0.47 ! [N: nat,X: a,Xs: list_a] :
% 0.17/0.47 ( ( append_a @ ( replicate_a @ N @ X ) @ ( cons_a @ X @ Xs ) )
% 0.17/0.47 = ( cons_a @ X @ ( append_a @ ( replicate_a @ N @ X ) @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % replicate_app_Cons_same
% 0.17/0.47 thf(fact_13_append_Oassoc,axiom,
% 0.17/0.47 ! [A: list_a,B: list_a,C: list_a] :
% 0.17/0.47 ( ( append_a @ ( append_a @ A @ B ) @ C )
% 0.17/0.47 = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append.assoc
% 0.17/0.47 thf(fact_14_append_Oassoc,axiom,
% 0.17/0.47 ! [A: list_nat,B: list_nat,C: list_nat] :
% 0.17/0.47 ( ( append_nat @ ( append_nat @ A @ B ) @ C )
% 0.17/0.47 = ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append.assoc
% 0.17/0.47 thf(fact_15_append__assoc,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 0.17/0.47 ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
% 0.17/0.47 = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_assoc
% 0.17/0.47 thf(fact_16_append__assoc,axiom,
% 0.17/0.47 ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
% 0.17/0.47 ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
% 0.17/0.47 = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_assoc
% 0.17/0.47 thf(fact_17_append__same__eq,axiom,
% 0.17/0.47 ! [Ys: list_a,Xs: list_a,Zs: list_a] :
% 0.17/0.47 ( ( ( append_a @ Ys @ Xs )
% 0.17/0.47 = ( append_a @ Zs @ Xs ) )
% 0.17/0.47 = ( Ys = Zs ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_same_eq
% 0.17/0.47 thf(fact_18_append__same__eq,axiom,
% 0.17/0.47 ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
% 0.17/0.47 ( ( ( append_nat @ Ys @ Xs )
% 0.17/0.47 = ( append_nat @ Zs @ Xs ) )
% 0.17/0.47 = ( Ys = Zs ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_same_eq
% 0.17/0.47 thf(fact_19_same__append__eq,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 0.17/0.47 ( ( ( append_a @ Xs @ Ys )
% 0.17/0.47 = ( append_a @ Xs @ Zs ) )
% 0.17/0.47 = ( Ys = Zs ) ) ).
% 0.17/0.47
% 0.17/0.47 % same_append_eq
% 0.17/0.47 thf(fact_20_same__append__eq,axiom,
% 0.17/0.47 ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
% 0.17/0.47 ( ( ( append_nat @ Xs @ Ys )
% 0.17/0.47 = ( append_nat @ Xs @ Zs ) )
% 0.17/0.47 = ( Ys = Zs ) ) ).
% 0.17/0.47
% 0.17/0.47 % same_append_eq
% 0.17/0.47 thf(fact_21_nonneg_Ohyps_I4_J,axiom,
% 0.17/0.47 ord_less_eq_a @ zero_zero_a @ xa ).
% 0.17/0.47
% 0.17/0.47 % nonneg.hyps(4)
% 0.17/0.47 thf(fact_22_list_Oinject,axiom,
% 0.17/0.47 ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
% 0.17/0.47 ( ( ( cons_a @ X21 @ X22 )
% 0.17/0.47 = ( cons_a @ Y21 @ Y22 ) )
% 0.17/0.47 = ( ( X21 = Y21 )
% 0.17/0.47 & ( X22 = Y22 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % list.inject
% 0.17/0.47 thf(fact_23_list_Oinject,axiom,
% 0.17/0.47 ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
% 0.17/0.47 ( ( ( cons_nat @ X21 @ X22 )
% 0.17/0.47 = ( cons_nat @ Y21 @ Y22 ) )
% 0.17/0.47 = ( ( X21 = Y21 )
% 0.17/0.47 & ( X22 = Y22 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % list.inject
% 0.17/0.47 thf(fact_24_append__replicate__commute,axiom,
% 0.17/0.47 ! [N: nat,X: a,K: nat] :
% 0.17/0.47 ( ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ K @ X ) )
% 0.17/0.47 = ( append_a @ ( replicate_a @ K @ X ) @ ( replicate_a @ N @ X ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_replicate_commute
% 0.17/0.47 thf(fact_25_append__replicate__commute,axiom,
% 0.17/0.47 ! [N: nat,X: nat,K: nat] :
% 0.17/0.47 ( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
% 0.17/0.47 = ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_replicate_commute
% 0.17/0.47 thf(fact_26_Cons__eq__appendI,axiom,
% 0.17/0.47 ! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
% 0.17/0.47 ( ( ( cons_a @ X @ Xs1 )
% 0.17/0.47 = Ys )
% 0.17/0.47 => ( ( Xs
% 0.17/0.47 = ( append_a @ Xs1 @ Zs ) )
% 0.17/0.47 => ( ( cons_a @ X @ Xs )
% 0.17/0.47 = ( append_a @ Ys @ Zs ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % Cons_eq_appendI
% 0.17/0.47 thf(fact_27_Cons__eq__appendI,axiom,
% 0.17/0.47 ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
% 0.17/0.47 ( ( ( cons_nat @ X @ Xs1 )
% 0.17/0.47 = Ys )
% 0.17/0.47 => ( ( Xs
% 0.17/0.47 = ( append_nat @ Xs1 @ Zs ) )
% 0.17/0.47 => ( ( cons_nat @ X @ Xs )
% 0.17/0.47 = ( append_nat @ Ys @ Zs ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % Cons_eq_appendI
% 0.17/0.47 thf(fact_28_assms_I1_J,axiom,
% 0.17/0.47 xs != nil_a ).
% 0.17/0.47
% 0.17/0.47 % assms(1)
% 0.17/0.47 thf(fact_29_assms_I2_J,axiom,
% 0.17/0.47 ( ( last_a @ xs )
% 0.17/0.47 != zero_zero_a ) ).
% 0.17/0.47
% 0.17/0.47 % assms(2)
% 0.17/0.47 thf(fact_30_replicate__eq__replicate,axiom,
% 0.17/0.47 ! [M: nat,X: a,N: nat,Y: a] :
% 0.17/0.47 ( ( ( replicate_a @ M @ X )
% 0.17/0.47 = ( replicate_a @ N @ Y ) )
% 0.17/0.47 = ( ( M = N )
% 0.17/0.47 & ( ( M != zero_zero_nat )
% 0.17/0.47 => ( X = Y ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % replicate_eq_replicate
% 0.17/0.47 thf(fact_31_replicate__eq__replicate,axiom,
% 0.17/0.47 ! [M: nat,X: nat,N: nat,Y: nat] :
% 0.17/0.47 ( ( ( replicate_nat @ M @ X )
% 0.17/0.47 = ( replicate_nat @ N @ Y ) )
% 0.17/0.47 = ( ( M = N )
% 0.17/0.47 & ( ( M != zero_zero_nat )
% 0.17/0.47 => ( X = Y ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % replicate_eq_replicate
% 0.17/0.47 thf(fact_32_le__zero__eq,axiom,
% 0.17/0.47 ! [N: nat] :
% 0.17/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.17/0.47 = ( N = zero_zero_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % le_zero_eq
% 0.17/0.47 thf(fact_33_not__gr__zero,axiom,
% 0.17/0.47 ! [N: nat] :
% 0.17/0.47 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.17/0.47 = ( N = zero_zero_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % not_gr_zero
% 0.17/0.47 thf(fact_34_append_Oright__neutral,axiom,
% 0.17/0.47 ! [A: list_nat] :
% 0.17/0.47 ( ( append_nat @ A @ nil_nat )
% 0.17/0.47 = A ) ).
% 0.17/0.47
% 0.17/0.47 % append.right_neutral
% 0.17/0.47 thf(fact_35_append_Oright__neutral,axiom,
% 0.17/0.47 ! [A: list_a] :
% 0.17/0.47 ( ( append_a @ A @ nil_a )
% 0.17/0.47 = A ) ).
% 0.17/0.47
% 0.17/0.47 % append.right_neutral
% 0.17/0.47 thf(fact_36_append__is__Nil__conv,axiom,
% 0.17/0.47 ! [Xs: list_nat,Ys: list_nat] :
% 0.17/0.47 ( ( ( append_nat @ Xs @ Ys )
% 0.17/0.47 = nil_nat )
% 0.17/0.47 = ( ( Xs = nil_nat )
% 0.17/0.47 & ( Ys = nil_nat ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_is_Nil_conv
% 0.17/0.47 thf(fact_37_append__is__Nil__conv,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.47 ( ( ( append_a @ Xs @ Ys )
% 0.17/0.47 = nil_a )
% 0.17/0.47 = ( ( Xs = nil_a )
% 0.17/0.47 & ( Ys = nil_a ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_is_Nil_conv
% 0.17/0.47 thf(fact_38_Nil__is__append__conv,axiom,
% 0.17/0.47 ! [Xs: list_nat,Ys: list_nat] :
% 0.17/0.47 ( ( nil_nat
% 0.17/0.47 = ( append_nat @ Xs @ Ys ) )
% 0.17/0.47 = ( ( Xs = nil_nat )
% 0.17/0.47 & ( Ys = nil_nat ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % Nil_is_append_conv
% 0.17/0.47 thf(fact_39_Nil__is__append__conv,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.47 ( ( nil_a
% 0.17/0.47 = ( append_a @ Xs @ Ys ) )
% 0.17/0.47 = ( ( Xs = nil_a )
% 0.17/0.47 & ( Ys = nil_a ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % Nil_is_append_conv
% 0.17/0.47 thf(fact_40_self__append__conv2,axiom,
% 0.17/0.47 ! [Ys: list_nat,Xs: list_nat] :
% 0.17/0.47 ( ( Ys
% 0.17/0.47 = ( append_nat @ Xs @ Ys ) )
% 0.17/0.47 = ( Xs = nil_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % self_append_conv2
% 0.17/0.47 thf(fact_41_self__append__conv2,axiom,
% 0.17/0.47 ! [Ys: list_a,Xs: list_a] :
% 0.17/0.47 ( ( Ys
% 0.17/0.47 = ( append_a @ Xs @ Ys ) )
% 0.17/0.47 = ( Xs = nil_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % self_append_conv2
% 0.17/0.47 thf(fact_42_append__self__conv2,axiom,
% 0.17/0.47 ! [Xs: list_nat,Ys: list_nat] :
% 0.17/0.47 ( ( ( append_nat @ Xs @ Ys )
% 0.17/0.47 = Ys )
% 0.17/0.47 = ( Xs = nil_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_self_conv2
% 0.17/0.47 thf(fact_43_append__self__conv2,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.47 ( ( ( append_a @ Xs @ Ys )
% 0.17/0.47 = Ys )
% 0.17/0.47 = ( Xs = nil_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_self_conv2
% 0.17/0.47 thf(fact_44_self__append__conv,axiom,
% 0.17/0.47 ! [Xs: list_nat,Ys: list_nat] :
% 0.17/0.47 ( ( Xs
% 0.17/0.47 = ( append_nat @ Xs @ Ys ) )
% 0.17/0.47 = ( Ys = nil_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % self_append_conv
% 0.17/0.47 thf(fact_45_self__append__conv,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.47 ( ( Xs
% 0.17/0.47 = ( append_a @ Xs @ Ys ) )
% 0.17/0.47 = ( Ys = nil_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % self_append_conv
% 0.17/0.47 thf(fact_46_append__self__conv,axiom,
% 0.17/0.47 ! [Xs: list_nat,Ys: list_nat] :
% 0.17/0.47 ( ( ( append_nat @ Xs @ Ys )
% 0.17/0.47 = Xs )
% 0.17/0.47 = ( Ys = nil_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_self_conv
% 0.17/0.47 thf(fact_47_append__self__conv,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.47 ( ( ( append_a @ Xs @ Ys )
% 0.17/0.47 = Xs )
% 0.17/0.47 = ( Ys = nil_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % append_self_conv
% 0.17/0.47 thf(fact_48_append__Nil2,axiom,
% 0.17/0.47 ! [Xs: list_nat] :
% 0.17/0.47 ( ( append_nat @ Xs @ nil_nat )
% 0.17/0.47 = Xs ) ).
% 0.17/0.47
% 0.17/0.47 % append_Nil2
% 0.17/0.47 thf(fact_49_append__Nil2,axiom,
% 0.17/0.47 ! [Xs: list_a] :
% 0.17/0.47 ( ( append_a @ Xs @ nil_a )
% 0.17/0.47 = Xs ) ).
% 0.17/0.47
% 0.17/0.47 % append_Nil2
% 0.17/0.47 thf(fact_50_nth__Cons__0,axiom,
% 0.17/0.47 ! [X: a,Xs: list_a] :
% 0.17/0.47 ( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
% 0.17/0.47 = X ) ).
% 0.17/0.47
% 0.17/0.47 % nth_Cons_0
% 0.17/0.47 thf(fact_51_nth__Cons__0,axiom,
% 0.17/0.47 ! [X: nat,Xs: list_nat] :
% 0.17/0.47 ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
% 0.17/0.47 = X ) ).
% 0.17/0.47
% 0.17/0.47 % nth_Cons_0
% 0.17/0.47 thf(fact_52_take0,axiom,
% 0.17/0.47 ( ( take_a @ zero_zero_nat )
% 0.17/0.47 = ( ^ [Xs2: list_a] : nil_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % take0
% 0.17/0.47 thf(fact_53_take__eq__Nil,axiom,
% 0.17/0.47 ! [N: nat,Xs: list_a] :
% 0.17/0.47 ( ( ( take_a @ N @ Xs )
% 0.17/0.47 = nil_a )
% 0.17/0.47 = ( ( N = zero_zero_nat )
% 0.17/0.47 | ( Xs = nil_a ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % take_eq_Nil
% 0.17/0.47 thf(fact_54_empty__replicate,axiom,
% 0.17/0.47 ! [N: nat,X: a] :
% 0.17/0.47 ( ( nil_a
% 0.17/0.47 = ( replicate_a @ N @ X ) )
% 0.17/0.47 = ( N = zero_zero_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % empty_replicate
% 0.17/0.47 thf(fact_55_empty__replicate,axiom,
% 0.17/0.47 ! [N: nat,X: nat] :
% 0.17/0.47 ( ( nil_nat
% 0.17/0.47 = ( replicate_nat @ N @ X ) )
% 0.17/0.47 = ( N = zero_zero_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % empty_replicate
% 0.17/0.47 thf(fact_56_replicate__empty,axiom,
% 0.17/0.47 ! [N: nat,X: a] :
% 0.17/0.47 ( ( ( replicate_a @ N @ X )
% 0.17/0.47 = nil_a )
% 0.17/0.47 = ( N = zero_zero_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % replicate_empty
% 0.17/0.47 thf(fact_57_replicate__empty,axiom,
% 0.17/0.47 ! [N: nat,X: nat] :
% 0.17/0.47 ( ( ( replicate_nat @ N @ X )
% 0.17/0.47 = nil_nat )
% 0.17/0.47 = ( N = zero_zero_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % replicate_empty
% 0.17/0.47 thf(fact_58_last__replicate,axiom,
% 0.17/0.47 ! [N: nat,X: a] :
% 0.17/0.47 ( ( N != zero_zero_nat )
% 0.17/0.47 => ( ( last_a @ ( replicate_a @ N @ X ) )
% 0.17/0.47 = X ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_replicate
% 0.17/0.47 thf(fact_59_last__replicate,axiom,
% 0.17/0.47 ! [N: nat,X: nat] :
% 0.17/0.47 ( ( N != zero_zero_nat )
% 0.17/0.47 => ( ( last_nat @ ( replicate_nat @ N @ X ) )
% 0.17/0.47 = X ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_replicate
% 0.17/0.47 thf(fact_60_append1__eq__conv,axiom,
% 0.17/0.47 ! [Xs: list_a,X: a,Ys: list_a,Y: a] :
% 0.17/0.47 ( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
% 0.17/0.47 = ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
% 0.17/0.47 = ( ( Xs = Ys )
% 0.17/0.47 & ( X = Y ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append1_eq_conv
% 0.17/0.47 thf(fact_61_append1__eq__conv,axiom,
% 0.17/0.47 ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
% 0.17/0.47 ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
% 0.17/0.47 = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
% 0.17/0.47 = ( ( Xs = Ys )
% 0.17/0.47 & ( X = Y ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % append1_eq_conv
% 0.17/0.47 thf(fact_62_nth__take,axiom,
% 0.17/0.47 ! [I: nat,N: nat,Xs: list_a] :
% 0.17/0.47 ( ( ord_less_nat @ I @ N )
% 0.17/0.47 => ( ( nth_a @ ( take_a @ N @ Xs ) @ I )
% 0.17/0.47 = ( nth_a @ Xs @ I ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % nth_take
% 0.17/0.47 thf(fact_63_last__appendL,axiom,
% 0.17/0.47 ! [Ys: list_nat,Xs: list_nat] :
% 0.17/0.47 ( ( Ys = nil_nat )
% 0.17/0.47 => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
% 0.17/0.47 = ( last_nat @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_appendL
% 0.17/0.47 thf(fact_64_last__appendL,axiom,
% 0.17/0.47 ! [Ys: list_a,Xs: list_a] :
% 0.17/0.47 ( ( Ys = nil_a )
% 0.17/0.47 => ( ( last_a @ ( append_a @ Xs @ Ys ) )
% 0.17/0.47 = ( last_a @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_appendL
% 0.17/0.47 thf(fact_65_last__appendR,axiom,
% 0.17/0.47 ! [Ys: list_nat,Xs: list_nat] :
% 0.17/0.47 ( ( Ys != nil_nat )
% 0.17/0.47 => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
% 0.17/0.47 = ( last_nat @ Ys ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_appendR
% 0.17/0.47 thf(fact_66_last__appendR,axiom,
% 0.17/0.47 ! [Ys: list_a,Xs: list_a] :
% 0.17/0.47 ( ( Ys != nil_a )
% 0.17/0.47 => ( ( last_a @ ( append_a @ Xs @ Ys ) )
% 0.17/0.47 = ( last_a @ Ys ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_appendR
% 0.17/0.47 thf(fact_67_nth__replicate,axiom,
% 0.17/0.47 ! [I: nat,N: nat,X: a] :
% 0.17/0.47 ( ( ord_less_nat @ I @ N )
% 0.17/0.47 => ( ( nth_a @ ( replicate_a @ N @ X ) @ I )
% 0.17/0.47 = X ) ) ).
% 0.17/0.47
% 0.17/0.47 % nth_replicate
% 0.17/0.47 thf(fact_68_nth__replicate,axiom,
% 0.17/0.47 ! [I: nat,N: nat,X: nat] :
% 0.17/0.47 ( ( ord_less_nat @ I @ N )
% 0.17/0.47 => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
% 0.17/0.47 = X ) ) ).
% 0.17/0.47
% 0.17/0.47 % nth_replicate
% 0.17/0.47 thf(fact_69_last__snoc,axiom,
% 0.17/0.47 ! [Xs: list_a,X: a] :
% 0.17/0.47 ( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
% 0.17/0.47 = X ) ).
% 0.17/0.47
% 0.17/0.47 % last_snoc
% 0.17/0.47 thf(fact_70_last__snoc,axiom,
% 0.17/0.47 ! [Xs: list_nat,X: nat] :
% 0.17/0.47 ( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
% 0.17/0.47 = X ) ).
% 0.17/0.47
% 0.17/0.47 % last_snoc
% 0.17/0.47 thf(fact_71_take__0,axiom,
% 0.17/0.47 ! [Xs: list_a] :
% 0.17/0.47 ( ( take_a @ zero_zero_nat @ Xs )
% 0.17/0.47 = nil_a ) ).
% 0.17/0.47
% 0.17/0.47 % take_0
% 0.17/0.47 thf(fact_72_last_Osimps,axiom,
% 0.17/0.47 ! [Xs: list_a,X: a] :
% 0.17/0.47 ( ( ( Xs = nil_a )
% 0.17/0.47 => ( ( last_a @ ( cons_a @ X @ Xs ) )
% 0.17/0.47 = X ) )
% 0.17/0.47 & ( ( Xs != nil_a )
% 0.17/0.47 => ( ( last_a @ ( cons_a @ X @ Xs ) )
% 0.17/0.47 = ( last_a @ Xs ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last.simps
% 0.17/0.47 thf(fact_73_last_Osimps,axiom,
% 0.17/0.47 ! [Xs: list_nat,X: nat] :
% 0.17/0.47 ( ( ( Xs = nil_nat )
% 0.17/0.47 => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
% 0.17/0.47 = X ) )
% 0.17/0.47 & ( ( Xs != nil_nat )
% 0.17/0.47 => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
% 0.17/0.47 = ( last_nat @ Xs ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last.simps
% 0.17/0.47 thf(fact_74_last__ConsL,axiom,
% 0.17/0.47 ! [Xs: list_a,X: a] :
% 0.17/0.47 ( ( Xs = nil_a )
% 0.17/0.47 => ( ( last_a @ ( cons_a @ X @ Xs ) )
% 0.17/0.47 = X ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_ConsL
% 0.17/0.47 thf(fact_75_last__ConsL,axiom,
% 0.17/0.47 ! [Xs: list_nat,X: nat] :
% 0.17/0.47 ( ( Xs = nil_nat )
% 0.17/0.47 => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
% 0.17/0.47 = X ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_ConsL
% 0.17/0.47 thf(fact_76_last__ConsR,axiom,
% 0.17/0.47 ! [Xs: list_a,X: a] :
% 0.17/0.47 ( ( Xs != nil_a )
% 0.17/0.47 => ( ( last_a @ ( cons_a @ X @ Xs ) )
% 0.17/0.47 = ( last_a @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_ConsR
% 0.17/0.47 thf(fact_77_last__ConsR,axiom,
% 0.17/0.47 ! [Xs: list_nat,X: nat] :
% 0.17/0.47 ( ( Xs != nil_nat )
% 0.17/0.47 => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
% 0.17/0.47 = ( last_nat @ Xs ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_ConsR
% 0.17/0.47 thf(fact_78_last__append,axiom,
% 0.17/0.47 ! [Ys: list_nat,Xs: list_nat] :
% 0.17/0.47 ( ( ( Ys = nil_nat )
% 0.17/0.47 => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
% 0.17/0.47 = ( last_nat @ Xs ) ) )
% 0.17/0.47 & ( ( Ys != nil_nat )
% 0.17/0.47 => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
% 0.17/0.47 = ( last_nat @ Ys ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_append
% 0.17/0.47 thf(fact_79_last__append,axiom,
% 0.17/0.47 ! [Ys: list_a,Xs: list_a] :
% 0.17/0.47 ( ( ( Ys = nil_a )
% 0.17/0.47 => ( ( last_a @ ( append_a @ Xs @ Ys ) )
% 0.17/0.47 = ( last_a @ Xs ) ) )
% 0.17/0.47 & ( ( Ys != nil_a )
% 0.17/0.47 => ( ( last_a @ ( append_a @ Xs @ Ys ) )
% 0.17/0.47 = ( last_a @ Ys ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % last_append
% 0.17/0.47 thf(fact_80_take__Nil,axiom,
% 0.17/0.47 ! [N: nat] :
% 0.17/0.47 ( ( take_a @ N @ nil_a )
% 0.17/0.47 = nil_a ) ).
% 0.17/0.47
% 0.17/0.47 % take_Nil
% 0.17/0.47 thf(fact_81_take__equalityI,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.47 ( ! [I2: nat] :
% 0.17/0.47 ( ( take_a @ I2 @ Xs )
% 0.17/0.47 = ( take_a @ I2 @ Ys ) )
% 0.17/0.47 => ( Xs = Ys ) ) ).
% 0.17/0.47
% 0.17/0.47 % take_equalityI
% 0.17/0.47 thf(fact_82_transpose_Ocases,axiom,
% 0.17/0.47 ! [X: list_list_a] :
% 0.17/0.47 ( ( X != nil_list_a )
% 0.17/0.47 => ( ! [Xss: list_list_a] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_list_a @ nil_a @ Xss ) )
% 0.17/0.47 => ~ ! [X2: a,Xs3: list_a,Xss: list_list_a] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_list_a @ ( cons_a @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % transpose.cases
% 0.17/0.47 thf(fact_83_transpose_Ocases,axiom,
% 0.17/0.47 ! [X: list_list_nat] :
% 0.17/0.47 ( ( X != nil_list_nat )
% 0.17/0.47 => ( ! [Xss: list_list_nat] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_list_nat @ nil_nat @ Xss ) )
% 0.17/0.47 => ~ ! [X2: nat,Xs3: list_nat,Xss: list_list_nat] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_list_nat @ ( cons_nat @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % transpose.cases
% 0.17/0.47 thf(fact_84_longest__common__suffix,axiom,
% 0.17/0.47 ! [Xs: list_nat,Ys: list_nat] :
% 0.17/0.47 ? [Ss: list_nat,Xs4: list_nat,Ys2: list_nat] :
% 0.17/0.47 ( ( Xs
% 0.17/0.47 = ( append_nat @ Xs4 @ Ss ) )
% 0.17/0.47 & ( Ys
% 0.17/0.47 = ( append_nat @ Ys2 @ Ss ) )
% 0.17/0.47 & ( ( Xs4 = nil_nat )
% 0.17/0.47 | ( Ys2 = nil_nat )
% 0.17/0.47 | ( ( last_nat @ Xs4 )
% 0.17/0.47 != ( last_nat @ Ys2 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % longest_common_suffix
% 0.17/0.47 thf(fact_85_longest__common__suffix,axiom,
% 0.17/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.47 ? [Ss: list_a,Xs4: list_a,Ys2: list_a] :
% 0.17/0.47 ( ( Xs
% 0.17/0.47 = ( append_a @ Xs4 @ Ss ) )
% 0.17/0.47 & ( Ys
% 0.17/0.47 = ( append_a @ Ys2 @ Ss ) )
% 0.17/0.47 & ( ( Xs4 = nil_a )
% 0.17/0.47 | ( Ys2 = nil_a )
% 0.17/0.47 | ( ( last_a @ Xs4 )
% 0.17/0.47 != ( last_a @ Ys2 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % longest_common_suffix
% 0.17/0.47 thf(fact_86_replicate__0,axiom,
% 0.17/0.47 ! [X: a] :
% 0.17/0.47 ( ( replicate_a @ zero_zero_nat @ X )
% 0.17/0.47 = nil_a ) ).
% 0.17/0.47
% 0.17/0.47 % replicate_0
% 0.17/0.47 thf(fact_87_replicate__0,axiom,
% 0.17/0.47 ! [X: nat] :
% 0.17/0.47 ( ( replicate_nat @ zero_zero_nat @ X )
% 0.17/0.47 = nil_nat ) ).
% 0.17/0.47
% 0.17/0.47 % replicate_0
% 0.17/0.47 thf(fact_88_zero__less__iff__neq__zero,axiom,
% 0.17/0.47 ! [N: nat] :
% 0.17/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.47 = ( N != zero_zero_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % zero_less_iff_neq_zero
% 0.17/0.47 thf(fact_89_gr__implies__not__zero,axiom,
% 0.17/0.47 ! [M: nat,N: nat] :
% 0.17/0.47 ( ( ord_less_nat @ M @ N )
% 0.17/0.47 => ( N != zero_zero_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % gr_implies_not_zero
% 0.17/0.47 thf(fact_90_not__less__zero,axiom,
% 0.17/0.47 ! [N: nat] :
% 0.17/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.17/0.47
% 0.17/0.47 % not_less_zero
% 0.17/0.47 thf(fact_91_gr__zeroI,axiom,
% 0.17/0.47 ! [N: nat] :
% 0.17/0.47 ( ( N != zero_zero_nat )
% 0.17/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.17/0.47
% 0.17/0.47 % gr_zeroI
% 0.17/0.47 thf(fact_92_zero__le,axiom,
% 0.17/0.47 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.17/0.47
% 0.17/0.47 % zero_le
% 0.17/0.47 thf(fact_93_strict__sorted_Oinduct,axiom,
% 0.17/0.47 ! [P: list_a > $o,A0: list_a] :
% 0.17/0.47 ( ( P @ nil_a )
% 0.17/0.47 => ( ! [X2: a,Ys3: list_a] :
% 0.17/0.47 ( ( P @ Ys3 )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ Ys3 ) ) )
% 0.17/0.47 => ( P @ A0 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % strict_sorted.induct
% 0.17/0.47 thf(fact_94_strict__sorted_Oinduct,axiom,
% 0.17/0.47 ! [P: list_nat > $o,A0: list_nat] :
% 0.17/0.47 ( ( P @ nil_nat )
% 0.17/0.47 => ( ! [X2: nat,Ys3: list_nat] :
% 0.17/0.47 ( ( P @ Ys3 )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ Ys3 ) ) )
% 0.17/0.47 => ( P @ A0 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % strict_sorted.induct
% 0.17/0.47 thf(fact_95_strict__sorted_Ocases,axiom,
% 0.17/0.47 ! [X: list_a] :
% 0.17/0.47 ( ( X != nil_a )
% 0.17/0.47 => ~ ! [X2: a,Ys3: list_a] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_a @ X2 @ Ys3 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % strict_sorted.cases
% 0.17/0.47 thf(fact_96_strict__sorted_Ocases,axiom,
% 0.17/0.47 ! [X: list_nat] :
% 0.17/0.47 ( ( X != nil_nat )
% 0.17/0.47 => ~ ! [X2: nat,Ys3: list_nat] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_nat @ X2 @ Ys3 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % strict_sorted.cases
% 0.17/0.47 thf(fact_97_map__tailrec__rev_Oinduct,axiom,
% 0.17/0.47 ! [P: ( a > a ) > list_a > list_a > $o,A0: a > a,A1: list_a,A2: list_a] :
% 0.17/0.47 ( ! [F: a > a,X_1: list_a] : ( P @ F @ nil_a @ X_1 )
% 0.17/0.47 => ( ! [F: a > a,A3: a,As: list_a,Bs: list_a] :
% 0.17/0.47 ( ( P @ F @ As @ ( cons_a @ ( F @ A3 ) @ Bs ) )
% 0.17/0.47 => ( P @ F @ ( cons_a @ A3 @ As ) @ Bs ) )
% 0.17/0.47 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % map_tailrec_rev.induct
% 0.17/0.47 thf(fact_98_map__tailrec__rev_Oinduct,axiom,
% 0.17/0.47 ! [P: ( nat > a ) > list_nat > list_a > $o,A0: nat > a,A1: list_nat,A2: list_a] :
% 0.17/0.47 ( ! [F: nat > a,X_1: list_a] : ( P @ F @ nil_nat @ X_1 )
% 0.17/0.47 => ( ! [F: nat > a,A3: nat,As: list_nat,Bs: list_a] :
% 0.17/0.47 ( ( P @ F @ As @ ( cons_a @ ( F @ A3 ) @ Bs ) )
% 0.17/0.47 => ( P @ F @ ( cons_nat @ A3 @ As ) @ Bs ) )
% 0.17/0.47 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % map_tailrec_rev.induct
% 0.17/0.47 thf(fact_99_map__tailrec__rev_Oinduct,axiom,
% 0.17/0.47 ! [P: ( a > nat ) > list_a > list_nat > $o,A0: a > nat,A1: list_a,A2: list_nat] :
% 0.17/0.47 ( ! [F: a > nat,X_1: list_nat] : ( P @ F @ nil_a @ X_1 )
% 0.17/0.47 => ( ! [F: a > nat,A3: a,As: list_a,Bs: list_nat] :
% 0.17/0.47 ( ( P @ F @ As @ ( cons_nat @ ( F @ A3 ) @ Bs ) )
% 0.17/0.47 => ( P @ F @ ( cons_a @ A3 @ As ) @ Bs ) )
% 0.17/0.47 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % map_tailrec_rev.induct
% 0.17/0.47 thf(fact_100_map__tailrec__rev_Oinduct,axiom,
% 0.17/0.47 ! [P: ( nat > nat ) > list_nat > list_nat > $o,A0: nat > nat,A1: list_nat,A2: list_nat] :
% 0.17/0.47 ( ! [F: nat > nat,X_1: list_nat] : ( P @ F @ nil_nat @ X_1 )
% 0.17/0.47 => ( ! [F: nat > nat,A3: nat,As: list_nat,Bs: list_nat] :
% 0.17/0.47 ( ( P @ F @ As @ ( cons_nat @ ( F @ A3 ) @ Bs ) )
% 0.17/0.47 => ( P @ F @ ( cons_nat @ A3 @ As ) @ Bs ) )
% 0.17/0.47 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % map_tailrec_rev.induct
% 0.17/0.47 thf(fact_101_list__nonempty__induct,axiom,
% 0.17/0.47 ! [Xs: list_a,P: list_a > $o] :
% 0.17/0.47 ( ( Xs != nil_a )
% 0.17/0.47 => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
% 0.17/0.47 => ( ! [X2: a,Xs3: list_a] :
% 0.17/0.47 ( ( Xs3 != nil_a )
% 0.17/0.47 => ( ( P @ Xs3 )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ Xs3 ) ) ) )
% 0.17/0.47 => ( P @ Xs ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % list_nonempty_induct
% 0.17/0.47 thf(fact_102_list__nonempty__induct,axiom,
% 0.17/0.47 ! [Xs: list_nat,P: list_nat > $o] :
% 0.17/0.47 ( ( Xs != nil_nat )
% 0.17/0.47 => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
% 0.17/0.47 => ( ! [X2: nat,Xs3: list_nat] :
% 0.17/0.47 ( ( Xs3 != nil_nat )
% 0.17/0.47 => ( ( P @ Xs3 )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ Xs3 ) ) ) )
% 0.17/0.47 => ( P @ Xs ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % list_nonempty_induct
% 0.17/0.47 thf(fact_103_successively_Oinduct,axiom,
% 0.17/0.47 ! [P: ( a > a > $o ) > list_a > $o,A0: a > a > $o,A1: list_a] :
% 0.17/0.47 ( ! [P2: a > a > $o] : ( P @ P2 @ nil_a )
% 0.17/0.47 => ( ! [P2: a > a > $o,X2: a] : ( P @ P2 @ ( cons_a @ X2 @ nil_a ) )
% 0.17/0.47 => ( ! [P2: a > a > $o,X2: a,Y2: a,Xs3: list_a] :
% 0.17/0.47 ( ( P @ P2 @ ( cons_a @ Y2 @ Xs3 ) )
% 0.17/0.47 => ( P @ P2 @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) )
% 0.17/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % successively.induct
% 0.17/0.47 thf(fact_104_successively_Oinduct,axiom,
% 0.17/0.47 ! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
% 0.17/0.47 ( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
% 0.17/0.47 => ( ! [P2: nat > nat > $o,X2: nat] : ( P @ P2 @ ( cons_nat @ X2 @ nil_nat ) )
% 0.17/0.47 => ( ! [P2: nat > nat > $o,X2: nat,Y2: nat,Xs3: list_nat] :
% 0.17/0.47 ( ( P @ P2 @ ( cons_nat @ Y2 @ Xs3 ) )
% 0.17/0.47 => ( P @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) )
% 0.17/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % successively.induct
% 0.17/0.47 thf(fact_105_remdups__adj_Oinduct,axiom,
% 0.17/0.47 ! [P: list_a > $o,A0: list_a] :
% 0.17/0.47 ( ( P @ nil_a )
% 0.17/0.47 => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
% 0.17/0.47 => ( ! [X2: a,Y2: a,Xs3: list_a] :
% 0.17/0.47 ( ( ( X2 = Y2 )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ Xs3 ) ) )
% 0.17/0.47 => ( ( ( X2 != Y2 )
% 0.17/0.47 => ( P @ ( cons_a @ Y2 @ Xs3 ) ) )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) )
% 0.17/0.47 => ( P @ A0 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % remdups_adj.induct
% 0.17/0.47 thf(fact_106_remdups__adj_Oinduct,axiom,
% 0.17/0.47 ! [P: list_nat > $o,A0: list_nat] :
% 0.17/0.47 ( ( P @ nil_nat )
% 0.17/0.47 => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
% 0.17/0.47 => ( ! [X2: nat,Y2: nat,Xs3: list_nat] :
% 0.17/0.47 ( ( ( X2 = Y2 )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ Xs3 ) ) )
% 0.17/0.47 => ( ( ( X2 != Y2 )
% 0.17/0.47 => ( P @ ( cons_nat @ Y2 @ Xs3 ) ) )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) )
% 0.17/0.47 => ( P @ A0 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % remdups_adj.induct
% 0.17/0.47 thf(fact_107_sorted__wrt_Oinduct,axiom,
% 0.17/0.47 ! [P: ( a > a > $o ) > list_a > $o,A0: a > a > $o,A1: list_a] :
% 0.17/0.47 ( ! [P2: a > a > $o] : ( P @ P2 @ nil_a )
% 0.17/0.47 => ( ! [P2: a > a > $o,X2: a,Ys3: list_a] :
% 0.17/0.47 ( ( P @ P2 @ Ys3 )
% 0.17/0.47 => ( P @ P2 @ ( cons_a @ X2 @ Ys3 ) ) )
% 0.17/0.47 => ( P @ A0 @ A1 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % sorted_wrt.induct
% 0.17/0.47 thf(fact_108_sorted__wrt_Oinduct,axiom,
% 0.17/0.47 ! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
% 0.17/0.47 ( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
% 0.17/0.47 => ( ! [P2: nat > nat > $o,X2: nat,Ys3: list_nat] :
% 0.17/0.47 ( ( P @ P2 @ Ys3 )
% 0.17/0.47 => ( P @ P2 @ ( cons_nat @ X2 @ Ys3 ) ) )
% 0.17/0.47 => ( P @ A0 @ A1 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % sorted_wrt.induct
% 0.17/0.47 thf(fact_109_remdups__adj_Ocases,axiom,
% 0.17/0.47 ! [X: list_a] :
% 0.17/0.47 ( ( X != nil_a )
% 0.17/0.47 => ( ! [X2: a] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_a @ X2 @ nil_a ) )
% 0.17/0.47 => ~ ! [X2: a,Y2: a,Xs3: list_a] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % remdups_adj.cases
% 0.17/0.47 thf(fact_110_remdups__adj_Ocases,axiom,
% 0.17/0.47 ! [X: list_nat] :
% 0.17/0.47 ( ( X != nil_nat )
% 0.17/0.47 => ( ! [X2: nat] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_nat @ X2 @ nil_nat ) )
% 0.17/0.47 => ~ ! [X2: nat,Y2: nat,Xs3: list_nat] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % remdups_adj.cases
% 0.17/0.47 thf(fact_111_shuffles_Oinduct,axiom,
% 0.17/0.47 ! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
% 0.17/0.47 ( ! [X_1: list_a] : ( P @ nil_a @ X_1 )
% 0.17/0.47 => ( ! [Xs3: list_a] : ( P @ Xs3 @ nil_a )
% 0.17/0.47 => ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
% 0.17/0.47 ( ( P @ Xs3 @ ( cons_a @ Y2 @ Ys3 ) )
% 0.17/0.47 => ( ( P @ ( cons_a @ X2 @ Xs3 ) @ Ys3 )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) ) )
% 0.17/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % shuffles.induct
% 0.17/0.47 thf(fact_112_shuffles_Oinduct,axiom,
% 0.17/0.47 ! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
% 0.17/0.47 ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
% 0.17/0.47 => ( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
% 0.17/0.47 => ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys3: list_nat] :
% 0.17/0.47 ( ( P @ Xs3 @ ( cons_nat @ Y2 @ Ys3 ) )
% 0.17/0.47 => ( ( P @ ( cons_nat @ X2 @ Xs3 ) @ Ys3 )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys3 ) ) ) )
% 0.17/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % shuffles.induct
% 0.17/0.47 thf(fact_113_min__list_Oinduct,axiom,
% 0.17/0.47 ! [P: list_a > $o,A0: list_a] :
% 0.17/0.47 ( ! [X2: a,Xs3: list_a] :
% 0.17/0.47 ( ! [X212: a,X222: list_a] :
% 0.17/0.47 ( ( Xs3
% 0.17/0.47 = ( cons_a @ X212 @ X222 ) )
% 0.17/0.47 => ( P @ Xs3 ) )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ Xs3 ) ) )
% 0.17/0.47 => ( ( P @ nil_a )
% 0.17/0.47 => ( P @ A0 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % min_list.induct
% 0.17/0.47 thf(fact_114_min__list_Oinduct,axiom,
% 0.17/0.47 ! [P: list_nat > $o,A0: list_nat] :
% 0.17/0.47 ( ! [X2: nat,Xs3: list_nat] :
% 0.17/0.47 ( ! [X212: nat,X222: list_nat] :
% 0.17/0.47 ( ( Xs3
% 0.17/0.47 = ( cons_nat @ X212 @ X222 ) )
% 0.17/0.47 => ( P @ Xs3 ) )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ Xs3 ) ) )
% 0.17/0.47 => ( ( P @ nil_nat )
% 0.17/0.47 => ( P @ A0 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % min_list.induct
% 0.17/0.47 thf(fact_115_min__list_Ocases,axiom,
% 0.17/0.47 ! [X: list_a] :
% 0.17/0.47 ( ! [X2: a,Xs3: list_a] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_a @ X2 @ Xs3 ) )
% 0.17/0.47 => ( X = nil_a ) ) ).
% 0.17/0.47
% 0.17/0.47 % min_list.cases
% 0.17/0.47 thf(fact_116_min__list_Ocases,axiom,
% 0.17/0.47 ! [X: list_nat] :
% 0.17/0.47 ( ! [X2: nat,Xs3: list_nat] :
% 0.17/0.47 ( X
% 0.17/0.47 != ( cons_nat @ X2 @ Xs3 ) )
% 0.17/0.47 => ( X = nil_nat ) ) ).
% 0.17/0.47
% 0.17/0.47 % min_list.cases
% 0.17/0.47 thf(fact_117_induct__list012,axiom,
% 0.17/0.47 ! [P: list_a > $o,Xs: list_a] :
% 0.17/0.47 ( ( P @ nil_a )
% 0.17/0.47 => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
% 0.17/0.47 => ( ! [X2: a,Y2: a,Zs2: list_a] :
% 0.17/0.47 ( ( P @ Zs2 )
% 0.17/0.47 => ( ( P @ ( cons_a @ Y2 @ Zs2 ) )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Zs2 ) ) ) ) )
% 0.17/0.47 => ( P @ Xs ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % induct_list012
% 0.17/0.47 thf(fact_118_induct__list012,axiom,
% 0.17/0.47 ! [P: list_nat > $o,Xs: list_nat] :
% 0.17/0.47 ( ( P @ nil_nat )
% 0.17/0.47 => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
% 0.17/0.47 => ( ! [X2: nat,Y2: nat,Zs2: list_nat] :
% 0.17/0.47 ( ( P @ Zs2 )
% 0.17/0.47 => ( ( P @ ( cons_nat @ Y2 @ Zs2 ) )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Zs2 ) ) ) ) )
% 0.17/0.47 => ( P @ Xs ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % induct_list012
% 0.17/0.47 thf(fact_119_splice_Oinduct,axiom,
% 0.17/0.47 ! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
% 0.17/0.47 ( ! [X_1: list_a] : ( P @ nil_a @ X_1 )
% 0.17/0.47 => ( ! [X2: a,Xs3: list_a,Ys3: list_a] :
% 0.17/0.47 ( ( P @ Ys3 @ Xs3 )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ Xs3 ) @ Ys3 ) )
% 0.17/0.47 => ( P @ A0 @ A1 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % splice.induct
% 0.17/0.47 thf(fact_120_splice_Oinduct,axiom,
% 0.17/0.47 ! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
% 0.17/0.47 ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
% 0.17/0.47 => ( ! [X2: nat,Xs3: list_nat,Ys3: list_nat] :
% 0.17/0.47 ( ( P @ Ys3 @ Xs3 )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ Ys3 ) )
% 0.17/0.47 => ( P @ A0 @ A1 ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % splice.induct
% 0.17/0.47 thf(fact_121_list__induct2_H,axiom,
% 0.17/0.47 ! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
% 0.17/0.47 ( ( P @ nil_a @ nil_a )
% 0.17/0.47 => ( ! [X2: a,Xs3: list_a] : ( P @ ( cons_a @ X2 @ Xs3 ) @ nil_a )
% 0.17/0.47 => ( ! [Y2: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys3 ) )
% 0.17/0.47 => ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
% 0.17/0.47 ( ( P @ Xs3 @ Ys3 )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
% 0.17/0.47 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % list_induct2'
% 0.17/0.47 thf(fact_122_list__induct2_H,axiom,
% 0.17/0.47 ! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
% 0.17/0.47 ( ( P @ nil_a @ nil_nat )
% 0.17/0.47 => ( ! [X2: a,Xs3: list_a] : ( P @ ( cons_a @ X2 @ Xs3 ) @ nil_nat )
% 0.17/0.47 => ( ! [Y2: nat,Ys3: list_nat] : ( P @ nil_a @ ( cons_nat @ Y2 @ Ys3 ) )
% 0.17/0.47 => ( ! [X2: a,Xs3: list_a,Y2: nat,Ys3: list_nat] :
% 0.17/0.47 ( ( P @ Xs3 @ Ys3 )
% 0.17/0.47 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys3 ) ) )
% 0.17/0.47 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % list_induct2'
% 0.17/0.47 thf(fact_123_list__induct2_H,axiom,
% 0.17/0.47 ! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
% 0.17/0.47 ( ( P @ nil_nat @ nil_a )
% 0.17/0.47 => ( ! [X2: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X2 @ Xs3 ) @ nil_a )
% 0.17/0.47 => ( ! [Y2: a,Ys3: list_a] : ( P @ nil_nat @ ( cons_a @ Y2 @ Ys3 ) )
% 0.17/0.47 => ( ! [X2: nat,Xs3: list_nat,Y2: a,Ys3: list_a] :
% 0.17/0.47 ( ( P @ Xs3 @ Ys3 )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
% 0.17/0.47 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % list_induct2'
% 0.17/0.47 thf(fact_124_list__induct2_H,axiom,
% 0.17/0.47 ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
% 0.17/0.47 ( ( P @ nil_nat @ nil_nat )
% 0.17/0.47 => ( ! [X2: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X2 @ Xs3 ) @ nil_nat )
% 0.17/0.47 => ( ! [Y2: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys3 ) )
% 0.17/0.47 => ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys3: list_nat] :
% 0.17/0.47 ( ( P @ Xs3 @ Ys3 )
% 0.17/0.47 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys3 ) ) )
% 0.17/0.47 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % list_induct2'
% 0.17/0.47 thf(fact_125_neq__Nil__conv,axiom,
% 0.17/0.47 ! [Xs: list_a] :
% 0.17/0.47 ( ( Xs != nil_a )
% 0.17/0.47 = ( ? [Y3: a,Ys4: list_a] :
% 0.17/0.47 ( Xs
% 0.17/0.47 = ( cons_a @ Y3 @ Ys4 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % neq_Nil_conv
% 0.17/0.47 thf(fact_126_neq__Nil__conv,axiom,
% 0.17/0.47 ! [Xs: list_nat] :
% 0.17/0.47 ( ( Xs != nil_nat )
% 0.17/0.47 = ( ? [Y3: nat,Ys4: list_nat] :
% 0.17/0.47 ( Xs
% 0.17/0.47 = ( cons_nat @ Y3 @ Ys4 ) ) ) ) ).
% 0.17/0.47
% 0.17/0.47 % neq_Nil_conv
% 0.17/0.47 thf(fact_127_list_Oinducts,axiom,
% 0.17/0.47 ! [P: list_a > $o,List: list_a] :
% 0.17/0.48 ( ( P @ nil_a )
% 0.17/0.48 => ( ! [X1: a,X23: list_a] :
% 0.17/0.48 ( ( P @ X23 )
% 0.17/0.48 => ( P @ ( cons_a @ X1 @ X23 ) ) )
% 0.17/0.48 => ( P @ List ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % list.inducts
% 0.17/0.48 thf(fact_128_list_Oinducts,axiom,
% 0.17/0.48 ! [P: list_nat > $o,List: list_nat] :
% 0.17/0.48 ( ( P @ nil_nat )
% 0.17/0.48 => ( ! [X1: nat,X23: list_nat] :
% 0.17/0.48 ( ( P @ X23 )
% 0.17/0.48 => ( P @ ( cons_nat @ X1 @ X23 ) ) )
% 0.17/0.48 => ( P @ List ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % list.inducts
% 0.17/0.48 thf(fact_129_list_Oexhaust,axiom,
% 0.17/0.48 ! [Y: list_a] :
% 0.17/0.48 ( ( Y != nil_a )
% 0.17/0.48 => ~ ! [X213: a,X223: list_a] :
% 0.17/0.48 ( Y
% 0.17/0.48 != ( cons_a @ X213 @ X223 ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % list.exhaust
% 0.17/0.48 thf(fact_130_list_Oexhaust,axiom,
% 0.17/0.48 ! [Y: list_nat] :
% 0.17/0.48 ( ( Y != nil_nat )
% 0.17/0.48 => ~ ! [X213: nat,X223: list_nat] :
% 0.17/0.48 ( Y
% 0.17/0.48 != ( cons_nat @ X213 @ X223 ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % list.exhaust
% 0.17/0.48 thf(fact_131_list_OdiscI,axiom,
% 0.17/0.48 ! [List: list_a,X21: a,X22: list_a] :
% 0.17/0.48 ( ( List
% 0.17/0.48 = ( cons_a @ X21 @ X22 ) )
% 0.17/0.48 => ( List != nil_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % list.discI
% 0.17/0.48 thf(fact_132_list_OdiscI,axiom,
% 0.17/0.48 ! [List: list_nat,X21: nat,X22: list_nat] :
% 0.17/0.48 ( ( List
% 0.17/0.48 = ( cons_nat @ X21 @ X22 ) )
% 0.17/0.48 => ( List != nil_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % list.discI
% 0.17/0.48 thf(fact_133_list_Odistinct_I1_J,axiom,
% 0.17/0.48 ! [X21: a,X22: list_a] :
% 0.17/0.48 ( nil_a
% 0.17/0.48 != ( cons_a @ X21 @ X22 ) ) ).
% 0.17/0.48
% 0.17/0.48 % list.distinct(1)
% 0.17/0.48 thf(fact_134_list_Odistinct_I1_J,axiom,
% 0.17/0.48 ! [X21: nat,X22: list_nat] :
% 0.17/0.48 ( nil_nat
% 0.17/0.48 != ( cons_nat @ X21 @ X22 ) ) ).
% 0.17/0.48
% 0.17/0.48 % list.distinct(1)
% 0.17/0.48 thf(fact_135_append_Oleft__neutral,axiom,
% 0.17/0.48 ! [A: list_nat] :
% 0.17/0.48 ( ( append_nat @ nil_nat @ A )
% 0.17/0.48 = A ) ).
% 0.17/0.48
% 0.17/0.48 % append.left_neutral
% 0.17/0.48 thf(fact_136_append_Oleft__neutral,axiom,
% 0.17/0.48 ! [A: list_a] :
% 0.17/0.48 ( ( append_a @ nil_a @ A )
% 0.17/0.48 = A ) ).
% 0.17/0.48
% 0.17/0.48 % append.left_neutral
% 0.17/0.48 thf(fact_137_append__Nil,axiom,
% 0.17/0.48 ! [Ys: list_nat] :
% 0.17/0.48 ( ( append_nat @ nil_nat @ Ys )
% 0.17/0.48 = Ys ) ).
% 0.17/0.48
% 0.17/0.48 % append_Nil
% 0.17/0.48 thf(fact_138_append__Nil,axiom,
% 0.17/0.48 ! [Ys: list_a] :
% 0.17/0.48 ( ( append_a @ nil_a @ Ys )
% 0.17/0.48 = Ys ) ).
% 0.17/0.48
% 0.17/0.48 % append_Nil
% 0.17/0.48 thf(fact_139_eq__Nil__appendI,axiom,
% 0.17/0.48 ! [Xs: list_nat,Ys: list_nat] :
% 0.17/0.48 ( ( Xs = Ys )
% 0.17/0.48 => ( Xs
% 0.17/0.48 = ( append_nat @ nil_nat @ Ys ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % eq_Nil_appendI
% 0.17/0.48 thf(fact_140_eq__Nil__appendI,axiom,
% 0.17/0.48 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.48 ( ( Xs = Ys )
% 0.17/0.48 => ( Xs
% 0.17/0.48 = ( append_a @ nil_a @ Ys ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % eq_Nil_appendI
% 0.17/0.48 thf(fact_141_psums_Ocases,axiom,
% 0.17/0.48 ! [X: list_a] :
% 0.17/0.48 ( ( X != nil_a )
% 0.17/0.48 => ( ! [X2: a] :
% 0.17/0.48 ( X
% 0.17/0.48 != ( cons_a @ X2 @ nil_a ) )
% 0.17/0.48 => ~ ! [X2: a,Y2: a,Xs3: list_a] :
% 0.17/0.48 ( X
% 0.17/0.48 != ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % psums.cases
% 0.17/0.48 thf(fact_142_psums_Ocases,axiom,
% 0.17/0.48 ! [X: list_nat] :
% 0.17/0.48 ( ( X != nil_nat )
% 0.17/0.48 => ( ! [X2: nat] :
% 0.17/0.48 ( X
% 0.17/0.48 != ( cons_nat @ X2 @ nil_nat ) )
% 0.17/0.48 => ~ ! [X2: nat,Y2: nat,Xs3: list_nat] :
% 0.17/0.48 ( X
% 0.17/0.48 != ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % psums.cases
% 0.17/0.48 thf(fact_143_rev__nonempty__induct,axiom,
% 0.17/0.48 ! [Xs: list_a,P: list_a > $o] :
% 0.17/0.48 ( ( Xs != nil_a )
% 0.17/0.48 => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
% 0.17/0.48 => ( ! [X2: a,Xs3: list_a] :
% 0.17/0.48 ( ( Xs3 != nil_a )
% 0.17/0.48 => ( ( P @ Xs3 )
% 0.17/0.48 => ( P @ ( append_a @ Xs3 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
% 0.17/0.48 => ( P @ Xs ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % rev_nonempty_induct
% 0.17/0.48 thf(fact_144_rev__nonempty__induct,axiom,
% 0.17/0.48 ! [Xs: list_nat,P: list_nat > $o] :
% 0.17/0.48 ( ( Xs != nil_nat )
% 0.17/0.48 => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
% 0.17/0.48 => ( ! [X2: nat,Xs3: list_nat] :
% 0.17/0.48 ( ( Xs3 != nil_nat )
% 0.17/0.48 => ( ( P @ Xs3 )
% 0.17/0.48 => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
% 0.17/0.48 => ( P @ Xs ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % rev_nonempty_induct
% 0.17/0.48 thf(fact_145_append__eq__Cons__conv,axiom,
% 0.17/0.48 ! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
% 0.17/0.48 ( ( ( append_a @ Ys @ Zs )
% 0.17/0.48 = ( cons_a @ X @ Xs ) )
% 0.17/0.48 = ( ( ( Ys = nil_a )
% 0.17/0.48 & ( Zs
% 0.17/0.48 = ( cons_a @ X @ Xs ) ) )
% 0.17/0.48 | ? [Ys5: list_a] :
% 0.17/0.48 ( ( Ys
% 0.17/0.48 = ( cons_a @ X @ Ys5 ) )
% 0.17/0.48 & ( ( append_a @ Ys5 @ Zs )
% 0.17/0.48 = Xs ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_eq_Cons_conv
% 0.17/0.48 thf(fact_146_append__eq__Cons__conv,axiom,
% 0.17/0.48 ! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
% 0.17/0.48 ( ( ( append_nat @ Ys @ Zs )
% 0.17/0.48 = ( cons_nat @ X @ Xs ) )
% 0.17/0.48 = ( ( ( Ys = nil_nat )
% 0.17/0.48 & ( Zs
% 0.17/0.48 = ( cons_nat @ X @ Xs ) ) )
% 0.17/0.48 | ? [Ys5: list_nat] :
% 0.17/0.48 ( ( Ys
% 0.17/0.48 = ( cons_nat @ X @ Ys5 ) )
% 0.17/0.48 & ( ( append_nat @ Ys5 @ Zs )
% 0.17/0.48 = Xs ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_eq_Cons_conv
% 0.17/0.48 thf(fact_147_Cons__eq__append__conv,axiom,
% 0.17/0.48 ! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
% 0.17/0.48 ( ( ( cons_a @ X @ Xs )
% 0.17/0.48 = ( append_a @ Ys @ Zs ) )
% 0.17/0.48 = ( ( ( Ys = nil_a )
% 0.17/0.48 & ( ( cons_a @ X @ Xs )
% 0.17/0.48 = Zs ) )
% 0.17/0.48 | ? [Ys5: list_a] :
% 0.17/0.48 ( ( ( cons_a @ X @ Ys5 )
% 0.17/0.48 = Ys )
% 0.17/0.48 & ( Xs
% 0.17/0.48 = ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % Cons_eq_append_conv
% 0.17/0.48 thf(fact_148_Cons__eq__append__conv,axiom,
% 0.17/0.48 ! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
% 0.17/0.48 ( ( ( cons_nat @ X @ Xs )
% 0.17/0.48 = ( append_nat @ Ys @ Zs ) )
% 0.17/0.48 = ( ( ( Ys = nil_nat )
% 0.17/0.48 & ( ( cons_nat @ X @ Xs )
% 0.17/0.48 = Zs ) )
% 0.17/0.48 | ? [Ys5: list_nat] :
% 0.17/0.48 ( ( ( cons_nat @ X @ Ys5 )
% 0.17/0.48 = Ys )
% 0.17/0.48 & ( Xs
% 0.17/0.48 = ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % Cons_eq_append_conv
% 0.17/0.48 thf(fact_149_rev__exhaust,axiom,
% 0.17/0.48 ! [Xs: list_a] :
% 0.17/0.48 ( ( Xs != nil_a )
% 0.17/0.48 => ~ ! [Ys3: list_a,Y2: a] :
% 0.17/0.48 ( Xs
% 0.17/0.48 != ( append_a @ Ys3 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % rev_exhaust
% 0.17/0.48 thf(fact_150_rev__exhaust,axiom,
% 0.17/0.48 ! [Xs: list_nat] :
% 0.17/0.48 ( ( Xs != nil_nat )
% 0.17/0.48 => ~ ! [Ys3: list_nat,Y2: nat] :
% 0.17/0.48 ( Xs
% 0.17/0.48 != ( append_nat @ Ys3 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % rev_exhaust
% 0.17/0.48 thf(fact_151_rev__induct,axiom,
% 0.17/0.48 ! [P: list_a > $o,Xs: list_a] :
% 0.17/0.48 ( ( P @ nil_a )
% 0.17/0.48 => ( ! [X2: a,Xs3: list_a] :
% 0.17/0.48 ( ( P @ Xs3 )
% 0.17/0.48 => ( P @ ( append_a @ Xs3 @ ( cons_a @ X2 @ nil_a ) ) ) )
% 0.17/0.48 => ( P @ Xs ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % rev_induct
% 0.17/0.48 thf(fact_152_rev__induct,axiom,
% 0.17/0.48 ! [P: list_nat > $o,Xs: list_nat] :
% 0.17/0.48 ( ( P @ nil_nat )
% 0.17/0.48 => ( ! [X2: nat,Xs3: list_nat] :
% 0.17/0.48 ( ( P @ Xs3 )
% 0.17/0.48 => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X2 @ nil_nat ) ) ) )
% 0.17/0.48 => ( P @ Xs ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % rev_induct
% 0.17/0.48 thf(fact_153_psums_Osimps_I1_J,axiom,
% 0.17/0.48 ( ( descar1375166517sums_a @ nil_a )
% 0.17/0.48 = nil_a ) ).
% 0.17/0.48
% 0.17/0.48 % psums.simps(1)
% 0.17/0.48 thf(fact_154_psums_Osimps_I1_J,axiom,
% 0.17/0.48 ( ( descar226543321ms_nat @ nil_nat )
% 0.17/0.48 = nil_nat ) ).
% 0.17/0.48
% 0.17/0.48 % psums.simps(1)
% 0.17/0.48 thf(fact_155_replicate__append__same,axiom,
% 0.17/0.48 ! [I: nat,X: a] :
% 0.17/0.48 ( ( append_a @ ( replicate_a @ I @ X ) @ ( cons_a @ X @ nil_a ) )
% 0.17/0.48 = ( cons_a @ X @ ( replicate_a @ I @ X ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % replicate_append_same
% 0.17/0.48 thf(fact_156_replicate__append__same,axiom,
% 0.17/0.48 ! [I: nat,X: nat] :
% 0.17/0.48 ( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
% 0.17/0.48 = ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % replicate_append_same
% 0.17/0.48 thf(fact_157_psums_Osimps_I2_J,axiom,
% 0.17/0.48 ! [X: a] :
% 0.17/0.48 ( ( descar1375166517sums_a @ ( cons_a @ X @ nil_a ) )
% 0.17/0.48 = ( cons_a @ X @ nil_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % psums.simps(2)
% 0.17/0.48 thf(fact_158_psums_Osimps_I2_J,axiom,
% 0.17/0.48 ! [X: nat] :
% 0.17/0.48 ( ( descar226543321ms_nat @ ( cons_nat @ X @ nil_nat ) )
% 0.17/0.48 = ( cons_nat @ X @ nil_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % psums.simps(2)
% 0.17/0.48 thf(fact_159_zero__reorient,axiom,
% 0.17/0.48 ! [X: a] :
% 0.17/0.48 ( ( zero_zero_a = X )
% 0.17/0.48 = ( X = zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % zero_reorient
% 0.17/0.48 thf(fact_160_zero__reorient,axiom,
% 0.17/0.48 ! [X: nat] :
% 0.17/0.48 ( ( zero_zero_nat = X )
% 0.17/0.48 = ( X = zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % zero_reorient
% 0.17/0.48 thf(fact_161_not__Cons__self2,axiom,
% 0.17/0.48 ! [X: a,Xs: list_a] :
% 0.17/0.48 ( ( cons_a @ X @ Xs )
% 0.17/0.48 != Xs ) ).
% 0.17/0.48
% 0.17/0.48 % not_Cons_self2
% 0.17/0.48 thf(fact_162_not__Cons__self2,axiom,
% 0.17/0.48 ! [X: nat,Xs: list_nat] :
% 0.17/0.48 ( ( cons_nat @ X @ Xs )
% 0.17/0.48 != Xs ) ).
% 0.17/0.48
% 0.17/0.48 % not_Cons_self2
% 0.17/0.48 thf(fact_163_append__eq__append__conv2,axiom,
% 0.17/0.48 ! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
% 0.17/0.48 ( ( ( append_a @ Xs @ Ys )
% 0.17/0.48 = ( append_a @ Zs @ Ts ) )
% 0.17/0.48 = ( ? [Us: list_a] :
% 0.17/0.48 ( ( ( Xs
% 0.17/0.48 = ( append_a @ Zs @ Us ) )
% 0.17/0.48 & ( ( append_a @ Us @ Ys )
% 0.17/0.48 = Ts ) )
% 0.17/0.48 | ( ( ( append_a @ Xs @ Us )
% 0.17/0.48 = Zs )
% 0.17/0.48 & ( Ys
% 0.17/0.48 = ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_eq_append_conv2
% 0.17/0.48 thf(fact_164_append__eq__append__conv2,axiom,
% 0.17/0.48 ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
% 0.17/0.48 ( ( ( append_nat @ Xs @ Ys )
% 0.17/0.48 = ( append_nat @ Zs @ Ts ) )
% 0.17/0.48 = ( ? [Us: list_nat] :
% 0.17/0.48 ( ( ( Xs
% 0.17/0.48 = ( append_nat @ Zs @ Us ) )
% 0.17/0.48 & ( ( append_nat @ Us @ Ys )
% 0.17/0.48 = Ts ) )
% 0.17/0.48 | ( ( ( append_nat @ Xs @ Us )
% 0.17/0.48 = Zs )
% 0.17/0.48 & ( Ys
% 0.17/0.48 = ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_eq_append_conv2
% 0.17/0.48 thf(fact_165_append__eq__appendI,axiom,
% 0.17/0.48 ! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
% 0.17/0.48 ( ( ( append_a @ Xs @ Xs1 )
% 0.17/0.48 = Zs )
% 0.17/0.48 => ( ( Ys
% 0.17/0.48 = ( append_a @ Xs1 @ Us2 ) )
% 0.17/0.48 => ( ( append_a @ Xs @ Ys )
% 0.17/0.48 = ( append_a @ Zs @ Us2 ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_eq_appendI
% 0.17/0.48 thf(fact_166_append__eq__appendI,axiom,
% 0.17/0.48 ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
% 0.17/0.48 ( ( ( append_nat @ Xs @ Xs1 )
% 0.17/0.48 = Zs )
% 0.17/0.48 => ( ( Ys
% 0.17/0.48 = ( append_nat @ Xs1 @ Us2 ) )
% 0.17/0.48 => ( ( append_nat @ Xs @ Ys )
% 0.17/0.48 = ( append_nat @ Zs @ Us2 ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_eq_appendI
% 0.17/0.48 thf(fact_167_append__Cons,axiom,
% 0.17/0.48 ! [X: a,Xs: list_a,Ys: list_a] :
% 0.17/0.48 ( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
% 0.17/0.48 = ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_Cons
% 0.17/0.48 thf(fact_168_append__Cons,axiom,
% 0.17/0.48 ! [X: nat,Xs: list_nat,Ys: list_nat] :
% 0.17/0.48 ( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
% 0.17/0.48 = ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_Cons
% 0.17/0.48 thf(fact_169_v__decompose,axiom,
% 0.17/0.48 ! [Xs5: list_a] :
% 0.17/0.48 ( ( v @ ( append_a @ Xs5 @ xsa ) )
% 0.17/0.48 = ( plus_plus_int @ ( v @ ( append_a @ Xs5 @ ( cons_a @ ( nth_a @ xsa @ p ) @ nil_a ) ) ) @ ( v @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % v_decompose
% 0.17/0.48 thf(fact_170_p__less__length,axiom,
% 0.17/0.48 ord_less_nat @ p @ ( size_size_list_a @ xsa ) ).
% 0.17/0.48
% 0.17/0.48 % p_less_length
% 0.17/0.48 thf(fact_171_neq0__conv,axiom,
% 0.17/0.48 ! [N: nat] :
% 0.17/0.48 ( ( N != zero_zero_nat )
% 0.17/0.48 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.17/0.48
% 0.17/0.48 % neq0_conv
% 0.17/0.48 thf(fact_172_less__nat__zero__code,axiom,
% 0.17/0.48 ! [N: nat] :
% 0.17/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.17/0.48
% 0.17/0.48 % less_nat_zero_code
% 0.17/0.48 thf(fact_173_bot__nat__0_Onot__eq__extremum,axiom,
% 0.17/0.48 ! [A: nat] :
% 0.17/0.48 ( ( A != zero_zero_nat )
% 0.17/0.48 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.17/0.48
% 0.17/0.48 % bot_nat_0.not_eq_extremum
% 0.17/0.48 thf(fact_174_order__refl,axiom,
% 0.17/0.48 ! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% 0.17/0.48
% 0.17/0.48 % order_refl
% 0.17/0.48 thf(fact_175_order__refl,axiom,
% 0.17/0.48 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.17/0.48
% 0.17/0.48 % order_refl
% 0.17/0.48 thf(fact_176_minus__poly__rev__list_Oinduct,axiom,
% 0.17/0.48 ! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
% 0.17/0.48 ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
% 0.17/0.48 ( ( P @ Xs3 @ Ys3 )
% 0.17/0.48 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
% 0.17/0.48 => ( ! [Xs3: list_a] : ( P @ Xs3 @ nil_a )
% 0.17/0.48 => ( ! [Y2: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys3 ) )
% 0.17/0.48 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % minus_poly_rev_list.induct
% 0.17/0.48 thf(fact_177_plus__coeffs_Oinduct,axiom,
% 0.17/0.48 ! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
% 0.17/0.48 ( ! [Xs3: list_a] : ( P @ Xs3 @ nil_a )
% 0.17/0.48 => ( ! [V: a,Va: list_a] : ( P @ nil_a @ ( cons_a @ V @ Va ) )
% 0.17/0.48 => ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
% 0.17/0.48 ( ( P @ Xs3 @ Ys3 )
% 0.17/0.48 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
% 0.17/0.48 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % plus_coeffs.induct
% 0.17/0.48 thf(fact_178_plus__coeffs_Oinduct,axiom,
% 0.17/0.48 ! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
% 0.17/0.48 ( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
% 0.17/0.48 => ( ! [V: nat,Va: list_nat] : ( P @ nil_nat @ ( cons_nat @ V @ Va ) )
% 0.17/0.48 => ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys3: list_nat] :
% 0.17/0.48 ( ( P @ Xs3 @ Ys3 )
% 0.17/0.48 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys3 ) ) )
% 0.17/0.48 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % plus_coeffs.induct
% 0.17/0.48 thf(fact_179_minf_I8_J,axiom,
% 0.17/0.48 ! [T: a] :
% 0.17/0.48 ? [Z: a] :
% 0.17/0.48 ! [X3: a] :
% 0.17/0.48 ( ( ord_less_a @ X3 @ Z )
% 0.17/0.48 => ~ ( ord_less_eq_a @ T @ X3 ) ) ).
% 0.17/0.48
% 0.17/0.48 % minf(8)
% 0.17/0.48 thf(fact_180_minf_I8_J,axiom,
% 0.17/0.48 ! [T: nat] :
% 0.17/0.48 ? [Z: nat] :
% 0.17/0.48 ! [X3: nat] :
% 0.17/0.48 ( ( ord_less_nat @ X3 @ Z )
% 0.17/0.48 => ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% 0.17/0.48
% 0.17/0.48 % minf(8)
% 0.17/0.48 thf(fact_181_minf_I6_J,axiom,
% 0.17/0.48 ! [T: a] :
% 0.17/0.48 ? [Z: a] :
% 0.17/0.48 ! [X3: a] :
% 0.17/0.48 ( ( ord_less_a @ X3 @ Z )
% 0.17/0.48 => ( ord_less_eq_a @ X3 @ T ) ) ).
% 0.17/0.48
% 0.17/0.48 % minf(6)
% 0.17/0.48 thf(fact_182_minf_I6_J,axiom,
% 0.17/0.48 ! [T: nat] :
% 0.17/0.48 ? [Z: nat] :
% 0.17/0.48 ! [X3: nat] :
% 0.17/0.48 ( ( ord_less_nat @ X3 @ Z )
% 0.17/0.48 => ( ord_less_eq_nat @ X3 @ T ) ) ).
% 0.17/0.48
% 0.17/0.48 % minf(6)
% 0.17/0.48 thf(fact_183_add__left__cancel,axiom,
% 0.17/0.48 ! [A: int,B: int,C: int] :
% 0.17/0.48 ( ( ( plus_plus_int @ A @ B )
% 0.17/0.48 = ( plus_plus_int @ A @ C ) )
% 0.17/0.48 = ( B = C ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_left_cancel
% 0.17/0.48 thf(fact_184_add__right__cancel,axiom,
% 0.17/0.48 ! [B: int,A: int,C: int] :
% 0.17/0.48 ( ( ( plus_plus_int @ B @ A )
% 0.17/0.48 = ( plus_plus_int @ C @ A ) )
% 0.17/0.48 = ( B = C ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_right_cancel
% 0.17/0.48 thf(fact_185_le0,axiom,
% 0.17/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.17/0.48
% 0.17/0.48 % le0
% 0.17/0.48 thf(fact_186_bot__nat__0_Oextremum,axiom,
% 0.17/0.48 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.17/0.48
% 0.17/0.48 % bot_nat_0.extremum
% 0.17/0.48 thf(fact_187_add_Oleft__neutral,axiom,
% 0.17/0.48 ! [A: int] :
% 0.17/0.48 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.17/0.48 = A ) ).
% 0.17/0.48
% 0.17/0.48 % add.left_neutral
% 0.17/0.48 thf(fact_188_add_Oleft__neutral,axiom,
% 0.17/0.48 ! [A: a] :
% 0.17/0.48 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.17/0.48 = A ) ).
% 0.17/0.48
% 0.17/0.48 % add.left_neutral
% 0.17/0.48 thf(fact_189_add_Oleft__neutral,axiom,
% 0.17/0.48 ! [A: nat] :
% 0.17/0.48 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.17/0.48 = A ) ).
% 0.17/0.48
% 0.17/0.48 % add.left_neutral
% 0.17/0.48 thf(fact_190_add_Oright__neutral,axiom,
% 0.17/0.48 ! [A: int] :
% 0.17/0.48 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.17/0.48 = A ) ).
% 0.17/0.48
% 0.17/0.48 % add.right_neutral
% 0.17/0.48 thf(fact_191_add_Oright__neutral,axiom,
% 0.17/0.48 ! [A: a] :
% 0.17/0.48 ( ( plus_plus_a @ A @ zero_zero_a )
% 0.17/0.48 = A ) ).
% 0.17/0.48
% 0.17/0.48 % add.right_neutral
% 0.17/0.48 thf(fact_192_add_Oright__neutral,axiom,
% 0.17/0.48 ! [A: nat] :
% 0.17/0.48 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.17/0.48 = A ) ).
% 0.17/0.48
% 0.17/0.48 % add.right_neutral
% 0.17/0.48 thf(fact_193_double__zero,axiom,
% 0.17/0.48 ! [A: int] :
% 0.17/0.48 ( ( ( plus_plus_int @ A @ A )
% 0.17/0.48 = zero_zero_int )
% 0.17/0.48 = ( A = zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % double_zero
% 0.17/0.48 thf(fact_194_double__zero,axiom,
% 0.17/0.48 ! [A: a] :
% 0.17/0.48 ( ( ( plus_plus_a @ A @ A )
% 0.17/0.48 = zero_zero_a )
% 0.17/0.48 = ( A = zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % double_zero
% 0.17/0.48 thf(fact_195_double__zero__sym,axiom,
% 0.17/0.48 ! [A: int] :
% 0.17/0.48 ( ( zero_zero_int
% 0.17/0.48 = ( plus_plus_int @ A @ A ) )
% 0.17/0.48 = ( A = zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % double_zero_sym
% 0.17/0.48 thf(fact_196_double__zero__sym,axiom,
% 0.17/0.48 ! [A: a] :
% 0.17/0.48 ( ( zero_zero_a
% 0.17/0.48 = ( plus_plus_a @ A @ A ) )
% 0.17/0.48 = ( A = zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % double_zero_sym
% 0.17/0.48 thf(fact_197_add__cancel__left__left,axiom,
% 0.17/0.48 ! [B: int,A: int] :
% 0.17/0.48 ( ( ( plus_plus_int @ B @ A )
% 0.17/0.48 = A )
% 0.17/0.48 = ( B = zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_left_left
% 0.17/0.48 thf(fact_198_add__cancel__left__left,axiom,
% 0.17/0.48 ! [B: a,A: a] :
% 0.17/0.48 ( ( ( plus_plus_a @ B @ A )
% 0.17/0.48 = A )
% 0.17/0.48 = ( B = zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_left_left
% 0.17/0.48 thf(fact_199_add__cancel__left__left,axiom,
% 0.17/0.48 ! [B: nat,A: nat] :
% 0.17/0.48 ( ( ( plus_plus_nat @ B @ A )
% 0.17/0.48 = A )
% 0.17/0.48 = ( B = zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_left_left
% 0.17/0.48 thf(fact_200_add__cancel__left__right,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( ( plus_plus_int @ A @ B )
% 0.17/0.48 = A )
% 0.17/0.48 = ( B = zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_left_right
% 0.17/0.48 thf(fact_201_add__cancel__left__right,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( ( plus_plus_a @ A @ B )
% 0.17/0.48 = A )
% 0.17/0.48 = ( B = zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_left_right
% 0.17/0.48 thf(fact_202_add__cancel__left__right,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( ( plus_plus_nat @ A @ B )
% 0.17/0.48 = A )
% 0.17/0.48 = ( B = zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_left_right
% 0.17/0.48 thf(fact_203_add__cancel__right__left,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( A
% 0.17/0.48 = ( plus_plus_int @ B @ A ) )
% 0.17/0.48 = ( B = zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_right_left
% 0.17/0.48 thf(fact_204_add__cancel__right__left,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( A
% 0.17/0.48 = ( plus_plus_a @ B @ A ) )
% 0.17/0.48 = ( B = zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_right_left
% 0.17/0.48 thf(fact_205_add__cancel__right__left,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( A
% 0.17/0.48 = ( plus_plus_nat @ B @ A ) )
% 0.17/0.48 = ( B = zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_right_left
% 0.17/0.48 thf(fact_206_add__cancel__right__right,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( A
% 0.17/0.48 = ( plus_plus_int @ A @ B ) )
% 0.17/0.48 = ( B = zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_right_right
% 0.17/0.48 thf(fact_207_add__cancel__right__right,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( A
% 0.17/0.48 = ( plus_plus_a @ A @ B ) )
% 0.17/0.48 = ( B = zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_right_right
% 0.17/0.48 thf(fact_208_add__cancel__right__right,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( A
% 0.17/0.48 = ( plus_plus_nat @ A @ B ) )
% 0.17/0.48 = ( B = zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_cancel_right_right
% 0.17/0.48 thf(fact_209_add__eq__0__iff__both__eq__0,axiom,
% 0.17/0.48 ! [X: nat,Y: nat] :
% 0.17/0.48 ( ( ( plus_plus_nat @ X @ Y )
% 0.17/0.48 = zero_zero_nat )
% 0.17/0.48 = ( ( X = zero_zero_nat )
% 0.17/0.48 & ( Y = zero_zero_nat ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_eq_0_iff_both_eq_0
% 0.17/0.48 thf(fact_210_zero__eq__add__iff__both__eq__0,axiom,
% 0.17/0.48 ! [X: nat,Y: nat] :
% 0.17/0.48 ( ( zero_zero_nat
% 0.17/0.48 = ( plus_plus_nat @ X @ Y ) )
% 0.17/0.48 = ( ( X = zero_zero_nat )
% 0.17/0.48 & ( Y = zero_zero_nat ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % zero_eq_add_iff_both_eq_0
% 0.17/0.48 thf(fact_211_add__le__cancel__right,axiom,
% 0.17/0.48 ! [A: int,C: int,B: int] :
% 0.17/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.17/0.48 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_cancel_right
% 0.17/0.48 thf(fact_212_add__le__cancel__right,axiom,
% 0.17/0.48 ! [A: a,C: a,B: a] :
% 0.17/0.48 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 0.17/0.48 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_cancel_right
% 0.17/0.48 thf(fact_213_add__le__cancel__right,axiom,
% 0.17/0.48 ! [A: nat,C: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.17/0.48 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_cancel_right
% 0.17/0.48 thf(fact_214_add__le__cancel__left,axiom,
% 0.17/0.48 ! [C: int,A: int,B: int] :
% 0.17/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.17/0.48 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_cancel_left
% 0.17/0.48 thf(fact_215_add__le__cancel__left,axiom,
% 0.17/0.48 ! [C: a,A: a,B: a] :
% 0.17/0.48 ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 0.17/0.48 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_cancel_left
% 0.17/0.48 thf(fact_216_add__le__cancel__left,axiom,
% 0.17/0.48 ! [C: nat,A: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.17/0.48 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_cancel_left
% 0.17/0.48 thf(fact_217_add__less__cancel__right,axiom,
% 0.17/0.48 ! [A: int,C: int,B: int] :
% 0.17/0.48 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.17/0.48 = ( ord_less_int @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_cancel_right
% 0.17/0.48 thf(fact_218_add__less__cancel__right,axiom,
% 0.17/0.48 ! [A: nat,C: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.17/0.48 = ( ord_less_nat @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_cancel_right
% 0.17/0.48 thf(fact_219_add__less__cancel__left,axiom,
% 0.17/0.48 ! [C: int,A: int,B: int] :
% 0.17/0.48 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.17/0.48 = ( ord_less_int @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_cancel_left
% 0.17/0.48 thf(fact_220_add__less__cancel__left,axiom,
% 0.17/0.48 ! [C: nat,A: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.17/0.48 = ( ord_less_nat @ A @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_cancel_left
% 0.17/0.48 thf(fact_221_append__eq__append__conv,axiom,
% 0.17/0.48 ! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs: list_nat] :
% 0.17/0.48 ( ( ( ( size_size_list_nat @ Xs )
% 0.17/0.48 = ( size_size_list_nat @ Ys ) )
% 0.17/0.48 | ( ( size_size_list_nat @ Us2 )
% 0.17/0.48 = ( size_size_list_nat @ Vs ) ) )
% 0.17/0.48 => ( ( ( append_nat @ Xs @ Us2 )
% 0.17/0.48 = ( append_nat @ Ys @ Vs ) )
% 0.17/0.48 = ( ( Xs = Ys )
% 0.17/0.48 & ( Us2 = Vs ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_eq_append_conv
% 0.17/0.48 thf(fact_222_append__eq__append__conv,axiom,
% 0.17/0.48 ! [Xs: list_a,Ys: list_a,Us2: list_a,Vs: list_a] :
% 0.17/0.48 ( ( ( ( size_size_list_a @ Xs )
% 0.17/0.48 = ( size_size_list_a @ Ys ) )
% 0.17/0.48 | ( ( size_size_list_a @ Us2 )
% 0.17/0.48 = ( size_size_list_a @ Vs ) ) )
% 0.17/0.48 => ( ( ( append_a @ Xs @ Us2 )
% 0.17/0.48 = ( append_a @ Ys @ Vs ) )
% 0.17/0.48 = ( ( Xs = Ys )
% 0.17/0.48 & ( Us2 = Vs ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % append_eq_append_conv
% 0.17/0.48 thf(fact_223_take__all__iff,axiom,
% 0.17/0.48 ! [N: nat,Xs: list_a] :
% 0.17/0.48 ( ( ( take_a @ N @ Xs )
% 0.17/0.48 = Xs )
% 0.17/0.48 = ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% 0.17/0.48
% 0.17/0.48 % take_all_iff
% 0.17/0.48 thf(fact_224_take__all,axiom,
% 0.17/0.48 ! [Xs: list_a,N: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 0.17/0.48 => ( ( take_a @ N @ Xs )
% 0.17/0.48 = Xs ) ) ).
% 0.17/0.48
% 0.17/0.48 % take_all
% 0.17/0.48 thf(fact_225_length__replicate,axiom,
% 0.17/0.48 ! [N: nat,X: nat] :
% 0.17/0.48 ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
% 0.17/0.48 = N ) ).
% 0.17/0.48
% 0.17/0.48 % length_replicate
% 0.17/0.48 thf(fact_226_length__replicate,axiom,
% 0.17/0.48 ! [N: nat,X: a] :
% 0.17/0.48 ( ( size_size_list_a @ ( replicate_a @ N @ X ) )
% 0.17/0.48 = N ) ).
% 0.17/0.48
% 0.17/0.48 % length_replicate
% 0.17/0.48 thf(fact_227_length__psums,axiom,
% 0.17/0.48 ! [Xs: list_a] :
% 0.17/0.48 ( ( size_size_list_a @ ( descar1375166517sums_a @ Xs ) )
% 0.17/0.48 = ( size_size_list_a @ Xs ) ) ).
% 0.17/0.48
% 0.17/0.48 % length_psums
% 0.17/0.48 thf(fact_228_length__psums,axiom,
% 0.17/0.48 ! [Xs: list_nat] :
% 0.17/0.48 ( ( size_size_list_nat @ ( descar226543321ms_nat @ Xs ) )
% 0.17/0.48 = ( size_size_list_nat @ Xs ) ) ).
% 0.17/0.48
% 0.17/0.48 % length_psums
% 0.17/0.48 thf(fact_229_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.17/0.48 ! [A: int] :
% 0.17/0.48 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.17/0.48 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.17/0.48
% 0.17/0.48 % zero_le_double_add_iff_zero_le_single_add
% 0.17/0.48 thf(fact_230_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.17/0.48 ! [A: a] :
% 0.17/0.48 ( ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
% 0.17/0.48 = ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% 0.17/0.48
% 0.17/0.48 % zero_le_double_add_iff_zero_le_single_add
% 0.17/0.48 thf(fact_231_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.17/0.48 ! [A: int] :
% 0.17/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.17/0.48 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % double_add_le_zero_iff_single_add_le_zero
% 0.17/0.48 thf(fact_232_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.17/0.48 ! [A: a] :
% 0.17/0.48 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
% 0.17/0.48 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % double_add_le_zero_iff_single_add_le_zero
% 0.17/0.48 thf(fact_233_le__add__same__cancel2,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.17/0.48 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % le_add_same_cancel2
% 0.17/0.48 thf(fact_234_le__add__same__cancel2,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( ord_less_eq_a @ A @ ( plus_plus_a @ B @ A ) )
% 0.17/0.48 = ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % le_add_same_cancel2
% 0.17/0.48 thf(fact_235_le__add__same__cancel2,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.17/0.48 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % le_add_same_cancel2
% 0.17/0.48 thf(fact_236_le__add__same__cancel1,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.17/0.48 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % le_add_same_cancel1
% 0.17/0.48 thf(fact_237_le__add__same__cancel1,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( ord_less_eq_a @ A @ ( plus_plus_a @ A @ B ) )
% 0.17/0.48 = ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % le_add_same_cancel1
% 0.17/0.48 thf(fact_238_le__add__same__cancel1,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.17/0.48 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % le_add_same_cancel1
% 0.17/0.48 thf(fact_239_add__le__same__cancel2,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.17/0.48 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_same_cancel2
% 0.17/0.48 thf(fact_240_add__le__same__cancel2,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ B )
% 0.17/0.48 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_same_cancel2
% 0.17/0.48 thf(fact_241_add__le__same__cancel2,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.17/0.48 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_same_cancel2
% 0.17/0.48 thf(fact_242_add__le__same__cancel1,axiom,
% 0.17/0.48 ! [B: int,A: int] :
% 0.17/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.17/0.48 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_same_cancel1
% 0.17/0.48 thf(fact_243_add__le__same__cancel1,axiom,
% 0.17/0.48 ! [B: a,A: a] :
% 0.17/0.48 ( ( ord_less_eq_a @ ( plus_plus_a @ B @ A ) @ B )
% 0.17/0.48 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_same_cancel1
% 0.17/0.48 thf(fact_244_add__le__same__cancel1,axiom,
% 0.17/0.48 ! [B: nat,A: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.17/0.48 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_le_same_cancel1
% 0.17/0.48 thf(fact_245_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.17/0.48 ! [A: int] :
% 0.17/0.48 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.17/0.48 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.17/0.48
% 0.17/0.48 % zero_less_double_add_iff_zero_less_single_add
% 0.17/0.48 thf(fact_246_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.17/0.48 ! [A: a] :
% 0.17/0.48 ( ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
% 0.17/0.48 = ( ord_less_a @ zero_zero_a @ A ) ) ).
% 0.17/0.48
% 0.17/0.48 % zero_less_double_add_iff_zero_less_single_add
% 0.17/0.48 thf(fact_247_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.17/0.48 ! [A: int] :
% 0.17/0.48 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.17/0.48 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % double_add_less_zero_iff_single_add_less_zero
% 0.17/0.48 thf(fact_248_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.17/0.48 ! [A: a] :
% 0.17/0.48 ( ( ord_less_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
% 0.17/0.48 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % double_add_less_zero_iff_single_add_less_zero
% 0.17/0.48 thf(fact_249_less__add__same__cancel2,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.17/0.48 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % less_add_same_cancel2
% 0.17/0.48 thf(fact_250_less__add__same__cancel2,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( ord_less_a @ A @ ( plus_plus_a @ B @ A ) )
% 0.17/0.48 = ( ord_less_a @ zero_zero_a @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % less_add_same_cancel2
% 0.17/0.48 thf(fact_251_less__add__same__cancel2,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.17/0.48 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % less_add_same_cancel2
% 0.17/0.48 thf(fact_252_less__add__same__cancel1,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.17/0.48 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % less_add_same_cancel1
% 0.17/0.48 thf(fact_253_less__add__same__cancel1,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( ord_less_a @ A @ ( plus_plus_a @ A @ B ) )
% 0.17/0.48 = ( ord_less_a @ zero_zero_a @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % less_add_same_cancel1
% 0.17/0.48 thf(fact_254_less__add__same__cancel1,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.17/0.48 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.17/0.48
% 0.17/0.48 % less_add_same_cancel1
% 0.17/0.48 thf(fact_255_add__less__same__cancel2,axiom,
% 0.17/0.48 ! [A: int,B: int] :
% 0.17/0.48 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.17/0.48 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_same_cancel2
% 0.17/0.48 thf(fact_256_add__less__same__cancel2,axiom,
% 0.17/0.48 ! [A: a,B: a] :
% 0.17/0.48 ( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ B )
% 0.17/0.48 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_same_cancel2
% 0.17/0.48 thf(fact_257_add__less__same__cancel2,axiom,
% 0.17/0.48 ! [A: nat,B: nat] :
% 0.17/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.17/0.48 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_same_cancel2
% 0.17/0.48 thf(fact_258_add__less__same__cancel1,axiom,
% 0.17/0.48 ! [B: int,A: int] :
% 0.17/0.48 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.17/0.48 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_same_cancel1
% 0.17/0.48 thf(fact_259_add__less__same__cancel1,axiom,
% 0.17/0.48 ! [B: a,A: a] :
% 0.17/0.48 ( ( ord_less_a @ ( plus_plus_a @ B @ A ) @ B )
% 0.17/0.48 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_same_cancel1
% 0.17/0.48 thf(fact_260_add__less__same__cancel1,axiom,
% 0.17/0.48 ! [B: nat,A: nat] :
% 0.17/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.17/0.48 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_less_same_cancel1
% 0.17/0.48 thf(fact_261_length__0__conv,axiom,
% 0.17/0.48 ! [Xs: list_a] :
% 0.17/0.48 ( ( ( size_size_list_a @ Xs )
% 0.17/0.48 = zero_zero_nat )
% 0.17/0.48 = ( Xs = nil_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % length_0_conv
% 0.17/0.48 thf(fact_262_length__greater__0__conv,axiom,
% 0.17/0.48 ! [Xs: list_a] :
% 0.17/0.48 ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
% 0.17/0.48 = ( Xs != nil_a ) ) ).
% 0.17/0.48
% 0.17/0.48 % length_greater_0_conv
% 0.17/0.48 thf(fact_263_nth__append__length,axiom,
% 0.17/0.48 ! [Xs: list_nat,X: nat,Ys: list_nat] :
% 0.17/0.48 ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
% 0.17/0.48 = X ) ).
% 0.17/0.48
% 0.17/0.48 % nth_append_length
% 0.17/0.48 thf(fact_264_nth__append__length,axiom,
% 0.17/0.48 ! [Xs: list_a,X: a,Ys: list_a] :
% 0.17/0.48 ( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
% 0.17/0.48 = X ) ).
% 0.17/0.48
% 0.17/0.48 % nth_append_length
% 0.17/0.48 thf(fact_265_less__eq__nat_Osimps_I1_J,axiom,
% 0.17/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.17/0.48
% 0.17/0.48 % less_eq_nat.simps(1)
% 0.17/0.48 thf(fact_266_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.17/0.48 ! [A: int,B: int,C: int] :
% 0.17/0.48 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.17/0.48 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % ab_semigroup_add_class.add_ac(1)
% 0.17/0.48 thf(fact_267_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.17/0.48 ! [I: int,J: int,K: int,L: int] :
% 0.17/0.48 ( ( ( I = J )
% 0.17/0.48 & ( K = L ) )
% 0.17/0.48 => ( ( plus_plus_int @ I @ K )
% 0.17/0.48 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_mono_thms_linordered_semiring(4)
% 0.17/0.48 thf(fact_268_le__0__eq,axiom,
% 0.17/0.48 ! [N: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.17/0.48 = ( N = zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % le_0_eq
% 0.17/0.48 thf(fact_269_impossible__Cons,axiom,
% 0.17/0.48 ! [Xs: list_nat,Ys: list_nat,X: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
% 0.17/0.48 => ( Xs
% 0.17/0.48 != ( cons_nat @ X @ Ys ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % impossible_Cons
% 0.17/0.48 thf(fact_270_impossible__Cons,axiom,
% 0.17/0.48 ! [Xs: list_a,Ys: list_a,X: a] :
% 0.17/0.48 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
% 0.17/0.48 => ( Xs
% 0.17/0.48 != ( cons_a @ X @ Ys ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % impossible_Cons
% 0.17/0.48 thf(fact_271_Ex__list__of__length,axiom,
% 0.17/0.48 ! [N: nat] :
% 0.17/0.48 ? [Xs3: list_a] :
% 0.17/0.48 ( ( size_size_list_a @ Xs3 )
% 0.17/0.48 = N ) ).
% 0.17/0.48
% 0.17/0.48 % Ex_list_of_length
% 0.17/0.48 thf(fact_272_neq__if__length__neq,axiom,
% 0.17/0.48 ! [Xs: list_a,Ys: list_a] :
% 0.17/0.48 ( ( ( size_size_list_a @ Xs )
% 0.17/0.48 != ( size_size_list_a @ Ys ) )
% 0.17/0.48 => ( Xs != Ys ) ) ).
% 0.17/0.48
% 0.17/0.48 % neq_if_length_neq
% 0.17/0.48 thf(fact_273_group__cancel_Oadd1,axiom,
% 0.17/0.48 ! [A4: int,K: int,A: int,B: int] :
% 0.17/0.48 ( ( A4
% 0.17/0.48 = ( plus_plus_int @ K @ A ) )
% 0.17/0.48 => ( ( plus_plus_int @ A4 @ B )
% 0.17/0.48 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % group_cancel.add1
% 0.17/0.48 thf(fact_274_group__cancel_Oadd2,axiom,
% 0.17/0.48 ! [B2: int,K: int,B: int,A: int] :
% 0.17/0.48 ( ( B2
% 0.17/0.48 = ( plus_plus_int @ K @ B ) )
% 0.17/0.48 => ( ( plus_plus_int @ A @ B2 )
% 0.17/0.48 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % group_cancel.add2
% 0.17/0.48 thf(fact_275_bot__nat__0_Oextremum__unique,axiom,
% 0.17/0.48 ! [A: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.48 = ( A = zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % bot_nat_0.extremum_unique
% 0.17/0.48 thf(fact_276_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.17/0.48 ! [A: nat] :
% 0.17/0.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.48 => ( A = zero_zero_nat ) ) ).
% 0.17/0.48
% 0.17/0.48 % bot_nat_0.extremum_uniqueI
% 0.17/0.48 thf(fact_277_add_Oassoc,axiom,
% 0.17/0.48 ! [A: int,B: int,C: int] :
% 0.17/0.48 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.17/0.48 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % add.assoc
% 0.17/0.48 thf(fact_278_add_Oleft__cancel,axiom,
% 0.17/0.48 ! [A: int,B: int,C: int] :
% 0.17/0.48 ( ( ( plus_plus_int @ A @ B )
% 0.17/0.48 = ( plus_plus_int @ A @ C ) )
% 0.17/0.48 = ( B = C ) ) ).
% 0.17/0.48
% 0.17/0.48 % add.left_cancel
% 0.17/0.48 thf(fact_279_add_Oright__cancel,axiom,
% 0.17/0.48 ! [B: int,A: int,C: int] :
% 0.17/0.48 ( ( ( plus_plus_int @ B @ A )
% 0.17/0.48 = ( plus_plus_int @ C @ A ) )
% 0.17/0.48 = ( B = C ) ) ).
% 0.17/0.48
% 0.17/0.48 % add.right_cancel
% 0.17/0.48 thf(fact_280_add_Ocommute,axiom,
% 0.17/0.48 ( plus_plus_int
% 0.17/0.48 = ( ^ [A5: int,B3: int] : ( plus_plus_int @ B3 @ A5 ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % add.commute
% 0.17/0.48 thf(fact_281_add_Oleft__commute,axiom,
% 0.17/0.48 ! [B: int,A: int,C: int] :
% 0.17/0.48 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.17/0.48 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.17/0.48
% 0.17/0.48 % add.left_commute
% 0.17/0.48 thf(fact_282_add__left__imp__eq,axiom,
% 0.17/0.48 ! [A: int,B: int,C: int] :
% 0.17/0.48 ( ( ( plus_plus_int @ A @ B )
% 0.17/0.48 = ( plus_plus_int @ A @ C ) )
% 0.17/0.48 => ( B = C ) ) ).
% 0.17/0.48
% 0.17/0.48 % add_left_imp_eq
% 0.17/0.48 thf(fact_283_add__right__imp__eq,axiom,
% 0.32/0.48 ! [B: int,A: int,C: int] :
% 0.32/0.48 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.48 = ( plus_plus_int @ C @ A ) )
% 0.32/0.48 => ( B = C ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_right_imp_eq
% 0.32/0.48 thf(fact_284_size__neq__size__imp__neq,axiom,
% 0.32/0.48 ! [X: list_a,Y: list_a] :
% 0.32/0.48 ( ( ( size_size_list_a @ X )
% 0.32/0.48 != ( size_size_list_a @ Y ) )
% 0.32/0.48 => ( X != Y ) ) ).
% 0.32/0.48
% 0.32/0.48 % size_neq_size_imp_neq
% 0.32/0.48 thf(fact_285_less__mono__imp__le__mono,axiom,
% 0.32/0.48 ! [F2: nat > nat,I: nat,J: nat] :
% 0.32/0.48 ( ! [I2: nat,J2: nat] :
% 0.32/0.48 ( ( ord_less_nat @ I2 @ J2 )
% 0.32/0.48 => ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
% 0.32/0.48 => ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.48 => ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_mono_imp_le_mono
% 0.32/0.48 thf(fact_286_le__neq__implies__less,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.48 => ( ( M != N )
% 0.32/0.48 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_neq_implies_less
% 0.32/0.48 thf(fact_287_less__or__eq__imp__le,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( ord_less_nat @ M @ N )
% 0.32/0.48 | ( M = N ) )
% 0.32/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_or_eq_imp_le
% 0.32/0.48 thf(fact_288_le__eq__less__or__eq,axiom,
% 0.32/0.48 ( ord_less_eq_nat
% 0.32/0.48 = ( ^ [M2: nat,N2: nat] :
% 0.32/0.48 ( ( ord_less_nat @ M2 @ N2 )
% 0.32/0.48 | ( M2 = N2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_eq_less_or_eq
% 0.32/0.48 thf(fact_289_less__imp__le__nat,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_nat @ M @ N )
% 0.32/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_imp_le_nat
% 0.32/0.48 thf(fact_290_ex__least__nat__le,axiom,
% 0.32/0.48 ! [P: nat > $o,N: nat] :
% 0.32/0.48 ( ( P @ N )
% 0.32/0.48 => ( ~ ( P @ zero_zero_nat )
% 0.32/0.48 => ? [K2: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ K2 @ N )
% 0.32/0.48 & ! [I3: nat] :
% 0.32/0.48 ( ( ord_less_nat @ I3 @ K2 )
% 0.32/0.48 => ~ ( P @ I3 ) )
% 0.32/0.48 & ( P @ K2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ex_least_nat_le
% 0.32/0.48 thf(fact_291_nat__less__le,axiom,
% 0.32/0.48 ( ord_less_nat
% 0.32/0.48 = ( ^ [M2: nat,N2: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.32/0.48 & ( M2 != N2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_less_le
% 0.32/0.48 thf(fact_292_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % comm_monoid_add_class.add_0
% 0.32/0.48 thf(fact_293_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.48 ! [A: a] :
% 0.32/0.48 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % comm_monoid_add_class.add_0
% 0.32/0.48 thf(fact_294_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % comm_monoid_add_class.add_0
% 0.32/0.48 thf(fact_295_add_Ocomm__neutral,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % add.comm_neutral
% 0.32/0.48 thf(fact_296_add_Ocomm__neutral,axiom,
% 0.32/0.48 ! [A: a] :
% 0.32/0.48 ( ( plus_plus_a @ A @ zero_zero_a )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % add.comm_neutral
% 0.32/0.48 thf(fact_297_add_Ocomm__neutral,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % add.comm_neutral
% 0.32/0.48 thf(fact_298_add_Ogroup__left__neutral,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % add.group_left_neutral
% 0.32/0.48 thf(fact_299_add_Ogroup__left__neutral,axiom,
% 0.32/0.48 ! [A: a] :
% 0.32/0.48 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % add.group_left_neutral
% 0.32/0.48 thf(fact_300_add__le__imp__le__right,axiom,
% 0.32/0.48 ! [A: int,C: int,B: int] :
% 0.32/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.48 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_le_imp_le_right
% 0.32/0.48 thf(fact_301_add__le__imp__le__right,axiom,
% 0.32/0.48 ! [A: a,C: a,B: a] :
% 0.32/0.48 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 0.32/0.48 => ( ord_less_eq_a @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_le_imp_le_right
% 0.32/0.48 thf(fact_302_add__le__imp__le__right,axiom,
% 0.32/0.48 ! [A: nat,C: nat,B: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.48 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_le_imp_le_right
% 0.32/0.48 thf(fact_303_add__le__imp__le__left,axiom,
% 0.32/0.48 ! [C: int,A: int,B: int] :
% 0.32/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.48 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_le_imp_le_left
% 0.32/0.48 thf(fact_304_add__le__imp__le__left,axiom,
% 0.32/0.48 ! [C: a,A: a,B: a] :
% 0.32/0.48 ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 0.32/0.48 => ( ord_less_eq_a @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_le_imp_le_left
% 0.32/0.48 thf(fact_305_add__le__imp__le__left,axiom,
% 0.32/0.48 ! [C: nat,A: nat,B: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.48 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_le_imp_le_left
% 0.32/0.48 thf(fact_306_le__iff__add,axiom,
% 0.32/0.48 ( ord_less_eq_nat
% 0.32/0.48 = ( ^ [A5: nat,B3: nat] :
% 0.32/0.48 ? [C2: nat] :
% 0.32/0.48 ( B3
% 0.32/0.48 = ( plus_plus_nat @ A5 @ C2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_iff_add
% 0.32/0.48 thf(fact_307_add__right__mono,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int] :
% 0.32/0.48 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.48 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_right_mono
% 0.32/0.48 thf(fact_308_add__right__mono,axiom,
% 0.32/0.48 ! [A: a,B: a,C: a] :
% 0.32/0.48 ( ( ord_less_eq_a @ A @ B )
% 0.32/0.48 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_right_mono
% 0.32/0.48 thf(fact_309_add__right__mono,axiom,
% 0.32/0.48 ! [A: nat,B: nat,C: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_right_mono
% 0.32/0.48 thf(fact_310_less__eqE,axiom,
% 0.32/0.48 ! [A: nat,B: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.48 => ~ ! [C3: nat] :
% 0.32/0.48 ( B
% 0.32/0.48 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_eqE
% 0.32/0.48 thf(fact_311_add__left__mono,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int] :
% 0.32/0.48 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.48 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_left_mono
% 0.32/0.48 thf(fact_312_add__left__mono,axiom,
% 0.32/0.48 ! [A: a,B: a,C: a] :
% 0.32/0.48 ( ( ord_less_eq_a @ A @ B )
% 0.32/0.48 => ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_left_mono
% 0.32/0.48 thf(fact_313_add__left__mono,axiom,
% 0.32/0.48 ! [A: nat,B: nat,C: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_left_mono
% 0.32/0.48 thf(fact_314_add__mono,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.48 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.48 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.48 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono
% 0.32/0.48 thf(fact_315_add__mono,axiom,
% 0.32/0.48 ! [A: a,B: a,C: a,D: a] :
% 0.32/0.48 ( ( ord_less_eq_a @ A @ B )
% 0.32/0.48 => ( ( ord_less_eq_a @ C @ D )
% 0.32/0.48 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono
% 0.32/0.48 thf(fact_316_add__mono,axiom,
% 0.32/0.48 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.48 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono
% 0.32/0.48 thf(fact_317_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.48 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.48 ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.48 & ( ord_less_eq_int @ K @ L ) )
% 0.32/0.48 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(1)
% 0.32/0.48 thf(fact_318_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.48 ! [I: a,J: a,K: a,L: a] :
% 0.32/0.48 ( ( ( ord_less_eq_a @ I @ J )
% 0.32/0.48 & ( ord_less_eq_a @ K @ L ) )
% 0.32/0.48 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(1)
% 0.32/0.48 thf(fact_319_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.48 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.48 & ( ord_less_eq_nat @ K @ L ) )
% 0.32/0.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(1)
% 0.32/0.48 thf(fact_320_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.48 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.48 ( ( ( I = J )
% 0.32/0.48 & ( ord_less_eq_int @ K @ L ) )
% 0.32/0.48 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(2)
% 0.32/0.48 thf(fact_321_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.48 ! [I: a,J: a,K: a,L: a] :
% 0.32/0.48 ( ( ( I = J )
% 0.32/0.48 & ( ord_less_eq_a @ K @ L ) )
% 0.32/0.48 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(2)
% 0.32/0.48 thf(fact_322_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.48 ( ( ( I = J )
% 0.32/0.48 & ( ord_less_eq_nat @ K @ L ) )
% 0.32/0.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(2)
% 0.32/0.48 thf(fact_323_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.48 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.48 ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.48 & ( K = L ) )
% 0.32/0.48 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(3)
% 0.32/0.48 thf(fact_324_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.48 ! [I: a,J: a,K: a,L: a] :
% 0.32/0.48 ( ( ( ord_less_eq_a @ I @ J )
% 0.32/0.48 & ( K = L ) )
% 0.32/0.48 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(3)
% 0.32/0.48 thf(fact_325_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.48 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.48 & ( K = L ) )
% 0.32/0.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_semiring(3)
% 0.32/0.48 thf(fact_326_add__less__imp__less__right,axiom,
% 0.32/0.48 ! [A: int,C: int,B: int] :
% 0.32/0.48 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.48 => ( ord_less_int @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_imp_less_right
% 0.32/0.48 thf(fact_327_add__less__imp__less__right,axiom,
% 0.32/0.48 ! [A: nat,C: nat,B: nat] :
% 0.32/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.48 => ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_imp_less_right
% 0.32/0.48 thf(fact_328_add__less__imp__less__left,axiom,
% 0.32/0.48 ! [C: int,A: int,B: int] :
% 0.32/0.48 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.48 => ( ord_less_int @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_imp_less_left
% 0.32/0.48 thf(fact_329_add__less__imp__less__left,axiom,
% 0.32/0.48 ! [C: nat,A: nat,B: nat] :
% 0.32/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.48 => ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_imp_less_left
% 0.32/0.48 thf(fact_330_add__strict__right__mono,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int] :
% 0.32/0.48 ( ( ord_less_int @ A @ B )
% 0.32/0.48 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_strict_right_mono
% 0.32/0.48 thf(fact_331_add__strict__right__mono,axiom,
% 0.32/0.48 ! [A: nat,B: nat,C: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A @ B )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_strict_right_mono
% 0.32/0.48 thf(fact_332_add__strict__left__mono,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int] :
% 0.32/0.48 ( ( ord_less_int @ A @ B )
% 0.32/0.48 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_strict_left_mono
% 0.32/0.48 thf(fact_333_add__strict__left__mono,axiom,
% 0.32/0.48 ! [A: nat,B: nat,C: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A @ B )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_strict_left_mono
% 0.32/0.48 thf(fact_334_add__strict__mono,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.48 ( ( ord_less_int @ A @ B )
% 0.32/0.48 => ( ( ord_less_int @ C @ D )
% 0.32/0.48 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_strict_mono
% 0.32/0.48 thf(fact_335_add__strict__mono,axiom,
% 0.32/0.48 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.48 ( ( ord_less_nat @ A @ B )
% 0.32/0.48 => ( ( ord_less_nat @ C @ D )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_strict_mono
% 0.32/0.48 thf(fact_336_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.48 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.48 ( ( ( ord_less_int @ I @ J )
% 0.32/0.48 & ( K = L ) )
% 0.32/0.48 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(1)
% 0.32/0.48 thf(fact_337_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.48 ( ( ( ord_less_nat @ I @ J )
% 0.32/0.48 & ( K = L ) )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(1)
% 0.32/0.48 thf(fact_338_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.48 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.48 ( ( ( I = J )
% 0.32/0.48 & ( ord_less_int @ K @ L ) )
% 0.32/0.48 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(2)
% 0.32/0.48 thf(fact_339_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.48 ( ( ( I = J )
% 0.32/0.48 & ( ord_less_nat @ K @ L ) )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(2)
% 0.32/0.48 thf(fact_340_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.48 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.48 ( ( ( ord_less_int @ I @ J )
% 0.32/0.48 & ( ord_less_int @ K @ L ) )
% 0.32/0.48 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(5)
% 0.32/0.48 thf(fact_341_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.48 ( ( ( ord_less_nat @ I @ J )
% 0.32/0.48 & ( ord_less_nat @ K @ L ) )
% 0.32/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(5)
% 0.32/0.48 thf(fact_342_list_Osize_I3_J,axiom,
% 0.32/0.48 ( ( size_size_list_a @ nil_a )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % list.size(3)
% 0.32/0.48 thf(fact_343_length__induct,axiom,
% 0.32/0.48 ! [P: list_a > $o,Xs: list_a] :
% 0.32/0.48 ( ! [Xs3: list_a] :
% 0.32/0.48 ( ! [Ys6: list_a] :
% 0.32/0.48 ( ( ord_less_nat @ ( size_size_list_a @ Ys6 ) @ ( size_size_list_a @ Xs3 ) )
% 0.32/0.48 => ( P @ Ys6 ) )
% 0.32/0.48 => ( P @ Xs3 ) )
% 0.32/0.48 => ( P @ Xs ) ) ).
% 0.32/0.48
% 0.32/0.48 % length_induct
% 0.32/0.48 thf(fact_344_nth__take__lemma,axiom,
% 0.32/0.48 ! [K: nat,Xs: list_a,Ys: list_a] :
% 0.32/0.48 ( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Xs ) )
% 0.32/0.48 => ( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Ys ) )
% 0.32/0.48 => ( ! [I2: nat] :
% 0.32/0.48 ( ( ord_less_nat @ I2 @ K )
% 0.32/0.48 => ( ( nth_a @ Xs @ I2 )
% 0.32/0.48 = ( nth_a @ Ys @ I2 ) ) )
% 0.32/0.48 => ( ( take_a @ K @ Xs )
% 0.32/0.48 = ( take_a @ K @ Ys ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % nth_take_lemma
% 0.32/0.48 thf(fact_345_add__nonpos__eq__0__iff,axiom,
% 0.32/0.48 ! [X: int,Y: int] :
% 0.32/0.48 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.32/0.48 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.32/0.48 => ( ( ( plus_plus_int @ X @ Y )
% 0.32/0.48 = zero_zero_int )
% 0.32/0.48 = ( ( X = zero_zero_int )
% 0.32/0.48 & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_nonpos_eq_0_iff
% 0.32/0.48 thf(fact_346_add__nonpos__eq__0__iff,axiom,
% 0.32/0.48 ! [X: a,Y: a] :
% 0.32/0.48 ( ( ord_less_eq_a @ X @ zero_zero_a )
% 0.32/0.48 => ( ( ord_less_eq_a @ Y @ zero_zero_a )
% 0.32/0.48 => ( ( ( plus_plus_a @ X @ Y )
% 0.32/0.48 = zero_zero_a )
% 0.32/0.48 = ( ( X = zero_zero_a )
% 0.32/0.48 & ( Y = zero_zero_a ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_nonpos_eq_0_iff
% 0.32/0.48 thf(fact_347_add__nonpos__eq__0__iff,axiom,
% 0.32/0.48 ! [X: nat,Y: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.32/0.48 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 0.32/0.48 => ( ( ( plus_plus_nat @ X @ Y )
% 0.32/0.48 = zero_zero_nat )
% 0.32/0.48 = ( ( X = zero_zero_nat )
% 0.32/0.48 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_nonpos_eq_0_iff
% 0.32/0.48 thf(fact_348_add__nonneg__eq__0__iff,axiom,
% 0.32/0.48 ! [X: int,Y: int] :
% 0.32/0.48 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.48 => ( ( ( plus_plus_int @ X @ Y )
% 0.32/0.48 = zero_zero_int )
% 0.32/0.48 = ( ( X = zero_zero_int )
% 0.32/0.48 & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_nonneg_eq_0_iff
% 0.32/0.48 thf(fact_349_add__nonneg__eq__0__iff,axiom,
% 0.32/0.48 ! [X: a,Y: a] :
% 0.32/0.48 ( ( ord_less_eq_a @ zero_zero_a @ X )
% 0.32/0.48 => ( ( ord_less_eq_a @ zero_zero_a @ Y )
% 0.32/0.48 => ( ( ( plus_plus_a @ X @ Y )
% 0.32/0.48 = zero_zero_a )
% 0.32/0.48 = ( ( X = zero_zero_a )
% 0.32/0.48 & ( Y = zero_zero_a ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_nonneg_eq_0_iff
% 0.32/0.48 thf(fact_350_add__nonneg__eq__0__iff,axiom,
% 0.32/0.48 ! [X: nat,Y: nat] :
% 0.32/0.48 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.48 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.48 => ( ( ( plus_plus_nat @ X @ Y )
% 0.32/0.48 = zero_zero_nat )
% 0.32/0.48 = ( ( X = zero_zero_nat )
% 0.32/0.48 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_nonneg_eq_0_iff
% 0.32/0.48
% 0.32/0.48 % Conjectures (1)
% 0.32/0.48 thf(conj_0,conjecture,
% 0.32/0.48 ( ( descar1375166517sums_a @ xsa )
% 0.32/0.48 = ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( descar1375166517sums_a @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 ------------------------
% 2.58/1.66 ---- Embedded in THF ---
% 2.58/1.66 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.58/1.66 %%% Generated on Tue Feb 24 08:24:29 EST 2026
% 2.58/1.66 %%% using '$$dhol' embedding, version 1.3.0.
% 2.58/1.66 %%% Logic specification used:
% 2.58/1.66 %%% thf(spec, logic, $$dhol).
% 2.58/1.66
% 2.58/1.66 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2DTF2527.p
% See solution above
% 2.58/1.66 ------------------------
% 2.58/1.69 ---- Cleaned THF ---
% 2.58/1.69 thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 2.58/1.69 list_list_nat: $tType ).
% 2.58/1.69
% 2.58/1.69 thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 2.58/1.69 list_list_a: $tType ).
% 2.58/1.69
% 2.58/1.69 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 2.58/1.69 list_nat: $tType ).
% 2.58/1.69
% 2.58/1.69 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 2.58/1.69 list_a: $tType ).
% 2.58/1.69
% 2.58/1.69 thf(ty_n_t__Nat__Onat,type,
% 2.58/1.69 nat: $tType ).
% 2.58/1.69
% 2.58/1.69 thf(ty_n_t__Int__Oint,type,
% 2.58/1.69 int: $tType ).
% 2.58/1.69
% 2.58/1.69 thf(ty_n_tf__a,type,
% 2.58/1.69 a: $tType ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Opsums_001t__Nat__Onat,type,
% 2.58/1.69 descar226543321ms_nat: list_nat > list_nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Opsums_001tf__a,type,
% 2.58/1.69 descar1375166517sums_a: list_a > list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 2.58/1.69 plus_plus_int: int > int > int ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 2.58/1.69 plus_plus_nat: nat > nat > nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
% 2.58/1.69 plus_plus_a: a > a > a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 2.58/1.69 zero_zero_int: int ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.58/1.69 zero_zero_nat: nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
% 2.58/1.69 zero_zero_a: a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 2.58/1.69 append_nat: list_nat > list_nat > list_nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Oappend_001tf__a,type,
% 2.58/1.69 append_a: list_a > list_a > list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 2.58/1.69 last_nat: list_nat > nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olast_001tf__a,type,
% 2.58/1.69 last_a: list_a > a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
% 2.58/1.69 cons_list_nat: list_nat > list_list_nat > list_list_nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
% 2.58/1.69 cons_list_a: list_a > list_list_a > list_list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 2.58/1.69 cons_nat: nat > list_nat > list_nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olist_OCons_001tf__a,type,
% 2.58/1.69 cons_a: a > list_a > list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
% 2.58/1.69 nil_list_nat: list_list_nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
% 2.58/1.69 nil_list_a: list_list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 2.58/1.69 nil_nat: list_nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Olist_ONil_001tf__a,type,
% 2.58/1.69 nil_a: list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 2.58/1.69 nth_nat: list_nat > nat > nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Onth_001tf__a,type,
% 2.58/1.69 nth_a: list_a > nat > a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 2.58/1.69 replicate_nat: nat > nat > list_nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Oreplicate_001tf__a,type,
% 2.58/1.69 replicate_a: nat > a > list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_List_Otake_001tf__a,type,
% 2.58/1.69 take_a: nat > list_a > list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 2.58/1.69 size_size_list_nat: list_nat > nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 2.58/1.69 size_size_list_a: list_a > nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 2.58/1.69 ord_less_int: int > int > $o ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.58/1.69 ord_less_nat: nat > nat > $o ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
% 2.58/1.69 ord_less_a: a > a > $o ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 2.58/1.69 ord_less_eq_int: int > int > $o ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.58/1.69 ord_less_eq_nat: nat > nat > $o ).
% 2.58/1.69
% 2.58/1.69 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 2.58/1.69 ord_less_eq_a: a > a > $o ).
% 2.58/1.69
% 2.58/1.69 thf(sy_v_p____,type,
% 2.58/1.69 p: nat ).
% 2.58/1.69
% 2.58/1.69 thf(sy_v_v,type,
% 2.58/1.69 v: list_a > int ).
% 2.58/1.69
% 2.58/1.69 thf(sy_v_xa____,type,
% 2.58/1.69 xa: a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_v_xs,type,
% 2.58/1.69 xs: list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_v_xs1____,type,
% 2.58/1.69 xs1: list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_v_xs2____,type,
% 2.58/1.69 xs2: list_a ).
% 2.58/1.69
% 2.58/1.69 thf(sy_v_xsa____,type,
% 2.58/1.69 xsa: list_a ).
% 2.58/1.69
% 2.58/1.69 thf(fact_0_nonneg_Ohyps_I1_J,axiom,
% 2.58/1.69 xsa != nil_a ).
% 2.58/1.69
% 2.58/1.69 thf(fact_1_p__nz,axiom,
% 2.58/1.69 ( ( nth_a @ xsa @ p )
% 2.58/1.69 != zero_zero_a ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_2_nonneg_Ohyps_I2_J,axiom,
% 2.58/1.69 ( ( last_a @ xsa )
% 2.58/1.69 != zero_zero_a ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_3_psums__0__Cons,axiom,
% 2.58/1.69 ! [Xs: list_nat] :
% 2.58/1.69 ( ( descar226543321ms_nat @ ( cons_nat @ zero_zero_nat @ Xs ) )
% 2.58/1.69 = ( cons_nat @ zero_zero_nat @ ( descar226543321ms_nat @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_4_psums__0__Cons,axiom,
% 2.58/1.69 ! [Xs: list_a] :
% 2.58/1.69 ( ( descar1375166517sums_a @ ( cons_a @ zero_zero_a @ Xs ) )
% 2.58/1.69 = ( cons_a @ zero_zero_a @ ( descar1375166517sums_a @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_5_xs1__def,axiom,
% 2.58/1.69 ( xs1
% 2.58/1.69 = ( replicate_a @ p @ zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_6_xs__decompose,axiom,
% 2.58/1.69 ( xsa
% 2.58/1.69 = ( append_a @ xs1 @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_7_psums__replicate__0__append,axiom,
% 2.58/1.69 ! [N: nat,Xs: list_nat] :
% 2.58/1.69 ( ( descar226543321ms_nat @ ( append_nat @ ( replicate_nat @ N @ zero_zero_nat ) @ Xs ) )
% 2.58/1.69 = ( append_nat @ ( replicate_nat @ N @ zero_zero_nat ) @ ( descar226543321ms_nat @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_8_psums__replicate__0__append,axiom,
% 2.58/1.69 ! [N: nat,Xs: list_a] :
% 2.58/1.69 ( ( descar1375166517sums_a @ ( append_a @ ( replicate_a @ N @ zero_zero_a ) @ Xs ) )
% 2.58/1.69 = ( append_a @ ( replicate_a @ N @ zero_zero_a ) @ ( descar1375166517sums_a @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_9_less__p__zero,axiom,
% 2.58/1.69 ! [Q: nat] :
% 2.58/1.69 ( ( ord_less_nat @ Q @ p )
% 2.58/1.69 => ( ( nth_a @ xsa @ Q )
% 2.58/1.69 = zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_10__092_060open_062take_Ap_Axs_A_061_Areplicate_Ap_A_I0_058_058_Ha_J_092_060close_062,axiom,
% 2.58/1.69 ( ( take_a @ p @ xsa )
% 2.58/1.69 = ( replicate_a @ p @ zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_11_replicate__app__Cons__same,axiom,
% 2.58/1.69 ! [N: nat,X: nat,Xs: list_nat] :
% 2.58/1.69 ( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
% 2.58/1.69 = ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_12_replicate__app__Cons__same,axiom,
% 2.58/1.69 ! [N: nat,X: a,Xs: list_a] :
% 2.58/1.69 ( ( append_a @ ( replicate_a @ N @ X ) @ ( cons_a @ X @ Xs ) )
% 2.58/1.69 = ( cons_a @ X @ ( append_a @ ( replicate_a @ N @ X ) @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_13_append_Oassoc,axiom,
% 2.58/1.69 ! [A: list_a,B: list_a,C: list_a] :
% 2.58/1.69 ( ( append_a @ ( append_a @ A @ B ) @ C )
% 2.58/1.69 = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_14_append_Oassoc,axiom,
% 2.58/1.69 ! [A: list_nat,B: list_nat,C: list_nat] :
% 2.58/1.69 ( ( append_nat @ ( append_nat @ A @ B ) @ C )
% 2.58/1.69 = ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_15_append__assoc,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 2.58/1.69 ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
% 2.58/1.69 = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_16_append__assoc,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
% 2.58/1.69 ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
% 2.58/1.69 = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_17_append__same__eq,axiom,
% 2.58/1.69 ! [Ys: list_a,Xs: list_a,Zs: list_a] :
% 2.58/1.69 ( ( ( append_a @ Ys @ Xs )
% 2.58/1.69 = ( append_a @ Zs @ Xs ) )
% 2.58/1.69 = ( Ys = Zs ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_18_append__same__eq,axiom,
% 2.58/1.69 ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
% 2.58/1.69 ( ( ( append_nat @ Ys @ Xs )
% 2.58/1.69 = ( append_nat @ Zs @ Xs ) )
% 2.58/1.69 = ( Ys = Zs ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_19_same__append__eq,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 2.58/1.69 ( ( ( append_a @ Xs @ Ys )
% 2.58/1.69 = ( append_a @ Xs @ Zs ) )
% 2.58/1.69 = ( Ys = Zs ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_20_same__append__eq,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
% 2.58/1.69 ( ( ( append_nat @ Xs @ Ys )
% 2.58/1.69 = ( append_nat @ Xs @ Zs ) )
% 2.58/1.69 = ( Ys = Zs ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_21_nonneg_Ohyps_I4_J,axiom,
% 2.58/1.69 ord_less_eq_a @ zero_zero_a @ xa ).
% 2.58/1.69
% 2.58/1.69 thf(fact_22_list_Oinject,axiom,
% 2.58/1.69 ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
% 2.58/1.69 ( ( ( cons_a @ X21 @ X22 )
% 2.58/1.69 = ( cons_a @ Y21 @ Y22 ) )
% 2.58/1.69 = ( ( X21 = Y21 )
% 2.58/1.69 & ( X22 = Y22 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_23_list_Oinject,axiom,
% 2.58/1.69 ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
% 2.58/1.69 ( ( ( cons_nat @ X21 @ X22 )
% 2.58/1.69 = ( cons_nat @ Y21 @ Y22 ) )
% 2.58/1.69 = ( ( X21 = Y21 )
% 2.58/1.69 & ( X22 = Y22 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_24_append__replicate__commute,axiom,
% 2.58/1.69 ! [N: nat,X: a,K: nat] :
% 2.58/1.69 ( ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ K @ X ) )
% 2.58/1.69 = ( append_a @ ( replicate_a @ K @ X ) @ ( replicate_a @ N @ X ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_25_append__replicate__commute,axiom,
% 2.58/1.69 ! [N: nat,X: nat,K: nat] :
% 2.58/1.69 ( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
% 2.58/1.69 = ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_26_Cons__eq__appendI,axiom,
% 2.58/1.69 ! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
% 2.58/1.69 ( ( ( cons_a @ X @ Xs1 )
% 2.58/1.69 = Ys )
% 2.58/1.69 => ( ( Xs
% 2.58/1.69 = ( append_a @ Xs1 @ Zs ) )
% 2.58/1.69 => ( ( cons_a @ X @ Xs )
% 2.58/1.69 = ( append_a @ Ys @ Zs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_27_Cons__eq__appendI,axiom,
% 2.58/1.69 ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
% 2.58/1.69 ( ( ( cons_nat @ X @ Xs1 )
% 2.58/1.69 = Ys )
% 2.58/1.69 => ( ( Xs
% 2.58/1.69 = ( append_nat @ Xs1 @ Zs ) )
% 2.58/1.69 => ( ( cons_nat @ X @ Xs )
% 2.58/1.69 = ( append_nat @ Ys @ Zs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_28_assms_I1_J,axiom,
% 2.58/1.69 xs != nil_a ).
% 2.58/1.69
% 2.58/1.69 thf(fact_29_assms_I2_J,axiom,
% 2.58/1.69 ( ( last_a @ xs )
% 2.58/1.69 != zero_zero_a ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_30_replicate__eq__replicate,axiom,
% 2.58/1.69 ! [M: nat,X: a,N: nat,Y: a] :
% 2.58/1.69 ( ( ( replicate_a @ M @ X )
% 2.58/1.69 = ( replicate_a @ N @ Y ) )
% 2.58/1.69 = ( ( M = N )
% 2.58/1.69 & ( ( M != zero_zero_nat )
% 2.58/1.69 => ( X = Y ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_31_replicate__eq__replicate,axiom,
% 2.58/1.69 ! [M: nat,X: nat,N: nat,Y: nat] :
% 2.58/1.69 ( ( ( replicate_nat @ M @ X )
% 2.58/1.69 = ( replicate_nat @ N @ Y ) )
% 2.58/1.69 = ( ( M = N )
% 2.58/1.69 & ( ( M != zero_zero_nat )
% 2.58/1.69 => ( X = Y ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_32_le__zero__eq,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.58/1.69 = ( N = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_33_not__gr__zero,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.58/1.69 = ( N = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_34_append_Oright__neutral,axiom,
% 2.58/1.69 ! [A: list_nat] :
% 2.58/1.69 ( ( append_nat @ A @ nil_nat )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_35_append_Oright__neutral,axiom,
% 2.58/1.69 ! [A: list_a] :
% 2.58/1.69 ( ( append_a @ A @ nil_a )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_36_append__is__Nil__conv,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ( ( ( append_nat @ Xs @ Ys )
% 2.58/1.69 = nil_nat )
% 2.58/1.69 = ( ( Xs = nil_nat )
% 2.58/1.69 & ( Ys = nil_nat ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_37_append__is__Nil__conv,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( ( append_a @ Xs @ Ys )
% 2.58/1.69 = nil_a )
% 2.58/1.69 = ( ( Xs = nil_a )
% 2.58/1.69 & ( Ys = nil_a ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_38_Nil__is__append__conv,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ( ( nil_nat
% 2.58/1.69 = ( append_nat @ Xs @ Ys ) )
% 2.58/1.69 = ( ( Xs = nil_nat )
% 2.58/1.69 & ( Ys = nil_nat ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_39_Nil__is__append__conv,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( nil_a
% 2.58/1.69 = ( append_a @ Xs @ Ys ) )
% 2.58/1.69 = ( ( Xs = nil_a )
% 2.58/1.69 & ( Ys = nil_a ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_40_self__append__conv2,axiom,
% 2.58/1.69 ! [Ys: list_nat,Xs: list_nat] :
% 2.58/1.69 ( ( Ys
% 2.58/1.69 = ( append_nat @ Xs @ Ys ) )
% 2.58/1.69 = ( Xs = nil_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_41_self__append__conv2,axiom,
% 2.58/1.69 ! [Ys: list_a,Xs: list_a] :
% 2.58/1.69 ( ( Ys
% 2.58/1.69 = ( append_a @ Xs @ Ys ) )
% 2.58/1.69 = ( Xs = nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_42_append__self__conv2,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ( ( ( append_nat @ Xs @ Ys )
% 2.58/1.69 = Ys )
% 2.58/1.69 = ( Xs = nil_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_43_append__self__conv2,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( ( append_a @ Xs @ Ys )
% 2.58/1.69 = Ys )
% 2.58/1.69 = ( Xs = nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_44_self__append__conv,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ( ( Xs
% 2.58/1.69 = ( append_nat @ Xs @ Ys ) )
% 2.58/1.69 = ( Ys = nil_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_45_self__append__conv,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( Xs
% 2.58/1.69 = ( append_a @ Xs @ Ys ) )
% 2.58/1.69 = ( Ys = nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_46_append__self__conv,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ( ( ( append_nat @ Xs @ Ys )
% 2.58/1.69 = Xs )
% 2.58/1.69 = ( Ys = nil_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_47_append__self__conv,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( ( append_a @ Xs @ Ys )
% 2.58/1.69 = Xs )
% 2.58/1.69 = ( Ys = nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_48_append__Nil2,axiom,
% 2.58/1.69 ! [Xs: list_nat] :
% 2.58/1.69 ( ( append_nat @ Xs @ nil_nat )
% 2.58/1.69 = Xs ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_49_append__Nil2,axiom,
% 2.58/1.69 ! [Xs: list_a] :
% 2.58/1.69 ( ( append_a @ Xs @ nil_a )
% 2.58/1.69 = Xs ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_50_nth__Cons__0,axiom,
% 2.58/1.69 ! [X: a,Xs: list_a] :
% 2.58/1.69 ( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
% 2.58/1.69 = X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_51_nth__Cons__0,axiom,
% 2.58/1.69 ! [X: nat,Xs: list_nat] :
% 2.58/1.69 ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
% 2.58/1.69 = X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_52_take0,axiom,
% 2.58/1.69 ( ( take_a @ zero_zero_nat )
% 2.58/1.69 = ( ^ [Xs2: list_a] : nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_53_take__eq__Nil,axiom,
% 2.58/1.69 ! [N: nat,Xs: list_a] :
% 2.58/1.69 ( ( ( take_a @ N @ Xs )
% 2.58/1.69 = nil_a )
% 2.58/1.69 = ( ( N = zero_zero_nat )
% 2.58/1.69 | ( Xs = nil_a ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_54_empty__replicate,axiom,
% 2.58/1.69 ! [N: nat,X: a] :
% 2.58/1.69 ( ( nil_a
% 2.58/1.69 = ( replicate_a @ N @ X ) )
% 2.58/1.69 = ( N = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_55_empty__replicate,axiom,
% 2.58/1.69 ! [N: nat,X: nat] :
% 2.58/1.69 ( ( nil_nat
% 2.58/1.69 = ( replicate_nat @ N @ X ) )
% 2.58/1.69 = ( N = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_56_replicate__empty,axiom,
% 2.58/1.69 ! [N: nat,X: a] :
% 2.58/1.69 ( ( ( replicate_a @ N @ X )
% 2.58/1.69 = nil_a )
% 2.58/1.69 = ( N = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_57_replicate__empty,axiom,
% 2.58/1.69 ! [N: nat,X: nat] :
% 2.58/1.69 ( ( ( replicate_nat @ N @ X )
% 2.58/1.69 = nil_nat )
% 2.58/1.69 = ( N = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_58_last__replicate,axiom,
% 2.58/1.69 ! [N: nat,X: a] :
% 2.58/1.69 ( ( N != zero_zero_nat )
% 2.58/1.69 => ( ( last_a @ ( replicate_a @ N @ X ) )
% 2.58/1.69 = X ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_59_last__replicate,axiom,
% 2.58/1.69 ! [N: nat,X: nat] :
% 2.58/1.69 ( ( N != zero_zero_nat )
% 2.58/1.69 => ( ( last_nat @ ( replicate_nat @ N @ X ) )
% 2.58/1.69 = X ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_60_append1__eq__conv,axiom,
% 2.58/1.69 ! [Xs: list_a,X: a,Ys: list_a,Y: a] :
% 2.58/1.69 ( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
% 2.58/1.69 = ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
% 2.58/1.69 = ( ( Xs = Ys )
% 2.58/1.69 & ( X = Y ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_61_append1__eq__conv,axiom,
% 2.58/1.69 ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
% 2.58/1.69 ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
% 2.58/1.69 = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
% 2.58/1.69 = ( ( Xs = Ys )
% 2.58/1.69 & ( X = Y ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_62_nth__take,axiom,
% 2.58/1.69 ! [I: nat,N: nat,Xs: list_a] :
% 2.58/1.69 ( ( ord_less_nat @ I @ N )
% 2.58/1.69 => ( ( nth_a @ ( take_a @ N @ Xs ) @ I )
% 2.58/1.69 = ( nth_a @ Xs @ I ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_63_last__appendL,axiom,
% 2.58/1.69 ! [Ys: list_nat,Xs: list_nat] :
% 2.58/1.69 ( ( Ys = nil_nat )
% 2.58/1.69 => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
% 2.58/1.69 = ( last_nat @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_64_last__appendL,axiom,
% 2.58/1.69 ! [Ys: list_a,Xs: list_a] :
% 2.58/1.69 ( ( Ys = nil_a )
% 2.58/1.69 => ( ( last_a @ ( append_a @ Xs @ Ys ) )
% 2.58/1.69 = ( last_a @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_65_last__appendR,axiom,
% 2.58/1.69 ! [Ys: list_nat,Xs: list_nat] :
% 2.58/1.69 ( ( Ys != nil_nat )
% 2.58/1.69 => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
% 2.58/1.69 = ( last_nat @ Ys ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_66_last__appendR,axiom,
% 2.58/1.69 ! [Ys: list_a,Xs: list_a] :
% 2.58/1.69 ( ( Ys != nil_a )
% 2.58/1.69 => ( ( last_a @ ( append_a @ Xs @ Ys ) )
% 2.58/1.69 = ( last_a @ Ys ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_67_nth__replicate,axiom,
% 2.58/1.69 ! [I: nat,N: nat,X: a] :
% 2.58/1.69 ( ( ord_less_nat @ I @ N )
% 2.58/1.69 => ( ( nth_a @ ( replicate_a @ N @ X ) @ I )
% 2.58/1.69 = X ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_68_nth__replicate,axiom,
% 2.58/1.69 ! [I: nat,N: nat,X: nat] :
% 2.58/1.69 ( ( ord_less_nat @ I @ N )
% 2.58/1.69 => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
% 2.58/1.69 = X ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_69_last__snoc,axiom,
% 2.58/1.69 ! [Xs: list_a,X: a] :
% 2.58/1.69 ( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
% 2.58/1.69 = X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_70_last__snoc,axiom,
% 2.58/1.69 ! [Xs: list_nat,X: nat] :
% 2.58/1.69 ( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
% 2.58/1.69 = X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_71_take__0,axiom,
% 2.58/1.69 ! [Xs: list_a] :
% 2.58/1.69 ( ( take_a @ zero_zero_nat @ Xs )
% 2.58/1.69 = nil_a ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_72_last_Osimps,axiom,
% 2.58/1.69 ! [Xs: list_a,X: a] :
% 2.58/1.69 ( ( ( Xs = nil_a )
% 2.58/1.69 => ( ( last_a @ ( cons_a @ X @ Xs ) )
% 2.58/1.69 = X ) )
% 2.58/1.69 & ( ( Xs != nil_a )
% 2.58/1.69 => ( ( last_a @ ( cons_a @ X @ Xs ) )
% 2.58/1.69 = ( last_a @ Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_73_last_Osimps,axiom,
% 2.58/1.69 ! [Xs: list_nat,X: nat] :
% 2.58/1.69 ( ( ( Xs = nil_nat )
% 2.58/1.69 => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
% 2.58/1.69 = X ) )
% 2.58/1.69 & ( ( Xs != nil_nat )
% 2.58/1.69 => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
% 2.58/1.69 = ( last_nat @ Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_74_last__ConsL,axiom,
% 2.58/1.69 ! [Xs: list_a,X: a] :
% 2.58/1.69 ( ( Xs = nil_a )
% 2.58/1.69 => ( ( last_a @ ( cons_a @ X @ Xs ) )
% 2.58/1.69 = X ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_75_last__ConsL,axiom,
% 2.58/1.69 ! [Xs: list_nat,X: nat] :
% 2.58/1.69 ( ( Xs = nil_nat )
% 2.58/1.69 => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
% 2.58/1.69 = X ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_76_last__ConsR,axiom,
% 2.58/1.69 ! [Xs: list_a,X: a] :
% 2.58/1.69 ( ( Xs != nil_a )
% 2.58/1.69 => ( ( last_a @ ( cons_a @ X @ Xs ) )
% 2.58/1.69 = ( last_a @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_77_last__ConsR,axiom,
% 2.58/1.69 ! [Xs: list_nat,X: nat] :
% 2.58/1.69 ( ( Xs != nil_nat )
% 2.58/1.69 => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
% 2.58/1.69 = ( last_nat @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_78_last__append,axiom,
% 2.58/1.69 ! [Ys: list_nat,Xs: list_nat] :
% 2.58/1.69 ( ( ( Ys = nil_nat )
% 2.58/1.69 => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
% 2.58/1.69 = ( last_nat @ Xs ) ) )
% 2.58/1.69 & ( ( Ys != nil_nat )
% 2.58/1.69 => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
% 2.58/1.69 = ( last_nat @ Ys ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_79_last__append,axiom,
% 2.58/1.69 ! [Ys: list_a,Xs: list_a] :
% 2.58/1.69 ( ( ( Ys = nil_a )
% 2.58/1.69 => ( ( last_a @ ( append_a @ Xs @ Ys ) )
% 2.58/1.69 = ( last_a @ Xs ) ) )
% 2.58/1.69 & ( ( Ys != nil_a )
% 2.58/1.69 => ( ( last_a @ ( append_a @ Xs @ Ys ) )
% 2.58/1.69 = ( last_a @ Ys ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_80_take__Nil,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ( ( take_a @ N @ nil_a )
% 2.58/1.69 = nil_a ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_81_take__equalityI,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ! [I2: nat] :
% 2.58/1.69 ( ( take_a @ I2 @ Xs )
% 2.58/1.69 = ( take_a @ I2 @ Ys ) )
% 2.58/1.69 => ( Xs = Ys ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_82_transpose_Ocases,axiom,
% 2.58/1.69 ! [X: list_list_a] :
% 2.58/1.69 ( ( X != nil_list_a )
% 2.58/1.69 => ( ! [Xss: list_list_a] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_list_a @ nil_a @ Xss ) )
% 2.58/1.69 => ~ ! [X2: a,Xs3: list_a,Xss: list_list_a] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_list_a @ ( cons_a @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_83_transpose_Ocases,axiom,
% 2.58/1.69 ! [X: list_list_nat] :
% 2.58/1.69 ( ( X != nil_list_nat )
% 2.58/1.69 => ( ! [Xss: list_list_nat] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_list_nat @ nil_nat @ Xss ) )
% 2.58/1.69 => ~ ! [X2: nat,Xs3: list_nat,Xss: list_list_nat] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_list_nat @ ( cons_nat @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_84_longest__common__suffix,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ? [Ss: list_nat,Xs4: list_nat,Ys2: list_nat] :
% 2.58/1.69 ( ( Xs
% 2.58/1.69 = ( append_nat @ Xs4 @ Ss ) )
% 2.58/1.69 & ( Ys
% 2.58/1.69 = ( append_nat @ Ys2 @ Ss ) )
% 2.58/1.69 & ( ( Xs4 = nil_nat )
% 2.58/1.69 | ( Ys2 = nil_nat )
% 2.58/1.69 | ( ( last_nat @ Xs4 )
% 2.58/1.69 != ( last_nat @ Ys2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_85_longest__common__suffix,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ? [Ss: list_a,Xs4: list_a,Ys2: list_a] :
% 2.58/1.69 ( ( Xs
% 2.58/1.69 = ( append_a @ Xs4 @ Ss ) )
% 2.58/1.69 & ( Ys
% 2.58/1.69 = ( append_a @ Ys2 @ Ss ) )
% 2.58/1.69 & ( ( Xs4 = nil_a )
% 2.58/1.69 | ( Ys2 = nil_a )
% 2.58/1.69 | ( ( last_a @ Xs4 )
% 2.58/1.69 != ( last_a @ Ys2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_86_replicate__0,axiom,
% 2.58/1.69 ! [X: a] :
% 2.58/1.69 ( ( replicate_a @ zero_zero_nat @ X )
% 2.58/1.69 = nil_a ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_87_replicate__0,axiom,
% 2.58/1.69 ! [X: nat] :
% 2.58/1.69 ( ( replicate_nat @ zero_zero_nat @ X )
% 2.58/1.69 = nil_nat ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_88_zero__less__iff__neq__zero,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.58/1.69 = ( N != zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_89_gr__implies__not__zero,axiom,
% 2.58/1.69 ! [M: nat,N: nat] :
% 2.58/1.69 ( ( ord_less_nat @ M @ N )
% 2.58/1.69 => ( N != zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_90_not__less__zero,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_91_gr__zeroI,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ( ( N != zero_zero_nat )
% 2.58/1.69 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_92_zero__le,axiom,
% 2.58/1.69 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_93_strict__sorted_Oinduct,axiom,
% 2.58/1.69 ! [P: list_a > $o,A0: list_a] :
% 2.58/1.69 ( ( P @ nil_a )
% 2.58/1.69 => ( ! [X2: a,Ys3: list_a] :
% 2.58/1.69 ( ( P @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ A0 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_94_strict__sorted_Oinduct,axiom,
% 2.58/1.69 ! [P: list_nat > $o,A0: list_nat] :
% 2.58/1.69 ( ( P @ nil_nat )
% 2.58/1.69 => ( ! [X2: nat,Ys3: list_nat] :
% 2.58/1.69 ( ( P @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ A0 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_95_strict__sorted_Ocases,axiom,
% 2.58/1.69 ! [X: list_a] :
% 2.58/1.69 ( ( X != nil_a )
% 2.58/1.69 => ~ ! [X2: a,Ys3: list_a] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_a @ X2 @ Ys3 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_96_strict__sorted_Ocases,axiom,
% 2.58/1.69 ! [X: list_nat] :
% 2.58/1.69 ( ( X != nil_nat )
% 2.58/1.69 => ~ ! [X2: nat,Ys3: list_nat] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_nat @ X2 @ Ys3 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_97_map__tailrec__rev_Oinduct,axiom,
% 2.58/1.69 ! [P: ( a > a ) > list_a > list_a > $o,A0: a > a,A1: list_a,A2: list_a] :
% 2.58/1.69 ( ! [F: a > a,X_1: list_a] : ( P @ F @ nil_a @ X_1 )
% 2.58/1.69 => ( ! [F: a > a,A3: a,As: list_a,Bs: list_a] :
% 2.58/1.69 ( ( P @ F @ As @ ( cons_a @ ( F @ A3 ) @ Bs ) )
% 2.58/1.69 => ( P @ F @ ( cons_a @ A3 @ As ) @ Bs ) )
% 2.58/1.69 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_98_map__tailrec__rev_Oinduct,axiom,
% 2.58/1.69 ! [P: ( nat > a ) > list_nat > list_a > $o,A0: nat > a,A1: list_nat,A2: list_a] :
% 2.58/1.69 ( ! [F: nat > a,X_1: list_a] : ( P @ F @ nil_nat @ X_1 )
% 2.58/1.69 => ( ! [F: nat > a,A3: nat,As: list_nat,Bs: list_a] :
% 2.58/1.69 ( ( P @ F @ As @ ( cons_a @ ( F @ A3 ) @ Bs ) )
% 2.58/1.69 => ( P @ F @ ( cons_nat @ A3 @ As ) @ Bs ) )
% 2.58/1.69 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_99_map__tailrec__rev_Oinduct,axiom,
% 2.58/1.69 ! [P: ( a > nat ) > list_a > list_nat > $o,A0: a > nat,A1: list_a,A2: list_nat] :
% 2.58/1.69 ( ! [F: a > nat,X_1: list_nat] : ( P @ F @ nil_a @ X_1 )
% 2.58/1.69 => ( ! [F: a > nat,A3: a,As: list_a,Bs: list_nat] :
% 2.58/1.69 ( ( P @ F @ As @ ( cons_nat @ ( F @ A3 ) @ Bs ) )
% 2.58/1.69 => ( P @ F @ ( cons_a @ A3 @ As ) @ Bs ) )
% 2.58/1.69 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_100_map__tailrec__rev_Oinduct,axiom,
% 2.58/1.69 ! [P: ( nat > nat ) > list_nat > list_nat > $o,A0: nat > nat,A1: list_nat,A2: list_nat] :
% 2.58/1.69 ( ! [F: nat > nat,X_1: list_nat] : ( P @ F @ nil_nat @ X_1 )
% 2.58/1.69 => ( ! [F: nat > nat,A3: nat,As: list_nat,Bs: list_nat] :
% 2.58/1.69 ( ( P @ F @ As @ ( cons_nat @ ( F @ A3 ) @ Bs ) )
% 2.58/1.69 => ( P @ F @ ( cons_nat @ A3 @ As ) @ Bs ) )
% 2.58/1.69 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_101_list__nonempty__induct,axiom,
% 2.58/1.69 ! [Xs: list_a,P: list_a > $o] :
% 2.58/1.69 ( ( Xs != nil_a )
% 2.58/1.69 => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a] :
% 2.58/1.69 ( ( Xs3 != nil_a )
% 2.58/1.69 => ( ( P @ Xs3 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) ) ) )
% 2.58/1.69 => ( P @ Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_102_list__nonempty__induct,axiom,
% 2.58/1.69 ! [Xs: list_nat,P: list_nat > $o] :
% 2.58/1.69 ( ( Xs != nil_nat )
% 2.58/1.69 => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat] :
% 2.58/1.69 ( ( Xs3 != nil_nat )
% 2.58/1.69 => ( ( P @ Xs3 )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Xs3 ) ) ) )
% 2.58/1.69 => ( P @ Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_103_successively_Oinduct,axiom,
% 2.58/1.69 ! [P: ( a > a > $o ) > list_a > $o,A0: a > a > $o,A1: list_a] :
% 2.58/1.69 ( ! [P2: a > a > $o] : ( P @ P2 @ nil_a )
% 2.58/1.69 => ( ! [P2: a > a > $o,X2: a] : ( P @ P2 @ ( cons_a @ X2 @ nil_a ) )
% 2.58/1.69 => ( ! [P2: a > a > $o,X2: a,Y2: a,Xs3: list_a] :
% 2.58/1.69 ( ( P @ P2 @ ( cons_a @ Y2 @ Xs3 ) )
% 2.58/1.69 => ( P @ P2 @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_104_successively_Oinduct,axiom,
% 2.58/1.69 ! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
% 2.58/1.69 ( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
% 2.58/1.69 => ( ! [P2: nat > nat > $o,X2: nat] : ( P @ P2 @ ( cons_nat @ X2 @ nil_nat ) )
% 2.58/1.69 => ( ! [P2: nat > nat > $o,X2: nat,Y2: nat,Xs3: list_nat] :
% 2.58/1.69 ( ( P @ P2 @ ( cons_nat @ Y2 @ Xs3 ) )
% 2.58/1.69 => ( P @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_105_remdups__adj_Oinduct,axiom,
% 2.58/1.69 ! [P: list_a > $o,A0: list_a] :
% 2.58/1.69 ( ( P @ nil_a )
% 2.58/1.69 => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
% 2.58/1.69 => ( ! [X2: a,Y2: a,Xs3: list_a] :
% 2.58/1.69 ( ( ( X2 = Y2 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) ) )
% 2.58/1.69 => ( ( ( X2 != Y2 )
% 2.58/1.69 => ( P @ ( cons_a @ Y2 @ Xs3 ) ) )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) )
% 2.58/1.69 => ( P @ A0 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_106_remdups__adj_Oinduct,axiom,
% 2.58/1.69 ! [P: list_nat > $o,A0: list_nat] :
% 2.58/1.69 ( ( P @ nil_nat )
% 2.58/1.69 => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
% 2.58/1.69 => ( ! [X2: nat,Y2: nat,Xs3: list_nat] :
% 2.58/1.69 ( ( ( X2 = Y2 )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Xs3 ) ) )
% 2.58/1.69 => ( ( ( X2 != Y2 )
% 2.58/1.69 => ( P @ ( cons_nat @ Y2 @ Xs3 ) ) )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) )
% 2.58/1.69 => ( P @ A0 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_107_sorted__wrt_Oinduct,axiom,
% 2.58/1.69 ! [P: ( a > a > $o ) > list_a > $o,A0: a > a > $o,A1: list_a] :
% 2.58/1.69 ( ! [P2: a > a > $o] : ( P @ P2 @ nil_a )
% 2.58/1.69 => ( ! [P2: a > a > $o,X2: a,Ys3: list_a] :
% 2.58/1.69 ( ( P @ P2 @ Ys3 )
% 2.58/1.69 => ( P @ P2 @ ( cons_a @ X2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_108_sorted__wrt_Oinduct,axiom,
% 2.58/1.69 ! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
% 2.58/1.69 ( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
% 2.58/1.69 => ( ! [P2: nat > nat > $o,X2: nat,Ys3: list_nat] :
% 2.58/1.69 ( ( P @ P2 @ Ys3 )
% 2.58/1.69 => ( P @ P2 @ ( cons_nat @ X2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_109_remdups__adj_Ocases,axiom,
% 2.58/1.69 ! [X: list_a] :
% 2.58/1.69 ( ( X != nil_a )
% 2.58/1.69 => ( ! [X2: a] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_a @ X2 @ nil_a ) )
% 2.58/1.69 => ~ ! [X2: a,Y2: a,Xs3: list_a] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_110_remdups__adj_Ocases,axiom,
% 2.58/1.69 ! [X: list_nat] :
% 2.58/1.69 ( ( X != nil_nat )
% 2.58/1.69 => ( ! [X2: nat] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_nat @ X2 @ nil_nat ) )
% 2.58/1.69 => ~ ! [X2: nat,Y2: nat,Xs3: list_nat] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_111_shuffles_Oinduct,axiom,
% 2.58/1.69 ! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
% 2.58/1.69 ( ! [X_1: list_a] : ( P @ nil_a @ X_1 )
% 2.58/1.69 => ( ! [Xs3: list_a] : ( P @ Xs3 @ nil_a )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
% 2.58/1.69 ( ( P @ Xs3 @ ( cons_a @ Y2 @ Ys3 ) )
% 2.58/1.69 => ( ( P @ ( cons_a @ X2 @ Xs3 ) @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_112_shuffles_Oinduct,axiom,
% 2.58/1.69 ! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
% 2.58/1.69 ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
% 2.58/1.69 => ( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys3: list_nat] :
% 2.58/1.69 ( ( P @ Xs3 @ ( cons_nat @ Y2 @ Ys3 ) )
% 2.58/1.69 => ( ( P @ ( cons_nat @ X2 @ Xs3 ) @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys3 ) ) ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_113_min__list_Oinduct,axiom,
% 2.58/1.69 ! [P: list_a > $o,A0: list_a] :
% 2.58/1.69 ( ! [X2: a,Xs3: list_a] :
% 2.58/1.69 ( ! [X212: a,X222: list_a] :
% 2.58/1.69 ( ( Xs3
% 2.58/1.69 = ( cons_a @ X212 @ X222 ) )
% 2.58/1.69 => ( P @ Xs3 ) )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) ) )
% 2.58/1.69 => ( ( P @ nil_a )
% 2.58/1.69 => ( P @ A0 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_114_min__list_Oinduct,axiom,
% 2.58/1.69 ! [P: list_nat > $o,A0: list_nat] :
% 2.58/1.69 ( ! [X2: nat,Xs3: list_nat] :
% 2.58/1.69 ( ! [X212: nat,X222: list_nat] :
% 2.58/1.69 ( ( Xs3
% 2.58/1.69 = ( cons_nat @ X212 @ X222 ) )
% 2.58/1.69 => ( P @ Xs3 ) )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Xs3 ) ) )
% 2.58/1.69 => ( ( P @ nil_nat )
% 2.58/1.69 => ( P @ A0 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_115_min__list_Ocases,axiom,
% 2.58/1.69 ! [X: list_a] :
% 2.58/1.69 ( ! [X2: a,Xs3: list_a] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_a @ X2 @ Xs3 ) )
% 2.58/1.69 => ( X = nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_116_min__list_Ocases,axiom,
% 2.58/1.69 ! [X: list_nat] :
% 2.58/1.69 ( ! [X2: nat,Xs3: list_nat] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_nat @ X2 @ Xs3 ) )
% 2.58/1.69 => ( X = nil_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_117_induct__list012,axiom,
% 2.58/1.69 ! [P: list_a > $o,Xs: list_a] :
% 2.58/1.69 ( ( P @ nil_a )
% 2.58/1.69 => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
% 2.58/1.69 => ( ! [X2: a,Y2: a,Zs2: list_a] :
% 2.58/1.69 ( ( P @ Zs2 )
% 2.58/1.69 => ( ( P @ ( cons_a @ Y2 @ Zs2 ) )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Zs2 ) ) ) ) )
% 2.58/1.69 => ( P @ Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_118_induct__list012,axiom,
% 2.58/1.69 ! [P: list_nat > $o,Xs: list_nat] :
% 2.58/1.69 ( ( P @ nil_nat )
% 2.58/1.69 => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
% 2.58/1.69 => ( ! [X2: nat,Y2: nat,Zs2: list_nat] :
% 2.58/1.69 ( ( P @ Zs2 )
% 2.58/1.69 => ( ( P @ ( cons_nat @ Y2 @ Zs2 ) )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Zs2 ) ) ) ) )
% 2.58/1.69 => ( P @ Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_119_splice_Oinduct,axiom,
% 2.58/1.69 ! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
% 2.58/1.69 ( ! [X_1: list_a] : ( P @ nil_a @ X_1 )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a,Ys3: list_a] :
% 2.58/1.69 ( ( P @ Ys3 @ Xs3 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) @ Ys3 ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_120_splice_Oinduct,axiom,
% 2.58/1.69 ! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
% 2.58/1.69 ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat,Ys3: list_nat] :
% 2.58/1.69 ( ( P @ Ys3 @ Xs3 )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ Ys3 ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_121_list__induct2_H,axiom,
% 2.58/1.69 ! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( P @ nil_a @ nil_a )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a] : ( P @ ( cons_a @ X2 @ Xs3 ) @ nil_a )
% 2.58/1.69 => ( ! [Y2: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys3 ) )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
% 2.58/1.69 ( ( P @ Xs3 @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_122_list__induct2_H,axiom,
% 2.58/1.69 ! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
% 2.58/1.69 ( ( P @ nil_a @ nil_nat )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a] : ( P @ ( cons_a @ X2 @ Xs3 ) @ nil_nat )
% 2.58/1.69 => ( ! [Y2: nat,Ys3: list_nat] : ( P @ nil_a @ ( cons_nat @ Y2 @ Ys3 ) )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a,Y2: nat,Ys3: list_nat] :
% 2.58/1.69 ( ( P @ Xs3 @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_123_list__induct2_H,axiom,
% 2.58/1.69 ! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
% 2.58/1.69 ( ( P @ nil_nat @ nil_a )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X2 @ Xs3 ) @ nil_a )
% 2.58/1.69 => ( ! [Y2: a,Ys3: list_a] : ( P @ nil_nat @ ( cons_a @ Y2 @ Ys3 ) )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat,Y2: a,Ys3: list_a] :
% 2.58/1.69 ( ( P @ Xs3 @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_124_list__induct2_H,axiom,
% 2.58/1.69 ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ( ( P @ nil_nat @ nil_nat )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X2 @ Xs3 ) @ nil_nat )
% 2.58/1.69 => ( ! [Y2: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys3 ) )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys3: list_nat] :
% 2.58/1.69 ( ( P @ Xs3 @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_125_neq__Nil__conv,axiom,
% 2.58/1.69 ! [Xs: list_a] :
% 2.58/1.69 ( ( Xs != nil_a )
% 2.58/1.69 = ( ? [Y3: a,Ys4: list_a] :
% 2.58/1.69 ( Xs
% 2.58/1.69 = ( cons_a @ Y3 @ Ys4 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_126_neq__Nil__conv,axiom,
% 2.58/1.69 ! [Xs: list_nat] :
% 2.58/1.69 ( ( Xs != nil_nat )
% 2.58/1.69 = ( ? [Y3: nat,Ys4: list_nat] :
% 2.58/1.69 ( Xs
% 2.58/1.69 = ( cons_nat @ Y3 @ Ys4 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_127_list_Oinducts,axiom,
% 2.58/1.69 ! [P: list_a > $o,List: list_a] :
% 2.58/1.69 ( ( P @ nil_a )
% 2.58/1.69 => ( ! [X1: a,X23: list_a] :
% 2.58/1.69 ( ( P @ X23 )
% 2.58/1.69 => ( P @ ( cons_a @ X1 @ X23 ) ) )
% 2.58/1.69 => ( P @ List ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_128_list_Oinducts,axiom,
% 2.58/1.69 ! [P: list_nat > $o,List: list_nat] :
% 2.58/1.69 ( ( P @ nil_nat )
% 2.58/1.69 => ( ! [X1: nat,X23: list_nat] :
% 2.58/1.69 ( ( P @ X23 )
% 2.58/1.69 => ( P @ ( cons_nat @ X1 @ X23 ) ) )
% 2.58/1.69 => ( P @ List ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_129_list_Oexhaust,axiom,
% 2.58/1.69 ! [Y: list_a] :
% 2.58/1.69 ( ( Y != nil_a )
% 2.58/1.69 => ~ ! [X213: a,X223: list_a] :
% 2.58/1.69 ( Y
% 2.58/1.69 != ( cons_a @ X213 @ X223 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_130_list_Oexhaust,axiom,
% 2.58/1.69 ! [Y: list_nat] :
% 2.58/1.69 ( ( Y != nil_nat )
% 2.58/1.69 => ~ ! [X213: nat,X223: list_nat] :
% 2.58/1.69 ( Y
% 2.58/1.69 != ( cons_nat @ X213 @ X223 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_131_list_OdiscI,axiom,
% 2.58/1.69 ! [List: list_a,X21: a,X22: list_a] :
% 2.58/1.69 ( ( List
% 2.58/1.69 = ( cons_a @ X21 @ X22 ) )
% 2.58/1.69 => ( List != nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_132_list_OdiscI,axiom,
% 2.58/1.69 ! [List: list_nat,X21: nat,X22: list_nat] :
% 2.58/1.69 ( ( List
% 2.58/1.69 = ( cons_nat @ X21 @ X22 ) )
% 2.58/1.69 => ( List != nil_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_133_list_Odistinct_I1_J,axiom,
% 2.58/1.69 ! [X21: a,X22: list_a] :
% 2.58/1.69 ( nil_a
% 2.58/1.69 != ( cons_a @ X21 @ X22 ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_134_list_Odistinct_I1_J,axiom,
% 2.58/1.69 ! [X21: nat,X22: list_nat] :
% 2.58/1.69 ( nil_nat
% 2.58/1.69 != ( cons_nat @ X21 @ X22 ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_135_append_Oleft__neutral,axiom,
% 2.58/1.69 ! [A: list_nat] :
% 2.58/1.69 ( ( append_nat @ nil_nat @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_136_append_Oleft__neutral,axiom,
% 2.58/1.69 ! [A: list_a] :
% 2.58/1.69 ( ( append_a @ nil_a @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_137_append__Nil,axiom,
% 2.58/1.69 ! [Ys: list_nat] :
% 2.58/1.69 ( ( append_nat @ nil_nat @ Ys )
% 2.58/1.69 = Ys ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_138_append__Nil,axiom,
% 2.58/1.69 ! [Ys: list_a] :
% 2.58/1.69 ( ( append_a @ nil_a @ Ys )
% 2.58/1.69 = Ys ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_139_eq__Nil__appendI,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ( ( Xs = Ys )
% 2.58/1.69 => ( Xs
% 2.58/1.69 = ( append_nat @ nil_nat @ Ys ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_140_eq__Nil__appendI,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( Xs = Ys )
% 2.58/1.69 => ( Xs
% 2.58/1.69 = ( append_a @ nil_a @ Ys ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_141_psums_Ocases,axiom,
% 2.58/1.69 ! [X: list_a] :
% 2.58/1.69 ( ( X != nil_a )
% 2.58/1.69 => ( ! [X2: a] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_a @ X2 @ nil_a ) )
% 2.58/1.69 => ~ ! [X2: a,Y2: a,Xs3: list_a] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_142_psums_Ocases,axiom,
% 2.58/1.69 ! [X: list_nat] :
% 2.58/1.69 ( ( X != nil_nat )
% 2.58/1.69 => ( ! [X2: nat] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_nat @ X2 @ nil_nat ) )
% 2.58/1.69 => ~ ! [X2: nat,Y2: nat,Xs3: list_nat] :
% 2.58/1.69 ( X
% 2.58/1.69 != ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_143_rev__nonempty__induct,axiom,
% 2.58/1.69 ! [Xs: list_a,P: list_a > $o] :
% 2.58/1.69 ( ( Xs != nil_a )
% 2.58/1.69 => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a] :
% 2.58/1.69 ( ( Xs3 != nil_a )
% 2.58/1.69 => ( ( P @ Xs3 )
% 2.58/1.69 => ( P @ ( append_a @ Xs3 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
% 2.58/1.69 => ( P @ Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_144_rev__nonempty__induct,axiom,
% 2.58/1.69 ! [Xs: list_nat,P: list_nat > $o] :
% 2.58/1.69 ( ( Xs != nil_nat )
% 2.58/1.69 => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat] :
% 2.58/1.69 ( ( Xs3 != nil_nat )
% 2.58/1.69 => ( ( P @ Xs3 )
% 2.58/1.69 => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
% 2.58/1.69 => ( P @ Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_145_append__eq__Cons__conv,axiom,
% 2.58/1.69 ! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
% 2.58/1.69 ( ( ( append_a @ Ys @ Zs )
% 2.58/1.69 = ( cons_a @ X @ Xs ) )
% 2.58/1.69 = ( ( ( Ys = nil_a )
% 2.58/1.69 & ( Zs
% 2.58/1.69 = ( cons_a @ X @ Xs ) ) )
% 2.58/1.69 | ? [Ys5: list_a] :
% 2.58/1.69 ( ( Ys
% 2.58/1.69 = ( cons_a @ X @ Ys5 ) )
% 2.58/1.69 & ( ( append_a @ Ys5 @ Zs )
% 2.58/1.69 = Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_146_append__eq__Cons__conv,axiom,
% 2.58/1.69 ! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
% 2.58/1.69 ( ( ( append_nat @ Ys @ Zs )
% 2.58/1.69 = ( cons_nat @ X @ Xs ) )
% 2.58/1.69 = ( ( ( Ys = nil_nat )
% 2.58/1.69 & ( Zs
% 2.58/1.69 = ( cons_nat @ X @ Xs ) ) )
% 2.58/1.69 | ? [Ys5: list_nat] :
% 2.58/1.69 ( ( Ys
% 2.58/1.69 = ( cons_nat @ X @ Ys5 ) )
% 2.58/1.69 & ( ( append_nat @ Ys5 @ Zs )
% 2.58/1.69 = Xs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_147_Cons__eq__append__conv,axiom,
% 2.58/1.69 ! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
% 2.58/1.69 ( ( ( cons_a @ X @ Xs )
% 2.58/1.69 = ( append_a @ Ys @ Zs ) )
% 2.58/1.69 = ( ( ( Ys = nil_a )
% 2.58/1.69 & ( ( cons_a @ X @ Xs )
% 2.58/1.69 = Zs ) )
% 2.58/1.69 | ? [Ys5: list_a] :
% 2.58/1.69 ( ( ( cons_a @ X @ Ys5 )
% 2.58/1.69 = Ys )
% 2.58/1.69 & ( Xs
% 2.58/1.69 = ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_148_Cons__eq__append__conv,axiom,
% 2.58/1.69 ! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
% 2.58/1.69 ( ( ( cons_nat @ X @ Xs )
% 2.58/1.69 = ( append_nat @ Ys @ Zs ) )
% 2.58/1.69 = ( ( ( Ys = nil_nat )
% 2.58/1.69 & ( ( cons_nat @ X @ Xs )
% 2.58/1.69 = Zs ) )
% 2.58/1.69 | ? [Ys5: list_nat] :
% 2.58/1.69 ( ( ( cons_nat @ X @ Ys5 )
% 2.58/1.69 = Ys )
% 2.58/1.69 & ( Xs
% 2.58/1.69 = ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_149_rev__exhaust,axiom,
% 2.58/1.69 ! [Xs: list_a] :
% 2.58/1.69 ( ( Xs != nil_a )
% 2.58/1.69 => ~ ! [Ys3: list_a,Y2: a] :
% 2.58/1.69 ( Xs
% 2.58/1.69 != ( append_a @ Ys3 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_150_rev__exhaust,axiom,
% 2.58/1.69 ! [Xs: list_nat] :
% 2.58/1.69 ( ( Xs != nil_nat )
% 2.58/1.69 => ~ ! [Ys3: list_nat,Y2: nat] :
% 2.58/1.69 ( Xs
% 2.58/1.69 != ( append_nat @ Ys3 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_151_rev__induct,axiom,
% 2.58/1.69 ! [P: list_a > $o,Xs: list_a] :
% 2.58/1.69 ( ( P @ nil_a )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a] :
% 2.58/1.69 ( ( P @ Xs3 )
% 2.58/1.69 => ( P @ ( append_a @ Xs3 @ ( cons_a @ X2 @ nil_a ) ) ) )
% 2.58/1.69 => ( P @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_152_rev__induct,axiom,
% 2.58/1.69 ! [P: list_nat > $o,Xs: list_nat] :
% 2.58/1.69 ( ( P @ nil_nat )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat] :
% 2.58/1.69 ( ( P @ Xs3 )
% 2.58/1.69 => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X2 @ nil_nat ) ) ) )
% 2.58/1.69 => ( P @ Xs ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_153_psums_Osimps_I1_J,axiom,
% 2.58/1.69 ( ( descar1375166517sums_a @ nil_a )
% 2.58/1.69 = nil_a ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_154_psums_Osimps_I1_J,axiom,
% 2.58/1.69 ( ( descar226543321ms_nat @ nil_nat )
% 2.58/1.69 = nil_nat ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_155_replicate__append__same,axiom,
% 2.58/1.69 ! [I: nat,X: a] :
% 2.58/1.69 ( ( append_a @ ( replicate_a @ I @ X ) @ ( cons_a @ X @ nil_a ) )
% 2.58/1.69 = ( cons_a @ X @ ( replicate_a @ I @ X ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_156_replicate__append__same,axiom,
% 2.58/1.69 ! [I: nat,X: nat] :
% 2.58/1.69 ( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
% 2.58/1.69 = ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_157_psums_Osimps_I2_J,axiom,
% 2.58/1.69 ! [X: a] :
% 2.58/1.69 ( ( descar1375166517sums_a @ ( cons_a @ X @ nil_a ) )
% 2.58/1.69 = ( cons_a @ X @ nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_158_psums_Osimps_I2_J,axiom,
% 2.58/1.69 ! [X: nat] :
% 2.58/1.69 ( ( descar226543321ms_nat @ ( cons_nat @ X @ nil_nat ) )
% 2.58/1.69 = ( cons_nat @ X @ nil_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_159_zero__reorient,axiom,
% 2.58/1.69 ! [X: a] :
% 2.58/1.69 ( ( zero_zero_a = X )
% 2.58/1.69 = ( X = zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_160_zero__reorient,axiom,
% 2.58/1.69 ! [X: nat] :
% 2.58/1.69 ( ( zero_zero_nat = X )
% 2.58/1.69 = ( X = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_161_not__Cons__self2,axiom,
% 2.58/1.69 ! [X: a,Xs: list_a] :
% 2.58/1.69 ( ( cons_a @ X @ Xs )
% 2.58/1.69 != Xs ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_162_not__Cons__self2,axiom,
% 2.58/1.69 ! [X: nat,Xs: list_nat] :
% 2.58/1.69 ( ( cons_nat @ X @ Xs )
% 2.58/1.69 != Xs ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_163_append__eq__append__conv2,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
% 2.58/1.69 ( ( ( append_a @ Xs @ Ys )
% 2.58/1.69 = ( append_a @ Zs @ Ts ) )
% 2.58/1.69 = ( ? [Us: list_a] :
% 2.58/1.69 ( ( ( Xs
% 2.58/1.69 = ( append_a @ Zs @ Us ) )
% 2.58/1.69 & ( ( append_a @ Us @ Ys )
% 2.58/1.69 = Ts ) )
% 2.58/1.69 | ( ( ( append_a @ Xs @ Us )
% 2.58/1.69 = Zs )
% 2.58/1.69 & ( Ys
% 2.58/1.69 = ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_164_append__eq__append__conv2,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
% 2.58/1.69 ( ( ( append_nat @ Xs @ Ys )
% 2.58/1.69 = ( append_nat @ Zs @ Ts ) )
% 2.58/1.69 = ( ? [Us: list_nat] :
% 2.58/1.69 ( ( ( Xs
% 2.58/1.69 = ( append_nat @ Zs @ Us ) )
% 2.58/1.69 & ( ( append_nat @ Us @ Ys )
% 2.58/1.69 = Ts ) )
% 2.58/1.69 | ( ( ( append_nat @ Xs @ Us )
% 2.58/1.69 = Zs )
% 2.58/1.69 & ( Ys
% 2.58/1.69 = ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_165_append__eq__appendI,axiom,
% 2.58/1.69 ! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
% 2.58/1.69 ( ( ( append_a @ Xs @ Xs1 )
% 2.58/1.69 = Zs )
% 2.58/1.69 => ( ( Ys
% 2.58/1.69 = ( append_a @ Xs1 @ Us2 ) )
% 2.58/1.69 => ( ( append_a @ Xs @ Ys )
% 2.58/1.69 = ( append_a @ Zs @ Us2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_166_append__eq__appendI,axiom,
% 2.58/1.69 ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
% 2.58/1.69 ( ( ( append_nat @ Xs @ Xs1 )
% 2.58/1.69 = Zs )
% 2.58/1.69 => ( ( Ys
% 2.58/1.69 = ( append_nat @ Xs1 @ Us2 ) )
% 2.58/1.69 => ( ( append_nat @ Xs @ Ys )
% 2.58/1.69 = ( append_nat @ Zs @ Us2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_167_append__Cons,axiom,
% 2.58/1.69 ! [X: a,Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
% 2.58/1.69 = ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_168_append__Cons,axiom,
% 2.58/1.69 ! [X: nat,Xs: list_nat,Ys: list_nat] :
% 2.58/1.69 ( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
% 2.58/1.69 = ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_169_v__decompose,axiom,
% 2.58/1.69 ! [Xs5: list_a] :
% 2.58/1.69 ( ( v @ ( append_a @ Xs5 @ xsa ) )
% 2.58/1.69 = ( plus_plus_int @ ( v @ ( append_a @ Xs5 @ ( cons_a @ ( nth_a @ xsa @ p ) @ nil_a ) ) ) @ ( v @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_170_p__less__length,axiom,
% 2.58/1.69 ord_less_nat @ p @ ( size_size_list_a @ xsa ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_171_neq0__conv,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ( ( N != zero_zero_nat )
% 2.58/1.69 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_172_less__nat__zero__code,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_173_bot__nat__0_Onot__eq__extremum,axiom,
% 2.58/1.69 ! [A: nat] :
% 2.58/1.69 ( ( A != zero_zero_nat )
% 2.58/1.69 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_174_order__refl,axiom,
% 2.58/1.69 ! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_175_order__refl,axiom,
% 2.58/1.69 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_176_minus__poly__rev__list_Oinduct,axiom,
% 2.58/1.69 ! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
% 2.58/1.69 ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
% 2.58/1.69 ( ( P @ Xs3 @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
% 2.58/1.69 => ( ! [Xs3: list_a] : ( P @ Xs3 @ nil_a )
% 2.58/1.69 => ( ! [Y2: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys3 ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_177_plus__coeffs_Oinduct,axiom,
% 2.58/1.69 ! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
% 2.58/1.69 ( ! [Xs3: list_a] : ( P @ Xs3 @ nil_a )
% 2.58/1.69 => ( ! [V: a,Va: list_a] : ( P @ nil_a @ ( cons_a @ V @ Va ) )
% 2.58/1.69 => ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
% 2.58/1.69 ( ( P @ Xs3 @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_178_plus__coeffs_Oinduct,axiom,
% 2.58/1.69 ! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
% 2.58/1.69 ( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
% 2.58/1.69 => ( ! [V: nat,Va: list_nat] : ( P @ nil_nat @ ( cons_nat @ V @ Va ) )
% 2.58/1.69 => ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys3: list_nat] :
% 2.58/1.69 ( ( P @ Xs3 @ Ys3 )
% 2.58/1.69 => ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys3 ) ) )
% 2.58/1.69 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_179_minf_I8_J,axiom,
% 2.58/1.69 ! [T: a] :
% 2.58/1.69 ? [Z: a] :
% 2.58/1.69 ! [X3: a] :
% 2.58/1.69 ( ( ord_less_a @ X3 @ Z )
% 2.58/1.69 => ~ ( ord_less_eq_a @ T @ X3 ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_180_minf_I8_J,axiom,
% 2.58/1.69 ! [T: nat] :
% 2.58/1.69 ? [Z: nat] :
% 2.58/1.69 ! [X3: nat] :
% 2.58/1.69 ( ( ord_less_nat @ X3 @ Z )
% 2.58/1.69 => ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_181_minf_I6_J,axiom,
% 2.58/1.69 ! [T: a] :
% 2.58/1.69 ? [Z: a] :
% 2.58/1.69 ! [X3: a] :
% 2.58/1.69 ( ( ord_less_a @ X3 @ Z )
% 2.58/1.69 => ( ord_less_eq_a @ X3 @ T ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_182_minf_I6_J,axiom,
% 2.58/1.69 ! [T: nat] :
% 2.58/1.69 ? [Z: nat] :
% 2.58/1.69 ! [X3: nat] :
% 2.58/1.69 ( ( ord_less_nat @ X3 @ Z )
% 2.58/1.69 => ( ord_less_eq_nat @ X3 @ T ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_183_add__left__cancel,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ A @ B )
% 2.58/1.69 = ( plus_plus_int @ A @ C ) )
% 2.58/1.69 = ( B = C ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_184_add__right__cancel,axiom,
% 2.58/1.69 ! [B: int,A: int,C: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ B @ A )
% 2.58/1.69 = ( plus_plus_int @ C @ A ) )
% 2.58/1.69 = ( B = C ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_185_le0,axiom,
% 2.58/1.69 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_186_bot__nat__0_Oextremum,axiom,
% 2.58/1.69 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_187_add_Oleft__neutral,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( plus_plus_int @ zero_zero_int @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_188_add_Oleft__neutral,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( plus_plus_a @ zero_zero_a @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_189_add_Oleft__neutral,axiom,
% 2.58/1.69 ! [A: nat] :
% 2.58/1.69 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_190_add_Oright__neutral,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( plus_plus_int @ A @ zero_zero_int )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_191_add_Oright__neutral,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( plus_plus_a @ A @ zero_zero_a )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_192_add_Oright__neutral,axiom,
% 2.58/1.69 ! [A: nat] :
% 2.58/1.69 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_193_double__zero,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ A @ A )
% 2.58/1.69 = zero_zero_int )
% 2.58/1.69 = ( A = zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_194_double__zero,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( ( plus_plus_a @ A @ A )
% 2.58/1.69 = zero_zero_a )
% 2.58/1.69 = ( A = zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_195_double__zero__sym,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( zero_zero_int
% 2.58/1.69 = ( plus_plus_int @ A @ A ) )
% 2.58/1.69 = ( A = zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_196_double__zero__sym,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( zero_zero_a
% 2.58/1.69 = ( plus_plus_a @ A @ A ) )
% 2.58/1.69 = ( A = zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_197_add__cancel__left__left,axiom,
% 2.58/1.69 ! [B: int,A: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ B @ A )
% 2.58/1.69 = A )
% 2.58/1.69 = ( B = zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_198_add__cancel__left__left,axiom,
% 2.58/1.69 ! [B: a,A: a] :
% 2.58/1.69 ( ( ( plus_plus_a @ B @ A )
% 2.58/1.69 = A )
% 2.58/1.69 = ( B = zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_199_add__cancel__left__left,axiom,
% 2.58/1.69 ! [B: nat,A: nat] :
% 2.58/1.69 ( ( ( plus_plus_nat @ B @ A )
% 2.58/1.69 = A )
% 2.58/1.69 = ( B = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_200_add__cancel__left__right,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ A @ B )
% 2.58/1.69 = A )
% 2.58/1.69 = ( B = zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_201_add__cancel__left__right,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( ( plus_plus_a @ A @ B )
% 2.58/1.69 = A )
% 2.58/1.69 = ( B = zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_202_add__cancel__left__right,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( ( plus_plus_nat @ A @ B )
% 2.58/1.69 = A )
% 2.58/1.69 = ( B = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_203_add__cancel__right__left,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( A
% 2.58/1.69 = ( plus_plus_int @ B @ A ) )
% 2.58/1.69 = ( B = zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_204_add__cancel__right__left,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( A
% 2.58/1.69 = ( plus_plus_a @ B @ A ) )
% 2.58/1.69 = ( B = zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_205_add__cancel__right__left,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( A
% 2.58/1.69 = ( plus_plus_nat @ B @ A ) )
% 2.58/1.69 = ( B = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_206_add__cancel__right__right,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( A
% 2.58/1.69 = ( plus_plus_int @ A @ B ) )
% 2.58/1.69 = ( B = zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_207_add__cancel__right__right,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( A
% 2.58/1.69 = ( plus_plus_a @ A @ B ) )
% 2.58/1.69 = ( B = zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_208_add__cancel__right__right,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( A
% 2.58/1.69 = ( plus_plus_nat @ A @ B ) )
% 2.58/1.69 = ( B = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_209_add__eq__0__iff__both__eq__0,axiom,
% 2.58/1.69 ! [X: nat,Y: nat] :
% 2.58/1.69 ( ( ( plus_plus_nat @ X @ Y )
% 2.58/1.69 = zero_zero_nat )
% 2.58/1.69 = ( ( X = zero_zero_nat )
% 2.58/1.69 & ( Y = zero_zero_nat ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_210_zero__eq__add__iff__both__eq__0,axiom,
% 2.58/1.69 ! [X: nat,Y: nat] :
% 2.58/1.69 ( ( zero_zero_nat
% 2.58/1.69 = ( plus_plus_nat @ X @ Y ) )
% 2.58/1.69 = ( ( X = zero_zero_nat )
% 2.58/1.69 & ( Y = zero_zero_nat ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_211_add__le__cancel__right,axiom,
% 2.58/1.69 ! [A: int,C: int,B: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 2.58/1.69 = ( ord_less_eq_int @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_212_add__le__cancel__right,axiom,
% 2.58/1.69 ! [A: a,C: a,B: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 2.58/1.69 = ( ord_less_eq_a @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_213_add__le__cancel__right,axiom,
% 2.58/1.69 ! [A: nat,C: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.58/1.69 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_214_add__le__cancel__left,axiom,
% 2.58/1.69 ! [C: int,A: int,B: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 2.58/1.69 = ( ord_less_eq_int @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_215_add__le__cancel__left,axiom,
% 2.58/1.69 ! [C: a,A: a,B: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 2.58/1.69 = ( ord_less_eq_a @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_216_add__le__cancel__left,axiom,
% 2.58/1.69 ! [C: nat,A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.58/1.69 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_217_add__less__cancel__right,axiom,
% 2.58/1.69 ! [A: int,C: int,B: int] :
% 2.58/1.69 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 2.58/1.69 = ( ord_less_int @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_218_add__less__cancel__right,axiom,
% 2.58/1.69 ! [A: nat,C: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.58/1.69 = ( ord_less_nat @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_219_add__less__cancel__left,axiom,
% 2.58/1.69 ! [C: int,A: int,B: int] :
% 2.58/1.69 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 2.58/1.69 = ( ord_less_int @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_220_add__less__cancel__left,axiom,
% 2.58/1.69 ! [C: nat,A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.58/1.69 = ( ord_less_nat @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_221_append__eq__append__conv,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs: list_nat] :
% 2.58/1.69 ( ( ( ( size_size_list_nat @ Xs )
% 2.58/1.69 = ( size_size_list_nat @ Ys ) )
% 2.58/1.69 | ( ( size_size_list_nat @ Us2 )
% 2.58/1.69 = ( size_size_list_nat @ Vs ) ) )
% 2.58/1.69 => ( ( ( append_nat @ Xs @ Us2 )
% 2.58/1.69 = ( append_nat @ Ys @ Vs ) )
% 2.58/1.69 = ( ( Xs = Ys )
% 2.58/1.69 & ( Us2 = Vs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_222_append__eq__append__conv,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a,Us2: list_a,Vs: list_a] :
% 2.58/1.69 ( ( ( ( size_size_list_a @ Xs )
% 2.58/1.69 = ( size_size_list_a @ Ys ) )
% 2.58/1.69 | ( ( size_size_list_a @ Us2 )
% 2.58/1.69 = ( size_size_list_a @ Vs ) ) )
% 2.58/1.69 => ( ( ( append_a @ Xs @ Us2 )
% 2.58/1.69 = ( append_a @ Ys @ Vs ) )
% 2.58/1.69 = ( ( Xs = Ys )
% 2.58/1.69 & ( Us2 = Vs ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_223_take__all__iff,axiom,
% 2.58/1.69 ! [N: nat,Xs: list_a] :
% 2.58/1.69 ( ( ( take_a @ N @ Xs )
% 2.58/1.69 = Xs )
% 2.58/1.69 = ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_224_take__all,axiom,
% 2.58/1.69 ! [Xs: list_a,N: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 2.58/1.69 => ( ( take_a @ N @ Xs )
% 2.58/1.69 = Xs ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_225_length__replicate,axiom,
% 2.58/1.69 ! [N: nat,X: nat] :
% 2.58/1.69 ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
% 2.58/1.69 = N ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_226_length__replicate,axiom,
% 2.58/1.69 ! [N: nat,X: a] :
% 2.58/1.69 ( ( size_size_list_a @ ( replicate_a @ N @ X ) )
% 2.58/1.69 = N ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_227_length__psums,axiom,
% 2.58/1.69 ! [Xs: list_a] :
% 2.58/1.69 ( ( size_size_list_a @ ( descar1375166517sums_a @ Xs ) )
% 2.58/1.69 = ( size_size_list_a @ Xs ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_228_length__psums,axiom,
% 2.58/1.69 ! [Xs: list_nat] :
% 2.58/1.69 ( ( size_size_list_nat @ ( descar226543321ms_nat @ Xs ) )
% 2.58/1.69 = ( size_size_list_nat @ Xs ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_229_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 2.58/1.69 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_230_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
% 2.58/1.69 = ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_231_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 2.58/1.69 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_232_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
% 2.58/1.69 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_233_le__add__same__cancel2,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 2.58/1.69 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_234_le__add__same__cancel2,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ A @ ( plus_plus_a @ B @ A ) )
% 2.58/1.69 = ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_235_le__add__same__cancel2,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 2.58/1.69 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_236_le__add__same__cancel1,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 2.58/1.69 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_237_le__add__same__cancel1,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ A @ ( plus_plus_a @ A @ B ) )
% 2.58/1.69 = ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_238_le__add__same__cancel1,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.58/1.69 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_239_add__le__same__cancel2,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 2.58/1.69 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_240_add__le__same__cancel2,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ B )
% 2.58/1.69 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_241_add__le__same__cancel2,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.58/1.69 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_242_add__le__same__cancel1,axiom,
% 2.58/1.69 ! [B: int,A: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 2.58/1.69 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_243_add__le__same__cancel1,axiom,
% 2.58/1.69 ! [B: a,A: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ ( plus_plus_a @ B @ A ) @ B )
% 2.58/1.69 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_244_add__le__same__cancel1,axiom,
% 2.58/1.69 ! [B: nat,A: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 2.58/1.69 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_245_zero__less__double__add__iff__zero__less__single__add,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 2.58/1.69 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_246_zero__less__double__add__iff__zero__less__single__add,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
% 2.58/1.69 = ( ord_less_a @ zero_zero_a @ A ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_247_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 2.58/1.69 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_248_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( ord_less_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
% 2.58/1.69 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_249_less__add__same__cancel2,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 2.58/1.69 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_250_less__add__same__cancel2,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( ord_less_a @ A @ ( plus_plus_a @ B @ A ) )
% 2.58/1.69 = ( ord_less_a @ zero_zero_a @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_251_less__add__same__cancel2,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 2.58/1.69 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_252_less__add__same__cancel1,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 2.58/1.69 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_253_less__add__same__cancel1,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( ord_less_a @ A @ ( plus_plus_a @ A @ B ) )
% 2.58/1.69 = ( ord_less_a @ zero_zero_a @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_254_less__add__same__cancel1,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.58/1.69 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_255_add__less__same__cancel2,axiom,
% 2.58/1.69 ! [A: int,B: int] :
% 2.58/1.69 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 2.58/1.69 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_256_add__less__same__cancel2,axiom,
% 2.58/1.69 ! [A: a,B: a] :
% 2.58/1.69 ( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ B )
% 2.58/1.69 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_257_add__less__same__cancel2,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.58/1.69 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_258_add__less__same__cancel1,axiom,
% 2.58/1.69 ! [B: int,A: int] :
% 2.58/1.69 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 2.58/1.69 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_259_add__less__same__cancel1,axiom,
% 2.58/1.69 ! [B: a,A: a] :
% 2.58/1.69 ( ( ord_less_a @ ( plus_plus_a @ B @ A ) @ B )
% 2.58/1.69 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_260_add__less__same__cancel1,axiom,
% 2.58/1.69 ! [B: nat,A: nat] :
% 2.58/1.69 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 2.58/1.69 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_261_length__0__conv,axiom,
% 2.58/1.69 ! [Xs: list_a] :
% 2.58/1.69 ( ( ( size_size_list_a @ Xs )
% 2.58/1.69 = zero_zero_nat )
% 2.58/1.69 = ( Xs = nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_262_length__greater__0__conv,axiom,
% 2.58/1.69 ! [Xs: list_a] :
% 2.58/1.69 ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
% 2.58/1.69 = ( Xs != nil_a ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_263_nth__append__length,axiom,
% 2.58/1.69 ! [Xs: list_nat,X: nat,Ys: list_nat] :
% 2.58/1.69 ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
% 2.58/1.69 = X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_264_nth__append__length,axiom,
% 2.58/1.69 ! [Xs: list_a,X: a,Ys: list_a] :
% 2.58/1.69 ( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
% 2.58/1.69 = X ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_265_less__eq__nat_Osimps_I1_J,axiom,
% 2.58/1.69 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_266_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 2.58/1.69 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_267_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.58/1.69 ! [I: int,J: int,K: int,L: int] :
% 2.58/1.69 ( ( ( I = J )
% 2.58/1.69 & ( K = L ) )
% 2.58/1.69 => ( ( plus_plus_int @ I @ K )
% 2.58/1.69 = ( plus_plus_int @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_268_le__0__eq,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.58/1.69 = ( N = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_269_impossible__Cons,axiom,
% 2.58/1.69 ! [Xs: list_nat,Ys: list_nat,X: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
% 2.58/1.69 => ( Xs
% 2.58/1.69 != ( cons_nat @ X @ Ys ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_270_impossible__Cons,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a,X: a] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
% 2.58/1.69 => ( Xs
% 2.58/1.69 != ( cons_a @ X @ Ys ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_271_Ex__list__of__length,axiom,
% 2.58/1.69 ! [N: nat] :
% 2.58/1.69 ? [Xs3: list_a] :
% 2.58/1.69 ( ( size_size_list_a @ Xs3 )
% 2.58/1.69 = N ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_272_neq__if__length__neq,axiom,
% 2.58/1.69 ! [Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( ( size_size_list_a @ Xs )
% 2.58/1.69 != ( size_size_list_a @ Ys ) )
% 2.58/1.69 => ( Xs != Ys ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_273_group__cancel_Oadd1,axiom,
% 2.58/1.69 ! [A4: int,K: int,A: int,B: int] :
% 2.58/1.69 ( ( A4
% 2.58/1.69 = ( plus_plus_int @ K @ A ) )
% 2.58/1.69 => ( ( plus_plus_int @ A4 @ B )
% 2.58/1.69 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_274_group__cancel_Oadd2,axiom,
% 2.58/1.69 ! [B2: int,K: int,B: int,A: int] :
% 2.58/1.69 ( ( B2
% 2.58/1.69 = ( plus_plus_int @ K @ B ) )
% 2.58/1.69 => ( ( plus_plus_int @ A @ B2 )
% 2.58/1.69 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_275_bot__nat__0_Oextremum__unique,axiom,
% 2.58/1.69 ! [A: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.58/1.69 = ( A = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_276_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.58/1.69 ! [A: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.58/1.69 => ( A = zero_zero_nat ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_277_add_Oassoc,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 2.58/1.69 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_278_add_Oleft__cancel,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ A @ B )
% 2.58/1.69 = ( plus_plus_int @ A @ C ) )
% 2.58/1.69 = ( B = C ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_279_add_Oright__cancel,axiom,
% 2.58/1.69 ! [B: int,A: int,C: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ B @ A )
% 2.58/1.69 = ( plus_plus_int @ C @ A ) )
% 2.58/1.69 = ( B = C ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_280_add_Ocommute,axiom,
% 2.58/1.69 ( plus_plus_int
% 2.58/1.69 = ( ^ [A5: int,B3: int] : ( plus_plus_int @ B3 @ A5 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_281_add_Oleft__commute,axiom,
% 2.58/1.69 ! [B: int,A: int,C: int] :
% 2.58/1.69 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 2.58/1.69 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_282_add__left__imp__eq,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ A @ B )
% 2.58/1.69 = ( plus_plus_int @ A @ C ) )
% 2.58/1.69 => ( B = C ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_283_add__right__imp__eq,axiom,
% 2.58/1.69 ! [B: int,A: int,C: int] :
% 2.58/1.69 ( ( ( plus_plus_int @ B @ A )
% 2.58/1.69 = ( plus_plus_int @ C @ A ) )
% 2.58/1.69 => ( B = C ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_284_size__neq__size__imp__neq,axiom,
% 2.58/1.69 ! [X: list_a,Y: list_a] :
% 2.58/1.69 ( ( ( size_size_list_a @ X )
% 2.58/1.69 != ( size_size_list_a @ Y ) )
% 2.58/1.69 => ( X != Y ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_285_less__mono__imp__le__mono,axiom,
% 2.58/1.69 ! [F2: nat > nat,I: nat,J: nat] :
% 2.58/1.69 ( ! [I2: nat,J2: nat] :
% 2.58/1.69 ( ( ord_less_nat @ I2 @ J2 )
% 2.58/1.69 => ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
% 2.58/1.69 => ( ( ord_less_eq_nat @ I @ J )
% 2.58/1.69 => ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_286_le__neq__implies__less,axiom,
% 2.58/1.69 ! [M: nat,N: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ M @ N )
% 2.58/1.69 => ( ( M != N )
% 2.58/1.69 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_287_less__or__eq__imp__le,axiom,
% 2.58/1.69 ! [M: nat,N: nat] :
% 2.58/1.69 ( ( ( ord_less_nat @ M @ N )
% 2.58/1.69 | ( M = N ) )
% 2.58/1.69 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_288_le__eq__less__or__eq,axiom,
% 2.58/1.69 ( ord_less_eq_nat
% 2.58/1.69 = ( ^ [M2: nat,N2: nat] :
% 2.58/1.69 ( ( ord_less_nat @ M2 @ N2 )
% 2.58/1.69 | ( M2 = N2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_289_less__imp__le__nat,axiom,
% 2.58/1.69 ! [M: nat,N: nat] :
% 2.58/1.69 ( ( ord_less_nat @ M @ N )
% 2.58/1.69 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_290_ex__least__nat__le,axiom,
% 2.58/1.69 ! [P: nat > $o,N: nat] :
% 2.58/1.69 ( ( P @ N )
% 2.58/1.69 => ( ~ ( P @ zero_zero_nat )
% 2.58/1.69 => ? [K2: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ K2 @ N )
% 2.58/1.69 & ! [I3: nat] :
% 2.58/1.69 ( ( ord_less_nat @ I3 @ K2 )
% 2.58/1.69 => ~ ( P @ I3 ) )
% 2.58/1.69 & ( P @ K2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_291_nat__less__le,axiom,
% 2.58/1.69 ( ord_less_nat
% 2.58/1.69 = ( ^ [M2: nat,N2: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.58/1.69 & ( M2 != N2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_292_comm__monoid__add__class_Oadd__0,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( plus_plus_int @ zero_zero_int @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_293_comm__monoid__add__class_Oadd__0,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( plus_plus_a @ zero_zero_a @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_294_comm__monoid__add__class_Oadd__0,axiom,
% 2.58/1.69 ! [A: nat] :
% 2.58/1.69 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_295_add_Ocomm__neutral,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( plus_plus_int @ A @ zero_zero_int )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_296_add_Ocomm__neutral,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( plus_plus_a @ A @ zero_zero_a )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_297_add_Ocomm__neutral,axiom,
% 2.58/1.69 ! [A: nat] :
% 2.58/1.69 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_298_add_Ogroup__left__neutral,axiom,
% 2.58/1.69 ! [A: int] :
% 2.58/1.69 ( ( plus_plus_int @ zero_zero_int @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_299_add_Ogroup__left__neutral,axiom,
% 2.58/1.69 ! [A: a] :
% 2.58/1.69 ( ( plus_plus_a @ zero_zero_a @ A )
% 2.58/1.69 = A ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_300_add__le__imp__le__right,axiom,
% 2.58/1.69 ! [A: int,C: int,B: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 2.58/1.69 => ( ord_less_eq_int @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_301_add__le__imp__le__right,axiom,
% 2.58/1.69 ! [A: a,C: a,B: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 2.58/1.69 => ( ord_less_eq_a @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_302_add__le__imp__le__right,axiom,
% 2.58/1.69 ! [A: nat,C: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.58/1.69 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_303_add__le__imp__le__left,axiom,
% 2.58/1.69 ! [C: int,A: int,B: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 2.58/1.69 => ( ord_less_eq_int @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_304_add__le__imp__le__left,axiom,
% 2.58/1.69 ! [C: a,A: a,B: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 2.58/1.69 => ( ord_less_eq_a @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_305_add__le__imp__le__left,axiom,
% 2.58/1.69 ! [C: nat,A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.58/1.69 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_306_le__iff__add,axiom,
% 2.58/1.69 ( ord_less_eq_nat
% 2.58/1.69 = ( ^ [A5: nat,B3: nat] :
% 2.58/1.69 ? [C2: nat] :
% 2.58/1.69 ( B3
% 2.58/1.69 = ( plus_plus_nat @ A5 @ C2 ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_307_add__right__mono,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ A @ B )
% 2.58/1.69 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_308_add__right__mono,axiom,
% 2.58/1.69 ! [A: a,B: a,C: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ A @ B )
% 2.58/1.69 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_309_add__right__mono,axiom,
% 2.58/1.69 ! [A: nat,B: nat,C: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.69 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_310_less__eqE,axiom,
% 2.58/1.69 ! [A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.69 => ~ ! [C3: nat] :
% 2.58/1.69 ( B
% 2.58/1.69 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_311_add__left__mono,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ A @ B )
% 2.58/1.69 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_312_add__left__mono,axiom,
% 2.58/1.69 ! [A: a,B: a,C: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ A @ B )
% 2.58/1.69 => ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_313_add__left__mono,axiom,
% 2.58/1.69 ! [A: nat,B: nat,C: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.69 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_314_add__mono,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int,D: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ A @ B )
% 2.58/1.69 => ( ( ord_less_eq_int @ C @ D )
% 2.58/1.69 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_315_add__mono,axiom,
% 2.58/1.69 ! [A: a,B: a,C: a,D: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ A @ B )
% 2.58/1.69 => ( ( ord_less_eq_a @ C @ D )
% 2.58/1.69 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_316_add__mono,axiom,
% 2.58/1.69 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.69 => ( ( ord_less_eq_nat @ C @ D )
% 2.58/1.69 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_317_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.58/1.69 ! [I: int,J: int,K: int,L: int] :
% 2.58/1.69 ( ( ( ord_less_eq_int @ I @ J )
% 2.58/1.69 & ( ord_less_eq_int @ K @ L ) )
% 2.58/1.69 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_318_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.58/1.69 ! [I: a,J: a,K: a,L: a] :
% 2.58/1.69 ( ( ( ord_less_eq_a @ I @ J )
% 2.58/1.69 & ( ord_less_eq_a @ K @ L ) )
% 2.58/1.69 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_319_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.58/1.69 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.58/1.69 ( ( ( ord_less_eq_nat @ I @ J )
% 2.58/1.69 & ( ord_less_eq_nat @ K @ L ) )
% 2.58/1.69 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_320_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.58/1.69 ! [I: int,J: int,K: int,L: int] :
% 2.58/1.69 ( ( ( I = J )
% 2.58/1.69 & ( ord_less_eq_int @ K @ L ) )
% 2.58/1.69 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_321_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.58/1.69 ! [I: a,J: a,K: a,L: a] :
% 2.58/1.69 ( ( ( I = J )
% 2.58/1.69 & ( ord_less_eq_a @ K @ L ) )
% 2.58/1.69 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_322_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.58/1.69 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.58/1.69 ( ( ( I = J )
% 2.58/1.69 & ( ord_less_eq_nat @ K @ L ) )
% 2.58/1.69 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_323_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.58/1.69 ! [I: int,J: int,K: int,L: int] :
% 2.58/1.69 ( ( ( ord_less_eq_int @ I @ J )
% 2.58/1.69 & ( K = L ) )
% 2.58/1.69 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_324_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.58/1.69 ! [I: a,J: a,K: a,L: a] :
% 2.58/1.69 ( ( ( ord_less_eq_a @ I @ J )
% 2.58/1.69 & ( K = L ) )
% 2.58/1.69 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_325_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.58/1.69 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.58/1.69 ( ( ( ord_less_eq_nat @ I @ J )
% 2.58/1.69 & ( K = L ) )
% 2.58/1.69 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_326_add__less__imp__less__right,axiom,
% 2.58/1.69 ! [A: int,C: int,B: int] :
% 2.58/1.69 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 2.58/1.69 => ( ord_less_int @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_327_add__less__imp__less__right,axiom,
% 2.58/1.69 ! [A: nat,C: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.58/1.69 => ( ord_less_nat @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_328_add__less__imp__less__left,axiom,
% 2.58/1.69 ! [C: int,A: int,B: int] :
% 2.58/1.69 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 2.58/1.69 => ( ord_less_int @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_329_add__less__imp__less__left,axiom,
% 2.58/1.69 ! [C: nat,A: nat,B: nat] :
% 2.58/1.69 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.58/1.69 => ( ord_less_nat @ A @ B ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_330_add__strict__right__mono,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( ord_less_int @ A @ B )
% 2.58/1.69 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_331_add__strict__right__mono,axiom,
% 2.58/1.69 ! [A: nat,B: nat,C: nat] :
% 2.58/1.69 ( ( ord_less_nat @ A @ B )
% 2.58/1.69 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_332_add__strict__left__mono,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int] :
% 2.58/1.69 ( ( ord_less_int @ A @ B )
% 2.58/1.69 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_333_add__strict__left__mono,axiom,
% 2.58/1.69 ! [A: nat,B: nat,C: nat] :
% 2.58/1.69 ( ( ord_less_nat @ A @ B )
% 2.58/1.69 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_334_add__strict__mono,axiom,
% 2.58/1.69 ! [A: int,B: int,C: int,D: int] :
% 2.58/1.69 ( ( ord_less_int @ A @ B )
% 2.58/1.69 => ( ( ord_less_int @ C @ D )
% 2.58/1.69 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_335_add__strict__mono,axiom,
% 2.58/1.69 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.58/1.69 ( ( ord_less_nat @ A @ B )
% 2.58/1.69 => ( ( ord_less_nat @ C @ D )
% 2.58/1.69 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_336_add__mono__thms__linordered__field_I1_J,axiom,
% 2.58/1.69 ! [I: int,J: int,K: int,L: int] :
% 2.58/1.69 ( ( ( ord_less_int @ I @ J )
% 2.58/1.69 & ( K = L ) )
% 2.58/1.69 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_337_add__mono__thms__linordered__field_I1_J,axiom,
% 2.58/1.69 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.58/1.69 ( ( ( ord_less_nat @ I @ J )
% 2.58/1.69 & ( K = L ) )
% 2.58/1.69 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_338_add__mono__thms__linordered__field_I2_J,axiom,
% 2.58/1.69 ! [I: int,J: int,K: int,L: int] :
% 2.58/1.69 ( ( ( I = J )
% 2.58/1.69 & ( ord_less_int @ K @ L ) )
% 2.58/1.69 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_339_add__mono__thms__linordered__field_I2_J,axiom,
% 2.58/1.69 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.58/1.69 ( ( ( I = J )
% 2.58/1.69 & ( ord_less_nat @ K @ L ) )
% 2.58/1.69 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_340_add__mono__thms__linordered__field_I5_J,axiom,
% 2.58/1.69 ! [I: int,J: int,K: int,L: int] :
% 2.58/1.69 ( ( ( ord_less_int @ I @ J )
% 2.58/1.69 & ( ord_less_int @ K @ L ) )
% 2.58/1.69 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_341_add__mono__thms__linordered__field_I5_J,axiom,
% 2.58/1.69 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.58/1.69 ( ( ( ord_less_nat @ I @ J )
% 2.58/1.69 & ( ord_less_nat @ K @ L ) )
% 2.58/1.69 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_342_list_Osize_I3_J,axiom,
% 2.58/1.69 ( ( size_size_list_a @ nil_a )
% 2.58/1.69 = zero_zero_nat ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_343_length__induct,axiom,
% 2.58/1.69 ! [P: list_a > $o,Xs: list_a] :
% 2.58/1.69 ( ! [Xs3: list_a] :
% 2.58/1.69 ( ! [Ys6: list_a] :
% 2.58/1.69 ( ( ord_less_nat @ ( size_size_list_a @ Ys6 ) @ ( size_size_list_a @ Xs3 ) )
% 2.58/1.69 => ( P @ Ys6 ) )
% 2.58/1.69 => ( P @ Xs3 ) )
% 2.58/1.69 => ( P @ Xs ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_344_nth__take__lemma,axiom,
% 2.58/1.69 ! [K: nat,Xs: list_a,Ys: list_a] :
% 2.58/1.69 ( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Xs ) )
% 2.58/1.69 => ( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Ys ) )
% 2.58/1.69 => ( ! [I2: nat] :
% 2.58/1.69 ( ( ord_less_nat @ I2 @ K )
% 2.58/1.69 => ( ( nth_a @ Xs @ I2 )
% 2.58/1.69 = ( nth_a @ Ys @ I2 ) ) )
% 2.58/1.69 => ( ( take_a @ K @ Xs )
% 2.58/1.69 = ( take_a @ K @ Ys ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_345_add__nonpos__eq__0__iff,axiom,
% 2.58/1.69 ! [X: int,Y: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 2.58/1.69 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 2.58/1.69 => ( ( ( plus_plus_int @ X @ Y )
% 2.58/1.69 = zero_zero_int )
% 2.58/1.69 = ( ( X = zero_zero_int )
% 2.58/1.69 & ( Y = zero_zero_int ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_346_add__nonpos__eq__0__iff,axiom,
% 2.58/1.69 ! [X: a,Y: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ X @ zero_zero_a )
% 2.58/1.69 => ( ( ord_less_eq_a @ Y @ zero_zero_a )
% 2.58/1.69 => ( ( ( plus_plus_a @ X @ Y )
% 2.58/1.69 = zero_zero_a )
% 2.58/1.69 = ( ( X = zero_zero_a )
% 2.58/1.69 & ( Y = zero_zero_a ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_347_add__nonpos__eq__0__iff,axiom,
% 2.58/1.69 ! [X: nat,Y: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 2.58/1.69 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 2.58/1.69 => ( ( ( plus_plus_nat @ X @ Y )
% 2.58/1.69 = zero_zero_nat )
% 2.58/1.69 = ( ( X = zero_zero_nat )
% 2.58/1.69 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_348_add__nonneg__eq__0__iff,axiom,
% 2.58/1.69 ! [X: int,Y: int] :
% 2.58/1.69 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 2.58/1.69 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 2.58/1.69 => ( ( ( plus_plus_int @ X @ Y )
% 2.58/1.69 = zero_zero_int )
% 2.58/1.69 = ( ( X = zero_zero_int )
% 2.58/1.69 & ( Y = zero_zero_int ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_349_add__nonneg__eq__0__iff,axiom,
% 2.58/1.69 ! [X: a,Y: a] :
% 2.58/1.69 ( ( ord_less_eq_a @ zero_zero_a @ X )
% 2.58/1.69 => ( ( ord_less_eq_a @ zero_zero_a @ Y )
% 2.58/1.69 => ( ( ( plus_plus_a @ X @ Y )
% 2.58/1.69 = zero_zero_a )
% 2.58/1.69 = ( ( X = zero_zero_a )
% 2.58/1.69 & ( Y = zero_zero_a ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(fact_350_add__nonneg__eq__0__iff,axiom,
% 2.58/1.69 ! [X: nat,Y: nat] :
% 2.58/1.69 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 2.58/1.69 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 2.58/1.69 => ( ( ( plus_plus_nat @ X @ Y )
% 2.58/1.69 = zero_zero_nat )
% 2.58/1.69 = ( ( X = zero_zero_nat )
% 2.58/1.69 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 2.58/1.69
% 2.58/1.69 thf(conj_0,conjecture,
% 2.58/1.69 ( ( descar1375166517sums_a @ xsa )
% 2.58/1.69 = ( append_a @ ( replicate_a @ p @ zero_zero_a ) @ ( descar1375166517sums_a @ ( cons_a @ ( nth_a @ xsa @ p ) @ xs2 ) ) ) ) ).
% 2.58/1.69 ------------------------
% 2.78/1.71 % (2636)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_2527 for (2999ds/27Mi)
% 2.78/1.71 % (2638)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_2527 for (2999ds/2Mi)
% 2.78/1.71 % (2639)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_2527 for (2999ds/275Mi)
% 2.78/1.71 % (2638)Instruction limit reached!
% 2.78/1.71 % (2638)------------------------------
% 2.78/1.71 % (2638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.71 % (2638)Termination reason: Unknown
% 2.78/1.71 % (2638)Termination phase: shuffling
% 2.78/1.71
% 2.78/1.71 % (2638)Memory used [KB]: 1407
% 2.78/1.71 % (2638)Time elapsed: 0.002 s
% 2.78/1.71 % (2638)Instructions burned: 3 (million)
% 2.78/1.71 % (2638)------------------------------
% 2.78/1.71 % (2638)------------------------------
% 2.78/1.71 % (2640)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/18Mi)
% 2.78/1.72 % (2635)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_2527 for (2999ds/4Mi)
% 2.78/1.72 % (2640)Instruction limit reached!
% 2.78/1.72 % (2640)------------------------------
% 2.78/1.72 % (2640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.72 % (2640)Termination reason: Unknown
% 2.78/1.72 % (2640)Termination phase: Property scanning
% 2.78/1.72
% 2.78/1.72 % (2640)Memory used [KB]: 1663
% 2.78/1.72 % (2640)Time elapsed: 0.007 s
% 2.78/1.72 % (2640)Instructions burned: 18 (million)
% 2.78/1.72 % (2640)------------------------------
% 2.78/1.72 % (2640)------------------------------
% 2.78/1.72 % (2641)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/3Mi)
% 2.78/1.72 % (2635)Instruction limit reached!
% 2.78/1.72 % (2635)------------------------------
% 2.78/1.72 % (2635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.72 % (2635)Termination reason: Unknown
% 2.78/1.72 % (2635)Termination phase: shuffling
% 2.78/1.72
% 2.78/1.72 % (2635)Memory used [KB]: 1407
% 2.78/1.72 % (2635)Time elapsed: 0.004 s
% 2.78/1.72 % (2635)Instructions burned: 4 (million)
% 2.78/1.72 % (2635)------------------------------
% 2.78/1.72 % (2635)------------------------------
% 2.78/1.72 % (2637)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/2Mi)
% 2.78/1.72 % (2637)Instruction limit reached!
% 2.78/1.72 % (2637)------------------------------
% 2.78/1.72 % (2637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.72 % (2637)Termination reason: Unknown
% 2.78/1.72 % (2637)Termination phase: shuffling
% 2.78/1.72
% 2.78/1.72 % (2637)Memory used [KB]: 1407
% 2.78/1.72 % (2637)Time elapsed: 0.002 s
% 2.78/1.72 % (2637)Instructions burned: 2 (million)
% 2.78/1.72 % (2637)------------------------------
% 2.78/1.72 % (2637)------------------------------
% 2.78/1.72 % (2641)Instruction limit reached!
% 2.78/1.72 % (2641)------------------------------
% 2.78/1.72 % (2641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.72 % (2641)Termination reason: Unknown
% 2.78/1.72 % (2641)Termination phase: shuffling
% 2.78/1.72
% 2.78/1.72 % (2641)Memory used [KB]: 1407
% 2.78/1.72 % (2641)Time elapsed: 0.003 s
% 2.78/1.72 % (2641)Instructions burned: 6 (million)
% 2.78/1.72 % (2641)------------------------------
% 2.78/1.72 % (2641)------------------------------
% 2.78/1.72 % (2634)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/183Mi)
% 2.78/1.73 % (2645)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_2527 for (2999ds/1041Mi)
% 2.78/1.73 % (2636)Instruction limit reached!
% 2.78/1.73 % (2636)------------------------------
% 2.78/1.73 % (2636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.73 % (2636)Termination reason: Unknown
% 2.78/1.73 % (2636)Termination phase: Property scanning
% 2.78/1.73
% 2.78/1.73 % (2636)Memory used [KB]: 1791
% 2.78/1.73 % (2636)Time elapsed: 0.016 s
% 2.78/1.73 % (2636)Instructions burned: 28 (million)
% 2.78/1.73 % (2636)------------------------------
% 2.78/1.73 % (2636)------------------------------
% 2.78/1.73 % (2643)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_2527 for (2999ds/15Mi)
% 2.78/1.73 % (2639)Refutation not found, incomplete strategy
% 2.78/1.73 % (2639)------------------------------
% 2.78/1.73 % (2639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.73 % (2639)Termination reason: Refutation not found, incomplete strategy
% 2.78/1.73
% 2.78/1.73
% 2.78/1.73 % (2639)Memory used [KB]: 6908
% 2.78/1.73 % (2639)Time elapsed: 0.025 s
% 2.78/1.73 % (2639)Instructions burned: 81 (million)
% 2.78/1.73 % (2639)------------------------------
% 2.78/1.73 % (2639)------------------------------
% 2.78/1.74 % (2643)Instruction limit reached!
% 2.78/1.74 % (2643)------------------------------
% 2.78/1.74 % (2643)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.74 % (2643)Termination reason: Unknown
% 2.78/1.74 % (2643)Termination phase: shuffling
% 2.78/1.74
% 2.78/1.74 % (2643)Memory used [KB]: 1663
% 2.78/1.74 % (2643)Time elapsed: 0.006 s
% 2.78/1.74 % (2643)Instructions burned: 15 (million)
% 2.78/1.74 % (2643)------------------------------
% 2.78/1.74 % (2643)------------------------------
% 2.78/1.74 % (2642)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_2527 for (2999ds/37Mi)
% 2.78/1.74 % (2644)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_2527 for (2999ds/3Mi)
% 2.78/1.74 % (2644)Instruction limit reached!
% 2.78/1.74 % (2644)------------------------------
% 2.78/1.74 % (2644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.74 % (2644)Termination reason: Unknown
% 2.78/1.74 % (2644)Termination phase: shuffling
% 2.78/1.74
% 2.78/1.74 % (2644)Memory used [KB]: 1407
% 2.78/1.74 % (2644)Time elapsed: 0.004 s
% 2.78/1.74 % (2644)Instructions burned: 4 (million)
% 2.78/1.74 % (2644)------------------------------
% 2.78/1.74 % (2644)------------------------------
% 2.78/1.75 % (2647)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/16Mi)
% 2.78/1.75 % (2648)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_2527 for (2999ds/3Mi)
% 2.78/1.75 % (2648)Instruction limit reached!
% 2.78/1.75 % (2648)------------------------------
% 2.78/1.75 % (2648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.75 % (2648)Termination reason: Unknown
% 2.78/1.75 % (2648)Termination phase: shuffling
% 2.78/1.75
% 2.78/1.75 % (2648)Memory used [KB]: 1407
% 2.78/1.75 % (2648)Time elapsed: 0.002 s
% 2.78/1.75 % (2648)Instructions burned: 3 (million)
% 2.78/1.75 % (2648)------------------------------
% 2.78/1.75 % (2648)------------------------------
% 2.78/1.75 % (2647)Instruction limit reached!
% 2.78/1.75 % (2647)------------------------------
% 2.78/1.75 % (2647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.75 % (2647)Termination reason: Unknown
% 2.78/1.75 % (2647)Termination phase: shuffling
% 2.78/1.75
% 2.78/1.75 % (2647)Memory used [KB]: 1663
% 2.78/1.75 % (2647)Time elapsed: 0.006 s
% 2.78/1.75 % (2647)Instructions burned: 16 (million)
% 2.78/1.75 % (2647)------------------------------
% 2.78/1.75 % (2647)------------------------------
% 2.78/1.75 % (2646)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/7Mi)
% 2.78/1.76 % (2642)Instruction limit reached!
% 2.78/1.76 % (2642)------------------------------
% 2.78/1.76 % (2642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.76 % (2642)Termination reason: Unknown
% 2.78/1.76 % (2642)Termination phase: shuffling
% 2.78/1.76
% 2.78/1.76 % (2642)Memory used [KB]: 2046
% 2.78/1.76 % (2642)Time elapsed: 0.020 s
% 2.78/1.76 % (2642)Instructions burned: 37 (million)
% 2.78/1.76 % (2642)------------------------------
% 2.78/1.76 % (2642)------------------------------
% 2.78/1.76 % (2646)Instruction limit reached!
% 2.78/1.76 % (2646)------------------------------
% 2.78/1.76 % (2646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.76 % (2646)Termination reason: Unknown
% 2.78/1.76 % (2646)Termination phase: shuffling
% 2.78/1.76
% 2.78/1.76 % (2646)Memory used [KB]: 1535
% 2.78/1.76 % (2646)Time elapsed: 0.006 s
% 2.78/1.76 % (2646)Instructions burned: 8 (million)
% 2.78/1.76 % (2646)------------------------------
% 2.78/1.76 % (2646)------------------------------
% 2.78/1.76 % (2650)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/7Mi)
% 2.78/1.76 % (2650)Instruction limit reached!
% 2.78/1.76 % (2650)------------------------------
% 2.78/1.76 % (2650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.76 % (2650)Termination reason: Unknown
% 2.78/1.76 % (2650)Termination phase: shuffling
% 2.78/1.76
% 2.78/1.76 % (2650)Memory used [KB]: 1407
% 2.78/1.76 % (2650)Time elapsed: 0.003 s
% 2.78/1.76 % (2650)Instructions burned: 7 (million)
% 2.78/1.76 % (2650)------------------------------
% 2.78/1.76 % (2650)------------------------------
% 2.78/1.76 % (2651)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/3Mi)
% 2.78/1.76 % (2651)Instruction limit reached!
% 2.78/1.76 % (2651)------------------------------
% 2.78/1.76 % (2651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.76 % (2651)Termination reason: Unknown
% 2.78/1.76 % (2651)Termination phase: shuffling
% 2.78/1.76
% 2.78/1.76 % (2651)Memory used [KB]: 1407
% 2.78/1.76 % (2651)Time elapsed: 0.002 s
% 2.78/1.76 % (2651)Instructions burned: 3 (million)
% 2.78/1.76 % (2651)------------------------------
% 2.78/1.76 % (2651)------------------------------
% 2.78/1.77 % (2649)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_2527 for (2999ds/3Mi)
% 2.78/1.77 % (2654)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/710Mi)
% 2.78/1.77 % (2649)Instruction limit reached!
% 2.78/1.77 % (2649)------------------------------
% 2.78/1.77 % (2649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.77 % (2649)Termination reason: Unknown
% 2.78/1.77 % (2649)Termination phase: shuffling
% 2.78/1.77
% 2.78/1.77 % (2649)Memory used [KB]: 1407
% 2.78/1.77 % (2649)Time elapsed: 0.004 s
% 2.78/1.77 % (2649)Instructions burned: 4 (million)
% 2.78/1.77 % (2649)------------------------------
% 2.78/1.77 % (2649)------------------------------
% 2.78/1.77 % (2655)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/6Mi)
% 2.78/1.77 % (2655)Instruction limit reached!
% 2.78/1.77 % (2655)------------------------------
% 2.78/1.77 % (2655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.77 % (2655)Termination reason: Unknown
% 2.78/1.77 % (2655)Termination phase: shuffling
% 2.78/1.77
% 2.78/1.77 % (2655)Memory used [KB]: 1407
% 2.78/1.77 % (2655)Time elapsed: 0.003 s
% 2.78/1.77 % (2655)Instructions burned: 6 (million)
% 2.78/1.77 % (2655)------------------------------
% 2.78/1.77 % (2655)------------------------------
% 2.78/1.77 % (2652)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/4Mi)
% 2.78/1.77 % (2653)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/18Mi)
% 2.78/1.78 % (2634)Instruction limit reached!
% 2.78/1.78 % (2634)------------------------------
% 2.78/1.78 % (2634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.78 % (2634)Termination reason: Unknown
% 2.78/1.78 % (2634)Termination phase: Saturation
% 2.78/1.78
% 2.78/1.78 % (2634)Memory used [KB]: 7291
% 2.78/1.78 % (2634)Time elapsed: 0.053 s
% 2.78/1.78 % (2634)Instructions burned: 183 (million)
% 2.78/1.78 % (2634)------------------------------
% 2.78/1.78 % (2634)------------------------------
% 2.78/1.78 % (2652)Instruction limit reached!
% 2.78/1.78 % (2652)------------------------------
% 2.78/1.78 % (2652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.78 % (2652)Termination reason: Unknown
% 2.78/1.78 % (2652)Termination phase: shuffling
% 2.78/1.78
% 2.78/1.78 % (2652)Memory used [KB]: 1407
% 2.78/1.78 % (2652)Time elapsed: 0.004 s
% 2.78/1.78 % (2652)Instructions burned: 5 (million)
% 2.78/1.78 % (2652)------------------------------
% 2.78/1.78 % (2652)------------------------------
% 2.78/1.78 % (2657)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_2527 for (2999ds/21Mi)
% 2.78/1.78 % (2653)Instruction limit reached!
% 2.78/1.78 % (2653)------------------------------
% 2.78/1.78 % (2653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.78 % (2653)Termination reason: Unknown
% 2.78/1.78 % (2653)Termination phase: Property scanning
% 2.78/1.78
% 2.78/1.78 % (2653)Memory used [KB]: 1791
% 2.78/1.78 % (2653)Time elapsed: 0.012 s
% 2.78/1.78 % (2653)Instructions burned: 19 (million)
% 2.78/1.78 % (2653)------------------------------
% 2.78/1.78 % (2653)------------------------------
% 2.78/1.79 % (2659)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_2527 for (2999ds/6Mi)
% 2.78/1.79 % (2656)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_2527 for (2999ds/902Mi)
% 2.78/1.79 % (2659)Instruction limit reached!
% 2.78/1.79 % (2659)------------------------------
% 2.78/1.79 % (2659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.79 % (2659)Termination reason: Unknown
% 2.78/1.79 % (2659)Termination phase: shuffling
% 2.78/1.79
% 2.78/1.79 % (2659)Memory used [KB]: 1407
% 2.78/1.79 % (2659)Time elapsed: 0.003 s
% 2.78/1.79 % (2659)Instructions burned: 7 (million)
% 2.78/1.79 % (2659)------------------------------
% 2.78/1.79 % (2659)------------------------------
% 2.78/1.79 % (2657)Instruction limit reached!
% 2.78/1.79 % (2657)------------------------------
% 2.78/1.79 % (2657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.79 % (2657)Termination reason: Unknown
% 2.78/1.79 % (2657)Termination phase: shuffling
% 2.78/1.79
% 2.78/1.79 % (2657)Memory used [KB]: 1791
% 2.78/1.79 % (2657)Time elapsed: 0.008 s
% 2.78/1.79 % (2657)Instructions burned: 22 (million)
% 2.78/1.79 % (2657)------------------------------
% 2.78/1.79 % (2657)------------------------------
% 2.78/1.80 % (2658)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/5Mi)
% 2.78/1.80 % (2661)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_2527 for (2999ds/779Mi)
% 2.78/1.80 % (2658)Instruction limit reached!
% 2.78/1.80 % (2658)------------------------------
% 2.78/1.80 % (2658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.80 % (2658)Termination reason: Unknown
% 2.78/1.80 % (2658)Termination phase: shuffling
% 2.78/1.80
% 2.78/1.80 % (2658)Memory used [KB]: 1407
% 2.78/1.80 % (2658)Time elapsed: 0.005 s
% 2.78/1.80 % (2658)Instructions burned: 6 (million)
% 2.78/1.80 % (2658)------------------------------
% 2.78/1.80 % (2658)------------------------------
% 2.78/1.80 % (2662)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_2527 for (2999ds/19Mi)
% 2.78/1.80 % (2662)Instruction limit reached!
% 2.78/1.80 % (2662)------------------------------
% 2.78/1.80 % (2662)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.80 % (2662)Termination reason: Unknown
% 2.78/1.80 % (2662)Termination phase: shuffling
% 2.78/1.80
% 2.78/1.80 % (2662)Memory used [KB]: 1663
% 2.78/1.80 % (2662)Time elapsed: 0.006 s
% 2.78/1.80 % (2662)Instructions burned: 19 (million)
% 2.78/1.80 % (2662)------------------------------
% 2.78/1.80 % (2662)------------------------------
% 2.78/1.80 % (2660)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/377Mi)
% 2.78/1.81 % (2664)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/17Mi)
% 2.78/1.82 % (2663)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_2527 for (2999ds/879Mi)
% 2.78/1.82 % (2664)Instruction limit reached!
% 2.78/1.82 % (2664)------------------------------
% 2.78/1.82 % (2664)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.82 % (2664)Termination reason: Unknown
% 2.78/1.82 % (2664)Termination phase: shuffling
% 2.78/1.82
% 2.78/1.82 % (2664)Memory used [KB]: 1791
% 2.78/1.82 % (2664)Time elapsed: 0.007 s
% 2.78/1.82 % (2664)Instructions burned: 18 (million)
% 2.78/1.82 % (2664)------------------------------
% 2.78/1.82 % (2664)------------------------------
% 2.78/1.83 % (2665)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_2527 for (2998ds/3Mi)
% 2.78/1.83 % (2665)Instruction limit reached!
% 2.78/1.83 % (2665)------------------------------
% 2.78/1.83 % (2665)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.83 % (2665)Termination reason: Unknown
% 2.78/1.83 % (2665)Termination phase: shuffling
% 2.78/1.83
% 2.78/1.83 % (2665)Memory used [KB]: 1407
% 2.78/1.83 % (2665)Time elapsed: 0.003 s
% 2.78/1.83 % (2665)Instructions burned: 5 (million)
% 2.78/1.83 % (2665)------------------------------
% 2.78/1.83 % (2665)------------------------------
% 2.78/1.84 % (2666)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_2527 for (2998ds/30Mi)
% 2.78/1.85 % (2666)Instruction limit reached!
% 2.78/1.85 % (2666)------------------------------
% 2.78/1.85 % (2666)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.85 % (2666)Termination reason: Unknown
% 2.78/1.85 % (2666)Termination phase: Property scanning
% 2.78/1.85
% 2.78/1.85 % (2666)Memory used [KB]: 1791
% 2.78/1.85 % (2666)Time elapsed: 0.011 s
% 2.78/1.85 % (2666)Instructions burned: 31 (million)
% 2.78/1.85 % (2666)------------------------------
% 2.78/1.85 % (2666)------------------------------
% 2.78/1.86 % (2656)Refutation not found, incomplete strategy
% 2.78/1.86 % (2656)------------------------------
% 2.78/1.86 % (2656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.86 % (2656)Termination reason: Refutation not found, incomplete strategy
% 2.78/1.86
% 2.78/1.86
% 2.78/1.86 % (2656)Memory used [KB]: 7419
% 2.78/1.86 % (2656)Time elapsed: 0.071 s
% 2.78/1.86 % (2656)Instructions burned: 122 (million)
% 2.78/1.86 % (2656)------------------------------
% 2.78/1.86 % (2656)------------------------------
% 2.78/1.86 % (2667)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_2527 for (2998ds/127Mi)
% 2.78/1.87 % (2668)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_2527 for (2998ds/100Mi)
% 2.78/1.90 % (2668)Instruction limit reached!
% 2.78/1.90 % (2668)------------------------------
% 2.78/1.90 % (2668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.90 % (2668)Termination reason: Unknown
% 2.78/1.90 % (2668)Termination phase: Property scanning
% 2.78/1.90
% 2.78/1.90 % (2668)Memory used [KB]: 2430
% 2.78/1.90 % (2668)Time elapsed: 0.028 s
% 2.78/1.90 % (2668)Instructions burned: 103 (million)
% 2.78/1.90 % (2668)------------------------------
% 2.78/1.90 % (2668)------------------------------
% 2.78/1.91 % (2669)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_2527 for (2998ds/3Mi)
% 2.78/1.91 % (2669)Instruction limit reached!
% 2.78/1.91 % (2669)------------------------------
% 2.78/1.91 % (2669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.91 % (2669)Termination reason: Unknown
% 2.78/1.91 % (2669)Termination phase: shuffling
% 2.78/1.91
% 2.78/1.91 % (2669)Memory used [KB]: 1407
% 2.78/1.91 % (2669)Time elapsed: 0.003 s
% 2.78/1.91 % (2669)Instructions burned: 4 (million)
% 2.78/1.91 % (2669)------------------------------
% 2.78/1.91 % (2669)------------------------------
% 2.78/1.92 % (2667)Instruction limit reached!
% 2.78/1.92 % (2667)------------------------------
% 2.78/1.92 % (2667)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.92 % (2667)Termination reason: Unknown
% 2.78/1.92 % (2667)Termination phase: Saturation
% 2.78/1.92
% 2.78/1.92 % (2667)Memory used [KB]: 7036
% 2.78/1.92 % (2667)Time elapsed: 0.061 s
% 2.78/1.92 % (2667)Instructions burned: 127 (million)
% 2.78/1.92 % (2667)------------------------------
% 2.78/1.92 % (2667)------------------------------
% 2.78/1.92 % (2670)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_2527 for (2997ds/20Mi)
% 2.78/1.93 % (2671)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_2527 for (2997ds/86Mi)
% 2.78/1.93 % (2670)Instruction limit reached!
% 2.78/1.93 % (2670)------------------------------
% 2.78/1.93 % (2670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.93 % (2670)Termination reason: Unknown
% 2.78/1.93 % (2670)Termination phase: Property scanning
% 2.78/1.93
% 2.78/1.93 % (2670)Memory used [KB]: 1663
% 2.78/1.93 % (2670)Time elapsed: 0.013 s
% 2.78/1.93 % (2670)Instructions burned: 21 (million)
% 2.78/1.93 % (2670)------------------------------
% 2.78/1.93 % (2670)------------------------------
% 2.78/1.94 % (2672)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_2527 for (2997ds/107Mi)
% 2.78/1.95 % (2671)Instruction limit reached!
% 2.78/1.95 % (2671)------------------------------
% 2.78/1.95 % (2671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.95 % (2671)Termination reason: Unknown
% 2.78/1.95 % (2671)Termination phase: Property scanning
% 2.78/1.95
% 2.78/1.95 % (2671)Memory used [KB]: 2302
% 2.78/1.95 % (2671)Time elapsed: 0.026 s
% 2.78/1.95 % (2671)Instructions burned: 90 (million)
% 2.78/1.95 % (2671)------------------------------
% 2.78/1.95 % (2671)------------------------------
% 2.78/1.96 % (2673)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_2527 for (2997ds/39Mi)
% 2.78/1.97 % (2672)Instruction limit reached!
% 2.78/1.97 % (2672)------------------------------
% 2.78/1.97 % (2672)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.97 % (2672)Termination reason: Unknown
% 2.78/1.97 % (2672)Termination phase: Saturation
% 2.78/1.97
% 2.78/1.97 % (2672)Memory used [KB]: 7036
% 2.78/1.97 % (2672)Time elapsed: 0.032 s
% 2.78/1.97 % (2672)Instructions burned: 110 (million)
% 2.78/1.97 % (2672)------------------------------
% 2.78/1.97 % (2672)------------------------------
% 2.78/1.98 % (2673)Instruction limit reached!
% 2.78/1.98 % (2673)------------------------------
% 2.78/1.98 % (2673)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.98 % (2673)Termination reason: Unknown
% 2.78/1.98 % (2673)Termination phase: shuffling
% 2.78/1.98
% 2.78/1.98 % (2673)Memory used [KB]: 2174
% 2.78/1.98 % (2673)Time elapsed: 0.013 s
% 2.78/1.98 % (2673)Instructions burned: 42 (million)
% 2.78/1.98 % (2673)------------------------------
% 2.78/1.98 % (2673)------------------------------
% 2.78/1.98 % (2674)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_2527 for (2997ds/448Mi)
% 2.78/1.99 % (2675)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_2527 for (2997ds/46Mi)
% 2.78/2.00 % (2654)Instruction limit reached!
% 2.78/2.00 % (2654)------------------------------
% 2.78/2.00 % (2654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/2.00 % (2654)Termination reason: Unknown
% 2.78/2.00 % (2654)Termination phase: Saturation
% 2.78/2.00
% 2.78/2.00 % (2654)Memory used [KB]: 8059
% 2.78/2.00 % (2654)Time elapsed: 0.228 s
% 2.78/2.00 % (2654)Instructions burned: 710 (million)
% 2.78/2.00 % (2654)------------------------------
% 2.78/2.00 % (2654)------------------------------
% 2.78/2.00 % (2675)Instruction limit reached!
% 2.78/2.00 % (2675)------------------------------
% 2.78/2.00 % (2675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/2.00 % (2675)Termination reason: Unknown
% 2.78/2.00 % (2675)Termination phase: Preprocessing 2
% 2.78/2.00
% 2.78/2.00 % (2675)Memory used [KB]: 1791
% 2.78/2.00 % (2675)Time elapsed: 0.015 s
% 2.78/2.00 % (2675)Instructions burned: 47 (million)
% 2.78/2.00 % (2675)------------------------------
% 2.78/2.00 % (2675)------------------------------
% 2.78/2.01 % (2676)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_2527 for (2996ds/98Mi)
% 4.83/2.02 % (2660)Instruction limit reached!
% 4.83/2.02 % (2660)------------------------------
% 4.83/2.02 % (2660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.83/2.02 % (2660)Termination reason: Unknown
% 4.83/2.02 % (2660)Termination phase: Saturation
% 4.83/2.02
% 4.83/2.02 % (2660)Memory used [KB]: 8443
% 4.83/2.02 % (2660)Time elapsed: 0.215 s
% 4.83/2.02 % (2660)Instructions burned: 377 (million)
% 4.83/2.02 % (2660)------------------------------
% 4.83/2.02 % (2660)------------------------------
% 4.83/2.02 % (2677)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_2527 for (2996ds/507Mi)
% 4.83/2.03 % (2678)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_2527 for (2996ds/149Mi)
% 4.83/2.03 % (2676)Instruction limit reached!
% 4.83/2.03 % (2676)------------------------------
% 4.83/2.03 % (2676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.83/2.03 % (2676)Termination reason: Unknown
% 4.83/2.03 % (2676)Termination phase: Property scanning
% 4.83/2.03
% 4.83/2.03 % (2676)Memory used [KB]: 2430
% 4.83/2.03 % (2676)Time elapsed: 0.027 s
% 4.83/2.03 % (2676)Instructions burned: 98 (million)
% 4.83/2.03 % (2676)------------------------------
% 4.83/2.03 % (2676)------------------------------
% 4.83/2.04 % (2679)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_2527 for (2996ds/39Mi)
% 4.83/2.05 % (2679)Instruction limit reached!
% 4.83/2.05 % (2679)------------------------------
% 4.83/2.05 % (2679)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.83/2.05 % (2679)Termination reason: Unknown
% 4.83/2.05 % (2679)Termination phase: Preprocessing 2
% 4.83/2.05
% 4.83/2.05 % (2679)Memory used [KB]: 1791
% 4.83/2.05 % (2679)Time elapsed: 0.013 s
% 4.83/2.05 % (2679)Instructions burned: 39 (million)
% 4.83/2.05 % (2679)------------------------------
% 4.83/2.05 % (2679)------------------------------
% 4.83/2.06 % (2680)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_2527 for (2996ds/6Mi)
% 4.83/2.07 % (2680)Instruction limit reached!
% 4.83/2.07 % (2680)------------------------------
% 4.83/2.07 % (2680)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.83/2.07 % (2680)Termination reason: Unknown
% 4.83/2.07 % (2680)Termination phase: shuffling
% 4.83/2.07
% 4.83/2.07 % (2680)Memory used [KB]: 1535
% 4.83/2.07 % (2680)Time elapsed: 0.004 s
% 4.83/2.07 % (2680)Instructions burned: 8 (million)
% 4.83/2.07 % (2680)------------------------------
% 4.83/2.07 % (2680)------------------------------
% 4.83/2.07 % (2677)First to succeed.
% 4.83/2.07 % (2677)Refutation found. Thanks to Tanya!
% 4.83/2.07 % SZS status Theorem for DTF2THF_2527
% 4.83/2.07 % SZS output start Proof for DTF2THF_2527
% 4.83/2.07 thf(type_def_5, type, list_nat: $tType).
% 4.83/2.07 thf(type_def_7, type, list_a: $tType).
% 4.83/2.07 thf(type_def_8, type, int: $tType).
% 4.83/2.07 thf(type_def_9, type, nat: $tType).
% 4.83/2.07 thf(type_def_10, type, a: $tType).
% 4.83/2.07 thf(type_def_11, type, list_list_nat: $tType).
% 4.83/2.07 thf(type_def_12, type, list_list_a: $tType).
% 4.83/2.07 thf(func_def_0, type, list_list_nat: $tType).
% 4.83/2.07 thf(func_def_1, type, list_list_a: $tType).
% 4.83/2.07 thf(func_def_2, type, list_nat: $tType).
% 4.83/2.07 thf(func_def_3, type, list_a: $tType).
% 4.83/2.07 thf(func_def_4, type, nat: $tType).
% 4.83/2.07 thf(func_def_5, type, int: $tType).
% 4.83/2.07 thf(func_def_6, type, a: $tType).
% 4.83/2.07 thf(func_def_7, type, descar226543321ms_nat: list_nat > list_nat).
% 4.83/2.07 thf(func_def_8, type, descar1375166517sums_a: list_a > list_a).
% 4.83/2.07 thf(func_def_9, type, plus_plus_int: int > int > int).
% 4.83/2.07 thf(func_def_10, type, plus_plus_nat: nat > nat > nat).
% 4.83/2.07 thf(func_def_11, type, plus_plus_a: a > a > a).
% 4.83/2.07 thf(func_def_12, type, zero_zero_int: int).
% 4.83/2.07 thf(func_def_13, type, zero_zero_nat: nat).
% 4.83/2.07 thf(func_def_14, type, zero_zero_a: a).
% 4.83/2.07 thf(func_def_15, type, append_nat: list_nat > list_nat > list_nat).
% 4.83/2.07 thf(func_def_16, type, append_a: list_a > list_a > list_a).
% 4.83/2.07 thf(func_def_17, type, last_nat: list_nat > nat).
% 4.83/2.07 thf(func_def_18, type, last_a: list_a > a).
% 4.83/2.07 thf(func_def_19, type, cons_list_nat: list_nat > list_list_nat > list_list_nat).
% 4.83/2.07 thf(func_def_20, type, cons_list_a: list_a > list_list_a > list_list_a).
% 4.83/2.07 thf(func_def_21, type, cons_nat: nat > list_nat > list_nat).
% 4.83/2.07 thf(func_def_22, type, cons_a: a > list_a > list_a).
% 4.83/2.07 thf(func_def_23, type, nil_list_nat: list_list_nat).
% 4.83/2.07 thf(func_def_24, type, nil_list_a: list_list_a).
% 4.83/2.07 thf(func_def_25, type, nil_nat: list_nat).
% 4.83/2.07 thf(func_def_26, type, nil_a: list_a).
% 4.83/2.07 thf(func_def_27, type, nth_nat: list_nat > nat > nat).
% 4.83/2.07 thf(func_def_28, type, nth_a: list_a > nat > a).
% 4.83/2.07 thf(func_def_29, type, replicate_nat: nat > nat > list_nat).
% 4.83/2.07 thf(func_def_30, type, replicate_a: nat > a > list_a).
% 4.83/2.07 thf(func_def_31, type, take_a: nat > list_a > list_a).
% 4.83/2.07 thf(func_def_32, type, size_size_list_nat: list_nat > nat).
% 4.83/2.07 thf(func_def_33, type, size_size_list_a: list_a > nat).
% 4.83/2.07 thf(func_def_34, type, ord_less_int: int > int > $o).
% 4.83/2.07 thf(func_def_35, type, ord_less_nat: nat > nat > $o).
% 4.83/2.07 thf(func_def_36, type, ord_less_a: a > a > $o).
% 4.83/2.07 thf(func_def_37, type, ord_less_eq_int: int > int > $o).
% 4.83/2.07 thf(func_def_38, type, ord_less_eq_nat: nat > nat > $o).
% 4.83/2.07 thf(func_def_39, type, ord_less_eq_a: a > a > $o).
% 4.83/2.07 thf(func_def_40, type, p: nat).
% 4.83/2.07 thf(func_def_41, type, v: list_a > int).
% 4.83/2.07 thf(func_def_42, type, xa: a).
% 4.83/2.07 thf(func_def_43, type, xs: list_a).
% 4.83/2.07 thf(func_def_44, type, xs1: list_a).
% 4.83/2.07 thf(func_def_45, type, xs2: list_a).
% 4.83/2.07 thf(func_def_46, type, xsa: list_a).
% 4.83/2.07 thf(func_def_60, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 4.83/2.07 thf(func_def_61, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 4.83/2.07 thf(func_def_63, type, sK2: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_64, type, sK3: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_65, type, sK4: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_66, type, sK5: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_67, type, sK6: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_68, type, sK7: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_69, type, sK8: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_70, type, sK9: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_71, type, sK10: nat > (nat > $o) > nat).
% 4.83/2.07 thf(func_def_72, type, sK11: (list_nat > list_a > $o) > list_nat).
% 4.83/2.07 thf(func_def_73, type, sK12: (list_nat > list_a > $o) > nat).
% 4.83/2.07 thf(func_def_74, type, sK13: (list_nat > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_75, type, sK14: (list_nat > list_a > $o) > a).
% 4.83/2.07 thf(func_def_76, type, sK15: (list_nat > list_a > $o) > nat).
% 4.83/2.07 thf(func_def_77, type, sK16: (list_nat > list_a > $o) > list_nat).
% 4.83/2.07 thf(func_def_78, type, sK17: (list_nat > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_79, type, sK18: (list_nat > list_a > $o) > a).
% 4.83/2.07 thf(func_def_80, type, sK19: nat > nat).
% 4.83/2.07 thf(func_def_81, type, sK20: list_nat > nat).
% 4.83/2.07 thf(func_def_82, type, sK21: list_nat > nat).
% 4.83/2.07 thf(func_def_83, type, sK22: list_nat > list_nat).
% 4.83/2.07 thf(func_def_84, type, sK23: list_nat > nat).
% 4.83/2.07 thf(func_def_85, type, sK24: list_list_a > list_list_a).
% 4.83/2.07 thf(func_def_86, type, sK25: list_list_a > list_a).
% 4.83/2.07 thf(func_def_87, type, sK26: list_list_a > list_list_a).
% 4.83/2.07 thf(func_def_88, type, sK27: list_list_a > a).
% 4.83/2.07 thf(func_def_89, type, sK28: list_nat > nat).
% 4.83/2.07 thf(func_def_90, type, sK29: list_nat > list_nat).
% 4.83/2.07 thf(func_def_91, type, sK30: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_92, type, sK31: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_93, type, sK32: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_94, type, sK33: (list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_95, type, sK34: ((nat > nat) > list_nat > list_nat > $o) > nat > nat).
% 4.83/2.07 thf(func_def_96, type, sK35: ((nat > nat) > list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_97, type, sK36: ((nat > nat) > list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_98, type, sK37: ((nat > nat) > list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_99, type, sK38: ((nat > nat) > list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_100, type, sK39: ((nat > nat) > list_nat > list_nat > $o) > nat > nat).
% 4.83/2.07 thf(func_def_101, type, sK40: ((a > a > $o) > list_a > $o) > a).
% 4.83/2.07 thf(func_def_102, type, sK41: ((a > a > $o) > list_a > $o) > a > a > $o).
% 4.83/2.07 thf(func_def_103, type, sK42: ((a > a > $o) > list_a > $o) > a > a > $o).
% 4.83/2.07 thf(func_def_104, type, sK43: ((a > a > $o) > list_a > $o) > a).
% 4.83/2.07 thf(func_def_105, type, sK44: ((a > a > $o) > list_a > $o) > a).
% 4.83/2.07 thf(func_def_106, type, sK45: ((a > a > $o) > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_107, type, sK46: ((a > a > $o) > list_a > $o) > a > a > $o).
% 4.83/2.07 thf(func_def_108, type, sK47: ((a > nat) > list_a > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_109, type, sK48: ((a > nat) > list_a > list_nat > $o) > a > nat).
% 4.83/2.07 thf(func_def_110, type, sK49: ((a > nat) > list_a > list_nat > $o) > list_a).
% 4.83/2.07 thf(func_def_111, type, sK50: ((a > nat) > list_a > list_nat > $o) > a > nat).
% 4.83/2.07 thf(func_def_112, type, sK51: ((a > nat) > list_a > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_113, type, sK52: ((a > nat) > list_a > list_nat > $o) > a).
% 4.83/2.07 thf(func_def_114, type, sK53: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_115, type, sK54: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_116, type, sK55: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_117, type, sK56: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_118, type, sK57: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_119, type, sK58: (list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_120, type, sK59: ((a > a > $o) > list_a > $o) > a > a > $o).
% 4.83/2.07 thf(func_def_121, type, sK60: ((a > a > $o) > list_a > $o) > a).
% 4.83/2.07 thf(func_def_122, type, sK61: ((a > a > $o) > list_a > $o) > a > a > $o).
% 4.83/2.07 thf(func_def_123, type, sK62: ((a > a > $o) > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_124, type, sK63: list_a > a).
% 4.83/2.07 thf(func_def_125, type, sK64: list_a > list_a).
% 4.83/2.07 thf(func_def_126, type, sK65: list_a > a).
% 4.83/2.07 thf(func_def_127, type, sK66: list_a > a).
% 4.83/2.07 thf(func_def_128, type, sK67: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_129, type, sK68: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_130, type, sK69: ((nat > a) > list_nat > list_a > $o) > list_nat).
% 4.83/2.07 thf(func_def_131, type, sK70: ((nat > a) > list_nat > list_a > $o) > nat > a).
% 4.83/2.07 thf(func_def_132, type, sK71: ((nat > a) > list_nat > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_133, type, sK72: ((nat > a) > list_nat > list_a > $o) > nat).
% 4.83/2.07 thf(func_def_134, type, sK73: ((nat > a) > list_nat > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_135, type, sK74: ((nat > a) > list_nat > list_a > $o) > nat > a).
% 4.83/2.07 thf(func_def_136, type, sK75: ((nat > nat > $o) > list_nat > $o) > nat > nat > $o).
% 4.83/2.07 thf(func_def_137, type, sK76: ((nat > nat > $o) > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_138, type, sK77: ((nat > nat > $o) > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_139, type, sK78: ((nat > nat > $o) > list_nat > $o) > nat > nat > $o).
% 4.83/2.07 thf(func_def_140, type, sK79: ((nat > nat > $o) > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_141, type, sK80: ((nat > nat > $o) > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_142, type, sK81: ((nat > nat > $o) > list_nat > $o) > nat > nat > $o).
% 4.83/2.07 thf(func_def_143, type, sK82: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_144, type, sK83: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_145, type, sK84: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_146, type, sK85: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_147, type, sK86: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_148, type, sK87: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_149, type, sK88: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_150, type, sK89: list_a > a).
% 4.83/2.07 thf(func_def_151, type, sK90: list_a > list_a).
% 4.83/2.07 thf(func_def_152, type, sK91: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_153, type, sK92: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_154, type, sK93: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_155, type, sK94: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_156, type, sK95: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_157, type, sK96: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_158, type, sK97: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_159, type, sK98: list_a > list_a).
% 4.83/2.07 thf(func_def_160, type, sK99: list_a > a).
% 4.83/2.07 thf(func_def_161, type, sK100: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_162, type, sK101: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_163, type, sK102: (list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_164, type, sK103: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_165, type, sK104: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_166, type, sK105: (list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_167, type, sK106: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_168, type, sK107: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_169, type, sK108: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_170, type, sK109: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_171, type, sK110: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_172, type, sK111: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_173, type, sK112: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_174, type, sK113: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_175, type, sK114: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_176, type, sK115: list_a > list_a > nat).
% 4.83/2.07 thf(func_def_177, type, sK116: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_178, type, sK117: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_179, type, sK118: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_180, type, sK119: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_181, type, sK120: list_nat > list_nat).
% 4.83/2.07 thf(func_def_182, type, sK121: list_nat > nat).
% 4.83/2.07 thf(func_def_183, type, sK122: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_184, type, sK123: (list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_185, type, sK124: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_186, type, sK125: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_187, type, sK126: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_188, type, sK127: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_189, type, sK128: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_190, type, sK129: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_191, type, sK130: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_192, type, sK131: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_193, type, sK132: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_194, type, sK133: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_195, type, sK134: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_196, type, sK135: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_197, type, sK136: ((a > a) > list_a > list_a > $o) > a > a).
% 4.83/2.07 thf(func_def_198, type, sK137: ((a > a) > list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_199, type, sK138: ((a > a) > list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_200, type, sK139: ((a > a) > list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_201, type, sK140: ((a > a) > list_a > list_a > $o) > a > a).
% 4.83/2.07 thf(func_def_202, type, sK141: ((a > a) > list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_203, type, sK142: list_list_nat > list_list_nat).
% 4.83/2.07 thf(func_def_204, type, sK143: list_list_nat > list_nat).
% 4.83/2.07 thf(func_def_205, type, sK144: list_list_nat > nat).
% 4.83/2.07 thf(func_def_206, type, sK145: list_list_nat > list_list_nat).
% 4.83/2.07 thf(func_def_207, type, sK146: ((nat > nat > $o) > list_nat > $o) > nat > nat > $o).
% 4.83/2.07 thf(func_def_208, type, sK147: ((nat > nat > $o) > list_nat > $o) > nat > nat > $o).
% 4.83/2.07 thf(func_def_209, type, sK148: ((nat > nat > $o) > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_210, type, sK149: ((nat > nat > $o) > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_211, type, sK150: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_212, type, sK151: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_213, type, sK152: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_214, type, sK153: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_215, type, sK154: (list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_216, type, sK155: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_217, type, sK156: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_218, type, sK157: list_nat > nat).
% 4.83/2.07 thf(func_def_219, type, sK158: list_nat > list_nat).
% 4.83/2.07 thf(func_def_220, type, sK159: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_221, type, sK160: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_222, type, sK161: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_223, type, sK162: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_224, type, sK163: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_225, type, sK164: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_226, type, sK165: (list_a > list_nat > $o) > a).
% 4.83/2.07 thf(func_def_227, type, sK166: (list_a > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_228, type, sK167: (list_a > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_229, type, sK168: (list_a > list_nat > $o) > list_a).
% 4.83/2.07 thf(func_def_230, type, sK169: (list_a > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_231, type, sK170: (list_a > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_232, type, sK171: (list_a > list_nat > $o) > a).
% 4.83/2.07 thf(func_def_233, type, sK172: (list_a > list_nat > $o) > list_a).
% 4.83/2.07 thf(func_def_234, type, sK173: (list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_235, type, sK174: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_236, type, sK175: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_237, type, sK176: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_238, type, sK177: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_239, type, sK178: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_240, type, sK179: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_241, type, sK180: (list_nat > list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_242, type, sK181: (list_nat > list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_243, type, sK182: list_a > list_a > nat > nat).
% 4.83/2.07 thf(func_def_244, type, sK183: list_nat > list_nat > list_nat).
% 4.83/2.07 thf(func_def_245, type, sK184: list_nat > list_nat > list_nat).
% 4.83/2.07 thf(func_def_246, type, sK185: list_nat > list_nat > list_nat).
% 4.83/2.07 thf(func_def_247, type, sK186: nat > nat).
% 4.83/2.07 thf(func_def_248, type, sK187: (list_nat > $o) > list_nat).
% 4.83/2.07 thf(func_def_249, type, sK188: (list_nat > $o) > nat).
% 4.83/2.07 thf(func_def_250, type, sK189: nat > list_a).
% 4.83/2.07 thf(func_def_251, type, sK190: list_a > a).
% 4.83/2.07 thf(func_def_252, type, sK191: list_a > list_a).
% 4.83/2.07 thf(func_def_253, type, sK192: list_a > a).
% 4.83/2.07 thf(func_def_254, type, sK193: list_a > a).
% 4.83/2.07 thf(func_def_255, type, sK194: list_a > list_a > list_a).
% 4.83/2.07 thf(func_def_256, type, sK195: list_a > list_a > list_a).
% 4.83/2.07 thf(func_def_257, type, sK196: list_a > list_a > list_a).
% 4.83/2.07 thf(func_def_258, type, sK197: (nat > nat) > nat).
% 4.83/2.07 thf(func_def_259, type, sK198: (nat > nat) > nat).
% 4.83/2.07 thf(func_def_260, type, sK199: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_261, type, sK200: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_262, type, sK201: nat > nat > nat).
% 4.83/2.07 thf(func_def_263, type, sK202: a > a).
% 4.83/2.07 thf(func_def_264, type, sK203: a > a).
% 4.83/2.07 thf(func_def_265, type, sK204: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_266, type, sK205: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_267, type, sK206: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_268, type, sK207: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_269, type, sK208: (list_a > list_a > $o) > a).
% 4.83/2.07 thf(func_def_270, type, sK209: (list_a > list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_271, type, sK210: list_a > list_a).
% 4.83/2.07 thf(func_def_272, type, sK211: list_a > a).
% 4.83/2.07 thf(func_def_273, type, sK212: list_a > a).
% 4.83/2.07 thf(func_def_274, type, sK213: list_a > list_a).
% 4.83/2.07 thf(func_def_275, type, sK214: list_nat > nat).
% 4.83/2.07 thf(func_def_276, type, sK215: list_nat > nat).
% 4.83/2.07 thf(func_def_277, type, sK216: list_nat > nat).
% 4.83/2.07 thf(func_def_278, type, sK217: list_nat > list_nat).
% 4.83/2.07 thf(func_def_279, type, sK218: list_nat > nat).
% 4.83/2.07 thf(func_def_280, type, sK219: list_nat > list_nat).
% 4.83/2.07 thf(func_def_281, type, sK220: (list_a > $o) > list_a).
% 4.83/2.07 thf(func_def_282, type, sK221: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_283, type, sK222: (list_a > $o) > a).
% 4.83/2.07 thf(func_def_285, type, ph224: !>[X0: $tType]:(X0)).
% 4.83/2.07 thf(f2057,plain,(
% 4.83/2.07 $false),
% 4.83/2.07 inference(trivial_inequality_removal,[status(thm)],[f2056])).
% 4.83/2.07 thf(f2056,plain,(
% 4.83/2.07 ((descar1375166517sums_a @ xsa) != (descar1375166517sums_a @ xsa))),
% 4.83/2.07 inference(superposition,[status(thm)],[f2054,f1896])).
% 4.83/2.07 thf(f1896,plain,(
% 4.83/2.07 (xsa = (append_a @ (replicate_a @ p @ zero_zero_a) @ (cons_a @ (nth_a @ xsa @ p) @ xs2)))),
% 4.83/2.07 inference(definition_unfolding,[status(thm)],[f1494,f1853])).
% 4.83/2.07 thf(f1853,plain,(
% 4.83/2.07 (xs1 = (replicate_a @ p @ zero_zero_a))),
% 4.83/2.07 inference(cnf_transformation,[status(thm)],[f1001])).
% 4.83/2.07 thf(f1001,plain,(
% 4.83/2.07 (xs1 = (replicate_a @ p @ zero_zero_a))),
% 4.83/2.07 inference(fool_elimination,[status(thm)],[f6])).
% 4.83/2.07 thf(f6,axiom,(
% 4.83/2.07 (xs1 = (replicate_a @ p @ zero_zero_a))),
% 4.83/2.07 file('/export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2THF_2527.p',fact_5_xs1__def)).
% 4.83/2.07 thf(f1494,plain,(
% 4.83/2.07 (xsa = (append_a @ xs1 @ (cons_a @ (nth_a @ xsa @ p) @ xs2)))),
% 4.83/2.07 inference(cnf_transformation,[status(thm)],[f837])).
% 4.83/2.07 thf(f837,plain,(
% 4.83/2.07 (xsa = (append_a @ xs1 @ (cons_a @ (nth_a @ xsa @ p) @ xs2)))),
% 4.83/2.07 inference(fool_elimination,[status(thm)],[f7])).
% 4.83/2.07 thf(f7,axiom,(
% 4.83/2.07 (xsa = (append_a @ xs1 @ (cons_a @ (nth_a @ xsa @ p) @ xs2)))),
% 4.83/2.07 file('/export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2THF_2527.p',fact_6_xs__decompose)).
% 4.83/2.07 thf(f2054,plain,(
% 4.83/2.07 ((descar1375166517sums_a @ xsa) != (descar1375166517sums_a @ (append_a @ (replicate_a @ p @ zero_zero_a) @ (cons_a @ (nth_a @ xsa @ p) @ xs2))))),
% 4.83/2.07 inference(superposition,[status(thm)],[f1785,f1645])).
% 4.83/2.07 thf(f1645,plain,(
% 4.83/2.07 ( ! [X0 : nat,X1 : list_a] : (((append_a @ (replicate_a @ X0 @ zero_zero_a) @ (descar1375166517sums_a @ X1)) = (descar1375166517sums_a @ (append_a @ (replicate_a @ X0 @ zero_zero_a) @ X1)))) )),
% 4.83/2.07 inference(cnf_transformation,[status(thm)],[f969])).
% 4.83/2.07 thf(f969,plain,(
% 4.83/2.07 ! [X0 : nat,X1 : list_a] : ((append_a @ (replicate_a @ X0 @ zero_zero_a) @ (descar1375166517sums_a @ X1)) = (descar1375166517sums_a @ (append_a @ (replicate_a @ X0 @ zero_zero_a) @ X1)))),
% 4.83/2.07 inference(fool_elimination,[status(thm)],[f968])).
% 4.83/2.07 thf(f968,plain,(
% 4.83/2.07 ! [X0 : nat,X1 : list_a] : ((append_a @ (replicate_a @ X0 @ zero_zero_a) @ (descar1375166517sums_a @ X1)) = (descar1375166517sums_a @ (append_a @ (replicate_a @ X0 @ zero_zero_a) @ X1)))),
% 4.83/2.07 inference(rectify,[status(thm)],[f9])).
% 4.83/2.07 thf(f9,axiom,(
% 4.83/2.07 ! [X1 : nat,X0 : list_a] : ((descar1375166517sums_a @ (append_a @ (replicate_a @ X1 @ zero_zero_a) @ X0)) = (append_a @ (replicate_a @ X1 @ zero_zero_a) @ (descar1375166517sums_a @ X0)))),
% 4.83/2.07 file('/export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2THF_2527.p',fact_8_psums__replicate__0__append)).
% 4.83/2.07 thf(f1785,plain,(
% 4.83/2.07 ((descar1375166517sums_a @ xsa) != (append_a @ (replicate_a @ p @ zero_zero_a) @ (descar1375166517sums_a @ (cons_a @ (nth_a @ xsa @ p) @ xs2))))),
% 4.83/2.07 inference(cnf_transformation,[status(thm)],[f1055])).
% 4.83/2.07 thf(f1055,plain,(
% 4.83/2.07 ((descar1375166517sums_a @ xsa) != (append_a @ (replicate_a @ p @ zero_zero_a) @ (descar1375166517sums_a @ (cons_a @ (nth_a @ xsa @ p) @ xs2))))),
% 4.83/2.07 inference(flattening,[status(thm)],[f413])).
% 4.83/2.07 thf(f413,plain,(
% 4.83/2.07 ~((descar1375166517sums_a @ xsa) = (append_a @ (replicate_a @ p @ zero_zero_a) @ (descar1375166517sums_a @ (cons_a @ (nth_a @ xsa @ p) @ xs2))))),
% 4.83/2.07 inference(fool_elimination,[status(thm)],[f353])).
% 4.83/2.07 thf(f353,negated_conjecture,(
% 4.83/2.07 ~((descar1375166517sums_a @ xsa) = (append_a @ (replicate_a @ p @ zero_zero_a) @ (descar1375166517sums_a @ (cons_a @ (nth_a @ xsa @ p) @ xs2))))),
% 4.83/2.07 inference(negated_conjecture,[status(cth)],[f352])).
% 4.83/2.07 thf(f352,conjecture,(
% 4.83/2.07 ((descar1375166517sums_a @ xsa) = (append_a @ (replicate_a @ p @ zero_zero_a) @ (descar1375166517sums_a @ (cons_a @ (nth_a @ xsa @ p) @ xs2))))),
% 4.83/2.07 file('/export/starexec/sandbox/tmp/tmp.eUe8aLBIZV/DTF2THF_2527.p',conj_0)).
% 4.83/2.07 % SZS output end Proof for DTF2THF_2527
% 4.83/2.07 % (2677)------------------------------
% 4.83/2.07 % (2677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.83/2.07 % (2677)Termination reason: Refutation
% 4.83/2.07
% 4.83/2.07 % (2677)Memory used [KB]: 7803
% 4.83/2.07 % (2677)Time elapsed: 0.045 s
% 4.83/2.07 % (2677)Instructions burned: 143 (million)
% 4.83/2.07 % (2677)------------------------------
% 4.83/2.07 % (2677)------------------------------
% 4.83/2.07 % (2633)Success in time 0.38 s
%------------------------------------------------------------------------------